Potentials and Partial Differential Equations: The Legacy of David R. Adams 9783110792720, 9783110792652

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Suzanne Lenhart, Jie Xiao (Eds.) Potentials and Partial Differential Equations

Advances in Analysis and Geometry

�

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 8

Potentials and Partial Differential Equations �

The Legacy of David R. Adams Edited by Suzanne Lenhart and Jie Xiao

Mathematics Subject Classification 2020 Primary: 35J50, 42B25, 46E35; Secondary: 53C56, 58E30 Editors Prof. Suzanne Lenhart University of Tennessee Mathematics Department Knoxville TN 37996-1300 USA [email protected]

Prof. Jie Xiao Memorial University of Newfoundland Department of Mathematics & Stat. 230 Elizabeth Ave. St. John’s NL A1C 5S7 Canada [email protected]

ISBN 978-3-11-079265-2 e-ISBN (PDF) 978-3-11-079272-0 e-ISBN (EPUB) 978-3-11-079278-2 ISSN 2511-0438 Library of Congress Control Number: 2023931740 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. © 2023 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Foreword This volume is dedicated to the legacy of David R. Adams (January 23, 1941–July 30, 2021) – one of the most outstanding pioneers in nonlinear potential analysis and partial differential equations.

He made significant contributions to the following areas: calculus of variations and optimal control, optimization, Fourier analysis, functional analysis, partial differential equations, potential theory, and real functions. David received his PhD from the University of Minnesota in 1969 under the direction of Norman Myers, and he worked in the areas of partial differential equations and potential theory and analysis. After postdocs at Consiglio Nazionale delle Ricerche in Rome, Italy, (supervised by Guido Stampacchia), Rice University, and University of Calihttps://doi.org/10.1515/9783110792720-201

VI � Foreword fornia, San Diego, he joined the faculty of the University of Kentucky in 1973, where he remained a professor until his retirement in 2015. This volume contains a dozen papers by some strong researchers in the areas of function spaces, potential theory, and partial differential equations in which David R. Adams excelled. The topics range from sharp functional-potential-trace inequalities to elliptic partial differential equations. It is our hope that this volume will not only reflect appropriately the unprecedented impact and diversity of David R. Adams’ profound accomplishments but also offer an exceptional resource for the graduate students and researchers interested in analysis-geometry of partial differential equations and beyond. Last but not least, Dr. Adams was a valuable mentor for many students and researchers, and so this volume may help others to continue these mentoring efforts. Suzanne Lenhart Jie Xiao

Contents Foreword � V Der-Chen Chang, Shu-Cheng Chang, Yingbo Han, and Jingzhi Tie Yau’s conjecture on the dimension of harmonic polynomials � 1 Nageswari Shanmugalingam On Carrasco Piaggio’s theorem characterizing quasisymmetric maps from compact doubling spaces to Ahlfors regular spaces � 23 Tuoc Phan On trace theorems for weighted mixed-norm Sobolev spaces and applications � 49 Juncheng Wei and Yuanze Wu Sharp stability of the logarithmic Sobolev inequality in the critical point setting � 77 Qinfeng Li and Changyou Wang On a variational problem of nematic liquid crystal droplets � 103 Petteri Harjulehto and Ritva Hurri-Syrjänen Estimates for the variable order Riesz potential with applications � 127 Yoshihiro Sawano and Kazuki Kobayashi A remark on the atomic decomposition in Hardy spaces based on the convexification of ball Banach spaces � 157 Bernhard Ruf A Bliss–Adams inequality � 179 Cristina Tarsi Trudinger-type inequalities in ℝN with radial increasing mass-weight � 197 Liguang Liu and Jie Xiao In response to David R. Adams’ October 12, 2001, letter � 215 Augusto C. Ponce and Daniel Spector Some remarks on capacitary integrals and measure theory � 235

VIII � Contents Javier Martínez Perales and Carlos Pérez Remarks on vector-valued Gagliardo and Poincaré–Sobolev-type inequalities with weights � 265 Index � 287

Der-Chen Chang, Shu-Cheng Chang, Yingbo Han, and Jingzhi Tie

Yau’s conjecture on the dimension of harmonic polynomials Dedicated to the memory of David R. Adams

Abstract: Let M be an m-dimensional complete noncompact Riemannian manifold with nonnegative Ricci curvature. In a celebrating article [32], S.-T. Yau conjectured that the dimension hd (M) of ℋd (M m ), the space of harmonic functions of polynomial growth of degree at most d is finite for each positive integer d and satisfies the estimate: hd (M) ≤ hd (ℝm ). In this article, we first calculate the exact formula of the dimension of the linear space of harmonic polynomials with degrees up to d on the Euclidean space ℝm and the 2d m−1 Heisenberg group Hn . For the case ℝm , we have hd (ℝm ) ≈ (m−1)! . In the second part of 2n

d . In the third part of the paper, we survey the the paper, we show that hd (Hn ) ≈ (2n)! results on Yau’s conjecture for a 2n + 1-dimension pseudo-Hermitian manifold M. In this case, there exists two universal constants C0 and C̃ depending on n only such that 2n ̃ hd (M) ≤ C0 d 2n+1 C . In addition, C̃ = 2n + 5 if the torsion vanishes.

Keywords: Generalized ultraspherical polynomials, polynomial growth harmonic functions, CR volume doubling property, Sobolev inequality and mean value inequality MSC 2020: Primary 32V05, Secondary 32V20, 53C56

1 Introduction In [14] and [32], S.-Y. Cheng and S.-T. Yau derived the well-known gradient estimate for positive harmonic functions and obtained the classical Liouville theorem, which states Acknowledgement: Der-Chen Chang is partially supported by an NSF grant DMS-1408839 and a McDevitt Endowment Fund at Georgetown University. Shu-Cheng Chang is partially supported in part by the MOST of Taiwan. Yingbo Han is partially supported by an NSFC grant 11971415, Nanhu Scholars Program for Young Scholars of Xinyang Normal University. Der-Chen Chang, Department of Mathematics and Statistics, Georgetown University, Washington, DC 20057-0001, USA; and Department of Mathematics, Fu Jen Catholic University, Taipei 24205, Taiwan, ROC, e-mail: [email protected] Shu-Cheng Chang, Department of Mathematics and Taida Institute for Mathematical Sciences (TIMS), National Taiwan University, Taipei 10617, Taiwan, ROC; and Mathematical Science Research Center, Chongqing University of Technology, Chongqing 400054, P.R. China, e-mail: [email protected] Yingbo Han, School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, Henan, P.R. China, e-mail: [email protected] Jingzhi Tie, Department of Mathematics, University of Georgia, Athens GA 30602-7403, USA, e-mail: [email protected] https://doi.org/10.1515/9783110792720-001

2 � D.-C. Chang et al. that any bounded harmonic function is constant in complete noncompact Riemannian manifolds with nonnegative Ricci curvature. Let ℋd (M m ) be the space of harmonic functions of polynomial growth of degree at most d in a complete noncompact Riemannian manifold M m . Yau conjectured that the dimension hd (M m ) of ℋd (M) is finite for each positive integer d and satisfies the estimate hd (M) ≤ hd (ℝm ). Colding and Minicozzi [15] affirmatively answered the first question and proved that hd (M) ≤ C0 d m−1 for manifolds of nonnegative Ricci curvature with C0 depending on the Neumann Poincaré inequality and the volume doubling constant. Later, Li [27] produced an elegant and shorter proof requiring only the manifold to satisfy the volume doubling property and the mean value inequality. For the latter question, the sharp upper bound estimate is still missing except for the special cases m = 2 or d = 1 obtained by Li–Tam [29, 30] and Kasue [24], and the rigidity part is only known for the special case d = 1 obtained by Li [26] and Cheeger–Colding–Minicozzi [13]. In [11], by modifying the arguments of Yau [32], Cheng–Yau [14], and Chang–Kuo–Lai [10], Chang et al. derived a subgradient estimate for positive pseudo-harmonic functions in a complete noncompact pseudo-Hermitian (2n + 1)-manifold (M, J, θ). This subgradient estimate can serve as the CR version of Yau’s gradient estimate. As an application of the subgradient estimate, the CR analog of Liouville-type theorem holds for positive pseudo-harmonic functions. In our paper [9], we studied the CR analog of Yau’s conjecture on the space ℋd (M) consisting of all pseudo-harmonic functions of polynomial growth of degree at most d in a complete noncompact pseudo-Hermitian (2n + 1)-manifold. We proved that the first part of CR Yau’s conjecture holds for pseudo-harmonic functions of polynomial growth via Li’s method, which is more adaptable to the case of pseudo-Hermitian geometry. We obtained the precise estimate of the dimension of the linear space of the pseudoHarmonic polynomials of degree at most d on the Heisenberg group. The main result in this note is the derivation of the exact formula of the dimension. The paper is organized as follows. We first derive the exact formula of hd (ℝm ). The result is classical and we include it here for completeness. Then we derive the exact formula for hd (Hn ). In [9], we did not have this exact formula and had to use complicated estimates to obtain the leading term of hd (Hn ). With the exact formula for hd (Hn ), we can see the leading term right away. We survey the main results of [9] and outline the main idea of their proofs in the last two sections.

Yau’s conjecture on the dimension of harmonic polynomials

� 3

2 Harmonic polynomials on ℝm We will start from ℝm , m

x = (x1 , x2 ⋅ ⋅ ⋅ , xm ) ∈ ℝ ,

m

|x| =

[∑ xj2 ] j=1

1 2

,

Δ=

𝜕2 𝜕2 𝜕2 + 2 + ⋅⋅⋅ + 2 . 2 𝜕xm 𝜕x1 𝜕x2

f is harmonic if Δf = 0. Define d

m

d

ℋ (ℝ ) = {f : Δf = 0 and |f | ≤ C0 |x| }.

One can prove that f ∈ ℋd (ℝm ) 󳨐⇒ Hence, if d < 2, then

𝜕f 𝜕xj

𝜕f ∈ ℋd−1 (ℝm ) 𝜕xj

must be a constant function for all xj . This implies f must be

a linear function spanned by the coordinate function xj , j = 1, 2, . . . , m and the constant functions. Therefore, dim(ℋd (ℝm )) := hd (ℝm ) = m + 1

for d < 2.

We know claim that any function f ∈ ℋd (ℝm ) must be a harmonic polynomial of degree at most d. To see this, we argue by induction and assume that this true for d ≥ 2. To prove this, it is also valid for d + 1. We consider f ∈ ℋd+1 (ℝm ), then 𝜕f ∈ ℋd (ℝm ) for 1 ≤ j ≤ m. 𝜕xj Then the induction hypothesis implies that 𝜕f = pj . 𝜕xj Here, pj is a polynomial with degree at most d. On other hand, x1

f (x1 , 0, . . . , 0) = ∫ p1 (t, 0, . . . , 0)dt + f (0, 0, . . . , 0). 0

Hence, f (x1 , 0, . . . , 0) is a polynomial at most degree d + 1 in x1 . Applying the above formula repeatedly, we have

4 � D.-C. Chang et al. xk

f (x1 , . . . , xk , 0, . . . , 0) = ∫ pk (x1 , . . . , xk−1 , t, 0, . . . , 0)dtk 0

+ f (x1 , . . . , xk−1 , 0 ⋅ ⋅ ⋅ , 0) We can argue inductively that f (x1 , . . . , xm ) is a polynomial of degree at most d + 1 and the claim is proved. Let dim(ℋd (ℝm )) := hd (ℝm ) and we will determine hd (ℝm ) for each d ∈ ℤ+ . We first introduce another space Vk (ℝm ), which is the vector space of homogeneous polynomials of degree k in ℝm , then each polynomial p ∈ Vk has the form p(x) = ∑ cα x α , |α|=k

where α = (α1 , α2 , . . . , αm ) ∈ ℕm , x = (x1 , x2 , . . . , xm ) ∈ ℝm , and α

αm cα x α = cα1 ,α2 ,...,αm x1 1 ⋅ ⋅ ⋅ xm

and

m

|α| = ∑ αj . j=1

Let σk,m be the dimension of Vk . Then σk,m is the number of m-tuples (α1 , . . . , αm ) with |α| = k. We can visualize the dimension as follows. Represent a given m-tuple (α1 , α2 , . . . , αm ) by a sequence of k dots and m − 1 vertical lines as in the example given below: ...|...|...|... There are α1 dots before the first line, α2 dots between the first line and the second line, and so on. Clearly, there is a one-to-one and onto correspondence between such sequences and the m-tuples. The total number of dots and lines is m + k − 1 and so the number of different arrangements of the dots and lines is σk,m = dim(Vk ) = (

m+k−1 m+k−1 )=( ). m−1 k

Next, let 𝒫 d (ℝm ) be the space of all polynomials of the degree at most d defined on ℝm and νd (ℝm ) = dim𝒫 d (ℝm ) be its dimension. We will determine νd (ℝm ) next. Any polynomial of degree up to d can be written as sum of polynomials pj (x) ∈ Vj for 0 ≤ j ≤ d, i. e., d

p(x) = ∑ pj (x) j=0

Hence, we have

with pj ∈ Vj .

Yau’s conjecture on the dimension of harmonic polynomials

d

m

d

m

d

d

j=0

j=0

� 5

m−1+j ). m−1

and νd (ℝm ) = ∑ σj,m = ∑ (

𝒫 (ℝ ) = ⨁ Vj (ℝ ) j=0

We apply a classical summation formula d

m−1+k m+d )=( ) m−1 m

(2.1)

∑(

k=0

to get d

m−1+j m+d )=( ). m−1 d

νd (ℝm ) = ∑ ( j=0

Finally, we will be able to find the number of independent harmonic polynomials of degree d in m variables. We start with the polynomials of homogeneous degree k. Let x = x1 and y = (x2 , . . . xm ). For each polynomial p ∈ Vk of homogeneous degree k, we can write it in the form k

p(x, y) = ∑ Ak−j (y)x j , j=0

where Ak−j (y) is homogeneous polynomial of degree k − j in y = (x2 , . . . , xn ). If furthermore we assume that p(x, y) is harmonic, apply the operator Δ=

m 𝜕2 𝜕2 𝜕2 + = + Δy ∑ 𝜕x12 j=2 𝜕xj2 𝜕x 2

to p(x, y) to get 0 = Δp = k

𝜕2 p + Δy p 𝜕x 2

k

= ∑ Ak−j (y)j(j − 1)x j−2 + ∑[Δy Ak−j (y)]x j j=2

j=0

k−2

= Δy A1 (y)x k−1 + Δy A0 (y)x k + ∑ [Ak−2−j (y)(j + 2)(j + 1) + Δy Ak−j (y)]x j j=0

k−2

= ∑ [Ak−2−j (y)(j + 2)(j + 1) + Δy Ak−j (y)]x j j=0

Note that since A0 (y) and A1 (y) are of degree 0 and 1, respectively, the first two terms vanish. Hence, we conclude −Ak−2−j (y)(j + 2)(j + 1) = Δy Ak−j (y) 󳨐⇒ Ak−2−j (y) = −

Δy Ak−j (y)

(j + 2)(j + 1)

,

6 � D.-C. Chang et al. for all 0 ≤ j ≤ k − 2. This gives an inductive formula for all the coefficients Aj (y) for 0 ≤ j ≤ k −2 once Ak (y) and Ak−1 (y) are fixed and arbitrary polynomials in y of homogeneous degree k and k − 1, respectively. Therefore, the number of linearly independent harmonic polynomials of homogeneous degree k in m variables is μk (ℝm ) = σk,m−1 + σk−1,m−1 . Finally, the dimension of the harmonic polynomials up to degree d is given by the sum as follows: d

d−1

k=0

j=0

hd (ℝm ) = ∑ μk (ℝm ) = σd,m−1 + 2 ∑ σj,m−1 m−1+d−1 ) + 2νd−1 (ℝm−1 ) d m−1+d−1 m−1+d−1 =( ) + 2( ) d d−1 m+d−2 m+d−2 m+d−2 )+( ) )+( =( d−1 d−1 d m+d−1 m+d−2 =( )+( ) = νd,m−1 + νd−1,m−1 d d−1 =(

Note that by Sterling’s formula hd (ℝm ) ≈

2 d m−1 . (m − 1)!

We will use the formula (2.1), d

m−1+k m+d )=( ) m−1 m

∑(

k=0

repeatedly, and we will give its proof here. We can prove it by induction with d. When d = 0, both sides equal 1. Assume it holds for d and we needs to show it holds for d + 1. Then d+1

d m−1+k m−1+k m−1+d+1 )= ∑( )+( ) m−1 m − 1 m−1 k=0

∑(

k=0

m+d m+d m+d+1 )+( )=( ) m m−1 m

=(

Here, we have used the classical formula

Yau’s conjecture on the dimension of harmonic polynomials

� 7

ℓ+1 ℓ ℓ )=( )+( ) m m m−1

( with ℓ = m + d.

3 The Heisenberg group The Heisenberg group Hn is the Lie group with underlying manifold ℂn × ℝ = {[z, t] : z ∈ ℂn , t ∈ ℝ} and multiplication law n

[z, t] ⋅ [w, s] = [z + w, t + s + 2 Im ∑ zj w̄ j ].

(3.1)

j=1

It is easy to check that the multiplication (3.1) does indeed make ℂn × ℝ into a group whose identity is the origin e = [0, 0], and where the inverse is given by [z, t]−1 = [−z, −t]. The Lie algebra hn of Hn is a vector space which, together with a Lie bracket operation defined on it, represents the infinitesimal action of Hn . Let hn denote the vector space of left-invariant vector fields on Hn . Note that this linear space is closed with respect to the bracket operation [V1 , V2 ] = V1 V2 − V2 V1 . The space hn , equipped with this bracket, is referred to as the Lie algebra of Hn . Lie algebra structure of hn is most readily understood by describing it in terms of the following basis: Xj =

𝜕 𝜕 + 2yj , 𝜕xj 𝜕t

Yj =

𝜕 𝜕 − 2xj 𝜕yj 𝜕t

and

T=

𝜕 ; 𝜕t

(3.2)

where j = 1, 2, . . . , n, z = (z1 , z2 , . . . , zn ) ∈ ℂn with zj = xj + iyj ; t ∈ ℝ. Note that we have the commutation relations [Yj , Xk ] = 4δjk T

for j, k = 1, 2, . . . , n.

(3.3)

Next, we define the complex vector fields 1 𝜕 𝜕 Z̄ j = (Xj + iYj ) = − izj ̄ 2 𝜕zj 𝜕t

1 𝜕 𝜕 Zj = (Xj − iYj ) = + iz̄j 2 𝜕zj 𝜕t

and (3.4)

8 � D.-C. Chang et al. for j = 1, 2, . . . , n. Here, as usual, 1 𝜕 𝜕 𝜕 = ( −i ) 𝜕zj 2 𝜕xj 𝜕yj

𝜕 1 𝜕 𝜕 = ( + i ). 𝜕z̄j 2 𝜕xj 𝜕yj

and

The commutation relations (3.3) then become [Z̄ j , Zk ] = 2iδjk T with all other commutators among the Zj , Z̄ k , and T vanishing. The Heisenberg sub-Laplacian is the differential operator 1 2

n

n

1 4

2

2

ℒγ = − ∑(Zj Z̄ j + Z̄ j Zj ) + γT = − ∑(Xj + Yj ) + iγT j=1

j=1

(3.5)

with Zj and Z̄ j given by (3.4). The operator ℒ2γ was first introduced by Folland and Stein [17] in the study of 𝜕̄b complex on a nondegenerate CR manifold. They found the fundamental solution of ℒ2γ . Gaveau [18] and Huanicki [23] derived the assoiated heat kernel. Greiner and Stein [22] studied ℒ2γ for a more gernal type CR manifold. Beals and Greiner [5] solved the non-isotropic case. We refer to [4], [6] for more details for the non-isotropic case.

4 Harmonics polynomials on Hn Greiner [20] [21] initiated the study of ℒγ -harmonic polynomials, i. e., ℒγ p = 0 on H1 .

He found a basis and proved that the linear space ℋm of H-homogeneous ℒγ -harmonic polynomials of degree m, m = 0, 1, 2, . . . has dimension m + 1. Dunkl [16] derived the general formulas of ℒγ -harmonic polynomials for the isotropic Heisenberg group Hn . (γ)

Below we will follow Dunkl’s formulation of ℋm . A basis of ℋm can be defined as fol(α,β) lows. First, we define the generalized ultraspherical polynomial Cν (z) of degree ν with the index (α, β) by its generating formula (γ)

(γ)

∞

(1 − ρz)̄ −α (1 − ρz)−β = ∑ r ν Cν(α,β) (z) for |ρz| < 1. ν=0

From the generating function (4.1), we have ν

Cν(α,β) (z) = ∑

j=0

(α)j (β)ν−j j!(ν − j)!

where the shifted factorial (a)n is defined by

z̄j zν−j ,

ν ∈ ℕ,

(4.1)

Yau’s conjecture on the dimension of harmonic polynomials

(a)0 = 1,

(a)j+1 = (a)j (a + j) = a(a + 1)(a + 2) ⋅ ⋅ ⋅ (a + j) =

� 9

Γ(a + j + 1) . Γ(a)

Then for k, l ∈ ℕ, we can define Vk,l to be the set of harmonic and homogeneous polynomials on ℂn of bidegree (k, l), i. e., p(z, z)̄ satisfies p(cz) = ck c̄l p(z) for ∀c ∈ ℂ,

and

n

∑ j=1

𝜕2 p =0 𝜕zj 𝜕z̄j

Vk,l is an irreducible U(n)-module of dimension μk,l =

k + l + n − 1 (n − 1)k (n − 1)l ⋅ . n−1 k!l!

For all p ∈ Vk,l , (α,β)

ℒγ ([p(z)Cν

(t + i|z|2 )]) = 0,

where α =

n−γ n+γ , β= , ν ∈ ℕ, 2 2

and every ℒγ -harmonic polynomial is a linear combination of such terms for all k, l, ν ∈ ℕ. (α,β) For p ∈ Vk,l , p(z)Cν (t + i|z|2 ) is H-homogeneous of degree 2ν + k + l. We will find (γ) the dimension of the linear space ℋm of H-homogeneous ℒγ -harmonic polynomials of degree m for n > 1 from the dimension of Vk,l . This question is to find how many triples (ν, k, l) ∈ ℕ3 so that 2ν + k + l = m for fixed m ∈ ℕ. Koranyi and Stanton [25] proved that if |γ| < n and a ℒγ -harmonic function f on Hn is majorized by a polynomial, then f must be a polynomial. We give an outline of the computations of the ℒγ -harmonic functions below. Because the space of ℒγ -harmonic polynomials is U(n)-module, it can be decomposed into copies of Vk,l , with k, l ∈ ℕ, so that every ℒγ -harmonic function is a sum of the terms like p(z)g1 (z, t) where p ∈ Vk,l and g1 is invariant under U(n). Hence, g1 (z, t) = g2 (|z|2 , t) for g2 (s, t). Furthermore, Greiner observed that g2 (|z|2 , t) = g(t + i|z|2 ) for some g. Then some simple calculations yield that for any p(z) ∈ Vk,l , ℒγ [p(z)g(t + i|z|2 )] = 0 if and only if g2 (t + i|z|2 ) satisfies ((ζ − ζ ̄ )

𝜕2

𝜕ζ 𝜕ζ ̄

− (α + l)

𝜕 𝜕 + (β + k) )g(ζ ) = 0, 𝜕ζ 𝜕ζ ̄

with α =

n−γ , 2

β=

n+γ . (4.2) 2

Equation (4.2) is derived from the change of variable ζ = t + i|z|2 . Polynomial solutions can be split by degree of homogeneity. Let ν

g(ζ ) = ∑ aj ζ j ζ ̄ n−j . j=0

Then (4.2) leads to the two term recurrence

10 � D.-C. Chang et al. (j + 1)(β + k + ν − j − 1)aj+1 − (ν − j)(α + l + j)aj = 0, which has unique solution aj = c

(α + l)j (β + k)ν−j j!(ν − j)!

,

for some constant c. This yields the polynomial solutions for g are arbitrary linear com(α,β) binations of Cν (ζ ) for ν ∈ ℕ. Equation (4.2) has an interesting nonpolynomial solution g(ζ ) = (c − ζ ̄ )−α−l (c − ζ )−β−k ,

c ∈ ℂ.

This can be verified by direct computations. This type solution is also not smooth. When n = 1, the dimension σm,1 of the linear space of ℒγ -harmonic polynomials of H-homogeneous degree m is m + 1. Then the dimension of ℒγ -harmonic polynomials of H-homogeneous degree less than or equal to m is 1 + 2 + 3 + ⋅ ⋅ ⋅ + (m + 1) =

(m + 1)(m + 2) m2 ≈ . 2 2

Next, we consider the case n ≥ 2. We first compute the dimension σm,n of the linear space of ℒγ -harmonic polynomials of H-homogeneous degree m. Since any ℒγ -harmonic

polynomial of H-homogeneous degree m is the linear combination of p(z)Cν with p(z) ∈ Vk,l and k, l, ν ∈ ℕ satisfying k + l + 2ν = m, we have

(α,β)

[ m2 ]

σm,n = ∑

[ m2 ]

∑

j=0 k+l=m−2j

μk,l = ∑

j=0

(n − 1)k (n − 1)l m − 2j + n − 1 ∑ n−1 k!l! k+l=m−2j

(t + i|z|2 )

(4.3)

where m

[

m m 2 ] = integer part of = { m−1 2 2 2

when m is even when m is odd

We will find the sum over k +l = p = m−2j first by applying a trick from the binomial formula for (1 − x)−(n−1) . First, we observe that (n − 1)n(n + 1) ⋅ (n − 2 + k) k ∞ (n − 1)k k x =∑ x . k! k! k=0 k=0 ∞

(1 − x)−(n−1) = ∑

This implies the sum over k + l = p = m − 2j in (4.3) is the coefficient of x p of the Taylor series of (1 − x)−2(n−1) , i. e., ∑

k+l=m−2j

(n − 1)k (n − 1)l (2n − 2)m−2j = . k!l! (m − 2j)!

Yau’s conjecture on the dimension of harmonic polynomials

� 11

This is because (1 − x)−(n−1) (1 − x)−(n−1) = (1 − x)−2(n−1) . The above formula is the result of the product formula for the Taylor series. Hence, we have [ m2 ]

σm,n = ∑

j=0

[ m2 ]

=∑

j=0

[ m2 ]

=∑

j=0

m − 2j + n − 1 (2n − 2)m−2j ⋅ n−1 (m − 2j)! m − 2j + n − 1 (2n − 2 + m − 2j − 1)! ⋅ n−1 (2n − 3)!(m − 2j)! (m − 2j) + n − 1 (m − 2j + 1)(m − 2j + 2) ⋅ ⋅ ⋅ (m − 2j + 2n − 3) ⋅ n−1 (2n − 3)!

In the above sum, we rewrite the first term as (m − 2j) + n − 1 = (m − 2j) + 2n − 2 − (n − 1) and the numerator as [(m − 2j) + n − 1][(m − 2j + 1)(m − 2j + 2) ⋅ ⋅ ⋅ (m − 2j + 2n − 3)]

= [(m − 2j) + 2n − 2 − (n − 1)][(m − 2j + 1)(m − 2j + 2) ⋅ ⋅ ⋅ (m − 2j + 2n − 3)] = [(m − 2j + 1)(m − 2j + 2) ⋅ ⋅ ⋅ (m − 2j + 2n − 2)]

− (n − 1)[(m − 2j + 1)(m − 2j + 2) ⋅ ⋅ ⋅ (m − 2j + 2n − 3)]

=

(m − 2j + 2n − 2)! (m − 2j + 2n − 3)! − (n − 1) (m − 2j)! (m − 2j)!

We can rewrite σm,n as [ m2 ]

σm,n = ∑

j=0

(m − 2j + 2n − 2)! (m − 2j + 2n − 3)! − (n − 1) ⋅ (2n − 3)!(m − 2j)! (2n − 3)!(m − 2j)!

[ m2 ]

m − 2j + 2n − 2 m − 2j + 2n − 3 = ∑ 2( )−( ) 2n − 2 2n − 3 j=0 Next, we apply the formula j+1 j j )−( )=( ) m m−1 m

( to obtain

m − 2j + 2n − 2 m − 2j + 2n − 3 m − 2j + 2n − 3 ( )−( )=( ) 2n − 2 2n − 3 2n − 2

12 � D.-C. Chang et al. This will yield that [ m2 ]

m − 2j + 2n − 2 m − 2j + 2n − 3 )+( ). 2n − 2 2n − 2

dm = ∑ ( j=0

This implies m

2m − 2j + 2n − 2 2m − 2j + 2n − 3 )+( ) 2n − 2 2n − 2

σ2m,n = ∑ ( j=0 m

2ℓ + 2n − 2 2ℓ + 2n − 3 )+( ) 2n − 2 2n − 2

= ∑( ℓ=0

m+n−1

2k 2k − 1 )+( ) 2n − 2 2n − 2

(let k = ℓ + n − 1)

= ∑ ( k=n−1

2m+2n−2

=

∑

k=2n−2

(let ℓ = m − j)

k ) 2n − 2

(

Similarly, we can compute σ2m+1,n to get σ2m+1,n =

2m+2n−1

∑

k=2n−2

k ) 2n − 2

(

We can conclude that σm,n =

m+2n−2

m k k + 2n − 2 )= ∑( ) ∑ ( 2n − 2 2n − 2 k=2n−2 k=0

For example, this will yield σ0,n = 1 and σ1,n = 1 + 2n − 1 = 2n, and σm,1 = m + 1. We apply the formula (2.1) p

m−1+k m+p )=( ) m−1 m

∑(

k=0

with p = m and m = 2n − 1 to get 2n − 1 + m ). 2n − 1

σm,n = (

Let hd (Hn ) be the dimension of the space of the harmonic polynomials of degree up to d. Then finally we have d

d

m=0

m=0

2n − 1 + m 2n + d (2n + d)! d 2n )=( )= ≈ . (2n)!d! (2n)! 2n − 1 2n

hd (Hn ) = ∑ σm,n = ∑ (

Yau’s conjecture on the dimension of harmonic polynomials

� 13

5 CR volume doubling property, Sobolev, and mean value inequalities In this section, we recall the CR curvature-dimension inequality and the heat kernel estimate in a pseudo-Hermitian (2n + 1)-manifold when the pseudo-Hermitian Ricci curvature tensor and the torsion tensor satisfy some specific conditions. Then we obtain the CR volume doubling property via the heat kernel estimate. Finally, by applying the volume doubling property and the heat kernel estimate, we prove the above CR Sobolev inequality. Then we obtain the mean value inequality. One of the key steps of Li–Yau’s proof of gradient estimates of the heat equation on a Riemannian manifold is the Bochner formula in terms of the Riemannian Ricci curvature tensors. In the CR analogue of the Li–Yau gradient estimate [12], the crucial step is the CR Bochner formula [19]. On the other hand, Bakry and Emery [2] pioneered the approach to generalize curvature in the context of gradient estimates by the so-called curvature-dimension inequality. We define the CR version of the curvature-dimension inequality, which was first introduced by Baudoin, Bonnefont, and Garofalo [3] in the context of sub-Riemannian geometry. Definition 5.1 ([9]). Let (M, J, θ) be a smooth pseudo-Hermitian (2n + 1)-manifold and a real frame {eβ , en+β , T} spanning the tangent space TM. For ρ1 ∈ ℝ, ρ2 > 0, d ≥ 0, and m > 0, we say that M satisfies the CR curvature-dimension inequality CD(ρ1 , ρ2 , d, m) if 1 d (Δ f )2 + (ρ1 − )Γ(f , f ) + ρ2 ΓZ (f , f ) ≤ Γ2 (f , f ) + νΓZ2 (f , f ) m b ν for any smooth function f ∈ C ∞ (M) and ν > 0. Here, we have Γ(f , f ) := ∑ |ej f |2 , j∈I2n

Z

Γ (f , f ) := |Tf |2 ,

1 Γ2 (f , f ) := [Δb (Γ(f , f )) − 2 ∑ (ej f )(ej Δb f )], 2 j∈I 2n

ΓZ2 (f , f )

1 := [Δb (ΓZ (f , f )) − 2(Tf )(TΔb f )]. 2

Note that we also have Γ2 (f , f ) = ∑ |ei ej f |2 + ∑ (ej f )([Δb , ej ]f ) i,j∈I2n

j∈I2n

and ΓZ2 (f , f ) = ∑ |ei Tf |2 + (Tf )([Δb , T]f ). i∈I2n

(5.1)

14 � D.-C. Chang et al. Now we proceed to derive a CR curvature-dimension inequality in a closed pseudoHermitian (2n + 1)-manifold under some specific assumptions on the pseudo-Hermitian Ricci curvature tensor and the torsion tensor. We will first prove the key lemma. Lemma 5.2 ([9]). Let (M, J, θ) be a complete pseudo-Hermitian (2n + 1)-manifold with (i) Ric(X, X) ≥ k0 ⟨X, X⟩Lθ , (ii)

sup |Aij,𝚤 |̄ 2 ≤ k2 < ∞,

sup |Aij | ≤ k1 < ∞ and

i,j∈In

i,j∈In

for X = X α Zα ∈ T1,0 M and k0 , k1 , k2 are constants with k1 , k2 ≥ 0. Then M satisfies the CR curvature-dimension inequality CD(ρ1 , ρ2 , 4, 2mn) for 1 < m < +∞, and N > 0 such that ρ1 := −κ and ρ2 :=

2 2 2 2n 8n3 N 2 2mn N k1 − − > 0, m ϵ m−1

for 0 < ν ≤ N and a positive constant ϵ with κ = 2[−k0 + |n − 2|k1 + ϵk2 ] ≥ 0. We now apply the CR curvature-dimension inequality CD as in Lemma 5.2 and the subgradient estimate in [3] for the semigroup solution u(x, t) = Pt f (x) of the heat flow, and prove the following crucial estimate for the symmetric heat kernel p(x, y, t) > 0 associated to the heat semigroup Pt . As usual, Vx (ρ) is the volume of the geodesic ball Bx (ρ) centered at x with radius ρ. We also refer to [7] and [8] for similar results for the general solution of the heat flow. Proposition 5.3 ([9]). Let (M, J, θ) be a complete pseudo-Hermitian (2n + 1)-manifold with Ric(X, X) ≥ k0 ⟨X, X⟩Lθ and sup |Aij | ≤ k1 < ∞ and

i,j∈In

sup |Aij,𝚤 |̄ 2 ≤ k2 < ∞,

i,j∈In

for X = X α Zα ∈ T1,0 M and k0 , k1 , k2 are constants with k1 , k2 ≥ 0. Then

Yau’s conjecture on the dimension of harmonic polynomials

� 15

(i) There exist positive constants C1 (ρ2 ), C2 (ρ2 ), C3 (ρ2 ) such that for x, y ∈ M, t > 0, p(x, y, t) ≤

C1 1 2

Vx (√t) Vy (√t)

1 2

exp(−C2

2 dcc (x, y) + C3 κt). t

(5.2)

(ii) There exist positive constants C4 (ρ2 ), C5 (ρ2 ), C6 (ρ2 ) such that for x, y ∈ M, t > 0, p(x, y, t) ≥

d 2 (x, y) C4 2 − C6 κ(t + dcc (x, y))]. exp[−C5 cc t Vx (√t)

(5.3)

(iii) There exist positive constants C7 (ρ2 ), C8 (ρ2 ) such that for 0 < s < t, C

p(x, x, s) t 7 ≤ ( ) eC8 κ(t−s) . p(x, x, t) s

(5.4)

Here, C7 = mn(1 + ρ6 ). ρ2 > 0 and κ = κ(k0 , k1 , k2 ) ≥ 0 are constants as in Lemma 5.2. 2 In addition, if the torsion is vanishing, C7 = n + 3

with ρ2 = 2n and m = 1.

As consequences of the previous estimate, we will derive the CR volume doubling property. Theorem 5.4 ([9]). Under the same hypothesis of Proposition 5.3, for any σ > 1, then there exist a positive constant C ′ such that ′ 2

Vx (σρ) ≤ C ′ σ 2C7 e(C σ

+C6 )κρ2

Vx (ρ).

(5.5)

The estimate (5.5) tells us that Vx (ρ) satisfies the volume doubling property. Let H B0 ,D (x, y, t) be the Dirichlet heat kernel on the geodesic ball B0 = Bx0 (r) with x, y ∈ Bx0 (r). Theorem 5.5 ([9]). Under the same hypothesis of Proposition 5.3, for r 2 < T, we have H B0 ,D (x, y, t) ≤ where Q = 3mn(1 +

C ′′ Q − Q2 C ′′ κr2 r t e , Vx0 (r)

6 ). ρ2

Finally, applying the volume doubling constant and upper bound estimate of the heat kernel, we can prove the CR analogue of the Sobolev inequality based on the method by Li [28] and Saloff-Coste [31]. Theorem 5.6 ([9]). Under the same hypothesis of Proposition 5.3, for any φ ∈ C0∞ (Bx (r)), x ∈ M, we have

16 � D.-C. Chang et al.

(

2Q 1 ∫ |φ| Q−2 dμ) Vx (r)

Bx (r)

2

≤ Cr 2 eCκr [

Q−2 Q

1 ( ∫ |∇b φ|2 dμ + r −2 ∫ φ2 dμ)], Vx (r) Bx (r)

where Q = 3mn(1 +

Bx (r)

6 ). ρ2

In the following, we apply the volume doubling estimate (5.5) and CR Sobolev inequality to obtain the following mean value inequality through the method of Moser’s iteration [31]. Theorem 5.7 ([9]). Under the same hypothesis of Proposition 5.3, then there exists a constant C > 0 such that for any ρ > 0, x ∈ M, and any nonnegative subpseudo-harmonic function f defined on M, it satisfies 2

2

[f (x)] ≤ CVx−1 (ρ)eCκρ

∫ f (y)2 dμ. Bx (ρ)

Since the mean value inequality holds for any ρ > 0 and x ∈ M, the right-hand side will induce the Morrey norm of f 2 (x). We refer interested readers to Chapter 17 of [1] for details.

6 Polynomial growth pseudoharmonic functions In this section, we will prove our main result. We first recall Lemma 28.3 of [28]. Lemma 6.1. Let K be a k-dimensional linear space of sections of a vector bundle E over M. Assume that M has polynomial volume growth at most of order μ, i. e., Vp (ρ) ≤ Cρμ for p ∈ M and ρ → ∞. Suppose each section u ∈ K are polynomial growth at most degree d, such that |u|(x) ≤ Cr d (x), where r(x) is the Carnot–Carathéodory distance to the fixed-point p ∈ M. For any β > 1, δ > 0, and ρ0 > 0, there exists ρ > ρ0 such that if {ui }ki=1 is an orthonormal basis of K with respect to the inner product Aβρ (u, v) = ∫B (βρ) ⟨u, v⟩dμ, then p

Yau’s conjecture on the dimension of harmonic polynomials

� 17

k

∑ ∫ |ui |2 dμ ≥ kβ−(2d+μ+δ) . i=1 B (ρ) p

In the following, we prove the main result by applying the volume double property and mean valued inequality. Theorem 6.2 ([9]). Assume the same hypothesis of Proposition 5.3 with κ = 0. Suppose E is a rank-m vector bundle over M. Let Sd (M, E) ⊂ Γ(E) be a linear subspace of sections of E satisfying (a) △b |u| ≥ 0, and (b) |u|(x) ≤ O(r d (x)) as r(x) → ∞, for all u ∈ Sd (M, E). Then the dimension of Sd (M, E) is finite. Moreover, there exists a constant C > 0 depending only on C7 such that 2n

dim Sd (M, E) ≤ mCCℳ d 2n+1 (2C7 −1)

(6.1)

for all d ≥ 1. Proof. From the volume double property, we have the comparison inequality Vp (ρ2 ) ≤ C ′ Vp (ρ1 )(

2C7

ρ2 ) ρ1

(6.2)

and then we have Vp (ρ) ≤ Cρ2C7 .

(6.3)

On the other hand, we also have the mean value inequality f 2 (x) ≤ Cℳ Vp−1 (ρ) ∫ f 2 (y)dμ. Bp (ρ)

Let K be a finite-dimensional linear subspace of Sd (M, E) with dim K = k and let {ui }ki=1 be any basis of K. Then for p ∈ M, ρ > 0 and any 0 < ϵ < 1, to complete the proof of the theorem, it suffices to show that k

∑ ∫ |ui |2 dμ ≤ mCCℳ i=1 B (ρ) p

sup

u∈{⟨A,U⟩}

∫

|u|2 dμ,

(6.4)

Bp ((1+ϵ)ρ)

where the supremum is taken over all u ∈ K of the form u = ⟨A, U⟩ for some unit vector A = (a1 , . . . , ak ) ∈ Rk with U = (u1 , . . . , uk ). We will prove (6.4) later. To complete the proof of Theorem 6.2, let {ui }ki=1 be an Aβρ -orthonormal basis of any finite-dimensional subspace K ⊂ Sd (M, E). By applying (6.3) and Lemma 6.1, there exists a ρ > 0 such that

18 � D.-C. Chang et al. k

∑ ∫ |ui |2 dμ ≥ kβ−(2d+2C7 +δ) .

(6.5)

i=1 B (ρ) p

Since ∫B((1+ϵ)ρ) |u|2 = 1 for all u ∈ {⟨A, U⟩}, it follows from the inequality (6.4) that by setting β = 1 + ϵ, we have

k

∑ ∫ |ui |2 dμ ≤ mCCℳ ϵ−(2C7 −1) . i=1 B (ρ) p

For d ≥ 1, setting 2n

ϵ = (2d)− 2n+1

(6.6)

combined with (6.5) gives us k

∑ ∫ |ui |2 dμ ≥ Ck. i=1 B (ρ) p

Therefore, the estimate (6.1) on k follows easily. Finally, we prove this claim (6.4) by following the method in [28]. For completeness, we will outline it here. We first observe that for any x ∈ Bp (ρ) there exists a subspace Kx ⊂ K, which is of at most codimension m, such that u(x) = 0 for all u ∈ Kx . Hence, by an orthonormal change of basis, we may assume that ui ∈ Kx for m + 1 ≤ i ≤ k and ∑ki=1 |ui |2 (x) = ∑ni=1 |ui |2 (x). Since △b |ui | ≥ 0, it follows from the CR mean valued inequality that k

n

i=1

i=1

∑ |ui |2 (x) = ∑ |ui |2 (x) m

≤ Cℳ Vx−1 ((1 + ϵ)ρ − r(x)) ∑

∫

i=1 B ((1+ϵ)ρ−r(x)) x

≤ Cℳ Vx−1 ((1 + ϵ)ρ − r(x)) sup

u∈{⟨A,U⟩}

|ui |2 dμ |u|2 dμ.

∫

(6.7)

Bx ((1+ϵ)ρ−r(x))

The volume double property (6.2) and the fact that r(x) ≤ ρ imply that C ′ Vp ((1 + ϵ)ρ − r(x)) ≥ Vp (2ρ)( ≥ Vp (ρ)(

2C7

(1 + ϵ)ρ − r(x) ) 2ρ

2C7

(1 + ϵ)ρ − r(x) ) 2ρ

.

(6.8)

Yau’s conjecture on the dimension of harmonic polynomials

� 19

From (6.7) and (6.8), we have k

∑ ∫ |ui |2 dμ ≤ i=1 B (ρ) p

mC ′ Cℳ 22C7 Vp (ρ)

sup

u∈{⟨A,U⟩}

u2 dμ ∫ (

∫ Bp ((1+ϵ)ρ)

Bp (ρ)

(1 + ϵ)ρ − r(x) ) ρ

−2C7

dμ. (6.9)

Now we define f (r) = ((1 + ϵ) − ρ−1 r)−2C7 . It follows that f ′ (r) ≥ 0 and then (1 + ϵ)ρ − r(x) ) ρ

−2C7

∫ ( Bp (ρ)

dμ

ρ

= ∫ Ap (t)f (t)dt 0 ρ

󵄨ρ = [f (t)Vp (t)]󵄨󵄨󵄨0 − ∫ f ′ (t)Vp (t)dt 0

ρ

󵄨ρ ≤ [f (t)Vp (t)]󵄨󵄨󵄨0 − ρ−2C7 Vp (ρ) ∫ f ′ (t)t 2C7 dt 0

ρ

󵄨ρ 󵄨ρ ≤ [f (t)Vp (t)]󵄨󵄨󵄨0 − ρ−2C7 Vp (ρ)([f (t)t 2C7 ]󵄨󵄨󵄨0 − 2C7 ∫ f (t)t 2C7 −1 dt) 0

ρ

≤ 2ρ−1 Vp (ρ)C7 ∫((1 + ϵ) − tρ−1 )

−2C7

dt

0

2C7 ≤ V (ρ)ϵ−(2C7 −1) . 2C7 − 1 p

(6.10)

Hence, (6.4) follows from (6.9) and (6.10). Corollary 6.3 ([9]). Under the same hypothesis of Theorem 6.2, we conclude that the dimension of ℋd (M) is finite. Moreover, there exists a constant C0 = C(Cℳ , C𝒱 ) > 0 such that 2n

hd (M) ≤ C0 d 2n+1 (2C7 −1) , for all d ≥ 1. Remark. From the result of Corollary 6.3, we know that the dimension hd (M) of ℋd (M) 2n is bounded above by C0 d 2n+1 (2C7 −1) . When the torsion is zero, then C7 = n + 3 by Proposition 5.3. Therefore, one has 2n+5

hd (M) ≤ C0 d (2n) 2n+1 .

20 � D.-C. Chang et al. 2n

d > 1, which is larger than hd (Hn ) ≈ (2n)! . Moreover, when M = Hn , the However, 2n+5 2n+1 topologically dimension m = 2n + 1 and our result shows that 2n = m − 1. In some sense, the bound for hd (Hn ) is sharp.

