Analysis and Partial Differential Equations on Manifolds, Fractals and Graphs 9783110700763, 9783110700633

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Alexander Grigor’yan, Yuhua Sun (Eds.) Analysis and Partial Differential Equations on Manifolds, Fractals and Graphs

Advances in Analysis and Geometry

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Editor-in-Chief Jie Xiao, Memorial University, Canada Editorial Board Der-Chen Chang, Georgetown University, USA Goong Chen, Texas A&M University, USA Andrea Colesanti, University of Florence, Italy Robert McCann, University of Toronto, Canada De-Qi Zhang, National University of Singapore, Singapore Kehe Zhu, University at Albany, USA

Volume 3

Analysis and Partial Differential Equations on Manifolds, Fractals and Graphs |

Edited by Alexander Grigor’yan and Yuhua Sun

Mathematics Subject Classification 2010 Primary: 28A80, 31E05, 35K08, 53C44, 58J35, 58J65; Secondary: 31C25, 35B65, 46E35, 60H30, 60J35, 60J60, 60J75 Editors Prof. Alexander Grigor’yan Department of Mathematics University of Bielefeld Universitätsstr. 25 33615 Bielefeld Germany [email protected]

Dr. Yuhua Sun School of Mathematical Sciences Nankai University 94 Weijin Road 300071 Tianjin China [email protected]

ISBN 978-3-11-070063-3 e-ISBN (PDF) 978-3-11-070076-3 e-ISBN (EPUB) 978-3-11-070085-5 ISSN 2511-0438 Library of Congress Control Number: 2020946859 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2021 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Contents Preface | VII

Part I: Fractals and graphs Zhen-Qing Chen, Takashi Kumagai, and Jian Wang Stability of heat kernel estimates and parabolic Harnack inequalities for general symmetric pure jump processes | 3 Alexander Grigor’yan, Eryan Hu, and Jiaxin Hu The pointwise existence and properties of heat kernel | 27 Qingsong Gu and Ka-Sing Lau Resistance estimates and critical exponents of Dirichlet forms on fractals | 71 Bobo Hua and Xueping Huang A survey on unbounded Laplacians and intrinsic metrics on graphs | 103 Naotaka Kajino Energy measures for diffusions on fractals: a survey | 119 Shi-Lei Kong, Ka-Sing Lau, Jun Jason Luo, and Xiang-Yang Wang Hyperbolic graphs induced by iterations and applications in fractals | 143 Tim Jaschek and Mathav Murugan Geometric implications of fast volume growth and capacity estimates | 183 Ryosuke Shimizu Parabolic index of an infinite graph and Ahlfors regular conformal dimension of a self-similar set | 201 Meng Yang Metrics and uniform Harnack inequality on the Strichartz hexacarpet | 275

Part II: Euclidean spaces and manifolds Alexander Grigor’yan Analysis on manifolds and volume growth | 299

VI | Contents Alexander Grigor’yan, Satoshi Ishiwata, and Laurent Saloff-Coste Geometric analysis on manifolds with ends | 325 Feida Jiang and Xinyi Shen A matrix Harnack estimate for a Kolmogorov type equation | 345 Songzi Li and Xiang-Dong Li Entropy power concavity inequality on Riemannian manifolds and Ricci flow | 359 Liguang Liu and Jie Xiao Fractional differential operators and divergence equations | 385 Li Ma and Cong-han Wang Interior gradient estimates for mean curvature type equations and related flows | 421 Lei Ni An alternate induction argument in Simons’ proof of holonomy theorem | 443 Simon Nowak Higher integrability for nonlinear nonlocal equations with irregular kernel | 459 Yuhua Sun and Fanheng Xu On nonexistence results of porous medium type equations and differential inequalities on Riemannian manifolds | 493 Index | 515

Preface This volume contains contributions written by the participants of the conference “Analysis and PDEs on Manifolds and Fractals” held at the Nankai University (Tianjin, China) in September 2019. A quite wide range of topics was presented at the conference from combinatorics of graphs to flows on manifolds. It was the main idea of the conference to bring together experts from different areas of Mathematics nevertheless working on similarly problems in different settings. The same spirit is reflected in the diversity of the papers included in this volume. The conference was possible due to a generous support of Nankai University, SFB1283 of the German Research Council via Bielefeld University (Germany), Tsinghua University, Tianjin University, Nanjing University of Information Science and Technology, the Key Laboratory of Pure Mathematics and Combinatorics, Ministry of Education (LPMC), and National Natural Science Foundation of China (NSFC). The conference was devoted to two remarkable anniversaries that occurred in the same year 2019: the centenary of the Nankai University and the fiftieth anniversary of the University of Bielefeld, and the publication of this proceedings consolidates the cooperation of the mathematics departments of these schools. Alexander Grigor’yan and Yuhua Sun October 2020

https://doi.org/10.1515/9783110700763-201

VIII | Preface

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Part I: Fractals and graphs

Zhen-Qing Chen, Takashi Kumagai, and Jian Wang

Stability of heat kernel estimates and parabolic Harnack inequalities for general symmetric pure jump processes Abstract: In this paper, we survey recent work on heat kernel estimates for general symmetric pure jump processes on a metric measure space (M, ρ, μ) generated by the following type of nonlocal Dirichlet forms: ℰ (f , g) = ∫ (f (x) − f (y))(g(x) − g(y))J(dx, dy), M×M

where J(dx, dy) is a symmetric Radon measure on M × M \ diag that may have different growth behaviors for small and large jumps. Under general volume doubling condition on (M, ρ, μ) and some mild quantitative assumptions on J(dx, dy) that are allowed to have light tails of polynomial decay at infinity, we present stability results for two-sided heat kernel estimates and heat kernel upper bounds, as well as the corresponding parabolic Harnack inequalities. The results extend considerably those for mixture of symmetric stable-like jump processes in metric measure spaces, and more interestingly, they have connections to these for symmetric diffusions with jumps. Keywords: Symmetric pure jump process, nonlocal Dirichlet form, heat kernel estimate, parabolic Harnack inequality, jumping kernel MSC 2010: Primary 60J35, 35K08, 60J75, Secondary 31C25, 60J25, 60J45

Acknowledgement: The first author was supported in part by Simons Foundation Grant 520542 and a Victor Klee Faculty Fellowship at UW. The second author was supported in part by JSPS KAKENHI Grant Number JP17H01093 and by the Alexander von Humboldt Foundation. The third author was supported in part by the National Natural Science Foundation of China (Nos. 11831014 and 12071076), the Program for Probability and Statistics: Theory and Application (No. IRTL1704) and the Program for Innovative Research Team in Science and Technology in Fujian Province University (IRTSTFJ). Zhen-Qing Chen, Department of Mathematics, University of Washington, Seattle, WA 98195, USA, e-mail: [email protected] Takashi Kumagai, Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan, e-mail: [email protected] Jian Wang, College of Mathematics and Informatics & Fujian Key Laboratory of Mathematical Analysis and Applications (FJKLMAA) & Center for Applied Mathematics of Fujian Province (FJNU), Fujian Normal University, 350007 Fuzhou, P.R. China, e-mail: [email protected] https://doi.org/10.1515/9783110700763-001

4 | Z.-Q. Chen et al. Contents 1 1.1 1.2 2 2.1 2.2 2.3 2.4 2.5 3 4

Preliminaries | 4 Mixture of symmetric stable-like processes | 4 Motivating example: beyond mixture of stable-like processes | 8 Heat kernel for symmetric pure jump processes | 11 Two scaling functions | 11 Formulas for heat kernel estimates | 13 Jumping kernel and functional inequalities | 14 Stable characterizations of two-sided heat kernel estimates | 16 Further remarks on Theorems 2.7 and 2.8 | 17 Stability of parabolic Harnack inequalities | 20 Examples | 22 Bibliography | 24

1 Preliminaries In this section, we first recall the history on heat kernel estimates for mixture of symmetric stable-like processes on metric measure spaces, and then present an interesting example, which is not of mixture of stable-like type and exhibits some new phenomena. In particular, this example will lead us to consider general symmetric pure jump Markov processes whose jumping kernel can have light tails at infinity, which is (a special but important case of) the subject of this paper.

1.1 Mixture of symmetric stable-like processes Two-sided heat kernel estimates and parabolic Harnack inequalities have a long history in the theory of partial differential equations and diffusion processes (which are strong Markov processes with continuous sample paths). There are many beautiful results including the De Giorgi–Nash–Moser theory and Aronson’s Gaussian estimates in these areas. Studies of transition density functions for Markov processes with discontinuous sample paths, or equivalently, heat kernels for nonlocal operators, are relatively recent. We start with a (rotationally) symmetric α-stable process Z = {Zt , t ≥ 0; ℙx , x ∈ ℝd } on ℝd with α ∈ (0, 2), which is a Lévy process such that α

𝔼x ei⟨Zt −x,ξ ⟩ = e−t|ξ |

for every x, ξ ∈ ℝd .

The infinitesimal generator of a symmetric α-stable process Z on ℝd is the fractional Laplacian Δα/2 := −(−Δ)α/2 , which is a prototype of nonlocal operators. The fractional

Heat kernel estimates for general symmetric pure jump processes | 5

Laplacian can be written in the form Δα/2 u(x) = c lim

ε→0

∫ {|x−y|>ε}

u(y) − u(x) dy |y − x|d+α

for some constant c = c(d, α) > 0. It is well-known (see, e. g., [11]) that the transition density function p(t, x, y) of the symmetric α-stable process has the following two sided estimates: p(t, x, y) ≃ t −d/α ∧

t , |x − y|d+α

x, y ∈ ℝd , t > 0.

In this paper, we write f ≃ g, if there exist constants c1 , c2 > 0 such that c1 g(x) ≤ f (x) ≤ c2 g(x) for some range of x. For a, b ∈ ℝ, set a ∨ b := max{a, b} and a ∧ b := min{a, b}. Note that, in contrast to the standard Gaussian tail for Brownian motions, heat kernel of symmetric α-stable processes has polynomial decay when the distance |x − y| → ∞. This is called the heavy tail phenomenon for stable processes. Due to this property, many systems in physics and economics can and have been modeled by non-Gaussian stable processes. Let (M, ρ, μ) be an Ahlfors d-regular set, that is, μ(Bρ (x, r)) ≃ r d for all r > 0 and x ∈ M, where Bρ (x, r) := {y ∈ M : ρ(x, y) < r}. Let J(x, y) be a positive symmetric function on M × M such that J(x, y) ≃ ρ(x, y)−(d+α) ,

x, y ∈ M

(1.1)

for some 0 < α < 2. Consider a nonlocal regular Dirichlet form (ℰ , ℱ ) on L2 (M; μ) given by ℰ (f , g) =

∫

(f (x) − f (y))(g(x) − g(y))J(x, y)μ(dx)μ(dy),

f , g ∈ ℱ,

M×M\diag

where diag denotes the diagonal set {(x, x) : x ∈ M}. The reader is referred to [12, 22] for terminology and theory of symmetric Dirichlet forms and their associated symmetric Markov processes. It was established in Chen and Kumagai [15] that the symmetric strong Markov process X associated with (ℰ , ℱ ) on L2 (M; μ) has infinite lifetime, and has a jointly Hölder continuous transition density function p(t, x, y) with respect to the measure μ, which enjoys the following two-sided estimates p(t, x, y) ≃ t −d/α ∧

t ρ(x, y)d+α

for any (t, x, y) ∈ (0, ∞) × M × M.

(1.2)

We call the above Hunt process X a symmetric α-stable-like process on M. Note that when M = ℝd and J(x, y) = c(x, y)|x − y|−(d+α) with 0 < c1 ≤ c(x, y) ≤ c2 < ∞ for some constants α ∈ (0, 2) and c1 , c2 > 0, X is a symmetric α-stable-like process on ℝd , and the associated infinitesimal generator can be viewed as an analog to a divergence form

6 | Z.-Q. Chen et al. operator for fractional Laplacians. Since J(x, y) is the weak limit of p(t, x, y)/t as t → 0, heat kernel estimate (1.2) implies (1.1). Therefore, the results from [15] give a stable characterization for α-stable-like heat kernel estimates when α ∈ (0, 2) and the metric measure space (M, ρ, μ) is a d-set for some constant d > 0. This result has later been extended in [16] to mixtures of stable-like processes on more general metric measure spaces, under some growth condition on the weighted function ϕ such as r

∫ 0

r2 s ds ≤ c ϕ(s) ϕ(r)

for r > 0.

(1.3)

For α-stable-like processes where ϕ(r) = r α , condition (1.3) corresponds exactly to 0 < α < 2. Some of the key methods used in [15] were inspired by a previous work [9] on symmetric random walks with stable-like long range jumps on the integer lattice ℤd . See [8, 28] for related works on long range random walks on graphs. The notion of a d-set arises in the theory of function spaces and in fractal geometry. Geometrically, self-similar sets are typical examples of d-sets. There are many self-similar fractals on which there exist subdiffusive processes with walk dimension dw > 2 (that is, diffusion processes with scaling relation time ≈ spacedw ). For example, the walk dimension of Brownian motions on the Sierpinski gasket in ℝn (n ≥ 2) is log(n+3)/ log 2; see [5]. A direct calculation shows that the β-subordination of the subdiffusive processes on these fractals are jump processes whose Dirichlet forms (ℰ , ℱ ) are of the form given above with α = βdw in (1.1), and their transition density functions have two-sided estimates (1.2). Note that as β ∈ (0, 1), α ∈ (0, dw ), so α can be larger than 2. When α > 2, the approach in [15] ceases to work as it is hopeless to construct good cut-off functions a priori in this case. A long standing open problem in the field was whether the estimate (1.2) holds for generic symmetric jump processes having jumping kernel of the form (1.1) for any α ∈ (0, dw ). A related open question was to characterize the heat kernel estimate (1.2) by conditions that are stable under “rough isometries”. These open problems have recently been solved affirmatively in [17]. Actually, in [17] we obtained stability of two-sided heat kernel estimates for mixtures of symmetric stable-like jump processes on general metric measure spaces that satisfy the volume doubling condition and the reverse volume doubling condition. In details, let (M, ρ, μ) be a locally compact separable metric space, and μ a positive Radon measure on M with full support. In what follows, we will refer to such a triple (M, ρ, μ) as a metric measure space. We assume μ(M) = ∞. Denote the open ball centered at x with radius r by B(x, r) and μ(B(x, r)) by V(x, r). Definition 1.1. (i) We say that (M, ρ, μ) satisfies the volume doubling property (VD), if there exists a constant Cμ ≥ 1 such that V(x, 2r) ≤ Cμ V(x, r) for all x ∈ M and r > 0.

Heat kernel estimates for general symmetric pure jump processes | 7

(ii) We say that (M, ρ, μ) satisfies the reverse volume doubling property (RVD), if there exist constants lμ , cμ > 1 such that V(x, lμ r) ≥ cμ V(x, r)

for all x ∈ M and r > 0.

We remark that under RVD, μ(M) = ∞ if and only if M has infinite diameter. We also note that when M is connected and unbounded, VD implies RVD. We consider the following regular nonlocal Dirichlet form (ℰ , ℱ ) on L2 (M; μ): ℰ (f , g) =

(f (x) − f (y))(g(x) − g(y))J(dx, dy),

∫

f , g ∈ ℱ,

(1.4)

M×M\diag

where J(dx, dy) is a symmetric Radon measure on M × M \ diag. Associated with the regular Dirichlet form (ℰ , ℱ ) on L2 (M; μ) is a μ-symmetric Hunt process X = {Xt , t ≥ 0; ℙx , x ∈ M \ 𝒩 }. Here 𝒩 ⊂ M is a properly exceptional set for (ℰ , ℱ ) in the sense that 𝒩 is nearly Borel, μ(𝒩 ) = 0 and M𝜕 \ 𝒩 is X-invariant. This Hunt process is unique up to a properly exceptional set; see [22, Theorem 4.2.8]. Since (ℰ , ℱ ) only has a nonlocal part, the associated Hunt process X is of the pure jump type. We fix X and 𝒩 , and write M0 = M\𝒩 . Let {Pt }t≥0 be the transition semigroup of the Markov process X (or of the Dirichlet form (ℰ , ℱ ) on L2 (M; μ)). The transition density function for the Markov process X is a measurable function p(t, x, y) : M0 × M0 → (0, ∞) for every t > 0 such that 𝔼x [f (Xt )] = Pt f (x) = ∫ p(t, x, y)f (y)μ(dy) p(t, x, y) = p(t, y, x)

for all x ∈ M0 , f ∈ L∞ (M; μ),

(1.5)

for all t > 0, x, y ∈ M0 ,

p(s + t, x, z) = ∫ p(s, x, y)p(t, y, z)μ(dy)

for all s, t > 0, x, z ∈ M0 .

In the literature, p(t, x, y) is also called the heat kernel of the process X or of the Dirichlet form (ℰ , ℱ ) on L2 (M; μ). Let ℝ+ := [0, ∞), and ϕ : ℝ+ → ℝ+ be a strictly increasing continuous function with ϕ(0) = 0 and ϕ(1) = 1 so that there exist constants c1 , c2 > 0 and β2 ≥ β1 > 0 such that β

β2

R R 1 ϕ(R) ≤ c2 ( ) c1 ( ) ≤ r ϕ(r) r

for all 0 < r ≤ R.

In [17, Theorem 1.13], the following are shown to be equivalent under the VD and RVD condition: (i) there exists a heat kernel p(t, x, y) associated with (ℰ , ℱ ), which has the following estimates for all t > 0 and all x, y ∈ M: p(t, x, y) ≃

1 t ∧ . V(x, ϕ−1 (t)) V(x, ρ(x, y))ϕ(ρ(x, y))

(1.6)

Here and in what follows, ϕ−1 (t) is the inverse function of the strictly increasing function t 󳨃→ ϕ(t).

8 | Z.-Q. Chen et al. (ii) Jϕ and a cut-off Sobolev inequality CSJ(ϕ) hold, where Jϕ means that there exists a nonnegative symmetric function J(x, y) so that for μ × μ-almost all x, y ∈ M, J(dx, dy) = J(x, y) μ(dx) μ(dy)

(1.7)

and J(x, y) ≃

1 . V(x, ρ(x, y))ϕ(ρ(x, y))

(1.8)

See Definition 2.3 below for the definition of CSJ(ϕ). (iii) Jϕ and Eϕ hold, where Eϕ means that 𝔼x τB(x,r) ≃ ϕ(r)

for all x ∈ M0 and r > 0,

(1.9)

where τA = inf{t > 0 : Xt ∉ A}. We emphasize that in the above result from [17, Theorem 1.13], the underlying metric measure space (M, ρ, μ) is only assumed to satisfy the general VD and RVD. Neither uniform VD nor uniform RVD property is assumed. We do not assume M to be connected or (M, ρ) to be geodesic. See related works for symmetric stable-like processes with metric measure spaces satisfying Ahlfors d-set condition [23], and for random walks with α-stable-like long range jumps on connected locally finite infinite graphs satisfying global Ahlfors d-set condition [29]. We point out that, for symmetric jump processes associated with nonlocal Dirichlet forms (ℰ , ℱ ) above, parabolic Harnack inequalities are strictly weaker than the twosided heat kernel estimates. It is established in [18, Corollary 1.21] that the parabolic Harnack inequalities, together with a suitable lower bound for the jumping kernel, are equivalent to the two-sided heat kernel estimates. (See [8, Theorem 1.4] for the corresponding result for random walks with α-stable-like long range jumps with 0 < α < 2 on graphs satisfying the global Ahlfors d-set condition.) This is in contrast to symmetric diffusion processes, where parabolic Harnack inequalities are equivalent to twosided heat kernel estimates. The reader is referred to [18] for the stability results of parabolic Harnack inequalities for symmetric pure jump Dirichlet forms, and to [19] for various characterizations of elliptic Harnack inequalities for symmetric pure jump processes.

1.2 Motivating example: beyond mixture of stable-like processes Consider the following example. Example 1.2. Let (M, ρ, μ) be a metric measure space such that the volume doubling (VD) condition holds. Suppose X := {Xt , t ≥ 0; ℙx , x ∈ M} is a conservative symmetric

Heat kernel estimates for general symmetric pure jump processes | 9

diffusion process on M that has a transition density function q(t, x, y) with respect to μ such that q(t, x, y) ≍

1/(1−β)

ρ(x, y)β 1 exp(−c( ) t V(x, t 1/β )

),

t > 0, x, y ∈ M

(1.10)

for some β ≥ 2. Here and in what follows, for two positive functions f (t, z) and g(t, z), notation f ≍ g means that there exist positive constants ci (i = 1, . . . , 4) such that c1 f (c2 t, z) ≤ g(t, z) ≤ c3 f (c4 t, z). For example, a celebrate result due to Aronson [2] says that a symmetric diffusion process X on ℝd associated with the uniformly elliptic operator d

𝜕 𝜕 (aij (x) ) 𝜕xi 𝜕xj i,j=1

ℒ= ∑

has the two-sided estimates (1.10) with β = 2, V(x, r) ≃ r d and ρ(x, y) = |x − y| being the Euclidean metric. This is also the case for Brownian motions on the Sierpinski gasket with β = log(n + 3)/ log 2 in (1.10), or on the Sierpinski carpet in ℝn (n ≥ 2) with β > 2 in (1.10); see [5, 26]. Let S := (St )t≥0 be a subordinator with S0 = 0 that is independent of X and has the Laplace exponent ∞

f (r) = ∫ (1 − e−rs )ν(s) ds,

r > 0,

0

where ν(s) =

1

1 s1+γ1 {01}

with γ1 ∈ (0, 1) and γ2 ∈ (1, ∞). Let Y := (Yt )t≥0 be the subordinate process defined by Yt := XSt for all t > 0. For a set A ⊂ M, define the exit time τAY = inf{t > 0 : Yt ∉ A}. The subordinate process Y has the following properties which are established in [21, Example 1.1]. Assertion 1. The subordinated process Y is a symmetric jump process such that (i) its jumping kernel J(dx, dy) has a density with respect to the product measure μ×μ on M × M \ diag given by 1

{ V(x,ρ(x,y))ρ(x,y)α1 , J(x, y) ≃ { 1 { V(x,ρ(x,y))ρ(x,y)α2 ,

ρ(x, y) ≤ 1, ρ(x, y) > 1,

where α1 = γ1 β and α2 = γ2 β, and diag stands for the diagonal of M × M.

(1.11)

10 | Z.-Q. Chen et al. (ii) for any x0 ∈ M and r > 0, Y 𝔼x [τB(x ] ≃ r α1 ∨ r β . 0 ,r)

(1.12)

Assertion 2. The process Y has a jointly continuous transition density function p(t, x, y) with respect to the measure μ on M so that p(t, x, y) ≃

1 ∧ (tJ(x, y)) for t ≤ 1 V(x, t 1/α1 )

and 1 t 1 1 1/β + 1/β ) V(x, t 1/β ) {ρ(x,y)≤t } V(x, ρ(x, y))ρ(x, y)α2 {ρ(x,y)>t } 1 ≤ p(t, x, y) ≤ c2 ( ∧ (tJ(x, y) + q(c3 t, x, y))) for t > 1, V(x, t 1/β )

c1 (

where q(t, x, y) is the transition density function for the diffusion process X of the form (1.10). If, in addition, (M, ρ, μ) is connected and satisfies the chain condition, then 1 { V(x,t 1/α1 ) ∧ (tJ(x, y)), p(t, x, y) ≍ { 1 { V(x,t 1/β ) ∧ (tJ(x, y) + q(t, x, y)),

t ≤ 1, t > 1.

(1.13)

The first property of Assertion 1 says that the scaling function of jumping kernel for the process Y is ϕj (r) := r α1 ∨ r α2 with α1 < β < α2 , while the second property of Assertion 1 indicates that the scaling function of the process Y is r α1 ∨ r β . The scaling function of the process Y is different from the associated jumping kernel at large scale (that is, when r > 1). Thus, comparing (1.11)–(1.12) with (1.8)–(1.9), we can see that the behavior of the symmetric jump process Y in Example 1.2 is different from that of symmetric α1 -stable-like or mixed stable-like processes studied in [15, 17], where the scale functions for large jumps are all assumed to be less than those for the diffusion processes if there is one (for example, r 󳨃→ r 2 in the Euclidean space case). Due to these differences, the process Y above has two-sided heat kernel estimates (1.13), which is of a different shape from (1.6). In particular, this may appear surprising at the first glance that in (1.13) there is the diffusive scaling ϕc (r) := r β when r > 1 is involved. But it becomes quite reasonable if one thinks more about it as the jumping kernel J(dx, dy) of Y has finite second moment in the case of β = 2. Note that in this example, ϕj (r) ≥ ϕc (r) for all r > 0. The purpose of this article is to summarize our recent work in [17, 18, 21] on stable characterizations of heat kernel estimates and parabolic Harnack inequalities for general symmetric pure jump Dirichlet forms, which include a large class of symmetric pure jump Dirichlet forms that have light jumping kernel at infinity and thus possibly exhibit diffusive behaviors.

Heat kernel estimates for general symmetric pure jump processes | 11

2 Heat kernel for symmetric pure jump processes In this section, we survey some recent developments on heat kernel estimates for general symmetric pure jump processes. The reader is referred to [21] for further details.

2.1 Two scaling functions Let (M, ρ, μ) be a metric measure space. We assume that all balls are relatively compact and assume for simplicity that μ(M) = ∞. We do not assume M to be connected or (M, ρ) to be geodesic. Throughout this paper, we always assume that VD holds, and occasionally we also assume RVD holds. We are concerned with regular nonlocal Dirichlet forms (ℰ , ℱ ) on L2 (M; μ) given by (1.4). Let X = {Xt , t ≥ 0; ℙx , x ∈ M \ 𝒩 } be the associated Hunt process. In order to consider heat kernel estimates for the Dirichlet form (ℰ , ℱ ) above, that may have light jumping kernel at infinity, we need to introduce two different scaling functions. Set ℝ+ := [0, ∞). Let ϕj : ℝ+ → ℝ+ and ϕc : [1, ∞) → ℝ+ be strictly increasing continuous functions with ϕj (0) = ϕc (0) = 0, ϕj (1) = ϕc (1) = 1 and satisfying that there exist constants c1,ϕj , c2,ϕj , c1,ϕc , c2,ϕc > 0, β2,ϕj ≥ β1,ϕj > 0 and β2,ϕc ≥ β1,ϕc > 1 such that β1,ϕj

R c1,ϕj ( ) r

≤

β1,ϕc

R c1,ϕc ( ) r

ϕj (R)

β2,ϕj

R ≤ c2,ϕj ( ) ϕj (r) r

for all 0 < r ≤ R,

β2,ϕc

≤

ϕc (R) R ≤ c2,ϕc ( ) ϕc (r) r

for all 0 < r ≤ R.

(2.1)

Since β1,ϕc > 1, we know from [10, Definition, p. 65; Definition, p. 66; Theorem 2.2.4, and its remark, p. 73] that there exists a strictly increasing function ϕ̄ c : ℝ+ → ℝ+ such that there is a constant c1 ≥ 1 with c1−1 ϕc (r)/r ≤ ϕ̄ c (r) ≤ c1 ϕc (r)/r

for all r > 0.

Clearly, by (2.1) and (2.2), there exist constants c1,ϕ̄ , c2,ϕ̄ > 0 such that c

β1,ϕc −1

R c1,ϕ̄ ( ) c r

≤

β −1 ϕ̄ c (R) R 2,ϕc ≤ c2,ϕ̄ ( ) c r ϕ̄ (r) c

c

for all 0 < r ≤ R.

Define β∗ := sup{β > 0 : there is a constant c∗ > 0 so that ϕj (R)/ϕj (r) ≥ c∗ (R/r)β for 0 < r < R ≤ 1}

(2.2)

12 | Z.-Q. Chen et al. and β∗ := sup{β > 0 : there is a constant c∗ > 0 so that ϕj (R)/ϕj (r) ≥ c∗ (R/r)β for all R > r ≥ 1}.

Throughout this paper, we assume that there is a constant c0 ≥ 1 so that ϕc (r) ≤ c0 ϕj (r) on [0, 1] if β∗ > 1 and ϕc (r) ≤ c0 ϕj (r) on (1, ∞) if β∗ > 1.

(2.3)

We point out that, by (2.1) with β1,ϕc > 1, ϕc is not comparable to ϕj on [0, 1] when β∗ ≤ 1, and ϕc is not comparable to ϕj on [1, ∞) when β∗ ≤ 1. Roughly speaking, the function ϕj plays the role of the scaling function in the jumping kernel; while ϕc is a scale function that should be intrinsically determined by ϕj and the metric measure space (M, ρ, μ) which will possibly appear in the expression of heat kernel estimates. However, we do not have a universal formula for ϕc . For example, when the state space M (such as ℝd or a nice fractal) has a nice diffusion process, ϕc can be the scaling function of the diffusion in some cases but can also be a different scale function in some other cases; see Examples 1.2 and 4.1. In some cases, ϕc can just be ϕj on a part or the whole of [0, ∞). To cover a wide spectrum of scenarios, in the formulation and characterization we allow ϕc to be any function that satisfies conditions (2.1) and (2.3). Given ϕc and ϕj as above, we set {ϕj (r)1{β∗ ≤1} + ϕc (r)1{β∗ >1} ϕ(r) := { {ϕj (r)1{β∗ ≤1} + ϕc (r)1{β∗ >1}

for 0 < r ≤ 1, for r > 1.

(2.4)

In view of the assumptions above, ϕ is strictly increasing on ℝ+ such that ϕ(0) = 0, ϕ(1) = 1 and there exist constants c1,ϕ , c2,ϕ > 0 so that β1,ϕ

R c1,ϕ ( ) r

β2,ϕ

≤

ϕ(R) R ≤ c2,ϕ ( ) ϕ(r) r

for all 0 < r ≤ R,

where β1,ϕ = β1,ϕc ∧ β1,ϕj and β2,ϕ = β2,ϕc ∨ β2,ϕj . Clearly, we have by (2.3) that ϕ(r) ≤ c0 ϕj (r)

for every r ≥ 0.

−1 ̄ −1 Denote by ϕ−1 j (t), ϕc (t) and ϕc (t) the inverse functions of the strictly increasing functions t 󳨃→ ϕ (t), t 󳨃→ ϕ (t) and t 󳨃→ ϕ̄ (t), respectively. Then the inverse function ϕ−1 j

c

of ϕ is given by

c

−1 −1 {ϕj (r)1{β∗ ≤1} + ϕc (r)1{β∗ >1} ϕ (r) := { −1 −1 {ϕj (r)1{β∗ ≤1} + ϕc (r)1{β∗ >1} −1

for 0 < r ≤ 1, for r > 1.

Throughout this paper, we will fix the notations for these functions ϕc , ϕj , ϕ, and ϕ̄ c . In particular, as we will see from the results below, ϕ is the “true” scaling function for the process X. For example, ϕ(r) = r α1 ∨ r β for the process Y in Example 1.2, and the scaling function for its jumping kernel is ϕj (r) = r α1 ∨ r α2 , where α1 < β < α2 .

Heat kernel estimates for general symmetric pure jump processes | 13

2.2 Formulas for heat kernel estimates In this subsection, we present formulas of heat kernel estimates for general symmetric pure jump processes. As we will see, the processes will enjoy heat kernel estimates with different forms so that the scaling functions ϕc , ϕj , and ϕ̄ c are fully or partly involved, according to different ranges of the indexes β∗ and β∗ . Recall that the function ϕ̄ c (r) is a strictly increasing function satisfying (2.2). For any t > 0 and x, y ∈ M0 , set p(j) (t, x, y) :=

t 1 ∧ V(x, ρ(x, y))ϕ V(x, ϕ−1 (t)) j (ρ(x, y)) j

(2.5)

ρ(x, y) 1 exp(− ). −1 ̄ V(x, ϕ−1 (t)) ϕc (t/ρ(x, y)) c

(2.6)

and p(c) (t, x, y) :=

Here, p(j) (t, x, y) follows from two-sided heat kernel estimates for mixtures of symmetric stable-like processes on metric measure space; see (1.6); while p(c) (t, x, y) is partly motivated by two-sided heat kernel estimates for strongly local Dirichlet forms, see [24, 25]. Definition 2.1. (i) We say that HK(ϕj , ϕc ) holds if there exists a heat kernel p(t, x, y) of the semigroup {Pt } associated with (ℰ , ℱ ) which has the following estimates for all x, y ∈ M0 : (j) { {p (t, x, y) { { { { { { 1 { { ∧ (p(c) (t, x, y) + p(j) (t, x, y))) {{ V(x,ϕ−1 c (t)) p(t, x, y) ≍ { { p(j) (t, x, y) { { { { { { { {{ 1 ∧ (p(c) (t, x, y) + p(j) (t, x, y))) {{ V(x,ϕ−1 c (t))

if β∗ ≤ 1, if β∗ > 1, if β∗ ≤ 1, if β∗ > 1,

for t ≤ 1,

for t > 1. (2.7)

(ii) We say HK− (ϕj , ϕc ) holds if the upper bound in (2.7) holds but the lower bound is replaced by the following statement: there are constants c1 , c2 > 0 so that 1 { { V(x,ϕ−1 (t)) , p(t, x, y) ≥ c1 { t { , { V(x,ρ(x,y))ϕj (ρ(x,y))

ρ(x, y) ≤ c2 ϕ−1 (t), ρ(x, y) > c2 ϕ−1 (t).

(iii) We say UHK(ϕj , ϕc ) holds if the upper bound in (2.7) holds. The scale function ϕc plays a role in the definitions of HK(ϕj , ϕc ) and HK− (ϕj , ϕc ) only for t ≤ 1 when β∗ > 1 and for t > 1 when β∗ > 1. The cut-off time 1 here is not

14 | Z.-Q. Chen et al. important – it can be replaced by any fixed constant T > 0. Furthermore, it follows from Theorem 3.3 below that HK− (ϕj , ϕc ) (and so HK(ϕj , ϕc )) is stronger than PHI(ϕ), which in turn yields the Hölder regularity of parabolic functions; see the proof of [20, Theorem 1.17]. In particular, this implies that if HK− (ϕj , ϕc ) (respectively, HK(ϕj , ϕc )) holds, then it can be strengthened to hold for all x, y ∈ M, and consequently the Hunt process X can be refined to start from every point in M. We note that the expression of HK(ϕj , ϕc ) takes different form depending on whether β∗ ≤ 1 (respectively, β∗ ≤ 1) or not. This is because when β∗ > 1 (respectively, β∗ > 1), the heat kernel estimate may involve another function ϕc that is intrinsically determined by ϕj , but we do not have a generic formula for it under our general setting. However, it does not necessarily mean that the heat kernel estimates for p(t, x, y) has a phase transition exactly at β∗ = 1 or β∗ = 1. For example, suppose (M, ρ, μ) is a connected metric measure space satisfying the chain condition on which there is a symmetric diffusion process enjoying the heat kernel estimate (1.10) with β > 1 as in Example 1.2. Consider a symmetric pure jump process Y on this space whose jumping density has bounds (1.11) with α1 < β and α2 > 0. Clearly, β∗ = α1 and β∗ = α2 . When α2 < β, the transition density function p(t, x, y) of Y has estimates (2.5) with ϕj (r) = r α1 1{01} as in [17], whereas for α2 > β, p(t, x, y) satisfies (1.13) as in Example 1.2. Hence for this example, phase transition for the expression of heat kernel estimates for p(t, x, y) occurs at α2 = β.

2.3 Jumping kernel and functional inequalities To state our stable characterizations of heat kernel estimates for general symmetric pure jump processes, we need four definitions. Definition 2.2. Let ψ : ℝ+ → ℝ+ . We say Jψ holds, if there exists a non-negative symmetric function J(x, y) so that (1.7) is satisfied, and (1.8) holds with ψ in place of ϕ for all x, y ∈ M, that is, c2 c1 ≤ J(x, y) ≤ . V(x, ρ(x, y))ψ(ρ(x, y)) V(x, ρ(x, y))ψ(ρ(x, y))

(2.8)

We say that Jψ,≤ (resp. Jψ,≥ ) if (1.7) holds and the upper bound (resp. lower bound) in (2.8) holds. Note that, since ϕ(r) ≤ c0 ϕj (r) for all r > 0, Jϕj ,≤ implies Jϕ,≤ , that is, Jϕ,≤ is weaker than Jϕj ,≤ . Let U ⊂ V be open sets of M with U ⊂ U ⊂ V. We say a nonnegative bounded measurable function φ is a cut-off function for U ⊂ V, if φ = 1 on U, φ = 0 on V c and 0 ≤ φ ≤ 1 on M. For f , g ∈ ℱ , we define the carré du-Champ operator Γ(f , g) for the

Heat kernel estimates for general symmetric pure jump processes | 15

symmetric pure jump Dirichlet form (ℰ , ℱ ) by Γ(f , g)(dx) = ∫ (f (x) − f (y))(g(x) − g(y))J(dx, dy). y∈M

Clearly, ℰ (f , g) = Γ(f , g)(M). We now introduce the following cut-off Sobolev inequality CSJ(ϕ) that controls the energy of cut-off functions. Definition 2.3. Let ℱb = ℱ ∩ L∞ (M, μ). We say that condition CSJ(ϕ) holds, if there exist constants C0 ∈ (0, 1] and C1 , C2 > 0 such that for every 0 < r ≤ R, almost all x0 ∈ M and any f ∈ ℱ , there exists a cut-off function φ ∈ ℱb for B(x0 , R) ⊂ B(x0 , R + r) such that f 2 dΓ(φ, φ)

∫ B(x0 ,R+(1+C0 )r)

2

≤ C1 ∫ (f (x) − f (y)) J(dx, dy) + U×U ∗

C2 ϕ(r)

∫

f 2 dμ,

(2.9)

B(x0 ,R+(1+C0 )r)

where U = B(x0 , R + r) \ B(x0 , R) and U = B(x0 , R + (1 + C0 )r) \ B(x0 , R − C0 r). ∗

Remark 2.4. (i) CSJ(ϕ) for symmetric pure jump Dirichlet forms is first introduced in [17] as a counterpart of CSA(ϕ) for strongly local Dirichlet forms (see [1, 6, 7]). A similar condition is called condition (AB) in [23] for the case ϕ(r) = r α . As pointed out in [17, Remark 1.6(ii)], the main difference between CSJ(ϕ) and CSA(ϕ) is that the integrals on the left-hand side and in the second term of the right-hand side of the inequality (2.9) are over B(x0 , R + (1 + C0 )r) instead of over B(x0 , R + r) in [1]. Note that the integral over B(x0 , R + r)c is zero on the left-hand side of (2.9) for the case of strongly local Dirichlet forms. As we see from the approach of [17] in the study of stability of heat kernel estimates for symmetric mixed stable-like processes, it is important to enlarge the ball B(x0 , R + r) and integrate over B(x0 , R + (1 + C0 )r) rather than over B(x0 , R + r). (ii) Denote by ℱloc the space of functions locally in ℱ ; that is, f ∈ ℱloc if and only if for any relatively compact open set U ⊂ M there exists g ∈ ℱ such that f = g μ-a. e. on U. Since each ball is relatively compact and (2.9) uses the property of f on B(x0 , R + (1 + C0 )r) only, CSJ(ϕ) also holds for any f ∈ ℱloc . ity.

We next introduce the Faber–Krahn inequality and the (weak) Poincaré inequal-

Definition 2.5. We say that the MMD space (M, ρ, μ, ℰ ) satisfies the Faber–Krahn inequality FK(ϕ) if there exist positive constants C and p such that for any ball B(x, r) and any open set D ⊂ B(x, r), λ1 (D) ≥

C p (V(x, r)/μ(D)) , ϕ(r)

16 | Z.-Q. Chen et al. where λ1 (D) = inf{ℰ (f , f ) : f ∈ ℱD with ‖f ‖2 = 1} and ℱD is defined to be the √ℰ (⋅, ⋅) + ‖ ⋅ ‖22 -closure in ℱ of ℱ ∩ Cc (D). Definition 2.6. We say that the (weak) Poincaré inequality PI(ϕ) holds if there exist constants C > 0 and κ ≥ 1 such that for any ball Br = B(x, r) with x ∈ M and for any f ∈ ℱb , ∫(f − f Br )2 dμ ≤ Cϕ(r)

Br

where f Br =

1 ∫ μ(Br ) Br

2

∫ (f (y) − f (x)) J(dx, dy), Bκr ×Bκr

f dμ is the average value of f on Br .

2.4 Stable characterizations of two-sided heat kernel estimates With the notations above, we can now state the following stable characterizations of two-sided heat kernel estimates and upper bounds of heat kernel for general symmetric pure jump process from [21]. Theorem 2.7. Assume that the metric measure space (M, ρ, μ) satisfies VD and RVD, and the functions ϕc and ϕj satisfy (2.1) and (2.3). Let ϕ(r) be defined by (2.4). The following are equivalent: (1) HK− (ϕj , ϕc ). (2) PI(ϕ), Jϕj and CSJ(ϕ). If, in addition, (M, ρ, μ) is connected and satisfies the chain condition, then each assertion above is equivalent to (3) HK(ϕj , ϕc ). We refer the reader to [21, Theorem 1.11] for more equivalent characterizations of HK− (ϕj , ϕc ) and HK(ϕj , ϕc ). We emphasize again that the connectedness and the chain condition of the underlying metric measure space (M, ρ, μ) are only used to derive optimal lower bounds in off-diagonal estimates for the heat kernel when time is small (i. e., from HK− (ϕj , ϕc ) to HK(ϕj , ϕc )), while for the equivalence between (1) and (2) in the result above, the metric measure space (M, ρ, μ) is only assumed to satisfy the general VD and RVD, that is, we do not assume M to be connected or (M, ρ) to be geodesic. (In fact, (2) 󳨐⇒ (1) in Theorem 2.7 holds true under VD and (2.1), without assuming RVD.) Furthermore, we do not assume the uniform comparability of volume of balls, that is, we do not assume the existence of a nondecreasing function V on [0, ∞) with V(0) = 0 so that μ(B(x, r)) ≍ V(r) for all x ∈ M and r > 0. We also have the following characterizations for UHK(ϕj , ϕc ) (see [21, Theorem 1.12] for more equivalent characterizations). In the following, we say (ℰ , ℱ ) on L2 (M; μ) is conservative if its associated Hunt process X has infinite lifetime. This is equivalent to Pt 1 = 1 on M0 for every t > 0. It follows from [17, Proposition 3.1] that any equivalent

Heat kernel estimates for general symmetric pure jump processes | 17

statement of Theorem 2.7 implies that the process X is conservative. We also point out that UHK(ϕj , ϕc ) alone does not imply the conservativeness of the associated Dirichlet form (ℰ , ℱ ) (see [17, Proposition 3.1 and Remark 3.2] for more details). Theorem 2.8. Assume that the metric measure space (M, ρ, μ) satisfies VD and RVD, and that the functions ϕc and ϕj satisfy (2.1) and (2.3). Let ϕ(r) be defined by (2.4). Then the following are equivalent: (1) UHK(ϕj , ϕc ) and (ℰ , ℱ ) on L2 (M; μ) is conservative. (2) FK(ϕ), Jϕj ,≤ and CSJ(ϕ). We emphasize that the above two theorems are equivalent characterizations and stability results. It is possible that none of the statements hold with a bad selection of ϕc .

2.5 Further remarks on Theorems 2.7 and 2.8 In this subsection, we make further comments on the formulations of HK− (ϕj , ϕc ) and HK(ϕj , ϕc ), and discuss relations of the main results above (Theorems 2.7 and 2.8) to those in the literature. Recall that the function ϕ is defined by (2.4) and p(j) (t, x, y), p(c) (t, x, y) are defined as in (2.5) and (2.6). Remark 2.9. (i) By simple calculations, we have p(c) (t, x, y) ⪯

t 1 ∧ −1 V(x, ϕc (t)) V(x, ρ(x, y))ϕc (ρ(x, y))

on (0, ∞) × M0 × M0 ,

where f ⪯ g means that there exists a constant c > 0 such that f (x) ≤ cg(x) for the specified range of x. Hence, when β∗ > 1 and ϕc = ϕj on [1, ∞), it holds that p(c) (t, x, y) ⪯

1 t ∧ V(x, ρ(x, y))ϕ V(x, ϕ−1 (t)) j (ρ(x, y)) j

on (1, ∞) × M0 × M0 .

Consequently, in this case we have 1 ∧ (p(c) (t, x, y) + p(j) (t, x, y)) ≍ p(j) (t, x, y) V(x, ϕ−1 (t)) c

on (1, ∞) × M0 × M0 .

Similarly, one can check that in the case of β∗ > 1 and ϕc = ϕj on [0, 1], 1 ∧ (p(c) (t, x, y) + p(j) (t, x, y)) ≍ p(j) (t, x, y) on (0, 1] × M0 × M0 . V(x, ϕ−1 (t)) c Therefore, when ϕ = ϕj on [0, ∞) (that is, when ϕc = ϕj on [0, 1] if β∗ > 1 and ϕc = ϕj on (1, ∞) if β∗ > 1), HK(ϕj , ϕc ) is just the heat kernel estimate p(j) (t, x, y). Thus, in this case, Theorems 2.7 and 2.8 are essentially the main results (Theorems 1.13 and 1.15) of [17]. We note that by the proof of [17, Lemma 4.1], Jϕj ,≥ implies

18 | Z.-Q. Chen et al. PI(ϕj ), so in the case of ϕ = ϕj on [0, ∞), we can drop condition PI(ϕ) from the statement of Theorem 2.7. (ii) When ϕ(r) = ϕc (r) on [0, ∞) (that is, when β∗ ∧ β∗ > 1, and ϕc (r) is a strictly increasing function satisfying (2.1) and (2.3)), HK− (ϕj , ϕc ) and HK(ϕj , ϕc ) are just HK(Φ, ψ) and SHK(Φ, ψ) in [4, Definition 2.8] with ψ = ϕj

and

Φ = ϕc .

In this case, our Theorems 2.7 and 2.8 have also been independently obtained in [4, Theorem 2.14, Corollary 2.15, Theorem 2.17, and Corollary 2.18]. (iii) When ϕ(r) = ϕc (r)1[0,1] +ϕj (r)1(1,∞) (that is, when β∗ > 1 and either β∗ ≤ 1 or β∗ > 1 with ϕc (r) = ϕj (r) for all r ∈ [1, ∞)), HK(ϕj , ϕc ) is reduced into 1 ∧ (p(c) (t, x, y) + p(j) (t, x, y))), { V(x,ϕ−1 c (t)) p(t, x, y) ≍ { (j) {p (t, x, y),

0 < t ≤ 1, t > 1.

(2.10)

In this case HK(ϕj , ϕc ) is of the same form as that of HK(ϕc , ϕj ) in [20] for symmetric diffusions with jumps, or equivalently, for symmetric Dirichlet forms that contain both the strongly local part and the pure jump part; see [20, Definition 1.11 and Remark 1.12] for details. However, there are differences between them. In [20] the function ϕc in HK(ϕc , ϕj ) is the scaling function of the diffusion (i. e., the strongly local part of Dirichlet forms), while in the present paper the function ϕc in HK(ϕj , ϕc ) is determined by ϕj and the underlying metric measure space (M, ρ, μ). In Remark 2.9, we have discussed the form of HK(ϕj , ϕc ) for the cases of ϕ(r) = ϕj (r) on [0, ∞), ϕ(r) = ϕc (r) on [0, ∞), and ϕ(r) = ϕc (r)1[0,1] + ϕj (r)1(1,∞) , respectively. We now discuss the remaining case of ϕ(r). Remark 2.10. This remark is concerned with the case that ϕ(r) = ϕj (r)1[0,1] +ϕc (r)1(1,∞) . It consists of two subcases of β∗ > 1 where either β∗ ≤ 1 or β∗ > 1 with ϕc (r) = ϕj (r) for all r ∈ (0, 1]. (i) In this case, we can rewrite the expression of HK(ϕj , ϕc ) in the following way. For 0 < t ≤ 1, 1 { , { V(x,ϕ−1 (t)) j p(t, x, y) ≃ p (t, x, y) ≃ { t { , { V(x,ρ(x,y))ϕj (ρ(x,y)) (j)

ρ(x, y) ≤ c1 ϕ−1 j (t), ρ(x, y) > c1 ϕ−1 j (t).

For t > 1, p(t, x, y) ≍

1 ∧ (p(c) (t, x, y) + p(j) (t, x, y)) V(x, ϕ−1 c (t))

1 , { V(x,ϕ−1 c (t)) ≍{ t + { V(x,ρ(x,y))ϕj (ρ(x,y))

ρ(x, y) ≤ c2 ϕ−1 c (t), 1 V(x,ϕ−1 c (t))

exp(−

ρ(x,y) ), ϕ̄ −1 c (t/ρ(x,y))

ρ(x, y) > c2 ϕ−1 c (t).

Heat kernel estimates for general symmetric pure jump processes | 19

In particular, for t ∈ (0, 1], the heat kernel estimates HK(ϕj , ϕc ) are completely dominated by the jump kernel for the associated Dirichlet form (ℰ , ℱ ). For t > 1, we have the following more explicit expression of HK(ϕj , ϕc ): 1 , ρ(x, y) ≤ c2 ϕ−1 { c (t), V(x,ϕ−1 { c (t)) { { { ρ(x,y) 1 −1 p(t, x, y) ≍ { V(x,ϕ−1 (t)) exp(− ϕ̄ −1 (t/ρ(x,y)) ), c2 ϕc (t) < ρ(x, y) < t∗ , c { c { { { t ρ(x, y) ≥ t∗ , { V(x,ρ(x,y))ϕ (ρ(x,y)) , j

where t∗ satisfies (β1,ϕc −1)/β2,ϕc −1 c3 ϕ−1 (ϕ−1 c (t) log c (t)/ϕj (t)) ≤ t∗

(β2,ϕc −1)/β1,ϕc −1 ≤ c4 ϕ−1 (ϕ−1 c (t) log c (t)/ϕj (t)),

and β1,ϕc and β2,ϕc are given in (2.1). In this case, one can check that only the information of ϕc on [1, ∞) is actually needed for the expression of HK(ϕj , ϕc ) because t/ρ(x, y) ≥ c3 > 0 holds when t > 1 and c2 ϕ−1 c (t) ≤ ρ(x, y) ≤ t∗ . Figure 1 indicates in this case which term is the dominant one for the estimate of p(t, x, y) in each region.

Figure 1: Dominant term in the heat kernel estimates HK(ϕj , ϕc ) for p(t, x, y) for the case in Remark 2.10.

(ii) The definition of HK(ϕj , ϕc ) in this case is different from HK(ϕc , ϕj ) in [20] for sym̄ x, y) in the case of metric diffusions with jumps. Denote by the heat kernel by p(t, symmetric diffusion with jumps. We say HK(ϕc , ϕj ) holds if (2.10) holds. Thus the expressions for HK(ϕj , ϕc ) for symmetric jump processes with lighter jump tails and HK(ϕc , ϕj ) for symmetric diffusions with jumps are exactly switched over the time interval (0, 1] and (1, ∞). The reason for this difference is as follows. For diffusions with jumps, due to the heavy tail property of jumps, the behaviors of the processes for large time are dominated by the pure jump part, and the behav-

20 | Z.-Q. Chen et al. iors for small time enjoy the continuous nature from diffusions, as well as some interactions with jumps. For symmetric pure jump processes with lighter tails considered in this case, the small time behavior of the processes is controlled by the jump kernels; however, since large jumps of the associated process are so light (as a typical example, one can consider symmetric jump processes on ℝd whose associated jump measure has finite second moments), it yields similar long time estimates as those for diffusions. (iii) The lower bound in HK− (ϕj , ϕc ) can be expressed more explicitly; namely, there are constants c1 , c2 > 0 so that p(j) (t, x, y), 0 < t ≤ 1, { { { { 1 t > 1, ρ(x, y) ≤ c2 ϕ−1 p(t, x, y) ≥ c1 { V(x,ϕ−1 (t)) , c (t), c { { { t −1 { V(x,ρ(x,y))ϕj (ρ(x,y)) , t > 1, ρ(x, y) > c2 ϕc (t). The next remark is on the formulation and proofs of Theorems 2.7 and 2.8. Remark 2.11. Motivated by Example 1.2, we start with two scaling functions ϕj and ϕc , which is quite natural. This allows us to treat the cases of 0 < r ≤ 1 and r > 1 separately with possibly different scaling indices. It also allows us to incorporate the heat kernel formulation considered in [3, 4]. The significance of this viewpoint is further illustrated by Example 4.1, where the local lower scaling index of the scale function ϕj on (1, ∞) is strictly larger than 1, while the local lower scaling index of the scale function ϕj on (0, 1) can take any value in (0, 2). Both Examples 1.2 and 4.1 are outside the main settings of [3, 4]. The most difficult case in the proof of Theorems 2.7 and 2.8 is when ϕ(r) = ϕj (r)1[0,1] +ϕc (r)1(1,∞) , that is, the case discussed in Remark 2.10. On the other hand, as mentioned in Remark 2.10(ii), the expressions of heat kernel estimates for the process studied in this case are the same as those for diffusions with jumps studied in [20] but with time interval t ≤ 1 and t > 1 switched. It turns out that some ideas and strategies from [20] on diffusions with jumps can be adapted in this case but there are also new ingredients needed to deal with symmetric pure jump processes having light tails in [21]. Our approach can yield more concise and explicit heat kernel estimate HK(ϕj , ϕc ) as mentioned in Remark 2.10(i). Furthermore, our approach also gives us the stable characterizations of parabolic Harnack inequalities (see Theorem 3.3 below), which are useful as indicated in Example 4.2.

3 Stability of parabolic Harnack inequalities Let Z := {Vs , Xs }s≥0 be the space–time process corresponding to X, where Vs = V0 − s. Denote the law of the space–time process Z starting from (t, x) by ℙ(t,x) . For every open subset D of [0, ∞) × M, define τD = inf{s > 0 : Zs ∉ D}.

Heat kernel estimates for general symmetric pure jump processes | 21

Definition 3.1. (i) We say that a Borel measurable function u(t, x) on [0, ∞) × M is parabolic (or caloric) on D = (a, b)×B(x0 , r) for the process X if there is a properly exceptional set 𝒩u associated with the process X so that for every relatively compact open subset U of D, u(t, x) = 𝔼(t,x) u(ZτU ) for every (t, x) ∈ U ∩ ([0, ∞) × (M\𝒩u )). (ii) We say that the parabolic Harnack inequality (PHI(ϕ)) holds for the process X, if there exist constants 0 < C1 < C2 < C3 < C4 , C5 > 1, and C6 > 0 such that for every x0 ∈ M, t0 ≥ 0, R > 0 and for every non-negative function u = u(t, x) on [0, ∞) × M that is parabolic on cylinder Q(t0 , x0 , ϕ(C4 R), C5 R) := (t0 , t0 + ϕ(C4 R)) × B(x0 , C5 R), ess supQ− u ≤ C6 ess infQ+ u,

(3.1)

where Q− := (t0 +ϕ(C1 R), t0 +ϕ(C2 R))×B(x0 , R) and Q+ := (t0 +ϕ(C3 R), t0 +ϕ(C4 R))× B(x0 , R). The next definition was introduced in [8] in the setting of graphs, and then extended in [13] to the general setting of metric measure spaces. Definition 3.2. We say that UJS holds if there is a symmetric function J(x, y) so that (1.7) holds, and there is a constant c > 0 such that for any x, y ∈ M and 0 < r ≤ ρ(x, y)/2, J(x, y) ≤

c ∫ J(z, y)μ(dz). V(x, r) B(x,r)

The following result is established in [21, Theorem 1.15], which extends the corresponding result in [18, Theorem 1.20 and Corollary 1.21] from jump kernels of mixture stable types to more general jump kernels. It gives the stable characterization of parabolic Harnack inequalities, as well as the relation between parabolic Harnack inequalities and two-sided heat kernel estimates. Theorem 3.3. Suppose that the metric measure space (M, ρ, μ) satisfies VD and RVD, and that the functions ϕc and ϕj satisfy (2.1) and (2.3). Let ϕ(r) be defined by (2.4). Then PHI(ϕ) ⇐⇒ PI(ϕ) + Jϕ,≤ + CSJ(ϕ) + UJS. Consequently, HK− (ϕj , ϕc ) ⇐⇒ PHI(ϕ) + Jϕj . If, additionally, the metric measure space (M, ρ, μ) is connected and satisfies the chain condition, then HK(ϕj , ϕc ) ⇐⇒ PHI(ϕ) + Jϕj .

22 | Z.-Q. Chen et al. Like [18, Theorem 1.20], we can obtain more equivalent statements for PHI(ϕ). But we will not go into details here for the sake of space consideration. We also note that similar to the setting of [18], the proof of Theorem 3.3 is proved under an additional assumption that the jumping measure J(dx, dy) in the nonlocal Dirichlet form (ℰ , ℱ ) in (1.4) is of the form J(dx, dy) = J(x, dy) μ(dx) on M × M \ diag. Under this assumption, it is shown in [18, Proposition 3.3] that PHI(ϕ) implies that J(x, dy) is absolutely continuous with respect to the measure μ(dy) for μ-a. e. x ∈ M and so J(dx, dy) is absolutely continuous with respect to μ(dx) μ(dy). Recently it was observed in [27] that the proof of [18, Proposition 3.3] can be refined to show that PHI(ϕ) always implies that J(dx, dy) is absolutely continuous with respect to the product measure μ(dx) μ(dy) on M × M \ diag.

4 Examples To illustrate the main results presented above, we give two more examples in this section. They are taken from Examples 5.2 and 5.3 from [21], where the reader can find details of the proofs for the assertions below. The first example shows that for Theorem 2.7, the scale function ϕc (r) does not need to be the scaling function corresponding to diffusion processes (e. g., see Example 1.2) even in the case of the Euclidean space. Example 4.1. Let M = ℝd and μ(dx) = dx. Consider the following jump kernel J(x, y) ≃

1

|x −

y|d ϕ

− y|)

j (|x

,

where ϕj is a strictly increasing continuous function with ϕj (0) = 0 and ϕj (1) = 1 so that (i) there are constants c1,ϕj , c2,ϕj > 0 and 0 < β1,ϕj ≤ β2,ϕj < 2 such that β1,ϕj

R c1,ϕj ( ) r

≤

β2,ϕj

ϕj (R)

R ≤ c2,ϕj ( ) ϕj (r) r

for all 0 < r ≤ R ≤ 1;

(4.1)

∗ ∗ ∗ ∗ (ii) there are constants c1,ϕ , c2,ϕ > 0 and 1 < β1,ϕ ≤ β2,ϕ < ∞ such that j j j j ∗ β1,ϕ

R ∗ c1,ϕ ( ) j r

j

≤

ϕj (R) ϕj (r)

∗ β2,ϕ

≤

R ∗ c2,ϕ ( ) j r

j

for all 1 ≤ r ≤ R.

(4.2)

Let (ℰ , ℱ ) be the regular symmetric pure jump Dirichlet form on L2 (ℝd ; dx) having above J(x, y) as its jump kernel, where ℱ = {f ∈ L2 (ℝd ; dx) : ℰ (f , f ) < ∞}. Define ϕ(r) := 1[0,1] (r)ϕj (r) + 1(1,∞) (r)ϕc (r),

Heat kernel estimates for general symmetric pure jump processes | 23

where 2 {r , ϕc (r) = { Φ(r) { Φ(1) ,

r ∈ (0, 1], r ∈ [1, ∞)

and Φ(r) =

r 2 ∫0

r2

s/ϕj (s) ds

,

r > 0.

It is shown in [21, Example 5.2] that HK(ϕj , ϕc ) holds for (ℰ , ℱ ) by Theorem 2.7.

The above assertion improves [3, Theorem 1.4 and Corollary 1.5], in which an extra condition that β1,ϕj > 1 is required. We also note that in this example, ϕc (r) does not

need to be comparable to the quadratic function r 󳨃→ r 2 for all r > 0. Nevertheless, we can treat symmetric pure jump forms with the growth order of ϕj (r) not necessarily strictly less than 2, as in [3]. By [10, Corollaries 2.6.2 and 2.6.4], ϕc (r) and ϕj (r) are ∗ comparable on [1, ∞) if and only if β2,β < 2, in which case, the heat kernel estimate j

HK(ϕj , ϕc ) is reduced to HK(ϕj ) in [17] (see Remark 2.9(i)). Therefore, the “diffusive

∗ scaling” appears in HK(ϕj , ϕc ) only when β2,ϕ ≥ 2. For example, when ϕj (r) = r α ∨ r 2 j for all r > 0 with α ∈ (0, 2), we can take

ϕc (r) := r 2 1{0≤r≤1} +

r2 1 log(e − 1 + r) {r>1}

and so ϕ(r) = r α 1{0≤r≤1} +

r2 1 . log(e − 1 + r) {r>1}

It holds that for any t > 1 and x, y ∈ ℝd , p(c) (t, x, y) ≍

ρ(x, y)2 1 exp(− ). t log(1 + t/ρ(x, y)) V(x, (t log(1 + t))1/2 )

This does not belong to the so-called (sub)-Gaussian estimates. Indeed, long time behavior of the process is superdiffusive, and its heat kernel estimates are “superGaussian”. Our second example is to illustrate our stable characterization of parabolic Harnack inequalities. The assertion of the example below is a counterpart of [13, Theorem 1.4], which is concerned with a local version of parabolic Harnack inequalities. One can use the assertion below to recover the parabolic Harnack inequalities for a large class of symmetric jump processes studied in [14].

24 | Z.-Q. Chen et al. Example 4.2. Let M = ℝd and μ(dx) = dx. Consider a nonnegative symmetric function J(x, y) on ℝd × ℝd such that 1 , |x − y|d ϕj (|x − y|) 1 J(x, y) ⪯ , |x − y|d+2

J(x, y) ≃

|x − y| ≤ 1, |x − y| > 1,

and sup

x∈ℝd

∫

|x − y|2 J(x, y) dy < ∞.

(4.3)

{|x−y|≥1}

Here, ϕj is a strictly increasing continuous function with ϕj (0) = 0 and ϕj (1) = 1 so that (4.1) holds with 0 < β1,ϕj ≤ β2,ϕj < 2.

Let (ℰ , ℱ ) be the regular symmetric pure jump Dirichlet form on L2 (ℝd ; dx) having the above J(x, y) as its jump kernel, where ℱ = {f ∈ L2 (ℝd ; dx) : ℰ (f , f ) < ∞}. It can be shown that PHI(ϕ) holds with ϕ(r) = ϕj (r)1{0≤r≤1} + r 2 1{r>1} , if and only if UJS holds for the jump kernel J(x, y) given above.

Bibliography [1]

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[12] Z.-Q. Chen, M. Fukushima, Symmetric Markov Processes, Time Change, and Boundary Theory (Princeton Univ. Press, Princeton and Oxford, 2012). [13] Z.-Q. Chen, P. Kim, T. Kumagai, On heat kernel estimates and parabolic Harnack inequality for jump processes on metric measure spaces. Acta Math. Sin. Engl. Ser. 25, 1067–1086 (2009). [14] Z.-Q. Chen, P. Kim, T. Kumagai, Global heat kernel estimates for symmetric jump processes. Trans. Am. Math. Soc. 363, 5021–5055 (2011). [15] Z.-Q. Chen, T. Kumagai, Heat kernel estimates for stable-like processes on d-sets. Stoch. Process. Appl. 108, 27–62 (2003). [16] Z.-Q. Chen, T. Kumagai, Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab. Theory Relat. Fields 140, 277–317 (2008). [17] Z.-Q. Chen, T. Kumagai, J. Wang, Stability of heat kernel estimates for symmetric non-local Dirichlet forms. To appear in Memoirs Amer. Math. Soc., available at arXiv:1604.04035. [18] Z.-Q. Chen, T. Kumagai, J. Wang, Stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms. To appear in J. European Math. Soc., Published online first: 2020-08-05, DOI: 10.4171/JEMS/996, available at arXiv:1609.07594. [19] Z.-Q. Chen, T. Kumagai, J. Wang, Elliptic Harnack inequalities for symmetric non-local Dirichlet forms. J. Math. Pures Appl. 125, 1–42 (2019). [20] Z.-Q. Chen, T. Kumagai, J. Wang, Heat kernel and parabolic Harnack inequalities for symmetric Dirichlet forms. To appear in Advances in Math., Published online first: 2020-7-07, 107269, https://doi.org/10.1016/j.aim.2020.107269, available at arXiv:1908.07650. [21] Z.-Q. Chen, T. Kumagai, J. Wang, Heat kernel estimates for general symmetric pure jump Dirichlet forms (2019). Preprint, available at arXiv:1908.07655. [22] M. Fukushima, Y. Oshima, M. Takeda, Dirichlet Forms and Symmetric Markov Processes, 2nd rev. and ext. edn. (de Gruyter, Berlin, 2011). [23] A. Grigor’yan, E. Hu, J. Hu, Two-sided estimates of heat kernels of jump type Dirichlet forms. Adv. Math. 330, 433–515 (2018). [24] A. Grigor’yan, A. Telcs, Two-sided estimates of heat kernels on metric measure spaces. Ann. Probab. 40, 1212–1284 (2012). [25] B. M. Hambly, T. Kumagai, Transition density estimates for diffusion processes on post critically finite self-similar fractals. Proc. Lond. Math. Soc. 78, 431–458 (1999). [26] T. Kumagai, Random Walks on Disordered Media and Their Scaling Limits. Lect. Notes Math., vol. 2101 (Springer, 2014). [27] G. Liu, M. Murugan, Parabolic Harnack inequality implies the existence of jump kernel (2020). Preprint, available at arXiv:2005.12450. [28] M. Murugan, L. Saloff-Coste, Transition probability estimates for long range random walks. N.Y. J. Math. 21, 723–757 (2015). [29] M. Murugan, L. Saloff-Coste, Heat kernel estimates for anomalous heavy-tailed random walks. Ann. Inst. Henri Poincaré Probab. Stat. 55, 697–719 (2019).

Alexander Grigor’yan, Eryan Hu, and Jiaxin Hu

The pointwise existence and properties of heat kernel Abstract: We consider a semigroup acting on the function space L1 based on a measure space. Assuming that the semigroup satisfies the L1 –L∞ ultracontractivity property, we prove that it possesses an integral kernel that is defined pointwise and has some nice properties, including the joint measurability and the continuity in one variable. We apply this result to a heat semigroup associated with a regular Dirichlet form on the space L2 . Keywords: Semigroup, ultracontractivity, heat kernel, Dirichlet form MSC 2010: Primary 35K08, Secondary 60J35

Contents 1 2 3 4 5 6

Introduction | 27 Main results | 29 Preliminaries | 33 Proof of Theorem 2.1 | 39 Applications | 51 Proof of Theorem 2.2 | 64 Bibliography | 69

1 Introduction In this paper we are concerned with the existence and properties of integral kernels of semigroups acting on the space L1 := L1 (M, μ) where (M, μ) is a measure space. Assume that a semigroup {Pt }t>0 is L1 –L∞ locally ultracontractive, that is, there exist an at most countable family 𝒮 of open sets with M = ⋃U∈𝒮 U and a function ϕ : 𝒮 × (0, ∞) 󳨃→ ℝ+ Acknowledgement: AG was supported by SFB1283 of the German Research Council. EH was supported by the National Natural Science Foundation of China (No. 11801403). JH was supported by NSFC (No. 11871296), by SFB 1283, and by Tsinghua University Initiative Scientific Research Program. Alexander Grigor’yan, Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany, e-mail: [email protected] Eryan Hu, Center for Applied Mathematics, Tianjin University, Tianjin 300072, China, e-mail: [email protected] Jiaxin Hu, Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China, e-mail: [email protected] https://doi.org/10.1515/9783110700763-002

28 | A. Grigor’yan et al. such that ‖Pt f ‖L∞ (U) ≤ ϕ(U, t)‖f ‖L1

(1.1)

for all U ∈ 𝒮 , f ∈ L1 and t > 0. The following two questions arise naturally: (1) Does the semigroup {Pt }t>0 possess an integral kernel that is a jointly measurable function pt (x, y) on M × M such that Pt f (x) = ∫ pt (x, y)f (y) dμ(y),

(1.2)

M

for all f ∈ L1 , t > 0 and μ-almost all x ∈ M? (2) Once the integral kernel exists, what further properties does it have? In the special case when 𝒮 = {M}, the existence of the integral kernel was dealt with in many papers but a common drawback of most of them is that they do not address the joint measurability of pt (x, y) in (x, y) without which the notion of integral kernel is unusable. For example, this question was neglected in the widely-cited paper [3], where diagonal upper bounds of pt (x, y) were obtained. Later on, some efforts have been made in this direction in [2, Theorem 3.1], [7, Theorem 2.10], [6, Corollary 3.8], [14, Theorems 2 and 1], [1, Proposition 4.14]. Although the joint measurability was mentioned in the paragraph following [7, Theorem 2.10], it was based on the result of [2, Theorem 3.1] that had a gap in the proof. In the setting of symmetric Dirichlet forms, the joint measurability of the integral kernel was proved in [6, Lemma 3.3, Corollary 3.8], but the proof uses a number of quite advanced tools, which makes it not self-contained. In this paper, we fix the problem of a joint measurability in a more general setting. More precisely, we start with a semigroup {Pt }t>0 in L1 satisfying the L1 –L∞ ultra-contractivity property, and show the existence of an integral kernel of the semigroup {Pt }t>0 that is defined pointwise and possesses some other nice properties, see Theorem 2.1. We apply our theorem to the most important and interesting class of L1 -semigroups that arise from Dirichlet forms and are referred to as heat semigroups. Their integral kernels are called heat kernels. We show in Theorem 2.2 the pointwise existence of the heat kernels. Heat kernels have been widely used in the literature for many purposes and in various settings, and our result provides a solid foundation for this concept. Notation. The term “for each x” means “for a fixed but an arbitrary x”. The term “for all (or any) x” means “for an arbitrary x” but the statement which follows is independent of the choice of x. For simplicity, set Lp := Lp (M, μ) for 1 ≤ p ≤ ∞ by omitting M, μ. The identities and inequalities between Lp -functions are understood μ-almost everywhere in M.

Existence of the heat kernel | 29

2 Main results In this section, we state the main results of this paper. Let (M, μ) be a measure space. Recall that the support of a measure ν on M is the smallest closed set outside which ν vanishes: supp[ν] := M \ ∪{O ⊂ M : O is open with ν(O) = 0}. For a nonnegative measurable function f , the induced measure f .ν is defined by d(f .ν)(x) = f (x)dν(x). In particular, for a measurable subset F of M, we set m := 1F .μ. Then by definition m(Ω) = ∫ dm = ∫ 1F dμ = ∫ dμ = μ(F ∩ Ω) Ω

Ω

F∩Ω

for any measurable subset Ω. A point x belongs to supp[1F .μ] = supp[m] if and only if there exists an open neighborhood Ux of x such that m(Ux ) = μ(Ux ∩ F) > 0. A subset F of M is said to be regular if supp[1F .μ] = F, that is, μ(Ux ∩ F) > 0 for any x ∈ F and any open neighborhood Ux of x. A regular set excludes any unnecessary isolated point in M with zero measure (called nonatomic point). For a sequence of subsets {Fk }∞ k=1 of M, denote by C({Fk }) := {u : u|Fk is continuous for each k}. An increasing sequence of closed subsets {Fk }∞ k=1 of M is called a μ-nest of M if lim μ(M \ Fk ) = 0,

k→∞

and is regular if each Fk is regular. We say that a function pt (x, y) pointwise defined in (0, ∞) × M × M satisfies condition (Ap ) for some 1 ≤ p ≤ ∞, if there exists a regular μ-nest {Fn }∞ n=1 of M such that the following properties are true: for each t, s > 0 and each x, y in M, (1) (measurability) pt (⋅, ⋅) is jointly measurable in M × M; (2) (continuity and integrability in one variable) pt (x, ⋅) ∈ C({Fn }) ∩ Lp

and pt (⋅, y) ∈ C({Fn }) ∩ Lp ;

(2.1)

(3) (continuity in integral forms) for each f ∈ Lp , ∫ pt (⋅, z)f (z) dμ(z) ∈ C({Fn }) M

and

∫ pt (z, ⋅)f (z) dμ(z) ∈ C({Fn }); M

(2.2)

30 | A. Grigor’yan et al. (4) (semigroup property) pt+s (x, y) = ∫ pt (x, z)ps (z, y) dμ(z).

(2.3)

M

Let T0 ∈ (0, ∞]. Recall that a family of linear bounded operators {Pt }t∈(0,T0 ) from Lp to Lp for 1 ≤ p ≤ ∞ is called a semigroup on Lp if for any t1 , t2 > 0 with t1 + t2 < T0 , Pt1 +t2 = Pt1 Pt2 ,

(2.4)

where (2.4) is understood such that for any f ∈ Lp , there is a null (or measure zero) set 𝒩t1 ,t2 ,f depending on t1 , t2 , f such that Pt1 +t2 f (x) = Pt1 (Pt2 f )(x) holds for any point x outside the set 𝒩t1 ,t2 ,f . Note that the operator Pt may not be defined at t = 0. A jointly measurable function pt (x, y) on (0, T0 ) × M × M is said to be an integral kernel of a semigroup {Pt }t∈(0,T0 ) on Lp for 1 ≤ p ≤ ∞ if Pt f (x) = ∫ pt (x, y)f (y) dμ(y)

(2.5)

M p

for μ-almost all x in M when f ∈ L . An integral kernel may not be defined pointwise. ̂ t }t∈(0,T ) be a family of operFor a semigroup {Pt }t∈(0,T0 ) on Lp for 1 ≤ p < ∞, let {P 0 ators defined by ̂ t g) (Pt f , g) = (f , P

(2.6)

p for any f ∈ Lp , g ∈ Lq , where (⋅, ⋅) is the usual inner product in the L2 space, and q := p−1 ̂ t defined in this way is linear. It is also bounded is the conjugate of p. Clearly, each P

from Lq to Lq :

̂ t ‖Lq →Lq := sup ‖P ̂ t g‖Lq ≤ ‖Pt ‖Lp →Lp , ‖P ‖g‖Lq =1

(2.7)

since for any f ∈ Lp , g ∈ Lq 󵄨󵄨 ̂ 󵄨 󵄨 󵄨 󵄨󵄨(Pt g, f )󵄨󵄨󵄨 = 󵄨󵄨󵄨(Pt f , g)󵄨󵄨󵄨 ≤ ‖Pt f ‖Lp ‖g‖Lq ≤ ‖Pt ‖Lp →Lp ‖f ‖Lp ‖g‖Lq ,

(2.8)

which gives that 󵄨󵄨 ̂ t g‖Lq = sup 󵄨󵄨󵄨(P ̂ ‖P 󵄨 t g, f )󵄨󵄨 ≤ ‖Pt ‖Lp →Lp ‖g‖Lq . ‖f ‖Lp =1

̂ t }t∈(0,T ) satisfies the semigroup property: Moreover, {P 0 ̂ t+s = P ̂t P ̂s P

for each t, s > 0,

(2.9)

Existence of the heat kernel | 31

since for any f ∈ Lp , g ∈ Lq , ̂ t+s g) = (Pt+s f , g) = (Ps (Pt f ), g) = (Pt f , P ̂ s g) = (f , P ̂t P ̂ s g), (f , P ̂ t+s g = P ̂t P ̂ s g almost everywhere in M for any g ∈ Lq . The family from which we see P ̂ t }t∈(0,T ) is called the dual semigroup of {Pt }t∈(0,T ) on Lp for 1 ≤ p < ∞. {P 0 0 We call the triple (M, d, μ) a metric measure space if (M, d) is a locally compact, separable metric space, and μ is a Radon measure on M with full support. We will work on a semigroup {Pt }t∈(0,T0 ) on the space L1 (M, μ). Theorem 2.1. Let T0 ∈ (0, ∞], {Pt }t∈(0,T0 ) be a semigroup on L1 (M, μ) for a metric meâ t }t∈(0,T ) be its dual semigroup defined by (2.6) such that sure space (M, d, μ), and let {P 0 ̂ each Pt (t ∈ (0, T0 )) is bounded from L1 to L1 . Assume that there exist a countable family 𝒮 of open sets with M = ⋃U∈𝒮 U and a function φ : 𝒮 × (0, T0 ) 󳨃→ ℝ+ such that, for each t ∈ (0, T0 ), U ∈ 𝒮 and each f ∈ L1 , ‖Pt f ‖L∞ (U) ≤ φ(U, t)‖f ‖L1 , ̂ t f ‖L∞ (U) ≤ φ(U, t)‖f ‖ 1 . ‖P L

(2.10) (2.11)

Then {Pt }t∈(0,T0 ) possesses an integral kernel pt (x, y) pointwise defined in (0, ∞) × M × M that satisfies condition (Ap ) with p = 1 for some regular μ-nest {Fn }∞ n=1 in M, and pt (x, y) = 0

for any t > 0

(2.12)

whenever one of points x, y lies outside ⋃∞ n=1 Fn . Moreover, for each t ∈ (0, T0 ) and each x ∈ U, y ∈ M, 󵄨󵄨 󵄨 󵄨󵄨pt (x, y)󵄨󵄨󵄨 ≤ φ(U, t)

󵄨 󵄨 and 󵄨󵄨󵄨pt (y, x)󵄨󵄨󵄨 ≤ φ(U, t).

(2.13)

We will prove Theorem 2.1 in Section 4. Below we turn to consider an interesting class of semigroups in L1 , the heat semigroup {Pt }t>0 , whose integral kernel is called a heat kernel. We are concerned with the existence of a pointwise defined heat kernel. A strongly continuous, contractive, symmetric, and sub-Markovian semigroup {Pt }t>0 on L2 is called a heat semigroup, that is, for any t > 0 and f , g ∈ L2 , – (strongly continuous) lim ‖Pt f − f ‖L2 = 0;

(2.14)

‖Pt f ‖L2 ≤ ‖f ‖L2 ;

(2.15)

t→0

–

– –

(contractive)

(symmetric) (Pt f , g) = (f , Pt g); (sub-Markovian) Pt f ≥ 0 when f ≥ 0, and Pt f ≤ 1 when f ≤ 1, where inequalities are understood in the sense of μ-almost everywhere.

32 | A. Grigor’yan et al. Recall that a Dirichlet form (ℰ , ℱ ) on L2 is a bilinear form such that, for any u, v ∈ ℱ , – ℱ is dense in L2 and complete in the norm of ℰ11/2 where 1/2

2

ℰ1 (u) := (‖u‖2 + ℰ (u))

– –

with ℰ (u) := ℰ (u, u);

(ℰ , ℱ ) is positive definite, ℰ (u) ≥ 0, and symmetric, ℰ (u, v) = ℰ (v, u); the function u+ ∧ 1 belongs to ℱ , and ℰ (u+ ∧ 1) ≤ ℰ (u).

A heat semigroup on L2 and a Dirichlet form on L2 are mutually corresponding (cf. [5, Theorem 1.4.1, p. 25]). Any heat semigroup {Pt }t>0 can be extended to be contractive both on L1 (cf. [5, p. 37]) and on L∞ , and therefore, is contractive on Lp for any 1 ≤ p ≤ ∞ by using the Riesz–Thorin interpolation theorem. For simplicity, we still denote its extension by {Pt }t>0 . A family of functions pt (x, y) on (0, ∞) × M × M is called a heat kernel (or a symmetric transition density) if the following conditions are satisfied: for any s, t > 0 and μ-almost all x, y ∈ M, (1) (measurability) pt (⋅, ⋅) is jointly measurable on M × M; (2) (Markovianity) pt (x, y) ≥ 0 and ∫ pt (x, y) dμ(y) ≤ 1; M

(3) (symmetry) pt (x, y) = pt (y, x); (4) (semigroup property) ps+t (x, y) = ∫ ps (x, z)pt (z, y) dμ(z). M

(5) (identity approximation) for any f ∈ L2 , L2

∫ pt (x, y)f (y)dμ(y)󳨀→f (x)

as t → 0 + .

M

For a Dirichlet form (ℰ , ℱ ) on L2 , an increasing sequence of closed subsets {Fk }∞ k=1 of M is called an ℰ -nest of M if lim cap(M \ Fk ) = 0,

k→∞

where cap(A) is the capacity of a measurable set A defined by cap(A) := inf{ℰ (u) + ‖u‖22 : u ∈ ℱ , u ≥ 1 a. e. on A}. Note that by definition μ(A) ≤ cap(A) for any measurable subset A of M.

(2.16)

Existence of the heat kernel | 33

Theorem 2.2. Let (ℰ , ℱ ) be a regular Dirichlet form on L2 (M, μ) for a metric measure space (M, d, μ), and let {Pt }t>0 be the associated heat semigroup on L2 . Fix T0 ∈ (0, ∞] and 1 ≤ p ≤ 2. Assume that there exist a countable family 𝒮 of open sets with M = ⋃U∈𝒮 U and a function φ : 𝒮 × (0, T0 ) 󳨃→ ℝ+ such that, for each t ∈ (0, T0 ), U ∈ 𝒮 and each f ∈ Lp ∩ L2 , ‖Pt f ‖L∞ (U) ≤ φ(U, t)‖f ‖Lp .

(2.17)

Then {Pt }t>0 possesses a heat kernel pt (x, y) pointwise defined in (0, ∞) × M × M that satisfies condition (Ap ) and some regular ℰ -nest {Fn }∞ n=1 of M, and pt (x, y) = 0

for any t > 0

(2.18)

whenever one of points x, y lies outside ⋃∞ n=1 Fn . Moreover, for each t ∈ (0, T0 ) and x ∈ U, 󵄩󵄩 󵄩 󵄩󵄩pt (x, ⋅)󵄩󵄩󵄩Lp󸀠 ≤ φ(U, t), where p󸀠 =

p p−1

(2.19)

is the Hölder conjugate of p, and for any 1 ≤ q ≤ p󸀠 , (q−1)(p−1) 󵄩󵄩 󵄩 . 󵄩󵄩pt (x, ⋅)󵄩󵄩󵄩Lq ≤ (φ(U, t))

(2.20)

We prove Theorem 2.2 in Section 6.

3 Preliminaries For a topological space X, let ℬ(X) be a collection of all Borel sets of X and 𝒳 a sigmaalgebra on X. The following proposition shows the joint measurability of a function on a product space, which is a modification of [9, Lemma 9.2, p. 122]. Similar results on measurability are addressed in [11, 10, 13]. We shall use this conclusion to prove the joint measurability of the integral kernel of a semigroup on L1 (see the proofs of Theorems 2.1 and 2.2 below). Proposition 3.1. Let (X, d, 𝒳 ) be a separable metric space with ℬ(X) ⊂ 𝒳 and (Y, 𝒴 ) a measurable space. For two sets A ∈ 𝒳 and B ∈ 𝒴 , assume that f is a real-valued function pointwise defined on X × Y satisfying the following conditions: (i) f (x, y) = 0 for any (x, y) ∈ (A × B)c ; (ii) f (x, ⋅) is measurable in B for each x ∈ A; (iii) f (⋅, y) restricted on A is continuous for each y ∈ B. Then f is jointly measurable with respect to (X × Y, 𝒳 × 𝒴 ).

34 | A. Grigor’yan et al. Proof. Without loss of generality, we assume that f (x, y) ≥ 0 for all (x, y) ∈ X × Y. Otherwise, we shall consider f+ := f ∨ 0 and f− := (−f ) ∨ 0 separately. Since X is separable, we choose a countable set {xn }∞ n=1 ⊂ A such that for each ∞ 1 k ≥ 1, A ⊂ ⋃n=1 B(xn , k ), that is, A can be covered by balls {B(xn , k1 )}n≥1 of the same radius k1 . For each k ≥ 1, let A1,k := B(x1 , k1 ) ∩ A and n−1 1 1 An,k := (B(xn , ) \ ⋃ B(xi , )) ∩ A k k i=1

for n ≥ 2. Some of these sets may be empty, but the following argument will still work. 1 Then the family of sets {An,k }∞ n=1 are disjoint, An,k ⊂ B(xn , k ), and ∞

A = ⨆ An,k .

(3.1)

n=1

Define a family of functions fk on X × Y by ∞

fk (x, y) := ∑ f (xn , y)1An,k (x), n=1

x ∈ X, y ∈ Y, k ≥ 1.

We claim that 1. For each k, the function fk is jointly measurable; 2. For each x ∈ X, y ∈ Y, we have lim fk (x, y) = f (x, y).

k→∞

(3.2)

Indeed, by assumptions (i), (ii), for each n ≥ 1 and each a ≥ 0, {y ∈ Y : f (xn , y) ≤ a} = {y ∈ B : f (xn , y) ≤ a} ∪ Bc ∈ 𝒴 , which implies that f (xn , y)1An,k (x) is jointly measurable because both sets Bc and {y ∈ B : f (xn , y) ≤ a} belong to the sigma-algebra 𝒴 . Therefore, the function fk , as the sum of products of two measurable functions f (xn , y) and 1An,k (x), is jointly measurable, thus proving our first claim. To show our second claim, note that if x ∈ Ac , y ∈ Y, or if x ∈ A, y ∈ Bc , by the definition of fk , we see that fk (x, y) = 0 = f (x, y), and thus (3.2) is obvious. On the other hand, let x ∈ A, y ∈ B. By assumption (iii), for each ε > 0, there exists some K(x, y, ε) > 0 such that for each k ≥ K(x, y, ε) and any x 󸀠 ∈ B(x, k1 ) ∩ A, 󵄨󵄨 󸀠 󵄨 󵄨󵄨f (x , y) − f (x, y)󵄨󵄨󵄨 < ε. Since x ∈ A = ⋃∞ n=1 An,k , we see that for each k ≥ K(x, y, ε), there exists some n ≥ 1 such that x ∈ An,k , that is, x ∈ B(xn , k1 ) ∩ A, which in turn gives that 1 xn ∈ B(x, ) ∩ A. k

Existence of the heat kernel | 35

It follows that |f (xn , y) − f (x, y)| < ε, and hence, by definition of fk , 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨fk (x, y) − f (x, y)󵄨󵄨󵄨 = 󵄨󵄨󵄨f (xn , y) − f (x, y)󵄨󵄨󵄨 < ε, thus showing that (3.2) is also true. This proves our second claim. Finally, by our claim, the function f , as a limit of jointly measurable functions fk , is also jointly measurable on X × Y. Let (M, μ) be a separable measure space. The following says that any μ-nest of M, after removing all the unnecessary points, each of which has an open neighborhood of zero measure, will become a regular μ-nest of M. Proposition 3.2. Given a μ-nest {Fk } of M, let Fk󸀠 = supp[1Fk .μ] for each k. Then Fk󸀠 ⊂ Fk for each k ≥ 1, and {Fk󸀠 } is a regular μ-nest. Proof. Note that each Fk󸀠 is closed. For each k, the fact that 1Fk .μ(Fkc ) = 0 implies Fkc ⊂ ∪{O ⊂ M : O is open with 1Fk .μ(O) = 0}, thus showing that Fk󸀠 = supp[1Fk .μ] = M \ ∪{O ⊂ M : O is open with 1Fk .μ(O) = 0} ⊂ Fk . For an open set O ⊂ M, if 1Fk+1 .μ(O) = 0 then 1Fk .μ(O) = 0, and, hence, the set ∪{O ⊂ M : O is open with 1Fk+1 .μ(O) = 0} is contained in ∪{O ⊂ M : O is open with 1Fk .μ(O) = 0}, thus showing that 󸀠 Fk+1 = supp[1Fk+1 .μ] ⊃ supp[1Fk .μ] = Fk󸀠 .

(3.3)

On the other hand, since M is separable, by definition of Fk󸀠 , there exists a countable ∞ 󸀠 c family of open sets {Oi }∞ i=1 with μ(Fk ∩ Oi ) = 0, i ≥ 1 such that (Fk ) ⊂ ⋃i=1 Oi . It follows that c

μ(Fkc ) ≤ μ((Fk󸀠 ) )

c

c

= μ(Fkc ∩ (Fk󸀠 ) ) + μ(Fk ∩ (Fk󸀠 ) ) ∞

≤ μ(Fkc ) + ∑ μ(Fk ∩ Oi ) =

μ(Fkc ),

i=1

(3.4)

which gives that μ((Fk󸀠 )c ) = μ(Fkc ) ↓ 0 as k ↑ ∞. This together with (3.3) shows that {Fk󸀠 } is a μ-nest of M.

36 | A. Grigor’yan et al. It remains to prove that {Fk󸀠 } is regular. Indeed, we have by (3.4) c

μ(Fk \ Fk󸀠 ) = μ((Fk󸀠 ) ) − μ(Fkc ) = 0, showing that for any open set O, μ(Fk ∩ O) = μ(Fk󸀠 ∩ O) + μ((Fk \ Fk󸀠 ) ∩ O) = μ(Fk󸀠 ∩ O). Hence, we see that the following two sets are identical: ∪{O ⊂ M : O is open with 1Fk .μ(O) = 0} and ∪{O ⊂ M : O is open with 1F 󸀠 .μ(O) = 0}, k

which yields that Fk󸀠 = supp[1Fk .μ] = supp[1F 󸀠 .μ]. Thus {Fk󸀠 } is regular. k

We introduce the notion of the μ-quasicontinuity of a function. Definition 3.3. For a measure μ, we say that a function u on M is μ-quasicontinuous (or nearly μ-continuous) if for any ε > 0 there is an open set G ⊂ M such that μ(G) < ε and u|M\G is finite continuous. Here u|M\G is the restriction of u to M \ G. Proposition 3.4. For a function u pointwise defined on M, the following two conditions are equivalent: (i) u is μ-quasicontinuous. (ii) There is a (regular) μ-nest {Fk } such that u ∈ C({Fk }). Proof. (i) ⇒ (ii). Since u is μ-quasicontinuous, for each k ≥ 1, there is an open set Gk such that μ(Gk ) < k1 , and u|M\Gk is continuous. Let k

c

k

F̃k := (⋂ Gj ) = ⋃ Gjc j=1

j=1

for k ≥ 1.

Then {F̃k }k≥1 is increasing, and k 1 μ(M \ F̃k ) = μ(⋂ Gj ) ≤ μ(Gk ) ≤ → 0 k j=1

as k ↑ ∞,

showing that {F̃k } is a μ-nest on M. Note that the restriction of function u on each set k

F̃k = ⋃ Gjc j=1

Existence of the heat kernel | 37

is continuous. Denote Fk := supp[1F̃ .μ]. k

By Proposition 3.2, the μ-nest {Fk } is regular. Note that u ∈ C({Fk }), since u restricted on F̃k (⊃ Fk ) is continuous. (ii) ⇒ (i). Assume u ∈ C({Fk }) for a μ-nest {Fk }. For any ε > 0, choose k to be large enough such that μ(M \ Fk ) < ε. For such k, let G := M \ Fk . Then G is open, μ(G) < ε, and u|Gc = u|Fk is continuous. Thus u is μ-quasicontinuous by definition. Lemma 3.5. The following statements are true: (i) Let S = {ul }l≥1 be a countable family of μ-quasicontinuous functions on M. Then there is a common regular μ-nest {Fk } such that S ⊂ C({Fk }). (ii) Let {Fk } be a regular μ-nest and u belongs to C({Fk }). If u ≥ 0 μ-almost everywhere on an open set U, then u(x) ≥ 0 for every point x ∈ U ∩ (⋃∞ k=1 Fk ). Lemma 3.5(i) says that any μ-quasicontinuous function belongs to C({Fk }) for some regular μ-nest {Fn } of M, when S contains only one function, and at the same time, all countable μ-quasicontinuous functions share a common regular μ-nest. Lemma 3.5(ii) says that an almost everywhere nonnegative function in an open set is nonnegative everywhere in a slightly smaller subset. Proof. (i) For each l ≥ 1, since ul is μ-quasicontinuous, by Proposition 3.4, one can (l) (l) c 1 choose a μ-nest {Fk(l) }∞ k=1 such that ul ∈ C({Fk }) and μ((Fk ) ) < 2l k . For k ≥ 1, let ∞

Fk := ⋂ Fk(l) .

(3.5)

l=1

Clearly, each Fk is closed because so is Fk(l) for any l, k ≥ 1. Moreover, {Fk }∞ k=1 is increas(l) ing because Fk(l) ⊂ Fk+1 for any k, l ≥ 1. Since by (3.5) ∞

c

∞

c

∞

μ(Fkc ) = μ(⋃(Fk(l) ) ) ≤ ∑ μ((Fk(l) ) ) ≤ ∑ l=1

l=1

l=1

1 1 = ↓0 l 2k k

as k ↑ ∞,

we see that {Fk } is a μ-nest. Note that S ⊂ C({Fk }) since ul |F (l) is continuous and Fk ⊂ Fk(l) k

for any l ≥ 1 and k ≥ 1. Let {Fk󸀠 } be the regularization of {Fk } as in Proposition 3.2. Clearly, Fk󸀠 ⊂ Fk ⊂ Fk(l)

for any k, l ≥ 1.

(3.6)

Then S ⊂ C({Fk󸀠 }), thus showing (i) by relabeling the notion Fk󸀠 by Fk . (ii) Suppose that there is a point x ∈ U ∩ Fk such that u(x) < 0. Since u|Fk is continuous on each Fk , there is an open neighborhood Ux ⊂ U of x such that u(y) < 0

38 | A. Grigor’yan et al. for any point y ∈ Ux ∩ Fk . Since {Fk } is regular, we have μ(Ux ∩ Fk ) > 0, thus showing that u is strictly negative in a subset of U with positive measure, a contraction to our assumption. Remark 3.6. For a regular μ-nest {Fk } of M and for a pointwise defined function u on M such that u ∈ C({Fk }) and u = 0 outside ∪∞ k=1 Fk , it is straightforward to see by Lemma 3.5(ii) that if u ≥ 0 μ-almost everywhere in M, then u ≥ 0 pointwise in M. We will frequently use this fact. The following says that any function in the space Lp for 1 ≤ p < ∞ admits some μ-quasicontinuous modification. Let C(M) be the collection of all continuous functions on M, and C0 (M) the collection of all continuous functions with compact supports on M. Lemma 3.7. Let (M, d, μ) be a metric measure space. Any function u from the space Lp ̃ , that is, there exists a regular for p ∈ [1, ∞) has a μ-quasicontinuous modification u ∞ ̃ ∈ C({Fk }) and u = u ̃ almost everywhere in M. μ-nest {Fk }k=1 of M such that u Proof. For any v ∈ C(M) ∩ Lp and any λ > 0, the set G := {x ∈ M : |v(x)| > λ} is open and μ(G) ≤ ∫ G

‖v‖pLp |v|p dμ ≤ . λp λp

(3.7)

Since C0 (M) is dense in Lp , for any u ∈ Lp , we can choose a sequence of functions {uk } ⊂ C0 (M) such that ‖uk − u‖Lp → 0 as k ↑ ∞. Without loss of generality, we assume that for each k, ‖uk+1 − uk ‖pLp ≤ 2−(p+1)k . Denote 󵄨 󵄨 Gk := {x ∈ M : 󵄨󵄨󵄨uk+1 (x) − uk (x)󵄨󵄨󵄨 > 2−k }. Clearly, the set Gk is open. Using the inequality (3.7) with G = Gk , λ = 2−k and v = uk+1 − uk , we obtain μ(Gk ) ≤ 2−k . For each k, let ∞

Ek := ⋂ Glc . l=k

Then {Ek } is a μ-nest of M, since each Ek is closed, {Ek } is increasing, and ∞

∞

∞

l=k

l=k

l=k

μ(Ekc ) = μ(⋃ Gl ) ≤ ∑ μ(Gl ) ≤ ∑ 2−l = 2−k+1 → 0

as k → ∞.

Existence of the heat kernel | 39

Moreover, we have for any x ∈ Ek and any n > m ≥ k + 1 ∞

󵄨 󵄨 󵄨 󵄨󵄨 −m+1 ≤ 2−k , 󵄨󵄨un (x) − um (x)󵄨󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨ul+1 (x) − ul (x)󵄨󵄨󵄨 = 2 l=m

thus showing that the functions un uniformly converge as n → ∞ on each Ek . Define ̃ (x) := lim un (x) u n→∞

∞

for x ∈ ⋃ Ek . k=1

̃ ∈ C({Ek }) and u = u ̃ almost everywhere in M. Finally, by regularization Then u ̃ is a in Proposition 3.2, there is a regular μ-nest {Fk } of M with Fk ⊂ Ek , and u μ-quasicontinuous modification of u. The proof is complete. We remark that Lemma 3.7 is different from Lusin’s theorem1 in that μ here is not necessarily finite. If μ is finite on M, that is, if μ(M) < ∞, then any measurable function u on M admits μ-quasicontinuous modification by directly applying Lusin’s theorem. In this case, Lusin’s theorem is sharper than Lemma 3.7, as the function u in Lusin’s theorem is assumed to be measurable (instead of u ∈ Lp ). Lemmas 3.5 and 3.7 are respectively motivated by [5, Theorem 2.1.2, p. 69] and [5, Theorem 2.1.3, p. 71], wherein a regular Dirichlet form is assumed to exist in L2 . Here we do not assume the existence of a Dirichlet form.

4 Proof of Theorem 2.1 In this section, we prove Theorem 2.1. Proof of Theorem 2.1. The proof is quite long. We divide the proof into four steps. ̂ t to all t > 0 when T0 < ∞. Step 1. We will extend the definitions of Pt and P Indeed, we need only to consider Pt . Let us first extend the definition of Pt to all t ∈ [T0 , 2T0 ). For t ∈ [T0 , 2T0 ), let Pt f = Pt/2 Pt/2 f ,

f ∈ L1 .

(4.1)

Note that in the above definition Pt is well defined since t/2 ∈ (0, T0 ) and Pt/2 is bounded from L1 to L1 . Moreover, ‖Pt ‖L1 󳨃→L1 ≤ (‖Pt/2 ‖L1 󳨃→L1 )2 < ∞. 1 Lusin’s theorem. Let M be a Hausdorff space, 𝒜 a σ-algebra containing ℬ(M), μ a regular measure on 𝒜, and let u be an 𝒜-measurable function on M. Let A ∈ 𝒜 with 0 < μ(A) < ∞. Then, for any ε > 0, there exists a compact set K ⊂ A such that μ(A \ K) < ε and u|K is continuous.

40 | A. Grigor’yan et al. Let us verify that {Pt }t∈(0,2T0 ) satisfies the semigroup property. Indeed, since {Pt }t∈(0,T0 ) satisfies the semigroup property, as, by (4.1), we have for any t1 , t2 ∈ (0, 2T0 ) with t1 + t2 < 2T0 and f ∈ L1 , Pt1 Pt2 f = Pt1 /2 Pt1 /2 Pt2 /2 Pt2 /2 f = Pt1 /2 P(t1 +t2 )/2 Pt2 /2 f

= Pt1 /2 Pt2 /2 Pt1 /2 Pt2 /2 f = P(t1 +t2 )/2 P(t1 +t2 )/2 f = Pt1 +t2 f .

Hence, we have extended the semigroup {Pt }t∈(0,T0 ) on L1 to the semigroup {Pt }t∈(0,2T0 ) . Repeating the above arguments, we can further extend it to {Pt }t>0 . Let us extend (2.10). Indeed, by (2.10), we have for any U ∈ 𝒮 and t ≥ T0 , ‖Pt f ‖L∞ (U) = ‖PT0 /2 Pt−T0 /2 f ‖L∞ (U)

≤ φ(U, T0 /2)‖Pt−T0 /2 f ‖L1

≤ φ(U, T0 /2)‖Pt−T0 /2 ‖L1 󳨃→L1 ‖f ‖L1 ,

f ∈ L1 .

This shows that for any t > 0, ̃(U, t)‖f ‖L1 , ‖Pt f ‖L∞ (U) ≤ φ

f ∈ L1 ,

(4.2)

where ̃(U, t) := { φ

φ(U, t), φ(U, T0 /2)‖Pt−T0 /2 ‖L1 󳨃→L1

for t ∈ (0, T0 ), for t ∈ [T0 , ∞).

̂ t }t∈(0,T ) to {P ̂ t }t>0 , and obtain an inequality similar to Similarly, we can extend {P 0 (4.2). Therefore, in the rest of the proof, it suffices to consider the case when T0 = ∞. ̂ f ) for P f (resp. P ̂t f ) Step 2. We will construct a pointwise realization Qt f (resp. Q t t 1 when t > 0, f ∈ L . In particular, we show that there exists a common regular μ-nest 1 1 ̂ ̂ {Fn }∞ n=1 such that Qt = Pt , Qt = Pt on L , and for all t > 0 and all f ∈ L , ̂ f ∈ C({F }). Qt f ∈ C({Fn }) and Q t n

(4.3)

Indeed, since L1 is separable, there exists a countable family {fk }∞ k=1 which is dense in L . Consider the countable family 1

̂ s fk : k ≥ 1, s ∈ ℚ+ }, {Ps fk , P ̂ s fk has where ℚ+ is the set of all positive rational numbers. By Lemma 3.7, each Ps fk , P ̂ a μ-quasicontinuous version, say, hs,k , hs,k , respectively. Moreover, by Lemma 3.5(i), there is a common regular μ-nest {Fn }∞ n=1 such that ̂ : k ≥ 1, s ∈ ℚ } ⊂ C({F }). {hs,k , h s,k + n

Existence of the heat kernel | 41

Let ∞

M0 := ⋃ Fn n=1

and 𝒩 = M \ M0 .

(4.4)

It is clear that μ(𝒩 ) = 0 since {Fn } is a μ-nest. Therefore, it follows that for each U ∈ 𝒮 , s ∈ ℚ+ and each k, j ≥ 1, 󵄨 󵄨 󵄨 󵄨 sup 󵄨󵄨󵄨hs,k (x) − hs,j (x)󵄨󵄨󵄨 = sup 󵄨󵄨󵄨hs,k (x) − hs,j (x)󵄨󵄨󵄨

x∈U∩M0

x∈U\𝒩

= ‖hs,k − hs,j ‖L∞ (U)

(using hs,k − hs,j ∈ C({Fn })

and Lemma 3.5(ii))

= ‖Ps fk − Ps fj ‖L∞ (U) ≤ φ(U, s)‖fk − fj ‖L1

(using (2.10)).

(4.5)

Since {fk } is dense in L1 , for each f ∈ L1 , there exists a sequence {fki } from the set {fk } such that ‖fki − f ‖L1 → 0

asi → ∞.

Thus by (4.5), for each s ∈ ℚ+ and U ∈ 𝒮 , the sequence {hs,ki }i≥1 converges uniformly to a function, say, Qs f , in U ∩ M0 . Clearly, the function Qs f is independent of the choice of {ki }. Since U ∈ 𝒮 is arbitrary, and 𝒮 covers M, for any s ∈ ℚ+ and f ∈ L1 , we can define the function Qs f on M by Qs f (x) = {

limi→∞ hs,ki (x) 0

for x ∈ M0 = ⋃U∈𝒮 (U ∩ M0 ), for x ∈ 𝒩 .

(4.6)

Moreover, it follows that Qs f |Fn ∩U is continuous for any n ≥ 1 since each hs,ki |Fn is continuous, and hence, Qs f |Fn = Qs f |Fn ∩(∪U∈𝒮 U) is continuous, that is, Qs f ∈ C({Fn })

for all s ∈ ℚ+ and all f ∈ L1 .

(4.7)

Let us show that for each s ∈ ℚ+ , Qs = Ps

in L1 ,

(4.8)

that is, Qs f = Ps f almost everywhere in M when f ∈ L1 . Indeed, using the fact that Ps is L1 󳨃→ L1 bounded, it follows that 󵄩 󵄩 lim ‖hs,ki − Ps f ‖L1 = lim ‖Ps fki − Ps f ‖L1 = lim 󵄩󵄩󵄩Ps (fki − f )󵄩󵄩󵄩L1 = 0, i→∞ i→∞

i→∞

42 | A. Grigor’yan et al. which implies that hs,ki converges to Ps f in measure as i → ∞. By the definition (4.6), a. e.

a. e.

we obtain that Qs f = Ps f , which is (4.8). Here the sign = is understood in the sense of almost everywhere in M. We will extend Qs f in (4.6) to any positive real number t (not only for positive rationals s) by using the semigroup property. To do this, we claim that for any fixed real number t > 0 and f ∈ L1 , Qs (Pt−s f )(x) = Qs󸀠 (Pt−s󸀠 f )(x)

for every x ∈ M and s, s󸀠 ∈ (0, t) ∩ ℚ+ .

(4.9)

Indeed, we see by (4.6) that for any t > 0, f ∈ L1 and s, s󸀠 ∈ (0, t) ∩ ℚ+ Qs (Pt−s f )(x) = 0 = Qs󸀠 (Pt−s󸀠 f )(x)

whenever x ∈ 𝒩 ,

(4.10)

since Pt−s f , Pt−s󸀠 f ∈ L1 . On the other hand, it follows from (4.8) and the semigroup property of {Pt } that for any t > 0, f ∈ L1 and s, s󸀠 ∈ (0, t) ∩ ℚ+ a. e.

a. e.

a. e.

a. e.

Qs (Pt−s f ) = Ps (Pt−s f ) = Pt f = Ps󸀠 (Pt−s󸀠 f ) = Qs󸀠 (Pt−s󸀠 f ).

(4.11)

This together with (4.7) and Lemma 3.5(ii) shows that Qs (Tt−s f )(x) = Qs󸀠 (Tt−s󸀠 f )(x)

whenever x ∈ M0 .

(4.12)

Combining (4.12) and (4.10), we obtain (4.9). Consequently, we can extend Qs f in (4.6) to any positive real number t by defining Qt f (x) = Qs (Pt−s f )(x),

f ∈ L1 , x ∈ M,

(4.13)

where s is a positive rational smaller than t. Note that the above formula is consistent when t is rational. Moreover, the family {Qt }t>0 possesses the following properties: – For all t > 0, f ∈ L1 , Qt f = Qs (Pt−s f ) ∈ C({Fn }) –

by using definition (4.13), (4.7), since Pt−s f ∈ L1 when f ∈ L1 and s ∈ (0, t) ∩ ℚ+ . For any t > 0, Qt = Pt

–

(4.14)

in L1

(4.15)

by using (4.11), (4.13). For each t > 0, x ∈ U ∈ 𝒮 , and each f ∈ L1 , we have 󵄨󵄨 󵄨 󵄨󵄨Qt f (x)󵄨󵄨󵄨 ≤ ‖Pt f ‖L∞ (U) ≤ φ(U, t)‖f ‖L1 .

(4.16)

Existence of the heat kernel | 43

Indeed, when x ∈ 𝒩 , we see by (4.10) and (4.13) that Qt f (x) = 0, and so (4.16) is true. On the other hand, when x ∈ U \ 𝒩 , we see that 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨Qt f (x)󵄨󵄨󵄨 ≤ sup 󵄨󵄨󵄨Qt f (z)󵄨󵄨󵄨 z∈U\𝒩

= ‖Qt f ‖L∞ (U)

= ‖Pt f ‖L∞ (U) –

(using (4.14) and Remark 3.6) (using (4.15)),

and so (4.16) is also true. The {Qt }t>0 satisfies the semigroup property. More precisely, for any real t1 , t2 > 0, f ∈ L1 and any x ∈ M, Qt1 +t2 f (x) = Qt1 (Qt2 f )(x).

(4.17)

Indeed, we see by (4.15) that a. e.

a. e.

a. e.

Qt1 +t2 f = Pt1 +t2 f = Pt1 (Pt2 f ) = Qt1 (Qt2 f ). From this and (4.14), it follows that a. e.

0 = Qt1 +t2 f − Qt1 (Qt2 f ) := u ∈ C({Fn }). Note that by (4.10), Qt1 +t2 f (x) = 0 = Qt1 (Qt2 f )(x)

–

for every x ∈ 𝒩 and f ∈ L1 . Therefore, we conclude that (4.17) is true by Remark 3.6. Note that (4.17) also means that the pointwise realization in above way does not change the semigroup property of operators. Each operator from {Qt }t>0 is bounded. This immediately follows from (4.15): ‖Qt ‖L1 󳨃→L1 = ‖Pt ‖L1 󳨃→L1 < ∞.

–

(4.18)

Each operator from {Qt }t>0 is linear as well. More precisely, for any a, b ∈ ℝ, f , g ∈ L1 , and any x ∈ M, Qt (af + bg)(x) = aQt f (x) + bQt g(x). Indeed, for any t > 0 and any f , g ∈ L1 , a. e.

Qt (af + bg) = Pt (af + bg) (using (4.15)) a. e.

= aPt f + bPt g

a. e.

= aQt f + bQt g

(using the linearity of Pt ) (using (4.15) again),

(4.19)

44 | A. Grigor’yan et al. that is, for μ-almost all x ∈ M, Qt (af + bg)(x) = aQt f (x) + bQt g(x). Since the functions on the both sides belong to C({Fn }) by virtue of (4.14), this equality indeed is true for every point x in M0 by using Lemma 3.5(ii). Clearly, (4.19) is also true for x in 𝒩 by (4.10), since the both sides equal to zero in this case. ̂ f of function P ̂ t f . Indeed, note that each We turn to introduce a pointwise realization Q t ̂ is the μ-quasicontinuous version of P ̂ f . Similar to (4.5), we have by (2.11) that for h s,k

t k

each U ∈ 𝒮 , s ∈ ℚ+ and each k, j ≥ 1,

󵄨̂ 󵄨󵄨 ̂ sup 󵄨󵄨󵄨h s,k (x) − hs,j (x)󵄨󵄨 ≤ φ(U, s)‖fk − fj ‖L1 .

x∈U\𝒩

As before, let {fki } be a sequence from the set {fk } such that ‖fki − f ‖L1 → 0 as i → ∞. ̂ f of function P ̂s f Hence, for any s ∈ ℚ+ , f ∈ L1 , we can define a pointwise realization Q s by

̂ ̂ f (x) = { limi→∞ hs,ki (x), Q s 0,

for x ∈ M0 , otherwise.

We similarly have that for any fixed real number t > 0 and f ∈ L1 , ̂ (P ̂ ̂ ̂ Q s t−s f )(x) = Qs󸀠 (Pt−s󸀠 f )(x)

for every x ∈ M and s, s󸀠 ∈ (0, t) ∩ ℚ+ .

̂ f (x) to any positive real Consequently, for any x in M and any f ∈ L1 , we can extend Q s number t by ̂ f (x) = Q ̂ (P ̂ Q t s t−s f )(x)

(4.20)

for some rational s > 0 strictly less than t. Clearly, for any t > 0, f ∈ L1 , ̂ f (x) = 0 Q t

whenever x ∈ 𝒩 .

(4.21)

̂ } possesses the following properties, whose proofs are the same to those Then {Q t t>0 for {Qt }t>0 : – For all t > 0 and all f ∈ L1 ,

–

̂ f ∈ C({F }). Q t n

(4.22)

in L1 .

(4.23)

For each t > 0, ̂ =P ̂t Q t

Existence of the heat kernel | 45

–

For each t > 0, x ∈ U ∈ 𝒮 , and each f ∈ L1 , 󵄨 󵄨󵄨̂ ̂ t f ‖L∞ (U) ≤ φ(U, t)‖f ‖ 1 . 󵄨󵄨Qt f (x)󵄨󵄨󵄨 ≤ ‖P L

–

̂ } satisfies the semigroup property, i. e., for any real t , t > 0, f ∈ L1 , and The {Q t t>0 1 2 any x ∈ M, ̂ ̂ ̂ Q t1 +t2 f (x) = Qt1 (Qt2 f )(x).

–

(4.25)

̂ } is bounded, that is, Each operator from {Q t t>0 ̂ ‖ 1 1 = ‖P ̂t ‖ 1 1 < ∞ ‖Q t L 󳨃→L L 󳨃→L

–

(4.24)

(4.26)

by using (4.23). ̂ } is linear, i. e., for any a, b ∈ ℝ, f , g ∈ L1 , and any x ∈ M, Each operator from {Q t t>0 ̂ (af + bg)(x) = aQ ̂ f (x) + bQ ̂ g(x). Q t t t

(4.27)

Moreover, for any f , g ∈ L1 ∩ L∞ , ̂ g) ̂ t g) = (f , Q (Qt f , g) = (Pt f , g) = (f , P t

(4.28)

by using (4.15), (2.6), and (4.23). ̂ } , and show Step 3. We will work on the pointwise realization semigroups {Qt }t>0 , {Q t t>0 the existence of their integral kernels qt (x, y) and q̂t (x, y), respectively. The functions qt (x, y) and q̂t (x, y) will be used to construct the desired pt (x, y). To do this, we have by (4.19), (4.16) that for any fixed x ∈ M, the map L1 ∋ f 󳨃→ Qt f (x) defines a bounded linear functional on L1 . Therefore, for each t > 0, x ∈ U ∈ 𝒮 , there is a function qt (x, ⋅) ∈ L∞ such that Qt f (x) = ∫ qt (x, z)f (z) dμ(z)

for any f ∈ L1 ,

(4.29)

M

and by (4.16), 󵄩󵄩 󵄩 󵄨 󵄨 󵄩󵄩qt (x, ⋅)󵄩󵄩󵄩L∞ = sup 󵄨󵄨󵄨Qt f (x)󵄨󵄨󵄨 ≤ φ(U, t). ‖f ‖L1 =1

(4.30)

Similarly, for each t > 0, y ∈ U ∈ 𝒮 , we have by (4.24), (4.27) that the map L1 ∋ f 󳨃→ ̂ f (y) defines a bounded linear functional on L1 . It follows that there is a function Q t q̂t (y, ⋅) ∈ L∞ such that each t > 0, y ∈ U ∈ 𝒮 ̂ f (y) = ∫ q̂ (y, z)f (z) dμ(z) Q t t M

for any f ∈ L1 ,

(4.31)

46 | A. Grigor’yan et al. 󵄩 󵄩󵄩 ̂ 󵄩󵄩qt (y, ⋅)󵄩󵄩󵄩L∞ ≤ φ(U, t).

(4.32)

Note that functions qt (x, ⋅) and q̂t (y, ⋅) in M are almost-everywhere defined for each t > 0 and x, y ∈ M. Moreover, when x, y ∈ 𝒩 , t > 0, we have qt (x, ⋅) = 0 = q̂t (y, ⋅) μ-a. e. in M.

(4.33)

We show that for any t > 0, 󵄩 󵄩 ̂t ‖ 1 1 , sup󵄩󵄩󵄩qt (x, ⋅)󵄩󵄩󵄩L1 ≤ ‖P L 󳨃→L

(4.34)

󵄩 󵄩 sup󵄩󵄩󵄩q̂t (y, ⋅)󵄩󵄩󵄩L1 ≤ ‖Pt ‖L1 󳨃→L1 .

(4.35)

x∈M

y∈M

Indeed, by (2.7) with p = 1, q = ∞, we have for each t > 0, f ∈ L∞ , ̂ t f ‖L∞ ≤ ‖Pt ‖ 1 1 ‖f ‖L∞ . ‖P L 󳨃→L

(4.36)

By duality, we also have for each t > 0, f ∈ L∞ , ̂ t ‖ 1 1 ‖f ‖L∞ . ‖Pt f ‖L∞ ≤ ‖P L 󳨃→L

(4.37)

It follows by Remark 3.6, (4.14), (4.15), and (4.37) that for any f ∈ L1 ∩ L∞ and x ∈ M, 󵄨󵄨 󵄨 ̂ t ‖ 1 1 ‖f ‖L∞ . 󵄨󵄨Qt f (x)󵄨󵄨󵄨 ≤ ‖Qt f ‖L∞ = ‖Pt f ‖L∞ ≤ ‖P L 󳨃→L

(4.38)

For a compact subset K of M, consider the layer-cake decomposition of |qt (x, y)| over K: 󵄨󵄨 󵄨 󵄨󵄨qt (x, y)󵄨󵄨󵄨 = qt (x, y)(1K∩Vt − 1K∩(Vt )c ), where the set Vt := {y ∈ M : qt (x, y) ≥ 0}. By (4.29), (4.38), we see for each t > 0, x ∈ M, 󵄨 󵄨 󵄨 󵄨 ∫󵄨󵄨󵄨qt (x, y)󵄨󵄨󵄨 dμ(y) = 󵄨󵄨󵄨Qt (1K∩Vt − 1K∩(Vt )c )(x)󵄨󵄨󵄨

K

̂ t ‖ 1 1 ‖1K∩V − 1K∩(V )c ‖L∞ ≤ ‖P ̂t ‖ 1 1 . ≤ ‖P L 󳨃→L L 󳨃→L t t

Passing to the limit as K ↑ M, we have (4.34). Similarly, inequality (4.35) also holds. Step 4. We construct the desired pt (x, y) by using functions qt (x, y), q̂t (x, y). Indeed, for t > 0 and x, y ∈ M, define pt (x, y) = ∫ qt/2 (x, z)q̂t/2 (y, z) dμ(z). M

(4.39)

Existence of the heat kernel | 47

Note that the integral in the right-hand side of (4.39) is well defined by (4.32) and (4.34). We can rewrite (4.39) as follows: ̂ q (x, ⋅)(y) = p (x, y) = Q q̂ (y, ⋅)(x), Q t/2 t/2 t t/2 t/2

t > 0, x, y ∈ M,

(4.40)

by using (4.29), (4.31), (4.34), (4.35). The function pt (x, y) is pointwise defined for (t, x, y) ∈ (0, ∞) × M × M. Let us prove that the regular μ-nest {Fn } and the function pt (x, y) defined above satisfy all the properties stated in Theorem 2.1. In fact, we have following: – Property (2.12) is true whenever x ∈ 𝒩 or y ∈ 𝒩 , by using definition (4.39) and (4.33). – For t > 0 and x, y ∈ M, pt (x, ⋅) ∈ C({Fn })

and pt (⋅, y) ∈ C({Fn }),

(4.41)

by using (4.40), (4.22), (4.35), and (4.14), (4.34). Consequently, for any n ≥ 1, the function pt (⋅, ⋅)1Fn ×Fn (⋅, ⋅) is jointly measurable by Proposition 3.1. Hence, the joint measurability of pt (⋅, ⋅) follows by noting that pt (x, y) = lim pt (x, y)1Fn ×Fn (x, y), n→∞

–

x, y ∈ M.

For t > 0, x ∈ M and f ∈ L1 , Qt f (x) = ∫ pt (x, z)f (z) dμ(z),

(4.42)

M

since we have ̂ q (x, ⋅), f ) (by (4.40)) (pt (x, ⋅), f ) = (Q t/2 t/2 = (qt/2 (x, ⋅), Qt/2 f ) (by (4.28))

= Qt/2 Qt/2 f (x) = Qt f (x)

(by (4.29))

(by (4.17)).

Similarly, for t > 0, y ∈ M and f ∈ L1 , ̂ f (y) = ∫ p (z, y)f (z) dμ(z), Q t t

(4.43)

M

–

by using (4.40), (4.28), (4.31), (4.25). Hence, property (2.2) follows from (4.42), (4.14) and (4.43), (4.22). For t > 0 and f ∈ L1 , a. e.

Pt f = Qt f = ∫ pt (⋅, z)f (z) dμ(z) M

48 | A. Grigor’yan et al. and ̂ f = ∫ p (z, ⋅)f (z) dμ(z), ̂ t f a.=e. Q P t t M

–

̂ t }t>0 by using (4.42), (4.15), and (4.43), (4.23). That is, the semigroups {Pt }t>0 and {P possess the integral kernels pt (x, ⋅) and pt (⋅, y), respectively. For any t > 0 and x ∈ U ∈ 𝒮 , 󵄩󵄩 󵄩 󵄩󵄩pt (x, ⋅)󵄩󵄩󵄩L∞ ≤ φ(U, t) by using (4.16) and (4.42). This, together with (4.41), Remark 3.6, and (2.12), yields that for any t > 0 and x ∈ U ∈ 𝒮 , y ∈ M, 󵄨󵄨 󵄨 󵄨󵄨pt (x, y)󵄨󵄨󵄨 ≤ φ(U, t). Similarly, for any t > 0 and x ∈ U ∈ 𝒮 , y ∈ M, 󵄨󵄨 󵄨 󵄨󵄨pt (y, x)󵄨󵄨󵄨 ≤ φ(U, t),

–

by using (4.24), (4.43), (4.41), Remark 3.6, (2.12). Hence, property (2.13) follows. Property (2.1) is true. Indeed, it follows from (4.29), (4.42), and (4.34) that for t > 0 and x ∈ M, 󵄩 󵄩 󵄩 󵄩 ̂ t ‖ 1 1 < ∞. sup󵄩󵄩󵄩pt (x, ⋅)󵄩󵄩󵄩L1 = sup󵄩󵄩󵄩qt (x, ⋅)󵄩󵄩󵄩L1 ≤ ‖P L 󳨃→L x∈M

x∈M

Similarly, it follows from (4.31), (4.43), and (4.35) that for t > 0 and x ∈ M, 󵄩 󵄩 󵄩 󵄩 sup󵄩󵄩󵄩pt (⋅, y)󵄩󵄩󵄩L1 = sup󵄩󵄩󵄩q̂t (y, ⋅)󵄩󵄩󵄩L1 ≤ ‖Pt ‖L1 󳨃→L1 < ∞. y∈M

–

y∈M

Consequently, property (2.1) follows from the above two formulas and (4.41). Finally, the semigroup property (2.3) is true. Indeed, note first that integral on the right-hand side of (2.3) is well defined by (2.1) and (2.13). By (2.12), we only need to consider the case when t > 0 and x, y ∈ M0 . Since pt (⋅, ⋅) is jointly measurable, we have by Fubini’s theorem that ∫ pt+s (x, y)f (y) dμ(y) = Qt+s f (x) = Qt Qs f (x)

(by (4.42) and (4.17))

M

= ∫ pt (x, z)(∫ ps (z, y)f (y) dμ(y)) dμ(z) M

M

= ∫(∫ pt (x, z)ps (z, y) dμ(z))f (y) dμ(y) M M

(by (4.42))

Existence of the heat kernel | 49

for t, s > 0 and x ∈ M0 . Hence, a. e.

pt+s (x, y) = ∫ pt (x, z)ps (z, y) dμ(z),

μ-a. a. y ∈ M.

M

This, together with (4.41), (2.2), (2.1), and Lemma 3.5(ii), gives that pt+s (x, y) = ∫ pt (x, z)ps (z, y) dμ(z),

y ∈ M0 .

M

Therefore, we obtain (2.3). The proof is complete. We call the special case of (1.1) when 𝒮 = {M} the global L1 -L∞ ultracontractivity of a semigroup {Pt }t>0 on L1 , which will guarantee that conditions (2.10) and (2.11) are both true. Indeed, the global ultracontractivity is just (2.10) with φ(U, t) = φ(t) whilst condition (2.11) is also true since ̂ t f ‖L∞ ≤ φ(t)‖f ‖ 1 ‖P L

(4.44)

for any t > 0, f ∈ L1 ; this is because by (2.6), (1.1), 󵄨󵄨 ̂ 󵄨 󵄨 󵄨 󵄨󵄨(Pt f , g)󵄨󵄨󵄨 = 󵄨󵄨󵄨(Pt g, f )󵄨󵄨󵄨 ≤ ‖Pt g‖L∞ ‖f ‖L1 ≤ φ(t)‖g‖L1 ‖f ‖L1 for any g ∈ L1 , and so 󵄨󵄨 ̂ ̂ t f ‖L∞ = sup 󵄨󵄨󵄨(P ‖P 󵄨 t f , g)󵄨󵄨 ≤ sup φ(t)‖g‖L1 ‖f ‖L1 = φ(t)‖f ‖L1 . ‖g‖L1 =1

‖g‖L1 =1

Therefore, by Theorem 2.1, we have the following. ̂ t }t∈(0,T ) be its dual semiCorollary 4.1. Let {Pt }t∈(0,T0 ) be a semigroup on L1 , and let {P 0 1 ̂ group defined by (2.6) such that each Pt is bounded from L to L1 , where T0 ∈ (0, ∞]. If condition (1.1) with 𝒮 = {M} and t ∈ (0, T0 ) holds, then {Pt }t∈(0,T0 ) possesses an integral kernel pt (x, y) pointwise defined in (0, ∞)×M ×M that satisfies condition (Ap ) with p = 1. Moreover, for each t ∈ (0, T0 ) and all x, y ∈ M, 󵄨󵄨 󵄨 󵄨󵄨pt (x, y)󵄨󵄨󵄨 ≤ φ(t).

(4.45)

Corollary 4.2. Assume that all the hypothesis of Theorem 2.1 are satisfied, and pt (x, y) is the corresponding integral kernel. Then the following statements are true: (1) If in addition, the semigroup {Pt }t∈(0,T0 ) is positive, that is, Pt f ≥ 0 μ-a. e. for any nonnegative function f ∈ L1 and t ∈ (0, T0 ), then pt (x, y) ≥ 0

for each x, y ∈ M, t > 0.

50 | A. Grigor’yan et al. ̂ t have continuous versions for any t ∈ (0, T0 ), then pt (x, ⋅) (2) If in addition, both Pt and P and pt (⋅, x) are continuous in M for any x ∈ M and t > 0. (3) If in addition, {Pt } is symmetric, i. e., (Pt f , g) = (f , Pt g) for any f , g ∈ L1 ∩ L∞ , t ∈ (0, T0 ), then pt (x, y) is also symmetric: pt (x, y) = pt (y, x)

for each x, y ∈ M, t > 0.

Proof. As shown in the proof of Theorem 2.1, it suffices to consider the case when T0 = ∞. (1) Let {Fn } be the regular μ-nest as in Theorem 2.1. Fix t > 0. By (2.12), we only need to consider the case when x, y ∈ M0 where M0 := ⋃∞ n=1 Fn . Indeed, since Pt is positive, we obtain by (4.15) that for any 0 ≤ f ∈ L1 , a. e.

Qt f = Pt f ≥ 0, where Qt is as in (4.13). This together with (4.14) and Lemma 3.5(ii) yields that Qt f (x) ≥ 0,

x ∈ M0 .

Hence, we obtain by (4.42) that for any x ∈ M0 , pt (x, y) ≥ 0,

μ-a. a. y ∈ M0 .

Combining this and (2.1), and using Lemma 3.5(ii), we have that pt (x, y) ≥ 0 for any x, y ∈ M0 . (2) Since Pt have continuous version for any t > 0, we can choose Qt = Pt in Step 1 of the proof of Theorem 2.1. Thus the μ-nest {Fn } can be taken as Fn := M for n ≥ 1, and the conclusion follows directly from (2.1). ̂ f (x) for any f ∈ L1 and ̂ t , and so Qt f (x) = Q (3) If {Pt } is symmetric, we have Pt = P t x ∈ M. Hence, q̂t (x, ⋅) = qt (x, ⋅) for each x ∈ M, t > 0. It follows from definition (4.39) that pt (x, y) = ∫ qt/2 (x, z)q̂t/2 (y, z) dμ(z) = ∫ qt/2 (x, z)qt/2 (y, z) dμ(z) = pt (y, x) M

M

for each x, y ∈ M, t > 0. The following result is an L2 -version of Theorem 2.1. Theorem 4.3. Let {Pt }t∈(0,T0 ) be a semigroup on L2 (M, μ) for a metric measure space ̂ t }t∈(0,T ) be its dual semigroup defined by (2.6), where T0 ∈ (0, ∞]. (M, d, μ), and let {P 0 Assume that there exists a countable family 𝒮 of open sets with M = ⋃U∈𝒮 U and a function φ : 𝒮 × (0, T0 ) 󳨃→ ℝ+ such that, for each t ∈ (0, T0 ), U ∈ 𝒮 , and each f ∈ L2 ,

Existence of the heat kernel | 51

̂ t f ‖L∞ (U) ≤ φ(U, t)‖f ‖ 2 . ‖Pt f ‖L∞ (U) ∨ ‖P L

(4.46)

Then {Pt }t∈(0,T0 ) possesses an integral kernel pt (x, y) pointwise defined in (0, ∞) × M × M that satisfies condition (Ap ) with p = 2 for some regular μ-nest {Fn }∞ n=1 in M, and pt (x, y) = 0

for any t > 0

whenever one of points x, y lies outside ⋃∞ n=1 Fn . Moreover, for each t ∈ (0, T0 ) and any U ∈ 𝒮, 󵄩 󵄩 󵄩 󵄩 sup(󵄩󵄩󵄩pt (x, ⋅)󵄩󵄩󵄩L2 ∨ 󵄩󵄩󵄩pt (⋅, x)󵄩󵄩󵄩L2 ) ≤ φ(U, t). x∈U

(4.47)

Proof. Note that by (2.7), for any t ∈ (0, T0 ), ̂ t ‖ 2 2 = ‖Pt ‖ 2 2 < ∞, ‖P L 󳨃→L L 󳨃→L ̂ t } is also a semigroup on L2 . and so {P The rest of the proof is similar to that of Theorem 2.1. We omit the details. Remark 4.4. In Theorem 4.3, consider the following condition instead of the assumption (4.46): there exists a function φ : M × ℝ+ × ℝ+ 󳨃→ ℝ+ such that, for any t > 0 and any ball B := B(x0 , r), ̂ t f ‖L∞ (B) ≤ φ(x0 , r, t)‖f ‖ 2 , ‖Pt f ‖L∞ (B) ∨ ‖P L

f ∈ L2 .

Then (4.47) in Theorem 4.3 becomes such that, for any t > 0 and any B := B(x0 , r), 󵄩 󵄩 󵄩 󵄩 sup(󵄩󵄩󵄩pt (x, ⋅)󵄩󵄩󵄩L2 ∨ 󵄩󵄩󵄩pt (⋅, x)󵄩󵄩󵄩L2 ) ≤ φ(x0 , r, t). x∈B

In particular, by the semigroup property, the above gives an on-diagonal upper estimate: 󵄨 󵄨 sup󵄨󵄨󵄨pt (x, x)󵄨󵄨󵄨 ≤ φ(x0 , r, t) x∈B

for any t > 0.

Remark 4.5. The parallel statements on the additional properties of pt (x, y) (positivity, continuity, symmetry) in Corollary 4.2 corresponding to Theorem 4.3 for a semigroup {Pt }t∈(0,T0 ) on the space L2 (instead of on L1 ) are also true.

5 Applications In this section we give two applications of Theorem 2.1. One application is to consider a family of moving semigroups {PtB }t>0 in L1 . We show the existence of pointwise integral

52 | A. Grigor’yan et al. kernel pBt (x, y) and then look at its limit when B is expanding to the whole space M. The other application is to consider an asymmetric Dirichlet form, and we obtain the existence of the asymmetric heat kernel and its diagonal upper estimate. Fix T0 ∈ (0, ∞]. Let {PtB }t∈(0,T0 ) be a semigroup in L1 such that it vanishes outside B in the sense that PtB f (x) = 0

(5.1)

for any t ∈ (0, T0 ) and f ∈ L1 and μ-almost all x ∈ Bc . A semigroup {PtB }t∈(0,T0 ) in L1 is monotone increasing in B if whenever B1 ⊂ B2 , we have B

B

Pt 1 f ≤ Pt 2 f

μ-a. e.

(5.2)

for any 0 ≤ f ∈ L1 and t ∈ (0, T0 ), and is positive if PtB f ≥ 0

μ-a. e.

(5.3)

for any 0 ≤ f ∈ L1 and t ∈ (0, T0 ). The following is the first application of Theorem 2.1. Lemma 5.1. Let T0 ∈ (0, ∞] and {PtB }t∈(0,T0 ) be a semigroup on L1 (M, μ) for a separable metric space (M, d), which is monotone increasing in B, positive, and satisfies (5.1). Let ̂ B }t∈(0,T ) be defined by (2.6) such that it is a semigroup on L1 . Assume that there exists {P t 0 a function φ : M × (0, T0 ) 󳨃→ ℝ+ such that, for any t ∈ (0, T0 ), f ∈ L1 , and any metric ball B in M, 󵄩󵄩 B 󵄩󵄩 󵄩󵄩Pt f 󵄩󵄩L∞ ≤ φ(B, t)‖f ‖L1 .

(5.4)

Then there exists a function pt (x, y) pointwise defined on (0, ∞) × M × M satisfying the following: (1) (Measurability) For any t > 0, function pt (⋅, ⋅) is jointly measurable in M × M. (2) (Positivity) For any t > 0 and any x, y ∈ M, pt (x, y) ≥ 0.

(5.5)

(3) (Semigroup property) For any t, s > 0 and any x, y ∈ M, we have pt+s (x, y) = ∫ pt (x, z)ps (z, y) dμ(z).

(5.6)

M

(4) (Limit kernel) For any t > 0 and any nonnegative f ∈ L1 , B lim P m f m→∞ t

= ∫ pt (⋅, y)f (y) dμ(y), M

where {Bm }∞ m=1 is any sequence of concentric balls tending to M.

(5.7)

Existence of the heat kernel | 53

In particular, if φ(B, t) in (5.4) is furthermore independent of B, φ(B, t) = φ(t), then pt (x, y) ≤ φ(t)

(5.8)

for any t ∈ (0, T0 ) and any x, y ∈ M. Proof. As shown in the proof of Theorem 2.1, it suffices to consider the case when T0 = ∞. We divide the proof into three steps. Step 1. Fix a sequence of concentric metric balls {Bm }∞ m=1 such that Bm → M as Bm 1 m → ∞. Consider the semigroup {Pt }t>0 on L . By assumption (5.4), for any t > 0, f ∈ L1 , 󵄩󵄩 Bm 󵄩󵄩 󵄩󵄩Pt f 󵄩󵄩L∞ ≤ φ(Bm , t)‖f ‖L1 .

(5.9)

From this, we have for any t > 0, f ∈ L1 , Bm 󵄨󵄨 󵄩󵄩 ̂ Bm 󵄩󵄩 󵄨 ̂ Bm 󵄨󵄨 󵄨󵄨 󵄩󵄩Pt f 󵄩󵄩L∞ = sup 󵄨󵄨󵄨(P t f , g)󵄨󵄨 = sup 󵄨󵄨(f , Pt g)󵄨󵄨 ‖g‖L1 =1

‖g‖L1 =1

󵄩 B 󵄩 ≤ sup 󵄩󵄩󵄩Pt m g 󵄩󵄩󵄩L∞ ‖f ‖L1 ≤ φ(Bm , t)‖f ‖L1 . ‖g‖L1 =1

(5.10) B

Therefore, all the hypotheses in Theorem 2.1 are satisfied for the semigroup {Pt m }t>0 ̂ B }t>0 . It follows that there exists a regular μ-nest {F (m) } of M and a funcand its dual {P t n Bm tion pt (x, y) pointwise defined on (0, ∞) × M × M satisfying the following properties: B (1) For each t > 0, the function pt m (⋅, ⋅) is jointly measurable in M × M. (2) For each t > 0 and each x, y in M, B

B

pt m (x, ⋅) ∈ C({Fn(m) }) ∩ L1

and pt m (⋅, y) ∈ C({Fn(m) }) ∩ L1 .

(5.11)

(3) For each t > 0 and each f ∈ L1 , B

∫ pt m (⋅, z)f (z) dμ(z) ∈ C({Fn(m) })

and

M

B

∫ pt m (z, ⋅)f (z) dμ(z) ∈ C({Fn(m) }). (5.12)

M

(4) For each t, s > 0 and each x, y in M, B

B

m pt+s (x, y) = ∫ pt m (x, z)pBs m (z, y) dμ(z).

(5.13)

M

(5) For each t > 0, we have B

pt m (x, y) = 0

(5.14)

(m) whenever one of points x, y lies outside ⋃∞ n=1 Fn . (6) For each t > 0 and each x, y ∈ M,

󵄨󵄨 Bm 󵄨 󵄨󵄨pt (x, y)󵄨󵄨󵄨 ≤ φ(Bm , t).

(5.15)

54 | A. Grigor’yan et al. (7) For each t > 0, f ∈ L1 , B

B

Pt m f (⋅) = ∫ pt m (⋅, y)f (y) dμ(y),

μ-a. e. in M.

(5.16)

M

(8) For each t > 0 and each x, y in M, B

pt m (x, y) ≥ 0.

(5.17)

All the properties (1)–(7) above are proved in Theorem 2.1 except for property (8), which follows from Corollary 4.2(i). Step 2. Without loss of generality, we assume that for all m, n ≥ 1 1 . 2m n

μ(M \ Fn(m) ) ≤

Otherwise, we can take a subsequence satisfying this inequality. By Lemma 3.5(i) and its proof, we can construct a regular μ-nest {Fn } such that ∞

Fn ⊂ ⋂ Fn(m) ,

n ≥ 1.

m=1

(5.18)

(m) The set Fn is not empty for large n, since the intersection ⋂∞ has almost the m=1 Fn same measure as M for large n by using the fact that the measure of its complement is small: ∞

c

∞

c

m=1

m=1

1

∞

μ( ⋃ (Fn(m) ) ) ≤ ∑ μ((Fn(m) ) ) ≤ ∑

m=1

2m n

=

1 →0 n

as n → ∞.

Since C({Fn(m) }) ⊂ C({Fn }) for all m ≥ 1, it follows from (5.11) that for any m ≥ 1, t > 0 and x, y ∈ M, B

pt m (x, ⋅) ∈ C({Fn })

B

(5.19)

B

(5.20)

and pt m (⋅, y) ∈ C({Fn }),

whilst by (5.12), for each f ∈ L1 , B

∫ pt m (⋅, z)f (z) dμ(z) ∈ C({Fn })

and

M

On the other hand, since {PtB }t>0 1

have for any 0 ≤ f ∈ L ,

∫ pt m (z, ⋅)f (z) dμ(z) ∈ C({Fn }).

M

is monotone increasing in B on L1 by assumption, we B

B

Pt m f (x) ≤ Pt m+1 f (x)

(5.21)

for each t > 0 and μ-almost all x ∈ M. By (5.16), (5.20), (5.21), and Lemma 3.5(ii), we obtain that for any f ∈ L1 , B

B

∫ pt m (⋅, z)f (z) dμ(z) ≤ ∫ pt m+1 (z, ⋅)f (z) dμ(z)

M

M

∞

for x ∈ M0 := ⋃ Fn . n=1

Existence of the heat kernel | 55

Therefore, for each t > 0 and every x ∈ M0 , B

B

pt m (x, y) ≤ pt m+1 (x, y),

μ-a. a. y ∈ M.

By (5.19) and using Lemma 3.5(ii) again, this inequality holds for every y ∈ M0 . We now define the function pt (x, y) for t > 0 by pt (x, y) = {

B

limm→∞ pt m (x, y), 0,

for x, y ∈ M0 , otherwise.

(5.22)

Step 3. We verify that pt (x, y) defined by (5.22) satisfies all the properties in Lemma 5.1. In fact, the joint measurability of pt (x, y), being a limit of the jointly B measurable functions pt m (x, y), is obvious. The positivity (5.5) of pt (x, y) follows from (5.17) and definition (5.22). Property (5.7) is also true by using (5.16) and the monotone convergence theorem. We show the semigroup property (5.6). Note that if one point x or y lies outside the set M0 , then pt+s (x, y) = 0 = ∫ pt (x, z)ps (z, y) dμ(z). M

If both x and y belong to M0 , then the semigroup property follows from (5.13) and the monotone convergence theorem. Finally, if φ(B, t) = φ(t) for any ball B, we see by (2.13) that B

pt m (x, y) ≤ φ(Bm , t) = φ(t) for all t > 0 and all x, y ∈ M0 , thus (5.8) follows by taking m → ∞. The proof is complete. Next we consider the coercive closed form (ℰ , 𝒟(ℰ )) in L2 introduced in [8, Definition 2.4, p. 16], and apply Theorem 2.1 to the semigroup corresponding to (ℰ , 𝒟(ℰ )). Recall that a coercive closed form (ℰ , 𝒟(ℰ )) in L2 is a bilinear form ℰ defined on 𝒟(ℰ ) × 𝒟(ℰ ) such that – 𝒟(ℰ ) is dense in L2 , and complete in the norm of ℰ11/2 where 2

1/2

ℰ1 (u) := (‖u‖2 + ℰ (u))

– –

with ℰ (u) := ℰ (u, u). (ℰ , 𝒟(ℰ )) is positive definite, that is, ℰ (u) ≥ 0 for any u ∈ 𝒟(ℰ ). The weak sector condition holds, i. e., there exists a constant K such that 󵄨󵄨 󵄨 1/2 1/2 󵄨󵄨ℰ1 (u, v)󵄨󵄨󵄨 ≤ K ℰ1 (u) ℰ1 (v) for any u, v ∈ 𝒟(ℰ ).

56 | A. Grigor’yan et al. For u, v ∈ 𝒟(ℰ ), we set ℰ̂(u, v) = ℰ (v, u).

(5.23)

In particular, we see ℰ̂(u) = ℰ (u) for any u ∈ 𝒟(ℰ ). Clearly, if (ℰ , 𝒟(ℰ )) is a coercive closed form (ℰ , 𝒟(ℰ )) on L2 then so is (ℰ̂, 𝒟(ℰ )). A coercive closed form (ℰ , 𝒟(ℰ )) on L2 is said to be symmetric if ℰ (u, v) = ℰ (v, u) for any u, v ∈ 𝒟(ℰ ), and to be Markovian if ℰ (u+ ∧ 1) ≤ ℰ (u)

for any u ∈ 𝒟(ℰ ).

A coercive closed form (ℰ , 𝒟(ℰ )) on L2 is called a Dirichlet form on L2 if it is symmetric and Markovian, see [8, Definition 4.5, p. 34] or [5, p. 5]. It turns out that a coercive closed form (ℰ , 𝒟(ℰ )) in L2 is uniquely corresponding to a strongly continuous contraction semigroup {Pt }t>0 on L2 (cf. [8, Diagram 2, p. 27]) by the relationship ℰ (u, v) = lim( t→0

u − Pt u , v) t

(5.24)

̂ t }t>0 is unique corresponding to the for any u, v ∈ 𝒟(ℰ ). Clearly, the dual semigroup {P ̂ coercive closed form (ℰ , 𝒟(ℰ )), since for any v, u ∈ 𝒟(ℰ ), ℰ̂(v, u) = ℰ (u, v) = lim( t→0

̂t v v−P u − Pt u , v) = lim( , u). t→0 t t

(5.25)

̂ t }t>0 is also strongly continuous, contractive on L2 . Moreover, {P The following says that the Nash inequality associated with a coercive closed form (ℰ , 𝒟(ℰ )) on L2 will imply conditions (2.10), (2.11) in Theorem 2.1. This conclusion was proved in [3, Theorem 2.1] when (ℰ , 𝒟(ℰ )) is a regular conservative Dirichlet form in L2 . The following says that this result is still valid for a more general setting. Lemma 5.2. Let {Pt }t>0 be a strongly continuous contraction semigroup on L2 and let (ℰ , 𝒟(ℰ )) be the corresponding coercive closed form on L2 determined by (5.24). Assume ̂ t }t>0 are bounded in the norm of L1 , that is, for all t > 0, f ∈ L1 , that {Pt }t>0 , {P ‖Pt f ‖L1 ≤ a‖f ‖L1 , ̂ t f ‖ 1 ≤ a‖f ‖ 1 ‖P L L

(5.26) (5.27)

for some constant a > 0. If (ℰ , 𝒟(ℰ )) satisfies the Nash inequality, that is, there exist three constants λ, ν > 0 and ρ ≥ 0 such that ‖f ‖2(1+ν) ≤ λ(ℰ (f ) + ρ‖f ‖2L2 )‖f ‖2ν L1 L2

for all f ∈ 𝒟(ℰ ) ∩ L1 ,

(5.28)

̂ t }t>0 and {Pt }t>0 satisfy the L1 –L∞ ultracontractivity property, that is, for all then both {P t > 0, f ∈ L1 , 1

ν 1 ̂ t f ‖L∞ } ≤ a ( λ ) eρt t − ν ‖f ‖ 1 . max{‖Pt f ‖L∞ , ‖P L ν

2

(5.29)

Consequently, all the hypotheses in Theorem 2.1 are satisfied for {Pt }t>0 and its dual ̂ t }t>0 , and thus {Pt }t>0 possesses an integral kernel pt (x, y) pointwise defined in (0, ∞)× {P

Existence of the heat kernel | 57

M × M satisfying condition (Ap ) with p = 1, and, moreover, 1

ν 󵄨 󵄨󵄨 2 λ ρt − 1 󵄨󵄨pt (x, y)󵄨󵄨󵄨 ≤ a ( ) e t ν ν

(5.30)

for all t > 0 and all x, y in M. Proof. Let 𝒜 be the domain of the infinitesimal generator of semigroup {Pt }t>0 on L2 , that is, 𝒜 is a subspace of L2 that consists of all functions f such that there exists some g ∈ L2 satisfying 󵄩 󵄩 lim󵄩󵄩s−1 (f − Ps f ) − g 󵄩󵄩󵄩L2 = 0.

(5.31)

s→0󵄩

By the Hille–Yosida theorem, the space 𝒜 is dense in L2 since the semigroup {Pt }t>0 on L2 is strongly continuous. For each t > 0, if f ∈ 𝒜 then Pt f ∈ 𝒜, see, for example, [8, Exercise 1.9, p. 10]. For t > 0, f ∈ 𝒜, we have d ‖P f ‖2 2 = lim s−1 ((Pt+s f , Pt+s f ) − (Pt f , Pt f )) s→0 dt t L = lim s−1 ((Pt+s f − Pt f , Pt+s f − Pt f ) + 2(Pt+s f − Pt f , Pt f )) s→0

= − 2ℰ (Pt f , Pt f ),

(5.32) (5.33)

since the first term on the right-hand side of (5.32) tends to zero, whilst for the second, we see by (5.24) and the fact that Pt f ∈ 𝒜, s−1 (Pt+s f − Pt f , Pt f ) = −s−1 (Pt f − Ps (Pt f ), Pt f ) → −ℰ (Pt f , Pt f )

as s → 0.

Temporally fix f ∈ 𝒜 with ‖f ‖L1 ≤ 1. By (5.26), ‖Pt f ‖L1 ≤ a‖f ‖L1 ≤ a. Let u(t) := e−2ρt ‖Pt f ‖2L2 ,

t > 0.

Applying (5.28) with f being replaced by Pt f , we see by (5.33) that −

d u(t) = 2e−2ρt (ℰ (Pt f ) + ρ‖Pt f ‖2L2 ) dt 1 ≥ 2e−2ρt ⋅ 2ν ‖Pt f ‖2(1+ν) L2 λa 2 −2ρt 2ρt 2 2 1+ν = 2ν e (e u(t)) = 2ν e2νρt u(t)1+ν ≥ 2ν u(t)1+ν . λa λa λa

58 | A. Grigor’yan et al. Integrating this inequality over (0, t), we obtain u(t)−ν ≥

2ν t. λa2ν

Therefore, ‖Pt f ‖2L2 = e2ρt u(t) ≤ e2ρt (

2ν t) λa2ν

−1/ν

For a general nonzero f ∈ 𝒜, we consider the function ‖Pt f ‖L2 ≤ eρt (

2ν t) λa2ν

f ‖f ‖L1

. and have

−1/(2ν)

‖f ‖L1 .

Since 𝒜 is dense in L2 , we conclude that for any t > 0, 2ν t) λa2ν

eρt .

(5.34)

2ν t) λa2ν

eρt .

(5.35)

−1/(2ν)

‖Pt ‖L1 →L2 ≤ ( Similarly, we have for any t > 0, ̂t ‖ 1 2 ≤ ( ‖P L →L

−1/(2ν)

We show (5.29) by using the semigroup property. Indeed, for any f , g ∈ L1 ∩ L2 , ̂ t g) (P2t f , g) = (Pt (Pt f ), g) = (Pt f , P ̂ t g‖ 2 ≤ ‖Pt f ‖ 2 ‖P L

L

̂ t ‖ 1 2 ‖g‖ 1 , ≤ ‖Pt ‖L1 →L2 ‖f ‖L1 ⋅ ‖P L L →L

which implies that 󵄨 󵄨 ‖P2t f ‖L∞ = sup 󵄨󵄨󵄨(P2t f , g)󵄨󵄨󵄨 ‖g‖L1 ≤1

̂t ‖ 1 2 , ≤ ‖Pt ‖L1 →L2 ‖f ‖L1 ⋅ ‖P L →L thus showing that ̂t ‖ 1 2 . ‖P2t ‖L1 →L∞ ≤ ‖Pt ‖L1 →L2 ‖P L →L Similarly, we have ̂ 2t ‖ 1 ∞ ≤ ‖P ̂ t ‖ 1 2 ‖Pt ‖ 1 2 . ‖P L →L L →L L →L

(5.36)

Existence of the heat kernel | 59

It follows from (5.36), (5.34), (5.35) that 2ν t) λa2ν

e2ρt ,

ν t) λa2ν

eρt .

−1/ν

‖P2t ‖L1 →L∞ ≤ ( which gives, after changing 2t by t, that

−1/ν

‖Pt ‖L1 →L∞ ≤ (

̂ t ‖ 1 ∞ is true. Thus (5.29) follows. The same bound for ‖P L →L Finally, the upper bound of pt (x, y) in (5.30) follows directly from (2.13) where U = M, φ(U, t) = (

ν t) λa2ν

−1/ν

eρt .

The proof is complete. In order to apply Lemma 5.2, one needs to verify the contractivity (5.26), (5.27) ̂ t }t>0 in the norm of L1 , respectively. The following of the semigroups {Pt }t>0 and {P provides a criterion in terms of the form (ℰ , 𝒟(ℰ )). Proposition 5.3 ([12, Theorem 1.1.5, p. 7]). Let (ℰ , 𝒟(ℰ )) be a coercive closed form on L2 . Then the following statements are equivalent: (1) For any u ∈ 𝒟(ℰ ), the function u+ ∧ 1 ∈ 𝒟(ℰ ) and ℰ (u+ ∧ 1, u − u+ ∧ 1) ≥ 0.

(5.37)

(2) {Pt }t>0 is sub-Markovian, i. e., for any f ∈ L2 with 0 ≤ f ≤ 1 μ-a. e., we have 0 ≤ Pt f ≤ 1 μ-a. e. for any t > 0. ̂ t } is positivity preserving and contractive in L1 , that is, if f ∈ L1 with f ≥ 0 μ-a. e., (3) {P ̂ t f ≥ 0 μ-a. e. and ‖P ̂ t f ‖ 1 ≤ ‖f ‖ 1 . then P L L Proof. Note that a coercive closed form (ℰ , 𝒟(ℰ )) is a special closed form introduced in [12, p. 1] with α0 = 0. Proposition 5.3 follows immediately from [12, Theorem 1.1.5, p. 7] ̂ t }t>0 . wherein the notions {Tt }t>0 , {T̂t }t>0 are used instead of {Pt }t>0 , {P We give an example where all the hypotheses in Lemma 5.2 are satisfied. Example 5.4. Consider the asymmetric operator ℒ = Δ − b ⋅ ∇ − c on ℝn for n ≥ 3, where the functions b : ℝn 󳨃→ ℝn belong to the Kato class Kn,2 : Kn,2 := {b : lim sup r→0 x∈ℝn

∫ |x−y|≤r

|b(y)| dy = 0}, |x − y|n−2

60 | A. Grigor’yan et al. and c is a positive constant. The Kato class Kn,2 is an extension of spaces Lp when p is large: Lp ⊂ Kn,2 Indeed, if u ∈ Lp with

n 2

if

n < p ≤ ∞. 2

(5.38)

< p ≤ ∞, then by Hölder’s inequality, for all x ∈ ℝn , 1/q

∫ B(x,r)

|u(y)| dy ≤ ‖u‖p ( ∫ |x − y|−(n−2)q dy) |x − y|n−2 B(x,r)

p ) p−1

1/q

r

= ‖u‖p (ωn−1 ∫ s

(with q =

−(n−2)q+n−1

ds)

0

= C(n, p)r 2−n/p ‖u‖p → 0

as r → 0,

since 2 − n/p > 0, where ωn−1 is the area of the unit sphere in ℝn , thus showing (5.38). Let ℱ = W01,2 (ℝn ) be the usual Sobolev space. The operator ℒ determines a bilinear form on ℱ × ℱ by ℰ (u, v) = ∫ ∇u ⋅ ∇v dx + ∫ b ⋅ ∇u ⋅ v dx + c ∫ uv dx, ℝn

ℝn

u, v ∈ ℱ .

ℝn

We claim that if the constant c is large enough, then (ℰ , ℱ ) is coercive closed form on L2 (ℝn ). To see this, applying [4, Theorem 3.25, p. 91], we have that there exists c0 = c0 (n, b) > 0 such that for all u ∈ ℱ , 1 󵄨 󵄨2 󵄨 󵄨2 ∫ 󵄨󵄨󵄨b(x)󵄨󵄨󵄨 󵄨󵄨󵄨u(x)󵄨󵄨󵄨 vdx ≤ ∫ |∇u|2 dx + c0 ∫ |u|2 dx. 4

ℝn

ℝn

(5.39)

ℝn

From this and using the elementary inequality ab ≤ 41 a2 + b2 for a, b ≥ 0, we have for any u, v ∈ ℱ , 1/2 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨2 󵄨󵄨 󵄨󵄨2 (b(x) ⋅ ∇u(x))v(x) dx b(x) ≤ ‖∇u‖ ( v(x) dx) ∫ ∫ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 2 󵄨󵄨 n 󵄨󵄨 n ℝ

ℝ

1/2

1 ≤ ‖∇u‖2 ( ‖∇v‖22 + c0 ‖v‖22 ) 4 1 1 ≤ ‖∇u‖22 + ‖∇v‖22 + c0 ‖v‖22 . 4 4

(5.40)

In particular, for any u ∈ ℱ , 󵄨󵄨 󵄨󵄨 1 󵄨󵄨 󵄨 2 2 󵄨󵄨 ∫ (b(x) ⋅ ∇u(x))u(x) dx󵄨󵄨󵄨 ≤ ‖∇u‖2 + c0 ‖u‖2 . 󵄨󵄨 󵄨󵄨 2 n ℝ

(5.41)

Existence of the heat kernel | 61

Denote by 𝔻1 (u, v) = 𝔻(u, v) + (u, v) where 1 ∫ ∇u ⋅ ∇v dx. 2

𝔻(u, v) =

ℝn

It follows by (5.41) that 2

ℰ (u, u) = 2𝔻(u, u) + ∫ (b(x) ⋅ ∇u(x))u(x) dx + c‖u‖2 ℝn

≥ 2𝔻(u, u) − (𝔻(u, u) + c0 ‖u‖22 ) + c‖u‖22 = 𝔻(u, u) + (c − c0 )‖u‖2L2 ≥ 𝔻(u, u) ≥ 0

(5.42)

whenever c ≥ c0 . From this and using (5.41), we have 2

2

ℰ1 (u, u) = ℰ (u, u) + ‖u‖2 ≥ 𝔻(u, u) + ‖u‖2 = 𝔻1 (u, u).

(5.43)

On the other hand, it follows from (5.40) that for any u, v ∈ ℱ , 󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨 󵄨󵄨ℰ (u, v)󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∫ ∇u∇vdx + ∫ (b(x) ⋅ ∇u(x))v(x) dx + c ∫ uv dx󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 n n n ℝ

ℝ

ℝ 1/2

1 ≤ ‖∇u‖2 ‖∇v‖2 + ‖∇u‖2 ( ‖∇v‖22 + c0 ‖v‖22 ) + c‖u‖2 ‖v‖2 4 1 ≤ ‖∇u‖2 (‖∇v‖2 + ‖∇v‖2 + √c0 ‖v‖2 ) + c‖u‖2 ‖v‖2 2 ≤ c1 𝔻1 (u, u)1/2 𝔻1 (v, v)1/2

(5.44)

for some constant c1 > 0. Combining this with (5.43), we obtain for all u ∈ ℱ , 𝔻1 (u, u) ≤ ℰ1 (u, u) ≤ c1 𝔻1 (u, u) + ‖u‖22 ≤ (c1 + 1)𝔻1 (u, u)

(5.45)

whenever c ≥ c0 . Therefore, ℱ is complete in the norm of √ℰ1 (u, u) if c ≥ c0 , since ℱ is complete in 𝔻1 -norm. Clearly, ℱ is dense in L2 , and the form (ℰ , ℱ ) is positive definite if c ≥ c0 . We need to verify the weak sector condition. Indeed, we have by (5.44), (5.45) that for all u, v ∈ ℱ , 󵄨󵄨 󵄨 󵄨 󵄨 1/2 1/2 󵄨󵄨ℰ1 (u, v)󵄨󵄨󵄨 = 󵄨󵄨󵄨ℰ (u, v) + (u, v)󵄨󵄨󵄨 ≤ c1 𝔻1 (u, u) 𝔻1 (v, v) + ‖u‖2 ‖v‖2 ≤ (c1 + 1)𝔻1 (u, u)1/2 𝔻1 (v, v)1/2 ≤ (c1 + 1)ℰ1 (u, u)1/2 ℰ1 (v, v)1/2 whenever c ≥ c0 , thus proving the weak sector condition. Therefore, the form (ℰ , ℱ ) is a coercive closed form in L2 (ℝn ) when c is large enough. In order to show the Nash inequality, note that for any u ∈ ℱ ∩ L1 (ℝn ), ‖u‖2(1+2/n) ≤ c2 𝔻(u, u)‖u‖4/n 1 . 2

62 | A. Grigor’yan et al. From this, we have by (5.42) that ‖u‖2(1+2/n) ≤ c2 ℰ (u, u)‖u‖4/n 1 , 2 thus showing that (ℰ , ℱ ) satisfies the Nash inequality (5.28) with λ = c2 , ρ = 0 and ν = n2 . ̂ t }t>0 is contractive in L1 (ℝn ). Indeed, for any u ∈ ℱ , we have We show that {P u+ ∧ 1 ∈ ℱ , and 1, { { u+ ∧ 1 = { u, { { 0, a. e.

∇(u+ ∧ 1) = {

u − 1, if u > 1, { { u − u+ ∧ 1 = { 0, if 0 < u ≤ 1, { u, if u ≤ 0, { if 0 < u ≤ 1, 0, if 0 < u ≤ 1, a. e. ∇(u − u+ ∧ 1) = { otherwise, ∇u, otherwise.

if u > 1, if 0 < u ≤ 1, if u ≤ 0, ∇u, 0,

If follows that ℰ (u+ ∧ 1, u − u+ ∧ 1) = 0 + 0 + c ∫ (u − 1) dx ≥ 0, {u>1}

̂ t } is contractive in L1 by Proposition 5.3. and condition (5.37) is true. Thus, {P It remains to show that ‖Pt ‖L1 →L1 is uniformly bounded in t. We will show that ‖Pt ‖L1 →L1 ≤ 1

for all t > 0

(5.46)

when the functions b ∈ Kn,2 further satisfies esup div b ≤ c.

(5.47)

ℝn

In fact, let u := Pt f for any nonnegative f ∈ L2 . Then 𝜕 u = ℒu = Δu − b ⋅ ∇u − cu, 𝜕t where 𝜕t𝜕 u is understood the Fréchet derivative with respect to the inner product of L2 . Integrating over ℝn , we have by (5.47) that d ∫ u(t, x) dx = ∫ (Δu(x) − b(x) ⋅ ∇u(x) − cu(x)) dx dt ℝn

ℝn

= 0 + ∫ (div b(x) − c)u(x) dx ≤ 0.

ℝn

(integration by parts)

Existence of the heat kernel | 63

From this, we see for all t > 0 ‖Pt f ‖L1 = ∫ u(t, x) dx ≤ ∫ u(0, x) dx = ‖f ‖L1 , ℝn

ℝn

thus showing (5.46). Therefore, all the hypotheses in Lemma 5.2 are satisfied where a = 1, and hence, the semigroup {Pt } associated with the coercive closed form (ℰ , ℱ ) in L2 possesses an integral kernel pt (x, y) satisfying condition (Ap ) with p = 1, and by (5.30) with a = 1, λ = c2 , ρ = 0, ν = n2 , n/2

cn 󵄨󵄨 󵄨 󵄨󵄨pt (x, y)󵄨󵄨󵄨 ≤ ( 2 ) 2

n

t− 2

for all t > 0 and all x, y in M. There is a plenty of examples in which functions b ∈ Kn,2 satisfies (5.47), for instance, b = (f1 , f2 , . . . , fn ) where fi (x) =

c exp(−|xi |) n

for x = (x1 , x2 , . . . , xn ) ∈ ℝn

(1 ≤ i ≤ n),

since each fi ∈ L∞ ⊂ Kn,2 by virtue of (5.38). In the remainder of this section, we briefly state another application of Theorem 2.1 related with Green function. Proposition 5.5. Let (M, d, μ) be a metric measure space. Let {Pt }t>0 be a semigroup on ̂ t = Pt . Assume that {Pt }t>0 possesses an integral kernel qt (x, y) such that for L1 and P each t > 0, 󵄨󵄨 󵄨 󵄨󵄨qt (x, y)󵄨󵄨󵄨 ≤ ϕ(t)

(5.48)

for μ-almost all x, y in M, where ϕ : ℝ+ 󳨃→ ℝ+ is a measurable function. Then there exist a regular μ-nest {Fn }∞ n=1 and a pointwise defined version pt (x, y) of qt (x, y) in (0, ∞)×M ×M such that the following properties are true: for each t, s > 0 and all x, y ∈ M, (1) pt (⋅, ⋅) is jointly measurable in M × M; (2) pt (x, ⋅) and pt (⋅, y) belong to C({Fn }), and |pt (x, y)| ≤ ϕ(t); (3) pt+s (x, y) = ∫M pt (x, z)ps (z, y) dμ(z); (4) Pt f (x) = ∫M pt (x, y)f (y) dμ(y) for μ-almost all x ∈ M. Proof. By (5.48), we have for any t > 0, f ∈ L1 , 󵄩󵄩 󵄩󵄩 ̂ t f ‖L∞ = ‖Pt f ‖L∞ = 󵄩󵄩󵄩∫ qt (⋅, y)f (y) dμ(y)󵄩󵄩󵄩 ≤ ϕ(t)‖f ‖ 1 . ‖P L 󵄩󵄩 󵄩󵄩 ∞ 󵄩 󵄩L It follows that all the hypothesis of Theorem 2.1 are satisfied, and Proposition 5.5 follows.

64 | A. Grigor’yan et al. Proposition 5.5 has the following advantage. One may define the Green function G(x, y) as the integral of a heat kernel qt (x, y) with respect to dt, that is, ∞

G(x, y) = ∫ qt (x, y) dt. 0

However, this integral may not be well defined, because qt (x, y) is defined for μ × μ-almost all (x, y) ∈ M × M where the null set may depend on t. Proposition 5.5 says that one can use a pointwise defined version pt (x, y), instead of qt (x, y) itself, to define the Green function by ∞

G(x, y) = ∫ pt (x, y) dt,

(5.49)

0

since we have a common measurable set 𝒩 independent of t with μ(𝒩 ) = 0, and the integral in (5.49) makes sense in this way.

6 Proof of Theorem 2.2 In this section we prove Theorem 2.2. We shall use the following results in the proof. Proposition 6.1 ([5, Lemma 2.1.3, p. 69]). Let (ℰ , ℱ ) be a Dirichlet form on L2 . Given an ℰ -nest {Fk } of M, let Fk󸀠 = supp[1Fk .μ] for each k. Then Fk󸀠 ⊂ Fk for each k ≥ 1, and {Fk󸀠 } is a regular ℰ -nest. For a Dirichlet form (ℰ , ℱ ), a function u is ℰ -quasicontinuous if for any ε > 0, there is an open set G ⊂ M such that cap(G) < ε and u|M\G is finite continuous (cf. [5, p. 69]). Lemma 6.2 ([5, Theorem 2.1.2, p. 69]). Let (ℰ , ℱ ) be a Dirichlet form on L2 . The following statements are true: (i) Let S = {ul , l ≥ 1} be a countable family of ℰ -quasicontinuous functions on M. Then there is a common regular ℰ -nest {Fk } of M such that S ⊂ C({Fk }). (ii) Let {Fk } be a regular ℰ -nest on M and u belongs to C({Fk }). If u ≥ 0 μ-almost everywhere on an open set U, then u(x) ≥ 0 for every point x ∈ U ∩ (⋃∞ k=1 Fk ). Lemma 6.3 ([5, Theorem 2.1.3, p. 71]). Let (ℰ , ℱ ) be a regular Dirichlet form on L2 . Then ̃ , that is, function u ̃ is each function u in ℱ has an ℰ -quasicontinuous modification u ̃ almost everywhere in M. ℰ -quasicontinuous and u = u We need to assume that (ℰ , ℱ ) is regular in Lemma 6.3. We begin to prove Theorem 2.2.

Existence of the heat kernel | 65

Proof of Theorem 2.2. As shown in the proof of Theorem 2.1, it suffices to consider the case when T0 = ∞. We sketch the proof, since the argument is similar to that for Theorem 2.1. In fact, one needs only to replace μ-nest in the proof of Theorem 2.1 by ℰ -nest here, and the rest argument keeps the same but much simpler since {Pt }t>0 is ̂ t . Let p ∈ [1, 2], and note that {Pt } is contractive on Lq (q ∈ [1, ∞]), symmetric, Pt = P that is, ‖Pt f ‖Lp ≤ ‖f ‖Lq

t > 0, f ∈ Lq .

Step 1. We show that there exists a pointwise realization Qt f for Pt f when f ∈ Lp , t > 0, and also a common regular ℰ -nest {Fn }∞ n=1 of M such that for all t > 0 and all p f ∈L , Qt f ∈ C({Fn }).

(6.1)

Indeed, note that if f ∈ L2 ∩ Lp (M), then Pt f ∈ ℱ

for any t > 0

(6.2)

(cf. [5, Lemma 1.3.3(i), p. 23]). Since (ℰ , ℱ ) is regular, the space ℱ ∩ C0 (M) is dense in C0 (M) in the supremum norm. Using the fact that C0 (M) is dense in Lp , we see that ℱ ∩ C0 (M) is dense in Lp . Since Lp is separable, there exists a sequence {fk }∞ k=1 from p ℱ ∩ C0 (M) dense in L . It follows from (6.2) that the function Pt fk ∈ ℱ for each t > 0, k ≥ 1, and thus it has an ℰ -quasicontinuous version ht,k by using Lemma 6.3. Consider the countable family {hs,k : s ∈ ℚ+ , k ≥ 1}, where ℚ+ is the set of all positive rational numbers as before. By Lemma 6.2(i), there exists a common regular ℰ -nest {Fn }∞ n=1 such that hs,k ∈ C({Fn })

for all s ∈ ℚ+ , k ≥ 1.

(6.3)

Set M0 := ⋃∞ n=1 Fn and 𝒩 = M \ M0 . Clearly, μ(𝒩 ) = cap(𝒩 ) = 0. We will extend (6.3) to any function f in Lp (not only for fk ) by using assumption (2.17), and then continue to extend it to any real positive t (not only for rationals s) by using the semigroup property. To do this, similar to (4.5), we have for each x ∈ U in 𝒮 and k, j ≥ 1, 󵄨 󵄨 󵄨 󵄨 sup 󵄨󵄨󵄨hs,k (x) − hs,j (x)󵄨󵄨󵄨 = sup 󵄨󵄨󵄨hs,k (x) − hs,j (x)󵄨󵄨󵄨

x∈U∩M0

x∈U\𝒩

= ‖hs,k − hs,j ‖L∞ (U) (using hs,k − hs,j ∈ C({Fn }) and Lemma 6.2(ii)) a. e.

= ‖Ps fk − Ps fj ‖L∞ (U) (using hs,k = Ps fk ∀k) ≤ φ(U, s)‖fk − fj ‖Lp (using (2.17)).

(6.4)

66 | A. Grigor’yan et al. For any f ∈ Lp , there is a sequence {fki }i≥1 from {fk }k≥1 such that ‖fki − f ‖Lp → 0 as i → ∞. Thus by (6.4), for each s ∈ ℚ+ and U ∈ 𝒮 , the sequence {hs,ki }i≥1 converges uniformly to a function, say, Qs f , in U ∩ M0 . Since U ∈ 𝒮 is arbitrary, and 𝒮 covers M, for any s ∈ ℚ+ and f ∈ Lp , we can define the function Qs f on M by Qs f (x) = {

limi→∞ hs,ki (x), 0,

for x ∈ M0 = ⋃U∈𝒮 (U ∩ M0 ), for x ∈ 𝒩 .

(6.5)

It follows by (6.3) that Qs f ∈ C({Fn })

(6.6)

for all s ∈ ℚ+ , k ≥ 1 and all f ∈ Lp . Similarly to (4.9), we can prove that Qs (Pt−s f )(x) = Qs󸀠 (Pt−s󸀠 f )(x)

for every x ∈ M and s, s󸀠 ∈ (0, t) ∩ ℚ+ .

Consequently, we can extend Qs f in (6.5) to any positive real number t by defining Qt f (x) = Qs (Pt−s f )(x),

f ∈ Lp , x ∈ M,

(6.7)

where s is a positive rational smaller than t. Note that the above formula is consistent when t is rational. Then {Qt }t>0 satisfies the following properties: – For each t > 0, we have Qt = Pt on Lp . Moreover, for any x ∈ 𝒩 and any f ∈ Lp , Qt f (x) = 0. –

For each t > 0, x ∈ U ∈ 𝒮 , and each f ∈ Lp , we have 󵄨󵄨 󵄨 󵄨󵄨Qt f (x)󵄨󵄨󵄨 ≤ ‖Pt f ‖L∞ (U) ≤ φ(U, t)‖f ‖Lp .

–

For all t > 0 and all f ∈ Lp , we have Qt f ∈ C({Fn }).

–

{Qt }t>0 satisfies the semigroup property, namely, for any real t1 , t2 > 0, f ∈ Lp , and any x ∈ M, Qt1 +t2 f (x) = Qt1 (Qt2 f )(x).

Existence of the heat kernel | 67

–

Qt is bounded and linear, that is, for each t > 0, ‖Qt ‖Lp 󳨃→Lp = ‖Pt ‖Lp 󳨃→Lp ≤ 1 < ∞,

Qt (af + bg)(x) = aQt f (x) + bQt g(x) for all x ∈ M, a, b ∈ ℝ, and f , g ∈ Lp , as well as

(Qt f , g) = (Pt f , g) = (f , Pt g) = (f , Qt g). Step 2. We show that the semigroup {Qt }t>0 possesses an integral kernel qt (x, y). More p precisely, let p󸀠 = p−1 ∈ [2, ∞] be the Hölder conjugate of p. Then, for each t > 0 and

x ∈ U ∈ 𝒮 , there exists a function qt (x, ⋅) in Lp such that for any f ∈ Lp , 󸀠

Qt f (x) = ∫ qt (x, y)f (y) dμ(y), M

󵄩󵄩 󵄩 󵄩󵄩qt (x, ⋅)󵄩󵄩󵄩Lp󸀠 ≤ φ(U, t),

(6.8) (6.9)

and 󵄩 󵄩 sup󵄩󵄩󵄩qt (x, ⋅)󵄩󵄩󵄩L1 ≤ 1. x∈M

(6.10)

Consequently, by (6.9), (6.10), and Hölder inequality, we obtain for any q ∈ [1, p󸀠 ], (q−1)(p−1) 󵄩 󵄩 sup󵄩󵄩󵄩qt (x, ⋅)󵄩󵄩󵄩Lq ≤ (φ(U, t)) . x∈U

(6.11)

In particular, for any t > 0, qt (x, ⋅) ∈ Lp since p ∈ [1, p󸀠 ]. Note that the function qt (x, ⋅) is defined for each t > 0 and each x ∈ M, and qt (x, ⋅) = 0

in M

(6.12)

whenever x ∈ 𝒩 and t > 0. Step 3. We construct the desired pt (x, y) by using function qt (x, y). Indeed, we can define pt (x, y) for any t > 0 and any x, y ∈ M by pt (x, y) = ∫ qt/2 (x, z)qt/2 (y, z) dμ(z).

(6.13)

M

Note that the integral in the right hand side of (6.13) is well defined by (6.9) and the fact that qt (x, ⋅) ∈ Lp . Similar to (4.42), we have for any t > 0 and any x ∈ M, Qt f (x) = ∫ pt (x, z)f (z) dμ(z). M

(6.14)

68 | A. Grigor’yan et al. Finally, we verify that the function pt (x, y) defined by (6.13) is a heat kernel. We only need to verify the symmetry and positivity of pt (x, y), and (2.20). Other properties can be verified as in Step 3 of the proof of Theorem 2.1. Indeed, symmetry follows directly from the definition (6.13). Positivity can be verified by the similar arguments in Corollary 4.2(1). It remains to prove (2.20). By (6.8) and (6.14), we obtain that for any t > 0 and x ∈ M, a. e.

pt (x, ⋅) = qt (x, ⋅). This together with (6.11) yields (2.20). The proof is complete. Remark 6.4. Consider the special case when 𝒮 = {M}, p = 1, and T0 = ∞ in Theorem 2.2. In this case, by the fact that pt (x, ⋅) is quasicontinuous, the inequality (2.19) becomes the diagonal upper estimate: for any t > 0, pt (x, y) ≤ φ(M, t),

x, y ∈ M.

This result in this special case was already addressed in [2, Theorem 3.1]. However, the authors used the joint measurability of the function p0 (t, x, y) (whose counterpart is qt (x, y) in (6.8) in our paper) in (x, y) without proof; see formula [2, (3.5)] and formulas following it, they did not prove the joint measurability of pt (x, y) in (x, y) for any fixed t > 0 either. Note also that the function M(t) in [2, Theorem 3.1] was assumed to be left continuous, while the function φ(M, t) in Theorem 2.2 is not assumed to be left continuous. Remark 6.5. Under the assumption in Theorem 2.2, when 𝒮 = {M}, one can prove by duality that condition (2.17) with p = 1 is equivalent to that with p = 2. While in the present settings, it is not clear if they are equivalent or not. Roughly speaking, if we denote the function in condition (2.17) by φp , then condition (2.17) with p = 1 implies that the heat kernel satisfies the diagonal upper estimate: pt (x, y) ≤ φ1 (U, t),

t ∈ (0, T0 ), x ∈ U ∈ 𝒮 , y ∈ M,

(6.15)

while (2.17) with p = 2 implies that the heat kernel satisfies the on-diagonal upper estimate: 󵄩 󵄩2 pt (x, x) = 󵄩󵄩󵄩pt/2 (x, ⋅)󵄩󵄩󵄩L2 ≤ φ2 (U, t/2)2 ,

t ∈ (0, T0 ), x ∈ U ∈ 𝒮 .

(6.16)

Clearly, (6.15) implies (6.16) with φ2 (U, t) := √φ1 (U, 2t), while it is not clear whether (6.16) implies (6.15) with some function φ1 determined by φ2 . Remark 6.6. Note that the number p in Theorem 2.2 is assumed to be as in [1, 2]. In fact, under the assumption of Theorem 2.2 but with p ∈ (2, ∞), we can follow the arguments of Lemma 5.1 to obtain all the results of Theorem 2.2 except that pt (x, ⋅) ∈ C({Fn }). The

Existence of the heat kernel | 69

p < 2 < p, then the reason is as follows. In the case when p ∈ (2, ∞), we have p󸀠 = p−1 p combination of (6.9) and (6.10) cannot guarantee that qt/2 (x, ⋅) ∈ L as done in the case when p ∈ [1, 2]. Consequently, by (6.13), the function pt (x, ⋅) = Qt/2 qt/2 (x, ⋅)(⋅) may not be in C({Fn }).

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Qingsong Gu and Ka-Sing Lau

Resistance estimates and critical exponents of Dirichlet forms on fractals Abstract: Let Bσ2,∞ , Bσ2,2 denote the Besov spaces defined on a compact set K ⊂ ℝd that is equipped with an α-regular measure μ (K is called an α-set). The critical exponent σ ∗ is the supremum of the σ such that Bσ2,2 ∩ C(K) is dense in C(K). It is known that Bσ2,2 is the domain of a nonlocal regular Dirichlet form, and for certain standard self-similar set, ∗ Bσ2,∞ is the domain of a local regular Dirichlet form, but it is not clear in general. In this

paper, we give a brief account on our investigation on the critical exponent and Bσ2,∞ through the resistance growth rates. Also we study the convergence of the Bσ2,2 -norm to ∗

the Bσ2,∞ -norm as σ ↗ σ ∗ and the associated Dirichlet forms using a recent technique of Grigor’yan and Yang [13]. The theorem extends a celebrated result of Bourgain, Brezis, and Mironescu [6] on classical domains. ∗

Keywords: Besov space, Dirichlet form, Γ-convergence, Harnack inequality, heat kernel, p. c. f. fractal, resistance estimate MSC 2010: Primary 28A80, Secondary 46E30, 46E35

Contents 1 2 3 3.1 3.2 4 4.1 4.2 5 5.1 5.2 5.3

Introduction | 72 Discretization of Besov spaces | 74 Resistance and critical exponents on p. c. f. sets | 76 Resistance and critical exponents | 78 Two examples | 81 Non-p. c. f. case: triangular carpets and diamonds | 84 Triangular carpet and resistance estimates | 84 Triangular diamonds and resistance estimates | 88 Uniform Harnack inequality | 91 P. c. f. sets with uniform resistance growth | 92 Triangular diamond | 93 Triangular carpet | 94

Acknowledgement: The research is supported in part by an HKRGC grant. The authors thank Professor S.-M. Ngai for going through the manuscript in detail and improving the presentation. Qingsong Gu, Department of Mathematics, Nanjing University, Nanjing 210093, China; and Department of Mathematics and Statistics, Memorial University, St. John’s, NL A1C 5S7, Canada, e-mail: [email protected] Ka-Sing Lau, Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong; and Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15213, USA, e-mail: [email protected] https://doi.org/10.1515/9783110700763-003

72 | Q. Gu and K.-S. Lau

6 7

Construction of Dirichlet forms | 96 Some open problems | 100 Bibliography | 101

1 Introduction It is well-known that on a smooth domain Ω in ℝd , the Sobolev space W 1,p , 1 < p < ∞, has a (semi-)norm defined by 󵄨 󵄨p ‖f ‖pW 1,p = ∫󵄨󵄨󵄨∇f (x)󵄨󵄨󵄨 dx. Ω

It has an equivalent expression given by (cf. [42, p. 139]) 󵄨 󵄨p ‖f ‖pW 1,p ≍ sup |u|−p ∫ 󵄨󵄨󵄨f (x) − f (x + u)󵄨󵄨󵄨 dx |u|>0

(1.1)

Ω

(by ϕ ≍ ψ, we mean that there exists C > 0 such that C −1 ϕ(u) ≤ ψ(u) ≤ Cϕ(u) for all u). The fractional Sobolev space W σ,p , 0 < σ < 1, has the (semi-)norm ‖f ‖pW σ,p = ∬

Ω×Ω

|f (x) − f (y)|p dx dy. |x − y|d+pσ

(1.2)

Note that W 1,p ⊂ W σ,p ⊂ W σ ,p for σ 󸀠 ≤ σ < 1, and for σ > 1, W σ,p contains only constants. We refer to the exponent 1 as the critical exponent of the family {W σ,p }σ>0 . Note that if f ∈ W σ,p , 0 < σ < 1, is a smooth nonconstant function, then ‖f ‖W σ,p → ∞ as σ → 1. In [6], Bourgain, Brezis, and Mironescu adjusted a factor to these norms and obtained 󸀠

Theorem 1.1. Assume f ∈ Lp (Ω). Then limσ↗1− (1 − σ)1/p ‖f ‖W σ,p = C‖f ‖W 1,p , where C depends on the dimension d and p. In this paper, we will give a brief account of our investigation on the corresponding function spaces of (1.1) and (1.2), the Besov spaces Bσ2,∞ and Bσ2,2 on fractal domains (we consider case p = 2 only). We study the critical exponent σ ∗ , and the behavior of Bσ2,∞ . We also prove an analog of Theorem 1.1. The details are in [14, 15, 17]. Let K be a closed subset in ℝd , and let μ be an α-regular measure on K, that is, there exists α > 0 such that for any Euclidean ball B(x, r) with 0 < r < diam(K), ∗

C −1 r α ≤ μ(B(x, r)) ≤ Cr α .

(1.3)

Fix σ > 0. For u ∈ L2 (K, μ), let [u]2Bσ

2,∞

󵄨 󵄨2 := sup r −2σ ∫ ∫ − 󵄨󵄨󵄨u(x) − u(y)󵄨󵄨󵄨 dμ(y) dμ(x) 0 σ ∗ , then Bσ2,∞ consists of constant functions only. For this we define another critical exponent ∗

σ # := sup{σ : Bσ2,∞ contains nonconstant functions}. Clearly, σ ∗ ≤ σ # , and for the standard examples, we have σ ∗ = σ # , but not in general. In [15], we attempted to get a better understanding of the LRDF through the critical exponents as well as the functional behaviors of the associated Besov spaces, which is not clear on more general fractal sets. Our approach in this study is to use a device of triangulation system by Jonsson [25, 26], extended by Bodin [5], to discretize the two Besov spaces Bσ2,∞ and Bσ2,2 . We use the electrical network techniques to estimate the effective resistances and the traces

74 | Q. Gu and K.-S. Lau on the discretized systems. This allows us to get better insight to the two critical ex∗ # ponents, and to get hold of the properties of Bσ2,∞ and Bσ2,∞ . We will discuss these in Sections 3 and 4 and provide some new examples. We prove an analog of Theorem 1.1 by employing a method introduced by Grigor’yan and Yang [13] using a uniform Harnack inequality and De Giorgi’s Γ-convergence of Dirichlet forms. Under appropriate conditions on the resistances, we show that as σ ↗ σ ∗ , (σ ∗ − σ)[u]2Bσ → [u]2Bσ∗ , and [u]2Bσ∗ is an LRDF. This provides us an alternate 2,2

2,∞

2,∞

way to prove the existence of such Dirichlet forms. This will be studied in Section 5. For convenience, we will use the notion f ≍ g (≲ g respectively) to denote the fact that there exists C > 0 such that C −1 f (x) ≤ g(x) ≤ Cf (x) (f (x) ≤ Cg(x) respectively) for the variable x.

2 Discretization of Besov spaces For a compact set K ⊂ ℝd , we set diam(K) := supx,y∈K |x − y| to be the diameter of K. We also define the in-radius r(K) of a compact convex set K with nonempty interior to be the largest possible radius of a ball contained in K. Following [26, 5], a d-simplex Δ is the closed convex hull of d + 1 affinely independent points {v1 , v2 , . . . , vd+1 } in ℝd . We denote by V(Δ) := {v1 , . . . , vd+1 } the vertices of Δ. A triangulation of a compact set K in ℝd is a finite set 𝒯 of d-simplices satisfying (A1) If Δ1 , Δ2 ∈ 𝒯 and Δ1 ≠ Δ2 , then Δ1 ∩ Δ2 is either empty or a common face of lower dimension. (A2) If v is a vertex of Δ ∈ 𝒯 , then v ∈ K. (A3) K ⊆ ⋃Δ∈𝒯 Δ. For a triangulation 𝒯 of K, we set δ(𝒯 ) = maxΔ∈𝒯 diam(Δ), and for a sequence of triangulations {𝒯n }∞ n=0 (𝒯0 is defined to be a seed set Δ), we set δn = δ(𝒯n ). d Definition 2.1. We call a sequence of triangulations {𝒯n }∞ n=0 of a compact set K ∈ ℝ a regular triangular system (RTS) if the following conditions hold: (T1) For each Δ ∈ 𝒯n+1 , there is Δ󸀠 ∈ 𝒯n such that Δ ⊆ Δ󸀠 . (T2) There is C1 > 0 such that for any n and for all Δ, Δ󸀠 ∈ 𝒯n ,

C1−1 diam(Δ) ≤ diam(Δ󸀠 ) ≤ C1 diam(Δ). (T3) There are constants c1 , c2 ∈ (0, 1) such that for all n ≥ 0, c1 δn ≤ δn+1 ≤ c2 δn . (T4) There is a constant c3 > 0 such that for all n ≥ 0 and Δ ∈ 𝒯n , r(Δ) ≥ c3 diam(Δ).

Resistance estimates and critical exponents of Dirichlet forms on fractals | 75

(T5) There are constants c4 , c5 > 0 such that for all Δ, Δ󸀠 ∈ 𝒯n and n ≥ 0, if x ∈ Δ, y ∈ Δ󸀠 and |x − y| ≤ c4 δn , then there is z ∈ Δ ∩ Δ󸀠 such that |z − x| ≤ c5 |x − y| and |z − y| ≤ c5 |x − y|. For a Besov space Bσ2,∞ on a compact set K with an α-regular measure, we need a continuity property of its functions (cf., for example, [12], where the following proposition is put under the assumption that a heat kernel exists, but it was not used in the proof). Proposition 2.2. For 2σ > α, the identity map ι : Bσ2,∞ → C (2σ−α)/2 (K) is a continuous embedding. (Here C β (K) denotes the class of Hölder functions on K of order β.) We let V(Δ) denote the vertices of Δ. For an RTS {𝒯n }∞ n=0 , let Vn = ∪{V(Δ) : Δ ∈ 𝒯n }, and define En (u) =

∑

x,y∈V(Δ),Δ∈𝒯n

󵄨2 󵄨󵄨 󵄨󵄨u(x) − u(y)󵄨󵄨󵄨 ,

u ∈ ℓ(Vn ),

where ℓ(Vn ) is the set of real-valued functions on Vn . We then have the following discretization theorem. Theorem 2.3. Let K be a connected α-set admitting an RTS {𝒯j }∞ j=0 . Then for 2σ > α, [u]2Bσ

2,∞

≍ sup{δj −(2σ−α) Ej [u]} j≥0

(2.1)

and ∞

[u]2Bσ ≍ ∑ δj −(2σ−α) Ej [u]. 2,2

j=0

(2.2)

The discretized version of the Besov spaces in Theorem 2.3 was first established by Jonsson [25] on the Sierpinski gasket. He introduced the notion of RTS on α-sets to study the piecewise linear bases of the Besov space [26]. It is known that polyhedred domains, some simple nested fractal, and the Sierpinski carpet admit RTS. However, there is no good characterization of the class of α-sets. In [5], Bodin extended Jonsson’s discretization theorem to the Besov spaces Bσp,q , 1 ≤ p, q ≤ ∞ for a d-set that admits an RTS. He stated without proof that similar to the RTS case, the discretization is also true for p. c. f. sets. Actually, there are technical steps that need to be justified (see Section 3 and [14]). We remark that in [43], Strichartz constructed several types of functional spaces (including the Besov type) on p. c. f. self-similar sets admitting a regular harmonic structure, in particular, on the Sierpinski gasket. His construction of the Besov spaces uses the discrete approximation of the resistance network of the self-similar energy identity [28]. It is quite different from the approach here on discretizing the continuous Besov spaces on which a regular harmonic structure is not assumed.

76 | Q. Gu and K.-S. Lau The fractals we are going to study are separated into two cases: the p. c. f. case and non-p. c. f. case. We will discuss these two cases separately in the next two sections.

3 Resistance and critical exponents on p. c. f. sets Let {Fi }Ni=1 be an iterated function system (IFS) on ℝd such that Fi (x) = ρ(x − bi ) + bi ,

1 ≤ i ≤ N,

(3.1)

where 0 < ρ < 1 and bi ∈ ℝd . Let K = ⋃Ni=1 Fi (K) be a self-similar set, and let μ be a self-similar measure defined by μ = N1 ∑Ni=1 μ ∘ Fi−1 . If the IFS satisfies the open set condition (OSC), i. e., there is a nonempty bounded open set O such that Fi (O) ⊂ O and N Fi (O) ∩ Fj (O) = 0 for i ≠ j, then the Hausdorff dimension of K is dimH (K) = α = −log , log ρ and μ is the α-Hausdorff measure normalized on K; it is α-regular in the sense of (1.1). Without loss of generality, we always assume that K is connected. We define the symbolic space of K as usual. Let Σ = {1, . . . , N} be the alphabet, Σn the set of words of length n, and Σ∞ the set of infinite words ω = ω1 ω2 . . . ; let π : Σ∞ → K be defined by {x} = {π(ω)} = ⋂n≥1 Kω1 ...ωn , a symbolic representation of x ∈ K by ω. Following Kigami [28], we define the critical set 𝒞 and the post-critical set 𝒫 for K by m

−1

𝒞 = π (⋃1≤i 0 such that for any integer m ≥ 1 and any two words ω and ω󸀠 with length m and Kω ∩ Kω󸀠 = 0 (if and) only if dist(Kω , Kω󸀠 ) ≥ c0 ρm . Equivalently, |x − y| < c0 ρm (if and) only if x and y lie in the same or neighboring m-cells. The property on the Sierpinski gasket is obvious and was used in [25]. It was also studied in [33] in another context, and there are non-p. c. f. examples for which condition (H) fails even with the open set condition. In [40], Pietruska-Pałuba and Stós introduced another separation condition, called property (P), which is similar to property (H). They proved that it holds on certain nested fractals, and was used to study the Hajłasz–Sobolev spaces on nested fractals via the Poincaré inequality. The following fact is proved in [14]: Proposition 3.1. Suppose the IFS {Fi }Ni=1 in (3.1) has the p. c. f. property. Then it satisfies condition (H). In comparison with the RTS, property (H) is analogous to (T5). We note that in the proof of Theorem 2.3 (cf. [5]), the d-simplices can be replaced by the cells Kw generated by the IFS with vertices Fw (V0 ) and the sequence of triangulations can be replaced n by the sequence {Kw : |w| = n}∞ n=0 . Then we see that in this situation, δn = ρ , the analogous conditions (T1)–(T3) hold trivially; (T4) is also true since each “simplex” is now a cell Fw (K) using Fw (V0 ) as the vertices of the cell. The condition analogous to (T5) is property (H). This is because (H) ensures that if |x − y| < c0 ρn for two points x and y in two n-cells Fw (V0 ) and Fv (V0 ), respectively, then Kw ∩Kv ≠ 0 and we have x = y or |x − y| ≍ ρn . Therefore there exists z ∈ Fw (V0 ) ∩ Fv (V0 ) satisfying |x − z| ≲ ρn ≲ |x − y| and |y − z| ≲ ρn ≲ |x − y|. Given this, we have the following discretization analogous to Theorem 2.3 [14]. We denote by En [u] = ∑x,y∈Vω ,|ω|=n (u(x) − u(y))2 and call it the primal energy of u at the nth level. Theorem 3.2. Let K be a p. c. f. self-similar set satisfying (3.1). Then for 2σ > α, [u]2Bσ

2,∞

≍ supj≥0 {ρ−(2σ−α)j Ej [u]}

and ∞

[u]2Bσ ≍ ∑j=0 ρ−(2σ−α)j Ej [u]. 2,2

78 | Q. Gu and K.-S. Lau

3.1 Resistance and critical exponents Let Gn := (Vn , rn ) denote the corresponding electrical network of En [u] with rn (x, y) = cn (x, y)−1 , x, y ∈ Vn as resistance. It is known that [28, Theorem 2.1.6] for any m < n, there is an induced network of Gn on Vm with resistance Rn,m (x, y) such that for u ∈ ℓ(Vm ), min{En [υ] : υ ∈ ℓ(Vn ), υ|Vm = u} = ∑

x,y∈Vm

1 󵄨2 󵄨󵄨 󵄨u(x) − u(y)󵄨󵄨󵄨 . Rn,m (x, y) 󵄨

(3.2)

We will refer the above υ that attains the minimum the harmonic extension of u on Vn . Let {Rn }∞ n=0 be a sequence of positive real numbers. Suppose there exists R > 1 such that for any ε > 0, there exists N(ε) such that for all n ≥ N(ε), R(1−ε)n ≤ Rn ≤ R(1+ε)n . Then we call R the asymptotic geometric growth rate of Rn . Definition 3.3. We call Rn,m (x, y), x, y ∈ Vm the trace (or the induced resistance) of Gn on Vm . In particular, for m = 0, we will use the notation Rn (p, q), p, q ∈ V0 for simplicity. We also use R(p, q) to denote the asymptotic geometric growth rate of Rn (p, q) if it exists. We state a general result for the critical exponents σ ∗ of the Besov spaces on the p. c. f. sets with respect to the primal energy [15, Theorem 4.1]. Lemma 3.4. For the primal energy En [u], n ≥ 0 as defined in (1.3), suppose σ satisfies 2σ > α, and there exists an integer N ≥ 1 such that ρ−(2σ−α)N ≤ RN (p, q),

∀p, q ∈ V0 , p ≠ q,

(3.3)

then u ∈ ℓ(V0 ) has an extension to K, and consequently, Bσ2,∞ is dense in C(K). Proof. Let u ∈ V0 . By using (3.2), we obtain min

v∈ℓ(VN ),v|V0

󵄨 󵄨2 {ρ−(2σ−α)N EN [v]} ≤ ∑ 󵄨󵄨󵄨u(p) − u(q)󵄨󵄨󵄨 . =u p,q∈V0

Let u1 be the harmonic extension of u on VN . We then use u1 as initial data on each Fω (V0 ), |ω| = N, and obtain u2 , the harmonic extension of u1 on V2N . Repeating this procedure to V2N , V3N , . . . , for k ≥ 1, 󵄨 󵄨2 ρ−(2σ−α)kN EkN [uk ] ≤ ∑ 󵄨󵄨󵄨u(p) − u(q)󵄨󵄨󵄨 . p,q∈V0

Hence there is ũ on V∗ = ⋃n≥0 Vn such that 󵄨 󵄨2 sup ρ−(2σ−α)kN EkN [u]̃ ≤ ∑ 󵄨󵄨󵄨u(p) − u(q)󵄨󵄨󵄨 . k≥1

p,q∈V0

Then we can extend ũ continuously to K with ũ ∈ Bσ2,∞ [15].

Resistance estimates and critical exponents of Dirichlet forms on fractals | 79

It follows that for any v ∈ C(K), if we let vn be the restriction of v on Vn , we can extend vn on each cell Kω , |ω| = n so that vn ∈ Bσ2,∞ (this vn is a piecewise harmonic σ function). The sequence {vn }∞ n=1 converges to v uniformly. This shows that B2,∞ is dense in C(K). Let K be a p. c. f. self-similar set with an IFS satisfying (3.1). We can show that there exists λ > 1 such that Rn (p, q) ≥ λn for all n and p ≠ q, p, q ∈ V0 . If further, the asymptotic geometric growth rate R(p, q) exists, we have the following useful theorem. Theorem 3.5. Let K be a p. c. f. self-similar set with an IFS satisfying (3.1). Assume R(p, q) (> 1), p, q ∈ V0 exist, and let R∗ = min{R(p, q) : p ≠ q, p, q ∈ V0 }. Then for the Besov spaces Bσ2,∞ defined on K, the critical exponent is σ ∗ = 21 ( −loglogR ρ + α). Furthermore at σ ∗ , ∗ (i) if R∗n ≤ Rn (p, q) for all n ≥ 0 and p, q ∈ V0 , then Bσ2,∞ is dense in C(K); ∗

(ii) if R∗ satisfies limn→∞

R∗n Rn (p,q)

= ∞ for some p, q ∈ V0 , then Bσ2,∞ is not dense in C(K). ∗

Proof. If σ < 21 ( −loglogR ρ + α), then, by the definition of R∗ , we can find N = N(σ) such that ∗

ρ−(2σ−α)N ≤ Rn (p, q)

∀p, q ∈ V0 .

Hence Lemma 3.4 implies Bσ2,∞ is dense in C(K). Therefore 21 ( −loglogR ρ + α) ≤ σ ∗ . ∗

Next consider σ > 21 ( −loglogR ρ + α). Let σ = 21 ( −loglogR ρ + α) + ε, and let R∗ε = ρ−2ε R∗ . For any u ∈ Bσ2,∞ , we restrict u ∈ ℓ(V0 ) with values u(pi ), pi ∈ V0 . From the trace formula in (3.2), we obtain ∗

min

∗

∗

{ρ

v∈ℓ(Vn ),v|V0 =u

R −( log +2ε)n − log ρ

En [v]} = ∑ i=j̸

R∗n 󵄨󵄨 󵄨2 ε 󵄨u(p ) − u(pj )󵄨󵄨󵄨 . Rn (pi , pj ) 󵄨 i

(3.4)

By the definition of R∗ , we have R∗ = R(p, q) for some p, q ∈ V0 , and R∗n ε ≥ ρ−εn → ∞ Rn (p, q)

as n → ∞.

(3.5)

As u ∈ Bσ2,∞ , the left-hand side of (3.4) is uniformly bounded for all n > 0. Hence by (3.5), u(p) = u(q) on the right-hand side of (3.4). This proves Bσ2,∞ is not dense in C(K), hence σ ∗ ≤ 21 ( −loglogR ρ + α). ∗

Assertion (i) follows from the Lemma 3.4. For (ii), if we replace (3.5) by the assump∗ tion, then the same argument applies and we conclude that u(p) = u(q), so that Bσ2,∞ is not dense in C(K). It is important to know the density of Bσ2,∞ in C(K) (for regularity). Part (i) covers the standard cases that renormalization factors exist [28]; there is an example in ∗

80 | Q. Gu and K.-S. Lau Subsection 3.2 (eyebolted Vicsek cross) that satisfies (ii). In addition, we have the situ∗ ation R∗ ≍ Rn (p, q) for some p, q ∈ V0 (Sierpinski sickle) and the density question Bσ2,∞ in C(K) is not covered by the above two situations. We develop a “quotient network” technique to handle this case. Definition 3.6. Let ∼ be an equivalence relation on V0 that contains at least two equivalence classes. We define the induced equivalence relation ∼n to be the smallest equivalence relation on Vn generated by (i) (embedding) for x ∼n−1 y in Vn−1 (⊂ Vn ), then x ∼n y in Vn ; (ii) (self-similar) for x ∼n−1 y in Vn−1 , then Fi (x) ∼n Fi (y) for 1 ≤ i ≤ N. We say that ∼ is a compatible (equivalence) relation if for any n ≥ 0 and any x, y ∈ Vn , x ∼n y in Vn if and only if x ∼n+1 y in Vn+1 . We call an equivalence class J of Vn (or V∗ ) a boundary class if J ∩ V0 ≠ 0, and a nonboundary class otherwise. To obtain some separation property of the boundary classes, we introduce a property on the compatible relation: (B) Any 1-cell can intersect at most one boundary class in V1 . Proposition 3.7. With the assumptions in Theorem 3.5, suppose there is a compatible relation with property (B), which satisfies (i) for each nontrivial boundary class J of V∗ and for any u on J ∩ V0 , there is an extension of u on J such that supn≥0 R∗n EJ,n (u) < ∞, (ii) for any two distinct equivalence classes Ji , Jj of V0 , R∼ (Ji , Jj ) exists and satisfies R∼ (Ji , Jj ) > R∗ . Then Bσ2,∞ is dense in C(K). ∗

The main idea behind condition (ii) is to use the equivalence class (i. e., shorting in the sense of electrical network) to get rid of the small Rn (p, q), and reduce to an expression analogous to Theorem 3.5(i); there are also sufficient conditions for (i) to hold. Since the proof is rather long and technical, and involves the use of graph directed systems on the quotients [36], we will refer the reader to [15] for details. We will demonstrate the theorem by the Sierpinski sickle in the next subsection. In the following, we will consider the second critical exponent σ # . It concerns with the upper bound of the asymptotic geometric growths of R(p, q), p, q ∈ V0 ; again we make use of the quotient network. Theorem 3.8. Let K be a p. c. f. self-similar set with an IFS satisfying (3.1). Assume R(p, q) (> 1), p, q ∈ V0 exist. Let R# = min {s :

∀ p ≠ q in V0 , ∃ a chain p = p1 , p2 , . . . , pm = q in V0 ∋ R(pi , pi+1 ) ≤ s, 1 ≤ i ≤ m − 1

}.

Resistance estimates and critical exponents of Dirichlet forms on fractals | 81

Suppose there is a compatible relation on V0 such that for any small ε > 0, (R# )(1−ε)n = 0, n→∞ R∼ (J , J ) n i j lim

∀Ji , Jj ∈ V0∼ , Ji ≠ Jj .

(3.6)

#

Then σ # = 21 ( −loglogR ρ + α). In the proof of the above theorem, we see that if u ∈ ℓ(Vn∼ ), then u can be extended to be in Bσ2,∞ , σ < σ # and u is constant on each equivalence class J ∈ V∗∼ . By using #

Theorem 3.7, we can prove the following proposition which gives the density of Bσ2,∞ in L2 (K, μ). Proposition 3.9. With the same assumption as in Theorem 3.8, suppose that max{dimH J ̄ : J is a boundary class} < dimH K.

(3.7)

Then for σ < σ # , Bσ2,∞ is dense in L2 (K, μ). If further, for any u on ℓ(V0∼ ), there is an extension of u on V∗∼ such that sup R#n En (u) < ∞. n≥0

(3.8)

#

Then Bσ2,∞ is dense in L2 (K, μ). The above two results will be demonstrated by the two examples in the next subsection.

3.2 Two examples We present two asymmetric p. c. f. sets to illustrate the previous results. In the examples, the major work is to estimate the growth rate of the traces Rn (p, q), p, q ∈ V0 , we also introduce a recursive construction to obtain a Dirichlet form. The first example is modified from the Vicsek cross by adding two eyebolts on the cross to produce the irregularity (see Figure 1), we call it the eyebolted Vicsek cross. It consists of 21 maps with contraction ratio 1/9, and has four boundary points V0 . By equipping the Vn with the primal energy, and using a generalized Δ–Y transform from electrical network theory [15, Section 2], we can show that the traces Rn (p, q) have different growth rates, but the same asymptotic growth rate 9n . By using this, we conclude the following Theorem 3.10. For the eyebolted Vicsek cross K in Figure 1, the critical exponents are 1 log 21 ). σ ∗ = σ # = (1 + 2 log 9

82 | Q. Gu and K.-S. Lau

Figure 1: The eyebolted Vicsek cross K.

Moreover,

(i) Bσ2,∞ (⊂ C(K)) is dense in L2 (K, μ), but not dense in C(K); ∗

(ii) there are two kinds of (nonprimal) local regular Dirichlet forms that can be constructed on K, one satisfies the energy self-similar identity; the other follows from a “reverse recursive construction” and the energy is not self-similar. Their domains ∗ are different from Bσ2,∞ . For statement (i), we carry out the estimate of Rn (p, q), p, q ∈ V0 , and apply The∗ orem 3.5 (ii) to show that Bσ2,∞ is not dense in C(K); apply Proposition 3.9 with the

equivalence classes V0 = J1 ∪ J2 ∪ J3 = {p1 } ∪ {p2 , p4 } ∪ {p3 } to show Bσ2,∞ is dense in L2 (K, μ). We see that we cannot have a regular (sufficiently many continuous func∗ tions) Dirichlet form from the renormalized limit of the primal energy and has Bσ2,∞ as its domain. On the other hand, in (ii), we can use different conductances to obtain energy forms that yield local regular Dirichlet forms on K. One construction gives an energy form that satisfies the energy self-similar identity [28]; it provides a concrete constructive proof to implement the abstract proof (fixed point theory) for the existence of such energy form on asymmetric p. c. f. sets [34, 35, 41, 21, 38]. The other construction, we call it reverse recursive method (RRM), is to fix an initial data at V0 , and iterate this to Vn to obtain a sequence of compatible networks. The Dirichlet form is not self-similar. This RRM is analogous to a probabilistic study by Hattori, Hattori and Watanabe [23] on the Sierpinski gasket K (abc-gasket), they showed that there is an asymptotically one-dimensional diffusion process on K. Some further development and extensions using the probability method can be found in [18, 20, 22] by Hambly et al. In [16], Qiu et al. give a characterization of all the local regular Dirichlet forms on the Sierpinski gasket based on the RRM. We call the second example a Sierpinski sickle. It is a connected p. c. f. set K generated by an IFS of 17 similitudes of contraction ratio 1/7 (see Figure 2); the boundary V0 has three points. By using the Δ–Y transform, we can show that the traces Rn (p1 , p2 ) ≍ Rn (p2 , p3 ) ≍ (17/2)n , Rn (p3 , p1 ) ≍ 7n . ∗

Resistance estimates and critical exponents of Dirichlet forms on fractals | 83

Figure 2: The Sierpinski sickle K.

Theorem 3.11. For the Sierpinski sickle (Figure 2), we have 1 log 17 σ ∗ = (1 + ), 2 log 7 Moreover,

1 2 log 17 − log 2 σ# = ( ). 2 log 7 #

(i) Bσ2,∞ (⊂ C(K)) is dense in C(K), and Bσ2,∞ is dense in L2 (K, μ); ∗

(ii) there are (nonprimal) Dirichlet forms on K that satisfy the energy self-similar identity; but the reverse recursive method does not yield a Dirichlet form on K.

The proof of (i) is by taking equivalence class V0 = J1 ∪ J2 = {p1 , p3 } ∪ {p3 }, and applying Theorems 3.5, 3.8, and Propositions 3.7, 3.9. We remark that not all asymmetric p. c. f. sets K will give inhomogeneous rate on the Rn (p, q)’s. In fact, in the above two examples, the construction is quite delicate; if we make small variances on the IFS, then the growth rate of the Rn (p, q)’s on K will have the same asymptotic growth rate. The following figures are such examples with simpler configuration on the handle part of the sickle (ρ = 1/7, 1/6, 1/5, respectively), and Rn (p, q) ≍ λn for some λ > 1 in each case. Remark. In view of this and the results in Subsection 3.1 and the examples, it is seen that for the existence of LRDF at critical exponents with respect to the primal energy, it is reasonable to assume the following trace (resistance) growth condition on the p. c. f. sets: (R) there exists λ > 1 and C > 0 such that for any two distinct p, q ∈ V0 , and for any n ≥ 1, C −1 λn ≤ Rn (p, q) ≤ Cλn .

(3.9)

(Note that for p. c. f. sets of (3.1), there always exists λ > 1 such that Rn (p, q) ≥ λn .) In particular, the three modified Sierpinski sickles in Figures 3, 4, 5 are examples satisfying condition (R). In Section 6, we will study the existence of LRDF on p. c. f. sets assuming condition (R), together with two non-p. c. f. sets with RTS that V0 can be defined and possess (R) (Section 4).

84 | Q. Gu and K.-S. Lau

Figure 3: K1 .

Figure 4: K2 .

Figure 5: K3 .

4 Non-p. c. f. case: triangular carpets and diamonds In this section, we study two classes of non-p. c. f. self-similar sets on which we can derive resistance growth rate satisfying condition (R) in (3.9). The two classes are named triangular carpets inspired by Barlow and Bass [2, 3] on the Sierpinski carpets, and the other class is triangular diamonds which are finitely ramified fractals (in the wide sense), generalizing the diamond fractals first studied by Kigami, Strichartz, and Walker [29].

4.1 Triangular carpet and resistance estimates The triangular carpets are triangular version of the generalized Sierpinski carpets. Let T be an equilateral triangle with side length 1 and three vertices Q0 = exp(πi/3), Q1 = 0, Q2 = 1. Let ℓ ≥ 4 be an integer. Divide each side of T into ℓ equal segments and draw line segments through these dividing points that are parallel to one of the three sides. This yields altogether ℓ2 subtriangles with side length 1/ℓ among them. For each subtriangle Δi , we assign a similitude Fi such that Fi (T) = Δi (contraction ratio is ρ = ℓ−1 ). We choose a subclass {Fi }i∈ℐ and let K be the invariant set. Reindex ℐ as Σ = {1, . . . , N} and follow the standard notations, with w ∼ w󸀠 meaning w, w󸀠 ∈ Σn and Fw (T) ∩ Fw󸀠 (T) is a line segment. Definition 4.1. We call (K, {Fi }i∈Σ ) a (symmetric) triangular carpet (T-carpet) if the following conditions hold: (i) (Symmetry) T1 = ⋃i∈Σ Fi (T) is invariant under the isometry group (dihedral group D3 ) of T. (ii) (Connectedness) K is connected. (iii) (Nondiagonality) Let n ≥ 1, and let u, w be two distinct elements in Σn . If Ku ∩ Kw = {x} is a singleton, let Σn (x) := {w ∈ Σn : x ∈ Kw }. Then there is a chain u = w(1), w(2), . . . , w(k) = w such that w(i) ∼ w(i + 1) and w(i) ∈ Σn (x) for i = 1, . . . , k − 1. (iv) (Borders included) The three sides of T are contained in T1 .

Resistance estimates and critical exponents of Dirichlet forms on fractals | 85

The following is an example of T-carpet with ℓ = 4 (i. e., ρ = 1/4), and #(ℐ ) = 15 (see Figure 6). Note that for ℓ > 4, there are more choices on the deleted triangle, for example, for ℓ = 6, for suitable choice of ℐ we can delete a hexagon in the center.

Figure 6: An example of T -carpet.

Let Tn be the nth iteration of T. Observe that the subtriangles in Tn can be systematically grouped together to form hexagons (or partial hexagons). Therefore we define the hexagon Ω = ⋃5k=0 T k where T k is a rotation of T with radian exp(kπi/3), 0 ≤ k ≤ 5 (see Figure 8). We do the same for Tn and K, and denote them by Ωn and H, respectively; we call the hexagonal carpet H an H-carpet. We will use H to study the resistance growth, and then reduce it back to the T-carpet. The advantage of using the H-carpet is that it has more symmetry like the Sierpinski carpet, and the techniques in [2, 3] can be applied. To proceed with the resistance estimates, we use the technique of potentials and flows on graphs (see [3, 9]). Let (G, g) be a finite graph of wire network with g(x, y) ≥ 0, g(x, y) = g(y, x) and g(x, x) = 0, x, y ∈ G, define the energy on (G, g) to be 2

ℰ [u] := ℰ (u, u) = ∑ g(x, y)(u(x) − u(y)) , x,y∈G

with u ∈ ℱ , the domain of ℰ , which is the collection of all the real functions on G. Let A and B be two disjoint nonempty subsets of G. The effective resistance [28] is defined as −1

ℛ(A, B)

:= inf{ℰ [u] : u = 0 on A, u = 1 on B}.

(4.1)

The infimum can be attained by a unique function u0 which is harmonic at any points in G\(A∪B), and is called the optimal potential (or realization) of the potential problem pot(G; A, B). To define the current in the network (G, g), we let N(x) = {y : g(x, y) > 0}. A flow I from A to B on G is a real-valued function such that I(x, y) = −I(y, x),

∑ I(x, y) = 0

y∈N(x)

for x ∈ G \ (A ∪ B).

86 | Q. Gu and K.-S. Lau The total flux of I is given by T(I; A, B) := ∑x∈A,y∈N(x) I(x, y) = − ∑x∈B,y∈N(x) I(x, y). Define the energy of I as E[I] := E(I, I) =

∑

g(x,y)>0

g(x, y)−1 I(x, y)2 .

Then we have [9] ℛ(A, B) = inf{E[I] : I is a flow from A to B on G with total flux 1}.

(4.2)

The unique I that attains the minimum in (4.2) is called an optimal flow of cur(G; A, B). Now for n ≥ 0, we introduce two types of wire networks on Ωn , namely Gn and Dn , to approximate Ω. The network Gn is defined on each such n-triangle in Ωn by joining the midpoint of each side of an n-level triangle to its center by a wire of resistance 1 (see Figure 7 for G0 and G1 ).

Figure 7: G0 and G1 .

The network Dn is defined by joining any two of the vertices of each n-level subtriangle by a wire of resistance 1 (see Figure 8 for D0 and D1 ).

Figure 8: D0 and D1 .

Resistance estimates and critical exponents of Dirichlet forms on fractals | 87

On Gn , let EnG [I] be the energy of a flow I on Gn , i. e., EnG [I] = ∑x∼y,x,y∈Gn |I(x, y)|2 . In view of (4.2), we denote the effective resistance by G

G

ℛn := min{En [I] : I is a flow from (L1 ∩ Gn ) to (L4 ∩ Gn ) with total flux 1},

(4.3)

where L1 = Q1 Q2 , and L4 = Q4 Q5 . By modifying a flow technique in [3], we prove a weak subadditivity property of ℛGn , i. e., there exists C > 0 such that G

G

G

ℛm+n ≤ C ℛm ⋅ ℛn

∀m, n ≥ 0.

For Dn , we let ℰnG [u] be the energy of a function u defined on Gn , i. e., ℰnG [u] = ∑x∼y,x,y∈Gn |u(x) − u(u)|2 . By (4.1), we denote (ℛDn )

−1

:= min{ℰnD [u] : u = 0 on (L1 ∩ Dn ), u = 1 on (L4 ∩ Dn )}.

(4.4)

Then through some construction of potentials we obtain a superadditivity estimate for RDn : there exists C > 0 such that D

D

D

ℛm+n ≥ C ℛm ⋅ ℛn

∀m, n ≥ 0.

We show that ℛGn and ℛDn are equivalent. Then by using the above two estimates of the effective resistance of the H-carpet H, we arrive at the following (see [17] for details). Theorem 4.2. Let H be an H-carpet, and let ℛGn and ℛDn be the effective resistance defined as in (4.3) and (4.4). Then there exists λ > 1 and C > 0 independent of n such that C −1 λn ≤ ℛGn ≤ Cλn ,

C −1 λn ≤ ℛDn ≤ Cλn .

(4.5)

For the above theorem, we can strengthen it to be a pointwise estimate of the effective resistances of the Qi , Qj in the H-carpet. Let ℛn (Qi , Qj ) be the effective resistance between Qi and Qj on Dn , i. e., −1

ℛn (Qi , Qj )

:= inf{En [u] : u ∈ ℓ(Dn ), u(Qi ) = 1, u(Qj ) = 0}.

Theorem 4.3. There exists C > 0 such that for i ≠ j, C −1 λn ≤ ℛn (Qi , Qj ) ≤ Cλn , where λ > 1 is the growth rate of the effective resistance in Theorem 4.2. Consequently, we have 6

2

C −1 λn ⋅ ℰn [u] ≤ ∑(u(Qi ) − u(Qi+1 )) ≤ Cλn ⋅ ℰn [u], i=1

(4.6)

where u is any harmonic function on Dn \ {Qi }6i=1 with boundary value {u(Qi )}6i=1 . By restricting the above to the T-carpet with vertices {Q0 , Q1 , Q2 }, we have the similar estimates.

88 | Q. Gu and K.-S. Lau Remark. In one step of the proof, we need to use the uniform Harnack estimates obtained in Section 5. It is easy to see that on the T-carpet, ℛn (Qi , Qj ) ≍ Rn (Qi , Qj ), and hence condition (R) in (3.9) is satisfied. This will be used in Theorem 6.2 for the construction of the LRDF.

4.2 Triangular diamonds and resistance estimates In this subsection, we consider another class of fractals based on the triangle as in Subsection 4.1, but selecting different family of subtriangles as an IFS. They are not necessarily finitely ramified but can be separated into clusters that are finitely ramified, i. e., they are finitely ramified in the wider sense of graph directed system [36]. Let p1 = 0, p2 = 1 and p3 = exp(πi/3) be the three vertices of a unit triangle T. Let ℓ ≥ 3 be an integer. Divide each side of T into ℓ equal pieces and draw line segments through these dividing points that are parallel to the three sides. This makes altogether ℓ2 small triangles with side length 1/ℓ. Denote by Fj : ℝ2 → ℝ2 the affine map such that Fj maps T onto the small triangle aj . We denote by O the centroid of T. Definition 4.4. Let ℐ ⫋ {1, 2, . . . , ℓ2 }, and let 𝒦 be the self-similar set generated by the iterated function system {Fi }i∈ℐ . We call such (𝒦, {Fi }i∈ℐ ) a triangular diamond (T-diamond) if the following conditions hold. (i) (Symmetry) ⋃i∈ℐ Fi (T) is invariant under the isometry group of T (i. e., the group D3 ). (ii) (Connectedness) 𝒦 is connected. (iii) (Diamond) ℐ = (⋃3i=1 ℐi ) ∪ ℐ0 is a disjoint union. We denote by G = ⋃j∈ℐ0 Fj (T) with O ∈ G, Ti = ⋃j∈ℐi Fj (T), i = 1, 2, 3 with pi ∈ Ti ; G ∩ 𝜕T = 0, G ∩ Ti = {qi } with ]; Ti , i = 1, 2, 3 is symmetric under the reflection on |pi − qi | = k √3/ℓ, 1 ≤ k ≤ [ ℓ−1 2 the perpendicular bisector of pi qi . Figure 9 is a triangular diamond with ℓ = 4, k = 1. Note that we allow ℐ0 = 0, i. e., G = {O}, as the diamond fractal in [29].

Figure 9: A triangular diamond.

Resistance estimates and critical exponents of Dirichlet forms on fractals | 89

Let Dn be the graph consisting of vertices of the n-level subtriangles in the nth iteration of T. It is observed that these subtriangles can be grouped together to consist of triangles, 6-pointed stars and (at most) eleven k-pointed partial stars, 1 ≤ k ≤ 5 (see Figures 10 and 11). We will use them to set up the investigation.

Figure 10: The 6-pointed star, and the first level cells which consists of 1, 4, 6-pointed stars (shaded areas) and G1 (in dotted areas).

Figure 11: The 11 partial stars.

For G defined in (iii), we let Gn = Dn ∩ G, 𝒢 = 𝒦 ∩ G. Note that T1 is contained in the cone with vertex at q1 , subtended by an angle π/3; we define S to be the union of the six

90 | Q. Gu and K.-S. Lau , k = 1, . . . , 5. It comes out to be a 6-pointed copies of T1 rotated around q1 by angles kπ 6 star S (see Figure 10); denote the fractal by 𝒮 , and denote the six boundary vertices of S by 𝒱0 = {P1 , . . . , P6 } and the center P0 = q1 . We let Sn , n ≥ 1 denote the graph on S induced by Dn , and use p ∼ q for p, q belonging to the same n-subtriangle. For the 11 k-pointed “partial” stars, 1 ≤ k ≤ 5, not all of them may appear in an individual case (e. g., in our example, only 1, 2, 4, 6-pointed stars show up). By considering symmetry, we can always require that P1 is in the partial star (see Figure 11). We denote by S̃n a subgraph in Sn of a partial star S,̃ and let ℒ ⊆ {1, 2, . . . , 6} be the collection of i such that the vertex Pi ∈ S̃n . According to the possible shapes of 𝒮 and 𝒢 at the boundaries, for the case ℒ = {1} (i. e., the first case in Figure 11), we include P0 as an additional vertex. And in the third and fifth cases, it is possible that the two situations with P0 as boundary and without P0 as boundary; we include both of them in the consideration. ̃ First we estimate the trace RGn , RSn , RSn on the respective sets. Recall we have defined the notation of trace in (3.2) and Definition 3.3. By symmetry, we see that all RGn (i, j) are equal on the three vertices, we denote it by RGn . For Sn , there are three different RSn (1, j), namely, RSn (1, j), j = 2, 3, 4. We let RSn := min {RSn (1, i)}. i=2,3,4

̃ We prove the following trace relations of S and the S’s. Proposition 4.5. With the above notations, we have: (i) The harmonic extension un of u ∈ 𝒱0 to Sn satisfies ℰn [un ] ≍ (RSn )−1 ∑6i=1 (u(Pi ) − u(Pi+1 ))2 , (P7 = P1 by convention) and the constant in ≍ does not depend on n and u. Consider the following cases of harmonic extensions u on S̃n : ̃ (ii) If ℒ = {i}, then for fixed values u(Pi ) and u(P0 ), ℰnS [u] ≍ (RSn )−1 (u(Pi ) − u(P0 ))2 . (iii) If ℒ ≠ {i}, then ̃ (a) for fixed values {u(Pi )}i∈ℒ on S̃n , ℰnS [u] ≍ (RSn )−1 ∑i,j∈ℒ (u(Pi ) − u(Pj ))2 ; ̃ (b) for fixed values {u(P )} on S̃ , ℰ S [u] ≍ (RS )−1 ∑ (u(P ) − u(P ))2 . i

i∈ℒ∪{0}

n

n

n

i∈ℒ

i

0

The proof of this proposition relies on the geometric structure of the sets and constructions of potentials, see [17]. To relate the growth rates of RGn and RSn , we need to have more structure of the graphs Gn , Sn , and S̃n . This can be obtained through the setup of graph directed systems [10, 36]. We give an outline in the following. Consider the finite family of seed sets {𝒮 (i) }0≤i≤6 where 𝒮 (0) = 𝒢 , 𝒮 (6) = 𝒮 , and (i) 𝒮 are the i-pointed partial stars, 1 ≤ i ≤ 5 (there are 11 of them; for k = 2, 3, 4, by slight abuse of notations, we let 𝒮 (k) be multiply represented for brevity). Let V = {0, 1, . . . , 6}. By using the IFS {Fi }i∈ℐ , we can decompose each 𝒮 (i) into families of cells Ae , e ∈ J(i, j) ⊂ V × V (the J(i, j) can be empty), such that:

Resistance estimates and critical exponents of Dirichlet forms on fractals | 91

(i) (Self-similar) Each cell Ae is a contractive image of an 𝒮 (k) for some k (we call it a G-cell if it is the image of 𝒢 , and an S-cell if it is an image of {𝒮 (i) }1≤i≤6 ). (ii) (Separation) Different cells can only intersect at the vertices; S-cells can only be connected through G-cells, and vice versa. We iterate the graph directed system until all the configurations of the partial stars appear, say at n̄ (in the example n̄ = 2), and then use the setup (i), (ii) to define “clusters” of Sn (= Sn(6) ), n ≥ n̄ as follows. For n = k n̄ + n󸀠 where 0 ≤ n󸀠 ≤ n,̄ we partition the 6-point 󸀠 star Sn to be “clusters” Sn ∩{A : A ∈ 𝒜n } (at n = k n,̄ there are two partitions 𝒜0 and 𝒜n̄ ). The partition has the following property inherited from the IFS {fe }e∈J(i,j) , 0 ≤ i, j ≤ 6: (i󸀠 ) The number of clusters in Sn is bounded by some integer N independent of n; each cluster in Sn is an fe -image of a cluster in Sn−1 . (Note that when n = k n,̄ we need to use the dual partition property to compare those in Sn−1 , Sn , and in Sn , Sn+1 ). (ii󸀠 ) Any two clusters intersect at most one vertex; any two S-clusters can only be connected by path through the G-clusters, and vice versa (see Figure 10). By using Proposition 4.5 and the graph directed system, we obtain the following resistance growth theorem. Theorem 4.6. On Gn and Sn , n ≥ 1, there exists C > 0 such that C −1 RGn−1 ≤ RSn ≤ CRGn+1 . Also, there exist λ > 1 and C 󸀠 > 0 such that C 󸀠 −1 λn ≤ RGn ≤ C 󸀠 λn ,

C 󸀠 −1 λn ≤ RSn ≤ C 󸀠 λn .

(4.7)

By Theorem 4.6, our main conclusion for the triangular diamond is the following. Theorem 4.7. Let Dn be the network approximating the triangular diamond 𝒦 with boundary points {p1 , p2 , p3 }. Let R󸀠n := R󸀠n (i, j) (i ≠ j) be the trace of Dn on the boundary. Then C −1 λn ≤ R󸀠n ≤ Cλn .

(4.8)

The theorem will be used in the proof of the Harnack inequality in Section 5, and the existence of a local regular Dirichlet form (Theorem 6.5).

5 Uniform Harnack inequality In this section, we prove the following uniform Harnack inequality (UHI) for the fractals considered in Sections 3 and 4. We will use ρ := ℓ−1 for the contraction ratio for the T-carpet and T-diamond. Note that we have proved in last section that both the T-carpet and T-diamond have property (R) (see (3.9)).

92 | Q. Gu and K.-S. Lau Theorem 5.1. Suppose K is either (i) a p. c. f. set satisfying (3.1) and the resistance growth condition (R) (with Dn := Vn ), or (ii) a T-carpet or T-diamond with Dn (as in Section 4) as its approximating graph. Then there exist C > 0 and δ ∈ (0, 1) such that for any nonnegative harmonic function u on B(x, r) ∩ Dn (≠ 0), where x ∈ K, 0 < r < 1, and B(x, r) = {y ∈ K : |x − y| < r}, max u ≤ C

B(x,δr)∩Dn

min

B(x,δr)∩Dn

u.

Our main application is to use it to prove the following proposition, which is essential in the construction of the Dirichlet form in Section 6 (Theorem 6.2). Proposition 5.2. Let K satisfy the assumption in Theorem 5.1, and let V0 be the boundary set if K is p. c. f.; or let V0 = {p1 , p2 , p3 } be the vertices of the triangle if K is a triangular carpet/diamond. Then for any small ξ > 0, let Kξ := K \ (⋃p∈V0 B(p, ξ )). There exists Cξ > 0 such that if n ≥ 1 and u is a function on Vn which is harmonic on Vn \ V0 , then for all x, y ∈ Vn ∩ Kξ , 󵄨󵄨 󵄨 γ 󵄨󵄨u(x) − u(y)󵄨󵄨󵄨 ≤ Cξ |x − y| max |u|, V n

where γ > 0 is independent of ξ , n and u. The proof of this Proposition is to use the Moser’s argument as in [2, Theorem 3.9] (see also [14, Lemma 6.4] for an outline).

5.1 P. c. f. sets with uniform resistance growth We turn to prove Theorem 5.1 for the p. c. f. case. It follows easily from the following lemmas. For m ≥ 1, for x ∈ Vm , we denote Ωm (x) := ∪{Kω : |ω| = m, x ∈ Kω },

ℒm (x) = {ω : |ω| = m, Kω ⊆ Ωm (x)}.

Also we let Ω󸀠m (x) := ∪{Kω : |ω| = m, Kω ∩ Ωm (x) ≠ 0}. Lemma 5.3. Let K be a p. c. f. set satisfying (3.1) and condition (R). Then there exists C0 > 0 such that for any x ∈ Vm , and any nonnegative harmonic function u on Ω󸀠m (x) ∩ Vn , n ≥ m, we have u(Fω (p)) ≤ C0 u(Fω (q))

for any ω ∈ ℒm (x), p, q ∈ V0 .

Consequently, there is C1 > 0 (independent of m, n and u) such that max u ≤ C1 min u.

Ωm (x)∩Vn

Ωm (x)∩Vn

Resistance estimates and critical exponents of Dirichlet forms on fractals | 93

Proof. Fix a pair p, q ∈ V0 , p ≠ q. Without loss of generality, we assume that u(Fω (p)) = 1. Clearly, we have Ωm (Fω (q)) ⊆ Ω󸀠m (x), so u is nonnegative and harmonic on Ωm (Fω (q)), and u is uniquely determined by the values of u on ⋃η∈ℒm {Fη (V0 )}\{Fω (q)}, where ℒm := ℒm (Fω (q)). Define v on Vm with 0 outside {Fω (p), Fω (q)}, 1 on Fω (p) and harmonic at Fω (q). It is clear that u(Fω (q)) ≥ v(Fω (q)). Let ṽ be the harmonic extension of v on Vn . Writing γ = v ∘ Fω (q), then we have En [ṽ] = ∑ En−m [v ∘ Fη ] = ∑ η∈ℒm

(1 − γ)2 = + ∑ Rn−m (p, q) η∈ℒ 2

2

m

∑

η∈ℒm x,y∈V0

1 2 (v ∘ Fη (x) − v ∘ Fη (y)) Rn−m (x, y)

γ2 + ∑ Rn−m (x, q) η∈ℒ

∑

x∈V0 ,x =p,q ̸

m

∑

x∈V0 ,x =p,q ̸

1 Rn−m (x, p)

=: μ1 (1 − γ) + μ2 γ + μ3 . Since v is harmonic on Fω (q) with boundary Vm \ Fω (q), the above energy expression μ1 has a minimum for v(Fω (q)) := γ = μ +μ . Now observe that C −1 λm−n ≤ Rn−m (x, y) ≤ 1

2

Cλm−n for x, y ∈ V0 by assumption, and that #{(η, x) : η ∈ ℒm , x ∈ V0 } is uniformly bounded. Hence there exists c0 independent of m, n, ω and u such that v(Fω (q)) ≥ c0 > 0. By taking C0 = c0−1 , we have u(Fω (p)) = 1 ≤ C0 v(Fω (q)) ≤ C0 u(Fω (q)),

and completes the proof of the first part. By the maximum principle of the harmonic function u on Ωm (x), u takes maximum and minimum on the set ⋃ω∈ℒm (x) {Fω (V0 )}\{x}. On the other hand, applying the above result to these Kω for ω ∈ ℒm (x), it follows that for any y ∈ ⋃ω∈ℒm (x) {Fω (V0 )}, C0−1 u(y) ≤ u(x) ≤ C0 u(y). This implies that max u ≤ C02 min u.

Ωm (x)∩Vn

Ωm (x)∩Vn

5.2 Triangular diamond For the triangular diamond, condition (R) is satisfied (Theorem 4.7). The proof of the uniform Harnack inequality is essentially the same as the above p. c. f. case. Let Dn be the graph on ⋃|w|=n Fw (V0 ), where V0 = {p1 , p2 , p3 }, defined by the vertices of Fw (V0 ). Observe that the triangular diamond 𝒦 consists of three 1-pointed stars 𝒮 (1) and the center part 𝒢 . By using the graph directed construction in Section 4.2 (apply on 𝒮 (1) ), we can partition 𝒦 into clusters. Let B0 be the set of boundary points of

94 | Q. Gu and K.-S. Lau these clusters, and let Bm be the mth iteration of B0 through the {fe }e∈J(i,j) , 0 ≤ i, j ≤ 6 of the maps defined there. We call these the m-clusters of 𝒦. They are subsets of Dn and have the separation property that different cells can intersect at most one vertex (property (ii) in the construction) on Bm . For n > m ≥ 0, and x ∈ Bm , we denote Ωm (x) := ∪{I : I is an m-cluster, x ∈ I},

ℒm (x) = {I : I is an m-cluster, I ⊆ Ωm (x)}.

Also we let Ω󸀠m (x) := ∪{I : I is an m-cluster, I∩Ωm (x) ≠ 0}. Then the proof of Theorem 5.1 is the same as the above lemma by dealing with the newly defined sets Ωm (x) and Ω󸀠m (x).

5.3 Triangular carpet As remarked in Theorem 4.3, we have used the UHI to prove the pointwise resistance growth, and hence we cannot use it to prove the UHI. Instead, we will adopt the technique in [2] on the H-carpet first, and then reduce to the triangular carpet easily. For this, we need to estimate two types of hitting probability of the simple random walk Xk on the graph Gn , namely, the “corner move” and the “knight move” as the moves on the Sierpinski carpet in [2], modifying certain technicalities. Let B be the boundaries of the two configurations in Figure 12, and let τB be the first time the random walk hits the boundary. For the corner move, we have

Figure 12: The corner move (left) and the knight move (right).

Proposition 5.4. For x ∈ P1 P0 ∩ Gn , we have Px (XτB ∈ P1 P2 ) ≥

1 . 8

The proof of this is simple; we only need to use the reflection principle on “first entrance” and “last exit”, which heavily depends on the symmetry of the set.

Resistance estimates and critical exponents of Dirichlet forms on fractals | 95

We adopt the approach in [2], which involves more delicate use of random walk. The main difference is that there are 12 subcases that need to be checked. The reader can refer to [17] for details. By using these two moves, we can prove the following form of uniform Harnack inequality of nonnegative harmonic functions on Gn by the same technique as in [2, Section 3]. Proposition 5.5. Let Ω be a hexagon in Gn centered at Q0 with side length 1/ℓ, and let Ω󸀠 be a hexagon such that Ω ⊂ Ω󸀠 \ 𝜕Ω󸀠 and Ω󸀠 ⊂ Gn \ 𝜕Gn . Then there exists C > 0 independent of n such that for any nonnegative harmonic function u on Ω󸀠 , max u ≤ C min u. Ω

(5.1)

Ω

Proof. The basic ideas are from Barlow–Bass [2, Section 3]. The following is an outline of the proof. First, we prove (5.1) for the functions u of the form u(x) = P x (Xτ 󸀠 = z) for some 𝜕Ω

z ∈ 𝜕Ω󸀠 , i. e.,

u(x) ≤ Cu(y)

for any x, y ∈ Ω,

(5.2)

where C is independent of the choice of z and n. To do this, we need to show that there is δ > 0 such that for any curve γ (i. e., a sequence of consecutive points) inside Gn starting from y to reach 𝜕Ω󸀠 , it holds that P x (Xk hits γ before τ𝜕Ω󸀠 ) ≥ δ.

(5.3)

The proof of (5.3) relies on the constructions of the corner and knight moves mentioned above. We can assume without loss of generality that x and y are located on 𝜕Ω. Then the corner and knight moves are sufficient to construct a sequence of consecutive moves inside Ω󸀠 \ Ω that loop around the point y until it crosses the curve γ (see Figure 13), where γ := γ(t), t ∈ Dn is some curve from y to 𝜕Ω󸀠 on which the values of u is nondecreasing on t; see [2] for the existence of such γ. Then the consecutive moves have a probability ≥ δ > 0 where δ is a finite product (the number is bounded independent of n and the curve) of 1/8 and 1/12, which represents the number of corner and knight moves we used. This will show Xk must hit γ before τΩ with probability ≥ δ by the planar topological property. Let Tγ be the first time Xk hits γ or 𝜕Ω󸀠 . Then the inequality (5.2) follows by conditioning the martingale u(Xk∧Tγ ) on Tγ , i. e., u(x) = Ex (u(Xk∧Tγ )) ≥ Ex (u(Xγ(s) )1{Xk hits γ before τ 󸀠 } ) ≥ δu(γ(s)) ≥ δu(y), Ω

where γ(s) is the point on γ that Xk hits. A general nonnegative harmonic function u on Gn can be represented as the linear combination of P x (Xn (τΩ ) = z) as u(x) = ∑ u(z)P x (Xτ𝜕Ω = z), z∈𝜕Ω

and this immediately shows (5.1).

96 | Q. Gu and K.-S. Lau

Figure 13: A path of corner and knight moves that satisfies (5.3).

By the equivalence of Gn and Dn , we can conclude the UHI holds also on Dn as in the following. Proof of Theorem 5.1 for T-carpet. In view of the equivalence of the networks, we use Dn instead of Gn . Note that for the given B(x, r), the size of Ω󸀠 in the above proposition can be adjusted accordingly, and the center Q0 of Ω can be adjusted to x ∈ Ω, and the δ can also be determined. By restricting the consideration on the H-carpet to the triangular carpet, the theorem follows.

6 Construction of Dirichlet forms We have the discretization of the p. c. f. sets in Theorem 3.2. For the T-carpet and T-diamond, we let ρ := ℓ−1 , V0 = {p1 , p2 , p3 } be the three vertices of T, and use the same notations, Vn , V∗ , En etc, as for the p. c. f. sets. By using the regular triangular system (Theorem 2.3), we see that the same discretization for Bσ2,∞ and Bσ2,2 holds for the p. c. f. fractals as in Theorem 3.2. Recall the definition of Dirichlet form [11], a Dirichlet form (ℰ , ℱ ) (DF) is a closed nonnegative bilinear form densely defined on L2 (K, μ) and has the Markov property (u ∈ ℱ ⇒ ũ := (u ∨ 0) ∧ 1 ∈ ℱ and ℰ [u]̃ ≤ ℰ [u]). It is called regular if ℱ ∩ C(K) is dense in C(K) with the supremum norm and dense in ℱ with the ℰ11/2 -norm. It is called local if ℰ (u, v) = 0 for u, v ∈ ℱ with disjoint compact supports. It is clear that for σ < σ ∗ , Bσ2,∞ and Bσ2,2 are dense in C(K); Bσ2,2 always supports a nonlocal regular DF because of its integral expression (by (1.3)). In the following we will show that on the fractals under consideration, local regular DFs exist with domain ∗ Bσ2,∞ , making use of Bσ2,2 , σ < σ ∗ as in Theorem 1.1. For 2σ − α > 0 and u ∈ ℓ(Vn ) (or in ℓ(V∗ )), we write σ

−(2σ−α)n

ℰn [u] = ρ

En [u].

(6.1)

Resistance estimates and critical exponents of Dirichlet forms on fractals | 97

We introduce property (E) for the fractals discussed above, which plays an important role in the construction of the Dirichlet forms (see [14, Section 5] for the p. c. f. case). For a function u on V0 , we say that v is an extension of u on Vn (or V∗ ) if v|V0 = u. Definition 6.1. We say that K satisfies property (E) if there exists σ > 0 with 2σ − α > 0, and C > 0 such that (i) for any u ∈ Bσ2,∞ and for all n ≥ 1, E0 [u] ≤ C ℰ σn [u]; ̃ ∈ Bσ2,∞ . (ii) for any u ∈ ℓ(V0 ), there exists an extension u Theorem 6.2. Let K be a p. c. f. set satisfying (3.1) and resistance growth condition (R) or a T-carpet or T-diamond (with the respective α and λ). Then K satisfies property (E) with log λ 1 ) σ̄ = (α + 2 − log ρ

and

λ = ρ−(2σ−α) . ̄

Moreover σ̄ is the critical exponent, i. e. σ̄ = σ ∗ and Bσ is dense in C(K). ∗

Proof. We prove this for the p. c. f. set. The T-diamond and the T-carpet cases are similar by replacing condition (R) with the pointwise estimate of the resistance growth rate (Theorem 4.3 and Theorem 4.7). ̄ Let λ = ρ−(2σ−α) (> 1). It follows that 2σ̄ − α > 0. That K satisfies property (E)(i) is ̄ simple. Because Rn ≍ λn , we have for any u ∈ Bσ2,∞ , and v the harmonic extension of u|V0 on Vn , E0 [u] ≤ Cλn En [v] ≤ Cλn En [u] = C ℰnσ [u]. ̄

To prove K satisfies property (E)(ii), let u ∈ ℓ(V0 ) with values u(p) for p ∈ V0 . We can extend un to be the harmonic function on Vn \ V0 with un |V0 = u. Then take η = 1/n, by Proposition 5.2, we have on K1/n := K \ ∪p∈V0 B(p, 1/n), 󵄨󵄨 󵄨 γ 󵄨󵄨un (x) − un (y)󵄨󵄨󵄨 ≤ Cn |x − y| max |u| ∀x, y ∈ K1/n . V0

(6.2)

By using the well-known Whitney extension and trace theorem on closed sets [27, §2.2, ̃ n on ℝd , and then restrict back on K Theorem 2], we can extend un from K1/n to be u preserving the bound Cn and the Hölder exponent. Then by using the same argument ̃ nk }k≥0 converging to some as in the proof of [14, Theorem 6.1], we have a subsequence {u ̃ in C(K). u ̃ is the required function in property (E)(ii). We fix m ≥ 0. As unk → u ̃ We show that u uniformly on any K1/n , we have (as finite sums) σ̄

m

̃ ] = lim λ Em [unk ]. ℰm [u k→∞

We claim that λm Em [unk ] ≤ C 2 E0 [u].

(6.3)

98 | Q. Gu and K.-S. Lau Indeed, for any nk ≥ m, since 2

En [un ] = ∑ Rn (p, q)−1 (un (p) − un (q)) ≤ Cλ−n E0 [u], p,q∈V0

(6.4)

we have Enk [unk ] ≤ Cλ−nk E0 [u]. Also by using condition (R) on each (nk − m)-cell, we have Enk [unk ] ≳ ∑

2

∑ Rnk −m (p, q)−1 (unk (Fω (p)) − unk (Fω (q))) ≥ C −1 λ−(nk −m) Em [unk ].

|ω|=m p,q∈V0

The claim follows from these two estimates. Substituting this into (6.3), and observing ̄ σ̄ ̃ ] ≤ C 2 E0 [u]. This implies u ̃ ∈ Bσ2,∞ that C does not depend on m, we have supm≥0 ℰm [u , and completes the proof of the first part. ̄ For the second part, we observe that Bσ2,∞ is a lattice in C(X). By using the same ̄ proof as the above, starting from u on Vn , we can extend u ∈ ℓ(Vn ) to ũ ∈ Bσ2,∞ . Hence σ̄ B2,∞ separates points of K. By the lattice version of the Stone–Weierstrass theorem, ̄ Bσ2,∞ is dense in C(K). Hence σ̄ ≤ σ ∗ . λ 1 ̄ (α+ −log )). Let u ∈ Bσ2,∞ . It follows from property On the other hand, for σ > σ(:= 2 log ρ (E)(i) that E0 [u] ≤ C ℰnσ [u] = Cρ−(σ−σ)n ℰnσ [u]. ̄

̄

̄ As limn→∞ ρ−(σ−σ)n = ∞, it implies E0 [u] = 0, i. e., u is constant function. Hence σ̄ ≥ # ∗ σ ≥ σ . We conclude that σ̄ = σ # = σ ∗ .

The proof of our main result makes use of the Γ-convergence of closed forms in the wide sense [7, 37]. The first attempt to use the Γ-limits to study a diffusive Dirichlet form can be found in [31]. Their setup is on a general metric measure space with volume doubling and some other assumptions; the Γ-convergence is based on a two-sided heat kernel of the Dirichlet form, discretizing the metric space, and the heat kernel to obtain an approximating sequence. Definition 6.3. Let M be a locally compact metric space, and μ be a regular Borel mea2 sure with supp(μ) = M. Let {ℰ n }∞ n=1 and ℰ be closed forms on L (M, μ) (in the wide sense [11], i. e., they are not necessarily dense in L2 (M, μ)). We say that ℰ n Γ-converges to ℰ if n 2 (i) for any {un }∞ n=1 ⊂ L (M, μ) converges strongly to u, we have limn→∞ ℰ [un ] ≥ ℰ [u];

(ii) for any u ∈ L2 (M, μ), there exist a sequence {un } ⊂ L2 (M, μ) converging strongly to u such that limn→∞ ℰ n [un ] ≤ ℰ [u]. Theorem 6.4 (De Giorgi [7, Theorem 8.5]). If {ℰ n }n is a sequence of closed forms on L2 (M, ν), then there exists a subsequence {ℰ nk }k that Γ-converges to a closed form ℰ on L2 (M, ν). σ σ σ −(2σ−α)n We let ℰ2,2 be the Dirichlet form defined by ∑∞ En [u] n=0 ℰn [u], where ℰn [u] = ρ σ σ (as in (6.1)). Then ℰ2,2 has domain B2,2 . Our main result is

Resistance estimates and critical exponents of Dirichlet forms on fractals | 99

Theorem 6.5. Let K be the same as in Theorem 6.2. Then K admits a local regular Dirich∗ ∗ let form (ℰ σ , ℱ ) on K with domain ℱ = Bσ2,∞ dense in C(K), and self-similar energy identity ℰ σ [u] = λ ∑i ℰ σ [u ∘ Fi ]. Moreover, there exists C > 0 such that for u ∈ ℱ , ∗

∗

σ C −1 ℰ σ [u] ≤ lim±σ↗σ∗ (σ ∗ − σ)ℰ2,2 [u] ≤ C ℰ σ [u], ∗

∗

where lim± means lim sup and lim inf. σ Proof. Consider the family of closed forms {(σ ∗ − σ)ℰ2,2 : σ < σ ∗ , 2σ − α > 0}. There is σ a sequence σn ↗ σ ∗ and {ℰ2,2n } Γ-converges to some closed form ℰ ∗ in L2 (K, μ) (Theorem 6.4). Let ℱ = {u ∈ L2 (K, μ) : ℰ ∗ (u) < ∞}. Observe that for any σ, σ 󸀠 > 0, σ

2(σ 󸀠 −σ)n

ℰn [u] = ρ

(ρ−2(σ −σ)n En [u]) = ρ2(σ −σ)n ℰnσ [u]. 󸀠

󸀠

󸀠

Hence for u ∈ Bσ2,∞ , by Definition 6.3(i), we have ∗

σ

∗

∗

∞

∗

2(σ ∗ −σn )j σ ∗ ℰj [u]

ℰ [u] ≤ lim (σ − σn )ℰ2,2n [u] = lim (σ − σn ) ∑ ρ n→∞

n→∞

j=0

≤ C1 sup ℰjσ [u]. ∗

j≥0

Therefore we have ℰ ∗ [u] ≤ C[u]2Bσ∗ , so that Bσ2,∞ ⊆ ℱ . ∗

2,∞

On the other hand, assume that u ∈ ℱ . By the Γ-convergence (Definition 6.3(ii)), for any u ∈ L2 (K, μ), there exists {un }n ⊂ L2 (K, μ) converging to u strongly such that σ

lim (σ ∗ − σn )ℰ2,2n [un ] ≤ ℰ ∗ [u] < ∞.

n→∞

Hence there is some n0 such that for n ≥ n0 , (in the case ℰ ∗ [u] = 0, then u ∈ Bσ2,2 trivially) ∗

σ

(σ ∗ − σn )ℰ2,2n [un ] ≤ 2ℰ ∗ [u], σ

σ

which implies un ∈ B2,2n for n ≥ n0 (Theorem 3.2). As B2,2n ⊂ C(K) (Proposition 2.2), un is pointwisely defined. We then have for m ≥ 0, ∞

2ℰ ∗ [u] ≥ (σ ∗ − σn ) ∑ ℓ−2(σ

∗

j=0 ∞

≥ (σ ∗ − σn ) ∑ ℓ−2(σ j=m

−σn )j σ ∗ ℰj [un ]

∗

−σn )j σ ∗ ℰj [un ]

≥

∞ ∗ 1 ∗ σ∗ (σ − σn ) ∑ ℓ−2(σ −σn )j ℰm [un ] C j=m

≥

1 −2(σ∗ −σn )m σ∗ 1 σn ℓ ℰm [un ] = ℰ [u ]. C2 C2 m n

(by property (E)(i)) (6.5)

100 | Q. Gu and K.-S. Lau From this and Theorem 3.2, there is a constant C > 0 such that for n ≥ n0 , σn [un ]2Bσn ≤ C sup ℰm [un ] ≤ C3 ℰ ∗ [u], 2,∞

where C3 = 2CC2 .

m≥0

σ

σ

It follows that limk→∞ ℰmn [unk ] = ℰmn [u] for m ≥ 0, n ≥ n0 . In view of (6.5), we see σ that ℰmn [u] ≤ C2 ℰ ∗ [u], and by letting n → ∞, and take supremum on m, we have σ sup ℰm [u] ≤ C2 ℰ ∗ [u]. ∗

m≥0

This gives [u]2Bσ∗ ≤ C2 ℰ ∗ (u), so that ℱ ⊆ Bσ2,∞ . Together with the first part, we have ∗

ℱ = Bσ2,∞ . ∗

2,∞

Next we can apply a standard argument ([13, 32]) to use ℰ ∗ to construct an equiva∗ lent closed form (ℰ σ , ℱ ) that has a self-similar energy identity (see [14] for detail). For ∗ ∗ ∗ this ℰ σ , the Markovian property of ℰ σ is clear, and the locality of ℰ σ is an immedi∗ ate consequence of self-similarity (see the proof in [28, Theorem 3.4.6]). Hence ℰ σ is a local regular Dirichlet form.

7 Some open problems In this study, we came across a number of problems that seem interesting: Q1. For the homogeneous p. c. f. sets in Section 3, find a better handling of the growth rate of the trace Rn (p, q), p, q ∈ V0 or the effective resistance ℛn (p, q). In particular, find a tractable sufficient condition for condition (R). Q2. Can we extend the consideration in Section 3 to more general p. c. f. sets? In the proof of Theorem 4.3 of the resistance growth of the H-carpet, we make use of the uniform Harnack inequality (UHI) in one technical step. Hence in the proof of the UHI, we need to use the more complicated probabilistic method, instead of the simpler analytic method of resistance growth (see [17, Section 7]). Q3. Can we prove Theorem 4.3 by using a pure analytic method? This will facilitate the extension to more general non-p. c. f. fractals, for example, on certain self-similar sets that satisfy the RTS and condition (R). The convergence theorem of BBM [6] (Theorem 1.1) is for the p-energy forms. Q4. How to extend our consideration to the convergence of Bσp,p to Bσp,∞ ? ∗

Resistance estimates and critical exponents of Dirichlet forms on fractals | 101

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Bobo Hua and Xueping Huang

A survey on unbounded Laplacians and intrinsic metrics on graphs Abstract: There has been intense research of discrete Laplacian on weighted graphs from various points of view. In this survey we summarize some recent analytic results related to intrinsic metrics. Keywords: Weighted graphs, intrinsic metrics, discrete Laplacian, Ricci curvature MSC 2010: Primary 05C81, 31C25, Secondary 31C05, 58J35

Contents 1 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

A brief introduction | 103 Graphs with unbounded Laplacians | 104 Weighted graphs | 104 Gamma calculus | 106 Intrinsic metrics | 107 Completeness of graphs | 108 Davies estimates | 109 Ancient solutions | 110 Uniqueness class and stochastic completeness | 112 Upper escape rate | 113 Final remarks | 114 Appendix A. A remark on finite jump size | 114 Bibliography | 115

1 A brief introduction Weighted graphs can be viewed as discrete analogs of Riemannian manifolds. They naturally appeared in discretization of manifolds, cf. [29, 40, 41, 28]. The analogue Acknowledgement: B. H. is supported by National Natural Science Foundation of China (Grant No. 11831004 and No. 11926313). X. H. is supported by National Natural Science Foundation of China (Grant No. 11601238), and The Startup Foundation for Introducing Talent of Nanjing University of Information Science and Technology (Grant No. 2015r053). We are grateful to R. Wojciechowski who kindly helped us improve the presentation. We also thank him for pointing out the reference [46]. Bobo Hua, School of Mathematical Sciences, LMNS, Fudan University, Shanghai 200433, China; and Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200433, China, e-mail: [email protected] Xueping Huang, Department of Mathematics, Nanjing University of Information Science and Technology, Jiangsu 210044, China, e-mail: [email protected] https://doi.org/10.1515/9783110700763-004

104 | B. Hua and X. Huang goes further in multiple ways, and frequently one can transfer the methods or even results between the two settings. A weighted graph is a locally finite, connected, undirected graph enriched with the data of weights both on vertices and edges. A discrete Laplace operator can be naturally defined using the weights. See Subsection 2.1 for details. Classical harmonic function theory thus has an analogue on weighted graphs. Together with the corresponding heat equation and heat kernel, the Laplace operator plays a fundamental role in analysis on graphs similar to that on manifolds. The discrete Laplacian canonically generates a Markov process, the so-called continuous time random walk/Markov chain, cf. [52, 1]. It is then natural to compare properties of random walks on graphs and Brownian motions on manifolds. To characterize the properties of discrete Laplacian in geometric terms such as volume growth of distance balls, the graph metric is typically not suitable. Indeed, the combinatorial graph metric does not see the weight structure which determines the Laplacian. It turns out in most cases that what one really needs is a control on gradient of cut-off functions constructed from the metric. In this way, the concept of intrinsic metrics becomes inevitable and has probabilistic interpretations related to Levy measures and Einstein relations, cf. [51, 21, 42]. In the following section, we briefly outline the basic theory of weighted graphs and summarize various recent results around the discrete Laplacian, typically related to intrinsic metrics.

2 Graphs with unbounded Laplacians 2.1 Weighted graphs Let (V, E) be a (possibly infinite) undirected graph with the set of vertices V and the set of edges E. Two vertices x, y are called neighbors, denoted by x ∼ y, if x and y are connected by an edge, i. e., {x, y} ∈ E. A graph is called locally finite if for any x ∈ V, #{y ∈ V : y ∼ x} < ∞. A graph (V, E) is called connected if for any x ≠ y ∈ V there is a finite sequence of vertices, {xi }ni=0 , such that x = x0 ∼ x1 ∼ ⋅ ⋅ ⋅ ∼ xn = y. In this paper, we always consider infinite graphs that are locally finite, connected and simple (without loops or multiedges). For x ≠ y, we denote by d(x, y) := inf{n|x = z0 ∼ ⋅ ⋅ ⋅ ∼ zn = y} the combinatorial graph distance between vertices x and y, i. e., the minimal number of edges in a path among all paths connecting x and y. Let w : E → (0, ∞),

{x, y} 󳨃→ wxy = wyx ,

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be an edge weight function, and m : V → (0, ∞),

x 󳨃→ mx ,

be a vertex weight function. We call the quadruple G = (V, E, m, w) a weighted graph. For convenience, we extend the function w on E to the total set V × V, w : V × V → [0, ∞), such that wxy = 0 if x ≁ y. In this case, we can write, for x ∈ V and f : V ×V → ℝ, ∑ wxy f (x, y) =

y∈V

∑

y∈V:y∼x

wxy f (x, y).

We denote by C(V) (resp. C0 (V)) the set of functions (resp. of finite support) on V. p For any p ∈ [1, ∞], we denote by ℓp (V, m), or simply ℓm , the space ℓp -summable functions on V with respect to the measure m, and by ‖ ⋅ ‖ℓmp the ℓp norm of a function. 2 Given a weighted graph (V, E, m, w), there is an associated Dirichlet form w. r. t. ℓm corresponding to the Dirichlet boundary condition, Q : D(Q) × D(Q) → ℝ 1 (f , g) 󳨃→ Q(f , g) := ∑ w (f (y) − f (x))(g(y) − g(x)), 2 x,y∈V xy where D(Q) is the completion of C0 (V) under the norm ‖ ⋅ ‖Q defined by ‖f ‖Q = √‖f ‖2ℓ2 + Q(f , f ), m

∀f ∈ C0 (V).

The infinitesimal generator L associated to the Dirichlet form Q is called the Lapla2 cian. The associated heat semigroup on ℓm is denoted by Pt = e−tL , which can be ex∞ tended to a semigroup on ℓ by monotonicity. For locally finite graphs, we define the formal Laplacian, denoted by Δ, as Δf (x) =

1 ∑ w (f (y) − f (x)) ∀f : V → ℝ. m(x) y∈X xy

This formal Laplacian can be used to identify the generators defined before. A result of Keller and Lenz [44, Theorem 9] states that Lf = −Δf ,

∀f ∈ D(L),

(2.1)

where D(L) is the domain of the operator L. For any subset Ω ⊂ V, 1Ω is the indicator function on Ω. The graph is called stochastically complete if Pt 1V = 1V for some (hence all) t > 0. Different choices for the measure m induce different Laplacians. Given the weight w on E, typical choices of m of particular interest are as follows:

106 | B. Hua and X. Huang – –

m(x) = ∑y∼x wxy for any x ∈ V and the associated Laplacian is called the normalized Laplacian. m(x) = 1 for any x ∈ V and the Laplacian is called combinatorial (or physical) Laplacian.

Define the weighted vertex degree Deg : V → [0, ∞) by Deg(x) =

1 ∑w , m(x) y∈V xy

x ∈ V.

Then it is proven in [43] that the Laplacian associated with the graph (V, E, m, w) is a 2 bounded operator on ℓm if and only if sup Deg(x) < ∞. x∈V

Note that normalized Laplacians have weighted degree Deg ≡ 1, and hence are bounded operators, so that these weighted graphs are always stochastically complete, see [17, 43]. Thus, for stochastic completeness, the only interesting cases are combinatorial Laplacians, or more general unbounded Laplacians. The measure m on V is called nondegenerate if δ := inf m(x) > 0. x∈V

(2.2)

The nondegeneracy of the measure m yields a very useful fact for ℓp (V, m) spaces: for any 1 ≤ p < q ≤ ∞, ℓp (V, m) 󳨅→ ℓq (V, m).

2.2 Gamma calculus We introduce the Γ-calculus and curvature dimension conditions on graphs following [54, 49]. First we define two natural bilinear forms associated to the Laplacian. Given f : V → ℝ and x, y ∈ V, we denote by ∇xy f := f (y) − f (x) the difference of the function f on the vertices x and y. Definition 2.1. The gradient form Γ, called the “carré du champ” operator, is defined by 1 Γ(f , g)(x) = (Δ(fg) − f Δg − gΔf )(x) 2 1 = ∑ w ∇ f ∇ g. 2m(x) y xy xy xy

For simplicity, we write Γ(f ) := Γ(f , f ). Moreover, the iterated gradient form, denoted by Γ2 , is defined as 1 Γ2 (f , g) = (ΔΓ(f , g) − Γ(f , Δg) − Γ(g, Δf )). 2

We write Γ2 (f ) := Γ2 (f , f ) = 21 ΔΓ(f ) − Γ(f , Δf ).

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Now we can introduce curvature dimension conditions on graphs. Definition 2.2 ([54, 49]). We say that a graph (V, E, m, w) satisfies the CD(K, ∞) condition, K ∈ ℝ, if for any x ∈ V and f ∈ C(V), Γ2 (f )(x) ≥ KΓ(f )(x).

2.3 Intrinsic metrics In order to deal with unbounded Laplacians, we need the following notion of intrinsic metrics on graphs introduced in [21]. A pseudometric ρ is a symmetric function, ρ : V × V → [0, ∞), with zero diagonal which satisfies the triangle inequality. Definition 2.3 (Intrinsic metric). A pseudometric ρ on V is called intrinsic if ∑ wxy ρ2 (x, y) ≤ m(x),

y∈V

for all x ∈ V.

For R ≥ 0, we write BR (x) := {y ∈ V : ρ(y, x) ≤ R} for the ball of radius R centered at x. In various situations, the combinatorial graph distance proves to be insufficient for the investigation of unbounded Laplacians, see [56, 57, 45]. For this reason the concept of intrinsic metrics received quite some attention as a candidate to overcome these problems. Indeed, intrinsic metrics already have been applied successfully to various problems on graphs [3, 5, 19, 20, 27, 36, 32]. For any weighted graph (V, E, m, w), intrinsic metrics always exist. There is a natural intrinsic metric introduced by [34, Lemma 1.6.4]. Example 2.4. For any given weighted graph there is an intrinsic path metric defined by δ(x, y) =

inf

x=x0 ∼⋅⋅⋅∼xn =y

n−1

−1

∑ (Deg(xi ) ∨ Deg(xi+1 )) 2 ,

i=0

x, y ∈ V, x ≠ y,

where the infimum is taken over all paths connecting x and y. An intrinsic metric is called admissible if the following hold: (1) there exists some (hence for all) x ∈ V, such that #BR (x) < ∞ for all R > 0, and (2) the jump size s := supx∼y ρ(x, y) is finite.

108 | B. Hua and X. Huang The existence of admissible intrinsic metrics leads to nice cut-off functions. For any 0 < r < R, we define a cut-off function on BR (x0 ) \ Br (x0 ) as 1, { { R−ρ(x,x ) 0 ηr,R (x) = { , R−r { 0, {

x ∈ Br (x0 ),

x ∈ BR (x0 ) \ Br (x0 ), x ∈ V \ BR (x0 ).

(2.3)

Since ρ is intrinsic, one can show that Γ(ηr,R ) ≤

1 . 2(R − r)2

(2.4)

2.4 Completeness of graphs We introduce a completeness condition for infinite graphs: A graph G = (V, E, m, w) is called complete if there exists a nondecreasing sequence of finitely supported functions {ηk }∞ k=1 such that lim ηk = 1V

k→∞

and

Γ(ηk ) ≤

1 . k

(2.5)

Without loss of generality, we may assume 0 ≤ ηk ≤ 1 for all k ∈ ℕ by considering min{max{ηk , 0}, 1} instead, if necessary. This completeness condition was introduced for Markov diffusion semigroups in [2, Definition 3.3.9], and is adapted to graphs in [33]. As is well-known, this condition is equivalent to the geodesic completeness for Riemannian manifolds, see [55]. In the nonlocal setting, an analogue first appeared in [21, Proposition 4.7]. For the graph case, it is explicitly stated in [33] that a graph with an admissible intrinsic metric is complete, which was used implicitly by various authors. Theorem 2.5. Let G = (V, E, m, w) be a graph and ρ be an admissible intrinsic metric on G. Then G is a complete graph. Proof. Set ηk := ηk,2k , as defined in (2.3). Then {ηk } is a nondecreasing sequence of finitely supported functions which converges to the constant function 1 pointwise. Moreover, by (2.4), Γ(ηk ) ≤

1 1 < . 2 k 2k

Remark 2.6. In a forthcoming version of [46], it is proven that a complete weighted graph always admits an intrinsic metric with finite balls (condition (1) of admissibility). Local finiteness is not assumed in [46], therefore condition (2) on finite jump size does not hold in general. Anyway, a simple argument shows that for a locally finite weighted graph, each intrinsic metric with finite balls can be deformed into one which additionally has finite jump size. Thus each locally finite complete weighted graph admits an admissible intrinsic metric. We refer to Appendix A for details.

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For a complete graph, the curvature dimension condition is equivalent to the semigroup gradient estimate. See [2] for corresponding results on diffusion semigroups. Theorem 2.7 ([33]). Let G = (V, E, m, w) be a complete graph and m be nondegenerate, i. e., infx∈V m(x) > 0. Then the following are equivalent: (1) G satisfies CD(K, ∞). (2) For any f ∈ C0 (V), Γ(Pt f ) ≤ e−2Kt Pt (Γ(f )). A direct consequence of the gradient bounds is the stochastic completeness for graphs satisfying the CD(K, ∞) condition. Theorem 2.8. Let G = (V, E, m, w) be a complete graph satisfying the CD(K, ∞) condition for some K ∈ ℝ. Suppose that the measure m is nondegenerate, then G is stochastically complete.

2.5 Davies estimates The Davies–Gaffney–Grigor’yan Lemma (DGG Lemma for short) is a powerful tool in geometric analysis that leads to strong heat kernel estimates and has many important applications. On manifolds one writes it in the following form (cf. [25, pp. 326]). Lemma 2.9 (Davies–Gaffney–Grigor’yan). Let M be a complete Riemannian manifold and pt (x, y) the minimal heat kernel on M. For any two measurable subsets B1 and B2 of M and t > 0, we have ∫ ∫ pt (x, y) dvol(x) dvol(y) ≤ √vol(B1 )vol(B2 ) exp(−λt − B1 B2

d2 (B1 , B2 ) ), 4t

(2.6)

where λ is the greatest lower bound, i. e., the bottom, of the L2 -spectrum of the Laplacian on M and d(B1 , B2 ) = infx1 ∈B1 ,x2 ∈B2 d(x1 , x2 ) the distance between B1 and B2 . A result of Coulhon and Sikora [14] implies in the graph case that the DGG Lemma is equivalent to the finite propagation speed property of the wave equation. However, Friedman and Tillich [22, pp. 249] showed that for graphs the wave equation does not have finite propagation speed. Thus, some modification is necessary to get the Davies– Gaffney–Grigor’yan estimate on graphs. Theorem 2.10 ([4]). Let (V, E, m, w) be a weighted graph with an intrinsic metric ρ with finite jump size s > 0. Let A, B be two subsets in V. Then ∑ ∑ pt (x, y)mx my ≤ √m(A)m(B) exp(−λt − ζs (t, ρ(A, B))),

y∈B x∈A

110 | B. Hua and X. Huang where λ is the bottom of the ℓ2 -spectrum of the Laplacian and ζs (t, r) =

rs 1 (rs ⋅ arcsinh − √t 2 + r 2 s2 + t), t s2

t > 0, r ≥ 0.

The function ζs (t, r), for s = 1, appeared already in a number of publications in the graph setting; see, for example, [15, 53, 16]. This result is sharp for the lattice graph ℤ. As a direct application, this yields the Davies’ heat kernel estimate, [15, Theorem 10], by setting A = {x}, B = {y}, for x, y ∈ V. Corollary 2.11 (Davies). For a weighted graph (V, E, m, w) with the normalized Laplacian, pt (x, y) ≤

1 exp(−λt − ζ1 (t, d(x, y))), √mx my

where d is the combinatorial distance. Here one needs the simple fact that the combinatorial distance is intrinsic for the normalized Laplacian case.

2.6 Ancient solutions For a Riemannian manifold M with nonnegative Ricci curvature, Yau [58] proved the Liouville theorem that any positive harmonic function on M is constant. Yau conjectured that for any k > 0 the space of polynomial growth harmonic functions of growth rate at most k, denoted by ℋk (M), is a finite-dimensional linear space, see, e. g., [59, 60]. This conjecture was settled in [9], see also [10, 12, 11, 47, 8, 48] for related results. A natural generalization is to consider ancient solutions, defined on the time interval (−∞, 0], of polynomial growth to heat equations. For a Riemannian manifold M and k > 0, we denote by 𝒫k (M) the space of ancient solutions u(x, t) for which there exist p ∈ M and a constant Cu > 0 such that BR

sup

(p)×[−R2 ,0]

|u| ≤ Cu (1 + R)k ,

∀R > 0.

Calle [6, 7] initiated the study of dimensional bounds for 𝒫k (M). For an n-dimensional Riemannian manifold M with nonnegative Ricci curvature, Lin and Zhang [50] proved that dim 𝒫k (M) ≤ C(n)k n+1 ,

k ≥ 1.

Colding and Minicozzi [13] proved the following general result, which yields an improvement of Lin and Zhang’s result: dim 𝒫k (M) ≤ C(n)k n ,

k ≥ 1.

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Theorem 2.12 ([13]). If M has polynomial volume growth, i. e., there exist p ∈ M and constants C, α such that Vol(BR (p)) ≤ C(1 + R)α ,

∀R > 0,

where Vol denotes the Riemannian volume, then dim 𝒫2k (M) ≤ (k + 1) dim ℋ2k (M),

∀k ≥ 1.

For any k > 0, we denote by ℋk (G) the space of harmonic functions of polynomial growth on G with the growth rate at most k, i. e., f ∈ ℋk (G) if f is a harmonic function on V and there exist x0 ∈ V and a constant Cf such that sup |f (x)| ≤ Cf (1 + R)k ,

x∈BR (x0 )

∀R > 0.

We say that G has polynomial volume growth with respect to ρ if there are x0 ∈ V and constants α, C such that m(BR (x0 )) ≤ C(1 + R)α ,

∀R > 0.

(2.7)

We consider ancient solutions of polynomial growth for continuous-time heat equations on graphs. Let ℝ− := (−∞, 0]. A function u(x, t) on V × ℝ− is called an ancient solution to the (continuous-time) heat equation if for any x ∈ V, u(x, ⋅) ∈ C 1 (ℝ− ) and it satisfies the heat equation 𝜕 u(x, t) = Δu(x, t), 𝜕t

∀x ∈ V, t ∈ ℝ− .

(2.8)

Positive ancient solutions were studied for parabolic Martin boundary theory in probability theory, see [18]. As is well-known in spectral theory, for any μ bounded above by the bottom of the Laplacian spectrum of the graph G, there exists a positive function h : V → ℝ such that −Δh = μh. One can verify that u(x, t) = e−μt h(x) is an ancient solution to the heat equation. Therefore, there are plenty of ancient solutions of exponential growth in time. We denote by 𝒫k (G) the space of ancient solutions of polynomial growth to the heat equation with the growth rate at most k, i. e., u ∈ 𝒫k (G) if u is an ancient solution to the heat equation and there are x0 ∈ V and a constant Cu such that sup (x,t)∈BR (x0

)×[−R2 ,0]

|u(x, t)| ≤ Cu (1 + R)k ,

∀R > 0.

Theorem 2.13 ([31]). Let G be a weighted graph admitting an admissible intrinsic metric. If G has polynomial volume growth, then for all k ≥ 1, dim 𝒫2k (G) ≤ (k + 1) dim ℋ2k (G).

112 | B. Hua and X. Huang

2.7 Uniqueness class and stochastic completeness If a weighted graph G = (V, E, m, w) is not stochastically complete, then 1V − Pt 1V (t ≥ 0) is a nontrivial bounded solution to the following Cauchy problem: 𝜕 { u(x, t) = Δu(x, t), { 𝜕t {u(⋅, 0) ≡ 0. Indeed, stochastic completeness is equivalent to the uniqueness of bounded solutions to the heat equation. More generally, fixing T > 0 and a reference point x,̄ one can study the growth condition of T

∫ ∑ u2 (x, t)m(x) dt, 0 x∈Br (x)̄

where Br (x)̄ is the ball of radius r centered at x̄ for an intrinsic metric ρ, under which the solution to the heat equation is unique. This is the so called uniqueness class problem. Theorem 2.14 ([35]). Let ρ be an admissible intrinsic metric on G. Let u solve 𝜕 { { u(x, t) = Δu(x, t), 𝜕t { { {u(⋅, 0) ≡ 0. Fix T > 0 and x̄ ∈ V. If T

∫ ∑ u2 (x, t)m(x) dt ≤ C exp(cr ln r) 0 x∈Br (x)̄

for some 0 < c < 21 , and all large r, then u ≡ 0. It is surprising that the above uniqueness class is sharp modulo constant c, very different from its counterpart in the manifold case, where the borderline is roughly exp(cr 2 ln r) (cf. [23, 24] for explicit results). From Theorem 2.14 it follows that volume growth of type exp(cr ln r) with respect to an intrinsic metric is sufficient for stochastic completeness. The sharp volume growth criterion for stochastic completeness is due to Folz [20] with some restrictions on weights. Its most general form is proven in [37]. Theorem 2.15 ([20, 37]). Let ρ be an admissible intrinsic metric on G. Denote V(r) = ̄ If m(Br (x)). ∞

∫ then G is stochastically complete.

r dr = ∞, ln V(r)

(♦)

A survey on unbounded Laplacians and intrinsic metrics on graphs | 113

Note that the borderline (♦) is the same as in the manifold case. Contrary to our discrete setting, on manifolds the optimal conditions for uniqueness class and stochastic completeness match. One possible explanation is that uniqueness class involves both local and global geometry, while stochastic completeness is merely related to large-scale asymptotic behavior. Graphs are good analogues of manifolds only from the global perspective. Folz’s argument was probabilistic and went by reduction to diffusion on a corresponding quantum graph. Simpler analytic proofs were given in [35]. Recently, Huang, Keller, and Schmidt [37] proved sharp uniqueness class criteria for a family of the socalled globally local graphs. By an argument in [39], stochastic completeness of a general weighted graph can be deduced from that of a related globally local one. In this way [37] yields yet another proof of Theorem 2.15, which avoids quantum graphs and runs more in the spirit of the original proof of Grigor’yan [23].

2.8 Upper escape rate Let G = (V, E, m, w) be a weighted graph. Suppose that G is stochastically complete and denote by (Xt )t≥0 the continuous time random walk associated with the heat semigroup (Pt )t≥0 . In other words, Pt f (x) = 𝔼x (f (Xt )), for each nonnegative function f on V. In this case, it is meaningful to wonder if the random walk at time t is asymptotically bounded in some ball of radius depending on t. Definition 2.16. Fix a reference point x̄ ∈ V. A nonnegative increasing function R(t) is called an upper escape rate function for G, if there is a random time T such that ℙx̄ (ρ(Xt , x)̄ ≤ R(t)

for all t ≥ T) = 1.

Theorem 2.17 ([39, 38]). Let ρ be an admissible intrinsic metric on G. Denote V(r) = ̄ Assume the condition (♦). Then ∃c > 0, R̂ > 0 such that the inverse function m(Br (x)). −1 ψ (t) of R

ψ(R) = c ∫ R̂

r dr ln V(r) + ln ln r

is an upper escape rate function for G. The above result is sharp in its generality, and is analogous to the results of Grigor’yan and Hsu [26], and Hsu and Qin [30] for manifolds. A direct adaption of analysis in manifold case only leads to nonoptimal results. The key idea is again that large time behavior of weighted graphs is similar to a diffusion. The techniques for

114 | B. Hua and X. Huang comparing the random walk on a weighted graphs with some related diffusion like processes are time change in Dirichlet form theory, and minimality of Feynman–Kac semigroups. See [39] for details.

2.9 Final remarks Intrinsic metrics are related to the analysis of unbounded Laplacians on graphs in various natural ways. Currently a lot of efforts are devoted to clarifying interconnections of many different notions of discrete Ricci curvature. We expect more results revealing role of intrinsic metrics in context of discrete curvature.

Appendix A. A remark on finite jump size Let G = (V, E, m, w) be a locally finite, connected simple weighted graph. Proposition A.1. Suppose that there is an intrinsic metric ρ on V such that balls with respect to ρ are all finite. Then there is an intrinsic metric δ with jump size bounded by 1 that satisfies the same property. Proof. Since G is connected, we can define a shortest path pseudometric δ on V by δ(x, y) =

n−1

inf

x=x0 ∼⋅⋅⋅∼xn =y

∑ (ρ(xi , xi+1 ) ∧ 1),

i=0

x, y ∈ V, x ≠ y,

where the infimum is taken over all paths connecting x and y. It is clear that for x ∼ y, δ(x, y) ≤ ρ(x, y). Therefore δ is an intrinsic metric and has jump size at most 1. We are left to show finiteness of balls with respect to δ. Fix x ∈ V and R > 0. Suppose y ≠ x satisfies δ(x, y) < R. By definition, there is a path x = x0 ∼ ⋅ ⋅ ⋅ ∼ xn = y such that n−1

∑ (ρ(xi , xi+1 ) ∧ 1) < R.

i=0

We rewrite the above as ∑

i:ρ(xi ,xi+1 )≤1

ρ(xi , xi+1 ) + ♯{i : ρ(xi , xi+1 ) > 1} < R.

(♣)

This clearly implies that #{i : ρ(xi , xi+1 ) > 1} < R. And for each consecutive sequence xi ∼ ⋅ ⋅ ⋅ ∼ xi+k with ρ(xl , xl+1 ) ≤ 1 (∀i ≤ l ≤ i + k − 1), we have ρ(xi , xi+k ) < R. Let A be a finite subset of V. We consider two type of operations on A: C(A) : = {z ∈ V : z ∈ A or ∃v ∈ A, z ∼ v},

A survey on unbounded Laplacians and intrinsic metrics on graphs | 115

U(A) : = ⋃ Bρ (v, R). v∈A

By local finiteness of G, C(A) is finite. Since balls with respect to ρ are all finite, U(A) is finite as well. Let N = [R] + 1 > R. By (♣), we observe that y ∈ U(CU)N ({x}). Hence Bδ (x, R) ⊆ U(CU)N ({x}) and is finite.

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Naotaka Kajino

Energy measures for diffusions on fractals: a survey Abstract: This article surveys the state of the art of the studies on singularity and other analytic and geometric properties of the energy measures associated with symmetric diffusions (or more precisely, strongly local, regular symmetric Dirichlet forms), especially when the state space is a fractal or when full off-diagonal sub-Gaussian estimates of the heat kernel hold. Keywords: Symmetric diffusion, regular symmetric Dirichlet form, Sierpiński gasket, generalized Sierpiński carpet, energy measure, singularity, heat kernel, sub-Gaussian estimate, Gaussian estimate, martingale dimension MSC 2010: Primary 28A80, 31E05, 60G30, Secondary 31C25, 35K08, 60G44, 60J60

Contents 1 2 3 3.1 3.2 4 5 6

Introduction | 119 The energy measures: their definition and basic properties | 121 The canonical Dirichlet forms on Sierpiński gaskets and carpets | 126 Sierpiński gaskets | 126 Sierpiński carpets | 128 Singularity of the energy measures | 130 Gaussian heat kernel estimates via time changes by energy measures | 134 Martingale dimension | 137 Bibliography | 141

1 Introduction This article surveys the state of the art of the studies on energy measures associated with symmetric diffusions (or more precisely, strongly local, regular symmetric Dirichlet forms). Given a strongly local, regular symmetric Dirichlet form (ℰ , ℱ ) over a locally compact separable metrizable topological space K equipped with a Radon measure m with full support, the ℰ -energy measure μ⟨u⟩ of u ∈ ℱ is a finite Borel measure on K which theoretically plays the role of the measure |∇u|2 dx in the general theory of regAcknowledgement: The author was supported in part by JSPS KAKENHI Grant Numbers JP18H01123, JP17H02849. Naotaka Kajino, Department of Mathematics, Graduate School of Science, Kobe University, Rokkodai-cho 1-1, Nada-ku, Kobe 657-8501, Japan, e-mail: [email protected] https://doi.org/10.1515/9783110700763-005

120 | N. Kajino ular symmetric Dirichlet forms as presented in [12, 11]. While μ⟨u⟩ is easily identified as |∇u|2 dx in the case of the Brownian motion on Riemannian manifolds, for symmetric diffusions on fractals μ⟨u⟩ usually admits no such simple expression and is typically singular with respect to the geometrically canonical volume measure, and its nature is a deep mystery in the field of analysis of Laplacians and symmetric diffusions on fractals. Here we summarize the basics of the energy measures associated with strongly local, regular symmetric Dirichlet forms and survey the known results on the singularity and other analytic and geometric properties of these measures when the state space is a fractal or when the heat kernel satisfies full off-diagonal sub-Gaussian estimates, typical forms of heat kernel estimates for diffusions on fractals. This article is organized as follows. First, in Section 2, we introduce the general framework of a strongly local, regular symmetric Dirichlet form and give the definition and basic properties of the associated energy measures. As the most typical classes of examples, some basics of self-similar Dirichlet forms on Sierpiński gaskets and Sierpiński carpets are summarized in Subsections 3.1 and 3.2, respectively, of Section 3. The known results on singularity of the energy measures are surveyed in Section 4. Section 5 treats the problem of analyzing the Dirichlet space obtained by changing the reference measure to energy measures, considered first by Kigami [26] and then by the author [19, 20] in the case of the two-dimensional Sierpiński gasket, and studied further in the author’s recent joint work [24] with Mathav Murugan. Then Section 6 introduces the martingale dimension (or the index), which could be considered as the maximum dimension of the “tangent space structure” induced by the Dirichlet form, and summarizes the known results on the problem of determining this dimension for Dirichlet forms on fractals, studied first by Kusuoka [30, 31] and more recently by Hino [15, 16, 17]. Notation. Throughout this paper, we use the following notation and conventions. (1) The symbols ⊂ and ⊃ for set inclusion allow the case of the equality. (2) ℕ := {n ∈ ℤ | n > 0}, i. e., 0 ∈ ̸ ℕ. (3) The cardinality (the number of elements) of a set A is denoted by #A. (4) We set sup 0 := 0 and inf 0 := ∞. We write a ∨ b := max{a, b}, a ∧ b := min{a, b}, a+ := a ∨ 0 and a− := −(a ∧ 0) for a, b ∈ [−∞, ∞], and we use the same notation also for [−∞, ∞]-valued functions and equivalence classes of them. All numerical functions in this paper are assumed to be [−∞, ∞]-valued. (5) Let K be a non-empty set. We define 1A = 1KA ∈ ℝK for A ⊂ K by 1A (x) := 1KA (x) := { 01 ifif xx∈A, ∈A, ̸ and set ‖u‖sup := ‖u‖sup,K := supx∈K |u(x)| for u : K → [−∞, ∞]. (6) Let K be a topological space. We set 𝒞 (K) := {u | u : K → ℝ, u is continuous} and 𝒞c (K) := {u ∈ 𝒞 (K) | K \ u−1 (0) is relatively compact (has compact closure) in K}. The interior of A ⊂ K in K is denoted by intK A and the Borel σ-field of K by B(K). (7) Let (K, B) be a measurable space and let μ, ν be σ-finite measures on (K, B). We write ν ≪ μ and ν ⊥ μ to mean that ν is absolutely continuous and singular,

Energy measures for diffusions on fractals: a survey | 121

respectively, with respect to μ. The μ-completion of B is denoted by Bμ . We set μ|A := μ|B|A for A ∈ B, where B|A := {B ∩ A | B ∈ B}. (8) Let n ∈ ℕ. The Euclidean inner product and norm on ℝn are denoted by ⟨⋅, ⋅⟩ and |⋅|, respectively. For each k ∈ {1, . . . , n}, the kth (classical or weak) partial derivative of a function u on ℝn is denoted by 𝜕k u.

2 The energy measures: their definition and basic properties In this section, we first introduce the general framework of a metric measure Dirichlet space, i. e., a strongly local, regular symmetric Dirichlet space equipped with a metric, then give the definition of its associated energy measures and mention their fundamental theoretical properties. Throughout this section, we consider a locally compact separable metric space (K, d), equipped with a Radon measure m with full support, i. e., a Borel measure m on K which is finite on any compact subset of K and strictly positive on any non-empty open subset of K, and we always assume #K ≥ 2 to exclude the trivial case of #K = 1. Such a triple (K, d, m) is referred to as a metric measure space. We set B(x, r) := {y ∈ K | d(x, y) < r} for (x, r) ∈ K × (0, ∞) and diam(K, d) := supx,y∈K d(x, y); note that #K ≥ 2 is equivalent to diam(K, d) ∈ (0, ∞]. Furthermore, let (ℰ , ℱ ) be a symmetric Dirichlet form on L2 (K, m); we refer to [12, 11] for details of the theory of symmetric Dirichlet forms. By definition, ℱ is a dense linear subspace of L2 (K, m), and ℰ : ℱ × ℱ → ℝ is a non-negative definite symmetric bilinear form which is closed (ℱ is a Hilbert space under the inner product ℰ1 := ℰ + ⟨⋅, ⋅⟩L2 (K,m) ) and Markovian (u+ ∧1 ∈ ℱ and ℰ (u+ ∧1, u+ ∧1) ≤ ℰ (u, u) for any u ∈ ℱ ). Recall that (ℰ , ℱ ) is called regular if and only if ℱ ∩ 𝒞c (K) is dense both in (ℱ , ℰ1 ) and in (𝒞c (K), ‖ ⋅ ‖sup ), and that (ℰ , ℱ ) is called strongly local if and only if ℰ (u, v) = 0 for any u, v ∈ ℱ with suppm [u], suppm [v] compact and suppm [u − a1K ] ∩ suppm [v] = 0 for some a ∈ ℝ. Here for a Borel measurable function u : K → [−∞, ∞] or an m-equivalence class u of such functions, suppm [u] denotes the support of the measure |u| dm, i. e., the smallest closed subset F of K with ∫K\F |u| dm = 0, which exists since K has a countable open base for its topology; note that suppm [u] coincides with the closure of K \ u−1 (0) in K if u is continuous. The pair (K, d, m, ℰ , ℱ ) of a metric measure space (K, d, m) and a strongly local, regular symmetric Dirichlet form (ℰ , ℱ ) on L2 (K, m) is termed a metric measure Dirichlet space, or an MMD space in abbreviation, and we fix an MMD space (K, d, m, ℰ , ℱ ) throughout the rest of this section. The central object of the discussions in this article is the energy measures associated with an MMD space, which are defined as follows. Note that uv ∈ ℱ for any u, v ∈ ℱ ∩ L∞ (K, m) by [12, Theorem 1.4.2-(ii)] and that {(−n) ∨ (u ∧ n)}∞ n=1 ⊂ ℱ and limn→∞ (−n) ∨ (u ∧ n) = u in norm in (ℱ , ℰ1 ) by [12, Theorem 1.4.2-(iii)].

122 | N. Kajino Definition 2.1 ([12, (3.2.13), (3.2.14) and (3.2.15)]). We define the ℰ -energy measure μ⟨u⟩ of u ∈ ℱ and the mutual ℰ -energy measure μ⟨u,v⟩ of u, v ∈ ℱ as follows. First, μ⟨u⟩ for u ∈ ℱ ∩ L∞ (K, m) is defined as the unique Borel measure on K such that ∫ f dμ⟨u⟩ = 2ℰ (u, fu) − ℰ (u2 , f )

for any f ∈ ℱ ∩ 𝒞c (K),

(2.1)

K

which exists and satisfies μ⟨u⟩ (K) ≤ 2ℰ (u, u) since 0 ≤ 2ℰ (u, f + u) − ℰ (u2 , f + ) ≤ 2‖f ‖sup ℰ (u, u) for any f ∈ ℱ ∩ 𝒞c (K) by [12, (3.2.13)] and hence the Riesz(–Markov– Kakutani) representation theorem [37, Theorem 2.14] is applicable. Then μ⟨u,v⟩ for u, v ∈ ℱ ∩ L∞ (K, m) is defined as the Borel signed measure on K given by μ⟨u,v⟩ := 1 (μ − μ⟨u−v⟩ ), so that μ⟨⋅,⋅⟩ is bilinear and symmetric and satisfies μ⟨u,u⟩ = μ⟨u⟩ and 4 ⟨u+v⟩ the Cauchy–Schwarz and triangle inequalities: for any A ∈ B(K), 󵄨󵄨 󵄨2 󵄨󵄨μ⟨u,v⟩ (A)󵄨󵄨󵄨 ≤ μ⟨u⟩ (A)μ⟨v⟩ (A),

μ⟨u+v⟩ (A)1/2 ≤ μ⟨u⟩ (A)1/2 + μ⟨v⟩ (A)1/2 .

(2.2) (2.3)

Finally, we define μ⟨u⟩ for general u ∈ ℱ by μ⟨u⟩ (A) := limn→∞ μ⟨(−n)∨(u∧n)⟩ (A), which exists and defines a Borel measure on K by (2.3) and μ⟨v⟩ (K) ≤ 2ℰ (v, v) for v ∈ ℱ ∩ L∞ (K, m), and define μ⟨u,v⟩ := 41 (μ⟨u+v⟩ − μ⟨u−v⟩ ) for general u, v ∈ ℱ , so that μ⟨⋅,⋅⟩ is again bilinear and symmetric and satisfies μ⟨u,u⟩ = μ⟨u⟩ , (2.2) and (2.3). Note that by [12, Lemma 3.2.3] and the strong locality of (ℰ , ℱ ), μ⟨u,v⟩ (K) = 2ℰ (u, v)

for any u, v ∈ ℱ .

(2.4)

Example 2.2. Let n ∈ ℕ. Consider the MMD space (ℝn , dEuc , dx, 21 D, H 1 (ℝn )) consisting of ℝn , the Euclidean metric dEuc on ℝn , the Lebesgue measure dx on ℝn , H 1 (ℝn ) := {u ∈ 󵄨 L2 (ℝn , dx) 󵄨󵄨󵄨 𝜕k u ∈ L2 (ℝn , dx) for any k ∈ {1, . . . , n}} and 1 1 D(u, v) := ∫ ⟨∇u, ∇v⟩ dx, 2 2

u, v ∈ H 1 (ℝn ),

(2.5)

ℝn

where ∇u := (𝜕1 u, . . . , 𝜕n u) for u ∈ H 1 (ℝn ). In this case, it follows by an elementary calculation using the Leibniz rule for the weak partial derivatives 𝜕k that 1 D(u, fu) − D(u2 , f ) = ∫ f |∇u|2 dx 2

for any u, f ∈ H 1 (ℝn ) ∩ L∞ (ℝn , dx),

ℝn

so that the 21 D-energy measure of u ∈ H 1 (ℝn ) coincides with the measure |∇u|2 dx. As an important consequence of the strong locality of (ℰ , ℱ ), we have the following counterpart of the classical chain rule in multivariable calculus. Here and in what ̃ denotes an arbitrary ℰ -quasi-continuous m-version of u ∈ ℱ , which exists follows, u by [12, Theorem 2.1.3] and is unique ℰ -q. e. (i. e., up to sets of capacity zero) by [12,

Energy measures for diffusions on fractals: a survey | 123

̃ are uniquely determined μ⟨v⟩ -a. e. for each v ∈ ℱ Lemma 2.1.4], so that the values of u since μ⟨v⟩ (N) = 0 for any N ∈ B(K) with capacity zero by [12, Lemma 3.2.4]; see [12, Section 2.1] for the definitions of the capacity and the ℰ -quasi-continuity of functions with respect to a regular symmetric Dirichlet form. Theorem 2.3 ([12, Theorem 3.2.2]). Let n ∈ ℕ, {uk }nk=1 ⊂ ℱ ∩L∞ (K, m) and let Φ : ℝn → ℝ be 𝒞 1 and satisfy Φ(0) = 0. Then Φ(u1 , . . . , un ) ∈ ℱ and n

̃1 , . . . , u ̃ n ) dμ⟨uk ,v⟩ dμ⟨Φ(u1 ,...,un ),v⟩ = ∑ 𝜕k Φ(u k=1

for any v ∈ ℱ .

(2.6)

An application of Theorem 2.3 further yields the following theorem. ̃ −1 on Theorem 2.4 ([11, Theorem 4.3.8]). Let u ∈ ℱ . Then the Borel measure μ⟨u⟩ ∘ u −1 −1 ̃ (A) := μ⟨u⟩ (u ̃ (A)) is absolutely continuous with respect to the ℝ defined by μ⟨u⟩ ∘ u Lebesgue measure on ℝ. Corollary 2.5. Let u, v ∈ ℱ . (1) μ⟨u⟩ ({x}) = 0 for any x ∈ K. ̃ = ṽ} := {x ∈ K | u ̃ (x) = ṽ(x)}. (2) μ⟨u⟩ |{̃u=̃v} = μ⟨v⟩ |{̃u=̃v} , where {u Proof. (1) Let x ∈ K. If {x} has zero capacity, then μ⟨u⟩ ({x}) = 0 by [12, Lemma 3.2.4], and ̃ (x) is independent of a particular choice of u ̃ and μ⟨u⟩ ({x}) ≤ μ⟨u⟩ ∘ otherwise u ̃ −1 ({u ̃ (x)}) = 0 by Theorem 2.4. u ̃ = ṽ}. Then by (2.3) and Theorem 2.4, (2) Let A ∈ B(K) satisfy A ⊂ {u 󵄨󵄨 1/2 1/2 󵄨2 ̃ = ṽ}) 󵄨󵄨μ⟨u⟩ (A) − μ⟨v⟩ (A) 󵄨󵄨󵄨 ≤ μ⟨u−v⟩ (A) ≤ μ⟨u−v⟩ ({u

̃ − ṽ)−1 (0)) = 0 = μ⟨u−v⟩ ((u

and thus μ⟨u⟩ (A) = μ⟨v⟩ (A). We next describe the probabilistic roles played by the ℰ -energy measures. For this purpose, we briefly recall some basics of an m-symmetric diffusion on K whose Dirichlet form is (ℰ , ℱ ) and of its additive functionals. Let KΔ := K ∪ {Δ} be the one-point compactification of K, so that B(KΔ ) = B(K) ∪ {A ∪ {Δ} | A ∈ B(K)}. In what follows, [−∞, ∞]-valued functions on K are always set to be 0 at Δ unless their values at Δ are already defined: u(Δ) := 0 for u : K → [−∞, ∞]. Let X = (Ω, M, {Xt }t∈[0,∞] , {ℙx }x∈KΔ ) be a diffusion without killing inside on (K, B(K)) with life time ζ and shift operators {θt }t∈[0,∞] . By definition, (Ω, M) is a measurable space, {Xt }t∈[0,∞] is a family of M/B(KΔ )-measurable maps Xt : Ω → KΔ such that [0, ∞) ∋ t 󳨃→ Xt (ω) ∈ KΔ is continuous and Xt (ω) = Δ for any t ∈ [ζ (ω), ∞] for each ω ∈ Ω, where ζ (ω) := inf{t ∈ [0, ∞) | Xt (ω) = Δ}, and {θt }t∈[0,∞] is a family of maps θt : Ω → Ω satisfying Xs ∘ θt = Xs+t for any s, t ∈ [0, ∞]. The pair X of such a

124 | N. Kajino stochastic process (Ω, M, {Xt }t∈[0,∞] ) and a family {ℙx }x∈KΔ of probability measures on (Ω, M) is then called a diffusion without killing inside on (K, B(K)) if and only if X is a normal Markov process on (K, B(K)) whose minimum completed admissible filtration F∗ = {Ft }t∈[0,∞] is right-continuous and which is strong Markov with respect to F∗ ; see [12, Section A.2, (M.2)–(M.5), the paragraph before Lemma A.2.2, and (A.2.3)] for the precise definitions of these notions. We further assume that X is m-symmetric with Dirichlet form (ℰ , ℱ ), i. e., Tt u = 𝔼(⋅) [u(Xt )] m-a. e. for each t ∈ (0, ∞) for any Borel measurable function u : K → [−∞, ∞] with ∫K u2 dm < ∞, where {Tt }t∈(0,∞) denotes the Markovian semigroup on L2 (K, m) corresponding to (ℰ , ℱ ) and 𝔼x [(⋅)] := ∫Ω (⋅) dℙx for x ∈ KΔ ; such a diffusion X without killing inside on (K, B(K)) exists by [12, Theorems 7.2.1 and 4.5.3]. For each σ-finite Borel measure ν on KΔ , the function KΔ ∋ x 󳨃→ ℙx [B] is B(KΔ )ν -measurable for any B ∈ F∞ by [11, Exercise A.1.20-(i)], then associated with ν is a measure ℙν on (Ω, F∞ ) given by ℙν [B] := ∫K ℙx [B] dν(x), and we set Δ

𝔼ν [(⋅)] := ∫Ω (⋅) dℙν . We also set σ̇ B (ω) := inf{t ∈ [0, ∞) | Xt (ω) ∈ B} for B ⊂ KΔ and ω ∈ Ω, so that σ̇ B is an F∗ -stopping time if B ∈ B(KΔ ) by [12, Theorem A.2.3]. We say that N ∈ B(K) is properly exceptional for X if and only if m(N) = 0 and ℙx [σ̇ N = ∞] = 1 for any x ∈ E \ N. Any such N has capacity zero by [12, Theorem 4.2.1-(ii)], and conversely any subset of K of capacity zero is included in some Borel properly exceptional set for X by [12, Theorem 4.1.1]. Definition 2.6. (1) A family A = {At }t∈[0,∞) of [−∞, ∞]-valued functions on Ω is called a continuous additive functional (CAF) of X if and only if At is Ft -measurable for each t ∈ [0, ∞) and there exist a set Λ ∈ F∞ and a properly exceptional set N ∈ B(K) for X such that the following two conditions hold: ℙx [Λ] = 1 for any x ∈ K \ N

and θt (Λ) ⊂ Λ for any t ∈ [0, ∞).

For each ω ∈ Λ, A(⋅) (ω) : [0, ∞) → [−∞, ∞] is continuous, A0 (ω) = 0, and for any s, t ∈ [0, ∞), As+t (ω) = At (ω) + As ∘ θt (ω), |At (ω)| < ∞ if t < ζ (ω), and At (ω) = Aζ (ω) (ω) if t ≥ ζ (ω).