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Nageswari Shanmugalingam

On Carrasco Piaggio’s theorem characterizing quasisymmetric maps from compact doubling spaces to Ahlfors regular spaces Dedicated to the memory of David R. Adams

Abstract: In this note, we deconstruct and explore the components of a theorem of Carrasco Piaggio, which relates Ahlfors regular conformal gauge of a compact doubling metric space to weights on Gromov-hyperbolic fillings of the metric space. We consider a construction of hyperbolic filling that is simpler than the one considered by Carrasco Piaggio, and we determine the effect of each of the four properties postulated by Carrasco Piaggio on the induced metric on the compact metric space. Keywords: Gromov hyperbolic filling, uniformization, metric space, quasisymmetry, Ahlfors regular, uniformly perfect, conformal change in metric MSC 2020: Primary 30L05, Secondary 30L10, 51F30, 53C23

1 Introduction Within the class of metric spaces, those that are Gromov hyperbolic possess the properties of negative curvature at large scale but are not concerned with small-scale behavior; and as such, Gromov hyperbolicity is stable under biLipschitz changes in the metric (unlike Alexandrov curvature conditions). First proposed as a structure useful in the study of Cayley graphs of hyperbolic groups [13], the study of Gromov hyperbolic spaces was subsequently found to be useful in the study of potential theory [4]. It is also connected to the study of metric geometry, as there is a close connection between Gromov hyperbolic spaces and uniform domains [5], and between rough quasiisometries between Gromov hyperbolic spaces and quasisymmetries between their visual boundaries. It is this latter connection that is explored further in [11], and is based on the fact Acknowledgement: This material is motivated by the series of learning seminars during the author’s stay at the Mathematical Sciences Research Institute (MSRI, Berkeley, CA) while she was resident there as a member of the program Analysis and Geometry in Random Spaces, which is supported by the National Science Foundation (NSF USA) under Grant No. 1440140, during the Spring of 2022. The author thanks MSRI for its kind hospitality, and Mario Bonk, Mathav Murugan, and Pekka Pankka for valuable discussions on [11] and for comments that helped improve the exposition of the paper. The author’s work is partially supported by the NSF (USA) grant DMS #2054960. Nageswari Shanmugalingam, Department of Mathematical Sciences, University of Cincinnati, P. O. Box 210025, Cincinnati 45221-0025, OH, USA, e-mail: [email protected] https://doi.org/10.1515/9783110792720-002

24 � N. Shanmugalingam that every compact doubling metric space is the boundary of a Gromov hyperbolic space, called hyperbolic filling, of the space. Now there is extensive literature on various uses of hyperbolic filling, dating back to the seminal paper of Gromov [13, p. 95], and made explicit in [3, 6–9, 11, 17–21, 23]; these are merely a sampling of current literature on the topic of Gromov hyperbolicity and hyperbolic filling. During the author’s stay at MSRI, there was an extensive discussion of the paper [11] characterizing metrics on a compact space that are quasisymmetrically equivalent and at least one of them an Ahlfors regular metric. The results of [11] were of great interest to many participants at MSRI. However, the complicated system of parameters used there made it difficult to see the underlying beautiful ideas in [11]. The goal of the current note is to deconstruct the role of some of the parameters used there, and to eliminate others, thus providing a simplified expository discourse on parts of [11]. The focus is on [11, Theorem 1.1]. The following theorem is the result of exploring the role of each of the conditions (H1)–(H4) assumed in [11]. Theorem 1.1. Let (Z, dZ ) be a compact doubling metric space. Fixing α ≥ 2, and τ ≥ 2α2 +1, we choose a hyperbolic filling X of Z associated with the parameters α and τ as in Definition 2.1. I. Suppose that ρ : X → (0, 1), and consider the function dρ on X × X associated with ρ as in Definition 3.2. (a) If ρ satisfies Condition (H1) of Definition 3.1, then dρ is a metric on X, with (X, dρ ) a locally compact, noncomplete metric space. Let 𝜕ρ X := X \ X, with X the completion of X with respect to the metric dρ . (b) If ρ satisfies Conditions (H1) and (H3) of Definition 3.1, then there is a homeomorphism Φ : Z → 𝜕ρ X and positive constants c, C such that for every x, y ∈ Z we have cdZ (x, y)τ− ≤ dρ (Φ(x), Φ(y)) ≤ CdZ (x, y)τ+ with τ− :=

log(η− ) , log(1/α)

τ+ :=

log(η+ ) . log(1/α)

(c) If ρ satisfies Conditions (H1), (H2), and (H3) of Definition 3.1, then the map Φ is a quasisymmetry. (d) If ρ satisfies Conditions (H1), (H2), and (H3) of Definition 3.1 and Condition (H4) of Definition 6.1, then (𝜕ρ X, dρ ) is Ahlfors p-regular. II. Conversely, suppose that Z is CU -uniformly perfect for some CU > 2, and α > CU3 with τ ≥ max{α2 + 1, 2CU3 (CU2 − 4)−1 }. If θ is any metric on Z for which (Z, θ) is Ahlfors p-regular and is quasisymmetric to (Z, dZ ), then there exists a function ρ : X → (0, 1) that satisfies Conditions (H1), (H2), (H3), and (H4).

Carrasco’s theorem on quasisymmetric maps

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Remark 1.1. Note in the above theorem that in Part I we do not require (Z, dZ ) to be uniformly perfect; then Conditions (H1)–(H3) do not imply uniform perfectness either (and indeed, the choice of ρ as the constant function ρ(x) = 1/α satisfies Conditions (H1)–(H3) with the resulting quasisymmetry a bi-Lipschitz map; see [3], claim I(b) of Theorem 1.1 above, or Theorem 4.2 below); however, Conditions (H1)–(H4) together imply that (Z, dZ ) must be uniformly perfect. Thus I(a)–(c) on their own are not explicitly covered in [11], for Carrasco Piaggio [11] does explicitly require Z to be uniformly perfect (see [11, Section 2.1]). That assumption is also implicit in the four conditions together (see Lemma 6.2 below), and conversely, if (Z, d) is quasisymmetric to (Z, θ) with θ Ahlfors p-regular, then necessarily (Z, d) is uniformly perfect as well. Interestingly also, in [11, p. 507, (2.8)], Carrasco Piaggio requires τ ≥ 32 (there τ is denoted λ) and then requires α ≥ 6κ2 max{τ, CU } (with α denoted as a and CU denoted as KP in [11]). The parameter κ is an additional one associated with the construction of hyperbolic filling as given in [11]; with the simplified construction as considered in this note and in [3], we have κ = 1. Thus, in [11] the parameter α depends on the choice of λ and CU , but in our note τ depends on the choice of α while in Part II, both α and τ depend on CU as well. As pointed out above, when considering only the conditions (H1)–(H3), the metric space (Z, d) need not be uniformly perfect, but still the quasisymmetry Φ obtained in Section 5 is necessarily a power quasisymmetry. Since there are compact doubling metric spaces and quasisymmetries on them that are not power quasisymmetries (see, e. g., the discussion in [14]), it follows that not all quasisymmetries on a doubling space are obtained using the method of Carrasco Piaggio [11]. Section 2 is devoted to describing the construction of hyperbolic filling, and the last five sections of this note are devoted to the proof of the claims of the theorem. We choose to use the construction of hyperbolic filling from [3] for its simplicity in relation to the one used in [11]. While the construction in [11] (see also [21]) gives greater flexibility to the choice of sets and vertices, it is perhaps this very flexibility that makes it difficult to see what the effect of the conditions (H1)–(H4) are, and so we chose the simpler version given in [3]. However, the ideas and basic premises are as in [11]. In Section 3, the conditions (H1)–(H3) are discussed and I(a) of Theorem 1.1 is proved, while in Section 4 the claim I(b) of the theorem is verified. Section 5 is devoted to the proof of I(c) of Theorem 1.1, and the discussion in Section 6 completes the proof of the part I of Theorem 1.1. The focus of Section 7 is to prove part II of Theorem 1.1. In Section 8, we list a set of four conditions that parallel the conditions of Carrasco Piaggio [11], but couched from the perspective of densities on a metric space that lead to conformal changes in the metric. We end that section by posing a query regarding an Adams-type inequality [1, 2, 24], which is known to hold in the case that the function ρ is the constant function ρ(x) = 1/α.

26 � N. Shanmugalingam

2 Construction of hyperbolic filling Recall that a metric space (Z, dZ ) is metric doubling if there is a positive integer N such that for each z ∈ Z and r > 0, if A ⊂ B(z, r) such that dZ (x, y) ≥ r/2 whenever x, y ∈ A with x ≠ y, then there are at most N number of elements in A. In this note, (Z, dZ ) is a compact metric space, such that it is a metric doubling space. Later we will also assume that Z is uniformly perfect, that is, there is some CU > 1 such that for each z ∈ Z and 0 < r < diam(Z)/2, the annulus BdZ (z, r) \ BdZ (z, r/CU ) is nonempty; however, for now we do not need this assumption. We will, however, also assume that 0 < diam(Z) < 1 without loss of generality (as we are not interested in singleton metric spaces). Constructions of hyperbolic fillings of compact doubling metric spaces can be found, for example, in [3, 6–9, 11]. The version we give here is that of [3]. The obtained graph in this construction, when equipped with the path metric, is Gromov hyperbolic; however, this fact is not essential for the discussion in this note, as we turn the graph into a metric graph by adding unit interval edges to connect neighboring pairs of vertices and then use path integrals to directly obtain a metric on the graph; hence, its boundary can be realized via a metric completion rather than as the visual boundary of a Gromov hyperbolic space. For this reason, we do not devote space to discussing Gromov hyperbolicity here. We refer the interested reader to the discussion in [3, Section 3]. Definition 2.1. By a rescaling of the metric if necessary, we may assume without loss of generality that 0 < diam(Z) < 1. We fix α ≥ 2 and τ > 1, and for each nonnegative integer n, we set An to be a maximal α−n -separated subset of Z, that is, if z, w ∈ Z with z ≠ w, then dZ (z, w) ≥ α−n , and Z = ⋃w∈An BdZ (w, α−n ). We can, via an inductive construction, ensure that An ⊂ An+1 for each nonnegative integer n. We set V = ⋃∞ n=0 An × {n}. The set V is the vertex set of the metric graph X to be constructed next. We do this construction as follows. The vertex w0 = (x0 , 0), with x0 ∈ A0 , will play the role of a root of the graph. (1) Two vertices v1 = (z1 , n1 ), v2 = (z2 , n2 ) ∈ V are neighbors, denoted v1 ∼ v2 , if v1 ≠ v2 and either n1 = n2 with BdZ (z1 , τα−n1 ) ∩ BdZ (z2 , τα−n2 ) ≠ 0, or else n1 = n2 ± 1 and BdZ (z1 , α−n1 ) ∩ BdZ (z2 , α−n2 ) ≠ 0. (2) We turn V into a metric graph X by gluing a unit-length interval to each pair of neighboring vertices. (3) We call a vertex v2 = (z2 , n2 ) a child of a vertex v1 = (z1 , n1 ) if v1 ∼ v2 and n2 = n1 + 1; we also then say that the edge [v1 , v2 ] is a vertical edge. If [v1 , v2 ] is a vertical edge, then necessarily dZ (z1 , z2 ) < α−n1 + α−n2 , and so with n = min{n1 , n2 }, we have that dZ (z1 , z2 ) < α1−n (we use our choice of α ≥ 2 here). (4) If v1 ∼ v2 with n1 = n2 , then we say that the edge [v1 , v2 ] is a horizontal edge. In this case, we have that dZ (z1 , z2 ) < τα1−n1 . (5) We say that a point x ∈ X is a descendant of a point y ∈ X if there is a vertically descending path from y to x.

Carrasco’s theorem on quasisymmetric maps

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(6) A vertex v is said to be a common ancestor of two points x, y ∈ X if there are two vertically descending paths, one from v to x and the other from v to y. (7) Also, given a vertex v = (z, n) ∈ V , we set Π1 (v) = z and

Π2 (v) = n.

(8) If τ ≥ 1 + 1/α and (z, n), (x1 , n − 1), (x2 , n − 1) ∈ V such that (z, n) ∼ (xi , n − 1) for i = 1, 2, then (x1 , n − 1) ∼ (x2 , n − 1). (9) Thanks to the doubling property, there is a constant C ≥ 1, depending only on the doubling constant related to the metric doubling property of (Z, dZ ) and the choice of α, τ, such that for each positive integer n we have ∑x∈An χBd (x,τα−n ) ≤ C pointwise Z everywhere on Z. (10) Suppose that ⋅ ⋅ ⋅ ∼ (xn+1 , n + 1) ∼ (xn , n) ∼ (yn , n) ∼ (yn+1 , n + 1) ∼ ⋅ ⋅ ⋅ is a path in the graph, allowing for the possibility that xn = yn by a slight abuse of notation above. We see that for each k ≥ n, dZ (xk , xk+1 ) ≤ α−k + α−k−1 ≤ α1−k (we use the choice α ≥ 2 here). With similar estimates holding for d(yk , yk+1 ), we see that the two sequences (xk )k≥n and (yk )k≥n are Cauchy sequences in Z, converging to points denoted x and y, respectively. We see that then for each j ≥ n, ∞

dZ (x, xj ) ≤ ∑ α1−n = n=j

α2−j , α−1

with a similar estimate holding for dZ (y, yj ). Suppose that x ≠ y. With nxy a nonnegative integer such that α−nxy < dZ (x, y) ≤ α1−nxy , and j0 a nonnegative integer such that α−j0 < τ − 1 ≤ α1−j0 , we have that α−nxy < dZ (x, y) ≤ dZ (x, xn ) + dZ (xn , yn ) + dZ (yn , y) ≤

2α2−n + 2τα−n ≤ α3+j0 −n . α−1

It follows that n ≤ 3 + j0 + nxy .

(2.2)

(11) Given a vertex v = (x, n) ∈ V , there is a vertically descending geodesic ray w0 = v0 ∼ v1 ∼ ⋅ ⋅ ⋅ ∼ vk ∼ ⋅ ⋅ ⋅ with vk = v for each k ≥ n. This is done by choosing vk = (xk , k) for k = 1, . . . , n − 1 such that xk ∈ Ak with dZ (x, xk ) ≤ α−k . Note that A0 has only one point by our hypothesis that diam(Z) < 1. The vertex w0 = (x0 , 0) plays a distinguished role in the graph corresponding to x0 ∈ A0 . If z ∈ An+1 \ An , then by the maximality of An there is a point wz ∈ An such that dZ (z, wz ) < α−n , and so (z, n + 1) ∼ (wz , n); therefore, it is easy to see that X is path-connected. While

28 � N. Shanmugalingam this construction is not exactly the one considered in [11], it is in the spirit of [11] and is the one used in [3]. From [3, Theorem 3.4], we know that X is Gromov hyperbolic, with hyperbolicity constant depending solely on α and τ. The larger the choice of τ, the greater the number of horizontal edges. Since Z is doubling, each vertex v ∈ V has a uniformly bounded degree, with the upper bound on the degree depending solely on the doubling constant associated with ν and the parameters α and τ. Henceforth, we will fix α ≥ 2 and τ ≥ 1 + α1 . The condition on τ ensures that the conclusion of (8) above holds.

3 Weighted uniformization metric and three conditions Since diam(Z) > 0, the graph X, equipped with the path metric dX , is necessarily unbounded. In this section, we consider a family of uniformizations, each dampening the metric dX at locations far from the root vertex w0 , so that the dampened metric on X turns X into a bounded noncomplete metric space. The principal object of study in this note is the boundary of the damped space, as it is in [11]. Definition 3.1. We consider a function ρ : V → ℝ that satisfies the following conditions (using the labels from [11]): (H1) There exist 0 < η− ≤ η+ < 1 such that ρ : X → [η− , η+ ]. (H2) There is a constant K0 > 0 so that if v1 , v2 ∈ V with v1 ∼ v2 , and if w0 ∼ w1 ∼ ⋅ ⋅ ⋅ ∼ wk = v1 and w0 = u0 ∼ u1 ∼ ⋅ ⋅ ⋅ ∼ un = v2 are vertical edges, then k

n

j=0

j=0

π(v1 ) := ∏ ρ(wj ) ≤ K0 ∏ ρ(uj ) =: K0 π(v2 ). This also defines π : V → (0, ∞). We extend π to all of X by setting π(x) = tπ(v1 ) + (1 − t)π(v2 ) when x is a nonvertex point in the edge [v1 , v2 ], and t denotes the distance from x to the vertex v1 . (H3) There is a constant K1 > 0 satisfying the following condition. Whenever x, y ∈ X with x, y belonging to different edges of X, there are two vertically descending paths w0 = v0 ∼ v1 ∼ ⋅ ⋅ ⋅ ∼ vk , w0 = u0 ∼ u1 ∼ ⋅ ⋅ ⋅ ∼ un with x ∈ [vk−1 , vk ], y ∈ [uk−1 , uk ]. Let vxy denote the vertex in the path w0 = v0 ∼ v1 ∼ ⋅ ⋅ ⋅ ∼ vk with largest possible value of Π2 (vxy ) such that either vxy = uΠ2 (vxy ) or else vxy ∼ uΠ2 (vxy ) . For every path γ in X with endpoints x and y, we must have ∫ π(γ(t)) dt ≥ K1−1 π(vxy ). γ

Carrasco’s theorem on quasisymmetric maps

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Remark 3.1. Note that in Condition (H2), if we have v2 = v1 instead of v2 ∼ v1 = (xv , n), then k = n and necessarily dZ (Π1 (wn−1 ), Π2 (un−1 )) ≤ dZ (Π1 (wn−1 ), xv ) + dZ (xv , Π2 (un−1 )) ≤ 2[α−(n−1) + α−n ] ≤ 4α1−n .

It follows that if α ≥ 2 and τ ≥ 2α2 + 1 > 4, then wn−1 ∼ un−1 . Hence, from (H2) we have that π(v), up to the ambiguity of the multiplicative constant K0 , is well-defined in that the choice of the descending path used to define π(v) is not crucial. Remark 3.2. If x ∈ X, we can find paths = w0 = v0 ∼ v1 ∼ ⋅ ⋅ ⋅ in X so that for each positive integer n we have that Π1 (vn ) ∈ An with dZ (x, Π1 (vn )) < α−n . Let w0 ∼ w1 ∼ ⋅ ⋅ ⋅ be another such path associated with a point y ∈ X, and let vxy be the vertex point in the path {vn : n = 0, 1, . . .} that is a neighbor of wΠ2 (vxy ) such that Π2 (vxy ) be the largest possible (i. e., the latest common ancestor). Then from (H3) above, when γ is the concatenation of the curves from vxy to x and to y, respectively, via the sequences (vn )n≥Π2 (vxy ) , vxy ∼ wΠ2 (vxy ) , and (wn )n≥Π2 (vxy ) , we have that ∫ π(γ(t)) dt ≥ K1−1 π(vxy ); γ

see point (10) of Definition 2.1 above. We will use the weight π as the conformal density to modify the metric on the graph X from the path metric to the metric dρ . In [11], a fourth condition is also required, but we will not consider that condition until the penultimate section of this note. We postpone its definition to that section; see Definition 6.1 below. Definition 3.2. Let dρ : X × X → [0, ∞) be given as follows. For x, y ∈ X, we set dρ (x, y) = inf ∫ π(γ(t)) dt, γ

γ

where the infimum is over all paths γ in X with endpoints x and y. We only consider paths that are arc-length parametrized with respect to the graph metric dX . Lemma 3.3. Suppose that ρ satisfies Condition (H1). Then dρ is a metric on X. Moreover, (X, dρ ) is a locally compact, noncomplete metric space. Proof. Let x, y ∈ X with x ≠ y and γ be a curve in X with endpoints x and y. If x and y belong to the same edge [v1 , v2 ] in X, then any curve γ connecting x to y has to contain a subcurve of dX -length at least dX (x, y) that lies in the subgraph obtained by adding the edges that have either v1 or v2 as a vertex-endpoint. Hence, with n = max{Π2 (v1 ), Π2 (v2 )}, we have that

30 � N. Shanmugalingam ∫ π(γ(t)) dt ≥ ηn+1 − dX (x, y) > 0, γ

and taking the infimum over all curves γ gives dρ (x, y) ≥ ηn+1 − dX (x, y) > 0. Next, suppose that x and y belong to different edges. Then any curve γ connecting x to y has to have a subcurve of positive dX -length that passes through a vertex v ≠ x such that v is a neighbor of one of the two vertices that make up the edge x lies in. It follows that Π2 (v) ≤ nx + 1, with nx a positive integer that depends solely on x. Hence, n +1

∫ π(γ(t)) dt ≥ π(v)dX (x, v) ≥ η−x γ

dX (x, v) > 0.

n +1

Taking the infimum over all γ gives dρ (x, y) ≥ η−x dX (x, v) > 0. Thus, in both cases we have that if x ≠ y, then dρ (x, y) > 0. The triangle inequality and symmetry follow immediately from the definition of dρ , and so dρ is a metric on X. From the first paragraph of this proof, we know that for each vertex v, the subgraph made up of all the edges that have v as an endpoint is a compact subset of (X, dρ ), and moreover, v is in the dρ -interior of this subgraph. Hence, (X, dρ ) is locally compact. Finally, for each nonnegative integer n, we set wn = (x0 , n). Then w0 ∼ w1 ∼ ⋅ ⋅ ⋅ ∼ wn ∼ wn+1 ∼ ⋅ ⋅ ⋅, and as the edge [wn , wn+1 ] is a path connecting the two vertices wn and wn+1 , we see that dρ (wn , wn+1 ) ≤ ηn+ .

(3.4)

As 0 < η+ < 1, it follows that (wn )n is a Cauchy sequence in (X, dρ ). This sequence does not converge to any element in X. Therefore, (X, dρ ) is noncomplete. If ρ is the constant function ρ(x) = 1/α, where α (together with τ) is the parameter used in constructing the hyperbolic filling X of Z, then π(v) ≈ α−n where n = Π2 (v). Therefore, from [3, Proposition 4.4] we know that 𝜕ρ X =: X \ X is bi-Lipschitz equivalent to Z. Here, the completion X is taken with respect to the metric dρ . We will not need this information for our discussion in this note, and so we do not elaborate on this further but refer the interested reader to [3]. In Lemma 3.3, only (H1) played a role. In the next section, Conditions (H1) and (H3) together will play a key role, but Condition (H2) will not.

4 Bi-Hölder property Recall from the pervious section that (X, dρ ) is locally compact but not complete. We set 𝜕ρ X := X \ X, where X is the completion of X with respect to dρ . As X is locally compact with respect to dρ (see Lemma 3.3), it follows that X is an open subset of X.

Carrasco’s theorem on quasisymmetric maps

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As shown in [3, Proposition 4.1], if η− < 1/α, then there is no guarantee that 𝜕ρ X is even homeomorphic to Z; hence, if η− < 1/α, then Condition (H3) becomes vital in obtaining that 𝜕ρ X is homeomorphic to Z. We now construct a natural map Φ : Z → 𝜕ρ X as follows. Definition 4.1. For z ∈ Z and for each positive integer n, we can find vn ∈ V such that with xn = Π1 (vn ) ∈ An and Π2 (vn ) = n, with dZ (xn , z) < α−n . Note that then z ∈ BdZ (xn , α−n ) ∩ BdZ (xn+1 , α−(n+1) ), and so vn ∼ vn+1 , and hence w0 = v0 ∼ v1 ∼ ⋅ ⋅ ⋅ ∼ vn ∼ vn+1 ∼ ⋅ ⋅ ⋅ is a vertically descending path in X, with π(vn ) ≤ ηn+ . Hence, the sequence (vn )n is a Cauchy sequence in (X, dρ ), for we have that dρ (vn , vn+1 ) ≤ 2ηn+ ; see (3.4). We set Φ(x) to be the class of all Cauchy sequences in (X, dρ ) that are equivalent to this Cauchy sequence. To see that Φ is well-defined, suppose that yn ∈ An for each positive integer n such that dZ (x, yn ) < α−n . Then (yn , n) ∼ vn , because z ∈ BdZ (xn , α−n ) ∩ BdZ (yn , α−n ). As above, the sequence ((yn , n))n is also Cauchy with respect to the metric dρ , but also dρ (vn , (yn , n)) ≤ ηn+ + ηn+1 + , and so the two Cauchy sequences are equivalent with respect to the metric dρ . Thus Φ : Z → 𝜕ρ X is well-defined. Theorem 4.2. Suppose that ρ satisfies Conditions (H1) and (H3). Then Φ is a homeomorphism with C −1 dZ (x, y)τ− ≤ dρ (Φ(x), Φ(y)) ≤ C dZ (x, y)τ+

(4.3)

for each x, y ∈ Z, where τ− :=

log(η− ) , log(1/α)

τ+ :=

log(η+ ) . log(1/α)

Moreover, dρ (Φ(x), Φ(y)) ≈ π(vxy ). We remind the reader that the root of X is denoted w0 = (x0 , 0). Proof. Let j0 be the unique integer such that α−j0 < τ − 1 ≤ α1−j0 . We first aim to prove (4.3). Let x, y ∈ Z, and choose a positive integer nxy such that −nxy α < dZ (x, y) ≤ α1−nxy . We fix a path w0 = v0 ∼ v1 ∼ ⋅ ⋅ ⋅ such that for each nonnegative integer n we have that Π1 (vn ) ∈ An with dZ (Π1 (vn ), x) ≤ α−n . Let v0 = w0 ∼ w1 ∼ ⋅ ⋅ ⋅ be a corresponding choice of descending sequence with respect to y. We claim that for each nonnegative integer n with n ≤ nxy − j0 − 1, either vn = wn or vn ∼ wn . To this end, we assume that vn ≠ wn and 1 ≤ n ≤ nxy − j0 − 1, for otherwise there is nothing to prove. Since dZ (x, Π1 (vn )) ≤ α−n and dZ (y, Π1 (wn )) ≤ α−n , and as n ≤ nxy − j0 − 1, it follows that dZ (x, Π1 (wn )) ≤ α−n + α1−nxy ≤ α−n (1 + α−j0 ) < τα−n . Consequently,

32 � N. Shanmugalingam x ∈ BdZ (Π1 (vn ), α−n ) ∩ BdZ (Π1 (wn ), τα−n ), and so vn ∼ wn . Next, we claim that if n is a positive integer with vn ∼ wn , then n ≤ nxy + (−j0 )+ + 3. Indeed, we have that α−nxy < d(x, y) ≤ d(x, Π1 (vn )) + d(y, Π1 (wn )) + 2τα−n ≤ 2(1 + τ)α−n ≤ α1−n (1 + τ), with α1−n (1+τ) ≤ α1−n (2+α1−j0 ) ≤ α3−n if j0 ≥ 0, and α1−n (1+τ) ≤ α3−n−j0 if j0 < 0. From this, we obtain n + (−j0 )+ − 3 < nxy . As n and nxy are integers, it follows that n ≤ nxy + (−j0 )+ + 3. We now fix a choice of sequences vn , wn , n = 0, 1, . . . as above corresponding to the points x, y ∈ Z, and let F[x, y] denote the collection of all vertices vn for which vn ∼ wn or vn = wn . Let vxy be the vertex in F for which Π2 (vxy ) = max{Π2 (v) : v ∈ F[x, y]}. For symmetry’s sake, we also set wxy to be from the sequence corresponding to y such that wxy = wΠ2 (vxy ) . Recall that j0 is the integer such that α−j0 < τ − 1 ≤ α1−j0 . From the above argument, we see that nxy − |j0 | − 1 ≤ Π2 (vxy ) = Π2 (wxy ) ≤ nxy + |j0 | + 1,

(4.4)

and that either wxy = vxy or wxy ∼ vxy . The curve β given by the path ⋅ ⋅ ⋅ ∼ vn ∼ vn−1 ∼ ⋅ ⋅ ⋅ ∼ vxy ∼ wxy ∼ ⋅ ⋅ ⋅ ∼ wn−1 ∼ wn ∼ ⋅ ⋅ ⋅ has Φ(x) and Φ(y) as its endpoints, and so dρ (Φ(x), Φ(y)) ≤ ∫ π(β(t)) dt = π(vxy ) β

∞

∑ [

j=Π2 (vxy )

π(vi ) π(wi ) + ]. π(vxy ) π(wxy )

Note that for j ≥ Π2 (vxy ), j−Π2 (vxy )

η−

≤

π(vi ) j−Π (v ) ≤ η+ 2 xy , π(vxy )

and

j−Π2 (vxy )

η−

≤

π(wi ) j−Π (v ) ≤ η+ 2 xy . π(vxy )

Therefore, dρ (Φ(x), Φ(y)) ≤

2 π(vxy ). 1 − η+

On the other hand, by (H3) we have that for all curves γ in X that have Φ(x) and Φ(y) as their endpoints (with respect to the metric dρ ), ∫ π(γ(t)) dt ≥ K1−1 π(vxy ). γ

It follows that K1−1 π(vxy ) ≤ dρ (Φ(x), Φ(y)) ≤

2 π(vxy ). 1 − η+

(4.5)

Carrasco’s theorem on quasisymmetric maps

� 33

Finally, we note from (4.4) that n

n −1−|j0 |

η−xy ≤ π(vxy ) ≤ η+xy

.

Recall that we choose nxy so that α−nxy < d(x, y) ≤ α1−nxy . Now the definition of τ+ and τ− , together with the choice of nxy above, gives us the validity of (4.3) with constant C depending only on η− , η+ , and j0 (which in turn depends only on τ and α). The last claim of the theorem follows from (4.5). Note that Z is compact. Therefore, to prove that Φ is a homeomorphism, it now suffices to prove surjectivity of Φ. Let (wk )k be a Cauchy sequence in (X, dρ ) that is not convergent in (X, dρ ). By replacing wk with its nearest vertex if necessary, we may assume without loss of generality that each wk is in the vertex set V (for this change in the sequence gives us a Cauchy sequence that is equivalent to the original sequence). By passing to a subsequence if necessary, we may also assume that for each positive integer k, – dρ (wk , wk+1 ) < (K12 α)−k , – Π2 (wk ) < Π2 (wk+1 ). Indeed, if there is some positive integer n0 such that Π2 (vk ) ≤ n0 for each positive integer k, then the sequence lies in the dX -ball {w ∈ X : dX (w, w0 ) ≤ n0 } where dX is the graph metric on X (obtained by considering path metric with each edge in X to be of unit length). In this case, we would have from the proof of Lemma 3.3 that d and dρ are bi-Lipschitz on this ball, and hence (wk )k would be convergent to a point in this dX -ball with respect to dX , and hence with respect to dρ , violating our assumption that the sequence is not convergent in (X, dρ ). Thus the above two conditions can be met by choosing a subsequence. For positive integers k, we set xk = Π1 (wk ). Then by the compactness of Z, we have that there is some x∞ ∈ Z and a subsequence of the sequence (xk )k , also denoted (xk )k , such that xk → x∞ with respect to the metric dZ . For each positive integer n, we choose vn ∈ V such that dZ (Π1 (vn ), x∞ ) < α−n . As in the construction of Φ, we know that (vk )k is a Cauchy sequence with respect to dρ , and that Φ(x∞ ) = [(vn )n ]ρ (where [(vn )n ]ρ denotes the collection of all Cauchy sequences in (X, dρ ) that are equivalent to the Cauchy sequence (vn )n ). We now show that (wk )k ∈ [(vn )n ]ρ , for this would conclude the proof of surjectivity of Φ. Since dZ (xk , x∞ ) → 0 as k → ∞, for each positive integer n we can find kn > n such that dZ (x∞ , xkn ) < α−n−1 . Then by the choice of vk we have that dZ (Π2 (vn ), xkn ) < α1−n . Then with un a common ancestor of vn and wkn with the largest value of Π2 (un ), we have from Condition (H3) and (H1) that Π (un )

dρ (vn , wkn ) ≈ π(un ) ≤ η+ 2

→0

as n → ∞,

the last assertion above following from (4.4). It follows that (wkn )n and (vn )n are equivalent Cauchy sequences in (X, dρ ), completing the proof of surjectivity of Φ.

34 � N. Shanmugalingam

5 Quasisymmetry Recall that a homeomorphism Ψ : W → Y , with (W , dW ) and (Y , dY ) two metric spaces, is quasisymmetric if there is a homeomorphism η : (0, ∞) → (0, ∞) with limt→0+ η(t) = 0 such that for every triple of distinct points x1 , x2 , x3 ∈ W we have dY (Ψ(x1 ), Ψ(x2 )) d (x , x ) ≤ η( W 1 2 ). dY (Ψ(x1 ), Ψ(x3 )) dW (x1 , x3 ) Given that η is a homeomorphism, it can be seen that Ψ−1 is also a quasisymmetry if Ψ is. In the event that W (and hence Y ) is uniformly perfect, then η can be chosen to be a power function; there are constants C ≥ 1 and 0 < Θ ≤ 1 such that the following choice of η works: η(t) = C max{t Θ , t 1/Θ }. We refer the interested reader to the discussion on quasisymmetric and quasiconformal maps found in [14]. We will see in Lemma 6.2 in the next section that when ρ satisfies Conditions (H1) through (H4), Z is necessarily uniformly perfect. We do not assume Condition (H4) here, and so the space Z need not be uniformly perfect; however, the quasisymmetric maps we obtain are still of the above-mentioned power function format. In this section, we will focus on quasisymmetric aspects of the map Φ defined in the previous section. Here, Condition (H2) plays a vital role. Theorem 5.1. Suppose that ρ satisfies all three of the conditions (H1), (H2), and (H3). Then the map Φ: Z → 𝜕ρ X constructed in Definition 4.1 is a quasisymmetric map. Proof. Let x, y, z be three distinct points in Z. Then by Theorem 4.2, and in particular, by (4.5), we have that dρ (Φ(x), Φ(y)) dρ (Φ(x), Φ(z))

≈

π(vxy ) π(vxz )

.

Suppose first that Π2 (vxy ) ≥ Π2 (vxz ). Let γ be a descending path from the root vertex w0 to x, passing through vxy , and let β be a descending path from w0 to x, passing through vxz . Then there is a vertex w in the path γ such that Π2 (w) = Π2 (vxz ); it follows that x ∈ BdZ (Π1 (vxz ), α−Π2 (vx,z ) ) ∩ BdZ (Π1 (w), α−Π2 (w) ), and so w ∼ vxz . Therefore, by Condition (H2) and Remark 3.1, we have that π(vxz ) ≈ π(w) with comparison constant K0 . Let γ be the path w0 = v0 ∼ v1 ∼ ⋅ ⋅ ⋅ ∼ vxy . It follows that

Carrasco’s theorem on quasisymmetric maps

dρ (Φ(x), Φ(y)) dρ (Φ(x), Φ(z))

≈

� 35

π(vxy ) π(w)

Π2 (vxy )

=

∏ ρ(wj )

j=Π2 (vxz )

Π (vxy )−Π2 (vxz )

≤ η+ 2

= α−τ+ (Π2 (vxy )−Π2 (vxz )) . Now by (4.4), we see that dρ (Φ(x), Φ(y)) dρ (Φ(x), Φ(z))

τ

≲(

dZ (x, y) + ) . dZ (x, z)

Now suppose that Π2 (vxy ) ≤ Π2 (vxz ). Then, reversing the roles of y and z in the above argument gives us (with β = (w0 = u0 ∼ u1 ∼ ⋅ ⋅ ⋅) and u the vertex in β such that Π2 (u) = Π2 (vxy )), dρ (Φ(x), Φ(z))

dρ (Φ(x), Φ(y))

≈

π(vxz ) π(u) Π2 (vxz )

=

∏ ρ(uj )

j=Π2 (vxy )

Π (vxz )−Π2 (vxy )

≳ η− 2

= α−τ− (Π2 (vxz )−Π2 (vxy )) . Invoking (4.4) again, we see that dρ (Φ(x), Φ(z))

dρ (Φ(x), Φ(y))

τ

≳(

dZ (x, z) − ) , dZ (x, y)

≲(

dZ (x, y) − ) . dZ (x, z)

from whence we obtain dρ (Φ(x), Φ(y)) dρ (Φ(x), Φ(z))

τ

Thus Φ is η-quasisymmetric with η(t) ≈ max{t τ+ , t τ− }. Up to now, we have made use of Conditions (H1), (H2), and (H3). In the next section, we introduce and use Condition (H4).

36 � N. Shanmugalingam

6 Ahlfors regularity For each nonnegative integer m and x ∈ Am , and for each positive integer n with n > m, we set Dn ((x, m)) to be the collection of all vertices (y, n) ∈ V such that there is a vertically descending path from the vertex (x, m) to (y, n). Observe that such a path is a subpath of a vertically descending path from the root vertex w0 to (y, n). Definition 6.1. We say that ρ : V → ℝ satisfies Condition (H4) if there exist p > 0 and K2 > 0 such that whenever x ∈ Am and n > m, we have p

K2−1 π((x, m)) ≤

∑

v∈Dn (x,m)

p

π(v)p ≤ K2 π((x, m)) .

For the rest of this section, we consider the Condition (H4) in addition to the three conditions given in Definition 3.1. Lemma 6.2. Suppose that ρ satisfies Conditions (H1) through (H4). Then (Z, dZ ) is uniformly perfect, and for each x ∈ An , diamdρ Φ((BdZ (x, α−n ))) ≈ π((x, n)). Proof. To prove uniform perfectness, it suffices to show that there is some positive integer N > 1 such that for each positive integer n and each x ∈ An , the annulus BdZ (x, α−n ) \ BdZ (x, α−n−N ) is nonempty. To this end, suppose that N > 2 is an integer and x ∈ An such that the annulus BdZ (x, α−n ) \ BdZ (x, α−n−N ) is empty. Then from Condition (H4) we see that p

p

p

N−1

p

Np

p

π((x, n)) ≤ K2 π((x, n + N)) = K2 π((x, n)) ∏ ρ((x, n + j)) ≤ K2 η+ π((x, n)) . j=0

Np

It follows that K2 η+ ≥ 1. Hence, if N>

1 log(K2 ) , p log(1/η+ )

then the annulus BdZ (x, α−n ) \ BdZ (x, α−n−N ) must be nonempty. It follows that (Z, dZ ) is uniformly perfect, with uniform perfectness constant CU = αN where N satisfies the above inequality. The second claim now follows from the restriction on N given above as well. Indeed, we can find z ∈ BdZ (x, α−n ) such that dZ (x, z) ≥ α−n−N . With vxz as in Condition (H3), we see from (4.4) that α−Π2 (vxz ) ≈ α−nxz ≈ α−n and that the graph-distance between the vertices vxy and (x, n) is bounded by a constant that depends only on the constants η+ , η− , K0 , and K1 . By (H2) and (H1), we have that π((v, n)) ≈ π(vxz ). Now by the last claim of Theorem 4.2, we have that π(vxz ) ≈ dρ (Φ(x), Φ(z)) ≤ diamdρ (Φ(BdZ (x, α−n ))).

Carrasco’s theorem on quasisymmetric maps

� 37

Now if we choose w ∈ BdZ (x, α−n ) such that 1 diamdρ (Φ(BdZ (x, α−n ))) ≤ dρ (Φ(x), Φ(w)), 2 then as there is a vertically descending path from the root w0 , through (x, n), ending at Φ(w), it follows that vxw is a descendant of (x, n); it follows that π(vxw ) ≤ π((x, n)), and so by Theorem 4.2 again, 1 diamdρ (Φ(BdZ (x, α−n ))) ≤ dρ (Φ(x), Φ(w)) ≈ π(vxw ) ≤ π((x, n)) ≈ π(vxz ). 2 The combination of the above two inequalities yields the final claim of this lemma. From now on, we will denote Φ(x) by x as well whenever x ∈ Z; thus we will also conflate Φ(BdZ (x, α−n )) with BdZ (x, α−n ), as this will not lead to confusion. Remark 6.1. We fix 0 < l ≤ L < ∞. A set E ⊂ Z is said to be an (L, l)-quasiball in (Z, θ) with center x ∈ E if there is some ρ > 0 such that Bθ (x, lρ) ⊂ E ⊂ Bθ (x, Lρ). Now, for x ∈ An , we set r :=

sup

y∈BdZ (x,α−n )

θ(x, y),

τ :=

inf

y∈X\BdZ (x,α−n )

θ(x, y).

By the quasisymmetry of (Z, θ) with respect to (Z, dZ ), we see that r ≤ η(1)τ. If τ ≥ r, then we have that BdZ (x, α−n ) = Bθ (x, r), and so we can take l = L = 1 and ρ = r. If τ < r, then we have that τ < r ≤ η(1)τ, and so Bθ (x, τ) ⊂ BdZ (x, α−n ) ⊂ Bθ (x, r) ⊂ Bθ (x, η(1)τ), and we can then take ρ = τ and l = 1, L = max{1, η(1)}. Thus for each x ∈ An we have that BdZ (x, α−n ) is (1, max{1, η(1)})-quasiball in (Z, θ) with center x. Theorem 6.3. Suppose that ρ satisfies Conditions (H1), (H2), (H3), and (H4). Then (𝜕ρ X, dρ ) is Ahlfors p-regular. Proof. To prove the claim, we construct an Ahlfors p-regular measure on Z as a weak limit of a sequence of measures on Z. We fix a positive integer n and set the measure μn on Z as follows: for each Borel set E ⊂ Z, we set μn (E) :=

p

∑ π((x, n)) .

x∈An ∩E

Note that, thanks to Condition (H4), there is a relationship between μn and μm for n > m given by K2−1 μn (Z) ≤ μm (Z) ≤ CK2 μn (Z). Here, the constant C is the bounded overlap constant mentioned in Definition 2.1 (9). It follows that for each positive integer n,

38 � N. Shanmugalingam 0 < (CK2 )−1 μ1 (Z) ≤ μn (Z) ≤ K2 μ1 (Z) < ∞, and hence the sequence of measures (μn )n is tight on Z, and so there is a subsequence (μnk )k and a Radon measure μ on Z such that μnk converges weakly to μ; moreover, K2−1 μ1 (Z) ≤ μ(Z) ≤ K2 μ1 (Z). Thus μ is nontrivial on Z. Note also that for each x ∈ Z = 𝜕ρ X, ∞

dρ (w0 , x) ≤ sup ∫ π(γ(t)) dt ≤ ∑ ηn+ = γ

n=0

γ

1 < ∞. 1 − η+

We now wish to show that μ is Ahlfors p-regular on Z with respect to the metric dρ . Since Φ is a quasisymmetric map from (Z, dZ ) to (Z, dρ ), it follows that balls in the metric dZ are quasiballs in the metric dρ ; see Remark 6.1 above. Hence, it suffices to verify the regularity condition for dZ -balls. Note first that if n is a positive integer and z ∈ An , then μn (BdZ (z, α−n )) = π((z, n))p . We fix x ∈ Z and 0 < r < 21 diamdZ (Z), and choose the unique positive integer nr such that α−nr −1 < r ≤ α−nr . Then, for integers m > nr + 3, by the definition of μm we have μm (BdZ (x, r)) =

∑

z∈Am ∩BdZ (x,r)

p

π((z, m)) .

With z0 ∈ Anr +2 such that d(x, z0 ) < α−nr −2 , note that necessarily z0 ∈ BdZ (x, r). Moreover, for m ≥ nr + 3, if z ∈ Am such that there is a vertically descending path from (z0 , nr + 3) to (z, m), then d(z, z0 ) < α−nr −2 , and so d(x, z) < α−nr −2 + α−nr −2 < α−nr −1 ≤ r, that is, z ∈ Am ∩ BdZ (x, r). Hence, Π1 (Dm (z0 , nr + 3)) ⊂ BdZ (x, r), whence we obtain from Condition (H4) that μm (BdZ (x, r)) ≥

∑

v∈Dm (z0 ,nr +3)

π(v)p ≥ K2−1 π(z0 , nr + 3)p .

Next, let us consider two points z, w ∈ Am ∩ BdZ (x, r). Then d(z, w) < α1−nr . Let v0 ∼ v1 ∼ ⋅ ⋅ ⋅ ∼ vm = (z, m) and v0 ∼ v′1 ∼ ⋅ ⋅ ⋅ ∼ v′m = (w, m) be two vertically descending paths from the root vertex w0 to the vertices (z, m) and (w, m), respectively. Then as m ≥ nr +3, we can find (z′ , nr − 1) in the first path and (w′ , nr − 1) in the second path. We will show that (z′ , nr − 1) ∼ (w′ , nr − 1). Indeed, d(z′ , w) ≤ d(z′ , z) + d(z, w)
CU3 , but this requirement is not needed in the first lemma below. We reserve the notation B(x, r) to denote balls in Z, centered at z ∈ Z, of radius r with respect to the metric dZ . Balls with respect to the metric θ will be denoted Bθ (x, r).

40 � N. Shanmugalingam Lemma 7.1. Let x, y ∈ An such that (x, n) ∼ (y, n). Then diamθ (B(x, α−n )) ≈ diamθ (B(y, α−n )) with the constant of comparison given by 2η(1)η(2τCU ). Moreover, if x, z ∈ Z and nxz is the positive integer with α−nxz < dZ (x, z) ≤ α1−nxz , then diamθ (B(x, α−nxz )) ≈ θ(x, z), with comparison constant max{2η(1), η(CU α)}. Finally, if x ∈ An , then there exists y ∈ B(x, α−n ) such that diamθ (B(x, α−n )) ≈ θ(x, y) with comparison constant 2η(CU ). Proof. The claim in the first part of the lemma follows immediately if x = y, so we assume without loss of generality that x ≠ y. Then we have that α−n ≤ dZ (x, y) ≤ 2τα−n . Let x1 ∈ B(x, α−n ) and ŷ1 ∈ B(y, α−n ) such that diamθ (B(x, α−n )) ≤ 2θ(x, x1 ),

dZ (ŷ1 , y) ≥ α−n /CU .

Then by the quasisymmetry of the metric θ with respect to dZ , we see that θ(x1 , x) d (x , x) α−n ≤ η( Z 1 ) ≤ η( −n ) = η(1). θ(y, x) dZ (y, x) α Therefore, by the choice of x1 , we have that diamθ (B(x, α−n )) ≤ 2η(1)θ(y, x).

(7.2)

Again by the quasisymmetry, θ(y, x) 2τα−n ≤ η( −n ) = η(2τCU ), θ(ŷ1 , y) α /CU and so θ(y, x) ≤ η(2τCU )θ(ŷ1 , y) ≤ η(2τCU ) diamθ (B(y, α−n )).