(CAF1) (CAF2)

Given any pair of such sets Λ and N, we call Λ a defining set and N an exceptional set, respectively, for A. (2) A CAF A = {At }t∈[0,∞) of X is called positive or finite, respectively, if and only if we can choose a defining set Λ for A so that for each ω ∈ Λ, the function t 󳨃→ At (ω) on [0, ∞) is [0, ∞]-valued or ℝ-valued, respectively. The collection of all positive continuous additive functionals (PCAFs) of X is denoted by A+c . (3) Let A = {At }t∈[0,∞) and B = {Bt }t∈[0,∞) be CAFs of X. We call A and B equivalent, and write A ∼A B, if and only if ℙm [At ≠ Bt ] = 0 for any t ∈ [0, ∞). By [11, Lemma A.3.2], A ∼A B if and only if there exist Λ ∈ F∞ and N ∈ B(K) which are a defining set and an exceptional set, respectively, for both A and B such that At (ω) = Bt (ω)

Energy measures for diffusions on fractals: a survey | 125

for any (t, ω) ∈ [0, ∞) × Λ. Equivalent CAFs of X are always identified henceforth, and any equality among CAFs of X will always mean the equivalence ∼A . It turns out that each PCAF A = {At }t∈[0,∞) of X admits a unique Borel measure μA on K capturing where and how fast t 󳨃→ At increases, as follows. Theorem 2.7 ([11, Theorem A.3.5-(i), (ii)], [12, Lemma 5.1.7]). Let A = {At }t∈[0,∞) ∈ A+c . Then there exists a unique Borel measure μA on K, called the Revuz measure of A, such that for any Borel measurable function f : K → [0, ∞], t

1 ∫ f dμA = lim 𝔼m [∫ f (Xs ) dAs ]. t↓0 t

K

(2.7)

0

Moreover, μA (N) = 0 for any N ∈ B(K) with capacity zero. Definition 2.8. We define the spaces ℳ of martingale additive functionals (MAF) of X and 𝒩c of continuous additive functionals of X of zero energy by 󵄨󵄨 M = {M } 2 󵄨󵄨 t t∈[0,∞) is a finite CAF of X, 𝔼x [Mt ] < ∞ }, 󵄨󵄨 and 𝔼 [M ] = 0 for ℰ -q. e. x ∈ K for each t ∈ (0, ∞) 󵄨 x t 󵄨󵄨 N = {N } 󵄨󵄨 t t∈[0,∞) is a finite CAF of X, eA (N) = 0, 𝒩c := {N 󵄨󵄨󵄨 }, 󵄨󵄨 𝔼 [|N |] < ∞ for ℰ -q. e. x ∈ K for each t ∈ (0, ∞) 󵄨 x t ℳ := {M 󵄨󵄨󵄨

(2.8) (2.9)

where the energy eA (A) of a CAF A = {At }t∈[0,∞) of X is defined by eA (A) := limt↓0 (2t)−1 𝔼m [A2t ] whenever the limit exists. Let M = {Mt }t∈[0,∞) ∈ ℳ. Then for ℰ -q. e. x ∈ K, M is an (F∗ , ℙx )-martingale with M0 = 0 ℙx -a. s. and 𝔼x [Mt2 ] < ∞ for any t ∈ [0, ∞). As stated in [12, the paragraph of (5.2.7)], there exists ⟨M⟩ = {⟨M⟩t }t∈[0,∞) ∈ A+c , unique up to the equivalence ∼A and called the quadratic variation of M, such that 𝔼x [Mt2 ] = 𝔼x [⟨M⟩t ] for any t ∈ [0, ∞) for ℰ -q. e. x ∈ K. We easily see that ⟨M⟩ is a finite CAF of X and that {Mt2 − ⟨M⟩t }t∈[0,∞) is an 2 (F∗ , ℙx )-martingale for ℰ -q. e. x ∈ K. Note also that 𝔼m [Ms+t ] ≤ 𝔼m [Ms2 ] + 𝔼m [Mt2 ] for any s, t ∈ [0, ∞) and hence that the energy eA (M) of M exists and is given by eA (M) = supt∈(0,∞) (2t)−1 𝔼m [Mt2 ]. Set ℳ∘ := {M ∈ ℳ | eA (M) < ∞}. The following theorem, known as the Fukushima decomposition theorem, is an extension of Itô’s formula and the semimartingale decomposition in the framework of regular symmetric Dirichlet forms. Theorem 2.9 ([12, Theorem 5.2.2 and Lemma 5.5.1-(ii)]). Let u ∈ ℱ . Then there exist M [u] = {Mt[u] }t∈[0,∞) ∈ ℳ∘ and N [u] = {Nt[u] }t∈[0,∞) ∈ 𝒩c , unique up to the equivalence ∼A , such that for ℰ -q. e. x ∈ K, ̃ (Xt ) − u ̃ (X0 ) = Mt[u] + Nt[u] u

for any t ∈ [0, ∞), ℙx -a. s.

(2.10)

126 | N. Kajino Theorem 2.10 ([12, Theorem 5.2.3]). Let u ∈ ℱ . Then μ⟨M [u] ⟩ = μ⟨u⟩ . Hence, for example, the question of whether μ⟨u⟩ is singular with respect to m, which is the theme of Section 4 below, could be considered as an analytical counterpart of that of whether ⟨M [u] ⟩ = {⟨M [u] ⟩t }t∈[0,∞) is singular as a function in t ∈ [0, ∞), although the actual relation between these two questions is not clear.

3 The canonical Dirichlet forms on Sierpiński gaskets and carpets Before surveying the known results on energy measures on fractal spaces in the following sections, we briefly summarize in this section some basics of the canonical self-similar Dirichlet forms on Sierpiński gaskets and Sierpiński carpets.

3.1 Sierpiński gaskets Let n ∈ ℕ, n ≥ 2, l := 2, set S := {0, 1, . . . , n} and let V0 := {qi | i ∈ S} ⊂ ℝn be the set of the vertices of a regular n-dimensional simplex △n ⊂ ℝn (△n is a compact convex subset of ℝn ). We further define fi : ℝn → ℝn by fi (x) := qi + 21 (x − qi ) for each i ∈ S, let K be the self-similar set associated with {fi }i∈S , i. e., the unique non-empty compact subset of ℝn such that K = ⋃i∈S fi (K), which exists and satisfies K ⊊ △n thanks to ⋃i∈S fi (△n ) ⊊ △n by [25, Theorem 1.1.4], and set Fi := fi |K for each i ∈ S. The set K is called the n-dimensional (standard) Sierpiński gasket (see Figure 1). Let d : K × K → [0, ∞) be the Euclidean metric on K given by d(x, y) := |x − y|. For k ∈ ℕ, we set Fw := Fw1 ∘ ⋅ ⋅ ⋅ ∘ Fwk and Kw := Fw (K) for each w = w1 . . . wk ∈ Sk and Vk := ⋃w∈Sk Fw (V0 ), so that Vk−1 ⊊ Vk for any k ∈ ℕ and the set V∗ := ⋃∞ k=0 Vk is dense in K.

Figure 1: The n-dimensional Sierpiński gaskets (n = 2, 3).

Energy measures for diffusions on fractals: a survey | 127

A natural self-similar Dirichlet form can be constructed on K in the following manner; see [25, Chapter 3] for details. First, we define ℰ (0) : ℝV0 × ℝV0 → ℝ by ℰ

(0)

(u, v) :=

1 ∑ (u(x) − u(y))(v(x) − v(y)), 2 x,y∈V

u, v ∈ ℝV0 .

(3.1)

0

Next, we set r := ℰ

n+1 n+3 (k)

and define ℰ (k) : ℝVk × ℝVk → ℝ for each k ∈ ℕ by

(u, v) := ∑

w∈Sk

1 (0) ℰ (u ∘ Fw |V0 , v ∘ Fw |V0 ), rk

u, v ∈ ℝVk .

(3.2)

Clearly, (ℰ (k) , ℝVk ) is an irreducible symmetric Dirichlet form on L2 (Vk , #) for any k ∈ ℕ ∪ {0}. Moreover, an elementary calculation shows the following proposition. Proposition 3.1. The pair (ℰ (0) , r) is a harmonic structure on (K, S, {Fi }i∈S ), i. e., for any j, k ∈ ℕ ∪ {0} with j ≤ k and any u ∈ ℝVj , ℰ (u, u) = min{ℰ (j)

(k)

󵄨 (v, v) 󵄨󵄨󵄨 v ∈ ℝVk , v|Vj = u}.

(3.3)

By Proposition 3.1, {ℰ (k) (u|Vk , u|Vk )}k=0 is non-decreasing and hence has the limit in [0, ∞] for any u ∈ 𝒞 (K). Then we define a linear subspace ℱ of 𝒞 (K) and a nonnegative definite symmetric bilinear form ℰ : ℱ × ℱ → ℝ by ∞

󵄨󵄨 󵄨 k→∞

ℱ := {u ∈ 𝒞 (K) 󵄨󵄨󵄨 lim ℰ ℰ (u, v) := lim ℰ k→∞

(k)

(k)

(u|Vk , u|Vk ) < ∞},

(u|Vk , v|Vk ) ∈ ℝ,

u, v ∈ ℱ ,

(3.4) (3.5)

so that (ℰ , ℱ ) is easily seen to possess the following self-similarity properties (note that ℱ ∩ 𝒞 (K) = ℱ in the present setting): ℱ ∩ 𝒞 (K) = {u ∈ 𝒞 (K) 󵄨󵄨󵄨 u ∘ Fi ∈ ℱ for any i ∈ S},

󵄨

1 r

ℰ (u, v) = ∑ ℰ (u ∘ Fi , v ∘ Fi ), i∈S

u, v ∈ ℱ .

(SSDF1) (SSDF2)

By [25, Proposition 2.2.4, Lemma 2.2.5, Theorem 2.2.6, Lemma 2.3.9, Theorems 2.3.10 and 3.3.4], (ℰ , ℱ ), called the standard resistance form on K, is a resistance form on K whose resistance metric Rℰ : K × K → [0, ∞) is a metric on K compatible with the topology of (K, d); here (ℰ , ℱ ) being a resistance form on K means that the following hold (see [25, Definition 2.3.1] or [28, Definition 3.1]): (RF1) {u ∈ ℱ | ℰ (u, u) = 0} = ℝ1K . (RF2) (ℱ /ℝ1K , ℰ ) is a Hilbert space. (RF3) {u|V | u ∈ ℱ } = ℝV for any non-empty finite subset V of K. (RF4) Rℰ (x, y) := supu∈ℱ \ℝ1K |u(x) − u(y)|2 /ℰ (u, u) < ∞ for any x, y ∈ K. (RF5) u+ ∧ 1 ∈ ℱ and ℰ (u+ ∧ 1, u+ ∧ 1) ≤ ℰ (u, u) for any u ∈ ℱ .

128 | N. Kajino See [25, Chapter 2] and [28, Part 1] for further details of resistance forms. Now we can turn (ℰ , ℱ ) into a Dirichlet form by equipping K with a measure, as follows. Theorem 3.2. Let ν be a Radon measure on K with full support. Then (ℰ , ℱ ) is an irreducible, strongly local, regular symmetric Dirichlet form on L2 (K, ν). Moreover, there exists a unique continuous function pν = pνt (x, y) : (0, ∞) × K × K → [0, ∞), called the (continuous) heat kernel of (K, d, ν, ℰ , ℱ ), such that ℙx [Xtν ∈ dy] = pνt (x, y) dν(y)

for any (t, x) ∈ (0, ∞) × K,

(3.6)

where X ν = (Ω, M, {Xtν }t∈[0,∞] , {ℙx }x∈KΔ ) denotes any ν-symmetric diffusion without killing inside on (K, B(K)) whose Dirichlet form is (ℰ , ℱ ). Proof. (ℰ , ℱ ) is a regular symmetric Dirichlet form on L2 (K, ν) by [28, Corollary 6.4 and Theorem 9.4], strongly local by the same argument as [14, Proof of Lemma 3.12] based on (SSDF1), (SSDF2) and ℰ (1K , 1K ) = 0, and irreducible by (RF1) and [11, Theorem 2.1.11]. The existence of pν is a special case of [28, Theorem 10.4], and its uniqueness is immediate from (3.6) and its continuity. The most typical choice of the measure ν is the self-similar measure m on (K, S, {Fi }i∈S ) with weight (1/#S)i∈S , i. e., the unique Borel measure on K such that m(Kw ) = (#S)−k for any k ∈ ℕ and any w ∈ Sk , which exists and is a constant multiple of the log2 (n + 1)-dimensional Hausdorff measure on (K, d) by [25, Proposition 1.5.8 and Theorem 1.5.7]. In this case we have the following heat kernel estimates, called subGaussian estimates in light of the fact that dw > 2. Theorem 3.3 ([9, Theorem 1.5]). Set df := log2 (n + 1) and dw := log2 (n + 3). Then there exist c1 , c2 , c3 , c4 ∈ (0, ∞) such that for any (t, x, y) ∈ (0, 1] × K × K, c1

t df /dw

1

1

c3 d(x, y)dw dw −1 d(x, y)dw dw −1 ) ) exp(−( ) ≤ pm ). t (x, y) ≤ df /dw exp(−( c2 t c4 t t

(3.7)

3.2 Sierpiński carpets Let n, l ∈ ℕ, n ≥ 2, l ≥ 3 and set Q0 := [0, 1]n . Let S ⊊ {0, 1, . . . , l − 1}n be non-empty, define fi : ℝn → ℝn by fi (x) := l−1 i + l−1 x for each i ∈ S and set Q1 := ⋃i∈S fi (Q0 ), so that Q1 ⊊ Q0 . Let K be the self-similar set associated with {fi }i∈S , i. e., the unique non-empty compact subset of ℝn such that K = ⋃i∈S fi (K), which exists and satisfies K ⊊ Q0 thanks to Q1 ⊊ Q0 by [25, Theorem 1.1.4], and set V0 := K \ (0, 1)n , Fi := fi |K for each i ∈ S and GSC(n, l, S) := (K, S, {Fi }i∈S ). Let d : K × K → [0, ∞) be the Euclidean metric on K given by d(x, y) := |x−y|, set df := logl #S, and let m be the self-similar measure on GSC(d, l, S) with weight (1/#S)i∈S , i. e., the unique Borel measure on K such that m(Kw ) = (#S)−k for any k ∈ ℕ and any w ∈ Sk , which exists by [25, Propositions 1.5.8, 1.4.3, 1.4.4 and Corollary 1.4.8]. Recall that m is a constant multiple of the df -dimensional Hausdorff

Energy measures for diffusions on fractals: a survey | 129

measure on (K, d); see, e. g., [25, Proposition 1.5.8 and Theorem 1.5.7]. Note that df < n by S ⊊ {0, 1, . . . , l − 1}n . The following definition is essentially due to Barlow and Bass [4, Section 2]. Definition 3.4 (Generalized Sierpiński carpet, [7, Subsection 2.2]). The triple GSC(n, l, S) = (K, S, {Fi }i∈S ) is called a generalized Sierpiński carpet if and only if the following four conditions are satisfied: (GSC1) (Symmetry) f (Q1 ) = Q1 for any isometry f of ℝn with f (Q0 ) = Q0 . (GSC2) (Connectedness) Q1 is connected. (GSC3) (Non-diagonality) intℝn (Q1 ∩ ∏nk=1 [(ik − εk )l−1 , (ik + 1)l−1 ]) is either empty or connected for any (ik )nk=1 ∈ ℤn and any (εk )nk=1 ∈ {0, 1}n .

(GSC4) (Borders included) [0, 1] × {0}n−1 ⊂ Q1 .

See [4, Remark 2.2] for a description of the meaning of each of the four conditions (GSC1), (GSC2), (GSC3) and (GSC4) in Definition 3.4. As special cases of Definition 3.4, GSC(2, 3, SSC ) and GSC(3, 3, SMS ) are called the Sierpiński carpet and the Menger 󵄨 sponge, respectively, where SSC := {0, 1, 2}2 \ {(1, 1)} and SMS := {(i1 , i2 , i3 ) ∈ {0, 1, 2}3 󵄨󵄨󵄨 3 ∑k=1 1{1} (ik ) ≤ 1} (see Figure 2).

Figure 2: The Sierpiński carpet, two other generalized Sierpiński carpets with n = 2 and the Menger sponge.

In the rest of this subsection, we assume that GSC(n, l, S) = (K, S, {Fi }i∈S ) is a generalized Sierpiński carpet, and recall some basics of the canonical Dirichlet form on GSC(n, l, S). There are two established ways of constructing a non-degenerate m-symmetric diffusion without killing inside on (K, B(K)), or equivalently, a nonzero, strongly local, regular symmetric Dirichlet form on L2 (K, m), one by Barlow and Bass [3, 4] using the reflecting Brownian motions on the domains approximating K, and the other by Kusuoka and Zhou [32] based on graph approximations. It had been a long-standing open problem to prove that the constructions in [3, 4] and in [32] give rise to the same diffusion on K, which Barlow, Bass, Kumagai and Teplyaev [7] have finally solved by proving the uniqueness of a non-zero conservative regular symmetric Dirichlet form on L2 (K, m) possessing certain local symmetry. As a consequence of the results in [7], after some additional arguments in [17, 21, 22] we have the unique

130 | N. Kajino existence of a canonical self-similar Dirichlet form (ℰ , ℱ ) on L2 (K, m) and the heat kernel estimates (3.7) with dw > 2 as follows. Definition 3.5. We define 𝒢0 := {f |K | f is an isometry of ℝn , f (Q0 ) = Q0 }, which forms a subgroup of the group of isometries of (K, d) by virtue of (GSC1). Theorem 3.6 ([7, Theorems 1.2 and 4.32], [17, Proposition 5.1], [21, Proposition 5.9]). There exists a unique (up to constant multiples of ℰ ) regular symmetric Dirichlet form (ℰ , ℱ ) on L2 (K, m) satisfying ℰ (u, u) > 0 for some u ∈ ℱ , ℰ (1K , 1K ) = 0, (SSDF1), (SSDF2) for some r ∈ (0, ∞), and the following: (GSCDF) If u ∈ ℱ ∩ 𝒞 (K) and g ∈ 𝒢0 then u ∘ g ∈ ℱ and ℰ (u ∘ g, u ∘ g) = ℰ (u, u). Definition 3.7. The regular symmetric Dirichlet form (ℰ , ℱ ) on L2 (K, m) as in Theorem 3.6 is called the canonical Dirichlet form on GSC(n, l, S), and we set dw := logl (#S/r). Note that (ℰ , ℱ ) is also strongly local by the same argument as [14, Proof of Lemma 3.12] based on (SSDF1), (SSDF2) and ℰ (1K , 1K ) = 0. Theorem 3.8 ([4, Remarks 5.4-1.], [22, Theorem 2.9]). dw > 2. Theorem 3.9 ([4, Theorem 1.3], [7, Theorem 4.30 and Remark 4.33]). There exists a unique continuous function pm = pm t (x, y) : (0, ∞) × K × K → [0, ∞), called the (continuous) heat kernel of (K, d, m, ℰ , ℱ ), such that (3.6) with ν = m holds for some m-symmetric diffusion X m = (Ω, M, {Xtm }t∈[0,∞] , {ℙx }x∈KΔ ) without killing inside on (K, B(K)) whose Dirichlet form is (ℰ , ℱ ). Moreover, there exist c1 , c2 , c3 , c4 ∈ (0, ∞) such that (3.7) holds for any (t, x, y) ∈ (0, 1] × K × K.

4 Singularity of the energy measures In the remaining three sections, we survey the existing studies in the literature which are closely related to the properties of the energy measures associated with canonical Dirichlet forms on fractals. First, in this section, we summarize the known results on singularity of the energy measures. As illustrated by the examples in the previous section, canonical Dirichlet forms on fractals usually do not admit simple expressions, unlike the classical Dirichlet form ( 21 D, H 1 (ℝn )) on ℝn in Example 2.2 and its counterpart on Riemannian manifolds. Then it becomes a fundamental and highly non-trivial problem to understand the actual nature of the associated energy measures deeply, and the first natural question to be asked in this direction is probably how different the energy measures are from the geometrically canonical volume measures on the fractal. The first answer to this question was given by Kusuoka in [30], where he proved the singularity of the energy measures with respect to the canonical self-similar measure on the fractal, for certain Dirichlet forms he constructed on some class of self-

Energy measures for diffusions on fractals: a survey | 131

similar fractals. In particular, he also showed that this result applies to the case of the Sierpiński gaskets as in Subsection 3.1 and yields the following theorem. Theorem 4.1 ([30, Example 1]). Let (K, d, m, ℰ , ℱ ) be as in Subsection 3.1. Then μ⟨u⟩ ⊥ m for any u ∈ ℱ . In [31, Corollary 7.16] Kusuoka extended his result in [30] to the case of the canonical diffusions on a class of nested fractals,1 although it is not clear how large the class of fractals is because of the complexity of the assumptions. Later in [10, Theorem 5.1] Ben-Bassat, Strichartz and Teplyaev obtained a similar result for self-similar Dirichlet forms on post-critically finite (in particular, finitely ramified) self-similar fractals under simpler assumptions and with a shorter proof. The best result known so far in this direction is due to Hino [14], who extended Theorem 4.1 to generalized Sierpiński carpets. To give the precise statement of the main result in [14], we need the notion of harmonic functions defined as follows. Definition 4.2. Let (K, d, m, ℰ , ℱ ) be an MMD space and U an open subset of K. Then h ∈ ℱ is said to be ℰ -harmonic on U if and only if either of the following two conditions, which are easily seen to be equivalent to each other, holds: ̃ ℰ -q. e. on K \ U}, ̃=h ℰ (h, h) = inf{ℰ (u, u) 󵄨󵄨󵄨 u ∈ ℱ , u

(4.1)

ℰ (h, v) = 0

(4.2)

󵄨

for any v ∈ ℱ ∩ 𝒞c (K) with suppm [v] ⊂ U, or equivalently, for any v ∈ ℱ with ṽ = 0 ℰ -q. e. on K \ U,

where the equivalence of the two conditions in (4.2) follows by [12, Corollary 2.3.1]. The main result in [14] can be stated as in the following theorem. Hino [14] has proved the same result in a general framework of self-similar Dirichlet forms on selfsimilar sets including general post-critically finite self-similar sets and generalized Sierpiński carpets, but for simplicity we state it only for the examples in Section 3. Theorem 4.3 (Special cases of [14, Theorems 2.1 and 2.2]). Let S, {Fi }i∈S , V0 and (K, d, m, ℰ , ℱ ) be either as in Subsection 3.1 or as in Subsection 3.2. Let α = (αi )i∈S ∈ (0, 1)S satisfy ∑i∈S αi = 1 and let να be the self-similar measure on (K, S, {Fi }i∈S ) with weight α, i. e., the unique Borel measure on K such that να (Kw ) = αw1 ⋅ ⋅ ⋅ αwk for any k ∈ ℕ and any w = w1 . . . wk ∈ Sk . Then either (i) μ⟨h⟩ = να for some h ∈ ℱ that is ℰ -harmonic on K \ V0 , or (ii) μ⟨u⟩ ⊥ να for any u ∈ ℱ . Hino has also proved in [14, Theorem 2.3] that the lower inequality in (3.7) with dw > 2 for the heat kernel pm t (x, y) excludes the possibility of case (i) in Theorem 4.3 for να = m, which in combination with Theorems 3.8, 3.9 and 4.3 yields the following theorem. 1 See the sentence following Theorem 6.4 below for some description of nested fractals.

132 | N. Kajino Theorem 4.4 ([14, Subsection 5.2]). Let (K, d, m, ℰ , ℱ ) be as in Subsection 3.2. Then μ⟨u⟩ ⊥ m for any u ∈ ℱ . To verify case (ii) in Theorem 4.3 for a wider range of α, Hino has given in [14, Theorem 2.4] another sufficient condition to exclude the possibility of case (i) in Theorem 4.3, which implies the following theorem for the examples in Section 3. Theorem 4.5 ([14, Theorems 5.1 and 5.3]). Let l, S, (K, d, m, ℰ , ℱ ), r be either as in Subsection 3.1 or as in Subsection 3.2, and let α = (αi )i∈S ∈ (0, 1)S satisfy ∑i∈S αi = 1 and maxi∈S αi < l−2 r −1 . Then μ⟨u⟩ ⊥ να for any u ∈ ℱ . Note that Theorems 4.1 and 4.4 are special cases of Theorem 4.5 since 1/#S < l−2 r −1 by dw = logl (#S/r) > 2. For self-similar Dirichlet forms on post-critically finite self-similar sets, in [18, Theorem 2] Hino and Nakahara have further provided simple topological criteria to exclude the possibility of case (i) in Theorem 4.3 for arbitrary α, one of which is applicable, for example, to the Sierpiński gaskets and yields the following result. Theorem 4.6 (Special cases of [18, Theorem 2]). Let S and (K, d, m, ℰ , ℱ ) be as in Subsection 3.1. Then μ⟨u⟩ ⊥ να for any u ∈ ℱ and any α = (αi )i∈S ∈ (0, 1)S with ∑i∈S αi = 1. In view of Theorem 4.6, it is natural to expect that the following conjecture would also hold. Conjecture 4.7. Let S and (K, d, m, ℰ , ℱ ) be as in Subsection 3.2. Then μ⟨u⟩ ⊥ να for any u ∈ ℱ and any α = (αi )i∈S ∈ (0, 1)S with ∑i∈S αi = 1. All the results on singularity of the energy measures described so far in this section heavily rely on the exact self-similarity of the state space and the Dirichlet form. In reality, however, even without the self-similarity the anomalous space–time scaling relation exhibited by the term d(x, y)dw /t with dw > 2 in (3.7) should still imply singular behavior of the sample paths of the quadratic variation ⟨M [u] ⟩ of the MAF M [u] in (2.10). Therefore it is natural to conjecture, as Barlow did in [2, Section 5, Remarks, 5.-1.], that the heat kernel estimates (3.7) with dw > 2 should imply the singularity of the energy measures with respect to the reference measure m of the MMD space. This long-standing open conjecture has been proved in the author’s recent joint work [23] with Mathav Murugan under the completely general framework of an MMD space (K, d, m, ℰ , ℱ ) with the metric d complete. To state the main theorem of [23], with a slightly simplified set of assumptions, we need the following generalization of (3.7) and another definition below. Definition 4.8 ((fHKE)Ψ ). Let (K, d, m, ℰ , ℱ ) be an MMD space, and let Ψ : [0, ∞) → [0, ∞) be a homeomorphism such that β0

−1 R cΨ ( ) r

β1

≤

Ψ(R) R ≤ cΨ ( ) Ψ(r) r

for any r, R ∈ (0, ∞) with r ≤ R

(4.3)

Energy measures for diffusions on fractals: a survey | 133

for some cΨ , β0 , β1 ∈ [1, ∞) with 1 < β0 ≤ β1 . We say that (K, d, m, ℰ , ℱ ) satisfies the full off-diagonal heat kernel estimates (fHKE)Ψ if and only if there exists a (unique) continuous function p = pt (x, y) : (0, ∞) × K × K → [0, ∞), called the (continuous) heat kernel of (K, d, m, ℰ , ℱ ), such that the following hold: ℙx [Xt ∈ dy] = pt (x, y) dm(y)

for any (t, x) ∈ (0, ∞) × K

(4.4)

for some m-symmetric diffusion X = (Ω, M, {Xt }t∈[0,∞] , {ℙx }x∈KΔ ) without killing inside on (K, B(K)) whose Dirichlet form is (ℰ , ℱ ), and there exist c1 , c2 , c3 , c4 ∈ (0, ∞) such that for any (t, x, y) ∈ (0, ∞) × K × K, c1 exp(−c2 tΦ(d(x, y)/t)) m(B(x, Ψ−1 (t)))

≤ pt (x, y) ≤

c3 exp(−c4 tΦ(d(x, y)/t)) m(B(x, Ψ−1 (t)))

,

(4.5)

where Φ(s) := ΦΨ (s) := supr∈(0,∞) (s/r −1/Ψ(r)) ∈ [0, ∞) for each s ∈ [0, ∞). If Ψ(r) = r β β

β

for any r ∈ [0, ∞) for some β ∈ (1, ∞), then Φ(s) = β β−1 (β − 1)s β−1 for any s ∈ [0, ∞), and (fHKE)Ψ with this function Ψ is written as (fHKE)β . −

ℰ Definition 4.9. Let (K, d, m, ℰ , ℱ ) be an MMD space. We define its intrinsic metric dm : K × K → [0, ∞] by

󵄨 ℰ dm (x, y) := sup{u(x) − u(y) 󵄨󵄨󵄨 u ∈ ℱloc ∩ 𝒞 (K), μ⟨u⟩ ≤ m},

(4.6)

󵄨󵄨 󵄨 u is an m-equivalence class of ℝ-valued Borel measurable } { { 󵄨󵄨󵄨 } := {u 󵄨󵄨󵄨 functions on K such that u1V = u# 1V m-a. e. for some } { 󵄨󵄨 # } 󵄨󵄨 u ∈ ℱ for each relatively compact open subset V of K 󵄨 { }

(4.7)

where ℱloc

and the ℰ -energy measure μ⟨u⟩ of u ∈ ℱloc is defined as the unique Borel measure on K such that μ⟨u⟩ (A) = μ⟨u# ⟩ (A) for any relatively compact A ∈ B(K) and any V, u# as in (4.7) with A ⊂ V; note that μ⟨u# ⟩ (A) is independent of a particular choice of such V, u# by [12, Lemmas 2.1.4 and 3.2.4] and Corollary 2.5-(2). Now we can state a slight simplification of the main theorem [23, Theorem 2.13] of [23] by combining it with some known implications of (fHKE)Ψ in [36, Theorem 2.11] and [5, 6, 1, 13, 8, 33], as follows. Theorem 4.10 (A simplification of [23, Theorem 2.13]). Let Ψ : [0, ∞) → [0, ∞) be a homeomorphism satisfying (4.3), let (K, d, m, ℰ , ℱ ) be an MMD space satisfying (fHKE)Ψ , and assume that (K, d) is complete. (1) (Singularity) If lim inf lim inf λ→∞

r↓0

λ2 Ψ(r/λ) = 0, Ψ(r)

ℰ then μ⟨u⟩ ⊥ m for any u ∈ ℱ . In this case, dm (x, y) = 0 for any x, y ∈ K.

(4.8)

134 | N. Kajino (2) (Absolute continuity) If lim sup r↓0

Ψ(r) > 0, r2

(4.9)

then {A ∈ B(K) | m(A) = 0} = {A ∈ B(K) | supu∈ℱ μ⟨u⟩ (A) = 0}, and in particular ℰ μ⟨u⟩ ≪ m for any u ∈ ℱ . In this case, dm is a geodesic metric on K, i. e., a metric on K admitting for any x, y ∈ K some γ : [0, 1] → K such that γ(0) = x, γ(1) = y and ℰ ℰ dm (γ(s), γ(t)) = |s − t|dm (x, y) for any s, t ∈ [0, 1], and there exist r1 ∈ (0, diam(K, d)) and c5 , c6 ∈ [1, ∞) such that c5−1 r 2 ≤ Ψ(r) ≤ c5 r 2

for any r ∈ (0, r1 ),

ℰ c6−1 d(x, y) ≤ dm (x, y) ≤ c6 d(x, y)

for any x, y ∈ K.

(4.10) (4.11)

If Ψ(r) = r β for any r ∈ [0, ∞) for some β ∈ (1, ∞), then (4.8) is equivalent to β > 2 and (4.9) is equivalent to β ≤ 2. In particular, Theorem 4.10-(1) recovers Theorems 4.1 and 4.4 as special cases by virtue of Theorems 3.3, 3.8 and 3.9. On the other hand, for general Ψ, the conditions (4.8) and (4.9) are not complementary to each other since there are examples of Ψ satisfying (4.3) but not either of (4.8) and (4.9); indeed, Ψ(r) = r 2 / log(e − 1 + r −1 ) is such an example. In view of Theorem 4.10, one might expect that the following conjecture would hold. Conjecture 4.11 (Energy measure singularity dichotomy; a simplification of [23, Conjecture 2.15]). Let Ψ : [0, ∞) → [0, ∞) be a homeomorphism satisfying (4.3), let (K, d, m, ℰ , ℱ ) be an MMD space satisfying (fHKE)Ψ , and assume that (K, d) is complete and that lim r↓0

Ψ(r) = 0. r2

(4.12)

Then μ⟨u⟩ ⊥ m for any u ∈ ℱ . There is a class of examples of MMD spaces which supports Conjecture 4.11; see [23, Remark 2.14] and the references therein for details.

5 Gaussian heat kernel estimates via time changes by energy measures Let (K, d, m, ℰ , ℱ ) be an MMD space with (K, d) complete. Theorem 4.10-(2) implies in particular that (fHKE)2 , the so-called Gaussian estimates, hold only if d is comparable ℰ to dm and m is equivalent to the ℰ -energy measures {μ⟨u⟩ }u∈ℱ in the sense that {A ∈ B(K) | m(A) = 0} = {A ∈ B(K) | supu∈ℱ μ⟨u⟩ (A) = 0}. Then it is natural to ask, in

Energy measures for diffusions on fractals: a survey | 135

the case when (K, d, m, ℰ , ℱ ) is “nicely fractal” in the sense that it satisfies (fHKE)β for some β ∈ (2, ∞), whether (K, d, m, ℰ , ℱ ) admits a Radon measure with full support μ on K equivalent to the ℰ -energy measures such that (K, dμℰ , μ, ℰ , ℱ μ ) satisfies (fHKE)2 .2 This problem was studied first in the case of the two-dimensional Sierpiński gasket by Kigami [26], where he proved the following theorem. Theorem 5.1 ([26]). Let V0 , (K, d, m, ℰ , ℱ ) be as in Subsection 3.1 with n = 2. Let h1 , h2 ∈ ℱ be ℰ -harmonic on K \ V0 and satisfy ℰ (h1 , h1 ) = ℰ (h2 , h2 ) = 1 and ℰ (h1 , h2 ) = 0, set μh1 ,h2 := μ⟨h1 ⟩ + μ⟨h2 ⟩ , and define dh1 ,h2 : K × K → [0, ∞] by 󵄨󵄨 󵄨 γ : [0, 1] → K, γ is continuous, dh1 ,h2 (x, y) := inf {ℓℝ2 ((h1 , h2 ) ∘ γ) 󵄨󵄨󵄨󵄨 }, 󵄨󵄨 γ(0) = x, γ(1) = y

(5.1)

where (h1 , h2 ) : K → ℝ2 is defined by (h1 , h2 )(x) := (h1 (x), h2 (x)) and ℓℝ2 ((h1 , h2 ) ∘ γ) denotes the length of (h1 , h2 ) ∘ γ with respect to the Euclidean norm | ⋅ |. Then μh1 ,h2 is a Radon measure on K with full support, dh1 ,h2 is a geodesic metric on K compatible with the topology of (K, d), and (K, dh1 ,h2 , μh1 ,h2 , ℰ , ℱ ) satisfies (fHKE)2 . The author studied the situation of Theorem 5.1 further in [19] and obtained the following results; see also [20] for a more thorough survey on this example. Theorem 5.2 ([19, Theorem 4.2-(1)]). In the situation of Theorem 5.1, the metric dh1 ,h2 coincides with the intrinsic metric dμℰh ,h of (K, dh1 ,h2 , μh1 ,h2 , ℰ , ℱ ). 1 2

Theorem 5.3 ([19, Theorem 4.2-(2) and Corollary 4.3]). Theorems 5.1 and 5.2 remain valid if h2 is chosen to be a constant function on K instead. The crucial observation made by Kigami in [26] for the proof of Theorem 5.1 is that μh1 ,h2 and dh1 ,h2 possess the following properties with respect to the Euclidean metric d on K. Definition 5.4. Let (K, d) be a metric space. 2 ℱ μ denotes the completion of ℱ ∩ 𝒞c (K) with respect to the inner product ℰ + ⟨⋅, ⋅⟩L2 (K,μ) , which

turns out to be canonically embedded in L2 (K, μ) so that (K, d, μ, ℰ, ℱ μ ) is an MMD space, and which can be identified as {̃ u | u ∈ ℱ } ∩ L2 (K, μ) if K is compact. We remark that the operation of replacing the reference measure m for the Dirichlet form (ℰ, ℱ ) with another Radon measure ν on K, which is ̃ to be uniquely deterrequired to satisfy ν(N) = 0 for any N ∈ B(K) with capacity zero in order for u mined ν-a. e. for each u ∈ ℱ , is called the time change of the MMD space (K, d, m, ℰ, ℱ ) by ν. The reason for this term is that the stochastic counterpart of this operation is considering the ν-symmetric (rightcontinuous strong) Markov process {Xtν }t∈[0,∞] given as a random time change Xtν := Xτt of {Xt }t∈[0,∞] by τt := inf{s ∈ [0, ∞) | As > t}; here X = (Ω, M, {Xt }t∈[0,∞] , {ℙx }x∈KΔ ) is an m-symmetric diffusion without killing inside on (K, B(K)) whose Dirichlet form is (ℰ, ℱ ), and A = {At }t∈[0,∞) is a PCAF of X with Revuz measure μA = ν, which exists by [12, Lemma 5.1.8] and is unique up to the equivalence ∼A by [12, Theorem 5.1.3]. We refer the reader to [12, Section 6.2] and [11, Chapter 5] for further details.

136 | N. Kajino (1) Let ν be a Borel measure on K. We say that ν is volume doubling with respect to d, or (VD)d in abbreviation, if and only if there exists cvd ∈ (0, ∞) such that 0 < ν(B(x, 2r)) ≤ cvd ν(B(x, r)) < ∞

for any (x, r) ∈ K × (0, ∞).

(5.2)

qs

(2) We call a metric ρ on K quasisymmetric to d, and write ρ ∼ d, if and only if there exists a homeomorphism η : [0, ∞) → [0, ∞) such that ρ(x, y) d(x, y) ≤ η( ) ρ(x, z) d(x, z)

for any x, y, z ∈ K with x ≠ z.

(5.3)

qs

Note that ∼ is an equivalence relation on the set of metrics on K, and that any qs metric ρ on K with ρ ∼ d is compatible with the topology of (K, d) and has the property that ν being (VD)ρ is equivalent to ν being (VD)d for any Borel measure ν on K; see, e. g., [28, Chapter 12] and [35, Lemma 1.2.18] for details. Theorem 5.5 ([26, 19], cf. [29]). Both in the situation of Theorem 5.1 and in that of Theorem 5.3, the following hold: (1) μh1 ,h2 is (VD)d . qs

(2) dh1 ,h2 ∼ d. In particular, μh1 ,h2 is (VD)dh ,h . 1 2

Proof. (1) This is [26, Theorem 3.2] in the situation of Theorem 5.1, and also in that of Theorem 5.3 this follows from [26, Proof of Theorem 3.2] on the basis of [19, Lemma 3.9] and [27, Theorem 1.3.5]. (2) This follows from [29, Theorem 13.6], whose assumptions can be easily verified, via [29, Proposition 6.8], on the basis of [26, Lemma 3.5, Proof of Theorem 3.2, and Theorem 5.11] in the situation of Theorem 5.1 and [19, Lemma 3.9 and Proposition 3.16-(1)] in the situation of Theorem 5.3. In [24], Mathav Murugan and the author have recently considered the problem of determining whether the same kind of results as Theorems 5.1, 5.3 and 5.5 can be obtained for (K, dμℰ , μ, ℰ , ℱ μ ) by taking a suitable Radon measure with full support μ on K equivalent to the ℰ -energy measures, for a general self-similar Dirichlet form (ℰ , ℱ ) on a self-similar set K. In this framework, it turns out that the existence of such μ implies the existence of h ∈ ℱ which is ℰ -harmonic on K \ V0 and whose ℰ -energy measure μ⟨h⟩ is another such μ. Namely we have the following theorem, which we state here only for the examples in Section 3 for simplicity. Theorem 5.6 (Special cases of [24, Subsections 6.2 and 6.4]). Let V0 , (K, d, m, ℰ , ℱ ) be either as in Subsection 3.1 or as in Subsection 3.2, and suppose that there exists a Radon measure μ on K such that {A ∈ B(K) | μ(A) = 0} = {A ∈ B(K) | supu∈ℱ μ⟨u⟩ (A) = qs

0}, dμℰ is a metric on K with dμℰ ∼ d and (K, dμℰ , μ, ℰ , ℱ μ ) satisfies (fHKE)2 . Then there

Energy measures for diffusions on fractals: a survey | 137

exists h ∈ ℱ which is ℰ -harmonic on K \ V0 such that μ ≪ μ⟨h⟩ , dμℰ⟨h⟩ is a metric on K with qs

dμℰ⟨h⟩ ∼ d and (K, dμℰ⟨h⟩ , μ⟨h⟩ , ℰ , ℱ μ⟨h⟩ ) satisfies (fHKE)2 .

In the case of the n-dimensional Sierpiński gasket with n ≥ 3, it turns out further that the conclusion of Theorem 5.6 does not hold and hence neither does the existence of μ as in Theorem 5.6. More precisely, while Theorem 5.5-(1) still holds, Theorem 5.5-(2) fails to hold, as stated in the following theorems. Theorem 5.7 (cf. [26, Theorem 3.2]). Let V0 , (K, d, m, ℰ , ℱ ) be as in Subsection 3.1 and let h ∈ ℱ \ ℝ1K be ℰ -harmonic on K \ V0 . Then μ⟨h⟩ is (VD)d . Proof. This can be proved in essentially the same way as [26, Proof of Theorem 3.2], which proves instead that ∑nk=1 μ⟨hk ⟩ is (VD)d for {hk }nk=1 ⊂ ℱ such that hk is ℰ -harmonic on K \ V0 , ℰ (hk , hk ) = 1 and ℰ (hj , hk ) = 0 for any j, k ∈ {1, . . . , n} with j ≠ k; a detailed proof of this fact will appear in [24, Subsubsection 6.3.2]. Theorem 5.8 ([24, Subsubsection 6.3.2]). Let V0 , (K, d, m, ℰ , ℱ ) be as in Subsection 3.1 with n ≥ 3 and let h ∈ ℱ be ℰ -harmonic on K \ V0 . Then either dμℰ⟨h⟩ is a metric on K which qs

does not satisfy dμℰ⟨h⟩ ∼ d, or dμℰ⟨h⟩ is not a metric on K.

Combining Theorem 5.8 with Theorem 5.6, we arrive at the following theorem. Theorem 5.9 ([24, Subsubsection 6.3.2]). Let V0 , (K, d, m, ℰ , ℱ ) be as in Subsection 3.1 with n ≥ 3. Then there does not exist a Radon measure μ on K such that {A ∈ B(K) | qs

μ(A) = 0} = {A ∈ B(K) | supu∈ℱ μ⟨u⟩ (A) = 0}, dμℰ is a metric on K with dμℰ ∼ d and (K, dμℰ , μ, ℰ , ℱ ) satisfies (fHKE)2 .3

For generalized Sierpiński carpets, very little is known beyond Theorem 5.6. The author has still observed in the case of the Sierpiński carpet GSC(2, 3, SSC ), SSC := {0, 1, 2}2 \ {(1, 1)}, that any h ∈ ℱ with h|[0,1]×{0} = 0 that is ℰ -harmonic on K \ V0 satisfies dμℰ⟨h⟩ (x, y) = 0 for any x, y ∈ [0, 1] × {0} and hence in particular that dμℰ⟨h⟩ fails to be a metric on K for any such h. In view of this observation and Theorem 5.6, one might expect that the following conjecture would also hold. Conjecture 5.10. Let V0 , (K, d, m, ℰ , ℱ ) be as in Subsection 3.2. Then there does not exist a Radon measure μ on K such that {A ∈ B(K) | μ(A) = 0} = {A ∈ B(K) | supu∈ℱ μ⟨u⟩ (A) = qs

0}, dμℰ is a metric on K with dμℰ ∼ d and (K, dμℰ , μ, ℰ , ℱ μ ) satisfies (fHKE)2 .

6 Martingale dimension In this last section, we introduce the notion of martingale dimension (or the index) of an MMD space, which could be considered as the maximum dimension of the “tan3 Note that in this case ℱ μ = ℱ by ℱ ⊂ 𝒞(K).

138 | N. Kajino gent space structure” induced by the Dirichlet form, and survey the known results on the problem of determining this dimension for Dirichlet forms on self-similar fractals. These notions were formulated first by Kusuoka [30, 31] for Dirichlet forms on selfsimilar sets and have been extended to arbitrary MMD spaces by Hino [16] in the form presented below. Throughout this section, (K, d, m, ℰ , ℱ ) denotes an arbitrary MMD space and X = (Ω, M, {Xt }t∈[0,∞] , {ℙx }x∈KΔ ) is an m-symmetric diffusion without killing inside on (K, B(K)) whose Dirichlet form is (ℰ , ℱ ), unless otherwise stated. We start with the definition of the index of (K, d, m, ℰ , ℱ ), which is stated solely in terms of the mutual ℰ -energy measures as follows. Definition 6.1 ([16, Definition 2.9]). We define indℰ ∈ ℕ ∪ {0, ∞} by n 󵄨󵄨 dμ⟨ui ,uj ⟩ 󵄨󵄨 } { 󵄨 { 󵄨󵄨 rank( ) ≤ p, ∑nk=1 μ⟨uk ⟩ -a. e., } } { d ∑nk=1 μ⟨uk ⟩ i,j=1 indℰ := inf {p ∈ ℕ ∪ {0} 󵄨󵄨󵄨󵄨 } } { 󵄨󵄨 } { 󵄨󵄨 n 󵄨 for any n ∈ ℕ and any {u } ⊂ ℱ } { 󵄨 k k=1

and call indℰ the index of (K, d, m, ℰ , ℱ ).

(6.1)

n

Note that the symmetric matrix (dμ⟨ui ,uj ⟩ /d ∑nk=1 μ⟨uk ⟩ )i,j=1 appearing in (6.1) could be considered as the “matrix representation of the inner product on the gradient vector fields of {uk }nk=1 ”, which allows us to interpret indℰ as the maximum dimension of the “tangent space structure” on K induced by (ℰ , ℱ ). We next give the definition of the martingale dimension of X, which requires some definitions concerning martingale additive functionals (MAFs) of X. Recall the definition (2.8) of the space ℳ of MAFs of X, that of the energy eA (M) of M ∈ ℳ given by eA (M) := limt↓0 (2t)−1 𝔼m [Mt2 ] = supt∈(0,∞) (2t)−1 𝔼m [Mt2 ], and that we set ℳ∘ := {M ∈ ℳ | eA (M) < ∞}. Then ℳ∘ clearly forms a real linear space, the mutual energy eA (M, L) of M = {Mt }t∈[0,∞) , L = {Lt }t∈[0,∞) ∈ ℳ∘ can be defined by eA (M, L) := 1 (e (M + L) − eA (M − L)) = limt↓0 (2t)−1 𝔼m [Mt Lt ], and (ℳ∘ , eA ) is a real Hilbert space 4 A by [12, Theorem 5.2.1]. On the other hand, for each M, L ∈ ℳ∘ we can define a finite CAF ⟨M, L⟩ = {⟨M, L⟩t }t∈[0,∞) of X by ⟨M, L⟩ := 41 (⟨M + N⟩ − ⟨M − L⟩) and a Borel signed measure μ⟨M,L⟩ on K by μ⟨M,L⟩ := 41 (μ⟨M+N⟩ − μ⟨M−L⟩ ), so that μ⟨M,L⟩ is the unique Borel signed measure on K satisfying (2.7) with A = ⟨M, L⟩ for any Borel measurable function f : K → ℝ with ‖f ‖sup < ∞. Then ⟨M, M⟩ = ⟨M⟩ and μ⟨M,M⟩ = μ⟨M⟩ for any M ∈ ℳ∘ , ℳ∘ × ℳ∘ ∋ (M, L) 󳨃→ ⟨M, L⟩ is bilinear and symmetric and hence so is ℳ∘ × ℳ∘ ∋ (M, L) 󳨃→ μ⟨M,L⟩ , which together imply (see [12, Lemma 5.6.1]) that 2

(∫ |fg| d|μ⟨M,L⟩ |) ≤ ∫ f 2 dμ⟨M⟩ ∫ g 2 dμ⟨L⟩ K

K

(6.2)

K

for any M, L ∈ ℳ∘ and any Borel measurable functions f , g : K → [−∞, ∞], where |μ⟨M,L⟩ | denotes the total variation measure of μ⟨M,L⟩ . Now since μ⟨L⟩ (K) =

Energy measures for diffusions on fractals: a survey | 139

limt↓0 t −1 𝔼m [⟨L⟩t ] = limt↓0 t −1 𝔼m [L2t ] = 2eA (L) for any L = {Lt }t∈[0,∞) ∈ ℳ by (2.7), it follows from (6.2) that each M ∈ ℳ∘ and each f ∈ L2 (K, μ⟨M⟩ ) define a bounded linear functional L 󳨃→ ∫K f dμ⟨M,L⟩ on the Hilbert space (ℳ∘ , eA ) and hence (see [12, Theorem 5.6.1]) yield a unique f ∙ M ∈ ℳ∘ such that eA (f ∙ M, L) =

1 ∫ f dμ⟨M,L⟩ 2

for any L ∈ ℳ∘ .

(6.3)

K

The MAF f ∙ M is called the stochastic integral of f with respect to M, which by [12, Lemma 5.6.2] satisfies dμ⟨f ∙M,L⟩ = f dμ⟨M,L⟩ for any L ∈ ℳ∘ and, if f ∈ 𝒞c (K), (f ∙ M)t = t

t

∫0 f (Xs ) dMs for any t ∈ [0, ∞) ℙx -a. s. for ℰ -q. e. x ∈ K, where {∫0 f (Xs ) dMs }t∈[0,∞) denotes the usual Itô-type stochastic integral. Now we can give the definition of the martingale dimension of X as follows. Definition 6.2 ([16, Definition 3.4]). We define dimM X ∈ ℕ ∪ {0, ∞} by 󵄨󵄨 p 󵄨󵄨 there exists {M (k) }k=1 ⊂ ℳ∘ such that } { 󵄨󵄨 { } { } 󵄨 dimM X := inf {p ∈ ℕ ∪ {0} 󵄨󵄨󵄨󵄨 the map (hk )pk=1 󳨃→ ∑pk=1 hk ∙ M (k) from } { } 󵄨󵄨 { } 󵄨󵄨 p 2 󵄨󵄨 ∏k=1 L (K, μ⟨M (k) ⟩ ) to ℳ∘ is surjective } {

(6.4)

and call dimM X the martingale dimension of X. In view of (6.4), dimM X can be considered as the maximum number of independent components of random noise involved in the diffusion X. A fundamental fact concerning dimM X is that it is analytically characterized as the index indℰ of (K, d, m, ℰ , ℱ ) as stated in the following theorem, which has been established by Hino [16] as a generalization of Kusuoka’s results formulated and proved for Dirichlet forms on self-similar sets in [30, 31]. Theorem 6.3 ([16, Theorem 3.4], cf. [30, Theorem 5.4], [31, Theorem 6.12]). The martingale dimension dimM X of X is equal to the index indℰ of (K, d, m, ℰ , ℱ ): dimM X = indℰ .

(6.5)

For symmetric diffusions on smooth spaces, the value of the index is easily identified as the dimension of the tangent space or the maximum rank of the diffusion matrix. For example, the index of the MMD space (ℝn , dEuc , dx, 21 D, H 1 (ℝn )) in Example 2.2 is n since rank(⟨∇ui , ∇uj ⟩)ki,j=1 ≤ n dx-a. e. for any k ∈ ℕ and any {uj }kj=1 ⊂ H 1 (ℝn ) with the equality holding when k = n and uj is (locally) the jth coordinate function. On the other hand, it is highly non-trivial to identify the value of the index indℰ for Dirichlet forms on fractals, since there is no simple expression of the associated energy measures. This problem was also studied first by Kusuoka in [30], where he proved the following theorem asserting that the index is one for the standard resistance form on the n-dimensional Sierpiński gasket for any n ≥ 2 even though its Hausdorff dimension df = log2 (n + 1) becomes arbitrarily large as n → ∞.

140 | N. Kajino Theorem 6.4 ([30, Example 1]). Let (K, d, m, ℰ , ℱ ) be as in Subsection 3.1. Then indℰ = 1. Kusuoka conjectured in [30, Example 2] that the martingale dimension is one also for the canonical diffusions on nested fractals, a large class of post-critically finite selfsimilar sets with good Euclidean-geometric symmetry, introduced by Lindstrøm [34] as the framework of his construction of canonical diffusions on self-similar sets. Of course the problem of identifying the index can be considered for the canonical Dirichlet forms on generalized Sierpiński carpets as well, and no progress had been made in either of these two cases for almost two decades since Kusuoka’s work [30]. The situation changed through the intensive studies on this problem by Hino in [15, 17]. First, he proved in [15, Theorem 4.4] that the index is one for the selfsimilar Dirichlet form (ℰ , ℱ ), constructed in the same way as in Subsection 3.1 from a pair (ℰ (0) , (ri )i∈S ) satisfying (3.3) and (ri )i∈S ∈ (0, 1)S ,4 on any post-critically finite self-similar set (K, S, {Fi }i∈S ) satisfying the technical condition that Fi (q) = q for some i ∈ S

and K \ {q} is connected for each q ∈ V0 .

(6.6)

This framework covers the canonical diffusions on nested fractals constructed by Lindstrøm [34], whose martingale dimension is thus, as Kusuoka conjectured, one. In his later paper [17], Hino established a more sophisticated method of bounding the index from above, thereby improved [15, Theorem 4.4] by dropping the technical condition (6.6) ([17, Theorem 4.10]) and proved further the following theorem for generalized Sierpiński carpets. Theorem 6.5 ([17, Theorem 4.16]). Let S, df , (K, d, m, ℰ , ℱ ), r, dw be as in Subsection 3.2 and set ds := 2df /dw = 2 log#S/r #S. Then 1 ≤ indℰ ≤ ds .

(6.7)

In particular, if ds < 2(, which holds if df ≤ 2 by Theorem 3.8), then indℰ = 1. Note that ds < 2 is equivalent to r < 1 in the setting of Theorem 6.5 and that this is analogous to the assumption that (ri )i∈S ∈ (0, 1)S in Hino’s identification of the index as one for connected post-critically finite self-similar sets in [15, Theorem 4.4] and [17, Theorem 4.10]. On the other hand, nothing more than (6.7) is known concerning the value of the index indℰ for generalized Sierpiński carpets with ds ≥ 2. For example, in the case of the Menger sponge GSC(3, 3, SMS ) (see Figure 2 above), we have 2 < ds < df < 3 by [4, (9.2)] and Theorem 3.8 and hence indℰ ∈ {1, 2} by (6.7), but it is not known whether indℰ = 1 or indℰ = 2. 4 As defined in [25, Definition 3.1.2], such a pair (ℰ (0) , (ri )i∈S ) is called a regular harmonic structure on (K, S, {Fi }i∈S ); see [25, Chapter 3] for the details of the construction of (ℰ, ℱ ).

Energy measures for diffusions on fractals: a survey | 141

Open Problem 6.6. Let S, df , (K, d, m, ℰ , ℱ ), r, dw be as in Subsection 3.2 and suppose that ds := 2df /dw = 2 log#S/r #S ≥ 2, or equivalently, r ≥ 1. Determine the value of indℰ , or provide any lower bound on indℰ better than indℰ ≥ 1.

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Shi-Lei Kong, Ka-Sing Lau, Jun Jason Luo, and Xiang-Yang Wang

Hyperbolic graphs induced by iterations and applications in fractals

Abstract: The paper is a survey of our recent and ongoing investigation on the class of Gromov hyperbolic graphs arising from iterated function systems (IFS) in the theory of fractals. The relations of the hyperbolic boundaries and the attractors of IFSs are discussed. The applications include Lipschitz equivalence of attractors, as well as the discrete potential theory of random walks on such graphs. Keywords: Hyperbolic graph, hyperbolic boundary, Lipschitz equivalence, random walk, self-similar set MSC 2010: Primary 28A80, Secondary 05C63, 60J15 Contents 1 2 3 4 5 6 7

Introduction | 143 Hyperbolic graphs | 144 IFS and augmented trees | 149 Lipschitz equivalence of self-similar sets | 155 Random walks on augmented trees | 162 Expansive hyperbolic graphs | 171 Remarks and future work | 177 Bibliography | 178

1 Introduction With the intension to carry the probabilistic potential theory to the attractors of iterated function systems (IFS), Denker and Sato [10] first constructed a special type of Acknowledgement: The research was supported in part by SFB 1283 of the German Research Council, NSFC 11971500, HKRGC grant and Natural Science Foundation of Chongqing (No. cstc2019jcyjmsxmX0030). Shi-Lei Kong, Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany, e-mail: [email protected] Ka-Sing Lau, Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong; and Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA (current address), e-mail: [email protected] Jun Jason Luo, College of Mathematics and Statistics, Chongqing University, 401331 Chongqing, China, e-mail: [email protected] Xiang-Yang Wang, School of Mathematics, Sun Yat-Sen University, Guangzhou, China, e-mail: [email protected] https://doi.org/10.1515/9783110700763-006

144 | S.-L. Kong et al. (nonreversible) Markov chain {Zn }∞ n=0 on the tree of symbolic space of the Sierpinski gasket (SG), and showed that the Martin boundary of {Zn }∞ n=0 is homeomorphic to the SG. Motivated by this, Kaimanovich [30] introduced an “augmented tree” by adding “horizontal” edges to the coding tree according to the neighboring cells in each level of the SG, which was proved to be a hyperbolic graph in the sense of Gromov [23]. These ideas turned out to be very inspiring and far reaching in view of the very rich contents in the involved topics. In a series of papers, the initiatives were carried out and investigated in detail by the authors and their collaborators [9, 29, 37, 38, 39, 36, 42, 45, 46, 47, 48, 49, 50, 51, 52, 58]. Related literature includes [2, 3, 4, 5, 6, 12, 15, 31, 33, 35, 34, 59]. In this paper, we attempt to give a brief account of our decade-long investigation, as well as some ongoing works on the project of hyperbolic graphs on fractals. In Section 2, we recall some basic definitions for hyperbolic graphs, and characterize the hyperbolicity for the class of graphs in our consideration. In Section 3, we introduce the “augmented tree” for an iterated function system (IFS). We study its hyperbolicity using the criterion obtained in Section 2, and the Hölder equivalence of the hyperbolic boundary with the attractor of the IFS, in particular for the IFS of contractive similitudes. This is used to study the bi-Lipschitz equivalence of some totally disconnected self-similar sets in Section 4. In Section 5, we consider certain reversible random walk on the augment tree of the IFS of contractive similitudes. The Martin boundary, the hyperbolic boundary, and the attractor are shown to be homeomorphic; the Martin kernel, the Naïm kernel, and the induced energy form on the attractor are analyzed. In Section 6, we define a class of expansive hyperbolic graphs and a concept of index map. These unify various formulations of augmented trees, and include cases that are not governed by the IFS, such as refinement systems. Many of the properties studied in Section 3 are extended in this new setting. In Section 7, we provide some concluding remarks and future work of this study.

2 Hyperbolic graphs We first define some basic notations for a graph. Let X be a countably infinite set. A (undirected) graph is a pair (X, ℰ ), where ℰ is a symmetric subset of X × X \ {(x, x) : x ∈ X}. We call x ∈ X a vertex and (x, y) ∈ ℰ (also denoted by x ∼ y) an edge. The degree of a vertex x is the total number of edges which connect to x and is denoted by deg(x). Throughout the paper, we assume that the graph is locally finite, i. e., deg(x) < ∞ for all x ∈ X. For x, y ∈ X, a path from x to y is a finite sequence {x0 , x1 , . . . , xn } such that x = x0 , xn = y and (xi , xi+1 ) ∈ ℰ , denoted by p(x, y); we call n = |p(x, y)| the length of the path ({x} is a path with length 0 by convention). Moreover, if the above path has minimal length among all possible paths from x to y, then we say that the path is a geodesic and denote it by π(x, y). We always assume that the graph is connected, that is, for any pair x, y ∈ X, there exists a path from x to y. Hence a graph induces

Hyperbolic graphs induced by iterations and applications in fractals | 145

an integer-valued metric d(x, y) on X, which is the length of geodesic π(x, y) from x to y. Choose a reference vertex ϑ ∈ X and call it the root of the graph. We use |x| to denote d(ϑ, x). We can decompose ℰ as ℰ = ℰh ∪ ℰv where ℰv = {(x, y) ∈ ℰ : |x| − |y| = ±1},

ℰh = {(x, y) ∈ ℰ : |x| = |y|}.

An edge (x, y) in ℰv is called a vertical edge (x ∼v y), and an edge in ℰh is called a horizontal edge; the notation x ∼h y means that (x, y) ∈ ℰh or x = y; we also use dh (x, y) to denote the graph distance of the subgraph (X, ℰh ) (if there are no horizontal paths joining x, y, we let dh (x, y) = ∞ by convention). We also decompose the vertex set X as X = ⋃∞ n=0 Xn , where the nth level Xn = {x ∈ X : |x| = n}. By the local finiteness, it is easy to show (by induction) that Xn is a finite set for all n ≥ 0. A (geodesic) ray π(x0 , x1 , . . . ) is an infinite sequence with x0 = ϑ, xn ∈ Xn and (xn , xn+1 ) ∈ ℰv for all n ≥ 0. For x, y ∈ X with |y| − |x| =: m > 0, we say y is an mth descendant of x, or x is an mth predecessor of y, if there exists a ray π(ϑ, . . . , x, . . . , y, . . . ) connecting them. Denote by 𝒥m (x) and 𝒥−m (x) the sets of the mth descendants and predecessors of x, respectively, and let 𝒥∗ (x) = ⋃ 𝒥m (x), m≥1

𝒥−∗ (x) = ⋃ 𝒥−m (x). m≥1

Throughout, we assume that 𝒥1 (x) ≠ 0 for all x ∈ X. We call (X, ℰ ) a tree if ℰ = ℰv and 𝒥−1 (x) is a singleton for all x ∈ X \ {ϑ}. Definition 2.1 ([23]). Let (X, ℰ ) be a graph with a root ϑ ∈ X. Define the Gromov product of two vertices x, y ∈ X by 1 (x|y) = (|x| + |y| − d(x, y)). 2 For δ ≥ 0, we say that (X, ℰ ) is δ-hyperbolic (with respect to ϑ) if (x|y) ≥ min{(x|z), (z|y)} − δ,

∀x, y, z ∈ X.

(2.1)

Also (X, ℰ ) is said to be hyperbolic if it is δ-hyperbolic for some δ ≥ 0. The following justifies the definition of hyperbolicity [7, 60]. Proposition 2.2. If X is δ-hyperbolic with respect to a particular ϑ ∈ X, then it is 2δ-hyperbolic for any other fixed root ϑ󸀠 ∈ X. Hence the hyperbolicity is independent of the choice of the root. It is easy to see that a tree is 0-hyperbolic and (x|y) is the distance from ϑ to z, the confluence of x and y. For a δ-hyperbolic graph, the Gromov product (x|y) is roughly the distance from ϑ to π(x, y) in the following sense: d(ϑ, π(x, y)) − 2δ −

1 ≤ (x|y) ≤ d(ϑ, π(x, y)). 2

(2.2)

146 | S.-L. Kong et al.

Figure 1: A δ-thin geodesic triangle.

Indeed, the second inequality is trivial without requiring the hyperbolicity. For the first inequality, using (2.1) for z ∈ π(x, y) we have 1 1 (x|y) ≥ min{(x|z), (z|y)} − δ = |z| + min{|x| − d(x, z), |y| − d(y, z)} − δ. 2 2 Note that the two terms in min{⋅ ⋅ ⋅} have the sum |x| + |y| − d(x, y) = 2(x|y), therefore we can choose z such that |x| − d(x, z) = |y| − d(y, z) = (x|y) if (x|y) is an integer; and |x| − d(x, z) = |y| − d(y, z) − 1 = (x|y) − 21 otherwise. It follows that 1 1 1 (x|y) ≥ |z| + ((x|y) − ) − δ, 2 2 2 which implies (x|y) ≥ |z| − 2δ − 21 ≥ d(ϑ, π(x, y)) − 2δ − 21 . Note that in inequality (2.2), we cannot omit − 21 : consider the simplest example, a triangle with X = {ϑ, x, y}. Since (ϑ|x) = (ϑ|y) = 0 and (x|y) = 21 , (2.1) holds for δ = 0, hence it is 0-hyperbolic. Note that d(ϑ, π(x, y)) = 1, thus d(ϑ, π(x, y)) − 2δ − 21 = (x|y). (In [60, Lemma 22.4], the − 21 is missing in his estimate of (2.2).) It is instructive to know that the notion of hyperbolicity is motivated by the “thin triangles” in the Poincaré disc model: a geodesic triangle in (X, ℰ ) consists of three points x, y, z ∈ X as vertices together with the three geodesic arcs π(x, y), π(y, z), π(z, x) as sides; the triangle is called δ-thin if every point on any one of the sides is at a distance at most δ to one of the other two sides (see Figure 1). The following is the geometric characterization of hyperbolicity. Proposition 2.3 ([7, 17, 60]). In a δ-hyperbolic graph (X, ℰ ), every geodesic triangle is 8δ-thin. Conversely, if every geodesic triangle in (X, ℰ ) is δ󸀠 -thin, then (X, ℰ ) is (3δ󸀠 + 21 )-hyperbolic. For a > 0 small (say eδa < √2), and x, y ∈ X, let θa (x, y) = {

e−a(x|y) , 0,

x ≠ y; x = y.

(2.3)

Then by (2.1), θa (x, y) ≤ eδa max{θa (x, z), θa (z, y)} for all x, y, z ∈ X. It is known that θa (⋅, ⋅) is not necessarily a metric (unless δ = 0), but is always Lipschitz equivalent to

Hyperbolic graphs induced by iterations and applications in fractals | 147

a metric ρa (⋅, ⋅) on X (i. e., C1−1 ρa ≤ θa ≤ C1 ρa holds for some constant C1 ≥ 1). Hence we can regard θa as a metric for convenience, and call it a Gromov metric. Note that if we choose another b > 0 for the Gromov metric, then θa = θba/b . By (2.3), it is clear that a sequence {xn }∞ n=0 in X is a θa -Cauchy sequence if and only if (xm |xn ) → ∞ as m, n → ∞. Definition 2.4. Denote by X̂ the θa -completion of a hyperbolic graph X. We call 𝜕X = X̂ \ X the hyperbolic boundary of X. The hyperbolic boundary 𝜕X is a compact set. It is useful to identify ξ ∈ 𝜕X with the class of geodesic rays in X that converge to ξ . It is known that two rays π(x0 , x1 , . . . ) and π(y0 , y1 , . . . ) are equivalent as θa -Cauchy sequences if and only if d(xn , yn ) ≤ 4δ

for all n ≥ 0.

(To prove the necessity, we let n, m ≥ 0. Using (2.1), we have 1 n − d(xn , yn ) = (xn |yn ) ≥ min{(xn |xm+n ), (xn+m |yn+m ), (yn+m |yn )} − 2δ 2 = min{n, (xn+m |yn+m )} − 2δ. Letting m → ∞, we have (xn+m |yn+m ) → ∞ as the two rays are equivalent, hence d(xn , yn ) ≤ 4δ follows.) We can extend the Gromov product to X ∪ 𝜕X by letting (x|ξ ) = inf{ lim (x|xn )}, n→∞

(ξ |η) = inf{ lim (xn |yn )}, n→∞

where x ∈ X, ξ , η ∈ 𝜕X, and the infimum is taking over all geodesic rays π(x0 , x1 , . . . ) and π(y0 , y1 , . . . ) converging to ξ and η, respectively; the Gromov metric on X ∪ 𝜕X is defined in the same way as in (2.3). In the following we will consider a specific class of rooted graphs, and give a useful criterion for hyperbolicity. For x, y ∈ X, we say that a geodesic π(x, y) = [x0 , x1 , . . . , xn ] is an h-geodesic if it consists of horizontal edges only, and a v-geodesic if xi+1 ∈ 𝒥1 (xi ) for all i, or xi+1 ∈ 𝒥−1 (xi ) for all i; it is called a convex geodesic if there exist u, v ∈ π(x, y) such that π(x, y) = π(x, u) ∪ π(u, v) ∪ π(v, y) (also denoted by π(x, u, v, y)) in which π(u, v) is an h-geodesic, and π(x, u), π(v, y) are v-geodesics with u ∈ 𝒥−∗ (x) and v ∈ 𝒥−∗ (y) (one or two parts may vanish). Also between x, y, the convex geodesic may not be unique; by convention, we use the one such that |u| = |v| is minimum (see Figure 2). Note that 1 1 (x|y) = (|x| + |y| − d(x, y)) = |u| − dh (u, v) = (u|v). 2 2

(2.4)

148 | S.-L. Kong et al.

Figure 2: Two convex geodesics.

Lemma 2.5. Let (X, ℰ ) be a rooted graph such that (X, ℰv ) is a tree. Suppose the following property is satisfied: for any x, y ∈ X, x ∼h y ⇒ x−1 ∼h y−1 ,

(“ ∼h ” includes “ = ”),

(*)

where x −1 is the unique 1st predecessor of x. Then any pair x, y ∈ X can be joined by a convex geodesic. Proof. The proof is quite simple: following [30], we can use the following moves repeatedly to change the geodesic without increasing the length: for u, v ∈ π(x, y) with (u, v) ∈ ℰh , [u, v, v−1 ] → [u, u−1 , v−1 ] and [u−1 , u, v] → [u−1 , v−1 , v]. Eventually we get a convex geodesic connecting x and y. The lemma allows us to have a good grasp of the geodesics in such graphs, and condition (*) is automatically satisfied for the augmented trees of the IFS in the next section. Our main theorem in this section is the following criterion for hyperbolicity, which relies on the convex geodesics. Theorem 2.6. Let (X, ℰ ) be a rooted graph such that (X, ℰv ) is a tree and property (*) is satisfied. Then the following are equivalent: (i) (X, ℰ ) is hyperbolic; (ii) ∃ L < ∞ such that the lengths of all h-geodesics are bounded by L. This was proved in [45, Theorem 2.3] based on the thin triangle characterization (Proposition 2.3). In the following, we use an algebraic argument by the Gromov products in Definition 2.1. Proof. (i) ⇒ (ii) Let π(x0 , x1 , . . . , xn ) be an h-geodesic in (X, ℰ ). Then for i ∈ {0, 1, . . . , n}, it follows from (2.1) and (2.4) that |x0 | −

n i n−i = (x0 |xn ) ≥ min{(x0 |xi ), (xi |xn )} − δ = |x0 | − max{ , } − δ. 2 2 2

Therefore n ≤ max{i, n − i} + 2δ. By choosing i = ⌊n/2⌋ we have n ≤ 4δ + 1.

Hyperbolic graphs induced by iterations and applications in fractals | 149

(ii) ⇒ (i) We prove that (X, ℰ ) is L2 -hyperbolic. For x, y, z ∈ X, consider the convex geodesics π(x, u, v, z), π(z, u󸀠 , v󸀠 , y) connecting x, z and z, y, respectively (see Figure 3). Without loss of generality, assume that |u| ≤ |u󸀠 |. Then v = u󸀠 if |u| = |u󸀠 |, or v ∈ 𝒥−∗ (u󸀠 ) if |u| < |u󸀠 |. We compute the length of the path from x to y passing through u, v, u󸀠 , v󸀠 successively as L󸀠 := d(x, u) + d(u, v) + d(v, u󸀠 ) + d(u󸀠 , v󸀠 ) + d(v󸀠 , y) 󵄨 󵄨 󵄨 󵄨 ≤ (|x| − |u|) + d(u, v) + (󵄨󵄨󵄨u󸀠 󵄨󵄨󵄨 − |v|) + L + (|y| − 󵄨󵄨󵄨u󸀠 󵄨󵄨󵄨) 1 = |x| + |y| − 2(|u| − d(u, v)) + L 2 = |x| + |y| − 2(x|z) + L

(by (2.4)).

On the other hand, using the definition of Gromov product, we have L󸀠 ≥ d(x, y) = |x| + |y| − 2(x|y). Combining these two inequalities, we conclude that (x|y) ≥ (x|z) − L2 . This implies that (2.1) is satisfied for δ = L2 .

Figure 3: Illustration for the proof of (ii) ⇒ (i).

3 IFS and augmented trees We call a finite set of contractive maps {Sj }Nj=1 (N ≥ 2) on ℝd an iterated function system

(IFS). It is well known [27, 13] that there exists a unique nonempty compact set K ⊂ ℝd such that K = ⋃Nj=1 Sj (K). We call the set K the invariant set (or attractor) of the IFS. In particular, for IFS of similitudes (also called self-similar IFS) (i. e., Sj are similitudes: |Sj (x) − Sj (y)| = rj |x − y|), we call K a self-similar set. Throughout the paper, we mainly consider the self-similar IFS {Sj }Nj=1 . Let Σ = n 0 {1, 2, . . . , N}, and Σ∗ = ⋃∞ n=0 Σ , where Σ = {ϑ} contains the empty word ϑ only. Given n x = i1 i2 . . . in ∈ Σ , we denote Sx = Si1 ∘ Si2 ∘ ⋅ ⋅ ⋅ ∘ Sin , Kx = Sx (K) (where Sϑ is the

150 | S.-L. Kong et al. identity map), and rx = ri1 ⋅ ⋅ ⋅ rin , the product of contractive ratios of the similitudes. Let r∗ = min{rj : j ∈ Σ}. The symbolic space Σ∗ has a nature tree structure from each x ∈ Σ∗ to its descendants. But for x ∈ Σn , the diameter of Kx may vary greatly. On the other hand, there are different ways to redefine the coding space for the attractor K of the IFS. A commonly used tree structure which regroups the x ∈ Σ∗ in a more tractable manner is as follows: For n ≥ 1, let Xn := {x = i1 i2 . . . ik ∈ Σ∗ : rx ≤ r∗n < ri1 i2 ...ik−1 }.

(3.1)

It can be proved that Xn ∩ Xn+1 = 0 for each n. All maps in {Sx : x ∈ Xn } have approximately equal contraction ratios (≈ r∗n ), and all cells in {Kx : x ∈ Xn } have approximately equal diameters, r∗n+1 |K| < |Kx | ≤ r∗n |K|. Let X = ⋃∞ n=0 Xn (as usual X0 = {ϑ}). Then X has a natural tree structure, denoted by ℰv . It is direct to check that the total number of the 1st level descendants of each x is ∗ less than N 1+log r∗ / log r (and ≥ N) where r ∗ = max{rj : j ∈ Σ}. Note also that X can be a proper subset of Σ∗ , but they define the same limit points on infinite paths. In order to describe the relation of the neighboring cells of Kx , x ∈ X, we augment the tree (X, ℰv ) by adding horizontal edges in the following two ways: Definition 3.1. On tree (X, ℰv ), we define a horizontal edge set by ∞

ℰh = ⋃ {(x, y) ∈ Xn × Xn : x ≠ y, Kx ∩ Ky ≠ 0}; n=1

for a fixed constant γ > 0, we define another horizontal edge set by ∞

n

ℰ̃h = ⋃ {(x, y) ∈ Xn × Xn : x ≠ y, dist(Kx , Ky ) ≤ γr∗ }. n=1

(3.2)

Denote ℰ = ℰv ∪ ℰh , ℰ̃ = ℰv ∪ ℰ̃h , and call (X, ℰ ) and (X, ℰ̃) augmented trees of (X, ℰv ). Since the constant γ in ℰ̃h has no real significance as long as it is positive, we omit it in the notation for brevity. It is easy to check that both augmented trees have property (*) by observing that dist(Kx , Ky ) ≤ dist(Kxi , Kyj ). The augmented tree (X, ℰ ) is more naturally defined: it is our original consideration in [45], and has important applications (see Sections 4 and 5). But the hyperbolic property is not immediate: it needs additional conditions to control the fine structure of the attractor K, for which we will discuss in the sequel. The graph (X, ℰ̃) is a relaxation of the intersection condition in (X, ℰ ), enlarging the horizontal edge set ℰh . This allows us to have a more complete result on the hyperbolicity of the graph and the boundaries. The following is one of our main theorems.

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Theorem 3.2. For self-similar IFS, the augmented tree (X, ℰ̃) is hyperbolic. Moreover, the hyperbolic boundary 𝜕X is Hölder equivalent to the self-similar set K, i. e., there exists a canonical bijection κ : 𝜕X → K such that 󵄨 󵄨 C −1 θa (ξ , η)α ≤ 󵄨󵄨󵄨κ(ξ ) − κ(η)󵄨󵄨󵄨 ≤ Cθa (ξ , η)α ,

∀ξ , η ∈ 𝜕X,

(3.3)

where C ≥ 1 is a constant and α = − log r∗ /a. The proof of the theorem is in [46, Theorems 1.2 and 1.3]. We will give another proof of this in Theorem 6.10 under a more general setup, which also includes the augmented tree (Σ∗ , ℰ̃) and some more general cases. For the augmented tree (X, ℰ ), we have Proposition 3.3. For self-similar IFS, if the augmented tree (X, ℰ ) is hyperbolic, then there exists a bijective map κ : 𝜕X → K such that 󵄨󵄨 󵄨 α 󵄨󵄨κ(ξ ) − κ(η)󵄨󵄨󵄨 ≤ Cθa (ξ , η) ,

∀ξ , η ∈ 𝜕X,

where the constant C > 0 and α = − log r∗ /a. In particular, the hyperbolic boundary 𝜕X is homeomorphic to the attractor K. We do not know if (X, ℰ ) is hyperbolic in general. However, we have simple conditions that guarantee sufficiently many interesting cases that (X, ℰ ) is hyperbolic. Proposition 3.4. Suppose self-similar IFS has the following property: (H) ∃C > 0 ∋ for any n > 0 and x, y ∈ Xn , either Kx ∩ Ky ≠ 0

or

dist(Kx , Ky ) ≥ Cr∗n .

Then the augmented tree (X, ℰ ) is hyperbolic. Moreover, the hyperbolic boundary 𝜕X is Hölder equivalent to the self-similar set K as in (3.3). Proof. Let 0 < γ < C, we define horizontal edge set ℰ ̃ by (3.2). Then it is clear that the property (H) implies ℰ = ℰ ,̃ and the assertion follows from Theorem 3.2. Property (H) was introduced in [45], and also used in other applications [21, 26]. The property is satisfied for IFS of similitudes where the maps and the translations are defined by integral entries [45], so are the homogeneous p. c. f. IFS of similitudes [21]. There are also examples constructed so that (H) fails [45, 46, 58], including one that (X, ℰ ) is hyperbolic (by Theorem 3.5 below), but 𝜕X is not Hölder equivalent to K [37]. One of the most fundamental conditions on the IFS of similitudes {Si }Ni=1 is the open set condition (OSC) [27], namely, there exists a nonempty bounded open set O ⊂ ℝd such that Si (O) ⊂ O and the Sj (O)’s are disjoint. In such case, the Hausdorff dimension α of K is given by ∑Ni=1 riα = 1. It is also known that under the OSC, we can choose the open set O such that O ∩ K ≠ 0 [56], which implies 0 < ℋα (K)(< ∞). Moreover, the OSC is equivalent to [13, 56, 46]

152 | S.-L. Kong et al. (S) for any c > 0, there exists ℓ = ℓ(c) > 0 such that any ball B of radius cr∗n can intersect Kx with at most ℓ of x ∈ Xn . Theorem 3.5. If the self-similar IFS satisfies either (i) the condition (S); or (ii) the self-similar set K has a positive Lebesgue measure; then the augmented tree (X, ℰ ) is hyperbolic. Proof. Assuming (i), to prove the hyperbolicity, it suffices to show that the lengths of the h-geodesics are bounded by some constant. Suppose otherwise, for any integer m > 0, there exists an h-geodesic π(x0 , x3m ) = [x0 , x1 , . . . , x3m ] with xi ∈ Xn . We consider the mth predecessor yi = xi[−m] . Let p(y0 , y3m ) = [yi0 , . . . , yik ]

(3.4)

with yij ∈ {y0 , . . . , y3m } be the shortest horizontal path connecting y0 and y3m . By the geodesic property of π(x0 , x3m ), it is clear that 󵄨 󵄨 󵄨 󵄨 k = 󵄨󵄨󵄨p(y0 , y3m )󵄨󵄨󵄨 ≥ 󵄨󵄨󵄨π(x0 , x3m )󵄨󵄨󵄨 − 2m = m.

(3.5)

Now choose m ≥ ℓ such that (3m + 1)r∗m ≤ 1, where ℓ = ℓ(|K|) is as in condition (S). Let n D = ⋃3m i=0 Kxi . From |Kxi | ≤ r∗ |K| (i = 0, 1, . . . , 3m), it is direct to show that |D| ≤ (3m + 1)r∗n |K| ≤ r∗n−m |K|.

(3.6)

Note that Kxi ⊂ Kyi , we see that Kyi ∩ D ≠ 0 for each j = 0, 1, . . . , k. It follows that j

#{y ∈ Xn−m : Ky ∩ D ≠ 0} ≥ k + 1 > m ≥ ℓ. It contradicts condition (S) and the proof is completed. For case (ii), we proceed as in (i). Let m be such that |D| ≤ r∗n−m |K| (as in (3.6)), and let D󸀠 = ⋃ki=0 Kyi . By using the horizontal path property of D, we see that D󸀠 is contained in the (r∗n−m |K|)-neighborhood of D, hence, 󵄨󵄨 󸀠 󵄨󵄨 n−m n−m 󵄨󵄨D 󵄨󵄨 ≤ 2r∗ |K| + |D| ≤ Cr∗ . Now consider (3.4). By the shortest path property, we see that for the even terms (or odd terms) of {yi0 , . . . , yik }, every pair are disjoint (otherwise, we can shorten the path). Observe that ℒ(D󸀠 ) ≤ C 󸀠 |D󸀠 |d , we hence have ⌊k/2⌋

CC 󸀠 r∗(n−m)d ≥ ℒ(D󸀠 ) ≥ ∑ ℒ(Ky2i ) ≥ (⌊k/2⌋ + 1)ℒ(K)r∗(n−m+1)d > 0. i=0

This is a contradiction, as k ≥ m (by (3.5)) and m can be arbitrary large.

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It follows that for the IFS of similitudes with OSC, the augmented tree (X, ℰ ) is hyperbolic; the second condition applies to the well-known class of self-similar tilings (see, e. g., [40, 25]). We now consider the bounded degree (i. e., sup{deg(x) : x ∈ X} < ∞) property of the graphs. This property is important, especially when we consider random walks on graphs (see Section 5). In the following we will discuss this property in connection with the OSC. We need a simple lemma which can be proved by contrapositive argument. Lemma 3.6. Let {Sj }Nj=1 be an IFS of contractive similitudes. Suppose that (X, ℰ̃) is of bounded degree, then Sx ≠ Sy for any x ≠ y in X. Theorem 3.7. Let {Sj }Nj=1 be an IFS of contractive similitudes. Then (X, ℰ̃) is of bounded degree if and only if the IFS satisfies the OSC. Proof. Assume that the IFS satisfies the OSC. Let x ∈ Xn (n ≥ 1), then |Kx | ≤ r∗n |K|. Let B be a ball centered at some point in Kx with radius (2γ + |K|)r∗n . Note that Ky ∩ B ≠ 0 if x ∼h y, or x ∼v y with y ∈ Xn+1 . Also there is only one vertex x [−1] ∈ 𝒥−1 (x). By making use of condition (S) from the OSC, we have the following estimate deg(x) ≤ #{y : x ∼h y} + #{y : x ∼v y} ≤ ℓ(2γ + |K|) + ℓ((2γ + |K|)r∗−1 ) + 1. Hence (X, ℰ̃) is of bounded degree. To prove the converse, it suffices to show that the condition (S) holds. Suppose otherwise, there exists a constant c > 0 such that for any ℓ > 0, there exist n and a ball B ⊂ ℝd with radius cr∗n satisfying #{x ∈ Xn : Kx ∩ B ≠ 0} > ℓ. Let Xn,B denote the set in the above inequality, and let D = ∪{Kx : x ∈ Xn,B }. Then |D| ≤ 2|K|r∗n + cr∗n = (2|K| + c)r∗n . We can choose k0 independent of n such that {B1 , B2 , . . . , Bk0 } is a family of open balls with radius γr∗n /2 and covers D (where γ is in the definition (3.2) of ℰ̃h ). There exists a Bi that intersects at least ℓ󸀠 = ⌊ℓ/k0 ⌋ of Kx (x ∈ Xn,B ), say, Kx1 , Kx2 , . . . , Kx 󸀠 . Then dist(Kxi , Kxj ) ≤ γr∗n for 1 ≤ i, j ≤ ℓ󸀠 . Hence xi ∼h xj if i ≠ j. It follows that deg(xi ) ≥ ℓ󸀠 − 1,

ℓ

i = 1, 2, . . . , ℓ󸀠 .

Since ℓ can be arbitrarily large and k0 is a fixed constant, we see that ℓ󸀠 can be arbitrarily large. This contradicts that the graph is of bounded degree, and the condition (S) follows. Hence the IFS satisfies the OSC. Note that ℰh ⊂ ℰ̃h , as a direct consequence of the above theorem, we have

154 | S.-L. Kong et al. Corollary 3.8. Let {Sj }Nj=1 be an IFS of contractive similitudes satisfying the OSC, then the graph (X, ℰ ) is of bounded degree. There are variations of the IFS, that also fall into this framework of augmented trees. An easy example is the IFS {Si }2i=1 , with S1 (x) = 21 x, S2 (x) = 21 (x + 1), then the augmented tree has boundary [0, 1]. If we identify the two end vertices on each level of the tree, then we get a new hyperbolic graph with the boundary homeomorphic to a circle. A less trivial one is S1 (x) = rx, S2 (x) = rx +(1−r) where r = (√5−1)/2 is the golden ratio (corresponding to the Bernoulli convolution). Note that for x, y ∈ Xn , x ≠ y, Sx can equal Sy (e. g., S122 = S211 ). We can identify these indices and form a new hyperbolic graph. In this case, the vertical part of the graph is not a tree. This has been discussed in detail in [58] as quotient graphs for the IFSs with a weak separation condition (WSC) defined in [41]. It is known that the OSC is equivalent to the WSC together with Sx ≠ Sy for all x ≠ y [62]. Also in [48], the Moran sets and the hyperbolicity were studied. In the following we will introduce yet another type of IFS, the weighted IFS, which is connected to the study of energy forms on fractals. All these considerations can be embraced in a general setup of expansive hyperbolic graphs in Section 6. To consider the weighted IFS, we start by defining another regrouping of indices in the symbolic space Σ∗ as in the following. Let s = (s1 , s2 , . . . , sN ) be a vector with 0 < si < 1 for all i ∈ Σ. Denote s∗ = min{si : i ∈ Σ} and s∗ = max{si : i ∈ Σ}. For x = i1 i2 . . . in ∈ Σ∗ , write sx = si1 si2 ⋅ ⋅ ⋅ sin . We regroup Σ∗ by setting X0 (s) = {ϑ}, and for n ≥ 1, let Xn (s) := {x = i1 i2 . . . ik ∈ Σ∗ : sx ≤ sn∗ < si1 si2 ⋅ ⋅ ⋅ sik−1 }. Then X(s) = ⋃∞ n=0 Xn (s) has a natural tree structure denoted by ℰv . We also define horizontal edge sets as in Definition 3.1 to get the augmented trees (X(s), ℰ ) and (X(s), ℰ̃). ̃ Theorem 6.10 in Section 6 will yield the hyperbolicity, as well as the For (X(s), ℰ ), Hölder equivalence of 𝜕X(s) and K. The more interesting and useful case is on (X(s), ℰ ). For this, we can only prove the hyperbolicity for a restrictive class of self-similar sets, called post critically finite (p. c. f.) sets, defined by Kigami [32]. The crucial property of p. c. f. set is that the intersection of two cells Ki , Kj , i ≠ j has at most finitely many points. We have the following theorem (see [37]). Theorem 3.9. Let {Sj }Nj=1 be a contractive IFS that satisfies the p. c. f. property. Then the augmented tree (X(s), ℰ ) is hyperbolic, and is of bounded degree. Moreover, the embedding κ : (𝜕X(s), θa ) → (K, | ⋅ |) is a Hölder continuous homeomorphism. Let θa be the Gromov metric on 𝜕X(s), then it induces a new metric θ̃a on K via the homeomorphism κ. We consider the metric space (K, θ̃a ). Let α be the positive number satisfying ∑Nj=1 sαj = 1. We define the self-similar measure μs to be the unique Borel probability measure which satisfies the following identity N

μs (⋅) = ∑ sαj μs (Sj−1 (⋅)). j=1

Hyperbolic graphs induced by iterations and applications in fractals | 155

The following proposition provides an interesting relation of above self-similar measure and the induced Gromov metric on K. Proposition 3.10. Let {Sj }Nj=1 be a contractive IFS that satisfies the p. c. f. property. For s ∈ (0, 1)N , the self-similar measure μs is Ahlfors-regular with exponent (−α log s∗ /a) with respect to the new metric space (K, θ̃a ), i. e., μs (Bθ̃ (ξ , r)) ≍ r −α log s∗ /a , a

∀ξ ∈ K, r ∈ (0, 1).

On a p. c. f. set, we consider the energy form (E, 𝒟) defined through a harmonic structure with weight s ∈ (0, 1)N on K [32, 55], which satisfies the following selfsimilarity: E[u] = ∑ s−1 j E[u ∘ Sj ], j∈Σ

u ∈ 𝒟,

where 𝒟 = {u ∈ C(K) : E[u] < ∞} is the domain of E, and C(K) is the set of continuous functions on K. Also define the effective resistance between two nonempty disjoint compact subsets F, G ⊂ K by R(F, G) = (inf{E[u] : u = 1 on F, u = 0 on G}) . −1

Then R(ξ , η)(:= R({ξ }, {η}) for ξ , η ∈ K) is a metric on K (resistance metric [32]). Theorem 3.11. Let K be a connected p. c. f. set that admits a harmonic structure with weight s ∈ (0, 1)N , and let R be the resistance metric of the associated self-similar energy form. Then the metric θ̃a on K induced by (X(s), ℰ ) satisfies θ̃a (ξ , η) ≍ R(ξ , η)−a/ log s∗ ,

∀ξ , η ∈ K.

The interested reader can refer to [37] for the details.

4 Lipschitz equivalence of self-similar sets Recall that two compact metric spaces (X, d1 ) and (Y, d2 ) are said to be Lipschitz equivalent, denoted by X ≃ Y, if there exists a bi-Lipschitz map σ : X → Y, i. e., σ is a bijection and there exists a constant C > 0 such that C −1 d1 (x, y) ≤ d2 (σ(x), σ(y)) ≤ Cd1 (x, y),

for all x, y ∈ X.

Lipschitz classification of fractals was first started by Falconer and Marsh [14] on Cantor-type sets under the strong separation condition. The recent interest was due to Rao, Ruan, and Xi [43] on their solution to an open question of David and Semmes,

156 | S.-L. Kong et al. the so-called {1, 3, 5}–{1, 4, 5} problem, i. e., subdivide [0, 1] into five equal size subintervals, and pick the 1st, 3rd, 5th subintervals and the 1st, 4th, 5th subintervals (the 4th and 5th subintervals have nonvoid intersection) to form the respective IFSs and self-similar sets K1 , K2 . They used a technique of graph directed system to show that K1 and K2 are Lipschitz equivalent. The result stimulates a lot of interest and generalizations. In this section, we discuss a different approach to this type of Lipschitz equivalence problem through the augmented trees, hyperbolic graphs and hyperbolic boundaries [9, 51]. More developments can be found in [48, 49, 50, 52]. We will consider the self-similar IFS {Si }Ni=1 with equal contractive ratio. Let Σ = n {1, 2, . . . , N}, Xn = Σn and X = ⋃∞ n=0 Σ . Let (X, ℰ ) be the augmented tree as in Definition 3.1. For convenience, we call (X, ℰ ) an N-ary augmented tree. We say that T is an Xn -horizontal component if T ⊂ Xn is a maximal connected subset with respect to ℰh . In this case, we denote by T𝒟 the set of all descendants of T (including T itself), i. e., T𝒟 = {x ∈ X : x|n ∈ T} where x|n is the initial segment of x with length n. Obviously, T𝒟 should be a connected subgraph of X, and if (X, ℰ ) is hyperbolic then T𝒟 is also hyperbolic. Indeed suppose x, y are children of T and let γ be a horizontal geodesic in (X, ℰ ) with end points at x, y, respectively. Then the parents of γ in T form a connected horizontal path (by property (*) in Lemma 2.5), hence is in T (because T is a connected component). This implies that γ is also a horizontal geodesic in T𝒟 . Applying this argument to each level in T𝒟 , we see that the lengths of horizontal geodesics in T𝒟 are uniformly bounded, inherited from (X, ℰ ). We let ℱn denote the family of all Xn -horizontal components, and let ℱ = ⋃n≥0 ℱn . 󸀠 Note that for distinct T, T 󸀠 ∈ ℱn , the subgraphs T𝒟 , T𝒟 are disjoint. Denote T by ⌊x⌋ for x ∈ T, we can define a graph structure on ℱ as follows: ⌊x⌋ and ⌊y⌋ are connected by an edge if and only if (u, v) ∈ ℰv for some u ∈ ⌊x⌋ and v ∈ ⌊y⌋; we denote this graph by XQ (see the left two graphs of Figure 4). It is clear that XQ defined above is a tree, and we call it the quotient tree of X.

Figure 4: A rooted graph X , the quotient tree XQ and the union of three copies of X .

Hyperbolic graphs induced by iterations and applications in fractals | 157

For T, T 󸀠 ∈ ℱ , we say that T and T 󸀠 are equivalent, denoted by T ∼ T 󸀠 , if there exists 󸀠 a graph isomorphism g : T𝒟 → T𝒟 , i. e., the map g and its inverse map both preserve 󸀠 the vertical and horizontal edges of T𝒟 and T𝒟 . We denote the equivalence class by [T]. Definition 4.1. We call an augmented tree (X, ℰ ) simple if the equivalence classes in ℱ is finite. Let [T1 ], . . . , [Tm ] be the equivalence classes in ℱ \ {ϑ}, and let aij , where 1 ≤ i, j ≤ m, denote the cardinality of the horizontal components of the 1st level descendants of T ∈ [Ti ] that belong to [Tj ]. We call A = [aij ] the incidence matrix of (X, ℰ ). Proposition 4.2. A simple augmented tree (X, ℰ ) is always hyperbolic. Proof. Note that for each horizontal geodesic π(x, y) in X, the horizontal part must be contained in a horizontal component of the augmented tree. Since there are finitely many equivalence classes [T] of horizontal components, and each T contains finitely many vertices, it follows that π(x, y) is uniformly bounded, and hence (X, ℰ ) is hyperbolic. Definition 4.3. Let X, Y be two hyperbolic graphs. We say that σ is a near-isometry from X to Y if there exist finite subsets E ⊂ X, F ⊂ Y, and c > 0 such that σ : X\E → Y \F is a bijection and satisfies 󵄨󵄨 󵄨 󵄨󵄨d(σ(x), σ(y)) − d(x, y)󵄨󵄨󵄨 < c. The following two propositions are easy consequences from the definitions [51, 9]. Proposition 4.4. Let X, Y be two hyperbolic augmented trees. Suppose there exists a near-isometry from X to Y, then 𝜕X ≃ 𝜕Y. It is clear that 𝜕(X, ℰv ) is an N-ary Cantor set. Our aim is to show that a simple augmented tree (X, ℰ ) is near-isometric to (X, ℰv ); by the above proposition, 𝜕(X, ℰv ) ≃ 𝜕(X, ℰ ). In the following we develop some tools to construct such near-isometry. Proposition 4.5. Let (X, ℰ ) be a simple N-ary augmented tree, let [T1 ], . . . , [Tm ] be the equivalence classes with incidence matrix A, and let u = [u1 , . . . , um ]t where ui = #Ti . Then Au = Nu. ̂ = (⋃ℓ Xi ) ∪ {ϑ} Suppose (Xi , ℰi ), 1 ≤ i ≤ ℓ are augmented trees with roots ϑi . Let X i=1 ̂ where ϑ is an additional vertex. We equip X with an edge set ℰ̂ that consists of all ℰi ̂ ℰ̂) forms a new connected graph and each and the new edges joining ϑ and ϑi . Then (X, ̂ ℰ̂) the union of {Xi }ℓ . It follows (Xi , ℰi ) becomes its subgraph (see Figure 4). We call (X, i=1 that [9, Proposition 2.8] Proposition 4.6. Let (X, ℰ ) be an N-ary augmented tree such that 𝜕(X, ℰ ) ≃ 𝜕(X, ℰv ). ̂ ℰ̂) is the union of {(Xi , ℰi )}ℓ . Then Suppose (Xi , ℰi ), 1 ≤ i ≤ ℓ, are copies of (X, ℰ ), and (X, i=1 ̂ ℰ̂) ≃ 𝜕(X, ℰ ). 𝜕(X,

158 | S.-L. Kong et al. The following notions of rearrangeable matrix [8, 51] and quasirearrangeable matrix [9] are the most important technical devices in constructing the near-isometry between (X, ℰv ) and (X, ℰ ). Definition 4.7. Let a = [a1 , . . . , am ] and u = [u1 , . . . , um ]t be in ℕm . For N > 0, we say that a is (N, u)-rearrangeable if there exist p > 0 and a nonnegative integral p × m matrix C (rearranging matrix) such that a = ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ [1, . . . , 1] C p

and Cu = ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ [N, . . . , N]t . p

(4.1)

(In this case au = pN, and ui ≤ N.) We say that a is (N, u)-quasirearrangeable if the second identity is replaced by Cu ≤ [N, . . . , N]t . A matrix A is said to be (N, u)-rearrangeable (quasirearrangeable) if each row vector in A is (N, u)-rearrangeable (quasirearrangeable). (Note that p and C in each row may be different.) To realize the above definition, let us assume that there are m different kinds of objects, each kind has cardinality ai and each one of the same kind has weight ui (assume that the gcd of the ui ’s is 1), hence the total weight is ∑i ai ui = pN. The rearranging matrix C is a way to divide these objects into p groups (first identity in (4.1)) such that every entry of a row represents the number of each kind in the group, and the total weight of the objects in the group is N (the second identity in (4.1)). Remark. The main purpose of this rearrangeable matrix A is to modify the horizontal edges of the offsprings of a component T so that each component Ti in the offsprings has the same parent x ∈ T (when gcd(u) = 1). This rearrangement gives a near-isometry of (X, ℰ ) to (X, ℰv ), and hence 𝜕(X, ℰ ) ≃ 𝜕(X, ℰv ) (by Proposition 4.4). In the following we also consider Ak and the same idea holds with the kth generation. Recall that a nonnegative matrix A is called primitive if An > 0 for some n, and is called irreducible if for any (i, j), there exists k > 0 such that the (i, j)-entry of Ak is positive. In [9], we proved Proposition 4.8. Let A be an m × m primitive matrix and u ∈ ℕm . Let u = gcd(u), (i) if Au = Nu, then there exists k > 0 such that Ak is (uN k , u)-rearrangeable; (ii) if Au ≤ Nu, then there exists k > 0 such that Ak is (uN k , u)-quasi-rearrangeable. In both cases, the corresponding matrix Ci for each row of Ak is of size (ui /u) × m. It is well-known that for any nonnegative matrix A, it can be brought into the form of the upper triangular block by a permutation matrix P, [ P AP = [ [ t

A1

[ 0

..

∗ .

] ] ]

Ar ]

Hyperbolic graphs induced by iterations and applications in fractals | 159

where each Ai is a square matrix that is either irreducible or zero, i = 1, . . . , r. We give a stronger result that for certain power Aℓ , the block matrices are primitive, if not zero. The lemma has independent interest and might be useful elsewhere. Lemma 4.9. Let A be a nonnegative matrix, then we have (i) if An is irreducible for any n ≥ 1, then A is primitive; (ii) there is ℓ ≥ 1 such that the block matrices lying in the diagonal of the canonical form of Aℓ are either primitive or 0. Proposition 4.10. Let (X, ℰ ) be a simple N-ary augmented tree, and assume that the incidence matrix A is primitive. Then 𝜕(T𝒟 , ℰ ) ≃ 𝜕(X, ℰv ) for any horizontal component T ∈ ℱ. Proof. Here we only sketch the main idea. Since the incidence matrix A of (X, ℰ ) is primitive, A is also an incidence matrix of the subgraph T𝒟 . Let {[T1 ], . . . , [Tm ]} be the equivalence classes that are in T𝒟 , let ui = #Ti be the number of vertices in Ti , and let u = gcd(u). By Propositions 4.5 and 4.8, there exists k such that Ak is (uN k , u)-rearrangeable. Without loss of generality, we can assume that k = 1. Hence for each Ti , there is a Ci rearranging its descendants Ti Σ into pi = ui /u groups consisting of components Tj ’s, which are denoted by 𝒱i,k , 1 ≤ k ≤ pi ; the number of vertices in 𝒱i,k is uN. We denote this process by Step I. Let ℓ = #T, and let Y be the union of ℓ copies of (X, ℰv ). Let ℰ 󸀠 be an augmented structure on Y by adding horizontal edges that joining u consecutive vertices in each level (see the right figure in Figure 5). We call this Step II. (Note that number of vertices in the nth level is ℓN n−1 and u divides ℓ.) Then 𝜕(Y, ℰ 󸀠 ) ≃ 𝜕(Y, ℰv ) ≃ 𝜕(X, ℰv ) as the first ≃ follows from a direct check that the identity map is a near-isometry, and the second ≃ follows from Proposition 4.6.

Figure 5: An illustration of σ : (T𝒟 , ℰ) → (Y , ℰ 󸀠 ) with u = 2, ℓ = 4, the ∙, ×, ∘, ◻ denote four kinds of components.

With this setup, we can define a map σ : (T𝒟 , ℰ ) → (Y, ℰ 󸀠 ) as follows. On the first level, let σ be any bijection from T to Y1 . Suppose we have defined σ on Ti of T𝒟 in the

160 | S.-L. Kong et al. nth level, we can define σ on Ti Σ by first applying Step I of rearrangement to obtain pi {𝒱i,k }k=1 , then assigning the vertices of 𝒱i,k to the descendants of σ(Ti ) and applying Step II (see Figure 5). It follows from the rearrangement property that each σ(𝒱i,k ) are descendants of u consecutive vertices in σ(Ti )(⊂ Yn ) (see Theorem 3.7 in [51] for detail). By the same proof as Theorem 3.7 in [51], that σ is a near-isometry, and hence 𝜕(T𝒟 , ℰ ) ≃ 𝜕(Y, ℰ 󸀠 ) ≃ 𝜕(X, ℰv ). We continue the construction of the near-isometry σ : (X, ℰ ) → (X, ℰv ) with the following incidence matrix A. Lemma 4.11. Let (X, ℰ ) be a simple N-ary augmented tree with equivalence classes {[T1 ], . . . , [Tm ]}, and the incidence matrix is of the form A=[

A1 0

A3 ] A2

where A1 , A2 are nonzero matrices with orders r and m − r, respectively. Let ui = #Ti , u1 = [u1 , . . . , ur ]t and u = gcd(u). Suppose (i) A1 is (uN, u1 )-quasi-rearrangeable; (ii) for i = r + 1, . . . , m, there exist near-isometries σi : ((Ti )𝒟 , ℰ ) → (Yi , ℰv ). Then there exists a near-isometry σ : (X, ℰ ) → (X, ℰv ), hence 𝜕(X, ℰ ) ≃ 𝜕(X, ℰv ). Proof. For convenience, we assume that A1 is (N, u1 )-quasirearrangeable, i. e., gcd(u1 ) = 1; the general case follows from the same argument of Step II in last proposition. We will use (i) and (ii) to construct a near-isometry σ : (X, ℰ ) → (X, ℰv ). We write X1 = (X, ℰ ) and X2 = (X, ℰv ). Let σ(ϑ) = ϑ and σ(i) = i, i ∈ Σ. Suppose σ has been defined on Σn such that (1) for component T ∈ [Ti ], i ≤ r, σ(T) has the same parent, i. e., σ(x)−1 = σ(y)−1 for all x, y ∈ T ⊂ Σn . (2) for component T ∈ [Ti ], i ≥ r + 1, σ(x) = σi (x) for x ∈ T𝒟 . To define the map σ on Σn+1 , we note that if T ⊂ Σn in (2), then σ is well-defined by σi . If T ⊂ Σn in (1), without loss of generality, we let T ∈ [T1 ]. Then T gives rise to horizontal components in Σn+1 , we group them into 𝒵1,j , j = 1, . . . , m according to the components belonging to [Tj ]. By the quasirearrangeable property of A1 (assumption (i)), for the row vector a1 = [a11 , . . . , a1r ], there exists a nonnegative integral matrix C = [csj ]u1 ×r such that a1 = 1C

and Cu1 ≤ [N, . . . , N]t .

By using this, we can decompose a1 into u1 groups as follows. Note that a1j denotes the number of horizontal components that belong to [Tj ]. For each 1 ≤ s ≤ u1 , we choose csj , 1 ≤ j ≤ r, of those components that are of size uj , respectively, and denote this

Hyperbolic graphs induced by iterations and applications in fractals | 161

collection by Λs . Then ⋃rj=1 𝒵1,j can be rearranged into u1 groups r

⋃ 𝒵1,j = Λ1 ∪ ⋅ ⋅ ⋅ ∪ Λu1 ,

(4.2)

j=1

and the total number of vertices in each group is ≤ N. For the component T = {i1 , . . . , iu1 } ⊂ Σn in (X, ℰ ), we have defined σ(T) = {j1 = σ(i1 ), . . . , ju1 = σ(iu1 )} in (X, ℰv ) by induction. In view of (4.2), we define σ on ⋃rj=1 𝒵1,j by assigning vertices in Λs (cardinality ≤ N) to the descendants of js (cardinality N) in 󸀠 a one-to-one manner; for the remaining T 󸀠 ∈ ⋃m j=r+1 𝒵1,j (maybe empty), say T ∈ [Tj ] and j ≥ r + 1, we define for x ∈ T 󸀠 , σ(x) to be any point in σ(T)Σ \ ⋃rj=1 σ(𝒵1,j ) to fill up the σ(T)Σ. We also use σi to induce a near-isometry σ : T𝒟 → (σ(T))𝒟 . We apply the same construction of σ on the offsprings of every component in Σn+1 . Inductively, σ can be defined from X1 to X2 . We omit the proof that σ is a near-isometry, the reader can refer to [9, Lemma 4.4] for details. Theorem 4.12. Let K be a self-similar set generated by an equicontractive IFS {Si }Ni=1 . If the augmented tree (X, ℰ ) is simple, then 𝜕(X, ℰ ) ≃ 𝜕(X, ℰv ). Proof. Let {[T1 ], . . . , [Tm ]} be the equivalence classes of horizontal components, ui = #Ti , and A the associated incidence matrix. By Lemma 4.9, there exist ℓ ≥ 1 and a permutation matrix P such that [ PA P=[ [ t ℓ

A1

[ 0

..

∗ .

] ] ]

Ak ]

where Ai are either 0 or primitive. From the definition of incidence matrix, we see that Ak ≠ 0, hence is primitive. Without loss of generality, we let ℓ = 1. If k = 1, then A = A1 is primitive. For any horizontal component T ⊂ Σ, T𝒟 has incidence matrix A also. Hence by Proposition 4.10 that 𝜕(T𝒟 , ℰ ) ≃ 𝜕(X, ℰv ). As Σ is the disjoint union of such T, it follows that 𝜕(X, ℰ ) = 𝜕(∪(T𝒟 , ℰ )) ≃ 𝜕(X, ℰv ). If k = 2, let A1 , A2 correspond to {[T1 ], . . . , [Tr ]}, and {[Tr+1 ], . . . , [Tr ]}, respectively. If A1 = 0, we can take A2 as the incidence matrix of (X, ℰ ) by removing finitely many vertices that belong to [Ti ], 1 ≤ i ≤ r. By Proposition 4.10, the result follows. If A1 ≠ 0, then Proposition 4.10 implies that assumption (ii) in Lemma 4.11 is satisfied; the other assumptions also follow readily, and the theorem follows. The general case that k ≥ 2 follows by applying the above argument inductively. By applying Proposition 3.4 and Theorem 4.12, we obtain Theorem 4.13. Let K and K 󸀠 be self-similar sets that are generated by two IFSs that have the same number of similitudes, the same contraction ratio, and satisfy condition (H) (in Proposition 3.4). Suppose further the two augmented trees are simple. Then K ≃ K 󸀠 .

162 | S.-L. Kong et al. As an illustration, we consider the following example with IFS {Si }4i=1 defined by as in Figure 6. Let r be the contraction ratio of the IFS. Then the self-similar set K is Lipschitz equivalent to the canonical 4-ary cantor set of contraction ratio r. (For detail, please see [51, Example 5.4]) {Ji }4i=1

Figure 6: The IFS is defined by {Ji }4i=1 ; the attractor K is Lipschitz equivalent to the canonical 4-ary Cantor set.

5 Random walks on augmented trees In this section we discuss a class of random walks on the augmented trees such that the Martin boundaries are identified with the hyperbolic boundaries, and the attractor K. This allows us to carry the discrete potential theory, as well as the induced energy forms (Dirichlet forms) on K [38, 39, 36]. We use the notation f ≍ g to mean that there exists a constant C > 0 such that C −1 g(x) ≤ f (x) ≤ Cg(x) for any variable x in a given domain. We recall some basic notions of discrete potential theory (see [11, 60]). Let (X, 𝒢 ) be a locally finite connected graph with the root ϑ ∈ X. A (reversible) random walk on (X, 𝒢 ) is a Markov chain {Zn }∞ n=0 with the state space X and the transition proba, x, y ∈ X, where the conductance c(⋅, ⋅) is a nonnegative bility given by P(x, y) = c(x,y) m(x) symmetric function on X × X that satisfies c(x, y) > 0 if and only if (x, y) ∈ 𝒢 , and m(x) := ∑y∈X c(x, y) is the total conductance at x. A function f : X → ℝ is called P-harmonic if ∑y∈X P(x, y)f (y) = f (x) for all x ∈ X; the graph energy of f is defined by EX [f ] =

1 󵄨 󵄨2 ∑ c(x, y)󵄨󵄨󵄨f (x) − f (y)󵄨󵄨󵄨 . 2 x,y∈X

n We assume that {Zn } is transient, i. e., the Green function G(x, y) = ∑∞ n=0 P (x, y) is n finite for all x, y ∈ X, where P is the n-step transition probability which can be defined inductively by P n+1 (x, y) = ∑z∈X P(x, z)P n (z, y) with P 0 being the identity matrix on X. Write ℙ(⋅ | Z0 = x) as ℙx (⋅) for short. We denote by F(x, y) = ℙx (∃n ≥ 0 such that Zn = y)

Hyperbolic graphs induced by iterations and applications in fractals | 163

the ever-visiting probability from x to y; it is known that G(x, y) = F(x, y)G(y, y), and F(x, y) ≥ F(x, z)F(z, y) for all x, y, z ∈ X. The Martin kernel is defined as K(x, y) =

G(x, y) F(x, y) = , G(ϑ, y) F(ϑ, y)

x, y ∈ X.

Following [11, 60], we define the Martin metric ρM (⋅, ⋅) on X by 󵄨 󵄨 󵄨 󵄨 ρM (x, y) = ∑ a(u)(󵄨󵄨󵄨K(u, x) − K(u, y)󵄨󵄨󵄨 + 󵄨󵄨󵄨χu (x) − χu (y)󵄨󵄨󵄨), u∈X

where a : X → (0, ∞) satisfies ∑u∈X In view of

a(u) F(ϑ,u)

K(u, x) =

< ∞, and χu is the indicator function at u.

F(u, x) 1 ≤ , F(ϑ, x) F(ϑ, u)

we see that ρM (⋅, ⋅) is well defined and is a metric on X. ̂M be the completion of (X, ρM ). We call ℳ = X ̂M \ X the Martin Definition 5.1. Let X boundary of {Zn }. Note that the completion coincides with the minimal compactification of X such ̂M [60]. Under this topology (or that for every x ∈ X, K(x, ⋅) extends continuously to X ̂ the Martin metric on XM ), the trajectory {Zn } converges to an ℳ-valued random variable Z∞ almost surely. Let ν denote the hitting distribution of Z∞ on ℳ when Z0 = ϑ. ξ For ξ ∈ ℳ, the ξ -process is defined as the random walk {Zn }∞ n=0 on (X, 𝒢 ) with the tranK(y,ξ ) ξ ξ sition probability P (x, y) = P(x, y) K(x,ξ ) , x, y ∈ X (P is a transition probability since K(⋅, ξ ) is P-harmonic), and the corresponding hitting distribution is denoted by νξ . The minimal Martin boundary ℳmin of {Zn } is the collection of all ξ ∈ ℳ such that νξ is the point mass at ξ ; it is known that ν(ℳ \ ℳmin ) = 0. To obtain the energy form on ℳ, we define the Naïm kernel by Θ(x, y) =

K(x, y) F(x, y) = , G(x, ϑ) F(x, ϑ)G(ϑ, ϑ)F(ϑ, y)

x, y ∈ X.

Clearly, Θ(⋅, ⋅) is symmetric on X × X, and can be extended continuously to X × ℳ as the Martin kernel K(⋅, ⋅) does. The extension on ℳ × ℳ \ Δ (here Δ := {(ξ , ξ ) : ξ ∈ ℳ}) is formulated by ξ Θ(ξ , η) = lim ∑ ℓm (z)Θ(z, η), m→∞

ξ

ξ

z∈Xm

ξ

ξ ≠ η ∈ ℳ,

ξ

(5.1)

where ℓm (z) = ℙϑ (∃n ≥ 0 ∋ Zn = z, Zk ∉ Xm ∀k > n) is the last-visiting probability on Xm of the ξ -process [57]; the limit exists as the sum is increasing in m. For a ν-integrable function u on ℳ, its Poisson integral Hu is given by (Hu)(x) = ∫ K(x, ξ )u(ξ ) dν(ξ ), ℳ

x ∈ X.

164 | S.-L. Kong et al. Note that K(⋅, ξ ) is P-harmonic for all ξ ∈ ℳ, so is Hu. For u ∈ L2 (ℳ, ν), we define the induced energy of u by Eℳ [u] = EX [Hu] =

1 󵄨2 󵄨 ∑ c(x, y)󵄨󵄨󵄨Hu(x) − Hu(y)󵄨󵄨󵄨 . 2 x,y∈X

The domain of the quadratic form Eℳ is 𝒟ℳ = {u ∈ L2 (ℳ, ν) : Eℳ [u] < ∞}. Theorem 5.2 (Silverstein, [57]). The induced energy has the expression Eℳ [u] =

m(ϑ) 2

󵄨󵄨 󵄨2 󵄨󵄨u(ξ ) − u(η)󵄨󵄨󵄨 Θ(ξ , η)dν(ξ )dν(η),

∬

u ∈ 𝒟ℳ .

ℳ×ℳ\Δ

For a hyperbolic graph (X, 𝒢 ), we need some hypotheses on {Zn } to identify ℳ with the hyperbolic boundary 𝜕X: (p0 ) p∗ := inf(x,y)∈ℰ P(x, y) > 0; m(F) : F is a finite subset of X} < ∞, where m(F) = (SI) (strong isoperimetry) sup{ c(𝜕F) ∑x∈F m(x) and c(𝜕F) = ∑x∈F,y∉F c(x, y). The (p0 ) implies that deg(x) ≤ (miny∈X:x∼y P(x, y))−1 ≤ p−1 ∗ for all x ∈ X, hence (X, 𝒢 ) has bounded degree. Also, it is known that (SI) yields the transience of {Zn } [60]. Theorem 5.3 (Ancona, [1]). Let (X, 𝒢 ) be a hyperbolic graph. Suppose {Zn } is a random walk on (X, 𝒢 ) satisfying (p0 ) and (SI). Then there exists a constant C0 ≥ 1 such that F(x, y) ≤ C0 F(x, z)F(z, y)

(5.2)

whenever x, y, z ∈ X and z lies on some π(x, y). Moreover, the Martin boundary ℳ equals ℳmin , and is homeomorphic to the hyperbolic boundary 𝜕X. To apply the above theorem, we provide a sufficient condition for {Zn } satisfying (SI). For x ∈ X \ {ϑ}, define the return ratio at x ∈ X by λ(x) :=

ℙx (|Z1 | = |x| − 1) ∑z∈𝒥−1 (x) c(x, z) = . ℙx (|Z1 | = |x| + 1) ∑y∈𝒥1 (x) c(x, y)

Following a similar proof as [38, Theorem 5.1], we have Proposition 5.4. Suppose a random walk {Zn } on a rooted graph (X, 𝒢 ) satisfies (p0 ) and supx∈X\{ϑ} λ(x) < 1. Then {Zn } has strong isoperimetry (SI). To obtain explicit estimates of Martin kernels and Naïm kernels, we will consider a class of random walks satisfying (Rλ )(constant return ratio) λ(x) ≡ λ ∈ (0, 1) for all x ∈ X \ {ϑ}. With condition (Rλ ), by counting the time instants n0 = 0 and nk = inf{ℓ > nk−1 : |Zℓ | ≠ |Zℓ−1 |} for k ≥ 1 inductively, the sequence {|Znk |}∞ k=0 is a birth and death chain on the

Hyperbolic graphs induced by iterations and applications in fractals | 165

̃ 1) = 1, P(m, ̃ nonnegative integers with the transition probability P(0, m − 1) = 1 ̃ P(m, m + 1) = 1+λ for m ≥ 1; it follows that ̃ F(x, ϑ) = F(|x|, 0) = λ−|x| ,

∀x ∈ X.

λ , and 1+λ

(5.3)

Definition 5.5. Let λ ∈ (0, 1). A random walk {Zn } on (X, 𝒢 ) is said to be λ-natural (λNRW) if it satisfies (p0 ) and (Rλ ). Remark. The above definition generalizes the NRW (and quasi-NRW) in [38], which was defined by self-similar measures of natural weight (with doubling property respectively). We will see in the following Theorem 5.6 that the new NRW is characterized by the doubling regular Borel measures. Note also that this definition is equivalent to the NRW in [39] by Theorem 5.6. From Proposition 5.4 we know that every λ-NRW satisfies (SI), and if (X, 𝒢 ) is hyperbolic, then Theorem 5.3 applies. In the rest of this section, we will consider the λ-NRW {Zn } on the augmented tree (X, ℰ̃) associated to a self-similar set K (Definition 3.1). Recall that a regular Borel measure μ on K is called (volume) doubling (VD) if there exists C ≥ 1 such that 0 < μ(B(ξ , 2r)) ≤ Cμ(B(ξ , r)) < ∞,

∀ξ ∈ K, r > 0.

The following theorem improves and strengthens [38, Theorem 4.8]. Theorem 5.6. Let {Sj }Nj=1 be an IFS of contractive similitudes with attractor K, and let (X, ℰ̃) be an associated augmented tree in Definition 3.1. Then (X, ℰ̃) admits a λ-NRW if and only if {Sj }Nj=1 satisfies the OSC. Moreover, the conductance of the λ-NRW satisfies c(x, x− ) = λ−|x| μ(Kx ),

∀x ∈ X \ {ϑ},

(5.4)

for some doubling measure μ on K with μ(Kx ∩ Ky ) = 0,

∀x ≠ y ∈ X with |x| = |y|.

(5.5)

Proof. For the first statement, as the (p0 ) implies the bounded degree property of (X, ℰ̃), by Theorem 3.7, the OSC on {Sj }Nj=1 is necessary for possessing a λ-NRW. To prove the sufficiency, we assume the OSC, and use the α-Hausdorff measure ℋα to construct a λ-NRW, where α is the Hausdorff dimension of K. It is known that under the OSC, 0 < ℋα (Kx ) = rxα ℋα (K) < ∞ holds for all x ∈ X, and ℋα (Kx ∩ Ky ) = 0 for all distinct x, y ∈ X with |x| = |y| [13, 56, 62]. For x ∈ X \ {ϑ} and (x, y) ∈ ℰ̃h , we let c(x, x − ) = λ−|x| ℋα (Kx ),

c(x, y) = λ−|x| √ℋα (Kx )ℋα (Ky ).

Then for x ∈ X \ {ϑ}, λ(x) =

λ−|x| ℋα (Kx ) c(x, x− ) = −(|x|+1) = λ. ∑y∈𝒥1 (x) c(x, y) λ ∑y∈𝒥1 (x) ℋα (Ky )

166 | S.-L. Kong et al. To verify condition (p0 ), as the OSC is equivalent to the bounded degree property of (X, ℰ̃) (Theorem 3.7), let ℓ = supx∈X #{z : (x, z) ∈ ℰ̃h } < ∞. Note that for x ∈ Xn , we have r∗n+1 < rx ≤ r∗n by (3.1). Therefore m(x) = c(x, x− ) + ∑ c(x, y) + y∈𝒥1 (x)

= λ−n ℋα (K) ⋅ (rxα + λ−1 rxα + ≤

λ−n r∗αn ℋα (K)

⋅ (1 + λ

−1

∑ z∈X:(x,z)∈ℰ̃h

∑ z∈X:(x,z)∈ℰ̃h

c(x, z) rxα/2 rzα/2 )

+ ℓ),

and c(x, y) ≥ λ−n r∗α(n+1) ℋα (K) min{1, λ−1 r∗α } for y ∈ X with (x, y) ∈ ℰ̃. ≥ r∗α (1 + λ−1 + ℓ)−1 min{1, λ−1 r∗α } for all This proves the (p0 ) by P(x, y) = c(x,y) m(x) (x, y) ∈ ℰ̃. Hence such conductance c defines a λ-NRW.

For the second part, let c󸀠 be the conductance of a λ-NRW. Write qx = c󸀠 (x, x − )λ|x| for x ∈ X \ {ϑ}, and qϑ = m󸀠 (ϑ)λ. By λ(x) ≡ λ, we have qx = ∑y∈𝒥1 (x) qy for all x ∈ X. For n ≥ 0 and a Borel set E ⊂ K, define ℓ

ℓ

i=1

i=1

μn (E) = inf{∑ qxi : E ⊂ ⋃ Kxi , xi ∈ Xn }. Clearly, μn (E) is decreasing in n. Let μ(E) = limn→∞ μn (E). Then it is standard to check that μ is a regular Borel measure on K, and from the OSC, it follows that μn (Kx ) =

∑

y∈𝒥n−|x| (x)

qy = qx ,

∀x ∈ X, n ≥ |x|,

therefore μ(Kx ) = qx for all x ∈ X. Denote Ux = K \ (⋃y∈X|x| \{x} Ky ). Then Ux ⊂ Kx , we have qx = μ(K) −

∑

y∈X|x| \{x}

μ(Ky ) ≤ μ(Ux ) ≤ μ(Kx ) = qx .

Hence μ(Ux ) = μ(Kx ) = qx , and this proves (5.5). Finally we show that μ is doubling. Let p∗ = inf(x,y)∈ℰ̃ P(x, y) (> 0 by the (p0 )). Note that for x ∈ X \ {ϑ}, qx− < m󸀠 (x− )λ|x|−1 =

q c󸀠 (x − , x)λ|x|−1 ≤ x , P(x − , x) λp∗

(5.6)

and for (x, y) ∈ ℰ̃h , qy < m󸀠 (y)λ|y| =

q c󸀠 (y, x)λ|y| m󸀠 (x)λ|x| c󸀠 (x, x − )λ|x| < = ≤ 2x . − P(y, x) P(y, x) P(y, x)P(x, x ) p∗

(5.7)

Hyperbolic graphs induced by iterations and applications in fractals | 167

Suppose ξ ∈ K and 0 < r ≤ |K|. Let n1 be the integer such that r∗n1 |K| < r ≤ r∗n1 −1 |K|. Choose x ∈ Xn1 such that ξ ∈ Kx . As |Kx | ≤ r∗n1 |K| < r, we have Kx ⊂ B(ξ , r), and μ(B(ξ , r)) ≥ μ(Kx ) = qx . Let γ > 0 be as in (3.2). If γ ≤ 2r, then from the choice of n1 we see that γ ≤ 2|K|r∗n1 −1 , + 1 := m0 . Using (5.6) repeatedly, we have therefore n1 ≤ log(γ/2|K|) log r ∗

μ(B(ξ , 2r)) ≤ μ(K) = qϑ < (λp∗ )−n1 qx ≤ (λp∗ )−m0 μ(B(ξ , r)). If γ > 2r, we let n2 be the maximal integer such that γ ⋅ r∗n2 ≥ 2r and n2 ≤ n1 . Then either γ ⋅ r∗n2 +1 < 2r ≤ 2|K|r∗n1 −1 or n2 = n1 holds true, which implies 0 ≤ n1 − n2 ≤ + 2} =: m1 . Let u be the unique (n1 − n2 )th predecessor of x. Denote max{0, log(γ/2|K|) log r∗ Tu = ⋃v∈X:u∼h v Kv . For η ∉ Tu , as |η − ξ | ≥ dist(η, Ku ) > γ ⋅ r∗n2 ≥ 2r, we see that B(ξ , 2r) ⊂ Tu . It follows from (5.7) and (5.6) that μ(B(ξ , 2r)) ≤ μ(Tu ) = qu +

∑ v∈X:(u,v)∈ℰ̃h

qv

−2 −(n1 −n2 ) < (1 + ℓp−2 qx ∗ )qu < (1 + ℓp∗ )(λp∗ ) −m1 ≤ (1 + ℓp−2 μ(B(ξ , r)). ∗ )(λp∗ )

Hence μ is doubling, and completes the proof. Without loss of generality, we will assume that the doubling measure μ in (5.4) satisfies μ(K) = 1. For a λ-NRW {Zn } on (X, ℰ̃), by applying Theorems 3.2 and 5.3, the Martin boundary ℳ = ℳmin , and is homeomorphic to the hyperbolic boundary 𝜕X as well as the attractor K. From now on we will identify K with ℳ, and regard the hitting distribution ν as a probability measure on K. Using (5.3) and a time reversal argument on {Zn }, we have F(ϑ, x) ≍ ℙϑ (Zτm = x) = μ(Kx ),

∀x ∈ Xm , m ≥ 0,

(5.8)

where τm = inf{n ≥ 0 : Zn ∈ Xm } is the first hitting time for Xm (see [38] for details). Theorem 5.7. Let {Zn } be a λ-NRW on an augmented tree (X, ℰ̃) associated to a selfsimilar set K. Then ℳ, 𝜕X and K are homeomorphic. Moreover, the hitting distribution ν equals the probability doubling measure μ given in (5.4). Proof. We only need to prove ν = μ. Let Ux and Tx be as in the proof of the above theorem. We fix a projection ι : X → K that satisfies ι(x) ∈ Ux for each x ∈ X, and let Tx(m) =

⋃

y∈𝒥m (x)

Ty ,

x ∈ X, m ≥ 0.

168 | S.-L. Kong et al. (m) Then it is clear that ⋂∞ = Kx . For any m fixed, as dist(K \ Tx(m) , Kx ) > 0, the m=0 Tx event Z∞ ∈ Kx implies that ι(Zn ) lies eventually in Tx(m) . Using Fatou’s lemma, together with (5.5) and (5.8), we have

ν(Kx ) = ℙϑ (Z∞ ∈ Kx ) ≤ 𝔼ϑ (lim inf χT (m) (ι(Zn ))) n→∞

x

≤ lim inf ℙϑ (ι(Zn ) ∈ Tx(m) ) n→∞

≤ lim inf ℙϑ (ι(Zτℓ ) ∈ Tx(m) ) = μ(Tx(m) ). ℓ→∞

Letting m → ∞, it follows ν(Kx ) ≤ μ(Kx ) for all x ∈ X, which implies ν(F) ≤ μ(F) for any Borel set F ⊂ K; the same “≤” holds for K \ F. Hence ν(F) = μ(F), and completes the proof. ̂ we define For distinct ξ , η ∈ X ∪ K(≈ X), pμ (ξ , η) = sup{μ(Kz ) : z ∈ X and lies on some geodesic π(ξ , η)}. It is easy to see that the supremum can be reached at a vertex on the horizontal segment of some convex geodesic between ξ and η. Moreover, this pμ (⋅, ⋅) satisfies the estimate pμ (ξ , η) ≍ V(ξ , η) := μ(B(ξ , |ξ − η|)),

∀ξ , η ∈ K, ξ ≠ η.

(5.9)

̃ Theorem 5.8. Let {Zn }∞ n=0 be a λ-NRW on an augmented tree (X, ℰ ). Then F(x, y) ≍ λ|x|−(x|y) μ(Ky )pμ (x, y)−1 ,

∀x, y ∈ X.

Consequently, the Martin kernel satisfies the estimate K(x, η) ≍ λ|x|−(x|η) pμ (x, η)−1 ,

∀x ∈ X, η ∈ X ∪ K.

Proof. For x, y ∈ X, let π(x, u, v, y) be a convex geodesic on which a vertex w lies on the horizontal segment π(u, v) with μ(Kw ) = pμ (x, y). By observing that F(x, z) ≥ F(x, z)F(z, y) for z ∈ X together with (5.2) in Theorem 5.3, we have F(x, y) ≍ F(x, u)F(u, v)F(v, y) ≍

F(ϑ, y) F(x, ϑ) ⋅ F(u, v) ⋅ . F(u, ϑ) F(ϑ, v)

As the length of π(u, v) does not exceed L (Theorem 2.6), we see that pL∗ ≤ F(u, v) ≤ 1 (here p∗ := inf(x,y)∈ℰ̃ P(x, y) > 0), and the volume doubling property of μ implies that μ(Kv ) ≍ μ(Kw ) = pμ (x, y). From the estimates (5.3) and (5.8), it follows that F(x, y) ≍ λ|x|−|u| μ(Ky )μ(Kv )−1 ≍ λ|x|−(x|y) μ(Ky )pμ (x, y)−1 . F(x,y) This leads to the estimate K(x, y) = F(ϑ,y) ≍ λ|x|−(x|y) pμ (x, y)−1 for x, y ∈ X, and passing the limit along some ray [yi ]i that converges to η, it can be extended to X × K.

Hyperbolic graphs induced by iterations and applications in fractals | 169

K(x,η)

From the above estimates, it is easy to see that Θ(x, η) = F(x,ϑ)G(ϑ,ϑ) ≍ λ−(x|η) pμ (x, η)−1 for all x ∈ X and η ∈ K. By using a similar technique as in [38, Theorem 6.3], we can analyze the limit in (5.1), and extend such Naïm kernel estimate to K × K. ̃ Theorem 5.9. Let {Zn }∞ n=0 be a λ-NRW on an augmented tree (X, ℰ ). Then the Naïm kernel satisfies the estimate Θ(ξ , η) ≍ λ−(ξ |η) pμ (ξ , η)−1 ,

∀ξ , η ∈ K, ξ ≠ η.

Consequently, by (3.3) and (5.9), we have Θ(ξ , η) ≍

1 , V(ξ , η)|ξ − η|β

∀ξ , η ∈ K, ξ ≠ η,

where μ is the doubling measure associated with the λ-NRW as in (5.4), V(ξ , η) := log λ μ(B(ξ , |ξ − η|)), and β = log . r ∗

In particular, if μ is chosen to be the normalized α-Hausdorff measure on K (where α is the Hausdorff dimension of K), then μ(B(ξ , r)) ≍ r α for all ball B(ξ , r) ⊂ K, and the above estimate becomes Θ(ξ , η) ≍ |ξ − η|−(α+β) [38]. Applying Silverstein’s theorem (Theorem 5.2) together with Theorems 5.7 and 5.9, we get ̃ Theorem 5.10. Let {Zn }∞ n=0 be a λ-NRW on an augmented tree (X, ℰ ). Then the induced energy form (EK , 𝒟K ) satisfies EK [u] := EX [Hu] ≍ ∬ K×K\Δ

|u(ξ ) − u(η)|2 dμ(ξ )dμ(η), V(ξ , η)|ξ − η|β

where 𝒟K := {u ∈ L2 (K, μ) : EK [u] < ∞}, and β =

∀u ∈ 𝒟K ,

log λ . log r∗

Remark. The above theorems also hold for (X, ℰ ) with property (H), as in this case ℰ = ℰ ̃ as in Proposition 3.4. β/2

The domain 𝒟K is equal to a Besov space Λ2,2 (see [38, 39]); it is decreasing in β, and can be trivial (i. e., consists of only constant functions) when β is large. As EK defines a symmetric bilinear form EK (⋅, ⋅) via the standard polarization, we are interested in the conditions for (EK , 𝒟K ) to be nontrivial, or becomes a regular (nonlocal) Dirichlet form on L2 (K, μ) [16]. The key for this is to determine the value of the critical exponents β/2

β♯ := sup{β > 0 : dim(Λ2,2 ∩ C(K)) > 1}, β := sup{β > 0 : ∗

β/2 Λ2,2

and

∩ C(K) is dense in C(K) with the supremum norm},

where C(K) is the family of all continuous functions on K. It is known that 2 ≤ β∗ ≤ β♯ in general; for classical domains in ℝd with Lebegue measure, β∗ = β♯ = 2; for Cantortype sets, β∗ = β♯ = ∞; for the d-dimensional Sierpiński gasket with α-Hausdorff

170 | S.-L. Kong et al. measure (here α =

log(d+1) log 2 ♯

is the Hausdorff dimension), β∗ = β♯ =

log(d+3) log 2

[28]; some

examples with β < β are provided in [20, 36]. Moreover, if K satisfies a chain condition in [18], then β∗ ≤ β♯ ≤ d̄ μ + 1, where d̄ μ is the upper dimension given by ∗

d̄ μ = inf{α > 0 : ∃c > 0 such that μ(B(ξ , r)) ≥ cr α ∀ξ ∈ K and r ∈ (0, 1)}. d It is also known that 𝒟K ⊂ C(K) when β > d̄ μ (i. e., λ < r∗μ ). As a consequence, in the ∗ case that β > d̄ μ , (EK , 𝒟K ) is a regular Dirichlet form for any β ∈ (d̄ μ , β∗ ). We provide an approach to these critical exponents by using the networks of NRWs (see details in [36]). For each λ ∈ (0, 1), in view of Theorem 5.6, we fix a conductance c(λ) (⋅, ⋅) on (X, ℰ̃) that defines a λ-NRW with a given doubling measure μ: ̄

c(λ) (x, x− ) = λ−|x| μ(Kx ), x ∈ X \ {ϑ},

c(λ) (x, y) = λ−|x| √μ(Kx )μ(Ky ), (x, y) ∈ ℰ̃h . (5.10)

For m ≥ 1, by restricting the graph energy to ⋃m i=0 Xi , we let E(λ) X,m [f ] =

1 󵄨 󵄨2 c(λ) (x, y)󵄨󵄨󵄨f (x) − f (y)󵄨󵄨󵄨 ∑ 2 x,y∈X:|x|,|y|≤m

for a real function f on X, and define the level-m resistance by (λ) R(λ) m (x, y) = (inf{EX,m [f ] : f (x) = 1, f (y) = 0}) , −1

x, y ∈ Xm .

To represent the resistance on K by a limit, we choose a sequence {κm }∞ m=0 , in which κm is a map from K to Xm , satisfying that for any ξ ∈ K(≈ 𝜕X), {κm (ξ )}∞ m=0 is a geodesic ray converging to ξ . Define R(λ) (ξ , η) := lim inf R(λ) m (κm (ξ ), κm (η)), m→∞

ξ , η ∈ K.

d̄

With the assumption λ < r∗μ , it can be proved that the above limit always exists (hence the “lim inf” can be replaced by “lim”), and is independent of the choice of {κm }; in this case, R(λ) (ξ , η) > 0 if and only if there exists u ∈ 𝒟K such that u(ξ ) ≠ u(η). We will further assume that the measure μ is self-similar, i. e., μ(⋅) = ∑j∈Σ pj μ(Sj−1 (⋅)) for some set {pj }j∈Σ of positive probability weights. For j ∈ Σ, denote by j∞ the unique fixed point of the contractive similitude Sj , i. e., {j∞ } = ⋂∞ n=0 Kjn . Theorem 5.11. Suppose {Sj }j∈Σ is an IFS of contractive similitudes satisfying the OSC, and μ is a doubling self-similar measure on the attractor K. For λ ∈ (0, 1), let {Zn } be the λ-NRW on (X, ℰ̃) defined by the conductance c(λ) (⋅, ⋅) as in (5.10). Then 𝒟K ∩C(K) consists of only constant functions if R(λ) (i∞ , j∞ ) = 0 for all i, j ∈ Σ, and the converse is also true d̄

for λ ∈ (0, r∗μ ). Consequently, β♯ =

log λ♯ log r∗

if d̄

λ♯ := sup{λ > 0 : R(λ) (i∞ , j∞ ) = 0, ∀i, j ∈ Σ} ∈ (0, r∗μ ), and β♯ = ∞ if the above set of λ is empty.

Hyperbolic graphs induced by iterations and applications in fractals | 171

We also have a result for β∗ when K is a p. c. f. set that satisfies (⋆) there exist constants r0 , C > 0 such that for any i, j ∈ Σ and ζ ∈ Ki ∩ Kj , |ξ − ζ | + |ζ − η| ≤ C|ξ − η|

whenever ξ ∈ Ki ∩ B(ζ , r0 ) and η ∈ Kj ∩ B(ζ , r0 ).

This condition (⋆) is fulfilled for most of familiar p. c. f. sets including all nested fractals. Let V0 denote the projection of the post critical set, known as the boundary of K. Theorem 5.12. With the same assumption as in Theorem 5.11, assume further that K is d̄

p. c. f. and satisfies (⋆). If λ ∈ (0, r∗μ ), and for some ε ∈ (0, λ), R(λ−ε) (ξ , η) > 0,

∀ξ , η ∈ V0 , ξ ≠ η,

then 𝒟K is dense in C(K), and hence (EK , 𝒟K ) is a regular nonlocal Dirichlet form. λ∗ if Consequently, β∗ = log log r ∗

d̄

λ∗ := inf{λ > 0 : R(λ) (ξ , η) > 0, ∀ξ , η ∈ V0 , ξ ≠ η} ∈ [0, r∗μ ), and β∗ ≤

log λ∗ log r∗

otherwise.

6 Expansive hyperbolic graphs We establish a class of graphs called expansive hyperbolic graphs, which covers the various augmented trees considered, and includes cases not governed by the IFS, like refinement systems. It has the potential to have broader applications. Definition 6.1. We call a rooted graph (X, ℰ ) an expansive graph if it satisfies for x, y ∈ X with |x| = |y|, dh (x, y) > 1 ⇒ dh (u, v) > 1,

∀u ∈ 𝒥1 (x), v ∈ 𝒥1 (y),

or equivalently, if each u ∼h v with u ∈ 𝒥1 (x) and v ∈ 𝒥1 (y) implies that x ∼h y. It is easy to see that the above condition is also equivalent to max{dh (u, v), 1} ≥ dh (x, y),

u ∈ 𝒥1 (x), v ∈ 𝒥1 (y).

(6.1)

Intuitively, in an expansive rooted graph the children are drifted farther apart than their non-neighboring parents. Note that the expansive property is also equivalent to property (*) in Lemma 2.5 if the vertical subgraph (X, ℰv ) is a tree. There are important cases that the vertical parts of expansive graphs are not trees: for example, the treatment of the IFS with a weak separation condition by taking quotients of vertices on the augmented tree (X, ℰ ) [58]. By the same argument as in Lemma 2.5, we see that any two vertices x, y ∈ X can be connected by a convex geodesic. To study the hyperbolicity of (X, ℰ ), we introduce one more definition.

172 | S.-L. Kong et al. Definition 6.2. Let m, k be two positive integers. A rooted graph (X, ℰ ) is said to be (m, k)-departing if for x, y ∈ X, dh (x, y) > k ⇒ dh (u, v) > 2k,

∀u ∈ 𝒥m (x), v ∈ 𝒥m (y).

It follows from the definitions that every (1, 1)-departing graph is expansive; every rooted tree is (m, k)-departing for any m, k. However, an infinite expansive graph may not be (m, k)-departing for any m, k. It is direct to check that (m, k)-departing ⇒ (m, ℓk)-departing, and (ℓ󸀠 m, k)-departing ∀ℓ, ℓ󸀠 > 1.

(6.2)

In particular, (1, 1)-departing implies (m, k)-departing for any m, k ≥ 1. As an example, we can show that the augmented tree (X, ℰ ) of the Sierpinski gasket (see [30]) is (1, 1)-departing. With a little more work, we can show that the augmented tree (X, ℰ ) of the Hata tree (see [32]) is (2, 1)-departing, but not (1, 1)-departing. The (m, k)-departing property provides very useful criteria to check the hyperbolicity. Theorem 6.3. Let (X, ℰ ) be an expansive graph. Then the following are equivalent: (i) (X, ℰ ) is hyperbolic; (ii) ∃L < ∞ such that the lengths of all h-geodesics are bounded by L; (iii) (X, ℰ ) is (m, k)-departing for some positive integers m, k. Proof. (i) ⇔ (ii) follows from a similar proof as in Theorem 2.6. (ii) ⇒ (iii) We claim that (X, ℰ ) is (L + 1, L + 2)-departing. Indeed, let x, y ∈ X, x󸀠 ∈ 𝒥L+1 (x) and y󸀠 ∈ 𝒥L+1 (y) satisfying L + 2 < dh (x 󸀠 , y󸀠 ) ≤ 2(L + 2) (see Figure 7). By the expansive property (as in Proposition 2.5), there exists a convex geodesic π(x 󸀠 , u, v, y󸀠 ) between x󸀠 and y󸀠 , and u ≠ x󸀠 (by the first inequality and (ii)). Let u, v ∈ Xj , then 󵄨 󵄨 󵄨 󵄨 2(L + 2) ≥ dh (x󸀠 , y󸀠 ) > d(x󸀠 , y󸀠 ) = 2(󵄨󵄨󵄨x 󸀠 󵄨󵄨󵄨 − |j|) + dh (u, v) ≥ 2(󵄨󵄨󵄨x 󸀠 󵄨󵄨󵄨 − j). As ℓ := |x󸀠 | − j ≤ L + 1 = |x 󸀠 | − |x|, we have j ≥ |x|. Let u󸀠 ∈ 𝒥∗ (x) ∩ 𝒥−∗ (x 󸀠 ) ∩ Xj and v󸀠 ∈ 𝒥∗ (y) ∩ 𝒥−∗ (y󸀠 ) ∩ Xj . Since x󸀠 ∈ 𝒥ℓ (u) ∩ 𝒥ℓ (u󸀠 ), u and u󸀠 are predecessors of x 󸀠 , and we have u ∼h u󸀠 by the expansive property. Similarly v ∼h v󸀠 . Hence by (6.1) and (ii), dh (x, y) ≤ max{dh (u󸀠 , v󸀠 ), 1} ≤ dh (u, v) + 2 ≤ L + 2. This proves the claim. (iii) ⇒ (ii) Suppose (X, ℰ ) is (m, k)-departing. If (ii) does not hold, then there exist x, y ∈ X, |x| = |y|, and horizontal geodesic π(x, y) such that |π(x, y)| = 2m+ℓk +1, where ℓ > (2m + 1)/k is an integer. It is clear that 2|x| = d(x, ϑ) + d(ϑ, y) ≥ d(x, y) > 2m, which implies |x| > m. Let x[−m] ∈ 𝒥−m (x) and y[−m] ∈ 𝒥−m (y). Computing the length of the path joining x and y with x to x[−m] , horizontally to y[−m] , then to y, we have m + dh (x[−m] , y[−m] ) + m ≥ d(x, y) = 2m + ℓk + 1.

Hyperbolic graphs induced by iterations and applications in fractals | 173

Figure 7: Illustration for the proof of (ii) ⇒ (iii).

It follows that dh (x[−m] , y[−m] ) > ℓk. Hence by (6.2), we have dh (x, y) = d(x, y) > 2ℓk, i. e., 2m + ℓk + 1 > 2ℓk, which leads to ℓ < (2m + 1)/k, a contradiction. We will call the graph in the above theorem an expansive hyperbolic graph. The (m, k)-departing property also provides a useful estimate of the Gromov product. Let ℛv = {x = [xi ]∞ i=0 : x0 = ϑ, and xi+1 ∈ 𝒥1 (xi ), ∀i ≥ 0} be the set of all rays in (X, ℰ ). For any two rays x, y ∈ ℛv , we define |x ∨ y|j = sup{i ≥ 0 : dh (xi , yi ) ≤ j}. Lemma 6.4. Suppose (X, ℰ ) is expansive and (m, k)-departing. Then there exists D0 > 0 (depending on m, k) such that |(x|y) − |x ∨ y|k | ≤ D0 ,

∀x, y ∈ ℛv .

Moreover, two rays are equivalent if and only if dh (xi , yi ) ≤ k for all i. On the hyperbolic boundary (𝜕X, θa ), we define 𝒥𝜕 (x) = {ξ ∈ 𝜕X : ∃ ray π(ϑ, . . . , x, . . . ) that converges to ξ },

x ∈ X,

to be the set of descendants of x in 𝜕X. Under the Gromov metric θa , |𝒥𝜕 (x)| ≤ Ce−a|x| and is compact. This 𝒥𝜕 (x) acts as the Kx in the augmented tree of the IFS. By using Lemma 6.4, we have an analog of the condition (H) in Section 3. Proposition 6.5. Let (X, ℰ ) be an (m, k)-departing expansive graph. Then there exists a constant γ > 0 (depending on a) such that for x, y ∈ Xn , n ≥ 1, dh (x, y) > k ⇒ distθa (𝒥𝜕 (x), 𝒥𝜕 (y)) > γe−an . A metric space (M, ρ) is called a doubling metric space [24] if there exists an integer ℓ > 0 such that for any ξ ∈ M and r > 0, the ball Bρ (ξ , r) can be covered by a union of not more than ℓ balls of radius r/2. We prove the following interesting theorem using Theorem 6.3, Lemma 6.4 and Proposition 6.5. Theorem 6.6. Suppose (X, ℰ ) is a hyperbolic expansive graph and has bounded degree. Then the hyperbolic boundary (𝜕X, θa ) is doubling.

174 | S.-L. Kong et al. In the rest of this section, we will consider a generalization of the augmented trees. Fix a complete metric space (M, ρ), and let 𝒞M denote the family of all nonempty compact subsets of M. By our convention in Section 2, 𝒥𝜕 (x) ≠ 0 for all x ∈ X. Definition 6.7. Let (X, ℰv ) be a vertical rooted graph. A map Φ : X → 𝒞M is called an index map (on (X, ℰv ) over (M, ρ)) if it satisfies (i) Φ(y) ⊂ Φ(x) for all x ∈ X and y ∈ 𝒥1 (x); (ii) ⋂∞ i=0 Φ(xi ) is a singleton for all x = [xi ]i ∈ ℛv . We call K := ⋂∞ n=0 (⋃x∈Xn Φ(x)) the attractor of Φ, and Kx := Φ(x) ∩ K a cell of K. We also call the index map saturated if Φ(x) = ⋃y∈𝒥1 (x) Φ(y). Remark 1. For an index map Φ, let Φ𝜕 (x) := ⋂∞ n=0 (⋃y∈𝒥n (x) Φ(y)), x ∈ X. Then Φ𝜕 is a saturated index map, and Φ𝜕 (x) ⊂ Φ(x). If Φ is saturated, then Φ(x) = Kx = Φ𝜕 (x). The index map Φ defines a mapping κ0 : ℛv → K by ∞

{κ0 (x)} = ⋂ Φ(xi ), i=0

∀x ∈ ℛv .

Using the local finiteness of (X, ℰv ) and a diagonal argument (see [60, 46]), we can show that the image of κ0 is equal to K. Note that for a hyperbolic expansive graph (X, ℰ ), the hyperbolic boundary 𝜕X can be identified with a quotient set of ℛv . Hence the induced κ : 𝜕X → K is well-defined if κ0 satisfies κ0 (x) = κ0 (y) provided that x and y are equivalent; furthermore, κ : 𝜕X → K is one-to-one if the converse is also satisfied. With these, we see that κ : 𝜕X → K is a well-defined bijection if x, y are equivalent ⇔ κ0 (x) = κ0 (y). Definition 6.8. We call (X, ℰ , Φ) an admissible index triple if (X, ℰ ) is an expansive hyperbolic graph, Φ : X → 𝒞M is an index map on (X, ℰv ) over (M, ρ), and κ : 𝜕X → K is well-defined and is a bijection. In such case, (X, ℰ ) is said to be an admissible graph (with respect to Φ). Remark 2. For an admissible index triple (X, ℰ , Φ), if Φ is saturated, then by Remark 1, we have Φ(x) = Φ𝜕 (x) = κ(𝒥𝜕 (x)), and the Gromov metric θa implies 󵄨󵄨 󵄨 󵄨 󵄨 −a|x| 󵄨󵄨Φ(x)󵄨󵄨󵄨θ̃a = 󵄨󵄨󵄨𝒥𝜕 (x)󵄨󵄨󵄨θa ≤ Ce where θ̃a is the metric on K induced by θa via the bijection κ : 𝜕X → K. For a subset A in (M, ρ), we denote the diameter of A by |A|ρ (or simply by |A|). In Definition 6.7, we see that the family {Φ(x)}x∈X satisfies limn→∞ supx∈Xn |Φ(x)|ρ = 0. For b ∈ (0, ∞), we say that {Φ(x)}x∈X (or Φ) is of exponential type-(b) (under ρ) if the diameter |Φ(x)|ρ is decreasing in a rate of e−b|x| , i. e., |Φ(x)|ρ = O(e−b|x| ) as |x| → ∞, and call Φ an exponential type if it is of type-(b) for some b ∈ (0, ∞).