(7.3)

Combining (7.2) with (7.3) gives diamθ (B(x, α−n )) ≤ 2η(1)η(2τCU ) diamθ (B(y, α−n )). The first part of the lemma is now proved by reversing the roles of x and y in the above argument.

Carrasco’s theorem on quasisymmetric maps

� 41

To prove the second part of the lemma, note that we can find x1 , x̂1 ∈ B(x, α−nxz ) such that dZ (x, x̂1 ) ≥ α−nxz /CU

and

diamθ (B(x, α−nxz )) ≤ 2θ(x1 , x).

Then θ(x, x1 ) α−nxz ≤ η( −n ) = η(1), θ(x, z) α xz and so diamθ (B(x, α−nxz )) ≤ 2η(1)θ(x, z). On the other hand, α1−nxz θ(x, z) ≤ η( −n ) = η(CU α). θ(x, x̂1 ) α xz /CU It follows that θ(x, z) ≤ θ(x, x̂1 ) ≤ diamθ (B(x, α−nxz )). η(CU α) The second claim of the lemma follows. To prove the final claim of the lemma, by the use of uniform perfectness we can find a point y ∈ B(x, α−n ) \ B(x, α−n /CU ). Let x̂ ∈ B(x, α−n ) such that θ(x̂, x) ≥ diamθ (B(x, α−n ))/2. Then α−n θ(x̂, x) ≤ η( −n ) = η(CU ). θ(y, x) α /CU It follows that 1 diamθ (B(x, α−n )) ≤ η(CU )θ(y, x) ≤ η(CU ) diamθ (B(x, α−n )). 2 Throughout the rest of this section, we also assume that CU > 2,

α > CU3 ,

τ ≥ max{α2 + 1,

2CU3

CU2 − 4

}.

(7.4)

The first of the above conditions can always be assumed without loss of generality, and the remaining two conditions merely give us control over the hyperbolic filling parameters α and τ in terms of CU . Note that these assumptions are independent of the quasisymmetric metric θ. We are now ready to construct the function ρ : V → [0, ∞) as follows.

42 � N. Shanmugalingam As in [11] and in the converse statement given in Theorem 1.1, we assume that (Z, θ) is also an Ahlfors p-regular space for some p > 0, and set μ to be the p-dimensional Hausdorff measure on Z induced by the metric θ. Definition 7.5. We fix a maximal spanning tree T ⊂ X of the graph X such that w0 is the root of the spanning tree made up solely of vertical edges, and so that if [v, w] is an edge in T with w a child of v, then dZ (Π1 (v), Π1 (w)) < α−Π2 (v) . We set ρ(w0 ) = μ(Z)1/p , and for each positive integer n and x ∈ An we set ρ((x, n)) = (

1/p

μ(B(x, τα−n )) ) μ(B(z, τα1−n ))

,

where z ∈ An−1 such that (z, n − 1) is the parent of (x, n) in T. While the definition of ρ uses the measure μ associated with the metric θ, the balls B(x, τα−n ) are with respect to the metric dZ . However, note that as θ is quasisymmetric with respect to dZ , balls with respect to the metric dZ are quasi-balls with respect to the metric θ, as seen in Remark 6.1. The reason for using the balls with respect to dZ is that we do not know what the θ-radius of corresponding balls should be. In [18, (2.11)] and [18, Proof of Theorem 3.14(b)], (see [18, (2.17)] for yet another variant) the authors propose an alternative construction of ρ in the absence of Ahlfors regularity: ρ((x, n)) :=

diamθ (B(x, α−n )) , diamθ (B(z, α1−n ))

where z ∈ An−1 such that (x, n) is a child of (z, n − 1) in T. However, their construction requires the choice of parameter α involved in the hyperbolic filling to be dependent on the quasisymmetry scaling function η in relation to the uniform perfectness constant CU , and so gives a slightly different result than that of [11]; see the comment at the top of page 33 of [18], where the uniform perfectness constant CU is denoted KP , τ is denoted by λ, and α is denoted by a. Lemma 7.6. The function ρ from Definition 7.5 satisfies Conditions (H1) of Definition 3.1. Proof. Let (x, n), (z, n − 1) ∈ V such that (x, n) ∼ (z, n − 1) in the tree T. Now, if w ∈ B(x, τα−n ), then by triangle inequality and (7.4), dZ (w, z) < τα−n + α−n + α1−n = (

τ τ+1 + 1)α1−n < α1−n . α 2

That is, B(x, τα−n ) ⊂ B(z, [1+ 1+τ ]α1−n ) ⊂ B(z, τα1−n ). By the uniform perfectness of (Z, dZ ) α τ 1−n we can find a point v0 ∈ B(z, 2 α ) \ B(z, 2Cτ α1−n ). We now show that U

B(v0 ,

τ 1−n α ) ⊂ B(z, τα1−n ) \ B(x, τα−n ). α3

Carrasco’s theorem on quasisymmetric maps

� 43

Indeed, if y ∈ B(v0 , ατ3 α1−n ), then by triangle inequality we have dZ (y, z)
α. Hence, μ(B(v0 , ατ3 α1−n )) μ(B(x, τα−n )) μ(B(z, τα1−n ) \ B(x, τα−n )) =1− ≤1− . μ(B(z, τα1−n )) μ(B(z, τα1−n )) μ(B(z, τα1−n )) As (Z, dZ ) is quasisymmetric to (Z, θ), we know that balls in the metric dZ are quasi-balls in the metric θ; see Remark 6.1 above. Hence, as in the proof of Lemma 7.1, by the Ahlfors regularity of μ with respect to theta, we have that μ(B(v0 , ατ3 α1−n )) μ(B(z, τα1−n ))

≥ c0

for some constant 0 < c0 < 1 that depends on the quasisymmetry function η and the Ahlfors regularity constant of μ, but does not depend on v0 , x, z, n. Hence, we have that ρ((x, n)) ≤ (1 − c0 )1/p . By the Ahlfors regularity of μ, we also have that ρ(x, n) ≥ c1 > 0 with c1 depending only on the quasisymmetry parameter η and the Ahlfors regularity constant of μ. Thus we can take η− = c1 and η+ = (1 − c0 )1/p to complete the proof. Lemma 7.7. The function ρ satisfies Conditions (H3) and (H2). Proof. From the definition of ρ, for each v ∈ V we have that 1/p

π(v) = μ(B(Π1 (v), α−Π2 (v) ))

.

Since dZ -balls are quasiballs in the metric θ (see Remark 6.1), we have by the Ahlfors p-regularity of μ with respect to θ that π(v) ≈ diamθ (B(Π1 (v), α−Π2 (v) )). Let u, w ∈ V be two distinct vertices, and let vuw ∈ V be as described in Condition (H3), and let γ be any curve in X with endpoints u, w. Denoting the vertices in γ by u = u0 ∼ u1 ∼ ⋅ ⋅ ⋅ ∼ uk = w, we have that k−1

k−1

j=0

j=0

∫ π(γ(t)) dt = ∑ π(uj ) ≈ ∑ diamθ (B(Π1 (uj ), τα−Π2 (uj ) )). γ

44 � N. Shanmugalingam For j = 0, . . . , k − 1, we have that B(Π1 (uj ), τα−Π2 (uj ) ) intersects B(Π1 (uj+1 ), τα−Π2 (uj+1 ) ); see the construction given in Definition 2.1. Hence, by the triangle inequality and the second part of Lemma 7.1, we have that ∫ π(γ(t)) dt ≳ θ(Π1 (u), Π1 (w)) ≈ diamθ (B(Π1 (vuw ), α−Π2 (vuw ) )), γ

from which the first claim of the lemma follows. Condition (H2) now follows from an application of the first part of Lemma 7.1. Lemma 7.8. The function ρ satisfies Condition (H4) given in Definition 6.1. Proof. Let m be a positive integer and x ∈ A. We fix a positive integer n such that n > m. Note that for each v ∈ Dn (x, m) we have that there is a vertically descending path (x, m) = v0 ∼ v1 ∼ ⋅ ⋅ ⋅ ∼ vm−n = v, and so we have that m−n

m−n

j=1

j=1

dZ (x, Π1 (v)) ≤ ∑ dZ (Π1 (vj−1 ), Π1 (vj )) ≤ ∑ α−m−j+1 + α−m−j ≤

2 −m α , α−1

and so we have that Π1 (Dn (x, m)) ⊂ B(x, Aα−m ), where A = 2/(α − 1). It follows from the pairwise disjointness property of the balls B(Π1 (v), α−n ) that ∑

v∈Dn (x,m)

π(v)p =

∑

v∈Dn (x,m)

μ(B(Π1 (v), α−n )) ≤ μ(B(x, (A + 1)α−m )) ≤ Cμ(B(x, α−m )) p

= Cπ((x, m)) . Here, we have used the fact that μ is Ahlfors regular, and also from the construction of ρ, for each z ∈ Am we have μ(B(z, α−m )) = π((z, m))p . On the other hand, for each y ∈ B(x, α−m /2) there exists z ∈ An such that dZ (z, y) ≤ α−n . It follows from the choice of α ≥ 2 that dZ (x, z) ≤ dZ (x, y) + dZ (y, z)
0 such that whenever v, w ∈ V with v ∼ w, we have ω(v) ≤ K0 ω(w). (H3-a) We extend ω to edges v ∼ w in X linearly by setting ω(tv + (1 − t)w) = tω(v) + (1 − t)ω(w), where tv + (1 − t)w is the point on the edge v ∼ w that is a distance t ∈ [0, 1] away from v. There is a constant K1 > 0 such that whenever γ is a curve in X connecting x, y ∈ X, then ∫ ω(γ(t)) ds ≥ K1 ω(vxy ). γ

(H4-a) There exist p > 0 and K2 > 0 such that whenever x ∈ Am and n > m, we have 1 p p ω((x, m)) ≤ ∑ ω(v)p ≤ K2 ω((x, m)) . K2 v∈D (x,m) n

ω(v) where w is any ancestor of v with w ∼ v, It is not difficult to see that setting ρ(v) = ω(w) we have the original four conditions with π ∼ ω. Indeed, Condition (H1-a) corresponds to Condition (H1) of [11], Condition (H2-a) corresponds to Condition (H2) of [11], Condition (H3-a) corresponds to Condition (H3) of [11], and Condition (H4-a) corresponds to

46 � N. Shanmugalingam Condition (H4) of [11]. This perspective allows us to see that dρ is actually a conformal change in the path-metric on the graph X. In this note, we chose to use the original formulation of the conditions as found in [11]; see Definition 3.1 and Definition 6.1, as the purpose of this note is to provide an analysis of [11, Theorem 1.1]. However, this perspective helps bridge the gap between the construction proposed in [11] and the conformal changes in metrics associated with a Harnack density ω : X → (0, ∞). A density ω is a Harnack density if there are constants C, A ≥ 1 such that for x, y ∈ X with d(x, y) < A we have 1 ω(x) ≤ ≤ C. C ω(y) Given such a density ω, we can equip the (not complete, but locally complete) metric space (X, d) with the new metric (X, dω ) given by dω (x, y) = inf ∫ ω ds, γ

γ

where x, y ∈ X and the infimum is over all rectifiable curves in X with endpoints x, y. The papers [4, 5, 10, 12, 15, 16] are some of the many papers in current literature using such transformations. Any density ω that satisfies the conditions listed at the beginning of this section is automatically a Harnack density, thanks to (H2-a). Concluding remarks The results of [11] link the quasisymmetric geometry of Z, the boundary of the hyperbolic filling, to the metrics on this filling. We note here that in potential theory as well there is a connection between nonlocal energy minimization problems on the boundary of compact doubling spaces and local energy minimization problems in the hyperbolic filling [10]; and this connection is given through the perspective of Adams inequality, on the compactification of the hyperbolic filling, via a measure supported on the boundary Z. In [3], it was shown that if Z is equipped with a doubling measure, then its hyperbolic filling, modified according to the density ωα (x) = α−dX (x,w0 ) , yields a uniform domain which can be equipped with a lift μω of the measure on Z so that the corresponding metric measure space Xα := (X, dωα , μωα ) is bounded, doubling, and supports a 1-Poincaré inequality, as does its metric completion (with the zero-extension of the measure μωα to 𝜕ωα X). Moreover, the trace of the Sobolev classes on Xα are certain Besov classes on Z. This fact was exploited in [10] to study the Neumann boundary value problem on Xα and link it to certain nonlocal fractional operators on Z, and one of the key motivating ideas behind that analysis was an Adams-type inequality [1, 2, 24], with the singular measure given by the doubling measure on Z = 𝜕ωα X. Such an inequality was possible because the measure on Z has a codimensional relationship with the measure μωα on X. If X is equipped with the metric dω corresponding to a general density function ω satisfying

Carrasco’s theorem on quasisymmetric maps

� 47

Conditions (H1-a)–(H3-a) and Z is equipped with a doubling measure, then it would be interesting to know whether it is possible to lift the measure on Z to X, so that a corresponding codimensional relationship between the lift and the measure on Z is valid and supports an Adams inequality, and would indicate a connection between the study done in [11] and nonlinear potential theory as in [22]. The author recently was able to prove the validity of Poincaré inequality, and as a consequence the Adams inequality (using [24]) under certain additional conditions on the parameters α and τ.

Bibliography [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

[11] [12] [13] [14] [15] [16] [17] [18] [19]

D. R. Adams, A trace inequality for generalized potentials. Stud. Math. 48 (1973), 99–105. D. R. Adams and L. Hedberg, Function spaces and potential theory. Grundlehren der Mathematischen Wissenschaften, 314. Springer, Berlin, 1996. A. Björn, J. Björn and N. Shanmugalingam, Extension and trace results for doubling metric measure spaces and their hyperbolic fillings. J. Math. Pures Appl. (9) 159 (2022), 196–249. A. Björn, J. Björn and N. Shanmugalingam, Bounded geometry and p-harmonic functions under uniformization and hyperbolization. J. Geom. Anal. 31(5) (2021), 5259–5308. M. Bonk, J. Heinonen and P. Koskela, Uniformizing Gromov hyperbolic spaces. Astérisque, 270, 2001, viii+99 pp. M. Bonk and E. Saksman, Sobolev spaces and hyperbolic fillings. J. Reine Angew. Math. 737 (2018), 161–187. M. Bonk and O. Schramm, Embeddings of Gromov hyperbolic spaces. Geom. Funct. Anal. 10(2) (2000), 266–306. M. Bourdon and H. Pajot, Cohomologie ℓp et espaces de Besov. J. Reine Angew. Math. 558 (2003), 85–108. S. Buyalo and V. Schroeder, Elements of asymptotic geometry. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich, 2007. L. Capogna, J. Kline, R. Korte, N. Shanmugalingam and M. Snipes, Neumann problems for p-harmonic functions, and induced nonlocal operators in metric measure spaces. Preprint 2022. https://arxiv.org/pdf/ 2204.00571.pdf. M. Carrasco Piaggio, On the conformal gauge of a compact metric space. Ann. Sci. Éc. Norm. Supér. (4) 46(3) (2013), 495–548. R. Gibara and N. Shanmugalingam, Conformal transformation of uniform domains under weights that depend on distance to the boundary. Preprint 2022. https://arxiv.org/pdf/2204.01920.pdf. M. Gromov, Hyperbolic groups. Essays in group theory, 75–263. Math. Sci. Res. Inst. Publ., 8. Springer, New York, 1987. J. Heinonen, Lecture notes on analysis in metric spaces. Springer, New York, 2001. D. Herron, Universal convexity for quasihyperbolic type metrics. Conform. Geom. Dyn. 20 (2016), 1–24. D. Herron and D. Minda, Comparing invariant distances and conformal metrics on Riemann surfaces. Isr. J. Math. 122 (2001), 207–220. Z. Ibragimov, A hyperbolic filling for ultrametric spaces. Comput. Methods Funct. Theory 14(2–3) (2014), 315–329. N. Kajino and M. Murugan, On the conformal walk dimension: Quasisymmetric uniformization for symmetric diffusions. Preprint 2021. https://arxiv.org/pdf/2008.12836.pdf. M. Kesseböhmer, T. Samuel and K. Sender, The Sierpiński gasket as the Martin boundary of a non-isotropic Markov chain. J. Fractal Geom. 7(2) (2020), 113–136.

48 � N. Shanmugalingam

[20] J. Kigami, Transitions on a noncompact Cantor set and random walks on its defining tree. Ann. Inst. Henri Poincaré Probab. Stat. 49(4) (2013), 1090–1129. [21] J. Kigami, Geometry and analysis of metric spaces via weighted partitions. Lecture Notes in Mathematics, 2265. Springer, Cham, 2020, viii+162 pp. [22] J. Lehrbäck and N. Shanmugalingam, Potential theory and fractional Laplacian on compact doubling metric measure spaces, with application to quasisymmetry (in preparation). https://arxiv.org/abs/2210. 01095. [23] J. Lindquist, Weak capacity and modulus comparability in Ahlfors regular metric spaces. Anal. Geom. Metric Spaces 4(1) (2016), 399–424. [24] T. Mäkeläinen, Adams inequality on metric measure spaces. Rev. Mat. Iberoam. 25(2) (2009), 533–558.

Tuoc Phan

On trace theorems for weighted mixed-norm Sobolev spaces and applications Dedicated to the memory of David R. Adams

Abstract: We prove trace theorems for weighted, mixed-norm, Sobolev spaces in the upper-half space where the weight is a power function of the vertical variable. The results show the differentiability order of the trace functions depends only on the power in the weight function and the integrability power for the integration with respect to the vertical variable but not on the integrability powers for the integration with respect to the horizontal ones. They recover classical results in the case of unmixed-norm spaces. The work is motivated by the study of regularity theory for solutions of elliptic and parabolic equations with anisotropic features and with nonhomogeneous boundary conditions. The results provide an essential ingredient to the study of fractional elliptic and parabolic equations in divergence form with measurable coefficients. Keywords: Trace theorems, weighted mixed-norm Sobolev spaces, mixed-norm Besov spaces, boundary value problems, singular and degenerate coefficients MSC 2020: Primary 46E35, Secondary 35J25, 35J70, 35J75

1 Introduction and main results 1.1 Introduction We study traces of functions in weighted mixed-norm Sobolev spaces. To put the study into perspective, let us quickly recall a relevant classical result. For a given open nonempty bounded domain Ω ⊂ ℝd with d ∈ ℕ, let Q = Ω × (0, 1). For p ∈ [1, ∞) and α ∈ ℝ, we denote Lp (Q, μ) the weighted Lebesgue space consisting of measurable functions f : Q → ℝ such that 1

1 p

󵄨 󵄨p ‖f ‖Lp (Q,μ) = [∫ ∫󵄨󵄨󵄨f (x, y)󵄨󵄨󵄨 yα dxdy] < ∞ 0 Ω

where μ( y) = yα for y ∈ ℝ+ . As usual, the weighted Sobolev space Wp1 (Q, μ) is defined by Acknowledgement: The author was partially supported by the Simons Foundation, grant no. 354889. Tuoc Phan, Department of Mathematics, University of Tennessee, Knoxville, 227 Ayres Hall, 1403 Circle Drive, Knoxville 37996, TN, USA, e-mail: [email protected] https://doi.org/10.1515/9783110792720-003

50 � T. Phan Wp1 (Q, μ) = {u ∈ Lp (Q, μ) : Du ∈ Lp (Q, μ)}, where Du = (Dx u, Dy u) denotes the gradient of u in the weak sense, and Wp1 (Q, μ) is endowed with the norm ‖u‖Wp1 (Q,μ) = ‖u‖Lp (Q,μ) + ‖Du‖Lp (Q,μ) . On the other hand, for s ∈ (0, 1) and p ∈ [1, ∞), a function v ∈ Lp (Ω) is said in the Sobolev space Wps (Ω) if 1

‖v‖Wps (Ω)

p |v(x) − v(z)|p dxdz) < ∞. = ‖v‖Lp (Ω) + (∫ ∫ d+ps |x − z|

Ω Ω

The following trace theorem for weighted Sobolev space Wp1 (Q, μ) is classical; see [21, Theorem 2.8], for example. See also [14, Theorem 2.10] for a more general domains and general cases, [20, Theorem 2.8, p. 100] for the unweighted case (i. e., α = 0), and [1] for further discussion on trace inequalities. Theorem 1.1. Let Ω ⊂ ℝd be a bounded nonempty set with boundary 𝜕Ω ∈ C 1 . Also, let p ∈ [1, ∞), and α ∈ (−1, p − 1). Then there exists a unique linear trace operator 1− 1+α p

T : Wp1 (Q, μ) → Wp

(Ω)

such that Tu(⋅) = u(⋅, 0) in Ω for u ∈ C(Ω × [0, 1)) ∩ Wp1 (Q, μ). Moreover, we have ‖Tu‖

1− 1+α p

Wp

(Ω)

≤ N‖u‖Wp1 (Q,μ) ,

for every u ∈ Wp1 (Q, μ), where μ( y) = yα , N = N(α, d, p) > 0, and Q = Ω × (0, 1). The main theme of this paper is to investigate the traces of functions in weighted 1 d+1 1 mixed-norm spaces Wp,q ⃗ (ℝ+ , μ) and Wp,q (Q, μ), which are defined below, where p⃗ ∈

[1, ∞)d and p are the powers in the integrability with respect to the horizontal variables, and q ∈ [1, ∞) is the power in the integrability with respect to the vertical variable. One obvious motivation is to understand the analysis properties of weighted mixed-norm Sobolev spaces and extend the classical result, Theorem 1.1, to the anisotropic setting. Another and more important motivation is to provide an essential ingredient for the study of partial differential equations with anisotropic structures and with nonhomogeneous boundary conditions. Regarding the partial differential equations with anisotropic features, one example is a class of elliptic and parabolic equations studied in [9–11] in which coefficients are singular or degenerate in one variable direction as in (1.3) below. Existence, uniqueness, and regularity estimates of solutions in weighted mixed-norm spaces are proved

On trace theorems for weighted mixed-norm Sobolev spaces and applications

� 51

in [9–11, 18, 24] for the problems with homogeneous boundary conditions. Using the developed trace theorem (Theorem 1.2 below), we extend [9, Theorem 2.8] to the study of nonhomogeneous boundary value problem (1.3), and our result (Corollary 1.4 below) gives the optimal regularity conditions on the boundary data for the problem. In this research line, interested readers can find [17, 29] for other results related to equations with singular-degenerate coefficients, and also [13, 15, 16] for interesting results on elliptic and 1 d+1 parabolic equations in weighted spaces similar to the space Wp,q ⃗ (ℝ+ , μ) considered in this paper. Next, we give another example about the partial differential equations with anisotropic structures. In [19], a class fractional elliptic equations with measurable coefficients in divergence form in bounded domain was studied. The main idea in [19] was to convert the nonlocal problem into the local one using the extension operator introduced in [7]. To this end, [19] developed regularity theory for a class of equations with the anisotropic structures as coefficients are singular or degenerate in one variable direction. In one of its main results, [19, Theorem 2.1], the trace Theorem 1.1 plays a central role in the establishment of the regularity estimates of solutions. Because of this, and due to the anisotropic properties due to the extension problem, it is clearly expected that [19, Theorem 2.1] is not optimal. In this paper, we establish the trace theorem for 1 the anisotropic weighted Sobolev space Wp,q (Q, μ), which will provide us an essential ingredient to derive the optimal regularity result for solutions of the class of fractional elliptic equations studied in [19]. As this application is comprehensive, the work will be carried out in our forthcoming paper. See also [6] for another interesting work in which Theorem 1.1 is useful, and also [7] for well-known results on equations with fractional Laplacian. Finally, we would like to point out that the interest in this work is also closely related to the recent work [4, 12] in which the traces of functions in unweighted mixednorm Lizorkin–Triebel spaces were studied; see also [2, 27] for other results on traces of functions in Sobolev spaces with radial weights and with Muckenhoupt weights. Also, note that from Theorem 1.2 and Corollary 1.5, the differentiability order of the traces of 1 d+1 1 functions in Wp,q ⃗ (ℝ+ , μ) or in Wp,q (Q, μ) only depends on the weight power α and the integrability power q of the vertical variable, but not on horizontal integrability powers p⃗ or p, respectively. This phenomena is not easily recognized in Theorem 1.1 and to the best of our knowledge, it seems to be newly discovered in this paper, even in the unweighted case with α = 0; see Remark 1.3 below for more details.

1.2 Main results Let us begin with recalling the definitions of the mixed-norm and weighted mixed-norm spaces. For each p⃗ = (p1 , p2 , . . . , pd ) ∈ [1, ∞)d , the mixed-norm Lebesgue space Lp⃗ (ℝd ) is defined to be the space consisting of all measurable functions f : ℝd → ℝ such that its Lp⃗ (ℝd )-norm,

52 � T. Phan p3

p2

‖f ‖Lp⃗ (ℝd )

1

p1 p2 pd 󵄨p 󵄨 = (∫ . . . (∫(∫󵄨󵄨󵄨f (x1 , x2 , . . . , xd )󵄨󵄨󵄨 1 dx1 ) dx2 ) . . . dxd )

ℝ

ℝ

ℝ

is finite. Similar definitions can be formed when pk = ∞ for some k = 1, 2, . . . , d. The mixed-norm Lebesgue space Lp⃗ (ℝd ) was introduced in [3] to capture the anisotropic behaviors of functions. Clearly, if p1 = p2 = ⋅ ⋅ ⋅ = pd = p, then Lp⃗ (ℝd ) = Lp (ℝd ), where the later is the usual Lebesgue space. However, in general, there is not a clear relation between the two functional spaces. For example, for a given p⃗ = (p1 , p2 , . . . , pd ) ∈ [1, ∞)d , and for fk ∈ Lpk (ℝ) for k = 1, 2, . . . , d, we observe that f ∈ Lp⃗ (ℝd ), where f (x) = f1 (x1 )f2 (x2 ) . . . fd (xd ),

x = (x1 , x2 , . . . , xd ) ∈ ℝd .

However, there may not exist any p ∈ [1, ∞) such that f ∈ Lp (ℝd ). See [1] for more discussion on mixed-norm spaces and mixed-norm Sobolev inequality. Next, let us denote ℝ+ = (0, ∞) and ℝd+1 = ℝd × ℝ+ . For p⃗ ∈ [1, ∞]d , α ∈ ℝ, and + d+1 q ∈ [1, ∞), the weighted mixed-norm Lebesgue space Lp,q ⃗ (ℝ+ , μ) is defined by d+1 d+1 Lp,q d+1 < ∞} ⃗ (ℝ+ , μ) = {f : ℝ+ → ℝ : ‖f ‖Lp,q ⃗ (ℝ+ )

where μ( y) = yα for y ∈ ℝ+ and 1/q

󵄩 󵄩q ‖f ‖Lp,q⃗ (ℝd+1 = ( ∫ 󵄩󵄩󵄩f (⋅, y)󵄩󵄩󵄩L + ,μ)

p⃗

yα dy) (ℝd )

.

ℝ+

A similar definition is easily formulated when q = ∞. As usual, we define the weighted 1 d+1 mixed-norm Sobolev space Wp,q ⃗ (ℝ+ , μ) to be the space consisting of functions f ∈

d+1 d+1 d+1 Lp,q , and ⃗ (ℝ+ , μ) such that their weak derivatives Df = (Dx f , Dy f ) ∈ Lp,q ⃗ (ℝ+ , μ) it is endowed with the norm

‖f ‖W 1

⃗ p,q

(ℝd+1 + ,μ)

= ‖f ‖Lp,q⃗ (ℝd+1 + ‖Df ‖Lp,q⃗ (ℝd+1 . + ,μ) + ,μ)

Now, we recall the definition of mixed-norm Besov spaces. For each h ∈ ℝd and for f : ℝd → ℝ, let us define the difference Δh f by x ∈ ℝd .

Δh f (x) = f (x + h) − f (x),

ℓ d For p⃗ ∈ [1, ∞]d , q ∈ [1, ∞) and ℓ ∈ (0, 1), the mixed-norm Besov space Bp,q ⃗ (ℝ ) is the

space consisting of all f ∈ Lp⃗ (ℝd ) such that its norm ‖f ‖Bℓ

⃗ p,q

where

(ℝd )

= ‖f ‖Lp⃗ (ℝd ) + ‖f ‖bℓ

⃗ p,q

(ℝd )

< ∞,

On trace theorems for weighted mixed-norm Sobolev spaces and applications

‖f ‖bℓ

⃗ p,q

(ℝd )

= [∫( ℝd

‖Δh f ‖Lp⃗ (ℝd ) |h|ℓ

q

)

� 53

1

dh q ] . |h|d

ℓ d ℓ d Observe that when p⃗ = (p, p, . . . , p), Bp,q ⃗ (ℝ ) = Bp,q (ℝ ), where the later is the usual Besov space. The following trace theorem is the main result of the paper.

Theorem 1.2 (Trace theorem). Let p⃗ ∈ [1, ∞)d , q ∈ [1, ∞), α ∈ (−1, q − 1), and ℓ = 1 − Then there exists N = N(d, p,⃗ q, α) > 0 such that the following assertions hold true: 1 d+1 ℓ d (i) There is a trace map T : u ∈ Wp,q ⃗ (ℝ+ , μ) → Bp,q ⃗ (ℝ ) such that 󵄩󵄩 󵄩 , 󵄩󵄩T(u)󵄩󵄩󵄩Bℓ (ℝd ) ≤ N‖u‖W 1 (ℝd+1 + ,μ) ⃗ p,q ⃗ p,q

1 d+1 ∀u ∈ Wp,q ⃗ (ℝ+ , μ)

1+α . q

(1.1)

d+1

1 d+1 and T(u)(⋅, 0) = u(⋅, 0) if u ∈ C(ℝ+ ) ∩ Wp,q ⃗ (ℝ+ , μ).

ℓ d 1 d+1 (ii) There exists an extension map E : Bp,q ⃗ (ℝ+ , μ) satisfying ⃗ (ℝ ) → Wp,q

󵄩󵄩 󵄩 󵄩󵄩E(g)󵄩󵄩󵄩W 1 (ℝd+1 ,μ) ≤ N‖g‖Bℓ (ℝd ) , + ⃗ p,q ⃗ p,q

ℓ d ∀g ∈ Bp,q ⃗ (ℝ ).

ℓ d d Moreover, for every g ∈ Bp,q ⃗ (ℝ ), spt(E(g)(x, ⋅)) ⊂ [0, 1) for all x ∈ ℝ and

󵄩 󵄩 lim y−ℓ 󵄩󵄩󵄩E(g)(⋅, y) − g 󵄩󵄩󵄩L ⃗ (ℝd ) = 0. p

y→0+

Remark 1.3. As ℓ only depends on α and q, but not on p,⃗ we see that the differentiability 1 d+1 order of the traces of functions in Wp,q ⃗ (ℝ+ , μ) only depends on α and q. This phenomena does not seem to be seen before even in the unweighted case (i. e., when α = 0) and with d = 1. When p⃗ = (q, q, . . . , q), Theorem 1.2 is reduced to the classical results; see [21, Theorem 2.8], [14, Theorem 2.10], and [20, Theorem 2.8, p. 100]. We also remark that when d+1 1 α ≤ −1, then it follows from Lemma 2.3 below that T(u) ≡ 0 for any u ∈ Wp,q ⃗ (ℝ+ , μ) for any p⃗ ∈ [1, ∞)d and q ∈ [1, ∞).

We next give two applications of Theorem 1.2, which also demonstrates some motivations for this study. In the first application, we study a class of second-order elliptic equations in divergence form with coefficients that can be singular or degenerate near 2 the boundary of the domains. Let (aij ) : ℝd+1 → ℝ(d+1) be the (d + 1) × (d + 1) matrix + of measurable functions satisfying the uniformly elliptic and boundedness condition: there is ν ∈ (0, 1) such that aij (x, y)ξi ξj ≥ ν|ξ|2

and

󵄨 󵄨󵄨 −1 󵄨󵄨aij (x, y)󵄨󵄨󵄨 ≤ ν ,

(1.2)

d+1 for all (x, y) ∈ ℝd+1 . For each λ > 0, let us denote + and for all ξ = (ξ1 , ξ2 , . . . , ξd+1 ) ∈ ℝ the operator

54 � T. Phan β

β

L u(x, y) = −Di ( y aij (x, y)Dj u(x, y)) + λy u(x, y),

where β ∈ (−∞, 1) is a fixed number, and 𝜕u (x, y) 𝜕xi

Di u(x, y) =

for i = 1, 2, . . . , d

and

Dd+1 u(x, y) =

𝜕u (x, y) 𝜕y

for (x, y) ∈ ℝd × ℝ+ . Note that in the above the Enstein’s summation convention is used. As β ∈ (−∞, 1), the leading coefficients yβ aij (x, y) in L can be singular or degend+1 erate on the boundary 𝜕ℝd+1 + of ℝ+ . When β = 0, the operator L is reduced to the standard second-order partial differential operator with bounded and uniformly elliptic coefficients. When β ∈ (−1, 1) and (aij ) is the identity matrix, the operator L appears in the study of fractional Laplace equations; see [7] for instance. Interested readers may find in [17, 29], for instance, for some other work related to problems in geometric and probability in which equations with singular and degenerate coefficients also appear. We study the nonhomogeneous boundary value problem L u(x, y) = Di ( yβ Fi ) + √λyβ f

{

(x, y) ∈ ℝd × ℝ+ x ∈ ℝd ,

u(x, 0) = g(x)

(1.3)

where Fi : ℝd+1 → ℝ is a given measurable function for i = 1, 2, . . . , d + 1, and f : + d+1 ℝ+ → ℝ is a given measurable function. To state our result for (1.3), we need to impose a weighted partial VMO condition on the coefficients (aij ). For every r > 0 and x0 ∈ ℝd , y0 ∈ [0, ∞), we denote Dr (x0 , y0 ) = (Br (x0 ) × ( y0 − r, y0 + r)) ∩ ℝd+1 + where Br (x0 ) is the ball in ℝd of radius r and centered at x0 . We also denote the measure μ1 (dxdy) = y−β dxdy and f (x, y)μ1 (dxdy) =

∫ — Dr (x0 ,y0 )

1 μ1 (Dr (x0 , y0 ))

∫

f (x, y)μ1 (dxdy)

Dr(x0 ,y0 )

where μ1 (Dr (x0 , y0 )) =

∫

μ1 (dxdy).

Dr (x0 ,y0 )

Also, we denote the average of aij on Br (x0 ) by [aij ]r,x0 ( y) =

1 ∫ aij (x, y)dx, |Br (x0 )| Br (x0 )

y ∈ ℝ+ .

On trace theorems for weighted mixed-norm Sobolev spaces and applications

� 55

Then the partially bounded mean oscillations of the matrix a = (aij ) with respect to the weight μ1 in Dr (x0 , y0 ) is defined by ar# (x0 , y0 ) =

max

i,j=1,2,...,d+1

󵄨 󵄨 ∫ 󵄨󵄨󵄨aij (x, y) − [aij ]r,x0 ( y)󵄨󵄨󵄨μ1 (dxdy). —

Dr (x0 ,y0 )

Recall that μ(dxdy) = yα dxdy. For every p⃗ ∈ (1, ∞)d , q ∈ (1, ∞), α ∈ (−1, q − 1), and with 1+α ℓ d 1 d+1 g ∈ Bp,q , μ) is a weak solution to (1.3) if ⃗ (ℝ ⃗ (ℝ ) with ℓ = 1 − q , we say that u ∈ Wp,q ∫ yβ [aij Dj uDj φ + Fi Di φ + λuφ]dxdy = √λ ∫ f (x, y)yβ dxdy ℝd+1 +

ℝd+1 +

for all smooth, compactly supported function φ : ℝd+1 + → ℝ, and u(x, 0) = g(x) in sense of trace. Now, our result regarding (1.3) is stated in the following theorem. Corollary 1.4. Let p⃗ ∈ (1, ∞)d , q ∈ (1, ∞), β ∈ (−∞, 1), α ∈ (−1, q−1), ℓ = 1− 1+α , ν ∈ (0, 1), q and R0 ∈ (0, ∞). There exist sufficiently small positive number δ = δ(d, p,⃗ q, β, α, ν) and a sufficiently large positive number λ0 = λ0 (d, p,⃗ q, β, α, ν) such that the following assertions hold. Suppose that (1.2) and sup

d+1 (x0 ,y0 )∈ℝ+

sup ar# (x0 , y0 ) ≤ δ

r∈(0,R0 )

(1.4)

1 d+1 d+1 d+1 hold. If u ∈ Wp,q , μ) is a weak solution of (1.3) with f ∈ Lp,q ⃗ (ℝ+ , μ), F ∈ Lp,q ⃗ (ℝ+ , ⃗ (ℝ ℓ d −2 μ)d+1 , g ∈ Bp,q ⃗ (ℝ ), and λ ≥ λ0 R0 , then

‖Du‖Lp,q⃗ (ℝd+1 + √λ‖u‖Lp,q⃗ (ℝd+1 + ,μ) + ,μ) ≤ N[‖F‖Lp,q⃗ (ℝd+1 + ‖f ‖Lp,q⃗ (ℝd+1 ] + ,μ) + ,μ) 󵄩 󵄩 󵄩 󵄩 + N[󵄩󵄩󵄩DE(g)󵄩󵄩󵄩L ⃗ (ℝd+1 ,μ) + √λ󵄩󵄩󵄩E(g)󵄩󵄩󵄩L ⃗ (ℝd+1 ,μ) ], p,q p,q + + where N = N(d, p,⃗ q, β, α, ν) > 0 and E is defined in Theorem 1.2. Moreover, for every d+1 d+1 d+1 ℓ d −2 f ∈ Lp,q , g ∈ Bp,q ⃗ (ℝ+ , μ), F ∈ Lp,q ⃗ (ℝ+ , μ) ⃗ (ℝ ), and λ ≥ λ0 R0 , there exists a unique 1 d+1 weak solution u ∈ Wp,q , μ) to (1.3). ⃗ (ℝ

We remark that the problem (1.3) when g ≡ 0 is studied in [9] for both elliptic and parabolic cases. Corollary 1.4 therefore recovers [9, Theorem 2.8] for (1.3) when g ≡ 0. Even when p⃗ = (p1 , p2 , . . . , pd ) with p1 = p2 = ⋅ ⋅ ⋅ = pd and β = 0, Corollary 1.4 seems to be new as it deals with nonhomogeneous boundary value problems in weighted mixed1 d+1 norm spaces. We also point out that when α ≤ −1, to search for a solution in Wp,q ⃗ (ℝ+ , μ) of (1.3), it requires that g ≡ 0 as pointed out in Remark 1.3. In fact, when α ∈ (qβ − 1, −1] 1 d+1 and g ≡ 0, the existence and uniqueness of weak solutions Wp,q ⃗ (ℝ+ , μ) can be obtained from [9].

56 � T. Phan 1 In the second application of Theorem 1.2, we prove the trace theorem for Wp,q (Ω ×

(0, 1), μ) where Ω ⊂ ℝd is open and bounded domain with Lipschitz boundary 𝜕Ω. As before, for p, q ∈ [1, ∞) and Q = Ω × (0, 1), let us denote Lp,q (Q, μ) be the weighed mixednormed Lebesgue space consisting of measurable functions f : Q → ℝ such that 1

‖f ‖Lp,q (Q,μ)

q

p 󵄨p 󵄨 = (∫(∫󵄨󵄨󵄨f (x, y)󵄨󵄨󵄨 dx) yα dy)

0

1/q

< ∞.

Ω

1 d+1 Similar to the weighted mixed-norm Sobolev space Wp,q ⃗ (ℝ+ , μ), let us denote the 1 weighted mixed-norm Sobolev space Wp,q (Q, μ) by

1 Wp,q (Q, μ) = {f ∈ Lp,q (Q, μ) : Df ∈ Lp,q (Q, μ)}

and it is endowed with the norm ‖f ‖Wp,q 1 (Q,μ) = ‖f ‖L (Q,μ) + ‖Df ‖L (Q,μ) . p,q p,q ℓ For ℓ ∈ (0, 1), and p, q ∈ [1, ∞), we define the Besov space Bp,q (Ω) as ℓ ℓ Bp,q (Ω) = {f ∈ Lp (Ω) : ∃g ∈ Bp,q (ℝd ) : g|Ω = f }

and it is endowed with the norm ℓ d ‖f ‖Bp,q ℓ (Ω) = inf{‖g‖ ℓ Bp,q (ℝd ) for g ∈ Bp,q (ℝ ) : g|Ω = f }.

Interested readers can find [8, 23, 25] and the monograph [26] for more details on Besov spaces in bounded domains. Note that it is possible to use the vector p⃗ instead of the ℓ scaler p in the definition Bp,q (Ω); however, we do not pursue this to avoid complication in writing the mixed-norm of functions in Lp⃗ (Ω) due to the geometry of Ω. 1 Our trace theorem for weighted mixed-norm space Wp,q (Q, μ) with Q = Ω × (0, 1) α and μ( y) = y for y ∈ (0, 1) is now stated below. Corollary 1.5. Assume Ω ⊂ ℝd is an open bounded domain with Lipschitz boundary 𝜕Ω . There and Q = Ω × (0, 1). Assume also that p, q ∈ [1, ∞), α ∈ (−1, q − 1) and ℓ = 1 − 1+α q exists a linear operator 1 ℓ TΩ : u ∈ Wp,q (Q, μ) → Bp,q (Ω) 1 such that TΩ (u)(⋅, 0) = u(⋅, 0) if u ∈ C(Ω × [0, 1)) ∩ Wp,q (Q, μ) and

󵄩󵄩 󵄩 1 (Q,μ) , 󵄩󵄩TΩ (u)󵄩󵄩󵄩Bℓ (Ω) ≤ N‖u‖Wp,q p,q

1 ∀u ∈ Wp,q (Q, μ),

where N = N(p, q, d, α, Ω) > 0 and μ( y) = yα for y ∈ (0, 1).

On trace theorems for weighted mixed-norm Sobolev spaces and applications

� 57

As previously mentioned, Corollary 1.5 provides key ingredient for the study of the regularity in Besov spaces of solutions to the class of nonlocal elliptic equations with measurable coefficients and this will be carried out in our forthcoming paper. When p = q, Corollary 1.5 is reduced to the classical result, Theorem 1.1. As in Theorem 1.2, it is interesting to note that ℓ only depends on α, q but not on p, the phenomena is not implied by Theorem 1.1. It is possible to extend Theorem 1.2 and Corollary 1.5 to higher-ordered k Wp,q -weighted mixed-norm Sobolev spaces with k ∈ ℕ and more general bounded domains as in [14, Theorem 2.10]. However, we do no pursue this direction to avoid the technical complexity in this paper. The remaining parts of the paper is organized as follows. In Section 2, several preliminary estimates and inequalities needed to prove Theorem 1.2 are recalled and proved. Section 3 is devoted to the proof of Theorem 1.2. The proofs of Corollary 1.4 and Corollary 1.5 will be given in Section 4.

2 Preliminary estimates and inequalities 2.1 Hardy’s inequalities We begin with the following weighted Hardy’s inequality, which is useful in the paper. Lemma 2.1 (Hardy’s inequality). Let q ∈ [1, ∞), a ∈ (0, ∞], and σ < 1 − q1 . Then we have 1

1

󵄨󵄨 t 󵄨󵄨q q q −1 a 󵄨󵄨 1 󵄨󵄨 󵄨󵄨q qσ 󵄨 (∫ t 󵄨󵄨 ∫ f (s)ds󵄨󵄨 dt) ≤ (1 − − σ) (∫󵄨󵄨f (t)󵄨󵄨 t dt) , 󵄨󵄨 t 󵄨󵄨 q 󵄨 0 󵄨 0 0 a

qσ 󵄨󵄨󵄨 1

for every measurable function f : (0, a) → ℝ. A similar statement also holds when q = ∞. Proof. The proof is standard and we give it here for completeness. We only consider the case q < ∞. Note that with the change of variable s = ty, we have 1

1

q a 1 󵄨󵄨 t 󵄨󵄨q q q 󵄨󵄨 󵄨 󵄨 qσ (∫ t 󵄨󵄨 ∫ f (s)ds󵄨󵄨󵄨 dt) ≤ (∫ t (∫󵄨󵄨󵄨f (ty)󵄨󵄨󵄨dy) dt) . 󵄨󵄨 t 󵄨󵄨 󵄨 0 󵄨 0 0 0 a

qσ 󵄨󵄨󵄨 1

Then it follows from Minkowski’s inequality that 1

1

1 a 󵄨󵄨 t 󵄨󵄨q q q 󵄨󵄨 󵄨󵄨 󵄨󵄨q qσ 󵄨 (∫ t 󵄨󵄨 ∫ f (s)ds󵄨󵄨 dt) ≤ ∫(∫󵄨󵄨f (ty)󵄨󵄨 t dt) dy 󵄨󵄨 t 󵄨󵄨 󵄨 0 󵄨 0 0 0 a

qσ 󵄨󵄨󵄨 1

1

= ∫y

−σ− q1

0

ay

󵄨 󵄨q dy(∫󵄨󵄨󵄨f (τ)󵄨󵄨󵄨 τ qσ dτ) 0

1 q

58 � T. Phan 1 q

a

1

−σ− 1 󵄨q 󵄨 ≤ (∫󵄨󵄨󵄨f (τ)󵄨󵄨󵄨 τ qσ dτ) ∫ y q dy 0

−1

a

0

1 q

1 󵄨q 󵄨 = (1 − − σ) (∫󵄨󵄨󵄨f (τ)󵄨󵄨󵄨 τ qσ dτ) , q 0

where we have used the change of variable τ = ty in the second step of the above estimate. The assertion is proved. The following consequence of Lemma 2.1 is needed in this paper. Corollary 2.2. Let θ ∈ [1, ∞), a ∈ (0, ∞] and β < d − θ1 . Then there is a constant N = N(d, β, θ) > 0 such that a

(∫(t

β−d

0

θ

1 θ

1

θ d−1 󵄨 󵄨 󵄨 󵄨θ ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨dx) dt) ≤ N( ∫ (|x|β− θ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨) dx) ,

|x|≤t

|x|≤a

for every measurable function f : ℝd → ℝ. A similar statement also holds when θ = ∞. Proof. We only provide the proof when θ < ∞. Using the polar coordinate and Minkowski’s inequality, we have a

(∫(t 0

θ

1 θ

a

θ

t

󵄨 󵄨 󵄨 󵄨 ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨dx) dt) = (∫( ∫ t β−d ∫󵄨󵄨󵄨f (rξ)󵄨󵄨󵄨r d−1 drdξ) dt)

β−d

0

|η|≤t

𝕊d−1

1 θ

0

1

a

󵄨󵄨 t 󵄨󵄨 θ 󵄨 d−1 󵄨θ (β−d+1)θ 󵄨󵄨󵄨 1 󵄨󵄨󵄨 ≤ ∫ (∫ t 󵄨󵄨 ∫󵄨󵄨f (rξ)󵄨󵄨󵄨r dr 󵄨󵄨󵄨 dt) dξ. 󵄨󵄨 t 󵄨󵄨 󵄨 0󵄨 𝕊d−1 0 Then, applying Lemma 2.1 for q = θ and σ = β − d + 1, we obtain a

(∫(t 0

β−d

θ

1 θ

a

󵄨 󵄨 󵄨 󵄨θ ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨dx) dt) ≤ N( ∫ ∫ r βθ 󵄨󵄨󵄨f (rξ)󵄨󵄨󵄨 drdξ) 𝕊d−1 0

|x|≤t

= N( ∫ (|x| |x|≤a

1 θ

1

θ 󵄨θ 󵄨󵄨f (x)󵄨󵄨󵄨) dx) ,

󵄨 β− d−1 θ 󵄨

for N = N(d, β, θ) > 0. The proof is completed. We now state and prove the following lemma, which provides the first step in the proof of Theorem 1.2.

On trace theorems for weighted mixed-norm Sobolev spaces and applications

� 59

Lemma 2.3. Let p⃗ ∈ [1, ∞)d , q ∈ [1, ∞), α ∈ (−q, q − 1). There exists N = N(q, α) > 0 such that 󵄩 󵄩󵄩 󵄩󵄩u(⋅, 0)󵄩󵄩󵄩Lp⃗ (ℝd ) ≤ N‖u‖W 1 (ℝd ×(0,1),μ) ⃗ p,q 1 d d for every u ∈ Wp,q ⃗ (ℝ × (0, 1), μ) ∩ C(ℝ × [0, 1)). Moreover, if α ≤ −1 then u(⋅, 0) = 0 in 1 d sense of trace for every u ∈ Wp,q ⃗ (ℝ × (0, 1), μ).

Proof. We prove the first assertion in the lemma. Using the density, we may assume that u ∈ C 1 (ℝd × [0, 1]). For each (x, y) ∈ ℝd × (0, 1), by the fundamental theorem of calculus, we have y

u(x, y) − u(x, 0) = ∫ Dd+1 u(x ′ , s)ds 0

Then, by Minkowski’s inequality, we obtain 󵄩󵄩 y 󵄩󵄩 󵄩󵄩 󵄨 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄨 󵄩󵄩 ′ 󵄩󵄩u(⋅, 0)󵄩󵄩Lp⃗ (ℝd ) ≤ 󵄩󵄩u(⋅, y)󵄩󵄩Lp⃗ (ℝd ) + 󵄩󵄩󵄩∫󵄨󵄨󵄨Dd+1 u(x , s)󵄨󵄨󵄨ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩0 󵄩Lp⃗ (ℝd ) y

󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩u(⋅, y)󵄩󵄩󵄩L ⃗ (ℝd ) + ∫󵄩󵄩󵄩Dd+1 u(⋅, s)󵄩󵄩󵄩L ⃗ (ℝd ) ds p

p

0

y

1− α+1 󵄩 󵄩q 󵄩 󵄩 ≤ 󵄩󵄩󵄩u(⋅, y)󵄩󵄩󵄩L ⃗ (ℝd ) + Ny q (∫󵄩󵄩󵄩Dd+1 u(⋅, s)󵄩󵄩󵄩L p

1− 󵄩 󵄩 = 󵄩󵄩󵄩u(⋅, y)󵄩󵄩󵄩L ⃗ (ℝd ) + Ny p

α+1 q

0

p⃗

sα ds) (ℝd )

1/q

‖Dd+1 u‖Lp,q⃗ (ℝd ×(0,y),μ) ,

(2.1)

where N = N(q, α) > 0, and we also used Hölder’s inequality and the fact that α < q − 1 in the third step. Now, from α + q > 0 it follows that 1

α

∫ y q dy = 0

q . α+q

Accordingly, we have 1

1

0

0

α 1− 1 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩u(⋅, 0)󵄩󵄩󵄩Lp⃗ (ℝd ) ≤ N ∫󵄩󵄩󵄩u(⋅, y)󵄩󵄩󵄩Lp⃗ (ℝd ) y q dy + N‖Dd+1 u‖Lp,q⃗ (Q,μ) ∫ y q dy

≤ N‖u‖Lp,q⃗ (ℝd ×(0,1),μ) + N‖Dd+1 u‖Lp,q⃗ (ℝd ×(0,1),μ) ≤ N‖u‖W 1

⃗ p,q

(ℝd ×(0,1),μ) ,

60 � T. Phan where N = N(q, α) > 0. This proves the first assertion of the lemma. Next, we prove the second assertion of the lemma. Let (x, y) ∈ ℝd × (0, 1). For every s ∈ (0, y), similar to (2.1), we have y

󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄨 󵄩 󵄩 󵄩 󵄩󵄩 󵄨 󵄩󵄩 󵄩󵄩u(⋅, y)󵄩󵄩󵄩Lp⃗ (ℝd ) ≤ 󵄩󵄩󵄩u(⋅, s)󵄩󵄩󵄩Lp⃗ (ℝd ) + 󵄩󵄩󵄩∫󵄨󵄨󵄨Dd+1 u(x, τ)󵄨󵄨󵄨dτ 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩s 󵄩Lp⃗ (ℝd ) 1+α 1− 󵄩 󵄩 ≤ 󵄩󵄩󵄩u(⋅, s)󵄩󵄩󵄩L ⃗ (ℝd ) + Ny q ‖Dd+1 u‖Lp,q⃗ (ℝd ×(0,y),μ) . p Then, integrating this last estimate with respect to s on (0, y) and then using the Hölder’s inequality, we have y

2− 1+α 󵄩 󵄩 󵄩 󵄩 y󵄩󵄩󵄩u(⋅, y)󵄩󵄩󵄩L ⃗ (ℝd ) ≤ ∫󵄩󵄩󵄩u(⋅, s)󵄩󵄩󵄩L ⃗ (ℝd ) ds + Ny q ‖Dd+1 u‖Lp,q⃗ (ℝd ×(0,y),μ) p p 0

1− 1+α q

≤ Ny

[‖u‖Lp,q⃗ (ℝd ×(0,y),μ) + y‖Dd+1 u‖Lp,q⃗ (ℝd ×(0,y),μ) ],

where N = N(α, q) > 0. Then − 1+α 󵄩󵄩 󵄩 󵄩󵄩u(⋅, y)󵄩󵄩󵄩Lp⃗ (ℝd ) ≤ Ny q [‖u‖Lp,q⃗ (ℝd ×(0,y),μ) + y‖Dd+1 u‖Lp,q⃗ (ℝd ×(0,y),μ) ].

From this, as α ≤ −1, and lim [‖u‖Lp,q⃗ (ℝd ×(0,y),μ) + y‖Dd+1 u‖Lp,q⃗ (ℝd ×(0,y),μ) ] = 0,

y→0+

we see that 󵄩 󵄩 lim 󵄩󵄩󵄩u(⋅, y)󵄩󵄩󵄩L ⃗ (ℝd ) = 0.

y→0+

p

The proof of the lemma is completed.

2.2 Mixed-norm Besov space equivalent norms For x0 ∈ ℝd and ρ > 0, we denote Bρ (x0 ) the ball in ℝd of radius ρ centered at x0 . We also write Bρ = Bρ (0). Let φ ∈ C0∞ (B1 ) be a fixed radial cut-off function satisfying 0≤φ≤1

and

∫ φ(x)dx = 1. ℝd

For each δ > 0, let φδ (x) = δ−d φ(x/δ),

x ∈ ℝd .

(2.2)

On trace theorems for weighted mixed-norm Sobolev spaces and applications

� 61

We also denote ω(δ, f )p⃗ = sup ‖Δh f ‖Lp⃗ (ℝd ) .

(2.3)

|h| 0 such that for every p⃗ ∈ [1, ∞)d , we have ‖φδ ∗ f − f ‖Lp⃗ (ℝd ) ≤ ω(δ, f )p⃗ ,

and ‖Di φδ ∗ f ‖Lp⃗ (ℝd ) ≤ Nω(δ, f )p⃗ ,

for δ > 0 and i = 1, 2, . . . , d, where ∗ denotes the convolution operator. Proof. We provide the proof of completeness. Observe that φδ ∗ f (x) − f (x) = ∫[f (x − δz) − f (x)]φ(z)dz = ∫ Δ−δz f (x)φ(z)dz, B1

B1

for x ∈ ℝd . Then, by Minkowski’s inequality, we obtain ‖φδ ∗ f − f ‖Lp⃗ (ℝd ) ≤ ∫ ‖Δ−δz f ‖Lp⃗ (ℝd ) φ(z)dz ≤ ω(δ, f )p⃗ , B1

which proves the first assertion of the lemma. As φ is radial, ∫ Di φ(z)dz = 0. B1

Then we also have Di φδ ∗ f (x) = ∫ Δ−δz f (x)Di φ(z)dz, B1

and the second assertion of the lemma follows in the same way. We now state and prove the following result on the equivalent norms of the mixednorm Besov spaces, which will be used later in the proof of Theorem 1.2. Lemma 2.5. Let ℓ ∈ (0, 1), p,⃗ θ ∈ [1, ∞)d , and a ∈ [1, ∞]. Define ‖f ‖(1) ℓ Bp,θ (ℝd ) ⃗ and

a

= ‖f ‖Lp⃗ (ℝd ) + (∫( 0

ω(t, f )p⃗ tℓ

θ

dt ) ) t

1 θ

(2.4)

62 � T. Phan

‖f ‖(2) ℓ (ℝd ) Bp,θ ⃗ Then the norms ‖ ⋅ ‖(1) Bℓ

⃗ p,θ

(ℝd )

∞

kℓ

−k

θ

1 θ

= ‖f ‖Lp⃗ (ℝd ) + ( ∑ (2 ω(2 , f )p⃗ ) ) .

(2.5)

k=1

, and ‖ ⋅ ‖Bℓ , ‖ ⋅ ‖(2) Bℓ (ℝd )

⃗ p,θ

⃗ p,θ

(ℝd )

are equivalent.

Proof. The result is well known in the unmixed-norm case, but does not seem to be recorded elsewhere in the mixed-norm case. In addition, the proof is not too long, hence we provide it for completeness. Note that ‖Δh f ‖Lp⃗ (ℝd ) ≤ ω(|h|, f )p⃗ . Therefore, it follows from the spherical coordinate and ω(t, f )p⃗ ≤ 2‖f ‖Lp⃗ (ℝd ) as well as

‖f ‖bℓ

⃗ p,θ

(ℝd )

= [∫(

‖Δh f ‖Lp⃗ (ℝd )

≤ [∫(

ω(|h|, f )p⃗

ℝd a

≤ N[∫( 0 a

≤ N[∫( 0 a

≤ N[∫( 0

dh θ ) ] |h|d

|h|ℓ

ℝd

|h|ℓ

ω(t, f )p⃗ tℓ

1

θ

dh θ ) ] |h|d θ

1 θ

θ

1 θ

θ

1 θ

∞

θ

ω(t, f )p⃗ dt dt ) ] + N[ ∫ ( ) ] t t tℓ

1 θ

a

ω(t, f )p⃗ tℓ

ω(t, f )p⃗ tℓ

1

θ

∞

dt ) ] + N‖f ‖Lp⃗ (ℝd ) ( ∫ t −θℓ−1 dt) t )

1 θ

1

dt ] + N‖f ‖Lp⃗ (ℝd ) , t

where N = N(d, θ, ℓ) > 0. From this and (2.4), there is N = N(d, θ, ℓ) > 0 such that ‖f ‖Bℓ

⃗ p,θ

(ℝd )

≤ N‖f ‖(1) Bℓ

⃗ p,θ

(ℝd )

.

This proves one direction in the assertion about the equivalence between ‖ ⋅ ‖Bℓ ‖ ⋅ ‖(1) Bℓ

⃗ p,θ

⃗ p,θ

(ℝd )

(ℝd )

and

. Now, we prove the other direction. By a simple manipulation, we see that Δh f (x) = Δη f (x) + Δh−η f (x + η),

∀η, h ∈ ℝd .

It then follows from the triangle inequality that ‖Δh f ‖Lp⃗ (ℝd ) ≤ ‖Δη f ‖Lp⃗ (ℝd ) + ‖Δh−η f ‖Lp⃗ (ℝd ) . Integrating this inequality with respect to η in the ball B|h|/2 (h/2), we obtain

On trace theorems for weighted mixed-norm Sobolev spaces and applications

‖Δh f ‖Lp⃗ (ℝd ) ≤

N ( |h|d

∫

B|h|/2 (h/2)

‖Δη f ‖Lp⃗ (ℝd ) dη +

∫ B|h|/2 (h/2)

� 63

‖Δh−η f ‖Lp⃗ (ℝd ) dη)

N ≤ d ∫ ‖Δη f ‖Lp⃗ (ℝd ) dη ≤ N ∫ |η|−d ‖Δη f ‖Lp⃗ (ℝd ) dη |h| B|h|

B|h|

where N = N(d) > 0, and we used the fact that B|h|/2 (h/2) ⊂ B|h| and h − η ∈ B|h| for all η ∈ B|h|/2 (h/2). As a consequence, we obtain ω(t, f )p⃗ ≤ N ∫ |η|−d ‖Δη f ‖Lp⃗ (ℝd ) dη,

∀t > 0

|η|≤t

and, therefore, a

1 θ

θ

a

1 θ

θ

1 ω(t, f ) dh [∫( ℓ ) ] ≤ N(∫[t −ℓ− θ ∫ |η|−d ‖Δη f ‖Lp⃗ (ℝd ) dη] dt) . t t

0

0

|η|≤t

Now, applying Corollary 2.2 with β = d − ℓ − θ1 , we obtain a

1 θ

θ

1

θ d ω(t, f ) dh θ ] ≤ N(∫ (|η|−ℓ− θ ‖Δη f ‖Lp⃗ (ℝd ) ) dη) [∫( ℓ ) t t

Ba

0

≤ N‖f ‖bℓ . ⃗ p,θ

Therefore, there is N = N(d, ℓ, θ) > 0 such that ‖f ‖(1) Bℓ

⃗ p,θ

(ℝd )

≤ N‖f ‖Bℓ

⃗ p,θ

(ℝd )

and ‖ ⋅ ‖Bℓ are equivalent. and this completes the proof that the norms ‖ ⋅ ‖(1) Bℓ Next, we consider the norm ‖ ⋅

‖(2) ℓ Bp,θ ⃗

⃗ p,θ

⃗ p,θ

is defined in (2.5). We will show that ‖ ⋅ ‖(2) Bℓ ⃗ p,θ

defined in (2.4). Observe that equivalent to ‖ ⋅ ‖(1) Bℓ ⃗ p,θ

a

(∫( 0

ω(t, f )p⃗ tℓ

θ

1 θ

1/2

θ

1 θ

a

θ

1 θ

ω(t, f )p⃗ dt ω(t, f )p⃗ dt dt ) ) = (∫( ) ) + (∫( ) ) . ℓ t t t t tℓ 0

1/2

It follows directly from the definition of ω(t, f )p⃗ and the triangle inequality that ω(t, f )p⃗ ≤ 2‖f ‖Lp⃗ (ℝd ) and, therefore,

64 � T. Phan a

(∫( 1/2

ω(t, f )p⃗ tℓ

1 θ

θ

∞

dt ) ) ≤ 2‖f ‖Lp⃗ (ℝd ) ( ∫ t −ℓθ−1 dt) t

1 θ

1/2

= N‖f ‖Lp⃗ (ℝd ) , for N = N(θ, ℓ) > 0. On the other hand, as ω(t, f )p⃗ is nondecreasing, we see that 1/2

(∫( 0

ω(t, f )p⃗ tℓ

1 θ

θ

2−k

θ

∞ ω(t, f )p⃗ dt dt ) ) ) = (∑ ∫ ( ) t t tℓ k=1 2−k−1

∞

1 θ

1 θ

θ

≤ N( ∑ (2kℓ ω(2−k , f )p⃗ ) ) . k=1

Then, from (2.4), it follows that ‖f ‖(1) Bℓ

⃗ p,θ

(ℝd )

≤ N‖f ‖(2) , Bℓ (ℝd ) ⃗ p,θ

where N = N(d, ℓ, θ) > 0. It now remains to prove the other direction of the inequality. By Minkowski’s inequality, it follows that ∞

θ

1 θ

∞

θ

1 θ

( ∑ (2kℓ ω(2−k , f )p⃗ ) ) = N[ω(2−1 , f )p⃗ + ( ∑ (2kℓ ω(2−k , f )p⃗ ) ) ] k=1

k=2

1/2

≤ N[‖f ‖Lp⃗ (ℝd ) + ( ∫ ( 0

ω(t, f )p⃗ tℓ

θ

1 θ

dt ) ) ]. t

Therefore, ‖f ‖(2) ≤ N‖f ‖(1) Bℓ (ℝd ) Bℓ ⃗ p,θ

⃗ p,θ

(ℝd )

where N = N(d, θ, ℓ) > 0. The proof of the lemma is completed.

3 Proof of Theorem 1.2 This section is to prove Theorem 1.2. The proof requires several lemmas and estimates. For k = 1, 2, . . . , d, and for h ∈ ℝ, let us denote the difference in k-variable direction of a function f as Δk,h f (x) = f (x + hek ) − f (x),

x ∈ ℝd ,

On trace theorems for weighted mixed-norm Sobolev spaces and applications

� 65

where ek is the Euclidean kth-unit coordinate vector. We start with the one-dimensional space as its proof contains important ideas and techniques. Lemma 3.1. Let p, q ∈ [1, ∞) and α ∈ (−1, q − 1). Then there is N = N(p, q, α) > 0 such that q

∞

p 󵄨p 󵄨 󵄩 󵄩󵄩 α 󵄩󵄩f (⋅, 0)󵄩󵄩󵄩bℓ (ℝ) ≤ N( ∫ (∫󵄨󵄨󵄨Df (x, y)󵄨󵄨󵄨 dx) y dy) p,q

0

1/q

ℝ

for all f ∈ C(ℝ × [0, ∞)) such that Df ∈ Lp,q (ℝd+ , μ) and ℓ = 1 −

1+α q

∈ (0, 1).

Proof. Let g(x) = f (x, 0) for x ∈ ℝ and note that Δh g(x) = [f (x + h, 0) − f (x + h, |h|)] + [f (x + h, |h|) − f (x, |h|)] + [f (x, |h|) − f (x, 0)]

= − Δ2,|h| f (x + h, 0) + Δ1,h f (x, |h|) + Δ2,|h| f (x, 0),

x ∈ ℝ.

Then, by triangle inequality, it follows that 1

1

‖Δh g‖Lp (ℝ)

p p 󵄨 󵄨p 󵄨 󵄨p ≤ (∫󵄨󵄨󵄨Δ2,|h| f (x + h, 0)󵄨󵄨󵄨 dx) + (∫󵄨󵄨󵄨Δ1,h f (x, |h|)󵄨󵄨󵄨 dx)

ℝ

󵄨 󵄨p + (∫󵄨󵄨󵄨Δ2,|h| f (x, 0)󵄨󵄨󵄨 dx) ℝ

ℝ

1 p

1

1

p p 󵄨 󵄨p 󵄨 󵄨p = 2(∫󵄨󵄨󵄨Δ2,|h| f (x, 0)󵄨󵄨󵄨 dx) + (∫󵄨󵄨󵄨Δ1,h f (x, |h|)󵄨󵄨󵄨 dx) .

ℝ

ℝ

Then, for ℓ = 1 −

1+α , q

we have

(∫( ℝ

‖Δh g‖Lp (ℝ) |h|ℓ

1

q

dh q ) ) |h| q

∞

p 󵄨 󵄨p ≤ N( ∫ h−1−qℓ (∫󵄨󵄨󵄨Δ2,h f (x, 0)󵄨󵄨󵄨 dx) dh)

0

ℝ ∞

+ N( ∫ h

−1−qℓ

0

= N[I1 + I2 ],

q

1/q

1/q

p 󵄨 󵄨p (∫󵄨󵄨󵄨Δ1,h f (x, h)󵄨󵄨󵄨 dx) dh)

ℝ

where N = N(q) > 0. We now control I1 . From the fundamental theorem of calculus, we have

66 � T. Phan h

Δ2,h f (x, 0) = f (x, h) − f (x, 0) = ∫ D2 f (x, ξ)dξ 0

and, therefore, it follows from Minkowski’s inequality that p

h

1

p 󵄨p 󵄨 󵄨 󵄨 (∫󵄨󵄨󵄨Δ2,h f (x, 0)󵄨󵄨󵄨 dx) ≤ (∫(∫󵄨󵄨󵄨D2 f (x, ξ)󵄨󵄨󵄨dξ) dx)

ℝ

h

0

ℝ

1/p

󵄨 󵄨p ≤ ∫(∫󵄨󵄨󵄨D2 f (x, ξ)󵄨󵄨󵄨 dx) 0

h

1 p

dξ

ℝ

= ∫ w(ξ)dξ, 0

where w(ξ) = (∫ℝ |D2 f (x, ξ)|p dx)1/p . Then we see that I1 ≤ ( ∫ h

−1−ℓq

q

h

∞

1/q

(∫ w(ξ)dξ) )

∞

= (∫ h 0

0

0

Now as 1 − ℓ − 1/q = in Lemma 2.1 that

α q


0 such that 󵄩󵄩 γ 󵄩󵄩 󵄩󵄩D ψk 󵄩󵄩L

∞ (ℝ)

≤ N0 2kγ ,

∀γ ∈ ℕ,

∀k ∈ ℤ

(3.3)

and ∑ ψk ( y) = 1,

k∈ℤ

∀y ∈ (0, ∞).

(3.4)

We note that the existence of such a family of partition of unity {ψk }k∈ℤ is well known; see [5, Lemma 5, p. 43] for instance. As in (2.2), let us recall that φ ∈ C0∞ (ℝd ) is a standard cut-off function, which is radial and it satisfies 0 ≤ φ ≤ 1,

∫ φ(x)dx = 1,

and

supp(φ) ⊂ B1 .

ℝd

We denote φk (x) = 2kd φ(2k x) and 𝒜k,φ (g)(x) = ∫ g(z)φk (x − z)dz,

k ∈ ℤ.

ℝd ℓ d For a given g ∈ Bp,q ⃗ (ℝ ), we define the following Burenkov’s extension map ∞

E(g)(x, y) = ∑ ψk ( y)𝒜k,φ (g)(x), k=1

(x, y) ∈ ℝd+1 + .

(3.5)

Lemma 3.3 (Boundedness of the extension map). Let p⃗ ∈ [1, ∞)d , q ∈ [1, ∞), α ∈ (−1, ℓ d 1 d+1 q − 1), ℓ = 1 − 1+α , and let E be defined as in (3.5). Then E : Bp,q ⃗ (ℝ+ , μ) and ⃗ (ℝ ) → Wp,q q there exists N = N(p,⃗ q, α, d) > 0 such that 󵄩󵄩 󵄩 󵄩󵄩E(g)󵄩󵄩󵄩W 1 (ℝd+1 ,μ) ≤ N‖g‖Bℓ (ℝd ) , + ⃗ p,q ⃗ p,q

ℓ d ∀g ∈ Bp,q ⃗ (ℝ ).

(3.6)

Moreover, 󵄩 󵄩 lim y−ℓ 󵄩󵄩󵄩E(g)(⋅, y) − g 󵄩󵄩󵄩L ⃗ (ℝd ) = 0 p

y→0+

and

spt(E(g)(x, ⋅)) ⊂ [0, 1) ∀x ∈ ℝd .

(3.7)

On trace theorems for weighted mixed-norm Sobolev spaces and applications

� 71

Proof. From (3.2), we note that spt(ψk ) ⊂ (0, 1) when k ∈ ℕ. Then ∀x ∈ ℝd .

spt(E(g)(x, ⋅)) ⊂ [0, 1],

(3.8)

On the other hand, from mixed-norm Young’s inequality (see [22, Theorem 2.1] for instance) that 󵄩󵄩 󵄩 󵄩󵄩𝒜k,φ (g)󵄩󵄩󵄩Lp⃗ (ℝd ) ≤ ‖φk ‖L1 (ℝd ) ‖g‖Lp⃗ (ℝd ) = ‖g‖Lp⃗ (ℝd ) . Therefore, for each y > 0, by Minkowski’s inequality and (3.4), ∞

󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩E(g)(⋅, y)󵄩󵄩󵄩Lp⃗ (ℝd ) ≤ ∑ ψk ( y)󵄩󵄩󵄩𝒜k,φ (g)󵄩󵄩󵄩Lp⃗ (ℝd ) k=1

∞

≤ ‖g‖Lp⃗ (ℝd ) ∑ ψk ( y) k=1

≤ ‖g‖Lp⃗ (ℝd ) . From this and due to (3.8) and α > −1, 1

󵄩󵄩 󵄩 α 󵄩󵄩E(g)󵄩󵄩󵄩Lp,q⃗ (ℝd+1 ,μ) ≤ ‖g‖Lp⃗ (ℝd ) (∫ y dy) +

1/q

0

= N‖g‖Lp⃗ (ℝd ) ≤ N‖g‖Bℓ

⃗ p,q

(3.9)

(ℝd ) ,

d+1 where N = N(α, q) > 0. Next, we control the Lp,q , μ)-norms of the derivatives of ⃗ (ℝ E(g). We note that from the definition of the sequence {Sk }k , the definition of μ, and (3.2), it follows that

E(g)(x, y) = ψ1 ( y)𝒜1,φ g(x),

x ∈ ℝd ,

y > 7/16

and 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩E(g)󵄩󵄩󵄩W 1 (ℝd ×(7/16,∞),μ) = N(α)󵄩󵄩󵄩E(g)󵄩󵄩󵄩W 1 (ℝd ×(7/16,1)) . ⃗ ⃗ p,q p,q We note that for all k = 0, 1, 2, . . ., 󵄩󵄩 k 󵄩 󵄩󵄩Dx E(g)󵄩󵄩󵄩L ⃗

d ×(7/16,1)

p,q (ℝ

= ‖ψ1 ‖Lq (7/16,1) ‖Dkx 𝒜1,φ (g)Lp⃗ (ℝd ) ≤ N‖g‖Lp⃗ (ℝd ) .

On the other hand, 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩Dy E(g)󵄩󵄩󵄩Lp,q⃗ (ℝd ×(7/16,∞) = ‖Dy ψ1 ‖Lq (7/16,1) 󵄩󵄩󵄩𝒜1,φ (g)󵄩󵄩󵄩Lp⃗ (ℝd ) ≤ N‖g‖Lp⃗ (ℝd ) .

72 � T. Phan Therefore, we conclude that 󵄩 󵄩󵄩 󵄩󵄩DE(g)󵄩󵄩󵄩Lp,q⃗ (ℝd ×(7/16,∞),μ) ≤ N‖g‖Lp⃗ (ℝd ) .

(3.10)

Next, we consider the case when 0 < y ≤ 7/16. We recall the definition of ω(δ, g)p⃗ in (2.3). Applying Lemma 2.4, we infer that ∞

󵄩 󵄩󵄩 󵄩 k󵄩 󵄩󵄩Dx E(g)(⋅, y)󵄩󵄩󵄩Lp⃗ (ℝd ) ≤ ∑ ψk ( y)2 󵄩󵄩󵄩𝒜k,Dφ (g)󵄩󵄩󵄩Lp⃗ (ℝd ) k=1 ∞

≤ N ∑ ψk ( y)2k ω(2−k , g)p⃗ . k=1

Then it follows from Lemma 2.5 that 1/q

∞

q q 󵄩󵄩 󵄩 kq −k 󵄩󵄩Dx E(g)󵄩󵄩󵄩Lp,q⃗ (ℝd ×(0,7/16),μ) ≤ N( ∑ ‖ψk ‖L ((0,1),μ) 2 ω(2 , g)p⃗ ) q k=1 ∞

q(1− 1+α )k q

≤ N( ∑ 2 ≤

k=1 N‖g‖(2) ℓ Bp,q ⃗

q

1/q

ω(2−k , g)p⃗ )

≤ N‖g‖Bℓ ,

(3.11)

⃗ p,q

where N = N(d, p,⃗ q, α) > 0. Now, we control ‖Dy E(g)‖Lp,q⃗ (ℝd ×(0,7/16),μ) . For y ∈ (0, 7/16), we see that ψk ( y) = 0 for all k ∈ ℤ and k ≤ 0. Then it follows from (3.4) that ∞

∑ ψk ( y) = 1

k=1

∞

∑ ψ′k ( y) = 0.

and then

k=1

This in turn implies ∞

∞

k=1

k=1

Dy E(g)(x, y) = ∑ ψ′k ( y)𝒜k,φ (g)(x) = ∑ ψ′k ( y)[𝒜k,φ (g)(x) − g(x)], for each (x, y) ∈ ℝd × (0, 7/16). From the last formula, (3.3), Lemma 2.4, and the first assertion in Lemma 2.5, it follows that 󵄩󵄩 󵄩 󵄩󵄩Dy E(g)(⋅, y)󵄩󵄩󵄩Lp,q⃗ (ℝd ×(0,7/16),μ)

1/q

∞

󵄩 󵄩q 󵄩 󵄩q ≤ N(q)( ∑ 󵄩󵄩󵄩ψ′k ( y)󵄩󵄩󵄩L ((0,7/16),μ) 󵄩󵄩󵄩𝒜k,φ (g) − g 󵄩󵄩󵄩L (ℝd ) ) q p⃗ k=1

∞

≤ N( ∑ 2 k=1

k(1− 1+α )q q

q

ω(2−k , g)p⃗ )

1/q

On trace theorems for weighted mixed-norm Sobolev spaces and applications

� 73

≤ N‖g‖(2) Bℓ

⃗ p,q

(3.12)

≤ N‖g‖Bℓ . ⃗ p,q

Now, by collecting (3.10), (3.11), and (3.12), we obtain (3.13)

󵄩 󵄩󵄩 󵄩󵄩DE(g)󵄩󵄩󵄩Lp,q⃗ (ℝd+1 ,μ) ≤ N‖g‖Bℓ⃗ + p,q

for N = N(p,⃗ q, α, d) > 0. Hence, (3.6) follows from the estimates (3.9) and (3.13). Now, observe that second assertion in (3.7) is verified in (3.8); it remains to prove the first assertion in (3.7). For each y ∈ (0, 1/4), let s = s( y) ∈ ℕ such that 2−(s+1) < y < 2−s . Then we see that 󵄩󵄩 s+1 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩E(g)(⋅, y) − g 󵄩󵄩󵄩Lp⃗ (ℝd ) = 󵄩󵄩󵄩 ∑ ψk ( y)𝒜k,φ (g) − g 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩k=s−1 󵄩Lp⃗ (ℝd ) s+1

󵄩 󵄩 ≤ ∑ ψk ( y)󵄩󵄩󵄩𝒜k,φ (g) − g 󵄩󵄩󵄩L ⃗ (ℝd ) p

k=s−1

s+1

≤ N ∑ ψk ( y)ω(2−k , g)p⃗ k=s−1

s+1

≤ Nyℓ ∑ 2kℓ ω(2−k , g)p⃗ . k=s−1

We observe from Lemma 2.5 that lim 2kℓ ω(2−k , g)p⃗ = 0.

k→∞

Moreover, as y → 0+ we have s → ∞. Therefore, 󵄩 󵄩 lim y−ℓ 󵄩󵄩󵄩E(g)(⋅, y) − g 󵄩󵄩󵄩L ⃗ (ℝd ) = 0, p

y→0+

and the proof of the lemma is completed. Proof of Theorem 1.2. From Lemma 2.3 and Lemma 3.2, it follows that the trace map T : 1 d ℓ d Wp,q ⃗ (ℝ+ , μ) → Bp,q ⃗ (ℝ ) is well-defined and (1.1) holds. Therefore, (i) of Theorem 1.2 is proved. On the other hand, (ii) of Theorem 1.2 follows from Lemma 3.3.

4 Proofs of Corollary 1.4 and Corollary 1.5 This section provides the proofs of Corollary 1.4 and Corollary 1.5, which are applications of Theorem 1.2. We first start with the proof of Corollary 1.4.

74 � T. Phan Proof of Corollary 1.4. Let v = E(g), where E is defined in Theorem 1.2. We have v ∈ 1 d+1 Wp,q ⃗ (ℝ+ , μ) and ‖v‖W 1

⃗ p,q

(ℝd+1 + ,μ)

≤ N‖g‖Bℓ

⃗ p,q

(ℝd )

1 d+1 where N = N(d, p,⃗ q, α) > 0. Let w = u − v, and we see that u ∈ Wp,q ⃗ (ℝ+ , μ) is a weak

1 d+1 solution of (1.3) if and only if w ∈ Wp,q ⃗ (ℝ+ , μ) is a weak solution of

{

L w(x, y) = Di ( yβ Gi ) + √λyβ f ̃

w(x, 0) = 0

(x, y) ∈ ℝd × ℝ+ ; x ∈ ℝd ,

(4.1)

where f ̃ = f − √λv

and

Gi = Fi + aij Dj v,

i = 1, 2, . . . , d + 1.

Now, we apply [9, Remark 2.7] and [9, Theorem 2.8] to (4.1) to find δ > 0 and λ0 > 0 1 d+1 such that the existence, uniqueness, and estimates of solutions w ∈ Wp,q ⃗ (ℝ+ , μ) to the equation for (4.1) follow when λ ≥ λ0 R−2 0 and (1.4) holds. In particular, we have ≤ N[‖G‖Lp,q⃗ (ℝd+1 + ‖f ̃‖Lp,q⃗ (ℝd+1 ] ‖Dw‖Lp,q⃗ (ℝd+1 + √λ‖w‖Lp,q⃗ (ℝd+1 + ,μ) + ,μ) + ,μ) + ,μ) for λ ≥ λ0 R−2 0 . From this and by the triangle inequality, we obtain ‖Du‖Lp,q⃗ (ℝd+1 + √λ‖u‖Lp,q⃗ (ℝd+1 + ,μ) + ,μ) ≤ N[‖F‖Lp,q⃗ (ℝd+1 + ‖f ‖Lp,q⃗ (ℝd+1 ] + ,μ) + ,μ) + N[‖Dv‖Lp,q⃗ (ℝd+1 + √λ‖v‖Lp,q⃗ (ℝd+1 ]. + ,μ) + ,μ) The proof of Corollary 1.4 is completed. Next, we prove Corollary 1.5 Proof of Corollary 1.5. Since Ω is a bounded Lipschitz domain, it follows from the Stein extension theorem (see [28, p. 181]) that there is a linear, bounded extension map E0 : Wp1 (Ω) → Wp1 (ℝd ) satisfying 󵄩󵄩 󵄩 󵄩󵄩E0 (u)󵄩󵄩󵄩W 1 (ℝd ) ≤ N(p, n, Ω)‖u‖Wp1 (Ω) , p

∀u ∈ Wp1 (Ω).

1 1 As E0 is linear, we see that E0 (u) ∈ Wp,q (ℝd × (0, 1), μ) for all u ∈ Wp,q (Q, μ) and

󵄩󵄩 󵄩 󵄩󵄩E0 (u)󵄩󵄩󵄩W 1

d ×(0,1),μ)

p,q (ℝ

≤ N(p, n, Ω)‖u‖Wp,q 1 (Q,μ) ,

1 ∀u ∈ Wp,q (Q, μ).

Next, let ϕ0 ∈ Cc∞ (ℝ) be the standard cut-off function such that ϕ0 ( y) = 0 for y ≥ 1 and ℓ 1 (Ω) as ϕ0 ( y) = 1 for y ∈ [0, 1/2]. We then define TΩ : Wp,q (Q, μ) → Bp,q

On trace theorems for weighted mixed-norm Sobolev spaces and applications

� 75

TΩ (u) = T ∘ E0 (u)̄ |Ω , 1 ̄ y) = u(x, y)ϕ0 ( y), x ∈ Ω, y ∈ ℝ+ , and T is defined as in for u ∈ Wp,q (Q, μ), where u(x, Theorem 1.2. It is then clear that the assertion in Corollary 1.5 follows.

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Juncheng Wei and Yuanze Wu

Sharp stability of the logarithmic Sobolev inequality in the critical point setting Dedicated to the memory of David R. Adams

Abstract: In this paper, we consider the Euclidean logarithmic Sobolev inequality 2 d 󵄨 󵄨2 󵄨 󵄨 ‖∇u‖2L2 (ℝd ) ), ∫ 󵄨󵄨󵄨u(x)󵄨󵄨󵄨 log󵄨󵄨󵄨u(x)󵄨󵄨󵄨dx ≤ log( 4 πde

ℝd

where u ∈ W 1,2 (ℝd ) with d ≥ 2 and ‖u‖L2 (ℝd ) = 1. It is well known that extremal functions of this inequality are precisely the Gaussians d |x|2 σ gσ,z (x) = (πσ)− 2 g∗ (√ (x − z)) with g∗ (x) = e− 2 . 2

We prove that if u ≥ 0 satisfying (ν − 21 )c0 < ‖u‖2H 1 (ℝd ) < (ν + 21 )c0 and ‖ − Δu + u −

2u log |u|‖H −1 ≤ δ, where c0 = ‖g1,0 ‖2H 1 (ℝd ) , ν ∈ ℕ, and δ > 0 sufficiently small, then 󵄩 󵄩 distH 1 (u, ℳν ) ≲ 󵄩󵄩󵄩−Δu + u − 2u log |u|󵄩󵄩󵄩H −1 ,

which is optimal in the sense that the order of the right-hand side is sharp, where ν

d

ℳ = {(g1,0 (⋅ − z1 ), g1,0 (⋅ − z2 ), . . . , g1,0 (⋅ − zν )) | zi ∈ ℝ }.

Our result provides an optimal stability of the Euclidean logarithmic Sobolev inequality in the critical point setting. Keywords: Euclidean logarithmic Sobolev inequality, optimal stability, critical point setting MSC 2020: Primary 35A23, Secondary 35B35

Acknowledgement: The first author is partially supported by NSERC of Canada. Juncheng Wei, Department of Mathematics, University of British Columbia, Vancouver V6T 1Z2, BC, Canada, e-mail: [email protected] Yuanze Wu, School of Mathematics, China University of Mining and Technology, Xuzhou 221116, P.R. China, e-mail: [email protected] https://doi.org/10.1515/9783110792720-004

78 � J. Wei and Y. Wu

1 Introduction A fundamental task in understanding functional inequalities arise in the calculus of variations, geometry, etc., is to study the best constants, the classification of extremal functions, as well as their qualitative properties for parameters in the full region, since such functional inequalities are crucial in understanding nonlinear partial differential equations (Nonlinear PDEs for short) by virtue of the complete knowledge of the best constants, extremal functions, and qualitative properties. The most well-studied functional inequality in the community of Nonlinear PDEs is the Sobolev inequality, whose classical one with exponent 2 states that for any u ∈ H 1 (ℝd ) with d ≥ 3, there holds S( ∫ |u|

2d d−2

dx)

d−2 d

≤ ∫ |∇u|2 dx,

(1.1)

ℝd

ℝd

where S > 0 is a constant, which is only dependent of the dimension and H 1 (ℝd ) is the classical Sobolev space given by 2d

H 1 (ℝd ) = {u ∈ L d−2 (ℝd ) | |∇u| ∈ L2 (ℝd )}. It has been proved in [3, 56] that S = πd(d − 2)(

Γ( d2 ) Γ(d)

)

2 d

is optimal and the extremal functions, which are called the Aubin–Talenti bubbles in the literature, are given by 1 ) Uλ,z,c (x) = c( 2 1 + λ |x − z|2

d−2 2

,

where λ > 0, c ∈ ℝ, and z ∈ ℝd . Once a functional inequality is well understood for its best constant and extremal functions, it is natural to concern its stability, which is increasingly interested in recent years by its important applications in understanding many Nonlinear PDEs, such as the fast diffusion equation, the Keller–Segel equation, and so on. The basic question one wants to address in this aspect is the following (cf. [31]): (Q) Suppose we are given a functional inequality for which minimizers are known. Can we prove, in some quantitative way, that if a function “almost attains the equality” then it is close (in some suitable sense) to one of the minimizers? Such study was first raised by Brezis and Lieb in [9] for the classical Sobolev inequality (1.1) as an open question, which was settled by Bianchi and Egnell in [7] by proving that

Sharp stability of the logarithmic Sobolev inequality in the critical point setting

dist2H 1 (u, 𝒵 ) ≲ ‖∇u‖2L2 (ℝd ) − S‖u‖2

2d

L d−2 (ℝN )

� 79

,

where ‖ ⋅ ‖Lp (ℝN ) is the usual norm in the Lebesgue space Lp (ℝd ) and 𝒵 = {Uλ,z,c | (λ, z, c) ∈ ℝ+ × ℝ

d+1

}.

Unlike Bianchi and Egnell’s study [7] in the functional inequality setting, in recent years, Figalli et al. initiated the study on the stability of the classical Sobolev inequality (1.1) in the critical point setting, that is, studying the stability of the Euler–Lagrange equation of the classical Sobolev inequality (1.1). The study on the stability of the classical Sobolev inequality (1.1) in the critical point setting is more challenging since the Euler–Lagrange equation of the classical Sobolev inequality (1.1) has sign-changing solutions. Moreover, the “almost” solutions of the Euler–Lagrange equation of the classical Sobolev inequality (1.1) may decompose into several parts at infinity (cf. [55]). By setting the study in a suitable way, Figalli et al. proved that (1) (Ciraolo–Figalli–Maggi [17]) Let d ≥ 3 and u ∈ D1,2 (ℝd ) be positive such that d 4 ‖∇u‖2L2 (ℝd ) ≤ 32 S 2 and ‖Δu + |u| d−2 u‖H −1 ≤ δ for some δ > 0 sufficiently small. Then 4 󵄩 󵄩 distD1,2 (u, ℳ0 ) ≲ 󵄩󵄩󵄩Δu + |u| d−2 u󵄩󵄩󵄩H −1 ,

where ℳ0 = {U[z, λ] | z ∈ ℝd , λ > 0}. (2) (Figalli–Glaudo [32]) Let u ∈ D1,2 (ℝd ) be nonnegative such that 1 d 1 d (ν − )S 2 < ‖u‖2D1,2 (ℝN ) < (ν + )S 2 2 2 4

and ‖Δu + |u| d−2 u‖H −1 ≤ δ for some δ > 0 sufficiently small. Then, for 3 ≤ d ≤ 5, 4 󵄩 󵄩 distD1,2 (u, ℳν0 ) ≲ 󵄩󵄩󵄩Δu + |u| d−2 u󵄩󵄩󵄩H −1

where ν

d

ℳ0 = {(U[z1 , λ1 ], U[z2 , λ2 ], . . . , U[zν , λν ]) | zi ∈ ℝ , λi > 0}.

All the above results are optimal in the sense that the orders of the right-hand sides in the above estimates are sharp, while the optimal stability of the classical Sobolev inequality (1.1) in the critical point setting for the case N ≥ 6 was left in [32] as an open problem, which was solved by Deng, Sun, and Wei in [22] very recently, by proving the following optimal stability: 1

distD1,2 (u, ℳν0 )

{‖Δu + |u|u‖H −1 | ln(‖Δu + |u|u‖H −1 )| 2 , d+2 ≲{ 4 2(d−2) d−2 {‖Δu + |u| u‖H −1 ,

d = 6; d ≥ 7.

80 � J. Wei and Y. Wu According to the important applications in understanding many Nonlinear PDEs, several other famous functional inequalities, such as the Gagliardo–Nirenberg–Sobolev inequality (cf. [12, 13, 26, 27, 50, 52, 53]), the Hardy–Littlewood–Sobolev inequality (cf. [6, 19, 35, 38, 41, 44, 51]), the Caffarelli–Kohn–Nirenberg inequality (cf. [10, 14, 18, 23–25, 29, 46, 58, 59, 62]), the Lp -Sobolev inequality (cf. [1, 15, 16, 33, 34, 36, 37, 49]), and so on, are also wildly studied on the best constants, the classification of extremal functions and stability. However, according to their multiparameters or no-Hilbert, most of these functional inequalities are far from well understood except for some special cases. It is worth pointing out that, besides the classical Sobolev inequality (1.1), the Euclidean logarithmic Sobolev inequality, ∫ |u|2 log |u|dx ≤ ℝd

2 d log( ‖∇u‖2L2 (ℝd ) ) 4 πde

(1.2)

where u ∈ W 1,2 (ℝd ) with d ≥ 2 and ‖u‖L2 (ℝd ) = 1 is also well understood for the best constants and the classification of extremal functions. It is well known (cf. [21, 61]) that this inequality is optimal and extremal functions of (1.2) are precisely the Gaussians d |x|2 σ gσ,z (x) = (πσ)− 2 g∗ (√ (x − z)) with g∗ (x) = e− 2 . 2

(1.3)

Moreover, it is equivalent to the Gross logarithmic inequality (cf. [39]) with respect to Gaussian weight ∫ |g|2 log |g|dμ ≤ ∫ |∇g|2 dμ ℝd

ℝd

d

with ∫ |g|2 dμ = 1 and dμ = (2π)− 2 e−

|x|2 2

.