Hyperbolic graphs induced by iterations and applications in fractals | 175

The following two classes of rooted graphs are our main consideration of the index triples, which generalize the two augmented trees in Definition 3.1: the exponential type-(b) corresponds to the r∗ in (3.2) with r∗ = e−b . Definition 6.9. Let Φ be an index map on the vertical rooted graph (X, ℰv ). We define a horizontal edge set by ∞

:= ⋃ {(x, y) ∈ Xn × Xn \ Δ : Φ(x) ∩ Φ(y) ≠ 0},

(∞)

ℰh

n=1

and let ℰ (∞) = ℰv ∪ ℰh(∞) . We call (X, ℰ (∞) ) an AI∞ -graph, augmented index graph of type-(∞) (or intersection type). Suppose in addition Φ is of exponential type-(b). Then for a fixed γ > 0, we define (b)

ℰh

∞

:= ⋃ {(x, y) ∈ Xn × Xn \ Δ : distρ (Φ(x), Φ(y)) ≤ γe−bn }, n=1

(6.3)

and let ℰ (b) = ℰv ∪ ℰh(b) . We call (X, ℰ (b) ) an AIb -graph, augmented index graph of type-(b). It is clear that both (X, ℰ (b) ) and (X, ℰ (∞) ) are expansive. First we consider the AIb -graphs. Theorem 6.10. For an index map Φ on (X, ℰv ) over (M, ρ) of exponential type-(b), the associated AIb -graph is (m, 1)-departing for some positive integer m, and is an admissible graph. Moreover, κ : (𝜕X, θa ) → (K, ρ) is a Hölder equivalence, i. e., a/b

ρ(κ(ξ ), κ(η))

≍ θa (ξ , η),

∀ξ , η ∈ 𝜕X.

(6.4)

Proof. To show that it is (m, 1)-departing for some m ≥ 1, let δ0 := supz∈X eb|z| |Φ(z)|. Let u ∈ 𝒥m (x) and v ∈ 𝒥m (y) with dh (u, v) = 2. Using the triangle inequality twice, we have dist(Φ(x), Φ(y)) ≤ dist(Φ(u), Φ(v)) ≤ (2γ + δ0 )e−b(|x|+m) ≤ γe−b|x| , where the positive integer m is chosen to give the last inequality, i. e., (2γ +δ0 )e−bm ≤ γ. Therefore x ∼h y, and this shows that (X, ℰ ) is (m, 1)-departing. The hyperbolicity of AIb -graph follows from Theorem 6.3. By Lemma 6.4 (with k = 1) and (6.3), we see that two rays x, y are equivalent if and only if dist(Φ(xi ), Φ(yi )) ≤ γe−bi for all i (i. e., κ0 (x) = κ0 (y)). This implies κ : 𝜕X → K is a well-defined bijection, and (X, ℰ (b) ) is an admissible graph. We now prove that κ is a Hölder equivalence. For distinct ξ , η ∈ 𝜕X, we take two rays x, y ∈ ℛv that converge to ξ , η, respectively, with (ξ |η) = (x|y). Let n = |x ∨ y|1 as in Lemma 6.4 with k = 1, i. e., dh (xn , yn ) ≤ 1 and dh (xn+1 , yn+1 ) ≥ 2. By Lemma 6.4, we have |(ξ |η) − n| = |(x|y) − n| ≤ D0 for some D0 > 0. As κ(ξ ) ∈ Φ(xn+1 ) ⊂ Φ(xn ) and

176 | S.-L. Kong et al. κ(η) ∈ Φ(yn+1 ) ⊂ Φ(yn ), we get the lower bound of (6.4) by ρ(κ(ξ ), κ(η)) ≥ distρ (Φ(xn+1 ), Φ(yn+1 ))

≥ γe−b(n+1) ≥ γe−b(D0 +1) e−b(ξ |η) ≥ C1 θa (ξ , η)b/a ,

and the upper bound by 󵄨 󵄨 󵄨 󵄨 ρ(κ(ξ ), κ(η)) ≤ 󵄨󵄨󵄨Φ(xn )󵄨󵄨󵄨 + dist(Φ(xn ), Φ(yn )) + 󵄨󵄨󵄨Φ(yn )󵄨󵄨󵄨

≤ (2δ0 + γ)e−bn ≤ (2δ0 + γ)ebD0 e−b(ξ |η) ≤ C2 θa (ξ , η)b/a .

This completes the proof. Now we turn to the study of the AI∞ -graphs. Unlike the AIb -graph, the AI∞ -graph is not always hyperbolic. Proposition 6.11. Suppose the index map Φ is of exponential type-(b), and the associated AI∞ -graph (X, ℰ (∞) ) is hyperbolic. Then (X, ℰ (∞) ) is an admissible graph, and κ : (𝜕X, θa ) → (K, ρ) is Hölder continuous, i. e., a/b

ρ(κ(ξ ), κ(η))

≤ Cθa (ξ , η),

∀ξ , η ∈ 𝜕X.

(6.5)

Proof. Note that (X, ℰ (∞) ) is a subgraph of (X, ℰ (b) ). On the AIb -graph (X, ℰ (b) ), let us denote its graph distance and Gromov product by d󸀠 (⋅, ⋅) and (⋅|⋅)󸀠 , respectively. By The󸀠 orem 6.10, κ󸀠 : (𝜕X 󸀠 , θa󸀠 ) → K is a bijection, and satisfies ρ(κ 󸀠 (ξ ), κ 󸀠 (η)) ≍ e−b(ξ |η) for all ξ , η ∈ 𝜕X 󸀠 . As κ = κ󸀠 on ℛv , it follows that κ = κ󸀠 on 𝜕X = 𝜕X 󸀠 is a well-defined bijection, and (X, ℰ (∞) ) is admissible. From ℰ (∞) ⊂ ℰ (b) , it follows that d(x, y) ≥ d󸀠 (x, y), and 1 1 (x|y) = (|x| + |y| − d(x, y)) ≤ (|x| + |y| − d󸀠 (x, y)) = (x|y)󸀠 2 2

∀x, y ∈ X.

Taking limits, we have (ξ |η) ≤ (ξ |η)󸀠 , and a/b

󸀠

θa (ξ , η) ≥ c1 e−a(ξ |η) ≥ c1 e−a(ξ |η) ≥ c2 ρ(κ󸀠 (ξ ), κ 󸀠 (η))

a/b

= c2 ρ(κ(ξ ), κ(η))

.

This verifies (6.5), the Hölder continuity of κ. The following is a characterization of the Hölder equivalence of 𝜕X to K for the AI∞ -graph associated to some saturated Φ. Theorem 6.12. Suppose Φ is a saturated index map on (X, ℰv ) over (M, ρ). Then for b ∈ (0, ∞) and an integer k > 0, the following assertions are equivalent. (i) The AI∞ -graph (X, ℰ (∞) ) is (m, k)-departing for some m > 0, and κ : (𝜕X, θa ) → (K, ρ) is a Hölder equivalence with exponent b/a, i. e., a/b

ρ(κ(ξ ), κ(η))

≍ θa (ξ , η),

∀ξ , η ∈ 𝜕X.

(ii) Φ is of exponential type-(b) under ρ, and there exists γ > 0 such that (X, ℰ (∞) ) satisfies for x, y ∈ X, |x| = |y| and

dh (x, y) > k ⇒ distρ (Φ(x), Φ(y)) > γe−b|x| .

Hyperbolic graphs induced by iterations and applications in fractals | 177

Remark 3. For k = 1, the above condition (ii) is just the condition (H) on self-similar sets in Section 3. In comparison with Theorem 3.2, the above theorem gives a more complete criterion for the AI∞ -graph on Hölder equivalence of 𝜕X to K. In the following, we give two sufficient conditions for hyperbolicity of the AI∞ -graph. We first define two separation conditions. Let Φ be an index map on a vertical graph (X, ℰv ) over a complete metric space (M, ρ) with attractor K. We call a map ι : X → K a projection (with respect to Φ) if it satisfies ι(x) ∈ Kx (:= Φ(x) ∩ K) for all x ∈ X. Definition 6.13. For b ∈ (0, ∞), we say that Φ (or {Kx }x∈X ) satisfies ̄ such that (i) condition (Sb ) if for any c > 0, there is a constant ℓ̄ = ℓ(c) #{x ∈ Xn : Kx ∩ F ≠ 0} ≤ ℓ,̄

∀n ≥ 0 and F ⊂ M with |F|ρ < ce−bn ;

(ii) condition (Bb ) if there exist a projection ι : X → K and c0 ∈ (0, ∞) such that Bρ (ι(x), c0 e−b|x| ) ∩ K ⊂ Kx ,

∀x ∈ X.

Note that (Sb ) is an analog of (S) in Section 3, and they are equivalent when M is ℝd (see [44, Theorem 2.1(iii), (iv)]) or any other doubling metric space. It is well-known that for the OSC on self-similar sets in ℝd , the desired open set O can be chosen to satisfy O ∩ K ≠ 0 [56]. Then by taking ξ ∈ O ∩ K, a ball Bρ (ξ , r) ⊂ O and ι(x) = Sx (ξ ) for x ∈ X, we see that the OSC implies condition (Bb ). Similar to Theorem 3.5, we have Theorem 6.14. Let Φ be an index map with attractor K, and is of exponential type-(b). If either (i) condition (Sb ) is satisfied; or (ii) the attractor (K, ρ) is doubling, and condition (Bb ) is satisfied; then the AI∞ -graph is hyperbolic and hence an admissible graph. By using (Sb ), we can also obtain an analog of Theorem 3.7. Theorem 6.15. Let Φ be an index map with attractor K, and is of exponential type-(b). Then the AIb -graph has bounded degree if and only if condition (Sb ) is satisfied. Also the (Sb ) is sufficient for the AI∞ -graph to have bounded degree.

7 Remarks and future work In Section 3, we provided some sufficient conditions for the hyperbolicity of augmented tree (X, ℰ ). It is interesting to find an example of a self-similar IFS that gives a

178 | S.-L. Kong et al. nonhyperbolic (X, ℰ ). (Note that in [37], we constructed an AI∞ -graph (Definition 6.7) that is not hyperbolic; however, the example is not self-similar.) In the application of Gromov hyperbolic graphs to the Lipschitz equivalence problem in Section 4, we only showed the case that the self-similar sets have equal contraction ratios. Actually, by some minor modification of the matrix rearrangeable technique, we can also use it to deal with more general self-similar sets, say, IFS with multiple contraction ratios, and even with substantial overlaps [49]. Furthermore, this technique can be used for classification of certain fractal squares with nice overlapping structures [52]. In Section 5, we discussed the λ-NRW on the hyperbolic graph (X, ℰ̃) and obtained a nonlocal regular Dirichlet form EK (i. e., the induced energy form) on the attractor K β/2 log λ with domain Λ2,2 where β = log (Theorem 5.10). There is further functional relationr∗ ship of the graph energy EX and the induced energy EK studied in [36, Section 3]. By varying the return ratio λ, we obtain a critical exponent λ∗ , which is of crucial importance: for the classical examples in analysis on fractals, this value gives another Besov β∗ /2 space Λ2,∞ as the domain of the local regular Dirichlet form (LRDF) (or equivalently, Laplacian) [28, 53, 54]. Recently, Grigor’yan and Yang [19] gave an analytic proof of the existence of the β/2 β/2 LRDF on the Sierpinski carpet using the Γ-convergence of the Λ2,2 -norm to Λ2,∞ -norm ∗ as β ↗ β . This approach was used in [19, 20, 22, 61] to p. c. f. sets, and some nonp. c. f. sets. It will be interesting to find out any limiting r. w. of λ-NRW as λ ↘ λ∗ that yields β ↗ β∗ . Also it is important to investigate the relationship of the Dirichlet form obtained from this approach of random walks on hyperbolic graphs and the one from the classical discrete approximation in analysis of fractals. In Section 6, we used index maps to establish the AIb -graphs as the generalization of the augmented tree (X, ℰ̃) in Section 3. It should be possible to consider the λ-NRW on the AIb -graph, and expect the same estimate for the induced energy forms. Also in here we have not yet discussed the near-isometry between hyperbolic graphs; in fact, the transform allows us to extend the scope to large classes of hyperbolic graphs, and enables us to consider the λ-NRW in these more general hyperbolic graphs. We expect the volume doubling property of the hitting distribution, and a similar estimate for the Naïm kernel in term of the Gromov metric. We will discuss this in detail in a forthcoming paper.

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Tim Jaschek and Mathav Murugan

Geometric implications of fast volume growth and capacity estimates Abstract: We obtain connectivity of annuli for a volume-doubling metric measure Dirichlet space which satisfies a Poincaré inequality, a capacity estimate, and a fast volume growth condition. This type of connectivity was introduced by Grigor’yan and Saloff-Coste in order to obtain stability results for Harnack inequalities and to study diffusions on manifolds with ends. As an application of our result, we obtain stability of the elliptic Harnack inequality under perturbations of the Dirichlet form with radial-type weights. Keywords: Annular connectivity, Poincaré inequality, capacity, Harnack inequality MSC 2010: Primary 54C40, 14E20, Secondary 46E25, 20C20

Contents 1 1.1 2 3 4

Introduction | 183 Main results | 185 Chain connectivity | 188 From chain connectivity to path connectivity | 192 Applications | 196 Bibliography | 198

1 Introduction In this work, we study geometric consequences of analytic properties in the context of metric measure spaces equipped with a Dirichlet form. We are interested in connectivity properties of metric spaces at various scales and locations for spaces that satisfy a Poincaré inequality, an upper bound on capacity, and certain conditions on volume growth. We obtain a connectivity condition on annuli introduced by Grigor’yan and Saloff-Coste [16]. A similar connectivity of annuli was used to obtain heat kernel bounds on a family of planar graphs in [23, Theorem 6.2(d)]. Acknowledgement: M. Murugan’s research partially supported by NSERC and the Canada research chairs program. Tim Jaschek, Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada; and 1QB Information Technologies (1QBit), Vancouver, BC V6E 4B1, Canada (current address), e-mail: [email protected] Mathav Murugan, Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada, e-mail: [email protected] https://doi.org/10.1515/9783110700763-007

184 | T. Jaschek and M. Murugan Much of the motivation for our work arises from analysis and probability on fractals. For a large class of fractal spaces (X, d), there exists a diffusion process which is symmetric with respect to some canonical measure m and exhibits strong subdiffusive behavior in the sense that its transition density (heat kernel) pt (x, y) satisfies the following sub-Gaussian estimates: 1

c1 d(x, y)β β−1 exp(−c ( ) ), pt (x, y) ≥ 2 t m(B(x, t 1/β )) 1

c3 d(x, y)β β−1 exp(−c ( ) ), pt (x, y) ≤ 4 t m(B(x, t 1/β ))

(1.1)

for all points x, y ∈ X and all t > 0, where c1 , c2 , c3 , c4 > 0 are some constants, d is a natural metric on X, B(x, r) denotes the open ball of radius r centered at x, and β ≥ 2 is an exponent describing the diffusion called the walk dimension. Often, m is a Hausdorff measure and is Ahlfors df -regular, that is, m(B(x, r)) ≍ r df for all x ∈ X and 0 < r < diam(X, d). The number df is called the volume growth exponent of the space. This result was obtained first for the Sierpiński gasket in [5], then for nested fractals in [19], for affine nested fractals in [8], and for Sierpiński carpets in [2]. We refer to [1] for a general introduction to diffusions on fractals. An important motivation for this work arises from a conjecture of Grigor’yan, Hu, and Lau [13, Conjecture 4.15], [14, p. 1495], see also [20, Open Problem III]. The conjecture is a characterization of the sub-Gaussian heat kernel estimate (1.1), in terms of the volume doubling property, a capacity upper bound, and a Poincaré inequality. The answer to this conjecture is known only in certain “low dimensional settings” (or strongly recurrent case) [3]. Recent progress had been made on a family of planar graphs [23] and on some transient graphs [25] but still under quite restricted assumptions. If we further assume that the measure m is Ahlfors df -regular, then the setting in [3] corresponds to df < β, where β is the walk dimension as described above. Roughly speaking, we consider spaces that are complementary to the “strongly recurrent” regime considered in [3]. In the case of polynomial volume growth as described above, our “fast volume growth” condition corresponds to the complementary case df ≥ β while [3] considers df < β; see Definition 1.5. Since this is the case where [13, Conjecture 4.15] is still open, we hope our work will simulate further progress on the conjecture of Grigor’yan, Hu, and Lau. We briefly survey some previous related works. A major milestone in the understanding of heat kernel bounds and Harnack inequalities is the characterization of the parabolic Harnack inequality by the combination of the volume-doubling property and the Poincaré inequality due to Grigor’yan and Saloff-Coste [10, 26]. Such a characterization implies the stability of the parabolic Harnack inequality under bounded perturbation of the Dirichlet form. More recently, the understanding of geometric consequences of analytic properties has played an important role in works on the stability of elliptic Harnack inequality and the singularity of energy measures. In particular, a

Geometric implications of fast volume growth and capacity estimates | 185

crucial step in obtaining the stability of elliptic Harnack inequality in [4] is the result that any geodesic space that satisfies the elliptic Harnack inequality admits a doubling measure. In [21], the chain condition plays a role in the proof of the singularity of the energy measures for spaces satisfying the sub-Gaussian heat kernel estimate. In [24], the chain condition was obtained as a consequence of the sub-Gaussian heat kernel estimate.

1.1 Main results Throughout this paper, we consider a complete, locally compact separable metric space (X, d), equipped with a Radon measure m with full support, that is, a Borel measure m on X which is finite on any compact set and strictly positive on any nonempty open set. Such a triple (X, d, m) is referred to as a metric measure space. In what follows, we set diam(X, d) := supx,y∈X d(x, y) and B(x, r) := {y ∈ X | d(x, y) < r} for x ∈ X and r > 0. Let (ℰ , ℱ ) be a symmetric Dirichlet form on L2 (X, m). In other words, the domain ℱ is a dense linear subspace of L2 (X, m), such that ℰ : ℱ × ℱ → ℝ is a nonnegative definite symmetric bilinear form which is closed (ℱ is a Hilbert space under the inner product ℰ1 (⋅, ⋅) := ℰ (⋅, ⋅) + ⟨⋅, ⋅⟩L2 (X,m) ) and Markovian (the unit contraction operates on ℱ , that is, (u ∨ 0) ∧ 1 ∈ ℱ and ℰ ((u ∨ 0) ∧ 1, (u ∨ 0) ∧ 1) ≤ ℰ (u, u) for any u ∈ ℱ ). Recall that (ℰ , ℱ ) is called regular if ℱ ∩ 𝒞c (X) is dense both in (ℱ , ℰ1 ) and in (𝒞c (X), ‖ ⋅ ‖sup ). Here 𝒞c (X) is the space of ℝ-valued continuous functions on X with compact support. For a function u ∈ ℱ , let suppm [u] ⊂ X denote the support of the measure |u| dm, that is, the smallest closed subset F of X with ∫X\F |u| dm = 0. Note that suppm [u] coin-

cides with the closure of X \ u−1 ({0}) in X if u is continuous. Recall that (ℰ , ℱ ) is called strongly local if ℰ (u, v) = 0 for any u, v ∈ ℱ with suppm [u], suppm [v] compact and v is constant m-almost everywhere in a neighborhood of suppm [u]. The pair (X, d, m, ℰ , ℱ ) of a metric measure space (X, d, m) and a strongly local, regular symmetric Dirichlet form (ℰ , ℱ ) on L2 (X, m) is termed a metric measure Dirichlet space, or an MMD space. We refer to [9, 6] for a comprehensive account of the theory of symmetric Dirichlet forms. We recall the notion of curves and path connectedness in a metric space. Definition 1.1 (Path connected). Let (X, d) be a metric space and let A ⊂ X. We say that γ is a curve in A from x to y if γ : [0, 1] → A is continuous, γ(0) = x and γ(1) = y. For two sets B1 ⊂ B2 ⊂ X we say that B1 is path connected in B2 if for all x, y ∈ B1 , there exists a curve in B2 from x to y. Henceforth, we fix a function Ψ : (0, ∞) → (0, ∞) to be a continuous increasing bijection of (0, ∞) onto itself, such that for all 0 < r ≤ R, β

β

R 1 Ψ(R) R 2 C −1 ( ) ≤ ≤ C( ) , r Ψ(r) r

(1.2)

186 | T. Jaschek and M. Murugan for some constants 0 < β1 < β2 and C > 1. Throughout this work, the function Ψ is meant to denote the space–time scaling of a symmetric diffusion process. Definition 1.2 (Volume doubling). We say that (X, d, m) satisfies the volume doubling property (VD) if there exists CD ≥ 1 such that m(B(x, 2r)) ≤ CD m(B(x, r)),

for all x ∈ X, r > 0.

(VD)

We recall the definition of energy measures associated to an MMD space. Note that fg ∈ ℱ for any f , g ∈ ℱ ∩L∞ (X, m) by [9, Theorem 1.4.2-(ii)] and that {(−n)∨(f ∧n)}∞ n=1 ⊂ ℱ and limn→∞ (−n) ∨ (f ∧ n) = f in norm in (ℱ , ℰ1 ) by [9, Theorem 1.4.2-(iii)]. Let (X, d, m, ℰ , ℱ ) be an MMD space. The energy measure Γ(f , f ) of f ∈ ℱ is defined, first for f ∈ ℱ ∩ L∞ (X, m) as the unique ([0, ∞]-valued) Borel measure on X with the property that 1 ∫ g dΓ(f , f ) = ℰ (f , fg) − ℰ (f 2 , g), 2

for all g ∈ ℱ ∩ 𝒞c (X),

(1.3)

X

and then by Γ(f , f )(A) := limn→∞ Γ((−n) ∨ (f ∧ n), (−n) ∨ (f ∧ n))(A) for each Borel subset A of X for general f ∈ ℱ . The notion of an energy measure can be extended to the local Dirichlet space ℱloc , which is defined as 󵄨󵄨 󵄨 For any relatively compact open subset V of X, there 2 ℱloc := {f ∈ Lloc (X, m) 󵄨󵄨󵄨󵄨 }. 󵄨󵄨 exists f # ∈ ℱ such that f 1V = f # 1V m-a. e. (1.4) For any f ∈ ℱloc and for any relatively compact open set V ⊂ X, we define Γ(f , f )(V) := Γ(f # , f # )(V), where f # is as in the definition of ℱloc . Since (ℰ , ℱ ) is strongly local, the value of Γ(f # , f # )(V) does not depend on the choice of f # , and is therefore well defined. Since X is locally compact, this defines a Radon measure Γ(f , f ) on X. Definition 1.3 (Poincaré inequality). We say that (X, d, m, ℰ , ℱ ) satisfies the Poincaré inequality PI(Ψ), if there exist constants CP , A ≥ 1 such that for all x ∈ X, r ∈ (0, ∞) and f ∈ ℱloc , ∫ (f − f )2 dm ≤ CP Ψ(r)

∫ dΓ(f , f ),

PI(Ψ)

B(x,Ar)

B(x,r)

where f = m(B(x, r))−1 ∫B(x,r) f dm. The following elementary observation will be used along with the Poincaré inequality: ∫ (f − f )2 dm = B(x,r)

1 ∫ 2m(B(x, r))

󵄨 󵄨2 ∫ 󵄨󵄨󵄨f (y) − f (z)󵄨󵄨󵄨 m(dy) m(dz).

B(x,r) B(x,r)

(1.5)

Geometric implications of fast volume growth and capacity estimates | 187

Definition 1.4 (Capacity estimate). Let (X, d, m, ℰ , ℱ ) be an MMD space. For disjoint subsets A, B ⊂ X, we define ℱ (A, B) := {f ∈ ℱ :

f ≡ 1 on a neighborhood of A and f ≡ 0 on a }, neighborhood of B

and the capacity Cap(A, B) as Cap(A, B) := inf{ℰ (f , f ) : f ∈ ℱ (A, B)}. We say that (X, d, m, ℰ , ℱ ) satisfies the capacity estimate cap(Ψ)≤ if there exist C1 , A1 > 1 such that for all 0 < r < diam(X, d)/A1 , x ∈ X, Cap(B(x, r), B(x, A1 r)c ) ≤ C1

m(B(x, r)) . Ψ(r)

(cap(Ψ)≤ )

Definition 1.5 (Fast volume growth). We say that (X, d, m, ℰ , ℱ ) satisfies the fast volume growth condition FVG(Ψ) if there exists a constant CF > 0 such that Ψ(R) m(B(x, R)) ≤ CF , Ψ(r) m(B(x, r))

(FVG(Ψ))

for all 0 < r ≤ R < diam(X, d) and x ∈ X. Our main result is the following path connectedness of annuli. Theorem 1.6. Let (X, d, m, ℰ , ℱ ) be an MMD space that satisfies (VD), PI(Ψ), cap(Ψ)≤ and FVG(Ψ), where Ψ satisfies (1.2). Then there exists C0 ≥ 2 such that for all x ∈ X, r > 0, B(x, 2r) \ B(x, r) is path connected in B(x, C0 r) \ B(x, r/C0 ). Remark 1.7. (1) Condition FVG(Ψ) is not necessary for the conclusion of Theorem 1.6. For example, the Brownian motion on the standard two-dimensional Sierpiński carpet satisfies all of the hypotheses except FVG(Ψ) and satisfies the conclusion. On the other hand, if the volume growth exponent df is strictly less than the walk dimension β, the gluing construction of Delmotte [7] shows that FVG(Ψ) is a sharp condition. In particular, if df < β, then by gluing two copies of the same space at a point, one obtains a space that satisfies (VD), PI(Ψ), cap(Ψ)≤ with Ψ(r) = r β , but fails to satisfy the conclusion. (2) Our argument is quite flexible and can be localized at different scales. For example, the cylinder 𝕊×ℝ satisfies the hypotheses and conclusion of Theorem 1.6 only at small enough scales. On the other hand, the cable system (graph with edges represented by copies of the unit interval) corresponding to ℤd , d ≥ 2, satisfies the hypotheses and conclusion only at large enough scales. (3) Theorem 1.6 can be applied to obtain stability of elliptic Harnack inequality under perturbations of the Dirichlet form (Theorem 4.6). Furthermore, our theorem can be combined with [22, Theorem 1.1] to obtain that the conformal dimension of (X, d) is strictly greater than one.

188 | T. Jaschek and M. Murugan Much of the proof involves estimating capacities between sets from above and below. Our proof is motivated by the arguments in [18, 12, 24]. The basic idea behind our approach is that if the capacity between two sets is strictly positive, then the two sets cannot belong to different connected components. Using lower bounds on capacities implied by the Poincaré inequality, we obtain a connectivity result in Proposition 2.3. This along with upper bounds on capacity leads to a more quantitative estimate in Theorem 3.1, which strengthens a recent result in [24] used to show the chain condition for spaces satisfying sub-Gaussian heat kernel bounds. In this work, Theorem 3.1 plays a crucial role in upgrading from connectivity to path connectivity.

2 Chain connectivity We recall the definition of an ϵ-chain in a metric space (X, d). Definition 2.1 (ϵ-chain). Let B ⊂ X. We say that a sequence {xi }Ni=0 of points in X is an ϵ-chain in B between points x, y ∈ X if x0 = x,

xN = y,

and d(xi , xi+1 ) ≤ ϵ

for all i = 0, 1, . . . , N − 1,

and xi ∈ B,

for all i = 0, 1, . . . , N.

For any ϵ > 0, B ⊂ X and x, y ∈ B, define Nϵ (x, y; B) = inf{n : {xi }ni=0 is ϵ-chain in B between x and y, n ∈ ℕ}, with the usual convention that inf 0 = +∞. We introduce a notion of connectedness based on the existence of ϵ-chains. Definition 2.2 (Chain connectedness). Let (X, d) be a metric space and let B1 ⊆ B2 ⊆ X. We say that B1 is chain connected in B2 if Nϵ (x, y; B2 ) < ∞,

for all x, y ∈ B1 and for all ϵ > 0.

Proposition 2.3. Let (X, d, m, ℰ , ℱ ) be an MMD such that balls are precompact. Assume that (X, d, m, ℰ , ℱ ) satisfies the Poincaré inequality PI(Ψ). Then for all x ∈ X, r > 0, B(x, r) is chain connected in B(x, Ar), where A ≥ 1 is the constant in PI(Ψ). Remark 2.4. We note if (X, d) is a complete metric space and if m is a Borel measure that satisfies (VD), all metric balls in (X, d) are precompact. We always apply Proposition 2.3, when the metric space is complete and admits a doubling measure.

Geometric implications of fast volume growth and capacity estimates | 189

Proof of Proposition 2.3. Let A ≥ 1 be the constant in PI(Ψ). Let x ∈ X, ϵ > 0 and Ux = {y ∈ B(x, Ar) : Nϵ (x, y; B(x, Ar)) < ∞}. By definition of Ux , we have d(y, w) > ϵ,

for all y ∈ Ux , w ∈ B(x, Ar) \ Ux .

(2.1)

Let N denote an ϵ/2-net in (X, d) such that N ∩ B(x, Ar) is an ϵ/2-net of B(x, Ar). By the precompactness of metric balls, the set N ∩ B(x, Ar) is finite. For each z ∈ N, choose a function ϕz ∈ Cc (X) ∩ ℱ such that 1 ≥ ϕz ≥ 0, ϕz |B(z,ϵ/2) ≡ 1, and supp(ϕz ) ⊂ B(z, ϵ). Define ϕ(y) = sup ϕz (y), z∈N∩Ux

for all y ∈ X.

Since N ∩ Ux is a finite set, ϕ ∈ ℱ . By (2.1), and since ⋃z∈N B(z, ϵ/2) = X, we obtain ϕ ≡ 1Ux ,

on B(x, Ar).

(2.2)

By [6, Theorem 4.3.8], the push-forward measure of Γ(ϕ, ϕ) by ϕ is absolutely continuous with respect to the 1-dimensional Lebesgue measure. Since {0, 1} has zero Lebesgue measure, by using (2.2) we obtain Γ(ϕ, ϕ)(B(x, Ar)) = 0.

(2.3)

Since ϕ is continuous, by PI(Ψ), ϕ is constant on the ball B(x, r). Therefore ϕ(y) = ϕ(x) = 1 for all y ∈ B(x, r), r > 0. Therefore ϕ ≡ 1 on B(x, r), which along with (2.2) implies that Ux ∩ B(x, r) = B(x, r). Since ϵ > 0 was arbitrary, we obtain that B(x, r) is chain connected in B(x, Ar) for all x ∈ X, r > 0. The following lemma records an useful consequence of the conditions cap(Ψ)≤ and FVG(Ψ). Lemma 2.5. Let (X, d, m, ℰ , ℱ ) be an MMD space that satisfies (VD), cap(Ψ)≤ , and FVG(Ψ), where Ψ satisfies (1.2). Then for any δ > 0, there exist C, Aδ > 1 such that Cap(B(x, R/C), B(x, R)c ) ≤ δ

m(B(x, R)) , Ψ(R)

(2.4)

for all x ∈ X, 0 < R < diam(X, d)/Aδ . Proof. Let C1 , A1 be the constants in cap(Ψ)≤ , and let CF denote the constant in FVG(Ψ). Therefore, for all x ∈ X, 0 < r ≤ R < diam(X, d)/A1 , we have Cap(B(x, r), B(x, A1 r)c ) ≤ C1

m(B(x, r)) m(B(x, R)) ≤ C1 CF . Ψ(r) Ψ(R)

(2.5)

190 | T. Jaschek and M. Murugan Therefore, by the strong locality (see [15, Lemma 2.5]) for any k ∈ ℕ, c Cap(B(x, A−k 1 R), B(x, R) )

k−1

≤ (∑

i=0

−1 −i−1 −i c −1 Cap(B(x, A1 R), B(x, A1 R) ) ) −1

k−1

Ψ(R) ) ≤ (∑ C C m(B(x, R)) i=0 1 F ≤

(by (2.5))

C1 CF m(B(x, R)) . k Ψ(R)

(2.6)

By choosing k ∈ ℕ large enough so that k > δ−1 C1 CF , we obtain the desired estimate. In the following proposition, we obtain the chain connectivity of annuli under the same assumption as in our main result in Theorem 1.6. Proposition 2.6. Let (X, d, m, ℰ , ℱ ) be an MMD space that satisfies (VD), PI(Ψ), cap(Ψ)≤ , and FVG(Ψ), where Ψ satisfies (1.2). Then there exist C, K ≥ 2 such that for all x ∈ X, 0 < r < diam(X, d)/K, B(x, 2r) \ B(x, r) is chain connected in B(x, Cr) \ B(x, r/C). Proof. Let A ≥ 1 denote the constant in PI(Ψ). Let C ≥ 3A be a constant whose value will be determined later in the proof. Assume by contradiction that B(x, 2r) \ B(x, r) is not chain connected in B(x, Cr) \ B(x, r/C). Then there exist y, z ∈ B(x, 2r) \ B(x, r) and ϵ > 0 such that Nϵ (y, z; B(x, Cr) \ B(x, r/C)) = ∞. Define Uy = {w ∈ B(x, Cr) \ B(x, r/C) : Nϵ (w, y; B(x, Cr) \ B(x, r/C)) < ∞} and Vy = (B(x, Cr) \ B(x, r/C)) \ Uy . By our assumption, z ∉ Uy . By Proposition 2.3, B(z, r/(2A)) is chain connected in B(z, r/2). Using this and B(z, r/2) ⊂ B(x, Cr) \ B(x, r/C)), we have Nϵ (w, z; B(x, Cr) \ B(x, r/C)) ≤ Nϵ (w, z; B(z, r/2)) < ∞,

for all w ∈ B(z, r/(2A)). (2.7)

Combining (2.7) with z ∉ Uy , we have B(z, r/(2A)) ∩ Uy = 0.

(2.8)

By the same argument as (2.7), we have B(y, r/(2A)) ⊂ Uy .

(2.9)

Geometric implications of fast volume growth and capacity estimates | 191

Let N be an ϵ/3-net of (X, d) such that N ∩ Uy ∩ (B(x, Cr) \ B(x, r/C)) is an ϵ/3-net of Uy ∩ B(x, Cr) \ B(x, r/C). For each z ∈ N, choose a function ψz ∈ Cc (X) ∩ ℱ such that 0 ≤ ψz ≤ 1, ψz |B(z,ϵ/3) ≡ 1, and supp(ψz ) ⊂ B(z, 2ϵ/3). Define ψ = sup ψz . z∈N∩Uy

We observe that ψ ∈ ℱ ∩ Cc (X),

0 ≤ ψ ≤ 1,

ψ|B(x,Cr)\B(x,r/C) ≡ 1Uy |B(x,Cr)\B(x,r/C) .

(2.10)

Similar to (2.3), we deduce that Γ(ψ, ψ)(B(x, Cr) \ B(x, r/C)) = 0.

(2.11)

Let h ∈ Cc (B(x, r/2)) ∩ ℱ be such that 0 ≤ h ≤ 1, h|B(x,2r/C) ≡ 1, and c

ℰ (h, h) ≤ 2 Cap(B(x, 2r/C), B(x, r/2) ).

(2.12)

Let g = max(h, ψ) ∈ Cc (X) ∩ ℱ . By (2.10) and the properties of h above, we have g≡1

on B(x, 2r/C) ∪ Uy ,

g≡0

on B(x, r/2)c ∩ Uyc ∩ B(x, Cr).

(2.13)

Furthermore, since d(z1 , z2 ) ≥ ϵ for all z1 ∈ Uy and z2 ∈ Vy , we have ψ ≡ 0 on ∪w∈Vy B(w, ϵ/3). Therefore g ≡ h on B(x, 2r/C) ⋃ ∪w∈Vy B(w, ϵ/3). Hence g ≡ h,

in a neighborhood of B(x, r/2) \ (B(x, 2r/C) ∪ Uy ).

(2.14)

By (2.11) and strong locality, Γ(g, g)(B(x, 3Ar)) = Γ(g, g)(B(x, r/2) \ (B(x, 2r/C) ∪ Uy )) (by (2.13)) = Γ(h, h)(B(x, r/2) \ (B(x, 2r/C) ∪ Uy )) (by (2.14)) ≤ ℰ (h, h) (by (2.10))

≤ 2 Cap(B(x, 2r/C), B(x, r/2)c ) (by (2.11)).

(2.15)

Let δ > 0 be an arbitrary constant. By (1.2), and Lemma 2.5, we can choose C ≥ 3A large enough so that Cap(B(x, 2r/C), B(x, r/2)c ) ≤ δ

m(B(x, r/2)) m(B(x, r)) ≤ C2 δ , Ψ(r/2) Ψ(r)

(2.16)

where C2 depends only the constants in (1.2). Combining (2.15) and (2.16), we obtain Γ(g, g)(B(x, 3Ar)) ≤ 2C2 δ

m(B(x, r)) . Ψ(r)

(2.17)

192 | T. Jaschek and M. Murugan Evidently, by (2.8), (2.9), and (2.13), g = max(h, ψ) satisfies g ≡ 1 on B(y, r/(2A)),

g ≡ 0 on B(z, r/(2A)).

(2.18)

By the triangle inequality, B(y, r/(2A)) ∪ B(z, r/(2A)) ⊂ B(x, 3r). Now, we use the Poincaré inequality PI(Ψ) along with (1.5) to derive the following estimate. Since g ∈ ℱ ∩ C(X) and satisfies (2.18), we have Γ(g, g)(B(x, 3Ar)) 1 ≥ 2CP m(B(x, 3r))Ψ(3r) 1 ≥ 2CP m(B(x, 3r))Ψ(3r)

∫

󵄨 󵄨2 ∫ 󵄨󵄨󵄨g(p) − g(q)󵄨󵄨󵄨 m(dp) m(dq)

B(x,3r) B(x,3r)

∫

∫

󵄨󵄨 󵄨2 󵄨󵄨g(p) − g(q)󵄨󵄨󵄨 m(dp) m(dq)

B(z,r/(2A)) B(y,r/(2A))

m(B(y, r/(2A))m(B(z, r/(2A)) (using (2.18)) ≥ 2CP m(B(x, 3r))Ψ(3r) m(B(x, r)) ≥ , (by (1.2) and (VD)) C1 Ψ(r)

(2.19)

where the constant C1 ≥ 1 depends only on the constants in (1.2), (VD), and PI(Ψ). By choosing δ < (2C1 C2 )−1 , the bounds (2.17) and (2.19) lead to the desired contradiction.

3 From chain connectivity to path connectivity The following result strengthens a bound on the length of chains given in [24]. The improvement is that the upper bound was on Nϵ (x, y; X) instead of Nϵ (x, y; B(x, A0 d(x, y))). Theorem 3.1. Let (X, d, m, ℰ , ℱ ) be an MMD space that satisfies (VD), PI(Ψ), and cap(Ψ)≤ , where Ψ satisfies (1.2). Then there exist C, A0 > 1 such that for all ϵ > 0 and for all x, y ∈ X that satisfy d(x, y) ≥ ϵ, we have 2

Nϵ (x, y; B(x, A0 d(x, y))) ≤ C

Ψ(d(x, y)) . Ψ(ϵ)

(3.1)

For two measures m, ν on (X, d), for R > 0, x ∈ X, we define a truncated maximal function MRm ν(x) = sup

0 0, x, y ∈ B(x0 , CP−1 R), and for all u ∈ 𝒞 (X) ∩ ℱloc , 󵄨2 󵄨󵄨 m m 󵄨󵄨u(x) − u(y)󵄨󵄨󵄨 ≤ CΨ(R)(MR Γ(u, u)(x) + MR Γ(u, u)(y)), where Γ(u, u) denotes the energy measure of u. Lemma 3.3 (Partition of unity, [24, Lemma 2.5]). Let (X, d, m, ℰ , ℱ ) be an MMD space that satisfies (VD) and cap(Ψ)≤ . Let ϵ > 0 and let V denote any ϵ-net. Let ϵ < diam(X, d)/A1 , where A1 ≥ 1 is the constant in cap(Ψ)≤ . Then, there exists a family of functions {ψz : z ∈ V} that satisfies the following properties: (a) {ψz : z ∈ V} is partition of unity, that is, ∑z∈V ψz ≡ 1. (b) For all z ∈ V, ψz ∈ Cc (X) ∩ ℱ with 0 ≤ ψz ≤ 1, ψz |B(z,ϵ/4) ≡ 1, and ψz |B(z,5ϵ/4)c ≡ 0. (c) For all z ∈ V, z 󸀠 ∈ V \ {z}, we have ψz 󸀠 |B(z,ϵ/4) ≡ 0. (d) There exists C > 1 such that for all z ∈ V, ℰ (ψz , ψz ) ≤ C

m(B(z, ϵ)) . Ψ(ϵ)

Proof of Theorem 3.1. Let A1 denote the constant in cap(Ψ)≤ . Since Nϵ (x, y; B) ≤ Nϵ󸀠 (x, y; B) whenever B ⊂ X and ϵ󸀠 ≤ ϵ, by replacing ϵ by ϵ/(2A1 ) if necessary and by using (1.2), we assume that ϵ < diam(X, d)/A1 . Fix x, y ∈ X, ϵ > 0 such that d(x, y) ≥ ϵ. Set ϵ󸀠 = ϵ/3. Let V be an ϵ󸀠 -net such that {x, y} ⊂ V. Let A0 > 0 be A0 = 2A(CP + 2),

(3.3)

where A is the constant in PI(Ψ) and CP is as given in Lemma 3.2. Define û : V ∩ B(x, 2(CP + 2)d(x, y)) → [0, ∞) as ̂ u(z) := Nϵ (x, z; B(x, A0 d(x, y))).

(3.4)

̃ = V ∩ B(x, 2(CP + 2)d(x, y)). By Proposition 2.3, û is finite. By definition, We set V 󵄨󵄨 ̂ ̂ 2 )󵄨󵄨󵄨󵄨 ≤ 1, 󵄨󵄨u(z1 ) − u(z

̃ such that d(z1 , z2 ) < ϵ. for all z1 , z2 ∈ V

(3.5)

Let {ψz : z ∈ V} denote the partition of unity defined in Lemma 3.3. Define u : X → [0, ∞) as ̂ u(p) := ∑ u(z)ψ z (p). ̃ z∈V

For any ball B(x0 , r), x0 ∈ X, r > 0, by Lemma 3.3(b) we have u(p) =

∑ z∈V∩B(x0 ,r+5ϵ󸀠 /4)

̂ u(z)ψ z (p),

for all p ∈ B(x0 , r).

(3.6)

194 | T. Jaschek and M. Murugan Since V ∩B(x0 , r+5ϵ󸀠 /4) is a finite set by (VD), we obtain that u ∈ ℱloc . By Lemma 3.3(b), ̂ we have u|B(z,ϵ󸀠 /4) ≡ u(z) for all z ∈ V. Therefore, by [6, Theorem 4.3.8], the pushforward measure of Γ(u, u) by u is absolutely continuous with respect to the 1-dimensional Lebesgue measure. Therefore, we obtain Γ(u, u)(B(z, ϵ󸀠 /4)) = 0,

for all z ∈ V.

(3.7)

By (3.6) and Lemma 3.3(a), we have ̂ + u(p) = u(z)

∑ w∈V∩B(z,9ϵ󸀠 /4)

̂ ̂ (u(w) − u(z))ψ w (p),

for all p ∈ B(z, ϵ󸀠 ), z ∈ V.

(3.8)

By (VD), there exits C1 > 1 such that supz∈V |V ∩ B(z, 9ϵ󸀠 /4)| ≤ C1 . By (3.8), and the Cauchy–Schwarz inequality, there exists C2 > 1 such that the following holds: for all z ∈ V ∩ B(x, 2(CP + 1)d(x, y)), we have Γ(u, u)(B(z, ϵ󸀠 )) ≤ C1 ≲

2

∑ w∈V∩B(z,9ϵ󸀠 /4)

̂ ̂ (u(w) − u(z)) ℰ (ψw , ψw )

m(B(w, ϵ󸀠 )) Ψ(ϵ󸀠 ) w∈V∩B(z,9ϵ󸀠 /4)

≤ C2

∑

m(B(z, ϵ󸀠 /2)) Ψ(ϵ)

(by (3.5) and Lemma 3.3(d))

(by (VD) and (1.2)).

(3.9)

In the second line above, we used the fact that z ∈ V ∩ B(x, 2(CP + 1)d(x, y)) and w ∈ V ∩ B(z, 9ϵ󸀠 /4) implies w ∈ B(x, 2(CP + 2)d(x, y)). By Lemma 3.3, (3.9), and (VD), there exists C3 > 0 such that for all z ∈ V ∩ B(x, 2d(x, y)), r ≤ 2CP d(x, y), we have Γ(u, u)(B(z, r)) ≤ C2

m(B(w, ϵ󸀠 /2)) m(B(z, r)) ≤ C3 . Ψ(ϵ) Ψ(ϵ) w∈B(z,r+5ϵ󸀠 /4) ∑

(3.10)

Combining (3.7) and (3.10), we obtain MRm Γ(u, u)(z) = sup r 0 such that 2

Nϵ (x, y; B(x, A0 d(x, y))) ≤ C4

Ψ(d(x, y)) Ψ(ϵ)

for all x, y ∈ X, ϵ ≤ d(x, y).

Thus we obtain (3.1). Corollary 3.4. Let (X, d, m, ℰ , ℱ ) be an MMD space which satisfies (VD), PI(Ψ), and cap(Ψ)≤ , where Ψ satisfies (1.2). Then there exists a curve from x to y in B(x, 2A0 d(x, y)) for all x, y ∈ X, where A0 is the constant in Theorem 3.1.

Geometric implications of fast volume growth and capacity estimates | 195

Proof. By (1.2) and Theorem 3.1, for any ϵ ∈ (0, 1/2), there exists N ∈ ℕ such that Nϵd(x,y) (x, y, B(x, A0 d(x, y))) ≤ N,

for all x, y ∈ X,

where A0 is the constant in Theorem 3.1. For the remainder of the proof, we fix ϵ ∈ (0, 1/2) and N ∈ ℕ as above. Let x, y ∈ X a pair of distinct points. For each k ∈ ℕ, we define γk : [0, 1] → X as follows. Let z0(1) , z1(1) , . . . , zN(1) be a sequence of points in B(x, A0 d(x, y)) such that (1) d(zi(1) , zi+1 ) < ϵd(x, y), with z0(1) = x, zN(1) = y. Let γ1 : [0, 1] → X be the piecewise constant function on intervals defined by γ1 (t) = zi(1) ,

for all i = 0, . . . , N − 1 and for all i/(N + 1) ≤ t < (i + 1)/(N + 1)

and γ1 (1) = y. Similarly, for all i = 0, . . . , N and we chose zj(2) , j = i(N + 1), i(N + 1) + (1) (2) (2) (1) (2) , zk ∈ B(zi(1) , A0 d(zi(1) , zi+1 )), for 1, . . . , i(N + 1) + N such that zi(N+1) = zi(1) , zi(N+1)+N = zi+1

(2) k = i(N + 1), . . . , i(N + 1) + N, d(zj(2) , zj+1 ) < ϵ2 d(x, y) and define

γ2 (t) = zj(2) , for all j = 0, 1, . . . , (N + 1)2 − 1 and for all j/(N + 1)2 ≤ t < (j + 1)/(N + 1)2 with γ2 (1) = y. We similarly define γk : [0, 1] → X that is piecewise constant on intervals [j/(N + 1)k , (j + 1)/(N + 1)k ), j = 0, 1, . . . , (N + 1)k − 1. Since for all t ∈ [0, 1], d(γk (t), γk+1 (t)) < A0 ϵk d(x, y), the sequence {γk (t), k ∈ ℕ} is Cauchy, and hence converges to say γ(t) ∈ X. This limit defines a function γ : [0, 1] → X. Note that ∞

d(x, γ(t)) ≤ ∑ ϵk d(x, y) = A0 d(x, y)/(1 − ϵ) < 2A0 d(x, y). k=0

If |t1 − t2 | ≤

1 Nk

for some k ∈ ℕ, we have

d(γ(t1 ), γ(t2 )) ≤ d(γk (t1 ), γ(t1 )) + d(γk (t2 ), γ(t2 )) + d(γk (t1 ), γk (t2 )) ∞

≤ 2(∑ A0 ϵl d(x, y)) + 2(A0 + 1)ϵk d(x, y) l=k

≤ 2(A0 + 1)(1 + (1 − ϵ)−1 )ϵk d(x, y), which implies the continuity of γ. Hence γ : [0, 1] → B(x, 2A0 d(x, y)) is continuous and γ(0) = x, γ(1) = y. We now have all tools at our disposal to prove the main result. Proof of Theorem 1.6. We choose C0 = 2C, where C is the constant in Proposition 2.6. r Let ϵ = 4CA , where A is the constant in PI(Ψ). By Proposition 2.6, for any x ∈ X, r > 0 and y, z ∈ B(x, 2r) \ B(x, r), Nϵ (y, z; B(x, Cr) \ B(x, r/C)) < ∞.

196 | T. Jaschek and M. Murugan Let {xi }Ni=1 be an ϵ-chain between y and z in B(x, Cr) \ B(x, r/C). By Corollary 3.4, there exist curves γi : [0, 1] → B(zi , r/(2C)), i = 0, . . . , N − 1 from zi to zi+1 . Since B(zi , r/(2C)) ⊂ B(x, C0 r)\B(x, r/C0 ) for all i = 0, . . . , N −1, by concatenating the curves γi , i = 0, . . . , N −1, we obtain a curve from y to z in B(x, C0 r) \ B(x, r/C0 ). Under the assumptions of Theorem 1.6, we can obtain a quantitative bound on Nϵ as follows. By the volume doubling property, the minimum number of balls of radii r required to cover a ball of radius R depends only on R/r. Using this property, we could obtain a uniform bound (that does not depend on x or r) on Nr/(4CA) (y, z, B(x, Cr)\ B(x, r/C)) in the proof of Theorem 1.6. As a consequence, using Theorem 3.1, for any ϵ > 0, x ∈ X, 0 < r < diam(X, d)/A, we have 2

Nr/(4CA) (y, z, B(x, Cr) \ B(x, r/C)) ≲

Ψ(r) , Ψ(ϵ)

for all y, z ∈ B(x, 2r) \ B(x, r).

4 Applications For simplicity, we will assume that our MMD space is unbounded, that is, diam(X, d) = ∞. We recall a condition introduced in [16, 17]. Definition 4.1 (Relatively connected annuli). We say that (X, d) satisfies (RCA) (this stands for relatively connected annuli) if there is a point o and a constant A such that for any r > A and any two points x, y ∈ X with d(o, x) = d(o, y) = r there is a continuous path in B(o, Ar) \ B(o, r/A) connecting x to y. A weighted manifold is a Riemannian manifold (M, g) equipped with a measure μ that has a smooth positive density with respect to the Riemannian measure. This space is equipped with a weighted Laplace operator that generalizes the Laplace–Beltrami operator and is symmetric with respect to the measure μ. Such spaces naturally arise in the context of Doob h-transforms and Schrödinger operators. We refer the reader to [11] for a comprehensive survey on weighted manifolds and applications. Motivated by weighted manifolds (and the corresponding weighted Laplace operator), we perturb an MMD space by a continuous function w : X → (0, ∞) by a family of admissible weights. Definition 4.2 (Admissible weight). Let (X, d, m, ℰ , ℱ ) be an MMD space and let w : X → (0, ∞) be continuous. We say that w is an admissible weight if the following conditions hold: (a) There exist o ∈ X, α1 , α2 ∈ ℝ, C > 1 such that C −1 (

α

α

d(o, x) 1 w(x) d(o, x) 2 ) ≤ C( ) , d(o, y) w(y) d(o, y)

for all x, y ∈ X such that d(o, y) ≥ d(o, x).

(4.1)

Geometric implications of fast volume growth and capacity estimates | 197

(b) There exists an MMD space (X, d, w dm, ℰw , ℱw ), where ℱw ⊂ ℱloc and ℰw (f , f ) = ∫ w dΓ(f , f ),

for all f ∈ ℱw ,

X

where ℱloc is as defined in (1.4). Furthermore, ℱ ∩ 𝒞c (X) forms a core for the Dirichlet form (ℰw , ℱw ) on L2 (w dm). In the context of manifolds, we refer to [17, 11] for examples of admissible weights. We recall the definition of a generalized capacity bound introduced in [14] based on a similar condition due to Andres and Barlow. Definition 4.3 (Generalized capacity estimate). For open subsets U, V of X with U ⊂ V, we say that a function φ ∈ ℱ is a cut-off function for U ⊂ V if 0 ≤ φ ≤ 1, φ = 1 on a neighborhood of U and suppm [φ] ⊂ V. Then we say that (X, d, m, ℰ , ℱ ) satisfies the generalized capacity estimate Gcap(Ψ), if there exists CS > 0 such that the following holds: for each x ∈ X and each R, r > 0, f ∈ ℱ there exists a cut-off function φ ∈ ℱ for B(x, R) ⊂ B(x, R + r) such that ∫ f 2 dΓ(φ, φ) ≤ CS X

φ2 dΓ(f , f ) +

∫ B(x,R+r)\B(x,R)

CS Ψ(r)

∫

f 2 dm.

Gcap(Ψ)

B(x,R+r)\B(x,R)

Here and in what follows, we always consider a quasicontinuous version of f ∈ ℱ , which exists by [9, Theorem 2.1.3] and is unique ℰ -q. e. (i. e., up to sets of capacity zero) by [9, Lemma 2.1.4], so that the values of f are uniquely determined Γ(g, g)-a. e. for each g ∈ ℱ since Γ(g, g)(N) = 0 for any Borel subset N of X of capacity zero by [9, Lemma 3.2.4]. By choosing a function f ∈ ℱ such that f ≡ 1 in a neighborhood of B(x, R + r) in the above definition, we note that Gcap(Ψ) implies cap(Ψ)≤ . We recall the definition of harmonic functions and the elliptic Harnack inequality. Definition 4.4 (Harmonic functions and elliptic Harnack inequality). Let (X, d, m, ℰ , ℱ ) be an MMD space. A function h ∈ ℱ is said to be ℰ -harmonic on an open subset U of X, if ℰ (h, f ) = 0,

(4.2)

for all f ∈ ℱ ∩ 𝒞c (X) with supp[f ] ⊂ U, where supp[f ] denotes the support of f . We say that an MMD space (X, d, m, ℰ , ℱ ) satisfies the elliptic Harnack inequality (abbreviated as EHI), if there exist C > 1, δ ∈ (0, 1) such that for all x ∈ X, r > 0 and for any nonnegative harmonic function h on the ball B(x, r), we have ess sup h ≤ C ess inf h. B(x,δr)

B(x,δr)

EHI

198 | T. Jaschek and M. Murugan We recall the definition of a remote ball. Definition 4.5 (Remote ball). Fix o ∈ X and ϵ ∈ (0, 1]. We say that a ball B(x, r) is ϵ-remote with respect to o, if r ≥ 21 ϵd(o, x). As an application of the annular connectivity result in Theorem 1.6, we obtain the following stability of elliptic Harnack inequality under perturbation by admissible weights. Theorem 4.6. Let (X, d, m, ℰ , ℱ ) be an unbounded MMD space that satisfies (VD), PI(Ψ), Gcap(Ψ), and FVG(Ψ), where Ψ satisfies (1.2). Let w ∈ 𝒞 (X), w : X → (0, ∞) be an admissible weight. Then the corresponding weighted MMD space (X, d, w dm, ℰw , ℱw ) satisfies the elliptic Harnack inequality. Proof. By Corollary 3.4, B(x, r) \ B(x, r/2) ≠ 0 for any ball B(x, r). This along with (VD) implies the following reverse volume doubling property: there exist C1 , α > 0 such that α

R m(B(x, R)) ≥ C1−1 ( ) , m(B(x, r)) r

for any x ∈ X, 0 < r ≤ R.

Let o ∈ X be the point as in Definition 4.2 and let μ = w dm denote the weighted measure. By (4.1), the weight w is comparable to a constant on any ϵ-remote ball with respect to o for any ϵ ∈ (0, 1]. By choosing ϵ ∈ (0, 1] small enough, the MMD space (X, d, w dm, ℰw , ℱw ) satisfies the volume doubling property, reverse volume doubling property, Gcap(Ψ) and FVG(Ψ) for all remote balls. By [14, proof of Theorem 1.2], the MMD space (X, d, w dm, ℰw , ℱw ) satisfies EHI for all remote balls. By applying [17, Lemma 6.4] for the metric measure space (X, d, m), we obtain the annuli covering condition in [17, Definition 6.2]. By Theorem 1.6 and [17, Lemma 6.3], we obtain that the MMD space (X, d, w dm, ℰw , ℱw ) satisfies EHI for all balls. Remark 4.7. We remark that the FVG(Ψ) condition in Theorem 4.6 is necessary. For example, for Brownian motion on ℝ, the weight w(x) = (1 + x 2 )α/2 where α > 1 fails to satisfy the Liouville property and hence the elliptic Harnack inequality. This is because the diffusion corresponding to the Dirichlet form (ℰw , ℱw ) has two transient ends at ±∞. Hence, the probability that the diffusion eventually ends up in one of them (say +∞) is a nonconstant positive harmonic function.

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Ryosuke Shimizu

Parabolic index of an infinite graph and Ahlfors regular conformal dimension of a self-similar set Abstract: We investigate relations between two critical indices determined through p-energies on graphs, namely, the Ahlfors regular conformal dimension of a selfsimilar set and the parabolic index of an infinite graph obtained as a “blow-up” of the self-similar set. First, we survey the discrete nonlinear potential theory developed by Yamasaki et al. in the 1970s. Especially, we describe p-parabolicity and p-hyperbolicity of infinite graphs, which generalize recurrence and transience of simple random walks on graphs, respectively. Parabolic index is defined as the threshold of these two-phases. Secondly, we construct infinite graphs called blow-ups of a self-similar set and show that their parabolic indices are no greater than the Ahlfors regular conformal dimension of the original self-similar set. This result gives a new aspect of the Ahlfors regular conformal dimension. Moreover, we see that parabolic indices of both the Sierpinski gasket and the diamond fractal coincide with their Ahlfors regular conformal dimensions and are equal to 1. Keywords: Ahlfors regular conformal dimension, parabolic index, p-energy, p-modulus MSC 2010: Primary 31E05, Secondary 30C65

Contents 1 2 3 3.1 3.2 3.3 4 5 6

Introduction | 202 p-energy and associated function spaces on a weighted graph | 206 Variational problems on weighted graphs | 214 p-conductance and p-modulus between two sets | 214 Nakamura–Yamasaki’s duality | 224 Another expression of the dual p-modulus | 230 Generalized resistance metrics | 236 p-conductance and p-modulus to infinity | 238 p-Laplacians on weighted graphs | 244

Acknowledgement: The author was supported by JSPS KAKENHI Grant Number JP20J20207. The author would like to express deep gratitude to his supervisor, Professor Jun Kigami, for invaluable guidance during the Master course, for suggesting interesting problems, helpful discussions, and careful readings of earlier versions of this paper. Ryosuke Shimizu, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan, e-mail: [email protected] https://doi.org/10.1515/9783110700763-008

202 | R. Shimizu

7 8 9 10 11 12

Parabolic index | 250 Stability under rough isometries of weighted graphs | 256 Self-similar sets and their blow-ups | 259 Ahlfors regular conformal dimension of a self-similar set | 266 Parabolic index of a blow-up | 268 Examples | 269 Bibliography | 273

1 Introduction Given a metric space (X, d), its Hausdorff dimension dimH (X, d) depends on the metric d. For example, the ϵth power of d, dϵ , is also a metric on X for all ϵ ∈ (0, 1] and dimH (X, dϵ ) = ϵ−1 dimH (X, d). Note that the balls with respect to dϵ are exactly the same balls with respect to d as subsets of the space X. Therefore, we can make the Hausdorff dimension arbitrarily large without changing the shapes of balls. How about lowering the Hausdorff dimension? In this case, we are going to allow a modification of the metric called quasisymmetry but ask the existence of a measure with a nice property called Ahlfors regularity. Roughly, two metrics on the same space are quasisymmetric to each other if and only if their balls are distorted in a uniformly bounded manner. See Definition 10.1 for the exact statement. A measure μ is said to have Ahlfors regularity with respect to a metric d if the measure of a ball is comparable to some power, which is independent of a ball, of the radius of the ball (see Definition 10.3). Under such restrictions, how low the Hausdorff dimension can be is the value of interest, which is called the Ahlfors regular conformal dimension. Namely, 󵄨 dimAR (X, d) = inf {dimH (X, ρ) 󵄨󵄨󵄨 ρ is quasisymmetric to d and ρ is Ahlfors regular } . For limited classes of metric spaces, their Ahlfors regular conformal dimensions are known, e. g., it is n for n-dimensional Euclidean space. However, in general, it may be very hard to calculate the exact value of this dimension. Carrasco Piaggio [4] has given a characterization of the Ahlfors regular conformal dimension in terms of a critical exponent related to the combinatorial modulus, which is one of fundamental tools in analysis and geometry of metric spaces. In the recent work [12], by using sequences of graphs which approximate a metric space, Kigami has modified Carrasco Piaggio’s characterization as a critical exponent of p of the p-energy on graphs instead of the combinatorial modulus. The notion of a p-energy has been playing essential roles in nonlinear potential theory. On infinite graphs, the study of non-linear potential theory was initiated by Yamasaki and coworkers in the 1970s. They have obtained quite a number of interesting results in [18, 20, 21, 23, 24, 25, 26]. However, their work has been forgotten for many years partly because their terminologies and notations were somewhat different from those used by the majority. In view of Kigami’s result, it is

Parabolic index and Ahlfors regular conformal dimension | 203

natural to expect some connection between those classical studies of the nonlinear potential theory and the recent study of Ahlfors regular conformal dimension. The aim of the present study is to make such a connection clear. In the first part of this paper, we give a self-contained survey of the nonlinear potential theory on infinite graphs. In particular, we are going to focus on the notion of a parabolic index. Let G = (V, E) be a nondirected, locally finite and connected graph, i. e., V and E are the vertices and the edges, respectively, of the graph. For a given function u : V → ℝ, its p-energy ℰp (u) is defined as ℰp (u) =

1 󵄨 󵄨p ∑ 󵄨󵄨󵄨u(y) − u(x)󵄨󵄨󵄨 . 2 (x,y)∈E

In the case p = 2, the study of the 2-energy on a graph is nothing but the discrete potential theory. This theory is closely related to electrical networks. For instance, attaching a resistor of resistance 1 to vertices x and y for each (x, y) ∈ E, we regard G as an electrical network. Then the actual effective resistance between x and y is given by R(x, y) = inf{ℰ2 (u) | u(x) = 1, u(y) = 0} .

(1.1)

−1

It is also known that R is a metric on V, which is called the resistance metric and plays essential roles in analysis on fractals (see [10], for example). As an analogue of (1.1), we can define −p󸀠 /p

Rp (x, y) = inf{ℰp (u) | u(x) = 1, u(y) = 0}

(1.2)

,

where p󸀠 is the conjugate index of p, i. e., it holds that 1/p + 1/p󸀠 = 1. Note that R2 coincides with the resistance metric R. Moreover, we obtain the following result: Theorem 1.1 (Theorem 4.3). Rp is a metric for any p > 1. In the recent work [2], the same result is proved in the setting of finite networks. In Section 4, we give a proof of Theorem 1.1 using a classical result due to Nakamura and Yamasaki [18], which covers the case of infinite networks. Electrical network theory can also be used to analyze random walks, and thus there are deep connections between discrete potential theory and random walks. Our focus here is a classification of infinite graphs based on behaviors of the associated random walks, i. e., whether random walks are recurrent or transient. If a random walk (on a connected graph) returns to its starting point with probability 1, then it is said to be recurrent. If not, it is said to be transient. It is well-known that this dichotomy can be described in terms of discrete potential theory. Indeed, simple random walks on G are recurrent if and only if R(x, ∞) = inf{ℰ2 (u) | u(x) = 1, u has a finite support}

−1

(1.3)

is equal to ∞ (we define 0−1 = ∞). One of the important observations is that the definition of R(x, ∞) uses only 2-energy. Similar to (1.2), it is natural to consider general-

204 | R. Shimizu izations in terms of the p-energy. We define −p󸀠 /p

Rp (x, ∞) = inf{ℰp (u, u) | u(x) = 1, u has a finite support}

,

(1.4)

and say that G is p-parabolic (resp. p-hyperbolic) if and only if Rp (x, ∞) = ∞ (resp. Rp (x, ∞) < ∞). This classification of infinite graphs has been introduced by Yamasaki [26]. Although this definition is different from what is given in Definition 5.4, these conditions will be shown to be equivalent to each other in Theorems 7.1 and 7.3. Furthermore, in [26], Yamasaki has observed that there exists p∗ (G) ∈ [1, ∞] such that p∗ (G) = inf{p | G is p-parabolic} = sup{p | G is p-hyperbolic}. This value p∗ (G) is called the parabolic index of G. It is important to note that both the Ahlfors regular conformal dimension and the parabolic index are determined through discrete p-energy. In the second part of this paper, we will see a connection between the parabolic index and the Ahlfors regular conformal dimension for a class of self-similar sets. More precisely, we are going to construct infinite graphs, called blow-ups, for a given self-similar set (K, d). Our result, Theorem 11.2, states that for any blow-up G of (K, d), one has Theorem 1.2 (Theorem 11.2). Let (K, d) be a self-similar set satisfying certain conditions. If G is a blow-up of K, then dimAR (K, d) ≥ p∗ (G).

(1.5)

For detailed conditions and statement, see Theorem 11.2. We remark that we can apply the above result for typical examples of self-similar sets such as the Sierpinski gasket, Sierpinski carpet, and nested fractals. An interesting problem is whether the equality holds or not in (1.5). We prove equalities between Ahlfors regular conformal dimensions and parabolic indices for the Sierpinski gasket and the diamond fractal (Figure 1). Furthermore, we obtain the following result:

Figure 1: Examples of self-similar sets. From left to right, Sierpinski gasket, Sierpinski carpet, and diamond fractal.

Parabolic index and Ahlfors regular conformal dimension | 205

Theorem 1.3 (Theorem 12.3, 12.5). If (K, d) is the Sierpinski gasket or the diamond fractal and G is a blow-up of K, then dimAR (K, d) = p∗ (G) = 1.

(1.6)

These results also give us new examples of infinite graphs whose parabolic indices can be obtained exactly. This paper is organized as follows. Section 2 is devoted to the introduction of the p-energy on weighted graphs. In Section 3, we consider some variational problems related to p-energies. Most of the results in this section have been obtained by Nakamura and Yamasaki [18]. After these preparations, Theorem 1.1 is proved in Section 4. From Section 5 to 8, we will study the parabolic index of an infinite graph. In particular, we give various conditions which are equivalent to the p-parabolicity in Section 7. In Section 9, we review basic notions and notations on a self-similar set and give a definition of its blow-ups. Also, our standing assumptions for self-similar sets are stated in this section. In Section 10, we are going to introduce basic notions in quasiconformal geometry including quasisymmetry and Ahlfors regularity, and briefly recall the result of [12] on the characterization of Ahlfors regular conformal dimensions. Sections 11 and 12 are devoted to the proofs of our main results and discussions of the cases of the Sierpinski gasket and the diamond fractal. Notation. Throughout this paper, we follow the following notations. (1) For any a, b ∈ ℝ, we write a ∨ b = max{a, b} and a ∧ b = min{a, b}. (2) Define sgn : ℝ → {−1, 0, 1} by 1 if t > 0, { { { sgn(t) = {0 if t = 0, { { {−1 if t < 0. (3) For any nonempty countable set X, we use ℓ(X) to denote the collection of realvalued functions on X, i. e., ℓ(X) = {u : X → ℝ}. Furthermore, ℓc (X) denotes the functions with finite supports, i. e., ℓc (X) = {u ∈ ℓ(X) | u(x) = 0 except for finite points}. (4) For any p ∈ (1, ∞), we use p󸀠 to denote the conjugate index of p, that is, 1/p + 1/p󸀠 = 1. Note that p󸀠 = p/(p − 1). (5) Let (X, d) be a metric space. We define the diameter of a subset A ⊆ X with respect to d, denoted by diam(A, d), by diam(A, d) = sup d(x, y). x,y∈A

206 | R. Shimizu Moreover, for each x ∈ X and r > 0, we set Bd (x, r) = {y ∈ X | d(x, y) < r}.

2 p-energy and associated function spaces on a weighted graph In the first part of this section, we introduce the basic notations and terminologies on a weighted graph. After preparing the framework of a weighted graph, we define p-energies and the p-Dirichlet space on a weighted graph and examine their properties. Definition 2.1. Let V be a countable set and let E be a subset of V ×V such that (x, y) ∈ E if and only if (y, x) ∈ E. A pair G = (V, E) is called a (nondirected) graph. The elements of V are called vertices, and the elements of E are called edges. If (x, x) ∈ ̸ E for all x ∈ V, then we say that G = (V, E) is simple. Definition 2.2. Let G = (V, E) be a simple graph. (1) A finite path is a sequence λ = [x0 , x1 , . . . , xn ] of vertices such that (xi−1 , xi ) is an edge for any i ∈ {1, . . . , n}. If n = 0, then λ is a single point. Here, the number n ∈ ℤ+ is called the length of a path λ, which is denoted by len(λ), and we also call λ a path from x0 to xn . If #{xi }len(λ) = len(λ) + 1, then λ is called a simple path. We say i=0 that a path λ0 is a subpath of a path λ = [x0 , x1 , . . . , xn ] if there exist k, l ∈ {0, . . . , n} with k ≤ l such that λ0 = [xk , xk+1 , . . . , xl ]. Let 𝒫 (G) be the collection of all paths in G, i. e., 𝒫 (G) = {λ = [x0 , x1 , . . . , xn ] | n ∈ ℤ+ , λ is a path}.

(2) The graph metric dG on V associated with the graph G = (V, E) is defined by dG (x, y) = inf{len(λ) | λ is a path from x to y}. (3) We say that G is connected if and only if dG (x, y) < ∞ for any x, y ∈ V. (4) Let x ∈ V. We define the neighbor of x, NG (x), by NG (x) = {y ∈ V | (x, y) ∈ E}. We say that G is locally finite if and only if #NG (x) < ∞ for any x ∈ V. The following notations are useful. Definition 2.3. Let G = (V, E) be a simple graph. (1) For any edge e = (x, y) ∈ E, we denote ě = (y, x) ∈ E, which is called the reverse edge of e.

Parabolic index and Ahlfors regular conformal dimension | 207

(2) For a pair of paths λ1 = [x0 , x1 , . . . , xn ], λ2 = [y0 , y1 , . . . , ym ] with xn = y1 , we define the concatenation of λ1 and λ2 , λ1 ∗ λ2 , by λ1 ∗ λ2 = [x0 , x1 , . . . , xn = y0 , y1 , . . . , ym ]. (3) For x ∈ V and λ = [x0 , x1 , . . . , xn ] ∈ 𝒫 (G), we define x ∩ λ = {x} ∩ {x0 , x1 , . . . , xn }. (4) Let A be a subset of V. We define A = {x ∈ V | dG (x, y) ≤ 1 for some y ∈ A}. Set also 𝜕A = A \ A, 𝜕i A = {x ∈ A | dG (x, y) = 1 for some y ∈ 𝜕A}, and 𝜕e A = {(x, y), (y, x) | (x, y) ∈ E, x ∈ A, y ∈ ̸ A}. Note that 𝜕e A = {(x, y), (y, x) | (x, y) ∈ E, x ∈ 𝜕i A, y ∈ 𝜕A}. (5) A sequence of finite subsets {Vn }n∈ℕ of V is called an exhaustion of V if and only if Vn ⊆ Vn+1 for any n ∈ ℕ and ⋃n∈ℕ Vn = V. We also say that an exhaustion {Vn }n∈ℕ is good if and only if Vn ⊆ Vn+1 for any n ∈ ℕ. Now we introduce weighted graphs. Definition 2.4. Let G = (V, E) be a simple graph. If μ = {μxy }x,y∈V satisfies the following conditions: (a) μxy ≥ 0 for any x, y ∈ V, (b) μxy = μyx for any x, y ∈ V, (c) μxy > 0 if and only if (x, y) ∈ E, then μ is called a weight on G and (G, μ) is called a weighted graph. A weight μ on G is called simple if and only if 1

if (x, y) ∈ E,

0

otherwise.

μxy = {

Following [3], we introduce some operations on weighted graphs. Definition 2.5. Let (G, μ) = (V, E, μ) be a simple weighted graph and let e0 = (x, y) ∈ E.

208 | R. Shimizu (1) Let H be a subset of V. Then the subgraph GH = (H, E H ) induced by H is the graph with the vertices H and the edges E H given by E H = {(x, y) ∈ E | x, y ∈ H}. If μ is a weight on G, the induced weight μH on the subgraph GH is given by μH xy = μxy ,

x, y ∈ H.

(2) The weighted graph (V, E, μ(cut) ) obtained by cutting the edge e0 is the weighted graph with the vertices V, the edges E, and the weights μ(cut) given by μe ,

μ(cut) ={ e

0,

e ≠ e0 ,

e = e0 .

(3) The weighted graph (V (short) , E (short) , μ(short) ) obtained by shorting the edge e0 is the weighted graph given by V (short) = (V \ {x, y}) ∪ {♦}, E (short) = E V\{x,y} ∪ {(z, ♦), (♦, z) | z ∈ (NG (x) ∪ NG (y)) \ {x, y}}, and μ(short) zw

μzw { { { = {μzx + μzy { { {μxw + μyw

if z, w ∈ V \ {x, y}, if w = ♦, if z = ♦.

Let p ∈ (1, ∞) and let (G, μ) = (V, E, μ) be a connected, locally finite, simple weighted graph. We recall the definition of an ℓp space and define some function spaces on a weighted graph. Definition 2.6. (1) Define ℓp (E, μ) = {θ ∈ ℓ(E) | ‖θ‖ℓp (E,μ) < ∞} where ‖ ⋅ ‖ℓp (E,μ) is given by 1/p

󵄨 󵄨p ‖θ‖ℓp (E,μ) = ( ∑ 󵄨󵄨󵄨θ(e)󵄨󵄨󵄨 μe ) e∈E

.

(2) Define ℓ# (E) = {θ ∈ ℓ(E) | θ(e) = −θ(e)̌ for any e ∈ E}

Parabolic index and Ahlfors regular conformal dimension | 209

and ℓsym (E) = {θ ∈ ℓ(E) | θ(e) = θ(e)̌ for any e ∈ E}. Furthermore, we define ℓ#p (E, μ) = ℓp (E, μ) ∩ ℓ# (E) and p ℓsym (E, μ) = ℓp (E, μ) ∩ ℓsym (E).

(3) We define μ(x) = ∑y∈V μxy for any x ∈ V, and extend μ to a measure on V by setting μ(A) = ∑ μ(x) x∈A

for any subset A of V. We also define ℓp (V, μ) = {u ∈ ℓ(V) | ‖u‖ℓp (V,μ) < ∞} where ‖ ⋅ ‖ℓp (V,μ) is given by 󵄨 󵄨p ‖u‖ℓp (V,μ) = ( ∑ 󵄨󵄨󵄨u(x)󵄨󵄨󵄨 μ(x))

1/p

x∈V

.

We use (⋅, ⋅)ℓ2 (V,μ) and ⟨⋅, ⋅⟩ℓ2 (E,μ) to denote the inner product of ℓ2 (V, μ) and ℓ2 (E, μ), respectively, that is, for any u, v ∈ ℓ2 (V, μ) and θ, φ ∈ ℓ2 (E, μ), (u, v)ℓ2 (V,μ) = ∑ u(x)v(x)μ(x), x∈V

and ⟨θ, φ⟩ℓ2 (E,μ) = ∑ θ(e)φ(e)μe . e∈E

Next we introduce the p-energy on a weighted graph. In our notation, the 2-energy coincides with the Dirichlet form associated with random walks induced by a weighted graph. For this reason, the p-energy can be viewed as a nonlinear generalization of the Dirichlet form in discrete settings. On the other hand, p-energy is not bilinear except in the case p = 2. Definition 2.7. (1) We define the discrete gradient ∇ : ℓ(V) → ℓ# (E) by ∇u((x, y)) = u(y) − u(x) for any u ∈ ℓ(V) and (x, y) ∈ E.

210 | R. Shimizu (2) For u, v ∈ ℓ(V), define (G,μ)

ℰp

(u, v) =

1 󵄨p−1 󵄨 ∑ ∑ sgn(v(y) − v(x))󵄨󵄨󵄨v(y) − v(x)󵄨󵄨󵄨 (u(y) − u(x))μxy 2 x∈V y∈V

whenever the sum converges absolutely. We also define (G,μ)

ℰp

1 (u) = ℰp(G,μ) (u, u) = ‖∇u‖pℓp (E,μ) . 2

The quantity ℰp (u) is called the p-energy of u. (3) For any nonempty subset A ⊆ V and u, v ∈ ℓ(V), define (G,μ)

ℰp,A (u, v) = (G,μ)

1 󵄨 󵄨p−1 ∑ ∑ sgn(v(y) − v(x))󵄨󵄨󵄨v(y) − v(x)󵄨󵄨󵄨 (u(y) − u(x))μxy 2 x∈A y∈A

whenever the sum converges absolutely. We also define ℰp,A (u) = ℰp,A (u, u) = (G,μ)

(G,μ)

1 󵄨 󵄨p ∑ ∑ 󵄨󵄨u(x) − u(y)󵄨󵄨󵄨 μxy . 2 x∈A y∈A󵄨

(4) Let o ∈ V. We define the p-Dirichlet space on (G, μ), 𝒟p (G, μ), by p

𝒟 (G, μ) = {u ∈ ℓ(V) | ℰp

(G,μ)

(u) < ∞}

with the norm 󵄨 󵄨 ‖u‖𝒟p = ℰp(G,μ) (u)1/p + 󵄨󵄨󵄨u(o)󵄨󵄨󵄨. If no confusion may occur, we omit (G, μ) in the above notations. For example, we (G,μ) (G,μ) write ℰp (u, v), ℰp (u), and 𝒟p instead of ℰp (u, v), ℰp (u), and 𝒟p (G, μ). The definition of the norm ‖ ⋅ ‖𝒟p depends on the base point o. However, the topology given by the norm is the same for any base point. Lemma 2.8. Any choice of the base point gives rise to an equivalent norm of 𝒟p , that is, for any x ∈ V there exists a constant Cx ∈ [1, ∞) such that 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 Cx−1 (ℰp (u)1/p + 󵄨󵄨󵄨u(x)󵄨󵄨󵄨) ≤ ℰp (u)1/p + 󵄨󵄨󵄨u(o)󵄨󵄨󵄨 ≤ Cx (ℰp (u)1/p + 󵄨󵄨󵄨u(x)󵄨󵄨󵄨) for any u ∈ 𝒟p . Proof. Let λo,x = [o = x0 , x1 , . . . , xn−1 , xn = x] be a simple path from o to x. By Hölder’s inequality, we have n

󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨u(o) − u(x)󵄨󵄨󵄨 ≤ ∑󵄨󵄨󵄨u(xi−1 ) − u(xi )󵄨󵄨󵄨 i=1

n

1/p

󵄨 󵄨p ≤ (∑󵄨󵄨󵄨u(xi−1 ) − u(xi )󵄨󵄨󵄨 μxi−1 xi ) i=1

1/p󸀠 −p󸀠 /p (∑ μxi−1 xi ) . i=1 n

Parabolic index and Ahlfors regular conformal dimension

| 211

This implies n

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 −p󸀠 /p 󵄨󵄨u(x)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨u(o) − u(x)󵄨󵄨󵄨 + 󵄨󵄨󵄨u(o)󵄨󵄨󵄨 ≤ (∑ μxi−1 xi )

1/p󸀠

i=1

1/p

󵄨 󵄨 + 󵄨󵄨󵄨u(o)󵄨󵄨󵄨

1/p

󵄨 󵄨 + 󵄨󵄨󵄨u(x)󵄨󵄨󵄨.

ℰp (u)

and n

󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 −p󸀠 /p 󵄨󵄨u(o)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨u(o) − u(x)󵄨󵄨󵄨 + 󵄨󵄨󵄨u(x)󵄨󵄨󵄨 ≤ (∑ μxi−1 xi )

1/p󸀠

ℰp (u)

i=1

Hence we obtain 1/p

󵄨 󵄨 󵄨 󵄨 + 󵄨󵄨󵄨u(x)󵄨󵄨󵄨 ≤ Cx (ℰp (u)1/p + 󵄨󵄨󵄨u(o)󵄨󵄨󵄨)

1/p

󵄨 󵄨 󵄨 󵄨 + 󵄨󵄨󵄨u(o)󵄨󵄨󵄨 ≤ Cx (ℰp (u)1/p + 󵄨󵄨󵄨u(x)󵄨󵄨󵄨),

ℰp (u)

and ℰp (u)

/p 1/p where Cx = ((∑ni=1 μ−p ) ∨ 1. This completes the proof. xi−1 xi ) 󸀠

󸀠

In the rest of this section, we prove basic properties of the p-energy and the p-Dirichlet space. Given a real-valued function u, we use u to denote (u ∨ 0) ∧ 1, which is called the unit contraction of u. The following properties of the p-energy are well-known if p = 2 (see [3, Proposition 1.21 and Lemma 1.27], for example). Proposition 2.9. (1) ℰp (u) ≤ ℰp (u) for any u ∈ ℓ(V).

(2) ℰp (u1 ∧ u2 ∧ ⋅ ⋅ ⋅ ∧ un ) ≤ ∑ni=1 ℰp (ui ) for any {ui }ni=1 ⊆ ℓ(V).

(3) ℰp (u) = 0 if and only if u is constant.

(4) ℰp (u) ≤ 2p−1 ‖u‖pℓp (V,μ) for any u ∈ 𝒟p . In particular, ℓp (V, μ) ⊆ 𝒟p . (5) For any u ∈ 𝒟p and (x, y) ∈ E, 󵄨 󵄨󵄨 −1/p 1/p 󵄨󵄨u(x) − u(y)󵄨󵄨󵄨 ≤ μxy ℰp (u) .

(2.1)

Proof. (1) Since |u(x) − u(y)| ≤ |u(x) − u(y)| for any x, y ∈ V, we have ℰp (u) ≤ ℰp (u). (2) Set v = u1 ∧ u2 ∧ ⋅ ⋅ ⋅ ∧ un ∈ ℓ(V). Let (x, y) ∈ E and assume v(x) ≤ v(y). By the definition of v, there exists j ∈ {1, . . . , n} such that v(x) = uj (x). We have 0 ≤ v(y) − v(x) = v(y) − uj (x) ≤ uj (y) − uj (x). In particular, we obtain n

󵄨󵄨 󵄨p 󵄨 󵄨p 󵄨 󵄨p 󵄨󵄨v(x) − v(y)󵄨󵄨󵄨 ≤ max 󵄨󵄨󵄨ui (x) − ui (y)󵄨󵄨󵄨 ≤ ∑󵄨󵄨󵄨ui (x) − ui (y)󵄨󵄨󵄨 i∈{1,...,n} i=1

212 | R. Shimizu for any (x, y) ∈ E. Summing over (x, y) ∈ E, we have ℰp (v) ≤ ∑ni=1 ℰp (ui ). (3) This statement is obvious since μ has a positive weight on any edge. (4) By Fubini’s theorem, we have ℰp (u) =

≤

(∗)

=

1 󵄨p 󵄨 ∑ ∑ 󵄨󵄨u(y) − u(x)󵄨󵄨󵄨 μxy 2 x∈V y∈V 󵄨

2p−1 󵄨p 󵄨p 󵄨 󵄨 ∑ ∑ (󵄨󵄨u(x)󵄨󵄨󵄨 + 󵄨󵄨󵄨u(y)󵄨󵄨󵄨 )μxy 2 x∈V y∈V 󵄨

2p−1 󵄨 󵄨 󵄨p 󵄨p ( ∑ 󵄨󵄨u(x)󵄨󵄨󵄨 ∑ μxy + ∑ 󵄨󵄨󵄨u(y)󵄨󵄨󵄨 ∑ μxy ) = 2p−1 ‖u‖pℓp (V,μ) , 2 x∈V 󵄨 y∈V y∈V x∈V

where we used an elementary inequality (a + b)p ≤ 2p−1 (ap + bp ) for any a, b ∈ ℝ+ in (∗). (5) Let (x, y) ∈ E and u ∈ 𝒟p . Then 1/p −1/p 󵄨󵄨 󵄨 󵄨 󵄨p 1/p −1/p 󵄨󵄨u(x) − u(y)󵄨󵄨󵄨 ≤ (󵄨󵄨󵄨u(x) − u(y)󵄨󵄨󵄨 μxy ) μxy ≤ ℰp (u) μxy .

This completes the proof. The following proposition states topological properties of 𝒟p . These properties are also known for the case p = 2. See [3, Proposition 1.21], for example. Proposition 2.10. (1) (𝒟p , ‖ ⋅ ‖𝒟p ) is a Banach space. (2) Convergence in 𝒟p implies pointwise convergence. (3) For any bounded sequence {un }n∈ℕ in 𝒟p , there exist a subsequence {unk }k and u ∈ 𝒟p such that unk converges pointwise to u as k → ∞ and ‖u‖𝒟p ≤ lim infk→∞ ‖unk ‖𝒟p . In particular, ℰp (u) ≤ lim inf ℰp (unk ). k→∞

(4) Let {Vn }n∈ℕ be an exhaustion with o ∈ V1 and let {un ∈ ℓ(Vn )}n∈ℕ be a sequence of functions with supn∈ℕ (ℰp,Vn (un ) + |un (o)|) < ∞. Then there exist a subsequence {unk }k∈ℕ and u ∈ 𝒟p such that unk converges pointwise to u as k → ∞ and ‖u‖𝒟p ≤ lim infk→∞ ‖unk ‖𝒟p |V , where nk

‖unk ‖𝒟p |V

nk

󵄨 󵄨 = ℰp,Vn (u)1/p + 󵄨󵄨󵄨unk (o)󵄨󵄨󵄨. k

In particular, ℰp (u) ≤ lim inf ℰp,Vn (unk ). k→∞

k

Parabolic index and Ahlfors regular conformal dimension

| 213

Proof. (1)–(2) Let {un }n∈ℕ be a Cauchy sequence in 𝒟p . Set A = {x ∈ V | {un (x)}n∈ℕ is convergent}. If {un }n∈ℕ is a Cauchy sequence in 𝒟p , then {un (o)}n∈ℕ ⊆ ℝ is also a Cauchy sequence. Thus A is nonempty. Fix x ∈ A. Let y ∈ V with (x, y) ∈ E. Then {un (y)}n∈ℕ ⊆ ℝ is a Cauchy sequence and y ∈ A. Indeed, we have 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨un (y) − um (y)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨un (y) − um (y) − (un (x) − um (x))󵄨󵄨󵄨 + 󵄨󵄨󵄨un (x) − um (x)󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 = 󵄨󵄨󵄨(un − um )(x) − (un − um )(y)󵄨󵄨󵄨 + 󵄨󵄨󵄨un (x) − um (x)󵄨󵄨󵄨 󵄨 1/p 󵄨󵄨 ≤ μ−1/p + 󵄨󵄨un (x) − um (x)󵄨󵄨󵄨 󳨀→ 0. xy ℰp (un − um ) n∧m→∞ Since (V, E) is connected, it follows that A = V. Define u ∈ ℓ(V) by u(x) = limn→∞ un (x) for any x ∈ V. By Fatou’s lemma, we have 󵄨 󵄨 ‖u − un ‖𝒟p = ℰp (u − un )1/p + 󵄨󵄨󵄨u(o) − un (o)󵄨󵄨󵄨 󵄨 󵄨 ≤ lim inf ℰp (um − un )1/p + 󵄨󵄨󵄨u(o) − un (o)󵄨󵄨󵄨 󳨀→ 0, m→∞ n→∞ which shows the completeness of (𝒟p , ‖ ⋅ ‖𝒟p ). (3) Let {un }n∈ℕ be a bounded sequence in 𝒟p , i. e., supn∈ℕ ‖un ‖𝒟p < ∞. Then {un (o)}n∈ℕ is also bounded. Using (2.1) and the connectedness of (V, E), we see that {un (x)}n∈ℕ is bounded for any x ∈ V. Thus the standard diagonal procedure yields a subsequence {unk }k∈ℕ that converges pointwise. Let u ∈ ℓ(V) be its limit. By Fatou’s lemma, we have 󵄨 󵄨 󵄨 󵄨 ‖u‖𝒟p = ℰp (u)1/p + 󵄨󵄨󵄨u(o)󵄨󵄨󵄨 ≤ lim inf ℰp (unk )1/p + lim 󵄨󵄨󵄨unk (o)󵄨󵄨󵄨 = lim inf ‖unk ‖𝒟p . k→∞ k→∞ k→∞ ̃ n ∈ ℓ(V) by (4) Define u un (x)

̃ n (x) = { u

0

if x ∈ Vn ,

otherwise,

for each n ∈ ℕ. By the same argument as in the proof of (3), we obtain a subsequence {unk }k∈ℕ converging pointwise and its limit u ∈ ℓ(V). By Fatou’s lemma, we have 󵄨 󵄨 ‖u‖𝒟p = ℰp (u)1/p + 󵄨󵄨󵄨u(o)󵄨󵄨󵄨

󵄨 󵄨 ≤ lim inf ℰp,Vn (unk )1/p + lim 󵄨󵄨󵄨unk (o)󵄨󵄨󵄨 = lim inf ‖unk ‖𝒟p |V . nk k k→∞ k→∞ k→∞

This completes the proof. Proposition 2.11. ∇ : 𝒟p → ℓp (E, μ) is a bounded linear operator. Proof. It is obvious that ∇ is linear. Let u ∈ 𝒟p . Then 󵄨 󵄨p ‖∇u‖pℓp (E,μ) = 2ℰp (u) ≤ 2(ℰp (u)1/p + 󵄨󵄨󵄨u(o)󵄨󵄨󵄨) = 2‖u‖p𝒟p . Taking the supremum over u, we obtain ‖∇‖𝒟p →ℓp (E,μ) ≤ 2.

214 | R. Shimizu Remark 2.12. Combining the above proposition with the fact that ℓp (E, μ) is a reflexive Banach space, we see that 𝒟p is also reflexive (see [18, Proposition 1.1]). Proposition 2.10 also follows from this fact.

3 Variational problems on weighted graphs This section is devoted to giving a detailed survey of the results in [18]. First, we introduce the notions of the p-conductance and the p-modulus, which will be of great importance throughout this paper. Nakamura and Yamasaki have considered essentially the same quantities in [18], although we are going to use different notations. These notions can be thought as discrete analogues of the notions, introduced by Heinonen and Koskela [7], of the p-capacity and the p-modulus on a metric space, respectively. In Theorem 3.24, we prove one of the main results in [18], Nakamura–Yamasaki’s duality, which is a generalization of Thompson’s principle in the theory of electrical networks (see [3, Theorem 2.31], for example). Throughout this section, (G, μ) is a connected, locally finite, simple weighted graph and p ∈ (1, ∞).

3.1 p-conductance and p-modulus between two sets First, we give the definition of the p-conductance and see its properties. This quantity is the same as in [18, p. 100]. Definition 3.1. Let A, B be nonempty subsets of V. Define the p-conductance between (G,μ) A and B, 𝒞p (A, B), by (G,μ)

𝒞p

(A, B) = inf{ℰp (u) | u ∈ ℓ(V) with u|A ≡ 1, u|B ≡ 0}.

(3.1)

We say that u is feasible for 𝒞p (A, B) if u ∈ ℓ(V) satisfies u|A ≡ 1 and u|B ≡ 0. Moreover, we say that ψ is a minimizer of 𝒞p (A, B) if ψ is feasible for 𝒞p (A, B) and ℰp (ψ) = 𝒞p (A, B). The reciprocal of 𝒞p (A, B) is denoted by ρp

(G,μ)

(A, B), i. e.,

ρ(G,μ) (A, B) = 𝒞p(G,μ) (A, B)−1 . p If no confusion can occur, we write 𝒞p (A, B) = 𝒞p (A, B) and ρp (A, B) = ρp (A, B). Furthermore, for any x, y ∈ V, we use 𝒞p (x, y) and ρp (x, y) to denote 𝒞p ({x}, {y}) and ρp ({x}, {y}), respectively. (G,μ)

Remark 3.2. (1) Note that 𝒞p (A, B) = ∞ if A ∩ B ≠ 0, and thus ρp (A, B) = 0 in this case.

(G,μ)

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| 215

(2) By Proposition 2.9-(1), we have 𝒞p (A, B) = inf{ℰp (u) | u ∈ ℓ(V) with u|A ≡ 1, u|B ≡ 0, 0 ≤ u ≤ 1}.

We prove existence and uniqueness of a minimizer of 𝒞p (A, B). We first see existence. Lemma 3.3. Let A, B be nonempty disjoint subsets of V. If 𝒞p (A, B) < ∞, then there exists a minimizer of 𝒞p (A, B). (n) (n) Proof. We can choose {ψ(n) p }n∈ℕ so that ψp is feasible for 𝒞p (A, B), 0 ≤ ψp ≤ 1 and

−1 (n) p ℰp (ψ(n) p ) ≤ 𝒞p (A, B) + n for each n ∈ ℕ. Since {ψp }n∈ℕ is a bounded sequence in 𝒟 ,

there exist a subsequence {ψp k }k∈ℕ converging pointwise and its limit ψp by Proposition 2.10-(3). Note that ψp is feasible for 𝒞p (A, B). Moreover, by Proposition 2.10-(3), we have (n )

ℰp (ψp ) ≤ lim inf ℰp (ψp k ) = 𝒞p (A, B). (n )

k→∞

Hence ψp is a minimizer of 𝒞p (A, B). To prove uniqueness, we need Clarkson’s inequality ([5]). For details and proofs, see [9, pp. 225 and 227], for example. Recall that p󸀠 denotes the conjugate index of p. Proposition 3.4 (Clarkson’s inequality in Lp ). Let (X, ℱ , μ) be a measure space and let f , g ∈ Lp (X, μ). (1) If p ∈ [2, ∞), then ‖f + g‖pLp + ‖f − g‖pLp ≤ 2p−1 (‖f ‖pLp + ‖g‖pLp ). (2) If p ∈ (1, 2], then ‖f + g‖pLp + ‖f − g‖pLp ≤ 2(‖f ‖pLp + ‖g‖pLp ) 󸀠

󸀠

p󸀠 /p

.

Corollary 3.5 (Clarkson’s inequality for ℰp ). Let u, v ∈ 𝒟p . (1) If p ∈ [2, ∞), then ℰp (u + v) + ℰp (u − v) ≤ 2

p−1

(3.2)

(ℰp (u) + ℰp (v)).

(2) If p ∈ (1, 2], then 1/(p−1)

ℰp (u + v)

+ ℰp (u − v)1/(p−1) ≤ 2(ℰp (u) + ℰp (v))

1/(p−1)

.

(3.3)

Proof. Apply Clarkson’s inequality in ℓp (E, μ) for f = ∇u and g = ∇v. Proposition 3.6. Let A, B be nonempty disjoint subsets of V. Suppose that 𝒞p (A, B) < ∞. Then there exists a unique minimizer of 𝒞p (A, B).

216 | R. Shimizu Proof. Existence has already been proved in Lemma 3.3. Let u, v ∈ 𝒟p be minimizers of 𝒞p (A, B). Set w = 21 (u + v). Then w is feasible for 𝒞p (A, B), and hence ℰp (w) ≤ 𝒞p (A, B). If p ≥ 2, then, by Clarkson’s inequality for ℰp , we have 0 ≤ ℰp (u − v) ≤ −ℰp (2w) + 2p−1 (ℰp (u) + ℰp (v)) ≤ −2p 𝒞p (A, B) + 2p 𝒞p (A, B) = 0. Thus u − v ∈ ℝ1V . If p ∈ (1, 2], then, by Clarkson’s inequality for ℰp , we have p󸀠 /p

0 ≤ ℰp (u − v)1/(p−1) ≤ − ℰp (2w)p /p + 2(ℰp (u) + ℰp (v)) 󸀠

= − 2p 𝒞p (A, B)p /p + 21+p /p 𝒞p (A, B)p /p = 0. 󸀠

󸀠

󸀠

󸀠

Thus u − v ∈ ℝ1V . Since u|A = v|A , it follows that u = v. Hereafter, we use ψp (A, B) to denote the minimizer of 𝒞p (A, B) for subsets A, B of V with 𝒞p (A, B) < ∞. Next, we prove basic properties of the p-conductance. Proposition 3.7. Let A, B be nonempty disjoint subsets of V. (1) (symmetry) 𝒞p (A, B) = 𝒞p (B, A). (2) (domain monotonicity) Let A󸀠 , B󸀠 be subsets of V. If A ⊆ A󸀠 and B ⊆ B󸀠 , then 𝒞p (A, B) ≤ 𝒞p (A󸀠 , B󸀠 ). (3) (monotonicity with respect to the parameter) If p, q ∈ [1, ∞) satisfy p ≤ q, then 𝒞q (A, B) ≤ 𝒞p (A, B). (4) If either 𝜕e A or 𝜕e B is finite, then 𝒞p (A, B) < ∞. In particular, if either A or B is finite, then 𝒞p (A, B) < ∞. Proof. (1) Let u ∈ ℓ(V) be feasible for 𝒞p (A, B). Then 1V − u is feasible for 𝒞p (B, A), and vice versa. Since ℰp (u) = ℰp (1V − u), it follows that 𝒞p (A, B) = 𝒞p (B, A). (2) If u ∈ ℓ(V) is feasible for 𝒞p (A󸀠 , B󸀠 ), then u is also feasible for 𝒞p (A, B). This observation proves the result. (3) By Remark 3.2, it holds that 𝒞p (A, B) = inf{ℰp (u) | u ∈ ℓ(V) with u|A ≡ 1, u|B ≡ 0, 0 ≤ u ≤ 1}.

If u ∈ ℓ(V) satisfies 0 ≤ u ≤ 1, then we have |u(x) − u(y)|q ≤ |u(x) − u(y)|p for any (x, y) ∈ E. Hence we obtain ℰq (u) ≤ ℰp (u) whenever u is feasible for 𝒞p (A, B) and 𝒞q (A, B). This proves 𝒞q (A, B) ≤ 𝒞p (A, B). (4) By (1), we may assume that 𝜕e A is finite. Clearly, 1A is feasible for 𝒞p (A, B). Thus we have 𝒞p (A, B) ≤ ℰp (1A ) = ∑e∈𝜕e A μe < ∞. The following proposition is inspired by the theory of resistance forms. See [10, Chapter 2], for example. In the case p = 2, this fact has been proved in [10, Proposition 2.1.16].

Parabolic index and Ahlfors regular conformal dimension

| 217

Proposition 3.8. For any x, y ∈ V, ρp (x, y) = sup{

|u(x) − u(y)|p 󵄨󵄨 p 󵄨󵄨 u ∈ 𝒟 \ ℝ1V }. ℰp (u)

(3.4)

In particular, |u(x) − u(y)|p ≤ ρp (x, y)ℰp (u) for any u ∈ ℓ(V) and x, y ∈ V. Proof. If x = y, then both sides of (3.4) are equal to 0. Let u ∈ 𝒟p \ ℝ1V and let x, y ∈ V u−u(y) with x ≠ y. Set v = u(x)−u(y) . Then v is feasible for 𝒞p (x, y). Thus we have |u(x) − u(y)|p ≤ ρp (x, y). ℰp (u) Taking the supremum over 𝒟p \ ℝ1V , we obtain sup{

|u(x) − u(y)|p 󵄨󵄨 p 󵄨󵄨 u ∈ 𝒟 \ ℝ1V } ≤ ρp (x, y). ℰp (u)

(3.5)

Let ψp be the minimizer of 𝒞p (x, y) with ψp (x) = 1 and ψp (y) = 0. It is obvious that ψp ∈ 𝒟p \ ℝ1V and |ψp (x) − ψp (y)|p /ℰp (ψp ) = 𝒞p (x, y)−1 = ρp (x, y). Hence ψp attains the equality of (3.5). The following lemma is useful for concrete examples. For p = 2, this lemma has been proved in [3, Proposition 2.18], for example. Lemma 3.9. Let A, B be nonempty finite subsets of V and let {An }n∈ℕ be an exhaustion of V with o ∈ A1 . Then lim 𝒞 (n) (A n→∞ p

∩ An , B ∩ An ) = 𝒞p (A, B),

where 𝒞p(n) (⋅, ⋅) is the p-conductance with respect to (An , E An , μAn ). Proof. Set Cn = 𝒞p(n) (A ∩ An , B ∩ An ) for each n ∈ ℕ. If u ∈ ℓ(V) is feasible for 𝒞p (A, B), then u|An is feasible for Cn for any n ∈ ℕ. In addition, if un+1 ∈ ℝAn+1 is feasible for Cn , then un+1 |An is feasible for Cn . Furthermore, ℰp,An (u|An ) ≤ ℰp,An+1 (u|An+1 ) ≤ ℰp (u). We conclude that Cn ≤ Cn+1 ≤ 𝒞p (A, B) for any n ∈ ℕ. Hence limn→∞ Cn ≤ 𝒞p (A, B). Set C = limn→∞ Cn . ̃ ∈ ℝAn be the For the converse inequality, we may assume that C is finite. Let ψ n ̃ ≤ 1 for any n ∈ ℕ. Define ψ ∈ ℓ(V) by minimizer of Cn . Note that 0 ≤ ψ n n ̃ (x) if x ∈ A , ψ n n ψn (x) = { 0 otherwise. ̃ ) = C for each n ∈ ℕ. Hence Then ℰp,An (ψn |An ) = ℰp(n) (ψ n n 󵄨 󵄨 sup(ℰp,An (ψn |An ) + 󵄨󵄨󵄨ψn (o)󵄨󵄨󵄨) ≤ C + 1 < ∞. n∈ℕ

218 | R. Shimizu By Proposition 2.10-(4), there exist a pointwise convergent subsequence {ψnk }k∈ℕ and its limit ψ ∈ ℓ(V). Furthermore, it follows that ℰp (ψ) ≤ lim inf ℰp,An (ψnk |An ) = lim inf Cnk = C. k→∞

k

k

k→∞

Since ψ is feasible for 𝒞p (A, B), we obtain 𝒞p (A, B) ≤ C. We have seen some basic properties of p-conductances. Next we introduce the notion of the p-modulus on graphs and see its properties. On graphs, the p-modulus is the reciprocal of the extremal length introduced by [18]. Recall that 𝒫 (G) is the collection of finite paths in G. Notation. Let θ ∈ ℓsym (E) and let λ = [x0 , x1 , . . . , xn ] ∈ 𝒫 (G). For simplicity, we use ∑λ θ to denote ∑ni=1 θ((xi−1 , xi )). Definition 3.10. Let Γ ⊆ 𝒫 (G) be a family of paths. We define the p-modulus of Γ, Mod(G,μ) (Γ), by p Mod(G,μ) (Γ) = p

1 inf ‖θ‖pℓp (E,μ) , 2 θ∈Ap (Γ)

p where Ap (Γ) = {θ ∈ ℓsym (E, μ) | ∑λ θ ≥ 1 for any λ ∈ Γ}. A function Ψ ∈ Ap (Γ) is said to

(Γ). If no confusion be a minimizer of Mod(G,μ) (Γ) if it satisfies 21 ‖Ψ‖pℓp (E,μ) = Mod(G,μ) p p can occur, we write Modp (Γ) = Mod(G,μ) (Γ). p Set also A (Γ) = {θ ∈ ℓsym (E) | ∑ θ ≥ 1 for any λ ∈ Γ}, λ

Ap (Γ) = {θ ∈ Ap (Γ) | θ ≥ 0}, +

and Ap (Γ) = {θ ∈ Ap (Γ) | 0 ≤ θ ≤ 1}. ∗

Lemma 3.11. Let Γ ⊆ 𝒫 (G) be a family of paths. Then Modp (Γ) =

1 1 inf+ ‖θ‖pℓp (E,μ) = inf ‖θ‖pℓp (E,μ) . 2 θ∈Ap (Γ) 2 θ∈Ap∗ (Γ)

Proof. It is obvious that Ap∗ (Γ) ⊆ Ap+ (Γ) ⊆ Ap (Γ) and thus inf

θ∈Ap∗ (Γ)

‖θ‖pℓp (E,μ) ≥

inf

θ∈Ap+ (Γ)

‖θ‖pℓp (E,μ) ≥ 2Modp (Γ).

For any θ ∈ Ap (Γ), it is obvious that θ ∈ Ap∗ (Γ). Since ‖θ‖ℓp (E,μ) ≤ ‖θ‖ℓp (E,μ) , we have the desired result.

Parabolic index and Ahlfors regular conformal dimension

| 219

As we mentioned in the beginning of this section, our definition of the p-modulus is a discrete analogue of the notion of the p-modulus on a metric space. The proposition below states basic properties of p-moduli. It is known that similar results hold in the metric space setting. See [8, Section 5.2], for example. Proposition 3.12. (1) Modp (0) = 0. (2) (monotonicity) Let Γ1 , Γ2 be families of paths in G. If Γ1 ⊆ Γ2 , then Modp (Γ1 ) ≤ Modp (Γ2 ). (3) (subadditivity) Let {Γn }n∈ℕ be a sequence of families of paths in G. Then ∞

Modp ( ⋃ Γn ) ≤ ∑ Modp (Γn ). n∈ℕ

n=1

(4) (majority) Let Γ, Γ0 be families of paths in G. If each path λ ∈ Γ has a subpath λ0 ∈ Γ0 , then Modp (Γ) ≤ Modp (Γ0 ). Proof. (1) Since 0 ⋅ 1E ∈ Ap (0), we have 0 ≤ Modp (0) ≤ ‖0 ⋅ 1E ‖ℓp (E,μ) = 0. (2) If Γ1 ⊆ Γ2 , then Ap (Γ1 ) ⊇ Ap (Γ2 ). Hence Modp (Γ1 ) ≤ Modp (Γ2 ). (3) It is enough to consider the case ∑∞ n=1 Modp (Γn ) < ∞. Fix ϵ > 0. For each n ∈ ℕ, choose a function θn ∈ Ap (Γn ) satisfying ‖θn ‖pℓp (E,μ) ≤ 2Modp (Γn ) + 2−n ϵ and set θ(e) = p 1/p p (∑∞ n=1 |θn (e)| ) . Then θ ∈ ℓ (E, μ) and θ ≥ θn for any n ∈ ℕ. Thus θ ∈ Ap (⋃n∈ℕ Γn ) and ∞

2Modp ( ⋃ Γn ) ≤ ‖θ‖pℓp (E,μ) ≤ 2 ∑ Modp (Γn ) + ϵ. n∈ℕ

n=1

Letting ϵ ↓ 0, we have Modp (⋃n∈ℕ Γn ) ≤ ∑∞ n=1 Modp (Γn ). (4) Fix θ ∈ Ap+ (Γ0 ). Let λ = [x0 , x1 , . . . , xn ] ∈ Γ. Since λ has a subpath λ0 = [xk , xk+1 , . . . , xl ] ∈ Γ0 , we obtain n

l

i=1

i=k+1

∑ θ((xi−1 , xi )) ≥ ∑ θ((xi−1 , xi )) ≥ 1. This implies θ ∈ Ap+ (Γ), and thus Ap+ (Γ0 ) ⊆ Ap+ (Γ). Hence Modp (Γ) ≤ Modp (Γ0 ). To prove existence and uniqueness of a minimizer of Modp (A, B), we are going to use a technique in convex analysis. The following Mazur’s lemma and the notion of uniformly convexity are indispensable in this technique. For a proof of Mazur’s lemma, see [8, Section 2.3], for example.

220 | R. Shimizu Lemma 3.13 (Mazur’s lemma). Let {vn }n∈ℕ be a sequence in a normed space V converging weakly to an element v ∈ V. Then there exist a subsequence {mk }k∈ℕ and a sem m quence of nonnegative numbers {λi(k) }i=kk satisfying ∑i=kk λi(k) = 1 for each k ∈ ℕ such m

that ∑i=kk λi(k) vi converges to v in the norm as k → ∞.

Definition 3.14. A Banach space (V, ‖ ⋅ ‖) is said to be uniformly convex if and only if, for any ϵ > 0, there exists δ > 0 such that ‖v‖ = ‖w‖ = 1

and ‖v − w‖ > ϵ

implies that 󵄩󵄩 󵄩󵄩 1 󵄩 󵄩󵄩 󵄩󵄩 (v + w)󵄩󵄩󵄩 < 1 − δ 󵄩󵄩 󵄩󵄩 2 for any v, w ∈ V. It is known that uniformly convex Banach spaces are reflexive (see, e. g., [8, Theorem 2.4.9]). What we need is the following theorem, which is a fundamental fact in convex analysis (see, e. g., [8, Corollary 2.4.16], [18, Theorem A]). Theorem 3.15. Let (V, ‖ ⋅ ‖) be a uniformly convex Banach space. If A is a nonempty closed convex subset of V, then there exists a unique element v ∈ A attaining the following minimum: ‖v‖ = min ‖w‖. w∈A

Proof. First, we prove existence of a minimizer of infw∈A ‖w‖. Since A ≠ 0, we have infw∈A ‖w‖ < ∞. Set M = infw∈A ‖w‖. We can choose a sequence {vn }n∈ℕ ⊆ A so that ‖vn ‖ ∈ [M, M + n−1 ) for each n ∈ ℕ. Obviously, it holds that supn∈ℕ ‖vn ‖ ≤ M + 1. Since (V, ‖ ⋅ ‖) is reflexive, there exist a subsequence {vnk }k∈ℕ and v∗ ∈ V such that vnk converges weakly to v∗ as k → ∞. By the lower semicontinuity of the norm with respect to weak limits, we have ‖v∗ ‖ ≤ lim inf ‖vnk ‖ ≤ M. k→∞

Applying Mazur’s lemma, we obtain a sequence {ṽk }k∈ℕ in A such that ṽk converges to v∗ in V as k → ∞. Since A is closed, it holds that v∗ ∈ A. Hence ‖v∗ ‖ = M. Suppose that w∗ ∈ A also has the minimal norm in A, i. e., ‖w∗ ‖ = M. Considering v∗ /M, w∗ /M instead of v∗ , w∗ , respectively, we may assume M = 1 without loss of generality. Suppose that v∗ ≠ w∗ . Because (V, ‖ ⋅ ‖) is uniformly convex, there exists δ > 0 such that 󵄩󵄩 󵄩󵄩 1 󵄩 󵄩󵄩 󵄩󵄩 (v∗ + w∗ )󵄩󵄩󵄩 < 1 − δ. 󵄩󵄩 󵄩󵄩 2

Parabolic index and Ahlfors regular conformal dimension

| 221

On the other hand, since 21 (v∗ + w∗ ) ∈ A, we have ‖ 21 (v∗ + w∗ )‖ ≥ M = 1. This contradiction implies v∗ = w∗ . Lemma 3.16 ([1, Lemma 2.1]). Assume that Modp (Γ) < ∞, i. e., Ap (Γ) ≠ 0. Then there exists a unique minimizer Ψ ∈ Ap∗ (Γ) of Modp (Γ). Proof. It is known that ℓp (E, μ) is uniformly convex (see [8, Proposition 2.4.19], for example). Thus, by Theorem 3.15 and Lemma 3.11, it is enough to show that Ap (Γ) is a closed convex subset of ℓp (E, μ). It is obvious that Ap (Γ) is convex. We prove that Ap (Γ) is closed in ℓp (E, μ). Let λ = [x0 , x1 , . . . , xn ] ∈ Γ and let θ, φ ∈ ℓp (E, μ). Then Hölder’s inequality shows 󵄨󵄨 n 󵄨󵄨 n n 󵄨󵄨 󵄨 󵄨󵄨∑ θ((xi−1 , xi )) − ∑ φ((xi−1 , xi ))󵄨󵄨󵄨 ≤ ∑󵄨󵄨󵄨θ((xi−1 , xi )) − φ((xi−1 , xi ))󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨i=1 󵄨󵄨 i=1 i=1 1/p󸀠 −p󸀠 /p (∑ μxi−1 xi ) ‖θ i=1 n

≤

− φ‖ℓp (E,μ) .

(3.6)

Let {θm }m∈ℕ ⊆ Ap (Γ) be a sequence converging to some θ ∈ ℓp (E, μ) with respect to ‖ ⋅ ‖ℓp (E,μ) . For any λ = [x0 , x1 , . . . , xn ] ∈ Γ, by (3.6), we have n

n

∑ θ((xi−1 , xi )) = lim ∑ θm ((xi−1 , xi )) ≥ 1. i=1

m→∞

i=1

This implies θ ∈ Ap (Γ). Hence Ap (Γ) is closed in ℓp (E, μ). Hereafter, we use Ψp (Γ) to denote the minimizer of Modp (Γ). Let A and B be nonempty disjoint subsets of V. We denote the collection of paths from A to B by Γ(A, B), i. e., Γ(A, B) = {λ = [x0 , . . . , xn ] ∈ 𝒫 (G) | x0 ∈ A, xn ∈ B, n ∈ ℕ}. We use Modp (A, B) to denote Modp (Γ(A, B)). The following theorem shows that the p-modulus Modp (A, B) is a geometric expression of the p-conductance 𝒞p (A, B). This is a discrete analogue of [6, Theorem 7.31] and has been proved for finite graphs in [1, Theorem 4.2]. Theorem 3.17. Suppose that Modp (A, B) < ∞. Let Ψp (A, B) be the minimizer of Modp (A, B) and let ψp (A, B) be the minimizer of 𝒞p (A, B). Then 󵄨 󵄨 Ψp (A, B) = 󵄨󵄨󵄨∇ψp (A, B)󵄨󵄨󵄨. In particular, Modp (A, B) = 𝒞p (A, B).

222 | R. Shimizu Proof. Set Ψ = Ψp (A, B) ∈ Ap∗ (Γ(A, B)). Define ψ0 ∈ ℓ(V) by ψ0 (x) = inf{∑ Ψ | λ ∈ Γ(B, {x})}1V\B (x) λ

for any x ∈ V. Since Ψ ∈ Ap (Γ(A, B)), we see that ψ0 |B ≡ 0 and ψ0 |A ≥ 1. Let ψ ∈ ℓ(V) be the unit contraction of ψ0 . Note that ψ|A ≡ 1 and ψ|B ≡ 0. For any λ = [x0 , x1 , . . . , xn ] ∈ Γ(A, B), we have n

󵄨 󵄨 󵄨 󵄨 ∑|∇ψ| = ∑󵄨󵄨󵄨ψ(xi ) − ψ(xi−1 )󵄨󵄨󵄨 ≥ 󵄨󵄨󵄨ψ(xn ) − ψ(x0 )󵄨󵄨󵄨 = 1. λ

i=1

This implies |∇ψ| ∈ A (Γ). Next we are going to prove that 0 ≤ |∇ψ0 | ≤ Ψ. Replacing (x, y) ∈ E by (y, x) if necessary, we may assume that 0 ≤ ∇ψ0 ((x, y)) = ψ0 (y) − ψ0 (x). In this way, we may exclude the case x ∈ ̸ B and y ∈ B. Now we have three cases: Case I: Assume that x, y ∈ B. In this case, ∇ψ0 ((x, y)) = 0 ≤ Ψ((x, y)). Case II: Assume that x ∈ B and y ∈ ̸ B. In this case, since (x, y) ∈ Γ(B, {y}), we have ∇ψ0 ((x, y)) = ψ0 (y) ≤ Ψ((x, y)). Case III: Assume that x ∈ ̸ B and y ∈ ̸ B. Let ϵ > 0. We can choose λϵ ∈ Γ(B, {x}) so that ψ0 (x) ≤ ∑ Ψ < ψ0 (x) + ϵ. λϵ

Considering the path λϵ ∗ (x, y) ∈ Γ(B, {y}), we have ψ0 (y) ≤

∑ Ψ ≤ ψ0 (x) + Ψ((x, y)) + ϵ.

λϵ ∗(x,y)

Letting ϵ ↓ 0, we see that 0 ≤ ψ0 (y)−ψ0 (x) ≤ Ψ((x, y)). Thus it holds that 0 ≤ |∇ψ0 (e)| ≤ Ψ(e) for any e ∈ E in all the cases. Since ψ is the unit contraction of ψ0 , we also have 0 ≤ |∇ψ(e)| ≤ Ψ(e) for all e ∈ E. This implies |∇ψ| ∈ Ap (Γ(A, B)) and ‖|∇ψ|‖pℓp (E,μ) ≤ ‖Ψ‖pℓp (E,μ) = 2Modp (A, B). Because the minimizer of Modp (A, B) is unique, we obtain |∇ψ| = Ψ. Clearly, ψ is feasible for 𝒞p (A, B). Hence 𝒞p (A, B) ≤ ℰp (ψ) =

1 ‖|∇ψ|‖pℓp (E,μ) = Modp (A, B). 2

Next we prove Modp (A, B) ≤ 𝒞p (A, B). Let u ∈ 𝒟p be feasible for 𝒞p (A, B) and let λ = [x0 , x1 , . . . , xn ] ∈ Γ(A, B). Since 1 = |u(x0 ) − u(xn )| ≤ ∑λ |∇u|, we see that |∇u| ∈ Ap (Γ(A, B)). Thus 1 Modp (A, B) ≤ ‖|∇u|‖pℓp (E,μ) = ℰp (u). 2

Parabolic index and Ahlfors regular conformal dimension

| 223

This implies Modp (A, B) ≤ 𝒞p (A, B). We conclude that ℰp (ψ) = Modp (A, B) = 𝒞p (A, B).

Uniqueness of the minimizer of 𝒞p (A, B) yields ψ = ψp (A, B). The following generalizations of the cutting-law and the shorting-law are immediate by Theorem 3.17 although these are not mentioned in [18, 20, 21, 23, 24, 25, 26]. For p = 2, these results are also known in electrical network theory (see [3, Corollary 2.27]). Recall the definitions of cuttings and shortings (Definition 2.5). Proposition 3.18. Let A, B be nonempty subsets of V with 𝒞p (A, B) < ∞. (1) Cutting decreases 𝒞p (A, B). (2) Shorting increases 𝒞p (A, B). Proof. (1) Let Mod(cut) (A, B) be the p-modulus on (V, E, μ(cut) ). Note that the definition p

of A (Γ(A, B)) does not depend on μ. Let θ ∈ ℓ(E). Since μ(cut) ≤ μe for any e ∈ E, it e holds that ‖θ‖ℓp (E,μ) ≥ ‖θ‖ℓp (E,μ(cut) ) . Thus we have that 1 inf ‖θ‖pℓp (E,μ) 2 θ∈A (Γ(A,B)) 1 inf ‖θ‖pℓp (E,μ(cut) ) = Mod(cut) (A, B). ≥ p 2 θ∈A (Γ(A,B))

Modp (A, B) =

By Theorem 3.17, we have the desired result. (2) Set ϒ = (V (short) , E (short) , μ(short) ). Let Mod(short) (A, B) be the p-modulus on ϒ and let p

Γ(short) (A, B) be the collection of paths in ϒ from A to B. Let λ = [x0 , x1 , . . . , xn ] ∈ Γ(A, B)S . If there exists k ∈ {1, . . . , n} such that V (short) = (V \ {xk−1 , xk }) ∪ {♦}, then we define λ(short) ∈ Γ(short) (A, B) by λ(short) = [x0 , . . . , xk−2 , ♦, xk+1 , . . . , xn ]. If (xi−1 , xi ) ≠ e for any i, then we define λ(short) = λ. Let θ∗ ∈ A (Γ(short) (A, B)). Define θ ∈ ℓ(E) by θ∗ ((z, w)) if (z, w) ∈ E V\{x,y} , { { { { { {θ∗ ((z, ♦)) if z ≠ x and w = y, θ((z, w)) = { { θ∗ ((♦, w)) if z = x and w ≠ y, { { { { if {z, w} = {x, y}. {0 Since ∑λ θ = ∑λ(short) θ∗ , we have θ ∈ A (Γ(A, B)). It is easy to see that ‖θ∗ ‖ℓp (E (short) ,μ(short) ) ≥ ‖θ‖ℓp (E,μ) .

224 | R. Shimizu Hence 1 inf ‖θ ‖p 2 θ∗ ∈A (Γ(short) (A,B)) ∗ ℓp (E (short) ,μ(short) ) 1 ≥ inf ‖θ‖pℓp (E,μ) = Modp (A, B), 2 θ∈A (Γ(A,B))

Mod(short) (A, B) = p

which completes the proof.

3.2 Nakamura–Yamasaki’s duality The aim of this subsection is to prove Nakamura–Yamasaki’s duality theorem (Theorem 3.24). To state this duality, we first define flows on a weighted graph, the dual p-energy of a flow, and some function spaces on the edge set. Definition 3.19. Let I ∈ ℓ# (E). For x ∈ V, define the divergence of I at x, Div I(x), by Div I(x) =

∑ I((x, y)).

y∈NG (x)

For a subset A of V, define the flux of I from A, Flux(I; A), by Flux(I; A) = ∑ Div I(x). x∈A

Throughout this subsection, we assume that A and B are nonempty finite subsets of V. Under this condition, Flux(⋅; A) and Flux(⋅; B) are linear. Definition 3.20. (1) Let i0 be a nonnegative constant. We say that I ∈ ℓ# (E) is a flow from A to B with strength i0 if and only if Div I(x) = 0 for any x ∈ V \ (A ∪ B) and Flux(I; A) = −Flux(I; B) = i0 . We also use ‖I‖ to denote the strength of I. If ‖I‖ = 1, then we call I a unit flow. (2) F(A, B) denotes the collection of flows from A to B, i. e., 󵄨󵄨 󵄨 Div I(x) = 0 for all x ∈ V \ (A ∪ B), F(A, B) = {I ∈ ℓ# (E) 󵄨󵄨󵄨󵄨 }. 󵄨󵄨 0 ≤ Flux(I; A) = −Flux(I; B) (3) Define μ

‖⋅‖ℓp (E,μ)

ℱp (A, B) = F(A, B) ∩ ℓc (E)

.

(3.7)

We define energies of flows as follows. Recall that p󸀠 is the conjugate index of p.

Parabolic index and Ahlfors regular conformal dimension

| 225

Definition 3.21. Let I ∈ ℓ# (E). We define the dual p-energy of I on (G, μ), Ep

(G,μ)

Ep(G,μ) (I) =

(I), by

󸀠 1 1 󵄨p󸀠 󵄨 ∑ 󵄨󵄨󵄨I(e)󵄨󵄨󵄨 μ∗e = ‖I‖pp󸀠 ∗ , 2 e∈E 2 ℓ (E,μ )

where μ∗ is the dual weight of μ given by 󸀠

μ∗e = (μe )−p /p for any e ∈ E. If no confusion can occur, we use Ep (I) to denote Ep

(G,μ)

(I).

Next we introduce the notion of the dual p-modulus and state its properties. Definition 3.22. Let A and B be nonempty subsets of V. The dual p-modulus between A and B, Mod(G,μ) (A, B)∗ , is defined by p μ∗

Mod(G,μ) (A, B)∗ = inf{Ep (I) | I ∈ ℱp󸀠 (A, B), ‖I‖ = 1}. p μ∗

We say that a unit flow J is feasible for Mod(G,μ) (A, B)∗ if and only if J ∈ ℱp󸀠 (A, B). p Moreover, we say that J is a minimizer of Mod(G,μ) (A, B)∗ if and only if J is feasible p

for Mod(G,μ) (A, B)∗ and Ep (J) = Mod(G,μ) (A, B)∗ . If no confusion can occur, we use p p Mod∗p (A, B) to denote Mod(G,μ) (A, B)∗ . p

Proposition 3.23 ([18, Propositions 4.1 and 4.2]). Let A, B be nonempty subsets of V. (1) Mod∗p (A, B) = Mod∗p (B, A). (2) If either A or B is a finite set, then Mod∗p (A, B) = inf{Ep (I) | I ∈ F(A, B) ∩ ℓc (E), ‖I‖ = 1}. (3) If A ∩ B = 0, then Mod∗p (A, B) < ∞. (4) Assume that A ∩ B = 0. If either A or B is a finite set, then there exists a unique minimizer of Mod∗p (A, B). In particular, Mod∗p (A, B) > 0. μ∗

μ∗

Proof. (1) For any unit flow I ∈ ℱp󸀠 (A, B), it holds that −I ∈ ℱp󸀠 (B, A), ‖ − I‖ = 1 and

Ep (I) = Ep (−I). Thus we have that Mod∗p (A, B) = Mod∗p (B, A). (2) Clearly, we have Mod∗p (A, B) ≤ inf{Ep (I) | I ∈ F(A, B) ∩ ℓc (E), ‖I‖ = 1}. Thus it is enough to prove the converse inequality. μ∗ By (1), we may assume that A is a finite set. Let I ∈ ℱp󸀠 (A, B). Then there exists

a sequence {In }n∈ℕ in F(A, B) ∩ ℓc (E) such that In converges to I in ℓp (E, μ∗ ). We can choose a subsequence {Ink }k∈ℕ so that Ink converges pointwise to I as k → ∞. Since A is finite, the pointwise convergence implies that 󸀠

‖Ink ‖ = Flux(Ink ; A) 󳨀→ Flux(I; A) = ‖I‖ = 1. k→∞

226 | R. Shimizu Hence there exists K ∈ ℕ such that ‖Ink ‖ > 0 for all k ≥ K. Set ̃Ink = Ink /‖Ink ‖ for each k ≥ K. Then ̃Ink is a unit flow from A to B. Therefore, 1/p󸀠

inf{(2Ep (J)) ≤ ‖̃In ‖ p󸀠 k

| J ∈ F(A, B) ∩ ℓc (E), ‖J‖ = 1} ̃ ∗ + ‖In − I‖ ∗ ≤ ‖In − In ‖ p󸀠

ℓ (E,μ )

k

k

ℓ (E,μ )

k

󸀠

ℓp (E,μ∗ )

+ ‖I‖ℓp󸀠 (E,μ∗ ) 1/p󸀠

= (1 − ‖Ink ‖−1 )‖Ink ‖ℓp󸀠 (E,μ∗ ) + ‖Ink − I‖ℓp󸀠 (E,μ∗ ) + (2Ep (I))

≤ sup ‖In ‖ℓp󸀠 (E,μ∗ ) (‖I‖ − ‖Ink ‖−1 ) + ‖Ink − I‖ℓp󸀠 (E,μ∗ ) + (2Ep (I)) n∈ℕ

1/p󸀠

󳨀→ (2Ep (I))

k→∞

1/p󸀠

.

μ∗

In particular, inf{Ep (I) | I ∈ F(A, B) ∩ ℓc (E), ‖I‖ = 1} ≤ Ep (I) for any I ∈ ℱp󸀠 (A, B). This proves the converse inequality. (3) Let (x, y) ∈ 𝜕e A and let (z, w) ∈ 𝜕e B with x ∈ A and z ∈ B. Since A ∩ B = 0, we have (x, y) ≠ (z, w). Define I ∈ ℓ# (E) by I = 1{(x,y)} − 1{(y,x)} − 1{(z,w)} + 1{(w,z)} . Then I ∈ F(A, B) ∩ ℓc (E). Hence Mod∗p (A, B) ≤ Ep (I) ≤ μ∗xy + μ∗zw < ∞. μ∗

(4) Clearly, {I ∈ ℱp󸀠 (A, B) | ‖I‖ = 1} is a closed convex subset of ℓp (E, μ∗ ). Thus, by 󸀠

μ∗

Theorem 3.15, there exists a unique unit flow I ∗ ∈ ℱp󸀠 (A, B) satisfying Mod∗p (A, B) = Ep (I ∗ ). Note that Ep (I) = 0 if and only if I ≡ 0. Since ‖I ∗ ‖ = 1, we have Ep (I ∗ ) > 0. Hereafter, we use Ip (A, B) to denote the minimizer of Mod∗p (A, B). Now we are ready to state Nakamura–Yamasaki’s duality obtained in [18]. Theorem 3.24 (Nakamura–Yamasaki’s duality [18, Theorem 5.1]). If A and B are nonempty finite disjoint subsets of V, then Modp (A, B)1/p Mod∗p (A, B)1/p = 1. 󸀠

(3.8)

To prove Nakamura–Yamasaki’s duality theorem, we need to prepare a few lemmas. We define gp (t) = |t|p−1 sgn(t) for t ∈ ℝ. The following lemma is immediate by the definition of gp , so we omit the proof. Lemma 3.25. gp satisfies the following properties: (1) |gp (t)| = |t|p−1 . (2) gp (−t) = −gp (t). (3) tgp (t) = |t|p . d (4) |t|p is differentiable and dt |t|p = pgp (t). The following three lemmas will be used to construct a flow playing an essential role in the proof of Nakamura–Yamasaki’s duality.

Parabolic index and Ahlfors regular conformal dimension

| 227

Lemma 3.26 ([18, Lemma 5.1]). Let A, B be nonempty finite disjoint subsets of V. Set μ∗ I = Ip (A, B). For any J ∈ ℱp󸀠 (A, B) satisfying ‖J‖ = 0, ∑ J(e)gp󸀠 (I(e))μ∗e = ⟨J, gp󸀠 (I)⟩ℓ2 (E,μ∗ ) = 0.

e∈E

μ∗

Proof. For any t ∈ ℝ, we have I + tJ ∈ ℱp󸀠 (A, B) and ‖I + tJ‖ = 1. Hence Mod∗p (A, B) = Ep (I) ≤ Ep (I + tJ) < ∞.

(3.9)

Set E(t) = Ep (I + tJ). Then we have 1 d󵄨 p󸀠 d 󵄨p󸀠 E(t) = ∑ 󵄨󵄨󵄨I(e) + tJ(e)󵄨󵄨󵄨 μ∗e = ∑ g 󸀠 (I(e) + tJ(e))J(e)μ∗e . dt 2 e∈E dt 2 e∈E p On the other hand, by (3.9), E(t) attains its minimum at t = 0. Thus it holds that d E(t)|t=0 = 0. Consequently, we obtain dt 0=

d p󸀠 E(t)|t=0 = ∑ g 󸀠 (I(e))J(e)μ∗e . dt 2 e∈E p

The lemma is now proved. Given a simple path λ = [x0 , x1 , . . . , xn ] ∈ 𝒫 (G)S , we define a natural associated flow I (λ) by

I

(λ)

1 if e = (xi−1 , xi ) for i ∈ {1, . . . , n}, { { { (e) = {−1 if e = (xi , xi−1 ) for i ∈ {1, . . . , n}, { { {0 otherwise.

Note that I (λ) ∈ ℓc (E) for any simple path λ. Lemma 3.27 ([18, Corollary 1, p. 109]). Let A, B be nonempty finite disjoint subsets of V. Set I = Ip (A, B). For any simple path λ ∈ Γ(A, B), Mod∗p (A, B) =

1 ∑ I (λ) (e)gp󸀠 (I(e))μ∗e . 2 e∈E

(3.10)

Proof. Set J = I − I (λ) . Then J satisfies the hypotheses of Lemma 3.26. Hence 0 = ∑ (I(e) − I (λ) (e))gp󸀠 (I(e))μ∗e = ∑ I(e)gp󸀠 (I(e))μ∗e − ∑ I (λ) (e)gp󸀠 (I(e))μ∗e . e∈E

Therefore, we obtain (3.10).

e∈E

e∈E

228 | R. Shimizu Lemma 3.28 ([18, Corollary 2, p. 109]). Let A, B be nonempty finite disjoint subsets of V. Set I = Ip (A, B). For any x, y ∈ V with x ≠ y and simple paths λ, γ ∈ Γ({x}, {y}), ∑ I (λ) (e)gp󸀠 (I(e))μ∗e = ∑ I (γ) (e)gp󸀠 (I(e))μ∗e .

e∈E

e∈E

Proof. Set J = I (λ) − I (γ) . Then J satisfies the hypotheses of Lemma 3.26, which gives the desired result immediately. Fix a path family {λxy }x∈A,y∈A̸ ⊆ 𝒫 (G), where λxy is a simple path from x to y. We assume that λxy = (x, y) for any (x, y) ∈ 𝜕e A. For each x ∈ A, we define vx ∈ ℓ(V) by 0 vx (y) = { 1 2

if y ∈ A, ∑e∈E I

(λxy )

(e)gp󸀠 (Ip (A, B)(e))μ∗e

if y ∈ ̸ A.

By Lemma 3.28, this is well-defined. Furthermore, we define v∗ ∈ ℓ(V) by 󵄨 󵄨 󵄨 󵄨 v∗ (y) = inf 󵄨󵄨󵄨vx (y)󵄨󵄨󵄨 = min󵄨󵄨󵄨vx (y)󵄨󵄨󵄨 x∈A x∈A for any y ∈ V. The following lemma is one of key ingredients of the proof of Nakamura– Yamasaki’s duality. Lemma 3.29 ([18, Lemma 5.2]). Let A, B be nonempty finite disjoint subsets of V. Set I = Ip (A, B). Then v∗ |B ≡ Mod∗p (A, B)

(3.11)

󵄨󵄨 󵄨 󵄨 󵄨p󸀠 −1 ∗ 󵄨󵄨∇v∗ (e)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨I(e)󵄨󵄨󵄨 μe

(3.12)

and

for all e ∈ E. Proof. By Lemma 3.27, it follows that vx |B ≡ Mod∗p (A, B) for any x ∈ A, and thus equation (3.11) holds. Let (x, y) ∈ E. If x, y ∈ A or x, y ∈ B, then it is obvious that ∇v∗ ((x, y)) = 0. Now we have three cases. Case I: Assume that x ∈ A. In this case, we have 0 ≤ ∇v∗ ((x, y)) = v∗ (y) 1󵄨 󵄨 󵄨 󵄨p󸀠 −1 ≤ 󵄨󵄨󵄨gp󸀠 (I((x, y)))μ∗xy − gp󸀠 (I((y, x)))μ∗yx 󵄨󵄨󵄨 = 󵄨󵄨󵄨I((x, y))󵄨󵄨󵄨 μ∗xy . 2 Thus we obtain (3.12) in this case. Case II: Assume that y ∈ A. By the same argument as in the Case I, we obtain (3.12). Case III: Assume that x ∈ ̸ A and y ∈ ̸ A. Replacing (x, y) by (y, x) if necessary, we may assume that ∇v∗ ((x, y)) ≥ 0. Choose a ∈ A so that v∗ (x) = |va (x)|. If y ∩ λax ≠ 0, then

Parabolic index and Ahlfors regular conformal dimension

| 229

there exist simple paths γay ∈ Γ({a}, {y}) and γyx ∈ Γ({y}, {x}) such that λax = γay ∗ γyx . Note that x ∩ γay = 0. Using Lemma 3.28 and the triangle inequality, we have 󵄨 󵄨 v∗ (x) = 󵄨󵄨󵄨va (x)󵄨󵄨󵄨 󵄨󵄨 1 󵄨󵄨󵄨 󵄨 = 󵄨󵄨󵄨 ∑ I (γay ∗γyx ) (e)gp󸀠 (I(e))μ∗e 󵄨󵄨󵄨 = 󵄨󵄨 2 󵄨󵄨e∈E

󵄨󵄨 1 󵄨󵄨󵄨󵄨 󵄨 (γ ∗(y,x)) (e)gp󸀠 (I(e))μ∗e 󵄨󵄨󵄨 󵄨󵄨 ∑ I ay 󵄨󵄨 2 󵄨󵄨e∈E

󵄨󵄨 1 󵄨󵄨󵄨 󵄨 = 󵄨󵄨󵄨 ∑ I (γay ) (e)gp󸀠 (I(e))μ∗e + gp󸀠 (I((y, x)))μ∗yx − gp󸀠 (I((x, y)))μ∗xy 󵄨󵄨󵄨 󵄨󵄨 2 󵄨󵄨e∈E 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 = 󵄨󵄨󵄨va (y) − gp󸀠 (I((x, y)))μ∗xy 󵄨󵄨󵄨 ≥ 󵄨󵄨󵄨va (y)󵄨󵄨󵄨 − 󵄨󵄨󵄨gp󸀠 (I((x, y)))μ∗xy 󵄨󵄨󵄨 󵄨 󵄨p󸀠 −1 ≥ v∗ (y) − 󵄨󵄨󵄨I((x, y))󵄨󵄨󵄨 μ∗xy . This implies ∇v∗ ((x, y)) = v∗ (y)−v∗ (x) ≤ |I((x, y))|p −1 μ∗xy . If y ∩λax = 0, then we define a simple path λ by λ = λax ∗(x, y) ∈ Γ({a}, {y}). By Lemma 3.28 and the triangle inequality, it follows that 󸀠

󵄨 󵄨 v∗ (y) ≤ 󵄨󵄨󵄨va (y)󵄨󵄨󵄨 󵄨󵄨 1 󵄨󵄨󵄨 󵄨 = 󵄨󵄨󵄨 ∑ I (λay ) (e)gp󸀠 (I(e))μ∗e 󵄨󵄨󵄨 = 󵄨󵄨 2 󵄨󵄨e∈E

󵄨 1 󵄨󵄨󵄨󵄨 (λ) ∗ 󵄨󵄨 󵄨󵄨 ∑ I (e)gp󸀠 (I(e))μe 󵄨󵄨󵄨 󵄨󵄨 2 󵄨󵄨e∈E

󵄨󵄨 1 󵄨󵄨󵄨 󵄨 = 󵄨󵄨󵄨 ∑ I (λax ) (e)gp󸀠 (I(e))μ∗e + gp󸀠 (I((x, y)))μ∗xy − gp󸀠 (I((y, x)))μ∗yx 󵄨󵄨󵄨 󵄨 󵄨󵄨 2 󵄨e∈E 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 = 󵄨󵄨󵄨va (x) + gp󸀠 (I((x, y)))μ∗xy 󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨va (x)󵄨󵄨󵄨 + 󵄨󵄨󵄨gp󸀠 (I((x, y)))μ∗xy 󵄨󵄨󵄨 󵄨 󵄨p󸀠 −1 = v∗ (x) + 󵄨󵄨󵄨I((x, y))󵄨󵄨󵄨 μ∗xy . This implies ∇v∗ ((x, y)) = v∗ (y) − v∗ (x) ≤ |I((x, y))|p −1 μ∗xy . 󸀠

Thus we have (3.12) in all the cases. Now the lemma is proved. μ∗

Proof of Theorem 3.24. Let I ∈ ℱp󸀠 (A, B) be a unit flow and let u ∈ ℓp (V, μ) be feasible for 𝒞p (A, B). Then 1 = Flux(I; A) = ∑ u(x) Div I(x) = ∑ u(x) ∑ I((x, y)) x∈V

x∈V

y∈NG (x)

1 = ( ∑ u(x) ∑ I((x, y)) + ∑ u(x) ∑ I((x, y))) 2 x∈V x∈V y∈N (x) y∈N (x) G

G

1 = ( ∑ u(x) ∑ I((x, y)) − ∑ u(x) ∑ I((y, x))) 2 x∈V x∈V y∈N (x) y∈N (x) G

=

G

1 1 ∑ ∑ (u(x) − u(y))I((x, y)) = − ∑ ∇u(e)I(e). 2 x∈V y∈N (x) 2 e∈E G

230 | R. Shimizu By Hölder’s inequality, 󵄨󵄨 1 󵄨󵄨󵄨 󵄨 1 = 󵄨󵄨󵄨 ∑ ∇u(e)I(e)󵄨󵄨󵄨 󵄨󵄨 2 󵄨󵄨e∈E 1/p

1 󵄨p 󵄨 ≤ ( ∑ 󵄨󵄨󵄨∇u(e)󵄨󵄨󵄨 μe ) 2 e∈E

1/p󸀠

󵄨p󸀠 󵄨 ( ∑ 󵄨󵄨󵄨I(e)󵄨󵄨󵄨 μ∗e ) e∈E

= ℰp (u)1/p Ep (I)1/p . 󸀠

This implies 1 ≤ 𝒞p (A, B)1/p Mod∗p (A, B)1/p = Modp (A, B)1/p Mod∗p (A, B)1/p . Next, we prove the converse inequality. Set I = Ip (A, B) and set u = 1 − v∗ / Mod∗p (A, B), where v∗ has appeared in Lemma 3.29. Then u is feasible for 𝒞p (A, B). Hence, by Lemma 3.29, 󸀠

󸀠

Modp (A, B) = 𝒞p (A, B) ≤ ℰp (u) = Mod∗p (A, B)−p ℰp (v∗ )

1 = Mod∗p (A, B)−p ‖∇v∗ ‖pℓp (E,μ) 2 1 󵄩 󸀠 󵄩p ≤ Mod∗p (A, B)−p 󵄩󵄩󵄩|I|p −1 μ∗ 󵄩󵄩󵄩ℓp (E,μ) 2 󸀠 1 = Mod∗p (A, B)−p ‖I‖pp󸀠 ∗ = Mod∗p (A, B)1−p . ℓ (E,μ ) 2

Thus we have that Modp (A, B)Mod∗p (A, B)p/p ≤ 1. 󸀠

3.3 Another expression of the dual p-modulus This subsection is devoted to giving a geometric expression of the dual p-modulus in Theorem 3.34. To prove Theorem 1.1, this expression will play a crucial role. The maxflows and min-cuts theorem, Theorem 3.33, is a key ingredient of this expression. To state this theorem, we need the following notions and notations. Definition 3.30. (1) Let A, B be nonempty disjoint subsets of V. A subset Π of V is said to separate A from B if and only if A ⊆ Π and B ⊆ Πc . The collection of sets separating A from B is denoted by sep(A; B). (2) Let θ ∈ ℓsym (E) be nonnegative. For any ℱ ⊆ ℓ(E), we define 󵄨 󵄨 M(θ; ℱ ) = sup{‖I‖ | I ∈ ℱ , 󵄨󵄨󵄨I(e)󵄨󵄨󵄨 ≤ θ(e) for any e ∈ E}.

(3.13)

(3) Let θ ∈ ℓsym (E) be nonnegative. For any finite subsets A, B of V, we define M ∗ (θ; A, B) =

inf

Π∈sep(A;B)

∑ θ(e).

e∈𝜕e Π

(3.14)

In the rest of this subsection, ν is a weight on G = (V, E), A, B are nonempty disjoint subsets of V and q ∈ (1, ∞).

Parabolic index and Ahlfors regular conformal dimension

| 231

Lemma 3.31 ([18, Lemma 3.2]). Suppose that A is a finite set. For any nonnegative θ∗ ∈ q ℓsym (E, ν), there exists I ∗ ∈ ℱqν (A, B) such that 󵄩󵄩 ∗ 󵄩󵄩 ∗ 󵄩󵄩I 󵄩󵄩 = M(θ ; F(A, B) ∩ ℓc (E))

(3.15)

󵄨󵄨 ∗ 󵄨󵄨 ∗ 󵄨󵄨I (e)󵄨󵄨 ≤ θ (e)

(3.16)

and

for all e ∈ E. Proof. Set Aθ∗ = {I ∈ ℱqν (A, B) | |I(e)| ≤ θ∗ (e) for any e ∈ E}. It is obvious that Aθ∗ is a convex subset of ℓq (E, ν). Since ℓq (E, ν)-convergence implies the pointwise convergence, Aθ∗ is closed with respect to the norm of ℓq (E, ν). Thus Aθ∗ is a closed convex subset of ℓq (E, ν). Furthermore, Aθ∗ is a bounded subset of ℓp (E, ν). Let {In }n∈ℕ be a sequence in (F(A, B) ∩ ℓc (E)) ∩ Aθ∗ satisfying lim ‖In ‖ = M(θ∗ ; F(A, B) ∩ ℓc (E)).

n→∞

Since {In }n∈ℕ is a bounded sequence in ℓq (E, ν), we can choose a subsequence {Ink }k∈ℕ so that Ink converges weakly as k → ∞. Let I ∗ ∈ ℱqν (A, B) be its limit. Applying Mazur’s lemma to {Ink }k∈ℕ , we deduce that I ∗ ∈ Aθ∗ . Since A is finite and (V, E) is locally finite, we obtain 󵄩󵄩 ∗ 󵄩󵄩 ∗ ∗ 󵄩󵄩I 󵄩󵄩 = Flux(I ; A) = ∑ Div I (x) x∈A

= lim ∑ Div Ink (x) = lim ‖Ink ‖ = M(θ∗ ; F(A, B) ∩ ℓc (E)). k→∞

k→∞

x∈A

Thus I ∗ has the required properties. q Suppose that A is a finite set. Fix a nonnegative function θ∗ ∈ ℓsym (E, ν). Let I ∗ be ν a flow in ℱq (A, B) satisfying (3.15) and (3.16). We define a subset 𝒱∗ by

󵄨󵄨 󵄨 for any λ = [x0 , x1 , . . . , xn ] ∈ Γ(A, {x}) there exists }. 󵄨󵄨 i ∈ {1, . . . , n} such that I ∗ ((xi−1 , xi )) = θ∗ ((xi−1 , xi ))

𝒱∗ = {x ∈ V \ A 󵄨󵄨󵄨󵄨

We also define a subset A∗ by A∗ = A ∪ (V \ 𝒱∗ ). The set A∗ has following properties. Proposition 3.32 ([23, Lemmas 8–10 and Corollary 2, p. 240]). (1) A∗ ∈ sep(A; B).

(3.17)

232 | R. Shimizu (2) For any e ∈ 𝜕e A∗ , ∇1(A∗ )c (e)I ∗ (e) = θ∗ (e).

(3.18)

(3) If #𝜕e A∗ < ∞, then M(θ∗ ; ℱqν (A, B)) = M(θ∗ ; F(A, B) ∩ ℓc (E)) =

inf

∑ θ∗ (e).

Π∈sep(A;B), #Π 0 and a path λ = [x0 , x1 , . . . , xl = x] ∈ Γ(A, {x}) so that I ∗ ((xi−1 , xi )) ≤ θ∗ ((xi−1 , xi )) − δ

(3.20)

for any i ∈ {1, . . . , l}. Let Aθ∗ be the same as in the proof of Lemma 3.31 and let {In }n∈ℕ be a sequence in (F(A, B) ∩ ℓc (E)) ∩ Aθ∗ such that In converges pointwise to I ∗ and limn→∞ ‖In ‖ = ‖I ∗ ‖. Let ϵ ∈ (0, δ/2). Then we can choose N ∈ ℕ so that 󵄨󵄨 󵄩 ∗ 󵄩󵄨 󵄨󵄨‖In ‖ − 󵄩󵄩󵄩I 󵄩󵄩󵄩󵄨󵄨󵄨 < ϵ and 󵄨󵄨 󵄨 ∗ 󵄨󵄨In ((xi−1 , xi )) − I ((xi−1 , xi ))󵄨󵄨󵄨 < ϵ for any i ∈ {1, . . . , l} and n ≥ N. By (3.20), for any i ∈ {1, . . . , l}, we have 󵄨󵄨 󵄨󵄨 δ (λ) 󵄨 󵄨󵄨 󵄨󵄨IN ((xi−1 , xi )) + I ((xi−1 , xi ))󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 2 󵄨 󵄨 󵄨 󵄨 δ ≤ 󵄨󵄨󵄨IN ((xi−1 , xi )) − I ∗ ((xi−1 , xi ))󵄨󵄨󵄨 + 󵄨󵄨󵄨I ∗ ((xi−1 , xi ))󵄨󵄨󵄨 + 2 δ ∗ ∗ < ϵ + θ ((xi−1 , xi )) − δ + < θ ((xi−1 , xi )). 2 Since IN + δ2 I (λ) ∈ F(A, B) ∩ ℓc (E) and ‖IN + δ2 I (λ) ‖ = ‖IN ‖ + δ2 ‖I (λ) ‖, we obtain 󵄩󵄩 ∗ 󵄩󵄩 ∗ 󵄩󵄩I 󵄩󵄩 = M(θ ; F(A, B) ∩ ℓc (E)) 󵄩󵄩 δ 󵄩󵄩󵄩 δ 󵄩 󵄩 δ 󵄩 󵄩 󵄩 ≥ 󵄩󵄩󵄩IN + I (λ) 󵄩󵄩󵄩 = ‖IN ‖ + ≥ 󵄩󵄩󵄩I ∗ 󵄩󵄩󵄩 − ϵ + > 󵄩󵄩󵄩I ∗ 󵄩󵄩󵄩. 2 2 2 󵄩󵄩 󵄩󵄩

Parabolic index and Ahlfors regular conformal dimension

| 233

This is a contradiction. Thus we have B ⊆ (A∗ )c . (2) Suppose that there exists e ∈ 𝜕e A∗ such that ∇1(A∗ )c (e)I ∗ (e) ≠ θ∗ (e).