ℝd

The Euclidean logarithmic Sobolev inequality has another two different forms: ∫ |v|2 log |v|2 dx ≤ ℝd

a2 ‖∇v‖2L2 (ℝd ) + (log ‖v‖2L2 (ℝd ) − d(1 + log a))‖v‖2L2 (ℝd ) π

(1.4)

for all a > 0 and any function v ∈ W 1,2 (ℝd ), and ∫ |w|2 log |w|2 dx + ℝd

d 1 (1 + log(2π)) ≤ ‖∇w‖2L2 (ℝd ) 2 2

(1.5)

for any function w ∈ W 1,2 (ℝd ), d ≥ 2, with ‖w‖L2 (ℝd ) = 1. Equations (1.4) and (1.5) are equivalent by the relation w(x) = v( √a x) and they are established in [45, Theorem 8.14] 2π and [60, Theorem 1.2], respectively. However, (1.2) is optimal since it can be obtained by minimizing the right-hand side of (1.4) for a > 0 (cf. [27]). The stability of the Euclidean logarithmic Sobolev inequality (1.2) in the functional inequality setting has also been widely studied; cf. [8, 11, 12, 27, 28, 30, 42, 43, 61] and

Sharp stability of the logarithmic Sobolev inequality in the critical point setting

� 81

the references therein. It is worth pointing out that most of these results are devoted to more general version of the logarithmic Sobolev inequality in the probability setting, which contains the Euclidean logarithmic Sobolev inequality (1.2) as a special case for |x|2

d

the Gaussians measure dμ = (2π)− 2 e− 2 . Thus it is natural to consider the stability of the Euclidean logarithmic Sobolev inequality (1.2) in the critical point setting, as that for the classical Sobolev inequality (1.1). Let ℰ (u) =

d 2 log( ‖∇u‖2L2 (ℝd ) ) − ∫ |u|2 log |u|dx. 4 πde ℝd

Then critical points of ℰ (u) in H 1 (ℝd ) with the finite energy on the smooth manifold 1

d

𝒮 = {u ∈ H (ℝ ) | ‖u‖L2 (ℝd ) = 1}

satisfy ∫ℝd |u|2 | log |u|2 |dx < +∞ and the following Euler–Lagrange equation: −(

d )Δu − (1 + 2σu )u = 2u log |u| in ℝd , 2‖∇u‖2L2 (ℝd )

(1.6)

where σu is a part of unknown and appears in (1.6) as the Lagrange multiplier. Thus (1.6) is the Euler–Lagrange equation of (1.2). Let 1

2 d ) , where λ = ( 2 2‖∇u‖L2 (ℝd )

d 2

uλ (x) = λ u(λx)

then by (1.6) and a direct calculation, uλ satisfies −Δuλ + (d log λ − 1 − 2σu )uλ = 2uλ log |uλ |

in ℝd .

(1.7)

Let uλ∗ (x) = αu uλ (x)

d

where αu = e 2 log λ−σu −1 ,

then by (1.7) and a direct calculation, uλ∗ satisfies the equation −Δu + u = 2u log |u|

in ℝd .

(1.8)

Thus the logarithmic Schrödinger equation (1.8) can be seen as the Euler–Lagrange equation of the Euclidean logarithmic Sobolev inequality (1.2). It has been proved in [20, 57] that the Gaussian g=e

1+d 2

g∗

(1.9)

82 � J. Wei and Y. Wu is the unique positive solution of the logarithmic Schrödinger equation (1.8), which satisfies u(x) → 0 as |x| → +∞, where g∗ is given by (1.3). We remark that any solution of (1.8) satisfying ∫ℝd |u|2 | log |u|2 |dx < +∞ must exponentially decay to zero as |x| → +∞ by the standard applications of the maximum principle. Moreover, it has been proved in [20, Theorem 1.3] (see also [40, Theorem 7.5]) that g is nondegenerate in the sense that Ker(ℒ) = span{𝜕xj g} where 2

ℒ = −Δ + |x| − (d + 2)

(1.10)

is the linearized operator of (1.8) at g. Note that (1.8) is invariant under translations, thus the smooth manifold d

ℳ = {g(⋅ − z) | z ∈ ℝ }

contains all positive solutions of (1.8) satisfying ∫ℝd |u|2 | log |u|2 |dx < +∞.

Let {un }, un ≥ 0 for all n, be bounded in H 1 (ℝd ) and almost solves (1.8), that is, ‖ − Δun + un − 2un log |un |‖H −1 → 0 as n → ∞. Then it is easy to see that {|un |2 | log |un |2 |} is bounded in L1 (ℝd ). Thanks to the Brezis–Lieb lemma [54, Lemma 3.1] and the positivity of the energy of the unique positive solution of (1.8) (cf. [54, Lemma 3.3]), it is standard (cf. [2, Proposition 3.1]) to prove the following Struwe’s decomposition of {un }. Proposition 1.1. There exists ν ∈ ℕ and y1,n , y2,n , . . . yν,n ⊂ ℝd such that 󵄩󵄩 󵄩󵄩 ν 󵄩󵄩 󵄩 󵄩󵄩un − ∑ g(⋅ − yi,n )󵄩󵄩󵄩 →0 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩H 1 (ℝd ) i=1

as n → ∞.

By Proposition 1.1, for nonnegative functions un with uniform H 1 -bound, we know that there exists ν ∈ ℕ such that {un } will be close to the manifold ν

d

ℳ = {(g(⋅ − z1 ), g(⋅ − z2 ), . . . , g(⋅ − zν )) | zi ∈ ℝ }

in the H 1 -topology. Thus it is natural to ask, if u ≥ 0 satisfy 1 1 (ν − )c0 < ‖u‖2H 1 (ℝd ) < (ν + )c0 2 2

󵄩 󵄩 and 󵄩󵄩󵄩−Δu + u − 2u log |u|󵄩󵄩󵄩H −1 ≤ δ,

where c0 = ‖g‖2H 1 (ℝd ) , ν ∈ ℕ, and δ > 0 sufficiently small, can we obtain a quantitative version of Proposition 1.1 by optimally controlling distH 1 (u, ℳν ) by ‖ − Δu + u − 2u log |u|‖H −1 ? The purpose of this paper is to give a positive answer to this natural question and our main result reads as follows. Theorem 1.2. Let u ≥ 0 satisfy 1 1 (ν − )c0 < ‖u‖2H 1 (ℝd ) < (ν + )c0 2 2

and

󵄩󵄩 󵄩 󵄩󵄩−Δu + u − 2u log |u|󵄩󵄩󵄩H −1 ≤ δ,

Sharp stability of the logarithmic Sobolev inequality in the critical point setting

� 83

where c0 = ‖g‖2H 1 (ℝd ) , ν ∈ ℕ and δ > 0 sufficiently small. Then 󵄩 󵄩 distH 1 (u, ℳν ) ≲ 󵄩󵄩󵄩−Δu + u − 2u log |u|󵄩󵄩󵄩H −1 ,

(1.11)

where ν

d

ℳ = {(g1,0 (⋅ − z1 ), g1,0 (⋅ − z2 ), . . . , g1,0 (⋅ − zν )) | zi ∈ ℝ }.

Moreover, (1.1) is optimal in the sense that the order of the right-hand side is sharp. Remark 1.3. The main ideas in proving Theorem 1.2 is similar to that of [22, 62], that is, we choose special y1,δ , y2,δ , . . . yν,δ ⊂ ℝd and decompose u − ∑νi=1 g(⋅ − yi,δ ) into two parts, where the first part is very regular, which can be well estimated and the second part is much smaller than the first part. However, the logarithmic nonlinearity is very different from the power-type one dealt with in [22, 62]; thus we need to be careful to estimate it and make sure that the strategy in [22, 62] works for (1.8). Notation Throughout this paper, C and C ′ are indiscriminately used to denote various absolutely positive constants. σ, σ ′ , σ ′′ are indiscriminately used to denote various absolutely positive constants, which can be taken arbitrary small. a ∼ b means that C ′ b ≤ a ≤ Cb and a ≲ b means that a ≤ Cb.

2 Preliminaries Let us consider the equation −Δu + u = 2u log |u| + f ,

in ℝd ,

(2.1)

where f ∈ H −1 and u ∈ H 1 (ℝd ) is nonnegative and satisfies 1 1 (ν − )c0 < ‖u‖2H 1 (ℝd ) < (ν + )c0 2 2

for some fixed ν ∈ ℕ

with c0 = ‖g‖2H 1 (ℝd ) . By multiplying (2.1) with u and integrating by parts, it is easy to see

that ∫ℝd |u|2 | log |u|2 |dx ≲ 1. Moreover, by Proposition 1.1, there exists z1,f , z2,f , . . . zν,f ⊂

ℝd such that

󵄩󵄩 󵄩󵄩 ν 󵄩󵄩 󵄩 󵄩󵄩u − ∑ g(⋅ − zi,f )󵄩󵄩󵄩 →0 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩H 1 (ℝd ) i=1 It follows that

as ‖f ‖H −1 → 0.

84 � J. Wei and Y. Wu 󵄩󵄩2 󵄩󵄩 ν 󵄩󵄩 󵄩󵄩 󵄩 →0 cf = inf 󵄩󵄩u − ∑ g(⋅ − zi )󵄩󵄩󵄩 d 󵄩󵄩 1 d 󵄩 zi ∈ℝ 󵄩 i=1 󵄩H (ℝ ) 󵄩

as ‖f ‖H −1 → 0.

(2.2)

Thus, by solving the minimizing problem (2.2) in a standard way (cf. [5]), we can write u = ∑νi=1 g(⋅ − yi,f ) + ρf , where {yi,f } is the solution of (2.2) and the remaining term ρf satisfies ‖ρf ‖2H 1 (ℝd ) = cf → 0

as ‖f ‖H −1 → 0

and the orthogonal conditions ⟨ρf , 𝜕xl gj,f ⟩H 1 (ℝd ) = 0,

l = 1, 2, . . . , d and j = 1, 2, . . . , ν.

(2.3)

For the sake of simplicity, we denote g(⋅ − yi,f ) by gi,f . Clearly, by (2.1), ρf satisfies 2(∑ν g +ρ )

ν f −1 i=1 i,f {−Δρf + ρf = (log(∑ν g +ρ ))−1 − 2 ∑i=1 gi,f log gi,f + f , in H , f i=1 i,f { {⟨ρf , 𝜕xl gi,f ⟩H 1 (ℝd ) = 0, l = 1, 2, . . . , d and i = 1, 2, . . . , ν.

(2.4)

It is convenient to write (2.4) as follows: {ℒf (ρf ) = E + N(ρf ) + f , in H −1 , { {⟨ρf , 𝜕xl gi,f ⟩H 1 (ℝd ) = 0, l = 1, 2, . . . , d and i = 1, 2, . . . , ν,

(2.5)

where ν

ℒf = −Δ − 1 − 2 log(∑ gi,f )

(2.6)

i=1

is the linear operator, ν

ν

ν

i=1

i=1

i=1

E = 2(∑ gi,f ) log(∑ gi,f ) − 2 ∑ gi,f log gi,f

(2.7)

is the error and ν

ν

ν

ν

i=1

i=1

i=1

N(ρf ) = 2(∑ gi,f + ρf ) log(∑ gi,f + ρf ) − 2(∑ gi,f ) log(∑ gi,f ) i=1

ν

− 2(1 + log(∑ gi,f ))ρf i=1

is the nonlinear part.

(2.8)

Sharp stability of the logarithmic Sobolev inequality in the critical point setting

� 85

3 Proof of (1.11) Let ηi = min ηi,j j=i̸

and

η = min ηi ,

(3.1)

i

where ηi,j = |yj,f − yi,f |. Then by (2.1), (2.2) and the uniqueness of g, we can easily see that η = min{|yi,f − yj,f |} → +∞ i=j̸

as δ → 0,

(3.2)

where we denote ‖f ‖H −1 by δ for the sake of simplicity. Let Ωi = {x ∈ ℝd | gi,f ≥ gj,f

for all j ≠ i}.

(3.3)

Then ℝd = ⋃νi=1 Ωi and ∑νj=1 gj,f ∼ gi,f in Ωi . Moreover, we also introduce Πc,i,j : 2(yi,f − yj,f )x + c = 0

and 𝕃i,j : (x − yi,f ) × (yi,f − yj,f ) = 0,

(3.4)

where c ∈ ℝ are constants. We take xc,i,j ∈ Πc.,i,j ∩ 𝕃i,j ∩ Ωi and denote |xc,i,j − yi,f | = ±αc,i,j ηi,j ,

(3.5)

where αc,i,j > − 21 with > 0, ⟨xc,i,j − yi,f , yi,f − yj,f ⟩ > 0, αc,i,j { < 0, ⟨xc,i,j − yi,f , yi,f − yj,f ⟩ < 0. Then |xc,i,j − yj,f | = (1 + αc,i,j )ηi,j . Lemma 3.1. We have ν

1

1

2

2

E ∼ ∑ χΩi gi,f (∑ χΠc,i,j e−(αc,i,j + 2 )ηi,j log(1 + e(αc,i,j + 2 )ηi,j )) i=1

j=i̸

(3.6)

as ‖f ‖H −1 → 0. Proof. By (2.7), we have ν

E = 2 ∑ gi,f log(1 + ∑ i=1

By (3.7), we write

j=i̸

gj,f gi,f

).

(3.7)

86 � J. Wei and Y. Wu

E = 2gi,f log(1 + ∑ l=i̸

=: I + II

gl,f gi,f

) + 2 ∑ gj,f log(1 + ∑ j=i̸

l=j̸

gl,f gj,f

)

in Ωi , where Ωi is given by (3.3). For I, we have I ∼ ∑ gj,f

in Ωi .

j=i̸

For gj,f log(1 + ∑l=j̸

gl,f gj,f

) with j ≠ i, we write

gj,f log(1 + ∑ l=j̸

gl,f gj,f

) = gj,f log(1 +

gi,f + ∑l=j,i ̸ gl,f gj,f

).

Then in Ωi , we have E ∼ II ∼ ∑ gj,f log(1 + j=i̸

gi,f gj,f

(3.8)

).

Let us consider the function gi,f gj,f

,

∀j ≠ i.

By (1.3), (1.9) and direct calculations, we have ∇(

gi,f gj,f

1

2

2

) = (yi,f − yj,f )e− 2 (|x−yi,f | −|x−yj,f | ) .

(3.9)

Then by (3.9), gi,f gj,f

≡ const. on the hyperplane Πc,i,j for all c ∈ ℝ,

where Πc,i,j is given by (3.4). Thus, we have gi,f gj,f

1

2

≡ e(αc,i,j + 2 )|yi,f −yj,f | ,

∀x ∈ Πc,i,j ∩ Ωi ,

(3.10)

where αc,i,j > − 21 is given by (3.5). It follows from (3.8), (3.10) and the fact that Πc,i,j ⊥ 𝕃i,j for all c ∈ ℝ that (3.6) holds. As that in [22, 62], we decompose ρf = ϕf + φf , where ϕf is the solution of the following equation:

Sharp stability of the logarithmic Sobolev inequality in the critical point setting

d ν ℒf (ϕf ) = E + N(ϕf ) − ∑j=1 ∑i=1 aj,i 𝜕xj gi,f , in H −1 , { ⟨ϕf , 𝜕xj g⟩H 1 (ℝd ) = 0, l = 1, 2, . . . , d

� 87

(3.11)

with aj,i being the Lagrange multipliers given by aj,i ∼ ⟨E + N(ϕf ), 𝜕xj gi,f ⟩L2 . To solve (3.11), let us first establish a good linear theory. Let gi,f ,d−1,j = e

1+d |zi,j |2 − 2 2

and 󵄨󵄨 yi,f + yj,f 󵄨 DR,i,j = {x ∈ Πc,i,j | 󵄨󵄨󵄨xc,i,j − 󵄨󵄨 2

󵄨󵄨 󵄨󵄨 󵄨󵄨 ≤ R}, 󵄨󵄨

where zi,j ⊥ 𝕃i,j and xc,i,j is given by (3.5). We define Ωi,j,d−1 = {x ∈ Ωi | gi,f ,d−1,j ≥ gi,f ,d−1,l , ∀l ≠ i, j}. Then as above, Ωi = ⋃j=i̸ Ωi,j,d−1 and ∑l=i̸ gi,f ,d−1,l ∼ gi,f ,d−1,j in Ωi,j,d−1 for all j, where Ωi is given by (3.3). We introduce the norms ν

‖u‖♮ = ∑ ∑( i=1 j=i̸

sup

Dcση,i,j ∩Ωi,j,d−1

|u| |u| ) + sup 1 2 −η2 ) 1−σ − (η gi,f Dση,i,j ∩Ωi,j,d−1 e 8 i,j gi,f ,d−1,j

and ν

‖u‖♯ = ∑ ∑(

sup

c i=1 j=i̸ Dση−1,i,j ∩Ωi,j,d−1

η2 |u| |u| + sup ). 1 2 2 g1−σ Dση−1,i,j ∩Ωi,j,d−1 e− 8 (ηi,j −η ) g i,f i,f ,d−1,j

Then 𝕏 = {u ∈ H 1 (ℝd ) | ‖u‖♯ < +∞}

and

𝕐 = {u ∈ L2 (ℝd ) | ‖u‖♮ < +∞},

are Banach spaces. Let us consider the following linear equation: ν

d

ℒf (ψ) = h − ∑i=1 ∑j=1 bj,i 𝜕xj gi,f ,

{

in ℝd ,

ϕ ∈ 𝕏⊥ ,

(3.12)

where ℒf is the linear operator given by (2.6), 𝕏⊥ = {u ∈ 𝕏 | ⟨u, 𝜕xj gi,f ⟩H 1 (ℝd ) = 0, j = 1, 2, . . . , d; i = 1, 2, . . . , ν}

(3.13)

88 � J. Wei and Y. Wu and bj,i are the Lagrange multipliers given by bj,i ∼ ⟨h, 𝜕xj gi,f ⟩L2 .

(3.14)

Lemma 3.2. As ‖f ‖H −1 → 0, (3.12) is unique solvable for every h ∈ 𝕐 with ‖ψ‖♯ +

∑νi=1 ∑dj=1 |bj,i | ≲ ‖h‖♮ .

Proof. By (1.3) and (1.9), it is easy to see that for R > 0 sufficiently large, ν

−(1 + 2 log(∑ gj,f )) ≥ 0 j=1

in (⋃νj=1 BR (yj,f ))c . It follows from ∑νj=1 gj,f ∼ gi,f in Ωi that −(1 + 2 log(∑νj=1 gj,f )) = |x − yi,f |2 + O(1) in Dση+2,i,j ∩ Ωi .

(3.15)

Note that by the definition of 𝕃i,j given by (3.4) and rotations, gi,f ,d−1,j is the unique solution of (1.8) in ℝd−1 . Thus, by (3.15) and rotations, the definitions of Πc,i,j and 𝕃i,j given by (3.4), ν

−Δgi,f ,d−1,j − (1 + 2 log(∑ gj,f ))gi,f ,d−1,j ≳ η2 gi,f ,d−1,j j=1

(3.16)

in Dση+2,i,j ∩ Ωi,j,d−1 . Similarly, ν

1−σ 1−σ −Δg1−σ i,f − (1 + 2 log(∑ gj,f ))gi,f ≳ gi,f j=1

(3.17)

in (⋃νj=1 BR (yj,f ))c . For every x ∈ Ωi,j,d−1 , by the fact that Πc,i,j ⊥ 𝕃i,j for all c ∈ ℝ, we can rewrite x = (αc,i,j , zi,j ), where αc,i,j > − 21 is given by (3.5) and zi,j ⊥ 𝕃i,j . Now, let 1

2

2

−2 − 8 (ηi,j −η ) ϕ(αc,i,j , zi,j ) = g1−σ gi,f ,d−1,j (1 − φ(αc,i,j )) i,f φ(αc,i,j ) + η e

where φ(αc,i,j ) is the unique solution of the following equation: ση−1 1 ση 1 ′′ ′ {−φ − φ + φ = 1, in ( ηi,j − 2 , ηi,j − 2 ), { ′ ση−1 1 φ ( η − 2 ) = φ′ ( ηση − 21 ) = 0. i,j i,j {

Then by (3.16) and (3.17), in (⋃νj=1 BR (yj,f ))c ,

Sharp stability of the logarithmic Sobolev inequality in the critical point setting − 1 (η2 −η2 )

{e 8 i,j −Δϕ − (1 + 2 log(∑ gj,f ))ϕ ≳ { 1−σ g , j=1 { i,f ν

gi,f ,d−1,j ,

αc,i,j ≤ αc,i,j ≥

� 89

ση−1 − 21 , ηi,j ση − 21 , ηi,j

which, together with the maximum principle, implies that |ψ| ≲ (‖h‖♮ + ‖ψ‖L∞ (𝜕(⋃νj=1 BR (yj,f )))c )ϕ(αc,i,j , zi,j )

(3.18)

in (⋃νj=1 BR (yj,f ))c for R > 0 sufficiently large. Based on the a priori estimate (3.18), we shall prove the a priori estimate ‖ψ‖♯ ≲ ‖h‖♮

uniformly as δ → 0.

(3.19)

Since the proofs for (3.19), based on the blow-up arguments, are standard nowadays (cf. [47, 48, 62]), we only sketch it here. We assume the contrary that there exists δn → 0, {ψn } solves (3.12) with {hn } ⊂ L2 (ℝd ) satisfying ‖ψn ‖♯ = 1, and ‖hn ‖♮ = on (1) as n → ∞. Since δn → 0, by (3.2), ν 1 d+1 log(∑ gj,f (x + yi,f )) → log g = − |x|2 + 2 2 j=1

in ℝd as n → ∞. Now, let ψi,n (x) = ϕn (x + yi,fn ), then by δn → 0 as n → ∞ and (3.14), it is standard to prove that ψi,n → ψi uniformly in every compact set of ℝd for every i, where ψi are bounded solutions of the following equation: −Δϕ + (|x|2 − d − 2)ϕ = 0

in ℝd .

By [20, Theorem 1.3] (see also [40, Theorem 7.5]), ψi = ∑dj=1 aj 𝜕xj g. On the other hand, by (1.3) and (1.9), we can pass to the limit in the orthogonal conditions of ψi,n given by (3.13), which implies that 0 = lim ⟨ψi,n , 𝜕xj g⟩H 1 (ℝd ) n→+∞

ν

= lim ∫ (2 + 2 log(∑ gj,fn ))ψn 𝜕xj gi,fn dx n→+∞

ℝd

j=1

= ∫ (3 + d − |x|2 )ψi 𝜕xj gdx ℝd

= ⟨ψi , 𝜕xj g⟩H 1 (ℝd )

90 � J. Wei and Y. Wu for all j = 1, 2, . . . , d. Since ⟨𝜕xi g, 𝜕xj g⟩H 1 (ℝd ) = ∫ (d + 3 + |x|2 )𝜕xi g𝜕xj gdx = 0, ℝd

we have ψi = 0, which implies that ψi,n → 0 uniformly in every compact set of ℝd for all i. Thus, by (3.18), we have ‖ψn ‖♮ = on (1), which is a contradiction. Thanks to the a priori estimate (3.19), by the Fredholm alternative, we know that the linear equation (3.12) is unique solvable for all h ∈ 𝕐. The estimate ∑νi=1 ∑dj=1 |bj,i | ≲ ‖h‖♮ comes from (3.1) and (3.14). By direct calculations, max e−(

2 αc,i,j 2

αc,i,j >− 21

+αc,i,j + 21 )η2i,j

1

1

2

2

log(1 + e(αc,i,j + 2 )ηi,j ) ∼ e− 8 ηi,j .

(3.20) 1

2

Thus, by taking σ > 0 sufficiently small and Lemma 3.1, we have ‖E‖♮ ≲ e− 8 η . We define 1

2

𝔹 = {ϕ ∈ 𝕏⊥ | ‖ϕ‖♯ ≤ Me− 8 η }

(3.21)

where M > 0 is a sufficiently large constant. Lemma 3.3. There exists M > 0 sufficiently large such that (3.11) has a unique solution 1 2 1 2 ϕf ∈ 𝔹 with ‖ϕf ‖♯ + ∑dj=1 ∑νi=1 |aj,i | ≲ e− 8 η as ‖f ‖H −1 → 0. Moreover, ‖ϕf ‖H 1 (ℝd ) ≲ e− 8 η as ‖f ‖H −1 → 0. Proof. The proof is standard nowadays, so we also sketch it here. For ϕ ∈ 𝔹, by (2.8), and the Taylor expansion, N(ϕ+ ) = 2 log(1 +

θϕ− )ϕ+ ∑νj=1 gj,f

and N(ϕ− ) = −2 log(1 −

θϕ− )ϕ− , ∑νj=1 gj,f

(3.22)

in Ωi , where Ωi is given by (3.3) and ϕ± = max{±ϕ, 0}. Note that for ϕ ∈ 𝔹, 1

2

2

ν e− 8 (ηi,j −η ) gi,f ,d−1,j ±ϕ± −σ g + sup ) ≲ ‖ϕ‖ ( sup ∑ ∑ ♯ i,f c η2 gi,f ∑νj=1 gj,f Dση−1,i,j ∩Ωi,j,d−1 i=1 j=i̸ Dση−1,i,j ∩Ωi

≲ η−2 .

(3.23)

Thus, by (3.22) and the symmetry of gi,f , ν |ϕ± | |ϕ± | 󵄩󵄩 󵄩 −2 + sup ) 󵄩󵄩N(ϕ± )󵄩󵄩󵄩♮ ≲ η ∑ ∑( sup 1−σ 1 2 −η2 ) − (η c Dση,i,j ∩Ωi,j,d−1 e 8 i,j i=1 j=i̸ Dση,i,j ∩Ωi gi,f gi,f ,d−1,j 1

2

≲ η−2 e− 8 η ,

Sharp stability of the logarithmic Sobolev inequality in the critical point setting

1

� 91

2

which implies that ‖N(ϕ)‖♮ ≲ η−2 e− 8 η . Now, we can solve (4.1) by the standard fixedpoint arguments in 𝔹 by choosing a sufficiently large M > 0. The estimate ‖ϕf ‖♯ + 1

2

∑dj=1 ∑νi=1 |aj,i | ≲ e− 8 η comes from Lemma 3.2. By multiplying (3.11) with ϕf on both sides 1

2

and integrating by parts and using the fact that ϕf ∈ 𝔹, we have ‖ϕf ‖2H 1 (ℝd ) ≲ e− 4 η as ‖f ‖H −1 → 0, which completes the proof. We recall that by (2.5) and (3.11), the remaining term φf = ρf − ϕf satisfies d ν ℒf (φf ) = N(ϕf + φf ) − N(ϕf ) + ∑j=1 ∑i=1 aj,i 𝜕xj gi,f + f , in H −1 , { ⟨φf , 𝜕xj gi,f ⟩H 1 (ℝd ) = 0, j = 1, 2, . . . , d and i = 1, 2, . . . , ν.

(3.24)

Moreover, it is well known (cf. [40, Theorem 7.5 and Remark 7.7]) that the eigenfunctions of the linear operator ℒ given by (1.10) forms an orthogonal basis in L2 (ℝd ), where the first eigenvalue is −2 with eigenspace span{g} and the second eigenvalue is 0 with eigenspace span{𝜕xj g}. Thus we can write ν

d

j=1

l=1

φf = ∑(cj gj,f + ∑ bl,j 𝜕xl gj,f ) + φ⊥ f ,

(3.25)

2 d where φ⊥ f is orthogonal to spanj,l {gj,f , 𝜕xl gj,f } in L (ℝ ).

Lemma 3.4. As ‖f ‖H −1 → 0, we have ν

󵄩 󵄩󵄩2 ‖φf ‖2L2 (ℝd ) ∼ ∑ |cj |2 + 󵄩󵄩󵄩φ⊥ f 󵄩 󵄩L2 (ℝd ) .

(3.26)

1 ⟨gj,f , ∇gj,f ⟩H 1 = − ∇yj,f ‖gj,f ‖2H 1 (ℝd ) = 0 2

(3.27)

j=1

Proof. Note that we have

for all j = 1, 2, . . . , ν. Thus, by (2.3), (3.25), and (3.27), d

0 = ∑(ci ⟨gi,f , 𝜕xl gj,f ⟩H 1 + ∑ bi,m ⟨𝜕xm gi,f , 𝜕xl gj,f ⟩H 1 ) + bl,j i=j̸

m=1

(3.28)

for all l = 1, 2, . . . , d and j = 1, 2, . . . , ν. Since |𝜕xl gi,f | ≲ ri,f gi,f ≲ g1−σ i,f for all l, where σ > 0 can be taken arbitrary small if necessary and ri,f = |x − yi,f |, by [4, Lemma 3.7],

92 � J. Wei and Y. Wu 󵄨 󵄨 󵄨 󵄨󵄨 1−σ ′ 1−σ 󵄨󵄨⟨gi,f , 𝜕xl gj,f ⟩H 1 󵄨󵄨󵄨 + 󵄨󵄨󵄨⟨𝜕xm gi,f , 𝜕xl gj,f ⟩H 1 󵄨󵄨󵄨 ≲ ∫ gi,f gj,f ℝd

≲ e−

1−σ ′′ 2

η2

(3.29)

for all i, j, l, m, which together with (3.28), implies that ν

d

∑ ∑ |bl,j | ≲ e−

1−σ ′′ 2

ν

η2

(3.30)

∑ |cj |.

j=1 l=1

j=1

Here, σ ′ , σ ′′ > 0 are constants, which can be taken arbitrarily small. Equation (3.26) then follows from (3.25) and (3.30). We denote N(ϕf +φf )−N(ϕf ) by Nϕf (φf ) for the sake of simplicity. Then by Lemma 3.3 and by multiplying (3.24) with φf and integrating by parts, we have 1

2

⟨ℒf (φf ), φf ⟩L2 ≲ ∫ Nϕf (φf )φf dx + (‖f ‖H −1 + e− 8 η )‖φf ‖H 1 (ℝd ) .

(3.31)

ℝd

Lemma 3.5. As ‖f ‖H −1 → 0, we have 3

2

‖φf ‖H 1 (ℝd ) ≲ ‖f ‖H −1 + e−( 8 −σ)η . Proof. For the sake of clarity, we divide the proof into several parts. Step 1. The estimate of Nϕf (φf )φf . By the Taylor expansion and u = ∑νj=1 gj,f + ϕf + φf ≥ 0, ν

ν

Nϕf (φf ) = 2(∑ gj,f + ϕf + φf ) log(∑ gj,f + ϕf + φf ) j=1

j=1

ν

ν

j=1

j=1

− 2(∑ gj,f + ϕf ) log(∑ gj,f + ϕf ) ν

− 2(1 + log(∑ gj,f ))φf

(3.32)

j=1

= 2 log(1 + = 2 log(1 +

ϕf + θφf ∑νj=1 gj,f ϕf

∑νj=1 gj,f

where θ, θ′ ∈ (0, 1). By (3.23), we have

(3.33)

)φf )φf +

φ2f

∑νi=1 gi,f + ϕf + θ′ φf

,

(3.34)

Sharp stability of the logarithmic Sobolev inequality in the critical point setting

󵄨󵄨 ϕf 󵄨󵄨 󵄨󵄨 ν 󵄨󵄨 ∑j=1 gj,f

󵄨󵄨 󵄨󵄨 −2 󵄨󵄨 ≲ η . 󵄨󵄨

� 93

(3.35)

Thus ν

ν

i=1

i=1

∑ gi,f + ϕf = (1 + O(η−2 )) ∑ gi,f . For every R > 0, let ν

ν

j=1

i=1

ϒR,α = {x ∈ 𝜕(⋃ BR (yj,f )) | φf = α ∑ gi,f } where α ≥ −1 + O(η−2 ). Since 0 ≤ α ≤ e

R2 −(d+5) 2

− 1 + O(η−2 ) implies that

2(1 + log(1 + O(η−2 ) + θα)) ≤ R2 − (d + 3) and −1 + O(η−2 ) + e−

R2 −(d+1) 2

≤ α < 0 implies that

−2(1 + log(1 + O(η−2 ) + θα)) ≤ R2 − (d + 3) by θ ∈ (0, 1), by (1.9), (3.33), and (3.35), for every R > 0 and every ν

x ∈ 𝜕(⋃ BR (yj,f )), j=1

one of the following cases must happen: (a) Nϕf (φf )φf ≤ (R2 − (d + 3))φ2f , (b) φf ≳ 1, (c) φf ∼ − ∑νi=1 gi,f . On the other hand, by (3.32), (3.34), and (3.35), in ϒR,α with −1 ≤ α < 0, 2((1 + O(η−2 ) + α) log(1 + O(η−2 ) + α) − α) = O(η−2 )α + It follows that θ′ →

1 2

α2 . 1 + O(η−2 ) + θ′ α

+ O(η−σ ) as α → −1. Thus, by (1.9), ν

Nϕf (φf )φf ≤ (max{−(2 + 2 log(∑ gj,f )), 0} + O(η−σ ))|φf |2 + O(φ3f ). j=1

2 Step 2. The estimate of ‖φ⊥ f ‖H 1 (ℝd ) .

(3.36)

94 � J. Wei and Y. Wu By (3.25), (3.30), (3.31), (3.36), and Lemma 3.4, ⟨ℒf (φf ), φf ⟩L2 ≲ ∫ Nϕf (φf )φf dx + (‖f ‖H −1 + e−

1−σ ′′ 2

η2

ν

∑ |cj |)‖φf ‖H 1 (ℝd ) j=1

ℝd ν

≤ ∫ max{−(2 + 2 log(∑ gj,f )), 0}φ2f dx j=1

ℝd

1−σ ′′ 2

+ O(‖φf ‖3H 1 (ℝd ) ) + (‖f ‖H −1 + e−

η2

ν

∑ |cj |)‖φf ‖H 1 (ℝd ) . j=1

It follows that ν

(|∇φf |2 − (1 + 2 log(∑ gj,f )))|φf |2 dx

∫

j=1

⋃νj=1 BR (yj,f )

+

∫

|∇φf |2 + |φf |2 dx

ℝd \(⋃νj=1 BR (yj,f ))

≤ O(‖φf ‖3H 1 (ℝd ) ) + (‖f ‖H −1 + e−

1−σ ′′ 2

η2

ν

∑ |cj |)‖φf ‖H 1 (ℝd ) , j=1

(3.37)

for a sufficiently large R > 0. For every j, ν

∫ (|∇φf |2 − (1 + 2 log(∑ gj,f )))|φf |2 dx j=1

BR (yj,f )

ν

2 − 2 log(∑ gj,f )]|φf |2 dx ∫ [(1 − σ)(d + 1) − ri,f

≥

j=1

BR (yj,f )

+

2 − d − 2)|φf |2 dx ∫ |∇φf |2 + (ri,f

(3.38)

BR (yj,f )

where σ > 0 is sufficient small. By [40, Theorem 7.5 and Remark 7.7], it is easy to show that 󵄨 ̃ ⊥ 󵄨󵄨2 󵄨󵄨 ̃ ⊥ 󵄨󵄨2 2 ∫ 󵄨󵄨󵄨∇φ f ,R 󵄨󵄨 + (rj,f − d − 2)󵄨󵄨φf ,R 󵄨󵄨 dx ≳

BR (yj,f )

󵄨 ̃ ⊥ 󵄨󵄨2 󵄨󵄨 ̃ ⊥ 󵄨󵄨2 ∫ 󵄨󵄨󵄨∇φ f ,R 󵄨󵄨 + 󵄨󵄨φf ,R 󵄨󵄨 dx

(3.39)

BR (yj,f )

⊥ ̃⊥ where R > 0 sufficiently large and φ f ,R = φf ψR with ψR being a smooth cut-off function such that ψR = 1 in B R (0) and ψR = 0 in BR (0)\B R (0). Note that 4

2

Sharp stability of the logarithmic Sobolev inequality in the critical point setting

� 95

󵄨 ̃ ⊥ 󵄨󵄨2 󵄨󵄨 ̃ ⊥ 󵄨󵄨2 2 ∫ 󵄨󵄨󵄨∇φ f ,R 󵄨󵄨 + (rj,f − d − 2)󵄨󵄨φf ,R 󵄨󵄨 dx

BR (yj,f )

󵄨󵄨 ⊥ 󵄨󵄨2 󵄨 󵄨󵄨2 2 ∫ 󵄨󵄨󵄨∇φ⊥ f 󵄨󵄨 + (ri,f − d − 2)󵄨󵄨φf 󵄨󵄨 dx

≲

BR (yj,f )

+

∫ BR (yj,f )\B R (yj,f )

󵄨󵄨 ̃ ⊥ 󵄨󵄨2 󵄨󵄨 ̃ ⊥ 󵄨󵄨2 󵄨󵄨∇φf ,R 󵄨󵄨 + 󵄨󵄨φf ,R 󵄨󵄨 dx

2

≲

󵄨 󵄨󵄨2 󵄨󵄨 ⊥ 󵄨󵄨2 2 ∫ 󵄨󵄨󵄨∇φ⊥ f 󵄨󵄨 + (ri,f − d − 2)󵄨󵄨φf 󵄨󵄨 dx

BR (yj,f )

and 󵄨 ̃ ⊥ 󵄨󵄨2 󵄨󵄨 ̃ ⊥ 󵄨󵄨2 ∫ 󵄨󵄨󵄨∇φ f ,R 󵄨󵄨 + 󵄨󵄨φf ,R 󵄨󵄨 dx

BR (yj,f )

=

󵄨 󵄨󵄨 ⊥ 󵄨󵄨2 󵄨󵄨 ⊥ 󵄨󵄨2 2 󵄨󵄨2 󵄨󵄨 ⊥ 󵄨󵄨2 ∫ 󵄨󵄨󵄨∇φ⊥ f 󵄨󵄨 + 󵄨󵄨φf 󵄨󵄨 dx + O(1) ∫ 󵄨󵄨∇φf 󵄨󵄨 + (ri,f − d − 2)󵄨󵄨φf 󵄨󵄨 dx,

BR (yj,f )

BR (yj,f )

thus, by (3.39), 󵄨 󵄨󵄨 ⊥ 󵄨󵄨2 2 󵄨󵄨2 ∫ 󵄨󵄨󵄨∇φ⊥ f 󵄨󵄨 + (ri,f − d − 2)󵄨󵄨φf 󵄨󵄨 dx ≳

BR (yj,f )

󵄨 󵄨󵄨2 󵄨󵄨 ⊥ 󵄨󵄨2 ∫ 󵄨󵄨󵄨∇φ⊥ f 󵄨󵄨 + 󵄨󵄨φf 󵄨󵄨 dx.

(3.40)

BR (yj,f )

On the other hand, we have |x − yi,f | ≥ η + O(1) for all j ≠ i in BR (yj,f ), which implies that η2

∑ gi,f ≲ e− 2 gj,f i=j̸

in BR (yj,f ).

It follows that ν

η2

2 −2 log(∑ gj,f ) = −(d + 1) + rj,f + O(e− 2 ) in BR (yj,f ), j=1

which implies that 󵄨󵄨 󵄨󵄨 ν 󵄨󵄨 󵄨 󵄨󵄨 ∫ [(1 − σ)(d + 1) − r 2 − 2 log(∑ gj,f )]|φf |2 dx 󵄨󵄨󵄨 ≲ σ‖φf ‖2 1 d . i,f 󵄨󵄨 󵄨󵄨 H (ℝ ) 󵄨󵄨 󵄨󵄨 j=1 BR (yj,f ) It follows from (3.37), (3.38), (3.40), and Lemma 3.4 that ν

󵄩󵄩 ⊥ 󵄩󵄩2 2 −(1−σ ′′ )η2 + ∑ |cj |2 . 󵄩󵄩φf 󵄩󵄩H 1 (ℝd ) ≲ ‖f ‖H −1 + e j=1

(3.41)

96 � J. Wei and Y. Wu Step 3. The estimate of ∑νj=1 |cj |2 . Let ̃ =g ψ ̂ ψ j j,f j , ̂ is a smooth cut-off function satisfying where ψ j |x − yj,f | ≤ ( 21 − σ)η,

̂ = {1, ψ j 0,

|x − yj,f | ≥ ( 21 − σ)η + 1.

Note that gj,f is the unique positive solution of (1.8); thus, by multiplying (3.24) with ̃ and integrating by parts, we have − sgn(cj )ψ j d

ν

̃ +fψ ̃ dx − sgn(cj ) ∫ (Nϕf (φf ) + ∑ ∑ aj,i 𝜕xj gi,f )ψ j j j=1 i=1

ℝd

̂ φ − 2φ ∇ψ ̂ ∇g − Δψ ̂ g φ dx = − sgn(cj ) ∫ −Δgj,f ψ j f f j j,f j j,f f ℝd ν

̂ g ϕ dx + sgn(cj ) ∫ (1 + 2 log(∑ gj,f ))ψ j j,f f j=1

ℝd

ν

2 ̂ g ϕ dx = sgn(cj ) ∫ (rj,f + 1 − d + 2 log(∑ gj,f ))ψ j j,f f j=1

ℝd

̂ ∇g + Δψ ̂ g ϕ dx. + sgn(cj ) ∫ 2ϕf ∇ψ j j,f j j,f f

(3.42)

ℝd

By (3.29) and similar estimates of (3.36), ν

̃ dx ≤ |c | ∫ max{−(2 + 2 log(∑ g )), 0}|g |2 ψ ̂ dx ∫ Nϕf (φf )ψ j j j,f j,f j ℝd

j=1

ℝd ν

ν

i=1

i=1

󵄩 󵄩󵄩 2 󵄩 󵄩 ⊥ 󵄩󵄩2 + o(∑ |ci | + 󵄩󵄩󵄩ϕ⊥ f 󵄩 󵄩H 1 (ℝd ) ) + ∑ |ci | + 󵄩󵄩ϕf 󵄩󵄩H 1 (ℝd ) . 2

̂ ) ⊂ Ω with ∑ g ≲ e−ση g in supp(ψ ̂ ); thus, by (1.9), (3.25) and Note that supp(ψ j j j,f j i=j̸ i,f Lemma 3.4, ν

̂ ∇g + Δψ ̂ g ϕ dx = o(∑ |c | + 󵄩󵄩󵄩ϕ⊥ 󵄩󵄩󵄩 1 d ) ∫ 2ϕf ∇ψ j j,f j j,f f i 󵄩 f 󵄩H (ℝ )

ℝd

and

i=1

Sharp stability of the logarithmic Sobolev inequality in the critical point setting

� 97

ν 󵄨󵄨 󵄨󵄨 2 ̃ dx 󵄨󵄨󵄨 ̂ g ϕ dx − 󵄨󵄨󵄨 ∫ N (φ )ψ sgn(cj ) ∫ (rj,f + 1 − d + 2 log(∑ gj,f ))ψ ϕf f j j j,f f 󵄨󵄨 󵄨󵄨󵄨 󵄨 j=1 d d ℝ

ℝ

ν

󵄩 󵄩󵄩 ∼ |cj | + o(∑ |ci | + 󵄩󵄩󵄩ϕ⊥ f 󵄩 󵄩H 1 (ℝd ) ), i=1

which, together with (3.42) and Lemma 3.3, implies that 󵄩 󵄩󵄩 −( 83 −σ)η2 |cj | + o(∑ ci + 󵄩󵄩󵄩ϕ⊥ ) + ‖f ‖H −1 . f 󵄩 󵄩H 1 (ℝd ) ) ≲ O(e

(3.43)

i=j̸

Since (3.43) holds for all j = 1, 2, . . . , ν and the linear part of this inequality is diagonally dominant, we have ν

3

2

∑ |cj |2 ≲ O(e−( 4 −2σ)η ) + ‖f ‖2H −1 .

(3.44)

j=1

Step 4. The estimate of ‖φf ‖H 1 (ℝd ) . Equation (3.26) comes from (3.41), (3.44), and Lemma 3.4. We also need the following lemma. 1

2

Lemma 3.6. As ‖f ‖H −1 → 0, we have e− 8 η ≲ ‖f ‖H −1 . Proof. Without loss of generality, we may assume that η = |y1,f − y2,f |. We define ̂ = g ψ, ψ 1,f

(3.45)

where ψ is a smooth cut-off function satisfying 2 log η {1, x ∈ B1 ( 21 (y1,f + y2,f ) + η2 (y2,f − y1,f ) + 5), ψ={ η 0, B2c ( 21 (y1,f + y2,f ) + 2 log (y2,f − y1,f ) + 5). η2 {

By similar estimates of (3.34), we know that N(ρf ) ≥ 0. Thus, similar to (3.42), by multî and integrating by parts and (3.34), we have plying (2.5) with −ψ ν

̂ 1 d ‖f ‖ −1 ≳ ∫ (r 2 + 1 − d + 2 log(∑ g ))ψg ρ dx ‖ψ‖ j,f 1,f f H H (ℝ ) 1,f j=1

ℝd

̂ + ∫ 2ρ ∇ψ∇g + Δψg ρ dx + ∫ (E + N(ρf ))ψdx f 1,f 1,f f ℝd

ℝd ν

2 ≳ ∫ (r1,f + 1 − d + 2 log(∑ gj,f ))ψg1,f ρf dx + ∫ Eψdx ℝd

j=1

ℝd

98 � J. Wei and Y. Wu ∫ 2ρf ∇ψ∇g1,f + Δψg1,f ρf dx.

(3.46)

ℝd

̂ ∼ e− 41 η2 . By Lemmas 3.3 and 3.5, By (1.9) and (3.20), ∫ℝd E ψdx 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∫ 2ρf ∇ψ∇g1,f + Δψg1,f ρf dx 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 d ℝ

󵄨󵄨 󵄨󵄨 ν 󵄨󵄨 󵄨󵄨 2 󵄨 + 󵄨󵄨 ∫ (r1,f + 1 − d + 2 log(∑ gj,f ))ψg1,f ρf dx 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 j=1 󵄨ℝd 󵄨 1

2

2

≲ η−2 e− 4 η + e−ση ‖f ‖H −1 . 1

2

1

2

̂ g1,f ‖H 1 (ℝd ) ≲ e− 8 η ; thus we obtain e− 8 η ≲ ‖f ‖H −1 . Note that ‖φ We close this section by the following proposition. Proposition 3.7. As ‖f ‖H −1 → 0, we have ‖ρf ‖H 1 (ℝd ) ≲ ‖f ‖H −1 . Proof. The conclusion follows immediately from Lemmas 3.3, 3.5, and 3.6.

4 Optimality of (1.11) In this section, we shall construct an example to show that (1.11) is optimal. Let g±L = g(x ± L2 e1 ) with e1 = (1, 0, . . . , 0) and we consider the following equation: d ℒL (ρL ) = EL + N(ρL ) − ∑j=1 (aj,+,L 𝜕xj gL + aj,−,L 𝜕xj g−L ), { ⟨ρL , 𝜕xj g±L ⟩H 1 (ℝd ) = 0,

in H −1 , j = 1, 2, . . . , d,

where ℒL = −Δ − 1 − 2 log(gL + g−L )

is the linear operator, EL = 2(gL + g−L ) log(gL + g−L ) − 2(gL log gL + g−L log g−L ) is the error, N(ρL ) = 2(gL + g−L + ρL ) log(gL + g−L + ρL ) − 2(gL + g−L ) log(gL + g−L ) − 2(1 + log(gL + g−L ))ρL

is the nonlinear part, and aj,±,L are Lagrange multipliers given by

(4.1)

Sharp stability of the logarithmic Sobolev inequality in the critical point setting

� 99

aj,±,L ∼ ⟨EL + N(ρL ), 𝜕xj g±L ⟩L2 . By Lemma 3.3, (4.1) is unique solvable in 𝔹, where 𝔹 is given by (3.21). Moreover, by Lemma 3.1, d

ν

1

2

‖ρL ‖♯ + ∑ ∑ |aj,±,L | ≲ e− 8 L . j=1 i=1

(4.2)

Let uL = gL + g−L + ρL . Then by similar estimates for (3.35), uL ∼ gL + g−L > 0. Moreover, by the classical regularity, ρL ∈ C 1,α (ℝd ) for all α ∈ (0, 1). It follows from (4.1) that fL := −ΔuL + uL − 2uL log |uL | d

= − ∑(aj,+,L 𝜕xj gL + aj,−,L 𝜕xj g−L ). j=1

(4.3)

Proposition 4.1. As L → +∞, we have 󵄩󵄩 󵄩󵄩2 2 󵄩󵄩 󵄩󵄩 󵄩 inf 󵄩󵄩uL − ∑ g(⋅ − zi )󵄩󵄩󵄩 ∼ ‖fL ‖2H −1 . d 󵄩 󵄩 zi ∈ℝ 󵄩 󵄩󵄩H 1 (ℝd ) i=1 󵄩 Proof. Thanks to Lemma 3.6 and (4.2), we have 1

2

‖fL ‖H −1 ∼ e− 8 L .

(4.4)

Now, we consider the minimizing problem 󵄩󵄩 󵄩󵄩2 2 󵄩󵄩 󵄩󵄩 󵄩 cL = inf 󵄩󵄩uL − ∑ g(⋅ − zi )󵄩󵄩󵄩 . d 󵄩 󵄩󵄩 1 d zi ∈ℝ 󵄩 i=1 󵄩 󵄩H (ℝ ) 1

(4.5)

2

Clearly, by (4.2), cL ≲ ‖ρL ‖2H 1 (ℝd ) ≲ e− 4 L . As before, we can also write 2

uL = ∑ g(⋅ − yi,L ) + ρ∗L i=1

(4.6)

where {yi,L } is the solution of (4.5) and the remaining term ρ∗L satisfies 󵄩󵄩 ∗ 󵄩󵄩2 − 1 L2 󵄩󵄩ρL 󵄩󵄩H 1 (ℝd ) = cL ≲ e 4 , and the orthogonal conditions

(4.7)

100 � J. Wei and Y. Wu ⟨ρ∗L , 𝜕xl gj,L ⟩H 1 (ℝd ) = 0,

l = 1, 2, . . . , d and j = 1, 2,

where gj,L = g(⋅ − yj,L ). By (4.6) and (4.7), we may assume that y1,L = y2,L = − L2 e1 + o(1). By (4.3) and (4.6), ρ∗L satisfies the equation

L e 2 1

+ o(1) and

−Δρ∗L + ρ∗L = 2uL log uL − 2(∑2j=1 gj,L log gj,L ) + fL , in ℝd ,

{

⟨ρ∗L , 𝜕xj gj,L ⟩H 1 (ℝd ) = 0,

l = 1, 2, . . . , d and j = 1, 2.

(4.8)

As before, we can write 2

2

j=1

j=1

2uL log uL − 2(∑ gj,L log gj,L ) = EL + NL (ρ∗L ) + 2(1 + log(∑ gj,L ))ρ∗L , which, together with (4.3), implies that we can rewrite (4.8) as follows: {

−ℒL (ρ∗L ) = EL + NL (ρ∗L ) − ∑dj=1 (aj,+,L 𝜕xj gL + aj,−,L 𝜕xj g−L ), ⟨ρ∗L , 𝜕xj gj,L ⟩H 1 (ℝd ) = 0,

in ℝd , l = 1, 2, . . . , d and j = 1, 2.

By Lemma 3.2 and similar estimates in the proof of Lemma 3.3, we have 󵄩󵄩 ∗ 󵄩󵄩 − 1 L2 󵄩󵄩ρL 󵄩󵄩♯ ≲ e 8 . ̂ given by (3.45), as the test function of (4.8) and estimating as that for (3.46), Now, using ψ, 1 2 1 2 ∗ we have ‖ρL ‖H 1 (ℝd ) ≳ e− 8 L . It follows from (4.7) that ‖ρ∗L ‖H 1 (ℝd ) ∼ e− 8 L , which together with (4.4) and (4.7), implies that ‖ρ∗L ‖H 1 (ℝd ) ∼ ‖fL ‖H −1 . It completes the proof. We close this section by the proof of Theorem 1.2. Proof of Theorem 1.2. The conclusion follows immediately from Propositions 3.7 and 4.1.

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Qinfeng Li and Changyou Wang

On a variational problem of nematic liquid crystal droplets Dedicated to the memory of David R. Adams

Abstract: Let μ > 0 be a fixed constant, and we prove that minimizers to the following energy functional: E(u, Ω) := ∫ |∇u|2 + μP(Ω) Ω

exist among pairs (Ω, u) such that Ω is an M-uniform domain with finite perimeter and fixed volume, and u ∈ H 1 (Ω, 𝕊2 ) with u = νΩ , the measure-theoretical outer unit normal, almost everywhere on the reduced boundary of Ω. The uniqueness of optimal configurations in various settings is also obtained. In addition, we consider a general energy functional given by 󵄨 󵄨2 Ef (u, Ω) := ∫󵄨󵄨󵄨∇u(x)󵄨󵄨󵄨 dx + ∫ f (u(x) ⋅ νΩ (x)) d ℋ2 (x), Ω

𝜕∗ Ω

where 𝜕∗ Ω is the reduced boundary of Ω and f is a convex positive function on ℝ. We prove that minimizers of Ef also exist among M-uniform outer-minimizing domains Ω with fixed volume and u ∈ H 1 (Ω, 𝕊2 ). Keywords: Liquid crystal droplets, M-uniform domains, outer minimal sets MSC 2020: Primary 35J50, Secondary 58E20, 58E30

1 Introduction In this paper, we study the existence of liquid crystal droplets (Ω0 , u0 ), consisting of a domain Ω0 ⊂ ℝ3 representing the shape of a liquid crystal drop and a unit vector field Acknowledgement: The first author is partially supported by the National Science Fund for Youth Scholars (No. 1210010723) and the Fundamental Research Funds for the Central Universities, Hunan Provincial Key Laboratory of intelligent information processing and Applied Mathematics. The second author is partially supported by NSF grants 1764417 and 2101224. Qinfeng Li, School of Mathematics, Hunan University, Changsha 410012, Hunan, P.R. China, e-mail: [email protected] Changyou Wang, Department of Mathematics, Purdue University, West Lafayette 47907, IN, USA, e-mail: [email protected] https://doi.org/10.1515/9783110792720-005

104 � Q. Li and C. Wang u0 ∈ H 1 (Ω, 𝕊2 ) representing the average orientation field of liquid crystal molecules within the liquid crystal drop Ω, that minimizes the total energy functional, including both the elastic energy in the bulk and the interfacial energy defined by 󵄨2 󵄨 Ef (u, Ω) := ∫󵄨󵄨󵄨∇u(x)󵄨󵄨󵄨 dx + ∫ f (u(x) ⋅ νΩ (x)) d ℋ2 (x), Ω

(1.1)

𝜕∗ Ω

among all pairs (Ω, u), where Ω is a domain of finite perimeter with a fixed volume that is compactly contained in the ball BR0 ⊂ ℝ3 with center 0 and radius R0 for some fixed constant R0 > 0, and u ∈ H 1 (Ω, 𝕊2 ), which is defined by 󵄨 󵄨 H 1 (Ω, 𝕊2 ) ≡ {v ∈ H 1 (Ω, ℝ3 ) : 󵄨󵄨󵄨v(x)󵄨󵄨󵄨 = 1 a. e. x ∈ Ω}. The functional Ef (u, Ω) should be understood in the sense that the surface integral is taken over the reduced boundary 𝜕∗ Ω of Ω, u⌊𝜕∗ Ω is the trace of u on 𝜕∗ Ω, νΩ is the measure theoretical outer unit normal of 𝜕∗ Ω, and f is usually assumed to have a nonnegative lower bound (with a typical choice of f (t) = μ(1 + wt 2 ), t ∈ [−1, 1], for some constants μ > 0 and −1 < w < 1). See [1] and [32] for basic facts on Sobolev spaces. We will study the following minimization problem of (1.1). Problem A Find a pair (Ω, u) that minimizes Ef (u, Ω) over all pairs (Ω, u) where Ω is a domain of finite perimeter in a fixed ball BR0 ⊂ ℝ3 , with a fixed volume V0 > 0, and u ∈ H 1 (Ω, 𝕊2 ), when f : [−1, 1] → ℝ is a nonnegative, continuous convex function. We are also interested in the case when there is a constant contact angle condition between the liquid crystal orientation field u and the reduced boundary of liquid crystal drop 𝜕∗ Ω, i. e., u ⋅ νΩ ≡ c on 𝜕∗ Ω, for some constant c ∈ [−1, 1]. In this case, the energy functional Ef (u, Ω) in (1.1) reduces to 󵄨 󵄨2 E(u, Ω) := ∫󵄨󵄨󵄨∇u(x)󵄨󵄨󵄨 dx + μℋ2 (𝜕∗ Ω)

(1.2)

Ω

for some constant μ ≥ 0. Problem A can be reformulated as follows. Problem B Find a pair (Ω, u) that minimizes E(u, Ω) over all pairs (Ω, u) where Ω is a domain of finite perimeter in a fixed ball BR0 ⊂ ℝ3 , with a fixed volume V0 > 0, and u ∈ H 1 (Ω, 𝕊2 ) satisfies u ⋅ νΩ ≡ c on 𝜕∗ Ω for some c ∈ [−1, 1]. We would like to mention that the contact angle condition in Problem B is referred as (i) the planar anchoring condition when the constant c = 0, and (ii) the homeotropic anchoring condition when the constant c = 1.

On a variational problem of nematic liquid crystal droplets

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We would like to point out that recently Geng and Lin in a very interesting paper [16] studied Problem B under the planar anchoring condition (i) in dimension two, and proved the existence of a minimizer (Ω, u) such that the optimal shape 𝜕Ω of the droplet 3 is a chord-arc curve with two cusps, which can be parametrized in H 2 and has its unit normal vector field νΩ belongs to VMO. See also [15] for related work. Because the homeotropic anchoring condition is an important physical condition, we are also interested in the following problem. Problem C Find a solution to Problem B when the contact angle condition corresponds to c = 1. Motivation The main difficulty of the minimization problems A, B, and C lies in showing the sequential lower semicontinuity of Ef (u, Ω) (or E(u, Ω)) when both domains Ω and vector fields u ∈ H 1 (Ω, 𝕊2 ) vary. It is even a difficult question to ask whether the configuration space is closed under weak convergence of liquid crystal pairs (Ω, u). In [28], under the assumption that all admissible domains Ω ⊂ BR0 are convex domains, Lin and Poon have proved that there exists a minimizing pair (Ω0 , u0 ) of Problem A. Moreover, u0 enjoys a partial regularity property similar to that of minimizing harmonic maps by Schoen and Uhlenbeck [33, 34]. It was further proven by [28] that, up x ) is a unique minimizer of Problem C among convex to translations, (Ω0 , u0 ) = (BR , |x| domains with |BR | = V0 . We would like to point out that the convexity assumption of admissible domains Ω plays a crucial role in [28], since a minimizing sequence (Ωi , ui ) of convex domains Ωi ⊂ BR0 with |Ωi | = V0 has a subsequence Ωik → Ω in L1 , for some bounded convex domain Ω ⊂ BR0 with |BR0 | = V0 , such that ℋ2 (𝜕Ωik ) → ℋ2 (𝜕Ω) and νΩi → νΩ almost everywhere k

with respect to a spherical coordinate system.1 Moreover, there exists u ∈ H 1 (Ω, 𝕊2 ) such that ∇uik χΩi → ∇uχΩ weakly in L2 (ℝ3 ) (see [11]). The uniqueness of minimizer of k Problem C among convex domains relies on the following important inequalities: 󵄨 󵄨2 ∫󵄨󵄨󵄨∇u(x)󵄨󵄨󵄨 dx ≥ ∫ H(x) d ℋ2 (x),

∀u ∈ H 1 (Ω, 𝕊2 ) with u = νΩ a. e. on 𝜕Ω,

(1.3)

∫ H(x) d ℋ2 (x) ≥ √4π ℋ2 (𝜕Ω) for convex Ω, equality holds iff Ω = BR ,

(1.4)

Ω

𝜕Ω

and

𝜕Ω

1 For example, one can parametrize 𝜕Ωik and 𝜕Ω over the unit sphere 𝕊2 .

106 � Q. Li and C. Wang where H denotes the mean curvature of 𝜕Ω. In [28], (1.3) is derived for any Ω ∈ W 2,1 , while (1.4) is proven by the Brunn–Minkowski inequality for convex domains. In this paper, we would like to relax the convexity assumption from [28] and investigate Problems A, B, and C over a larger class of domains possibly containing nonconvex domains with less regular boundaries. The class of domains contains Sobolev extension domains with some uniform parameters, as well outer minimal domains. The main theorems of this paper arose from the PhD thesis of the first author [26]. The interested reader can refer to [26] for more related results. Outline of this paper In Section 2, we will review certain classes of domains in ℝn , including M-uniform domains, which are Sobolev extension domains with constants depending on M and n; the outer minimal domains, which are a generalization of convex domains. In Section 3, we will show in Theorem 3.5 that, up to a set of measure zero, the L1 limit of M-uniform domains is M-uniform. A few other results on the relation between L1 -convergence and Hausdorff convergence are also derived. In Section 4, we will establish the weak lower semicontinuity of bulk elastic energy of (Ω, u) for two classes of domains: (a) the admissible sets of M-uniform domains, and (b) the admissible sets of outer minimal M-uniform domains. It is more subtle to prove the lower semicontinuity of surface energy for Problem A. We will only consider outer minimal sets and our proof is inspired by Reshetnyak’s lower semicontinuity theorem (see [30, Theorem 20.11]) and the perimeter convergence Lemma 4.1. Thus combining the compactness of M-uniform domain results and the lower-semicontinuity results, the existence Theorem 4.2 on Problems A, B, and C is proved among these classes of admissible sets. x ) is the unique In Section 5, we will apply results by [9, 14, 23, 24] to show (BR , |x| minimizer of Problem C over strictly star-shaped mean convex C 1,1 -domains, C 1,1 -outer minimal sets, and C 1,1 -revolutionary domains; see Theorem 5.5 and Remark 5.6.

2 Prerequisite: sets of finite perimeter and traces of functions We first stipulate some notation. Let V0 > 0 be the fixed volume in the Problems A, B, C. Since the admissible domains in the problems have this fixed volume, we will use the convention that any minimizing sequences have their diameters larger than a universal constant c0 = c0 (V0 ) > 0 because of the isodiametric inequality (see [12, Theorem 2.2.1]). We will denote by Br (x) := {y ∈ ℝn : |y − x| < r}

and Br := Br (0).

On a variational problem of nematic liquid crystal droplets

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Throughout this paper, all sets under consideration are contained in a large ball BR0 , where R0 > 0 is fixed. For any set A ⊂ ℝn , denote by Aϵ the interior ϵ-neighborhood {x ∈ A : Bϵ (x) ⊂ A}, and Aϵ the exterior ϵ-neighborhood ⋃x∈A Bϵ (x). Denote by int(A) the topological interior part of A, Ac = ℝn \ A, and diam(A) the diameter of A. For 0 ≤ d ≤ n, ℋd denotes the d-dimensional Hausdorff measure in ℝn . Let d H (⋅, ⋅) denote the Hausdorff distance in ℝn . P(A; D) denotes the distributional perimeter of A in D ⊂ ℝn . For a set A of finite perimeter, let νA denote the measure theoretical outer unit normal of the reduced boundary 𝜕∗ A, and μA denotes the Gauss–Green measure of A, that is, μA = νA ⋅ ℋn−1 ⌊𝜕∗ A . Denote by ωn the volume of unit ball in ℝn and |A| the Lebesgue measure of A. For any open set Ω ⊂ ℝn and u ∈ BV (Ω), denote by Du the distributional derivative of u, that is, a vector-valued Radon measure, and ‖Du‖(Ω) the total variation of u on Ω. In this paper, “≲c ” denotes an inequality up to constant multiplier c > 0. For any measurable set E and 0 ≤ α ≤ 1, we define E α = {x ∈ ℝn : lim

r→0

|E ∩ Br (x)| = α}, |Br (x)|

and refer E 1 and E 0 as the measure theoretical interior and exterior part of E, respectively. Denote by 𝜕∗ E := ℝn \ (E 0 ∪ E 1 ) the measure theoretical boundary of E, which is also called the essential boundary. In this paper, we will need the following theorem, due to Federer (see [12, Chapter 5]). Theorem 2.1. For any measurable set E, if ℋn−1 (𝜕∗ E) < ∞, then E is a set of finite perimeter. Furthermore, if E is a set of finite perimeter, then ℝn = E 0 ∪E 1 ∪𝜕∗ E, 𝜕∗ E ⊂ E (1/2) ⊂ 𝜕∗ E, and 𝜕∗ E = 𝜕∗ E (mod ℋn−1 ). Next, we recall the definition of M-uniform domains. Definition 2.2. For M ≥ 1, a domain Ω ⊂ ℝn is called an M-uniform domain, if for any two points x, y ∈ Ω, there is a rectifiable curve γ : [0, 1] → Ω such that γ(0) = x, γ(1) = y, and 1

ℋ (γ([0, 1])) ≤ M|x − y|,

1 󵄨 󵄨󵄨 󵄨 min{󵄨󵄨󵄨γ(t) − x 󵄨󵄨󵄨, 󵄨󵄨󵄨γ(t) − y󵄨󵄨󵄨}, d(γ(t), 𝜕Ω) ≥ M

(2.1) ∀t ∈ [0, 1].

(2.2)

Remark 2.3. P. Jones [25] introduced the notion of (ϵ, δ)-domain. One can check that any (ϵ, ∞)-domain is an M-domain, with M = ϵ2 . On the other hand, any M-uniform domain is a ( M1 2 , ∞)-domain.2 It was also proven by [25] that any (ϵ, δ) domain is a Sobolev ex2 Since (2.1) and (2.2) imply d(γ(t), 𝜕Ω) ≥

1 |γ(t) − x||γ(t) − y| 1 |γ(t) − x||γ(t) − y| ≥ 2 , M ℋ1 (γ([0, 1])) |x − y| M

∀t ∈ [0, 1].

108 � Q. Li and C. Wang tension domain, and the converse is true when n = 2. We refer to [17] and [25] for more details on M-uniform domains. Since we will study minimization problems involving traces of bounded H 1 vector fields in this paper, we will need the following Gauss–Green formula (see also [8]. Theorem 2.4. Let Ω be a bounded uniform domain of finite perimeter in ℝn and u ∈ H 1 (Ω) ∩ L∞ (Ω). Then for any ϕ ∈ C01 (ℝn , ℝn ), we have ∫ udivϕ + ∫ ϕDu = ∫ (ϕ ⋅ νΩ )u∗ d ℋn−1 , Ω

Ω

(2.3)

𝜕∗ Ω

where νΩ is the measure-theoretic unit outer normal to 𝜕∗ Ω, and u∗ is given by the formula lim

r→0

∫B (x)∩Ω |u − u∗ (x)| r

rn

= 0,

ℋ

n−1

-a. e. x ∈ 𝜕∗ Ω.

(2.4)

Proof. According to [25] (see also [22]), we may let û ∈ H01 (ℝn ) ∩ L∞ (ℝn ) be an extension of u such that û = u in Ω and ‖u‖̂ H 1 (ℝn ) ≤ C(n, Ω)‖u‖H 1 (Ω) . Hence, û ∈ BV (ℝn ), and thus according to [2, Theorem 3.77], the interior trace of u,̂ denoted by û ∗ here, is well-defined for ℋn−1 -a. e. on 𝜕∗ Ω, and equals to u∗ , given by (2.4), ̂ Ω . Since û is bounded, u∗ ∈ L1 (𝜕∗ Ω), and thus by [2, for ℋn−1 -a. e. on 𝜕∗ Ω. Let ũ = uχ n ̂ Ω ∈ BV (ℝ ), with Theorem 3.84], ũ = uχ Dũ = Du⌊̂ Ω1 −u∗ νΩ ℋn−1 ⌊𝜕∗ Ω . Hence, for any ϕ ∈ C01 (ℝn , ℝn ), we have ∫ ϕDũ = ∫ ϕDû − ∫ (ϕ ⋅ νΩ )u∗ d ℋn−1 . ℝn

(2.5)

𝜕∗ Ω

Ω1

Since ∫ ϕDũ = − ∫ ũ ÷ ϕ = − ∫ u ÷ ϕ, ℝn

Ω

ℝn

from (2.5) we have ∫ u ÷ ϕ + ∫ ϕDû = ∫ (ϕ ⋅ νΩ )u∗ d ℋn−1 . Ω

Ω1

𝜕∗ Ω

(2.6)

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Since Ω is equivalent to Ω1 up to a set of Lebesgue measure zero and û ∈ H 1 (ℝn ), we have Du⌊̂ Ω1 = Du⌊̂ Ω = Du⌊Ω

(2.7)

Hence, (2.6) and (2.7) imply (2.3). For the purpose later in this paper, we also introduce the following definition. Definition 2.5. For any c > 0, we denote by 𝒟c the class of bounded sets in ℝn such that for any set E ∈ 𝒟c , 󵄨󵄨 󵄨 n 󵄨󵄨Br (x) ∩ E 󵄨󵄨󵄨 > cr

(2.8)

holds for any x ∈ 𝜕E and 0 < r < diam(E). Recall that two sets E, F ⊂ ℝn are said to be ℋn -equivalent, denoted by E ≈ F, if EΔF = (E \ F) ∪ (F \ E) has zero Lebesgue measure. Note that by the Lebesgue density theorem, if E ∈ 𝒟c , then |𝜕E ∩ E c | = 0. Hence, 𝜕E ⊂ E (mod ℋn ) and E ≈ E. In particular, we have the following. Remark 2.6. Any E ∈ 𝒟c is equivalent to its closure E. We also have the following. Remark 2.7. For c > 0, if E ∈ 𝒟c is a set of finite perimeter, then there is c′ > 0 depending only on c and n such that for any x ∈ E and 0 < r < diam(E), |Br (x) ∩ E| ≥ c′ r n . Proof. For x ∈ E and 0 < r < diam(E), there are two cases: (a) If r ≥ 2d(x, 𝜕E), then there is z ∈ 𝜕E such that B r (z) ⊂ Br (x). Hence, 2

n

c n r 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨Br (x) ∩ E 󵄨󵄨󵄨 ≥ 󵄨󵄨󵄨B r (z) ∩ E 󵄨󵄨󵄨 ≥ c( ) = n r . 2 2 2 (b) If r ≤ 2d(x, 𝜕E), then B r (x) ⊂ E, and hence 2

󵄨󵄨 󵄨 ω n 󵄨 󵄨 󵄨󵄨Br (x) ∩ E 󵄨󵄨󵄨 ≥ 󵄨󵄨󵄨B r (x)󵄨󵄨󵄨 = nn r . 2 2 Hence, the conclusion holds with c′ = min{ 2cn ,

ωn }. 2n

The following proposition shows that any M-uniform domain belongs to 𝒟c for some c > 0. Proposition 2.8. For any M ≥ 1 and c0 > 0, if Ω ⊂ ℝn is an M-uniform domain, with diam(Ω) ≥ c0 > 0, then Ω ∈ 𝒟c for some c > 0 depending only on M, n, and c0 .

110 � Q. Li and C. Wang Proof. For any x ∈ 𝜕Ω and 0 < r < diam(Ω), we claim that there is a constant c1 = c1 (M) > 0 such that Br (x) ∩ Ω contains a ball of radius c1 r. Indeed, since 0 < r < diam(Ω), there is y ∈ Ω \ B r (x). Let γ be the curve joining x and y given by the definition of 2 M-uniform domain. Choose z ∈ 𝜕B r (x) ∩ γ. Then we have that z ∈ Ω and 3

d(z, 𝜕Ω) ≥

1 1 r r r r min{|z − x|, |z − y|} ≥ min{ , − } = . M M 3 2 3 6M

Hence, Bc1 r (z) ⊂ Ω, with c1 = r < diam(Ω),

1 . 6M

From this claim, we see that for any x ∈ 𝜕Ω and any

󵄨󵄨 󵄨 󵄨 󵄨 n n 󵄨󵄨Br (x) ∩ Ω󵄨󵄨󵄨 ≥ 󵄨󵄨󵄨Bc1 r (z)󵄨󵄨󵄨 ≥ ωn c1 r . This completes the proof. The following remark will be used in the proof of compactness of M-uniform domains. Remark 2.9. For M > 0 and c0 > 0, if Ω ⊂ ℝn is an M-uniform domain, with |Ω| ≥ c0 , then there is r0 > 0 depending only on M, n, c0 such that Ω contains a ball of radius r0 . Proof. It follows directly from the isodiametric inequality and Proposition 2.8. Similar to 𝒟c , we also define the class 𝒟c as follows. Definition 2.10. For c > 0, the set class 𝒟c consists of all bounded set E ⊂ ℝn such that 󵄨󵄨 c󵄨 n 󵄨󵄨Br (x) ∩ E 󵄨󵄨󵄨 > cr

(2.9)

holds for any x ∈ 𝜕E and 0 < r < diam(E). The following proposition from [30, Proposition 12.19] yields that we can always find an ℋn -equivalent set Ẽ of any set E of finite perimeter with slightly better topological boundary. Proposition 2.11. For any Borel set E ⊂ ℝn , there exists an ℋn -equivalent set Ẽ of E such that for any x ∈ 𝜕Ẽ and any r > 0, 󵄨 󵄨 0 < 󵄨󵄨󵄨Ẽ ∩ Br (x)󵄨󵄨󵄨 < ωn r n .

(2.10)

̃ In particular, sptμE = sptμẼ = 𝜕E. In order to illustrate the construction of such an equivalent set, which is needed in later sections, we will sketch the proof. Proof. First, we define two disjoint open sets 󵄨 󵄨 A1 := {x ∈ ℝn | there exists r > 0 such that 󵄨󵄨󵄨E ∩ Br (x)󵄨󵄨󵄨 = 0},

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and 󵄨 󵄨 A2 := {x ∈ ℝn | there exists r > 0 such that 󵄨󵄨󵄨E ∩ Br (x)󵄨󵄨󵄨 = ωn r n }. Then by simple covering arguments we have that |E ∩ A1 | = 0 and |A2 \ E| = 0. Set Ẽ = (A2 ∪ E) \ A1 . Then ̃ |EΔE| ≤ |A2 \ E| + |E ∩ A1 | = 0. ̃ and Ẽ ⊂ ℝn \ A1 , we have that 𝜕Ẽ ⊂ ℝn \ (A1 ∪ A2 ), and Moreover, since A2 ⊂ int(E) hence (2.10) holds. We now recall the notion of outer minimal sets, which can be viewed as a subsolution of area minimizing sets. It is a generalization of convex sets; see, for example, [19, Definition 15.6] and related results therein. Definition 2.12. A set E ⊂ ℝn of finite perimeter is an outer minimal set, if P(E) ≤ P(F) holds for any set F ⊃ E. We would like to point out that an outer-minimal set is also called as a pseudoconvex set by [27]. Thus by [27, Corollary 7.16] we have the following. Remark 2.13. If E ⊂ ℝn is outer-minimizing and sptμE = 𝜕E, then E ∈ 𝒟c , for some c > 0 depending only on n and E. Consequently, E = int(E)(modℋn ). Remark 2.14. Since the boundary of an outer minimal set (domain) can have positive ℋn measure (see [4]), an outer minimal domain may not be an M-uniform domain for any M ≥ 1. Combining Proposition 2.8 and Remark 2.13, we have the following. Remark 2.15. Let Ω be an M-uniform outer minimal domain with sptμΩ = 𝜕Ω, then Ω ∈ 𝒟c ∩ 𝒟c for some c > 0, and hence 𝜕∗ Ω = 𝜕Ω. We would like to state the following proposition, which is a consequence of [21, Corollary 1.10], since for any E ∈ 𝒟c , ℋn−1 (𝜕E ∩ E 0 ) = 0. Proposition 2.16. Let c > 0 and E ∈ 𝒟c . Then there exists bounded smooth sets Ei such that Ei ⋑ E, Ei → E in L1 and P(Ei ) → P(Ei ).

3 Compactness of M-uniform domains In this section, we will establish in Theorem 3.5 the L1 -compactness property of M-uniform domains. We begin with the following.

112 � Q. Li and C. Wang Lemma 3.1. For c > 0, suppose that {Di } ⊂ 𝒟c satisfies Di → D in L1 (ℝn ) as i → ∞. Then after modifying over a set of Lebesgue measure zero, D ∈ 𝒟c . Moreover, for any ϵ > 0, there is N = N(ϵ) > 0 such that for any i > N, the following properties hold: (i) D ⊂ Dϵi . (ii) (Di )ϵ ⊂ D. (iii) Di ⊂ Dϵ . In particular, d H (Di , D) → 0 as i → ∞. Proof. We first identify D with its ℋn -equivalent set in the sense of Proposition 2.11. We argue by contradiction. If (i) were false, then there would exist ϵ0 > 0, x0 ∈ D and a sequence k → ∞ such that Bϵ0 (x0 ) ∩ Dk = 0. Hence, by the hypothesis and Proposition 2.11, we obtain that 󵄨 󵄨 󵄨 󵄨 0 = 󵄨󵄨󵄨Bϵ (x0 ) ∩ Dk 󵄨󵄨󵄨 → 󵄨󵄨󵄨Bϵ (x0 ) ∩ D󵄨󵄨󵄨 > 0; this is impossible. If (ii) were false, then there would exist ϵ0 > 0 and a sequence of points xi ∈ (Di )ϵ0 \D. Assume that xi → x0 . Then x0 ∈ 𝜕D ∪ Dc . Hence, by the proof of Proposition 2.11, we have that ωn ϵ0n > |Bϵ0 (x0 ) ∩ D|. On the other hand, since Bϵ0 (xi ) ⊂ Di , we have that 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨Bϵ0 (x0 ) ∩ D󵄨󵄨󵄨 = lim 󵄨󵄨󵄨Bϵ0 (xi ) ∩ D󵄨󵄨󵄨 ≥ lim inf(󵄨󵄨󵄨Bϵ0 (xi ) ∩ Di 󵄨󵄨󵄨 − |Di ΔD|) i→∞ i→∞ = ωn ϵ0n − lim sup |Di ΔD| = ωn ϵ0n . i→∞

We get a desired contradiction. If (iii) were false, then there would exist ϵ0 > 0 and a subsequence of xi ∈ Di \ Dϵ0 . Without loss of generality, assume xi → x0 , and thus x0 ∈ ℝn \ Dϵ0 . By Remark 2.7, there is a c′ > 0 depending only on c and n such that 󵄨 󵄨 c′ ϵ0n ≤ 󵄨󵄨󵄨Bϵ0 (xi ) ∩ Di 󵄨󵄨󵄨. On the other hand, it follows from |Bϵ0 (x0 ) ∩ D| = 0 that 󵄨 󵄨 󵄨 󵄨 lim inf󵄨󵄨󵄨Bϵ0 (xi ) ∩ Di 󵄨󵄨󵄨 ≤ lim sup(󵄨󵄨󵄨Bϵ0 (xi ) ∩ D󵄨󵄨󵄨 + |DΔDi |) i→∞

i→∞

󵄨 󵄨 ≤ 󵄨󵄨󵄨Bϵ0 (x0 ) ∩ D󵄨󵄨󵄨 + lim sup |Di ΔD| = 0. i→∞

This yields a desired contradiction. It remains to show D ∈ 𝒟c . Indeed, by Proposition 2.11, x ∈ 𝜕D implies that x ∈ sptμD . ∗ Note Di → D in L1 (ℝn ) implies that μDi ⇀ μD as convergence of Radon measures. Hence, there exists xi ∈ sptμDi ⊂ 𝜕Di such that xi → x so that for any r > 0, it holds that

On a variational problem of nematic liquid crystal droplets

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󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 n 󵄨󵄨Br (x) ∩ D󵄨󵄨󵄨 = lim󵄨󵄨󵄨Br (xi ) ∩ D󵄨󵄨󵄨 ≥ lim inf󵄨󵄨󵄨Br (xi ) ∩ Di 󵄨󵄨󵄨 − lim sup |Di ΔD| ≥ cr . i i i

This implies D ∈ 𝒟c . The following remark follows directly from (i) and (iii). Remark 3.2. If Di and D satisfy the same assumptions as in Lemma 3.1, and if int(D) ≠ 0, then int(D) is connected. Similar to Lemma 3.1, for a set in the class 𝒟c we have the following. Lemma 3.3. For c > 0, if {Di } ⊂ 𝒟c and Di → D in L1 (ℝn ), then after modifying a set of zero ℋn -measure, D ∈ 𝒟c . Moreover, for any ϵ > 0, there is N = N(ϵ) > 0 such that if i > N, the following properties hold: (i) D ⊂ Dϵi . (ii) (Di )ϵ ⊂ D. (iii) Dϵ ⊂ Di . The following corollary follows directly from Lemma 3.3. Corollary 3.4. For any c > 0 and a sequence {Di } ⊂ 𝒟c with uniformly bounded perimeters, there is an open set D ∈ 𝒟c such that Di → D in L1 (ℝn ). Moreover, D and Di satisfy the properties (i), (ii), and (iii) of Lemma 3.3. Now we are ready to prove the main theorem of this section. Theorem 3.5. For M > 0, R0 > 0, and c0 > 0, if {Ωi } is a sequence of M-uniform domains in BR0 such that |Ωi | ≥ c0 > 0 and Ωi → D in L1 (ℝn ), then there is an M-uniform domain Ω such that Ωi → Ω in L1 (ℝn ). Proof. As in Proposition 2.11, we assume sptμD = 𝜕D. We first prove that int(D) ≠ 0. Indeed, notice that by Remark 2.9, there exists a r0 > 0 depending only on c0 , n, and M r such that each Ωi contains a ball of radius r0 . Therefore, for each Ωi , if ϵ < 20 , then by r0 definition (Ωi )ϵ contains a ball of radius 2 . By Lemma 3.1(ii), D also contains a ball of r radius 20 , and hence int(D) ≠ 0. Set Ω = int(D). It suffices to show that Ω is an M-uniform domain, since the L1 convergence of Ωi to Ω follows directly from Remark 2.6, Proposition 2.8, and the fact Ω ⊂ D ⊂ Ω. Fix any x, y ∈ Ω, then given any N >> M, say N > 2M, we may choose 0 < ϵ < N1 so small that kϵ < d(x, 𝜕Ω) ≤ (k +1)ϵ, k >> N (say k > (1+1/M)(N +1)), and |x−y| > 2(N +1)ϵ. From Lemma 3.1(i) and (iii), and since int(Ω) ≠ 0, we know that d H (Ωi , Ω) → 0, hence we may choose xi , yi ∈ Ωi ∩ Ω, with |xi − x| < ϵ, |yi − y| < ϵ for i large. By Lemma 3.1(ii), we may also choose i large such that (Ωi )ϵ ⊂ Ω.

(3.1)

114 � Q. Li and C. Wang Also, we choose γi ⊂ Ωi to be the rectifiable curve connecting xi and yi in Ωi as in the definition of M-uniform domain. For any p ∈ γi , if p ∈ BNϵ (xi ) ∪ BNϵ (yi ), then clearly p ∈ B(N+1)ϵ (x) ∪ B(N+1)ϵ (y) ⊂ Ω. Moreover, this implies d(p, 𝜕Ω) ≥ kϵ − (N + 1)ϵ >

1 1 (N + 1)ϵ ≥ min{|p − x|, |p − y|}. M M

(3.2)

Clearly, (3.2) also holds for any p on the line segment between xi and x, and between yi and y. If p ∉ BNϵ (xi ) ∪ BNϵ (yi ), then d(p, 𝜕Ωi ) ≥

1 1 min{|p − xi |, |p − yi |} > Nϵ, M M

thus p ∈ (Ωi )Nϵ/M ⊂ (Ωi )ϵ ⊂ Ω ∩ Ωi . Moreover, let r = d(p, 𝜕((Ωi )ϵ )), then by (3.1) we get Br (p) ⊂ Ω, so d(p, 𝜕Ω) ≥ r = d(p, 𝜕((Ωi )ϵ )) ≥ d(p, 𝜕Ωi ) − ϵ. Therefore, d(p, 𝜕Ωi ) − ϵ d(p, 𝜕Ω) ≥ min{|p − xi |, |p − yi |} min{|p − xi |, |p − yi |} ϵ 1 1 1 − ≥ − . ≥ M Nϵ M N

(3.3)

Hence, by the choice of ϵ and N we have that 1 1 − )(min{|p − x|, |p − y|} − ϵ) M N 1 1 1 ≥ ( − )(min{|p − x|, |p − y|}) − . M N MN

d(p, 𝜕Ω) ≥ (

(3.4)

Therefore, we may let γN be the curve with three parts. The first part connects x and xi with a line segment, the second part connects xi and yi with γi as above, and the third part connects yi and y with a line segment. It is clear that γN ⊂ Ω and γN connects x and y, then from (3.2) and (3.4) and the choice of ϵ, we obtain (i) 1

N

ℋ (γ ) ≤ M|x − y| + 2

M +1 ; N

(ii) d(p, 𝜕Ω) ≥ (

1 1 1 − ) min{|p − x|, |p − y|} − M N MN

∀p ∈ γN .

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Then by compactness of (Ω, d H ), and since γN is connected, there is a compact connected set E ⊂ Ω such that d H (γN , E) → 0 as N → ∞. Consequently, by [13, Theorem 3.18], 1

1

N

ℋ (E) ≤ lim inf ℋ (γ ) ≤ M|x − y|. N→∞

Moreover, by [13, Lemma 3.12], E is path connected, thus we can choose a curve γ ⊂ E joining x and y. For any p ∈ γ, we can choose sequence pN ∈ γN , pN → p. Since d(pN , 𝜕Ω) ≥ (

1 1 1 − ) min{|pN − x|, |pN − y|} − , M N 2MN

we have, after sending N → ∞, d(p, 𝜕Ω) ≥

1 min{|p − x|, |p − y|}, M

which also clearly implies γ ⊂ int Ω. Then γ satisfies both properties in the definition of M-uniform domain, thus Ω is M-uniform. By Remark 3.2 and Proposition 2.8, Ω is a domain. This completes the proof. Remark 3.6. The full generality of compactness of M-uniform domains is obtained in [10, Theorem 1.2], where it is shown that any sequence of M-uniform domains with fixed volume must have uniformly bounded fractional perimeters, and thus have an L1 limit up to a subsequence, and the limit is also M-uniform.

4 Existence of equilibrium liquid crystal droplets in Problems A–C In this section, we will study the existence of minimizers to Problems A–C, which can be extended in n-dimensions. We begin with the following lemma, which plays a crucial role in Problems A–C over outer minimal sets. See [5] and [6] for some basic properties on Hausdorff metric and convergence in measure. Lemma 4.1. For c > 0, let {Ei }∞ i=1 ∈ 𝒟c be a sequence of outward-minimizing sets such that Ei → E in L1 as i → ∞. Then E ∈ 𝒟c is also an outward-minimizing set. Moreover, P(Ei ) → P(E) and ℋn−1 (𝜕∗ Ei ) → ℋn−1 (𝜕∗ E) as i → ∞. Proof. Let F ⊃ E. Then by [2, Proposition 3.38(d)] and the outward-minimality of Ei , we have P(Ei ∩ F) ≤ P(F) + P(Ei ) − P(Ei ∪ F) ≤ P(F). This implies

116 � Q. Li and C. Wang P(E) = P(E ∩ F) ≤ lim inf P(Ei ∩ F) ≤ F(F). i

Hence, E is outward-minimizing. By Lemma 3.1 and Remark 2.13, E ∈ 𝒟c ∩ 𝒟c . It follows from Proposition 2.16 that for any ϵ > 0, there exists a smooth open set Oϵ ⋑ E such that P(Oϵ ) ≤ P(E) + ϵ. Applying Lemma 3.1(iii), we have that there exists a sufficiently large i0 ≥ 1 such that Ei ⊂ Oϵ ,

∀i ≥ i0 .

This, combined with the outward minimality of Ei , implies P(Ei ) ≤ P(Oϵ ) ≤ P(E) + ϵ,

∀i ≥ i0 .

Thus lim sup P(Ei ) ≤ P(E). i

On the other hand, by lower semicontinuity we have P(E) ≤ lim inf P(Ei ). i

Therefore, P(Ei ) → P(E) as i → ∞. Since Ei , E ∈ 𝒟c ∩ 𝒟c , the last statement follows from Theorem 2.1. Now we are ready to state the main theorem of this section. Theorem 4.2. The following statements hold: (i) For M ≥ 1, the infimum of Problem C in the class of M-uniform domains of finite perimeter is attained. (ii) For M > 1, the infimum of Problems A, B, C can be attained in the class of M-uniform outer minimal domains. Proof. We first prove (i). For a minimizing sequence (Ωi , ui ), where Ωi are M-uniform domains with finite perimeter and ui ∈ H 1 (Ωi , 𝕊2 ). Let û i ∈ H 1 (BR0 , ℝ3 ) be an extension of ui such that ‖û i ‖H 1 (BR ) ≤ C(n, M)‖ui ‖H 1 (Ωi ) . 0

Hence, there is a û ∈ H 1 (BR0 , ℝ3 ) such that û i ⇀ û

in H 1 (BR0 ).

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By Theorem 3.5, there is an M-uniform domain Ω ⊂ BR0 such that Ωi → Ω in L1 . Since ∇û i ⇀ ∇û in L2 (BR0 ) and χΩi → χΩ in L1 (BR0 ), by the lower semicontinuity, we have that ∫ |∇u|̂ 2 ≤ lim inf ∫ |∇û i |2 = lim inf ∫ |∇ui |2 . i→∞

Ω

i→∞

Ωi

(4.1)

Ωi

Denote u = u|̂ Ω . Then it is not hard to see |u| = 1 for a. e. x ∈ Ω so that u ∈ H 1 (Ω, 𝕊2 ). In order to show (Ω, u) is a minimizer of Problem C among M-uniform domains of finite perimeter, we have to verify that u∗ = νΩ for ℋn−1 -a. e. on 𝜕∗ Ω. In fact, it follows from χΩi → χΩ in L2 (BR0 ) and div(û i ) ⇀ div(u)̂ in L2 (BR0 ) and Theorem 2.4 that P(Ωi ) = ∫ div(ui ) = ∫ χΩi div(û i ) → ∫ χΩ div(u)̂ = ∫ div(u) BR0

Ωi

BR0

Ω

= ∫ u∗ ⋅ νΩ d ℋn−1 ≤ P(Ω). 𝜕∗ Ω

This, combined with the lower semicontinuity property of perimeter, implies that u∗ = νΩ for ℋn−1 -a. e. on 𝜕∗ Ω. Hence, the proof of (i) is complete. Next, we prove (ii). For Problem A in part (ii), let (Ωh , uh ) be a minimizing sequence among M-uniform, outer minimal domains and H 1 -unit vector fields on Ωh . Since Ωh are outward-minimizing sets in BR0 , P(Ωh ) are uniformly bounded. By Lemma 4.1 and Theorem 3.5, we may assume that there exists an M-uniform, outer minimal domain Ω such that up to a subsequence, Ωh → Ω in L1 and P(Ωh ) → P(Ω). As in the proof of (i) above, we may extend uh in BR0 , still denoted as uh , so that uh ⇀ u in H 1 (BR0 , ℝ3 ) for some u ∈ H 1 (BR0 , ℝ3 ). Thus we have ∫ |∇u|2 ≤ lim inf ∫ |∇uh |2 , h

Ω

Ωh

and u(x) ∈ 𝕊2 for a. e. x ∈ Ω. Since f is convex, we can write f (x) = sup(ai x + bi ). i

In the following, we do not distinguish u with u∗ on 𝜕∗ Ω, and we do not distinguish 𝜕∗ Ωh , 𝜕∗ Ω with 𝜕Ωh , 𝜕Ω due to Remark 2.15. Define τh (A) := ℋn−1 (𝜕∗ Ωh ∩ A),

τ(A) := ℋn−1 (𝜕∗ Ω ∩ A),

and

μh (A) := ∫ f (uh ⋅ νh )dτh , A

for any measurable A ⊂ ℝn , where νh is the measure theoretical outer unit normal of Ωh . Then Lemma 4.1 implies that

118 � Q. Li and C. Wang τh (A) → τ(A)

as h → ∞.

(4.2)

Since f is bounded and nonnegative, μh are nonnegative Radon measures so that we may assume there is a nonnegative Radon measure μ such that after passing to a subsequence, μh ⇀ μ as h → ∞ as weak convergence of Radon measures. Decompose μ as μ = (Dτ μ)τ + μs , μs ⊥ τ, and μs ≥ 0. Then lim inf μh (A) ≥ μ(A) ≥ ∫ Dτ μdτ. h→∞

(4.3)

A

It follows from Theorem 2.1 that x ∈ 𝜕∗ Ω holds for τ-a. e. x ∈ BR0 . Now any such x ∈ 𝜕∗ Ω, we claim that there exists rj → 0 such that for Bj = Brj (x), it holds that (a) ℋn−1 (𝜕Bj ∩ 𝜕Ω) = 0 and ℋn−1 (𝜕Bj ∩ 𝜕Ωh ) = 0, ∀h ≥ 1.