(3.21)

Choose a ∈ A∗ and b ∈ (A∗ )c so that e ∩ A∗ = {a} and e ∩ (A∗ )c = {b}. Then ∇1(A∗ )c ((a, b)) = 1 and thus, by (3.21), we have I ∗ ((a, b)) ≠ θ∗ ((a, b)). Since I ∗ (e󸀠 ) ≤ θ∗ (e󸀠 ) for any e󸀠 ∈ E, we obtain I ∗ ((a, b)) < θ∗ ((a, b)). Now we have two cases. Case I: Assume a ∈ A. Set δ = θ∗ ((a, b)) − I ∗ ((a, b)) > 0. Then (a, b) ∈ Γ(A, {b}) and I ∗ ((a, b)) = θ∗ ((a, b)) − δ. Hence we obtain b ∈ A∗ . This contradicts b ∈ (A∗ )c . Case II: Assume a ∈ ̸ A. Since a ∈ A∗ , there exist δ > 0 and a path λ = [x0 , x1 , . . . , xn = a] ∈ Γ(A, {a}) such that I ∗ ((xi−1 , xi )) ≤ θ∗ ((xi−1 , xi )) − δ

(3.22)

for any i ∈ {1, . . . , n}. Set δ0 = δ ∧ (θ∗ ((a, b)) − I ∗ ((a, b))) > 0 and define a path γ ∈ Γ(A, {b}) by γ = λ ∗ (a, b). By (3.22), we obtain I ∗ ((a, b)) ≤ θ∗ ((a, b)) − δ0 . Furthermore, for any i ∈ {1, . . . , n}, I ∗ ((xi−1 , xi )) ≤ θ∗ ((xi−1 , xi )) − δ ≤ θ∗ ((xi−1 , xi )) − δ0 . Hence we have b ∈ A∗ , which contradicts b ∈ (A∗ )c . (3) Let Π ∈ sep(A; B) and let I ∈ F(A, B). If ∇1Π ∈ ℓc (E) or I ∈ ℓc (E), then ∑ ∇1Πc (e)I(e) = − ∑ I(e)∇1Π (e)

e∈𝜕e Π

e∈𝜕e Π

1 = − ∑ ∑ I((x, y))(1Π (y) − 1Π (x)) 2 x∈V y∈N (x) G

1 = − (∑ I((x, y))(−1) + ∑ ∑ ∑ I((x, y))) 2 x∈Π y∈Π;(x,y)∈E y∈Π ̸ x ∈Π;(x,y)∈E ̸ = ∑

∑

x∈Π y∈Π;(x,y)∈E ̸

I((x, y)) = ∑ Div I(x) = ∑ Div I(x) = ‖I‖. x∈Π

x∈A

Since #𝜕e A∗ < ∞, we have ∇1A∗ ∈ ℓc (E). By (3.18), it holds that 󵄩 󵄩 M(θ∗ ; F(A, B) ∩ ℓc (E)) = 󵄩󵄩󵄩I ∗ 󵄩󵄩󵄩 = =

∑ ∇1(A∗ )c (e)I ∗ (e)

e∈𝜕e A∗

∑ θ∗ (e) ≥

e∈𝜕e

Π∗

inf

∑ θ∗ (e).

Π∈sep(A;B), #Π 0. Then we can pick nonnegative functions θ1∗ , θ2∗ ∈ ℓsym (E, μ∗ ) satisfying 󸀠

(

inf

Π∈sep({x};{y})

Ep (θ1∗ )

∑ θ1∗ (e)) ∧ (

e∈𝜕e Π

< Rp (x, y) + ϵ,

inf

Π󸀠 ∈sep({y};{z})

∑ θ2∗ (e)) ≥ 1,

e∈𝜕e Π󸀠

(4.3)

238 | R. Shimizu and Ep (θ2∗ ) < Rp (y, z) + ϵ. p Set θ∗ = {(θ1∗ )p +(θ2∗ )p }1/p ∈ ℓsym (E, μ∗ ). Since θ∗ ≥ θ1∗ ∨θ2∗ it holds that ∑e∈𝜕e Π θ∗ (e) ≥ 1 for any Π ∈ S(x, y, z). Hence 󸀠

󸀠

󸀠

󸀠

Rp (x, z) ≤ Ep (θ∗ ) = Ep (θ1∗ ) + Ep (θ2∗ ) < Rp (x, y) + Rp (y, z) + 2ϵ. Letting ϵ ↓ 0, we obtain Rp (x, z) ≤ Rp (x, y) + Rp (y, z). This completes the proof.

5 p-conductance and p-modulus to infinity So far in this paper, we have seen properties of the p-conductance 𝒞p (A, B). In this section, taking a proper limit of B, we define the p-conductance to infinity, 𝒞p (A, ∞), and give various expressions of 𝒞p (A, ∞). We also introduce the notions of p-parabolicity and p-hyperbolicity. The p-conductance to infinity will be used to characterize p-parabolicity and p-hyperbolicity in Section 7. Throughout this section, (G, μ) is a connected, locally finite, simple weighted graph and p ∈ (1, ∞). Definition 5.1. Let A be a nonempty subset of V. Define the p-conductance from A to infinity, 𝒞p

(G,μ)

(A, ∞), by (G,μ)

𝒞p

(A, ∞) = inf{ℰp (u) | u ∈ ℓc (V), u|A ≡ 1}.

We say that u is feasible for 𝒞p (A, ∞) if u ∈ ℓ(V) satisfies u ∈ ℓc (V) and u|A ≡ 1. The

reciprocal of 𝒞p

(G,μ)

(A, ∞) is denoted by ρp

𝒞p (A, ∞) and ρp (A, ∞) to denote

(G,μ)

(A, ∞). If no confusion can occur, we use

(G,μ) 𝒞p (A, ∞)

and ρp

(G,μ)

(A, ∞), respectively.

Remark 5.2. (1) If G = (V, E) is a finite graph, then 1V is feasible for 𝒞p (A, ∞). Hence 𝒞p (A, ∞) = 0 for any subset A ⊆ V. (2) Since G is locally finite, we have 𝒞p (A, ∞) ≤ ℰp (1A ) < ∞ for any finite subset A ⊆ V. By the same reasons as in Remark 3.2-(2) and the proof of Proposition 3.7-(3), the following proposition is obvious. Proposition 5.3. Let A be a nonempty subset of V. (1) 𝒞p (A, ∞) = inf{ℰp (u) | u ∈ ℓc (V), u|A ≡ 1, 0 ≤ u ≤ 1}.

(2) If q ∈ [p, ∞), then 𝒞q (A, ∞) ≤ 𝒞p (A, ∞).

Parabolic index and Ahlfors regular conformal dimension

| 239

Definition 5.4. (1) We define 𝒟0p (G, μ) by p

‖⋅‖𝒟p

𝒟0 (G, μ) = ℓc (V)

.

If no confusion may occur, we write 𝒟0p instead of 𝒟0p (G, μ). (2) We say that (G, μ) is p-parabolic (resp. p-hyperbolic) if and only if 1V ∈ 𝒟0p (G, μ) (resp. 1V ∈ ̸ 𝒟0p (G, μ)). We will make frequent use of the following properties of 𝒞p (A, ∞) in part II. The statements (1) and (2) are mentioned in [20, p. 137] and (3) is essentially the same as [3, Theorem 2.2]. Our proof of (3) is inspired by [3, Proposition 2.34]. Proposition 5.5. Let A be a nonempty finite subset of V. (1) 𝒞p (A, ∞) = inf{ℰp (u) | u ∈ 𝒟0p , u|A ≡ 1}. (2) There exists a unique function ψ ∈ 𝒟0p such that ψ|A ≡ 1 and (5.1)

𝒞p (A, ∞) = ℰp (ψ).

(3) Let {Vn }n∈ℕ be an exhaustion of V with A ⊆ V1 . Then c

𝒞p (A, ∞) = lim 𝒞p (A, (Vn ) ). n→∞

Proof. (1) It is obvious that 𝒞p (A, ∞) ≥ inf{ℰp (u) | u ∈ 𝒟0p , u|A ≡ 1}. We prove the converse inequality. Let u ∈ 𝒟0p with u|A ≡ 1. From the definition of 𝒟0p , there exists a sequence {un }n∈ℕ in ℓc (V) such that un converges to u in 𝒟p as n → ∞. By Proposition 2.10-(2), un also converges pointwise to u. Set an = minx∈A un (x) for each n ∈ ℕ. Since limn→∞ an = 1, there exists N ∈ ℕ such that an > 0 for any n ≥ N. Set vn = (un /an ) ∧ 1 for each n ≥ N. Then vn is feasible for 𝒞p (A, ∞) for any n ≥ N. By the same argument as in the proof of Proposition 3.23-(1), we have that 𝒞p (A, ∞) ≤ ℰp (u). Thus it holds that 𝒞p (A, ∞) ≤ inf{ℰp (u) | u ∈ 𝒟0p , u|A ≡ 1}. (2) Set A0 = {u ∈ 𝒟0p | u|A ≡ 1}. By Proposition 2.10-(2), we see that A0 is a convex closed subset of 𝒟p . By Lemma 2.8, we may assume that o ∈ A. Theorem 3.15 implies the desired result. (3) Since Vn ⊆ Vn+1 , we have 𝒞p (A, (Vn )c ) ≥ 𝒞p (A, (Vn+1 )c ) for each n ∈ ℕ. In addition, because Vn is a finite set, any feasible function for 𝒞p (A, (Vn )c ) is also feasible for 𝒞p (A, ∞). Hence we have 𝒞p (A, (Vn )c ) ≥ 𝒞p (A, ∞). Letting n → ∞, we obtain limn→∞ 𝒞p (A, (Vn )c ) ≥ 𝒞p (A, ∞). Next we prove the converse inequality. Let u ∈ ℓc (V) with u|A ≡ 1. Since u has a finite support, we have that 𝒞p (A, (Vn )c ) ≤ ℰp (u) for sufficiently large n ∈ ℕ. Thus the result follows. In what follows, we use ψp (A, ∞) to denote the unique function ψ ∈ 𝒟0p satisfying ψ|A ≡ 1 and (5.1).

240 | R. Shimizu In the previous section, the p-modulus Modp (A, B) has given a geometrical expression of 𝒞p (A, B). The rest of this section is devoted to giving similar results to this fact for 𝒞p (A, ∞). We start by introducing the notion of the p-modulus to infinity. Definition 5.6. Let A be a nonempty subset of V. (1) An infinite path is an infinite sequence λ = [x0 , x1 , . . . , xn , . . . ] of vertices such that (xi−1 , xi ) ∈ E for any i ∈ ℕ. We define the collection of infinite paths from A to infinity, Γ(A, ∞), by Γ(A, ∞) = {λ = [x0 , x1 , . . . , xn , . . . ] | x0 ∈ A, dG (x0 , xn ) 󳨀→ ∞}. n→∞

(2) We define the p-modulus from A to infinity, Mod(G,μ) (A, ∞), by p 1 inf ‖θ‖pℓp (E,μ) , 2 θ∈Ap (Γ(A,∞))

Mod(G,μ) (A, ∞) = p where p

󵄨󵄨 ∞ 󵄨 ∑i=1 θ((xi−1 , xi )) ≥ 1 for any }. 󵄨󵄨 λ = [x0 , x1 , . . . , xn , . . . ] ∈ Γ(A, ∞)

Ap (Γ(A, ∞)) = {θ ∈ ℓsym (E, μ) 󵄨󵄨󵄨󵄨

If no confusion can occur, we use Modp (A, ∞) to denote Mod(G,μ) (A, ∞). p Remark 5.7. (1) If (V, E) is a finite graph, then Γ(A, ∞) = 0 for any subset A ⊆ V. Hence Modp (A, ∞) = 0. (2) By the same argument as in the proof of Lemma 3.11, we have Modp (A, ∞) =

1 1 inf ‖θ‖pℓp (E,μ) = inf ‖θ‖pℓp (E,μ) , 2 θ∈Ap+ (Γ(A,∞)) 2 θ∈Ap∗ (Γ(A,∞))

where Ap (Γ(A, ∞)) = {θ ∈ Ap (Γ(A, ∞)) | θ ≥ 0} +

and Ap (Γ(A, ∞)) = {θ ∈ Ap (Γ(A, ∞)) | 0 ≤ θ ≤ 1}. +

Notation. Let θ ∈ ℓ(E) and let λ = [x0 , x1 , . . . , xn , . . . , ] be an infinite path. We write n ∑λ θ instead of ∑∞ i=1 θ((xi−1 , xi )). Recall that we also write ∑γ θ = ∑i=1 θ((yi−1 , yi )) for any γ = [y0 , y1 , . . . , yn ] ∈ 𝒫 (G). Next we give an approximation of the p-modulus to infinity as in the case of p-conductances (Proposition 5.5-(3)). To this end, we need a lemma.

Parabolic index and Ahlfors regular conformal dimension | 241

Lemma 5.8 ([18, Lemma 2.4]). Let A be a nonempty finite subset of V and let θ be a nonnegative function on E. If {An }n∈ℕ is an exhaustion of V with A ⊆ A1 , then lim

inf

n→∞ λ∈Γ(A,(An )c )

∑θ = λ

inf

λ󸀠 ∈Γ(A,∞)

∑ θ. λ󸀠

Moreover, there exists an infinite path λ∗ = [x0 , x1 , . . . , xn , . . . ] ∈ Γ(A, ∞) such that ∑θ = λ∗

inf

λ󸀠 ∈Γ(A,∞)

∑ θ. λ󸀠

Proof. Set ln = inf{∑λ θ | λ ∈ Γ(A, (An )c )} for each n ∈ ℕ. Since {An }n∈ℕ is nondecreasing and θ is nonnegative, {ln }n∈ℕ is nondecreasing. For any path λ ∈ Γ(A, ∞) and n ∈ ℕ, we can pick a subpath λ(n) of λ so that λ(n) ∈ Γ(A, (An )c ). Clearly, ∑λ θ ≥ ∑λ(n) θ ≥ ln . Hence ∑λ θ ≥ limn→∞ ln for any λ ∈ Γ(A, ∞). This implies that lim l n→∞ n

≤

inf

λ󸀠 ∈Γ(A,∞)

∑ θ.

(5.2)

λ󸀠

Next, we prove the converse inequality. Since An is a finite set, we can choose a (n) simple path λn = [x0(n) , x1(n) , . . . , xk(n) ] ∈ Γ(A, (An )c ) satisfying ln = ∑λn θ for each n ∈ ℕ.

Since A is a finite set, we can pick y0 ∈ A so that {n ∈ ℕ | x0(n) = y0 } is an infinite set. Furthermore, because NG (y0 ) is also a finite set, there exists y1 ∈ NG (y0 ) such that {n ∈ ℕ | x0(n) = y0 , x1(n) = y1 } is an infinite set. By induction, we obtain a simple path λ∗ = [y0 , y1 , . . . , yk , . . . ] ∈ Γ(A, ∞) satisfying #{n ∈ ℕ | λn is a subpath of λ∗ } = ∞. Thus for any k ∈ ℕ there exists N(k) ∈ ℕ such that [y0 , y1 , . . . , yk ] is a subpath of λN(k) . Since θ is nonnegative, it follows that k

∑ θ((yi−1 , yi )) ≤ ∑ θ = lN(k) ≤ lim ln . i=1

λN(k)

n→∞

Letting k → ∞, we obtain infλ󸀠 ∈Γ(A,∞) ∑λ󸀠 θ ≤ ∑λ∗ θ ≤ limn→∞ ln . In conjunction with (5.2), we have that infλ󸀠 ∈Γ(A,∞) ∑λ󸀠 θ = ∑λ∗ θ = limn→∞ ln . Theorem 5.9 ([18, Theorem 2.2]). Let A be a nonempty finite subset of V and let {An }n∈ℕ be an exhaustion of V with A ⊆ A1 . Then Modp (A, ∞) = lim Modp (A, (An )c ). n→∞

Proof. Since {An }n∈ℕ is an exhaustion, Ap+ (Γ(A, (An )c )) ⊆ Ap+ (Γ(A, ∞)) and {Modp (A, (An )c )}n∈ℕ is nonincreasing. Thus, by Lemma 3.11 and Remark 5.7-(2), we obtain limn→∞ Modp (A, (An )c ) ≥ Modp (A, ∞).

242 | R. Shimizu Next we prove the converse inequality. Let θ ∈ Ap+ (Γ(A, ∞)). By Lemma 5.8, we have that lim l n→∞ n

=

inf

λ∈Γ(A,∞)

∑ θ ≥ 1, λ

where ln is the same as in the proof of the Lemma 5.8. Thus there exists N ∈ ℕ such that ln > 0 for all n ≥ N. Set θn = θ/ln for any n ≥ N. Then θn ∈ Ap (A, (An )c ). Hence it follows that l−p 1 Modp (A, (An )c ) ≤ ‖θn ‖pℓp (E,μ) = n ‖θ‖pℓp (E,μ) . 2 2 Letting n → ∞ and taking the infimum over θ in Ap+ (Γ(A, ∞)), we obtain the desired inequality. We also define an “infinity” version of dual p-moduli. Definition 5.10. Let A be a nonempty finite subset of V. We say that Π separates A if and only if Π is a finite subset of V and A ⊆ Π. Moreover, we write sep(A) = {Π | Π ⊆ V, #Π < ∞, A ⊆ Π}. Definition 5.11. Let A be a nonempty finite subset of V. We define the dual p-modulus from A to infinity, Mod(G,μ) (A, ∞)∗ , by p Mod(G,μ) (A, ∞)∗ p 󵄩 󵄩p p󸀠 = inf{󵄩󵄩󵄩θ∗ 󵄩󵄩󵄩ℓp (E,μ∗ ) | θ∗ ∈ ℓsym (E, μ∗ ), θ∗ ≥ 0, ∑ θ∗ (e) ≥ 1 for any Π ∈ sep(A)}. e∈𝜕e Π

If no confusion can occur, we use Mod∗p (A, ∞) to denote Mod(G,μ) (A, ∞)∗ . p Theorem 5.12 ([18, Theorem 4.2]). Let A be a nonempty finite subset of V and let {An }n∈ℕ be an exhaustion of V with A ⊆ A1 . Then Mod∗p (A, ∞) = lim Mod∗p (A, (An )c ). n→∞

Proof. Since An ⊆ An+1 , it follows that sep(A; (An )c ) ⊆ sep(A; (An+1 )c ) for any n ∈ ℕ. Thus {Mod∗p (A, (Vn )c )}n∈ℕ is nondecreasing. It is obvious that sep(A; (Vn )c ) ⊆ sep(A). Hence we obtain lim Mod∗p (A, (Vn )c ) ≤ Mod∗p (A, ∞).

n→∞

Next we prove the converse inequality. If limn→∞ Mod∗p (A, (An )c ) = ∞, then the desired inequality is trivial. Suppose that limn→∞ Mod∗p (A, (An )c ) < ∞. Let Π ∈ sep(A).

Parabolic index and Ahlfors regular conformal dimension | 243

Since Π is a finite set, Π ∈ sep(A; (An )c ) for all n ∈ ℕ. By Proposition 3.23-(3), there p󸀠 exists a sequence of nonnegative functions {θn∗ }n∈ℕ in ℓsym (E, μ∗ ) satisfying inf{ ∑ θn∗ (e) | Π(n) ∈ sep(A, (An )c )} ≥ 1 e∈𝜕e Π(n)

and Mod∗p (A, (An )c ) = Ep (θn∗ ) for each n ∈ ℕ. Since supn∈ℕ Ep (θn∗ ) ≤ Mod∗p (A, ∞) < ∞, there exists a subsequence {θn∗k }k∈ℕ converging pointwise to some nonnegative function. Let θ∗ be the limit. Then p it is obvious that θ∗ ∈ ℓsym (E, μ∗ ). Since Π is a finite set, it follows that ∑e∈𝜕e Π θ∗ (e) ≥ 1. Therefore, by Fatou’s lemma, 󸀠

Mod∗p (A, ∞) ≤ Ep (θ∗ ) ≤ lim inf Ep (θn∗k ) = lim Mod∗p (A, (An )c ). k→∞

n→∞

This completes the proof. Remark 5.13. The function θ∗ appearing in the above proof attains the minimum of Mod∗p (A, ∞), i. e., Mod∗p (A, ∞) = Ep (θ∗ ). The following lemma gives us a computation of the dual p-modulus as a limit of dual p-moduli between two sets. Lemma 5.14 ([18, Lemma 5.3]). Let A be a nonempty finite subset of V and let {An }n∈ℕ be a good exhaustion of V with A ⊆ A1 . If Π ∈ sep(A, An+1 \ An ), then there exists a finite set Π0 ∈ sep(A, (An )c ) such that 𝜕e Π0 ⊆ 𝜕e Π. In particular, it holds that Mod∗p (A, An+1 \ Vn ) = Mod∗p (A, (An )c ). Proof. Let n ∈ ℕ and let Π ∈ sep(A, An+1 \ An ). Set Π0 = Π \ An . Then it is obvious that Π0 is a finite set. Furthermore, since A ⊆ An , Π0 ∈ sep(A; (An )c ). Let e = (x, y) ∈ 𝜕e Π0 . Changing roles of x and y, we may assume that x ∈ Π0 ⊆ Π and y ∈ ̸ Π0 . Suppose y ∈ Π. Then y ∈ Π0 ∩ (An+1 )c . Since An ⊆ An+1 , we have y ∈ Π0 ∩ (An+1 )c ⊆ An ∩ (An+1 )c = 0. This is a contradiction. Hence it follows that y ∈ (Π)c . This implies e = (x, y) ∈ 𝜕e Π. We obtain 𝜕e Π0 ⊆ 𝜕e Π. By a similar argument as in the proof of Proposition 3.12-(4), we have Mod∗p (A, An+1 \ An ) ≤ Mod∗p (A, (An )c ). The converse of this inequality is immediate by An+1 \ An ⊆ (An )c . We complete the proof. Now, we have Nakamura–Yamsaki’s duality theorem for the p-modulus to infinity and the coincidence of 𝒞p (A, ∞) and Modp (A, ∞).

244 | R. Shimizu Theorem 5.15 ([18, Theorem 5.3]). Let A be a nonempty finite subset of V. Then Modp (A, ∞)1/p Mod∗p (A, ∞)1/p = 1. 󸀠

Proof. Let {An }n∈ℕ be a good exhaustion of V with A ⊆ A1 . It is obvious that any path in Γ(A, An+1 \ Vn ) has a subpath in Γ(A, (An )c ). Also, we immediately have Γ(A, An+1 \ An ) ⊆ Γ(A, (An )c ). By Proposition 3.12-(2) and (4), it follows that Modp (A, (An )c ) = Modp (A, An+1 \ An ). On the other hand, by Lemma 5.14, we have Mod∗p (A, An+1 \ An ) = Mod∗p (A, (An )c ). Applying Theorem 3.24, we obtain 1 = Modp (A, An+1 \ An )1/p Mod∗p (A, An+1 \ An )1/p = Modp (A, (An )c ) 󸀠

1/p

Mod∗p (A, (An )c )

1/p󸀠

.

Letting n → ∞ and applying Theorem 5.9 and Theorem 5.12, we finish the proof. Theorem 5.16. Let A be a nonempty finite subset of V. Then 𝒞p (A, ∞) = Modp (A, ∞).

Proof. By Proposition 3.7-(4), we have 𝒞p (A, ∞) < ∞. Let {An }n∈ℕ be an exhaustion of V with A ⊆ A1 . Then, by Theorem 3.17, we have c

c

𝒞p (A, (An ) ) = Modp (A, (An ) )

for any n ∈ ℕ. Applying Proposition 5.5-(3) and Theorem 5.9, we obtain the desired result.

6 p-Laplacians on weighted graphs In this section, we will consider a natural difference operator, which we call the p-Laplacian, associated with the p-energy on weighted graphs, and introduce the notion of p-harmonic functions. Although there are many works on p-Laplacians by Yamasaki and coworkers, we focus on structures of the p-Dirichlet space here. Throughout this section, (G, μ) ia a connected, locally finite, simple weighted graph and p ∈ (1, ∞). Recall that ℰp (u, v) =

1 󵄨 󵄨p−1 ∑ ∑ sgn(v(y) − v(x))󵄨󵄨󵄨v(y) − v(x)󵄨󵄨󵄨 (u(y) − u(x))μxy 2 x∈V y∈V

for u, v ∈ ℓ(V). Note that ℰp (u, v) = 21 ⟨∇u, gp (∇v)⟩ℓ2 (E,μ) and that ℰp (u, v) is linear with respect to u.

Parabolic index and Ahlfors regular conformal dimension

Definition 6.1. We define the p-Laplacian Δp

: ℓ(V) → ℓ(V) by

(G,μ)

Δ(G,μ) u(x) = p

| 245

1 󵄨p−1 󵄨 ∑ sgn(u(y) − u(x))󵄨󵄨󵄨u(y) − u(x)󵄨󵄨󵄨 μxy μ(x) y∈V

for any u ∈ ℓ(V) and x ∈ V whenever the sum converges absolutely. If no confusion (G,μ) can occur, we use Δp to denote Δp . Remark 6.2. Δp is not a linear operator when p ≠ 2. The next lemma and theorem, which is the discrete version of Gauss–Green’s formula for the p-Laplacian, are inspired by [3, Theorem 1.24]. Lemma 6.3. ℰp (u, v) converges absolutely if either of the following conditions (i) or (ii) holds: (i) u, v ∈ 𝒟p , (ii) u ∈ ℓc (V) or v ∈ ℓc (V), Proof. (i) For any u, v ∈ 𝒟p , Hölder’s inequality shows 󵄨 󵄨p−1 󵄨 󵄨 ∑ ∑ 󵄨󵄨󵄨v(y) − v(x)󵄨󵄨󵄨 󵄨󵄨󵄨u(y) − u(x)󵄨󵄨󵄨μxy

x∈V y∈V

1/p󸀠

󵄨 󵄨p ≤ ( ∑ ∑ 󵄨󵄨󵄨v(y) − v(x)󵄨󵄨󵄨 μxy ) x∈V y∈V

1/p󸀠

≤ 2ℰp (v)

1/p

ℰp (u)

1/p

󵄨 󵄨p ( ∑ ∑ 󵄨󵄨󵄨u(y) − u(x)󵄨󵄨󵄨 μxy ) x∈V y∈V

< ∞.

(ii) Since (V, E) is locally finite, the sum appearing in ℰp (u, v) is a finite sum. Theorem 6.4 (Gauss–Green’s formula for Δp ). Let u, v ∈ ℓ(V). Suppose that ℰp (u, v) converges absolutely. Then ℰp (u, v) = −(u, Δp v)ℓ2 (V,μ) .

Proof. By Fubini’s theorem, 󵄨

󵄨p−1

ℰp (u, v) = ∑ ∑ sgn(v(y) − v(x))󵄨󵄨󵄨v(y) − v(x)󵄨󵄨󵄨 x∈V y∈V

u(y)μxy

󵄨 󵄨p−1 = ∑ u(y) ∑ sgn(v(y) − v(x))󵄨󵄨󵄨v(y) − v(x)󵄨󵄨󵄨 μxy y∈V

x∈V

= ∑ u(y)(−Δp v(y)μ(y)). y∈V

This completes the proof. Now, on the analogy of harmonic functions, we define the notion of p-harmonic functions.

246 | R. Shimizu Definition 6.5. Let u ∈ ℓ(V) and let A be a nonempty subset of V. We say that u is p-subharmonic (resp. p-superharmonic) at x ∈ V if Δp u(x) ≥ 0 (resp. Δp u(x) ≤ 0). If u is both p-subharmonic and p-superharmonic at x ∈ V, then we say that u is p-harmonic at x. If u is p-subharmonic (resp. p-superharmonic) at all x ∈ A, then we say that u is p-subharmonic (resp. p-superharmonic) in A. We say that u is p-harmonic in A if and only if u is p-harmonic at all x ∈ A. We denote the collection of p-harmonic functions on V by ℋp (G, μ), i. e. p

ℋp (G, μ) = {u ∈ 𝒟 (G, μ) | u is p-harmonic in V}.

Remark 6.6. ℋp (G, μ) may not be a linear subspace of 𝒟p (G, μ). In the following two propositions, we are going to prove that the minimizer of

𝒞p (A, B) (resp. 𝒞p (A, ∞)) is a solution of the Dirichlet problem with respect to Δp :

u(x) = 1A (x)

{

Δp u(x) = 0

if x ∈ A ∪ B (resp. x ∈ A), if x ∈ ̸ A ∪ B (resp. x ∈ ̸ A)

Proposition 6.7. Let A, B be nonempty disjoint subsets of V with A ∪ B ≠ V. Suppose

𝒞p (A, B) < ∞. Then ψp (A, B) is p-harmonic in V \ (A ∪ B).

Proof. Set ψ = ψp (A, B). Let x ∈ V \ (A ∪ B). Since (V, E) is locally finite, it follows that ψ + t1x is feasible for 𝒞p (A, B) for any t ∈ ℝ. Hence we have that ℰp (ψ) = 𝒞p (A, B) ≤ ℰp (ψ + t1x ). This implies that ℰp (ψ + t1x ) attains its minimum at t = 0. Thus 0=

d ℰ (ψ + t1x )|t=0 dt p

󵄨 󵄨p−1 = p ∑ sgn(ψ(x) − ψ(y))󵄨󵄨󵄨ψ(x) − ψ(y)󵄨󵄨󵄨 μxy = pμ(x)Δp ψ(x). y∈V

Hence Δp ψ(x) = 0. Proposition 6.8. Let A be a nonempty finite subset of V with A ≠ V. Then ψp (A, ∞) ∈ 𝒟0p is p-harmonic in V \ A. Proof. We write ψ = ψp (A, ∞) for simplicity. Let x ∈ V \ A. Since (V, E) is locally finite, it holds that 1x ∈ 𝒟0p . By Proposition 5.5-(1), we have that ℰp (ψ) = 𝒞p (A, ∞) ≤ ℰp (ψ+t1x ) for any t ∈ ℝ. Thus ℰp (ψ + t1x ) attains its minimum at t = 0. The rest of the arguments go exactly the same as in the proof of the previous lemma. Now we are ready to state Royden’s decomposition, which is the main goal of this section. Recall that (G, μ) is p-hyperbolic if and only if 1V ∈ ̸ 𝒟0p (G, μ) (see Definition 5.4-(2)). Theorem 6.9 (Royden’s decomposition [24]). If (G, μ) is p-hyperbolic, then for any u ∈ p 𝒟p (G, μ) there exist a unique u0 ∈ 𝒟0 (G, μ) and a unique h ∈ ℋp (G, μ) such that u = u0 + h.

Parabolic index and Ahlfors regular conformal dimension | 247

The rest of this section is devoted to the proof of the theorem above. Lemma 6.10 ([24, Lemma 2.1]). For any θ1 , θ2 ∈ ℓp (E, μ), ⟨gp (θ1 ) − gp (θ2 ), θ1 − θ2 ⟩ℓ2 (E,μ) ≥ 0. The equality holds if and only if θ1 = θ2 . Proof. Define f : ℝ → ℝ+ by f (t) = ‖θ1 + t(θ2 − θ1 )‖pℓp (E,μ) . Since |t|p is a convex function on ℝ, f is also a convex function on ℝ. Furthermore, if θ1 ≠ θ2 , then f is strictly convex, that is, for any λ ∈ [0, 1] and t1 , t2 ∈ ℝ, f (λt1 + (1 − λ)t2 ) < λf (t1 ) + (1 − λ)f (t2 ). On the other hand, by Hölder’s inequality, we have that 󵄨 󵄨󵄨 d 󵄨p 󵄨󵄨 󵄨 󵄨 ∑ 󵄨󵄨󵄨 󵄨󵄨󵄨θ1 (e) + t(θ2 (e) − θ1 (e))󵄨󵄨󵄨 󵄨󵄨󵄨μe 󵄨󵄨 󵄨 dt e∈E 󵄨 󵄨 󵄨p−1 󵄨 󵄨 = p ∑ 󵄨󵄨󵄨θ1 (e) + t(θ2 (e) − θ1 (e))󵄨󵄨󵄨 󵄨󵄨󵄨θ2 (e) − θ1 (e)󵄨󵄨󵄨μe e∈E

1/p󸀠

󵄨 󵄨p ≤ p( ∑ 󵄨󵄨󵄨θ1 (e) + t(θ2 (e) − θ1 (e))󵄨󵄨󵄨 μe ) e∈E

1/p

󵄨 󵄨p ( ∑ 󵄨󵄨󵄨θ2 (e) − θ1 (e)󵄨󵄨󵄨 μe ) e∈E

< ∞.

Hence f is differentiable whose derivative is given by f 󸀠 (t) = p ∑ sgn(θ1 (e) + t(θ2 (e) − θ1 (e)))|θ1 (e) + t(θ2 (e) − θ1 (e)))|p−1 (θ2 (e) − θ1 (e))μe . e∈E

The convexity of f implies that Thus we obtain

f (t+h)−f (t) h

is an increasing function with respect to h.

f (1) − f (0) ≥ f 󸀠 (0) and f (1) − f (0) ≤ f 󸀠 (1). These inequalities yield p⟨gp (θ1 )−gp (θ2 ), θ1 −θ2 ⟩ℓ2 (E,μ) = f 󸀠 (1)−f 󸀠 (0) ≥ 0, which implies the desired inequality. Lemma 6.11 ([24, Lemma 2.3]). For any v ∈ 𝒟0p (G, μ) and h ∈ ℋp (G, μ), it holds that ℰp (v, h) = 0. Proof. Since v ∈ 𝒟0p , there exists a sequence {vn }n∈ℕ ⊆ ℓc (V) such that ‖v − vn ‖𝒟p 󳨀→ 0. n→∞ By Theorem 6.4, ℰp (vn , h) = −(vn , Δp h)ℓ2 (V,μ) = 0

248 | R. Shimizu for each n ∈ ℕ. Hölder’s inequality implies that 󵄨 󵄨󵄨 1/p󸀠 1/p 󵄨󵄨ℰp (v − vn , h)󵄨󵄨󵄨 ≤ ℰp (h) ℰp (v − vn ) for each n ∈ ℕ. Letting n → ∞, we obtain ℰp (v, h) = limn→∞ ℰp (vn , h) = 0. Proof of Theorem 6.9. Let u ∈ 𝒟p . Set α = inf{ℰp (u − f ) | f ∈ 𝒟0p } ≤ ℰp (u) < ∞. Then we can choose a sequence {fn }n∈ℕ ⊆ 𝒟0p so that limn→∞ ℰp (u − fn ) = α. Claim 1. {fn }n∈ℕ is a Cauchy sequence with respect to ℰp (⋅) and lim ℰ (u n∧m→∞ p

−

fn + fm ) = α. 2

In particular, {ℰp (fn )}n∈ℕ is bounded. Proof of the Claim 1. We split arguments into two cases. Case I: Suppose p ∈ [2, ∞). For any n, m ∈ ℕ, by Clarkson’s inequality, α ≤ ℰp (u −

fn + fm f +f f −f ) ≤ ℰp (u − n m ) + ℰp ( n m ) 2 2 2 u − fm u − fn u − fm u − fn + ) + ℰp ( − ) 2 2 2 2

= ℰp (

≤ 2p−1 (ℰp (

u − fn u − fm ) + ℰp ( )) 2 2

1 = (ℰp (u − fn ) + ℰp (u − fm )). 2 Letting n ∧ m → ∞, we obtain lim ℰ (u n∧m→∞ p

−

f +f f −f fn + fm ) = lim {ℰp (u − n m ) + ℰp ( n m )} = α. n∧m→∞ 2 2 2

In particular, limn∧m→∞ ℰp (fn − fm ) = 0 and limn∧m→∞ ℰp (u −

fn +fm ) 2

= α.

Case II: Suppose p ∈ (1, 2]. For any n, m ∈ ℕ, by Clarkson’s inequality, α1/(p−1) ≤ ℰp (u −

1/(p−1)

fn + fm ) 2

1/(p−1)

+ ℰp (

fn − fm ) 2

1/(p−1)

= ℰp (

u − fm u − fn + ) 2 2

≤ 2(ℰp (

1/(p−1)

+ ℰp (

u − fm u − fn − ) 2 2

1/(p−1)

u − fn u − fm ) + ℰp ( )) 2 2

1/(p−1)

= 2−1/(p−1) (ℰp (u − fn ) + ℰp (u − fm ))

.

Parabolic index and Ahlfors regular conformal dimension | 249

Letting n ∧ m → ∞, we obtain 1/(p−1)

fn + fm ) n∧m→∞ 2 1/(p−1) 1/(p−1) f +f f −f = lim {ℰp (u − n m ) + ℰp ( n m ) } = α1/(p−1) . n∧m→∞ 2 2 lim ℰp (u −

In particular, limn∧m→∞ ℰp (fn − fm ) = 0 and limn∧m→∞ ℰp (u − Claim 2. {fn (o)}n∈ℕ is bounded.

fn +fm ) 2

= α.

Proof of the Claim 2. Suppose that {fn (o)}n∈ℕ is not bounded. Then we can choose a subsequence {fnk (o)}k∈ℕ so that limk→∞ |fnk (o)| = ∞ and |fnk (o)| > 0 for any k ∈ ℕ. Set gk = fnk /fnk (o). Since gk (o) = 1 and gk ∈ 𝒟0p for any k ∈ ℕ, it follows that 󵄨 󵄨−p 󵄨 󵄨−p ‖gk − 1V ‖𝒟p = ℰp (gk ) = 󵄨󵄨󵄨fnk (o)󵄨󵄨󵄨 ℰp (fnk ) ≤ 󵄨󵄨󵄨fnk (o)󵄨󵄨󵄨 sup ℰp (fn ) 󳨀→ 0. n∈ℕ

k→∞

Hence we obtain 1V ∈ 𝒟0p . This is a contradiction.

By Claims 1 and 2, we can choose a subsequence {fnk }k∈ℕ so that {fnk }k∈ℕ is a Cauchy sequence in 𝒟p . Let v be its limit. Then v ∈ 𝒟0p . Set h = u − v ∈ 𝒟p . Claim 3. h ∈ ℋp (G, μ).

Proof of the Claim 3. Since (fnk + fnl )/2 󳨀→ v in 𝒟p , we have k∧l→∞

ℰp (h) = ℰp (u − v) = lim ℰp (u −

fnk + fnl

k∧l→∞

2

) = α.

Thus, for any t ∈ ℝ and g ∈ 𝒟0p , it follows that α = ℰp (h) ≤ ℰp (h + tg). Since ℰp (h + tg) attains its minimum at t = 0, we obtain d ℰ (h + tg)|t=0 = pℰp (g, h) = 0. dt p Let z ∈ V. If we choose g = 1z , then 0 = ℰp (1z , h) = Δp h(z). This implies that h is p-harmonic in V. We conclude that h ∈ ℋp (G, μ). Claim 3 implies the existence of the desired decomposition. Finally, we prove the uniqueness of this decomposition. Suppose that vi ∈ 𝒟0p and hi ∈ ℋp (G, μ) satisfy u = vi + hi for i = 1, 2. Then we obtain 0 = ⟨0, gp (∇h1 ) − gp (∇h2 )⟩ = ⟨∇u − ∇u, gp (∇h1 ) − gp (∇h2 )⟩

= ⟨∇v1 − ∇v2 , gp (∇h1 ) − gp (∇h2 )⟩ + ⟨∇h1 − ∇h2 , gp (∇h1 ) − gp (∇h2 )⟩,

250 | R. Shimizu where we write ⟨⋅, ⋅⟩ = ⟨⋅, ⋅⟩ℓ2 (E,μ) for simplicity. By Lemma 6.11, it follows that ⟨∇v1 − ∇v2 , gp (∇h1 ) − gp (∇h2 )⟩ = −ℰp (v2 − v1 , h1 ) + ℰp (v2 − v1 , h2 ) = 0. Hence we obtain ⟨∇h1 − ∇h2 , gp (∇h1 ) − gp (∇h2 )⟩ = 0. Lemma 6.10 implies ∇h1 = ∇h2 . Thus h1 − h2 ∈ ℝ1V . Since h1 − h2 = v2 − v1 ∈ 𝒟0p and 1V ∈ ̸ 𝒟0p , we have h1 = h2 . This completes the proof.

7 Parabolic index Throughout this section, (G, μ) ia a connected, locally finite, simple weighted graph and p ∈ (1, ∞). In Definition 5.4-(2), we have introduced the notion of the p-parabolicity. It is known that the random walk associated with (G, μ) is recurrent if and only if 1V ∈ 𝒟02 (G, μ), that is, (G, μ) is 2-parabolic. There are various discrete potentialtheoretic conditions which are equivalent to the recurrence (see [3, Theorems 2.36 and 2.37], for example). Following [21] and [26], in this section, we prove nonlinear analogues of these results. The following theorem due to Yamasaki [26] gives a generalization of the recurrence from the viewpoint of the nonlinear potential theory. Theorem 7.1 ([26, Theorems 3.1 and 3.2]). The following are equivalent: (1) (G, μ) is p-parabolic. (2) 𝒟0p (G, μ) = 𝒟p (G, μ). (3) 𝒞p (A, ∞) = 0 for some nonempty finite subset A ⊆ V. (4) 𝒞p (A, ∞) = 0 for any nonempty finite subset A ⊆ V. (5) ρp (A, ∞) = ∞ for some nonempty finite subset A ⊆ V. (6) ρp (A, ∞) = ∞ for any nonempty finite subset A ⊆ V. Proof. We immediately see that (4) ⇔ (6) ⇒ (5) ⇔ (3) and (2) ⇒ (1). Thus it is enough to show that (3) ⇒ (1) ⇒ (4) and (1) ⇒ (2). (3) ⇒ (1) By condition (3), there exists a sequence {un }n∈ℕ in ℓc (V) such that un |A ≡ 1 for any n ∈ ℕ and limn→∞ ℰp (un ) = 0. Then we have 󵄩󵄩 󵄩 󳨀→ 0. 󵄩󵄩un − un (o) ⋅ 1V 󵄩󵄩󵄩𝒟p ≤ ℰp (un ) n→∞ By Proposition 2.10-(2), {un − un (o)}n∈ℕ converges pointwise to 0 as n → ∞. Since (un − un (o))|A = 1 − un (o), it follows that limn→∞ un (o) = 1. Hence 󵄨 󵄨 ‖1V − un ‖𝒟p ≤ ℰp (un )1/p + 󵄨󵄨󵄨1 − un (0)󵄨󵄨󵄨 󳨀→ 0. n→∞

This implies 1V ∈ 𝒟0p (G, μ).

Parabolic index and Ahlfors regular conformal dimension

| 251

(1) ⇒ (4) Let A be a nonempty finite subset of V. From 1V ∈ 𝒟0p , there exists a sequence {un }n∈ℕ in ℓc (V) satisfying limn→∞ ‖1V − un ‖𝒟p = 0. For each n ∈ ℕ, we set uAn = un 1Ac + 1A . Since un converges to 1V in 𝒟p , it follows that un − uAn converges pointwise to 0 and limn→∞ ℰp (un ) = 0. Therefore, since A is a finite set, we have 1 󵄨 󵄨 ∑ ∑ 󵄨󵄨󵄨(un − uAn )(y) − (un − uAn )(x)󵄨󵄨󵄨μxy 󳨀→ 0. n→∞ 2

A

ℰp (un − un ) =

x∈A y∈A

On the other hand, uAn is feasible for 𝒞p (A, ∞). Hence A

𝒞p (A, ∞) ≤ ℰp (un ) =

1 󵄩󵄩 A 󵄩󵄩p 1 󵄩 p 󵄩 ≤ (󵄩󵄩∇uA − ∇un 󵄩󵄩󵄩ℓp (E,μ) + ‖∇un ‖ℓp (E,μ) ) 󵄩∇u 󵄩 p 2 󵄩 n 󵄩ℓ (E,μ) 2 󵄩 n ≤ 2p−1 (ℰp (uAn − un ) + ℰp (un )).

Letting n → ∞, we obtain 𝒞p (A, ∞) = 0. To prove (1) ⇒ (2), we need the following lemma. Lemma 7.2 ([26, Lemma 3.1]). Let u ∈ 𝒟p be nonnegative and let {vn }n∈ℕ be a sequence of nonnegative functions in 𝒟p . If limn→∞ vn (x) = ∞ for any x ∈ V and limn→∞ ℰp (vn ) = ∞, then u ∧ vn converges to u in 𝒟p as n → ∞. Proof of Lemma 7.2. Set un = u ∧ vn , An = {x ∈ V | un (x) = vn (x)} and En = {(x, y) ∈ E | x ∈ An or y ∈ An }. Since limn→∞ vn (x) = ∞ for any x ∈ V, un converges pointwise to u and limn→∞ μ(En ) = 0. Note that {An }n∈ℕ is a decreasing sequence. Thus {En }n∈ℕ is also decreasing. By the definition of En , we have ∇(u − un )(e) = 0 for any e ∈ ̸ En . On the other hand, we have 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨∇(u − un )(e)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨∇u(e)󵄨󵄨󵄨 + 󵄨󵄨󵄨∇vn (e)󵄨󵄨󵄨

(7.1)

for any e ∈ En . Indeed, if e = (x, y) and x, y ∈ An , then (7.1) is obvious. If x ∈ An and y ∈ ̸ An , then un (y) = u(y) and vn (y) − u(y) ≥ 0, and hence 󵄨󵄨 󵄨 󵄨󵄨∇(u − un )((x, y))󵄨󵄨󵄨 = u(x) − vn (x) ≤ u(x) − vn (x) + vn (y) − u(y) 󵄨 󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨u(y) − u(x)󵄨󵄨󵄨 + 󵄨󵄨󵄨vn (y) − vn (x)󵄨󵄨󵄨. If x ∈ ̸ An and y ∈ An , similar arguments as above imply (7.1). By (7.1), we have that 󵄨 󵄨p 2ℰp (u − un ) = ∑ 󵄨󵄨󵄨∇(u − un )(e)󵄨󵄨󵄨 μe e∈En

󵄨 󵄨p 󵄨 󵄨p ≤ 2p−1 ( ∑ 󵄨󵄨󵄨∇u(e)󵄨󵄨󵄨 μe + ∑ 󵄨󵄨󵄨∇un (e)󵄨󵄨󵄨 μe ) p−1

=2

p−1

≤2

e∈En

e∈En

󵄨 󵄨p ∑ 󵄨󵄨󵄨∇u(e)󵄨󵄨󵄨 μe + 2p ℰp (un )

e∈En

󵄨 󵄨 (max󵄨󵄨󵄨∇u(e)󵄨󵄨󵄨μe )μ(En ) + 2p ℰp (un ). e∈E1

252 | R. Shimizu Letting n → ∞, we obtain limn→∞ ℰp (u − un ) = 0. Since limn→∞ un (o) = u(o), we conclude that un converges to u in 𝒟p . Now we prove the rest of the statement of Theorem 7.1. (1) ⇒ (2) Assume that (G, μ) is p-parabolic. Then there exists a sequence {un }n∈ℕ in ℓc (V) such that ‖1V − un ‖𝒟p < n−2 for any n ∈ ℕ. Since un converges to 1V in 𝒟p , taking a subsequence if necessary, we may assume that un ≥ 0 for all n ∈ ℕ. Set vn = nun . Then ℰp (vn ) ≤ np ‖1V − vn ‖p𝒟p ≤ n−p for any n ∈ ℕ and limn→∞ vn (x) = ∞ for any x ∈ V. Let u be a nonnegative function in 𝒟p . Since u ∧ vn ∈ ℓc (V), Lemma 7.2 implies u ∈ 𝒟0p (G, μ). Therefore, we have 𝒟p (G, μ) ∩ {u ∈ ℓ(V) | u ≥ 0} ⊆ 𝒟0p (G, μ). Considering the decomposition u = u+ − u− for u ∈ 𝒟p (G, μ), we obtain 𝒟0p (G, μ) = 𝒟p (G, μ). The following theorem is a collection of characterizations of the p-hyperbolicity. For the proof we follow Soardi and Yamasaki [20]. Theorem 7.3 ([20, Theorem 3.2], [26, Theorem 4.3]). The following are equivalent: (1) (G, μ) is p-hyperbolic. (2) 𝒟0p (G, μ) ≠ 𝒟p (G, μ). (3) 𝒞p (A, ∞) > 0 for some nonempty finite subset A ⊆ V. (4) 𝒞p (A, ∞) > 0 for any nonempty finite subset A ⊆ V. (5) ρp (A, ∞) < ∞ for some nonempty finite subset A ⊆ V. (6) ρp (A, ∞) < ∞ for any nonempty finite subset A ⊆ V. (7) Let x ∈ V. There exists a constant K ∈ (0, ∞) such that 󵄨󵄨 󵄨p 󵄨󵄨u(x)󵄨󵄨󵄨 ≤ K ℰp (u) for any u ∈ 𝒟0p (G, μ).

(8) For some nonempty finite subset A ⊆ V there exists I ∈ ℓ#p (E, μ∗ ) such that Div I(x) = 0 for any x ∈ ̸ A and Flux(I; A) = 1. 󸀠 (9) For any nonempty finite subset A ⊆ V there exists I ∈ ℓ#p (E, μ∗ ) such that Div I(x) = 0 for any x ∈ ̸ A and Flux(I; A) = 1. 󸀠

Proof. By Theorem 7.1, (1) through (6) are equivalent to each other. Thus, it is enough to show that (1), (7), (8), and (9) are equivalent. It is obvious that (9) implies (8). Now we are going to prove (4) ⇒ (7), (7) ⇒ (3), (8) ⇒ (3), and (1) ⇒ (9). (4) ⇒ (7) Let x ∈ V. We assume condition (4). Then 𝒞p ({x}, ∞) > 0. Suppose that there exists a sequence {un }n∈ℕ ⊆ 𝒟0p (G, μ) with ℰp (un ) = 1 such that limn→∞ |un (x)| = ∞. Since limn→∞ |un (x)| = ∞, there exists N ∈ ℕ such that |un (x)| > 0 for any n ≥ N. Set vn = un /un (x) for each n ≥ N. Then vn ∈ 𝒟0p (G, μ) and vn (x) = 1. In particular, we have 󵄨

󵄨−p

󵄨

󵄨−p

𝒞p ({x}, ∞) ≤ ℰp (vn ) = 󵄨󵄨󵄨un (x)󵄨󵄨󵄨 ℰp (un ) = 󵄨󵄨󵄨un (x)󵄨󵄨󵄨

󳨀→ 0.

n→∞

This is a contradiction. Hence there exists a constant K > 0 such that |u(x)| ≤ K for any u ∈ 𝒟0p (G, μ) with ℰp (u) = 1. This completes the proof of (4) ⇒ (7).

Parabolic index and Ahlfors regular conformal dimension

| 253

(7) ⇒ (3) Let x ∈ V. Suppose 𝒞p ({x}, ∞) = 0. Then there exists a sequence {un }n∈ℕ in ℓc (V) with un (x) = 1 such that limn→∞ ℰp (un ) = 0. Condition (7) implies 󵄨p 󵄨 1 = 󵄨󵄨󵄨un (x)󵄨󵄨󵄨 ≤ K ℰp (un ) 󳨀→ 0. n→∞

This is a contradiction. We conclude that 𝒞p ({x}, ∞) > 0, i. e., (3) holds for A = {x}. (8) ⇒ (3) Let I be a flow satisfying the conditions in (8) and let Π ∈ sep(A; ∞). Since Div I(x) = 0 for any x ∈ ̸ A and I ∈ ℓ# (E), we obtain 1 = Flux(I; A) = ∑

∑ Ixy

x∈A y∈NG (x)

= ∑ 1Π (x) ∑ Ixy = ∑ x∈V

y∈NG (x)

∑ (1Π (y) − 1Π (x))Ixy .

x∈V y∈NG (x)

This implies 󵄨 󵄨󵄨 󵄨 󵄨 󵄨 1 ≤ ∑ 󵄨󵄨󵄨∇1Π (e)󵄨󵄨󵄨󵄨󵄨󵄨I(e)󵄨󵄨󵄨 = ∑ 󵄨󵄨󵄨I(e)󵄨󵄨󵄨. e∈E

e∈𝜕e Π

Hence we have Mod∗p (A, ∞) ≤ Ep (|I|) < ∞. By Theorem 5.15, it follows that −p/p󸀠

𝒞p (A, ∞) = Modp (A, ∞) = Modp (A, ∞) ∗

> 0.

(1) ⇒ (9) Let A be a nonempty finite subset of V and let a ∈ A. We write ψ = ψp ({a}, ∞) for simplicity. Suppose that (G, μ) is p-hyperbolic. Then we have Δp ψ(a) ≠ 0. Indeed, if Δp ψ|A ≡ 0, then ψ ∈ ℋp (G, μ). By Theorem 6.9, it holds that ℋp (G, μ)∩𝒟0p (G, μ) = {0⋅1V }. Thus we have that ψ ≡ 0. This contradicts the fact that ψ(a) = 1. Set a Ixy =

gp (∇ψ((x, y)))μxy μ(a)Δp ψ(a)

for (x, y) ∈ E. Then I a ∈ ℓ#p (E, μ∗ ) for each a ∈ A and 󸀠

Div I a (x) = =

1 󵄨 󵄨p−1 ∑ sgn(ψ(y) − ψ(x))󵄨󵄨󵄨ψ(y) − ψ(x)󵄨󵄨󵄨 μxy μ(a)Δp ψ(a) y∈N (x) μ(x)Δp ψ(x)

μ(a)Δp ψ(a)

G

= 1a (x).

Hence I a satisfies the conditions in (9). Note that conditions (8) and (9) are generalizations of the flow criterion in [14], which has been appeared in many literatures for the case p = 2 as a criterion of the transience of a random walk. The p-parabolicity of (G, μ) is monotonic in the following sense:

254 | R. Shimizu Theorem 7.4 ([26, Theorem 5.1]). Let p, q ∈ (1, ∞) with p ≤ q. If (G, μ) is p-parabolic, then (G, μ) is also q-parabolic. Proof. Since, by Proposition 5.3-(2), 𝒞q (A, ∞) ≤ 𝒞p (A, ∞) for any nonempty finite subset A, we see that 𝒞p (A, ∞) = 0 implies 𝒞q (A, ∞) = 0. In light of Theorem 7.4, we give the following definition of the critical value of p-parabolicity. Definition 7.5 (Parabolic index). We define the parabolic index of (G, μ), p∗ (G, μ), by p∗ (G, μ) = inf{p ∈ (1, ∞) | (G, μ) is p-parabolic}. Remark 7.6. If V is finite, then (V, E, μ) is p-parabolic for any p ∈ (1, ∞). Therefore p∗ (V, E, μ) = 1. The following theorem is a useful criterion for p-parabolicity. In the case p = 2, this theorem is well-known as Nash-Williams’ criterion for recurrence. Theorem 7.7 ([26, Lemma 4.2]). If there exists a good exhaustion {An }n∈ℕ satisfying ∞

(An , (An )c ) = ∞, ∑ R(G,μ) p

n=1

then (G, μ) is p-parabolic. Proof. Since An ⊆ An+1 , we have 𝜕e An ∩𝜕e An+1 = 0. Pick I ∈ ℓ# (E) so that Div I(x) = 0 for any x ∈ ̸ A and Flux(I; A) = 1. Since A ⊆ An and Flux(I; A) = 1, we have ∑e∈𝜕e An |I(e)| ≥ 1. Note that 𝒞p (An , (An )c ) = ∑e∈𝜕e An μe . By Hölder’s inequality, p󸀠

p󸀠 /p 󵄨 󵄨 󵄨 󵄨p󸀠 1 ≤ ( ∑ 󵄨󵄨󵄨I(e)󵄨󵄨󵄨) ≤ ( ∑ 󵄨󵄨󵄨I(e)󵄨󵄨󵄨 μ∗e )𝒞p (An , (An )c ) . e∈𝜕e An

e∈𝜕e An

Hence, for any n ∈ ℕ, 󵄨 󵄨p󸀠 ∑ 󵄨󵄨󵄨I(e)󵄨󵄨󵄨 μ∗e ≥ Rp (An , (An )c ).

e∈𝜕e Vn

Summing over n ∈ ℕ, we get I ∈ ̸ ℓp (E, μ∗ ). Thus 󸀠

ℓ#p (E, μ∗ ) ∩ {I ∈ ℓ# (E) | Div I(x) = 0 for any x ∈ ̸ A, Flux(I; A) = 1} = 0. 󸀠

Condition (8) in Theorem 7.3 implies that (G, μ) is p-parabolic. The following criterion is also useful for concrete examples.

Parabolic index and Ahlfors regular conformal dimension

| 255

Lemma 7.8. Let {An }n∈ℕ be an exhaustion of V. If there exists l ∈ ℕ such that lim inf 𝒞p (An , (An+l )c ) = 0, n→∞

then (G, μ) is p-parabolic. Proof. Since ⋃n∈ℕ An = V, there exists N ∈ ℕ such that AN ≠ 0. Set A = AN . By Proposition 3.7-(2), we have c

c

𝒞p (A, (An+l ) ) ≤ 𝒞p (An , (An+l ) )

for any n ≥ N. By Proposition 5.5-(3), it follows that c

c

𝒞p (A, ∞) = lim inf 𝒞p (A, (An+l ) ) ≤ lim inf 𝒞p (An , (An+l ) ) = 0. n→∞

n→∞

Condition (3) in Theorem 7.1 implies that (G, μ) is p-parabolic. Example 7.9 (Trees). Let N ∈ ℕ with N ≥ 2. Define Tm = {1, . . . , N}m , where we set T0 = {0} (empty word), and T = ⋃ Tm . m∈ℤ+

For any v = v1 . . . vm−1 vm ∈ T \ {0}, we define [v]−1 ∈ T by [v]−1 = v1 . . . vm−1 if m ∈ ℕ. Set E = {(v, w) ∈ T × T | [v]−1 = w or [w]−1 = v}. Then 𝒯N = (T, E) is a connected, locally finite, simple weighted graph equipped with the simple weight; 𝒯N is called the N-ary tree with root ϕ. It is easy to see that p∗ (𝒯N ) = ∞, that is, 𝒯N is p-hyperbolic for any p > 1 as stated in [20]. Indeed, define I ∈ ℓ# (E) by I((v, w)) = N −m−1 and I((w, v)) = −Ivw for any (v, w) ∈ E with v ∈ Tm and w ∈ Tm+1 . Then I is a unit flow from {0} to infinity. Let p ∈ (1, ∞). Note that p󸀠 > 1. Since ∞

∞

∑ N m+1 N −p (m+1) = ∑ N −(p −1)(m+1) =

m=0

󸀠

m=0

󸀠

we have I ∈ ℓ#p (E). By Theorem 7.3-(8), 𝒯N is p-hyperbolic. 󸀠

1

󸀠 N p −1

−1

< ∞,

256 | R. Shimizu Example 7.10 (d-dimensional integer lattice). Let ℤd be provided the standard graph structure, i. e., ℤd = {x = (x1 , . . . , xd ) | xi ∈ ℤ for any i ∈ {1, . . . , d}} and the edges, E, be given by 󵄨󵄨 󵄨 x = (x1 , . . . , xd ) and y = (y1 , . . . , yd ) }. E = {(x, y) ∈ ℤd × ℤd 󵄨󵄨󵄨󵄨 󵄨󵄨 with ∑di=1 |xi − yi | = 1 In [16], Maeda has shown that p∗ (ℤd ) = d. It is easy to see that we can apply Theorem 7.7 for any p ≥ d. Maeda has constructed a flow satisfying condition (8) in Theorem 7.3 for any p < d.

8 Stability under rough isometries of weighted graphs In this section, we prove that p-parabolicity and p-hyperbolicity are stable under certain class of perturbations of weighted graphs, which are called rough isometries. This result has been originally proved in [21, Theorem 2.4] for weighted graphs equipped with simple weights which have bounded degrees. First, following [3, Chapter 1], we recall some additional conditions on weights. Definition 8.1. Let (G, μ) be a simple weighted graph. (1) We say that G has bounded geometry if sup #NG (x) < ∞. x∈V

(BG)

(2) We say that (G, μ) satisfies the p0 condition if there exists p0 ∈ (0, 1) such that μxy ≥ p0 μ(x)

(p0 )

for any (x, y) ∈ E. Definition 8.2 (rough isometry of weighted graphs). Let (G1 , μ1 ), (G2 , μ2 ) be connected simple weighted graphs that satisfy (p0 ) and let di be the graph metric of Gi for i = 1, 2. We denote by Vi the vertex set of Gi for each i. We say that φ : V1 → V2 is a rough isometry from (G1 , μ1 ) to (G2 , μ2 ) if and only if φ satisfies the following conditions: (a) There exist C1 ∈ [1, ∞) and C2 ∈ (0, ∞) such that C1−1 d1 (x, y) − C2 ≤ d2 (φ(x), φ(y)) ≤ C1 d1 (x, y) + C2 for any x, y ∈ V1 .

| 257

Parabolic index and Ahlfors regular conformal dimension

(b) There exists C3 ∈ (0, ∞) such that ⋃ Bd2 (φ(x), C3 ) = V2 .

x∈V1

(c) There exists C4 ∈ [1, ∞) such that C4−1 μ1 (x) ≤ μ2 (φ(x)) ≤ C4 μ1 (x) for any x ∈ V1 . If there exists a rough isometry from (G1 , μ1 ) to (G2 , μ2 ), then (G1 , μ1 ) is said to be roughly isometric to (G2 , μ2 ). Remark 8.3. Under condition (p0 ), if (G1 , μ1 ) is roughly isometric to (G2 , μ2 ), then (G2 , μ2 ) is roughly isometric to (G1 , μ1 ) as well. Moreover, being roughly isometric is an equivalence relation among the weighted graphs satisfying (p0 ). The following lemma shows that the p-energies are comparable under rough isometries. This fact is essentially obtained in [20, Section 2]. We will give a proof using a similar argument in [3, Proposition 2.56], where the case p = 2 is considered. Lemma 8.4. Let (V1 , E1 , μ1 ), (V2 , E2 , μ2 ) be weighted graphs satisfying (p0 ) and let φ : V1 → V2 be a rough isometry. Then there exists a constant K ∈ (0, ∞) such that ℰp (u ∘ φ) ≤ K ℰp (u) (1)

(2)

for any u ∈ ℓc (V2 ), where ℰp(i) is the p-energy with respect to (Vi , Ei , μi ) for i = 1, 2. Proof. Let u ∈ ℓc (V2 ) and let (x, y) ∈ E1 . Set x󸀠 = φ(x) and y󸀠 = φ(y). Since φ is a rough isometry, we have 0 ≤ d2 (x󸀠 , y󸀠 ) ≤ C1 + C2 ≤ k, where k = ⌈C1 + C2 ⌉ ∈ ℕ. Thus there exist l ∈ {0, 1, . . . , k} and a path λx󸀠 y󸀠 = [x󸀠 = z0 , z1 , . . . , zl = y󸀠 ] ∈ 𝒫 (V2 , E2 ) from x󸀠 to y󸀠 . By Hölder’s inequality, we obtain l

󵄨󵄨 󵄨 󵄨 󸀠 󵄨 󵄨 󸀠 󵄨 󵄨󵄨u ∘ φ(y) − u ∘ φ(x)󵄨󵄨󵄨 = 󵄨󵄨󵄨u(x ) − u(y )󵄨󵄨󵄨 ≤ ∑󵄨󵄨󵄨u(zi−1 ) − u(zi )󵄨󵄨󵄨 i=1

≤k

1/p󸀠

l

1/p

󵄨 󵄨p (∑󵄨󵄨󵄨u(zi−1 ) − u(zi )󵄨󵄨󵄨 ) i=1

.

In addition, for any i ∈ {1, . . . , l}, μ1 ((x, y)) ≤ μ1 (x) ≤ C4 μ2 (x 󸀠 ) ≤ C4 p−1 0 μ2 ((z0 , z1 ))

−k ≤ C4 p−i 0 μ2 ((zi−1 , zi )) ≤ C4 p0 μ2 ((zi−1 , zi )).

258 | R. Shimizu Hence we have l

󵄨p 󵄨p 󵄨 󵄨󵄨 −k p/p󸀠 ∑󵄨󵄨󵄨u(zi−1 ) − u(zi )󵄨󵄨󵄨 μ2 ((zi−1 , zi )). 󵄨󵄨u ∘ φ(y) − u ∘ φ(x)󵄨󵄨󵄨 μ1 ((x, y)) ≤ C4 p0 k i=1

(8.1)

Fix a family of paths {λx󸀠 y󸀠 }x,y∈V1 in 𝒫 (V2 , E2 ) satisfying len(λx󸀠 y󸀠 ) ≤ k for any (x, y) ∈ E1 . Set M(v,w) = #{(x, y) ∈ E1 | λx󸀠 y󸀠 ∩ (v, w) ≠ 0} for (v, w) ∈ E2 and set 𝒜w = φ−1 (Bd2 (w, k)). Since len(λx󸀠 y󸀠 ) ≤ k for any (x, y) ∈ E, we see that {(x, y) ∈ E1 | λx󸀠 y󸀠 ∩ (v, w) ≠ 0} ⊆ 𝒜w × 𝒜w . For any a, b ∈ 𝒜w , it follows that C1−1 d1 (a, b) − C2 ≤ d2 (a󸀠 , b󸀠 ) ≤ d2 (a󸀠 , w) + d2 (b󸀠 , w) ≤ 2k. This implies diam(𝒜w , d1 ) = maxa,b∈𝒜w d1 (a, b) ≤ D, where D is a positive constant depending only on C1 and C2 . Therefore, we can choose z ∈ V1 so that 𝒜w ⊆ Bd1 (z, D+1). Since (V2 , E2 ) satisfies (BG), we have sup #Bd1 (z, D + 1) < ∞. z∈V1

In particular, it holds that #𝒜w ≤ M, where M = supz∈V1 #Bd1 (z, D + 1), and so sup(v,w)∈E2 M(v,w) ≤ M 2 < ∞. Summing (8.1) over (x, y) ∈ E1 , we obtain −k p/p󸀠

ℰp (u ∘ φ) ≤ C4 p0 k (1)

M 2 ℰp(2) (u).

This completes the proof. Finally, we prove that p-parabolicity and p-hyperbolicity are stable under rough isometries. Theorem 8.5 ([21, Theorem 2.4]). p-parabolicity and p-hyperbolicity are stable under rough isometries of weighted graphs. Proof. Let (V1 , E1 , μ1 ), (V2 , E2 , μ2 ) be weighted graphs satisfying (p0 ) and suppose that (V1 , E1 , μ1 ) and (V2 , E2 , μ2 ) are roughly isometric to each other. Let φ : (V1 , E1 , μ1 ) → (V2 , E2 , μ2 ) be a rough isometry. Since being roughly isometric is an equivalence relation, it is enough to show that (V2 , E2 , μ2 ) is p-hyperbolic if (V1 , E1 , μ1 ) is p-hyperbolic. Let u ∈ ℓc (V2 ). Now, by the same argument as in the proof of Lemma 8.4, we see that #φ−1 (A) is a finite subset of V1 for any finite subset A ⊆ V2 . Thus u ∘ φ ∈ ℓc (V1 ). Let x1 ∈ V1 and set x2 = φ(x1 ) ∈ V2 . By Theorem 7.3-(7), there exists K1 > 0 such that |u1 (x1 )|p ≤ K1 ℰp(1) (u1 ) for any u1 ∈ ℓc (V1 ). Thus Lemma 8.4 implies that 󵄨󵄨 󵄨p 󵄨 󵄨p (1) (2) 󵄨󵄨u(x2 )󵄨󵄨󵄨 = 󵄨󵄨󵄨u ∘ φ(x1 )󵄨󵄨󵄨 ≤ K1 ℰp (u ∘ φ) ≤ K1 K ℰp (u) for any u ∈ ℓc (V2 ). Hence, by Theorem 7.3-(7), (V2 , E2 , μ2 ) is p-hyperbolic. By the above theorem, we have the stability of the parabolic index under rough isometries. Corollary 8.6 ([21, Theorem 2.4]). The parabolic index is an invariant of rough isometries of weighted graphs.

Parabolic index and Ahlfors regular conformal dimension

| 259

9 Self-similar sets and their blow-ups In this section, we introduce the notions concerning self-similar sets and consider their graph approximation. We also construct an infinite graph associated with this approximation. Following [10, Chapter 1], we start with a brief introduction of the notion of selfsimilar sets. Definition 9.1 ([10, Definitions 1.1.1 and 1.1.2]). (1) Let (X, dX ) and (Y, dY ) be metric spaces. A map f : X → Y is said to be uniformly Lipschitz continuous on X with respect to dX and dY if Lip(f ) = sup

x =y∈X ̸

dY (f (x), f (y)) < ∞. dX (x, y)

The above constant Lip(f ) is called the Lipschitz constant of f . (2) Let (X, d) be a metric space. If f : X → X is Lipschitz continuous on X with respect to d and Lip(f ) < 1, then f is called a contraction with respect to d with contraction ratio Lip(f ). In particular, a contraction f with contraction ratio r ∈ (0, 1) is called a r-similitude if d(f (x), f (y)) = rd(x, y) for any x, y ∈ X. It is known that any r-similitude f on (ℝn , dE ), where dE is the Euclidean metric, is given by f (x) = rUx + a for some a ∈ ℝn and U ∈ O(n) (see [10, Exercise 1.1]). The following theorem is an immediate application of the contraction principle. Theorem 9.2 ([10, Theorem 1.1.4]). Let (X, d) be a nonempty complete metric space. Let S be a nonempty finite set and let {Fi }i∈S be a family of contractions on (X, d). Then there exists a unique nonempty compact subset K of X that satisfies K = ⋃ Fi (K). i∈S

(9.1)

Definition 9.3 (Self-similar sets). Let (X, d) be a nonempty complete metric space. Let S be a nonempty finite set and let {Fi }i∈S be a family of contractions on (X, d). Then the compact subset K of X satisfying (9.1) is called the self-similar set with respect to {Fi }i∈S . If Fi is injective for any i ∈ S, then we also call the triple (K, S, {Fi }i∈S ) a selfsimilar structure. Remark 9.4. The notion of self-similar structures has been introduced by Kigami with the aim of giving topological descriptions of self-similar sets. Our definition above is a special case of the original definition. See [10, Section 1.3] for the definition with full generality. Next, we define basic notions on the word space and the shift space. We mostly follow the notations in [11]. Let S be a nonempty finite set.

260 | R. Shimizu Definition 9.5 (Words and shift space). (1) We define Wm (S) = Sm = {w1 . . . wm | wi ∈ S for i ∈ {1, . . . , m}} for m ∈ ℕ, and W0 (S) = {0}, where 0 is an element called the empty word. We also set W# (S) = ⋃m∈ℕ Wm (S) and W∗ (S) = W# (S) ∪ {0} = ⋃m∈ℤ+ Wm (S). For w ∈ W∗ (S), the length of w, which is denoted by |w|, is defined to be the unique m ∈ ℤ+ satisfying w ∈ Wm (S). (2) For w = w1 . . . wm ∈ W∗ (S) and v = v1 . . . vn ∈ W∗ (S), we set wv = w1 . . . wm v1 . . . vn . For w ∈ W∗ and n ∈ ℤ+ , we define wn = ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ww . . . w ∈ Wn|w| . Moreover, we set ×n

w ⋅ Wn (S) = {wv | v ∈ Wn (S)} and w ⋅ W∗ (S) = {v ∈ W∗ (S) | [v]|w| = w} for n ∈ ℤ+ . (3) For w = w1 . . . wm ∈ W∗ and l ∈ {0, . . . , m}, we set [w]l = w1 . . . wl ∈ Wl . Also for m ∈ ℤ+ and i ∈ S, we define σi(m) : Wm → Wm+1 by σi(m) (w) = iw for any w ∈ Wm . Moreover for w = w1 . . . wl ∈ W∗ , we write σw(m) = σw(m+l−1) ∘⋅ ⋅ ⋅∘σw(m) , i. e., σw(m) (v) = wv 1 l

for any v ∈ Wm . In addition for m ∈ ℕ, we define a map σ (m) : Wm → Wm−1 by σ (m) (w1 . . . wm ) = w2 . . . wm for any w = w1 . . . wm ∈ Wm . If no confusion can occur, we use σ to denote σ (m) . (4) The (one-sided) shift space with symbols S is defined by Σ(S) = Sℕ = {ω = ω1 ω2 ω3 . . . | ωi ∈ S for any i ∈ ℕ}. For each i ∈ S, we define σi : Σ(S) → Σ(S) by σi (ω1 ω2 ω3 . . . ) = iω1 ω2 ω3 . . . We also define σ : Σ(S) → Σ(S) by σ(ω1 ω2 ω3 . . . ) = ω2 ω3 ω4 . . . For w = w1 . . . wm ∈ W∗ (S), we write σw = σw1 ∘ ⋅ ⋅ ⋅ ∘ σwm and Σw (S) = σw (Σ(S)). Notice that Σw (S) = {wω1 ω2 ω3 . . . | ωi ∈ S for each i ∈ ℕ}. For ω = ω1 ω2 ω3 . . . ∈ Σ(S) and m ∈ ℤ+ , we define [ω]m = ω1 . . . ωm ∈ Wm (S). (5) For w ∈ W# (S), we define w∞ = www . . . ∈ Σ(S). If no confusion may occur, we omit S in the above notations. For example, we write Wm , W∗ and Σ instead of Wm (S), W∗ (S), and Σ(S), respectively. Definition 9.6. Let (K, S, {Fi }i∈S ) be a self-similar structure. For w = w1 w2 . . . wm ∈ W∗ , we define Fw = Fw1 ∘ Fw2 ∘ ⋅ ⋅ ⋅ ∘ Fwm . We also define Kw = Fw (K).

Parabolic index and Ahlfors regular conformal dimension | 261

Remark 9.7. A family of subsets {Kw }w∈W∗ is a partition parametrized by the #S-ary tree with root ϕ of K in the sense of [12]. Example 9.8. (1) (Sierpinski gasket (Figure 2)) Let p1 = (1/2, √3/2), p2 = (0, 0), p3 = (1, 0) ∈ ℝ2 . Define Fi (x) = (x − pi )/2 + pi for i = 1, 2, 3. Then the (planar) Sierpinski gasket is the self-similar set K with respect to {Fi }3i=1 . Let q1 = F3 (p2 ), q2 = F1 (p3 ), and q3 = F2 (p1 ). Then it follows that ⋃ i,j∈S with i=j̸

Ki ∩ Kj = {q1 , q2 , q3 }.

(9.2)

By (9.2), K \ {q1 , q2 , q3 } is disconnected. Such a self-similar set is often called a finitely ramified self-similar set.

Figure 2: The planar Sierpinski gasket.

(2) (Diamond fractal (Figure 3)) The diamond fractal has been introduced in [13]. For details, we refer to [11, Example 1.7.7]. Let p1 , p2 , p3 ∈ ℂ be given by p1 = 1/2 + √−3/2, p2 = 0, p3 = 1. Define q = (p1 + p2 + p3 )/3. For i ∈ {1, 2, 3}, define Fi (z) =

1 (z − pi ) + pi 3

262 | R. Shimizu

Figure 3: The diamond fractal.

and Fi+3 (z) = −

1 qi (z − pi+3 ) + pi+3 3 qi

where qi = pi −q and pi+3 = (3q+pi )/4. The diamond fractal is a nonempty compact set K satisfying K = ⋃6i=1 Fi (K). Note that K \ {q} is disconnected, so the diamond fractal has a property similar to finitely ramified self-similar sets. However, the intersections K1 ∩ K4 , K2 ∩ K5 , and K3 ∩ K6 are the ternary Cantor set. Now we introduce graphs associated with a self-similar structure. Hereafter (K, S, {Fi }i∈S ) is a self-similar structure and assume that K is connected. For any m ∈ ℕ, we set Em = {(v, w) ∈ Wm × Wm | v ≠ w, Kv ∩ Kw ≠ 0}. We obtain a sequence of finite graphs {Gm }m∈ℕ , where Gm = (Wm , Em ). We denote this sequence of graphs by 𝒮K . Since K is connected, it follows that Gm is a connected graph for each m ∈ ℤ+ . Remark 9.9. (Wm , Em ) is called a horizontal graph in [12]. 𝒮K has a self-similarity in the following sense.

Parabolic index and Ahlfors regular conformal dimension | 263

Definition 9.10. Let (V1 , E1 , μ1 ), (V2 , E2 , μ2 ) be simple weighted graphs. A map φ : V1 → V2 is called a graph isomorphism between (V1 , E1 ) and (V2 , E2 ) if and only if φ is bijective and (x, y) ∈ E1 if and only if (φ(x), φ(y)) ∈ E2 for any x, y ∈ V1 . If (V1 , E1 ) = (V2 , E2 ), then we say that φ is a graph automorphism of (V1 , E1 ). Lemma 9.11. 𝒮K satisfies the following conditions: (SSG1) For any i ∈ S and m ∈ ℤ+ , σi(m) : Wm → Wm+1 is a graph isomorphism between (Wm , Em ) and (i ⋅ Wm , (Em+1 )i⋅Wm ). ̃ (m) , Ẽ (m) ) be a graph given by (SSG2) Let (W n n ̃ (m) = {w ⋅ Wm | w ∈ Wn }, W n Ẽn(m) = {(w ⋅ Wm , v ⋅ Wm ) | (w ⋅ Wm ) ∩ 𝜕(v ⋅ Wm ) ≠ 0} ̃ (m) , Ẽ (m) ) through the map w 󳨃→ for n, m ∈ ℤ+ . Then (Wn , En ) is isomorphic to (W n n w ⋅ Wm for any n, m ∈ ℤ+ . Proof. Let v, w ∈ Wn . Since Fi is injective, we have Kiv ∩ Kiw = Fi (Kv ) ∩ Fi (Kw ) = Fi (Kv ∩ Kw ). This implies that Kv ∩ Kw ≠ 0 if and only if Kiv ∩ Kiw ≠ 0. Hence (SSG1) holds. Let v, w ∈ Wn . Assume (w ⋅ Wm ) ∩ 𝜕(v ⋅ Wm ) ≠ 0. Then there exist z 1 , z 2 ∈ Wm such that (vz 1 , wz 2 ) ∈ En+m . It follows that 0 ≠ Kvz 1 ∩ Kwz 2 ⊆ Kv ∩ Kw and hence (v, w) ∈ En . On the other hand, suppose (v, w) ∈ En . Since Kv = ⋃z 1 ∈Wm Kvz 1 and Kw = ⋃z 2 ∈Wm Kwz 2 by [10, Theorem 1.2.3], we have ( ⋃ Kvz 1 ) ∩ ( ⋃ Kwz 2 ) ≠ 0. z 1 ∈Wm

z 2 ∈Wm

This implies that there exist z 1 , z 2 ∈ Wm satisfying Kvz 1 ∩ Kwz 2 ≠ 0. Thus (w ⋅ Wm ) ∩ 𝜕(v ⋅ Wm ) ≠ 0. Hence (SSG2) holds. By (SSG1), 𝒮K can be considered as an increasing sequence. Next, we construct an infinite graph as an “increasing limit” of 𝒮K = {(Wm , Em )}m∈ℤ+ . Definition 9.12 (Blow-up). Let ω = ω0 ω1 ω2 . . . ∈ Σ. For m ∈ ℤ+ , define an injection ιω m : Wm → Wm+1 by ιω m (w) = ωm w for any w ∈ Wm . For n, m ∈ ℤ+ with n > m, we also define an injection ιω m,n : Wm → Wn by ω ω ω ιω m,n = ιn−1 ∘ ιn−2 ∘ ⋅ ⋅ ⋅ ∘ ιm .

264 | R. Shimizu We identify (Wm , Em ) as the subgraph of (Wm+1 , Em+1 ) through ιω m . By this identification, {(Wm , Em )}m∈ℤ+ is an increasing sequence of finite graphs. We use Gω = (V ω , E ω ) to denote the increasing limit of this sequence. Gω is called the blow-up of 𝒮K associated with ω. We can identify (Wm , Em ) as the subgraph of Gω . We use ιω m,∞ to denote the ω injection giving this identification and use Gm = (Wm,ω , Em,ω ) to denote this subgraph of Gω . Remark 9.13. The idea of blow-ups has already appeared in the study of self-similar sets. See [22], for example. Since Gω is the increasing limit of connected, locally finite, simple graph, the following proposition is obvious. Proposition 9.14. Let ω = ω0 ω1 ω2 . . . ∈ Σ. Then Gω is a connected, locally finite, simple graph. Furthermore, under the open set condition, Gω satisfies (BG). This fact is essentially due to Moran [17]. For a proof, see [10, Proposition 1.5.8], for example. Proposition 9.15. Let ω ∈ Σ and let (K, S, {Fi }i∈S ) be a self-similar structure. Assume that K is a subset of ℝn and (K, S, {Fi }i∈S ) satisfies the open set condition, i. e., there exists a bounded nonempty open set O ⊆ ℝn such that ⋃ Fi (O) ⊆ O i∈S

and Fi (O) ∩ Fj (O) = 0 for i, j ∈ S with i ≠ j. Then Gω satisfies (BG). In Figure 4, we present (modified) blow-ups of self-similar sets defined in Example 9.8. We choose ω = 2∞ in both cases of the Sierpinski gasket and the diamond fractal. In the case of the diamond fractal, we remark that the figure represents a graph made by deleting edges from the blow-up in order to have a simpler shape but keep the essential structure, i. e., the deleted graph and the original blow-up are roughly isometric. Next we state our standing assumptions in the rest of this paper. Suppose that (K, S, {Fi }i∈S ) is a self-similar structure with #S ≥ 2 and let r∗ ∈ (0, 1). Hereafter, we always assume that (K, S, {Fi }i∈S ) satisfies the following conditions: (S1) K is connected subset of ℝn . (S2)r∗ Fi : ℝn → ℝn is r∗ -similitude for any i ∈ S. (S3) (K, S, {Fi }i∈S ) satisfies the open set condition. (S4) Let dE be the Euclidean metric of ℝn . Then inf{r∗−m

inf

x∈Kv ,y∈Kw

dE (x, y) | m ∈ ℤ+ , v, w ∈ Wm with Kv ∩ Kw = 0} > 0.

Parabolic index and Ahlfors regular conformal dimension | 265

Figure 4: Blow-ups of the Sierpinski gasket and the diamond fractal.

(S5)

For any w ∈ W∗ , it holds that Kw \ (

⋃

v∈W|w| \{w}

Kv ) ≠ 0.

Remark 9.16. (1) If condition (S5) holds, then (K, S, {Fi }i∈S ) is said to be minimal. See [10, Theorem 1.3.8] for details on this notion. (2) The Sierpinski gasket and the diamond fractal satisfy (S1) through (S5). Furthermore, other typical examples of self-similar sets such as the Sierpinski carpet and nested fractals also satisfy (S1) through (S5). In fact, if Fi (x) = rx + ai for all i ∈ S, then (S1) and (S3) imply (S4). Lastly, we prove that the Euclidean metric is 1-adapted in the sense of [12] under the conditions (S1), (S2)r∗ , (S3), (S4), and (S5). See [12] for details on the adaptedness of metrics. Lemma 9.17. Let (K, S, {Fi }i∈S ) be a self-similar structure satisfying (S1), (S2)r∗ , (S3), (S4), and (S5). For any x ∈ K and s ∈ (0, 1], we set U1 (x, s) = ∪{Kw | w ∈ Wm and there exists v ∈ Wm such that Kv ∩ Kw ≠ 0}, where m ∈ ℤ+ is the unique integer which satisfies s ∈ [r∗m , r∗m−1 ). Then there exists α1 , α2 > 0 such that U1 (x, α1 r) ⊆ BdE (x, r) ⊆ U1 (x, α2 r) for any x ∈ K and r > 0.

266 | R. Shimizu Proof. If v, w ∈ Wm and Kv ∩ Kw ≠ 0, then we have sup

x∈Kv ,y∈Kw

dE (x, y) ≤ diam(Kv , dE ) + diam(Kw , dE ) ≤ 2diam(K, dE )r∗m .

This inequality implies U1 (x, α1 r) ⊆ BdE (x, r) for any x ∈ X and r > 0, where α1 = (2diam(K, dE ))−1 . On the other hand, if v, w ∈ Wm and Kv ∩ Kw = 0, then condition (S4) implies that inf

x∈Kv ,y∈Kw

dE (x, y) ≥ c∗ r∗m ,

where c∗ = inf{r∗−m

inf

x∈Kv ,y∈Kw

dE (x, y) | m ∈ ℤ+ , v, w ∈ Wm with Kv ∩ Kw = 0} > 0.

Thus we have that BdE (x, r) ⊆ U1 (x, α2 r) for any x ∈ X and r > 0, where α2 = (c∗ r∗ )−1 .

10 Ahlfors regular conformal dimension of a self-similar set As we have mentioned in the introduction, Kigami has obtained a characterization of the Ahlfors regular conformal dimension using p-energies with respect to a sequence of graphs in [12]. To state his result in our settings, we first recall definitions related to the Ahlfors regular conformal dimension. Definition 10.1 (Quasisymmetry). Let X be a nonempty set and let d, ρ be metrics on X. Then ρ is said to be quasisymmetric, d ∼ ρ for simplicity, with respect to d if and only QS

if there exists a homeomorphism η : [0, ∞) → [0, ∞) such that η(0) = 0 and, for any t ∈ [0, ∞), ρ(x, z) ≤ η(t)ρ(x, y) whenever d(x, z) ≤ td(x, y). Remark 10.2. (1) It follows that d ∼ ρ if and only if QS

ρ(x, a) d(x, a) ≤ η( ) ρ(x, b) d(x, b)

(10.1)

for any x, a, b ∈ X with x ≠ b. (2) It is known that ∼ is an equivalence relation on metrics on X. For a proof, see [6, QS

Proposition 10.6], for example.

Parabolic index and Ahlfors regular conformal dimension | 267

(3) The notion of quasisymmetry is closely related to the notion of quasiconformality. Indeed, a quasisymmetry uniformly controls distortions of annuli in metric spaces in the following sense: If (10.1) holds, then for any λ > 1, r > 0, and x ∈ X there exists s > 0 such that Bρ (x, s) ⊆ Bd (x, r) ⊆ Bd (x, λr) ⊆ Bρ (x, η(λ)s). For a proof, see [15, Lemma 1.2.18], for example. Definition 10.3 (Ahlfors regularity). Let (X, d) be a metric space and let α > 0. We say that d is α-Ahlfors regular if there exist a Borel regular measure μ on (X, d) and some constant C ∈ [1, ∞) such that C −1 r α ≤ μ(Bd (x, r)) ≤ Cr α for all x ∈ X and r ∈ (0, diam(X, d)] ∩ ℝ. We also say that d is Ahlfors regular if and only if d is α-Ahlfors regular for some α > 0, Remark 10.4. Let (X, d) be a metric space. If d is α-Ahlfors regular, then the Hausdorff dimension of (X, d) is equal to α. The Ahlfors regular conformal dimension of a metric spaces is defined as follows. Definition 10.5 (Ahlfors regular conformal dimension). Let (X, d) be a metric space. The Ahlfors regular conformal dimension of (X, d) is defined by dimAR (X, d) = inf{dimH (X, ρ) | ρ ∼ d, ρ is Ahlfors regular}, QS

(10.2)

where dimH (X, ρ) denotes the Hausdorff dimension of (X, ρ). Finally, we can provide Kigami’s result for self-similar sets. Let (K, S, {Fi }i∈S ) be a self-similar structure satisfying (S1) through (S5) and let 𝒮K = {Gm }m∈ℤ+ be the associated sequence of finite graphs. For each n ∈ ℤ+ , k ∈ ℕ and w ∈ W∗ , we set ℬn (w, k) = ∪{v ⋅ Wn | v ∈ W|w| with dG|w| (v, w) ≤ k}.

Furthermore, we define G

c

𝒞p (k) = sup 𝒞p n+|w| (w ⋅ Wn , ℬn (w, k) ). (n)

w∈W∗

Then Ahlfors regular conformal dimensions of self-similar sets are characterized as follows: Theorem 10.6 (A special case of [12, Theorem 4.6.9]). For any k ∈ ℕ, dimAR (K, dE ) = inf{p | lim inf 𝒞p(n) (k) = 0} = inf{p | lim sup 𝒞p(n) (k) = 0}. n→∞

n→∞

268 | R. Shimizu

11 Parabolic index of a blow-up In this section, we prove that the Ahlfors regular conformal dimension of a self-similar set gives an upper bound of parabolic indices of its blow-ups (Theorem 11.2). This is the second main result of this paper. We first prove a lemma needed later. Let (G, μ) = (V, E, μ) be a connected, locally finite, simple weighted graph and let p ∈ (1, ∞). Lemma 11.1. Let A and B be nonempty subsets of V with A ⊆ B. If V∗ is a subset of V with B ⊆ V∗ , then (GV∗ ,μV∗ )

c

𝒞p (A, B ) = 𝒞p

(A, Bc ∩ V∗ ).

Proof. Let u ∈ ℓ(V) be feasible for 𝒞p (A, Bc ). Since u|Bc ≡ 0 and B ⊆ V∗ , we see that ℰp (u) = ℰp,V∗ (u|V∗ ). (GV∗ ,μV∗ )

Clearly, u|V∗ is feasible for 𝒞p

(GV∗ ,μV∗ ) 𝒞p (A, Bc

(A, Bc ∩ V∗ ). Therefore, it holds that ℰp (u) ≥ (GV∗ ,μV∗ )

∩ V∗ ). This implies 𝒞p (A, Bc ) ≥ 𝒞p

(A, Bc ∩ V∗ ).

(GV∗ ,μV∗ )

For the converse inequality, let u∗ ∈ ℝV∗ be feasible for 𝒞p u ∈ ℓ(V) by

(A, Bc ∩V∗ ). Define

u∗ (x) if x ∈ V∗ , u(x) = { 0 if x ∈ ̸ V∗ . Since A ∪ B ⊆ V∗ , u is feasible for 𝒞p (A, Bc ). Furthermore, since B ⊆ V∗ , we have c

𝒞p (A, B ) ≤ ℰp (u) = ℰp,V∗ (u∗ ). (GV∗ ,μV∗ )

Thus 𝒞p (A, Bc ) ≤ 𝒞p

(A, Bc ∩ V∗ ). We complete the proof.

Before we state and prove the main result of this section, we introduce a few notations. Let ω = ω0 ω1 ω2 . . . ∈ Σ and let Gω be the blow-up of (K, S, {Fi }i∈S ) associated with ω. For any w ∈ W∗ and k ∈ ℕ, we define ω Wω n (w) = ιn+|w|,∞ (w ⋅ Wn ), Gω

ω

ω

𝒞ℋ1 (w) = {v ∈ W∗ | Wn (w) ∩ 𝜕Wn (v) ≠ 0} ∪ {w}

and Bω n (w, 1) =

⋃

ω

v∈𝒞ℋG1 (w)

Wω n (v).

If no confusion can occur, we omit ω in the above notations. Note that Wω n (ωn ) = Wn,ω ω for each n ∈ ℤ+ . Furthermore, ιω ( ℬ (w, k)) ⊆ B (w, k) for any n ∈ ℤ n + , k ∈ ℕ and n n+|w|,∞ w ∈ W∗ .

Parabolic index and Ahlfors regular conformal dimension | 269

Theorem 11.2. Let (K, S, {Fi }i∈S ) be a self-similar structure satisfying (S1) through (S5). For any ω = ω0 ω1 ω2 . . . ∈ Σ, dimAR (K, dE ) ≥ p∗ (Gω ).

(11.1)

Proof. Let p > dimAR (K, dE ). It is enough to show that Gω is p-parabolic. We set o∗ = ω ιω 0,∞ (0) ∈ V . Choose an increasing sequence of positive integers {rm }m∈ℤ+ satisfying limm→∞ rm = ∞ and Bω m (ωm , 1) ⊆ BdGω (o∗ , rm )

(11.2)

for any m ∈ ℤ+ . Set Vm = BdGω (o∗ , rm ) for each m ∈ ℤ+ . Then {Vm }m∈ℤ+ is an exhaustion. Since ⋃n∈ℤ+ Wn,ω = V ω , for any m ∈ ℤ+ , there exists n(m) ∈ ℤ+ such that Vm ⊆ Wn(m),ω . Thus Lemma 11.1 implies that Gω

Gω

c

c

(11.3)

𝒞p ({o∗ }, (Vm ) ) = 𝒞p n(m) ({o∗ }, (Vm ) ) ω for each m ∈ ℤ+ . Since Gn(m) is graph isomorphic to Gn(m) , we have Gω

c

G

ω

c

𝒞p n(m) ({o∗ }, (Vm ) ) = 𝒞p n(m) ({ωn(m)−1 . . . ω1 ω0 }, (ιn(m),∞ ) ((Vm ) )). −1

On the other hand, by (11.2) and the fact that Vm ⊆ Wn(m),ω , it follows that ω ιω n(m),∞ (ℬm (ωn(m)−1 . . . ωm , 1)) = Bm (ωm , 1).

Thus, by Proposition 3.7-(2), we have G

ω

c

𝒞p n(m) ({ωn(m)−1 . . . ω1 ω0 }, (ιn(m),∞ ) ((Vm ) )) −1

G

≤ 𝒞p n(m) ({ωn(m)−1 . . . ωm ⋅ Wm }, ℬm (ωn(m)−1 . . . ωm , 1)c ). ω

Combining this with (11.3), we have 𝒞pG ({o∗ }, (Vm )c ) ≤ 𝒞p(m) (1) for any m ∈ ℤ+ . Note that, by Theorem 10.6, limm→∞ 𝒞p(m) (1) = 0. By Proposition 5.5-(3), it follows that ω lim 𝒞 G ({o∗ }, (Vm )c ) m→∞ p

ω

= 𝒞pG ({o∗ }, ∞) = 0.

Thus, by Theorem 7.1-(3), Gω is p-parabolic.

12 Examples In this section, we are going to apply Theorem 11.2 to examples. Moreover, in the cases of the Sierpinski gasket and the diamond fractal, we will see that the Ahlfors regular conformal dimension coincides with parabolic index of a blow-up and this value is equal to 1. At first, we prove a lemma needed in examples below.

270 | R. Shimizu Lemma 12.1. Let {Vn }n∈ℕ be an exhaustion of G and let A be a nonempty subset of V1 \ 𝜕i V1 . Then, for any n ∈ ℕ, 𝒞p (A, 𝜕i Vn ) = 𝒞p (A, (Vn )c ).

Proof. By Proposition 3.7-(2), we have 𝒞p (A, 𝜕i Vn ) ≤ 𝒞p (A, (Vn )c ). On the other hand, let ̃ ∈ ℓ(V) by n ∈ ℕ and let u ∈ ℓ(V) be feasible for 𝒞p (A, 𝜕i Vn ). Define u u(x) ̃ (x) = { u 0

if x ∈ Vn , if x ∈ ̸ Vn .

̃ is also feasible for 𝒞p (A, 𝜕i Vn ). Since u|𝜕i Vn ≡ 0, we have ℰp (u ̃ ) ≤ ℰp (u). Letting Then u u = ψp (A, 𝜕i Vn ) and using Proposition 3.6, we have ψp (A, 𝜕i Vn )|(Vn )c ≡ 0. In particular, ψp (A, 𝜕i Vn ) is feasible for 𝒞p (A, (Vn )c ). Thus 𝒞p (A, 𝜕i Vn ) = ℰp (ψp (A, 𝜕i Vn )) ≤ 𝒞p (A, (Vn )c ).

This completes the proof. Example 12.2 (Sierpinski gasket). Let us consider the Sierpinski gasket defined in Example 9.8-(1). We use GSG = (V SG , E SG ) to denote the blow-up of the Sierpinski gasket associated with 2∞ . Set Wm = Wm,2∞ for each m ∈ ℤ+ . Note that we immediately obtain supv∈V SG #NGSG (v) ≤ 3. Moreover, it follows that {Wm }m∈ℤ+ is a good exhaustion of GSG , SG that is, Wm ⊆ Wm+1 for any m ∈ ℤ+ . The subgraph induced by Wm is denoted by Gm . Let dimAR (SG) denote the Ahlfors regular conformal dimension of the Sierpinski gasket with respect to the Euclidean metric and let p∗ (SG) denote the parabolic index of GSG . Theorem 12.3. dimAR (SG) = p∗ (SG) = 1. Proof. By Theorem 11.2, we immediately verify dimAR (SG) ≥ p∗ (SG). We prove dimAR (SG) ≤ p∗ (SG). Let w ∈ W∗ and set SG

Cp(n) = 𝒞pG ({ιn,∞ (2n )}, {ιn,∞ (1n ), ιn,∞ (3n )}) for each n ∈ ℤ+ . Since 𝜕i Wn (0) = {ιn,∞ (1n ), ιn,∞ (3n )}, Lemma 12.1 implies c

SG

Cp(n) = 𝒞pG ({ιn,∞ (2n )}, (Wn (0)) ). By Proposition 3.7-(2), we have that GSG

n

GSG

n

c

𝒞p ({ιn,∞ (2 )}, (Wn )c ) ≤ 𝒞p ({ιn,∞ (2 )}, (Wn−1 ) ) SG

= 𝒞pG ({ι0,∞ (0)}, (Wn−1 )c ),

Parabolic index and Ahlfors regular conformal dimension

| 271

for all n ∈ ℕ. Thus limn→∞ Cp(n) = 0 for all p > p∗ (SG). Let p > p∗ (SG). Since {Wm }m∈ℤ+ is a good exhaustion of GSG , Lemma 11.1 implies Cp(n) = 𝒞pGn ({2n }, {1n , 3n }). Applying Proposition 3.7-(2), we obtain G

c

G

𝒞p n+|w| (w ⋅ Wn , ℬn (w, 1) ) ≤ 𝒞p n+|w| (w ⋅ Wn , ℬn (w, 1)c ).

(12.1)

Set A(n) (w) = {v ∈ W|w| | w ⋅ Wn ∩ 𝜕(v ⋅ Wn ) ≠ 0}. Then, by (SSG2), we see that #A(n) (w) does not depend on n and #A(n) (w) ≤ 3. Set l = #A(n) (w) ≤ 3, A(n) (w) = {w1 , . . . , wl } and B(n) (w) = 𝜕(w ⋅ Wn ) ∩ wi ⋅ Wn for i ∈ {1, . . . , l}. Note that, in this case, #B(n) (w) = 1 for i i any i ∈ {1, . . . , l} and n ∈ ℤ+ . Furthermore, it follows that l

G

i

(Gn+|w| )w ⋅Wn

𝒞p n+|w| (w ⋅ Wn , ℬn (w, 1)c ) = ∑ 𝒞p i=1

i (n) (B(n) i (w), 𝜕(w ⋅ Wn ) \ Bi (w)).

By symmetries of both the Sierpinski gasket and its approximation {Gm }m∈ℤ+ , we have i

(Gn+|w| )w ⋅Wn

𝒞p

i (n) (n) (B(n) i (w), 𝜕(w ⋅ Wn ) \ Bi (w)) = Cp

(12.2)

for any i ∈ {1, . . . , l}. Combining this with (12.1), we obtain c

𝒞p (w ⋅ Wn , ℬn (w, 1) ) ≤ 3Cp . (n)

(n)

(12.3)

Thus it follows that 𝒞p(n) (1) ≤ 3Cp(n) for any n ∈ ℤ+ . Letting n → ∞, we have

limn→∞ 𝒞p(n) (1) = 0 for any p > p∗ (SG). Hence we obtain p∗ (SG) ≥ dimAR (SG). We conclude that dimAR (SG) = p∗ (SG). Next, we prove p∗ (SG) = 1. Let a ∈ [0, 1]. Set GSG

n n n n ψ(n) p = ψp ({ιn,∞ (2 )}, {ιn,∞ (1 ), ιn,∞ (3 )})

for each n ∈ ℤ+ . We set u(n+1) p,a ∈ ℓ(VSG ) by u(n+1) p,a (ι|w|,∞ (w)) (1 − a) + aψ(n) { p (ι|w|−1,∞ (σ(w))) if ι|w|,∞ (w) ∈ Wn (2), { { (n) = {(1 − a)ψp (ι|w|−1,∞ (σ(w))) if ι|w|,∞ (w) ∈ Wn (1) ∪ Wn (3), { { if ι|w|,∞ (w) ∈ ̸ Wn+1 (0). {0 GSG

(n+1) Since Cp(n+1) = 𝒞p n+1 ({ιn+1,∞ (2n+1 )}, {ιn+1,∞ (1n+1 ), ιn+1,∞ (3n+1 )}), u(n+1) . p,a is feasible for Cp Moreover, n+1 u(n+1) )) = 1, p,a (ιn+1,∞ (2

n (n+1) n (n+1) n u(n+1) p,a (ιn+1,∞ (21 )) = up,a (ιn+1,∞ (12 )) = up,a (ιn+1,∞ (23 ))

272 | R. Shimizu n = u(n+1) p,a (ιn+1,∞ (32 )) = 1 − a,

n+1 n (n+1) n u(n+1) )) = u(n+1) p,a (ιn+1,∞ (1 p,a (ιn+1,∞ (13 )) = up,a (ιn+1,∞ (31 )) n+1 = u(n+1) )) = 0, p,a (ιn+1,∞ (3

and u(n+1) p,a |Wn+1 (0)c ≡ 0. In particular, we obtain SG

SG

p (n) p p G (n) p Cp(n+1) ≤ ℰpG (u(n+1) p,a ) = (a + 2(1 − a) )ℰp (ψp ) = (a + 2(1 − a) )Cp .

This implies, for all n ∈ ℕ, n−1 (1) Cp .

Cp(n) ≤ (ap + 2(1 − a)p )

(12.4)

Since ap + 2(1 − a)p attains its minimum at a∗ = 1/(1 + 2−p /p ) ∈ (0, 1), we have ap∗ + 2(1 − a∗ )p ∈ (0, 1). Letting a = a∗ and n → ∞ in (12.4), we see that limn→∞ Cp(n) = 0 for any p > 1. Condition (3) in Theorem 7.1 implies that GSG is p-parabolic for any p > 1. We have completed the proof. 󸀠

Example 12.4 (Diamond fractal). Let us consider the diamond fractal in Example 9.8-(2). Let GDia = (V Dia , E Dia ) be the blow-up of the diamond fractal associated with 2∞ . Set Wm = Wm,2∞ for each m ∈ ℤ+ . Note that supv∈V Dia #NGDia (v) ≤ 5 and {Wm }m∈ℤ+ is a good exhaustion of GDia . The subgraph induced by Wm is denoted Dia by Gm . Let dimAR (Dia) denote the Ahlfors regular conformal dimension of the diamond fractal with respect to the Euclidean metric and let p∗ (Dia) denote the parabolic index of GDia . Let 𝒞p(n) be the p-conductances with respect to Gn . Theorem 12.5. dimAR (Dia) = p∗ (Dia) = 1. Proof. By Theorem 11.2, we immediately verify dimAR (Dia) ≥ p∗ (Dia). To prove dimAR (Dia) ≤ p∗ (Dia), it is enough to show that limn→∞ 𝒞p(n) (L) for some L ∈ ℕ. An easy observation tells us that there exist L, M ∈ ℕ satisfying the following property: for any w ∈ W∗ , there exist Aw ⊆ K such that #Aw ≤ M and ⋃{Kv | v ∈ W|w| , dG|w| (v, w) ≤ L} \ Aw is disconnected. This fact, in conjunction with an argument similar to that used to establish (12.3), implies (n+m)

𝒞p

(w ⋅ Wn , ℬn (w, L)c ) ≤ M( sup NGDia (v))Cp(n) v∈VDia

(12.5)

for all w ∈ Wm and sufficiently large m ∈ ℤ+ . Hence we have limn→∞ 𝒞p(n) (L) = 0 for any p > p∗ (Dia) and conclude that dimAR (Dia) = p∗ (Dia) by applying Theorem 10.6.

Parabolic index and Ahlfors regular conformal dimension

| 273

Next, we prove p∗ (Dia) = 1. Let a ∈ [0, 1] and set GDia

n n ψ(n) p = ψp ({ιn,∞ (2 )}, 𝜕i Wn (0))

for each n ∈ ℤ+ . We set u(n+1) p,a ∈ ℓ(VDia ) by (1 − a) + aψ(n) { p (ι|w|−1,∞ (σ(w))) { { { { {1 − a u(n+1) p,a (ι|w|,∞ (w)) = { { {(1 − a)ψ(n) p (ι|w|−1,∞ (σ(w))) { { { {0

if w ∈ Wn (2),

if w ∈ Wn (5),

if w ∈ Wn (4) ∪ Wn (6), otherwise.

GDia

(n+1) Then u(n+1) = 𝒞p n+1 ({ιn+1,∞ (2n+1 )}, 𝜕i Wn+1 (0)). Moreover, we obtain p,a is feasible for Cp Dia

Dia

p (n) p p G (n) p Cp(n+1) ≤ ℰpG (u(n+1) p,a ) = (a + 2(1 − a) )ℰp (ψp ) = (a + 2(1 − a) )Cp .

The rest of the proof is similar to the case of the Sierpinski gasket.

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Meng Yang

Metrics and uniform Harnack inequality on the Strichartz hexacarpet Abstract: We construct intrinsic metrics on the Strichartz hexacarpet using weight functions and show that these metrics do not satisfy the chain condition. We give uniform Harnack inequality on the approximating graphs of the Strichartz hexacarpet with respect to the intrinsic metrics instead of graph metrics. Keywords: Strichartz hexacarpet, intrinsic metric, uniform Harnack inequality, chain condition MSC 2010: 28A80

Contents 1 2 3 4 5

Introduction | 275 Statement of the main results | 276 Proof of Theorem 2.1 | 280 Proof of Proposition 2.3 | 288 Proof of Theorem 2.4 | 291 Bibliography | 295

1 Introduction A big open question in the analysis on fractals is to construct a Brownian motion, or equivalently, a local regular Dirichlet form on any given fractal. This has been done on many fractals, for example, the Sierpiński gasket (SG) [4, 10] and more general post critically finite (p. c. f.) self-similar sets [11, 12, 7] and finitely ramified fractals [15], the Sierpiński carpet (SC) [1, 14] and higher-dimensional SCs [2]. Recently, Grigor’yan and the author [6, 16] gave a unified analytic construction on the SG and the SC. On p. c. f. self-similar sets and finitely ramified fractals, the most intrinsically essential ingredient in the construction of Brownian motion is the so-called compatible condition. However, on non-p. c. f. self-similar sets and infinitely ramified fractals, Acknowledgement: The author was supported by SFB1283 of the German Research Council (DFG). The author is very grateful to Prof. Alexander Grigor’yan, Prof. Alexander Teplyaev, Prof. Jun Kigami and Dr. Qingsong Gu for very helpful discussions. Meng Yang, Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany; and CNRS UMR 5582, Institut Fourier, Université Grenoble Alpes, Gières, France (current address), e-mail: [email protected] https://doi.org/10.1515/9783110700763-009

276 | M. Yang compatible condition does not hold and uniform Harnack inequality is a key ingredient which provides compactness results for appropriate approximating sequences. But uniform Harnack inequality is not easy to verify and was obtained only on the SC and higher-dimensional SCs. The main purpose of this paper is to consider another concrete non-p. c. f. selfsimilar set and infinitely ramified fractal, that is, the Strichartz hexacarpet. The group of Teplyaev [3, 8] has given some results on this fractal mainly on the approximating graphs, but the existence of Brownian motion still remains a conjecture. Since the Strichartz hexacarpet is defined in a very abstract way, there was not even a canonical metric, needless to say uniform Harnack inequality. In this paper, we construct intrinsic metrics on the Strichartz hexacarpet using weight functions and give uniform Harnack inequality on the approximating graphs of the Strichartz hexacarpet with respect to the intrinsic metrics instead of graph metrics. We will see that the intrinsic metrics behave very differently from graph metrics due to the unusual connectedness property of the Strichartz hexacarpet. The construction of metrics using weight functions was initiated by Kameyama [9] and developed by Kigami [13]. Recently, Gu, Qiu, and Ruan [5] constructed metrics on the SC using weight functions with two parameters a and b. They showed that the weight functions give metrics if and only if a, b ∈ (0, 1) satisfy 2a + b ≥ 1 and a + 2b ≥ 1. They showed that the metrics satisfy the chain condition if and only if 2a + b = 1 or a + 2b = 1, that is, the point (a, b) lies on part of the boundary of the admissible region to give metrics. On the Strichartz hexacarpet, we will construct metrics using weight functions with one parameter μ. We will show that the weight functions give metrics if and only if μ ∈ [1/2, 1). However, we will show that for any μ ∈ [1/2, 1), the metrics do not satisfy the chain condition. Hence, unlike the case on the SC, one cannot obtain a metric satisfying the chain condition by adjusting the parameter on the Strichartz hexacarpet.

2 Statement of the main results Let W = {0, 1, 2, 3, 4, 5}. Let W0 = {0} and Wn = W n = {w = w1 . . . wn : wi ∈ W, i = 1, . . . , n} for any n ≥ 1. ∞ n Let W∗ = ⋃∞ n=0 Wn = ⋃n=0 W and

W∞ = W ∞ = {w = w1 w2 . . . : wi ∈ W, i = 1, 2, . . .}.

Strichartz hexacarpet |

277

For any n ≥ 0, for any w ∈ Wn , denote |w| = n. We use the convention that |0| = 0. For any n ≥ 1, for any w = w1 . . . wn−1 wn ∈ Wn , denote w− = w1 . . . wn−1 ∈ Wn−1 . (1) For any w(1) = w1(1) . . . wm ∈ Wm and w(2) = w1(2) . . . wn(2) ∈ Wn , denote (1) (2) w(1) w(2) = w1(1) . . . wm w1 . . . wn(2) ∈ Wm+n . (1) For any w(1) = w1(1) . . . wm ∈ Wm and w(2) = w1(2) w2(2) . . . ∈ W∞ , denote (1) (2) (2) w(1) w(2) = w1(1) . . . wm w1 w2 . . . ∈ W∞ .

For any i ∈ W, denote in = ⏟⏟⏟⏟⏟⏟⏟ i . . . i ∈ Wn , n times

i

∞

= ii . . . ∈ W∞ .

For any w(1) = w1(1) w2(1) . . . , w(2) = w1(2) w2(2) . . . ∈ W∞ , define s(w(1) , w(2) ) = min{i ≥ 1 : wi(1) ≠ wi(2) }, with the convention that min 0 = +∞. It is obvious that s(w(1) , w(2) ) ≥ min {s(w(1) , w(3) ), s(w(3) , w(2) )}

for any w(1) , w(2) , w(3) ∈ W∞ .

Fix arbitrary r ∈ (0, 1), for any w(1) , w(2) ∈ W∞ , let δr (w(1) , w(2) ) = r s(w

(1)

,w(2) )

,

with the convention that r +∞ = 0. It is obvious that for any w(1) , w(2) , w(3) ∈ W∞ , we have δr (w(1) , w(2) ) ≤ max{δr (w(1) , w(3) ), δr (w(3) , w(2) )}. Hence δr is an ultrametric on W∞ . By [12, Theorem 1.2.2], (W∞ , δr ) is a compact metric space. For any i ∈ W, define σi : W∞ → W∞ by w = w1 w2 . . . 󳨃→ σi (w) = iw1 w2 . . . .

278 | M. Yang For any w ∈ W∗ , for any i ∈ W, let j = i + 1 (mod 6), for any v ∈ {0, 5}∞ = {w = w1 w2 . . . : wi = 0, 5, i = 1, 2, . . .}. If i is even, then define wi1v ∼ wj1v

and wi2v ∼ wj2v.

wi3v ∼ wj3v

and wi4v ∼ wj4v.

If i is odd, then define

It is obvious that ∼ is an equivalence relation on W∞ . Let K = W∞ / ∼ be equipped with the quotient topology and π : W∞ → K the quotient map. Since at most two elements in W∞ are mapped to the same point in K, a simple topological argument gives that K is a compact Hausdorff space. For any i ∈ W, for any w(1) , w(2) ∈ W∞ , since w(1) ∼ w(2) if and only if σi (w(1) ) ∼ σi (w(2) ), there exists a unique map fi : K → K such that π ∘ σi = fi ∘ π. Therefore, K is a topological self-similar set, see [9, Definition 0.3]. By [9, Theorem 1.5], K is metrizable; K is called the Strichartz hexacarpet, see [3, Figures 2 and 4] for related figures. We use w ∈ W∞ also to denote the corresponding point π(w) ∈ K. For any w = w1 . . . wn ∈ W∗ , let fw = fw1 ∘ ⋅ ⋅ ⋅ ∘ fwn ,

Kw = fw1 ∘ ⋅ ⋅ ⋅ ∘ fwn (K), where f0 = id is the identity map. We say that Kw is an n-cell. We introduce a pseudometric given in [9] as follows. Let μ ∈ (0, 1), for any w ∈ W∗ , let gμ (w) = μ|w| . We say that {w(1) , . . . , w(m) } is a chain if w(i) ∈ W∗ for any i = 1, . . . , m and Kw(i) ∩ Kw(i+1) ≠ 0 for any i = 1, . . . , m − 1. We say that a chain {w(i) , . . . , w(j) } is a subchain of {w(1) , . . . , w(m) } for any i ≤ j. Denote 𝒞 as the set of all chains. (i) (1) (m) We say that ∑m }. i=1 gμ (w ) is the weight of the chain {w , . . . , w (1) (m) For any x, y ∈ K, we say that {w , . . . , w } is a chain connecting x and y if it is a chain satisfying x ∈ Kw(1) and y ∈ Kw(m) . Denote by 𝒞 (x, y) the set of all chains connecting x and y. For any x, y ∈ K, let m

dμ (x, y) = inf{∑ gμ (w(i) ) : {w(1) , . . . , w(m) } ∈ 𝒞 (x, y)}. i=1

Then dμ is a pseudometric by the remark in [9, Definition 1.10], that is, dμ (x, y) ≥ 0, dμ (x, y) = dμ (y, x) and dμ (x, y) ≤ dμ (x, z) + dμ (z, y) for any x, y, z ∈ K. The main results of this paper are as follows.

Strichartz hexacarpet | 279

Theorem 2.1. dμ is a metric if and only if μ ∈ [1/2, 1). For any μ ∈ [1/2, 1), for any i ∈ W, for any x, y ∈ K, we have dμ (fi (x), fi (y)) = μdμ (x, y). For any w ∈ W∗ , we have diamμ (Kw ) := sup{dμ (x, y) : x, y ∈ Kw } = μ|w| . The Hausdorff dimension of (K, dμ ) is α = − log 6/ log μ and the normalized Hausdorff measure ν of dimension α exists. Remark 2.2. If μ ∈ [1/2, 1), then by [9, Proposition 1.11], dμ is compatible with the topology of K. Hence (K, dμ ) is a compact metric space. For any x ∈ K, for any r ∈ (0, 1), denote Bμ (x, r) = {y ∈ K : dμ (x, y) < r}. Proposition 2.3. For any μ ∈ [1/2, 1), dμ does not satisfy the chain condition, for any θ ∈ (0, − log μ/ log 2), dμ does satisfy the θ-chain condition. Let V0 = {ij∞ : i ∈ W, j = 0, 5} and Vn+1 = ⋃ σi (Vn ) = {wij∞ : w ∈ Wn+1 , i ∈ W, j = 0, 5} for any n ≥ 0. i∈W

For any n ≥ 0, let Hn be the graph with vertex set Vn and edge set given by {(w(1) , w(2) ) : w(1) = wv(1) , w(2) = wv(2) , w ∈ Wn , v(1) , v(2) ∈ V0 , v(1) ≠ v(2) }. There are two metrics on Hn , one is the usual graph metric, the other is the metric induced from the intrinsic metric dμ on K. Theorem 2.4. For any μ ∈ [1/2, 1), there exists some positive constant C such that for any x ∈ K, for any r ∈ (0, 1), for any nonnegative harmonic function u in Vn ∩ Bμ (x, r), we have max

Vn ∩Bμ (x,μr)

u≤C

min

Vn ∩Bμ (x,μr)

u.

Remark 2.5. The harmonicity is defined using graphs. The balls are defined using the intrinsic metric instead of graph metrics. This paper is organized as follows. In Section 3, we prove Theorem 2.1. In Section 4, we prove Proposition 2.3. In Section 5, we prove Theorem 2.4. Notation. The letters c, C will always refer to some positive constants and may change at each occurrence. The sign ≍ means that the ratio of the two sides is bounded from above and below by positive constants. The sign ≲ (≳) means that the LHS is bounded by positive constant times the RHS from above (below).

280 | M. Yang

3 Proof of Theorem 2.1 We consider the case μ ∈ (0, 1/2) as follows. Lemma 3.1. For any μ ∈ (0, 1/2), we have dμ (0∞ , 10∞ ) = 0, hence dμ is not a metric. n

Proof. For any n ≥ 1, we construct a chain {w(n,1) , . . . , w(n,2 ) } ⊆ Wn connecting 0∞ and 10∞ as follows. For n = 1, let w(1,1) = 0 and w(1,2) = 1. n Assume that we have constructed a chain {w(n,1) , . . . , w(n,2 ) } ⊆ Wn connecting 0∞ and 10∞ . Then for n + 1, for any i = 1, . . . , 2n , let w(n+1,i) = 0w(n,i) , – – –

n

w(n+1,2

+i)

= 1w(n,2

n

+1−i)

.

The following facts are obvious from the above construction: n For any n ≥ 1, we have {w(n,1) , . . . , w(n,2 ) } ⊆ Wn . n For any n ≥ 1, we have w(n,1) = 0n and w(n,2 ) = 10n−1 , hence 0∞ ∈ Kw(n,1) and 10∞ ∈ Kw(n,2n ) . n

w(n+1,2 ) = 010n−1 and w(n+1,2

n

+1)

= 110n−1 .

n+1

To show that {w(n+1,1) , . . . , w(n+1,2 ) } ∈ 𝒞 (0∞ , 10∞ ), we only need to show that Kw(n+1,2n ) ∩ Kw(n+1,2n +1) ≠ 0, that is, K010n−1 ∩ K110n−1 ≠ 0. Indeed, 010∞ ∈ K010n−1 and 110∞ ∈ K110n−1 . By definition, we have 010∞ ∼ 110∞ , that is, 010∞ and 110∞ are indeed the same point in K, hence K010n−1 ∩ K110n−1 ≠ 0. n By induction principle, we obtain a chain {w(n,1) , . . . , w(n,2 ) } ⊆ Wn connecting 0∞ and 10∞ for any n ≥ 1. Hence dμ (0∞ , 10∞ ) ≤ 2n μn → 0 as n → +∞, hence dμ (0∞ , 10∞ ) = 0. Since 0∞ and 10∞ are distinct points in K, we have dμ is not a metric. We assume that μ ∈ [1/2, 1) hereafter. We need do some preparations as follows. We say that a chain {w(1) , . . . , w(m) } satisfies only adjacent intersection (OAI) condition if the following conditions are satisfied: – There exists no |i − j| ≥ 2 such that Kw(i) ∩ Kw(j) ≠ 0. – There exists no i ≠ j such that Kw(i) ⊆ Kw(j) . Lemma 3.2. dμ (x, y) m

= inf{∑ gμ (w(i) ) : {w(1) , . . . , w(m) } ∈ 𝒞 (x, y) satisfies (OAI) condition}. i=1

Proof. It is obvious that the LHS ≤ the RHS. Assume that {w(1) , . . . , w(m) } ∈ 𝒞 (x, y). If there exist i + 2 ≤ j such that Kw(i) ∩ Kw(j) ≠ 0, then {w(1) , . . . , w(i−1) , w(i) , w(j) , w(j+1) , . . . , w(m) } ∈ 𝒞 (x, y)

Strichartz hexacarpet |

281

and i

m

m

k=1

k=j

k=1

∑ gμ (w(k) ) + ∑ gμ (w(k) ) < ∑ gμ (w(k) ).

If there exist j + 2 ≤ i such that Kw(j) ∩ Kw(i) ≠ 0, then {w(1) , . . . , w(j−1) , w(j) , w(i) , w(i+1) , . . . , w(m) } ∈ 𝒞 (x, y) and j

m

m

k=1

k=i

k=1

∑ gμ (w(k) ) + ∑ gμ (w(k) ) < ∑ gμ (w(k) ).

If there exist i < j such that Kw(i) ⊆ Kw(j) , then {w(1) , . . . , w(i−1) , w(j) , . . . , w(m) } ∈ 𝒞 (x, y) and i−1

m

m

k=1

k=j

k=1

∑ gμ (w(k) ) + ∑ gμ (w(k) ) < ∑ gμ (w(k) ).

If there exist j < i such that Kw(i) ⊆ Kw(j) , then {w(1) , . . . , w(j) , w(i+1) , . . . , w(m) } ∈ 𝒞 (x, y) and j

m

m

k=1

k=i+1

k=1

∑ gμ (w(k) ) + ∑ gμ (w(k) ) < ∑ gμ (w(k) ).

Repeating the above procedure finitely many times, we eventually obtain a chain still in 𝒞 (x, y) satisfying (OAI) condition with less weight than the origin chain. Hence the RHS ≤ the LHS. Therefore, we obtain the desired result. For any w ∈ W∗ , the boundary 𝜕Kw is given by 𝜕Kw = {wiv : i ∈ W, v ∈ {0, 5}∞ }, the interior int(Kw ) is given by int(Kw ) = Kw \𝜕Kw . We collect some basic facts as follows.

282 | M. Yang Lemma 3.3. (1) For any w ∈ W∗ , 𝜕Kw is the disjoint union of 𝜕Kw ∩ Kw0 , . . . , 𝜕Kw ∩ Kw5 , that is, 𝜕Kw = ∐ (𝜕Kw ∩ Kwi ), i∈W

where for any i ∈ W, 𝜕Kw ∩ Kwi = {wiv : v ∈ {0, 5}∞ }. (2) For any n ≥ 1, for any w ∈ Wn , there exist at most three elements v ∈ Wn with v ≠ w such that Kv ∩ Kw ≠ 0. More precisely, there exist two elements v ∈ Wn with v ≠ w and v− = w− such that Kv ∩ Kw ≠ 0 and there exists at most one element v ∈ Wn with v ≠ w and v− ≠ w− such that Kv ∩ Kw ≠ 0. For any w ∈ W∗ , we say that {w(1) , . . . , w(m) } is a chain going through Kw if it is a chain satisfying Kw(i) ⊆ Kw for any i = 1, . . . , m, Kw(1) ∩ 𝜕Kw ≠ 0 and Kw(m) ∩ 𝜕Kw ≠ 0. Denote 𝒞 (Kw ) as the set of all chains going through Kw . Moreover, if there exist j1 , j2 ∈ W with j1 ≠ j2 such that 0 ≠ 𝜕Kw ∩ Kw(1) ⊆ 𝜕Kw ∩ Kwj1 and 0 ≠ 𝜕Kw ∩ Kw(m) ⊆ 𝜕Kw ∩ Kwj2 , then we say that {w(1) , . . . , w(m) } is a chain going through Kw with different entries, denoted as {w(1) , . . . , w(m) } ∈ 𝒞 (Kw ) with different entries, it is obvious that |w(i) | ≥ |w| + 1 for any i = 1, . . . , m. Lemma 3.4. For any w ∈ W∗ , for any {w(1) , . . . , w(m) } ∈ 𝒞 (Kw ) with different entries, we have m

∑ gμ (w(i) ) ≥ μ|w| . i=1

Proof. Denote n = |w|. By the proof of Lemma 3.2, we may assume that {w(1) , . . . , w(m) } satisfies (OAI) condition. Let j1 , j2 ∈ W with j1 ≠ j2 satisfy 0 ≠ 𝜕Kw ∩ Kw(1) ⊆ 𝜕Kw ∩ Kwj1 and 0 ≠ 𝜕Kw ∩ Kw(m) ⊆ 𝜕Kw ∩ Kwj2 . Let 󵄨 󵄨 k = max{󵄨󵄨󵄨w(i) 󵄨󵄨󵄨 : i = 1, . . . , m}. If k = n + 1 or k = n + 2, then direct calculation gives the desired result. Assume that this result holds for n + 1, n + 2, . . . , k − 1. For k > n + 2, we only need to find some {v(1) , . . . , v(l) } ∈ 𝒞 (Kw ) with different entries satisfying 0 ≠ 𝜕Kw ∩ Kv(1) ⊆ 𝜕Kw ∩ Kwj1 and 0 ≠ 𝜕Kw ∩ Kv(l) ⊆ 𝜕Kw ∩ Kwj2 and 󵄨 󵄨 max{󵄨󵄨󵄨v(i) 󵄨󵄨󵄨 : i = 1, . . . , l} < k such that l

m

i=1

i=1

∑ gμ (v(i) ) ≤ ∑ gμ (w(i) ).

Strichartz hexacarpet | 283

Then by induction assumption, we have m

l

∑ gμ (w(i) ) ≥ ∑ gμ (v(i) ) ≥ μn . i=1

i=1

If |w | = k, then |w | = k and (w ) = (w(2) )− . Indeed, suppose that |w(2) | < k, since Kw(1) ⊈ Kw(2) and 0 ≠ 𝜕Kw ∩ Kw(1) ⊆ 𝜕Kw ∩ Kwj1 , by Lemma 3.3, we have Kw(2) ∩ Kw ⊆ 𝜕Kw , contradicting to the fact that Kw(2) ⊆ Kw . Suppose that |w(2) | = k and (w(1) )− ≠ (w(2) )− , since 0 ≠ 𝜕Kw ∩Kw(1) ⊆ 𝜕Kw ∩Kwj1 , by Lemma 3.3 again, we have Kw(2) ∩Kw ⊆ 𝜕Kw , contradicting to the fact that Kw(2) ⊆ Kw . Let (1)

(2)

(1) −

− − 󵄨 󵄨 󵄨 󵄨 j = max{j : 󵄨󵄨󵄨w(1) 󵄨󵄨󵄨 = ⋅ ⋅ ⋅ = 󵄨󵄨󵄨w(j) 󵄨󵄨󵄨, (w(1) ) = ⋅ ⋅ ⋅ = (w(j) ) },

then j ≥ 2. Hence we have {(w(1) )− , w(j+1) , . . . , w(m) } ∈ 𝒞 (Kw ) with different entries satisfying 0 ≠ 𝜕Kw ∩ K(w(1) )− ⊆ 𝜕Kw ∩ Kwj1 and 0 ≠ 𝜕Kw ∩ Kw(m) ⊆ 𝜕Kw ∩ Kwj2 . Noting that j

j

i=1

i=1

gμ ((w(1) ) ) = μk−1 ≤ 2μk ≤ ∑ μk = ∑ gμ (w(i) ), −

we have m

m

i=j+1

i=1

−

gμ ((w(1) ) ) + ∑ gμ (w(i) ) ≤ ∑ gμ (w(i) ). Moreover, we have |(w(1) )− | = k − 1 < k. If |w(m) | = k, then by similar argument to the above, we have another chain going through Kw with different entries and with the last element (w(m) )− satisfying |(w(m) )− | = k − 1 < k. For a possibly new chain, denoted by {v(1) , . . . , v(l) }, that satisfies |v(1) | < k and (l) |v | < k. If 󵄨 󵄨 max{󵄨󵄨󵄨v(i) 󵄨󵄨󵄨 : i = 1, . . . , l} < k, then this is our desired chain. Otherwise, let 󵄨 󵄨 j = min{j : 󵄨󵄨󵄨v(j) 󵄨󵄨󵄨 = k}. By similar argument to the above, let − − 󵄨 󵄨 󵄨 󵄨 p = max{p : 󵄨󵄨󵄨v(j) 󵄨󵄨󵄨 = ⋅ ⋅ ⋅ = 󵄨󵄨󵄨v(p) 󵄨󵄨󵄨, (v(j) ) = ⋅ ⋅ ⋅ = (v(p) ) },

then l − 1 ≥ p ≥ j + 1. Hence we have {v(1) , . . . , v(j−1) , (v(j) )− , v(p+1) , . . . , v(l) } ∈ 𝒞 (Kw ) with different entries satisfying j−1

−

l

l

i=p+1

i=1

∑ gμ (v(i) ) + gμ ((v(j) ) ) + ∑ gμ (v(i) ) ≤ ∑ gμ (v(i) ). i=1

284 | M. Yang Repeating the above consideration finitely many times, we eventually obtain the desired chain. By induction principle, we have the desired result. Remark 3.5. By the above proof, 1/2 is critically important. Corollary 3.6. For any n ≥ 1, for any w(1) , w(2) ∈ Wn . If Kw(1) ∩ Kw(2) = 0, then dμ (Kw(1) , Kw(2) ) := inf{dμ (x, y) : x ∈ Kw(1) , y ∈ Kw(2) } ≥ μn . Proof. For any x ∈ Kw(1) , y ∈ Kw(2) , for any {v(1) , . . . , v(m) } ∈ 𝒞 (x, y), there exists w(3) ∈ Wn with w(3) ≠ w(1) and w(3) ≠ w(2) , either there exists i = 1, . . . , m such that Kv(i) ⊇ Kw(3) or there exist i1 ≤ i2 such that {v(i1 ) , . . . , v(i2 ) } ∈ 𝒞 (Kw(3) ) with different entries. For the first case, we have m

∑ gμ (v(i) ) ≥ gμ (v(i) ) = μ|v

(i)

|

≥ μ|w

i=1

(3)

|

= μn .

For the second case, by Lemma 3.4, we have m

i2

i=1

i=i1

∑ gμ (v(i) ) ≥ ∑ gμ (v(i) ) ≥ μ|w

(3)

|

= μn .

Hence dμ (x, y) ≥ μn , hence dμ (Kw(1) , Kw(2) ) ≥ μn . Lemma 3.7. For any w ∈ W∗ , for any x, y ∈ Kw , we have m

dμ (x, y) = inf{∑ gμ (w(i) ) : {w(1) , . . . , w(m) } ∈ 𝒞 (x, y), Kw(i) ⊆ Kw for any i = 1, . . . , m}. i=1

Proof. If w = 0, then this result is trivial. We may assume that |w| ≥ 1. It is obvious that the LHS ≤ the RHS. Since {w} ∈ 𝒞 (x, y), we have the RHS ≤ μ|w| . We only need to show that for arbitrary {w(1) , . . . , w(m) } ∈ 𝒞 (x, y), we have m

∑ gμ (w(i) ) ≥ RHS. i=1

If there exists i = 1, . . . , m such that |w(i) | ≤ |w|, then m

∑ gμ (w(i) ) ≥ gμ (w(i) ) = μ|w i=1

(i)

|

≥ μ|w| ≥ RHS.

We may assume that |w(i) | ≥ |w| + 1 for any i = 1, . . . , m.

Strichartz hexacarpet | 285

If Kw(i) ⊆ Kw for any i = 1, . . . , m, then it is trivial to have m

∑ gμ (w(i) ) ≥ RHS. i=1

Otherwise, there exists i = 1, . . . , m such that Kw(i) ⊈ Kw . Then there exists v ∈ W|w| with v ≠ w and Kw ∩ Kv ≠ 0, there exist i1 ≤ i2 such that Kw(i) ⊆ Kv for any i = i1 , . . . , i2 and exact one of the following conditions holds: (a) i2 = m. (b) i1 = 1, i2 < m and Kw(i2 +1) ⊆ Kw . (c) i1 = 1, i2 < m and Kw(i2 +1) ⊆ Ku for some u ∈ W|w| with u ≠ w and u ≠ v. (d) i1 > 1, Kw(i1 −1) ⊆ Kw , i2 < m and Kw(i2 +1) ⊆ Kw . (e) i1 > 1, Kw(i1 −1) ⊆ Kw , i2 < m and Kw(i2 +1) ⊆ Ku for some u ∈ W|w| with u ≠ w and u ≠ v. For (c) and (e). We have {w(i1 ) , . . . , w(i2 ) } ∈ 𝒞 (Kv ) with different entries. By Lemma 3.4, we have m

i2

i=1

i=i1

∑ gμ (w(i) ) ≥ ∑ gμ (w(i) ) ≥ μ|v| = μ|w| ≥ RHS. For (a), (b) and (d). By reflection, we replace w(i1 ) , . . . , w(i2 ) by v(i1 ) , . . . , v(i2 ) that are symmetric about Kw ∩ Kv , see Figure 1, then Kv(i1 ) , . . . , Kv(i2 ) ⊆ Kw and gμ (v(i) ) = gμ (w(i) ) for any i = i1 , . . . , i2 .

Figure 1: The reflection.

Repeat the above consideration to the chain {w(1) , . . . , w(i1 −1) , v(i1 ) , . . . , v(i2 ) , w(i2 +1) , . . . , w(m) } finitely many times, exact one of the following cases occurs. (i) We obtain a chain denoted by {v(1) , . . . , v(m) } ∈ 𝒞 (x, y) with Kv(i) ⊆ Kw for any i = m (i) (i) 1, . . . , m and ∑m i=1 gμ (v ) = ∑i=1 gμ (w ). (ii) Either (c) or (e) holds.

286 | M. Yang For (i), we have m

m

i=1

i=1

∑ gμ (w(i) ) = ∑ gμ (v(i) ) ≥ RHS. For (ii), we have m

∑ gμ (w(i) ) ≥ μ|w| ≥ RHS. i=1

Hence, we have the LHS ≥ the RHS. Proof of Theorem 2.1. The case μ ∈ (0, 1/2) has already been considered in Lemma 3.1. We may assume that μ ∈ [1/2, 1). We only need to show that for arbitrary fixed x, y ∈ K with x ≠ y, we have dμ (x, y) > 0. Since π −1 (x) contains at most two elements in W∞ for any x ∈ K, there exist unique w ∈ W∗ and j1 , j2 ∈ W with j1 ≠ j2 such that x ∈ Kwj1 \Kwj2 and y ∈ Kwj2 \Kwj1 . If Kwj1 ∩ Kwj2 = 0, then by Corollary 3.6, we have dμ (x, y) ≥ dμ (Kwj1 , Kwj2 ) ≥ μ|w|+1 > 0. If Kwj1 ∩ Kwj2 ≠ 0, then without lose of generality, we may assume that j1 = 0 and j2 = 1, then Kw0 ∩ Kw1 = π({w01v ∼ w11v, w02v ∼ w12v : v ∈ {0, 5}∞ }), there exist k (1) , k (2) ∈ W, v(1) = v1(1) v2(1) . . . , v(2) = v1(2) v2(2) . . . ∈ W∞ such that w0k (1) v(1) ∈ π −1 (x), w1k (2) v(2) ∈ π −1 (y). If k (1) ≠ k (2) or k (1) ∈ {0, 3, 4, 5} or k (2) ∈ {0, 3, 4, 5}, then for any {w(1) , . . . , w(m) } ∈ 𝒞 (x, y), either there exists i = 1, . . . , m such that Kw(i) contains a (|w| + 2)-cell or there exists some sub-chain passing through a (|w| + 2)-cell with different entries, hence m

∑ gμ (w(i) ) ≥ μ|w|+2 , i=1

hence dμ (x, y) ≥ μ|w|+2 > 0. Hence we may assume that k (1) = k (2) ∈ {1, 2}, without lose of generality, we may assume that k (1) = k (2) = 1.

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287

Since x ∈ Kw0 \Kw1 and y ∈ Kw1 \Kw0 , we have v(1) , v(2) ∈ ̸ {0, 5}∞ . Let n(1) = min{n : vn(1) ∈ ̸ {0, 5}}, n(2) = min{n : vn(2) ∈ ̸ {0, 5}}. For any {w(1) , . . . , w(m) } ∈ 𝒞 (x, y), for any j = 1, 2, either there exists i = 1, . . . , m such that Kw(i) contains a (|w| + 2 + n(j) )-cell or there exists some subchain passing through a (|w| + 2 + n(j) )-cell with different entries, hence m

(2)

(1)

∑ gμ (w(i) ) ≥ μ|w|+2+n + μ|w|+2+n , i=1

hence (1)

(2)

dμ (x, y) ≥ μ|w|+2+n + μ|w|+2+n

> 0.

Therefore, we have dμ (x, y) > 0 for any x, y ∈ K with x ≠ y. For any j ∈ W, for any x, y ∈ K, we have dμ (fj (x), fj (y)) m

= inf{∑ gμ (w(i) ) : {w(1) , . . . , w(m) } ∈ 𝒞 (fj (x), fj (y)), Kw(i) ⊆ Kj for any i = 1, . . . , m} i=1 m

= inf{∑ gμ (jw(i) ) : {jw(1) , . . . , jw(m) } ∈ 𝒞 (fj (x), fj (y)), Kjw(i) ⊆ Kj for any i = 1, . . . , m} i=1

m

= μ inf{∑ gμ (w(i) ) : {w(1) , . . . , w(m) } ∈ 𝒞 (x, y)} i=1

= μdμ (x, y), where we use Lemma 3.7 in the first equality, we use the fact that {jw(1) , . . . , jw(m) } ∈ 𝒞 (fj (x), fj (y)) if and only if {w(1) , . . . , w(m) } ∈ 𝒞 (x, y) in the third equality. For any x, y ∈ K, since {0} ∈ 𝒞 (x, y), we have dμ (x, y) ≤ gμ (0) = 1, hence diamμ (K) ≤ 1. For any x ∈ K0 , y ∈ K3 , for any {w(1) , . . . , w(m) } ∈ 𝒞 (x, y). Observe that

288 | M. Yang (a) Either there exists i = 1, . . . , m such that Kw(i) {w(i1 ) , . . . , w(i2 ) } ∈ 𝒞 (K1 ) with different entries. (b) Either there exists i = 1, . . . , m such that Kw(i) {w(i1 ) , . . . , w(i2 ) } ∈ 𝒞 (K2 ) with different entries. (c) Either there exists i = 1, . . . , m such that Kw(i) {w(i1 ) , . . . , w(i2 ) } ∈ 𝒞 (K4 ) with different entries. (d) Either there exists i = 1, . . . , m such that Kw(i) {w(i1 ) , . . . , w(i2 ) } ∈ 𝒞 (K5 ) with different entries.

⊇ K1 or there exist i1 ≤ i2 such that ⊇ K2 or there exist i1 ≤ i2 such that ⊇ K4 or there exist i1 ≤ i2 such that ⊇ K5 or there exist i1 ≤ i2 such that

Then either (a) and (b) hold or (c) and (d) hold. In both cases, we have m

∑ gμ (w(i) ) ≥ μ + μ = 2μ ≥ 1, i=1

hence dμ (x, y) ≥ 1, hence diamμ (K) = 1. By the contraction property of f0 , . . . , f5 , we have diamμ (Kw ) = μ|w| . By Lemma 3.3 and Corollary 3.6, we have the conditions in [12, Theorem 1.5.7] hold, hence the Hausdorff dimension of (K, dμ ) is α = − log 6/ log μ, the normalized Hausdorff measure ν of dimension α exists and is given by a self-similar measure.

4 Proof of Proposition 2.3 Recall that a metric space (K, d) satisfies the chain condition or the θ-chain condition if there exists a positive constant C such that for any x, y ∈ K, for any n ≥ 1, there exists a sequence {x0 , x1 , . . . , xn } in K with x0 = x and xn = y such that d(xi , xi+1 ) ≤ C

d(x, y) n

for any i = 0, . . . , n − 1,

d(xi , xi+1 ) ≤ C

d(x, y) nθ

for any i = 0, . . . , n − 1.

or (4.1)

For any n ≥ 1, let Gn be the graph with vertex set Wn and edge set given by {(w(1) , w(2) ) : w(1) , w(2) ∈ Wn , w(1) ≠ w(2) , Kw(1) ∩ Kw(2) ≠ 0}. For any w(1) , w(2) ∈ Wn , we denote w(1) ∼n w(2) if (w(1) , w(2) ) is an edge in Gn . Let dn be the graph metric on Gn , that is, dn (w(1) , w(2) ) is the minimum of the lengths of all paths joining w(1) and w(2) . Denote the diameter of Gn as diam(Gn ) := sup{dn (w(1) , w(2) ) : w(1) , w(2) ∈ Wn }.

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Lemma 4.1. There exists some positive constant C such that for any n ≥ 1, we have 1 (n ⋅ 2n ) ≤ diam(Gn ) ≤ C(n ⋅ 2n ). C Proof. For arbitrary fixed n ≥ 1. Obviously, Gn is a planer graph. Denote the outer circumference path Outn as in [3, Definition 5.1]. By [3, Proposition 5.2], we have |Outn | = 3n ⋅ 2n . For any w ∈ Wn , we have dn (w, Outn ) := inf{dn (w, v) : v ∈ Outn } ≲ n ⋅ 2n . For any w(1) , w(2) ∈ Wn , we have dn (w(1) , w(2) ) ≤ dn (w(1) , Outn ) + dn (w(2) , Outn ) + |Outn | ≲ n ⋅ 2n , hence diam(Gn ) ≲ n ⋅ 2n . By the graph structure of Gn , there exists some positive constant c such that for any n ≥ 1 diam(Gn+1 ) ≥ 2diam(Gn ) + c2n . By recursion, we have diam(Gn ) ≳ n ⋅ 2n . Therefore, we have diam(Gn ) ≍ n ⋅ 2n . Remark 4.2. It was conjectured in [3, Conjecture 5.4] an explicit formula for diam(Gn ). Proof of Proposition 2.3. Suppose that dμ satisfies the chain condition. Let C be the constant in the definition of the chain condition, take k1 ≥ 1 satisfying μ−k1 > C, let c be the constant in Lemma 4.1. For any k > 2cμ−k1 . Take w, v ∈ Wk such that dk (w, v) = diam(Gk ), take x ∈ Kw , y ∈ Kv , then there exists a sequence {x0 , . . . , x⌈μ−(k+k1 ) ⌉ } in K with x0 = x and x⌈μ−(k+k1 ) ⌉ = y such that dμ (xi , xi+1 ) ≤ C −(k+k1 )

Take w(0) , . . . , w(⌈μ i = 0, . . . , ⌈μ−(k+k1 ) ⌉.

⌉)

dμ (x, y) ⌈μ−(k+k1 ) ⌉

≤

C < μk . μ−(k+k1 ) −(k+k1 ) ⌉)

∈ Wk with w(0) = w, w(⌈μ

= v and xi ∈ Kw(i) for any

290 | M. Yang For any i = 0, . . . , ⌈μ−(k+k1 ) ⌉−1, we have Kw(i) ∩Kw(i+1) ≠ 0, otherwise, by Corollary 3.6, we have dμ (xi , xi+1 ) ≥ dμ (Kw(i) , Kw(i+1) ) ≥ μk , contradiction! Hence for any i = 0, . . . , ⌈μ−(k+k1 ) ⌉ − 1, either w(i) = w(i+1) or w(i) ∼k w(i+1) . Hence k+k1

diam(Gk ) = dk (w, v) = dk (w(0) , w(2

)

) ≤ ⌈μ−(k+k1 ) ⌉.

By Lemma 4.1, we have diam(Gk ) ≥ c1 (k ⋅ 2k ), hence 1 (k ⋅ 2k ) ≤ ⌈μ−(k+k1 ) ⌉ ≤ 2μ−(k+k1 ) , c that is, k≤

2c −k1 μ ≤ 2cμ−k1 , (2μ)k

contradiction! We only need to show that equation (4.1) holds for a sequence {nk }k≥1 with sup k≥1

nk+1 < +∞ nk

for any x, y ∈ K with dμ (x, y) < 1/2. Let c be the constant in Lemma 4.1. Let nk = 2([ck2k ] + 1). It is obvious that supk≥1 nk+1 /nk < +∞. For any x, y ∈ K with x ≠ y and dμ (x, y) < 1/2, there exists some integer N ≥ 1 such that 1 1 ≤ dμ (x, y) < N . 2N+1 2 There exist w, v ∈ WN such that x ∈ Kw , y ∈ Kv , then Kw ∩ Kv ≠ 0, otherwise, by Corollary 3.6, we have dμ (x, y) ≥ dμ (Kw , Kv ) ≥ μN ≥

1 , 2N

contradiction! Since diam(Gk ) ≤ ck2k by Lemma 4.1, there exist w(0) , . . . , w(nk ) ∈ Wk with x ∈ Kww(0) , y ∈ Kvw(nk ) satisfying Kww(nk /2) ∩ Kvw(nk /2+1) ≠ 0, Kww(i) ∩ Kww(i+1) ≠ 0 Kvw(i) ∩ Kvw(i+1) ≠ 0

n for any i = 0, . . . , k − 1, 2 n for any i = k + 1, . . . , nk − 1. 2

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291

Take arbitrary xi ∈ Kww(i) for any i = 1, . . . , nk /2 and xi ∈ Kvw(i) for any i = nk /2 + 1, . . . , nk , then 2 dμ (xi , xi+1 ) ≤ 2μN+k ≤ μk dμ (x, y). μ Take a constant C satisfying 21+2θ cθ k θ ≤ Cμ then dμ (xi , xi+1 ) ≤

1 (2θ μ)k

for any k ≥ 1,

dμ (x, y) 2 k μ dμ (x, y) ≤ C μ nθ k

for any i = 0, . . . , nk − 1.

5 Proof of Theorem 2.4 The following result states that an n-cell is comparable to a ball with radius μn with respect to the intrinsic metric dμ . Proposition 5.1. For any n ≥ 0, for any w ∈ Wn , we have the following results: (1) For any x ∈ Kw , we have Kw ⊆ Bμ (x, 2μn ). (2) There exists x ∈ Kw such that Bμ (x, μn+2 ) ⊆ Kw . Proof. (1) Since diamμ (Kw ) = μn , for any x ∈ Kw , we have Kw ⊆ Bμ (x, 2diamμ (Kw )) = Bμ (x, 2μn ). (2) Take x ∈ Kw22 ⊆ int(Kw ), see [3, Figure 2], for any v ∈ Wn+2 with Kw22 ∩ Kv = 0, by Corollary 3.6, we have dμ (Kw22 , Kv ) ≥ μn+2 . In particular, for any y ∈ ̸ Kw , we have dμ (x, y) ≥ μn+2 , hence Bμ (x, μn+2 ) ⊆ Kw . For any n ≥ 0, let X (n) be the simple random walk on Hn , let τB be the first exit time of X (n) from a subset B of Vn . We use the knight move technique developed by Barlow and Bass [1], but first we need some preparations. First, we have a corner move as follows. Lemma 5.2. For any n ≥ 1, for any w(1) , w(2) ∈ Wn with w(1) ≠ w(2) and Kw(1) ∩ Kw(2) ≠ 0. Each of 𝜕Kw(1) , 𝜕Kw(2) consists of six disjoint parts, 𝜕Kw(1) ∪ 𝜕Kw(2) consists of ten disjoint parts, 𝜕Kw(1) ∩𝜕Kw(2) consists of two disjoint parts. Denote by L0 one part of 𝜕Kw(1) ∩𝜕Kw(2) , denote by L1 , . . . , L8 the eight parts of (𝜕Kw(1) \𝜕Kw(2) )∪(𝜕Kw(2) \𝜕Kw(1) ), where L1 , L8 are two parts adjacent to L0 . Let B = (Kw(1) ∪Kw(2) )\(L1 ∪⋅ ⋅ ⋅∪L8 ), see Figure 2. Then for any k ≥ n, for any x ∈ L0 ∩ Vk , we have ℙx [Xτ(k) ∈ L1 ] ≥ B

1 . 8

292 | M. Yang

Figure 2: Corner move.

Second, we have knight move I as follows. Lemma 5.3. For any n ≥ 0, for any w ∈ Wn , 𝜕Kw ∩(𝜕Kw0 ∪⋅ ⋅ ⋅∪𝜕Kw5 ) consists of 12 disjoint parts, 𝜕Kw0 ∩ 𝜕Kw1 consists of two disjoint parts. Denote by L0 one part of 𝜕Kw0 ∩ 𝜕Kw1 which is not adjacent to 𝜕Kw , denote by L1 , . . . , L12 the 12 parts of 𝜕Kw ∩(𝜕Kw0 ∪⋅ ⋅ ⋅∪𝜕Kw5 ), where L1 , L12 are two parts adjacent to 𝜕Kw0 ∩ 𝜕Kw1 . Let B = int(Kw ), see Figure 3. Then for any k ≥ n, for any x ∈ L0 ∩ Vk , we have ℙx [Xτ(k) ∈ L1 ] ≥ B

1 . 12

Figure 3: Knight move I.

Third, we have knight move II as follows. Lemma 5.4. For any n ≥ 1, for any w(1) , w(2) ∈ Wn with w(1) ≠ w(2) and Kw(1) ∩ Kw(2) ≠ 0, there exist i(1) , i(2) , j(1) , j(2) ∈ W with i(1) ≠ i(2) and j(1) ≠ j(2) such that Kw(1) i(1) ∩ Kw(2) j(1) ≠ 0, Kw(2) j(1) ∩ Kw(2) j(2) ≠ 0, Kw(1) i(2) ∩ Kw(2) j(2) ≠ 0 and Kw(1) i(1) ∩ Kw(1) i(2) ≠ 0. Let v(1) = w(1) i(1) ,

v(2) = w(2) j(1) , v(3) = w(2) j(2) and v(4) = w(1) i(2) . Then ⋃4k=1 (𝜕Kv(k) \(⋃l=k̸ 𝜕Kv(l) )) consists of eight disjoint parts, 𝜕Kv(1) ∩ 𝜕Kv(2) consists of two disjoint parts. Denote by L0 one part of 𝜕Kv(1) ∩ 𝜕Kv(2) which is not adjacent to ⋃4k=1 (𝜕Kv(k) \(⋃l=k̸ 𝜕Kv(l) )), denote by L1 , . . . , L8 the eight parts of ⋃4k=1 (𝜕Kv(k) \(⋃l=k̸ 𝜕Kv(l) )),

Strichartz hexacarpet | 293

where L1 , L8 are two parts adjacent to 𝜕Kv(1) ∩ 𝜕Kv(2) . Let B = (⋃4k=1 Kv(k) )\(⋃8k=1 Lk ), see Figure 4. Then for any k ≥ n, for any x ∈ L0 ∩ Vk , we have ℙx [Xτ(k) ∈ L1 ] ≥ B

1 . 8

Figure 4: Knight move II.

Proof of Lemmas 5.2, 5.3, and 5.4. Denote pi = ℙx [Xτ(k) ∈ Li ]. B Using the reflection principle several times, we have that p1 is the largest among all the pi ’s, then we have the desired results. Proposition 5.5. For any n ≥ 0, for any w ∈ Wn . For any k ≥ n, for any x, y ∈ Kw53 ∩ Vk , for any path γ in Vk from y to 𝜕Kw ∩ Vk , see Figure 5 and [3, Figure 2], we have ℙx [X (k) hits γ before τint(Kw ) ] ≥

1 . 1241

Proof. Starting from x ∈ Kw53 ∩ Vk , X (k) hits the inner thick hexagon in Figure 5 almost surely. We only need to construct a closed curve starting from the inner thick hexagon and surrounding the inner thick hexagon. By symmetry, we only need to consider the cases x ∈ L1 ∩ Vk and x ∈ L2 ∩ Vk . If x ∈ L1 ∩Vk , then using 25 times corner moves, 7 times knight move I and 7 times knight move II, we obtain a closed curve surrounding the inner thick hexagon. If x ∈ L2 ∩ Vk , then using one more time knight move II and one more time corner move, we return to the case x ∈ L1 ∩ Vk . Therefore, using at most 41 times moves, we obtain a closed curve surrounding the inner thick hexagon. Combining Lemmas 5.2, 5.3, and 5.4, we obtain the desired result. Proof of Theorem 2.4. By Proposition 5.1, we only need to prove the following result.

294 | M. Yang

Figure 5: X (n) hits γ before τ.

There exists some positive constant C such that for any n ≥ 0, for any w ∈ Wn , for any k ≥ n, for any nonnegative harmonic function u in Vk ∩ int(Kw ), we have max u ≤ C min u. Vk ∩Kw53

Vk ∩Kw53

For any subset A of 𝜕Kw , denote hk (x, A) = ℙx [Xτ(k) int(K

w)

∈ A].

We only need to show that there exists some universal positive constant δ such that hk (x, A) ≥ δhk (y, A) (k) Indeed, let Ml = hk (Xl∧τ

For any η ∈ (0, 1), let

int(Kw )

for any x, y ∈ Vk ∩ Kw53 .

, A), then Ml is a martingale.

T = inf{l ≥ 0 : Ml < ηhk (y, A)} ∧ τint(Kw ) . Then hk (y, A) = 𝔼y hk (XT(k) , A) = 𝔼y [hk (XT(k) , A)1T=τint(K ) ] + 𝔼y [hk (XT(k) , A)1T 0, hence there exists some path γ = {γ(0), . . . , γ(l0 )} from y to 𝜕Kw such that hk (γ(l), A) ≥ ηhk (y, A) for any l = 0, . . . , l0 . Let S = inf{l ≥ 0 : Xl(k) ∈ γ}, then by Proposition 5.5, we have ℙx [S < τint(Kw ) ] ≥ 12−41 , hence hk (x, A) (k) = 𝔼x hk (XS∧τ

int(Kw )

(k) ≥ 𝔼x [hk (XS∧τ

, A)

int(Kw )

, A)1S 0 [18]. For example, in ℝn we have pt (x, y) =

|x − y|2 1 exp(− ). 4t (4πt)n/2

(1.1)

The heat kernel satisfies the semigroup identity pt+s (x, y) = ∫ pt (x, z)ps (z, y) dμ(z) M

and, hence, can be used as a transition density for constructing a diffusion process on M (see [17]). This diffusion process is called a Brownian motion on M. If M = ℝn then one obtains in this way the classical Brownian motion in ℝn with the time scaled by the factor 2.

2 Parabolicity and recurrence A function u ∈ C 2 (M) is called superharmonic if Δu ≤ 0. A manifold M is called parabolic if any positive superharmonic function on M is constant, and nonparabolic otherwise. For any compact set K ⊂ M, define its capacity by cap(K) =

inf

φ∈C0∞ (M), φ|K ≡1

∫ |∇φ|2 dμ. M

The following theorem gives equivalent characterizations of the parabolicity Theorem 2.1 ([16, Theorem 5.1]). The following properties are equivalent:

Analysis on manifolds and volume growth

– – – –

| 301

M is parabolic. Any bounded superharmonic function on M is constant. There exists no positive fundamental solution of −Δ on M. For all/some x, y ∈ M we have ∞

∫ pt (x, y) dt = ∞.

(2.1)

1

– –

For any compact set K ⊂ M, we have cap(K) = 0. Brownian motion on M is recurrent. The Green function of Δ is defined by ∞

g(x, y) = ∫ pt (x, y) dt. 0

Condition (2.1) is equivalent to the fact that g(x, y) ≡ ∞. If M is nonparabolic then g(x, y) < ∞ for all x ≠ y and, moreover, g(x, y) is the minimal positive fundamental solution of −Δ. A celebrated theorem of Polya (1921) says that Brownian motion in ℝn is recurrent if and only if n ≤ 2. Indeed, one can see from the explicit formula (1.1) for the heat kernel that condition (2.1) holds if and only if n ≤ 2. Surprisingly enough, there exist rather good sufficient conditions for the recurrence of Brownian motion in terms of the volume function. Let us fix a reference point x0 and set V(r) = V(x0 , r). Theorem 2.2 (Cheng–Yau, [4]). If there exists a sequence rk → ∞ such that, for some C > 0 and all k, V(rk ) ≤ Crk2 ,

(2.2)

then M is parabolic. Theorem 2.3 ([8, 29, 37]). If ∞

∫

r dr =∞ V(r)

(2.3)

then M is parabolic. One can show that (2.2) implies (2.3) so that Theorem 2.2 follows from Theorem 2.3.

302 | A. Grigor’yan Condition (2.3) is sharp: if f (r) is a smooth convex function such that f 󸀠 (r) > 0 and ∞

∫

r dr < ∞, f (r)

then there is a nonparabolic manifold such that V(r) = f (r) for large r. On the other hand, condition (2.3) is not necessary for parabolicity: there exist parabolic manifolds with arbitrarily large volume function V(r) as it follows from [16, Proposition 3.1].

3 Stochastic completeness A manifold M is called stochastically complete if for all x ∈ M and t > 0, ∫ pt (x, y) dμ(y) = 1. M

Here are some equivalent characterizations of the stochastic completeness. Theorem 3.1 ([16, Theorem 6.2]). The following conditions are equivalent: – M is stochastically complete. – For some/any λ > 0, any bounded solution u to Δu − λu = 0 on M is identically zero. – For some/any T ∈ (0, ∞], the Cauchy problem {

–

𝜕u 𝜕t

= Δu in M × (0, T),

u|t=0 = 0

(3.1)

has the only bounded solution u ≡ 0. The lifetime of Brownian motion on M is equal to ∞ almost surely. The following theorem provides a volume test for stochastic completeness.

Theorem 3.2 ([9]). If ∞

∫

r dr =∞ log V(r)

(3.2)

then M is stochastically complete. In particular, M is stochastically complete provided V(r) ≤ exp(Cr 2 )

(3.3)

V(rk ) ≤ exp(Crk2 )

(3.4)

or even if

for a sequence rk → ∞. That (3.3) implies the stochastic completeness was also proved by different methods in [6, 30, 36].

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| 303

Condition (3.2) is sharp: if f (r) is a smooth convex function such that f 󸀠 (r) > 0 and ∞

∫

r dr 0) implies the L1 -Liouville property for superharmonic functions. It may be interesting to investigate further relations between different types of Liouville properties. However, the main open questions in this area are these. Open Questions. Find optimal conditions in geometric terms for (a) L1 -Liouville property for harmonic functions; (b) L∞ -Liouville property for harmonic functions.

5 Bounded solutions of Schrödinger equations Let Q(x) be a nonnegative continuous function on M, Q ≢ 0. Consider the equation Δu − Qu = 0

(5.1)

and ask if (5.1) has a nontrivial (≡nonzero) bounded solution, that is, if L∞ -Liouville property holds for (5.1). In the case Q = const, the L∞ -Liouville property for (5.1) is equivalent to the stochastic completeness of M. If Q is compactly supported then one can show that the L∞ -Liouville property for (5.1) is equivalent to the parabolicity of M. In general, one can prove that (5.1) has a nontrivial bounded solution if and only if it has a positive solution. Set |x| = d(x, x0 ) and denote r/2

q(r) = inf Q(x) |x|=r

and

F(r) = ∫ √q(t) dt. 0

Theorem 5.1 ([11]). If there is a sequence rk → ∞ such that for some C > 0 and all k, V(rk ) ≤ Crk2 exp(CF(rk )2 ),

(5.2)

then (5.1) has no bounded solution except for u ≡ 0. Example 5.2. Let Q ≡ 1. Then we have q ≡ 1, F(r) = r/2, and (5.2) becomes V(rk ) ≤ exp(Crk2 ), which coincides with condition (3.4) for the stochastic completeness.

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Example 5.3. Let Q have a compact support. Since q(r) = 0 for large enough r, we obtain that F(r) = const for large r, and (5.2) becomes V(rl ) ≤ Crk2 , which coincides with the sufficient condition (2.2) for the parabolicity. Example 5.4. Assume that, for all large |x| and some c > 0, Q(x) ≥

c . |x|2 log |x|

Then r/2

F(r) ≥ ∫ 2

c

t √log t

dt ≃ √log r,

so that (5.2) is satisfied provided V(r) ≤ Cr N for some C, N > 0 and all large r. Hence, in this case (5.1) has no bounded solution except for zero. For example, this is the case for M = ℝn . On the other hand, if in ℝn , Q(x) ≤

C

|x|2 log1+ε

|x|

n

then (5.1) has a positive bounded solution in ℝ .

6 Semilinear PDEs Consider on M the inequality Δu + uσ ≤ 0

(6.1)

and ask if it has a nonnegative solution u on M except for u ≡ 0. Here σ > 1 is a given parameter. Note that any nonnegative solution of (6.1) is superharmonic. Hence, if M is parabolic then u must be identical zero. In particular, this is the case if V(r) ≤ Cr 2 . Otherwise, (6.1) may have positive solutions. For example, in ℝn with n > 2 inn equality (6.1) has a positive solution if and only if σ > n−2 (cf. [33]). Theorem 6.1 ([25]). Assume that, for all large r, V(r) ≤ Cr p logq r, where p=

2σ σ−1

and

q=

1 . σ−1

Then any nonnegative solution of (6.1) is identically zero.

(6.2)

(6.3)

306 | A. Grigor’yan 2σ or p = The values of the exponents p and q in (6.3) are sharp: if either p > σ−1 1 and q > σ−1 then there is a manifold satisfying (6.2) where inequality (6.1) has a positive solution. Theorem 6.1 can be equivalently reformulated as follows: if, for some α > 2, 2σ σ−1

V(r) ≤ Cr α log

α−2 2

r,

(6.4)

α then, for any σ ≤ α−2 , any nonnegative solution of (6.1) is identically zero. In this form it contains the aforementioned result for ℝn as in ℝn (6.4) is satisfied with α = n.

Conjecture ([26]). If ∞

∫

r 2σ−1 dr =∞ V(r)σ−1

(6.5)

then any nonnegative solution of (6.1) is identically zero. α . In particular, the function in (6.4) satisfies (6.5) with σ = α−2 σ Similar results for a more general inequality Δu + Qu ≤ 0 with Q(x) ≥ 0 were obtained in [34]. In view of the results of Section 5, it may be interesting to investigate the question of the existence of positive solutions for a semilinear equation Δu − Quσ = 0. Analogous problems for semilinear heat equation were addressed in [35].

7 Escape rate Let {Xt }t≥0 be a Brownian motion on M. An increasing positive function R(t) of t ∈ ℝ+ is called an upper rate function for Brownian motion if we have |Xt | < R(t) for all t large enough with probability 1. Similarly, an increasing positive function r(t) is called a lower rate function if we have |Xt | > r(t) for all t large enough with probability 1. So for large enough t, Xt is contained in the annulus B(x0 , R(t)) \ B(x0 , r(t)) almost surely, as in Figure 1. Note that an upper rate function may exist only on stochastically complete manifolds, and a lower rate function may exist only on nonparabolic manifolds. For example, in ℝn the function R(t) = √(4 + ε)t log log t is an upper rate function for any ε > 0 as it follows from Khinchin’s law of iterated logarithm. By the theorem of Dvoretzky–Erdös, if r(t)/√t is decreasing then r(t) is a lower rate function in ℝn , n > 2, if and only if ∞

∫(

n−2

r(t) ) √t

dt 0. Then the following function is an upper rate function: R(t) = √2Nt log t.

(7.3)

Under assumption (7.2), the upper rate function (7.3) is almost optimal (cf. [22]). A similar result holds for simple random walks on graphs: it was proved by Hardy and Littlewood in 1914 for ℤ, and in [3] for arbitrary graphs. To state the next result, we need the notion of an isoperimetric inequality. We say that a manifold M satisfies the isoperimetric inequality if there exists c > 0 such that for any bounded domain Ω ⊂ M with smooth boundary, σ(𝜕Ω) ≥ cμ(Ω)

n−1 n

,

(7.4)

where n = dim M and σ is the (n − 1)-dimensional Riemannian measure on the hypersurface 𝜕Ω. For example, (7.4) holds in ℝn and, more generally, on any Cartan– Hadamard manifold that is a complete noncompact simply connected manifold with nonpositive sectional curvature (cf. [27]). It is easy to see that (7.4) implies that V(x, r) ≥ c󸀠 r n for some c󸀠 > 0 and all r > 0.

308 | A. Grigor’yan Theorem 7.2 ([20]). Assume that M satisfies the isoperimetric inequality (7.4) and that ∞

∫

r dr = ∞. log V(r)

(7.5)

Define a function φ(t) as follows: φ(t)

t= ∫ r0

r dr . log V(r)

Then R(t) = φ(Ct) is an upper rate function. Example 7.3. If V(r) = Cr N then t≃

φ2 (t) , log φ(t)

whence R(t) ≃ φ(t) ≃ √t log t which matches (7.3). Example 7.4. If V(r) = exp(r α ) where 0 < α < 2 then t ≃ φ(t)2−α 1

whence R(t) = Ct 2−α . Example 7.5. If V(r) = exp(r 2 ) then t ≃ log φ(t) whence R(t) = exp(Ct). The next result holds on any complete Riemannian manifold without assumption about an isoperimetric inequality. Theorem 7.6 ([28]). On any complete manifold M satisfying (7.5), define function φ(t) as follows: φ(t)

t= ∫ r0

r dr . log V(r) + log log r

Then R(t) = Cφ(Ct) is an upper rate function. Example 7.7. Let V(r) ≤ C log r. Then t≃

φ2 (t) , log log φ(t)

and we obtain an upper rate function R(t) = C √t log log t.

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To state the next results about the lower rate function, we need the notion of a Faber–Krahn inequality. For any precompact domain Ω ⊂ M, set λmin (Ω) =

inf ∞

∫ |∇f |2 dμ

f ∈C0 (M)\{0}

∫ f 2 dμ

.

In fact, λmin (Ω) is the minimal eigenvalue of the Laplace operator in Ω with the Dirichlet boundary value on 𝜕Ω. By the theorem of Faber and Krahn, for Ω ⊂ ℝn we have λmin (Ω) ≥ cn μ(Ω)−2/n

(7.6)

where cn > 0, and the equality is attained if Ω is a ball. We say that a manifold M satisfies a relative Faber–KrahnFaber-Krahn inequality inequality if there exist c, ν > 0 such that, for any ball B(x, r) and any open set Ω ⊂ B(x, r), λmin (Ω) ≥

ν

c μ(B(x, r)) ( ) . μ(Ω) r2

(7.7)

As it follows from (7.6), in ℝn the relative Faber–Krahn inequality (7.7) holds with ν = 2/n and c = cn vn−2/n where vn is the volume of the unit ball in ℝn . More generally, the relative Faber–Krahn inequality holds on any complete manifold with nonnegative Ricci curvature (see [12]). Theorem 7.8 ([16]). If M satisfies the relative Faber–Krahn inequality then M is nonparabolic if and only if ∞

∫

r dr 2. We obtain from (7.9) γ(t) ≃ t N−2 , and (7.10) amounts to ∞

∫

r N−2 (t) dt < ∞, t N/2

which coincides with the Dvoretzky–Erdös condition (7.1).

8 Heat kernel lower bounds Here we show some results on pointwise lower bounds of the heat kernel that use only the volume function. Recall that x0 is a fixed point of M and V(r) = V(x0 , r). Theorem 8.1 ([5]). Assume that, for all r ≥ r0 > 0, V(r) ≤ Cr α ,

(8.1)

for some C, α > 0. Then, for all large enough t, pt (x0 , x0 ) ≥

1/4

V(√Kt log t)

,

(8.2)

where K = K(x0 , r0 , C, α) > 0. Consequently, for some c > 0, pt (x0 , x0 ) ≥

c . (t log t)α/2

(8.3)

If M has nonnegative Ricci curvature then, by the theorem of Li–Yau [32], the heat kernel satisfies on the diagonal the following two-sided estimate: pt (x, x) ≃

1 V(x, √t)

(8.4)

for all x ∈ M and t > 0. Hence, the lower bound (8.3) differs from the best possible estimate (8.4) by a log-factor. However, under the hypothesis (8.1) alone, the lower bound (8.2) is optimal and cannot be essentially improved (cf. [22]). Theorem 8.2 ([5]). Assume that the function V(r) is doubling, that is, V(2r) ≤ CV(r), and that, for all t > 0, pt (x0 , x0 ) ≤

C . V(√t)

pt (x0 , x0 ) ≥

c . V(√t)

Then, for all t > 0,

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Let Ω be an end of M, that is, an open connected proper subset of M such that Ω is noncompact but 𝜕Ω is compact. Moreover, assume that 𝜕Ω is a smooth hypersurface. Set also BΩ (x, r) = B(x, r) ∩ Ω

and VΩ (x, r) = μ(BΩ (x, r)).

We consider the closure Ω as a manifold with boundary 𝜕Ω and apply to this manifold the notion of parabolicity. We say that a function u ∈ C 2 (Ω) is superharmonic if Δu ≤ 0 in Ω and 𝜕u | ≥ 0 where ν is the exterior unit normal vector field on 𝜕Ω. 𝜕ν 𝜕Ω Then Ω is called parabolic if every positive superharmonic function in Ω is constant. Brownian motion in Ω can be constructed by using the heat kernel in Ω with the Neumann boundary condition, or, equivalently, from Brownian motion in M by imposing reflecting conditions on 𝜕Ω. Similarly one handles other notions used in Section 2 so that Theorems 2.1, 2.2, and 2.3 remain true also for Ω in place of M (see [8], [16, Section 5]). In the same way one extends to Ω the definition of stochastic completeness, which is equivalent to the fact that the lifetime of the reflected Brownian motion in Ω is equal to ∞. All the results of Section 3 remain true for Ω in place of M as well. The following theorem is new. Theorem 8.3. Let Ω be an end of M that satisfies the following two assumptions: – there exists x0 ∈ Ω, C > 0 and N > 2 such that for all large enough r, VΩ (x0 , r) ≤ Cr N –

(8.5)

Ω is nonparabolic.

Then, for any x ∈ M, there exist tx > 0 and cx > 0 such that pt (x, x) ≥

cx (t log t)N/2

for all t ≥ tx .

(8.6)

Conjecture. If (8.5) is satisfied with N ≤ 2 (and, hence, Ω is parabolic) then, for all x ∈ M and t ≥ tx , pt (x, x) ≥

cx tα

, whereas for N = 2 the value with some α > 0. It is expected that if N < 2 then α = 4−N 2 α can be taken arbitrarily close to 2 (cf. [24, Example 6.11]). Proof. Consider in Ω all functions v ∈ C 2 (Ω) ∩ C(Ω) that satisfy the following conditions:

312 | A. Grigor’yan – – –

v is harmonic in Ω, 0 ≤ v ≤ 1 in Ω, v = 0 on 𝜕Ω.

Denote by vΩ the maximal function that satisfies all these conditions – it exists as the supremum of all such v. This function is called a subharmonic potential of Ω (see [16, Definition 4.2]). In fact, vΩ (x) is equal to the probability of the event that Brownian motion started at x ∈ Ω never hits 𝜕Ω. For example, if Ω is the exterior of the unit ball in ℝn then Ω is nonparabolic provided n > 2 and vΩ (x) = 1 − |x|2−n (cf. [18, Exercise 8.15]). The set Ω is called massive if vΩ > 0 in Ω. The set Ω can be regarded as the exterior of a compact set 𝜕Ω on the manifold Ω. By [16, Corollary 4.6], Ω is massive provided Ω is nonparabolic, which is the case now. Hence, we have vΩ > 0 in Ω. Let pΩ t (x, y) denote the heat kernel in Ω with the Dirichlet boundary condition on 𝜕Ω. It follows from [23, Remark 2.1] that if M is stochastically complete then, for all x ∈ Ω, ∫ pΩ t (x, y) dμ(y) ↘ vΩ (x)

as t → ∞.

(8.7)

Ω

We apply this result with Ω in place of M considering Ω is an open subset of Ω. By Theorem 3.2 and (8.5), Ω is stochastically complete so that (8.7) is satisfied. The rest of the argument is a modification of that in [5] and [18, Theorem 16.5]. We have, for all t > 0, x, y ∈ Ω, and r > 0, Ω 2 pΩ 2t (x, x) = ∫ pt (x, y) dμ(y) ≥ Ω

≥

2 ∫ pΩ t (x, y) dμ(y) BΩ (x,r)

1 ( ∫ pΩ t (x, y) dμ(y)) VΩ (x, r)

2

BΩ (x,r)

=

1 (∫ pΩ t (x, y) dμ(y) − VΩ (x, r) Ω

2

∫

pΩ t (x, y) dμ(y)) .

(8.8)

Ω\BΩ (x,r)

Recall that ∫ pΩ t (x, y) dμ(y) ≥ vΩ (x).

Ω

If r is chosen so large that ∫ Ω\BΩ (x,r)

1 pΩ t (x, y) dμ(y) ≤ vΩ (x), 2

(8.9)

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| 313

then it follows from (8.8) that pΩ 2t (x, x) ≥

vΩ2 (x) . 4VΩ (x, r)

(8.10)

Let us specify r = r(t) that satisfies (8.9). Using pΩ t ≤ pt , we obtain pΩ t (x, y) dμ(y) ≤

∫ Ω\BΩ (x,r)

pt (x, y)e

∫

d2 (x,y) 8t

e−

d2 (x,y) 8t

Ω\BΩ (x,r)

≤ (∫ p2t (x, y)e

d2 (x,y) 4t

dμ(y))

M

×(

e−

∫

d2 (x,y) 4t

dμ(y)

1/2

1/2

dμ(y)) .

(8.11)

Ω\BΩ (x,r)

Next use the fact that, on any manifold M, the function E(x, t) := ∫ p2t (x, y)e

d2 (x,y) 4t

dμ(y)

M

is finite and monotone decreasing in t ([13], [18, Theorem 12.1 and Corollary 15.9]). Hence, assuming t ≥ 1, we obtain ∫ p2t (x, y)e

d2 (x,y) 4t

dμ(y) ≤ E(x, 1) =: E(x).

M

The integral in (8.11) can be estimated as follows: e−

∫

d2 (x,y) 4t

∞

dμ(y) = ∑

k=0

Ω\BΩ (x,r)

e−

∫

d2 (x,y) 4t

dμ(y)

BΩ (x,2k+1 r)\BΩ (x,2k r)

∞

≤ ∑ exp(− k=0

(2k r)2 )VΩ (x, 2k+1 r). 4t

The hypothesis (8.5) implies that, for R > 1, VΩ (x, R) ≤ Cx RN

(8.12)

with a constant Cx depending on x. Hence, we obtain, for r > 1 and t > 1, ∫ Ω\BΩ (x,r)

e−

d2 (x,y) 4t

∞

dμ(y) ≤ ∑ Cx exp(− k=0

4k r 2 N )(2k+1 r) . 4t

Assume now that r2 ≥ 1. 4t

(8.13)

314 | A. Grigor’yan Observing that exp(−

N/2

4k r 2 4k r 2 4k r 2 N )(2k+1 r) = 2N t N/2 exp(− )( ) 4t 4t t ≤ const t N/2 exp(−

4k r 2 ) 8t

and summing up the geometric series, we obtain e−

∫

d2 (x,y) 4t

N

dμ(y) ≤ Cx t 2 exp(−

Ω\BΩ (x,r)

r2 ). 8t

It follows that N

2 pΩ t (x, y) dμ(y) ≤ (E(x)Cx t exp(−

∫ Ω\BΩ (x,r)

1/2

r2 )) . 8t

Hence, to ensure (8.9) it suffices to have N

(E(x)Cx t 2 exp(−

r2 )) 8t

1/2

1 ≤ vΩ (x), 2

that is, 4E(x)Cx N r2 ≥ ln + ln t. 8t 2 vΩ (x)2 If t ≥ tx where tx is large enough, then this inequality is satisfied provided r2 = N ln t, 8t that is, for r = √8Nt ln t.

(8.14)

For this r, we clearly have also (8.13). Consequently, we obtain (8.9) and, hence, (8.10). Substituting (8.12) and (8.14) into (8.9), we obtain pΩ 2t (x, x) ≥

vΩ2 (x) , 4Cx (8Nt ln t)N/2

which yields pΩ t (x, x) ≥

cx (t log t)N/2

for all t ≥ tx .

(8.15)

Since pt (x, y) ≥ pΩ t (x, y), we obtain (8.6) for all x ∈ Ω. Finally, by means of a local Harnack inequality, (8.6) extends to all x ∈ M.

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9 Recurrence revisited For any α ∈ (0, 2), the operator (−Δ)α/2 is the generator of a jump process on M that is called the α-process. It is a natural generalization of the symmetric stable Levy process of index α in ℝd . By a general semigroup theory, the Green function g (α) (x, y) of (−Δ)α/2 is given by ∞

g

(α)

(x, y) = ∫ t α/2−1 pt (x, y) dt, 0

and the recurrence of the α-process is equivalent to g (α) ≡ ∞, that is, to ∞

∫ t α/2−1 pt (x, x) dt = ∞.

(9.1)

Theorem 9.1 ([16, Theorem 16.2]). If for all large enough r, V(r) ≤ Cr α ,

(9.2)

then the α-process is recurrent. Proof. Indeed, by Theorem 8.1 we have pt (x0 , x0 ) ≥

c

t α/2 logα/2 t

.

Substituting into (9.1), we see that the integral diverges.

10 Heat kernel upper bounds We say that a manifold M has bounded geometry if there exists ε > 0 such that all balls B(x, ε) are uniformly quasiisometric to a Euclidean ball Bε of radius ε, that is, there are a constant C and, for any x ∈ M, a diffeomorphism φx : B(x, ε) → Bε such that φx changes the Riemannian metric at most by the factor C. In particular, M has bounded geometry if its injectivity radius is positive and the Ricci curvature is bounded from below. Theorem 10.1 ([2]). Let M be a manifold of bounded geometry. Assume that, for all x ∈ M and r ≥ r0 > 0, V(x, r) ≥ cr N ,

(10.1)

where c > 0. Then, for all x ∈ M and large enough t, N

pt (x, x) ≤ Ct − N+1 .

(10.2)

316 | A. Grigor’yan For any N ≥ 1, there exists an example of a manifold with V(x, r) ≃ r N and N

pt (x, x) ≃ t − N+1 , for all x ∈ M and t ≥ 1. Indeed, take any fractal graph where the volume function is of the order r α and the on-diagonal decay of the heat kernel is of the order t −α/β . It is known that such a graph exists for any couple α, β satisfying 2≤β≤α+1 (see [1]). Choose α = N and β = N + 1 and then inflate the graph into a manifold. One of such graphs, the Vicsek tree, is shown in Figure 2.

Figure 2: Vicsek tree.

For this fractal, we have α=

log 5 log 3

and

β=α+1=

log 15 . log 3

It is possible to prove that on any manifold of bounded geometry there exists c > 0 such that V(x, r) ≥ cr, for all x ∈ M and large enough r, that is, (10.1) holds with N = 1 (this follows, for example, from [14, Theorem 2.1]). Hence, on any manifold of bounded geometry we have pt (x, x) ≤

C , √t

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| 317

for all x ∈ M and large enough t. Note for comparison that, for a cylinder M = K × ℝ where K is a compact manifold, we have V(x, r) ≃ r

and pt (x, x) ≃ t −1/2

for all x ∈ M and large enough r, t.

11 Biparabolic manifolds A function u ∈ C 4 (M) is called bisuperharmonic if Δu ≤ 0 and Δ2 u ≥ 0. For example, let M be nonparabolic and consider the Green operator ∞

Gf = ∫ g(x, y)f (y) dμ(y), 0

where g(x, y) the Green function. If f is nonnegative and superharmonic then the function u = Gf is bisuperharmonic provided it is finite. Here is another example of bisuperharmonic functions in a precompact domain Ω ⊂ M. Let τΩ be the first exit time from Ω of Brownian motion Xt . If f is a nonnegative continuous function on 𝜕Ω then the function u(x) = 𝔼x (τΩ f (XτΩ )) solves the following boundary value problem Δ2 u = 0 in Ω, { { Δu|𝜕Ω = −f , { { { u|𝜕Ω = 0, and, hence, is bisuperharmonic in Ω. A manifold M is called biparabolic, if any positive bisuperharmonic function on M is harmonic, that is, Δu = 0. Note that the notion of parabolicity also admits a similar equivalent definition: M is parabolic if and only if any positive superharmonic function on M is harmonic. One can show that ℝn is biparabolic if and only if n ≤ 4. For example, if n > 4 then u(x) = |x|−(n−4) is bisuperharmonic but not harmonic. Theorem 11.1 ([19]). If for all large enough r, V(r) ≤ C then M is biparabolic.

r4 log r

(11.1)

318 | A. Grigor’yan Condition (11.1) is not far from optimal in the following sense: for any β > 1, there exists a manifold M with V(r) ≤ Cr 4 logβ r that is not biparabolic. Conjecture. If ∞

4

V(r) ≤ Cr log r

or even ∫

r 3 dr = ∞, V(r)

then M is biparabolic. Recall that M is parabolic if and only if G ≡ ∞, that is, Gf ≡ ∞ for any nonzero f ≥ 0. For the proof of Theorem 11.1, we use the following lemma. Lemma 11.2. The following conditions are equivalent: (i) M is biparabolic. (ii) G2 ≡ ∞ (that is, G2 f ≡ ∞ for any nonzero function f ≥ 0). Proof of Theorem 11.1. Assuming (11.1), we prove that G2 f ≡ ∞ for any nonnegative nonzero function f . It is easy to compute that ∞

∞

0

0 M

G2 f (x) = ∫ tPt f (x) dt = ∫ ∫ tpt (x, y)f (y) dμ(y) dt. Fix an arbitrary x ∈ M and choose R > 0 so big that the ball B(x0 , R) contains both supp f and x. By the local Harnack inequality, we have, for all x, y ∈ B(x0 , R) and t > 2R2 , pt (x, y) ≥ cpt−R2 (x0 , x0 ) ≥ cpt (x0 , x0 ), where c = c(x0 , R) > 0. Hence, we obtain, for large enough t0 , ∞

G2 f (x) ≥ ∫

∞

∫ tpt (x, y)f (y) dμ(y) dt ≥ c‖f ‖L1 ∫ tpt (x0 , x0 ) dt.

t0 B(x0 ,R)

t0

By Theorem 8.1, we have, for large t, pt (x0 , x0 ) ≥

1/4

V(√Kt log t)

≥

where v(r) :=

r4 . log r

c

v(√t log t)

,

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For large t, we have v(√t log t) ≃ t 2 log t, whence ∞

∞

∫ tpt (x0 , x0 ) dt ≥ c ∫ t0

t0

∞

t dt dt ≃∫ = ∞. t log t v(√t log t) t0

We conclude that G2 f (x) = ∞, which was to be proved. Now let us construct for any β > 1 an example of a manifold M that is not biparabolic and satisfies V(r) ≤ Cr 4 logβ r. Fix n ≥ 2 and consider a smooth manifold M = ℝ × 𝕊n−1 , where any point x ∈ M is represented in the polar form as (r, θ) where r ∈ ℝ and θ ∈ 𝕊n−1 . Define the Riemannian metric g on M by g = dr 2 + ψ2 (r)dθ2 ,

(11.2)

where dθ2 is the standard Riemannian metric on 𝕊n−1 and ψ(r) is a smooth positive function on ℝ. Define the area function S(r), r ∈ ℝ, by S(r) = ωn ψ(r)n−1 , where ωn is the volume of 𝕊n−1 . We choose S(r) as follows: α

S(r) = {

e−r , |r|3 logβ |r|,

r > 2, r < −2,

(11.3)

where α, β are arbitrary real numbers such that α>2

and β > 1.

(11.4)

The manifold M looks as in Figure 3. Proposition 11.3. Under the hypotheses (11.3) and (11.4), the manifold M is not biparabolic, and the volume growth function of M satisfies V(r) ≤ Cr 4 logβ r.

(11.5)

320 | A. Grigor’yan

Figure 3: Manifold M with two ends.

Proof. Fix a reference point x0 = (0, 0). The volume estimate (11.5) follows from r

V(r) ≃ ∫ S(t) dt. −r

In order to prove that M is not biparabolic, it suffices to construct a positive harmonic function h on M such that the function u := Gh is finite at least at one point. Indeed, in this case we have u ∈ C ∞ (M) and Δu = −h. Hence, Δu < 0 and Δ2 u = Δh = 0 so that u is bisuperharmonic, but not harmonic; hence, M is not biparabolic. Choose h as follows: r

dt . S(t)

h(r) = ∫ −∞

(11.6)

It is finite by (11.3) and harmonic on M because it depends only on r and Δh =

1 𝜕 𝜕h 𝜕2 h S󸀠 (r) 𝜕h + = (S(r) ) = 0. 2 S(r) 𝜕r S(r) 𝜕r 𝜕r 𝜕r

Then one proves that, for any x = (r, θ), g(x0 , x) ≃ {

h(r), 1,

if r < −2, if r > 2.

We have −2

2

∞

Gh(x0 ) = ∫ g(x0 , x)h(x) dμ(x) ≃ 1 + ∫ h (r)S(r) dr + ∫ h(r)S(r) dr. M

−∞

2

Analysis on manifolds and volume growth

| 321

For r < −2, we have 3

r

β

dt

S(r) = |r| log |r| and h(r) ≃ ∫ −∞

|t|3 logβ

|t|

≃

1

|r|2 logβ

|r|

.

Since β > 1, we obtain −2

−2

2

∫ h (r)S(r) dr ≃ ∫ −∞

−∞

1

|r| logβ |r|

dr < ∞.

For r > 2, S(r) = e−r

α

r

α

and h(r) ≃ ∫ et dt ≃ 0

α

er . r α−1

Since α > 2, we have +∞

∞

∫ h(r)S(r) dr ≃ ∫ 2

2

dr < ∞. r α−1

Hence, Gh(x0 ) < ∞, which was to be proved.

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[9]

M. T. Barlow, Which values of the volume growth and escape time exponent are possible for graphs? Rev. Mat. Iberoam. 40, 1–31 (2004). M. T. Barlow, T. Coulhon, A. Grigor’yan, Manifolds and graphs with slow heat kernel decay. Invent. Math. 144, 609–649 (2001). M. T. Barlow, E. A. Perkins, Symmetric Markov chains in ℤd : how fast can they move? Probab. Theory Relat. Fields 82, 95–108 (1989). S. Y. Cheng, S.-T. Yau, Differential equations on Riemannian manifolds and their geometric applications. Commun. Pure Appl. Math. 28, 333–354 (1975). T. Coulhon, A. Grigor’yan, On-diagonal lower bounds for heat kernels and Markov chains. Duke Math. J. 89(1), 133–199 (1997). E. B. Davies, Heat kernel bounds, conservation of probability and the Feller property. J. Anal. Math. 58, 99–119 (1992). A. Dvoretzky, P. Erdös, Some problems on random walk in space, in Proc. Second Berkeley Symposium on Math. Stat. and Probability (University of California Press, 1951), pp. 353–368. A. Grigor’yan, On the existence of positive fundamental solution of the Laplace equation on Riemannian manifolds. Mat. Sb. 128(3), 354–363 (1985) (in Russian), Engl. transl.: Math. USSR Sb. 56, 349–358 (1987). A. Grigor’yan, On stochastically complete manifolds. DAN SSSR 290(3), 534–537 (1986) (in Russian), Engl. transl.: Sov. Math. Dokl. 34(2), 310–313 (1987).

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[10] A. Grigor’yan, Stochastically complete manifolds and summable harmonic functions. Izv. Akad. Nauk SSSR, Ser. Mat. 52(5), 1102–1108 (1988) (in Russian), Engl. transl.: Math. USSR Izv. 33(2), 425–432 (1989). [11] A. Grigor’yan, Bounded solutions of the Schrödinger equation on non-compact Riemannian manifolds. J. Sov. Math. 51(14), 66–77 (1989) (in Russian), Engl. transl.: Tr. Semin. Im. I. G. Petrovskogo 3, 2340–2349 (1990).

[12] A. Grigor’yan, The heat equation on non-compact Riemannian manifolds. Mat. Sb. 182(1), 55–87 (1991) (in Russian), Engl. transl.: Math. USSR Sb. 72(1), 47–77 (1992). [13] A. Grigor’yan, Heat kernel upper bounds on a complete non-compact manifold. Rev. Mat. Iberoam. 10(2), 395–452 (1994).

[14] A. Grigor’yan, Heat kernel on a manifold with a local Harnack inequality. Commun. Anal. Geom. 2(1), 111–138 (1994). [15] A. Grigor’yan, Escape rate of Brownian motion on weighted manifolds. Appl. Anal. 71(1–4), 63–89 (1999). [16] A. Grigor’yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Am. Math. Soc. 36, 135–249 (1999).

[17] A. Grigor’yan, Heat kernels on weighted manifolds and applications. Contemp. Math. 398, 93–191 (2006).

[18] A. Grigor’yan, Heat Kernel and Analysis on Manifolds. AMS-IP Studies in Advanced Mathematics, vol. 47 (AMS – IP, 2009).

[19] A. Grigor’yan, F. Sh, On biparabolicity of Riemannian manifolds. Rev. Mat. Iberoam. 35(7), 2025–2034 (2019).

[20] A. Grigor’yan, E. P. Hsu, Volume growth and escape rate of Brownian motion on a Cartan–Hadamard manifold, in Sobolev Spaces in Mathematics II, ed. by V. Maz’ya. International Mathematical Series, vol. 9 (Springer, 2009), pp. 209–225.

[21] A. Grigor’yan, M. Kelbert, Range of fluctuation of Brownian motion on a complete Riemannian manifold. Ann. Probab. 26, 78–111 (1998).

[22] A. Grigor’yan, M. Kelbert, On Hardy–Littlewood inequality for Brownian motion on Riemannian manifolds. J. Lond. Math. Soc. (2) 62, 625–639 (2000). [23] A. Grigor’yan, L. Saloff-Coste, Hitting probabilities for Brownian motion on Riemannian manifolds. J. Math. Pures Appl. 81, 115–142 (2002).

[24] A. Grigor’yan, L. Saloff-Coste, Heat kernel on manifolds with ends. Ann. Inst. Fourier 59, 1917–1997 (2009).

[25] A. Grigor’yan, Y. Sun, On non-negative solutions of the inequality Δu + uσ ≤ 0 on Riemannian manifolds. Commun. Pure Appl. Math. 67, 1336–1352 (2014).

[26] A. Grigor’yan, Y. Sun, I. Verbitsky, Superlinear elliptic inequalities on manifolds. J. Funct. Anal. 278, 108444 (2020). [27] D. Hoffman, J. Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds. Commun. Pure Appl. Math. 27(6), 715–727 (1974). See also “A correction to: Sobolev and isoperimetric inequalities for Riemannian submanifolds” Commun. Pure Appl. Math. 28, 765–766 (1975).

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[31] Y. T. Kuz’menko, S. A. Molchanov, Counterexamples to Liouville-type theorems. Vestn. Mosk. Univ., Ser. I Mat. Mekh. 34(6), 39–43 (1979) (in Russian), Engl. transl.: Mosc. Univ. Math. Bull. 35–39 (1979). [32] P. Li, S.-T. Yau, On the parabolic kernel of the Schrödinger operator. Acta Math. 156(3–4), 153–201 (1986). [33] E. Mitidieri, S. I. Pohozaev, Absence of global positive solutions of quasilinear elliptic inequalities. Dokl. Akad. Nauk 359(4), 456–460 (1998). [34] Y. Sun, Uniqueness result on nonnegative solutions of a large class of differential inequalities on Riemannian manifolds. Pac. J. Math. 280(1), 241–254 (2016). [35] Y. Sun, The absence of global positive solutions to semilinear parabolic differential inequalities in exterior domain. Proc. Am. Math. Soc. 145(8), 3455–3464 (2017). [36] M. Takeda, On a martingale method for symmetric diffusion process and its applications. Osaka J. Math. 26, 605–623 (1989). [37] N.Th. Varopoulos, Potential theory and diffusion of Riemannian manifolds, in Conference on Harmonic Analysis in Honor of Antoni Zygmund. Vols. I, II. Wadsworth Math. Ser. (Wadsworth, Belmont, Calif., 1983), pp. 821–837. [38] S.-T. Yau, Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry. Indiana Math. J. 25, 659–670 (1976).

Alexander Grigor’yan, Satoshi Ishiwata, and Laurent Saloff-Coste

Geometric analysis on manifolds with ends

Abstract: Consider a complete non-compact (weighted) Riemannian manifold M which can be decomposed as a connected sum of finitely many complete non-compact manifolds, each of which satisfies a two-sided Gaussian heat kernel bound. We review what is known concerning heat kernel estimates on M, including some recent progresses in the case M is parabolic, and discuss the difficulties remaining to obtain a complete result. We also discuss implication for the lowest eigenvalue (with Neumann boundary condition) of large balls centered at a fixed central point. Keywords: Manifold with ends, heat kernel, Poincaré constant MSC 2010: Primary 58-02, Secondary 35K08, 58J65, 58J35

Contents 1 2 2.1 2.2 2.2.1 2.2.2 2.2.3 2.3 3 3.1 3.2 3.3

Introduction | 326 The state-of-the-art | 328 Setting | 328 Heat kernel estimates | 329 Off-diagonal estimates | 329 Nonparabolic case | 330 Parabolic case | 331 Poincaré constant estimates | 333 Manifold with ends with oscillating volume functions | 335 Preliminaries | 335 Example 1 | 339 Example 2 | 340 Bibliography | 342

Acknowledgement: A. Grigor’yan was partially supported by SFB 1283 of the German Research Council. S. Ishiwata was partially supported by JSPS, KAKENHI 17K05215. L. Saloff-Coste was partially supported by NSF grant DMS-1707589. The second author would like to thank Professor Gilles Carron for telling him how to construct manifolds with oscillating volume function with (PHI). Alexander Grigor’yan, Department of Mathematics, University of Bielefeld, Bielefeld, 33501, Germany, e-mail: [email protected] Satoshi Ishiwata, Department of Mathematical Sciences, Yamagata University, Yamagata, 990-8560, Japan, e-mail: [email protected] Laurent Saloff-Coste, Department of Mathematics, Cornell University, Ithaca, NY, 14853-4201, USA, e-mail: [email protected] https://doi.org/10.1515/9783110700763-011

326 | A. Grigor’yan et al.

1 Introduction In this article, we discuss some recent progress on geometric analysis on manifold with ends. In the final section, we construct manifolds with ends with oscillating volume functions which may turn out to have different heat kernel estimates from those provided by known results. Throughout the history of geometric analysis, manifolds with ends have appeared in several contexts. For example, Cai [2], Kasue [20] and Li and Tam [23] studied manifolds with nonnegative Ricci (sectional) curvature outside a compact set where manifolds with ends play an important role. It should be pointed out that there are other recent works on manifolds with ends. See, for instance, Carron [3], Doan [6], Duong, Li, and Sikora [7], Hassel, Nix, and Sikora [18], Hassel and Sikora [19]. Because of the bottleneck structure inherent to most manifolds with ends, geometric and analytic properties of manifolds with ends are very different from a manifold such as ℝn . For example, in 1979, Kuz’menko and Molchanov [22] proved the following: Theorem 1.1. On M = ℝ3 #ℝ3 , the connected sum of two copies of ℝ3 , the weak Liouville property does not hold. Namely, there exists a nontrivial bounded harmonic function. It is a well-known fact that the parabolic Harnack inequality ((PHI) in short) implies the weak Liouville property. See [15, Section 2.1] and [29, 5.4.5] for details. By a contraposition argument, the above theorem implies that (PHI) does not hold on ℝ3 #ℝ3 . Denote by p(t, x, y) the heat kernel of a noncompact weighted manifold (M, d, μ), that is, the minimal positive fundamental solution of the heat equation 𝜕t u = Δu, where Δ is the weighed Laplacian. In 1986, Li and Yau proved in [24] that (LY) p(t, x, y) ≍

2 c e−bd (x,y)/t V(x, √t)

holds on noncompact manifolds with nonnegative Ricci curvature. Here V(x, r) := μ(B(x, r)) is the measure of the open geodesic ball B(x, r) = {y ∈ M : d(x, y) < r} and the sign ≍ means that both ≤ and ≥ hold but with different values of the positive constants C and b. We call this estimate a Li–Yau-type bound and write (LY) in short. The following theorem is a combined result of [8, 28] based on previous contributions of Moser [26], Kusuoka and Stroock [21], etc. Theorem 1.2. On a geodesically complete, noncompact weighted manifold M, the following conditions are equivalent: (1) The Li–Yau-type heat kernel estimates (LY), (2) The parabolic Harnack inequality (PHI),

Geometric analysis on manifolds with ends | 327

(3) The Poincaré inequality (PI), i. e., there exists C, κ > 0 such that for any x ∈ M, r > 0 and f ∈ C ∞ (B(x, r)), (PI)

∫ |f − fB(x,r) |2 dμ ≤ Cr 2 ∫ |∇f |2 dμ, B(x,r)

B(x,r)

1 where fB(x,r) = V(x,r) ∫B(x,r) f dμ, and the volume doubling property (VD), that is, there exists C > 0 such that for any x ∈ M, r > 0,

V(x, 2r) ≤ CV(x, r). Combining the above two theorems, the connected sum M = M1 #M2 = ℝ3 #ℝ3 satisfies neither (PI) nor (LY). Indeed, the function f (x) = {

1, −1,

x ∈ M1 , x ∈ M2

implies that fB(o,r) = 0 for a central reference point o ∈ M and for any large r > 0 ∫ |f − fB(o,r) |2 dμ ≃ r 3 ,

∫ |∇f |2 dμ ≃ const,

B(o,r)

B(o,r)

which fails (PI). Here f ≃ g means cf ≤ g ≤ Cf with some positive constants 0 < c ≤ C on a suitable range of functions f , g. Moreover, Benjamini, Chavel, and Feldman [1] obtained in 1996 the following heat kernel estimate. Theorem 1.3. Let M = M1 #M2 = ℝn #ℝn with n ≥ 3. There exists ε > 0 such that for x ∈ M1 , y ∈ M2 with |x| ≃ |y| ≃ √t, p(t, x, y) ≤

1

t

n+ε 2

≪

1 . t n/2

This theorem asserts that the heat kernel between two different ends is significantly smaller than that on one end because of a bottleneck effect. In view of the above facts, it is natural to ask on a manifold with ends for the behavior of the heat kernel p(t, x, y) and the estimate of Λ(B(x, r)) :=

sup

f ∈C 1 (B(x,r)) f =const ̸

∫B(x,r) |f − fB(x,r) |2 dμ ∫B(x,r) |∇f |2 dμ

,

(1.1)

which is called the Poincaré constant. Notation. Throughout this article, the letters c, c󸀠 , C, C 󸀠 , C 󸀠󸀠 denote positive constants whose values may be different at different instances. When the value of a constant is significant, it will be explicitly stated.

328 | A. Grigor’yan et al.

2 The state-of-the-art 2.1 Setting First of all, we begin with the definition of what we call a manifold with finitely many ends. For a fixed integer k ≥ 2, let M1 , . . . , Mk be a sequence of geodesically complete, noncompact weighted manifolds of the same dimension. Definition 2.1. We say that a weighted manifold M is a manifold with k ends M1 , M2 , . . . , Mk and write M = M1 # ⋅ ⋅ ⋅ #Mk

(2.1)

if there is a compact set K ⊂ M so that M \ K consists of k connected components E1 , E2 , . . . , Ek such that each Ei is isometric (as a weighted manifold) to Mi \ Ki for some compact set Ki ⊂ Mi (see Figure 1). Each Ei (or Mi ) will be referred to as an end of M.

Figure 1: Manifold with ends.

Here we remark that the definition of an end given above is different from the usual notion defined as a connected component of the ideal boundary. We say that a manifold M is parabolic if any positive superharmonic function on M is constant, and nonparabolic otherwise. See [9] for details. Throughout this article, we always assume that each end Mi satisfies (VD) and (PI). Moreover, if the end Mi is parabolic, then we also assume that Mi satisfies the relatively connected annuli condition defined as follows. Definition 2.2 ((RCA)). A weighted manifold M satisfies relatively connected annuli condition ((RCA) in short) with respect to a reference point o ∈ M if there exists a positive constant A > 1 such that for any r > A2 and all x, y ∈ M with d(o, x) = d(o, y) = r,

Geometric analysis on manifolds with ends | 329

there exists a continuous path from x to y staying in B(o, Ar)\B(o, A−1 r). See Figures 2 and 3 for typical positive and negative examples.

Figure 2: Manifold with (RCA).

Figure 3: Manifold without (RCA).

The assumption (RCA) seems technical but it makes it possible to obtain optimal estimates of the first exit (hitting) probability and the Dirichlet heat kernel in the exterior of a compact set of a parabolic manifold satisfying (LY). See [12], [13] and [14] for details.

2.2 Heat kernel estimates 2.2.1 Off-diagonal estimates Let M = M1 # ⋅ ⋅ ⋅ #Mk be a manifold with k ends. For t > 0, x ∈ Ei and y ∈ M, let pDEi (t, x, y) be the extended Dirichlet heat kernel on an end Ei , that is, the Dirichlet heat kernel in y ∈ Ei and extension to 0 if y ∈ ̸ Ei . Let τEi be the first exit time of the Brownian motion from Ei and then ℙx (τEi < t) is the first exit probability starting from x by time t from Ei . We will use the following theorem to estimate the off-diagonal heat kernel estimates. Theorem 2.3 (Grigor’yan and Saloff-Coste [16, Theorem 3.5]). Let M = M1 # ⋅ ⋅ ⋅ #Mk be a manifold with k ends and fix a central reference point o ∈ K. For x ∈ Ei , y ∈ Ej , and

330 | A. Grigor’yan et al. t > 1, p(t, x, y) ≃ pDEi (t, x, y) + p(t, o, o)ℙx (τEi < t)ℙy (τEj < t) t

+ ∫ p(s, o, o) ds(𝜕t ℙx (τEi < t)ℙy (τEj < t) + ℙx (τEi < t)𝜕t ℙy (τEj < t)).

(2.2)

0

Under the assumption of (PI), (VD) on each end, and, in addition, (RCA) on each parabolic end, applying results in [13] and [14], the quantities pDEi (t, x, y), ℙx (τEi < t), ℙy (τEj < t), 𝜕t ℙx (τEi ), and 𝜕t ℙy (τEj < t) can be estimated. Hence, estimating p(t, o, o) becomes the key missing task to obtain off-diagonal bounds on manifolds with ends.

2.2.2 Nonparabolic case First, we consider heat kernel estimates on M = M1 # ⋅ ⋅ ⋅ #Mk , where M is nonparabolic, namely, at least one end Mi is nonparabolic. For a fixed reference point oi ∈ Ki ⊂ Mi , let Vi (r) := μ(B(oi , r)) and r

hi (r) := 1 + (∫ 1

s ds ) . Vi (s) +

Here we remark that, under the assumption (LY), Mi is parabolic if and only if lim h (r) r→∞ i

= ∞.

In 2009, Grigor’yan and Saloff-Coste [16] obtained the following (see also [12] and [17]). Theorem 2.4. Let M = M1 # ⋅ ⋅ ⋅ #Mk be a manifold with k ends. Assume that each end Mi satisfies (PI) and (VD) and that each parabolic end satisfies (RCA). Assume also that M is nonparabolic. Then for all t > 0, p(t, o, o) ≃

1 . mini Vi (√t)h2i (√t)

If all ends M1 , . . . , Mk are nonparabolic, then all functions h1 , . . . , hk are bounded. Hence, the above theorem implies that p(t, o, o) ≃

1 , mini Vi (√t)

Geometric analysis on manifolds with ends | 331

namely, the behavior of the heat kernel at the central reference point is determined by the smallest end! As a typical example, let M be M1 #M2 = ℝn #ℝn , the connected sum of two copies of ℝn with n ≥ 3. Then the above theorem implies that p(t, o, o) ≃

1 . t n/2

Substituting this estimate into Theorem 2.3, we obtain that for x ∈ M1 , y ∈ M2 , p(t, x, y) ≃

1

t n/2

(

2 1 1 + n−2 )e−bd (x,y)/t , n−2 |x| |y|

where |x| = e + d(x, K). Remark 2.5. Recently, Lou [25] and Ooi [27] proved two-sided matching heat kernel estimates on non-parabolic spaces with varying dimension where the long time behavior is same as in [16]. 2.2.3 Parabolic case Next, we consider the case of manifolds with ends, M = M1 # ⋅ ⋅ ⋅ #Mk , which are parabolic, that is, for which all ends M1 , . . . , Mk are parabolic. To prove optimal heat kernel estimates, we need the following assumptions on each end. Definition 2.6 ([11]). An end Mi is called subcritical if hi (r) ≤ C

r2 Vi (r)

(∀r > 1)

and regularif there exist γ1 , γ2 > 0 satisfying 2γ1 + γ2 < 2 such that 2−γ2

R c( ) r

2+γ1

≤

Vi (R) R ≤ C( ) Vi (r) r

(∀1 < r ≤ R).

(2.3)

For example, an end Mi with volume function Vi (r) = r α (log r)β is parabolic if and only if either α < 2 or α = 2 and β ≤ 1. Moreover, Mi is subcritical if α < 2 and regular if α = 2 and β ≤ 1. We remark that if Mi satisfies (VD), then the reverse doubling property holds, implying that for any subcritical end there exists δ > 0 such that Vi (r) ≤ Cr 2−δ

(∀r > 0).

(2.4)

For r > 0, let m = m(r) be a number so that Vm (r) = max Vi (r). i

We can now state the following result.

(2.5)

332 | A. Grigor’yan et al. Theorem 2.7 (Grigor’yan, Ishiwata, Saloff-Coste [11]). Let M = M1 # ⋅ ⋅ ⋅ #Mk be a manifold with k parabolic ends. Assume that each end Mi satisfies (PI), (VD), (RCA), and is either subcritical or regular. If there exist both subcritical and regular ends, assume also that the constant δ in (2.4) satisfies δ > γ2 , namely, for any subcritical volume function Vi (r) and any regular volume function Vj (r), Vi (r) ≤ Cr 2−δ ≤ C 󸀠 r 2−γ2 ≤ C 󸀠󸀠 Vj (r)

(∀r > 0).

Moreover, assume that there exists an end Mm such that for all i = 1, . . . , k and for all r > 0, Vm (r) ≥ cVi (r) and

Vm (r)h2m (r) ≤ CVi (r)h2i (r).

(2.6)

Then for t > 0, p(t, o, o) ≃

1 . Vm (√t)

(2.7)

This means that the on-diagonal heat kernel estimates at the central reference point are determined by the largest end! Remark 2.8. In our approach, we require the existence of a fixed dominating end given by (2.6) for the optimal estimates in (2.7) to hold. Indeed, more generally, on a manifold with either regular or subcritical ends, we obtain for t > 1 (see [11] for the details), p(t, o, o) ≤ C

mini h2i (√t) . mini Vi (√t)h2i (√t)

(2.8)

The assumption in (2.6) implies that for all r > 1, mini h2i (r) C , ≤ mini Vi (r)h2i (r) Vm (r)

(2.9)

which allows applying [5, Theorem 7.2] for the matching lower bound. In Section 3, we construct manifolds with ends without a fixed dominating end and, in such cases, the estimates in (2.9) do not hold. As an illustrative example, let M = M1 # ⋅ ⋅ ⋅ #Mk be a manifold with parabolic ends, where each end Mi satisfies (PI), (VD), and (RCA). Let αi and βi be sequences satisfying (α1 , β1 ) ≥ (α2 , β2 ) ≥ ⋅ ⋅ ⋅ ≥ (αk , βk ) > (0, +∞) in the sense of lexicographical order, namely (αi , βi ) > (αj , βj ) means that αi > αj

or

αi = αj

and βi > βj

Geometric analysis on manifolds with ends | 333

and we assume that Vi (r) ≃ r αi (log r)βi ,

r > 2.

Here we need (α1 , β1 ) ≤ (2, 1) so that all ends M1 , . . . , Mk are parabolic. Then the above theorem implies that p(t, o, o) ≃

1 , t α1 /2 (log t)β1

t > 2.

As an explicit example, suppose that k = 2 and (α1 , β1 ) = (2, 0) and (α2 , β2 ) = (1, 0). Substituting the above estimates into (2.2), we obtain for x ∈ E1 , y ∈ E2 , and t > 1, { { { { p(t, x, y) ≃ { { { { {

1 −bd2 (x,y)/t e t √ |y| 1 (1 + √ log e|x|t ) t t 2 √ 1 (log e|x|t )e−bd (x,y)/t t

if |x| > √t, if |x|, |y| ≤ √t,

(2.10)

if |x| ≤ √t < |y|.

Remark 2.9. Assume that all ends of a manifold M = M1 # ⋅ ⋅ ⋅ #Mk are subcritical. Then for x ∈ Ei and y ∈ Ej with i ≠ j and t > 1, p(t, x, y) ≍

2 C e−bd (x,y)/t √ Vm ( t)

(see [10, Theorem 2.3]). Remark 2.10. Recently, Chen and Lou [4] proved two-sided matching heat kernel estimates on parabolic spaces with varying dimension where the long time behavior is same as in (2.10).

2.3 Poincaré constant estimates In this section, we consider the estimates of the Poincaré constant defined in (1.1). Recall that M = M1 # ⋅ ⋅ ⋅ #Mk is a manifold with ends M1 , . . . , Mk , where each end satisfies (VD) and (PI). Let o ∈ K be a central reference point. Our main interest is to obtain the Poincaré constant Λ(B(o, r)) at the central point o. In fact, by the monotonicity of Λ together with a Whitney covering argument (see [11]), for r > 2|x|, Λ(B(x, r)) ≃ Λ(B(o, r)). For r > 0, let n = n(r) be the number so that Vn (r) = max Vi (r), i=m ̸

where m = m(r) is the number of the largest end (see (2.5)). Then we obtain the following.

334 | A. Grigor’yan et al. Theorem 2.11 (Grigor’yan, Ishiwata, Saloff-Coste [11]). Let M = M1 # ⋅ ⋅ ⋅ #Mk be a manifold with k nonparabolic ends. Assume that each end Mi satisfies (VD) and (PI). Then for sufficiently large r > 1, Λ(B(o, r)) ≤ CVn (r). Moreover, if for all r > 0, rV 󸀠 (r) ≤ CV(r), then for sufficiently large r > 1, Λ(B(o, r)) ≃ Vn (r). When M has at least one parabolic end, we assume the following additional condition (see [11] for details). Definition 2.12 ((COE)). We say that a manifold with ends M = #i∈I Mi has critically ordered ends and write (COE) in short if there exist ε, δ, γ1 , γ2 > 0 such that γ1 < ε,

γ1 + γ2 < δ < 2,

2γ1 + γ2 < 2,

and a decomposition I = Isuper ⊔ Imiddle ⊔ Isub such that the following conditions are satisfied: (a) For each i ∈ Isuper and all r ≥ 1, Vi (r) ≥ cr 2+ϵ . (b) For each i ∈ Isub , Vi is subcritical (see Definition 2.6) and Vi (r) ≤ Cr 2−δ . (c) For each i ∈ Imiddle , Vi is regular with parameter γ1 and γ2 (see (2.3)). Moreover, for any pair i, j ∈ Imiddle we have either Vi ≥ cVj or Vj ≥ cVi (i. e., the ends in Imiddle can be ordered according to their volume growth uniformly over r ∈ [1, ∞)) and Vi ≥ cVj implies that Vi hi ≥ c󸀠 Vj hj . Besides, if M is parabolic (i. e., all ends are parabolic) then Vi ≥ CVj also implies Vi h2i ≤ C 󸀠 Vj h2j . Theorem 2.13 ([11]). Let M = M1 # ⋅ ⋅ ⋅ #Mk be a manifold with k ends, where each end satisfies (VD) and (PI). Suppose that there exists at least one parabolic end and each parabolic end satisfies (RCA). If M admits (COE), then for sufficiently large r > 1, Λ(B(o, r)) ≤ CVn (r)hn (r).

Geometric analysis on manifolds with ends | 335

If, in addition, each Vi satisfies rVi󸀠 (r) ≤ CVi (r)

(∀r > 1),

then, for sufficiently large r > 1, Λ(B(o, r)) ≃ Vn (r)hn (r). These results say that the Poincaré constant Λ(B(o, r)) is determined by the second largest end! As an explicit example, let M = ℝn #ℝn with n ≥ 2. Then Theorems 2.11 and 2.13 imply that Λ(B(o, r)) ≃ {

rn ,

n ≥ 3,

r log r,

n = 2.

2

Let M = M1 # ⋅ ⋅ ⋅ #Mk be a manifold with ends. Assume that each end Mi satisfies (VD), (PI) and that each parabolic end satisfies (RCA). Suppose that for i = 1, . . . , k, Vi (r) ≃ r αi (log r)βi , where (α1 , β1 ) ≥ (α2 , β2 ) ≥ ⋅ ⋅ ⋅ ≥ (αk , βk ) in the lexicographical order. Then Λ(B(o, r)) ≃ V2 (r)h2 (r) { { { { { { ≃{ { { { { { {

r α2 (log r)β2

r 2 log r(log log r)2 2

r log r r

if (α2 , β2 ) > (2, 1), if (α2 , β2 ) = (2, 1), if (2, −∞) < (α2 , β2 ) < (2, 1),

2

if (α2 , β2 ) < (2, −∞).

3 Manifold with ends with oscillating volume functions 3.1 Preliminaries The purpose of this section is to construct manifolds with ends for which the estimate in (2.8) might not give an optimal bound. To obtain such a manifold, we need a manifold with (VD) and (PI) together with oscillating volume function. First, let us recall the following theorem. Theorem 3.1 (Grigor’yan and Saloff-Coste [15, Theorem 5.7]). Let (M, μ) be a complete noncompact weighted manifold with (PHI) and (RCA) at a reference point o ∈ M. If a positive valued smooth function W : [0, ∞) → ℝ satisfies for all r > 0, sup W ≤ C inf W,

[r,2r]

[r,2r]

(3.1)

336 | A. Grigor’yan et al. r

∫ W 2 (s)s ds ≤ CW 2 (r)r 2 ,

(3.2)

0

then the weighted manifold (M, W 2 (d(o, ⋅))μ) also satisfies (PHI). Let (M1 , μ1 ) be the 2-dimensional Euclidean space ℝ2 with the Euclidean measure. We denote by (M2 , μ2 ) a weighted manifold (ℝ2 , W 2 (d(o, ⋅))μ1 ), where the positive valued function W : [0, ∞) → ℝ is defined as follows. For α > 2 and 0 < β < 2, define a function W so that for all k ∈ ℕ, { { r { { { 2 ∫ W (s)s ds = { { { { 0 { {

r2 , ( br )α b2k , k

r 2 log r, ( dr )β dk2 log dk , k

0 < r < a1 , ak ≤ r < bk , bk ≤ r < ck , ck ≤ r < dk , dk ≤ r < ak+1 ,

(3.3)

where the sequences ak ≤ bk < ck ≤ dk < ak+1 satisfy a1 > e and bk =

ck

1

(log ck ) α−2

, 1

ak+1 = dk (log dk ) 2−β

(3.4) (3.5)

(see Figure 4). These sequences will be fixed later. Then the function W satisfies { { { { { { { W(r) ≃ { { { { { { { {

1,

0 < r < a1 , ak ≤ r < bk ,

α−2 ( br ) 2 , k

bk ≤ r < ck ,

√log r,

2−β √log dk ( dr )− 2 , k

0, { { { { α−4 − α−2 { { { r 2 bk 2 , { W 󸀠 (r) ≃ { 1 1 { , { √log r r { { { 2−β { 4−β { − 2 √log dk dk 2 , { −r

ck ≤ r < dk , dk ≤ r < ak+1 , 0 < r < a1 , ak < r < bk , bk < r < ck ,

ck < r < dk , dk < r < ak+1 .

Since 0, { { { { 1 { W (r) { r , ≃{ 1 { W(r) { , r log r { { { 1 { −r, 󸀠

0 < r < a1 , ak < r < bk ,

bk < r < ck ,

ck < r < dk , dk < r < ak+1 ,

(3.6)

Geometric analysis on manifolds with ends | 337

Figure 4: Oscillating volume function (thick line).

the condition in (3.1) holds. Indeed, the estimates in (3.6) imply that for any 0 < r1 < r2 , r2

r2

r2

r1

r1

r1

r W(r2 ) r C ds W 󸀠 (s) C ds −C log 2 = − ∫ ≤∫ ds = log ≤∫ = C log 2 . r1 s W(s) W(r1 ) s r1 Then we obtain for any 0 < r1 ≤ r2 ≤ 2r1 , W(r1 ) ≃ W(r2 ). Taking r1 ≤ r2 , r3 ≤ 2r1 so that W(r2 ) = sup W [r1 ,2r1 ]

and W(r3 ) = inf W, [r1 ,2r1 ]

we conclude the condition in (3.1). Moreover, we see that { { { { { { 2 2 W (r)r ≃ { { { { { { {

r2 ,

r α b2−α k , 2

r log r, 2−β r β dk

log dk ,

0 < r < a1 , ak ≤ r < bk , bk ≤ r < ck , ck ≤ r < dk , dk ≤ r < ak+1 ,

which satisfies the condition in (3.2). Applying Theorem 3.1, the weighted manifold (M2 , μ2 ) = (ℝ2 , W 2 (d(o, ⋅))μ1 ) satisfies (PHI).1 1 We need a smooth modification of the function W satisfying (3.3) to apply Theorem 3.1. However, we omit the smoothing argument for simplicity.

338 | A. Grigor’yan et al. Let Z : [0, ∞) → ℝ be a positive-valued smooth function satisfying Z(r) = √1 + 2 log r

(r ≥ 1).

Let (M3 , μ3 ) be a weighted manifold (ℝ2 , Z 2 (d(o, ⋅))μ1 ). Then Theorem 3.1 implies also that (M3 , μ3 ) satisfies (PHI). For i = 1, 2, 3, we denote by Vi (r) the volume function on (Mi , μi ) at o ∈ Mi = ℝ2 . Then we obtain V1 (r) = πr 2 ,

r

V2 (r) = 2π ∫ W 2 (s)s ds, 0

1

V3 (r) = 2π ∫ Z 2 (s)s ds + 2πr 2 log r

(r ≥ 1).

0

It is easy to obtain for sufficiently large r > 1, h1 (r) ≃ log r

and h3 (r) ≃ log log r.

Now we estimate the function h2 (r). Observe that bk

∫

ak

b s ds = log k , V2 (s) ak

ck

ck

bk

bk

dk

dk

ck

ck

s ds 1 1 1−α ) ≃ 1, = ∫ bα−2 (1 − ds = ∫ k s V2 (s) α−2 log ck

log dk sds ds =∫ = log( ), ∫ V2 (s) s log s log ck

ak+1

∫ dk

ak+1

1 1 s ds s1−β ds = ∫ 2−β = (1 − ) ≃ 1. V2 (s) (2 − β) log dk d log d k k d k

Then we obtain a1

h2 (an ) = 1 + ∫ 1

ak+1

n b log dk s ds n−1 s ds +∑ ∫ ≃ ∑ (log k + log + 1), V2 (s) k=1 V2 (s) k=1 ak log ck

(3.7)

ak

which shows that the behavior of h2 (r) depends on the choice of sequences ak ≤ bk < ck ≤ dk < ak+1 satisfying (3.4) and (3.5).

Geometric analysis on manifolds with ends | 339

3.2 Example 1 For the first case, let us choose sequences ak ≤ bk < ck ≤ dk < ak+1 so that for any k ∈ ℕ, ak ≃ bk ,

ck ≃ dk .

(3.8)

In this case, we obtain the following. Lemma 3.2. If a1 is large enough and the sequences ak ≤ bk < ck ≤ dk < ak+1 satisfy (3.4), (3.5), and (3.8), then for sufficiently large r > 1, h2 (r) ≃

log r . log log r

Proof. By the estimate in (3.7), we obtain h2 (an ) ≃ n.

(3.9)

Let us consider the behavior an . By the assumption in (3.4), we always have 1

ck ≃ bk (log bk ) α−2 .

(3.10)

Assumptions in (3.4) and (3.5) imply that for any k ∈ ℕ, 1

1

ak+1 = dk (log dk ) 2−β ≃ ck (log ck ) 2−β 1

1

1

≃ bk (log bk ) α−2 (log(bk (log bk ) α−2 )) 2−β = bk (log bk )

1 α−2 1

≃ bk (log bk ) α−2 where γ =

1 α−2

+

1 . 2−β

1 + 2−β

= bk (log bk )γ ≃ ak (log ak )γ ,

Taking a1 large enough, there exist positive constants c, C, and

γ > γ such that for any n ∈ ℕ, 󸀠

1

2−β 1 log log bk ) (log bk + α−2

cnγn ≤ an ≤ Cnγ n . 󸀠

This implies that for sufficiently large n ∈ ℕ, log an ≃ n log n,

log log an ≃ log n.

Then we obtain for sufficiently large n ∈ ℕ, log an ≃ n ≃ h2 (an ), log log an which concludes the lemma.

340 | A. Grigor’yan et al. Now we estimate the heat kernel on M = M1 #M2 . By the definition of V1 (r), V2 (r) and by Lemma 3.2, we obtain for sufficiently large r > 1, V1 (r) = r 2 , h1 (r) ≃ log r,

V2 (r) ≃ { h2 (r) ≃

r2 ,

r 2 log r,

ak ≤ r < bk , ck ≤ r < dk ,

log r , log log r 2

V1 (r)h21 (r)

2

2

≃ r (log r) ,

V2 (r)h22 (r)

2 (log r) { r (log log r)2 , ≃{ 2 (log r)3 { r (log log r)2 ,

ak ≤ r < bk , ck ≤ r < dk .

According to the heat kernel upper estimate in (2.8), we obtain for ck ≤ √t < dk , t 2 ( logloglog √ ) mini h2i (√t) 1 t . p(t, o, o) ≤ ≃ ≃ 2 2 t(log log t)2 mini Vi (√t)hi (√t) t(log √t) √

Since V(r) ≃ maxi Vi (r) = r 2 log r in this interval, the above upper estimate is much larger than 1 1 ≃ , √ t log t V( t) which makes it difficult to obtain a matching lower bound by using [5, Theorem 7.2].

3.3 Example 2 Next, let us choose sequences ak ≤ bk < ck ≤ dk < ak+1 so that for some δ > 1 and for any k ∈ ℕ, ak ≃ bk ,

ckδ ≃ dk .

(3.11)

In this case, we obtain the following. Lemma 3.3. If a1 is large enough and the sequences ak ≤ bk < ck ≤ dk < ak+1 satisfy (3.4), (3.5), and (3.11), then for sufficiently large r > 1, h2 (r) ≃ log log r. Proof. The estimate in (3.7) yields that h2 (an ) ≃ n. By the estimates in (3.10) and (3.11), we obtain 1

δ

δ

dk ≃ ckδ ≃ (bk (log bk ) α−2 ) ≃ aδk (log ak ) α−2 .

Geometric analysis on manifolds with ends | 341

Assumptions in (3.4) and (3.5) imply that 1

ak+1 = dk (log dk ) 2−β ≃

δ aδk (log ak ) α−2 (δ log ak

≃ aδk (log ak )θ , δ + where θ = α−2 any k ∈ ℕ,

1

2−β δ log log ak ) + α−2

1 . Hence, there exist positive constants c, C, and η 2−β

> δ such that for

η

caδk ≤ ak+1 ≤ Cak . This implies that for any n ∈ ℕ, c

δ(n−1) −1 δ−1

aδ1

(n−1)

≤ an ≤ C

η(n−1) −1 η−1

η(n−1)

a1

.

Then we obtain η(n−1) − 1 δn−1 − 1 log c + δ(n−1) log a1 ≤ log an ≤ log C + η(n−1) log a1 . δ−1 η−1

Taking a1 large enough, we obtain for sufficiently large n ∈ ℕ, log log an ≃ n ≃ h2 (an ), which concludes the lemma.

Now let us consider the estimate of the heat kernel on M = M2 #M3 . By the definition of V2 (r), V3 (r) and by Lemma 3.3, we obtain for sufficiently large r > 1, V2 (r) ≃ {

r2 ,

ak ≤ r < bk ,

r log r,

ck ≤ r < dk ,

2

h2 (r) ≃ log log r, V2 (r)h22 (r) ≃ {

h3 (r) ≃ log log r,

r 2 (log log r)2 , 2

V3 (r) ≃ r 2 log r,

2

r log r(log log r) ,

ak ≤ r < bk , ck ≤ r < dk .

V3 (r)h23 (r) ≃ r 2 log r(log log r)2 .

Substituting above into (2.8), we obtain for sufficiently large k ∈ ℕ and ak ≤ √t < bk , p(t, o, o) ≤ C

(log log √t)2 1 ≃ . t(log log √t)2 t

Since V(r) ≃ maxi Vi (r) ≃ r 2 log r for all r > 1, the above upper estimate is much larger than 1 1 ≃ , V(√t) t log t which makes it difficult to obtain matching lower bound by using [5, Theorem 7.2]. We hope to prove matching heat kernel lower bounds in forthcoming work.

342 | A. Grigor’yan et al.

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[23] P. Li, L-F. Tam, Positive harmonic functions on complete manifolds with non-negative curvature outside a compact set. Ann. Math. 125, 171–207 (1987). [24] P. Li, S.-T. Yau, On the parabolic kernel of the Schrödinger operator. Acta Math. 156(3–4), 153–201 (1986). [25] S. Lou, Explicit heat kernels of a model of distorted Brownian motion on spaces with varying dimension. arXiv:2001.09226. [26] J. Moser, A Harnack inequality for parabolic differential equations. Commun. Pure Appl. Math. 17, 101–134 (1964). Correction: Commun. Pure Appl. Math. 20, 231–236 (1967). [27] T. Ooi, Heat kernel estimates on spaces with varying dimension. arXiv:2003.06760. [28] L. Saloff-Coste, A note on Poincaré, Sobolev, and Harnack inequalities. Int. Math. Res. Not. 2, 27–38 (1992). [29] L. Saloff-Coste, Aspects of Sobolev Type Inequalities. London Math. Soc. Lecture Notes Series, vol. 289 (Cambridge Univ. Press, 2002).

Feida Jiang and Xinyi Shen

A matrix Harnack estimate for a Kolmogorov type equation Abstract: In this paper, we derive a matrix Harnack estimate for a Kolmogorov type equation that satisfies the Hörmander condition, which is inspired by the estimates of Hamilton [4] and Huang [7]. Some properties of the positive solutions are obtained through the applications of the matrix Harnack estimate. Keywords: Kolmogorov type equation, matrix Harnack estimate, maximum principle MSC 2010: 35K70

Contents 1 2 3

Introduction | 345 Proof of Theorem 1.1 | 349 Proofs of corollaries | 355 Bibliography | 358

1 Introduction This paper aims to study a matrix Harnack estimate of a Kolmogorov type equation which is ultraparabolic. The simplest example of this equation is given by Kolmogorov [8] (Hamilton used this equation as castaway equation in [4]) ft = fxx − xfy .

(1.1)

Equation (1.1) is strongly degenerate due to the only existing term fxx and the lack of the term fyy . In 1934, Kolmogorov [8] constructed an explicit fundamental solution, which implies (1.1) is ultraparabolic. Hörmander pointed out in his famous paper on ultraparabolic differential equations [6] that the Kolmogorov method could also be applied to the more general operator L = div(AD) + ⟨x, BD⟩ − 𝜕t

(1.2)

Acknowledgement: The first author was supported in part by National Natural Science Foundation of China (No. 11771214, No. 11926418). Feida Jiang, Xinyi Shen, College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, P. R. China, e-mails: [email protected], [email protected] https://doi.org/10.1515/9783110700763-012

346 | F. Jiang and X. Shen where D = (𝜕x1 , . . . , 𝜕xN )T , and ⟨⋅, ⋅⟩ denote, respectively, the gradient and the inner product in ℝN ; div represents the divergence operator in ℝN ; A = (aij ) and B = (bij ) are N × N constant real matrices, and A is symmetric and nonnegative definite; 𝜕t is the gradient in time. In this paper, we study the following Kolmogorov type equation: n

k

mi

i=1

i=1

j=1

ft = ∑ αi fxi xi + ∑(βi xi ∑ fxi ) on ℝN × (0, T),

(1.3)

j

which has applications in diffusion theory and in finance, see [10]. Here in (1.3), 1 ≤ k ≤ n and 0 < T ≤ ∞, αi (i ∈ {1, . . . , n}) are positive constants, βi (i ∈ {1, . . . , k}) are nonzero constants, mi (i ∈ {1, . . . , k}) are natural numbers satisfying ∑ki=1 mi = N − n, and ij = n + ∑i−1 q=1 mq + j (i ∈ {1, . . . , k}, j ∈ {1, . . . , mi }). In (1.3), since xi can multiply mi terms, we label the terms that multiply xi as fxi (j ∈ {1, . . . , mi }) for the convenience j

of labeling the terms. The subscripts ij (i ∈ {1, . . . , k}, j ∈ {1, . . . , mi }) of x denote the numbers between n + 1 to N. The subscripts of fxi in (1.3) can be explicitly written in j

the following way

x11 = xn+1 , x21 = xn+m1 +1 ,

x12 = xn+2 ,

...,

x22 = xn+m1 +2 ,

x1m = xn+m1 , 1

x2m = xn+m1 +m2 ,

...,

2

.. .

xk1 = xn+m1 +m2 +⋅⋅⋅+mk−1 +1 = xn+∑k−1 m

q +1

q=1

= xN−mk +1 , xk2 = xN−mk +2 ,

...,

xkm = xN . k

Equation (1.3) can be written as Lf = 0, where L is the operator in (1.2), with A=(

A0 0

0 ), 0

B=(

0 0

B1 ) 0

being two constant real N × N matrices, where A0 = diag(α1 , α2 , . . . , αn ) > 0 and m1

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ β1 ⋅ ⋅ ⋅ β1

( ( ( ( B1 = ( 0 ( ( ( ( 0

⋅⋅⋅ .. . ⋅⋅⋅

0

0

0

⋅⋅⋅ m2

0

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ β2 ⋅ ⋅ ⋅ β2 .. . 0

⋅⋅⋅

0

0 0

0

⋅⋅⋅ ⋅⋅⋅ .. . ⋅⋅⋅

0

0

0

0

⋅⋅⋅ ⋅⋅⋅ .. . mk

0

0

) ) ) 0 ) ) ) ) )

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ βk ⋅ ⋅ ⋅ βk )k×(N−n)

Here αi (i ∈ {1, . . . , n}) in A0 and βi (i ∈ {1, . . . , k}) in B1 are the same as in (1.3).

.

A matrix Harnack estimate for a Kolmogorov type equation

| 347

Indeed, if we set Xi = αi 𝜕xi

for i ∈ {1, . . . , n},

Xij = βi xi 𝜕xi − 𝜕t j

Y = 𝜕t , i−1

for ij = n + ∑ mq + j,

i ∈ {1, . . . , k},

q=1

j ∈ {1, . . . , mi },

(1.4)

then Ker(A) does not contain nontrivial subspaces. In other words, X1 , . . . , XN and Y satisfy the following Hörmander condition: Rank[Lie(X1 , . . . , XN , Y)] = N + 1,

(1.5)

see Remark 3.5. Li and Yau were the first pioneers to study differential Harnack inequalities. They proved in [11] an estimate for a positive solution to the heat equation by the maximum principle and derived a sharp Harnack estimate by integrating the proven estimate on space–time paths. Hamilton obtained a matrix Harnack estimate for the heat equation in [3], whose trace form was of great significance. Hamilton also worked out a remarkable matrix Harnack estimate for Ricci flow in [5]. Besides, Harnack inequalities for evolving hypersurfaces were proved by Cao [2] and the result was established by Andrews [1] for a general class of hypersurface flows. Hamilton [4] was able to extend his matrix Harnack estimate for the heat equation in [3] to the simple Kolmogorov type equation (1.1). In equation (1.1) of [4], the solution f (x, y, t) denotes the probability that the castaway is adrift at the point (x, y) at time t. Huang [7] generalized Hamilton’s matrix Harnack estimate in [4] to the following more general equations in higher dimensions: n

k

i=1

i=1

ft = ∑ fxi xi + ∑ xi fxn+i ,

(1.6)

where 1 ≤ k ≤ n. In equation (1.3), the coefficients αi and βi can be any positive constants and any constants, respectively, which are always 1 in [4] and [7]. Moreover, the last term of (1.3) is more general than that of (1.6). In the case of B1 in (1.2) when n = k = 1, α1 = 1, in A0 , β1 = −1, m1 = 1, equation (1.3) becomes equation (1.1), and when αj = 1 (j = 1, . . . , n) in A0 , βj = 1 (j = 1, . . . , k), mj = 1 (j = 1, . . . , k), equation (1.3) becomes equation (1.6). Therefore, equation (1.3) is more general than those in [4] and [7], see Remark 3.1 for more explanations. We now state a theorem with a matrix Harnack estimate for equation (1.3), which is the main result of this paper. Theorem 1.1. Suppose that f is a positive solution to the Kolmogorov type equation (1.3) with 1 ≤ k ≤ n and 0 < T ≤ +∞. Assume f and its derivatives (with respect to the space

348 | F. Jiang and X. Shen variables) up to the second order are bounded on any compact subinterval of (0, T). If ℓ = log f , then we have the matrix Harnack estimate M=(

M1 M3

M2 ) ≥ 0, M4

(1.7)

where M1 = (ℓxi xj )n×n + diag( M11 M2 = ( ... Mn1

M12 .. . Mn2

2 1 1 2 ,..., , ,..., ), tα1 tαk 2tαk+1 2tαn

... .. . ...

M1(N−n) .. ), . Mn(N−n)

M3 = M2T ,

(1.8)

M4 = (ℓxn+i xn+j )(N−n)×(N−n) m1

mk

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ 6 6 6 6 + diag( 3 ,..., 3 ), , . . . , , . . . , t α1 m21 β12 t 3 α1 m21 β12 t 3 αk m2k βk2 t αk m2k βk2 and Mpj ℓxp xn+j + { { { { { { { { { { { = { ℓxp xn+j , { { { { { { { { { { ℓ , { xp xn+j

3 , t 2 αp mk βp

for p ∈ {1, . . . , k}, p and j ∈ {∑p−1 q=1 mq + 1, . . . , ∑q=1 mq },

for p ∈ {1, . . . , k},

p and j ∈ {1, . . . , ∑p−1 q=1 mq } ∪ {∑q=1 mq + 1, . . . , N − n},

for p ∈ [k + 1, n] and j ∈ [1, N − n].

Since the principal diagonal elements of matrix M are nonnegative, the matrix estimate (1.7) contains much information. We will give three consequences of Theorem 1.1. Corollary 1.2. Under the assumptions in Theorem 1.1, we have n

k

∑ ℓxi xi + ∑ i=1

i=1

n 2 1 + ∑ ≥ 0. tαi i=k+1 2tαi

(1.9)

Integrating estimate (1.9) in Corollary 1.2 along an optimal path, we have the following Harnack inequality.

A matrix Harnack estimate for a Kolmogorov type equation

| 349

Corollary 1.3. Under the assumptions in Theorem 1.1, for any points (z1 , . . . , zN , t1 ) and (z1̃ , . . . , zÑ , t2 ) with 0 < t1 < t2 < T, we have t f (z1̃ , . . . , zÑ , t2 ) ≥ ( 1 ) t2 ⋅e

n+3k 2

− ∑ni=1

f (z1 , . . . , zN , t1 )

(zĩ −zi )2 − 3 4αi (t2 −t1 ) (t −t )3 2 1

∑ki=1

1 αi

(1.10)

m

[ m1β ∑i=1i (zĩ j −zij )+ 21 (zĩ +zi )(t2 −t1 )]2 i i

.

The Harnack inequalities in [4, 7] for the simple Kolmogorov equation and equation (1.6) are special cases of the Harnack inequality (1.10). By fully tracing the matrix M estimate in Theorem 1.1, we have Corollary 1.4. With the same assumption as in Theorem 1.1, we have k

Δℓ + ∑ i=1

n k 2 1 6 + ∑ +∑ 3 ≥ 0, tαi i=k+1 2tαi i=1 t αi mi βi2

(1.11)

where Δ = ∑Ni=1 𝜕x2i . In view of Theorem 1.1 and Corollaries 1.2 and 1.4, we can draw a broader conclusion. With the same assumption as in Theorem 1.1, the sum of arbitrary principal diagonal elements is nonnegative. Note that in this paper, the conclusions from [7] have been extended to the more general equation (1.3). However, it is still far from resolving the second conjecture in Section 4 of [7], which also aims to extend the matrix Harnack estimate in [7] to a more general Kolmogorov type equation. It will be interesting to further investigate the second conjecture in Section 4 of [7]. The organization of this paper is as follows. In Section 2, we present the detailed proof of Theorem 1.1. The matrix Harnack estimate of M is proved in a precise way. In Section 3, we give the proofs of Corollaries 1.2, 1.3, and 1.4. Corollaries 1.2 and 1.4 are proved by using the conclusion M ≥ 0 in Theorem 1.1. The Harnack inequality in Corollary 1.3 is proved by integrating the estimate in Corollary 1.2 along an optimal path. Finally, some remarks are made as further explanations.

2 Proof of Theorem 1.1 This section expounds the method we use to prove Theorem 1.1 under the assumption that f and its derivatives (with respect to the space variables) up to the second order are bounded on any compact subinterval of (0, T). Our proof of Theorem 1.1 is based on a partial differential equation for the matrix M in (1.7). Although the proof is similar to that in [4, 7], our matrix M is more complex than that in [4, 7]. By decomposing the matrix M into blocks, the main complexities are the blocks M2 in (1.8) and N2 in (2.5). Below an explicit way to prove Theorem 1.1 is shown in detail.

350 | F. Jiang and X. Shen Proof of Theorem 1.1. By direct calculations, the evolution of ℓ = log f is given by n

k

mi

i=1

i=1

j=1

n

k

mi

i=1

i=1

j=1

ℓt = ∑ αi (ℓxi xi + ℓx2i ) + ∑(βi xi ∑ ℓxi ),

(2.1)

j

and M in (1.7) satisfies the equation Mt = ∑ αi (Mxi xi + 2ℓxi Mxi ) + ∑(βi xi ∑ Mxi ) + 𝒩 ,

(2.2)

N1 N3

(2.3)

j

where 𝒩 =(

N2 ). N4

Note that we use 𝒩 in (2.3) to denote the matrix and use N in (1.3) to denote the dimension of the Euclidean space ℝN . In (2.3), N1 , N2 , N3 , and N4 are given by N1 = (

Q1 Q3

N11 N2 = ( ... Nn1

Q2 ), Q4 N12 .. . Nn2

(2.4) ... .. . ...

N1(N−n) .. ), . Nn(N−n)

N3 = N2T ,

(2.5)

and 2 ∑ni=1 ℓx2n+1 xi − t 4 α 18m2 β2 1 1 1 .. N4 = ( . 2 ∑ni=1 ℓxN xi ℓxN xi

...

..

. ...

2 ∑ni=1 ℓxn+1 xi ℓxN xi .. ). . n 18 2 2 ∑i=1 ℓxN xi − t 4 α m2 β2 k

k k

In (2.4), the blocks Q1 , Q2 , Q3 , and Q4 of the matrix N1 are derived from 2 ∑ni=1 ℓx21 xi + 2ℓxn+1 x1 − t 22α 1 .. Q1 = ( . 2 ∑ni=1 ℓxk xi ℓxk xi + ℓxn+k x1 + ℓxk xn+1 Q2 = ( Q3 = QT2 ,

2 ∑ni=1 ℓx1 xi ℓxk+1 xi .. .

+ ℓxn+1 xk+1

2 ∑ni=1 ℓxk xi ℓxk+1 xi + ℓxn+k xk+1

... .. . ...

... .. . ...

2 ∑ni=1 ℓx1 xi ℓxk xi + ℓxn+1 xk + ℓx1 xn+k .. ), . n 2 2 2 ∑i=1 ℓxk xi + 2ℓxn+k xk − t 2 α

2 ∑ni=1 ℓx1 xi ℓxn xi

+ ℓxn+1 xn .. ), . n 2 ∑i=1 ℓxk xi ℓxn xi + ℓxn+k xn

k

A matrix Harnack estimate for a Kolmogorov type equation

| 351

and 2 ∑ni=1 ℓx2k+1 xi − 2t 2 α1 k+1 .. Q4 = ( . 2 ∑ni=1 ℓxn xi ℓxk+1 xi

... .. . ...

2 ∑ni=1 ℓxk+1 xi ℓxn xi .. ). . n 1 2 2 ∑i=1 ℓxn xi − 2t 2 α n

In (2.5), the entries Npj of the matrix N2 are given by 2 ∑ni=1 ℓxp xi ℓxn+j xi + ℓxn+p xn+j − { { { { { { { { { { { { { 2 ∑ni=1 ℓxp xi ℓxn+j xi + ℓxn+p xn+j , { { { Npj = { { { { { { { { { { { { { 2 ∑ni=1 ℓxp xi ℓxn+j xi + ℓxn+p xn+j , { { { {

6 , t 3 αp mp βp

for p ∈ {1, . . . , k} and p j ∈ {∑p−1 q=1 mq + 1, . . . , ∑q=1 mq },

for p ∈ {1, . . . , k} and j ∈ {1, . . . , ∑p−1 q=1 mq }∪

{∑pq=1 mq + 1, . . . , N − n}, for p ∈ [k + 1, n] and j ∈ [1, N − n].

To show that M ≥ 0 is preserved, we need an eigenvector V = {v1 , . . . , vN }T ≠ 0 corresponding to a zero eigenvalue, such that if MV = 0, then 𝒩 (V, V) = 0. Indeed, from MV = 0, we have ∑Nj=1 ℓxi xj vj = − tα2 vi − t 2 α 3β m vn+1+p(i−1) − ⋅ ⋅ ⋅ − t 2 α 3β m vn+p(i) , for 1 ≤ i ≤ k, { i i i i i i i { { { N 1 for k + 1 ≤ i ≤ n, ∑j=1 ℓxi xj vj = − 2tα vi , { i { { { N 3 6 for 1 ≤ i ≤ k, { ∑j=1 ℓxn+p(i−1)+q(i) xj vj = − t 2 αi βi mi vq(i) − t 3 αi m2i βi2 vn+p(i−1)+q(i) ,

(2.6)

where p(i) = ∑ip=1 mp and q(i) = {1 + mi−1 , . . . , mi }. Then by (2.6) it is easy to get 2

N

N

i=1

j=1

k

n+k

i=1

j=1

𝒩 (V, V) = 2 ∑(∑ ℓxi xj vj ) + 2 ∑ vi ( ∑ ℓxn+i xj vj ) −

2 vi vn+p(i−1)+q(i) 18 N−n vn+i 12 1 − − ∑ ∑ ∑ αi mi βi t 4 i=1 αi m2i βi2 2t 2 i=k+1 αi t 3 i=i

vi2

n

−

2 2 k vi ∑ 2 t i=1 αi

k

(2.7)

= 0. Next, in order to make the proof rigorous, we replace f by k

f ̃ = f + ε{t 2 ∑ i=1

k

+ 2t[∑( i=1

mi n k xi2 x2 1 + ∑ i + ∑( ∑ xi2j ) 2 αi i=1 αi i=1 mi βi j=1 m

i 2kt 3 1 xi } ∑ xij ) + n] + αi mi βi j=1 3

(2.8)

352 | F. Jiang and X. Shen with ε a small positive constant, which is also a solution of the equation, so that ℓx̃ i xj → 0 for i, j = 1, . . . , N as |x| → ∞ uniformly in t, where l ̃ = log f ̃. This allows us to only look at a compact set. Let γi = 1 + γ4 λi (i = 1, 2, 3), where γ4 is a small positive constant, and λi (i = 1, 2, 3) will be chosen later. We then modify M to be ̃ := ( M

̃1 M

̃2 M

̃3 M

̃4 M

),

where ̃1 = (ℓx̃ x )n×n + diag( 2γ1 , . . . , 2γ1 , 1 + γ4 , . . . , 1 + γ4 ), M i j tα1 tαk 2tαk+1 2tαn ̃12 . . . M ̃1(N−n) ̃11 M M . . .. . ̃3 = M ̃T , ̃2 = ( . .. .. ), M M 2 . . ̃n2 . . . M ̃n(N−n) ̃n1 M M ̃4 = (ℓx̃ x )N−n×N−n M n+i n+j m1

mk

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ 6γ3 6γ3 6γ3 6γ3 + diag( 3 ,..., 3 ), ,... 3 ,..., 3 t α1 m21 β12 t α1 m21 β12 t αk m2k βk2 t αk m2k βk2 and ℓx̃ p xn+j + { { { { { { { { { { { { { { ℓx̃ p xn+j , ̃pj = M { { { { { { { { { { { { { { ̃ { ℓxp xn+j ,

3γ2 , t 2 αp mp βp

for p ∈ {1, . . . , k}, p and j ∈ {∑p−1 q=1 mq + 1, . . . , ∑q=1 mq },

for p ∈ {1, . . . , k},

and j ∈ {1, . . . , ∑p−1 q=1 mq }∪ {∑pq=1 mq + 1, . . . , N − n},

for p ∈ [k + 1, n] and j ∈ [1, N − n].

̃ satisfies the same equation (2.2) with ℓ and 𝒩 replaced by ℓ̃ and Then M ̃ N

̃=( 1 𝒩 ̃ N

3

̃2 N ̃4 ) , N

(2.9)

namely, n

k

mi

i=1

i=1

j=1

̃t = ∑ αi (M ̃x x + 2ℓx̃ M ̃x ) + ∑(βi xi ∑ M ̃x ) + 𝒩 ̃. M i i i i i j

(2.10)

̃1 , N ̃2 , N ̃3 and N ̃4 are given by In (2.9), N ̃ ̃1 = ( Q1 N ̃ Q

3

̃ Q 2 ̃ ), Q 4

(2.11)

| 353

A matrix Harnack estimate for a Kolmogorov type equation

̃11 N ̃2 = ( .. N . ̃n1 N

̃12 N .. . ̃n2 N

... .. . ...

̃1(N−n) N .. ), . ̃ Nn(N−n)

̃3 = N ̃T , N 2

(2.12)

and 18γ

2 ∑ni=1 ℓx2̃ n+1 xi − t 4 α m42 β2 1 1 1 .. ̃4 = ( N . 2 ∑ni=1 ℓx̃ N xi ℓx̃ N xi

...

..

.

...

2 ∑ni=1 ℓx̃ n+1 xi ℓx̃ N xi .. ). . 18γ4 n 2̃ 2 ∑i=1 ℓxN xi − t 4 α m2 β2 k

k k

̃ ,Q ̃ ,Q ̃ , and Q ̃ of the matrix N ̃1 are given by In (2.11), the blocks Q 1 2 3 4 1 2 ∑ni=1 ℓx2̃ 1 xi + 2ℓx̃ n+1 x1 − t2γ 2α 1 .. ̃ =( Q 1 . n ̃ ̃ 2 ∑i=1 ℓxk xi ℓxk xi + ℓx̃ n+k x1 + ℓx̃ k xn+1

̃ =( Q 2 ̃ =Q ̃T , Q 3 2

2 ∑ni=1 ℓx̃ 1 xi ℓx̃ k+1 xi .. .

+ ℓx̃ n+1 xk+1

2 ∑ni=1 ℓx̃ k xi ℓx̃ k+1 xi + ℓx̃ n+k xk+1

... .. . ...

... .. . ...

2 ∑ni=1 ℓx̃ 1 xi ℓx̃ k xi + ℓx̃ n+1 xk + ℓx̃ 1 xn+k .. ), . 2γ n 2 ∑i=1 ℓx2̃ k xi + 2ℓx̃ n+k xk − t 2 α1

2 ∑ni=1 ℓx̃ 1 xi ℓx̃ n xi + ℓx̃ n+1 xn .. ), . n 2 ∑i=1 ℓx̃ k xi ℓx̃ n xi + ℓx̃ n+k xn

k

and 1+γ

2 ∑ni=1 ℓx2̃ k+1 xi − 2t 2 α 4 k+1 .. ̃ =( Q 4 . 2 ∑n ℓx̃ x ℓx̃ x i=1

n i

k+1 i

... .. . ...

2 ∑ni=1 ℓx̃ k+1 xi ℓx̃ n xi .. ). . 1+γ4 n 2̃ 2 ∑i=1 ℓxn xi − 2t 2 α n

̃pj of the matrix N ̃2 are given by In (2.12), the entries N 2 ∑ni=1 ℓx̃ p xi ℓx̃ n+j xi + ℓx̃ n+p xn+j − { { { { { { { { { { { { { 2 ∑ni=1 ℓx̃ p xi ℓx̃ n+j xi + ℓx̃ n+p xn+j , { { { ̃pj = N { { { { { { { { { { { { { 2 ∑ni=1 ℓx̃ p xi ℓx̃ n+j xi + ℓx̃ n+p xn+j , { { { {

6γ2 , t 3 αp mp βp

for p ∈ {1, . . . , k} and p j ∈ {∑p−1 q=1 mq + 1, . . . , ∑q=1 mq },

for p ∈ {1, . . . , k} and j ∈ {1, . . . , ∑p−1 q=1 mq }∪

{∑pq=1 mq + 1, . . . , N − n}, for p ∈ [k + 1, n] and j ∈ [1, N − n].

354 | F. Jiang and X. Shen Then we have 4γ 2 − γ1 − 3γ2 k vi2 6(2γ1 γ2 − γ2 − γ3 ) k vi vn+p(i−1)+q(i) 1̃ 𝒩 (V, V) = 1 − ∑ ∑ 2 α αi mi βi t3 t2 i=i i=1 i 2 γ 2 + γ4 n vi2 9(γ22 − γ3 ) N−n vn+i + . + 4 2 ∑ ∑ 4 2 2 α t 2t i=1 αi mi βi i=k+1 i

(2.13)

̃ = 0 for a nonzero vector V = {v1 , . . . , vN }T if and only if ̃(V, V) > 0 if MV Therefore 𝒩 2 2 F = (4γ1 − γ1 − 3γ2 )(γ2 − γ3 ) − (2γ1 γ2 − γ2 − γ3 )2 > 0. Let γ1 = 1 + λ1 ,

γ2 = 1 + λ2 ,

and γ3 = 1 + λ3 ,

where 0 < λ1 , λ2 , λ3 < 1. Then F = (−4λ12 + 10λ1 λ2 − 7λ22 − 3λ1 λ3 + 5λ2 λ3 − λ32 )γ42 + R(γ4 ) 1 = (λ1 , λ2 , λ3 )F0 (λ1 , λ2 , λ3 )T γ42 + R(γ4 ), 2 where −8 F0 = ( 10 −3

10 −14 5

−3 5 ), 2

R(γ4 ) denotes the terms of γ4 with the order greater than 2. Clearly, F0 has a positive eigenvalue and two negative eigenvalues, and det F0 > 0. Then by choosing (λ1 , λ2 , λ3 )T to be an eigenvector of F0 corresponding to its positive eigenvalue, there is a positive constant γ4󸀠 such that for any 0 < γ4 < γ4󸀠 , we have F > 0. Hence, by our choices of λ1 , λ2 , λ3 , and γ4 , we have proved that ̃(V, V) > 0 𝒩

(2.14)

̃ = 0 for a nonzero vector V = {v1 , . . . , vN }T . if MV ̃ it is readily checked that M ̃ > 0 if t > 0 is sufficiently From the definition of M, ̃ > 0 for all t > 0. Suppose that there is a first time and a small. We aim to prove M ̃ point, such that M has a zero eigenvalue and the corresponding eigenvector is V. At this point, we have ̃t (V, V) ≤ 0, M

̃x (V, V) ≤ 0 M i

for i = 1, . . . , N,

and

n

̃x x ≥ 0. ∑M i i i=1

(2.15)

̃ at this time t and Then (2.14) and (2.15) lead to a contradiction to equation (2.10) for M, point x for the above V. Thus, we have proved that ̃>0 M

for all t > 0.

(2.16)

Letting γ4 → 0 and ε → 0, from (2.16) we have proved M ≥ 0 rigorously. Now, we have completed the proof of Theorem 1.1.

A matrix Harnack estimate for a Kolmogorov type equation

| 355

3 Proofs of corollaries In this section, we prove the corollaries by controlling the derivatives of ℓ. By tracing the submatrix M1 and matrix M, we can get Corollaries 1.2 and 1.4, respectively. Using Corollary 1.2, through some integration, we get Corollary 1.3. We now give the proofs of Corollaries 1.2, 1.3, and 1.4 successively. Proof of Corollary 1.2. From Theorem 1.1, we have the entire matrix M ≥ 0, so we have M1 ≥ 0, which implies that the trace of M1 is nonnegative, namely n

k

∑ ℓxi xi + ∑ i=1

i=1

n 1 2 + ∑ ≥ 0. tαi i=k+1 2tαi

Proof of Corollary 1.3. From equation (2.1), and (1.9) in Corollary 1.2, we have m

k

ℓt ≥ −(∑ i=1

Along any path with

dxij dt

n n k i 2 1 + ∑ ) + ∑ αi ℓx2i + ∑(βi xi ∑ ℓxi ). j tαi i=k+1 2tαi i=1 i=1 j=1

= −xi (0 ≤ i ≤ k), we can compute

N n dx dx dℓ n + 3k = ℓt + ∑ ℓxi i ≥ ∑(αi ℓx2i + ℓxi i ) − dt dt dt 2t i=1 i=1 2

≥−

n + 3k 1 n 1 dxi − ∑ ( ). 2t 4 i=1 αi dt

Integrating along such path from (z1 , . . . , zN , t1 ) at time t1 to (z1̃ , . . . , zÑ , t2 ) at time t2 , we get t2

2

n t n + 3k 1 1 dx log 2 − ∫ ∑ ( i ) dt. ℓ(z1̃ , . . . , zÑ , t2 ) ≥ ℓ(z1 , . . . , zN , t1 ) − 2 t1 4 i=1 αi dt

(3.1)

t1

The optimal path can minimize the integral t2

n

∫∑

t1 i=1

2

1 dxi ( ) dt αi dt

with the constraints that t2

∫ xi dt = −

t1

m

i 1 ∑(zĩ j − zij ) βi mi j=1

(1 ≤ i ≤ k).

The rest of the computations are similar to those of Huang [7] and Hamilton [4]. The dx Euler–Lagrange equations give that along the optimal path dti there are constants

356 | F. Jiang and X. Shen independent of t for 1 ≤ i ≤ k, and that dxi , dt are constants independent of t for k + 1 ≤ i ≤ n. So such path should have the form xi = 3ai t 2 + 2bi t + ci , 1 ≤ i ≤ k, xi = di t + ei , k + 1 ≤ i ≤ n, and xn+ij = −(ai t 3 + bi t 2 + ci t + fi ), where ai , bi , ci , di , and fi are constants. As in Hamilton [4], we compute the optimal path from (x1 , . . . , xN , t1 ) at time t1 to (y1 , . . . , yN , t2 ) at time t2 , using the substitution xi = xî +

zĩ − zi zi t2 − zĩ t1 + t2 − t1 t2 − t1

(1 ≤ i ≤ n).

So we should minimize t2

n

∫∑

t1 i=1

2

1 dxî ( ) dt αi dt

with the constraints that t2

∫ xî dt = −

t1

m

i 1 1 ∑(zĩ j − zij ) − (zĩ + zi )(t2 − t1 ) βi mi j=1 2

(1 ≤ i ≤ k),

and the boundary conditions xî = 0

at t = t1 and at t = t2

for 1 ≤ i ≤ n.

The solution is given by 2

m

k i 6 1 1 1 xî = [ (zĩ j − zij ) + (zĩ + zi )(t2 − t1 )] (t2 − t)(t − t1 ), ∑ ∑ 3 2 (t2 − t1 ) i=1 αi mi βi j=1

for 1 ≤ i ≤ k, and xî = 0, for k + 1 ≤ i ≤ N. Then we have t2

n

∫∑

t1 i=1

2

n (z̃ − zi )2 1 dxi ( ) dt = ∑ i αi dt α (t − t1 ) i=1 i 2 m

2

k i 12 1 1 1 + [ (zĩ j − zij ) + (zĩ + zi )(t2 − t1 )] . ∑ ∑ 3 2 (t2 − t1 ) i=1 αi mi βi j=1

(3.2)

A matrix Harnack estimate for a Kolmogorov type equation

| 357

Inserting (3.2) into (3.1), we obtain ℓ(z1̃ , . . . , zÑ , t2 ) ≥ ℓ(z1 , . . . , zN , t1 ) −

t n + 3k 1 n (z̃ − zi )2 log 2 − ∑ i 2 t1 4 i=1 αi (t2 − t1 ) m

2

k i 1 3 1 1 − ∑ [ ∑(zĩ j − zij ) + (zĩ + zi )(t2 − t1 )] . 3 2 (t2 − t1 ) i=1 αi mi βi j=1

(3.3)

Thus the Harnack inequality (1.10) in Corollary 1.3 readily follows by exponentiating (3.3). Proof of Corollary 1.4. The matrix Harnack estimate (1.7) indicates that the trace of M is nonnegative, namely k

Δℓ + ∑ i=1

n k 1 6 2 + ∑ +∑ 3 ≥ 0. tαi i=k+1 2tαi i=1 t αi mi βi2

Remark 3.1. As referred in the introduction, our equation (1.3) is reduced to the special form (1.6) in [7], when αj = 1 (j = 1, . . . , n) in A0 , βj = 1 (j = 1, . . . , k), mj = 1 (j = 1, . . . , k) in B1 . In this case, our main results in this paper correspond to those in [7]. We now give some examples of equation (1.3), which cannot be included in (1.6). Example 3.2. Let N = 8, α1 = 1, α2 = 2, α3 = 3, n = 3 in A0 , β1 = −1, m1 = 2, β2 = 1/2, m2 = 2, β3 = −5, m3 = 1, and k = 3 in B1 in (1.2), equation (1.3) becomes 1 ft = fx1 x1 + 2fx2 x2 + 3fx3 x3 − x1 (fx4 + fx5 ) + x2 (fx6 + fx7 ) − 5x3 fx8 . 2

(3.4)

Example 3.3. Let N = 7, αj = 1 (j = 1, 2, 3), n = 3 in A0 , βj = 1 (j = 1, 2), m1 = 3, m2 = 1 and k = 2 in B1 in (1.2), equation (1.3) becomes ft = fx1 x1 + fx2 x2 + fx3 x3 + x1 (fx4 + fx5 + fx6 ) + x2 fx7 .

(3.5)

By permutating x1 , x2 , x3 as x3 , x2 , x1 , and x4 , x5 , x6 , x7 as x5 , x6 , x7 , x4 , respectively, equation (3.4) can also be written as ft = fx1 x1 + fx2 x2 + fx3 x3 + x2 fx4 + x3 (fx5 + fx6 + fx7 ).

(3.6)

Equations (3.4), (3.5), and (3.6) are more complicated and different from the following example of equation (1.6): ft = fx1 x1 + fx2 x2 + fx3 x3 + x1 fx4 + x2 fx5 + x3 fx6 .

(3.7)

Remark 3.4. By straightforward calculations, we can find the following solution of equation (1.3) with a pole at the origin: f ∗ (x, t) =

c

t

n+3k 2

e

− 4t1 ∑ni=1

xi2 αi

−

3 t3

∑ki=1

1 αi

m

( m1β ∑j=1i xij + 21 txi )2 i i

,

(3.8)

358 | F. Jiang and X. Shen where x = (x1 , . . . , xN ), c is a constant. Note that f ∗ > 0 holds if the constant c > 0. Then it is easy to check that the solution f ∗ (x, t) in (3.8) with c > 0 satisfies the Harnack inequality in (1.10) in Corollary 1.3. Note that the form of the solution in (3.8) is a generalization of those in [6, 4, 7, 9]. With this fundamental solution (3.8) in hand, we can interpret the matrix Harnack inequality (1.7) in Theorem 1.1 as D2 (ℓ − ℓ∗ ) ≥ 0, where ℓ = log f and ℓ∗ = log f ∗ . Remark 3.5. Here we verify the Hörmander condition (1.5) when the vector fields X1 , . . . , XN , Y are defined in (1.4). By direct calculations, we have [Xi , Xj ] = αi 𝜕xi αj 𝜕xj − αj 𝜕xj αi 𝜕xi =0

for 1 ≤ i, j ≤ n,

[Xi , Y] = 0

for 1 ≤ i ≤ N,

and [Xi , Xij ] = αi 𝜕xi (βi xi 𝜕xi − 𝜕t ) − (βi xi 𝜕xi − 𝜕t )αi 𝜕xi j

= αi βi 𝜕xi

j

j

for 1 ≤ i ≤ k, 1 ≤ j ≤ mi .

Then it is readily checked that the Hörmander condition (1.5) holds for such vector fields X1 , . . . , XN and Y satisfying (1.4).

Bibliography [1]

B. Andrews, Contraction of convex hypersurfaces in Euclidean space. Calc. Var. Partial Differ. Equ. 2, 151–171 (1994). [2] H. D. Cao, On Harnack inequalities for evolving hypersurfaces. Math. Z. 217, 179–197 (1992). [3] R. Hamilton, A matrix Harnack estimate for the heat equation. Commun. Anal. Geom. 1(1), 113–126 (1993). [4] R. Hamilton, Li–Yau estimate and their Harnack inequalities. Geom. Anal. I(ALM 17), 329–362 (2011). [5] R. Hamilton, The Harnack estimate for Ricci flow. J. Differ. Geom. 37(1), 225–243 (1993). [6] L. Hörmander, Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967). [7] H. Huang, A matrix differential Harnack estimate for a class of ultraparabolic equation. Potential Anal. 41(3), 771–782 (2014). [8] A. Kolmogrov, Zufällige Bewegungen (Zur Theorie der Brownschen Bewegung). Ann. Math. 35(1), 116–117 (1934). [9] E. Lanconelli, S. Polidoro, On a class of hypoelliptic evolution operators, partial differential equations, II (Turin, 1993). Rend. Semin. Mat. (Torino) 52, 29–63 (1993). [10] E. Lanconelli, A. Pascucci, S. Plidoro, Linear and nonlinear ultraparabolic equations of Kolmogorov type arising in diffusion theory and in finance, in Nonlinear Problems in Mathematical Physics and Related Topics, II. International Mathematical Series, vol. 2 (Kluwer/Plenum, New York, 2002), pp. 243–265. [11] P. Li, S. T. Yau, On the parabolic kernel of the Schrödinger operator. Acta Math. 156(3–4), 153–201 (1986).

Songzi Li and Xiang-Dong Li

Entropy power concavity inequality on Riemannian manifolds and Ricci flow Abstract: In our recent works [17, 21, 22], we prove the concavity of the Shannon and Renyi entropy powers for the heat equation and the nonlinear diffusion equation associated with the usual Laplacian or the Witten Laplacian on Riemannian manifolds with CD(K, m)-condition and (K, m)-super-Ricci flows, m ∈ [n, ∞) and K ∈ ℝ. The rigidity models are the Einstein and quasi-Einstein manifolds. Inspired by Perelman’s work, we prove the convexity of the Shannon entropy power for the conjugate heat equation on Ricci flow. The corresponding rigidity models are the shrinking Ricci solitons. As an application, we prove the entropy isoperimetric inequality on complete Riemannian manifolds with nonnegative (Bakry–Emery) Ricci curvature and maximal volume growth condition. The purpose of this paper is to give a survey on our results obtained in [17, 21, 22]. Keywords: Entropy power, Einstein manifolds, NIW formula, W-entropy, Ricci flow MSC 2010: Primary 53C44, 58J35, 58J65, Secondary 60J60, 60H30

Contents 1 2 3 3.1 3.2 3.3 4 4.1

Introduction | 360 Notations and preliminary results | 363 Shannon entropy power on complete Riemannian manifolds | 367 EDI and EPCI on heat equation | 367 Proof of EDI and EPCI | 370 NIW formula and rigidity theorem | 371 Shannon entropy power on super-Ricci flows and Ricci flow | 374 Entropy dissipation formulae | 374

Acknowledgement: Research of S. Li has been supported by NSFC No. 11901569. Research of X.-D. Li has been supported by NSFC No. 11771430 and Key Laboratory RCSDS, CAS, No. 2008DP173182. X.-D. Li would like to thank Prof. N. Mok for his suggestion which led us to study Shannon and Rényi entropy powers on manifolds and Ricci flow, and is grateful to Prof. A. Grigor’yan and Prof. Y.-H. Sun for inviting him to present this work at the Nankai International Conference on Analysis and PDEs on Manifolds and Fractals. Finally, we would like to thank Prof. Guangyue Han and Prof. Fengyu Wang for helpful discussions in the beginning stage of this work, and to express our gratitude to Dr. Yu-Zhao Wang for helpful discussion, his constant interest and careful reading on our work. Songzi Li, School of Mathematics, Renmin University of China, Beijing 100872, China, e-mail: [email protected] Xiang-Dong Li, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, No. 55, Zhongguancun East Road, Beijing, 100190, China; and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China, e-mail: [email protected] https://doi.org/10.1515/9783110700763-013

360 | S. Li and X.-D. Li

4.2 4.3 5 6 7

Shannon entropy power on super-Ricci flows | 375 Shannon entropy power on conjugate heat equation for a Ricci flow | 375 Rényi entropy power for nonlinear diffusion equation on manifolds | 376 Entropy isoperimetric inequality on manifolds | 378 Remarks and further works | 382 Bibliography | 383

1 Introduction In his 1948 seminal paper [31], Shannon introduced the notion of entropy power for continuous random vectors and stated that the entropy power of the sum of two independent random vectors is not less than the sum of the entropy powers of each random vector. More precisely, let X be an n-dimensional continuous random vector with probability distribution f (x)dx, and let H(X) = H(f ) = − ∫ f (x) log f (x) dx ℝn

be the differential entropy of X or f . The (Shannon) entropy power of X or f is defined by 2

N(X) = N(f ) = e n H(X) . The entropy power inequality (EPI) says that if X and Y are two independent continuous random vectors on ℝn , then N(X + Y) ≥ N(X) + N(Y).

(1.1)

Equivalently, if f and g denote the probability density functions of X and Y respectively on ℝn , we have N(f ∗ g) ≥ N(f ) + N(g),

(1.2)

where f ∗ g denotes the convolution of f and g, i. e., the probability density function of the law of X + Y. For the first complete proof of the entropy power inequality (1.1) or (1.2), see Stam [32] and Blachman [2]. Moreover, it is known that the equality holds in (1.1) if and only if X and Y are normally distributed with proportional covariance matrices. See [32, 2] and also [6, 35]. According to Lieb [10], Dembo [7], and Dembo–Cover–Thomas [8], the Shannon– Stam entropy power inequality (1.1) is equivalent to the following inequalities for H and for N: for any λ ∈ (0, 1), it holds H(√λX + √1 − λY) ≤ λH(X) + (1 − λ)H(Y),

(1.3)

Uniqueness of positive solutions |

361

or N(√λX + √1 − λY) ≥ λN(X) + (1 − λ)N(Y).

(1.4)

Moreover, the Shannon–Stam entropy power inequality (1.1) is an analogue of the Brunn–Minkowski isoperimetric inequality in geometry: Given two compact sets A and B in ℝn , and letting A + B := {x + y, x ∈ A, y ∈ B}, one has Vol1/n (A + B) ≥ Vol1/n (A) + Vol1/n (B). Indeed, EPI (1.1) is equivalent to e

2H(X+Y) n

≥e

2H(X) n

+e

2H(Y) n

.

Let Y = √tZ, where t ≥ 0, Z is a standard Gaussian random vector on ℝn with covariance matrix In . Let pt be the standard Gaussian heat kernel on ℝn , i. e., pt (x) =

2 1 − ‖x‖ 4t , e (4πt)n/2

∀x ∈ ℝn , t > 0.

Let u(t, x) = f ∗ pt (x) = ∫ f (y) ℝn

2 1 − ‖x−y‖ 4t e dy. (4πt)n/2

Then the law of X + √tZ is given by u(t, x)dx, and u(t, x) is the unique solution to the heat equation 𝜕t u = Δu

(1.5)

with the initial datum u(0, ⋅) = f . In [5], Costa proved the concavity of the Shannon entropy power along the heat equation (1.5) on ℝn . More precisely, let H(u(t)) = − ∫ u log u dx ℝ

be the differential entropy associated to the heat distribution u(x, t)dx at time t, and 2

N(u(t)) = e n H(u(t)) be the Shannon entropy power of u(x, t)dx at time t, or equivalently, N(u(t)) = N(X + √tZ) be the Shannon entropy power of X + √tZ. Then d2 N(u(t)) ≤ 0. dt 2

(1.6)

362 | S. Li and X.-D. Li Using an argument based on the Blachman–Stam inequality [2], the original proof of the entropy power concavity inequality of (1.6) has been simplified by Dembo et al. [7, 8] and Villani [34]. In [34], Villani pointed out the possibility of extending (1.6) to Riemannian manifolds with nonnegative Ricci curvature using the Bakry–Emery Γ2 -calculation. The concavity property of the Shannon entropy power for the heat equation (1.5) has been extended to the Rényi entropy power by Savaré and Toscani [30], when evaluated along the solution to a nonlinear diffusion equation 𝜕t u = Δup .

(1.7)

When p > 1, the nonlinear diffusion equation (1.7) is called the porous medium equation, while for 0 < p < 1, it is called the fast diffusion equation. For the detailed description of Savaré and Toscani’s result, see Section 5 below. The Shannon–Stam entropy power inequality (1.1) or (1.2) and its equivalent formulation (1.3) or (1.4) essentially require the linear structure (i. e., scaling and addition of vectors) of the Euclidean spaces. Due to the lack of scaling and addition on manifolds, one cannot extend the Shannon–Stam entropy power inequality and its equivalent formulations in a straightforward way to Riemann manifolds. However, as the heat equation (1.5) and the nonlinear diffusion equation (1.7) can be formulated on Riemannian manifolds, super-Ricci flows, and more general metric measure spaces, we can study the entropy power concavity inequality (EPCI) in these settings. In [17], we proved the concavity of the Shannon entropy power along the heat equation associated with the Witten Laplacian on complete Riemannian manifolds with the Bakry–Emery curvature-dimension CD(0, m)-condition. In particular, we proved that the Shannon entropy power is concave along the heat equation 𝜕t u = Δu on complete Riemannian manifolds with nonnegative Ricci curvature. In 2019, we proved the so-called NIW formula which indicates the relationship between the Shannon entropy power N, the Fisher information I, and the Perelman W-entropy functional for the heat equation associated with the Witten Laplacian on complete Riemannian manifolds. Under the curvature-dimension CD(K, m)-condition, we proved that the rigidity models of the Shannon entropy power are Einstein or quasi-Einstein manifolds with Hessian solitons. Moreover, we proved the convexity of the Shannon entropy power along the conjugate heat equation introduced by G. Perelman for the Ricci flow. Inspired by Perelman’s beautiful W-entropy formula, we proved the NFW formula which indicates the relationship between the Shannon entropy power N, the F-functional, and the W-entropy functional introduced in Perelman’s work [29] for the Ricci flow. Similarly to Perelman’s result, we proved that the corresponding rigidity models of the Shannon entropy power are the shrinking Ricci solitons. In [17], we also proved the concavity of the Rényi entropy power along the nonlinear diffusion equation associated with the Witten Laplacian on compact Riemannian

Uniqueness of positive solutions |

363

manifolds with the Bakry–Eemry curvature-dimension CD(0, m)-condition. In particular, we proved that the Rényi entropy power is concave along the nonlinear diffusion equation 𝜕t u = Δup on compact Riemannian manifolds with nonnegative Ricci curvature. In 2020, we proved the NIW formula which indicates the relationship between the Rényi entropy power N, the Fisher information I, and the Perelman W-entropy functional for the nonlinear diffusion equation associated with the Witten Laplacian on compact Riemannian manifolds. Under the curvature-dimension CD(K, m)-condition, we proved that the rigidity models of the Rényi entropy power are Einstein or quasiEinstein manifolds with Hessian solitons. Our works have been recently completed in two papers [21, 22] which have been posted on arXiv in 2020. The purpose of this paper is to give a survey of our results obtained in [17, 21, 22]. We hope that our works can bring some new insights into the study of geometry and analysis on Riemannian manifolds, super-Ricci flows, or more general metric measure spaces via the information-theoretic approach.

2 Notations and preliminary results Let (M, g) be an n-dimensional complete Riemannian manifold, ϕ ∈ C 2 (M) and dμ = e−ϕ dv, where v is the Riemannian volume measure on (M, g). The Witten Laplacian acting on smooth functions is defined by L = Δ − ∇ϕ ⋅ ∇. For any u, v ∈ C0∞ (M), the integration by parts formula holds ∫⟨∇u, ∇v⟩ dμ = − ∫ Luv dμ = − ∫ uLv dμ. M

M

M

Thus, L is the infinitesimal generator of the Dirichlet form ℰ (u, v) = ∫⟨∇u, ∇v⟩ dμ, M

u, v ∈ C0∞ (M).

By Itô’s theory, the Stratonovich SDE on M dXt = √2Ut ∘ dWt − ∇ϕ(Xt )dt,

∇∘dXt Ut = 0,

where Ut is the stochastic parallel transport along the trajectory of Xt , with initial data X0 = x and U0 = IdTx M , defines a diffusion process Xt on M with infinitesimal generator L. Moreover, the transition probability density of the L-diffusion process Xt with respect to μ, i. e., the heat kernel pt (x, y) of the Witten Laplacian L, is the fundamental solution to the heat equation 𝜕t u = Lu.

(2.1)

364 | S. Li and X.-D. Li In [1], Bakry and Emery proved the generalized Bochner formula 󵄩2 󵄩 L|∇u|2 − 2⟨∇u, ∇Lu⟩ = 2󵄩󵄩󵄩∇2 u󵄩󵄩󵄩HS + 2Ric(L)(∇u, ∇u),

(2.2)

where u ∈ C 2 (M), ∇2 u denotes the Hessian of u, ‖∇2 u‖HS is its Hilbert–Schmidt norm, and Ric(L) = Ric + ∇2 ϕ is now called the infinite-dimensional Bakry–Emery Ricci curvature associated with the Witten Laplacian L. For m ∈ [n, ∞), the m-dimensional Bakry–Emery Ricci curvature associated with the Witten Laplacian L is defined by Ricm,n (L) = Ric + ∇2 ϕ −

∇ϕ ⊗ ∇ϕ . m−n

In view of this, we have L|∇u|2 − 2⟨∇u, ∇Lu⟩ ≥

2|Lu|2 + 2Ricm,n (L)(∇u, ∇u). m

Here we make a convention that m = n if and only if ϕ is a constant. By definition, we have Ric(L) = Ric∞,n (L). Following [1], we say that (M, g, ϕ) satisfies the curvature-dimension CD(K, m)-condition for a constant K ∈ ℝ and m ∈ [n, ∞] if and only if Ricm,n (L) ≥ Kg. Note that, when m = n, ϕ = 0, we have that L = Δ is the usual Laplacian on (M, g), and the CD(K, n)-condition holds if and only if the Ricci curvature on (M, g) is bounded from below by K, i. e., Ric ≥ Kg. On the other hand, if M = ℝn , ϕ(x) =

‖x‖2 , 2

dμ(x) =

‖x‖2

e− 2 . (2π)n/2

Then

L=Δ−x⋅∇ is the Ornstein–Uhlenbeck operator, and its Bakry–Emery Ricci curvature is given by Ric(L) = I. Thus the CD(1, ∞)-condition holds for the Ornstein–Uhlenbeck operator on a Euclidean space. Indeed, this remains true for the Ornstein–Uhlenbeck operator on the infinite-dimensional Wiener space equipped with the Wiener measure.

Uniqueness of positive solutions |

365

In the case of Riemannian manifolds with a family of time-dependent metrics and potentials, we call (M, g(t), ϕ(t), t ∈ [0, T]) a (K, m)-super-Ricci flow if the metric g(t) and the potential function ϕ(t) satisfy 1 𝜕g + Ricm,n (L) ≥ Kg, 2 𝜕t

(2.3)

where L = Δg(t) − ∇g(t) ϕ(t) ⋅ ∇g(t) is the time dependent Witten Laplacian on (M, g(t), ϕ(t), t ∈ [0, T]), and K ∈ ℝ is a constant. When m = ∞, i. e., if the metric g(t) and the potential function ϕ(t) satisfy the following inequality: 1 𝜕g + Ric(L) ≥ Kg, 2 𝜕t we call (M, g(t), ϕ(t), t ∈ [0, T]) a (K, ∞)-super-Ricci flow or a K-super-Perelman Ricci flow. Indeed the (K, ∞)-Ricci flow (called also the K-Perelman Ricci flow) 1 𝜕g + Ric(L) = Kg 2 𝜕t is a natural extension of the modified Ricci flow 𝜕g = −2Ric(L) 𝜕t introduced by Perelman [29] as the gradient flow of 2

ℱ (g, ϕ) = ∫(R + |∇ϕ| )e

−ϕ

dv

M

on ℳ × C ∞ (M) under the constraint condition that the measure dμ = e−ϕ dv is preserved. For the study of the Li–Yau or Hamilton differential Harnack inequalities, W-entropy formulas, and related functional inequalities on (K, m) or (K, ∞)-superRicci flows, see [15, 16, 17, 20, 18, 19] and references therein. For super-Ricci flows on metric and measure spaces, see [33] and references therein. Now we recall two preliminary results which play an important rôle in the proof of the main results in this paper. The first one is the Li–Yau differential Harnack inequality and the other is the Li–Yau–Hamilton differential Harnack inequality for the heat equation associated with the Witten Laplacian on complete Riemannian manifolds with CD(−K, m)-condition. When m = n, ϕ = 0, L = Δ and Ric ≥ −Kg, where K ≥ 0 is a constant, they are due to Li–Yau [25] and Hamilton [9].

366 | S. Li and X.-D. Li Theorem 2.1 ([25, 9, 12, 18]). Let (M, g) be a complete Riemannian manifold, ϕ ∈ C 2 (M). Suppose that there exist some constants m ∈ [n, ∞) and K ≥ 0 such that Ricm,n (L) ≥ −Kg. Let u be a positive solution of the heat equation 𝜕t u = Lu. Then the Li–Yau differential Harnack inequality holds: for all α > 1, t > 0, 𝜕t u mα2 mα2 K |∇u|2 ≤ + , − α u 2t 2(α − 1) u2

(2.4)

and the Li–Yau–Hamilton differential Harnack inequality holds, 𝜕u m |∇u|2 − e2Kt t ≤ e4Kt . 2 u 2t u

(2.5)

In particular, if Ricm,n (L) ≥ 0, then the Li–Yau differential Harnack inequality holds, |∇u|2 𝜕t u m − ≤ . u 2t u2 According to [13, 14], we say that (M, g, ϕ) satisfies the bounded geometry condition if the Riemannian curvature tensor and its covariant derivatives are uniformly bounded up to the third order, and ϕ ∈ C 4 (M) such that ∇k ϕ are uniformly bounded on M for k = 1, 2, 3, 4. The following entropy dissipation formulae will play an important rôle in this paper. When M is compact, it is a well-known result due to Bakry and Emery [1]. Theorem 2.2. Let (M, g, ϕ) be a weighted complete Riemannian manifold satisfying the bounded geometry condition. Let u be the fundamental solution to the heat equation 𝜕t u = Lu for the Witten Laplacian L = Δ − ∇ϕ ⋅ ∇. Let H(u) = − ∫ u log u dμ.

(2.6)

M

Then d H(u) = ∫ |∇ log u|2 u dμ, dt

(2.7)

M

d2 H(u) = −2 ∫ Γ2 (log u, log u)u dμ, dt 2

(2.8)

M

where 󵄩 󵄩2 Γ2 (log u, log u) := 󵄩󵄩󵄩∇2 log u󵄩󵄩󵄩HS + Ric(L)(∇ log u, ∇ log u).

(2.9)

Indeed, for the validity of the first-order entropy dissipation formula (2.7), we need only to assume Ricm,n (L) ≥ Kg for some constants m ∈ [n, ∞) and K ∈ ℝ; see Theorem

Uniqueness of positive solutions |

367

4.1 in [13]. However, as explained in the proof of Theorem 4.3 in [13], for the validity of the second-order entropy dissipation formula (2.8), we need to verify that ∫( M

|Lu|2 |∇Lu|2 + ) dμ < ∞. u u

(2.10)

In [13, 14], it is proved that if (M, g, ϕ) satisfies the bounded geometry condition, then for any T > 0 we have that k

d(x, y) 1 󵄨󵄨 k 󵄨 + ) pt (x, y), 󵄨󵄨∇ pt (x, y)󵄨󵄨󵄨 ≤ Ck (K, m, T)( √t t

(2.11)

holds for all x, y ∈ M and t ∈ (0, T], where Ck (K, m, T) is a constant depending only on k, m, K and T, k = 1, 2, 3. This further implies that (2.10) holds, and hence the second entropy dissipation formula (2.8) can be derived. We would like to remark that the uniform boundedness condition required on ∇ϕ is not sharp, as we can see when 2 M = ℝn and ϕ(x) = K‖x‖ . Then L = Δ − Kx ⋅ ∇ is the Ornstein–Uhlenbeck operator. The 2 heat kernel to 𝜕t u = Lu with respect to the Lebesgue measure on ℝn is given by pt (x, y) = (

n/2

K ) 2π(1 − e−2Kt )

exp(−

K‖y − e−Kt x‖2 ), 2(1 − e−2Kt )

see [20]. In this case, the entropy dissipation formulae still hold but ∇ϕ = Kx is not uniformly bounded on ℝn . So, we would like to say that the “bounded geometry condition” on (M, g, ϕ) is only a geometric condition which ensures that the entropy dissipation formulae hold on complete Riemannian manifolds.

3 Shannon entropy power on complete Riemannian manifolds In this section, we study the Shannon entropy power for the heat equation associated with the Witten Laplacian on complete Riemannian manifolds.

3.1 EDI and EPCI on heat equation Now we state the main results of this section. Theorem 3.1. Let (M, g, ϕ) be an n-dimensional complete Riemannian manifold satisfying the bounded geometry condition. Suppose that Ricm,n (L) ≥ Kg for some constants m ≥ n and K ∈ ℝ. Let u(t) =

e−f (4πt)m/2

be the fundamental solution to the heat equation

368 | S. Li and X.-D. Li 𝜕t u = Lu. Let 2

Nm (u(t)) = e m H(u(t)) .

H(u(t)) = − ∫ u log u dμ, M

Then the entropy differential inequality EDI(K, m) holds, H 󸀠󸀠 +

2H 󸀠 2 + 2KH 󸀠 ≤ 0, m

with the initial boundary condition limt→0+ tH 󸀠 (u(t)) ≤ power concavity inequality EPCI(K, m) holds,

(3.1) m . 2

Equivalently, the entropy

d2 Nm dN ≤ −2K m , dt dt 2

(3.2)

or equivalently, d 2Kt d (e N (u(t))) ≤ 0. dt dt m Moreover, the equality in (3.1) or (3.2) holds on (0, T] for some T > 0 if and only if (M, g) is a quasi-Einstein manifold with Hessian soliton Ricm,n (L) = Kg,

∇2 f =

HK󸀠 (t) g, m

where HK󸀠 is the solution to the entropy differential equation HK󸀠󸀠 +

2 󸀠2 H + 2KHK󸀠 = 0, m K

lim tHK󸀠 (t) ≤

t→0+

m . 2

In particular, if Ricm,n (L) ≥ 0, then Nm (u(t)) is concave on (0, ∞), i. e., d2 Nm ≤ 0. dt 2

(3.3)

Moreover, the equality in (3.3) holds on (0, T] for some T > 0 if and only if M is isometric to ℝn , ϕ is a constant, m = n, and f (x, t) =

‖x‖2 , 4t

∀x ∈ ℝn , t > 0.

That is to say, the equality in (3.3) holds on (0, T] for some T > 0 if and only if M is the Euclidean space with the Gaussian Ricci soliton. In particular, when m = n, ϕ = C, and L = Δ, we have the following

Uniqueness of positive solutions |

369

Theorem 3.2. Let M be an n-dimensional complete Riemannian manifold satisfying the e−f bounded geometry condition. Suppose that Ric ≥ Kg for some K ∈ ℝ. Let u(t) = (4πt) n/2 be the fundamental solution to the heat equation 𝜕t u = Δu. Let 2

Nn (u(t)) = e n H(u(t)) .

H(u(t)) = − ∫ u log u dμ, M

Then the entropy differential inequality EDI(K, n) holds, H 󸀠󸀠 +

2H 󸀠 2 + 2KH 󸀠 ≤ 0, n

with the initial boundary condition limt→0+ tH 󸀠 (u(t)) ≤ power concavity inequality EPCI(K, n) holds,

(3.4) n . 2

Equivalently, the entropy

dN d2 Nn ≤ −2K n , 2 dt dt

(3.5)

d 2Kt d (e N (u(t))) ≤ 0. dt dt n

(3.6)

or equivalently,

Moreover, the equality in (3.4) or (3.5) holds on (0, T] for some T > 0 if and only if (M, g) is an Einstein manifold with Hessian soliton Ric = Kg,

∇2 f =

HK󸀠 (t) g, n

(3.7)

where HK󸀠 is the solution to the entropy differential equation HK󸀠󸀠 +

2 󸀠2 H + 2KHK󸀠 = 0, n K

lim tHK󸀠 (t) ≤

t→0+

n . 2

In particular, if Ric ≥ 0, then N(u(t)) is concave on (0, ∞), i. e., d 2 Nn ≤ 0. dt 2

(3.8)

Moreover, the equality in (3.8) holds on (0, T] for some T > 0 if and only if M is isometric to ℝn , and f (x, t) =

‖x‖2 , 4t

∀x ∈ ℝn , t > 0.

That is to say, the equality in (3.8) holds if and only if M is the Euclidean space ℝn with the Gaussian Ricci soliton.

370 | S. Li and X.-D. Li

3.2 Proof of EDI and EPCI In this subsection, we give a proof of the entropy differential inequality and the entropy power concavity inequality in Theorems 3.2 and 3.1. By Theorem 2.2, and using |L log u|2 󵄩2 󵄩󵄩 2 + Ricm,n (L)(∇ log u, ∇ log u), 󵄩󵄩∇ log u󵄩󵄩󵄩HS + Ric(L)(∇u, ∇u) ≥ m we have 1 |L log u|2 − H 󸀠󸀠 (u) ≥ ∫[ + Ricm,n (L)(∇ log u, ∇ log u)]u dμ. 2 m M

By the Cauchy–Schwarz inequality and integration by parts, we have 2

2

∫ |L log u|2 u d ≥ (∫ L log uu dμ) = (∫ |∇ log u|2 u dμ) = H 󸀠 (u)2 . M

M

M

Thus, under the CD(K, m)-condition, we can derive EDI(K, m), i. e., 1 1 − H 󸀠󸀠 (u) ≥ H 󸀠 (u)2 + KH 󸀠 (u). 2 m Note that 2H 󸀠 N , m m 󸀠󸀠 2Nm 2H 4H 󸀠 2 2H 󸀠 2 󸀠󸀠 󸀠󸀠 Nm = Nm + N = (H + ). m m m m2 m 󸀠 Nm =

(3.9)

From (3.9) we see that EDI(K, m) is equivalent to EPCI(K, m), i. e., 󸀠󸀠 󸀠 Nm ≤ −2KNm .

In particular, under the CD(0, m)-condition, we have 󸀠󸀠 Nm ≤ 0.

On the other hand, by Theorem 2.2, we have H 󸀠 (u(t)) = ∫ M

|∇u|2 dμ. u

By [12], under the condition Ricm,n (L) ≥ Kg, the solution to the heat equation 𝜕t u = Lu is unique in L∞ , and ∫M 𝜕t u dμ = ∫M Lu dμ = 0. Thus, for any function α : [0, ∞) → ℝ, we have H 󸀠 (u(t)) = ∫[ M

𝜕u |∇u|2 − α(t) t ]u dμ. u u2

Uniqueness of positive solutions | 371

Let K − = max{0, −K}. By the Li–Yau Harnack estimate (2.4), for any α > 1 and t > 0, we have H 󸀠 (u(t)) ≤

mα2 mα2 K − + , 2t 2(α − 1)

or, by the Hamilton-type Harnack estimate (2.5), for any t > 0, we have H 󸀠 (u(t)) ≤

m 4K − t e . 2t

From each of the above differential Harnack inequalities, we can derive that lim tH 󸀠 (u(t)) ≤

t→0+

m . 2

3.3 NIW formula and rigidity theorem In [29], Perelman introduced the ℱ -functional and reformulated the Ricci flow as the gradient flow of the ℱ -functional. He then introduced the 𝒲 -entropy functional and proved its monotonicity along the conjugate equation coupled with the Ricci flow. The ℱ -functional has been used by Perelman to characterize the steady gradient Ricci solitons, and the 𝒲 -entropy has been used to characterize the shrinking gradient Ricci solitons. As an application of the 𝒲 -entropy formula, Perelman proved the nonlocal collapsing theorem for the Ricci flow, which plays an important rôle for ruling out cigars, the one part of the singularity classification for the final resolution of the Poincaré and geometrization conjectures. Since Perelman’s preprint [29] was published on arXiv in 2002, many people have studied the 𝒲 -type entropy for other geometric flows on Riemannian manifolds. In [27, 28], Ni proved the W-entropy formula for the heat equation 𝜕t u = Δu on compact Riemannian manifolds with nonnegative Ricci curvature, In [26], Li and Xu extended Ni’s W-entropy formula to compact Riemannian manifolds with Ricci curvature bounded from below by a negative constant. In [13, 14], the second author extended Ni’s W-entropy formula to the heat equation 𝜕t u = Lu associated with the Witten Laplacian on complete Riemannian manifolds with CD(0, m)-condition. In [15, 18, 19, 20], the authors of this paper extended the W-entropy formula to the heat equation 𝜕t u = Lu associated with the Witten Laplacian on complete Riemannian manifolds with CD(K, m)-condition and on compact (K, m)-super Ricci flows, where K ∈ ℝ and m ∈ [n, ∞]. In [13, 14], a probabilistic interpretation of the W-entropy for the Ricci flow was given. See also [14, 15, 18, 19]. In [11], the monotonicity and the rigidity of the W-entropy were extended to the so-called RCD(0, N) metric measure spaces. In [29], Perelman pointed out that there is an essentially close connection between the Li–Yau–Hamilton–Perelman differential Harnack quantity and the 𝒲 -entropy for

372 | S. Li and X.-D. Li the conjugate heat equation on Ricci flow. In [27, 28, 26, 13, 14, 15, 18, 19], such a connection has been further developed between the W-entropy an the Li–Yau or the Li– Yau–Hamilton type differential Harnack quantity for the heat equation, 𝜕t u = Δu or 𝜕t u = Lu, associated with the Witten Laplacian on compact and complete Riemannian manifolds with Ric ≥ Kg or Ricm,n (L) ≥ Kg. On the other hand, at least from the technical point of view, we see that from the proof of EDI and EPCI, the EDI(K, m) is equivalent to EPCI(K, m), and the initial conditions limt→0+ tH 󸀠 (u(t)) ≤ n2 and limt→0+ tH 󸀠 (u(t)) ≤ m2 are derived from the Li–Yau or the Li–Yau–Hamilton type differential Harnack inequality for the heat equation, 𝜕t u = Δu or 𝜕t u = Lu. In this subsection, we find an interesting formula between the Shannon entropy power N, the Fisher information I, and the W-entropy for the heat equation associated with the Laplacian or the Witten Laplacian on complete Riemannian manifolds. We call it the NIW formula. It leads us to derive an explicit formula for the second derivative of the Shannon entropy power, which allows us to prove the rigidity theorem of the Shannon entropy power. Let Hm (u(t)) = H(u(t)) −

m log(4πet). 2

(3.10)

Inspired by Perelman [29], we introduce the W-entropy by the Boltzmann entropy formula (see [13, 14, 15]) Wm (u(t)) = Let u(t) =

e−f . (4πt)m/2

d (tHm (u(t))). dt

(3.11)

By [13, 14, 15], we have Wm (u(t)) = ∫(t|∇f |2 + f − m)u dμ. M

Moreover, the following W-entropy formula has been proved in [13, 14, 15]: 2 󵄨󵄨 d g 󵄨󵄨󵄨 󵄨 Wm (u(t)) = −2 ∫(t 󵄨󵄨󵄨∇2 f − 󵄨󵄨󵄨 + Ricm,n (L)(∇f , ∇f ))u dμ 󵄨󵄨 dt 2t 󵄨󵄨 M

−

2

m−n 2 ) u dμ. ∫ t(∇ϕ ⋅ ∇f + m−n 2t

(3.12)

M

In the case m = n, ϕ = 0, and L = Δ, this formula is due to Ni [27]. Note that Wm (u(t)) = H(u(t)) −

m m log(4πet) + tH 󸀠 (u(t)) − 2 2

(3.13)

Uniqueness of positive solutions | 373

and

Hence

m d W (u(t)) = tH 󸀠󸀠 (u(t)) + 2H 󸀠 (u(t)) − . dt m 2t H 󸀠󸀠 +

(3.14)

2

2 󸀠2 2 m 1 d H = (H 󸀠 − ) + W (u(t)). m m 2t t dt m

By the fact that 󸀠󸀠 Nm =

2Nm 2 (H 󸀠󸀠 + H 󸀠 2 ), m m

we have the following NIW formula which has its own interest. Theorem 3.3. Let (M, g, ϕ) be a complete Riemannian manifold satisfying the bounded e geometry condition, and u(t) = (4πt) m/2 be the fundamental solution to the heat equation 𝜕t u = Lu. Then the following NIW formula holds: −f

2

d2 Nm 2Nm 2 m 1 dWm = [ (I − ) + ], 2 m m 2t t dt dt

(3.15)

where I(u) = ∫M |∇ log u|2 u dμ = H 󸀠 (u) is the Fisher information. Moreover, 2

2 m d 2 Nm = − 2 ∫ Ricm,n (L)(∇f , ∇f )u dμ − ∫[Lf − ∫ Lfu dμ] u dμ 2Nm dt 2 m M

M

2

M

2 󵄩󵄩 m−n n Δf 󵄩󵄩󵄩󵄩 󵄩 − 2 ∫[ (Δf + ∇ϕ ⋅ ∇f ) + 󵄩󵄩󵄩∇2 f − g 󵄩󵄩 ]u dμ. 󵄩󵄩 mn m−n n 󵄩󵄩HS

(3.16)

M

In particular, when m = n and ϕ = C, we have the following NIW formula for the heat equation associated with the usual Laplacian on complete Riemannian manifolds. Theorem 3.4. Let M be a complete Riemannian manifold satisfying the bounded gee−f ometry condition, and u(t) = (4πt) n/2 be the fundamental solution to the heat equation 𝜕t u = Δu. Then the following NIW formula holds: 2

d2 Nn 2Nn 2 n 1 dWn [ (I − ) + ], = n n 2t t dt dt 2

(3.17)

where I(u) = ∫M |∇ log u|2 u dv = H 󸀠 (u) is the Fisher information. Moreover, 2 󵄩󵄩 4Nn d 2 Nn I 󵄩󵄩󵄩󵄩 󵄩󵄩 2 = − g (Ric(∇f , ∇f ) + )u dv. ∇ f − ∫ 󵄩 󵄩 󵄩󵄩 n n 󵄩󵄩󵄩HS dt 2 󵄩

(3.18)

M

Based on the explicit formula of the second derivative of the Shannon entropy power, we can prove the rigidity part of Theorem 3.2 and Theorem 3.1. For details, see [21].

374 | S. Li and X.-D. Li

4 Shannon entropy power on super-Ricci flows and Ricci flow 4.1 Entropy dissipation formulae Let (M, g(t), ϕ(t), t ∈ [0, T]) be a compact manifold with a family of time dependent Riemannian metrics g(t) and C 2 -smooth potentials ϕ(t), t ∈ [0, T]. Assume that the conjugate heat equation holds 𝜕ϕ 𝜕g = Tr( ). 𝜕t 𝜕t

Then

(4.1)

dμ = e−ϕ(t) d volg(t) is kept invariant in t, i. e., 𝜕 dμ = 0. 𝜕t

Let

L = Δg(t) − ∇g(t) ϕ(t) ⋅ ∇g(t) be the time-dependent Witten Laplacian on (M, g(t), ϕ(t), t ∈ [0, T]). Let u be a positive solution to the heat equation 𝜕t u = Lu, with ∫M u(0)dμ = 1. Then. ∫ u(t) dμ = 1,

∀t ∈ [0, T].