(b) ∫𝜕B ∩Ω uh ⋅ νBj d ℋn−1 → ∫𝜕B ∩Ω u ⋅ νBj d ℋn−1 as h → ∞. j

h

j

(c) μ(𝜕Bj ) = 0.

(d) Dτ μ(x) = limj

μ(Bj ) τ(Bj )

and limj→∞

∫B u⋅νBj dτ j

τ(Bj )

= u(x) ⋅ ν(x).

Indeed, (a) and (c) are true because τ, τh , and μ are nonnegative Radon measures. (d) follows from the Lebesgue differentiation Theorem. To see (b), let ũ h = uh χΩh and ũ = uχΩ . Since ũ h → ũ in L1 , we have 1

̃ ℋn−1 dr → 0 ∫ |uh̃ − u|̃ = ∫ ∫ |ũ h − u|d

as h → ∞.

0 𝜕Br (x)

B1 (x)

Therefore, by Fatou’s lemma, 1

̃ ℋn−1 dr = 0, ∫ lim inf ∫ |ũ h − u|d h→∞

0

𝜕Br (x)

hence for almost every r ∈ (0, 1) and for a subsequence of h → ∞, 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨

uh ⋅ νBr (x) d ℋn−1 −

∫

𝜕Br (x)∩Ωh

̃ ℋ ≤ ∫ |ũ h − u|d

∫ 𝜕Br (x)∩Ω

n−1

󵄨󵄨 󵄨 u ⋅ νBr (x) d ℋn−1 󵄨󵄨󵄨 󵄨󵄨

→ 0.

𝜕Br (x)

This completes the proof of (b). Now we return to the proof of (ii). By (c), μ(Bj ) = lim μh (Bj ) = lim h→∞

h→∞

∫ f (uh ⋅ νh ) d ℋn−1 . 𝜕Ωh ∩Bj

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Also as h → ∞, up to a subsequence we have ∫ uh ⋅ νh d ℋn−1 𝜕Ωh ∩Bj

=

uh ⋅ νΩh ∩Bj d ℋn−1 −

∫ 𝜕(Ωh ∩Bj )

∫ uh ⋅ νBj d ℋn−1 , 𝜕Bj ∩Ωh

= ∫ divuh − ∫ uh ⋅ νBj d ℋn−1 , Ωh ∩Bj

𝜕Bj ∩Ωh

→ ∫ divu − ∫ u ⋅ νBj d ℋn−1 , Ω∩Bj

=

𝜕Bj ∩Ω

u ⋅ νΩ∩Bj d ℋn−1 − ∫ u ⋅ νBj d ℋn−1

∫

𝜕Bj ∩Ω

𝜕(Ω∩Bj )

= ∫ u ⋅ νΩ d ℋn−1 . 𝜕Ω∩Bj

Therefore, for τ-a. e. x ∈ BR0 , it follows: Dτ μ(x) = lim j

μ(Bj ) τ(Bj )

= lim lim j

= lim

h ∩Bj

∫𝜕Ω

h ∩Bj

h

f (uh ⋅ νh ) d ℋn−1

ℋn−1 (𝜕Ω ∩ Bj )

h

≥ lim lim j

∫𝜕Ω

(ai uh ⋅ νh + bi ) d ℋn−1 ℋn−1 (𝜕Ω ∩ Bj )

∫𝜕Ω∩B (ai u ⋅ νΩ + bi ) d ℋn−1

j

j

ℋn−1 (𝜕Ω ∩ Bj )

also by (4.2)

,

= ai u(x) ⋅ νΩ (x) + bi .

(4.4)

Hence, Dτ μ ≥ f (u ⋅ νΩ ) for τ-a. e. x ∈ BR0 , and lim inf ∫ f (uh ⋅ νh ) d ℋn−1 = lim inf μh (BR0 ) ≥ ∫ Dτ μdτ h

h

𝜕Ωh

BR

≥ ∫ f (u ⋅ ν)dτ = ∫ f (u ⋅ ν) d ℋn−1 . BR0

Therefore, (Ω, u) is a minimizer.

𝜕Ω

(4.5)

120 � Q. Li and C. Wang To complete the proof of statements in (ii), it remains to show if (Ωi , ui ) are a minimizing sequence in Problem (B) and converges weakly to (Ω, u), then u ⋅ ν = c for ℋn−1 a. e. on 𝜕∗ Ω. This can be seen from lim inf ∫ f (ui ⋅ νi )d ℋn−1 ≥ ∫ f (u ⋅ ν)d ℋn−1 . i→∞

𝜕∗ Ωi

𝜕∗ Ω

In fact, by choosing f (t) = μ(t − c)2 we have that ∫ (u ⋅ ν − c)2 d ℋn−1 ≤ lim inf ∫ (ui ⋅ νi − c)2 d ℋn−1 = 0. i→∞

𝜕∗ Ω

(4.6)

𝜕∗ Ωi

Hence, u ⋅ ν ≡ c for ℋn−1 -a. e. on 𝜕∗ Ω. This completes the proof.

5 On the uniqueness of Problem C In this section, we will show the uniqueness of Problem C in the class of C 1,1 -star-shaped, mean convex domains in ℝ3 . We will assume the domains have volume V0 = |B1 |, where B1 ⊂ ℝ3 is the unit ball centered at 0. We begin with the following. Lemma 5.1. For any bounded C 1,1 -domain Ω ⊂ ℝ3 , inf{∫ |∇u|2 | u ∈ H 1 (Ω, 𝕊2 ), u = νΩ on 𝜕Ω} ≥ ∫ H𝜕Ω d ℋ2 , Ω

𝜕Ω

where H𝜕Ω is the mean curvature of 𝜕Ω. Proof. Let u ∈ H 1 (Ω, 𝕊2 ), with u = νΩ on 𝜕Ω, be such that ∫ |∇u|2 = inf{∫ |∇u|2 | u ∈ H 1 (Ω, 𝕊2 ), u = νΩ on 𝜕Ω}. Ω

Ω

Then by [33, 34], u ∈ C ∞ (Ω \ {ai }Ni=1 , 𝕊2 ) for a finite set ⋃Ni=1 {ai } ⋐ Ω. Observe that N

2

(div(u)) − tr(∇u)2 = div(div(u)u − (∇u)u)

in Ω \ ⋃{ai }. i=1

By [29, Proposition 2.2.1], we have that |∇u|2 ≥ (divu)2 − tr(∇u)2

N

in Ω \ ⋃{ai }. i=1

(5.1)

On a variational problem of nematic liquid crystal droplets

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x−a

By [3, Theorem 1.9], near each ai , u(x) ∼ R( |x−ai | ) for some rotation R ∈ O(3). In particui lar, one has that for r > 0 sufficiently small, 󵄨 󵄨󵄨 󵄨󵄨 2 󵄨󵄨 󵄨󵄨 ∫ (div(u)u − (∇u)u) ⋅ νBr (ai ) d ℋ 󵄨󵄨󵄨 = O(r). 󵄨󵄨 󵄨󵄨 𝜕Br (ai )

Hence, ∫ |∇u|2 ≥ Ω

∫

2

(div(u)) − tr(∇u)2

Ω\⋃Ni=1 Br (ai )

=

∫ Ω\⋃Ni=1

div((div u)u − (∇u)u)

Br (ai )

= ∫ (div(u)u − (∇u)u) ⋅ νΩ d ℋ2 𝜕Ω

n

− ∑ ∫ (div(u)u − (∇u)u) ⋅ νBr (ai ) d ℋ2 i=1 𝜕B (a ) r i

≥ ∫ (div(u) − ((∇u)νΩ ) ⋅ νΩ ) d ℋ2 − CNr 𝜕Ω

= ∫ (div𝜕Ω νΩ ) d ℋ2 − CNr 𝜕Ω

= ∫ H𝜕Ω d ℋ2 − CNr. 𝜕Ω

This implies (5.1) after sending r → 0. The inequality (5.1) leads us to study the minimization of the total mean curvatures. It is well known that ∫ H𝜕Ω d ℋ2 ≥ 4√πP(Ω)

(5.2)

𝜕Ω

is true if Ω is convex, and the equality holds if and only if Ω is a ball. Very recently, Dalphin–Henrot–Masnou–Takahashi [9] proved that if Ω is a revolutionary solid and H ≥ 0, then (5.2) is true, and the equality holds if and only if Ω is a ball. Without the mean convexity, (5.2) is false; see [9]. In the next lemma, we present a proof that (5.2) is true if Ω is a C 1,1 star-shaped and mean convex domain. The key ingredient of the proof is based on the result by Gerhardt [18]. We remark that a more general version of (5.2) has been proven by Guan–Li [20]. Here, we will sketch the proof, since it is elementary in ℝ3 . Lemma 5.2. The inequality (5.2) holds, if Ω is C 1,1 -strictly star-shaped and mean convex.

122 � Q. Li and C. Wang Proof. By the remark below, we may assume Ω ∈ C ∞ . By a standard argument, we can perturb Ω so that H > 0 everywhere. Indeed, represent 𝜕Ω as an embedding F 0 : 𝕊2 → ℝ3 and consider the mean curvature flow {Ft : 𝕊2 → ℝ3 : t ∈ [0, T)}, which is a family of embeddings so that 𝜕F = Hνt 𝜕t

F0 = F 0 ,

0 < t < T;

where νt is the inward unit normal of the embedding Ft . It is well known that the solution exists for a short time T > 0. If t > 0 is small, then Ft (𝕊2 ) remains to be star-shaped. The evolution of the mean curvature H of Ft (𝕊2 ) is given by 𝜕H = ΔH + |A|2 H, 𝜕t where A is the second fundamental form of Ft (𝕊2 ). Then the strong maximum principle implies that H > 0 everywhere on Ft (𝕊2 ) for t > 0. It is clear that after a small perturbation in C 1 -norm, Ω is still strictly star-shaped. Hence, it suffices to prove (5.2) by assuming H > 0 everywhere on 𝜕Ω. We argue it by contradiction. Suppose there were a strictly star-shaped domain Ω with H > 0 everywhere on 𝜕Ω such that ∫𝜕Ω H d ℋ2 4√πP(Ω)

< 1.

Representing 𝜕Ω as an embedding G0 : 𝕊2 → ℝ3 . Now consider the inverse mean curvature flow {Gt : 𝕊2 → ℝ3 : t ∈ [0, ∞)}, which is a family of embeddings that solves 𝜕G 1 = νt , 𝜕t H where νt is the inward unit normal of the embedding Gt . It has been shown by Gerhardt [18] that St := Gt (𝜕Ω) converges to the unit sphere 𝕊2 , up to rescalings by e−t/2 , as t → ∞. Set y(t) =

∫S H d ℋ2 t

4√π Area(St )

,

t > 0.

Observe that y(t) is scaling-invariant. Therefore, y(0) < 1 and y(t) → 1 as t → ∞. On the other hand, using the evolution equations under the inverse mean curvature flow we have that

On a variational problem of nematic liquid crystal droplets

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d |A|2 H = −ΔH − , dt H and d √g = √g, dt where Δ is the surface Laplacian and g is the metric on surface St induced by Euclidean metric in ℝ3 . Direct calculations imply 2 ∫S H d ℋ2 d ∫St H d ℋ |A|2 1 t ( ) = (∫(H − ) d ℋ2 ) − dt 4√πP(Ω) H 4√π Area(St ) 8√π Area(St ) St

=

1 2K 1 (∫ d ℋ2 − ∫ H d ℋ2 ) H 2 4√π Area(St ) St

St

4K − H 2 1 d ℋ2 ≤ 0, = ∫ 2H 4√π Area(St ) St

since H 2 ≥ 4K, here K is the Gauss curvature of St . Therefore, y(t) ≤ y(0) < 1 for all t > 0. We get a desired contradiction. Remark 5.3. (5.2) is actually true for any C 1 -strictly star-shaped surface with bounded nonnegative generalized mean curvature, in particular for a C 1,1 -mean convex surface. Indeed, by [24, Lemma 2.6], we can find a family of smooth strictly star-shaped mean convex hypersurfaces converging to the surface uniformly in C 1,α ∩ W 2,p for 0 < α < 1 and 1 < p < ∞ so that the total mean curvature of the smooth surfaces converges to the total mean curvature of the original surface. We refer the reader to [24] for the detail. By Lemma 5.2 and the isoperimetric inequality, P(Ω) ≥ 4π(

2/3

3 |Ω|) 4π

,

we immediately have the following. Corollary 5.4. It holds that inf{∫ |∇u|2 : Ω is C 1,1 -star-shaped, mean convex, |Ω| = |B1 |, u ∈ H 1 (Ω, 𝕊2 ), Ω

u = νΩ on 𝜕Ω} ≥ 8π, and the equality holds if and only if Ω = B1 , up to translation and rotation.

124 � Q. Li and C. Wang As a consequence, we have the following. Theorem 5.5. Problem C over C 1,1 -star-shaped and mean convex domains is uniquely x . achieved at Ω = B1 and u(x) = |x| Proof. By direct calculations, 2 󵄨󵄨 x 󵄨󵄨󵄨 2 󵄨 ∫󵄨󵄨󵄨∇( )󵄨󵄨󵄨 = ∫ 2 = 8π. 󵄨󵄨 |x| 󵄨󵄨 |x|

B1

B1

x Hence, by the first statement in Corollary 5.4, (4.2) is attained at (B1 , |x| ). The uniqueness follows from the last statement of Corollary 5.4 and [7, Theorem 7.1].

Remark 5.6. Huisken first proved that (5.2) holds if Ω is C 1,1 -outer minimal (not necessarily connected), though it seems that he did not publish it; see also Freire–Schwartz [14, Theorem 5]. Hence, the same result as in Theorem 5.5 holds in the class of C 1,1 -outer minimal open sets. By [9], the same result as in Theorem 5.5 holds in the class of smooth domains of revolution. See also [31].

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[15] A. Figalli and F. Maggi, On the shape of liquid drops and crystals in the small mass regime. Arch. Ration. Mech. Anal. 201 (2011), 143–207. [16] Z. Y. Geng and F. H. Lin, The two-dimensional liquid crystal droplet problem with a tangential boundary condition. Arch. Ration. Mech. Anal. 243 (2022), 1181–1221. [17] F. W. Gehring and B. G. Osgood, Uniform domains and the quasihyperbolic metric. J. Anal. Math. 36 (1979), 50–74. [18] C. Gerhardt, Flow of nonconvex surfaces into spheres. J. Differ. Geom. 32 (1990), 299–314. [19] E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, 80, 1984. [20] P. F. Guan and J. Y. Li, The quermassintegral inequalities for k-convex starshaped domains. Adv. Math. 221 (2009), 1725–1732. [21] C. F. Gui, Y. Y. Hu and Q. F. Li, On smooth interior approximation of sets of finite perimeter. Proc. Amer. Math. Soc. (in press). arXiv:2210.11734. [22] P. Harjulehto, Traces and Sobolev extension domains. Proc. Am. Math. Soc. 134 (2006), 2373–2382. [23] G. Huisken and T. Ilmanen, The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality. J. Differ. Geom. 59 (2001), 353–437. [24] G. Huisken and T. Ilmanen, Higher regularity of the inverse mean curvature flow. J. Differ. Geom. 80 (2008), 433–451. [25] P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147 (1981), 71–88. [26] Q. F. Li, Geometric Measure Theory with Applications to Shape Optimization Problems. Thesis (Ph. D.)-Purdue University. 2018. 249 pp. ISBN: 978-0438-01844-0, ProQuest LLC. [27] Q. F. Li and M. Torres, Morrey spaces and generalized Cheeger set. Adv. Calc. Var. 12 (2019), 111–133. [28] F. H. Lin and C. C. Poon, On nematic liquid crystal droplets. In: Elliptic and Parabolic Methods in Geometry (Minneapolis, MN, 1994), 91–121. A K Peters, Wellesley, MA, 1996. [29] F. H. Lin and C. Y. Wang, The Analysis of Harmonic Maps and Their Heat Flows. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008, xii+267 pp. [30] F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems. An Introduction to Geometric Measure Theory. Cambridge Studies in Advanced Mathematics, 135. Cambridge University Press, Cambridge, 2012. [31] P. Sternberg, G. Williams and W. P. Ziemer, C 1,1 -regularity of constrained area minimizing hypersurfaces. J. Differ. Equ. 94 (1991), 83–94. [32] E. M. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, N. J., 1970. [33] R. Schoen and K. Uhlenbeck, A regularity theory for harmonic maps. J. Differ. Geom. 17 (1982), 307–335. [34] R. Schoen and K. Uhlenbeck, Boundary regularity and the Dirichlet problem for harmonic maps. J. Differ. Geom. 18 (1983), 253–268.

Petteri Harjulehto and Ritva Hurri-Syrjänen

Estimates for the variable order Riesz potential with applications Dedicated to the memory of David R. Adams

Abstract: We study weak-type estimates and exponential integrability for the variableorder Riesz potential. As an application, we prove an exponential integrability result with respect to the Hausdorff content for functions from variable exponent Sobolev spaces. In particular, we generalize the earlier Trudinger-type estimates by taking the integration in the sense of Choquet integrals when John domains are considered. Moreover, new exponential integrability results also for domains with outward cusps are obtained. Keywords: Exponential integrability, Hausdorff content, nonsmooth domain, pointwise estimate, Poincaré inequality, Riesz potential, variable exponent MSC 2020: Primary 46E35, 46E30, Secondary 31C15, 42B20

1 Introduction If Ω is a bounded, open set in the Euclidean space ℝn , n ≥ 2, and α is a continuous function, which satisfies 0 < α(x) < n for every x ∈ Ω, then we write for the operator Iα(⋅) acting on locally integrable functions f in Ω, Iα(⋅) f (x) := ∫ Ω

|f (y)| dy. |x − y|n−α(x)

(1.1)

This is called the variable-order Riesz potential. By defining the variable dimensional β(⋅) Hausdorff content ℋ∞ based on [21, 2.10.1, p. 169], in Definition 4.1 we prove that the level sets of the variable order Riesz potential Iα(⋅) f are exponentially decaying with respect to the variable-dimensional Hausdorff content. We show the following. Theorem 1.1. Let Ω be an open, bounded set in ℝn and let α : Ω → (0, n) be a log-Hölder continuous function with 0 < α− := ess inf α(x) ≤ α+ := ess sup α(x) < n. x∈Ω

x∈Ω

Petteri Harjulehto, Ritva Hurri-Syrjänen, Department of Mathematics and Statistics, University of Helsinki, FI-00014 Helsinki, Finland, e-mails: [email protected], [email protected] https://doi.org/10.1515/9783110792720-006

128 � P. Harjulehto and R. Hurri-Syrjänen Then there exist constants c1 and c2 independent of the function f such that the inequality n−α(⋅)

ℋ∞

n

({x ∈ Ω : Iα(⋅) f (x) > t}) ≤ c1 exp(−c2 t n−α− )

holds for all f ∈ Ln/α(⋅) (Ω)

with ‖f ‖Ln/α(⋅) (Ω) ≤

1 . 2(1 + |Ω|)

This result could be seen as a generalization of the work of Ángel D. Martínez and Daniel Spector [38, Theorem 1.3] to the variable-order case. As an application of Theorem 1.1, we prove exponential integrability estimates for variable-exponent Sobolev functions defined on bounded domains. This result is our second main theorem, Theorem 5.9 where we generalize the earlier known results of the exponential integrability, which were with respect to the Lebesgue measure by taking the integration with respect to the Hausdorff content as the Choquet integral as Martínez and Spector stated and proved for the usual Riesz potential in [38]. We state an important corollary of Theorem 5.9, which gives new results to the s-John domains. Examples of these domains are convex domains and domains with Lipschitz boundaries but also domains, which are allowed to have outward cusps of the order s. Corollary 1.2. Let D be an s-John domain in ℝn , n ≥ 2, with 1 ≤ s < positive constants a and b such that

n . n−1

Then there exist

n

s(n−1) ≤b ∫ exp(a|u(x) − uB | s(n−1) ) d ℋ∞ D

for all continuous u ∈ L1n/(n−s(n−1)) (D) with ‖∇u‖Ln/(n−s(n−1)) (D) ≤ 1, where B := B(x0 , dist(x0 , 𝜕D)). In particular, this corollary generalizes [48, Theorem 2] and [18, Theorem 3.3] to the Choquet integral case when s = 1, that is, for John domains. Corollary 1.2 gives a complete new result whenever 1 < s < n/(n − 1). We also prove a weak-type estimate for the variable-order Riesz potential in Theorem 3.2. By this result, Poincaré-type inequalities with respect to the Luxemburg norm on the variable exponent Lebesgue space are proved to be valid for L1p -functions defined on a new class of domains in Theorems 5.5 and 5.6. The new class of domains suits well to the questions for variable exponent Lebesgue and Sobolev spaces. This paper is organized as follows. Definitions for variable exponent Lebesgue spaces and Sobolev spaces are recalled in Section 2. Estimates for the variable- order

Estimates for the variable order Riesz potential with applications

� 129

Riesz potential acting on functions from the variable-order Lebesgue spaces are given and proved in Section 3. The variable-dimensional Hausdorff content is defined and its basic properties are proved in Section 4. The variable-order maximal function is recalled there and pointwise estimates for the Riesz potential are proved there, too. Our main weak-type estimate Theorem 1.1 is proved in Section 4. Applications are considered in Section 5. We start by defining a new class of functions, which includes s-John domains and then state and prove Poincaré-type inequalities of Theorems 5.5 and 5.6. there. Our main theorem in Section 5 is on the exponential integrability, Theorem 5.9, which yields Corollary 1.2.

2 Notation Let U in ℝn be any bounded, open set. For any measurable function g : U → ℝ and measurable set A ⊂ U, we define gA+ := ess sup g(x) x∈A

and gA− := ess inf g(x). x∈A

In the case A = U, we write g + := gU+ and g − := gU− . We say that the function g : U → ℝ satisfies the log-Hölder continuity condition if there is a constant c1 > 0 such that |g(x) − g(y)| ≤

c1 log(e + 1/|x − y|)

(2.1)

for all x, y ∈ U, x ≠ y. Note that g is a log-Hölder continuous function if and only if there − + exists a constant c > 0 such that |B|gB∩U −gB∩U ≤ c for all open balls B ∩ U ≠ 0. By a variable exponent, we mean a measurable function p : U → [1, ∞) such that 1 ≤ p− ≤ p+ < ∞. The set of all variable exponents is denoted by 𝒫 (U). By 𝒫 log (U), we denote a subset consisting of all log-Hölder continuous variable exponents. We define a modular on the set of Lebesgue measurable functions f by setting ϱLp(⋅) (U) (f ) := ∫ |f (x)|p(x) dx. U

The variable exponent Lebesgue space Lp(⋅) (U) consists of all measurable functions f : U → ℝ for which the modular ϱLp(⋅) (U) (f ) is finite. We write f ∈ Lp(⋅) (U). The Luxemburg–Nakano norm on this space is defined as ‖f ‖Lp(⋅) (U) := inf{λ > 0: ϱLp(⋅) (U) (f /λ) ≤ 1}. Equipped with this norm, Lp(⋅) (U) is a Banach space. We use the abbreviation ‖f ‖p(⋅) to denote the norm in the whole space.

130 � P. Harjulehto and R. Hurri-Syrjänen For open sets U, the variable exponent Sobolev space L1p(⋅) (U) consists of functions

u ∈ L1loc (U) for which the absolute value of the distributional gradient |∇u| belongs to Lp(⋅) (U): 1,1 L1p(⋅) (U) := {u ∈ Wloc (U) : |∇u| ∈ Lp(⋅) (U)},

1,1 where Wloc (U) is the classical local Sobolev space. More information and proofs for these facts can be found in [17, Chapters 2, 4, 8, and 9] or from [13].

3 Strong- and weak-type estimates Let Ω be a bounded, open set in the Euclidean space ℝn , n ≥ 2. We assume that a continuous function α satisfies 0 < α(x) < n for every x ∈ Ω. We recall the definition (1.1) from the Introduction. The potential in (1.1) is called the variable-order Riesz potential of a function f . If α is a log-Hölder continuous function, then there exists a constant c > 0 depending only the log-Hölder constant of α such that c−1 Iα(⋅) f (x) ≤ ∫ Ω

|f (y)| dy ≤ cIα(⋅) f (x), |x − y|n−α(y)

we refer to [32, p. 270]. Let us define 1 1 α(x) := − p(x) n p#α (x)

i. e.

p#α (x) =

np(x) . n − α(x)p(x)

The variable-order Riesz potential has been studied extensively; see, for example, [19, 35, 39, 43, 44] and references therein. S. Samko has proved the following result. Lemma 3.1 (Theorem 3.2 of [44]). Let Ω ⊂ ℝn be a bounded, open set. Let p ∈ 𝒫 log (Ω) satisfy 1 < p− ≤ p+ < ∞. Assume that α− > 0 and (αp)+ < n. Then Iα(⋅) : Lp(⋅) (Ω) → #

Lpα (⋅) (Ω) is bounded, where p#α (x) =

np(x) . n−α(x)p(x)

Let us consider the case p− = 1. It is well known that I1 , where α ≡ 1, does not map L (Ω) → L1 (Ω). Thus, instead of the strong-type estimate, we can have only a weak-type estimate. For that, we need to assume that α is log-Hölder continuous. The following theorem is a modification of [10, Theorem 4.3]. We consider a bounded set and obtain a better control for the extra term than in [10, Theorem 4.3]. 1

Theorem 3.2. Let Ω ⊂ ℝn be a bounded, open set. Let p ∈ 𝒫 log (Ω) satisfy 1 ≤ p− ≤ p+ < ∞. Assume that α is a log-Hölder continuous function with α− > 0 and (αp)+ < n.

Estimates for the variable order Riesz potential with applications

� 131

np(x) . Then there exists a constant c such that for every f ∈ Lp(⋅) (Ω) with Let p#α (x) = n−α(x)p(x) ‖f ‖p(⋅) ≤ 1 and for every t > 0 the inequality,

∫ {x∈Ω:Iα(⋅) f (x)>t}

# 󵄨 󵄨 t pα (x) dx ≤ c ∫ |f (y)|p(y) dy + c󵄨󵄨󵄨{x ∈ Ω : 0 < |f (x)| ≤ 1}󵄨󵄨󵄨,

Ω

holds. The constant c depends only n, diam(Ω), α− , (αp)+ , p+ , and log-Hölder constants of p and α. Proof. As a part of the proof of [44, Theorem 3.2, p. 278 ], S. Samko proved the following pointwise inequality: p(x) #

(3.1)

Iα(⋅) f (x) ≤ c(ℳf (x)) pα (x) for almost all x ∈ Ω. Here, ℳ is the Hardy–Littlewood maximal operator ℳf (x) := sup r>0

1 |B(x, r)|

|f (y)| dy,

∫ B(x,r)∩Ω

and the constant c depends only on the dimension n, α− , (αp)+ and p+ . The proof is based on Hedberg’s trick, [31]. Thus we have p(x) #

{x ∈ Ω : Iα(⋅) f (x) > t} ⊂ {x ∈ Ω : c[ℳf (x)] pα (x) > t} =: E. Let us write AB f :=

1 ∫ |f (y)| dy. |B] B∩Ω

p(z) #

For every z ∈ E, we choose Bz := B(z, rz ) such that c(ABz f ) pα (z) > t where c is the constant from (3.1). Let x ∈ Bz and let us raise this inequality to the power p#α (x). Assume first that get ABz f

p#α (x)p(z) ≥ p(x). From now on, c p#α (z) −1 ≤ c|Bz | , and thus we obtain

may vary from line to line. Since ‖f ‖p(⋅) ≤ 1, we

#

t pα (x) ≤ c(ABz f )p(x) (ABz f ) ≤ c(ABz f )p(x) |Bz | = c(ABz f )p(x) |Bz |

p#α (x)p(z) −p(x) p#α (z)

p(x)−

p#α (x)p(z) p#α (z)

p(x)p#α (z)−p#α (x)p(z) p#α (z) p(z) #

= c(ABz f )p(x) |Bz |p(x)−p(z) |Bz | pα (z)

(p#α (z)−p#α (x))

.

132 � P. Harjulehto and R. Hurri-Syrjänen If p(x) − p(z) > 0, then |Bz |p(x)−p(z) is uniformly bounded by c(n) diam(Ω)np . Otherwise, we use log-Hölder continuity of p and obtain that |Bz |p(x)−p(z) is uniformly bounded by [17, Lemma 4.1.6, p. 101]. Let us look at the last term in the previous estimate. A short calculation gives that +

p#α (z) − p#α (x) =

n2 (p(z) − p(x)) + np(x)p(z)(α(z) − α(x)) . (n − α(z)p(z))(n − α(x)p(x))

Since p and α are bounded log-Hölder continuous functions and 0 < c1 ≤ (n − α(z)p(z))(n − α(x)p(x)) ≤ n2 < ∞, p(z) #

(p#α (z)−p#α (x))

with some positive constant c1 , the term |Bz | pα (z) is uniformly bounded. By [17, Theorem 4.2.4, p. 108], we can move the exponent p(x) inside the integral and obtain #

t pα (x) ≤ cABz (|f |p(⋅) + χ{00

holds. Remark 3.4. In Section 5, we need the operator ̃ f (x) := ∫ Is(⋅) Ω

|f (y)| dy, |x − y|s(x)(n−1)

n ̃ , where α(x) := n−s(x)(n−1). We where 1 ≤ s(x) < n−1 for every x ∈ Ω. We have Iα(⋅) = Is(⋅) see that α is log-Hölder continuous if and only if s is log-Hölder continuous. We recover the assumptions for s:

α− > 0 { (αp)+ < n

if and only if s+
0, and choose a covering that satisfies E ⊂ ⋃∞ i=1 B(xi , ri ) and ∞

β(xi )

∑ ri i=1

β(⋅) < ℋ∞ (E) + ε.

Estimates for the variable order Riesz potential with applications

� 135

We choose that V := ⋃∞ i=1 B(xi , ri ) and note that it is open. Thus ∞

β(xi )

β(⋅) β(⋅) inf ℋ∞ (U) ≤ ℋ∞ (V ) ≤ ∑ ri U

i=1

β(⋅) < ℋ∞ (E) + ε.

β(⋅) β(⋅) Since this holds for all ε > 0, we obtain infU ℋ∞ (U) ≤ ℋ∞ (E). Hence, the claim (3) is proved. Let us then prove (4). By (2), we have β(⋅)

∞

β(⋅)

ℋ∞ (⋂ Ki ) ≤ lim ℋ∞ (Ki ), i=1

i→∞

and the limit exists by (2) since the sequence of (Ki ) is decreasing. Let ε > 0, and choose by (3) an open set V that satisfies ⋂∞ i=1 Ki ⊂ V and β(⋅)

β(⋅)

∞

ℋ∞ (V ) < ℋ∞ (⋂ Ki ) + ε. i=1

Since V is open, we find j0 ∈ ℕ such that Kj ⊂ V for all j ≥ j0 . Thus for all j ≥ j0 we have β(⋅)

β(⋅)

∞

β(⋅)

ℋ∞ (Kj ) ≤ ℋ∞ (V ) < ℋ∞ (⋂ Ki ) + ε i=1

and, furthermore, ∞

β(⋅) β(⋅) lim ℋ∞ (Ki ) ≤ ℋ∞ (⋂ Ki ) + ε.

i→∞

i=1

Since this holds for all ε > 0, we obtain the inequality for the other direction. Hence, also the claim (4) is proved. β(⋅) Remark 4.3. ℋ∞ is not a capacity in the sense of Choquet [12], that is, if (Ei ) is a increasing sequence of sets then β(⋅)

∞

β(⋅)

ℋ∞ (⋃ Ei ) = lim ℋ∞ (Ei ). i=1

i→∞

If β is a constant function, then this has been proved in [14]; see also [15, 45]. We recall the definition of the variable-order fractional maximal function [35]. Definition 4.4 (Variable-order fractional maximal function). Let α : Ω → [0, n) any measurable function and f ∈ L1loc (Ω). Then

136 � P. Harjulehto and R. Hurri-Syrjänen

ℳα(⋅) f (x) := sup r>0

r α(x) |B(x, r)|

∫

|f (y)| dy,

x ∈ Ω.

B(x,r)∩Ω

We give a proof for the lower semicontinuity of this maximal function for the reader’s convenience. Lemma 4.5. Let Ω ⊂ ℝn be an open set, and α : Ω → (0, n] be continuous. Then ℳα(⋅) f is lower semicontinuous. Proof. Let us write Et := {x ∈ Ω : ℳα(⋅) f (x) > t}. We need to show that Et is open in ℝn . So, let x ∈ Et . By the definition of ℳα(⋅) for every x ∈ Et , there exists a radius rx > 0 such that rxα(x) |B(x, rx )|

|f (y)| dy > t.

∫ B(x,rx )∩Ω

By the properties of the Lebesgue measure, we obtain lim

R↘rx

Rα(x) |B(x, R)|

∫

|f (y)| dy =

B(x,rx )∩Ω

rxα(x) |B(x, rx )|

∫

|f (y)| dy.

B(x,rx )∩Ω

Hence, there exists R > rx such that λ :=

Rα(x) |f (y)| dy ∫ |B(x,R)| B(x,rx )∩Ω

t

> 1.

Let x ′ be such that |x − x ′ | < R − rx and assume that |α(x ′ ) − α(x)|
t. Hence, the set Et is open in Ω, and thus it is open also in ℝn . The following lemma is a generalization of [38, Lemma 3.5] and [41, Theorem (ii)] to the case of the variational-dimensional Hausdorff content. Lemma 4.6. Let Ω ⊂ ℝn be a bounded, open set. Let α : Ω → [0, n) be a measurable function such that α+ < n. Then there exists a constant c, depending only on the dimension n, such that the inequality n−α(⋅)

ℋ∞

({x ∈ Ω : ℳα(⋅) f (x) > t}) ≤

c(n) ‖f ‖L1 (Ω) t

holds for all f ∈ L1 (Ω). Proof. Let us write Et := {x ∈ Ω : ℳα(⋅) f (x) > t}. By the definition of ℳα(⋅) for every x ∈ Et , there exists a radius rx > 0 such that rxα(x) |B(x, rx )|

∫

|f (y)| dy > t.

B(x,rx )∩Ω

This inequality yields that rxn−α(x)
0 and (αp)+ < n. Let f ∈ Lp(⋅) (ℝn ) with ‖f ‖p(⋅) ≤ 1. Then there exists a constant c such that for every x ∈ Ω and every r > 0 the inequality,

∫ Ω\B(x,r)

|f (y)| p(x) dy ≤ c max{1, } n − α(x)p(x) |x − y|n−α(x)

p+ −1 p+

r

−

n−α(x)p(x) p(x)

,

holds. Here, the constant c depends only on the dimension n, the log-Hölder constant of p and diam(Ω). Proof. Let us denote by A(x, r) the annulus (B(x, r) \ B(x, r/2)) ∩ Ω and write I := {i ∈ ℕ : r ≤ 2i ≤ diam(Ω)}. Let us first note that Lemma 4.16 and Theorem 4.5.7 of [17] yield that ‖1‖Lp(⋅) (B) ≤ c|B|1/p(x) for all x ∈ B ∩ Ω and all ball B with diam(B) ≤ diam(Ω). Here, the constant c depends only on n, log-Hölder constant of p and diam(Ω). When we use in (4.2) Hölder’s inequality for the second inequality, for the norm of the constant one for the third inequality, and finally Hölder’s inequality again, we conclude that ∫ Ω\B(x,r)

|f (y)| dy ≤ ∑ 2i(α(x)−n) ∫ |f (y)| dy |x − y|n−α(x) i∈I A(x,2i )

≤ 2∑2

i(α(x)−n)

i∈I

≤ c∑2 i∈I

‖f ‖Lp(⋅) (A(x,2i )) ‖1‖Lp′ (⋅) (B(x,2i )∩Ω)

i(α(x)−n+

n ) p′ (x)

‖f ‖Lp(⋅) (A(x,2i ))

Estimates for the variable order Riesz potential with applications

−in

≤ c(∑ 2

(p+ )′ p#α (x)

i∈I

1 (p+ )′

)

1 p+

p+

(∑ ‖f ‖Lp(⋅) (A(x,2i )) ) i∈I

� 139

(4.2)

p+

np(x) for x ∈ Ω where p#α (x) = n−α(x)p(x) . Since ‖f ‖p(⋅) ≤ 1, we have ‖f ‖p(⋅) ≤ ϱp(⋅) (u) by [17, Lemma 3.2.5. p. 75], and so p+

∑ ‖f ‖Lp(⋅) (A(x,2i )) ≤ ∑ ∫ |f (y)|p(y) dy ≤ ∫ |f (y)|p(y) dy ≤ 1. i∈I

i∈I

Ω

A(x,2i )

The first term on the last line of (4.2) is a geometric sum. Thus we obtain that (∑ 2

−in

(p+ )′ p#α (x)

i∈I

1 (p+ )′

)

≤r

−

n p#α (x)

(1 − 2

−n

(p+ )′ p#α (x)

−

)

1 (p+ )′

.

p(x) Let us write that k(x) := max{1, n−α(x)p(x) }. Now n/p#α (x) ≥ 1/k(x) and k(x) ≥ 1. Thus by

the inequality x a ≤ ax + 1 − a (which follows from Bernoulli’s inequality) with x = 2−(p and a = k1 , we obtain

+ ′

−n

(1 − 2

(p+ )′ p#α (x)

)

−1

(p+ )′

−1

≤ (1 − 2− k(x) )

+ ′

)

−1

≤ (1 − 2−(p ) ) k(x).

Hence, we have −in

(∑ 2 i∈I

(p+ )′ p#α (x)

1 (p+ )′

)

+ ′

≤ (1 − 2−(p ) )

−

1 (p+ )′

1

k(x) (p+ )′ r

−

n p#α (x)

.

Finally, we note by (p+ )′ ∈ (1, ∞) that (1 − 2−(p ) ) ∈ ( 21 , 1). Hence, we have the inequality + ′

−

(1 − 2−(p ) ) + ′

1 (p+ )′

< 2.

Hedberg proofed his famous pointwise estimate for the Riesz in [31]. As far as we know, the fractional maximal function was used first in this kind of estimate by Adams in [1]. In the variable exponent case, the Hedberg-type estimates are well known and variants have been used and proved, for example, in [16, Theorem 3.8], [17, Proposition 6.1.6], [24, Theorem 3.3], [30, (4.7)], [39, p. 429], [40, Lemma 4.6], [44, p. 279]. Here, instead of the standard maximal operator we have the variable-dimension fractional, maximal operator, and we calculate how the constant depends on p and α. Lemma 4.8 (Hedberg-type estimate). Let Ω ⊂ ℝn be a bounded, open set. Let p ∈ 𝒫 log (Ω) satisfy the inequalities 1 ≤ p− ≤ p+ < ∞. Assume that α : Ω → (0, n) satisfies α− > 0 and (αp)+ < n. If δ(x) := n−α(x)p(x) , then there exists a constant c such that for every p(x) ε : Ω → (0, ∞), with ε(x) ≤ α(x) for all x ∈ Ω, the inequality

140 � P. Harjulehto and R. Hurri-Syrjänen

1 Iα(⋅) f (x) ≤ c max{1, } δ(x)

p+ −1 p+

δ(x)

(ℳα(⋅)−ε(⋅) f (x)) δ(x)+ε(x)

holds for all f ∈ Lp(⋅) (Ω) with ‖f ‖p(⋅) ≤ 1. Here, c depends only on the dimension n, ε− , the log-Hölder constant of p, and diam(Ω). Proof. For a ball B(x, r), we write Iα(⋅) f (x) =

∫ B(x,r)∩Ω

|f (y)| dy + |x − y|n−α(x)

∫ Ω\B(x,r)

|f (y)| dy |x − y|n−α(x)

=: I + II. For the first term, we obtain ∞

I≤∑

j=0

∫ B(x,r2−j )\B(x,r2−j−1 )∩Ω

|f (y)| dy |x − y|n−α(x)

(r2−j )n (r2−j )α(x)−ε(x) 1 −j α(x)−ε(x) (r2 ) |B(x, r2−j )| j=0 ∞

≤ c(n) ∑

∞

B(x,r2−j )∩Ω

≤ c(n)r ε(x) 2n−α(x) ∑ 2−jε(x) ℳα(⋅)−ε(⋅) f (x) = j=0

|f (y)|

∫

(r2−j−1 )n−α(x)

dy

c(n)2ε(x) ε(x) r ℳα(⋅)−ε(⋅) f (x). 2ε(x) − 1

Next, we estimate the second term. Since ‖f ‖p(⋅) ≤ 1, we use Lemma 4.7 in order to obtain that 1

n−α(x)p(x) (p+ )′ p(x) − II ≤ c max{1, } r p(x) n − α(x)p(x) 1

(p+ )′ 1 = c max{1, } r −δ(x) , δ(x)

and the constant depends only on n, the log-Hölder constant of p, and diam(Ω). If 1 − δ(x)+ε(x)

(ℳα(x)−ε f (x))

< diam(Ω)

we choose 1 − δ(x)+ε(x)

r(x) = (ℳα(x)−ε f (x))

.

Hence, 1

δ(x) (p+ )′ 1 Iα(⋅) f (x) ≤ c max{1, } (ℳα(x)−ε(x) f (x)) δ(x)+ε(x) , δ(x)

Estimates for the variable order Riesz potential with applications

� 141

where the constant depends only on the dimension n, the log-Hölder constant of p, ε− , and diam(Ω). If (ℳα(x)−ε f (x))

1 − δ(x)+ε(x)

≥ diam(Ω)

we choose r(x) = diam(Ω). Thus we obtain δ(x)

Iα(⋅) f (x) ≤ I ≤ c diam(Ω)ε(x) ℳα(⋅)−ε(⋅) f (x) ≤ c diam(Ω)n (ℳα(x)−ε f (x)) δ(x)+ε(x) , where the constant depends only on the dimension n, ε− , and diam(Ω). Next, we prove our main Theorem 1.1, which is a generalization of [38, Theorem 1.2] to the variable-order case. We clarify dependences of the final constants of the given parameters in Remark 4.9. Proof of Theorem 1.1. Let p ∈ 𝒫 log (Ω) satisfy the inequality p(x) < by the assumption ‖f ‖

Lemma 4.8, we obtain

n L α(⋅)

(Ω)

≤

1 2(1+|Ω|)

n . α(x)

Then ‖f ‖p(⋅) ≤ 1

and Corollary 3.3.4 of [17]. By Hedberg’s lemma, 1

δ(x) (p+ )′ 1 (ℳα(⋅)−ε(⋅) f (x)) δ(x)+ε(x) , } Iα(⋅) f (x) ≤ c max{1, δ(x)

where c depends only on the dimension n, ε− , the log-Hölder constant of p, and diam(Ω). Let 1 < r < min{2, αn+ }. By Hölder’s inequality, we obtain r

1

ℳα(⋅)−ε(⋅) f (x) ≤ (ℳr(α(⋅)−ε(⋅)) |f | (x)) r .

Thus we have 1

δ(x) (p+ )′ 1 } (ℳr(α(⋅)−ε(⋅)) |f |r (x)) r(δ(x)+ε(x)) . Iα(⋅) f (x) ≤ c max{1, δ(x)

These estimates yield n−r(α(⋅)−ϵ(⋅))

ℋ∞

({x ∈ Ω : Iα(⋅) f (x) > t})

n−r(α(⋅)−ϵ(⋅)) ≤ ℋ∞ ({x ∈ Ω :

ℳr(α(⋅)−ε(⋅)) |f |r (x) 1

(c min{1, δ(x)} (p+ )′ t)

r(δ(x)+ε(x)) δ(x)

> 1}),

(4.3)

where c depends only on the dimension n, ε− , and the log-Hölder constant of p, and diam(Ω). Let us recall that δ(x) = n−α(x)p(x) . Let σ ∈ (0, 1) be a small number and choose p(x) n pσ (x) := α(x) − σ. Note that the function pσ is log-Hölder continuous provided that α is

142 � P. Harjulehto and R. Hurri-Syrjänen log-Hölder continuous, and the log-Hölder constant of pσ is independent of σ. Moreover, n . With this the values of the function pσ can be chosen to be near the critical value α(x) choice, we have δ(x) =

n − α(x)pσ (x) α(x)2 σ (α− )2 (α+ )2 = ∈[ σ, σ]. pσ (x) n − α(x)σ n n − α+

Hence, δ(x) → 0+ uniformly as σ → 0+ . Assume that t0 is such that ct0 = e, where c is from (4.3). For every t > t0 , we choose σ so small that 1 + ′

c min{1, δ(x)} (pσ ) t ≈ e,

(4.4)

where we denote A ≈ B if is there exists a positive constant c independent of A and B such that c−1 A ≤ B ≤ cA. Thus we have n−r(α(⋅)−ϵ(⋅))

ℋ∞

({x ∈ Ω : Iα(⋅) f (x) > t})

r(δ(x) + ε(x)) )}) δ(x) rε− n−r(α(⋅)−ϵ(⋅)) ≤ ℋ∞ ({x ∈ Ω : ℳr(α(⋅)−ε(⋅)) |f |r (x) > c exp( + )}). δ n−r(α(⋅)−ϵ(⋅)) ≤ ℋ∞ ({x ∈ Ω : ℳr(α(⋅)−ε(⋅)) |f |r (x) > c exp(

Hence, by Lemma 4.6 we obtain n−r(α(⋅)−ϵ(⋅))

ℋ∞

({x ∈ Ω : Iα(⋅) f (x) > t}) ≤ c exp(−

rε− ). δ+

For the exponent, we obtain δ+ = sup x∈Ω

n − α(x)pσ (x) α(x)2 σ (α+ )2 σ = sup = , pσ (x) n − α+ σ x∈Ω n − α(x)σ

and thus δ(x) n − α+ σ (α− )2 (n − α+ ) n α(x)2 σ ⋅ ∈ [ , = ]. + 2 δ+ n − α(x)σ (α+ )2 σ n − α+ n(α ) This yields that δ(x) ≈ δ+ for every x ∈ Ω, and hence by (4.4) we have n−σα− 1 1 (p+σ )′ n−(1+σ)α− . ≈ ≈ t = t δ+ δ(x)

This implies that n−r(α(⋅)−ϵ(⋅))

ℋ∞

n−σα−

({x ∈ Ω : Iα(⋅) f (x) > t}) ≤ c1 exp(−c2 rε− t n−(1+σ)α− ),

Estimates for the variable order Riesz potential with applications

� 143

where c1 depend only on n, α− , α+ , and log-Hölder constant of α, and diam(Ω), and c2 depend only on n, α− , α+ , and log-Hölder constant of α. Finally, we choose ε(x) := (r − 1)α(x) and obtain n−α(⋅)

ℋ∞

n−σα−

({x ∈ Ω : Iα(⋅) f (x) > t}) ≤ c1 exp(−c2 r(r − 1)α− t n−(1+σ)α− ),

whenever t > t0 . Since the left-hand side and the constants are independent of σ, we take σ → 0+ and obtain n−α(⋅)

ℋ∞

n

({x ∈ Ω : Iα(⋅) f (x) > t}) ≤ c1 exp(−c2 r(r − 1)α− t n−α− ).