M

Therefore the Boltzmann–Shannon entropy H(u(t)) = − ∫ u log u dμ M

is well defined on [0, T]. Theorem 4.1 ([15, 18]). The following entropy dissipation formulae hold: d H(u(t)) = ∫ |∇ log u|2 u dμ, dt

(4.2)

M

d2 H(u(t)) = −2 ∫ Γ2 (∇ log u, ∇ log u)u dμ, dt 2 M

where

1 𝜕g 󵄨 󵄨2 Γ2 (∇ log u, ∇ log u) = 󵄨󵄨󵄨∇2 log u󵄨󵄨󵄨 + ( + Ric(L))(∇ log u, ∇ log u). 2 𝜕t

(4.3)

Uniqueness of positive solutions | 375

4.2 Shannon entropy power on super-Ricci flows Similarly to Theorem 3.3, we can prove the NIW formula for the heat equation associated with the time-dependent Witten Laplacian on compact manifolds equipped with time-dependent metrics and potentials. This leads us to prove to the following Theorem 4.2. Let (M, g(t), ϕ(t), t ∈ [0, T]) be a compact (K, m)-super-Ricci flow in the sense that 1 𝜕g + Ricm,n (L) ≥ Kg, 2 𝜕t

𝜕ϕ 1 𝜕g = Tr( ). 𝜕t 2 𝜕t

e Let u(t) = (4πt) m/2 be a positive solution to the heat equation 𝜕t u = Lu. Then the entropy differential inequality (3.1) and the entropy power concavity inequality (3.2) hold on (0, T]. Moreover, the equality in (3.1) or (3.2) holds if and only if (M, g(t), ϕ(t)) is a (K, m)-Ricci flow −f

𝜕ϕ 1 𝜕g = Tr( ) = 0, 𝜕t 2 𝜕t

1 𝜕g + Ricm,n (L) = Kg, 2 𝜕t and f is a Hessian soliton ∇2 f =

HK󸀠 (t) g, m

where HK󸀠 is the solution to the entropy differential equation HK󸀠󸀠 +

2 󸀠2 H + 2KHK󸀠 = 0, m K

lim tHK󸀠 (t) ≤

t→0+

m . 2

For the details of the proof, see [21].

4.3 Shannon entropy power on conjugate heat equation for a Ricci flow The following result extends Theorems 3.2 and 3.1 to the conjugate heat equation introduced by Perelman [29] for the Ricci flow. It provide us a new understanding for Perelman’s mysterious 𝒲 -entropy for Ricci flow and can be used to characterize the shrinking Ricci solitons. Theorem 4.3. Let (M, g(t), t ∈ [0, T]) be a compact Ricci flow 𝜕t g = −2Ricg(t) . Let u(t) be the fundamental solution to the conjugate heat equation 𝜕t u = −Δu + Ru,

376 | S. Li and X.-D. Li where R is the scalar curvature on (M, g(t)). Let 2

H(u(t)) = − ∫ u log u dv,

𝒩 (u(t)) = e n

H(u(t))

.

M

Then the 𝒩 ℱ𝒲 formula holds, 2

d2 𝒩 2𝒩 2 n 1 d𝒲 = [ (ℱ − ) + ], 2 n n 2τ τ dt dt where 2

ℱ = ∫(R + |∇ log u| )u dv, M

󵄨󵄨 󵄨 󵄨󵄨

2

𝒲 = 2τ ∫󵄨󵄨󵄨Ric + ∇ f − M

2

g 󵄨󵄨󵄨󵄨 󵄨 u dv. 2τ 󵄨󵄨󵄨

In particular, the Shannon entropy power is convex on (0, T], i. e., d2 𝒩 (u(τ)) ≥ 0. dt 2 2

d Moreover, dt 2 𝒩 (u(τ)) = 0 holds at some τ = τ0 ∈ (0, T] if and only if (M, g(τ), u(τ)) is a shrinking Ricci soliton

Ric + ∇2 log u =

g . 2τ

For the details of the proof, see [21].

5 Rényi entropy power for nonlinear diffusion equation on manifolds For p > 0, p ≠ 1, σ(n, p) = p − 1 + n2 . The pth Rényi entropy of a probability density f on ℝn is defined by Hp (f ) =

1 log ∫ f p (x) dx, 1−p ℝn

and the pth Rényi entropy power on ℝn is given by Nn,p (f ) = exp(σ(n, p)Hp (f )).

(5.1)

Uniqueness of positive solutions | 377

The Rényi entropy for p = 1 is defined as the limit of Hp (f ) as p → 1. H1 (f ) = lim Hp (f ) = H(f ) = − ∫ f (x) log f (x) dx. p→1

ℝn

Therefore, the Shannon entropy can be identified with the Rényi entropy of index p = 1, and the Shannon entropy power Nn (f ) is the limit of Nn,p (f ) as p → 1. In [30], Savaré and Toscani proved the concavity of the Rényi entropy power along the nonlinear diffusion equation on ℝn . More precisely, let p > 1 − 2/n and let u be a probability density on ℝn solving the nonlinear diffusion equation 𝜕t u = Δup . Then the pth Rényi entropy power on ℝn is concave in t ∈ (0, ∞), i. e., d2 N (u(t)) ≤ 0. dt 2 n,p In [17, 22], we proved the following result which extends Savaré and Toscani’s result to compact Riemannian manifolds. Theorem 5.1. Let M be an n-dimensional compact Riemannian manifold, ϕ ∈ C 2 (M). Suppose that Ricm,n (L) ≥ 0. Let p > 1 − 2/m and u be a probability density function on M solving the nonlinear diffusion equation 𝜕t u = Lup . Let σ(m, p) = p − 1 +

2 , m

(5.2)

and define the pth Rényi entropy power for (5.2), by Nm,p (u) = exp(σ(m, p)Hp (u)).

Then Nm,p (u(t)) is concave in t ∈ (0, ∞), i. e., d2 N (u(t)) ≤ 0. dt 2 m,p In particular, when m = n, ϕ = C be a constant and L = Δ, we have Theorem 5.2. Let M be an n-dimensional compact Riemannian manifold with nonnegative Ricci curvature. Let p > 1 − 2/n and u be a probability density function on M solving the nonlinear diffusion equation 𝜕t u = Δup .

(5.3)

378 | S. Li and X.-D. Li Let σ(n, p) = p − 1 + n2 , and define the pth Rényi entropy power for (5.3), by Nn,p (u) = exp(σ(n, p)Hp (u)). Then Nn,p (u(t)) is concave in t ∈ (0, ∞), i. e., d2 N (u(t)) ≤ 0. dt 2 n,p Moreover, we can prove the concavity inequality of the Rényi entropy power along the nonlinear diffusion equation associated with the Witten Laplacian on compact Riemannian manifolds with the Bakry–Eemry curvature-dimension CD(K, m)-condition. The NIW formula can be also established for the Rényi entropy N, the Fisher information I and the W-entropy for the nonlinear diffusion equation associated with the Witten Laplacian on compact Riemannian manifolds. The rigidity theorem can be also proved for the Rényi entropy power on compact Riemannian manifolds with the curvature-dimension CD(K, m)-condition. For the details, see [17, 22].

6 Entropy isoperimetric inequality on manifolds Taking g = pt in the Shannon–Stam entropy power inequality (1.2), where pt (x) =

‖x‖2 1 e− 4t n/2 (4πt)

is the density of the n-dimensional Gaussian N(0, tIn ) distribution with variance tIn , denoting u(t) = f ∗ pt , and using the fact H(pt ) =

n log(4πet), 2

we have N(u(t)) ≥ N(u(0)) + 4πet, which yields d 󵄨󵄨󵄨󵄨 N(u(t)) − N(u(0)) ≥ 4πe. 󵄨 N(u(t)) = lim t→0 dt 󵄨󵄨󵄨t=0 t Note that d 󵄨󵄨󵄨󵄨 2 󵄨 N(u(t)) = N(f )I(f ), dt 󵄨󵄨󵄨t=0 n

Uniqueness of positive solutions | 379

where I(f ) = ∫ ℝn

|∇f |2 dx f

is the Fisher information of the probability distribution f (x)dx on ℝn . Thus, we have the following entropy isoperimetric inequality (EII): N(f )I(f ) ≥ 2πen.

(6.1)

As an application of the concavity of the Shannon entropy power on Riemannian manifolds, we can extend the entropy isoperimetric inequality to complete Riemannian manifolds with nonnegative Ricci curvature and with maximal volume growth condition. Theorem 6.1. Let M be an n-dimensional complete Riemannian manifold with Ric ≥ 0. Suppose that there exists a constant Cn > 0 such that the maximal volume growth condition holds, Vol(B(x, r)) ≥ Cn r n ,

∀x ∈ M, r > 0.

Let u be the fundamental solution to the heat equation 𝜕t u = Δu. Then the following isoperimetric inequality for Shannon entropy power holds: for any probability distribution fdv on M such that I(f ) and H(f ) are well-defined, we have 2

I(f )N(f ) ≥ γn := 2πenκ n ,

(6.2)

where κ := lim inf

r→∞ x∈M

V(B(x, r)) , ωn r n

and ωn denotes the volume of the unit ball in ℝn . Equivalently, the Stam-type logarithmic Sobolev inequality holds: for any smooth function f such that ∫M f 2 dv = 1 and ∫M |∇f |2 dv < ∞, we have ∫ f 2 log f 2 dv ≤ M

4 n log( ∫ |∇f |2 dv). 2 γn M

Indeed, on complete Riemannian manifolds with nonnegative Ricci curvature, the entropy power concavity inequality EPCI(0, n) (see Theorem 3.2) holds d2 N(u(t)) ≤ 0. dt 2

380 | S. Li and X.-D. Li d d N(u(t)) is nonincreasing in t and limt→∞ dt N(u(t)) exists. Moreover, using the Thus, dt first-order entropy dissipation formula in Theorem 2.2, we have

2 d d 󵄨󵄨󵄨 N(u(t)) ≤ 󵄨󵄨󵄨 N(u(t)) = I(f )N(f ). dt dt 󵄨󵄨t=0 n Thus 2 d I(f )N(f ) ≥ lim N(u(t)). t→∞ dt n

(6.3)

Again, the first-order entropy dissipation formula in Theorem 2.2 implies d 2 N(u(t)) = I(u(t))N(u(t)) ≥ 0. dt n Thus N(u(t)) is nondecreasing in t, and we have lim

t→∞

N(u(t)) d N(u(t)) = lim . t→∞ dt t

(6.4)

It remains to prove that limt→∞ N(u(t)) is finite and to find its exact value in terms of t geometric constant of manifolds. Let Hn (u(t)) = H(u(t)) −

n log(4πet). 2

Then 2 N(u(t)) = (4πe)e n Hn (u(t)) . t

(6.5)

Under the condition Ric ≥ 0, the Li–Yau Harnack inequality yields d n H (u(t)) = − ∫[Δ log u + ]u dμ ≤ 0. dt n 2t M

Thus the limit limt→∞ Hn (u(t)) exists, and we need only to prove that limt→∞ Hn (u(t)) is finite and to find its exact value in terms of geometric constant of manifolds. The following result was proved by L. Ni [27] using sharp heat kernel estimate on complete Riemannian manifolds with nonnegative Ricci curvature. Proposition 6.2 ([27]). Let M be an n-dimensional complete Riemannian manifold with Ric ≥ 0 and the maximal volume growth condition: there exists a constant Cn > 0 such that Vol(B(x, r)) ≥ Cn r n ,

∀x ∈ M, r > 0.

Uniqueness of positive solutions | 381

Let u(t, x) = pt (x, o) be the fundamental solution to the heat equation 𝜕t u = Δu, where o ∈ M is fixed. Then lim Hn (u(t)) = log κ,

t→∞

(6.6)

where κ := lim inf

r→∞ x∈M

V(B(x, r)) , ωn r n

and ωn denotes the volume of the unit ball in ℝn . Proof of Theorem 6.1. The entropy isoperimetric inequality (6.2) follows from (6.3), (6.4), (6.5), and (6.6) in Proposition 6.2. In general, let γn be a positive constant and assume the following entropy isoperimetric inequality holds: I(f )N(f ) ≥ γn .

(6.7)

Then, equivalently, the Stam logarithmic Sobolev inequality (LSI) holds, ∫ f log f dv ≤

M

M 2

|∇f |2 1 n log( ∫ dv). 2 γn f

2

2

Replacing f by f with ∫M f dv = 1 and ∫M |∇f | dv < ∞, the above LSI is equivalent to the Stam-type logarithmic Sobolev inequality ∫ f 2 log f 2 dx ≤ M

n 4 log( ∫ |∇f |2 dv). 2 γn

(6.8)

M

Remark 6.3. We would like to mention that, if M is an n-dimensional complete Riemannian manifold with nonnegative Ricci curvature on which the logarithmic Sobolev inequality (6.8) holds with the same constant γn = 2πen as on the n-dimensional Euclidean space, then M must be isometric to ℝn ; see [4, 27]. The above argument can be extended to the general case of weighted complete Riemannian manifolds with the CD(0, m) and maximal volume growth conditions. Indeed, by similar argument as used for the proof of Proposition 6.2, and based on twosided heat kernel estimates and the maximal volume growth property, we can prove the following isoperimetric inequality for Shannon entropy power on weighted complete Riemannian manifolds. For details, see [21]. Theorem 6.4. Let M be an n-dimensional complete Riemannian manifold with Ricm,n (L) ≥ 0, where m ∈ ℕ and m > n. Suppose that there exists a constant Cm > 0 such that the maximal volume growth condition holds, i. e., μ(B(x, r)) ≥ Cm r m ,

∀x ∈ M, r > 0.

382 | S. Li and X.-D. Li Let u be the fundamental solution to the heat equation 𝜕t u = Lu. Then lim Hm (u(t)) = log κ,

t→∞

where κ := lim inf

r→∞ x∈M

V(B(x, r)) , ωm r m

and ωm denotes the volume of the unit ball in ℝm . Furthermore, the following entropy isoperimetric inequality holds: for any probability distribution fdμ such that I(f ) and H(f ) are well-defined, we have 2

I(f )N(f ) ≥ γm := 2πemκ m . Equivalently, the Stam-type logarithmic Sobolev inequality holds: for any smooth function f such that ∫M f 2 dμ = 1 and ∫M |∇f |2 dμ < ∞, we have ∫ f 2 log f 2 dμ ≤ M

m 4 log( ∫ |∇f |2 dμ). 2 γm M

In [22], we have used the concavity inequality of the Rényi entropy power to derive the entropy isoperimetric inequality and the Gagliardo–Nirenberg–Sobolev inequality on Riemannian manifolds with CD(0, m)-condition and maximal volume growth condition. To save the length of the paper, we refer the reader to [22] for the details.

7 Remarks and further works After we finished the earlier version of our paper [21] in September 2019, we found that the entropy differential inequality EDI(K, m) in Theorem 3.1 has been already proved by D. Bakry [3] on compact Riemannian manifolds, but he did not introduce the notion of the Shannon entropy power on manifolds and did not study the rigidity problem. The idea of using the Shannon and Rényi entropy powers to characterize the rigidity models is new. Theorem 4.3 uses the equilibrium state of the Shannon entropy power to characterize the shrinking Ricci solitons. It can be regarded as the natural correspondence of Perelman’s result on the characterization of the shrinking Ricci solitons as the stationary point of the W-entropy for the Ricci flow. The main results of this paper can be extended to the Shannon entropy power for the heat equation and the Rényi entropy power for the nonlinear diffusion equation associated with the usual Laplacian or the Witten Laplacian with the Neumann

Uniqueness of positive solutions | 383

boundary condition on Riemannian manifolds with convex boundary; see [24]. Moreover, we can further prove the concavity inequality of the Shannon and Rényi entropy powers along the geodesic flow on the L2 -Wasserstein space over compact Riemannian manifolds with CD(K, m)-condition; see [17, 23]. In [36], Wang and Zhang proved the entropy power concavity inequality for the p-Laplacian equation on compact Riemannian manifolds with nonnegative Ricci curvature. We would like to mention that the concavity inequalities of the Shannon and Rényi entropy powers in Theorems 3.2, 3.1, 5.1, and 5.2 had been obtained in our 2017 paper [17] and are independent of [36], while the NIW formulae and the rigidity theorems are proved in our 2019–2020 papers [21, 22].

Bibliography [1] [2] [3]

[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

D. Bakry, M. Emery, Diffusion hypercontractives, in Sém. Prob. XIX. Lect. Notes in Maths., vol. 1123 (1985), pp. 177–206. N. M. Blachman, The convolution inequality for entropy powers. IEEE Trans. Inf. Theory 2, 267–271 (1965). D. Bakry, L’hypercontractivité et son utilisation en théorie des semigroups, in Lectures on Probability Theory, Ecole d’Eté de Probabilités de Saint-Flour XXII-1992, ed. by P. Bernard. Lecture Notes in Mathematics, vol. 1581 (Springer, Berlin, 1994), pp. 1–114. D. Bakry, D. Concordet, M. Ledoux, Optimal heat kernel bounds under logarithmic Sobolev inequalities. ESAIM Probab. Stat. 1, 391–401 (1997). M. H. M. Costa, A new entropy power inequality. IEEE Trans. Inf. Theory 31(6), 751–760 (1985). T. M. Cover, J. A. Thomas, Elements of Information Theory, 2nd edn. (Wiley InterScience, A John Wiley Sons, Inc., Publication, 2006). A. Dembo A, Simple proof of the concavity of the entropy power with respect to the variance of additive normal noise. IEEE Trans. Inf. Theory 35, 887–888 (1989). A. Dembo, T. M. Cover, J. A. Thomas, Information theoretic inequalities. IEEE Trans. Inf. Theory 37(6), 1501–1518 (1991). R. S. Hamilton, A matrix Harnack estimate for the heat equation. Commun. Anal. Geom. 1(1), 113–126 (1993). E. H. Lieb, Proof of an entropy conjecture of Wehrl. Commun. Math. Phys. 62(1), 35–41 (1978). K. Kuwada, X.-D. Li, Monotonicity and rigidity of the W -entropy on RCD(0, N) spaces. Manuscr. Math. (2020). https://doi.org/10.1007/s00229-019-01177-y. X.-D. Li, Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds. J. Math. Pures Appl. 84, 1295–1361 (2005). X.-D. Li, Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry–Emery Ricci curvature. Math. Ann. 353, 403–437 (2012). X.-D. Li, Hamilton’s Harnack inequality and the W -entropy formula on complete Riemannian manifolds. Stoch. Process. Appl. 126, 1264–1283 (2016). S. Li, X.-D. Li, W -entropy formula for the Witten Laplacian on manifolds with time dependent metrics and potentials. Pac. J. Math. 278(1), 173–199 (2015). S. Li, X. D. Li, On Harnack inequalities for Witten Laplacian on Riemannian manifolds with super Ricci flows. Asian J. Math. 22, 577–598 (2018). S. Li, X.-D. Li, Entropy differential inequality and entropy power inequality on Riemannian manifolds and super-Ricci flows. Preprint (2017).

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[18] S. Li, X.-D. Li, Hamilton differential Harnack inequality and W -entropy for Witten Laplacian on Riemannian manifolds. J. Funct. Anal. 274, 3263–3290 (2018). [19] S. Li, X.-D. Li, W -entropy formulas on super-Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds. Sci. China Math. 61, 1385–1406 (2018). [20] S. Li, X.-D. Li, W -entropy, super Perelman Ricci flows and (K, m)-Ricci solitons. J. Geom. Anal. 30, 3149–3180 (2020). [21] S. Li, X.-D. Li, On the Shannon entropy power on Riemannian manifolds and Ricci flows (2020). arXiv:2001.00410v1. [22] S. Li, X.-D. Li, On the Rényi entropy power and the Gagliardo–Nirenberg–Sobolev inequality on Riemannian manifolds (2020). arXiv:2001.11184v1. [23] S. Li, X.-D. Li, On Shannon and Rényi entropy powers on Wasserstein space over Riemannian manifolds (2020, in preparation). [24] S. Li, X.-D. Li, On the entropy power concavity inequality on Riemannian manifolds with boundary (2020, in preparation). [25] P. Li, S.-T. Yau, On the parabolic kernel of the Schrödinger operator. Acta Math. 156, 153–201 (1986). [26] J. Li, X. Xu, Differential Harnack inequalities on Riemannian manifolds I: linear heat equation. Adv. Math. 226(5), 4456–4491 (2011). [27] L. Ni, The entropy formula for linear equation. J. Geom. Anal. 14(1), 87–100 (2004). [28] L. Ni, Addenda to “The entropy formula for linear equation”. J. Geom. Anal. 14(2), 329–334 (2004). [29] G. Perelman, The entropy formula for the Ricci flow and its geometric applications. http: //arXiv.org/abs/maths0211159. [30] G. Savaré, G. Toscani, The concavity of Rényi entropy power. IEEE Trans. Inf. Theory 60, 2687–2693 (2014). [31] C. E. Shannon, A mathematical theory of communication. Bell Syst. Tech. J. 27, 623–656 (1948). [32] A. J. Stam, Some inequalities satisfied by the quantities of information of Fisher and Shannon. Inf. Control 2, 101–112 (1959). [33] K.-Th. Sturm, Super-Ricci flows for metric measure spaces. J. Funct. Anal. 275(12), 3504–3569 (2018). [34] C. Villani, A short proof of the concavity of entropy power. IEEE Trans. Inf. Theory 46(4), 1695–1696 (2000). [35] C. Villani, Topics in Mass Transportation. Grad. Stud. Math. (Amer. Math. Soc., Providence, RI, 2003). [36] Y.-Z. Wang, X. Zhang, The concavity of p-entropy power and applications in functional inequalities. Nonlinear Anal. 179, 1–14 (2019).

Liguang Liu and Jie Xiao

Fractional differential operators and divergence equations Abstract: Motivated by the Meerschaert–Bebson–Bäumer’s fractional (0, 1) ∋ s-advection–dispersion equation: 𝜕t u(x, t) + v⃗ ⋅ ∇u(x, t) = κ(−Δ) = =

1+s 2

u(x, t)

±κ divs± (∇± u(x, t)) ±κ div± (∇±s u(x, t)),

which models anomalous dispersion in groundwater flow, in this paper we present innovative sharp Hardy–Rellich, Adams–Moser, and Morrey–Sobolev estimates for the fractional differential operators ∇±s (cf. Theorem 3.2) by sharply controlling Riesz’s Iα -operators (cf. Theorem 3.1), and optimally determine the distributional Lebesgue’s (cf. Theorem 5.1) or Lipschitz–John–Nirenberg–Morrey (cf. Theorem 5.2) solutions to the factional divergence equations ± divs± u = [∇±s ]∗ u = μ or f for a nonnegative Radon measure μ or a real-valued Morrey function f with the L1loc -integrability. Keywords: Fractional differentials, function spaces, divergence equations MSC 2010: 35Q35, 42B30, 46E35

Contents 1 2 2.1 2.2 3 3.1 3.2 4 4.1 4.2 5 5.1

Introduction | 386 Fractional differential operators | 387 ∇±s -operators | 387 Adjoint [∇±s ]∗ -operators | 390 Optimal differential-integral inequalities | 392 Controlling Riesz’s Iα -operators | 392 Dominating ∇±s -operators | 397 Hardy–Sobolev-type spaces | 402 s,p Hardy–Sobolev spaces {Hs,p , H± } | 402 Lebesgue–Hardy–Sobolev spaces H1,∞>p≥1 | 406 Distributional solutions of divergence equations | 408 Lebesgue’s solutions of ± divs± u = μ | 408

Acknowledgement: LL was supported by the National Natural Science Foundation of China (# 11771446); JX was supported by NSERC of Canada (# 202979463102000). Liguang Liu, School of Mathematics, Renmin University of China, Beijing 100872, China, e-mail: [email protected] Jie Xiao, Department of Mathematics and Statistics, Memorial University, St. John’s, NL A1C 5S7, Canada, e-mail: [email protected] https://doi.org/10.1515/9783110700763-014

386 | L. Liu and J. Xiao

5.2

Lipschitz–John–Nirenberg–Morrey’s solutions of ± divs± u = f | 410 Bibliography | 417

1 Introduction Upon recognizing the fractional vector calculus (FVC) and partial differential equations (PDE) utilized in Herbst’s study (cf. [13]) of the Klein–Gordon equation for a Coulomb potential and Meerschaert–Mortensen–Wheatcraft’s investigation (cf. [21]) of the particle mass density u(x, t) of a contaminant in some fluid at a point x ∈ ℝn at time t > 0, which solves the fractional (0, 1) ∋ s-advection–dispersion equation with a constant average velocity v,⃗ which maybe different from the fluid velocity, of contaminant particles, and a positive diffusivity constant κ (which was introduced by Meerschaert–Bebson–Bäumer in [20] to model anomalous dispersion in groundwater flow), namely 𝜕t u(x, t) + v⃗ ⋅ ∇u(x, t) = κ(−Δ) = =

1+s 2

u(x, t)

±κ divs± (∇± u(x, t)) ±κ div± (∇±s u(x, t)),

combining a fractional Fick’s law for flux with a classic mass balance – and conversely – combining a fractional mass balance with a classic Fickian flux, we are here seeking further applications of FVC in other areas (cf. [21, Conclusion]) to not only understand intrinsically the fractional s-gradient operators s

∇+s = (−Δ) 2

and

s

⃗ 2 ∇−s = R(−Δ)

via the sharp Hardy–Sobolev-type inequalities with R⃗ being the Riesz vector, but also determine optimally when the fractional s-divergence equations s

divs+ u = [∇+s ] u = (−Δ) 2 u = F ∗

and

s

⃗ 2u = F − divs− u = [∇−s ] u = −R(−Δ) ∗

are distributionally solvable for F which is either a nonnegative Radon measure μ or a real-valued Morrey function f with the L1loc -integrability. In short, the rest of this paper consists of two parts with the second as a PDE-application of the first. ▷ The first part is about the sharp inequalities based on the fractional operators ∇±s : – §2 collects some fundamental facts on ∇±s and their duals [∇±s ]∗ = ± divs± (cf. Propositions 2.1–2.2) through the Stein–Weiss–Hardy p-inequality and the Fefferman–Stein BMO-decomposition [8, 5, 19]; – §3 utilizes Theorem 3.1 – an optimal embedding principle for the Riesz potentials – to discover the existence of the fractional extensions of the Hardy– Rellich [4], Adams–Moser [2], and Morrey–Sobolev [39] inequalities (cf. Theorem 3.2).

Fractional differential operators and divergence equations | 387

▷

The second part is about the fractional divergence equations ± divs± u = [∇±s ]∗ u = F: – §4 handles the fractional Hardy–Sobolev (1, ∞) ∋ p-spaces H s,p and H±s,p and their end-points – the Lebesgue–Hardy–Sobolev spaces H 1,∞>p≥1 (cf. Propositions 4.1–4.2); – §5 discusses the distributional Lebesgue or Lipschitz–John–Nirenberg–Morrey solutions of the fractional divergence equations ± divs± u = [∇±s ]∗ u = μ or f for a nonnegative Radon measure μ or a Morrey function f of the L1loc -integrability (cf. Theorems 5.1–5.2).

Notation. In what follows, U ≲ V (resp. U ≳ V) means U ≤ cV (resp. U ≥ cV) for a positive constant c and U ≈ V amounts to U ≳ V ≳ U.

2 Fractional differential operators 2.1 ∇±s -operators For (n, p) ∈ ℕ × [1, ∞) let H p be the real Hardy space of all functions u in the Lebesgue space Lp on the Euclidean space ℝn with ⃗ Lp < ∞, ‖u‖H p = ‖u‖Lp + ‖Ru‖ where R⃗ = (R1 , . . . , Rn ) is the vector-valued Riesz transform on ℝn , with ⃗ { {Ru = (R1 u, . . . , Rn u) n+1 { {R u(x) = ( Γ( 2 ) ) p. v. ∫ n+1 j ℝn { π 2

yj −xj u(y) dy |x−y|n+1

a. e. x ∈ ℝn .

Also, for a vector-valued function f ⃗ = (f1 , . . . , fn ) we have n

‖f ⃗‖Lp = ‖|f ⃗|‖Lp ≈ ∑ ‖fj ‖Lp . j=1

Note that H p coincides with the classical Lebesgue space Lp whenever p ∈ (1, ∞) and the (0, 1) ∋ s-th order Riesz potential operator Is acting on a suitable function u is defined by Is u(x) = (

π

) Γ( n−s 2 n 2

2s Γ( 2s )

) ∫ |x − y|s−n u(y) dy

a. e. x ∈ ℝn .

ℝn

We refer the reader to Stein’s seminal texts [36, 37] for more about these basic notions. The Stein–Weiss–Hardy p-inequality (cf. [38] for p > 1 and (5.4) in §5 for p = 1) states

388 | L. Liu and J. Xiao that whenever 0 < s < 1 ≤ p
1 and (2.4) give (cf. [31, Lemma 2.4]) 1

p 󵄨p 󵄩 󵄩 ( ∫ (|x| 󵄨󵄨u(x)󵄨󵄨󵄨) dx) ≲ 󵄩󵄩󵄩∇+s u󵄩󵄩󵄩Lp

−s 󵄨󵄨

ℝn

∀u ∈ Is (Cc∞ ∩ H p ).

One the other hand, [31, Theorems 1.8–1.9] provides 1

p 󵄩 󵄩 󵄨p ( ∫ (|x| 󵄨󵄨u(x)󵄨󵄨󵄨) dx) ≲ 󵄩󵄩󵄩∇−s u󵄩󵄩󵄩Lp

−s 󵄨󵄨

ℝn

▷

∀u ∈ Is (Cc∞ ∩ H p ).

If 0 < s < p = 1 ≤ n, then according to Spector’s theorem [35, Theorem 1.4], the right-hand side of (2.4), except for n = 1, can be replaced by ‖∇−s u‖L1 , i. e., 󵄨 󵄨 󵄩 󵄩 ∫ |x|−s 󵄨󵄨󵄨u(x)󵄨󵄨󵄨 dx ≲ 󵄩󵄩󵄩∇−s u󵄩󵄩󵄩L1

under

n ≥ 2 and

ℝn

(u, |∇−s u|) ∈ Cc∞ × L1 ,

which may be viewed as a rough extension of Shieh–Spector’s theorem [32, Theorem 1.2]: 2−s

) 󵄩 s 󵄩 π 2 21−s Γ( n−s 󵄨 󵄨 2 )󵄩󵄩󵄩∇− |u|󵄩󵄩󵄩L1 ∫ |x|−s 󵄨󵄨󵄨u(x)󵄨󵄨󵄨 dx ≤ ( s n−1 (n − 1)Γ( 2 )Γ( 2 ) n

󵄨 󵄨 ∀󵄨󵄨󵄨∇−s |u|󵄨󵄨󵄨 ∈ L1

ℝ

and the classic optimal Hardy’s inequality (cf. [9]) whenever n ≥ 2: −1 −1 {∫ℝn |x| |u(x)| dx ≤ (n − 1) ‖∇u‖L1 { −1 −1 s {∫ℝn |x| |I1−s u(x)| dx ≤ (n − 1) ‖∇− u‖L1

∀u ∈ Cc∞ , ∀u ∈ I1−s (Cc∞ ).

However, the right-hand side of (2.4) cannot be replaced by ‖∇+s u‖L1 (cf. [36, p. 119], [29, Section 3.3], and [32, Section 1.1]). Along this way, if s+n > t > s > 0, then the Fubini theorem and [15, p. 132, (3)] produce a constant cn,s,t such that limt→s cn,s,t = ∞ and 󵄨󵄨󵄨 󵄨󵄨󵄨 󵄨 󵄨 ∫ |x|−t 󵄨󵄨󵄨 ∫ f (x)|y − x|s−n dy󵄨󵄨󵄨 dx ≤ ∫ ( ∫ |x|−t |y − x|s−n dx)󵄨󵄨󵄨f (y)󵄨󵄨󵄨 dy 󵄨 󵄨 󵄨 󵄨 n n n n ℝ

ℝ

ℝ

ℝ

󵄨 󵄨 = cn,s,t ∫ 󵄨󵄨󵄨f (y)󵄨󵄨󵄨|y|s−t dy, ℝn

390 | L. Liu and J. Xiao and hence 󵄨 󵄨 󵄨 󵄨 ∫ |x|−t 󵄨󵄨󵄨u(x)󵄨󵄨󵄨 dx ≤ cn,s cn,s,t ∫ 󵄨󵄨󵄨∇+s u(x)󵄨󵄨󵄨|x|s−t dx ℝn

ℝn

holds for 0n is sharp in the sense that if Ω is a Euclidean ball then ‖Iα f ‖L∞

sup

f ∈Lpc (Ω)

|Ω|

αp−n pn

‖f ‖Lp

= cαp>n .

Proof. (i) This is regarded as the sharp Stein–Weiss–Hardy p-inequality. The constant cαp 0 imply 󸀠

∫ |x − y|(α−n)p dy Ω ∞ 󸀠

= (n − α)p ∫( ∫ r (α−n)p −1 dr) dy 󸀠

Ω

|x−y|

∞

= (n − α)p󸀠 ∫ ( 0

󸀠

dy)r (α−n)p −1 dr

∫

B(x,r)∩Ω

∞

≤ (n − α)p󸀠 ∫ min{ 0

󸀠 ωn−1 n r , |Ω|}r (α−n)p −1 dr n 1

ω = (n − α)p ( n−1 n 󸀠

( ωn|Ω| ) n n−1

∫

∞

r

(α−n)p󸀠 +n−1

dr + |Ω|

0

1

( ωn|Ω| ) n n−1

ω 1 1 = (n − α)p ( + )( n−1 ) (α − n)p󸀠 + n (n − α)p󸀠 n 󸀠

n(p − 1) ωn−1 =( )( ) αp − n n

󸀠

r (α−n)p −1 dr)

∫

(n−α)p󸀠 n

|Ω|

(α−n)p󸀠 +n n

(n−α)p󸀠 n

|Ω|

.

Thus we arrive at the desired inequality 1

n(p − 1) p󸀠 ωn−1 󵄨󵄨 󵄨 ) ( ) 󵄨󵄨Iα f (x)󵄨󵄨󵄨 ≤ ‖f ‖Lp ( αp − n n

n−α n

|Ω|

(α−n)p󸀠 +n np󸀠

.

(α−n)p󸀠 +n n

396 | L. Liu and J. Xiao In order to prove that cαp>n

ω = ( n−1 ) n

n−α n

n(p − 1) ( ) αp − n

p−1 p

is sharp, let us consider Ω = B(x0 , r0 ) ∀(x0 , r0 ) ∈ ℝn × (0, ∞), { { { { { n ℝ ∋ x 󳨃→ fβ (x) = 1B(x0 ,r0 ) |x − x0 |β , { { { { { β + pn > 0. { On the one hand, a direct calculation gives βp

|x − x0 | dx)

‖fβ ‖Lp = ( ∫

1 p

B(x0 ,r0 ) r0

= (ωn−1 ∫ r 0

βp+n−1

1 p

dr)

1

1

p p β+ n ω n = ( n−1 ) ( ) r0 p . n βp + n

n p

On the other hand, from the fact α + β > α − 󵄨󵄨 󵄨 󵄨󵄨Iα fβ (x0 )󵄨󵄨󵄨 =

> 0, we get r0

|x − x0 |α−n+β dx = ωn−1 ∫ r α+βp−1 dr = (

∫

0

B(x0 ,r0 )

Combining the last two formulae gives cαp>n ≥

sup

x∈B(x0 ,r0 )

|Iα fβ (x)| |B(x0 , r0 )|

αp−n np

|Iα fβ (x0 )|

≥ (

α− n ωn−1 αp−n ) np r0 p ‖fβ ‖Lp n

ω = ( n−1 ) n

n−α n

n

1− p1

β∈(− pn ,∞)

1

βp + n p ( ) . (α + β)p

Accordingly, we reduce the problem to calculating sup

‖fβ ‖Lp

βp + n . (α + β)p

ωn−1 α+β )r . α+β 0

Fractional differential operators and divergence equations | 397

Consider the function −

βp + n n < β 󳨃→ h(β) = . p (α + β)p

Note that h󸀠 (β) = −p(α + β)−p−1 (β(p − 1) + n − α), { { { { { 󸀠 , h (β) ≥ 0 if β ≤ − n−α { p−1 { { { { 󸀠 if β ≥ − n−α . p−1 {h (β) ≤ 0 So this, combined with limβ→− n h(β) = 0, shows that h attains its optimal value at the

. Consequently, point β = − n−α p−1

p

1−p

βp + n αp − n =( ) p p−1 β∈(− n ,∞) (α + β) sup

.

p

This in turn implies cαp>n ≥

|Iα fβ (x)|

sup

|B(x0 , r0 )|

x∈B(x0 ,r0 )

ω = sup ( n−1 ) n n β∈(− ,∞) p

=(

ωn−1 ) n

n−α n

(

n−α n

αp−n np

‖fβ ‖Lp 1

βp+n

( (α+β)p ) p n

n(p − 1) ) αp − n

1−p p

p−1 p

.

Accordingly, when Ω is a Euclidean ball of ℝn , it holds that sup

f ∈Lpc (Ω)

‖Iα f ‖L∞ |Ω|

αp−n pn

‖f ‖Lp

= cαp>n .

3.2 Dominating ∇±s -operators Interestingly and naturally, with m

m

m∈{even} = (−1) 2 (−Δ) 2 , {∇ m−1 m−1 m−1 m { m∈{odd} ⃗ 2 = (−1) 2 ∇(−Δ) 2 = (−1) 2 R(−Δ) {∇

being replaced by the fractional version s

s {∇+ = (−Δ) 2 , s−1 s { s ⃗ 2 2 {∇− = ∇(−Δ) = R(−Δ) ,

Theorem 3.1 induces the following assertion.

398 | L. Liu and J. Xiao Theorem 3.2. Let 0 < s < 1 < p < ∞, Cc∞ (Ω) be the class of all infinitely differentiable functions with compact support contained in a given domain Ω ⊆ ℝn , and for ∇+s ,

∞ {Is (Cc (Ω))

ℱs,± (Ω) = {

{(−Δ)

1−s 2

for ∇−s .

(Cc∞ (Ω))

(i) If sp < p < n, then 1

sup

(∫ℝn (|x|−s |g(x)|)p dx) p ‖∇±s g‖Lp

g∈Cc∞

= κsp κsp=n,± then the last integral inequality cannot hold without forcing csp=n,± to depend only on s and p. (iii) If sp > n and Ω ⊆ ℝn is a domain with volume |Ω| < ∞, then Γ( n−s )

sp−n pn

|Ω| ‖g‖L∞ sup ≤ κsp>n,± ‖∇±s g‖Lp g∈ℱs,± (Ω)

c ( n2 ) { { { sp>n 2s π 2 Γ( 2s ) ={ Γ( n−s+1 ) { 2 {c sp>n ( s n2 1+s ) 2 π Γ( 2 ) {

for ∇+s , for ∇−s .

Moreover, the constant κsp>n,± is sharp in the sense that if Ω is a Euclidean ball then sup

g∈ℱs,± (Ω)

‖g‖L∞

|Ω|

sp−n pn

‖∇±s g‖Lp

= κsp>n,± .

Proof. The precise inequalities in (i), (ii), and (iii) are suitably called the sharp Hardy– Rellich, Adams–Moser, and Morrey–Sobolev inequalities for the fractional order twin

Fractional differential operators and divergence equations | 399

gradients ∇±0 0 there is a sufficiently small r0 > 0 such that 1 󵄨󵄨 󵄨󵄨 −1 󵄨 󵄨󵄨 󵄨󵄨 1 󵄨󵄨 1 󵄨󵄨 dt 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 sup 󵄨󵄨T(|x|, r)(log ) 󵄨󵄨 ≤ sup 󵄨󵄨 ∫󵄨U(t) − U(0)󵄨󵄨 󵄨󵄨 < ϵ 󵄨󵄨 x∈𝔹n 󵄨󵄨 log 1 󵄨 r t 󵄨󵄨󵄨 x∈𝔹nr 󵄨󵄨 r󵄨 r

∀0 < r ≤ r0 .

0

Thus, we have 1−s 󵄨󵄨 󵄨 󵄨󵄨(−Δ) 2 ur (x)󵄨󵄨󵄨 ≥ κ−s (1 − ϵ)

∀(x, r) ∈ 𝔹nr × (0, r0 ].

This, along with (3.8) and the second formula of (3.9), gives n

1−s

csp=n,−

κ|(−Δ) 2 ur (x)| n−s dx ≥ ∫ exp( ) ‖|∇ur |‖Lp |𝔹n | 𝔹nr

≥ r n exp(

κκ−s (1 − ϵ)

(ωn−1 log 1r )

s−n n

n n−s

,

)

which in turns implies that if 0 < r ≤ r0 then κκ−s (1 − ϵ) ≤ (log

csp=n,− rn

n−s n

)

1 (ωn−1 log ) r

s−n n

=

csp=n,− rn ( ωn−1 log 1r

log

n−s n

)

.

Letting ϵ ↓ 0 and r ↓ 0 yields κκ−s

▷

n ) ≤( ωn−1

n−s n

, i. e., κ ≤ κsp=n,−

n ) = (κ−s ) ( ωn−1 −1

n−s n

,

as desired. sp > n. Let (x0 , r0 ) ∈ ℝn × (0, ∞), { { { { { { { Ω = B(x0 , r0 ), { { { { { β = − n−s , { p−1 { { { { β+1 {u (x) = (β + 1)−1 1 { , { β B(x0 ,r0 ) |x − x0 | { { { 1−s { 2 {gβ (x) = (−Δ) uβ (x). Notice that uβ can be approximated by functions in Cc∞ and ∇−s gβ (x) = ∇uβ (x) = 1B(x0 ,r0 ) |x − x0 |β

x − x0 . |x − x0 |

So, by (3.7) and the calculations in the proof of Theorem 3.1(iii), we obtain 󵄩󵄩 s 󵄩󵄩 󵄩󵄩∇− gβ 󵄩󵄩Lp = ( ∫

B(x0 ,r0 )

1 p

1

1

p p β+ n ω n |x − x0 | dx) = ( n−1 ) ( ) r0 p n βp + n

βp

402 | L. Liu and J. Xiao and 󵄨󵄨 h ⋅ ∇uβ (x0 + h) 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨 dh󵄨󵄨󵄨 󵄨󵄨gβ (x0 )󵄨󵄨󵄨 = κ−s 󵄨󵄨󵄨 ∫ n−s+1 󵄨󵄨 󵄨󵄨 |h| n ℝ

= κ−s ∫ |h|β+s−n dh |h|n,− ≥ ≥

‖g‖L∞ (B(x0 ,r0 ))

sup

|B(x0 , r0 )| |gβ (x0 )|

g∈ℱs,− (B(x0 ,r0 ))

|B(x0 , r0 )|

sp−n pn

sp−n pn

‖∇±s g‖Lp

‖∇±s gβ ‖Lp

= κ−s csp>n = κsp>n,− , and so κsp>n,− is sharp.

4 Hardy–Sobolev-type spaces s,p

4.1 Hardy–Sobolev spaces {H s,p , H± } Suppose 0 < s < 1 ≤ p < ∞. Since both ∇+s u and ∇−s u are well defined when u ∈ 𝒮s󸀠 , the study for the case p = 1 of (2.4) in [19] motivates us to consider the fractional Hardy– Sobolev space s 󵄩 󵄩 H s,p = {u ∈ 𝒮s󸀠 : [u]H s,p = 󵄩󵄩󵄩(−Δ) 2 u󵄩󵄩󵄩H p < ∞}.

Note that u1 − u2 = constant ⇐⇒ [u1 ]H s,p = [u2 ]H s,p . So, [⋅]H s,p is properly a norm on quotient space of H s,p modulo the space of all real constants, and consequently this quotient space is a Banach space. Upon introducing 󵄩 󵄩 H±s,p = {u ∈ 𝒮s󸀠 : [u]H±s,p = 󵄩󵄩󵄩∇±s u󵄩󵄩󵄩Lp < ∞},

Fractional differential operators and divergence equations | 403

we find immediately H s,p = H+s,p ∩ H−s,p . Indeed, as shown in the next theorem, when s ∈ (0, 1) and p ∈ (1, ∞), these three spaces are equal to each other and they all contain the Schwartz class 𝒮 and 𝒮∞ = {ϕ ∈ 𝒮 : the Fourier transform of ϕ is 0 near the origin}

as two dense subspaces. Proposition 4.1. Let 0 < s < 1 < p < ∞ and p󸀠 =

p . p−1

(i) Then 𝒮∞ ⊆𝒮 ⊆ H s,p = H+s,p = H−s,p . Moreover, both 𝒮∞ and 𝒮 are dense in H s,p and H±s,p . (ii) For any distribution T ∈ 𝒮 󸀠 , the following three assertions are equivalent: (ii-a) T ∈ [H s,p ]∗ = [H+s,p ]∗ = [H−s,p ]∗ ; 󸀠 (ii-b) ∃ T0 ∈ Lp such that T = [∇+s ]∗ T0 in 𝒮 󸀠 ; (ii-c) ∃ (T1 , . . . , Tn ) ∈ (Lp )n such that T = [∇−s ]∗ (T1 , . . . , Tn ) in 𝒮 󸀠 . 󸀠

s

Proof. (i) Notice that any u ∈ 𝒮 satisfies (−Δ) 2 u ∈ 𝒮s (cf. [34]). Of course, any function in 𝒮s belongs to L1 n. According to Proposition 4.1, we only need to validate that such a 󸀠 p measure μ induces a bounded linear functional on H+s,p , where p󸀠 = p−1 . To this end, s

for any ϕ ∈ 𝒮 , by the fact ϕ = Is (−Δ) 2 ϕ and the Fubini theorem, we write s

s

∫ ϕ dμ = ∫ Is (−Δ) 2 ϕ(x) dμ(x) = ∫ (−Δ) 2 ϕ(x)Is μ(x) dx, ℝn

ℝn

ℝn

so the Hölder inequality gives 󵄨󵄨 󵄨󵄨 s 󵄨󵄨 󵄩 󵄨 󵄩 󵄨󵄨 ∫ ϕ dμ󵄨󵄨󵄨 ≤ ‖Is μ‖Lp 󵄩󵄩󵄩(−Δ) 2 ϕ󵄩󵄩󵄩Lp󸀠 = ‖Is μ‖Lp [ϕ]H s,p󸀠 . 󵄨󵄨 󵄨󵄨 + n ℝ

Combining this with the density of 𝒮 in H+s,p (cf. Proposition 4.1) leads to that μ can be 󸀠

extended to a bounded linear functional on H+s,p . 󸀠

s

5.2 Lipschitz–John–Nirenberg–Morrey’s solutions of ± div± u = f In accordance with the basic identity [∇±s ] (∇±s u) = ±(−Δ)s u ∀u ∈ Cc∞ ∗

Fractional differential operators and divergence equations | 411

and [30, Theorem 1.1], if Ω is an open subset of ℝn , (p, s) ∈ (2 − n−1 , ∞) × (0, 1], and u ∈ H s,p is a distributional solution to the following fractional p-Laplace equation with a natural variation structure: 󵄨 󵄨p−2 divs± (󵄨󵄨󵄨∇±s u󵄨󵄨󵄨 ∇±s u) = 0

in

Ω,

namely, 󵄨 󵄨p−2 ∫ 󵄨󵄨󵄨∇±s u󵄨󵄨󵄨 ∇±s u ⋅ ∇±s ϕ dx = 0 ℝn

∀ϕ ∈ Cc∞ (Ω),

then u ∈ C s+α loc (Ω) for some positive constant α depending on p only, we are led to the distributional Lipschitz–John–Nirenberg–Morrey’s solution of ± divs± u = [∇±s ] u = f . ∗

For any (p, κ) ∈ [1, ∞) × (0, n], the Morrey space Lp,κ was introduced by Morrey [22] and used to study some quasilinear elliptic partial differential equations, where Lp,κ comprises all Lebesgue measurable functions f on ℝn with 1

‖f ‖Lp,κ =

sup

(x,r)∈ℝn ×(0,∞)

(r

κ−n

p 󵄨 󵄨p ∫ 󵄨󵄨󵄨f (y)󵄨󵄨󵄨 dy) < ∞.

B(x,r)

Of particular importance is that functions in L1,κ have L1loc -integrability, and Lp,n is just the classical Lebesgue space Lp . For (p, κ) ∈ (1, ∞) × (0, n), let Hp,κ be the space of all Lebesgue measurable functions f on ℝn such that 1

‖f ‖Hp,κ

p 1−p 󵄨 󵄨p = inf( ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 (ω(x)) dx) < ∞, ω

ℝn

where the infimum is taken over all nonnegative functions ω on ℝn satisfying ∞

n ‖ω‖L1 (Λn−κ ) = ∫ Λn−κ (∞) ({x ∈ ℝ : ω(x) > t}) dt ≤ 1. (∞)

0

Here and hereafter, for any given α ∈ (0, n), the symbol Λα(∞) (E) denotes the αth order Hausdorff capacity of a set E ⊆ ℝn , determined by Λα(∞) (E) = inf{∑ rjα : E ⊆ ⋃ B(xj , rj ) with xj ∈ ℝn and rj ∈ (0, ∞)}. j

j

412 | L. Liu and J. Xiao According to [3], we have the duality [Hp ,κ ]∗ = Lp,κ . From [26, (5.1)] and [1, Corollary and Proposition 5], we have that if ‖|μ‖|n−κ < ∞ then 󸀠

∫ |Iκ u| dμ ≲ ‖Iκ u‖L1 (Λn−κ ) ≲ ‖u‖H 1 (∞)

ℝn

∀u ∈ H 1 .

(5.3)

Consequently, if dνκ (x) = |x|−κ dx then ‖|νκ ‖|n−κ < ∞, and hence (5.3) is used to produce the Stein–Weiss–Hardy p-inequality at the endpoint p = 1: 󵄨 󵄨 ∫ |x|−κ 󵄨󵄨󵄨Iκ u(x)󵄨󵄨󵄨 dx ≲ ‖u‖H 1

∀u ∈ H 1 .

(5.4)

ℝn

This, along with (cf. [19, (1.3)–(1.4)]) [u]H s,1 ≲ [u]W s,1 = ∫ ∫ ℝn

ℝn

|u(x) − u(y)| dy dx |x − y|n+s

∀u ∈ 𝒮 ,

implies 󵄨 󵄨 ∫ |x|−s 󵄨󵄨󵄨u(x)󵄨󵄨󵄨 dx ≲ [u]H s,1 ≲ [u]W s,1

(5.5)

∀u ∈ 𝒮 ,

ℝn

which may be viewed as an improvement of the case p = 1 of [9, Theorem 1.1]. Upon taking 0 ≤ φ ∈ 𝒮, { { { { { ∫ℝn φ(x) dx = 1, { { { { { −n −1 {φt (x) = t φ(t x)

∀(t, x) ∈ (0, ∞) × ℝn ,

we extend the real Hardy space Hp from p ∈ [1, ∞) to p ∈ (0, ∞) via defining (cf. [37]) 󵄩󵄩 󵄩󵄩 Hp = {f ∈ 𝒮 󸀠 : ‖f ‖Hp = 󵄩󵄩󵄩 sup |φt ∗ f |󵄩󵄩󵄩 p < ∞} for 󵄩t∈(0,∞) 󵄩L

0 < p < ∞.

Then (cf. [8, 37]) BMO

[Hp ] = { ∗

if p = 1,

Lipn(p−1 −1)

n , 1). if p ∈ ( n−1

Here and henceforth, Lip0 n, implies 1 󸀠

q 1−q󸀠 󵄩󵄩 s 󵄩󵄩 󸀠 κ 󵄨 s 󵄨q󸀠 dx) 󵄩󵄩∇j ϕ󵄩󵄩ℋq ,q( p −s) ≲ ( ∫ 󵄨󵄨󵄨∇j ϕ(x)󵄨󵄨󵄨 (w0 (x))

ℝn

−q󸀠 (n+s)+λ(q󸀠 −1)

≲ ( ∫ (1 + |x|) ℝn

< ∞.

dx)

1 q󸀠

Fractional differential operators and divergence equations | 417

This proves 𝒮 ⊆ X. Though we do not know if 𝒮 is dense in X, we use the space X̊ which is the closure of 𝒮 in X. Still we consider the operator A : X̊ → Y via u 󳨃→ A(u) = ∇−s u, and can show that A is injective and has a continuous inverse from A(X)̊ (the close range of A) to X.̊ Consequently, the closed range theorem (cf. [42, p. 208, Corollary 1]) can be applied to ̊ ∗ → (X)̊ ∗ defined via derive that the adjoint operator A∗ : [A(X)] ̊ ∗ × X̊ ⟨A∗ F,⃗ u⟩ = ⟨F,⃗ Au⟩ ∀(F,⃗ u) ∈ [A(X)] is surjective. Next, we validate that any f ∈ Lp,κ belongs to (X)̊ ∗ . Applying [18, Proposition 5.1] q,q( κ −s) gives the continuity of the mapping Is : Lp,κ → L p . Note that the boundedness of q󸀠 ,q( κ −s)

p Rj on H is given in [1, Chapter 8]. So, upon using (5.7) and the Fubini theorem, we derive that any ϕ ∈ 𝒮 satisfies

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 n 󵄨󵄨 󵄨󵄨 󵄨 󵄨 s 󵄨󵄨 ∫ ϕ(x)f (x) dx󵄨󵄨󵄨 = 󵄨󵄨󵄨∑ ∫ Is (Rj ∇j ϕ)(x)f (x) dx 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 n 󵄨󵄨 󵄨󵄨󵄨j=1 n 󵄨 ℝ ℝ 󵄨󵄨 n 󵄨󵄨 󵄨󵄨 󵄨󵄨 = 󵄨󵄨󵄨∑ ∫ Rj ∇js ϕ(x)Is f (x) dx 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨j=1 ℝn 󵄨 n

󵄩 󵄩 ≤ ∑󵄩󵄩󵄩Rj ∇js ϕ󵄩󵄩󵄩 j=1

Hq

󸀠 ,q( κ −s) p

‖Is f ‖

κ

Lq,q( p −s)

≲ ‖ϕ‖X ‖f ‖Lp,κ . This implies that f can be extended to a bounded linear functional on X,̊ that is, f ∈ (X)̊ ∗ . ̊ ∗ → (X)̊ ∗ , we again Because of f ∈ (X)̊ ∗ and the surjective property of A∗ : [A(X)] use the Hahn–Banach extension theorem to find κ

q, q( −s) F⃗ = (F1 , . . . , Fn ) ∈ Y ∗ = (L p )n

such that A∗ F⃗ = f .

Thus, for all ϕ ∈ 𝒮 , we have ⟨f , ϕ⟩ = ⟨A∗ F,⃗ ϕ⟩ = ⟨F,⃗ Aϕ⟩ = ⟨F,⃗ ∇−s ϕ⟩ = −⟨divs− F,⃗ ϕ⟩, thereby reaching − divs− F⃗ = f in 𝒮 󸀠 .

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418 | L. Liu and J. Xiao

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Li Ma and Cong-han Wang

Interior gradient estimates for mean curvature type equations and related flows This paper is dedicated to 100th anniversary of Nankai University.

Abstract: To understand the compactness of space of graphic minimal surfaces, it is important to get gradient estimates for such minimal surfaces. In this paper, we study interior gradient estimates for the mean curvature type equation and the mean curvature flow equation (δij −

ui uj

1 + |∇u|2

)uij = H(x)√1 + |∇u|2 + αu in Br (0),

and ut = √1 + |∇u|2 div(

∇u ) − H(x)√1 + |∇u|2 √1 + |∇u|2

in Br (0) × (0, T],

respectively, where H(x) ∈ C 1 (Br (0)), r > 0, and α ∈ ℝ is a real constant. Under the assumption that u(x) = o(|x|) at infinity, we also derive a Liouville-type result for the singular minimal surface equation div(

∇u √1 +

|∇u|2

)=

α

u√1 + |∇u|2

in ℝn ,

where α ∈ (− 161 , 0). Keywords: Gradient estimates, singular mean curvature equations, perturbation, mean curvature flows, Liouville-type theorem MSC 2010: Primary 53C20, Secondary 35B65, 58J50

Acknowledgement: The first author was supported in part by the National Natural Science Foundation of China (No. 11771124, No. 11271111). Li Ma wants to express his gratitude to conference organizers, in particular to Dr. Yuhua Sun for the invitation. Li Ma, School of Mathematics and Physics, University of Science and Technology Beijing, 30 Xueyuan Road, Haidian District, Beijing 100083, P. R. China; and College of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan 453007, P. R. China, e-mail: [email protected] Cong-han Wang, Department of Mathematics, College of Sciences, Shanghai University, 99 Shangda Road, BaoShan District, Shanghai 200444, P. R. China, e-mail: [email protected] https://doi.org/10.1515/9783110700763-015

422 | L. Ma and C.-h. Wang Contents 1 2 3 4

Introduction | 422 The proof of Theorem 1.1 | 425 The proofs of Theorems 1.2 and 1.3 | 429 The proof of Theorem 1.4 | 436 Bibliography | 440

1 Introduction In this paper, we study gradient estimates for the mean curvature type equations and mean curvature type flows. In the literature for minimal surface and mean curvature flow, such estimates have been obtained by carrying out the analysis on the graphs of solutions to the minimal surface equation. From the view point of partial differential equations [14], such estimates should hold true for equations of similar structural conditions to the minimal surface equation. In [5, 2], the gradient estimates have been obtained for minimal surface equation and certain fully nonlinear elliptic equations under various conditions. Different proofs for the gradient estimates for mean curvature type equations have been given in [16, 3, 32]. The gradient estimates for various elliptic equations on Riemannian manifolds and translating solitons of mean curvature flow in Euclidian spaces are important subjects in geometric analysis [23, 24, 25, 26, 27, 28, 4, 11, 12, 13, 17, 18, 19, 20, 21, 31]. The motivation for this paper is to understand the compactness of space of singular graphic minimal surfaces in the space ℝn × ℝ with the metric ds2 = |y|2α/n (dx2 + dy2 ), where α is a real constant, and this subject is largely open even for various existence results. The singular region for the metric is {y = 0}. This singular metric also comes from Levy’s process and plays an important role in the very recent study of fractional Laplacian problems. Note that for a smooth positive function u(x) ∈ C 2 (Ω), where Ω is a bounded domain in ℝn , the graph Σ(u) over the domain Ω is defined by Σ(u) = {(x, u(x)) ∈ ℝn × ℝ; x ∈ Ω} and its area is A(Σ(u)) = ∫ uα √1 + |∇u|2 dx. Ω

We call the critical point of the area functional A(Σ(u)) the singular minimal surface. One famous example is the hyperbolic space form where α = −n. In the paper [22], Ma has studied the existence of minimal graph evolution in the hyperbolic space and has

Interior gradient estimates for mean curvature type equations and related flows | 423

proved that there exists a unique global smooth solution to the flow with given regular initial data. We remark that, motivated from the generalization of minimal surfaces in Euclidean space where α = 0, L. Simon derived very general results, Theorems 3 and 4 in his paper [30], by different methods. When α = 0, using the Grim reaper as barriers, T. Colding and Minicozzi II [7] have obtained a sharp gradient estimate for mean curvature flow of graphs, which improves the early estimates obtained in [14] (see also [32]). One may refer to [29] for more results. The sharp estimates for singular minimal surfaces for α ≠ 0 are still under investigation. In [8], Ulrich Dierkes has studied the singular minimal surface equation div(

α ∇u )= √1 + |∇u|2 u√1 + |∇u|2

in Ω ⊂ ℝn ,

(1.1)

for function u : Ω → ℝn and domain Ω ⊂ ℝn , or Ω = ℝn . See also his recent work [9] for an interesting Bernstein theorem for singular minimal surfaces. For the solution to equation (1.1), the (singular) mean curvature of the graph is given by H(x) =

α , √ u 1 + |∇u|2

x ∈ ℝn .

As pointed out in [8], equation (1.1) is the Euler–Lagrange equation of the variational integral ∫ uα √1 + |∇u|2 dx, which, in case α = 1, might be interpreted as the last coordinate of the center of gravity of the surface N := graph(u) = {(x, u(x)) ∈ ℝn × ℝ; x ∈ ℝn }, where the surface N is supposed to have constant mass density. We remark that equation (1.1) had already been derived by Lagrange [19, pp. 158–162], as a model equation for “heavy surfaces in vertical gravitational fields”, see also Cisa de Gresy [10, pp. 274– 276] and Jellett [15, pp. 349–354]. To state our result, we let Br (0) be the ball of radius r in ℝn and let T > 0. We now consider the mean curvature type flow equation ut = √1 + |∇u|2 div(

∇u √1 + |∇u|2

) − H(x)√1 + |∇u|2

in Q,

(1.2)

where Q = Br (0) × (0, T], 0 < t ≤ T, and H(x) is a given smooth function on Q. We may write equation (1.2) in the following form: (δij −

ui uj

1 + |∇u|2

)uij = ut + H(x)√1 + |∇u|2 .

(1.3)

This is the view-point from the book [14] for doing gradient estimates. We prove the following result.

424 | L. Ma and C.-h. Wang Theorem 1.1. Let u ∈ C 3 (Q) be a nonnegative solution of (1.2). Let M = supQ u(x). Suppose |H(x)| ≤ C0 and |∇H(x)| ≤ C0 for x ∈ Br (0). Then, for t ∈ (0, T), we have some uniform constants Cj , 1 ≤ j ≤ 3, where C1 depends only on n, M and C0 , while C2 and C3 depend only on n and C0 , such that 2

2

TM TM 󵄨 󵄨󵄨 2 + C3 2 )/t]. 󵄨󵄨∇u(0, t)󵄨󵄨󵄨 ≤ exp[(C1 TM + C2 r r We consider the local gradient estimate of the positive solution to the singular minimal surface equation (1.1) when α ∈ (− 161 , 0). We write (1.1) in the form (δij −

ui uj

1+

|∇u|2

)uij =

α . u(x)

(1.4)

Theorem 1.2. Let α ∈ (− 161 , 0). Let u ∈ C 3 (Br (0)) be a positive solution to the singular minimal surface equation (1.1). Then we have some uniform constants Cj , 1 ≤ j ≤ 2, such that 2

M M 󵄨󵄨 󵄨 󵄨󵄨∇u(0)󵄨󵄨󵄨 ≤ exp(C1 + C2 2 ). r r We may derive the following Liouville-type result for the singular minimal surface equation (see also [8, 19] for related results) via the gradient estimate above. Theorem 1.3. Let α ∈ (− 161 , 0). There is no entire nonnegative solution u ∈ C 0,1 (ℝn ) of the singular minimal surface equation (1.1) which has sublinear growth in the sense that u(x) = o(|x|) as |x| → ∞. We are also interested in the linear perturbation of minimal surface equations in Euclidean spaces. For u ∈ C 3 (Br (0)), we let aij = (δij −

ui uj

1 + |∇u|2

),

where the summation convention is used. We now consider the linear perturbation problem to the mean curvature type equation aij uij = H(x)√1 + |∇u|2 + αu,

(1.5)

where H ∈ C 0,1 (Br (0)), uij = uxi xj , α ∈ ℝ and α ≤ 0 is a real parameter. It seems to us that there are very few results about this equation (see the remark after Theorem 1.4). We consider here the simple case when u ∈ C 3 (Br (0)) is a nonnegative solution of (1.5). We will prove the following result. Theorem 1.4. Let u ∈ C 3 (Br (0)) be a nonnegative solution of (1.5). Let M = supBr (0) u(x). Suppose |H(x)| ≤ C0 and |∇H(x)| ≤ C0 . Then there are uniform constants Cj , 1 ≤ j ≤ 4,

Interior gradient estimates for mean curvature type equations and related flows | 425

with C1 , C2 depending only on n, M and C0 , while C3 , C4 depend only on n and C0 , such that 2

2

M M 󵄨 󵄨󵄨 + C4 2 ). 󵄨󵄨∇u(0)󵄨󵄨󵄨 ≤ exp(C1 + C2 |α| + C3 r r We make a remark here. In the paper [32], X.J. Wang has remarked (see remark (3) after the proof of Theorem 1.1 in [32]) that his gradient estimate method works for the class of H(x, u) with Hu ≥ 0. Our case for α < 0, H(x, u) = H(x) + αu is not included in his class and our result Theorem 1.4 above is, of course, new. Our estimates above are based on suitable choices of auxiliary functions involving the logarithm function (see also X.J. Wang [32]), and then we can avoid geometric computations on the graphs of solutions. The first log function used in the estimation of derivatives of solutions to Monge–Ampere equation on compact Kaehler manifolds may be in the paper of Th. Aubin in 1976 (see p. 151 in the book [1]). Gradient estimates for mean curvature type equations have been the main topic in the classical book [14]. In the paper [6], L. Caffarelli, L. Nirenberg, and J. Spruck have studied the gradient estimates for nonlinear second-order elliptic equations and the Dirichlet problem for Weingaten hypersurfaces. Applying the maximum principle, they have used a different Harnack quantity to get the gradient estimate for solutions of their equations. In the rest of this paper, we will use Ci to denote various uniform constants, which may change line by line. The plan of this paper is as follows. Theorem 1.1 is proved in Section 2. In Section 3, we prove Theorems 1.2 and 1.3. In Section 4, we give the proof of Theorem 1.4.

2 The proof of Theorem 1.1 We define the Harnack quantity by G(x, t, ξ ) = tρ(x)φ(u) log uξ (x), on Q × Sn−1 , where ρ(x) = 1 − |x|2 /r 2 ,

φ(u) = 1 +

u , M

M = sup u(x). Q

Suppose that G(x, t, ξ ) attains its maximum at (x0 , t0 , ξ ) and ξ = (1, 0, . . . , 0). Then we have the first-order condition at (x0 , t0 ), 0 = (log G)i = 0 = (log G)t =

ρi φ󸀠 u1i + ui + , ρ φ u1 log u1 u1t φ󸀠 1 u + + , φ t u1 log u1 t

(2.1) (2.2)

426 | L. Ma and C.-h. Wang and the second-order condition at (x0 , t0 ), {(log G)ij } ≤ 0, where ρij

ρi ρj

φ󸀠󸀠 φ󸀠 2 φ󸀠 − 2 )ui uj + uij ρ φ φ φ u1ij u u 1 1i 1j − (1 + ) . + u1 log u1 log u1 u21 log u1

(log G)ij = (

−

ρ2

)+(

By (2.1), we have u1i u1j 2 u1 log u21

=

ρi ρj

+

ρ2

φ󸀠 2 φ󸀠 ui uj + (ρ u + ρj ui ). 2 φρ i j φ

Then, (log G)ij =

ρij

φ󸀠󸀠 φ󸀠 φ󸀠 ui uj + uij + (ρ u + ρj ui ) ρ φ φ φρ i j u1i u1j u1ij 2 − (1 + ) . + u1 log u1 log u1 u21 log u1

(2.3)

1 , aii 1+u21

= 1 for i ≥ 2, and

+

Since ui (x0 , t0 ) = 0 for i ≥ 2, we have at (x0 , t0 ), a11 = |∇u|2 .

aij = 0 for i ≠ j, aij uij = ut + f (x)√1 + Differentiating equation (1.3) gives

aii uii1 = (ut + f (x)√1 + |∇u|2 )1 − aij1 uij u1 u11 1/2 = u1t + f1 (x)(1 + u21 ) + f − aij1 uij , (1 + u21 )1/2 from which we obtain that aij1 = −(

ui uj

1+

|∇u|2

) =− 1

u1i uj

1+

u21

−

ui u1j

1+

u21

+

ui uj

(1 + u21 )2

⋅ 2u1 u11 .

Then, aii uii1 = (ut + f (x)√1 + |∇u|2 )1 − aij1 uij = (ut + f (x)√1 + |∇u|2 )1 + = (ut + f (x)√1 + |∇u|2 )1 + 1/2

= u1t + f1 (x)(1 + u21 )

+f

u1i uj

2ui uj u1 u11 uij − uij 2 1 + u1 (1 + u21 )2 2u u2 2u31 u211 + ∑ 1 1i2 2 2 (1 + u1 ) i≥2 1 + u1 2u u2 2u1 u211 u1 u11 + + ∑ 1 1i2 . 2 2 2 1/2 (1 + u1 ) + u1 ) i≥2 1 + u1

uij 1 + u21 2u1 u211 − 1 + u21 (1

+

ui u1j

Interior gradient estimates for mean curvature type equations and related flows | 427

Hence, if u1 (x0 , t0 ) is suitably large, we have aii (uii1 − (1 +

u2 2 ) 1i ) log u1 u1

= u1t + f1 (x)(1 + u21 ) −

1/2

+f

2u1 u211 u1 u11 + (1 + u21 )1/2 (1 + u21 )2

2u1 u21i u211 1 2 ) + (1 + ∑ log u1 u1 i≥2 1 + u21 1 + u21

≥ u1t + f1 (x)(1 + u21 ) +∑ i≥2

1/2

+f

u1 u21i . 2(1 + u21 )

Note that φ(u) = 1 +

u , M

u1 u211 u1 u11 + (1 + u21 )1/2 2(1 + u21 )2

φ󸀠 (u) =

1 , M

φ󸀠󸀠 (u) = 0.

We have ρii φ󸀠󸀠 2 φ󸀠 2φ󸀠 + ui + uii + ρu} ρ φ φ φρ i i a ρ a ρ φ󸀠 1 2φ󸀠 ≥ aii uii + 11 11 + ∑ ii ii + ρ1 u1 φ ρ ρ 1 + u21 φρ i≥2

aii {

≥ ≥ ≥

φ󸀠 2(n − 1) 2φ󸀠 ρ1 u1 1 −2 1/2 ) − + (ut + f (x)(1 + u21 ) ) + ( φ φρ 1 + u21 r2 ρ 1 + u21 r 2 ρ φ󸀠 −2 2(n − 1) 4φ󸀠 x1 u1 1/2 (ut + f (x)(1 + u21 ) ) + ( 2 ) − − φ φρ r 2 1 + u21 r ρ r2 ρ φ󸀠 2n 4φ󸀠 u1 1/2 (ut + f (x)(1 + u21 ) ) − 2 − . φ r ρ rφρ 1 + u21

We may suppose that G(x0 , t0 , ξ ) is suitably large so that gu1 ≥ 1. Then by (2.3) we obtain 0 ≥ aii (log G)ii ≥

u211 φ󸀠 2n 4φ󸀠 u1 1/2 (ut + f (x)(1 + u21 ) ) − 2 − + φ r ρ rφρ 1 + u21 2(1 + u21 )2 log u1 +

≥

u1 u11 u1t + f1 (x)(1 + u21 )1/2 + f (1+u 2 )1/2

u1 log u1

1

u211 φ󸀠 2n 4 1/2 (ut + f (x)(1 + u21 ) ) − 2 − + φ r ρ Mr 2(1 + u21 )2 log u1 +

u1 u11 u1t + f1 (x)(1 + u21 )1/2 + f (1+u 2 )1/2

u1 log u1

1

.

428 | L. Ma and C.-h. Wang We may also assume ρu1 ≥ 1 and so have φ󸀠 1 1 1 ≤ = ≤ , 2M φ M+u M

u1 1 ≤ . 1 + u21 u1

Then, −

4φ󸀠 u1 4 1 4 ≥− ≥− . rφρ 1 + u21 Mr ρu1 Mr

By (2.1) and (2.2), we get u1 u11 u1t + f1 (x)(1 + u21 )1/2 + f (1+u 2 )1/2 φ󸀠 2 1/2 1 [ut + f (x)(1 + u1 ) ] + φ u1 log u1

u1 u11 f1 (x)(1 + u21 )1/2 f (x) (1+u21 )1/2 u1t φ󸀠 φ󸀠 2 1/2 + + u + f (x)(1 + u1 ) + = φ t φ u1 log u1 u1 log u1 u1 log u1

=

f1 (x)(1 + u21 )1/2 u1t f (x)u1 u11 φ󸀠 φ󸀠 1/2 + f (x)(1 + u21 ) + ut + + φ u1 log u1 φ u1 log u1 (1 + u21 )1/2 u1 log u1

f1 (x)(1 + u21 )1/2 ρ f (x)u1 φ󸀠 1 φ󸀠 1/2 ≥ − + f (x)(1 + u21 ) + (− 1 − u1 ) + t φ ρ φ u1 log u1 (1 + u21 )1/2

f (x)(1 + u21 )1/2 u21 φ󸀠 1 ρ f (x)u1 1/2 + f (x)[(1 + u21 ) − ]+ 1 ≥− − 1 2 2 1/2 1/2 t ρ (1 + u1 ) φ u1 log u1 (1 + u1 ) 1 ≥− − t 1 ≥− − t

ρ1 f (x) + ρ ρ1 f (x) + ρ

f1 φ󸀠 1 + f (x) φ (1 + u21 )1/2 log u1 f1 f (x) 1 + , M + u (1 + u21 )1/2 log u1

where |f (x)| ≤ C0 with the constant C0 > 0, which implies that

|f (x)| M+u

≤

C f (x) ≥− 0 M+u M

and

C 1 f (x) ≥ − 0. 2 1/2 M + u (1 + u1 ) M

From |∇f (x)| ≤ C0 again, we may also get

f1 log u1

≥ −C0 and then

u1 u11 u1t + f1 (x)(1 + u21 )1/2 + f (1+u 2 )1/2 φ󸀠 2 1/2 1 [ut + f (x)(1 + u1 ) ] + φ u1 log u1 1 2f (x) C0 ≥− − − − C0 t rρ M 1 2C ≥ − − 0 − C0󸀠 , t rρ

where C0󸀠 =

C0 M

+ C0 .

C0 . M

Then

Interior gradient estimates for mean curvature type equations and related flows | 429

ρ

If G(x0 , t0 , ξ ) is large enough so that | ρ1 | ≤ ρ

φ󸀠 u 2φ 1

at (x0 , t0 ), then by (2.1),

φ󸀠

( ρ1 + φ u1 )2 u211 = u2 log2 u1 (1 + u21 )2 (1 + u21 )2 1 φ󸀠

≥ ≥

( 2φ u1 )2

(1 + u21 )2

u21 log2 u1

φ󸀠 2 log2 u1 . 8φ2

It follows that at (x0 , t0 ), u211 2C 1 4 2n 0 ≥ aii (log G)ii ≥ − − C0󸀠 − 0 − − 2 + t rρ Mr r ρ 2(1 + u21 )2 log u1 1 ≥ − − C1 − t 1 ≥ − − C1 − t

C2 C3 φ󸀠 2 − 2 + log u1 rρ r ρ 16φ2 C2 C3 log u1 − + . rρ r 2 ρ 64M 2

Then C C 1 log u1 ≤ 64M 2 ( + C1 + 2 + 23 ). t rρ r ρ Hence, at (x0 , t0 ), tM 2 tM 2 + C3 2 r r TM 2 TM 2 2 ≤ C1 TM + C2 + C3 2 . r r

G(x, t, ξ ) = tρφ log u1 ≤ C1 tρM 2 + C2

Then at x = 0, log u1 ≤ (C1 TM 2 + C2

TM 2 TM 2 + C3 2 )/t. r r

Finally 2

2

TM TM 󵄨󵄨 󵄨 2 + C3 2 )/t], 󵄨󵄨∇u(0, t)󵄨󵄨󵄨 ≤ exp[(C1 TM + C2 r r which is the desired estimate. We have thus completed the proof of Theorem 1.1.

3 The proofs of Theorems 1.2 and 1.3 We now prove Theorem 1.2 using a similar argument as in the previous section.

430 | L. Ma and C.-h. Wang Proof of Theorem 1.2. Define the Harnack quantity G(x, ξ ) = g(x)φ(u) log uξ (x) for x ∈ Br (0), ξ ∈ Sn−1 , where g(x) = 1 − |x|2 /r 2 ,

φ(u) = 1 +

u , M

M = sup u(x). Br (0)

Suppose the supremum sup{G(x, ξ ), x ∈ Br (0), ξ ∈ Sn−1 } is attained at some point x0 and ξ = e1 the x1 direction. Then at x0 , 0 = (log G)i =

u1i gi φ󸀠 + ui + , g φ u1 log u1

(3.1)

and the matrix {(log G)ij } ≤ 0, where (log G)ij = (

gij g

−

gi gj g2

)+(

u1ij

+

φ󸀠󸀠 φ󸀠 2 φ󸀠 − 2 )ui uj + uij φ φ φ

− (1 +

u1 log u1

u1i u1j 1 ) 2 . log u1 u1 log u1

By (3.1), we have u1i u1j

u21 log u21

=

gi gj g2

+

φ󸀠 2 φ󸀠 u u + (g u + gj ui ). φ2 i j φg i j

Then, (log G)ij =

gij g

+

+

φ󸀠󸀠 φ󸀠 φ󸀠 ui uj + uij + (g u + gj ui ) φ φ φg i j u1ij

u1 log u1

− (1 +

Since ui (x0 ) = 0 for i ≥ 2, we have at x0 , a11 = Differentiating the relation

aij = (δij − uu

u u

i j 1i j we can get aij1 = −( 1+|∇u| 2 )1 = − 1+u2 − 1

ui u1j 1+u21

+

u1i u1j 2 ) . log u1 u21 log u1

1 , aii 1+u21

ui uj

1 + |∇u|2 ui uj (1+u21 )2

(3.2)

= 1 for i ≥ 2, and aij = 0 for i ≠ j.

),

⋅ 2u1 u11 .

Interior gradient estimates for mean curvature type equations and related flows | 431

By (1.4), we have αu1 − aij1 uij u2 u1i uj ui u1j 2ui uj u1 u11 αu = − 21 + uij + uij − uij 2 2 u 1 + u1 1 + u1 (1 + u21 )2

aii uii1 = −

=− =−

2u1 u21i 2u31 u211 αu1 2u1 u211 − + + ∑ u2 1 + u21 (1 + u21 )2 i≥2 1 + u21 2u1 u21i 2u1 u211 αu1 + + . ∑ u2 (1 + u21 )2 i≥2 1 + u21

(3.3)

Then, if u1 (x0 ) > 0 is suitably large, we have u2 2 ) 1i ) log u1 u1 2u1 u21i 2u1 u211 u211 αu 2 1 = − 21 + (1 + ) + − ∑ 2 2 log u1 u1 i≥2 1 + u21 u (1 + u1 )2 1 + u1

aii (uii1 − (1 +

≥−

u1 u21i u1 u211 αu1 + + . ∑ u2 2(1 + u21 )2 i≥2 2(1 + u21 )

Note that φ(u) = 1 +

u , M

φ󸀠 (u) =

1 , M

φ󸀠󸀠 (u) = 0.

We have gii φ󸀠󸀠 2 φ󸀠 2φ󸀠 + ui + uii + gu} g φ φ φg i i a g a g φ󸀠 1 2φ󸀠 ≥ aii uii + 11 11 + ∑ ii ii + g1 u1 φ g g 1 + u21 φg i≥2

aii {

≥ ≥ ≥

φ󸀠 α 1 2(n − 1) 2φ󸀠 g1 u1 −2 + ) − + ( φ u 1 + u21 r 2 g φg 1 + u21 r2 g φ󸀠 α −2 2(n − 1) 4φ󸀠 x1 u1 +( 2 )− − φ u φg r 2 1 + u21 r g r2 g 1 α 2n 4φ󸀠 u1 − − . M + u u r 2 g rφg 1 + u21

We may suppose that G(x0 ) is suitably large so that gu1 ≥ 1. Then by (3.2), we obtain 0 ≥ aii (log G)ii ≥

1 − αu u2

u1 log u1

≥−

+

u211 1 α 2n 4φ󸀠 u1 − 2 − + 2 M + u u r g rφg 1 + u1 2(1 + u21 )2 log u1

u211 α 1 α 2n C + − − + . u2 log u1 M + u u r 2 g Mr 2(1 + u21 )2 log u1

432 | L. Ma and C.-h. Wang Since gu1 ≥ 1 and

φ󸀠 φ

1 M+u

=

−

≤

1 , u1 M 1+u21

≤

1 , u1

one has

4φ󸀠 u1 C C 1 ≥− . ≥− rφg 1 + u21 Mr gu1 Mr

If G(x0 ) is large enough so that | gg1 | ≤

φ󸀠 u 2φ 1

at x0 , then, by (3.1),

φ

( gg1 + φ u1 )2 u211 = u2 log2 u1 (1 + u21 )2 (1 + u21 )2 1 󸀠

φ󸀠

( 2φ u1 )2

u2 log2 u1 (1 + u21 )2 1 φ󸀠 2 ≥ log2 u1 . 8φ2

≥

It follows that

φ󸀠 2 log u1 1 α 2n C − 2 − + M + u u r g Mr 16φ2 1 log u1 C 1 α α C + . ≥− 2 − − 2 + u log u1 M + u u Mr r 2 g 64M 2

0 ≥ aii (log G)ii ≥ −

α

u2 log u

+

Note, for α ≤ 0, √log u1 1 α α = M + u u u√log u1 M + u α log u1 α + ≥ 2 u log u1 4(M + u)2 α log u1 α ≥ 2 + . u log u1 4M 2 Hence, −

α

u2 log u

1

+

α log u1 1 α α α ≥− 2 + 2 + M+uu u log u1 u log u1 4M 2 α log u1 ≥ . 4 M2

For α ∈ (− 161 , 0), by (3.4), we have log u1 1 C α C ( + )− − 2 ≤ 0, Mr r 2 g M 2 64 4 which implies g log u1 ≤ C1

M M2 + C2 2 . r r

Then G = gφ log u1 ≤ C1

M M2 + C2 2 . r r

(3.4)

Interior gradient estimates for mean curvature type equations and related flows | 433

Finally, 2

M M 󵄨 󵄨󵄨 󵄨󵄨∇u(0)󵄨󵄨󵄨 ≤ exp(C1 + C2 2 ). r r We have thus proved Theorem 1.2. We now prove Theorem 1.3, and again we use the gradient estimate method as above. At this time, we add one more parameter in the choice of Harnack quantity. Proof of Theorem 1.3. Assume we have a nontrivial nonnegative global solution to the singular minimal surface equation (1.1). By the maximum principle, we may assume u > 0 in ℝn and u is smooth in ℝn . By Theorem 1.2 and the sublinear growth condition on u, we have that there is a uniform constant C > 0 such that in ℝn .

|∇u| ≤ C

(3.5)

We now want to prove ∇u(0) = 0. We suppose by contradiction that |∇u(0)| ≥ δ > 0. Let G(x, ξ ) = g(x)φ(u)uξ for ξ ∈ Sn−1 and x ∈ Br (0), where g(x) = 1 − |x|2 /r 2 ,

φ(u) = (1 −

−β

u ) , M

󵄨 󵄨 M = Mr = 4 sup{󵄨󵄨󵄨u(x)󵄨󵄨󵄨, x ∈ Br (0)},

and the new parameter β ∈ (0, 1) will be determined below. Suppose G(x, ξ ) attains its maximum at x0 ∈ Br (0) and in the x1 direction. Then, by (3.5), we have g(x0 ) ≥ δ1

and u1 (x0 ) ≥ δ1

for some δ1 > 0 depending only on δ. We have at x0 0 = (log G)i =

u1i gi φ󸀠 + + ui , u1 g φ

(3.6)

and the matrix {(log G)ij } ≤ 0, where (log G)ij =

u1ij

φ󸀠󸀠 φ󸀠 2 φ󸀠 − 2 2 )ui uj + uij u1 φ φ φ 󸀠 gij gi gj φ +( −2 2 )− (g u + gj ui ). g gφ i j g +(

Hence, 0 ≥ aii (log G)ii = aii {

u1ij u1

+(

gij

−2

gi2 φ󸀠󸀠 φ󸀠 2 2 ) + ( − 2 )ui φ g2 φ2

g φ φ󸀠 + uii − 2 gi ui }. φ gφ 󸀠

(3.7)

434 | L. Ma and C.-h. Wang By (3.3), we have αu1 − aij1 uij u2 2u u2 2u1 u211 αu = − 21 + + ∑ 1 1i2 . 2 2 u (1 + u1 ) i≥2 1 + u1

aii uii1 = −

Hence, aii

2u21i 2u211 uii1 α =− 2 + +∑ 2 2 2 u1 u (1 + u1 ) i≥2 1 + u1 2

≥−

u11 α + . u2 2(1 + u21 )2

Note that φ(u) = (1 −

−β

u ) , M

φ󸀠 β = , φ M−u

φ󸀠󸀠 β(β + 1) . = φ (M − u)2

We have aii { ≥ ≥

g2 gii φ󸀠󸀠 φ󸀠 2 φ󸀠 φ󸀠 − 2 i2 + ( − 2 2 )u2i + u11 − 2 gi ui } g φ φ φg g φ

g2 a g g φ󸀠 2 φ󸀠 φ󸀠󸀠 2φ󸀠 aii uii + ∑ ii ii + a11 { 11 − 2 12 + ( − 2 2 )u21 − gu} φ g g φ φg 1 1 g φ i≥2

g12 1 φ󸀠 α 2(n − 1) β(β + 1) 2β2 1 −2 + ( ) − 2 +[ − ] − 2 2 2 2 2 2 φ u r g g 1 + u1 (M − u) (M − u)2 1 + u1 r g

u21 2φ󸀠 g1 u1 − 1 + u21 φg 1 + u21 ≥ ≥

g12 u21 φ󸀠 α 2(n − 1) β − β2 4φ󸀠 x1 u1 −2 − + ( ) − 2 + + 2 2 2 2 2 φ u φg r 2 1 + u21 r g r g g (M − u) 1 + u1

g2 φ󸀠 α 2n β 4φ󸀠 u1 − 2 − 2 12 + − . φ u r g g (M − u)2 rφρ 1 + u21

Then, by (3.7), we obtain 0 ≥ aii (log G)ii ≥− ≥− ≥−

u211 g12 φ󸀠 α 2n β 4φ󸀠 u1 α + + − − 2 + − u2 2(1 + u21 )2 φ u r 2 g g 2 (M − u)2 rφg 1 + u21 g2 u211 β α φ󸀠 α 2n C + − 2 − 2 12 + − + 2 2 φ u r g r(M − u) 2(1 + u21 )2 u g (M − u)

u211 β α φ󸀠 α 2n 8 C + − − + − + . u2 φ u r 2 g r 2 g 2 (M − u)2 r(M − u) 2(1 + u21 )2

Interior gradient estimates for mean curvature type equations and related flows | 435

By assumption, for r large enough,

M r

→ 0 as r → ∞. Then we have at x0 ,

󵄨󵄨 g 󵄨󵄨 βδ βu1 φ󸀠 2 󵄨󵄨 1 󵄨󵄨 ≤ 1 ≤ = u. 󵄨󵄨 󵄨󵄨 ≤ 󵄨󵄨 g 󵄨󵄨 δ1 r M 2(M − u) 2φ 1 By (3.6), we have φ󸀠

( gg1 + φ u1 )2 u211 = u2 (1 + u21 )2 (1 + u21 )2 1 φ󸀠

≥

( 2φ u1 )2

(1 + u21 )2

≥[

u21 2

u4 β ] 14 , 2(M − u) ω

where ω2 = 1 + u21 . Recall α ∈ (− 161 , 0). We now choose β ∈ (0, 1) small enough such that Then we have −

1 u

≥

β . M−u

β 1 α φ󸀠 α α = ( − ) ≥ 0. + 2 φ u u M−u u u

It follows that 2

[

u4 β 2n 8 C ] 14 ≤ 2 + 2 2 + . 2(M − u) ω r(M − u) r g r g

Letting r → ∞, we obtain u41 2n 8 C ≤ + + . ω4 r 2 g[ β ]2 r 2 g 2 [ β ]2 r(M − u)[ β ]2 2(M−u) 2(M−u) 2(M−u) Then at x0 , 2

2

9M (M − u) 󵄨 󵄨 G = gφ󵄨󵄨󵄨u1 (x0 )󵄨󵄨󵄨 ≤ C ≤C 2 . r2 r Hence, 2

9M 󵄨󵄨 󵄨 󵄨󵄨u1 (0)󵄨󵄨󵄨 ≤ C 2 → 0 r

as r → ∞.

We conclude u1 (0) = 0, which is a contradiction to our assumption. Since 0 can be any point in ℝn , we then have ∇u = 0 on ℝn , which implies that u is a positive constant function, which is absurd since a positive constant function is not a solution to the singular minimal surface equation. We have thus proved Theorem 1.3.

436 | L. Ma and C.-h. Wang Remark 3.1. Very recently, we found that a similar result to Theorem 1.3 had been obtained in [9] for a large range of α, but we had not known the paper [9] before we had finished our paper. As a comparison, we prefer to mention the following result of X. J. Wang in the article [32], which is a Liouville-type result for the mean curvature equation (δij −

ui uj

1 + |∇u|2

)uij = H(x)√1 + |∇u|2

(3.8)

under different conditions. Theorem 3.2. Let u ∈ C 3 (Br (0)) be an admissible solution of (3.8) with H(x) = 0. Suppose u(x) = o(1 + |x|)

as |x| → ∞.

Then u ≡ constant.

4 The proof of Theorem 1.4 In this section, we prove Theorem 1.4. We let f (x) = H(x)√1 + |∇u|2 + αu. Define the Harnack quantity by G(x, ξ ) = g(x)φ(u) log uξ (x) for x ∈ Br (0), ξ ∈ Sn−1 , where g(x) = 1 − |x|2 /r 2 , Note φ󸀠u =

1 . M

φ(u) = 1 +

u , M

M = sup u(x). Br (0)

Suppose the supremum sup{G(x, ξ ), x ∈ Br (0), ξ ∈ Sn−1 }

is attained at some point x0 and in the x1 direction as before. Then at x0 , for i = 1, . . . , n, 0 = (log G)i =

u1i gi φ󸀠 + ui + , g φ u1 log u1

and the matrix {(log G)ij } ≤ 0, where gij

gi gj

φ󸀠󸀠 φ󸀠 2 φ󸀠 − )u u + u i j g φ φ ij g2 φ2 u1ij u1i u1j 1 + − (1 + ) . u1 log u1 log u1 u21 log u1

(log G)ij = (

−

)+(

(4.1)

Interior gradient estimates for mean curvature type equations and related flows | 437

By (4.1), we have u1i u1j

=

u21 log u21

gi gj g2

φ󸀠 2 φ󸀠 u u + (g u + gj ui ). i j φg i j φ2

+

Hence, we have (log G)ij =

gij

φ󸀠󸀠 φ󸀠 φ󸀠 ui uj + uij + (g u + gj ui ) g φ φ φg i j u1ij u1i u1j 2 + − (1 + ) 2 . u1 log u1 log u1 u1 log u1 +

Since ui (x0 ) = 0 for i ≥ 2, we have at x0 , a11 = Differentiating equation (1.5), namely

aij uij = (δij −

1 , aii 1+u21

ui uj

1 + |∇u|2

= 1 for i ≥ 2, and aij = 0 for i ≠ j.

)uij = f (x),

we can get aij1 = −(

ui uj

1+

|∇u|2

) =− 1

u1i uj

1+

u21

−

ui u1j

1+

u21

+

ui uj

(1 + u21 )2

⋅ 2u1 u11 .

Then aii uii1 = f1 − aij1 uij u1i uj ui u1j 2ui uj u1 u11 = f1 + uij + uij − uij 2 2 1 + u1 1 + u1 (1 + u21 )2 = f1 + = f1 +

2u1 u21i 2u1 u211 2u31 u211 − + ∑ 1 + u21 (1 + u21 )2 i≥2 1 + u21 2u1 u21i 2u1 u211 + . ∑ (1 + u21 )2 i≥2 1 + u21

Hence, if u1 (x0 ) is suitably large, we have aii (uii1 − (1 + = f1 + ≥ f1 +

u2 2 ) 1i ) log u1 u1

2u1 u21i 2u1 u211 u211 1 2 − (1 + ) + ∑ log u1 u1 i≥2 1 + u21 (1 + u21 )2 1 + u21 u1 u21i u1 u211 + . ∑ 2(1 + u21 )2 i≥2 2(1 + u21 )

Note that φ(u) = 1 +

u , M

φ󸀠 (u) =

1 , M

(4.2)

φ󸀠󸀠 (u) = 0.

438 | L. Ma and C.-h. Wang We have aii {

gii φ󸀠󸀠 2 φ󸀠 2φ󸀠 + ui + uii + gu} g φ φ φg i i

≥

a g a g φ󸀠 1 2φ󸀠 aii uii + 11 11 + ∑ ii ii + g1 u1 φ g g 1 + u21 φg i≥2

≥

φ󸀠 2(n − 1) 2φ󸀠 g1 u1 1 −2 + f+ ( 2 )− 2 φ φg 1 + u21 r2 g 1 + u1 r g

≥ ≥

φ󸀠 2(n − 1) 4φ󸀠 x1 u1 −2 − f +( 2 )− φ φg r 2 1 + u21 r g r2 g

φ󸀠 2n 4φ󸀠 u1 f− 2 − . φ r g rφg 1 + u21

We may suppose G(x0 ) is suitably large so that gu1 ≥ 1. Then, by (4.2), we obtain 0 ≥ aii (log G)ii ≥ ≥

u211 f1 φ󸀠 2n 4φ󸀠 u1 + f− 2 − + u1 log u1 φ r g rφg 1 + u21 2(1 + u21 )2 log u1 u211 f1 φ󸀠 2n 4 . + f− 2 − + u1 log u1 φ r g Mr 2(1 + u21 )2 log u1

Based on gu1 ≥ 1,

φ󸀠 1 1 = ≤ , φ M+u M

u1 1 ≤ , 2 1 + u1 u1

we get −

4φ󸀠 u1 4 1 4 ≥− ≥− . rφg 1 + u21 Mr gu1 Mr

Recall that f (x) = H(x)√1 + |∇u|2 + αu. Then we have f1 (x) = H1 (x)√1 + |∇u|2 + αu1 +

H(x)u1 u11 . √1 + |∇u|2

By (4.1), we have f1 φ󸀠 + f u1 log u1 φ = ≥

H1 (x)(1 + u21 )1/2 αu1 H(x)u1 u11 φ󸀠 1/2 + + + [H(x)(1 + u21 ) + αu] 2 u1 log u1 u1 log u1 u1 log u1 (1 + u1 )1/2 φ H1 (x) g H(x)u1 φ󸀠 φ󸀠 α 1/2 + + (− 1 − u1 ) + [H(x)(1 + u21 ) + αu] 2 1/2 log u1 log u1 g φ φ (1 + u1 )

Interior gradient estimates for mean curvature type equations and related flows | 439

u21 −C0 g H(x)u1 φ󸀠 φ󸀠 2 1/2 +α− 1 + H(x)[(1 + u ] + αu ) − 1 log u1 g (1 + u21 )1/2 φ φ (1 + u21 )1/2 g H(x) H(x) 1 αu ≥ −C0 + α − 1 + + , g M + u (1 + u21 )1/2 M + u ≥

(4.3)

where √1 + |∇u|2 = (1 + u21 )1/2 ≥ u1 , g1 = | 2xr21 | ≤ 2r . From |H(x)| ≤ C0 , C0 > 0, we have H(x) ≥ −C0 . Then C H(x) ≥− 0 M+u M and

1 H(x) M+u (1+u21 )1/2

C

≥ − M0 . From

Then we obtain

u M+u

≤ 1, α ≤ 0, we also get

αu M+u

≥ α.

C f1 φ󸀠 2H(x) + f ≥ −C0 − 0 + 2α − u1 log u1 φ M rg 2C ≥ −C̄ 1 − 2|α| − 0 , rg where C̄ 1 = C0 +

C0 . M

If G(x0 ) is large enough so that | gg1 | ≤

φ󸀠 u 2φ 1

at x0 , then, by (4.1),

φ󸀠

( gg1 + φ u1 )2 u211 = u2 log2 u1 (1 + u21 )2 (1 + u21 )2 1 φ󸀠

≥ ≥

( 2φ u1 )2

(1 + u21 )2

u21 log2 u1

φ󸀠 2 log2 u1 . 8φ2

It follows that 2C φ󸀠 2 2n 4 + 0 ≥ aii (log G)ii ≥ −C̄ 1 − 2|α| − 0 − 2 − log u1 rg r g Mr 16φ2 2C ≥ −C̄ 1 − 2|α| − 0 − rg 2C ≥ −C̄ 1 − 2|α| − 0 − rg

φ󸀠 2 2n + log u1 r 2 g 16φ2 2n log u1 + . r 2 g 64M 2

Hence, g log u1 ≤ C1 + C2 |α| + C3

M2 M2 + C4 2 . r r

Then G = gφ log u1 ≤ C1 + C2 |α| + C3

M2 M2 + C4 2 . r r

440 | L. Ma and C.-h. Wang Finally, 2

2

M M 󵄨 󵄨󵄨 + C4 2 ). 󵄨󵄨∇u(0)󵄨󵄨󵄨 ≤ exp(C1 + C2 |α| + C3 r r We have thus proved Theorem 1.4.

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[21] Y. Y. Li, Interior gradient estimates for solutions of certain fully nonlinear elliptic equations. J. Differ. Equ. 90, 172–185 (1991). [22] L. Ma, On minimal graph evolutions in the hyperbolic space. Acta Math. 15, 371–374 (1999). [23] L. Ma, Gradient estimates for a simple nonlinear heat equation on manifolds. Appl. Anal. 96(2), 225–230 (2017). [24] L. Ma, Gradient estimates for a simple elliptic on complete non-compact Riemannian manifold. J. Funct. Anal. 241, 374–382 (2006). [25] L. Ma, Volume growth and Bernstein theorems for translating solitons. J. Math. Anal. Appl. 473, 1244–1252 (2019). https://doi.org/10.1016/j.jmaa.2019.01.019. [26] L. Ma, Convexity and the Dirichlet problem of translating mean curvature flows. Kodai Math. J. 41(2), 348–358 (2018). [27] L. Ma, M. Vicente, Bernstein theorem for translating solitons of hypersurfaces. Manuscr. Math. (2019). https://doi.org/10.1007/s00229-019-01112-1. [28] X. Ma, J. Xu, Gradient estimates of mean curvature equations with Neumann boundary value problems. Adv. Math. 290, 1010–1039 (2016). [29] W. Sheng, X.-J. Wang, Regularity and singularity in the mean curvature flow, in Trends in Partial Differential Equations. Adv. Lect. Math. (ALM), vol. 10 (Int. Press, Somerville, MA, 2010), pp. 399–436. [30] L. Simon, Interior gradient bounds for non-uniformly elliptic equations. Indiana Univ. Math. J. 25, 821–855 (1976). [31] N. S. Trudinger, Gradient estimates and mean curvature. Math. Z. 131, 165–175 (1973). [32] X. J. Wang, Interior gradient estimates for mean curvature equations. Math. Z. 228, 73–81 (1998).

Lei Ni

An alternate induction argument in Simons’ proof of holonomy theorem Abstract: The paper gives an exposition and an alternate argument of Simons algebraic proof [7] of the holonomy theorem via the holonomy system. Keywords: Curvature, Lie groups, holonomy groups, symmetric spaces MSC 2010: 53C29, 53C05, 58D19, 57S25

Contents 1 2 3 4 5

Introduction | 443 Preliminaries | 445 A derivation of Theorem 1.1 | 449 Simons’ constructions of flats and totally geodesic subspaces | 450 An alternate proof via the induction on dim(G(R)) | 455 Bibliography | 457

1 Introduction Berger’s classification [2] of Riemannian holonomy groups is very important in Riemannian geometry. The proof utilized Cartan’s classification of simple Lie groups. An intrinsic proof was later discovered by J. Simons [7]. In fact, Simons proved the following result without appealing to Cartan’s classification results. Theorem 1.1 (Berger). Assume that Hp0 , the restricted holonomy group of a Riemannian manifold (M n , g), acts irreducibly on the tangent space Mp . Then either Hp0 acts transitively on 𝕊n−1 ⊂ Mp or (M n , g) is a locally symmetric space with rank greater than or equal to 2. This result implies Berger’s list for possible holonomy groups of Riemannian manifolds which are not locally symmetric due to the earlier work of [4] on transformation Acknowledgement: We would like to thank the organizers of Nankai conference “Analysis and PDEs on Manifolds and Fractals” for the invitation to the event and this contribution. The author thanks Professors N. Wallach and H. Wu for their interests, Y. Niu, F. Zheng for helpful comments. He also thanks the participating students of the graduate course Math 250C (UCSD, Spring of 2020) for their patience. The alternate argument was cleaned up during the course. Lei Ni, Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA, e-mail: [email protected] https://doi.org/10.1515/9783110700763-016

444 | L. Ni groups of the sphere (cf. [3]). Simons obtained the above result by solving an algebraic problem in terms of the holonomy system: Theorem 1.2 (Simons). Assume that S is an irreducible Riemannian holonomy system. Assume that G acts nontransitively on 𝕊n−1 ⊂ V. Then S is symmetric. Here a Riemannian holonomy system S = {V, R, G} consists of a Euclidean space V of dimension n (we call it the degree of S) endowed with an inner product, a connected compact subgroup G of SO(n), and an algebraic curvature operator R (defined on V) satisfying the first Bianchi identity and such that Rx,y ∈ g, ∀x, y ∈ V with g ⊂ so(n) being the Lie algebra of G. The system S is called irreducible if G acts irreducibly on V. That a holonomy system S = {V, R, G} is symmetric means g(R) = R, ∀g ∈ G. Let SB2 (∧2 V) denote the space of algebraic curvature operators after identifying so(n) with ∧2 V. Here S2 (∧2 V) denotes the symmetric transformations of ∧2 V. The SB2 (⋅) denotes the subspace satisfying the first Bianchi identity. The action g(R) is the natural extension of the action of SO(n) on V to SB2 (∧2 V) (see Section 2 for details). What was proved in [7] is slightly stronger. To state that result, we need to introduce additional notions. Let G(R) ⊂ SB2 (∧2 V) be the linear subspace spanned by {g(R), g ∈ G}. Namely, G(R) is the subspace generated by the orbit of R under the action of G (also see Section 2 for more details). Define gR ⊂ g as the subspace spanned by {Q(∧2 V), Q ∈ G(R)}. One may check that gR (see Lemma 4.1) is an ideal of g. Now GR ⊂ G is defined as the Lie (closed) subgroup of G generated by gR . Clearly the nontransitivity of G implies the nontransitivity of GR . Theorem 1.3 (Simons). Let S = {V, R, G} be an irreducible Riemannian holonomy system. Assume that GR acts nontransitively on 𝕊n−1 . Then S is symmetric with rank ≥ 2. The proof of Simons [7] is via a double induction on dim(V) and dim(gR ). The purpose of this paper is to give an exposition of Simons’ proof via an alternate induction on dim(G(R)). Since it was believed that (cf. [5]) “... the proof of Simons is long and involved, except for the first general part. At some step he used case by case arguments, combined with induction on the dimension. Few mathematicians went through all the details of this proof...”, our hope is that the exposition here and this alternate induction can offer some enhancement in understanding the important work [7] which contains many ingenious ideas. There exist several expositions, e. g., [3, 6, 10], on holonomy theorem and Simons’ proof. Due to the importance of Theorem 1.1, our presentation includes basic definitions and a derivation of Theorem 1.1 using Theorem 1.2. The argument here does not use any result from the theory of symmetric spaces. Precisely, it does not use the correspondence between the orthogonal symmetric Lie algebras and symmetric spaces, or the full Ambrose–Singer’s theorem [1]. The presentation is completely self-contained except very basic results, such as the Schur’s lemma.

Simons’ theorem

| 445

2 Preliminaries In this section we recall basic concepts and definitions. A fiber bundle is a triple (E, F, M) with a projection map π : E → M such that π is regular with π −1 (x) (denoted as Ex ) being diffeomorphic to the space F such that for any point p ∈ M, there exists a neighborhood Uα and a diffeomorphism φα : Uα × F → π −1 (Uα ) such that φα (x, f ) ∈ π −1 (x). We say that it has a structure group G, if the transition functions Tαβ (x) (where φ−1 β ∘ φα (x, f ) = (x, Tαβ (x)(f ))) is in G. Here E, F, M are all smooth manifolds and we also require Tαβ (x) be smooth in x. A connexion of (E, F, M) is a mapping P, defined for any piecewisely smooth path γ : (0, 1) → M, a Pγ : Eγ(0) → Eγ(1) such that it satisfies that (i) Pγ depends on γ smoothly, (ii) Pγ1 ∘γ2 = Pγ1 ∘ Pγ2 , and (iii) Pγ−1 = (Pγ )−1 . Such Pγ is called the parallel transport along γ. In general, Pγ is in Diff(F). When F is a linear space and G is a subgroup of general linear transformations, namely (E, M) is a vector bundle, Pγ is required to be a linear map. Let Ω(x0 , M) be the loop space at x0 . Then P(⋅) : Ω(x0 , M) → Diff(Ex0 ) (or G) defined by Pγ is a homomorphism. The image (denoted by Hx0 ) is called the holonomy group. For most of our discussion, emphasis is given to the image of the connected component of the trivial loop γ(t) ≡ x0 , namely the loops which are homotopically trivial. The corresponding image is called the restricted holonomy group, denoted by Hx00 . Its Lie algebra is denoted by h. It is easy to see that for a different base point x1 , if γ is a path from x1 to x0 , then Hx0 = Pγ Hx1 Pγ−1 , and Hx00 = Pγ Hx01 Pγ−1 . A covariant derivative at point p is a map ∇ : Tp M × 𝒯p M → Tp M (𝒯p M denotes the germs of tangent vectors) satisfying axioms: (i) ∇αξ +βη Y = α∇ξ Y +β∇η Y; (ii) linear in the second component; (iii) ∇ξ (fY) = (ξf )Y + f ∇ξ Y. This is also called an affine connection. A global affine connection is that defined for all p ∈ M and such that if X, Y are smooth ∇X Y is smooth. Once M is endowed with a global affine connection, one can define the covariant derivative along a curve c(t) : (a, b) → M for a vector field X(t) along c(t) D by dt X(c(t)) = ∇c(t) ̇ X, if X is defined on c(t). This leads to a connexion defined above via the parallel transport along c(t) by solving an ordinary differential equation: For any Xx0 ∈ Tx0 M and a curve γ(t) with γ(0) = x0 and γ(1) = x1 , X(t) ∈ Tγ(t) M can be D X(t) = 0, X(0) = Xx0 . Then one defines Pγ (Xx0 ) ≑ X(1). In constructed by solving dt t ,t

t ,t

general, Pγ1 2 : Tγ(t1 ) M → Tγ(t2 ) M can be defined as Pγ1 2 (ξ ) = X(t2 ) with X(t) being the parallel vector along γ(t) with X(t1 ) = ξ . Note that the above discussion makes sense for any smooth vector bundle (E, M) of rank k as well. A basic result below asserts that a connexion on a vector bundle (with linear structure group) is equivalent to an affine connection. Lemma 2.1.

󵄨 D X(t)󵄨󵄨󵄨󵄨t dt 0

= limt→t0

t,t

Pγ 0 (X(t))−X(t0 ) . t−t0

We focus on the case that F = Ex = π −1 (x) is a vector space endowed with a smoothly depended inner product. Now Pγ is required to preserve this inner product (namely the metric is invariant w. r. t. the D above). Theorem 1.1 concerns the Levi-

446 | L. Ni Civita connection, namely the canonical affine connection of the Riemannian structure (M n , g) on its tangent bundle. For p ∈ M, let γ be a loop at p or a path towards p. We also use γ to denote the parallel transport along γ, which is an isometry of Mp , the tangent space at p. In this setting Hp0 ⊂ SO(n) is compact. Let h ⊂ so(n) be its Lie algebra. Let R be the curvature tensor of Levi-Civita connection. First we show that Lemma 2.2. ∀x, y ∈ Mp and ∀γ from q to p, γ(Rq )x,y ∈ h. Here ⟨γ(Rq )x,y z, w⟩ ≑ ⟨Rγ−1 (x),γ−1 (y) γ −1 (z), γ −1 (w)⟩,

∀x, y, z, w ∈ Mp .

(2.1)

Proof. We start with the case that γ is trivial, namely γ = {p}. Extend x and y to a neighborhood of p and denote them by X, Y. We can extend in such a way that [X, Y] = 0. For a vector field Z, recall that ∇X Z at p can be computed by lim

Pφt,0p (Z(φp (t))) − Z(p) t

t→0

.

Here φq (t) denotes the integral curve of X originated at q (also abbreviated as αq ), Pφt,0 denotes the parallel transport from φp (t) to φp (0) = p. Similarly, we let ψq (s) denote the integral curve of Y originated from q (also denoted as βq ). The assumption [X, Y] = 0 ensures that φt and ψs commute, namely ψφp (t) (s) = φψp (s) (t). For any z ∈ Mp , let Z(t, s) ≑ Pβ0,s ⋅ Pα0,tp (z). αp (t)

From the definition it is easy to see that ∇ 𝜕 Z|(t,s) = 0 and ∇ 𝜕 Z|(t,0) = 0. Also define the 𝜕s 𝜕t mapping Ψ(t, s) : Mp → Mp as Ψ(t, s) = Pβs,0 ⋅ Pαt,0 ⋅ Pβ0,s ⋅ Pα0,tp . β (s) p

αp (t)

p

For sufficiently small t and s, we have that Ψ(t, s) ∈ Hp0 . We shall show that Rx,y z can be expressed in terms of derivatives of Ψ(η)(z) for Ψ(η) = Ψ(√η, √η). We claim that Ψ(t, s)(z) − z 𝜕2 󵄨󵄨󵄨󵄨 = Ψ(t, s)(z). 󵄨 t,s→0 ts 𝜕t𝜕s 󵄨󵄨󵄨(0,0)

∇ 𝜕 ∇ 𝜕 Z|(0,0) = lim 𝜕s

𝜕t

(2.2)

󵄨 Noting that the left-hand side is Rx,y z ≑ −∇X ∇Y Z + ∇Y ∇X Z + ∇[X,Y] Z 󵄨󵄨󵄨p , letting t =

√η, s = √η, by claim (2.2), we have that Rx,y z = limη→0 (2.2), note that ∇ 𝜕 ∇ 𝜕 Z|(0,0) = lim 𝜕s

𝜕t

s→0

(∇ 𝜕 Z)(0, s) = lim 𝜕t

t→0

Ψ(η)(z)−z . For the proof of claim η

Pβs,0 ((∇ 𝜕 Z)(0, s)) − (∇ 𝜕 Z)(0, 0) p

𝜕t

𝜕t

s Pαt,0 (Z(t, s)) − Z(0, s) β (s) p

t

,

and

, Pβs,0 (Z(0, s)) = z. p

Claim (2.2) follows by putting the above three identities together.

Simons’ theorem | 447

The general case follows from the observation that γHq0 γ −1 = Hp0 , where γ is a path joining q to p. Hence γ(Rq )x,y = γ ⋅ Rγ−1 (x),γ−1 (y) ⋅ γ −1 lies in the Lie algebra of Hp0 since Rγ−1 (x),γ−1 (y) is in the Lie algebra of Hq0 . More precisely, if Ψ(η) is the element in Hq0 corresponding to γ −1 (x), γ −1 (y), recall from (2.1) that γ(Rq )x,y z = γ(Rqγ−1 (x),γ−1 (y) γ −1 (z)), thus Ψ(η)(γ −1 (z)) − γ −1 (z) γ ⋅ Ψ(η) ⋅ γ −1 (z) − z ) = lim . η→0 η→0 η η

γ(Rq )x,y z = γ(lim

The result follows since γ ⋅ Ψ ⋅ γ −1 ∈ Hp0 . Lemma 2.2 is the easy part of Ambrose–Singer’s theorem [1], perhaps known to Cartan. The second part of Ambrose–Singer’s theorem asserts that γ(Rq )x,y is all that is needed to generate h if γ runs through all possible paths. This part, however, is not needed/used for our discussion. Note that the argument above proves for a Riemannian connection on any Riemannian vector bundle E that γ(Rq )x,y ∈ h with h being the Lie algebra of Hp0 (E). Ambrose–Singer’s theorem also asserts that {γ(Rq )x,y }, ∀x, y ∈ Mp , with γ exhausting all possible paths, generates the Lie algebra of Hp0 (E). Given S = {V, R, G}, g(R) can be defined algebraically. First recall the action of SO(n) (hence G) on ∧2 (V). Let x⊗y(z) ≑ ⟨y, z⟩x. Then g(x⊗y) ≑ gx⊗gy. Direct calculation then shows that g(x ∧ y) = g ⋅ x ∧ y ⋅ g −1 (g ∈ SO(n) is used). (Note that x ∧ y can be identified with an element in so(n), and we have identified ∧2 V, ∧2 V, and Hom(V, V) using the metric on V.) Hence g(x ∧ y) = Adg (x ∧ y). Since g tr = g −1 , note that Adg acts on ∧2 (V) isometrically with respect to the metric on gl(V): ⟨A, B⟩ ≑

1 1 ∑⟨A(ei ), B(ei )⟩ = trace(Btr A) 2 i 2

since (Adg )tr = Adg tr , which is Adg −1 for g ∈ O(n). It is easy to see for A ∈ so(n) that 2⟨A, x ∧ y⟩ = ∑⟨A(ei ), (x ∧ y)(ei )⟩ = ∑(⟨A(ei ), ⟨y, ei ⟩x⟩ − ⟨A(ei ), ⟨x, ei ⟩y⟩) = − ∑⟨ei , A(x)⟩ ⋅ ⟨y, ei ⟩ + ∑⟨ei , A(y)⟩ ⋅ ⟨x, ei ⟩ = −⟨y, A(x)⟩ + ⟨x, A(y)⟩ = −2⟨y, A(x)⟩ = 2⟨x, A(y)⟩.

(2.3)

Here {ei } is an orthonormal basis of V. With this convention R(x ∧ y) is identified with −Rx,y . Recall that R can be viewed as a symmetric tensor of ∧2 (V) with ⟨R(x ∧ y), z ∧ w⟩ = ⟨Rx,y z, w⟩ = ⟨−Rx,y w, z⟩ = R(x, y, z, w). To be compatible with (2.1) when V = Mp , G = Hp0 , we define g(R)x,y ≑ g ⋅ Rg −1 (x),g −1 (y) ⋅ g −1 .

448 | L. Ni Lemma 2.2 asserts that {Mp , R, Hp0 } is a holonomy system. If S = {V, R, G} is a holonomy system, g(R)x,y ∈ g for all g ∈ G since g ⋅ A ⋅ g −1 = Adg (A) ∈ g if A = Rg −1 (x),g −1 (y) ∈ g. In the mean time, ⟨g(R)(x ∧ y), z ∧ w⟩ = ⟨g(R)x,y z, w⟩ = ⟨Rg −1 (x),g −1 (y) g −1 (z), g −1 (w)⟩ = ⟨R(Adg −1 (x ∧ y)), Adg −1 (z ∧ w)⟩ = ⟨Adg ⋅ R ⋅ Adg −1 (x ∧ y), z ∧ w⟩. Namely, g(R) = Adg ⋅ R ⋅ Adg −1 in SB2 (∧2 V). Viewing it as a (4, 0) tensor, one can check directly that g(R)(x, y, z, w) = R(g −1 x, g −1 y, g −1 z, g −1 w). It is easy to see that g(R) satisfies the first Bianchi identity. Similarly, ∀R1 , R2 ∈ SB2 (∧2 V), we define the inner product by ⟨R1 , R2 ⟩ ≑ ∑α ⟨R1 (bα ), R2 (bα )⟩ with {bα } being an orthonormal basis of ∧2 (V). It is easy to see that ⟨g(R1 ), g(R2 )⟩ = ⟨R1 , R2 ⟩, namely the action is an isometry. The Ricci curvature of R is defined as RicR (x, y) ≑ ∑⟨R(ei , x, ei , y), where {ei } is an orthonormal basis of V. g (R)−R For A ∈ g (or so(n)), let gt = exp(tA). Define A(R) ≑ limt→0 t t . Since gt (R) ∈ SB2 (∧2 V), ∀gt ∈ SO(n), A(R) ∈ SB2 (∧2 V), ∀A ∈ so(n). Direct calculation shows that g (x∧y)−x∧y A(x ∧ y) ≑ limt→0 t t = [A, x ∧ y]. Alternatively, A(x ∧ y) ≑ lim

t→0

gt (x ∧ y) − x ∧ y = A(x) ∧ y + x ∧ A(y) ≑ 2(A ∧ id)1 (x ∧ y). t

Denote [A, x ∧ y] also as adA (x ∧ y). Then A(R) = adA ⋅ R − R ⋅ adA and A(R) = 2(A ∧ id ⋅R − R ⋅ A ∧ id) = −2((A ∧ id)tr ⋅ R + R ⋅ A ∧ id), noting that (A ∧ B)tr = Atr ∧ Btr . Recalling R(x ∧ y) = −Rx,y , we also have that A(R)x,y = −A(R)(x ∧ y) = (A ⋅ Rx,y − Rx,y ⋅ A − RAx,y − Rx,Ay ), A(R)(x, y, z, w) = −(R(Ax, y, z, w) + R(x, Ay, z, w) + R(x, y, Az, w) + R(x, y, z, Aw)),

viewing as (4, 0) tensors.

From this it is easy to confirm again that A(R) satisfies the first Bianchi identity. For a holonomy system, it is easy to see that A(R)x,y ∈ g, ∀g ∈ G, A ∈ g. The conclusion of Theorem 1.2, namely S being symmetric (i. e., g(R) = R, ∀g ∈ G), is equivalent to A(R) = 0 for any A ∈ g. 1 Note here that A ∧ id only satisfies the first Bianchi identity if A is symmetric. Hence A ∧ id ∉ SB2 if A ∈ g (which is skew-symmetric).

Simons’ theorem | 449

3 A derivation of Theorem 1.1 Our derivation of Theorem 1.1 from Theorem 1.2 here follows the argument of [7]. The main difference is that no result from the theory of symmetric spaces, or the full Ambrose–Singer’s theorem, is needed. As in [7], it starts with a result of Kostant. First, let P be the projection from ∧2 (V) onto g, and T : g → g be the symmetric isomorphism corresponding to the negative definite bilinear form on g, B(A, A󸀠 ) ≑ K(A, A󸀠 ) − 2⟨A, A󸀠 ⟩,

(3.1)

with K being the Killing form of g (defined as K(A, A󸀠 ) = trace(adA ⋅ adA󸀠 )). Namely, T is defined by B(A, A󸀠 ) = ⟨T(A), A󸀠 ⟩. Theorem 3.1 (Kostant). Assume that S = {V, R, G} is an irreducible symmetric holonomy system. Then there exists a constant λ such that Rz,w = −λ(T −1 ⋅ P)(z ∧ w). Moreover, Rx,y = 0 if and only if R(x, y, x, y) = 0, and if R ≠ 0 (hence λ ≠ 0), RicR (x, x) = ∑ λ1 B([x, ei ], [x, ei ]). Proof. Assume that R ≠ 0 (otherwise the conclusion is obvious). A construction of a Lie algebra J (due to Cartan) is the key: Let J = g ⊕ V (orthogonal sum with the inner product of V and ⟨A, B⟩ on g as elements in so(n)) and define a Lie algebra structure of J by letting [A, A󸀠 ] ≑ [A, A󸀠 ],

[x, y] ≑ Rx,y ,

[A, x] ≑ A(x), ∀A, A󸀠 ∈ g, x, y ∈ V.

Since A(R) = 0, ∀A ∈ g, it is easy to check that the bracket so defined satisfies the Jacobi identity, namely J is a Lie algebra.2 That R satisfies the first Bianchi identity is also needed in checking the Jacobi identity for J. Let B󸀠 be the Killing form of J. It is a basic result of Lie algebra that B󸀠 is adJ -invariant (see, for example, page 180 of [6]). The proof now follows from the following claims: (i) B󸀠 |g is given by B defined by (3.1), hence is negative definite; (ii) B󸀠 (A, x) = 0; (the proofs of (i) and (ii) are computational and shall be given at the very end), and (iii) B󸀠 |V is adg -invariant, hence G-invariant, which, together with the irreducibility of G-action on V, implies that B󸀠 (x, y) = λ⟨x, y⟩ for some λ. Moreover, λ ≠ 0. Otherwise, B󸀠 ([x, y], [x, y]) = B󸀠 (x, [y, [x, y]]) = 0 since [y, [x, y]] ∈ V. On the other hand, by (i), which implies B󸀠 |g is negative definite, we have that [x, y] = Rx,y = 0, ∀x, y ∈ V. This contradicts R ≠ 0. Now observe that (a) ⟨[[x, y], z], w⟩ = −⟨[x, y], (z ∧ w)⟩ (using (2.3), namely ⟨A, z ∧ w⟩ = −⟨A(z), w⟩, ∀A ∈ so(n)) and (b) ⟨[[x, y], z], w⟩ = λ1 B󸀠 ([[x, y], z], w) = 1 󸀠 B ([x, y], [z, w]) which equals to λ1 ⟨[x, y], T([z, w])⟩. Theorem 3.1 now follows from (a) λ 2 A result of Borel, whose proof is also a by-product of the proof of Theorem 3.1, asserts that J is semisimple.

450 | L. Ni and (b) above, together with claim (c): span{Rx,y } ≑ gR = g, since (a)–(c) together imply that −λ⟨Rx,y , P(z ∧ w)⟩ = ⟨Rx,y , T(Rz,w )⟩,

∀Rx,y 󳨐⇒ T(Rz,w ) = −λP(z ∧ w).

For claim (c), note that gR is an ideal (cf. Lemma 4.1), let a be its orthogonal complement (w. r. t. B󸀠 ) in g. It is easy to see that ∀A ∈ a, y ∈ V, B󸀠 ([A, y], [A, y]) = B󸀠 (A, [y, [A, y]]) = 0 since [y, [A, y]] = Ry,A(y) ∈ gR . Hence due to B󸀠 |V = λ⟨⋅, ⋅⟩ this implies that [A, y] = A(y) = 0, ∀y ∈ V. Thus A = 0, ∀A ∈ a, namely a = 0. Finally, we prove (i) and (ii). By the definition B󸀠 (A, B) = trace(adA ⋅ adB ) = n ∑i=1 ⟨adA ⋅ adB (ei ), ei ⟩ + ∑α adA ⋅ adB (Aα ), Aα ⟩ where {ei } ({Aα }) is an orthonormal frame of V (g respectively). The second summand is K(A, B). By the definition of the Lie bracket, the first summand is −⟨B(ei ), A(ei )⟩ = −2⟨A, B⟩. This proves (i). The proof of (ii) is by a similar straightforward computation. Note that (b) above implies that ⟨Rx,y z, y⟩ = λ1 B(Rx,y , Rx,y ). Hence the sectional curvature K(x, y) = 0 ⇐⇒ Rx,y = 0. The formula for the Ricci curvature is via direct computations. In fact, Ric(x, x) = − 21 B󸀠 (x, x) (cf. page 182 of [6]). The following argument deriving Berger’s theorem, namely M has ∇R = 0, using Theorem 1.2 is the same as in [7]. Proof. We assume n ≥ 3 since n = 2 case is obvious. The goal is to show that M is a locally symmetric space. Assume that Hp0 acts on 𝕊n−1 nontransitively. By Lemma 2.2, S = {Mp , R, Hp0 } is a holonomy system. By the assumption of Theorem 1.1, S is irreducible. It is easy to see that Ricg(R) (x, y) = RicR (g −1 (x), g −1 (y)). Also use RicR to denote the corresponding symmetric automorphism of V. Namely, ⟨RicR (x), y⟩ ≑ RicR (x, y). Then Ricg(R) = g ⋅ RicR ⋅ g −1 . By the irreducibility of the system S, I. Schur’s lemma implies that RicR = f (p) id (i. e., R is Einstein at Mp ). Now for any q, pick a path γ from q to p. Consider the system Sγ = {Mp , γ(Rq ), Hp0 }. By Lemma 2.2 again, Sγ is an irreducible Riemannian holonomy system. By Theorems 1.2 and 3.1, we conclude that γ(Rq ) = cRp for some constant c. This implies that Ricγ(Rq ) = cRicRp = cf (p) id. On the other hand, by F. Schur’s lemma f (x) = β for a constant β. Namely, Ric p󸀠 = β id for all p󸀠 ∈ M. Thus R

Ricγ(Rq ) = γRicRq γ −1 = β id, since RicRq = β id. This implies c = 1, hence ∇R = 0. The claim rank ≥ 2 follows from Proposition 4.2 below.

4 Simons’ constructions of flats and totally geodesic subspaces Let S be an irreducible Riemannian holonomy system. The induction assumption is that Theorem 1.2 holds for S with degree smaller than n and dim(G(R)) ≤ k. The goal is to prove it for dim(V) = n and dim(G(R)) ≤ k + 1. The constructions in this section

Simons’ theorem | 451

are almost the same as those of [7]. The only difference is the alternate argument for Proposition 4.2 which was shown to us by Nolan Wallach. The original proof in [7] was due to I. Singer. We start with a lemma on gR of [7]. Lemma 4.1. (i) gR is an ideal in g. (ii) If g = gR ⊕ gR is an orthogonal decomposition then A(Q) = 0 for any A ∈ gR and Q ∈ G(R). Proof. By the definition, ∀A ∈ g, [A, ∑i,α gi (R)(bα )] = ∑ adA ⋅ gi (R)(bα ) with gi ∈ G, bα ∈ ∧2 (V). But A(gi (R)) = adA ⋅gi (R)−gi (R)⋅adA with A(gi (R))(bα ) ∈ gR since A(G(R)) ⊂ G(R). Part (i) follows from the facts that gi (R)(adA (bα )) ∈ gR and A(R)(bα ) ∈ gR . For (ii), observe that for any x, y, z, w, A ∈ gR , Q ∈ G(R), ⟨A(Q)(x ∧ y), z ∧ w⟩ = ⟨adA ⋅ Q(x ∧ y), z ∧ w⟩ − ⟨Q ⋅ adA (x ∧ y), z ∧ w⟩. The first term vanishes since [gR , gR ] = 0, in particular [A, Qx,y ] = 0. The second term is ⟨Q ⋅ adA (x ∧ y), z ∧ w⟩ = ⟨Q(z ∧ w), adA (x ∧ y)⟩ = −⟨adA ⋅ Q(z ∧ w), x ∧ y⟩ = 0, by reducing to the first. Putting them together, the lemma is proved. A subspace W ⊂ V (with dim(W) ≥ 2) is called a flat if Q(x ∧y) = 0 for any x, y ∈ W, for any Q ∈ G(R), or equivalently, Adg −1 (∧2 W) ∈ ker(R) for any g ∈ G. Clearly, W being a flat implies that g(W) is a flat. For a flat W, we have that ∀x, y ∈ W, ∀z, w ∈ V, ⟨Qx,y z, w⟩ = ⟨Qz,w x, y⟩ = 0. Hence W is a flat if and only if gR (W) ⊂ W ⊥ and W is maximal if and only if W is a maximal subspace such that gR (W) ⊂ W ⊥ . The main ingredients of Simons’ proof are the construction of flats out of the nontransitivity, and of total geodesic subspaces out of the maximal flats. A subspace E ⊂ V is called totally geodesic if for any Q ∈ G(R), any x, y, z ∈ E, Qx,y z ∈ E. Note that if E is totally geodesic, one then can view R as a curvature operator on the space E, which provides a possible reduction on dim(V). Proposition 4.2. If GR is nontransitive on 𝕊n−1 then there exists a flat W (dim(W) ≥ 2). In fact, ∀u ∈ V, there exists a flat W with u ∈ W. Proof. Assume R ≠ 0, otherwise the claim is true. The nontransitivity implies that there exist u, v ∈ 𝕊n−1 such that u ≠ g(v), ∀g ∈ GR . Now consider the function f (g) = ⟨u, g(v)⟩. Note f (g) ∈ [−1, 1). Since GR is compact, f (g) attains its maximum somewhere, say at g0 ∈ GR . We then have that for any A ∈ g, ⟨u, Ag0 (v)⟩ = 0. In particular, we have that ∀Q ∈ G(R), ⟨u, Qx,y g0 (v)⟩ = ⟨Qg0 (v),u x, y⟩ = 0,

∀x, y ∈ V.

This implies that Qu,g0 (v) = 0. Since g0 (v) ≠ u nor g0 (v) = −u, we conclude that W = span{u, g0 (v)} is a flat. The part g0 (v) ≠ −u is due to that f (g0 ) attains its maximum which cannot be −1, unless v = −u and g(−u) = −u, ∀g ∈ GR , which then implies that the eigenspace E(1) (with eigenvalue 1) of GR is nonempty and invariant under the action of G (since GR is a normal subgroup of G by, say Theorem 2.13.4 of Varadarajan’s

452 | L. Ni book [8]), thus either GR = {id} or E(1) ≠ V, contradicting the assumption that V is irreducible. The argument effectively shows that for any u ∈ 𝕊n−1 , there exists a flat W such that u ∈ W since the nontransitivity assumption implies that ∀u, there exists v such that u ≠ g(v) for any g ∈ GR . Note that Proposition 4.2 implies Theorem 1.2 for n = 2, for it implies that V is a flat, hence R = 0. Clearly, any flat is totally geodesic.3 The existence of flats leads to some totally geodesic subspaces via the Jacobi curvatures (named after the curvature term in the equation J 󸀠󸀠 + Rγ󸀠 ,J γ 󸀠 = 0 defining a Jacobi field J along a geodesic γ). Given any Q ∈ G(R) and x, y ∈ W, consider the linear x,y x,y transformation TQ : V → V (the Jacobi curvature) as ⟨TQ (z), w⟩ ≑ Q(x, z, y, w). By the first Bianchi identity and since Qx,y = 0, x,y

y,x

⟨TQ (z), w⟩ = Q(y, z, x, w) = ⟨TQ (z), w⟩ = Q(x, w, y, z) = ⟨T x,y (w), z⟩.

(4.1)

x,y

In particular, TQ : V → V is symmetric. More importantly, ∀x, y, s, t ∈ W, x,y

x,y

TQ ⋅ TQs,t = TQs,t ⋅ TQ ,

x,y

x,y

more generally TQ ⋅ TPs,t = TQs,t ⋅ TP , ∀P, Q ∈ G(R).

(4.2)

Since s, t, x, y ∈ W, we have that QAx,y = −Qx,Ay for any A ∈ g. Hence we have x,y

TQ ⋅ TPs,t (z) = Qx,Ps,z t y = −QPs,z x,t y = −QPx,z s,t y

x,y

= Qt,Px,z s y = Qy,Px,z s t = −QPx,z y,s t = TQs,t ⋅ TP (z). x,y

When Q = P, we have (4.2), namely {TQ }x,y∈W forms a family of commutative symmetric operators on V, which hence can be diagonalized simultaneously (cf. [9], Section 5 of Chapter 1 for an illuminating proof). Thus there exist unit X1Q , . . . , XnQ (when there is x,y no confusion we omit the superscript) such that they are eigenvectors of TQ (for any x,y x, y ∈ W). Since clearly TQ (W) = 0, we may assume that {Xi }1≤i≤μ forms an orthonormal basis of W (μ = dim(W)). We denote the corresponding eigenvalues by ΛkQ (x, y) x,y with 1 ≤ k ≤ n. Namely, TQ Xk = ΛkQ (x, y)Xk . Clearly, ΛkQ (x, y) is a bilinear form of W.

Equation (4.1) also implies that ΛkQ (x, y) is symmetric, which can also be viewed as a symmetric linear map of W via ⟨ΛkQ (x), y⟩ = ΛkQ (x, y). For 1 ≤ k ≤ μ, ΛkQ = 0. For μ + 1 ≤ k ≤ n, we shall show that either ΛkQ ≡ 0 (write as ΛkQ = 0) or it is of rank one. First we observe that if ΛkQ (x, y) = 0, ∀x, y ∈ W and for all 1 ≤ k ≤ n, we then have that Qx,z y = 0, ∀z ∈ V, x, y ∈ W. In particular, Q(x, z, x, z) = 0, ∀x ∈ W, z ∈ V. On the other hand, Proposition 4.2 asserts that for any x ∈ V there exists a flat W with x ∈ W. Hence if ΛkQ = 0 for all 1 ≤ k ≤ n and for all flats, Q(x, z, x, z) = 0, for any x, z ∈ V, hence Q = 0, and R = 0. Thus R ≠ 0 implies that for any Q, there exists at least one flat W and one k with ΛkQ ≠ 0 on W.

3 The flats and total geodesic subspaces are Lie algebra analogues of the maximum toruses and the centralizers of the torus subgroup of the Lie group of isometries.

Simons’ theorem | 453

Lemma 4.3. Let W be a flat. Assume for some k ΛkQ ≠ 0. (i) The rank of ΛkQ : W → W is one; (ii) Ps,Xk = 0,

for any s ∈ Uk,Q , P ∈ G(R), with Uk,Q = ker(ΛkQ ).

(4.3)

Proof. Pick x ∈ W with ΛkQ (x, x) ≠ 0. Then TQx,x Xk = ΛkQ (x, x)Xk ≠ 0. Let Uk,Q ≑ {s ∈

W | ΛkQ (x, s) = 0}, which defines a hypersurface in W. Since Px,Ay = −PAx,y , for any A ∈ g, x, y ∈ W, we have that ΛkQ (x, x)Ps,Xk = Ps,Qx,X

k

x

= −PQx,X

k

s,x

= −ΛkQ (x, s)PXk ,x = 0.

This proves (4.3). In particular, for any s ∈ Uk,Q , t ∈ W, ΛkQ (s, t)Xk = TQs,t Xk = Qs,Xk t = 0, 󳨐⇒ ΛkQ (s, t) = 0,

∀s ∈ Uk,Q , t ∈ W.

Thus ΛkQ |Uk,Q ≡ 0, which implies the rank one assertion. The above shows that Jacobi curvatures associated with vectors from a flat are special, and Uk,Q = ker(ΛkQ ) is independent of the choice of x ∈ W. Note that for Xk ∈ W ⊥ , ΛkQ (x, x) = KΣQ , the sectional curvature of Σ = span{x, Xk } for x ∈ W, |x| = 1. For ΛkQ ≠ 0, let λkQ ≠ 0 and xk ∈ W be the nonzero eigenvalue and an eigenvector of ΛkQ : W → W (with ΛkQ (xk ) = λkQ xk ). If ΛkQ = 0, let λkQ = 0 and pick any unit xk ∈ W. Order k by |λkQ |, Q Q Q |λμ+1 | ≤ |λμ+2 | ≤ ⋅ ⋅ ⋅ ≤ |λn−1 | ≤ |λnQ |,

let

with λjQ ≑ Q(xj , Xj , xj , Xj ).

The first construction of totally geodesic subspaces: For a maximal flat W, ΛkQ ≠ 0, Ek,Q ≑ {m ∈ V | Ps,m = 0, for all P ∈ G(R), s ∈ Uk,Q }.

By (4.3), we have that Xk ∈ Ek,Q and Uk,Q ⫋ W ⫋ Ek,Q . We show that Ek,Q ⫋ V. Proposition 4.4. Consider W, Q with ΛkQ ≠ 0. Then: (i) Ek,Q is totally geodesic; (ii) Ek,Q ≠ V unless R = 0; (iii) For a maximal W, Ek,Q ∩ Ek󸀠 ,Q󸀠 = W,

if Ek,Q ≠ Ek󸀠 ,Q󸀠 .

(4.4)

Proof. For any A ∈ g, P ∈ G(R), since g(P)s,m = 0 for any g ∈ G, s ∈ Uk,Q , m ∈ Ek,Q , 0=

d 󵄨󵄨󵄨󵄨 = −PAs,m − Ps,Am = −(P ⋅ adA )s,m . 󵄨 P dt 󵄨󵄨󵄨t=0 exp(−tA)s,exp(−tA)m

(4.5)

󸀠 (i) Let u, v, m ∈ Ek,Q and s ∈ Uk,Q . For any P, P 󸀠 ∈ G(R), Ps,P = −PP󸀠 u,v s,m = u,v m

PP󸀠 v,s u+Ps,u v,m = 0. Hence Pu,v m ∈ Ek,Q .

454 | L. Ni (ii) Note that V 󸀠 = {x | Px,y = 0, for all y ∈ V, P ∈ G(R)} is a G-invariant subspace. If Ek,Q = V then Uk,Q ⊂ V 󸀠 󳨐⇒ V 󸀠 = V. It then implies that V is a flat and R = 0. (iii) For any m ∈ Ek,Q ∩ Ek󸀠 ,Q󸀠 ≑ Ik,k󸀠 , we have that for any P ∈ G(R), Pm,v = 0 for v ∈ Uk,Q ∪Uk󸀠 ,Q󸀠 . If Ek,Q ≠ Ek󸀠 ,Q󸀠 , then Uk,Q ≠ Uk󸀠 ,Q󸀠 . Hence Pm,v = 0, ∀v ∈ W. This implies m ∈ W (hence W = Ik,k󸀠 ) by the maximality of W, otherwise W 󸀠 = span{W, m} is a flat, contradicting that W is maximal. The second construction of totally geodesic subspaces: For a fixed flat W, clearly W ⊂ ZQ ≑ span{Xk , ΛkQ = 0}. Define Z(W) ≑ ⋂Q∈G(R) ZQ . Since W ⊂ ZQ , ∀Q, W ⊂ Z(W). Hence V = Z(W) ⊕ N, where N is spanned by the eigenvectors XkQ with ΛkQ (x, x) ≠ 0 for some Q ∈ G(R), x ∈ W. Proposition 4.5. Z(W) is totally geodesic, namely for any Q, P ∈ G(R), Pw,Qx,y z w = 0, ∀w ∈ W, x, y, z ∈ Z(W). Proof. Note that x ∈ Z(W) if and only if Qw,x v = 0, ∀Q ∈ G(R), ∀w, v ∈ W. Now note that if x ∈ Z(W), then ∀P ∈ G(R), A ∈ g, gt = exp(tA), Pw,x v = 0 󳨐⇒ gt (P)w,x v = 0,

∀w, v ∈ W 󳨐⇒ −Pw,x ⋅ Av − PAw,x v − Pw,Ax v = 0.

Hence Pw,Qx,y z v = −Pw,z Qx,y v − PQx,y w,z v. If x ∈ W, Qx,y v = ΛkQ (x, v)y = 0 = Qx,y w =

ΛkQ (x, w)y, hence Pw,Qx,y z v = 0, which proves Qx,y z ∈ Z(W). For the general case, since Qy,v x, Qx,v y ∈ Z(W), the first Bianchi identity implies that Qx,y v = −Qy,v x + Qx,v y ∈ Z(W). Similarly, Qx,y w ∈ Z(W). Therefore applying the above special case, Pw,z Qx,y v, PQx,y w,z v ∈ Z(W). The first Bianchi identity again implies Pw,Qx,y z v ∈ Z(W). Now for any z 󸀠 ∈ Z(W), we have

⟨Pw,Qx,y z v, z 󸀠 ⟩ = ⟨Pv,z 󸀠 w, Qx,y z⟩ = 0. This implies that Pw,Qx,y z v = 0, hence Qx,y z ∈ Z(W). Proposition 4.6. Assume that Z(W) ≠ V.4 Then ∀w ∈ W, x ∈ Z(W), Qw,x |Z(W) = 0, ∀Q ∈ G(R). Proof. This is the place where the induction on dim(V) is applied by letting Q󸀠 = Q|Z(W) . Note that gZ(W) ≑ span{Px,y , x, y ∈ Z(W), P ∈ G(R)} is a subalgebra of gR by the proof of Lemma 4.1. Let GZ(W) ⊂ G be the closed subgroup generated by gZ(W) . Denote it by G󸀠 . Clearly, G󸀠 (Z(W)) ⊂ Z(W). Consider the holonomy system S󸀠 = (Z(W), Q󸀠 , G󸀠 ). By the definition, ∀h ∈ G󸀠 , h(Q󸀠 )w,x w = 0, ∀w ∈ W, x ∈ Z(W). If S󸀠 is irreducible, by induction we can assert that either it is symmetric or the action is transitive. In either case, we show that Q󸀠w,x = 0. (i) If S󸀠 = (Z(W), Q󸀠 , G󸀠 ) is symmetric, by Theorem 3.1, ⟨Q󸀠x,w x, w⟩ = 0 ⇐⇒ Q󸀠x,w = 0. Hence we have Q󸀠x,w = 0 from w ∈ W, x ∈ Z(W). 4 A priori for a fixed W it is possible that all λjQ = 0 (namely ΛkQ = 0, ∀k ∈ {μ + 1, . . . , n}). If it is the case for all Q, then Z(W) = V. Since this cannot be precluded, we assume Z(W) ≠ V. However, Theorem 1.2 eventually implies that λjQ ≠ 0 for μ + 1 ≤ j ≤ n.

Simons’ theorem | 455

(ii) If S󸀠 is transitive (on the unit sphere of Z(W)), then for a fixed w ∈ W, |w| = 1, for any z ∈ Z(W), |z| = 1, there exists h ∈ G󸀠 such that h(w) = z. Then from ⟨h−1 (Q󸀠 )w,x w, x⟩ = 0, ∀x ∈ Z(W) we have that ⟨(h−1 (Q󸀠 ))h−1 (z),x h−1 (z), x⟩ = 0 ⇐⇒ ⟨Q󸀠z,h(x) z, h(x)⟩ = 0, ∀z, x ∈ Z(W), |z| = 1. Since G󸀠 acts transitively on Z(W), this implies that the curvature tensor Q󸀠 |Z(W) = 0. When S󸀠 is reducible we split Z(W) = ⊕ℓi=1 Zi into orthogonal subspaces with {Zi } being irreducible G󸀠 -invariant subspaces. Observe that Q󸀠w,x (Zi ) ⊂ Zi . It is easy to see that Q󸀠w,x = ∑ℓi=1 Q󸀠wi ,xi . Moreover, for w󸀠 ∈ W, 0 = Q󸀠w,x w󸀠 = ∑ Q󸀠wi ,xi wi󸀠 . Hence Q󸀠wi ,xi wi󸀠 = 0 on Zi with wi , wi󸀠 , xi being the orthogonal projections of w, w󸀠 , x into Zi . The arguments (i) and (ii) above show that Q󸀠wi ,xi |Zi = 0. This proves Q󸀠w,x = 0 for the reducible case.

5 An alternate proof via the induction on dim(G(R)) We start with extending a result of [7] on the totally geodesic subspaces. Theorem 5.1. Assume that E ⫋ V is totally geodesic w. r. t. S = {V, R, G}. Define J ⊂ g, a subspace of g, and K ⊂ G(R), a subspace of G(R) as J ≑ {A ∈ g | A(Q)|E = 0, ∀Q ∈ G(R)}; K ≑ {Q ∈ G(R) | Qx,y z = 0, ∀x, y, z ∈ E}. Then A ∈ J if and only if A(G(R)) ⊂ K and the following statements hold: (i) For all A ∈ g, A(K) ⊂ K, hence G(K) ⊂ K; (ii) J is an ideal; (iii) gR ⊂ J; (iv) A(E) ⊂ E ⊥ implies that A ∈ J; (v) Qx,y ∈ J, if x ∈ E and y ∈ E ⊥ . Proof. Since E is totally geodesic, ∀x, y, z ∈ E, A(Q)x,y z ∈ E. Hence ⟨A(Q)x,y z, w⟩ = 0, ∀w ∈ E ⇐⇒ A(Q)|E = 0 ⇐⇒ A(Q)x,y z = 0, ∀x, y, z ∈ E. This shows that A ∈ J if and only if A(G(R)) ⊂ K. Let Q ∈ K, we claim that A(Q)x,y z = 0 for any x, y, z ∈ E. This implies (i). The claim follows from noting adA (z ∧ w) = A(z) ∧ w + z ∧ A(w) and ∀x, y, z, w ∈ E, −⟨A(Q)x,y z, w⟩ = ⟨A(Q)(x ∧ y), z ∧ w⟩

= ⟨adA Q(x ∧ y), z ∧ w⟩ − ⟨QadA (x ∧ y), z ∧ w⟩

(5.1)

= −⟨Q(x ∧ y), adA (z ∧ w)⟩ − ⟨adA (x ∧ y), Q(z ∧ w)⟩ = 0. For (ii), observe that for A ∈ J, B ∈ g, A(B(Q)) ∈ K by the equivalent definition of J, and A(Q) ∈ K implies that B(A(Q)) ∈ K by (i). Hence [A, B](Q) ∈ K. Claim (iii) follows from Lemma 4.1. Equation (5.1) also proves (iv) since if A(E) ⊂ E ⊥ , ∀Q ∈ G(R), ⟨Q(x ∧ y), adA (z ∧ w)⟩ = ⟨Qx,y w, A(z)⟩ − ⟨Qx,y z, A(w)⟩ = 0. Similarly, ⟨adA (x ∧ y), Q(z ∧ w)⟩ = 0. Hence ⟨A(Q)x,y z, w⟩ = 0, ∀x, y, z, w ∈ E. Claim (v) follows from (iv) and that Qx,y (E) ⊂ E ⊥ , since ∀x ∈ E, y ∈ E ⊥ ⟨Qx,y z, w⟩ = ⟨Qz,w x, y⟩ = 0, ∀z, w ∈ E. Corollary 5.2. For A = Qx,y with x ∈ E, y ∈ E ⊥ , Adg (A) ∈ J, and G(A(R)) ⊂ K. In particular, it holds for E = El,Q with ΛlQ ≠ 0.

456 | L. Ni Proof. Integrating part (ii) of the above theorem Adg (A) ∈ J follows from A ∈ J. By (v) of the above theorem, we have that A ∈ J, which implies that A(R) ∈ K. By part (i) of Theorem 5.1, G(A(R)) ∈ K. The alternate induction on dim(G(R)) below seems a more efficient way of applying the theorem and its corollary. The following identity (5.2) (Lemma 10 of [7]) is the key step. Proposition 5.3. 5 Let W be a fixed maximal flat with Z(W) ≠ V. Then (i) there are 󸀠 nonzero ΛkQ and ΛkQ󸀠 such that Ek,Q ≠ Ek󸀠 ,Q󸀠 ; (ii) In particular, W⊥ =

∑

ΛkQ =0,Q∈G(R) ̸

⊥ Ek,Q ,

or equivalently,

W=

⋂

ΛkQ =0,Q∈G(R) ̸

Ek,Q .

(5.2)

Proof. By Proposition 4.4 (particularly (4.4)), (5.2) follows from (i), namely the existence of two totally geodesic Ek,Q , Ek󸀠 ,Q󸀠 with Ek,Q ≠ Ek󸀠 ,Q󸀠 . We prove (i) by contradiction. Assume that all the totally geodesic subspaces Ek,Q (with ΛkQ ≠ 0) are the same. We denote it by E, which is totally geodesic by Proposition 4.4. Hence the respective kernels of ΛkQ , Uk,Q , are also the same by the duality. We denote the common Uk,Q by U. By Proposition 4.4 and the discussion after it, we deduce that E ⊥ ≠ 0 consists of vecx,y tors which are in the null space of TQ , ∀x, y ∈ W, ∀Q ∈ G(R). Namely, E ⊥ ⫋ Z(W), V = E + Z(W). (Note that W ⊂ E ∩ Z(W).) Pick x ∈ E ⊥ . Clearly, x ∉ W. Below we show that Pw,x = 0, ∀w ∈ W, ∀P ∈ G(R). This implies (i) since it is a contradiction to the maximality of W. Observe that ∀z ∈ E, w ∈ W, x ∈ E ⊥ , Pw,x z ∈ E (namely Pw,x (E) ⊂ E), since ∀u ∈ U,

−Qu,Pw,x z = QPw,x u,z = ΛkP (w, u)Qx,z = 0, ∀P, Q ∈ G(R),

by (4.5) and ΛkP (u, w) = 0. Now for any z 󸀠 ∈ V write it as z 󸀠 = e + e⊥ with respect to V = E ⊕ E ⊥ . Then Pw,x z ∈ E implies that ⟨Pw,x z, e⊥ ⟩ = 0. On the other hand, ⟨Pw,x z, e⟩ = ⟨Pz,e w, x⟩ = 0, since Pz,e w ∈ E, x ∈ E ⊥ . Thus Pw,x |E = 0. On the other hand, since x ∈ Z(W), Proposition 4.6 implies that Pw,x |Z(W) = 0. For any z ∈ V, we may write z = z1 + z2 with z1 ∈ E, z2 ∈ Z(W). We then have that Pw,x = 0, ∀P ∈ G(R), since Pw,x z = Pw,x z1 + Pw,x z2 = 0. Below we assume R ≠ 0 (otherwise nothing needs to be proved). Then it implies R(x, y, x, y) ≠ 0 for some x, y ∈ V. Pick {xi } a basis of V in a neighborhood x with R(xi , y, xi , y) ≠ 0. Now choose maximal flats W, Wi such that x ∈ W and xi ∈ Wi . By Proposition 4.2 and for W (and Wi ) so chosen, there exists some k with ΛkR ≠ 0. Hence Z(W) ≠ V (Z(Wi ) ≠ V). 5 The argument effectively implies that V = ∑ Ek,Q (Lemma 9 of [7]), since for any z, which belongs the orthogonal complement of the right-hand side, the proof implies Pw,z = 0, ∀w ∈ W.

Simons’ theorem | 457

Lemma 5.4. Let W and Ek = Ek,Q be as those of the last section. Let x ∈ Ek , y ∈ Ek⊥ . Then (i) for A = Qx,y , S󸀠 = {V, A(R), G} is a holonomy system such that dim(G(A(R))) < dim(G(R)); (ii) There exists a basis {Aℓ } of gR such that Sℓ = {V, Aℓ (R), G} satisfies (i). Proof. For part (i), Corollary 5.2 implies that G(A(R)) ⊂ K. On the other hand, the definition of Ek ensures that there exists Xk ∈ Ek and a unit eigenvector ek ∈ W of ΛkQ with Qek ,Xk ek = ΛkQ (ek , ek )Xk = λkQ Xk ≠ 0. Hence Q ∉ K, and we have that dim(G(A(R))) < dim(G(R)). To obtain a basis of gR with (i), we first pick g α such that {g α (R)} (with finitely many α) generates G(R). Then let {xi } be a basis of V chosen as above with R(xi , y, xi , y) ≠ 0. Now it is clear that {Aαi,j } with Aαi,j = g α (R)xi ,xj generates gR , hence contains a subset as a basis of gR . Write Qα = g α (R). Now for each xi , there exists a flat (which can be made into a maximal one) Wi with xi ∈ Wi . Since Wi is a flat Qαxi ,xj = Qαxi ,x⊥ j

with xj⊥ denotes the orthogonal projection of xj into Wi⊥ . Now apply the part (i) to Wi . The way of choosing Wi ensures that Z(Wi ) ≠ V, hence, by equation (5.2), we have xj⊥ = ∑ bl yl with yl ∈ (El,Q󸀠 )⊥ . This effectively expresses Aαi,j as a linear combination of Qαxi ,yl with xi ∈ Wi and yl ∈ (El,Q󸀠 )⊥ . Applying (i) with A = Qαxi ,yl , E = El,Q󸀠 we have dim(G(Qαxi ,yl (R))) < dim(G(R)). By the way {Qαxi ,yl } is constructed, it is easy to see that we can select a basis of gR out of this family.

Now we prove Theorem 1.2. If dim(G(R)) = 1, clearly, g(R) = R, ∀g ∈ G. Assume that Theorem 1.3 holds for any holonomy system with dim(G(R)) ≤ k, k ∈ ℤ, k > 0. Now we α prove it for the system with dim(G(R)) ≤ k+1. Apply Theorem 1.3 to Si,l = {V, Qαxi ,yl (R), G} α with Q , xi ∈ Wi and yl ∈ El as in Lemma 5.4. Then by Lemma 5.4, dim(G(Qαxi ,yl )) < α dim(G(R)). Thus Si,l is symmetric. Namely, we have that A(Qαxi ,yl (R)) = 0 for any A ∈ g.

Writing A = A1 + A2 , A1 ∈ gR , A2 ∈ gR with A1 = ∑ aαi,l Qαxi ,yl , this and Lemma 4.1 imply that A2 (R) = 0, hence A(R) = 0. This proves Theorem 1.3 for dim(G(R)) ≤ k + 1. Our approach avoids Lemmata 6, 9, 11, 12 of [7]. Since dim(GR (R)) = dim(G(R)), the same argument proves Theorem 1.3. The rank of S could be defined as the maximum dim(W) for all maximal flat W. It is the same as the rank of the symmetric space corresponding to the Cartan algebra J.

Bibliography [1] [2] [3] [4] [5]

W. Ambrose, I. M. Singer, A theorem of holonomy. Trans. Am. Math. Soc. 79, 428–443 (1953). M. Berger, Sur les groupes d’holonomie homogène des variétés à connexion affine et des variétés riemanniennes. Bull. Soc. Math. Fr. 83, 279–330 (1955) (in French). A. L. Besse, Einstein Manifolds (Springer, Berlin, 1978), xii+516 pp. D. Montgomery, H. Samelson, Transformation groups of spheres. Ann. Math. 44, 454–470 (1943). C. Olmos, A geometric proof of the Berger holonomy theorem. Ann. Math. 161(1), 579–588 (2005).

458 | L. Ni

[6] [7] [8]

T. Sakai, Riemannian Geometry (AMS, Providence, RI, 1996), xiv+358 pp. J. Simons, On the transitivity of holonomy systems. Ann. Math. 76, 213–234 (1962). V. S. Varadarajan, Lie Groups, Lie Alegeba and Their Representations (Springer, New York, 1984), xiii+430 pp. [9] H. Weyl, The Theory of Groups and Quantum Mechanics (Dover Publ., 1931). [10] H. Wu, W.-H. Chen, Selected Topics on Riemannian Geometry (Peking Univ. Press, Beijing, 1993) (in Chinese).

Simon Nowak

Higher integrability for nonlinear nonlocal equations with irregular kernel Abstract: We prove a higher regularity result for weak solutions to nonlinear nonlocal equations along the integrability scale of Bessel potential spaces H s,p under a mild continuity assumption on the kernel. By embedding, this also yields regularity in Sobolev–Slobodeckij spaces W s,p . Our approach is based on a characterization of Bessel potential spaces in terms of a certain nonlocal gradient-type operator and a perturbation approach commonly used in the context of local elliptic equations in divergence form. Keywords: Nonlocal equations, Sobolev regularity, Dirichlet problem MSC 2010: 35R09, 35B65, 35D30, 46E35, 47G20 Contents 1 1.1 1.2 2 2.1 2.2 2.3 3 3.1 3.2 4 5 6

Introduction | 459 Basic setting and main result | 459 Approach | 464 Preliminaries | 464 Some notation | 464 Some tools from real analysis | 465 Fractional Sobolev spaces | 467 Some preliminary estimates | 470 Tail estimates | 470 Higher Hölder regularity | 471 The Dirichlet problem | 473 Higher integrability of ∇s u | 478 Proof of the main result | 488 Bibliography | 490

1 Introduction 1.1 Basic setting and main result In this paper, we consider nonlinear nonlocal equations of the form LΦ Au = F

in Ω ⊂ ℝn ,

(1.1)

Acknowledgement: Supported by SFB 1283 of the German Research Foundation. Simon Nowak, Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501, Bielefeld, Germany, e-mail: [email protected] https://doi.org/10.1515/9783110700763-017

460 | S. Nowak where s ∈ (0, 1), Ω ⊂ ℝn is a domain (= open set), while A : ℝn × ℝn → ℝ is a coefficient and Φ : ℝ → ℝ is a nonlinearity with properties to be specified below. Moreover, the nonlocal operator LΦ A is formally given by LΦ A u(x) = p.v. ∫ ℝn

A(x, y) Φ(u(x) − u(y)) dy. |x − y|n+2s

We assume that the right-hand side F of (1.1) is formally of the form F(x) = p. v. ∫ ℝn

g(x, y) dy + f (x), |x − y|n+2s

x ∈ Ω,

(1.2)

where f : ℝn → ℝ and g : ℝn × ℝn → ℝ are given functions. The aim of this work is to generalize an approach introduced in [24], in order to prove a higher regularity result for weak solutions of the equation (1.1) along the integrability scale of Bessel potential spaces H s,p , in the case when the coefficient A exhibits a potentially very irregular behavior. Throughout the paper, for simplicity we assume that n > 2s. Moreover, we assume that A is a measurable function and that there exists some λ ≥ 1 such that λ−1 ≤ A(x, y) ≤ λ

for almost all x, y ∈ ℝn .

(1.3)

Furthermore, we require A to be symmetric, i. e., A(x, y) = A(y, x) for almost all x, y ∈ ℝn .

(1.4)

We call such a function A a kernel coefficient and define ℒ0 (λ) as the class of all such measurable kernel coefficients A that satisfy (1.3) and (1.4). Moreover, in our main results Φ : ℝ → ℝ is assumed to be a continuous function satisfying Φ(0) = 0 and the following Lipschitz continuity and monotonicity assumptions, namely 󵄨󵄨 󵄨 󸀠 󵄨 󸀠󵄨 󸀠 󵄨󵄨Φ(t) − Φ(t )󵄨󵄨󵄨 ≤ λ󵄨󵄨󵄨t − t 󵄨󵄨󵄨 for all t, t ∈ ℝ

(1.5)

and (Φ(t) − Φ(t 󸀠 ))(t − t 󸀠 ) ≥ λ−1 (t − t 󸀠 )

2

for all t, t 󸀠 ∈ ℝ,

(1.6)

where for simplicity we use the same constant λ ≥ 1 as in (1.3). In particular, Φ could be any C 1 function with Φ(0) = 0 such that the first derivative Φ󸀠 of Φ satisfies im Φ󸀠 ⊂ [λ−1 , λ]. In the case when Φ(t) = t, the operator LΦ A reduces to a linear nonlocal operator widely considered in the literature. The following nonlocal analogue of the Euclidean norm of the gradient of a function plays a key role in this paper.

Higher integrability for nonlocal equations | 461

Definition. Let s ∈ (0, 1). For any measurable function u : Ω → ℝ, we define the s-gradient ∇s u : ℝn → [0, ∞] by 1

2 (u(x) − u(y))2 dy) . ∇ u(x) = ( ∫ |x − y|n+2s n

s

ℝ

For any p ∈ [2, ∞), define the space 󵄨 󵄨 󵄨p 󵄨 󵄨p H s,p (Ω|ℝn ) = {u : ℝn → ℝ measurable 󵄨󵄨󵄨 ∫󵄨󵄨󵄨u(x)󵄨󵄨󵄨 dx + ∫󵄨󵄨󵄨∇s u(x)󵄨󵄨󵄨 dx < ∞}. Ω

Ω

s,p Moreover, by Hloc (Ω|ℝn ) we denote the set of all functions u : ℝn → ℝ that belong to s,p 󸀠 n H (Ω |ℝ ) for any relatively compact open subset Ω󸀠 of Ω. The main relevance of these spaces is due to the fact that they are closely related to the classical Bessel potential spaces H s,p (Ω) and Sobolev–Slobodeckij spaces W s,p (Ω). In fact, for any p ≥ 2 we have the inclusions s,p s,p s,p H s,p (ℝn ) ⊂ Hloc (Ω|ℝn ) ⊂ Hloc (Ω) ⊂ Wloc (Ω),

(1.7)

see Section 3. Denote by Hcs,2 (Ω) the set of all functions that belong to H s,2 (Ω|ℝn ) and are compactly supported in Ω. For all measurable functions u, φ : ℝn → ℝ, we define Φ

ℰA (u, φ) = ∫ ∫ ℝn ℝn

A(x, y) Φ(u(x) − u(y))(φ(x) − φ(y)) dy dx, |x − y|n+2s

provided the above expression is well-defined and finite, this is, for example, true if s,2 u ∈ Hloc (Ω|ℝn ) and φ ∈ Hcs,2 (Ω). Furthermore, throughout this paper we assume that the function g is measurable and symmetric in the sense of (1.4). In addition, by a slight abuse of notation we define the s-gradient ∇s g : ℝn → [0, ∞] of g by 1

∇s g(x) = ( ∫ ℝn

2 g(x, y)2 dy) . n+2s |x − y|

Also, for any such function g that satisfies ∇s g ∈ L2loc (Ω) and any φ ∈ Hcs,2 (Ω), we define ℰ (g, φ) = ∫ ∫ ℝn ℝn

g(x, y) (φ(x) − φ(y)) dy dx. |x − y|n+2s

The notation introduced above allows us to define our notion of weak solutions to the equation (1.1) as follows.

462 | S. Nowak 2n

n+2s (Ω) and a measurable symmetric function g : ℝn × ℝn → ℝ Definition. Given f ∈ Lloc s,2 s 2 with ∇ g ∈ Lloc (Ω), assume that F is given as in (1.2). We say that u ∈ Hloc (Ω|ℝn ) is a Φ weak solution of the equation LA u = F in Ω, if

Φ

ℰA (u, φ) = ℰ (g, φ) + (f , φ)L2 (Ω)

∀φ ∈ Hcs,2 (Ω).

In our main result, we need to impose the following additional continuity assumption on A: 󵄨 󵄨 lim sup 󵄨󵄨󵄨A(x + h, y + h) − A(x, y)󵄨󵄨󵄨 = 0

h→0 x,y∈K

for any compact set K ⊂ Ω.

(1.8)

Condition (1.8) was introduced in the recent paper [25] in the context of obtaining higher Hölder regularity. In particular, it is satisfied if A is either continuous in Ω × Ω or if A is translation invariant inside of Ω, that is, if there exists a measurable function a : ℝn → ℝ such that A(x, y) = a(x − y) for all x, y ∈ Ω. In addition, the condition (1.8) is also satisfied by some more general choices of kernel coefficients, for example, if A(x, y) = A󸀠 (x, y)A0 (x, y), 1

where A󸀠 ∈ ℒ0 (λ 2 ) is continuous in Ω × Ω and A0 is translation invariant inside of Ω, but is not required to satisfy any continuity or smoothness assumption. Furthermore, we stress that condition (1.8) only restricts the behavior of A inside of Ω × Ω, while outside of Ω × Ω a more general behavior is possible. We are now in the position to state our main result. Theorem 1.1. Let Ω ⊂ ℝn be a domain, s ∈ (0, 1), λ ≥ 1, and p ∈ (2, ∞). Moreover, let g : ℝn × ℝn → ℝ be a measurable symmetric function with ∇s g ∈ Lploc (Ω) and assume p⋆ np that f ∈ Lloc (Ω), where p⋆ = max{ n+sp , 2}. If A ∈ ℒ0 (λ) satisfies condition (1.8) and if Φ satisfies conditions (1.5) and (1.6) with respect to λ, then for F given as in (1.2) and any s,2 weak solution u ∈ Hloc (Ω|ℝn ) of the equation LΦ Au = F

in Ω,

s,p we have u ∈ Hloc (Ω|ℝn ). Moreover, for all open sets U ⋐ V ⋐ Ω, we have

󵄩󵄩 s 󵄩󵄩 󵄩 s 󵄩 󵄩 s 󵄩 󵄩󵄩∇ u󵄩󵄩Lp (U) ≤ C(‖f ‖Lp⋆ (V) + 󵄩󵄩󵄩∇ g 󵄩󵄩󵄩Lp (V) + 󵄩󵄩󵄩∇ u󵄩󵄩󵄩L2 (V) ),

(1.9)

where C = C(p, n, s, λ, U, V) > 0. Remark 1.2. In view of (1.7), under the assumptions of Theorem 1.1, weak solutions of s,p (1.1) in particular belong to the Bessel potential space Hloc (Ω) and also to the Sobolev– s,p Slobodeckij space Wloc (Ω). Moreover, the condition ∇s g ∈ Lploc (Ω) is, for example, satisfied if g has the form m

g(x, y) = ∑ Di (x, y)(gi (x) − gi (y)), i=1

(1.10)

Higher integrability for nonlocal equations | 463

s,p (Ω|ℝn ) for all i = 1, . . . , m. By (1.7), the where m ∈ ℕ, Di ∈ L∞ (ℝn × ℝn ) and gi ∈ Hloc latter condition is in particular satisfied if all gi belong to the Bessel potential space H s,p (ℝn ).

Remark 1.3. An interesting feature of the estimate (1.9) is that it is not a purely local estimate, in the sense that due to the nonlocal nature of the s-gradient ∇s , the lefthand side also depends on the values of u outside the domain Ω. In other words, we also gain some control on u outside the domain where the equation holds. For the sake of providing some context, let us briefly consider local elliptic equations in divergence form of the type div(B∇u) = div h + f

in Ω,

(1.11)

where the matrix of coefficients B = {bij }ni,j=1 is assumed to be uniformly elliptic and bounded, while h : ℝn → ℝn and f : ℝn → ℝ are given functions. Equation (1.11) can in some sense be thought of as a local analogue of the nonlocal equation (1.1) corresponding to the limit case s = 1. It is known that if the coefficients bij are continuous np

n+p 1,2 and h ∈ Lploc (Ω, ℝn ), f ∈ Lloc (Ω) for some p > 2, then weak solutions u ∈ Wloc (Ω) of 1,p the equation (1.11) belong to Wloc (Ω). This corresponds to our main result in the sense that we obtain local W s,p regularity for nonlocal equations of the type (1.1) in the case 1,p when A satisfies the continuity assumption (1.8). We note that this Wloc (Ω) regularity for solutions of equation (1.11) also holds if more generally the coefficients bij belong to the space VMO of functions with vanishing mean oscillation, cf. [5] or [1]. Therefore, an interesting question is if the conclusion of Theorem 1.1 remains true for kernel coefficients A that belong to VMO in a suitable sense. Regarding related previous results, in [24] Theorem 1.1 was proved in the linear case when Φ(t) = t and under the stronger assumption that A is translation invariant in the whole space ℝn and in the special case when g is of the form (1.10). Another very interesting result in this direction was recently proved in [23], where again in the linear case when Φ(t) = t it was in particular shown that if A ∈ ℒ0 (λ) is Hölder continuous with some arbitrary Hölder exponent and for some 2 ≤ p < ∞ we have n f ∈ Lp (ℝn ), then weak solutions u ∈ H s,2 (ℝn ) of the equation LΦ A u = f in ℝ beα,p n long to Hloc (ℝ ) for any α < min{2s, 1}, gaining not only integrability, but also differentiability, while for local equations of the type (1.11) no comparable gain of differentiability is attainable. Another interesting question is therefore if such a gain of differentiability is also achievable for possibly nonlinear equations of the type (1.1) that might only hold in some domain Ω with kernel coefficients that satisfy condition (1.8) or even for kernels of VMO-type. We plan to investigate this direction in the future. More results concerning Sobolev regularity for nonlocal equations are, for example, proved in [2, 17, 20, 28, 22, 3, 9, 12], while various results on Hölder regularity are proved in [14, 13, 4, 25, 27, 8, 15, 10, 16, 32, 21]. Furthermore, for some regularity results

464 | S. Nowak concerning nonlocal equations similar to (1.1) in the more general setting of measure data, we refer to [19].

1.2 Approach Our approach is inspired by an approach introduced by Caffarelli and Peral in [6] in the context of obtaining W 1,p estimates for local elliptic equations of the type (1.11). The philosophy of the approach is as follows. The first step is to locally approximate the gradient a weak solutions u of (1.11) by the gradient of a weak solution v to a suitable homogeneous equation for which an in some sense good enough estimate is already known. More precisely, in the context of local equations, one exploits the fact that the approximate solution v is already known to satisfy a local C 0,1 estimate in order to transfer some regularity to u. In fact, a real-variable argument based on the Vitali covering lemma, the Hardy–Littlewood maximal function, and an alternative characterization of Lp spaces then allows proving an Lp estimate for the gradient ∇u corresponding to our estimate (1.9), which then implies the desired local W 1,p estimate. The main idea in order to prove Theorem 1.1 is to apply a similar strategy with the gradient ∇u replaced by the nonlocal s-gradient ∇s u. In particular, in our nonlocal setting the local C 0,1 estimate for the approximate solution has to be replaced by a local C s+γ estimate for some γ > 0. Such an estimate was recently proved in [25] for equations of the type (1.1) with kernel coefficients that satisfy condition (1.8), opening the way towards obtaining our Theorem 1.1. This estimate is used in an adaptation of the real-variable argument described above in order to obtain the desired estimate (1.9) from Theorem 1.1. In contrast to [24], additional difficulties also arise due to the presence of the nonlinearity Φ, which are dealt with by careful applications of the conditions (1.5) and (1.6) throughout the paper and using the theory of monotone operators in order to prove existence and uniqueness for the corresponding Dirichlet problem.

2 Preliminaries 2.1 Some notation For convenience, let us fix some notation which we use throughout the paper. By C and Ci , i ∈ ℕ, we always denote positive constants, while dependencies on parameters of the constants will be shown in parentheses. As usual, by Br (x0 ) := {x ∈ ℝn | |x − x0 | < r} we denote the open ball with center x0 ∈ ℝn and radius r > 0. Moreover, if E ⊂ ℝn is measurable, then by |E| we denote the n-dimensional Lebesgue-measure of E. If

Higher integrability for nonlocal equations | 465

0 < |E| < ∞, then for any u ∈ L1 (E) we define uE := ∫ − u(x) dx :=

1 ∫ u(x) dx. |E| E

E

2.2 Some tools from real analysis In this section, we discuss some results from real analysis that are at the core of the real-variable argument mentioned in Section 1.2. The following result is an application of the well-known Vitali covering lemma, cf. [5, Theorem 2.7]. Lemma 2.1. Assume that E and F are measurable sets in ℝn that satisfy E ⊂ F ⊂ B1 . Assume further that there exists some ε ∈ (0, 1) such that |E| < ε|B1 |, and that for all x ∈ B1 and any r ∈ (0, 1) with |E ∩ Br (x)| ≥ ε|Br (x)|, we have Br (x) ∩ B1 ⊂ F. Then we have |E| ≤ 10n ε|F|. Another tool we use is the Hardy–Littlewood maximal function. Definition. Let f ∈ L1loc (ℝn ). Then the Hardy–Littlewood maximal function ℳf : ℝn → [0, ∞] of f is defined by ℳf (x) := ℳ(f )(x) := sup ∫ − 󵄨󵄨󵄨f (y)󵄨󵄨󵄨 dy. ρ>0

󵄨

󵄨

Bρ (x)

Moreover, for any domain Ω ⊂ ℝn and any function f ∈ L1 (Ω), consider the zero extension of f to ℝn f (x), if x ∈ Ω,

fΩ (x) := {

0,

if x ∉ Ω.

We then define ℳΩ f := ℳfΩ .

The following lemma contains the scaling and translation invariance of the Hardy–Littlewood maximal function and can be proved by using a change of variables.

466 | S. Nowak Lemma 2.2. Let f ∈ L1loc (ℝn ), r > 0 and y ∈ ℝn . Then for the function fr,y (x) := f (rx + y) and any x ∈ ℝn we have ℳfr,y (x) = ℳf (rx + y).

Similarly, for any domain Ω ⊂ ℝn , any function f ∈ L1 (Ω), and any x ∈ Ω, we have ℳΩ󸀠 fr,y (x) = ℳΩ f (rx + y),

| x ∈ Ω}. where Ω󸀠 := { x−y r We remark that for any f ∈ L1loc (ℝn ), ℳf is Lebesgue-measurable. The probably most important properties of the Hardy–Littlewood maximal function are contained in the following result, see [31]. Proposition 2.3. Let Ω ⊂ ℝn be a domain. (1) (weak 1–1 estimate) If f ∈ L1 (Ω) and t > 0, then 󵄨󵄨 󵄨 C 󵄨󵄨{x ∈ Ω | ℳΩ (f )(x) > t}󵄨󵄨󵄨 ≤ ∫ |f | dx, t Ω

where C = C(n) > 0. (2) (strong p–p estimates) If f ∈ Lp (Ω) for some p ∈ (1, ∞], then ‖f ‖Lp (Ω) ≤ ‖ℳΩ f ‖Lp (Ω) ≤ C‖f ‖Lp (Ω) , where C = C(n, p) > 0. (3) If f ∈ Lp (Ω) for some p ∈ [1, ∞], then the function ℳΩ f is finite almost everywhere. We conclude this section by giving an alternative characterization of Lp spaces, see [7, Lemma 7.3]. It can be proved by using the well-known formula ‖f ‖pLp (Ω)

∞

󵄨 󵄨 = p ∫ t p−1 󵄨󵄨󵄨{x ∈ Ω | f (x) > t}󵄨󵄨󵄨dt. 0

Lemma 2.4. Let 0 < p < ∞. Furthermore, suppose that f is a nonnegative and measurable function in a bounded domain Ω ⊂ ℝn and let τ > 0, β > 1. Then for ∞

󵄨 󵄨 S := ∑ βkp 󵄨󵄨󵄨{x ∈ Ω | f (x) > τβk }󵄨󵄨󵄨, k=1

we have C −1 S ≤ ‖f ‖pLp (Ω) ≤ C(|Ω| + S) for some constant C = C(τ, β, p) > 0. In particular, we have f ∈ Lp (Ω) if and only if S < ∞.

Higher integrability for nonlocal equations | 467

2.3 Fractional Sobolev spaces The following type of fractional Sobolev spaces is probably the most common type of such spaces in the literature concerned with nonlocal equations similar to (1.1). Definition. Let Ω ⊂ ℝn be a domain. For p ∈ [1, ∞) and s ∈ (0, 1), we define the Sobolev–Slobodeckij space p

|u(x) − u(y)| 󵄨 W s,p (Ω) := {u ∈ Lp (Ω) 󵄨󵄨󵄨 ∫ ∫ dy dx < ∞} |x − y|n+sp Ω Ω

with norm p

1/p

|u(x) − u(y)| 󵄨 󵄨p dy dx) ‖u‖W s,p (Ω) := (∫󵄨󵄨󵄨u(x)󵄨󵄨󵄨 dx + ∫ ∫ |x − y|n+sp Ω

.

Ω Ω

Moreover, we also define the corresponding local versions of these spaces by s,p Wloc (Ω) := {u ∈ Lploc (Ω) | u ∈ W s,p (Ω󸀠 ) for any domain Ω󸀠 ⋐ Ω}.

In addition, we also use the space W0s,2 (Ω) := {u ∈ W s,2 (ℝn ) | u = 0 in ℝn \ Ω}

= {u ∈ H s,2 (Ω|ℝn ) | u = 0 in ℝn \ Ω}.

Remark 2.5. The space W s,2 (Ω) is a separable Hilbert space with respect to the inner product (u, v)W s,2 (Ω) := (u, v)L2 (Ω) + ∫ ∫ Ω Ω

(u(x) − u(y))(v(x) − v(y)) dy dx. |x − y|n+2s

Furthermore, the space W0s,2 (Ω) clearly is a closed subspace of W s,2 (ℝn ) and is therefore also a separable Hilbert space with respect to the inner product (⋅, ⋅)W s,2 (ℝn ) . We often use the following fractional Poincaré-type inequalities. Lemma 2.6 (Fractional Poincaré inequality). Let s ∈ (0, 1) and R > 0. For any u ∈ W s,2 (BR ), we have |u(x) − u(y)|2 󵄨2 󵄨 dy dx, ∫ 󵄨󵄨󵄨u(x) − uBR 󵄨󵄨󵄨 dx ≤ CR2s ∫ ∫ |x − y|n+2s

BR

where C = C(n, s) > 0.

BR BR

468 | S. Nowak Proof. Using Jensen’s inequality, for any x ∈ BR we obtain 2 󵄨2 󵄨 󵄨 󵄨 󵄨2 󵄨󵄨 − 󵄨󵄨󵄨u(x) − u(y)󵄨󵄨󵄨 dy − 󵄨󵄨󵄨u(x) − u(y)󵄨󵄨󵄨 dy) ≤ ∫ 󵄨󵄨u(x) − uBR 󵄨󵄨󵄨 ≤ ( ∫ BR

BR

≤ CR2s ∫ BR

|u(x) − u(y)|2 dy, |x − y|n+2s

where C = C(n, s) > 0. The claim now follows by integrating both sides over BR . For a proof of the following inequality we refer to [25, Lemma 2.3]. Lemma 2.7 (Fractional Friedrichs–Poincaré inequality). Let s ∈ (0, 1) and consider a bounded domain Ω ⊂ ℝn . For any u ∈ W0s,2 (Ω), we have 2s |u(x) − u(y)|2 󵄨 󵄨2 dy dx, ∫ 󵄨󵄨󵄨u(x)󵄨󵄨󵄨 dx ≤ C|Ω| n ∫ ∫ |x − y|n+2s n n n

ℝ

(2.1)

ℝ ℝ

where C = C(n, s) > 0. We also use the following type of fractional Sobolev spaces. Definition. For p ∈ [1, ∞) and s ∈ ℝ, consider the Bessel potential space s

H s,p (ℝn ) := {u ∈ Lp (ℝn ) | ℱ −1 [(1 + |ξ |2 ) 2 ℱ u] ∈ Lp (ℝn )}, where ℱ denotes the Fourier transform and ℱ −1 denotes the inverse Fourier transform. We equip H s,p (ℝn ) with the norm s 󵄩 󵄩 ‖u‖H s,p (ℝn ) := 󵄩󵄩󵄩ℱ −1 [(1 + |ξ |2 ) 2 ℱ u]󵄩󵄩󵄩Lp (ℝn ) .

Moreover, for any domain Ω ⊂ ℝn we define H s,p (Ω) := {v|Ω | v ∈ H s,p (ℝn )} with norm ‖u‖H s,p (Ω) := inf{‖v‖H s,p (ℝn ) | v|Ω = u} and also the corresponding local Bessel potential spaces by s,p Hloc (Ω) := {u ∈ Lploc (Ω) | u ∈ H s,p (Ω󸀠 ) for any domain Ω󸀠 ⋐ Ω}.

The following result gives some relations between Bessel potential spaces and Sobolev–Slobodeckij spaces. Proposition 2.8. Let Ω ⊂ ℝn be a domain.

Higher integrability for nonlocal equations | 469

(1) If Ω is a bounded Lipschitz domain or Ω = ℝn , then for all s ∈ (0, 1), p ∈ (1, 2] we have W s,p (Ω) 󳨅→ H s,p (Ω). (2) For any s ∈ (0, 1) and any p ∈ [2, ∞), we have H s,p (Ω) 󳨅→ W s,p (Ω). For a proof of Proposition 2.8 in the case when Ω = ℝn , we refer to Theorem 5 in Chapter V of [31]. For a brief explanation on how to obtain the result for general domains, we refer to [24, Section 3]. We now generalize the notion of the s-gradient which was introduced in the introduction. Definition. Let s ∈ (0, 1). For any domain Ω ⊂ ℝn and any measurable function u : Ω → ℝ, we define the s-gradient ∇Ωs u : Ω → [0, ∞] by 1

∇Ωs u(x)

2 (u(x) − u(y))2 dy) . := (∫ |x − y|n+2s

Ω

In particular, note that we have ∇s u = ∇ℝs n u. As mentioned in the introduction, the notion of the s-gradient is closely related with the Bessel potential spaces H s,p . The precise relation is given by the following result. 2n Proposition 2.9. Let s ∈ (0, 1), p ∈ ( n+2s , ∞), and assume that Ω ⊂ ℝn is a bounded n Lipschitz domain or that Ω = ℝ . Then we have u ∈ H s,p (Ω) if and only if u ∈ Lp (Ω) and ∇Ωs u ∈ Lp (Ω). Moreover, we have

󵄩 󵄩 ‖u‖H s,p (Ω) ≃ ‖u‖Lp (Ω) + 󵄩󵄩󵄩∇Ωs u󵄩󵄩󵄩Lp (Ω) in the sense of equivalent norms. This characterization was first given by Stein in [30] in the case when Ω = ℝn . For the case when Ω is a bounded Lipschitz domain, we refer to [26, Theorem 1.3], where this characterization is proved in the more general context of Triebel–Lizorkin spaces and the so-called uniform domains. Remark 2.10. In view of Propositions 2.9 and 2.8, for any bounded Lipschitz domain Ω ⊂ ℝn and all s ∈ (0, 1), p ∈ [2, ∞), we have the inclusions H s,p (ℝn ) ⊂ H s,p (Ω|ℝn ) ⊂ H s,p (Ω) ⊂ W s,p (Ω). In the case when Ω ⊂ ℝn is an arbitrary domain, this implies the inclusions (1.7) from the introduction. We also use the following standard embedding theorems of Bessel potential spaces. For precise references, see [24, Section 3]. Theorem 2.11. Let 1 < p ≤ p1 < ∞, s, s1 ≥ 0, and assume that Ω ⊂ ℝn is a domain.

470 | S. Nowak np ] we have (1) If sp < n, then for any q ∈ [p, n−sp

H s,p (Ω) 󳨅→ Lq (Ω). (2) More generally, if s −

n p

= s1 −

n , p1

then

H s,p (Ω) 󳨅→ H s1 ,p1 (Ω). (3) If sp = n, then for any q ∈ [p, ∞) we have H s,p (Ω) 󳨅→ Lq (Ω). (4) If sp > n, then we have H s,p (Ω) 󳨅→ C α (Ω), where α = s − pn .

3 Some preliminary estimates For the rest of this paper, we fix real numbers s ∈ (0, 1) and λ ≥ 1.

3.1 Tail estimates The following lemma relates the tails of a function to the L2 norm of its s-gradient. For a proof, we refer to [24, Lemma 4.1]. Lemma 3.1. For all r, R > 0 and any u ∈ H s,2 (BR |ℝn ), we have ∫ ℝn \Br

u(y)2 󵄩 󵄩2 dy ≤ C(󵄩󵄩󵄩∇s u󵄩󵄩󵄩L2 (B ) + ‖u‖2L2 (BR ) ), R |y|n+2s

(3.1)

where C = C(n, s, r, R) > 0. Finally, the following result can be proved in the same way as [24, Corollary 4.4], by using the L∞ estimate from [25, Theorem 2.11] instead of the one from [24, Theorem 4.2]. It shows that if a function satisfies a homogeneous nonlocal equation, then the tails of its s-gradient can be controlled nicely, so that we can focus on estimating the local part of the s-gradient. Proposition 3.2. Consider a kernel coefficient A ∈ ℒ0 (λ) and assume that Φ satisfies (1.5) and (1.6). Then for all 0 < r < R and any weak solution u ∈ H s,2 (BR |ℝn ) of the equation LΦ Au = 0

in BR ,

Higher integrability for nonlocal equations | 471

we have the estimate 󵄩 s 󵄩 󵄩󵄩 s 󵄩 󵄩󵄩∇ℝn \BR u󵄩󵄩󵄩L∞ (Br ) ≤ C 󵄩󵄩󵄩∇ u󵄩󵄩󵄩L2 (BR ) ,

(3.2)

where C = C(n, s, r, R, λ) > 0.

3.2 Higher Hölder regularity In the basic case when A ∈ ℒ0 (λ), it is known that any weak solution to a corresponding homogeneous nonlocal equation is locally C α for some α > 0, cf. [10, Theorem 1.2]. The following result shows that if A ∈ ℒ0 (λ) additionally satisfies condition (1.8), then such weak solutions enjoy better Hölder regularity than in general. Proposition 3.3. Consider a kernel coefficient A ∈ ℒ0 (λ) that satisfies condition (1.8) in B5 and suppose that Φ satisfies (1.5) and (1.6) with respect to λ. Moreover, assume that u ∈ H s,2 (B5 |ℝn ) is a weak solution of the equation LΦ A u = 0 in B5 . Then for any 0 < α < min{2s, 1}, we have 󵄩 󵄩 [u]Cα (B3 ) ≤ C 󵄩󵄩󵄩∇s u󵄩󵄩󵄩L2 (B ) , 5 where C = C(n, s, λ, α) > 0 and [u]Cα (B3 ) := sup

x,y∈B3 x =y̸

|u(x) − u(y)| . |x − y|α

We will derive Proposition 3.3 from Theorem 3.4 below, which is proved in [25, Theorem 1.1]. In order to state the result, we need the following definitions. First, we define the tail space 󵄨 L12s (ℝn ) := {u ∈ L1loc (ℝn ) 󵄨󵄨󵄨 ∫ ℝn

|u(y)| dy < ∞}. 1 + |y|n+2s

The most important property of this space is that for any function u ∈ L12s (ℝn ), the quantity ∫ ℝn \BR (x0 )

|u(y)| dy |x0 − y|n+2s

is finite for all R > 0, x0 ∈ ℝn . s,2 Definition. We say that u ∈ Wloc (Ω) ∩ L12s (ℝn ) is a local weak solution of the equation Φ LA u = 0 in Ω, if Φ

ℰA (u, φ) = 0

∀φ ∈ Hcs,2 (Ω).

(3.3)

472 | S. Nowak Theorem 3.4. Let Ω ⊂ ℝn be a domain. Consider a kernel coefficient A ∈ ℒ0 (λ) that satisfies condition (1.8) in Ω and suppose that Φ satisfies (1.5) and (1.6) with respect s,2 (Ω)∩L12s (ℝn ) is a local weak solution of the equation to λ. Moreover, assume that u ∈ Wloc α LΦ A u = 0 in Ω. Then for any 0 < α < min{2s, 1}, we have u ∈ Cloc (Ω). n Furthermore, for all R > 0, x0 ∈ ℝ such that BR (x0 ) ⋐ Ω and any σ ∈ (0, 1), we have [u]Cα (BσR (x0 )) ≤

n C (R− 2 ‖u‖L2 (BR (x0 )) + R2s Rα

∫ ℝn \BR (x0 )

|u(y)| dy), |x0 − y|n+2s

(3.4)

where C = C(n, s, λ, α, σ) > 0. In order to derive Proposition 3.3 from Theorem 3.4, we need to ensure that as the terminology suggests, any weak solution as defined in the introduction is also a local weak solution. This is essentially a consequence of the following lemma. Lemma 3.5. Let R > 0 and s ∈ (0, 1). Then for any function u ∈ H s,2 (BR |ℝn ) and any R > 0, we have u ∈ L12s (ℝn ) and ∫ ℝn

|u(y)| 󵄩 󵄩 dy ≤ C(‖u‖L2 (BR ) + 󵄩󵄩󵄩∇s u󵄩󵄩󵄩L2 (B ) ), R 1 + |y|n+2s

where C = C(n, s, R) > 0. In particular, we have H s,2 (BR |ℝn ) ⊂ L12s (ℝn ). Proof. First of all, integration in polar coordinates yields dz = C1 R−2s , |z|n+2s

∫ ℝn \B

R

(3.5)

where C1 = C1 (n, s) > 0. We split the integral in question as follows: ∫ ℝn

|u(y)| |u(y)| 󵄨 󵄨 dy ≤ ∫ 󵄨󵄨󵄨u(y)󵄨󵄨󵄨 dy + ∫ dy n+2s 1 + |y|n+2s |y| n BR

ℝ \BR

1

1

2 2 |u(y)|2 󵄨 󵄨2 dy) , ≤ C2 (∫ 󵄨󵄨󵄨u(y)󵄨󵄨󵄨 dy) + C3 ( ∫ n+2s |y| n

BR

ℝ \BR

where C2 = C2 (n, R) > 0 and C3 = C3 (n, s, R) > 0. Here we used the Cauchy–Schwarz inequality and (3.5) in order to obtain the last inequality. In view of Lemma 3.1, we also have 1

( ∫ ℝn \BR

2 |u(y)|2 󵄩 󵄩 dy) ≤ C4 (‖u‖L2 (BR ) + 󵄩󵄩󵄩∇s u󵄩󵄩󵄩L2 (B ) ), n+2s R |y|

where C4 = C4 (n, s, R) > 0. The claim now follows by combining the above two estimates.

Higher integrability for nonlocal equations | 473

Proof of Proposition 3.3. Since in view of Lemma 3.5 the function u0 := u − uB5 ∈ s,2 (B5 ) ∩ L12s (ℝn ) is a local weak solution of H s,2 (B5 |ℝn ) ⊂ Wloc LΦ A u0 = 0

in B5 ,

by Theorem 3.4, (3.5), Lemma 3.1, and the fractional Poincaré inequality (Lemma 2.6), we have [u]Cα (B3 ) = [u0 ]Cα (B3 ) ≤ C1 (‖u0 ‖L2 (B4 ) + ∫

ℝn \B4

|u0 (y)| dy) |y|n+2s 1

2 |u0 (y)|2 ≤ C2 (‖u0 ‖L2 (B5 ) + ( ∫ dy) ) n+2s |y| n

ℝ \B4

󵄩 󵄩 󵄩 󵄩 ≤ C3 (‖u0 ‖L2 (B5 ) + 󵄩󵄩󵄩∇s u󵄩󵄩󵄩L2 (B ) ) ≤ C4 󵄩󵄩󵄩∇s u󵄩󵄩󵄩L2 (B ) , 5 5 where all constants depend only on n, s, λ and α. This finishes the proof.

4 The Dirichlet problem In this section, we are mainly concerned with the existence and uniqueness of weak solutions to nonlocal Dirichlet problems. Although in this paper we only use the existence of weak solutions in the space H s,2 (Ω|ℝn ), for future reference and for the sake of generality we also include some other solution spaces. Throughout this section, we fix a bounded domain Ω ⊂ ℝn and let X be a vector space that satisfies W s,2 (ℝn ) ⊂ X ⊂ H s,2 (Ω|ℝn ).

(4.1)

In particular, possible choices for X are X = H s,2 (Ω|ℝn ) and X = W s,2 (ℝn ). 2n

Definition. Suppose that X satisfies (4.1). Moreover, let h ∈ X and f ∈ L n+2s (Ω). We say that u ∈ X is a weak solution of the problem LΦ Au = f

{

u=h

in Ω,

a. e. in ℝn \ Ω,

(4.2)

if we have ℰAΦ (u, φ) = (f , φ)L2 (Ω) for all φ ∈ W0s,2 (Ω) and u = h a. e. in ℝn \ Ω. Proposition 4.1. Let Ω ⊂ ℝn be a bounded domain and suppose that X is a vector space that satisfies (4.1). Consider a kernel coefficient A ∈ ℒ0 (λ) and suppose that Φ satis2n fies (1.5) and (1.6). Moreover, let h ∈ X and f ∈ L n+2s (Ω). Then there exists a unique weak solution u ∈ X of the Dirichlet problem (4.2).

474 | S. Nowak Proof. We use an argument inspired by [18] based on the theory of monotone operators. Fix h ∈ X and consider the operator 𝒜 : W0s,2 (Ω) → (W0s,2 (Ω))⋆ defined by ⟨𝒜(v), φ⟩ := ⟨𝒜1 (v), φ⟩ + ⟨𝒜2 (v), φ⟩, where ⟨𝒜1 (v), φ⟩ := ∫ ∫ Ω ℝn

A(x, y) Φ(v(x) + h(x) − v(y) − h(y))(φ(x) − φ(y)) dy dx |x − y|n+2s

and A(x, y) Φ(v(x) + h(x) − v(y) − h(y))(φ(x) − φ(y)) dy dx. |x − y|n+2s

⟨𝒜2 (v), φ⟩ := ∫ ∫ ℝn \Ω Ω

Here by (W0s,2 (Ω))⋆ we denote the dual space of W0s,2 (Ω) consisting of all bounded linear functionals on W0s,2 (Ω). We split the further proof into a few observations. Observation 1: 𝒜 is well-defined. Let us show that for any v ∈ W0s,2 (Ω), 𝒜(v) is indeed a bounded linear functional and thus belongs to (W0s,2 (Ω))⋆ . For all v, φ ∈ W0s,2 (Ω), by (1.3), (1.5), and the Cauchy–Schwarz inequality, we have |v(x) − v(y)| 󵄨󵄨 󵄨󵄨 󵄨 󵄨 2 󵄨󵄨⟨𝒜(v), φ⟩󵄨󵄨󵄨 ≤ λ ∫ ∫ 󵄨φ(x) − φ(y)󵄨󵄨󵄨 dy dx n+2s 󵄨 |x − y| n n ℝ ℝ

+ 2λ2 ∫ ∫ Ω ℝn

|h(x) − h(y)| 󵄨󵄨 󵄨 󵄨φ(x) − φ(y)󵄨󵄨󵄨 dy dx |x − y|n+2s 󵄨 1

1

2 2 |φ(x) − φ(y)|2 |v(x) − v(y)|2 ≤ λ (∫ ∫ dy dx) dy dx) ( ∫ ∫ |x − y|n+2s |x − y|n+2s n n n n

2

ℝ ℝ

ℝ ℝ 1

1

2 2 |φ(x) − φ(y)|2 |h(x) − h(y)| ( + 2λ (∫ ∫ dy dx) dy dx) ∫ ∫ n+2s n+2s |x − y| |x − y| n n n

2

2

Ωℝ

ℝ ℝ

󵄩 󵄩 ≤ λ2 (‖v‖W s,2 (ℝn ) + 2󵄩󵄩󵄩∇s h󵄩󵄩󵄩L2 (Ω) )‖φ‖W s,2 (ℝn ) .

Thus, since by (4.1) we have X ⊂ H s,2 (Ω|ℝn ) and therefore ∇s h ∈ L2 (Ω), 𝒜(v) is indeed a bounded linear functional and therefore belongs to (W0s,2 (Ω))⋆ . Observation 2: 𝒜 is monotone. By (1.3) and (1.6), for all v, w ∈ W0s,2 (Ω) we have ⟨𝒜1 (v) − 𝒜1 (w), v − w⟩ =∫∫ Ω ℝn

A(x, y) (Φ(v(x) + h(x) − v(y) − h(y)) − Φ(w(x) + h(x) − w(y) − h(y))) |x − y|n+2s

× ((v(x) + h(x) − v(y) − h(y)) − (w(x) + h(x) − w(y) − h(y))) dy dx

≥ λ−2 ∫ ∫ Ω ℝn

((v(x) − v(y)) − (w(x) − w(y)))2 dy dx ≥ 0. |x − y|n+2s

Higher integrability for nonlocal equations | 475

By the same reasoning, we also have ⟨𝒜2 (v) − 𝒜2 (w), v − w⟩ ≥ 0, and therefore ⟨𝒜(v) − 𝒜(w), v − w⟩ ≥ 0, so that 𝒜 is monotone. s,2 Observation 3: 𝒜 is weakly continuous. Let {vj }∞ j=1 be a sequence in W0 (Ω) that converges to some function v ∈ W0s,2 (Ω) in W s,2 (ℝn ). By (1.3), (1.5), and the Cauchy– Schwarz inequality, for any φ ∈ W0s,2 (Ω) we obtain 󵄨󵄨 󵄨 󵄨󵄨⟨𝒜(vj ) − 𝒜(v), φ⟩󵄨󵄨󵄨 A(x, y) 󵄨󵄨 ≤∫∫ 󵄨Φ(vj (x) + h(x) − vj (y) − h(y)) |x − y|n+2s 󵄨 n n ℝ ℝ

󵄨 − Φ(v(x) + h(x) − v(y) − h(y))‖φ(x) − φ(y)󵄨󵄨󵄨 dy dx

≤ λ2 ∫ ∫ ℝn ℝn

|(vj (x) − vj (y)) − (v(x) − v(y))| 󵄨 󵄨󵄨φ(x) − φ(y)󵄨󵄨󵄨 dy dx 󵄨 󵄨 |x − y|n+2s

≤ λ2 ( ∫ ∫

|(vj (x) − v(x)) − (vj (y) − v(y))|2 |x − y|n+2s

ℝn ℝn

dy dx)

1 2

1

2 |φ(x) − φ(y)|2 × (∫ ∫ dy dx) n+2s |x − y| n n

ℝ ℝ

j→∞

≤ λ2 ‖vj − v‖W s,2 (ℝn ) ‖φ‖W s,2 (ℝn ) 󳨀󳨀󳨀󳨀→ 0. Therefore, we obtain lim ⟨𝒜(vj ) − 𝒜(v), φ⟩ = 0,

j→∞

which means that 𝒜 is weakly continuous. Observation 4: 𝒜 is coercive. By (1.3), (1.6), and (1.5), for any v ∈ W0s,2 (Ω) we have ⟨𝒜1 (v), v⟩ = ⟨𝒜1 (v), v + h⟩ − ⟨𝒜1 (v), h⟩ = ∫∫ Ω ℝn

A(x, y) Φ(v(x) + h(x) − v(y) − h(y))(v(x) + h(x) − v(y) − h(y)) dy dx |x − y|n+2s

−∫∫ Ω ℝn

A(x, y) Φ(v(x) + h(x) − v(y) − h(y))(h(x) − h(y)) dy dx |x − y|n+2s

1 (v(x) − v(y))2 ≥ λ−2 ( ∫ ∫ dy dx 2 |x − y|n+2s n n ℝ ℝ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ =:J1

|v(x) − v(y)||h(x) − h(y)| (h(x) − h(y))2 dy dx − dy dx). ∫ ∫ |x − y|n+2s |x − y|n+2s Ω ℝn Ω ℝn ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

−∫ ∫

=:J2

476 | S. Nowak By using Lemma 2.7, we estimate J1 further from below as follows; 2s

C −1 |Ω|− n (v(x) − v(y))2 1 ‖v‖2L2 (ℝn ) + ∫ ∫ dy dx J1 ≥ 1 4 4 |x − y|n+2s n n ≥

c‖v‖2W s,2 (ℝn ) ,

ℝ ℝ

where C1 = C1 (n, s) > 0 is given by Lemma 2.7 and c = c(n, s, λ, |Ω|) > 0. By the Cauchy– Schwarz inequality, for J2 we have J2 ≥ − ∫ ∫ Ω ℝn

|h(x) − h(y)||v(x) − v(y)| 󵄩 󵄩 dy dx ≥ −󵄩󵄩󵄩∇s h󵄩󵄩󵄩L2 (Ω) ‖v‖W s,2 (ℝn ) . |x − y|n+2s

Since by a similar reasoning as above, we have ⟨𝒜2 (v), v⟩ = ⟨𝒜2 (v), v + h⟩ − ⟨𝒜2 (v), h⟩ ≥ λ−2 ( ∫ ∫ ℝn \Ω Ω

− ∫ ∫ ℝn \Ω Ω

(v(x) − v(y))2 dy dx |x − y|n+2s

|v(x) − v(y)||h(x) − h(y)| (h(x) − h(y))2 dy dx − dy dx) ∫ ∫ |x − y|n+2s |x − y|n+2s n

≥ λ−2 (− ∫ ∫ Ω ℝn

ℝ \Ω Ω

|v(x) − v(y)||h(x) − h(y)| (h(x) − h(y))2 dy dx − dy dx), ∫ ∫ |x − y|n+2s |x − y|n+2s n Ωℝ

by combining the last four displays, we obtain 󵄩 󵄩 󵄩 󵄩2 ⟨𝒜(v), v⟩ ≥ λ−2 (c‖v‖2W s,2 (ℝn ) − 2󵄩󵄩󵄩∇s h󵄩󵄩󵄩L2 (Ω) ‖v‖W s,2 (ℝn ) − 2󵄩󵄩󵄩∇s h󵄩󵄩󵄩L2 (Ω) ). Therefore, we conclude that ⟨𝒜(v), v⟩ ‖v‖W s,2 (ℝn )

s

2

‖∇ h‖L2 (Ω) ‖v‖W s,2 (ℝn ) →+∞ 󵄩 󵄩 ) 󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀→ +∞, ≥ λ (c‖v‖W s,2 (ℝn ) − 2󵄩󵄩󵄩∇s h󵄩󵄩󵄩L2 (Ω) − 2 ‖v‖W s,2 (ℝn ) −2

which proves that 𝒜 is coercive. Therefore, since by Remark 2.5 W0s,2 (Ω) is a separable Hilbert space and thus in particular a separable reflexive Banach space, and by the above observations 𝒜 is monotone, weakly continuous and coercive, by the standard theory of monotone operators (see, e. g., [29, Corollary 2.2]), the operator 𝒜 is surjective. Therefore, it remains to prove that the linear functional φ 󳨃→ (f , φ)L2 (Ω) ,

φ ∈ W0s,2 (Ω)

Higher integrability for nonlocal equations | 477

belongs to (W0s,2 (Ω))⋆ . Indeed, by Hölder’s inequality and the fractional Sobolev inequality (cf. [11, Theorem 6.5]), for any φ ∈ W0s,2 (Ω) we have (f , φ)L2 (Ω)

󵄨 2n 󵄨 ≤ (∫󵄨󵄨󵄨f (x)󵄨󵄨󵄨 n+2s dx)

n+2s 2n

Ω

≤ C2 ‖f ‖

2n

L n+2s (Ω)

󵄨 2n 󵄨 ( ∫ 󵄨󵄨󵄨φ(x)󵄨󵄨󵄨 n−2s dx)

n−2s 2n

ℝn

‖φ‖W s,2 (ℝn ) ,

where C2 = C2 (n, s) > 0. Therefore, the above functional is indeed bounded and thus belongs to (W0s,2 (Ω))⋆ . Hence, by the surjectivity of 𝒜 there exists some v ∈ W0s,2 (Ω) such that ⟨𝒜(v), φ⟩ = (f , φ)L2 (Ω) for all φ ∈ W0s,2 (Ω). Since by (4.1) we have W0s,2 (Ω) ⊂ W s,2 (ℝn ) ⊂ X, we in particular have v ∈ X. Since also h ∈ X and X is a vector space, the function u := v + h also belongs to X and satisfies (f , φ)L2 (Ω) = ∫ ∫ Ω ℝn

A(x, y) Φ(u(x) − u(y))(φ(x) − φ(y)) dy dx |x − y|n+2s

+ ∫ ∫ ℝn \Ω Ω

= ∫∫ ℝn ℝn

A(x, y) Φ(u(x) − u(y))(φ(x) − φ(y)) dy dx |x − y|n+2s

A(x, y) Φ(u(x) − u(y))(φ(x) − φ(y)) dy dx, |x − y|n+2s

for all φ ∈ W0s,2 (Ω). Here we used that φ vanishes outside of Ω in order to obtain the last equality. Since by construction we also have u = h a. e. in ℝn \ Ω, u is a weak solution of the Dirichlet problem (4.2). Let us prove that this weak solution is unique. Assume that u1 , u2 ∈ X both solve the Dirichlet problem (4.2) weakly, so that we have u1 − u2 = h − h = 0

in ℝn \ Ω.

Since, moreover, by (4.1) the function u1 − u2 belongs to H s,2 (Ω|ℝn ), we clearly have u1 − u2 ∈ W0s,2 (Ω). Therefore, we can use u1 − u2 as a test function in (4.2) for both u1 and u2 , so that by subtracting the resulting equalities, along with (1.3), (1.6), and Lemma 2.7, we obtain 0= ∫ ∫ ℝn ℝn

A(x, y) (Φ(u1 (x) − u1 (y)) − Φ(u2 (x) − u2 (y))) |x − y|n+2s

× ((u1 (x) − u2 (x)) − (u1 (y) − u2 (y))) dy dx

≥ λ−2 ∫ ∫ ℝn ℝn

((u1 (x) − u1 (y)) − (u2 (x) − u2 (y)))2 dy dx |x − y|n+2s 2s

≥ λ−2 C1−1 |Ω|− n ‖u1 − u2 ‖2L2 (ℝn ) ≥ 0. This implies that ‖u1 − u2 ‖L2 (ℝn ) = 0 and therefore u1 = u2 a. e., so that there is exactly one weak solution to the Dirichlet problem (4.2) that belongs to X.

478 | S. Nowak

5 Higher integrability of ∇s u For the rest of this paper, we fix some kernel coefficient A ∈ ℒ0 (λ) and some function Φ : ℝ → ℝ satisfying Φ(0) = 0, (1.5), and (1.6). Moreover, we fix some f ∈ L2 (B6 ) and a measurable symmetric function g : ℝn × ℝn → ℝ with ∇s g ∈ L2 (B6 ). In addition, for notational clarity we define Lg := p. v. ∫ ℝn

g(x, y) dy, |x − y|n+2s

so that the function F defined in (1.2) has the form F = Lg + f . A crucial tool for the proof of the higher integrability of ∇s u is given by the following approximation lemma, which shows that any weak solution u of equation (1.1) is in some sense locally close to a weak solution of a corresponding homogeneous equation that satisfies the Hölder estimate from Proposition 3.3. Lemma 5.1. Let M > 0 and assume that A satisfies condition (1.8) in B5 . Then for any ε0 ∈ (0, 1), there exists some δ = δ(ε0 , n, s, λ, M) > 0, such that for any weak solution u ∈ H s,2 (B5 |ℝn ) of the equation LΦ A u = Lg + f

in B5

(5.1)

under the assumptions that A satisfies (1.8) in B5 , 󵄨 󵄨2 − 󵄨󵄨󵄨∇s u󵄨󵄨󵄨 dx ≤ M, ∫

(5.2)

󵄨 󵄨2 − (f 2 + 󵄨󵄨󵄨∇s g 󵄨󵄨󵄨 ) dx ≤ Mδ2 , ∫

(5.3)

B5

and

B5

there exists a weak solution v ∈ H s,2 (B5 |ℝn ) of the equation LΦ A v = 0 in B5

(5.4)

󵄩󵄩 s 󵄩 󵄩󵄩∇ (u − v)󵄩󵄩󵄩L2 (B5 ) ≤ ε0

(5.5)

󵄩󵄩 s 󵄩󵄩 󵄩󵄩∇ v󵄩󵄩L∞ (B2 ) ≤ N0

(5.6)

that satisfies

and the estimate

for some constant N0 = N0 (n, s, λ, M).

Higher integrability for nonlocal equations | 479

Proof. Fix ε0 ∈ (0, 1) and let δ > 0 to be chosen. Let v ∈ H s,2 (B5 |ℝn ) be the unique weak solution of the problem LΦ Av = 0

{

weakly in B5 ,

(5.7)

a. e. in ℝn \ B5 ,

v=u

note that v exists by Proposition 4.1. In view of (1.6), (1.5), (1.3), and using w := u − v ∈ W0s,2 (B5 ) as a test function in (5.7) and (5.1), we obtain 2

((u(x) − u(y)) − (v(x) − v(y))) 󵄩󵄩 s 󵄩󵄩2 dy dx 󵄩󵄩∇ w󵄩󵄩L2 (B5 ) ≤ λ ∫ ∫ A(x, y) |x − y|n+2s ℝn ℝn

≤ λ2 ( ∫ ∫ A(x, y) ℝn ℝn

Φ(u(x) − u(y))(w(x) − w(y)) dy dx |x − y|n+2s

Φ(u(x) − u(y))(w(x) − w(y)) dy dx) − ∫ ∫ A(x, y) |x − y|n+2s n ℝn ℝ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ =0

g(x, y)(w(x) − w(y)) = λ2 ( ∫ ∫ dy dx |x − y|n+2s n ℝn ℝ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ =:I1

+ ∫ f (x)w(x) dx). B⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 5 =:I2

By the Cauchy–Schwarz inequality and taking into account that w = 0 in ℝn \ B5 , for I1 we have I1 ≤ 2 ∫ ∫ B5 ℝn

|g(x, y)||w(x) − w(y)| dy dx |x − y|n+2s

󵄩 󵄩 󵄩 󵄩 ≤ 2󵄩󵄩󵄩∇s g 󵄩󵄩󵄩L2 (B ) 󵄩󵄩󵄩∇s w󵄩󵄩󵄩L2 (B ) , 5

5

while by additionally using Lemma 2.7, we deduce 󵄩 󵄩 I2 ≤ ‖f ‖L2 (B5 ) ‖w‖L2 (B5 ) ≤ C1 ‖f ‖L2 (B5 ) 󵄩󵄩󵄩∇s w󵄩󵄩󵄩L2 (B ) , 5

where C1 = C1 (n, s) > 0. Therefore, by combining the last three displays, we arrive at 󵄩󵄩 s 󵄩2 󵄩 s 󵄩2 󵄩 s 󵄩2 󵄩󵄩∇ (u − v)󵄩󵄩󵄩L2 (B5 ) ≤ C2 (󵄩󵄩󵄩∇ g 󵄩󵄩󵄩L2 (B5 ) + 󵄩󵄩󵄩∇ f 󵄩󵄩󵄩L2 (B5 ) ) ≤ 2C2 |B5 |Mδ2 ≤ ε02 ,

(5.8)

where the last inequality follows by choosing δ sufficiently small and C2 = C2 (n, s, λ) > 0. This completes the proof of (5.5).

480 | S. Nowak Let us now proof the estimate (5.6). For almost every x ∈ B2 , by Proposition 3.2, we have ∫ ℝn \B3

(v(z) − v(y))2 (v(x) − v(y))2 dy ≤ C3 ∫ ∫ dy dz, n+2s |x − y| |x − y|n+2s n B3 ℝ

where C3 = C3 (n, s, λ). Now choose γ > 0 small enough such that γ < s and s + γ < 1. In view of the assumption that A satisfies (1.8) in B5 , by Proposition 3.3, we have 󵄩 󵄩 [v]Cs+γ (B3 ) ≤ C4 󵄩󵄩󵄩∇s v󵄩󵄩󵄩L2 (B ) 5 for some constant C4 = C4 (n, s, λ, γ). Thus, for almost every x ∈ B2 we obtain ∫ B3

(v(x) − v(y))2 dy dy ≤ [v]2Cs+γ (B3 ) ∫ |x − y|n+2s |x − y|n−2γ B3

= C5 [v]2Cs+γ (B3 ) ≤ C6 ∫ ∫ B5 ℝn

(v(z) − v(y))2 dy dz, |x − y|n+2s

where C5 = C5 (n, γ) < ∞ and C6 = C6 (n, s, λ, γ) > 0. By combining the above estimates, along with (5.8) and (5.2), we conclude that for almost every x ∈ B2 , 2

(∇s v) (x) = ∫ ℝn \B3

(v(x) − v(y))2 (v(x) − v(y))2 dy + ∫ dy n+2s |x − y| |x − y|n+2s

≤ C7 ∫ ∫ B5 ℝn

B3

2

(v(z) − v(y)) dy dz |x − y|n+2s

≤ 2C7 (∫ ∫ B5 ℝn

≤ 2C7 (∫ ∫ B5 ℝn

(u(z) − u(y))2 (w(z) − w(y))2 dy dz + ∫ ∫ dy dz) n+2s |x − y| |x − y|n+2s n 2

B5 ℝ

(u(z) − u(y)) dy dz + ε02 ) ≤ 2C7 (|B5 |M + 1), |x − y|n+2s 1

where C7 = C7 (n, s, γ, λ). Therefore, (5.6) holds with N0 = (2C7 (|B5 |M + 1)) 2 . The following result is an application of the above approximation lemma and, roughly speaking, shows that if the maximal functions of ∇s u, ∇s f , and ∇s g are small enough in some point, then the set where the maximal function of ∇s u is large has to be very small. Lemma 5.2. There is a constant N1 = N1 (n, s, λ) > 1, such that the following is true. If A satisfies condition (1.8) in B6 , then for any ε > 0 there exists some δ = δ(ε, n, s, λ) > 0,

Higher integrability for nonlocal equations | 481

such that for any z ∈ B1 , any r ∈ (0, 1], and any weak solution u ∈ H s,2 (B5r (z)|ℝn ) of the equation LΦ A u = Lg + f

in B5r (z)

with 󵄨2 󵄨 {x ∈ Br (z) | ℳB6 (󵄨󵄨󵄨∇s u󵄨󵄨󵄨 )(x) ≤ 1} 󵄨 󵄨2 ∩ {x ∈ Br (z) | ℳB6 (|f |2 + 󵄨󵄨󵄨∇s g 󵄨󵄨󵄨 )(x) ≤ δ2 } ≠ 0, we have 󵄨󵄨 󵄨 s 󵄨2 2 󵄨 󵄨󵄨{x ∈ Br (z) | ℳB6 (󵄨󵄨󵄨∇ u󵄨󵄨󵄨 )(x) > N1 }󵄨󵄨󵄨 < ε|Br |.

(5.9)

Proof. Let ε0 ∈ (0, 1) and M > 0 to be chosen and consider the corresponding δ = δ(ε0 , n, s, λ, M) > 0 given by Lemma 5.1. Fix r ∈ (0, 1] and z ∈ ℝn . Define ̃ y) := A(rx + z, ry + z), A(x,

̃ (x) := r −s u(rx + z), u

g̃ (x, y) := r −s g(rx + z, ry + z),

̃f (x) := r s f (rx + z)

̃ belongs to the class ℒ0 (λ) and satisfies and note that under the above assumptions A s,2 n ̃ ∈ H (B5 |ℝ ) satisfies condition (1.8) in B5 , and that u ̃ = Lg̃ + ̃f weakly in B5 . LΦ ̃u A Therefore, by Lemma 5.1, there exists a weak solution ṽ ∈ H s,2 (B5 |ℝn ) of LÃ ṽ = 0

in B5

such that 󵄨 ̃ ̃ 󵄨󵄨2 − v)󵄨󵄨 dx ≤ ε02 , ∫ 󵄨󵄨󵄨∇s (u

(5.10)

B2

provided that conditions (5.2) and (5.3) are satisfied. By assumption, there exists a point x ∈ Br (z) such that 󵄨 s 󵄨2 ℳB6 (󵄨󵄨󵄨∇ u󵄨󵄨󵄨 )(x) ≤ 1,

2 󵄨 s 󵄨2 2 ℳB6 (|f | + 󵄨󵄨󵄨∇ g 󵄨󵄨󵄨 )(x) ≤ δ .

By the scaling and translation invariance of the Hardy–Littlewood maximal function ∈ B1 we thus have (Lemma 2.2), for x0 := x−z r 󵄨

s

s

󵄨2

󵄨2

s

󵄨2

̃ 󵄨󵄨󵄨 )(x0 ) = ℳB6 (󵄨󵄨󵄨∇ u󵄨󵄨󵄨 )(x) ≤ 1 ℳB6/r (−z) (󵄨󵄨󵄨∇ u 󵄨

and 2

2s

2

s

󵄨2

2

ℳB6/r (−z) (|̃f | + 󵄨󵄨󵄨∇ g̃ 󵄨󵄨󵄨 )(x0 ) = ℳB6 (r |f | + 󵄨󵄨󵄨∇ g 󵄨󵄨󵄨 )(x) ≤ δ .

󵄨

󵄨

482 | S. Nowak Therefore, for any ρ > 0 we have 󵄨 ̃ 󵄨󵄨2 − 󵄨󵄨󵄨∇s u ∫ 󵄨󵄨 dx ≤ 1,

Bρ (x0 )

󵄨2 󵄨 − (|̃f |2 + 󵄨󵄨󵄨∇s g̃ 󵄨󵄨󵄨 ) dx ≤ δ2 , ∫

(5.11)

Bρ (x0 )

̃ , ∇s g̃ and ̃f outside of B6/r (−z) are replaced by 0, which we also where the values of ∇s u do for the rest of the proof. Since B5 ⊂ B6 (x0 ), by (5.11) we have n

|B | 6 󵄨 ̃ 󵄨󵄨2 󵄨 ̃ 󵄨󵄨2 − 󵄨󵄨󵄨∇s u − 󵄨󵄨󵄨∇s u ∫ 󵄨󵄨 dx ≤ 6 ∫ 󵄨󵄨 dx ≤ ( ) |B5 | 5

B5

B6 (x0 )

and n

|B | 6 󵄨 󵄨 󵄨2 󵄨2 − (|̃f |2 + 󵄨󵄨󵄨∇s g̃ 󵄨󵄨󵄨 ) dx ≤ 6 ∫ − (|̃f |2 + 󵄨󵄨󵄨∇s g̃ 󵄨󵄨󵄨 ) dx ≤ ( ) δ2 . ∫ |B5 | 5

B5

B6 (x0 )

̃ , g̃ , and ̃f satisfy conditions (5.2) and (5.3) Since also B5 ⊂ B6/r (−z), we obtain that u 6 n ̃ and the corresponding approximate with M = ( 5 ) . Therefore, (5.10) is satisfied by u solution ṽ. Considering the function v ∈ H s,2 (B6 |ℝn ) given by v(x) := r s ṽ( x−z ) and r rescaling back yields 󵄨 󵄨2 󵄨 ̃ ̃ 󵄨󵄨2 − v)󵄨󵄨 dx ≤ ε02 r n . ∫ 󵄨󵄨󵄨∇s (u − v)󵄨󵄨󵄨 dx = r n ∫ 󵄨󵄨󵄨∇s (u

(5.12)

B2

B2r (y)

By Lemma 5.1, there is a constant N0 = N0 (n, s, λ) > 0 such that 󵄩󵄩 s ̃󵄩󵄩2 2 󵄩󵄩∇ v󵄩󵄩L∞ (B2 ) ≤ N0 .

(5.13)

Next, we define N1 := (max{4N02 , 3n })1/2 > 1 and claim that 󵄨 ̃ 󵄨󵄨2 2 {x ∈ B1 | ℳB6/r (−z) (󵄨󵄨󵄨∇s u 󵄨󵄨 )(x) > N1 } 󵄨 ̃ ̃ 󵄨󵄨2 ⊂ {x ∈ B1 | ℳB2 (󵄨󵄨󵄨∇s (u − v)󵄨󵄨 )(x) > N02 }.

(5.14)

In order to see this, assume that 󵄨 ̃ ̃ 󵄨󵄨2 x1 ∈ {x ∈ B1 | ℳB2 (󵄨󵄨󵄨∇s (u − v)󵄨󵄨 )(x) ≤ N02 }.

(5.15)

For ρ < 1, we have Bρ (x1 ) ⊂ B1 (x1 ) ⊂ B2 , so that together with (5.15) and (5.13) we deduce 󵄨 ̃ 󵄨󵄨2 󵄨 ̃ ̃ 󵄨󵄨2 󵄨󵄨 s ̃󵄨󵄨2 − 󵄨󵄨󵄨∇s u − (󵄨󵄨󵄨∇s (u − v)󵄨󵄨 + 󵄨󵄨∇ v󵄨󵄨 ) dx ∫ 󵄨󵄨 dx ≤ 2 ∫

Bρ (x1 )

Bρ (x1 )

󵄨 ̃ ̃ 󵄨󵄨2 󵄩 󵄩2 ≤2 ∫ − 󵄨󵄨󵄨∇s (u − v)󵄨󵄨 dx + 2󵄩󵄩󵄩∇s ṽ󵄩󵄩󵄩L∞ (B (x )) ρ 1 Bρ (x1 )

󵄨 ̃ ̃ 󵄨󵄨2 󵄩 󵄩2 ≤ 2ℳB2 (󵄨󵄨󵄨∇s (u − v)󵄨󵄨 )(x1 ) + 2󵄩󵄩󵄩∇s ṽ󵄩󵄩󵄩L∞ (B ) ≤ 4N02 . 2

Higher integrability for nonlocal equations | 483

On the other hand, for ρ ≥ 1 we have Bρ (x1 ) ⊂ B3ρ (x0 ), so that (5.11) implies |B3ρ | 󵄨 ̃ 󵄨󵄨2 − 󵄨󵄨󵄨∇s u ∫ 󵄨󵄨 dx ≤ |Bρ |

Bρ (x1 )

󵄨 ̃ 󵄨󵄨2 n − 󵄨󵄨󵄨∇s u ∫ 󵄨󵄨 dx ≤ 3 .

B3ρ (x0 )

Thus, we have 󵄨 ̃ 󵄨󵄨2 2 x1 ∈ {x ∈ B1 | ℳB6/r (−z) (󵄨󵄨󵄨∇s u 󵄨󵄨 )(x) ≤ N1 }, which implies (5.14). In view of Lemma 2.2, (5.14) is equivalent to 󵄨 󵄨2 {x ∈ Br (z) | ℳB6 (󵄨󵄨󵄨∇s u󵄨󵄨󵄨 )(x) > N12 } 󵄨 󵄨2 ⊂ {x ∈ Br (z) | ℳB2r (z) (󵄨󵄨󵄨∇s (u − v)󵄨󵄨󵄨 )(x) > N02 }.

(5.16)

For any ε > 0, using (5.16), the weak 1–1 estimate from Proposition 2.3 and (5.12), we conclude that there exists some constant C1 = C1 (n) > 0 such that 󵄨󵄨 󵄨 s 󵄨2 2 󵄨 󵄨󵄨{x ∈ Br (z) | ℳB6 (󵄨󵄨󵄨∇ u󵄨󵄨󵄨 )(x) > N1 }󵄨󵄨󵄨 󵄨 󵄨 󵄨2 󵄨 ≤ 󵄨󵄨󵄨{x ∈ Br (z) | ℳB2r (z) (󵄨󵄨󵄨∇s (u − v)󵄨󵄨󵄨 )(x) > N02 }󵄨󵄨󵄨 C 󵄨 󵄨2 ≤ 12 ∫ 󵄨󵄨󵄨∇s (u − v)󵄨󵄨󵄨 dx N0 B2r (z)

≤

C1 2 n C2 2 ε r = 2 ε0 |Br | < ε|Br |, N02 0 N0

where C2 = C2 (n) > 0 and the last inequality is obtained by choosing ε0 and thus also δ sufficiently small. This finishes our proof. Corollary 5.3. There is a constant N1 = N1 (n, s, λ) > 1, such that the following is true. If A satisfies the condition (1.8) in B6 , then for any ε > 0 there exists some δ = δ(ε, n, s, λ) > 0, such that for any z ∈ B1 , any r ∈ (0, 1), and any weak solution u ∈ H s,2 (B6 |ℝn ) of the equation LΦ A u = Lg + f

in B6

with 󵄨󵄨 󵄨 s 󵄨2 󵄨 2 󵄨󵄨{x ∈ Br (z) | ℳB6 (󵄨󵄨󵄨∇ u󵄨󵄨󵄨 )(x) > N1 } ∩ B1 󵄨󵄨󵄨 ≥ ε|Br |,

(5.17)

󵄨 󵄨2 Br (z) ∩ B1 ⊂ {x ∈ B1 | ℳB6 (󵄨󵄨󵄨∇s u󵄨󵄨󵄨 )(x) > 1} 󵄨 󵄨2 ∪ {x ∈ B1 | ℳB6 (|f |2 + 󵄨󵄨󵄨∇s g 󵄨󵄨󵄨 )(x) > δ2 }.

(5.18)

we have

484 | S. Nowak Proof. Let N1 = N1 (n, s, λ) > 1 be given by Lemma 5.2. Fix ε > 0, r ∈ (0, 1), z ∈ ℝn and consider the corresponding δ = δ(ε, n, s, λ) > 0 given by Lemma 5.2. We now argue by contradiction. Assume that (5.17) is satisfied but that (5.18) is false, so that there exists some x0 ∈ Br (z) ∩ B1 such that 󵄨2 󵄨 x0 ∈ Br (z) ∩ {x ∈ B1 | ℳB6 (󵄨󵄨󵄨∇s u󵄨󵄨󵄨 )(x) ≤ 1} 󵄨2 󵄨 ∩ {x ∈ B1 | ℳB6 (|f |2 + 󵄨󵄨󵄨∇s g 󵄨󵄨󵄨 )(x) ≤ δ2 } 󵄨 󵄨2 ⊂ {x ∈ Br (z) | ℳB6 (󵄨󵄨󵄨∇s u󵄨󵄨󵄨 )(x) ≤ 1} 󵄨 󵄨2 ∩ {x ∈ Br (z) | ℳB6 (|f |2 + 󵄨󵄨󵄨∇s g 󵄨󵄨󵄨 )(x) ≤ δ2 }. Since in addition we have B5r (z) ⊂ B6 , by Lemma 5.2 we arrive at 󵄨󵄨 󵄨 s 󵄨2 󵄨 2 󵄨󵄨{x ∈ Br (z) | ℳB6 (󵄨󵄨󵄨∇ u󵄨󵄨󵄨 )(x) > N1 } ∩ B1 󵄨󵄨󵄨 󵄨 󵄨 󵄨2 󵄨 ≤ 󵄨󵄨󵄨{x ∈ Br (z) | ℳB6 (󵄨󵄨󵄨∇s u󵄨󵄨󵄨 )(x) > N12 }󵄨󵄨󵄨 < ε|Br |, which contradicts (5.17). The following decay of level sets will be the main key to proving the higher integrability of ∇s u. Lemma 5.4. Let N1 = N1 (n, s, λ) > 1 be given by Corollary 5.3 and assume that A satisfies condition (1.8) in B6 . Moreover, let k ∈ ℕ, ε ∈ (0, 1), set ε1 := 10n ε and consider the corresponding δ = δ(ε, n, s, λ) > 0 given by Corollary 5.3. Then for any weak solution u ∈ H s,2 (B6 |ℝn ) of the equation LΦ A u = Lg + f

in B6

with 󵄨󵄨 󵄨 s 󵄨2 2 󵄨 󵄨󵄨{x ∈ B1 | ℳB6 (󵄨󵄨󵄨∇ u󵄨󵄨󵄨 )(x) > N1 }󵄨󵄨󵄨 < ε|B1 |,

(5.19)

we have 󵄨󵄨 󵄨 s 󵄨2 2k 󵄨 󵄨󵄨{x ∈ B1 | ℳB6 (󵄨󵄨󵄨∇ u󵄨󵄨󵄨 )(x) > N1 }󵄨󵄨󵄨 k

󵄨 󵄨2 j󵄨 2(k−j) 󵄨󵄨 ≤ ∑ ε1 󵄨󵄨󵄨{x ∈ B1 | ℳB6 (|f |2 + 󵄨󵄨󵄨∇s g 󵄨󵄨󵄨 )(x) > δ2 N1 }󵄨󵄨 j=1

󵄨 󵄨 󵄨2 󵄨 + ε1k 󵄨󵄨󵄨{x ∈ B1 | ℳB6 (󵄨󵄨󵄨∇s u󵄨󵄨󵄨 )(x) > 1}󵄨󵄨󵄨. Proof. We prove this lemma by induction on k. In view of (5.19) and Corollary 5.3, the case k = 1 is a direct consequence of Lemma 2.1 applied to the sets 󵄨 󵄨2 E := {x ∈ B1 | ℳB6 (󵄨󵄨󵄨∇s u󵄨󵄨󵄨 )(x) > N12 }

Higher integrability for nonlocal equations | 485

and 󵄨2 󵄨 󵄨2 󵄨 F := {x ∈ B1 | ℳB6 (󵄨󵄨󵄨∇s u󵄨󵄨󵄨 )(x) > 1} ∪ {x ∈ B1 | ℳB6 (|f |2 + 󵄨󵄨󵄨∇s g 󵄨󵄨󵄨 )(x) > δ2 }. ̂ := Φ(N1 t)/N1 , Next, assume that the conclusion is valid for some k ∈ ℕ. Define Φ(t) ̂ ̂ ̂ := u/N1 , ĝ := g/N1 , and f := f /N1 . Then Φ clearly satisfies conditions (1.5) and (1.6) u with respect to λ, and we have ̂ ̂ = Lĝ + ̂f weakly in B6 . LΦ Au

Moreover, since N1 > 1, we have 󵄨󵄨 󵄨 s ̂ 󵄨󵄨2 2 󵄨 󵄨󵄨{x ∈ B1 | ℳB6 (󵄨󵄨󵄨∇ u 󵄨󵄨 )(x) > N1 }󵄨󵄨󵄨 󵄨 󵄨 󵄨2 󵄨 = 󵄨󵄨󵄨{x ∈ B1 | ℳB6 (󵄨󵄨󵄨∇s u󵄨󵄨󵄨 )(x) > N14 }󵄨󵄨󵄨 󵄨 󵄨 󵄨2 󵄨 ≤ 󵄨󵄨󵄨{x ∈ B1 | ℳB6 (󵄨󵄨󵄨∇s u󵄨󵄨󵄨 )(x) > N12 }󵄨󵄨󵄨 < ε|B1 |. Thus, using the induction assumption yields 󵄨󵄨 󵄨 s 󵄨2 2(k+1) 󵄨󵄨 }󵄨󵄨 󵄨󵄨{x ∈ B1 | ℳB6 (󵄨󵄨󵄨∇ u󵄨󵄨󵄨 )(x) > N1 󵄨󵄨 󵄨󵄨 s ̂ 󵄨󵄨2 󵄨 = 󵄨󵄨{x ∈ B1 | ℳB6 (󵄨󵄨∇ u󵄨󵄨 )(x) > N12k }󵄨󵄨󵄨 k

󵄨 󵄨2 j󵄨 2(k−j) 󵄨󵄨 ≤ ∑ ε1 󵄨󵄨󵄨{x ∈ B1 | ℳB6 (|̂f |2 + 󵄨󵄨󵄨∇s ĝ 󵄨󵄨󵄨 )(x) > δ2 N1 }󵄨󵄨 j=1

󵄨 󵄨 ̂ 󵄨󵄨2 󵄨 + ε1k 󵄨󵄨󵄨{x ∈ B1 | ℳB6 (󵄨󵄨󵄨∇s u 󵄨󵄨 )(x) > 1}󵄨󵄨󵄨 k

󵄨 󵄨2 j󵄨 2(k+1−j) 󵄨󵄨 = ∑ ε1 󵄨󵄨󵄨{x ∈ B1 | ℳB6 (|f |2 + 󵄨󵄨󵄨∇s g 󵄨󵄨󵄨 )(x) > δ2 N1 }󵄨󵄨 j=1

󵄨 󵄨 󵄨2 󵄨 + ε1k 󵄨󵄨󵄨{x ∈ B1 | ℳB6 (󵄨󵄨󵄨∇s u󵄨󵄨󵄨 )(x) > N12 }󵄨󵄨󵄨.

Moreover, by using the case k = 1, we obtain k

󵄨 󵄨2 j󵄨 2(k+1−j) 󵄨󵄨 = ∑ ε1 󵄨󵄨󵄨{x ∈ B1 | ℳB6 (|f |2 + 󵄨󵄨󵄨∇s g 󵄨󵄨󵄨 )(x) > δ2 N1 }󵄨󵄨 j=1

󵄨 󵄨 󵄨2 󵄨 + ε1k 󵄨󵄨󵄨{x ∈ B1 | ℳB6 (󵄨󵄨󵄨∇s u󵄨󵄨󵄨 )(x) > N12 }󵄨󵄨󵄨 k

󵄨 󵄨2 j󵄨 2(k+1−j) 󵄨󵄨 ≤ ∑ ε1 󵄨󵄨󵄨{x ∈ B1 | ℳB6 (|f |2 + 󵄨󵄨󵄨∇s g 󵄨󵄨󵄨 )(x) > δ2 N1 }󵄨󵄨 j=1

󵄨 󵄨 󵄨2 󵄨 + ε1k (ε1 󵄨󵄨󵄨{x ∈ B1 | ℳB6 (|f |2 + 󵄨󵄨󵄨∇s g 󵄨󵄨󵄨 )(x) > δ2 }󵄨󵄨󵄨 󵄨 󵄨 󵄨2 󵄨 + ε1 󵄨󵄨󵄨{x ∈ B1 | ℳB6 (󵄨󵄨󵄨∇s u󵄨󵄨󵄨 )(x) > 1}󵄨󵄨󵄨)

k+1

󵄨 󵄨2 j󵄨 2(k+1−j) 󵄨󵄨 = ∑ ε1 󵄨󵄨󵄨{x ∈ B1 | ℳB6 (|f |2 + 󵄨󵄨󵄨∇s g 󵄨󵄨󵄨 )(x) > δ2 N1 }󵄨󵄨 j=1

󵄨 󵄨 󵄨2 󵄨 + ε1k+1 󵄨󵄨󵄨{x ∈ B1 | ℳB6 (󵄨󵄨󵄨∇s u󵄨󵄨󵄨 )(x) > 1}󵄨󵄨󵄨,

486 | S. Nowak so that, by combining the last two displays, we see that the conclusion is valid for k +1, which completes the proof. We are now set to prove the desired higher integrability of ∇s u in the case of balls. The main tools are Lemmas 5.4 and 2.4 as well as Proposition 2.3. Theorem 5.5. Let 2 < p < ∞. Moreover, let g : ℝn × ℝn → ℝ be a measurable symmetric function with ∇s g ∈ Lp (B6 ) and f ∈ Lp (B6 ). If A ∈ ℒ0 (λ) satisfies condition (1.8) in B6 and if Φ satisfies and assumptions (1.5) and (1.6) with respect to λ, then for any weak solution u ∈ H s,2 (B6 |ℝn ) of the equation LΦ A u = Lg + f

in B6 ,

we have ∇s u ∈ Lp (B1 ). Moreover, there exists a constant C = C(p, n, s, λ) > 0 such that 󵄩󵄩 s 󵄩󵄩 󵄩 s 󵄩 󵄩 s 󵄩 󵄩󵄩∇ u󵄩󵄩Lp (B1 ) ≤ C(‖f ‖Lp (B6 ) + 󵄩󵄩󵄩∇ g 󵄩󵄩󵄩Lp (B6 ) + 󵄩󵄩󵄩∇ u󵄩󵄩󵄩L2 (B6 ) ).

(5.20)

Proof. Fix p > 2 and let N1 = N1 (n, s, λ) > 1 be given by Lemma 5.4. Moreover, select ε ∈ (0, 1) small enough such that 1 N1p 10n ε ≤ . 2

(5.21)

Consider also the corresponding δ = δ(ε, n, s, λ) > 0 given by Corollary 5.3. If ∇s u = 0 a. e. in B6 , then the assertion is trivially satisfied, so that we can assume ‖∇s u‖L2 (B6 ) > 0. Next, we let γ > 0 to be chosen independently of u, g, and f and define uγ :=

u , Mγ

gγ :=

g , Mγ

and

fγ :=

f , Mγ

where Mγ := ‖∇s u‖L2 (B6 ) /γ. In addition, we define Φγ (t) := satisfies (1.5) and (1.6) with respect to λ, and have Φ

LA γ uγ = Lgγ + fγ

1 Φ(Mγ t), Mγ

note that Φγ

weakly in B6 .

Moreover, we have 󵄨 󵄨2 ∫ 󵄨󵄨󵄨∇s uγ 󵄨󵄨󵄨 dx = γ 2 .

B6

By combining this observation with the weak 1–1 estimate from Proposition 2.3, it follows that there is a constant C1 = C1 (n) > 0 such that C γ2 C 󵄨 s 󵄨2 󵄨󵄨 󵄨 s 󵄨2 2 󵄨 󵄨󵄨{x ∈ B1 | ℳB6 (󵄨󵄨󵄨∇ uγ 󵄨󵄨󵄨 )(x) > N1 }󵄨󵄨󵄨 ≤ 12 ∫ 󵄨󵄨󵄨∇ uγ 󵄨󵄨󵄨 dx = 1 2 < ε|B1 |, N1 N1 B6

Higher integrability for nonlocal equations | 487

where the last inequality is obtained by choosing γ small enough. Therefore, all assumptions of Lemma 5.4 are satisfied by uγ . Furthermore, by Proposition 2.3 and Lemma 2.4 applied with τ = 1, β = N12 and with p replaced by p/2, we deduce that 󵄨 s 󵄨2 󵄩p/2 󵄩 󵄩󵄩 s 󵄩󵄩p 󵄩󵄩∇ uγ 󵄩󵄩Lp (B1 ) ≤ 󵄩󵄩󵄩ℳB6 (󵄨󵄨󵄨∇ uγ 󵄨󵄨󵄨 )󵄩󵄩󵄩Lp/2 (B1 ) ∞

󵄨2 󵄨 󵄨 󵄨 ≤ C2 ( ∑ N1pk 󵄨󵄨󵄨{x ∈ B1 | ℳB6 (󵄨󵄨󵄨∇s uγ 󵄨󵄨󵄨 )(x) > N12k }󵄨󵄨󵄨 + |B1 |), k=1

where C2 = C2 (n, s, p, λ) > 0. Setting ε1 := 10n ε, by (5.21) we see that j

∞ ∞ 1 j ∑(N1p ε1 ) ≤ ∑( ) = 1. 2 j=1 j=1

(5.22)

Using Lemma 5.4, the Cauchy product, and (5.22), we compute ∞

󵄨 󵄨 󵄨2 󵄨 ∑ N1pk 󵄨󵄨󵄨{x ∈ B1 | ℳB6 (󵄨󵄨󵄨∇s uγ 󵄨󵄨󵄨 )(x) > N12k }󵄨󵄨󵄨

k=1

∞

k

k=1

j=1

󵄨 󵄨2 j󵄨 2(k−j) 󵄨󵄨 ≤ ∑ N1pk (∑ ε1 󵄨󵄨󵄨{x ∈ B1 | ℳB6 (|fγ |2 + 󵄨󵄨󵄨∇s gγ 󵄨󵄨󵄨 )(x) > δ2 N1 }󵄨󵄨 󵄨2 󵄨 󵄨 󵄨 + ε1k 󵄨󵄨󵄨{x ∈ B1 | ℳB6 (󵄨󵄨󵄨∇s uγ 󵄨󵄨󵄨 )(x) > 1}󵄨󵄨󵄨) ∞

∞

j 󵄨 󵄨 󵄨2 󵄨 = ( ∑ N1pk 󵄨󵄨󵄨{x ∈ B1 | ℳB6 (|fγ |2 + 󵄨󵄨󵄨∇s gγ 󵄨󵄨󵄨 )(x) > δ2 N12k }󵄨󵄨󵄨)(∑(N1p ε1 ) ) j=1

k=0

∞

k 󵄨 󵄨 󵄨2 󵄨 + ( ∑ (N1p ε1 ) )󵄨󵄨󵄨{x ∈ B1 | ℳB6 (󵄨󵄨󵄨∇s uγ 󵄨󵄨󵄨 )(x) > 1}󵄨󵄨󵄨 k=1

∞

󵄨 󵄨 󵄨2 󵄨 ≤ ∑ N1pk 󵄨󵄨󵄨{x ∈ B1 | ℳB6 (|fγ |2 + 󵄨󵄨󵄨∇s gγ 󵄨󵄨󵄨 )(x) > δ2 N12k }󵄨󵄨󵄨 + |B1 |. k=1

Next, by combining the previous two displays with Lemma 2.4 applied with τ = δ2 , β = N12 and with p replaced by p/2, and also taking into account the strong p–p estimates from Proposition 2.3, we deduce that 󵄩󵄩 s 󵄩󵄩p 󵄩󵄩∇ uγ 󵄩󵄩Lp (B1 ) ∞

󵄨 󵄨 󵄨2 󵄨 ≤ C2 ( ∑ N1pk 󵄨󵄨󵄨{x ∈ B1 | ℳB6 (|fγ |2 + 󵄨󵄨󵄨∇s gγ 󵄨󵄨󵄨 )(x) > δ2 N12k }󵄨󵄨󵄨 + 2|B1 |) k=1

󵄩 󵄨 󵄨2 󵄩p/2 ≤ C3 (󵄩󵄩󵄩ℳB6 (|fγ |2 + 󵄨󵄨󵄨∇s gγ 󵄨󵄨󵄨 )󵄩󵄩󵄩Lp/2 (B ) + 1) 6 󵄩󵄩 s 󵄩󵄩p p ≤ C4 (‖fγ ‖Lp (B ) + 󵄩󵄩∇ gγ 󵄩󵄩Lp (B ) + 1), 6 6

488 | S. Nowak where C3 = C3 (n, s, p, λ) > 0 and C4 = C4 (n, p) > 0. It follows that 󵄩 s 󵄩 󵄩󵄩 s 󵄩󵄩 1/p 󵄩󵄩∇ uγ 󵄩󵄩Lp (B1 ) ≤ C4 (‖fγ ‖Lp (B6 ) + 󵄩󵄩󵄩∇ gγ 󵄩󵄩󵄩Lp (B6 ) + 1), so that ‖∇s u‖L2 (B6 ) 󵄩 s 󵄩 󵄩󵄩 s 󵄩󵄩 1/p ) 󵄩󵄩∇ u󵄩󵄩Lp (B1 ) ≤ C4 (‖f ‖Lp (B6 ) + 󵄩󵄩󵄩∇ g 󵄩󵄩󵄩Lp (B6 ) + γ 󵄩 󵄩 󵄩 󵄩 ≤ C(‖f ‖Lp (B6 ) + 󵄩󵄩󵄩∇s g 󵄩󵄩󵄩Lp (B ) + 󵄩󵄩󵄩∇s u󵄩󵄩󵄩L2 (B ) ), 6 6 which proves the estimate (5.20).

6 Proof of the main result We are now set to prove our main result by using scaling and covering arguments. Proof of Theorem 1.1. Fix p ∈ (2, ∞). We first assume that f ∈ Lploc (Ω). Fix relatively compact bounded open sets U ⋐ V ⋐ Ω. Moreover, fix a Lipschitz domain U⋆ such that U ⋐ U⋆ ⋐ V. For any z ∈ V, fix some small enough rz ∈ (0, 1) such that B6rz (z) ⋐ V. Define Az (x, y) := A(rz x + z, rz y + z),

gz (x) := rz−s g(rz x + z, rz y + z),

uz (x) := rz−s u(rz x + z),

fz (x) := rzs f (rz x + z)

and note that for any z ∈ V, Az belongs to the class ℒ0 (λ) and satisfies condition (1.8) in B6 , and that uz ∈ H s,2 (B6 ) satisfies LΦ Az uz = Lgz + fz

weakly in B6 .

By Theorem 5.5, we obtain the estimate n

󵄩󵄩 s 󵄩󵄩 󵄩 s 󵄩 󵄩󵄩∇ u󵄩󵄩Lp (Br (z)) = rzp 󵄩󵄩󵄩∇ uz 󵄩󵄩󵄩Lq (B1 ) z n

󵄩 󵄩 󵄩 󵄩 ≤ rzp C1 (‖fz ‖Lp (B6 ) + 󵄩󵄩󵄩∇s gz 󵄩󵄩󵄩Lp (B ) + 󵄩󵄩󵄩∇s uz 󵄩󵄩󵄩L2 (B ) ) 6

=

C1 (rzs ‖f ‖Lp (B6r (z)) z n

≤ C1 max{1, rzp

− n2

n

n

6

− 󵄩 󵄩 󵄩 󵄩 + 󵄩󵄩󵄩∇s g 󵄩󵄩󵄩Lp (B (z)) + rzp 2 󵄩󵄩󵄩∇s u󵄩󵄩󵄩L2 (B (z)) ) 6rz 6rz

}(‖f ‖Lq (B6r

z

(z))

󵄩 󵄩 + 󵄩󵄩󵄩∇s g 󵄩󵄩󵄩Lp (B

6rz (z))

󵄩 󵄩 + 󵄩󵄩󵄩∇s u󵄩󵄩󵄩L2 (B

6rz (z))

),

where C1 = C1 (p, n, s, λ) > 0. Since {Brz (z)}z∈U is an open covering of U ⋆ and U ⋆ is com⋆

pact, there is a finite subcover {Brz (zi )}ki=1 of U ⋆ and hence of U⋆ . Let {ϕi }ki=1 be a partii

tion of unity subordinate to the covering {Brz (zi )}ki=1 of U ⋆ , that is, the ϕi are nonnegai

tive functions on ℝn , we have ϕi ∈ C0∞ (Brx (xi )) for all i = 1, . . . , k, ∑ki=1 ϕj ≡ 1 in an open i

Higher integrability for nonlocal equations | 489 n

neighborhood of U ⋆ , and ∑ki=1 ϕj ≤ 1 in ℝn . Setting C2 := C1 max{1, maxi=1,...,k rzpi summing the above estimates over i = 1, . . . , k, we conclude

− n2

} and

󵄨󵄨 󵄨󵄨󵄨󵄨 k 󵄨󵄨󵄨󵄨 󵄨 s 󵄨 󵄨󵄨󵄨󵄨󵄨󵄨 󵄩󵄩 s 󵄩󵄩 󵄩󵄩∇ u󵄩󵄩Lp (U⋆ ) = 󵄨󵄨󵄨󵄨󵄨󵄨∑󵄨󵄨󵄨∇ u󵄨󵄨󵄨ϕi 󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 q 󵄨󵄨󵄨󵄨 󵄨󵄨L (U⋆ ) 󵄨󵄨i=1 k

󵄨 󵄩 󵄩󵄨 ≤ ∑󵄩󵄩󵄩󵄨󵄨󵄨∇s u󵄨󵄨󵄨ϕi 󵄩󵄩󵄩Lp (B

rz

i=1

i

(zi ))

k

󵄩 󵄩 ≤ ∑󵄩󵄩󵄩∇s u󵄩󵄩󵄩Lp (B i=1

rz (zi ) ) i

k

≤ ∑ C2 (‖f ‖Lp (B6r i=1

z

(z))

󵄩 󵄩 + 󵄩󵄩󵄩∇s g 󵄩󵄩󵄩Lp (B

6rz (z))

󵄩 󵄩 + 󵄩󵄩󵄩∇s u󵄩󵄩󵄩L2 (B

6rz (z))

)

󵄩 󵄩 󵄩 󵄩 ≤ C2 k(‖f ‖Lp (V) + 󵄩󵄩󵄩∇s g 󵄩󵄩󵄩Lp (V) + 󵄩󵄩󵄩∇s u󵄩󵄩󵄩L2 (V) ), so that for C3 = C2 k we have 󵄩󵄩 s 󵄩󵄩 󵄩 s 󵄩 󵄩 s 󵄩 󵄩󵄩∇ u󵄩󵄩Lp (U⋆ ) ≤ C3 (‖f ‖Lp (V) + 󵄩󵄩󵄩∇ g 󵄩󵄩󵄩Lp (V) + 󵄩󵄩󵄩∇ u󵄩󵄩󵄩L2 (V) ).

(6.1)

p

pn ⋆ Next, consider the general case when f ∈ Lloc (Ω), where p⋆ = max{ n+ps , 2}. Consider the function fV : ℝn → ℝ defined by

f (x),

fΩ (x) := {

0,

if x ∈ V,

if x ∈ ℝn \ V

and note that fV ∈ Lp⋆ (ℝn )∩L2 (ℝn ). By [24, Proposition 5.1], there exists a unique weak s,2 solution h ∈ H s,2 (ℝn ) ⊂ Hloc (Ω|ℝn ) of the equation (−Δ)s h + h = fV

in ℝn ,

(6.2)

where (−Δ)s h(x) = Cn,s ∫ ℝn

h(x) − h(y) dy |x − y|n+2s

is the fractional Laplacian of h. In view of Proposition 2.9, Theorem 2.11, using the classical H 2s,p⋆ estimates for the fractional Laplacian on the whole space ℝn (cf. [17, ̃ y) := C (h(x) − h(y)), we obtain Lemma 3.5]) and setting h(x, n,s 󵄩 ̃ 󵄩󵄩 󵄩 s 󵄩 ‖h‖Lp (ℝn ) + 󵄩󵄩󵄩∇s h 󵄩󵄩Lp (ℝn ) = C4 (‖h‖Lp (ℝn ) + 󵄩󵄩󵄩∇ h󵄩󵄩󵄩Lp (ℝn ) ) ≤ C5 ‖h‖H s,p (ℝn )

≤ C6 ‖h‖H 2s,p⋆ (ℝn )

≤ C7 ‖fV ‖Lp⋆ (ℝn ) = C6 ‖f ‖Lp⋆ (V) ,

(6.3)

490 | S. Nowak where all constants depend only on n, s and p. Furthermore, u is a weak solution of ̃ + h in V, LΦ A u = Lg where ̃ y). g̃ (x, y) := g(x, y) + h(x, Therefore, by combining the estimates (6.1) and (6.3), we arrive at 󵄩󵄩 s 󵄩󵄩 󵄩 s 󵄩 󵄩 s 󵄩 󵄩󵄩∇ u󵄩󵄩Lp (U⋆ ) ≤ C3 (‖h‖Lp (V) + 󵄩󵄩󵄩∇ g̃ 󵄩󵄩󵄩Lp (V) + 󵄩󵄩󵄩∇ u󵄩󵄩󵄩L2 (V) ) 󵄩 ̃ 󵄩󵄩 󵄩 s 󵄩 󵄩 s 󵄩 ≤ C3 (‖h‖Lp (V) + 󵄩󵄩󵄩∇s h 󵄩󵄩Lp (V) + 󵄩󵄩󵄩∇ g 󵄩󵄩󵄩Lp (V) + 󵄩󵄩󵄩∇ u󵄩󵄩󵄩L2 (V) ) 󵄩 󵄩 󵄩 󵄩 ≤ C(‖f ‖Lp⋆ (V) + 󵄩󵄩󵄩∇s g 󵄩󵄩󵄩Lp (V) + 󵄩󵄩󵄩∇s u󵄩󵄩󵄩L2 (V) ) for some constant C = C(p, n, s, λ) > 0, which proves the estimate (1.9). In particular, since by assumption we have f , ∇s g ∈ Lp (V), we obtain that ∇s u ∈ Lp (U⋆ ). Let us now prove that u ∈ H s,p (U⋆ |ℝn ). For any r ∈ [2, p], define rn , p}, if rs < n, min{ n−rs

r ⋆ := {

p,

if rs ≥ n.

By Proposition 2.9 and Theorem 2.11, for any r ≥ 2 we have H s,r (U⋆ |ℝn ) ⊂ H s,r (U⋆ ) 󳨅→ Lr (U⋆ ). ⋆

Since u ∈ H s,2 (U⋆ |ℝn ), we have u ∈ L2 (U⋆ ). If p = 2⋆ , we have u, ∇s u ∈ Lp (U⋆ ), ⋆ and therefore u ∈ H s,p (U⋆ |ℝn ). If p > 2⋆ , then we have u, ∇s u ∈ L2 (U⋆ ), so that ⋆ ⋆⋆ ⋆ u ∈ H s,2 (U⋆ |ℝn ). We therefore arrive at u ∈ L2 (U⋆ ). If 2⋆ = p, then we have u, ∇s u ∈ ⋆ Lp (U⋆ ), and therefore u ∈ H s,p (U⋆ |ℝn ). If 2⋆ > p, then iterating the above procedure s,p n also yields u ∈ H (U⋆ |ℝ ), and therefore u ∈ H s,p (U|ℝn ) at some point. Since U is an s,p arbitrary relatively compact open subset of Ω, we conclude that u ∈ Hloc (Ω|ℝn ). This finishes the proof. ⋆

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Yuhua Sun and Fanheng Xu

On nonexistence results of porous medium type equations and differential inequalities on Riemannian manifolds Abstract: We study nonexistence of global positive solutions to porous medium type equation on manifolds, and nonexistence of nonnegative solution to differential inequalities on manifolds. We provide a universal method to deal with nonexistence results especially to porous medium type equations. Our nonexistence results are new even for the Euclidean spaces. Keywords: Porous medium type equation, differential inequalities, manifolds, nonexistence MSC 2010: Primary 58J05, Secondary 35J70

Contents 1 2 3 4

Introduction | 493 Statements | 497 Proof of Theorem 2.4 | 499 Proof of Theorem 2.6 | 507 Bibliography | 512

1 Introduction Our first aim in this paper is to study the nonexistence of global positive solutions to porous medium type equation {

𝜕t u = ∇(|∇u|σ ∇um ) + up , u(x, 0) = u0 (x),

in M × (0, ∞), in M,

(1.1)

where p > min{m + σ, 1}, m + σ > 0, σ > −2, m ≠ 0, and u0 is a continuous bounded function and not identically equal to zero, and also M is a geodesically complete noncompact connected Riemannian manifold. Acknowledgement: Sun was supported by the National Natural Science Foundation of China (No. 11501303, No. 11761131002), and Tianjin Natural Science Foundation (No. 19JCQNJC14600). Yuhua Sun, School of Mathematical Sciences and LPMC, Nankai University, 300071 Tianjin, P. R. China, e-mail: [email protected] Fanheng Xu, School of Mathematics (Zhuhai), Sun Yat-Sen University (Zhuhai Campus), 519082 Zhuhai, P. R. China, e-mail: [email protected] https://doi.org/10.1515/9783110700763-018

494 | Y. Sun and F. Xu Now let us recall some previously known result in this area. Problem (1.1) in ℝN and its variations have been widely addressed in the literature, see [3, 4, 11, 37, 38], and also a rather complete survey paper by Levine [24]. As far as we know, the first celebrated result in this area is due to the classical paper by Fujita [9] when studying (1.1) of special case σ = 0, m = 1, and M = ℝN : {

𝜕t u = Δu + up , u(x, 0) = u0 (x),

in ℝN × (0, ∞), in ℝN ,

(1.2)

He found that the exponent 1 + N2 is critical, namely, if 1 < p < 1 + N2 , then problem (1.2) possesses no global positive solution in ℝN ; while if p > 1 + N2 , (1.2) possesses positive solutions for a class of “small” initial data u0 . The number 1 + N2 is called Fujita’s exponent. The question of nonexistence of a positive solution in the borderline case p = 1 + N2 was solved by Hayakawa [18] for dimension N = 1, 2 and by Kobayashi, Sirao and Tanaka [23] for all N, one can also refer to papers [2, 38] for other methods. These proofs for the critical case essentially rely on a linear structure of the differential operator in the semilinear heat equation with the critical exponent, and hence cannot be directly applied to more general equations like (1.1). Problem (1.1) when σ = 0 and m = 1 was frequently investigated by many mathematicians, for example, without claim of completeness, Levine, Mochizuki, Mukai, Galaktionov, Kawango, Suzuki, Andreucci, and Tedneev, we refer to the papers [10, 11, 12, 13, 30, 22, 29, 34, 1]. Usually, their proofs rely on the comparison principle, and the well-known Zel’dovich–Kompaneez–Barsenblatt solution for porous medium equation, see [20]. Let us present some of their results. When M = ℝN , m = 1, (1.1) becomes {

𝜕t u = div(|∇u|σ ∇u) + up , u(x, 0) = u0 (x),

in ℝN × (0, ∞), in ℝN ,

(1.3)

and here div(|∇u|σ ∇u) = Δσ+2 u is the classical quasilinear operator. Galaktionov [10, 12] proved that if 1 < p ≤ σ + 1 + σ+2 and σ > 0, then (1.3) possesses no global nonN negative nontrivial solution; while if p > σ +1+ σ+2 , (1.3) possesses global nonnegative N nontrivial solutions for a class of “small” initial data u0 . And for the case of σ = 0, and M = ℝN , (1.1) becomes {

𝜕t u = Δum + up , u(x, 0) = u0 (x),

in ℝN × (0, ∞), in ℝN ,

(1.4)

Galaktionov [12], Kawanago [22], Mochizuki–Suzuki [30] independently showed that if 1 ≤ m < p ≤ m + N2 , then problem (1.1) possesses no global nonnegative nontrivial solution. When M = ℝN , in [11] assuming that m > 0 and σ > 0, Galaktionov and Levine obtained that if 1 < p ≤ m + σ + σ+2 , then (1.1) possesses no global nonnegative nonN trivial solution, while if p > m + σ + σ+2 , then (1.1) possesses a global positive solution N for some small initial value u0 .

Uniqueness of positive solutions |

495

As far as we know, since the complicated interaction of the gradient term in (1.1), very little literature is devoted to studying the nonexistence results for general σ and m on manifolds. In this paper, we would like to obtain the nonexistence result of (1.1) for general σ and m on manifolds. Now we recall some results of regarding (1.1) on manifolds. Throughout the paper, we let μ denote the Riemannian measure on M, d be the geodesic distance on M, and x0 be some fixed reference point on M. Let B(x0 , r) denote the geodesic ball centered at x0 with radii r, and denote V(r) := μ(B(x0 , r)). The manifold version of problem (1.4) was systematically investigated by Zhang in paper [39], where he dealt with a class of manifolds satisfying the following assumptions: (i) Assume that μ(B(x, r)) ≤ Cr α for α > 2, when r is large and for all x ∈ M. (ii)

1

𝜕 log g 2 𝜕r 1 2

≤

C , r

where r = d(x0 , x) is smooth. Here x0 is a fixed reference point, and

g is the volume density of the manifold. Zhang proved the following: Assuming conditions (i) and (ii) on a manifold are satisfied, and α > 2, (i) when m ≥ 1, if m < p ≤ m + α2 , then problem (1.1) possesses no global positive solution to (1.4). (ii) when 1 − α2 < m < 1, then problem (1.4) possesses no global positive solution. Here conditions (i)–(ii) are satisfied for a large class of manifolds, such as those with nonnegative Ricci curvature. The approach used by Zhang is firstly to construct a suitable integral functional to show that the integral functional in a selected fixed domain will blow-up or will be identically equal to zero, then one can derive the blow-up results of nonlinear parabolic equations on manifolds. Zhang’s approach is quite powerful, and can be also used to deal with other types of equation on manifolds like nonlinear inhomogeneous porous medium equation, and even with the blow-up problems in exterior domains [40]. However, after a very careful examination of Zhang’s paper [39], one can find that assumptions (i) and (ii) on manifolds are essential in his approach, neither can be relaxed nor dropped. Moreover, the paper [39] needs to deal with the critical case in a separate way to obtain the blow-up results. In [25], Mastrolia, Monticelli, and Punzo improved Zhang’s result in the case of m = 1, σ = 0 by dropping condition (ii), and relaxing condition (i) to α

V(r) ≤ Cr α (ln r) 2 ,

(1.5)

for all large enough r. Then if 1 < p ≤ 1 + α2 , (1.1) possesses no global positive solution. However, the sharpness of exponent α2 is not shown in [25].

496 | Y. Sun and F. Xu Recently, in [17], we showed the nonexistence of global positive solutions to a semilinear parabolic equation with potential terms on manifolds; as a byproduct, we also showed the sharpness of exponent α2 in (1.5), that is, if the volume condition (1.5) fails, there exists some class of manifolds which admit positive global solution for small initial data u0 . The second aim of this paper is concerned with the nonexistence of nonnegative solutions to the following differential inequality: div(|∇u|σ ∇um ) + up ≤ 0,

on M,

(1.6)

where p > m + σ > 0, σ > −2, and m ≠ 0, and M is still a geodesically complete noncompact connected Riemannian manifold. We want to characterize p, σ, m, and M to imply the nonexistence of positive solution to (1.6). There is a huge of literature devoted to the study of nonlinear elliptic differential inequalities similar to (1.6), especially those arising naturally in geometry and physics, see [6, 5, 7, 15, 26, 27, 28, 31, 36], and the references therein. When M = ℝN and m = 1, (1.6) has the form of Δσ󸀠 u + up ≤ 0,

(1.7)

where σ 󸀠 = σ + 2. Mitidieri and Pohozaev [27] proved that if 1 σ󸀠 ,

then the only nonnegative solution to (1.7) is identically equal to zero. The study on the nonexistence of solution of a differential inequality can also be generalized on Riemannian manifolds. As far as we know, Cheng and Yau probably were first to use the upper bound of the volume to derive the Liouville’s uniquenesstype result. In [8], they proved that if V(r) ≤ Cr 2 , for all large enough r, then any nonnegative superharmonic function on M is constant. Similar integral versions of the volume condition were obtained by Grigor’yan, Karp, Holopainen, and Varopoulos, respectively, to study superharmonic functions and p-superharmonic functions, see [14, 21, 35, 19]. Later in [15], Grigor’yan and the author studied the semilinear differential inequality Δu + up ≤ 0,

(1.8)

which is a special case of (1.6) with σ = 0, m = 1. They obtained that if V(r) ≤ Cr Θ1 (ln r)Θ2 ,

(1.9)

Uniqueness of positive solutions |

497

for all large enough r, then the only nonnegative solution to (1.8) is identically equal 2p 1 , Θ2 = p−1 . We emphasize that this uniqueness result does not to zero. Here Θ1 = p−1 need any additional assumption on the curvature of manifolds, or other assumption on the behavior of a solution. Moreover, condition (1.9) is sharp, namely, Θ1 , Θ2 cannot be relaxed, otherwise, one can find some manifold which satisfies (1.9) but admits a positive solution to (1.8) on M. In [33], the author generalized the results of [15] to problem (1.7), and obtained that if V(r) ≤ Cr Θ3 (ln r)Θ4 , for all large enough r, then the only nonnegative solution to (1.7) is identically equal to zero. Here Θ3 =

pσ 󸀠 , p−σ 󸀠 +1

Θ4 =

σ 󸀠 −1 , p−σ 󸀠 +1

and p > σ 󸀠 − 1 (or equivalently, p > σ + 1).

Simultaneously, the exponents Θ3 , Θ4 are sharp here in the same sense as above. Recently, some new advances in the study of superlinear differential inequalities on Riemannian manifolds were obtained by Grigor’yan, Verbitsky, and the author [16]. By using the Green function of the Laplacian, one of the results they obtained is that a necessary and sufficient condition for the existence of positive solution on manifold with nonnegative Ricci curvature to (1.8) becomes ∞ ∞

q−1

t dt ] ∫[∫ V(t)

r0

r dr < ∞,

(1.10)

r

for some r0 > 0. Condition (1.10) can be considered as a integral version of (1.9) on manifolds with nonnegative Ricci curvature; however, the question about whether condition (1.10) is still necessary and sufficient for the existence of a positive solution to (1.8) on general manifolds remains open.

2 Statements Before stating our main results, let us introduce some spaces. Let 1,q Wloc (M) = {f : M → ℝ|f ∈ Lqloc (M), ∇f ∈ Lqloc (M)},

(2.1)

where ∇f is understood in the distributional sense. Denote by Wc1,q (M) the subspace 1,q of Wloc (M) of functions with compact support, and BC(M) means that bounded con1,q tinuous function spaces; Wc1,q (M × [0, ∞)), Wloc (M × [0, ∞)) are defined similarly. Firstly, we begin with the definition of a weak solution of (1.1) and (1.6). 1,p 1,2 Definition 2.1. A function 0 ≤ u ∈ Wloc (M ×(0, ∞))∩Wloc (M ×(0, ∞)) is called a global 1,2 σ+1 m−1 1 nonnegative solution to (1.1) with |∇u| u ∈ Lloc (M × (0, ∞)) and u0 ∈ Wloc (M) ∩

498 | Y. Sun and F. Xu BC(M) provided that for any φ ∈ Cc∞ (M × [0, ∞)) there holds ∞

∞

∞

0 M

0 M

0 M

∫ ∫ up ψ dμ dt = ∫ ∫ m|∇u|σ um−1 ∇u ⋅ ∇ψ dμ dt − ∫ ∫ uψt dμ dt − ∫ u0 ψ(⋅, 0) dμ M

Remark 2.2. From definition (2.1), since u0 , ψ(⋅, 0) ≥ 0, we have that ∞

∞

p

σ m−1

∫ ∫ u ψ dμ dt ≤ m ∫ ∫ |∇u| u 0 M

∞

∇u ⋅ ∇ψ dμ dt − ∫ ∫ uψt dμ dt,

0 M

(2.2)

0 M

holds for all ψ ∈ Cc∞ (M × [0, ∞)). By taking the closure in Wc1,p (M × [0, ∞)), we obtain 󸀠

that (2.2) holds for all ψ ∈ Wc1,p (M × [0, ∞)), where p󸀠 = 󸀠

p . p−1

1,p 1,2 (M) is called a weak solution to (1.6) (M) ∩ Wloc Definition 2.3. A function 0 ≤ u ∈ Wloc

if |∇u|σ+1 um−1 ∈ L1loc (M) and for every φ ∈ Wc1,p (M), the inequality 󸀠

∫ up φ dμ ≤ m ∫ |∇u|σ um−1 ∇u ⋅ ∇φ dμ M

(2.3)

M

is satisfied. Define Γ1 =

σ+2 , p−m−σ

Γ2 = min{

1 m+σ , }. p−1 p−m−σ

(2.4)

Theorem 2.4. If V(r) ≤ Cr Γ1 (ln r)Γ2 ,

(2.5)

holds for all large enough r, then there exists no global positive solution to (1.1). Remark 2.5. Let us compare our results to those in the Euclidean space. When M = ℝN , from Theorem 2.4, we know that if N≤

σ+2 , p−m−σ

which means m+σ 0 and m < 0. For the case of m > 0, let us define 󸀠

ψ(x, t) = u−a φb , where a will take arbitrarily small positive value, and b will be a fixed large enough positive constant. For the case of m < 0, define ψ(x, t) = ua φb , where a, b are the same as above. We present the proof for m > 0, while the case of m < 0 can be dealt with similarly. It follows that ∇ψ = bu−a φb−1 ∇φ − au−a−1 φb ∇u and 𝜕t ψ = bφb−1 u−a 𝜕t φ − au−a−1 φb 𝜕t u.

500 | Y. Sun and F. Xu Substituting ψ(x, t) = u−a ψb into (2.2), we obtain ∞

∞

∫ ∫ up−a φb dμ dt + ma ∫ ∫ |∇u|σ+2 um−a−2 φb dμ dt 0 M

0 M ∞

∞

≤ mb ∫ ∫ |∇u|σ um−a−1 φb−1 ∇u ⋅ ∇φ dμ dt − ∫ ∫ u𝜕t ψ dμ dt.

(3.1)

0 M

0 M

Noting that ∞

mb ∫ ∫ |∇u|σ um−a−1 φb−1 ∇u ⋅ ∇φ dμ dt 0 M ∞

≤ mb ∫ ∫ |∇u|σ+1 um−a−1 φb−1 |∇φ| dμ dt 0 M ∞

≤

∞

ma Cmbσ+2 ∫ ∫ um−a+σ φb−σ−2 |∇φ|σ+2 dμ dt, ∫ ∫ |∇u|σ+2 um−a−2 φb dμ dt + 2 aσ+1 0 M

0 M

(3.2)

where we have used Young’s inequality. Combining (3.2) with (3.1), we obtain ∞

∫ ∫ up−a φb dμ dt + 0 M

0 M

σ+2

Cmb ≤ aσ+1

σ+2

≤

∞

ma ∫ ∫ |∇u|σ+2 um−a−2 φb dμ dt 2

Cmb aσ+1

∞

∫∫u

m−a+σ

b−σ−2

φ

|∇φ|

σ+2

∞

dμ dt − ∫ ∫ uψt dμ dt 0 M

0 M ∞

∫ ∫ um−a+σ φb−σ−2 |∇φ|σ+2 dμ dt 0 M

∞

− ∫ ∫[bφb−1 u1−a 𝜕t φ − au−a φb 𝜕t u] dμ dt.

(3.3)

0 M

Substituting ψ(x, t) = φ(x, t)b into (2.2), we obtain ∞

∞

0 M

0 M ∞

∫ ∫ up φb dμ dt ≤ mb ∫ ∫ |∇u|σ um−1 φb−1 ∇u ⋅ ∇φ dμ dt − b ∫ ∫ uφb−1 φt dμ dt. 0 M

(3.4)

Uniqueness of positive solutions |

Since ∞

mb ∫ ∫ |∇u|σ um−1 φb−1 ∇u ⋅ ∇φ dμ dt 0 M

∞

≤ mb ∫ ∫ |∇u|σ+1 um−1 φb−1 |∇φ| dμ dt 0 M ∞

≤

ma ∫ ∫ |∇u|σ+2 um−a−2 φb dμ dt 2 0 M

∞

Cmbσ+2 + ∫ ∫ um+σ+aσ+a φb−σ−2 |∇φ|σ+2 dμ dt. aσ+1 0 M

Hence, from (3.4), we have ∞

∞

ma ∫ ∫ u φ dμ dt ≤ ∫ ∫ |∇u|σ+2 um−a−2 φb dμ dt 2 p b

0 M

0 M

∞

Cmbσ+2 + ∫ ∫ um+σ+aσ+a φb−σ−2 |∇φ|σ+2 dμ dt aσ+1 0 M

∞

− b ∫ ∫ uφb−1 φt dμ dt 0 M

Combining with (3.3), we obtain ∞

∫ ∫ up φb dμ dt ≤ 0 M

∞

Cmbσ+2 ∫ ∫ um−a+σ φb−σ−2 |∇φ|σ+2 dμ dt aσ+1 ∞

0 M

− ∫ ∫[bφb−1 u1−a 𝜕t φ − au−a φb 𝜕t u] dμ dt 0 M

∞

Cmbσ+2 + ∫ ∫ um+σ+aσ+a φb−σ−2 |∇φ|σ+2 dμ dt aσ+1 ∞

0 M

− b ∫ ∫ uφb−1 φt dμ dt. 0 M

For convenience, let us denote ∞

I := ∫ ∫ up φb dμ dt, 0 M ∞

Cmbσ+2 K1 := ∫ ∫ um−a+σ φb−σ−2 |∇φ|σ+2 dμ dt, aσ+1 0 M

501

502 | Y. Sun and F. Xu ∞

K2 := − ∫ ∫[bφb−1 u1−a 𝜕t φ − au−a φb 𝜕t u] dμ dt, 0 M ∞

Cmbσ+2 K3 := ∫ ∫ um+σ+aσ+a φb−σ−2 |∇φ|σ+2 dμ dt, aσ+1 0 M

∞

K4 := −b ∫ ∫ uφb−1 φt dμ dt. 0 M

Define ∞

∞

J(δ) := ∫ ∫ |∇φ|δ dμ dt,

L(δ) := ∫ ∫ |𝜕t φ|δ dμ dt.

0 M

(3.5)

0 M

For K1 , we have K1 ≤

Cmbσ+2 (∬ up φb dμ dt) aσ+1 c

m−a+σ p

DR

∞

p(σ+2) b− p−m+a−σ

× (∫ ∫ φ

|∇φ|

p(σ+2) p−m+a−σ

dμ dt)

p−m+a−σ p

.

0 M

For K2 , we have ∞

K2 = − ∫ ∫ u1−a 𝜕t φb dμ dt +

∞

a ∫ ∫ φb 𝜕t u1−a dμ dt 1−a 0 M

0 M ∞

= − ∫ ∫ u1−a 𝜕t φb dμ dt − 0 M

a b ∫ u1−a 0 φ(x, 0) dμ 1−a M

∞

−

a ∫ ∫ u1−a 𝜕t φb dμ dt 1−a 0 M ∞

=−

1 a b ∫ ∫ u1−a 𝜕t φb dμ dt − ∫ u1−a 0 φ(x, 0) dμ 1−a 1−a 0 M ∞

≤−

1 ∫ ∫ u1−a 𝜕t φb dμ dt 1−a 0 M ∞

=−

b ∫ ∫ u1−a φb−1 𝜕t φ dμ dt. 1−a 0 M

M

(3.6)

503

Uniqueness of positive solutions |

Applying Hölder inequality further, we have ∞

K2 ≤

b ∫ ∫ u1−a φb−1 |𝜕t φ| dμ dt 1−a 0 M

b ≤ (∬ up φb dμ dt) 1−a

1−a p

DcR

∞

× (∫ ∫ φ

p b− p−1+a

|𝜕t φ|

p p−1+a

dμ dt)

p−1+a p

(3.7)

.

0 M

For K3 , we have ∞

Cmbσ+2 ∫ ∫ um+σ+aσ+a φb−σ−2 |∇φ|σ+2 dμ dt aσ+1

K3 =

0 M

σ+2

Cmb ≤ (∬ up φb dμ dt) aσ+1 c

m+σ+aσ+a p

DR

∞

p(σ+2) b− p−m−σ−aσ−a

× (∫ ∫ φ

|∇φ|

p(σ+2) p−m−σ−aσ−a

dμ, dt)

p−m−σ−aσ−a p

.

(3.8)

.

(3.9)

0 M

For K4 , we have ∞

K4 = −b ∫ ∫ uφb−1 φt dμ dt 0 M 1 p

p b

∞

p b− p−1

≤ b(∬ u φ dμ dt) ( ∫ ∫ φ

|𝜕t φ|

p p−1

dμ dt)

p−1 p

0 M

DcR

Combining (3.6), (3.7), (3.8), and (3.9), and noting that 0 ≤ φ ≤ 1 on M, we obtain I≤

C

aσ+1 +

I

C

m−a+σ p

aσ+1

I

p(σ + 2) J( ) p−m+a−σ

m+σ+aσ+a p

+I

p(σ + 2) J( ) p − m − σ − aσ − a

p + CI L( ) p−1 1 p

p−m+a−σ p

p−1 p

,

1−a p

p L( ) p−1+a

p−1+a p

p−m−σ−aσ−a p

(3.10)

where we have used that the terms involving m and b are absorbed into constants.

504 | Y. Sun and F. Xu We claim that there exists a constant C0 such that ∞

∫ ∫ up dμ dt < C0 < ∞.

(3.11)

0 M

Otherwise, there exists ∞

∫ ∫ up dμ dt > 1, 0 M

and we can find a large enough R such that ∬ up dμ dt > 1. DR

Noting that p > m + σ, if a is small enough close to zero, then there exists a constant β such that max{

m − a + σ 1 − a m + σ + aσ + a 1 , , , } < β < 1. p p p p

Hence, from (3.10), we obtain 1

p(σ + 2) I ≤ CI [ σ+1 J( ) p−m+a−σ a β

p−m+a−σ p

1

p(σ + 2) + σ+1 J( ) p − m − σ − aσ − a a

p + L( ) p−1+a

p−m−σ−aσ−a p

p−1+a p

p + L( ) p−1

p−1 p

]

(3.12)

and I 1−β ≤ C[

1

p(σ + 2) J( ) aσ+1 p − m + a − σ

p−m+a−σ p

1

p(σ + 2) + σ+1 J( ) p − m − σ − aσ − a a

+ L(

p ) p−1+a

p−m−σ−aσ−a p

p−1+a p

p + L( ) p−1

p−1 p

].

(3.13)

Let g ∈ C ∞ [0, ∞) be a nonnegative function satisfying g(t) = 1 on [0, 1];

g(t) = 0 on [2, ∞);

|g 󸀠 | ≤ C1 < ∞.

Let {ηk }k∈ℕ , {γk }k∈ℕ ∈ C ∞ [0, ∞) be two sequences of functions defined respectively by ηk (t) = g(

t ) 2kθ

(3.14)

Uniqueness of positive solutions |

505

and γk (x) = g(

r(x) ), 2k

(3.15)

where θ = (σ+2)(p−1) and r(x) = d(x0 , x). p−m−σ From (3.14) and (3.15), we have |𝜕t ηk | {

≤

C , 22k

t ∈ [22k , 22k+1 ],

C , 2k

x ∈ B2k+1 \ B2k ,

= 0,

otherwise,

(3.16)

and |∇γk | {

≤

= 0,

otherwise.

(3.17)

Let us define a sequence of functions {φi (x, t)}i∈ℕ by φi (x, t) =

1 2i ∑ η (t)γk (x). i k=i+1 k

(3.18)

It follows that φi (x, t) = 1 when (x, t) ∈ B2i × [0, 2iθ ]. Moreover, for distinct k, noting that supp(𝜕t ηk ) and supp(∇γk ) are disjoint, we obtain for any α > 0, 2i

|𝜕t φi |α = i−α ∑ |γk 𝜕t ηk |α k=i+1

(3.19)

and 2i

|∇φi |α = i−α ∑ |ηk ∇γk |α . k=i+1

(3.20)

Hence φi ∈ Wc1,p (M × [0, ∞)). 󸀠

Let 1 a= . i

(3.21)

506 | Y. Sun and F. Xu Substituting φi into (3.12), let us estimate term by term. Firstly, note that ∞

J(

p(σ+2) p(σ + 2) ) = ∫ ∫ |∇φi | p−m+a−σ dμ dt p−m+a−σ

0 M

≤

≤

≤

∞

1

i

p(σ+2) p−m+a−σ

i

2i

2kθ+1

k=i+1 0 2i

p(σ+2) p−m+a−σ

p(σ+2)

|∇γk | p−m+a−σ dμ/

∫ B2k+1 \B2k

p(σ+2)

Γ

∑ 2kθ (2−k ) p−m+a−σ (2k ) 1 k Γ2 ,

(3.22)

k=i+1

where we have used (3.20). Noting that θ = (σ+2)(p−1) and Γ1 = p−m−σ kθ −

p(σ+2)

∑ ∫ ηkp−m+a−σ dt

C

i

p(σ+2)

0 M k=i+1

1

p(σ+2) p−m+a−σ

2i

∫ ∫ ∑ |ηk ∇γk | p−m+a−σ dμ dt

σ+2 , p−m−σ

ap(σ + 2) kp(σ + 2) + kΓ1 = k. p−m+a−σ (p − m − σ)(p − m + a − σ) 1 i

Combining with the above (3.22), and noting that a = we obtain J(

(3.23)

is small enough close to zero,

p(σ + 2) Γ +1− p(σ+2) ) ≤ Ci 2 p−m+a−σ . p−m+a−σ

(3.24)

Secondly, ∞

p p L( ) = ∫ ∫ |𝜕t φi | p−1+a dμ dt p−1+a

0 M

=i

≤i

2k

p − p−1+a

∞

p

p

∑ ∫ |𝜕t ηk | p−1+a dt ∫ |γk | p−1+a dμ

k=i+1 0 2k

p − p−1+a

p − p−1+a

B2k+1

2kθ

2i

kθp − p−1+a kθ

∑ 2

k=i+1

p 󸀠 − p−1+a +Γ2 +1

≤C i

p

∑ ∫ |𝜕t ηk | p−1+a dt ∫ dμ

k=i+1

≤ Ci

M

2kθ+1

Γ

2 (2k ) 1 k Γ2

,

(3.25)

where (3.19) has been used. Similarly, we have J(

p(σ + 2) − p(σ+2) +Γ +1 ) ≤ Ci p−m−σ−aσ−a 2 p − m − σ − aσ − a

(3.26)

Uniqueness of positive solutions |

507

and L(

p − p +Γ +1 ) ≤ Ci p−1 2 . p−1

(3.27)

Combining with (3.24), (3.25), (3.26), (3.27), and (3.13), we obtain I 1−β ≤ C[i +i

− m−a+σ Γ2 p−m+a−σ p p

+i

Γ2 p−1+a − 1−a p p

Γ2 p−m−σ−aσ−a − m+σ+aσ+a p p

+i

Γ2 p−1 − p1 p

].

(3.28)

Recalling that Γ2 = min{

m+σ 1 , }, p−1 p−m−σ

1 a= , i

and letting i → ∞, recalling definition of I, we obtain 2iθ

p

1−β

( ∫ ∫ u dμ dt)

< ∞,

(3.29)

B2i 0

which means ∬

up dμ dt < ∞.

M×(0,∞)

Substituting φi into (3.12), and repeating the same procedures as above, we obtain 2iθ

2iθ+1

p

∫ ∫ u dμ dt ≤ C ∫ ∫ up dμ dt. B2i 0

Bci 2iθ 2

Letting i → ∞ and combining with (3.29), we obtain ∬

up dμ dt = 0,

M×(0,∞)

which contradicts with the positiveness of u, hence there exists no global positive solution to (1.1).

4 Proof of Theorem 2.6 Proof. Without loss of generality, let us assume that u is some positive solution to (1.6), otherwise we can replace u by u + ϵ for some small ϵ > 0. Let φ be a Lipschitz function on M with compact support, such that φ = 1 in a neighborhood of Br . We divide the proof into two cases: m > 0 and m < 0.

508 | Y. Sun and F. Xu For the case of m > 0, let ψ = φs u−t , where s, t are two variable positive exponents to be chosen later. Noting that ∇ψ = −tu−t−1 φs ∇u + su−t φs−1 ∇φ, and substituting into (2.3) with ψ, we obtain ∫ up ψ dμ ≤ m ∫ |∇u|σ um−1 ∇u ⋅ ∇ψ dμ, M

M

which is ∫ up−t φs dμ + mt ∫ um−t−2 |∇u|σ+2 φs dμ ≤ ms ∫ um−1−t |∇u|σ ∇u ⋅ ∇φφs−1 dμ. M

M

(4.1)

M

Applying Cauchy–Schwarz inequality to the right-hand side of (4.1), we obtain ms ∫ um−1−t |∇u|σ ∇u ⋅ ∇φφs−1 dμ M

≤ ms ∫ um−1−t |∇u|σ+1 |∇φ|φs−1 dμ M (m−t−2)(σ+1) σ+2

= ms ∫[u

|∇u|σ+1 φ

s(σ+1) σ+2

][u

m−t−2 +1 σ+2

s

|∇φ|φ σ+2 −1 ] dμ

M

mt Cmsσ+2 ≤ ∫ um−t−2 |∇u|σ+2 φs dμ + σ+1 ∫ um−t+σ |∇φ|σ+2 φs−σ−2 dμ. 2 t

(4.2)

M

M

Combining with (4.1), we obtain ∫ up−t φs dμ + M

mt Cmsσ+2 ∫ um−t−2 |∇u|σ+2 φs dμ ≤ σ+1 ∫ um−t+σ |∇φ|σ+2 φs−σ−2 dμ. 2 t M

(4.3)

M

Applying Cauchy–Schwarz inequality once more to the right-hand side of (4.3), we have Cmsσ+2 ∫ um−t+σ |∇φ|σ+2 φs−σ−2 dμ t σ+1 M

s(m+σ−t) Cmsσ+2 s−σ−2− s(m+σ−t) p−t = σ+1 ∫[um−t+σ φ p−t ][φ |∇φ|σ+2 ] dμ t

M

p−t

(p−t)(σ+2) 1 msσ+2 p−m−σ s− (p−t)(σ+2) ≤ ∫ up−t φs dμ + C 󸀠 ( σ+1 ) ∫ φ p−m−σ |∇φ| p−m−σ dμ 2 t

M

M

Substituting the above into (4.3), we obtain 1 mt ∫ up−t φs dμ + ∫ um−t−2 |∇u|σ+2 φs dμ 2 2 M

≤ C(

M

σ+2

ms ) t σ+1

p−t p−m−σ

∫φ M

s− (p−t)(σ+2) p−m−σ

|∇φ|

(p−t)(σ+2) p−m−σ

dμ.

(4.4)

Uniqueness of positive solutions |

509

Now substituting ψ = φs into (2.3), we have ∫ up φs dμ ≤ ms ∫ |∇u|σ+1 um−1 φs−1 |∇φ| dμ M

M m−t−2

≤ ms(∫ u

σ+2 s

φ dμ)

|∇u|

σ+1 σ+2

(∫ φ

M

s−σ−2 σt+σ+t+m

u

σ+2

|∇φ|

dμ)

1 σ+2

.

(4.5)

M

Combining with (4.4), we obtain p−t

σ+1

σ+2 (p−t)(σ+2) 1 msσ+2 p−m−σ s− (p−t)(σ+2) ∫ φ p−m−σ |∇φ| p−m−σ dμ] ∫ u φ dμ ≤ Cms[ ( σ+1 ) mt t

p s

M

M

× (∫ φ

s−σ−2 σt+σ+t+m

u

σ+2

|∇φ|

dμ)

1 σ+2

(4.6)

.

M

Applying Hölder inequality to the last integral of (4.6), and noting that φ = 1 on Br , ∫ φs−σ−2 uσt+σ+t+m |∇φ|σ+2 dμ M s p

≤ ( ∫ φ u dμ)

σt+σ+t+m p

(∫ φ

p(σ+2) s− p−σt−σ−t−m

|∇φ|

p(σ+2) p−σt−σ−t−m

dμ)

p−σt−σ−t−m p

.

M

M\Br

Combining with the above (4.6), we obtain 2

(p−t)(σ+1) σ+1 − (p−m−σ)(σ+2) 󸀠 − σ+2

s p

∫ φ u dμ ≤ C t

s p

( ∫ φ u dμ)

M

σt+σ+t+m p(σ+2)

M\Br

× (∫ φ

s− (p−t)(σ+2) p−m−σ

|∇φ|

(p−t)(σ+2) p−m−σ

dμ)

σ+1 σ+2

M

× (∫ φ

p(σ+2) s− p−σt−σ−t−m

|∇φ|

p(σ+2) p−σt−σ−t−m

dμ)

p−σt−σ−t−m p(σ+2)

,

(4.7)

M

where we have used that the terms involving s and m are absorbed into constant C 󸀠 . Recalling that 0 ≤ φ ≤ 1 on M, and φ = 1 on Br , we have p

1− σt+σ+t+m p(σ+2)

(∫ u dμ)

2

(p−t)(σ+1) σ+1 − (p−m−σ)(σ+2) 󸀠 − σ+2

≤C t

(∫ |∇φ|

Br

(p−t)(σ+2) p−m−σ

dμ)

σ+1 σ+2

M

× (∫ |∇φ| M

p(σ+2) p−σt−σ−t−m

dμ)

p−σt−σ−t−m p(σ+2)

.

(4.8)

510 | Y. Sun and F. Xu Now in the following, we use the same technique as in [15, 32]. Let {φ̃ k }k∈ℕ be a sequence satisfying the following conditions: each φ̃ k is a Lipschitz function such that supp(φ̃ k ) ⊂ B2k , and φ̃ k = 1 in a neighborhood of B2k−1 , and |∇φ̃ k | {

≤

C , 2k−1

for x ∈ B2k \ B2k−1 ,

= 0,

otherwise,

where C does not depend on k. Fix some n ∈ ℕ and set t=

1 , n

(4.9)

(4.10)

and φn =

∑2n k=n+1 φ̃ k . n

(4.11)

Note that φn = 1 on B2n , φ22n = 0 outside B22n , 0 ≤ φn ≤ 1 on M. Note also that for any a ≥ 1, using that all supp(∇φ̃ k ) are disjoint with each other, we have |∇φn |a =

a ∑2n k=n+1 |∇φ̃ k | . na

(4.12)

It is easy to see that 1,p φn ∈ Wloc (M). 󸀠

Consider the integral Jn (a) = ∫ |∇φn |a dμ,

(4.13)

M

where a takes values

(p−t)(σ+2) p−m−σ

and

p(σ+2) , and a is uniformly bounded for small t. p−σt−σ−t−m

Substituting (4.11) into (4.13), then applying (4.12) and (4.9), we obtain Jn (a) = ∫ |∇φn |a dμ M

=∫ M

a ∑2n k=n+1 |∇φ̃ k | dμ na

2n

= ∑

∫

k=n+1 B \B 2k 2k−1 2n

≤C ∑ ( k=n+1 2n

a

21−k ) μ(B2k ) n

≤ C󸀠 ∑ ( k=n+1

|∇φ̃ k |a dμ na

a

2−k ) μ(B2k ), n

(4.14)

Uniqueness of positive solutions | 511

where we have used that a is uniformly bounded. Noting that a = a1 =

(p − t)(σ + 2) , p−m−σ

a2 =

p(σ+2) p−m−σ

+ bt, with

p(σ + 2) , p − σt − σ − t − m

and respectively b takes the values b1 = −

σ+2 , p−m−σ

b2 =

p(σ + 2)(σ + 1) . (p − σt − σ − t − m)(p − m − σ)

Noting n + 1 ≤ k ≤ 2n and substituting t = n1 , we have that if b > 0, a

(

p

bt

2−k 2−k 2−k ) =( ) ( ) n n n ≤ C1 (

p

2−k ) , n

while if b < 0, a

(

p

2−k 2−k ) ≤ C2 ( ) . n n

Thus, we have p(σ+2)

2n 2−k p−m−σ (p − t)(σ + 2) )≤C ∑ ( ) Jn ( μ(B2k ) p−m−σ n k=n+1 p(σ+2)

2n

2−k p−m−σ k ∧1 ∧2 ≤C ∑ ( ) (2 ) (k) n k=n+1 p(σ+2) +∧2 +1 − p−m−σ

≤ C󸀠 n

p(σ+1) − p−m−σ

≤ C󸀠 n p(σ+2) , ∧2 p−m−σ

where we have used that ∧1 = Similarly, Jn (

=

(4.15)

,

m+σ . p−m−σ

p(σ + 2) − p(σ+1) ) ≤ Cn p−m−σ . p − σt − σ − t − m

(4.16)

Substituting (4.15) and (4.16) into (4.7), and noting t = n1 , r = 2n , we obtain p

( ∫ u dμ)

1− σt+σ+t+m p(σ+2)

σ+1

≤ Cn σ+2

(p− 1 )(σ+1)2

n + (p−m−σ)(σ+2)

p(σ+1) − p−m−σ

(n

σ+1

) σ+2

B2n p(σ+1) − p−m−σ

× (n

)

1 p− σ n −σ− n −m p(σ+2)

.

(4.17)

512 | Y. Sun and F. Xu Since (p − n1 )(σ + 1)2 (σ + 1)(p − σn − σ − n1 − m) p(σ + 1)2 σ+1 + − − =0 σ + 2 (p − m − σ)(σ + 2) (p − m − σ)(σ + 2) (p − m − σ)(σ + 2) It follows from (4.17) that 1− σt+σ+t+m p(σ+2)

p

( ∫ u dμ)

≤ C < ∞.

B2n

As n → ∞, we have ∫ up dμ < ∞.

(4.18)

M

Repeating the same procedure in (4.7), we obtain that ∫ up dμ ≤ C( ∫ up dμ) B2n

σ +σ+ 1 +m n n p(σ+2)

,

(4.19)

M\Br

which yields ∫ up dμ = 0, M

and hence u ≡ 0, thus contradicting the positiveness of u. For the case of m < 0, one can replace ψ = φs ut , where s, t are variable positive exponents, and repeat the same procedure as in the case of m > 0, by replacing m with |m| in each estimate. Hence, we complete the proof of Theorem 2.6.

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Index μ-nest 29 μ-quasicontinuous 36 ∇+s 388 ∇−s 388 ∇±s 390 qs

∼ 136

additive functional see also continuous additive functional Ahlfors regular conformal dimension 267 Ahlfros regular 267 alpha-process 315 Ambrose–Singer 444 – Theorem 447 ancient solutions 111 annular connectivity 198 approximation lemma 478 augmented tree 144, 148–154, 156, 157, 159–162, 165, 167–169, 171–175, 177, 178 Bakry–Emery Ricci curvature 364 Berger 443, 449 – holonomy theorem 443 Bessel potential space 468 biparabolic manifold 317 bisuperharmonic function 317 blow-up 263 bounded geometry 315 Brownian motion 300 CAF 124 – energy of 125 capacity 32, 183, 187, 188, 197 capacity upper bound 184 Cartan–Hadamard manifold 307 chain condition 288 Clarkson’s inequality 215 coercive closed form 55 combinatorial graph distance 104 continuous additive functional 124 – energy of 125 – equivalent 124 – finite 124 – of zero energy 125 – positive 124 contraction 259 curvature-dimension condition 362–364, 378

Davies–Gaffney–Grigor’yan Lemma 109 diamond fractal 261 diffusion – symmetric — with given Dirichlet form 124 diffusion without killing inside 124 Dirichlet form 32, see also symmetric Dirichlet form Dirichlet problem 473 discrete gradient 209 divs+ 391 divs− 391 divergence of a flow 224 doubling 165–170, 173, 177, 178 dual p-energy of a flow 225 dual p-modulus 225 – to infinity 242 dual weight 225 ℰ-nest 32 ℰ-quasicontinuous 64 Einstein manifolds 359, 362, 363, 368, 369 end of a manifold 311 energy measure 122 entropy 360, 361, 366–372, 374, 375, 380, 382 entropy isoperimetric inequality 359, 378, 379, 381, 382 entropy power 359–362, 368–370, 375, 378, 379, 383 escape rate 113 – volume growth criteria 113 exhaustion 207 – good 207 feasible function 214, 225, 238 (fHKE)Ψ 133 flow on a graph 224 flux of a flow 224 fractional Hardy–Sobolev space 402 fractional Laplacian 489 fractional Poincaré inequality 467 fractional Sobolev inequality 477 fractional Sobolev space 467 Fukushima decomposition theorem 125 full off-diagonal heat kernel estimates 133 Gauss–Green’s formula for p-Laplacian 245 Gaussian estimate 134

516 | Index

generalized p-resistance 237 generalized resistance metric 237 generalized Sierpiński carpet 129 – canonical Dirichlet form on 130 geodesic metric 134 gradient estimates 421–423, 425 graph 104, 206 – bounded geometry 256 – connected 104, 206 – locally finite 104, 206 – metric 206 – simple 104, 206 – subgraph 208 graph automorphism 263 graph isomorphism 263 Green function 301, 497, 513 Hardy space 387 Hardy–Littlewood maximal function 465 harmonic function 131 harmonic structure 127 – regular 140 Harnack inequality 183–185, 187, 197, 198, 348 heat equation 359, 361–363, 365–367, 369–375, 379, 381, 382 heat kernel 32, 128, 130, 133, 300 heat kernel estimate 3, 4, 6, 8, 10–17, 19–21, 23 heat semigroup 31 Hessian solitons 362, 363 higher integrability 486 Hölder regularity 471 holonomy group 443 holonomy system 444 horizontal graph 262 hyperbolic boundary 144, 147, 151, 164, 167, 173, 174 hyperbolic graph 143–147, 154, 156, 157, 164, 171, 173, 174, 178 index – of MMD space 138 induced measure 29 integral kernel 30 intrinsic metric – of MMD space 133 intrinsic metrics 107 – admissible intrinsic metrics 107 isoperimetric inequality 307 jumping kernel 4, 6, 8–12, 14

Kolmogorov type equation 345, 346 Kostant 449 Liouville property 303 Liouville-type theorem 421 Lipschitz equivalence 143, 144, 155, 156, 178 local weak solution 471 locally ultracontractive 27 lower rate function 306 Lusin’s theorem 39 MAF 125 – mutual energy of 138 – stochastic integral with respect to 139 manifold – biparabolic 317 – parabolic 328 – with ends 328 Martin boundary 144, 163, 164, 167 martingale additive functional 125 – mutual energy of 138 – stochastic integral with respect to 139 martingale dimension 139 massive set 312 matrix Harnack estimate 345, 347 max-flows and min-cuts theorem 235 Mazur’s lemma 219 mean curvature flow 421–423 Menger sponge 129 metric measure Dirichlet space 121 metric measure space 31, 121 minimizer 214, 218, 225 MMD space 121 monotone operators 474 mutual energy measure 122 N-ary tree 255 Nakamura–Yamasaki’s duality 226 Nash inequality 56 Nash-Williams criterion for p-parabolicity 254 near-isometry 157–161, 178 nested fractal 140 NIW formula 362, 363, 371–373, 375, 378, 383 nonlinear diffusion equation 362, 363, 377, 378, 382 nonlocal Dirichlet form 3, 8, 11, 22 nonlocal equation 459 nonlocal operator 460 p-conductance 214 – to infinity 238

Index | 517

p-Dirichlet space 210 p-energy 210 p-harmonic 246 p-hyperbolic graph 239 p-Laplacian – on a graph 245 p-modulus – to infinity 240 p-modulus of a family of curves on a graph 218 p-parabolic graph 239 p-subharmonic 246 p-superharmonic 246 p0 condition 256 parabolic Harnack inequality 21 parabolic index 254 parabolic manifold 300 path on a graph 206 – concatenation 207 – infinity 240 – length of 206 – simple 206 – subpath 206 PCAF 124 perturbation 424 Poincaré constant 327 Poincaré inequality 183, 184, 186, 188, 192, 327 pointwise realization 40 porous medium equation 494, 495 quadratic variation – of MAF 125 quasisymmetric 136, 266 random walk 143, 144, 153, 162–165, 178 rearrangeable matrix 158 recurrence 301, 315 relatively connected annuli condition 328 Rényi entropy 359, 362, 363, 376–378, 382, 383 resistance form 127 – standard — on Sierpiński gasket 127 Revuz measure 125 Ricci flow 359, 362, 365, 371, 372, 374, 375, 382 Riesz potential operator 387 Riesz transform 387 rough isometry 256 Royden’s decomposition of a p-Dirichlet function 246 s-gradient 461, 469

self-similar set 144, 149, 151, 152, 154–156, 161, 162, 165, 167, 177, 178, 259 self-similar structure 259 separate 230, 242 Shannon entropy 359, 361, 362, 367, 372–377, 379, 381, 382 shift space 260 Sierpiński carpet 129 – generalized 129 Sierpiński gasket 261 – n-dimensional (standard) 126 similitude 259 Simons 443 – theorem of holonomy system 444 singular mean curvature equations 421 Sobolev–Slobodeckij space 467 stochastic completeness 105, 302 – volume growth criteria 112 Strichartz hexacarpet 276 sub-Gaussian estimate 128 subharmonic potential 312 super Ricci flow 359, 362, 363, 365, 371, 374, 375 superharmonic function 300 symmetric Dirichlet form 121 – regular 121 – strongly local 121 symmetric pure jump process 3, 8, 11, 13, 14, 16, 20 time change – of MMD space 135 uniformly convex 220 uniqueness class problem 112 unit – flow on a graph 224 upper rate function 306 (VD)d 136 Vicsek tree 316 Vitali covering lemma 465 volume doubling 136 volume doubling property 327 volume function 301 W -entropy 362, 363, 365, 371, 372, 375, 378, 382

518 | Index

Wallach 451 weak solution 461, 473 weighted graph 105, 207 – carré du champ operator 106 – combinatorial Laplacian 106 – complete 108 – curvature dimension conditions 107 – Dirichlet form 105

– formal Laplacian 105 – normalized Laplacian 106 – physical Laplacian 106 – polynomial volume growth 111 – weighted degree 106 word 260 – empty 260 – length of 260