The claim holds also if 0 < t ≤ t0 . Indeed, for 0 < t ≤ t0 n−α(⋅)

ℋ∞

n−α(⋅) ({x ∈ Ω : Iα(⋅) f (x) > t}) ≤ ℋ∞ (Ω)

n

n

n−α(⋅) = ℋ∞ (Ω) exp(ct n−α− ) exp(−ct n−α− ) n

n

n−α(⋅) ≤ ℋ∞ (Ω) exp(ct0n−α ) exp(−ct n−α− ), −

where n−α(⋅)

ℋ∞

n

(Ω) ≤ (1 + diam(Ω))

by the definition of the Hausdorff content. Hence, the theorem is proved. Remark 4.9. The estimates in the previous proof yield Theorem 1.1 with a constant c1 , which depends only on diam(Ω), n, α− , α+ , and log-Hölder constant of α, and a constant c2 , which depends only on n, α− , α+ , and log-Hölder constant of α.

5 Applications to nonsmooth domains The definition of a bounded John domain goes back to F. John [33, Definition, p. 402] who defined an inner radius and an outer radius domain, and later this domain was renamed as a John domain in [37, 2.1]. We generalize this definition so that the shape of the John cusp can depend on the point. If s is a constant function, we have a so called s-John domain studied in [46]. For other studies and generalizations of John domains, we refer to [24, 26, 28]. Definition 5.1. Let D ⊂ ℝn , n ≥ 2, be a bounded domain, and s : D → [1, ∞) a function. The domain D is an s(⋅)-John domain if there exist constants 0 < α ≤ β < ∞ and a point x0 ∈ D such that each point x ∈ D can be joined to x0 by a rectifiable curve γx : [0, ℓ(γx )] → D, parametrized by its arc length, such that γx (0) = x, γx (ℓ(γx )) = x0 , ℓ(γx ) ≤ β, and

144 � P. Harjulehto and R. Hurri-Syrjänen t s(x) ≤

β dist(γx (t), 𝜕D) for all t ∈ [0, ℓ(γx )]. α

The point x0 is called a John center of D and γx is called a John curve of x. Example 5.2. We construct a mushrooms-type domain. Let (rm ) be a decreasing sequence of positive real numbers converging to zero. Let Qm , m = 1, 2, . . . , be a closed cube in ℝn with side length 2rm . Let φ : [0, ∞) → [0.∞) be an increasing function with limt→0+ φ(t) = φ(0) = 0 and φ(t) > 0 for t > 0. Let Pm , m = 1, 2, . . . , be a closed rectangle in ℝn , which has side length rm for one side and 2φ(rm ) for the remaining n − 1 sides. Let Q := [0, 12] × [0, 12]. We attach Qm and Pm together creating “mushrooms,” which we then attach, as pairwise disjoint sets, to the side {(0, x2 , . . . , xn ) : x2 , . . . , xn > 0} of Q so that the distance from the mushroom to the origin is at least 1 and at most 4; see Figure 1. We have to assume here also that φ(rm ) ≤ rm . We need copies of the mushrooms. By an isometric mapping we transform these mushrooms onto the side ∗ ∗ {(x1 , 0, . . . , xn ) : x1 , x3 , . . . , xn > 0} of Q and denote them by Qm and Pm . So, again the distance from the mushroom to the origin is at least 1 and at most 4. We define ∞

∗ ∗ D := int(Q ∪ ⋃ (Qm ∪ Pm ∪ Qm ∪ Pm )). m=1

See Figure 1.

Figure 1: s(⋅)-John domain.

(5.1)

Estimates for the variable order Riesz potential with applications

� 145

3

We set that φ(t) := t 2 . We define that s(x) := 1

in Q

and

s(x) :=

3 2

∞

∗ in ⋃ (Qm ∪ Qm ), m=1

∗ . Then D is an s(⋅)-John domain, and grows linearly from 1 to 32 in each Pm and each Pm which can be seen as in [28, Lemma 6.2].

Next, we prove a chaining results for s(⋅)-John domains. We refer to [23, Theorem 9.3] for the proof in the classical case, and [28, Lemma 3.5] and [25, 4.3. Lemma] for generalizations. Lemma 5.3. Let D ⊂ ℝn , n ≥ 2, be a s(⋅)-John domain with John constants α and β. Let x0 ∈ D the John center. Then for every x ∈ D \ B(x0 , dist(x0 , 𝜕D)) there exists a sequence of balls (B(xi , ri )) such that B(xi , 2ri ) is in D for each i = 0, 1, . . . , and for some constants K = K(α, β, dist(x0 , 𝜕D)), N = N(n), and M = M(n): (1) B0 = B(x0 , 21 dist(x0 , 𝜕D)); (2) dist(x, Bi )s(x) ≤ Kri , and ri → 0 as i → ∞; (3) no point of the domain D belongs to more than N balls B(xi , ri ); and (4) |B(xi , ri ) ∪ B(xi+1 , ri+1 )| ≤ M|B(xi , ri ) ∩ B(xi+1 , ri+1 )|. Proof. Let γ be a John curve joining x to x0 . Let us write B0′ := B(x0 ,

1 dist(x0 , 𝜕D)). 4

Let us consider the balls B0′ and B(γ(t),

1 dist(γ(t), 𝜕D ∪ {x})), 4

when t ∈ (0, l), here l stands for the length of γ. By the Besicovitch covering theorem, there is a sequence of closed balls B1′ , B2′ , … and B0′ that cover {γ(t) : t ∈ [0, l]} \ {x} and have a uniformly bounded overlap depending on n only [36, 2.7]. Let us define open balls Bi := 2Bi′ with center at xi := γ(ti ) and radius ri := 21 dist(xi , 𝜕D ∪ {x}), i = 1, 2, . . . , that is Bi = B(xi , ri ), i = 1, 2, . . . . For a ball B0 := B(x0 , 21 dist(x0 , 𝜕D)), we obtain by the definition of s(⋅)-John domain dist(x, B0 )s(x) ≤ ℓ(γx )s(x) ≤

2βr0 β dist(x0 , 𝜕D) = . α α

Assume then that i ≥ 1. If ri = 21 dist(xi , x), then dist(x, Bi ) ≤ 2ri . If ri = the definition of a s(⋅)-John domain gives that

1 2

dist(xi , 𝜕D), then

146 � P. Harjulehto and R. Hurri-Syrjänen dist(x, Bi )s(x) ≤ dist(x, xi )s(x) ≤ tis(x) ≤

2βri β dist(γ(ti ), 𝜕D) ≤ . α α

Thus, the first part of property (2) holds. Let us renumerate the balls. Let B0 be as before. If we have chosen balls Bi , i = 0, 1, . . . , m, then we choose that the ball Bm+1 is the ball for which xj ∈ Bm and tj < tm , by remembering that γ(tj ) = xj and γ(tm ) = xm . Hence, ri → 0 and xi → x, as i → ∞. Thus the second part of property (2) holds. The point x does not belong to any ball. Let x ′ be any other point in the domain D. The point x ′ cannot belong to the balls Bi with 3ri < dist(x ′ , x). If x ′ ∈ Bi , then 2ri ≤ dist(x, xi ) ≤ dist(x, x ′ ) + ri . Thus, we obtain that x ′ ∈ Bi if and only if 1 dist(x ′ , x) ≤ ri ≤ dist(x, x ′ ). 3 The Besicovitch covering theorem implies that the balls with radius of balls are disjoint. Hence, x ′ belongs to less than or equal to N

|B(x ′ , 2r)|

|B(0, 121 r)|

1 4

of the original

= 24n N

balls Bi , where the constant N is from the Besicovitch covering theorem and depends on the dimension n only. Hence, property (3) holds. If ri = 21 dist(xi , 𝜕D) and ri+1 = 21 dist(xi+1 , 𝜕D), then ri+1 ≥ 21 ri (since xi+1 ∈ Bi ), and thus we obtain n

|Bi | r ≤ ( 1 i ) = 2n . |Bi+1 | ri 2

If ri =

1 dist(xi , x) 2

and

ri+1 =

1 dist(xi+1 , x), 2

then ri+1 ≥ 21 ri , and thus we obtain |Bi |/|Bi+1 | ≤ 2n . If ri =

1 dist(xi , 𝜕D) and 2

ri+1 =

1 dist(xi+1 , x), 2

then ri ≤ 21 dist(xi , x) and we obtain the same ratio as before. Similarly, in the case when ri = 21 dist(xi , x) and ri+1 = 21 dist(xi+1 , 𝜕D). We have shown |Bi | ≤ 2n |Bi+1 |. In the same manner, we obtain 2ri+1 ≤ 3ri , and hence 2n |Bi | ≥ |Bi+1 |. These yield property (4).

Estimates for the variable order Riesz potential with applications

� 147

The previous lemma yields the following lemma. Lemma 5.4. Let D ⊂ ℝn , n ≥ 2, be an s(⋅)-John domain. Then ̃ |∇u|(x) |u(x) − uB(x0 ,dist(x0 ,𝜕D)) | ≤ cIs(⋅) for all u ∈ L11 (D) and almost all x ∈ D. Here, the constant c depends only on n, α, β, s+ , and dist(x0 , 𝜕D). Moreover, if u is additionally continuous then the pointwise inequality holds for all x ∈ D. Proof. If x ∈ B(x0 , dist(x0 , 𝜕D)), then |u(x) − uB(x0 ,dist(x0 ,𝜕D)) | ≤

diam(B(x0 , dist(x0 , 𝜕D)))n n|B(x0 , dist(x0 , 𝜕D))|

|∇u(y)| dy |x − y|n−1

∫ B(x0 ,dist(x0 ,𝜕D))

by [20, Lemma 7.16]. Since the domain is bounded, we have |x − y|s(x)(n−1) < diam(D)(s(x)−1)(n−1) |x − y|n−1 (sup s−1)(n−1)

≤ (1 + diam(D))

|x − y|n−1

≤ (1 + 2β)(sup s−1)(n−1) |x − y|n−1 ,

which gives the claim. Let us then assume that x ∈ D \ B(x0 , dist(x0 , 𝜕D)) and let (Bi )∞ i=0 be a sequence of balls constructed in Lemma 5.3. Property (2) gives that dist(x, Bi ) → 0 as i → ∞. Thus, property (2) and the Lebesgue differentiation theorem [47, Section 1, Corollary 1] imply that uBi → u(x) when i → ∞ for almost every x. Moreover, if u is continuous, then the convergence holds for every x. We obtain ∞

|u(x) − uB0 | ≤ ∑ |uBi − uBi+1 | i=0 ∞

≤ ∑(|uBi − uBi ∩Bi+1 | + |uBi+1 − uBi ∩Bi+1 |) i=0 ∞

≤ ∑( ∫ − |u(y) − uBi | dy + i=0 B ∩B i i+1

− |u(y) − uBi+1 | dy). ∫

Bi ∩Bi+1

By property (4), ∞

|u(x) − uB0 | ≤ 2C ∑ ∫ − |u(y) − uBi | dy. i=0 B

i

Using the (1, 1)-Poincaré inequality in a ball Bi , [20, Section 7.8], we obtain

148 � P. Harjulehto and R. Hurri-Syrjänen ∞

|u(x) − uB0 | ≤ C ∑ ri ∫ − |∇u(y)| dy. i=0

Bi

Thus, for each z ∈ Bi we obtain by property (2) that 1

1

|x − z| ≤ dist(x, Bi ) + 2ri ≤ (Cri ) s(x) + 2ri ≤ Cris(x) , where in the last inequality we used that D is bounded. Hence, we have C|x − z|s(x) ≤ ri . Using this, we obtain by property (3) that ∞

∞

|u(x) − uB0 | ≤ C ∑ ri ∫ − |∇u(y)| dy ≤ C ∑ ∫ i=0

i=0 B

Bi

i

|∇u(y)| dy rin−1

∞

|∇u(y)| |∇u(y)| ≤ C∑∫ dy ≤ C ∫ dy. s(x)(n−1) |x − y| |x − y|s(x)(n−1) i=0 D

Bi

Now Lemma 3.1 and Remark 3.4 yield the following theorem, where we do not need continuity of s. Note that α− > 0 if and only if s+ < if and only if p
1). Let D ⊂ ℝn , n ≥ 2, be an s(⋅)John domain. Assume that 1 ≤ s− ≤ s+
c, then a > 1 c or c1 Is(⋅) 2 b > 21 c. Thus

∞

∑ ∫ 2(j+1)q(x) dx

j=−∞ D

j

∞

⩽ ∑

j=−∞

2(j+1)q(x) dx

∫ ̃ |∇vj |(x)+c1 >2j−1 } {x∈Dj :c1 Is(⋅)

∞

∞

⩽ ∑

j=−∞

2(j+1)q(x) dx + ∑

∫

j=−∞

̃ |∇vj |(x)>2j−2 } {x∈D:c1 Is(⋅)

∫

2(j+1)q(x) dx.

{x∈Dj :c1 >2j−2 }

Since ‖∇u‖L1 (D) ⩽ 1, we obtain by Theorem 3.2 and Remark 3.4 for the first term on the right-hand side that ∞

∑

j=−∞

2(j+1)q(x) dx

∫ ̃ |∇vj |(x)>2j−2 } {x∈D:c1 Is(⋅) ∞

≤ 23n ∑

j=−∞

2(j−2)q(x) dx

∫ ̃ |∇vj |(x)>2j−2 } {x∈D:c1 Is(⋅)

∞

⩽ c ∑ (∫ |∇vj (y)| dy + |{0 < |∇vj | ≤ 1}|) j=−∞ D ∞

⩽ c ∑ (∫ |∇u(y)| dy + |Dj |) = c ∫ |∇u(y)| dy + c|D|. j=−∞ D j

D

Let j0 be the largest integer satisfying c1 > 2j0 −2 . Hence, j0

∞

∑

j=−∞

∫ {x∈Dj :c1 >2j−2 }

2(j+1)q(x) dx ≤ ∫ ∑ 2(j+1)q(x) dx ≤ c|D|. D j=−∞

Then we conclude the proof by the scaling argument: Since ‖u − uB ‖q(⋅) ≤ c for all ‖∇u‖L1 (D) ≤ 1, we obtain the claim by applying this to u/‖∇u‖L1 (D) . Remark 5.7. (1) If s ≡ 1 is chosen, then by Theorems 5.5 and 5.6 the classical Sobolev–Poincaré inequality is recovered for 1-John domains, [11].

Estimates for the variable order Riesz potential with applications

� 151

n

(2) Let s be a constant function and 1 < s < n/(n − 1). The target spaces L s(n−1) (D) in Theorem 5.6 is optimal, while the target space np

L n−sp+pn(s−1) (D) in Theorem 5.5 is not the best possible; see [22, 34]. (3) Let s be a constant function and 1 ≤ s < n/(n − 1), then the classical (1, 1)-Poincaré inequality in an s-John domain is recovered. And this yields the (p, p)-Poincaré inequality for all 1 < p < ∞. Recall that it has been known that the (p, p)-Poincaré inequality holds for all p ∈ [1, ∞) whenever 1 ≤ s < n/(n − 1), [46]. Let us recall the following result, Theorem 5.8 for bounded John domains. The result was proved for domains with a fixed-cone condition by N. S. Trudinger [48]. Domains with a fixed-cone condition are examples of John domains. But John domains form a strictly larger class of domains than domains with a fixed-cone condition. Theorem 5.8 ([18]). Let D be a 1-John domain in ℝn , n ≥ 2. There exists a constant a > 0 such that n

|u(x) − uD | n−1 ) dx < ∞ ∫ exp(a ‖∇u‖nLn (D) D

for all u ∈ L1n (D). Next, we improve Theorem 5.8. In the next theorem, the Hausdorff content is sharper than the Lebesgue measure in the following sense: the equation n 󵄨󵄨 󵄨 󵄨󵄨{x ∈ D : a|u(x) − uB | s(n−1) > t}󵄨󵄨󵄨 = 0

does not imply the equation s(n−1)

ℋ∞

n

({x ∈ D : a|u(x) − uB | s(n−1) > t}) = 0,

but the latter equation implies the former one. When the integration is with respect to the Hausdorff content, the integration is taken as a Choquet integral. This integral goes back to Choquet [12], and it has been developed and used by Adams and his collaborators; see [2–4, 7, 8]. Let us define that ∞

n−α(⋅) n−α(⋅) := ∫ ℋ∞ ({x ∈ D : |u(x)| > t}) dt. ∫ |u(x)| d ℋ∞ D

0

n−α(⋅) Note that ℋ∞ is monotone by Lemma 4.2, i. e., if A ⊂ B then n−α(⋅)

ℋ∞

n−α(⋅) (A) ≤ ℋ∞ (B).

(5.4)

152 � P. Harjulehto and R. Hurri-Syrjänen Hence, the function n−α(⋅) t 󳨃→ ℋ∞ ({x ∈ D : |u(x)| > t})

is a decreasing function ℝ → [0, ∞) for every u : D → ℝ. By the decreasing property, the function n−α(⋅) t 󳨃→ ℋ∞ ({x ∈ D : |u(x)| > t})

is measurable. Thus ∞

n−α(⋅) ({x ∈ D : |u(x)| > t}) dt ∫ ℋ∞ 0

is well-defined as a Lebesgue integral. The right-hand side of (5.4) can be understood also as an improper Riemann integral. For the notion of Choquet integral in applications, we refer to [3, 27]. There are recent interesting results for Choquet integrals in [42]. Theorem 5.9 (Sobolev–Poincaré inequality in the limit case). Let D be an s(⋅)-John domain n . When α(x) := n −s(x)(n − in ℝn , n ≥ 2. Assume that s ∈ 𝒫 log (D) satisfies 1 ≤ s− ≤ s+ < n−1 1), then there exist positive constants a and b > 0 such that n

s(⋅)(n−1) ≤b ∫ exp(a|u(x) − uB | s+ (n−1) ) d ℋ∞ D

for all continuous u ∈ L1n/α(⋅) (D) with ‖∇u‖

n

L α(⋅) (D)

≤ 1,

here B := B(x0 , dist(x0 , 𝜕D)). Theorem 5.9 yields Corollary 1.2 as a special case when s is a constant function. Proof of Theorem 5.9. Let m > 0 be a constant that we will fix later. By Lemma 5.4, there exists a constant c1 such that n

s+ (n−1) m s(⋅)(n−1) |u(x) − uB |) ) d ℋ∞ ∫ exp(( 2(1 + |D|)

D

n

s+ (n−1) mc1 s(⋅)(n−1) ̃ |∇u|(x)) ≤ ∫ exp(( ) d ℋ∞ . Is(⋅) 2(1 + |D|)

D

By using the definition (5.4) and splitting the interval of integration to two parts, we obtain

Estimates for the variable order Riesz potential with applications

� 153

n

s+ (n−1) m s(⋅)(n−1) |u(x) − uB |) ) d ℋ∞ ∫ exp(( 2(1 + |D|)

D

n

∞

≤

s(⋅)(n−1) ({x ∫ ℋ∞ 0 1

=

s+ (n−1) mc1 ̃ |∇u|(x)) Is(⋅) ) > t}) dt ∈ D : exp(( 2(1 + |D|) n

s(⋅)(n−1) ({x ∫ ℋ∞ 0

̃ |∇u|(x) s+ (n−1) Is(⋅) log(t) ) > }) dt ∈D:( + 2(1 + |D|) (mc1 )n/(s (n−1)) n

∞

+

s(⋅)(n−1) ({x ∫ ℋ∞ 1

̃ |∇u|(x) s+ (n−1) Is(⋅) log(t) ) }) dt. ∈D:( > + 2(1 + |D|) (mc1 )n/(s (n−1))

n−α(⋅) The integral over the unit interval is estimated by ℋ∞ (D). We estimate the second integral over the unbounded interval. Let us first note that

󵄩󵄩 |∇u| 󵄩󵄩 1 󵄩󵄩 󵄩󵄩 . ≤ 󵄩󵄩 󵄩󵄩 n 󵄩󵄩 2(1 + |D|) 󵄩󵄩L α(⋅) (D) 2(1 + |D|) ̃ = Iα(⋅) where α(x) = n − s(x)(n − 1). The condition α− > 0 By Remark 3.4, we have Is(⋅) n n n + holds if and only if s < n−1 and α+ ≤ 1 < n since s− ≥ 1. Moreover, n−α − = s+ (n−1) . By Theorem 1.1, there exist constants c2 and c3 such that n−α(⋅) ℋ∞ ({x

̃ ( ∈ D : Is(⋅)

≤ c2 exp(−c3

log(t) |∇u| )(x) > ( ) + 2(1 + |D|) (mc1 )n/(s (n−1)) c

∞

̃ ( ∈ D : Is(⋅)

1 ∞

≤ ∫ c2 t

−

})

3 − + log(t) (mc1 )n/(s (n−1)) . ) = c t + 2 (mc1 )n/(s (n−1))

Whenever we choose m > 0 to be so small that n−α(⋅) ({x ∫ ℋ∞

s+ (n−1) n

c3 + (mc1 )n/(s (n−1))

c3 + (mc1 )n/(s (n−1))

> 1, we obtain

log(t) |∇u| )(x) > ( ) + 2(1 + |D|) (mc1 )n/(s (n−1))

s+ (n−1) n

dt =: c4 < ∞.

1

Now the claim follows by choosing n

s+ (n−1) m a=( ) 2(1 + |D|)

and

n−α(⋅) b = ℋ∞ (D) + c4 .

}) dt

154 � P. Harjulehto and R. Hurri-Syrjänen

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Yoshihiro Sawano and Kazuki Kobayashi

A remark on the atomic decomposition in Hardy spaces based on the convexification of ball Banach spaces Dedicated to the memory of David R. Adams

Abstract: The purpose of the present note is to slightly shorten the proof of the atomic decomposition based on the paper by Dekel et al. The atomic decomposition in the present paper is applicable to Hardy spaces based on the convexification of ball Banach spaces. The decomposition is rather canonical although it does not depend linearly on functions. Also, this decomposition is applicable under a rather weak condition as we will see. Keywords: Hardy spaces, variable exponents, atomic decomposition MSC 2020: Primary 41A17, 42B35, Secondary 26A33

1 Introduction The goal of the present paper is to consider the atomic decomposition of the Hardy space H p (ℝn ) for p ∈ (0, ∞). Recall that the Hardy space H p (ℝn ), 0 < p < ∞, collects all f ∈ 𝒮 ′ (ℝn ) for which 󵄩󵄩 󵄨 tΔ 󵄨󵄩󵄩 󵄩󵄩sup󵄨󵄨e f 󵄨󵄨󵄩󵄩 < ∞, 󵄩󵄩 󵄨 󵄨󵄩󵄩Lp t>0 where {etΔ }t>0 stands for the heat semigroup. We use the following notation in the present paper: Let ℕ0 ≡ {0, 1, . . .}. A function f ∈ L∞ (ℝn ) with compact support is said to have moment of order L ∈ ℕ0 ∪ {−1} if ∫ x α f (x)dx = 0 ℝn

for all α ∈ ℕn0 with |α| ≤ L. Let A, B ≥ 0. Then A ≲ B means that there exists a constant C > 0 such that A ≤ CB, where C depends only on the parameters of importance. The symbol A ∼ B means that A ≲ B and B ≲ A happen simultaneously. The index σp is given by σp ≡

n −n min(1, p)

for 0 < p < ∞.

Yoshihiro Sawano, Kazuki Kobayashi, Department of Mathematics, Chuo University, 1-13-27, Kasuga, Tokyo 112-8551, Japan, e-mails: [email protected], [email protected] https://doi.org/10.1515/9783110792720-007

158 � Y. Sawano and K. Kobayashi The goal of the present note is to provide a short proof of a well-known theorem based on the paper [5]. To this end, we set up some notation. Let x ∈ ℝn and r > 0. We denote by B(x, r) the ball centered at x of radius r. Namely, we write B(x, r) ≡ {y ∈ ℝn : |x − y| < r}. In particular, B(r) ≡ B(0, r). The set of all balls is denoted by ℬ. Theorem 1.1. Let 0 < p ≤ 1. Let f ∈ H p (ℝn ) and L ∈ ℤ ∩ [[σp ], ∞). Then there exist ∞ a countable collection {fj }∞ j=1 of Lc -functions having moment of order L and a countable collection {Bj }∞ j=1 ⊂ ℬ such that ∞

in 𝒮 ′ (ℝn );

(1.1)

for all j ∈ ℕ;

(1.2)

f = ∑ fj j=1

supp(fj ) ⊂ 5Bj ∞

1 p

p

(1.3)

(∑ ‖fj ‖L∞ |Bj |) ≲ ‖f ‖H p . j=1

Here, aBj stands for the a-times expansion of Bj for a > 0. As in [19], the proof of Theorem 1.1 uses some Hilbert spaces and estimates as in Lemma 2.1 to control the grand maximal function. Recently, Dekel, Kerkyacharian, Kyriazis, and Petrushev significantly reduced this argument [5]. The goal of the present paper is to reexamine their proof and expand it to other Hardy spaces based on ball Banach function spaces. In order to extend Theorem 1.1 to other Hardy spaces such as the one based on variable Lebesgue spaces, we slightly generalize Theorem 1.1. To this end, we recall an equivalent definition of H p (ℝn ). We will use the notation ⟨x⟩ ≡ √1 + |x|2 for x ∈ ℝn . To simplify the notation, for N ∈ ℕ0 , we define 󵄨 󵄨 pN (ϕ) ≡ ∑ (sup ⟨x⟩N 󵄨󵄨󵄨𝜕α ϕ(x)󵄨󵄨󵄨), α∈ℕn0

x∈ℝn

ϕ ∈ 𝒮 (ℝn ).

(1.4)

|α|≤N

We define the unit ball ℱN with respect to pN by n

ℱN ≡ {ϕ ∈ 𝒮 (ℝ ) : pN (ϕ) ≤ 1}.

(1.5)

For j ∈ ℤ and ϕ ∈ 𝒮 (ℝn ), we write ϕj ≡ 2jn ϕ(2j ⋅). Let f ∈ 𝒮 ′ (ℝn ). We define the grand maximal operator ℳN f by ℳN f (x) ≡

󵄨 󵄨 sup 󵄨󵄨󵄨ϕk ∗ f (x)󵄨󵄨󵄨 (x ∈ ℝn ).

k∈ℤ,ϕ∈ℱN

(1.6)

Atomic decomposition in Hardy spaces

� 159

Let 0 < p ≤ 1. We can say that the Hardy space H p (ℝn ) is the set of all f ∈ 𝒮 ′ (ℝn ) for which the quantity ‖f ‖H p ≡ ‖ℳN f ‖Lp is finite; this definition coincides with the one above as long as N ≫ 1 [19, p. 91]. Denote by χE the indicator function of a set E. We refine Theorem 1.1 based on the spirit of Miyachi [13]. Theorem 1.2. Let 0 < p ≤ 1. Let f ∈ H p (ℝn ) and L ∈ ℤ ∩ [[σp ], ∞). Then there exist ∞ a countable collection {fj }∞ j=1 of Lc -functions having moment of order L and a countable ∞ collection {Bj }j=1 ⊂ ℬ satisfying (1.1), (1.2), and ∞

u

1 u

(∑(‖fj ‖L∞ χ 1 B ) ) ≲ ℳN f j=1

2

j

(1.7)

for all 0 < u < ∞ with the implicit constant depends only on n, N, and u. Once Theorem 1.2 is proved, we can prove Theorem 1.1 with ease. In fact, letting r = p ∈ (0, 1], we integrate (1.7) to have (1.3). So, we concentrate on (re)proving Theorem 1.2 in the present note after stating some preliminary facts in Section 2. The proof of Theorem 1.2 is quite akin to the one in [5]. Since the conclusion gets tighter as L is larger, we may assume that L ≫ 1. However, we start the proof from scratching. We clarify what is actually needed for the decomposition. We prove Theorem 1.2 with the spirit of [5]. We actually prove Theorem 1.2 in Section 3. Section 4 expands the argument presented within Section 3. As the starting point, we consider weighted Hardy spaces with weights in A1 . After that, we investigate other function spaces based on weighted Hardy spaces with weights in A1 .

2 Preliminaries A distribution f ∈ 𝒮 ′ (ℝn ) is said to vanish weakly at infinity if ψj ∗ f → 0 in 𝒮 ′ (ℝn ) as j → −∞ for all ψ ∈ 𝒮 (ℝn ). Since jn 󵄩󵄩 j 󵄩 󵄩󵄩ψ ∗ f 󵄩󵄩󵄩L∞ = O(2 p ‖f ‖H p )

for all f ∈ H p (ℝn ), as j → −∞, any element in H p (ℝn ) vanishes weakly at infinity. By taking advantage of the class ℱN , we use the following observation. Lemma 2.1. There exists A > 1 such that 󵄨 󵄨 󵄨 󵄨 sup 󵄨󵄨󵄨ϕk ∗ f (x)󵄨󵄨󵄨 ≤ A sup 󵄨󵄨󵄨ϕk ∗ f (y)󵄨󵄨󵄨

ϕ∈ℱN

ϕ∈ℱN

for all f ∈ 𝒮 ′ (ℝn ) and k ∈ ℤ if x, y ∈ ℝn satisfy |x − y| ≤ 22−k .

(2.1)

160 � Y. Sawano and K. Kobayashi Proof. Let ϕ ∈ ℱN . We calculate ϕk ∗ f (x) = ⟨f , ϕk (x − ⋅)⟩ = ⟨f , ϕk ((x − y) + (y − ⋅))⟩. Let A > 1 be the constant in Lemma 2.1. Set ϕk,x,y (z) ≡ ϕ(2k (x − y) + z)

(z ∈ ℝn ).

Then we have pN (ϕk,x,y ) ≤ ApN (ϕ) with the constant A > 1 depending on N. Thus 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 sup 󵄨󵄨󵄨ϕk ∗ f (x)󵄨󵄨󵄨 = A sup 󵄨󵄨󵄨A−1 (ϕk,x,y )k ∗ f (y)󵄨󵄨󵄨 ≤ A sup 󵄨󵄨󵄨ϕk ∗ f (y)󵄨󵄨󵄨,

ϕ∈ℱN

ϕ∈ℱN

ϕ∈ℱN

proving (2.1). We also need the well-known Whitney covering lemma. Lemma 2.2. Let Ω be a proper open set in ℝn . Write ρ(x) ≡ dist(x, 𝜕Ω) for x ∈ ℝn . Suppose ρ that {B(ξj , 5j )}∞ j=1 is a maximal disjoint family, where ρj ≡ ρ(ξj ) for j ∈ ℕ. ∞

(1) Ω = ⋃ B(ξj , j=1

ρj ). 2

(2) For each j ∈ ℕ, let 3 4

3 4

𝒥j ≡ {ν ∈ ℕ ∩ (j, ∞) : B(ξj , ρj ) ∩ B(ξν , ρν ) ≠ 0}.

Then ♯𝒥j ≤ 300n and 7−1 ρν ≤ ρj ≤ 7ρν for all ν ∈ 𝒥j . Proof. This is essentially contained in [5]. However, the number 300 did not appear in [5]. For the sake of convenience, we clarify why this number appears. Notice that ∑ χB(ξ

ν∈𝒥j

ρj ν , 35 )

≤ ∑ χB(ξν , ρν ) ≤ χB(ξ , 37 ρ ) , ν∈𝒥j

5

j 5

j

since 3 3 1 7 37 ρ + ρ + ρ ≤ 6ρj + ρj = ρj . 4 j 4 ν 5 ν 5 5 Thus ♯𝒥j × implying ♯𝒥j ≤ 259n ≤ 300n .

1 37n ≤ n , n 35 5

Atomic decomposition in Hardy spaces

� 161

3 Proof of Theorem 1.2 We transform Theorem 1.2 to the following equivalent form. Proposition 3.1. Let 0 < p ≤ 1. Let f ∈ H p (ℝn ) and L ∈ ℤ ∩ [[σp ], ∞). Then there exists a countable collection {Fj,r }j∈ℕ,r∈ℤ of L∞ c -functions having moment of order L with the following properties: (1) In 𝒮 ′ (ℝn ), f =

∑

(j,r)∈ℕ×ℤ

Fj,r .

(3.1)

(2) For all j ∈ ℕ and r ∈ ℤ, there exist ξj,r ∈ ℝn and ρj,r > 0 such that supp(Fj,r ) ⊂ B(ξj,r , 5ρj,r ).

(3.2)

(3) For all 0 < u < ∞, (

∑

u

1 u

(‖Fj,r ‖L∞ χB(ξj,r ,2−1 ρj,r ) ) ) ≲ ℳN f ,

(j,r)∈ℕ×ℤ

where the implicit constant depends on u, N, and n. The following is devoted to the proof of Proposition 3.1 assuming that f ≠ 0. For each k, r ∈ ℤ, we set Ωr ≡ {x ∈ ℝn : ℳN f (x) > 2r } and Vk,r ≡ {x ∈ ℝn : B(x, 2−k+1 ) ⊂ Ωr }. Notice that each Ωr is an open set, and hence ∞

Ωr = ⋃ Vk,r . k=−∞

If f ∈ 𝒮 ′ (ℝn ) \ {0}, then ∞

⋃ Ωr = ℝn .

r=−∞

Here is a geometric observation we need.

(3.3)

162 � Y. Sawano and K. Kobayashi Lemma 3.2. Let l0 , l1 , k, r ∈ ℤ;

{

x ∈ (Vl0 +1,r \ Vl0 ,r ) ∩ (Vl1 +1,r+1 \ Vl1 ,r+1 ).

(1) We have l0 ≤ l1 . (2) If B(x, 2−k ) ∩ (Vk,r \ Vk,r+1 ) ≠ 0, then l0 ≤ k ≤ l1 + 1. (3) If l0 + 2 ≤ k ≤ l1 − 1, then B(x, 2−k ) ⊂ Vk,r \ Vk,r+1 . Proof. We remark that x ∈ (Vl0 +1,r \ Vl0 ,r ) ∩ (Vl1 +1,r+1 \ Vl1 ,r+1 ) if and only if 2−l0 ≤ dist(x, 𝜕Ωr ) < 2−l0 +1

and

2−l1 ≤ dist(x, 𝜕Ωr+1 ) < 2−l1 +1 .

(1) Since Ωr ⊃ Ωr+1 , dist(x, 𝜕Ωr+1 ) ≤ dist(x, 𝜕Ωr ). Thus, in view of the above observation, the result follows immediately. (2) Let y ∈ B(x, 2−k ) ∩ (Vk,r \ Vk,r+1 ). Since y ∈ Vk,r , 2−l0 +1 > dist(x, 𝜕Ωr ) ≥ dist(y, 𝜕Ωr ) − |x − y| ≥ 21−k − 2−k = 2−k , implying k ≥ l0 . Likewise, since y ∉ Vk,r+1 , 2−l1 ≤ dist(x, 𝜕Ωr+1 ) ≤ dist(y, 𝜕Ωr+1 ) + |x − y| ≤ 21−k + 2−k < 22−k , implying k ≤ l1 + 1. (3) Let z ∈ B(x, 2−k ). Then since x ∈ Vl0 +1,r and k ≥ l0 + 2, dist(z, 𝜕Ωr ) ≥ dist(x, 𝜕Ωr ) − |x − z| ≥ 2−l0 − 2−k ≥ 21−k . Hence, B(x, 2−k ) ⊂ Vk,r . Likewise, since x ∉ Vl1 ,r+1 , dist(z, 𝜕Ωr+1 ) ≤ dist(x, 𝜕Ωr+1 ) + |x − z| < 21−l1 + 2−k ≤ 21−k . Hence, B(x, 2−k ) ∩ Vk,r+1 = 0. n Fix an integer L > 2p here and below. Let Φ, Ψ, Θ ∈ Cc∞ (ℝn ) be even functions supported in the unit ball and satisfy

Ψ = Φ1 − Φ = ΔL Θ,

∫ Φ(x)dx = 1. ℝn

(3.4)

Atomic decomposition in Hardy spaces

� 163

The pair (Φ, Ψ, Θ) is known to exist [18]. Write Ψ̃ ≡ Φ1 + Φ. Let f ∈ 𝒮 ′ (ℝn ) \ {0} be a distribution vanishing weakly at infinity. Also, let k, r ∈ ℤ. We set fk,r ≡ Ψk ∗ (χVk,r \Vk,r+1 ⋅ Ψ̃ k ∗ f ). A geometric observation shows that fk,r is supported on Ωr . We also need the L∞ -bound for the function of this type. Lemma 3.3. Let Γ, Γ̃ ∈ Cc∞ (ℝn )

with supp(Γ), supp(Γ)̃ ⊂ B(1).

Also let E ⊂ ℝn be a measurable set. Then 󵄨󵄨 k 󵄨 k r 󵄨󵄨Γ ∗ (χ(Vk,r \Vk,r+1 )∩E ⋅ Γ̃ ∗ f )(x)󵄨󵄨󵄨 ≲ 2 for all x ∈ ℝn . Proof. Since 󵄨󵄨 k 󵄨 k 󵄨󵄨Γ ∗ (χ(Vk,r \Vk,r+1 )∩E ⋅ Γ̃ ∗ f )(x)󵄨󵄨󵄨 󵄨 󵄨 ≤ ∫ 󵄨󵄨󵄨Γk (x − y)Γ̃ k ∗ f (y)󵄨󵄨󵄨dy Vk,r \Vk,r+1

≤A

∫ Vk,r \Vk,r+1

󵄨󵄨 k 󵄨 󵄨 k 󵄨 󵄨󵄨Γ (x − y)󵄨󵄨󵄨( inf 󵄨󵄨󵄨Γ̃ ∗ f (z)󵄨󵄨󵄨)dy z∈B(y,22−k )

thanks to Lemma 2.1, we have 󵄨󵄨 k 󵄨 k r 󵄨󵄨Γ ∗ (χ(Vk,r \Vk,r+1 )∩E ⋅ Γ̃ ∗ f )(x)󵄨󵄨󵄨 ≲ 2

∫ Vk,r \Vk,r+1

󵄨󵄨 k 󵄨 r 󵄨󵄨Γ (x − y)󵄨󵄨󵄨dy ≲ 2

by the definition of ℳN f , Vk,r+1 and Ωr . We decompose ∞

∞

∞

f = ∑ Ψk ∗ Ψ̃ k ∗ f = ∑ ( ∑ fk,r ). k=−∞

k=−∞ r=−∞

(3.5)

We need to pay attention to the order of the summation in (3.5). However, if f is good enough, then we can interchange the order of the summation. Lemma 3.4. Assume that f ∈ H p (ℝn ) with 0 < p ≤ 1 and that the integer L in (3.4) n satisfies L ∈ ℤ ∩ ( 2p , ∞). Then

164 � Y. Sawano and K. Kobayashi f = ∑ fk,r k,r∈ℤ

in the sense of absolute convergence in 𝒮 ′ (ℝn ). Namely, 󵄨 󵄨 ∑ 󵄨󵄨󵄨⟨fk,r , φ⟩󵄨󵄨󵄨 < ∞

k,r∈ℤ

for all φ ∈ 𝒮 (ℝn ). Proof. Fix k, r ∈ ℤ. Recall that Ψ is an even function. We calculate ⟨fk,r , φ⟩ =

Ψk ∗ φ(y)Ψ̃ k ∗ f (y)dy.

∫ Vk,r \Vk,r+1

Thanks to (3.4), by using integration by parts, we have −2n− np −1 󵄨󵄨 k 󵄨 󵄨 L k 󵄨 − max(0,2kL) ⟨y⟩ 󵄨󵄨Ψ ∗ φ(y)󵄨󵄨󵄨 = 󵄨󵄨󵄨(Δ Θ) ∗ φ(y)󵄨󵄨󵄨 ≲ 2

(y ∈ ℝn ),

if k ∈ ℤ. Meanwhile, if y ∈ Vk,r \ Vk,r+1 , we have kn kn 󵄨󵄨 ̃ k 󵄨 ̃ inf ℳN f (z) ≲ 2 p ‖ℳN f ‖Lp = 2 p ‖f ‖H p 󵄨󵄨Ψ ∗ f (y)󵄨󵄨󵄨 ≤ ApN (Ψ) −k z∈B(y,2 )

(3.6)

thanks to Lemma 2.1. As a consequence, kn −max(0,2kL) 󵄨󵄨 󵄨 ‖f ‖H p 󵄨󵄨⟨fk,r , φ⟩󵄨󵄨󵄨 ≲ 2 p

∫ Vk,r \Vk,r+1

dy

⟨y⟩

2n+ np +1

.

If we add this inequality over r ∈ ℤ, then we obtain kn −max(0,2kL) 󵄨 󵄨 ‖f ‖H p ∫ ∑ 󵄨󵄨󵄨⟨fk,r , φ⟩󵄨󵄨󵄨 ≲ 2 p

r∈ℤ

kn

∼2p

ℝn

−max(0,2kL)

dy

⟨y⟩

2n+ np +1

(3.7)

‖f ‖H p .

n If L > 2p , then this estimate is summable over k ∈ ℤ. Once we can prove that the series converges absolutely, we see that the series converges back to f thanks to (3.5).

Remark that the power 2n + np + 1 in the above proof (see (3.7), for example) seems superfluous: This number will turn out important in Section 4. From Lemma 3.4, ∞

∞

f = ∑ ( ∑ fk,r ) in 𝒮 ′ (ℝn ). r=−∞ k=−∞

(3.8)

Atomic decomposition in Hardy spaces

� 165

We analyze the summand with r fixed. Lemma 3.5. Let r ∈ ℤ. Then 󵄨󵄨 󵄨󵄨 ∞ 󵄨 󵄨󵄨 󵄨󵄨 ∑ fk,r (x)󵄨󵄨󵄨 ≲ 2r 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨k=−∞

for all x ∈ ℝn .

Proof. Since each fk,r is supported on Ωr , we may assume that x ∈ Ωr . We distinguish two cases: – Let x ∈ Ωr+1 . Choose l0 , l1 ∈ ℤ so that x ∈ (Vl0 +1,r \ Vl0 ,r ) ∩ (Vl1 +1,r+1 \ Vl1 ,r+1 ). Thanks to Lemma 3.2(1), l0 ≤ l1 . Fix x ∈ ℝn and k ∈ ℤ so that fk,r (x) ≠ 0. Then B(x, 2−k ) ∩ (Vk,r \ Vk,r+1 ) ≠ 0. Thus l0 ≤ k ≤ l1 + 1 according to Lemma 3.2(2). We further assume that l0 + 3 ≤ l1 ; otherwise we may simply use Lemma 3.3. Due to Lemma 3.2(3), we have fk,r (x) = Ψk ∗ Ψ̃ k ∗ f (x) = Φk+1 ∗ Φk+1 ∗ f (x) − Φk ∗ Φk ∗ f (x)

if l0 + 2 ≤ k ≤ l1 − 1.

Hence, thanks to Lemma 3.3, l1 −1

∑ fk,r (x) = Φl1 ∗ Φl1 ∗ f (x) − Φl0 +2 ∗ Φl0 +2 ∗ f (x) = O(2r ).

k=l0 +2

–

We do not have to take into account the terms for k ≥ l1 + 2 or k ≤ l0 − 1 since they vanish according to Lemma 3.2(2). If we handle the terms for l0 ≤ k ≤ l0 + 1 and l1 ≤ k ≤ l1 + 1 using Lemma 3.3 again, then we obtain the desired result. Let x ∈ Ωr \ Ωr+1 . Then let l1 = ∞ and x ∈ Vl0 +1,r \ Vl0 ,r with l0 ∈ ℤ in the above and go through the same argument. We can generalize Lemma 3.5, whose proof we omit.

Lemma 3.6. Let l0 , l1 , r ∈ ℤ satisfy l0 < l1 . Then 󵄨󵄨 l1 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∑ fk,r (x)󵄨󵄨󵄨 ≲ 2r 󵄨󵄨 󵄨󵄨 󵄨󵄨k=l0 󵄨󵄨 for all x ∈ ℝn , where the implicit constant does not depend on l0 and l1 .

166 � Y. Sawano and K. Kobayashi For an arbitrary set S, define an open set Sk by Sk ≡ {y ∈ ℝn : dist(y, S) < 21−k }. Lemma 3.7. Let l ∈ ℤ and x ∈ Sl \ Sl+1 . (1) Whenever k < l, B(x, 2−k ) ⊂ Sk . (2) Whenever k ≥ l + 2, B(x, 2−k ) ∩ Sk = 0. Proof. Since x ∈ Sl \ Sl+1 , we have 2−l ≤ dist(x, S) < 21−l . Let y ∈ B(x, 2−k ). (1) Using the triangle inequality, we obtain dist(y, S) ≤ |x − y| + dist(x, S) ≤ 2−k + 21−l ≤ 21−k , implying y ∈ Sk . (2) Using the triangle inequality again, we obtain dist(y, S) ≥ −|x − y| + dist(x, S) > −2−k + 2−l ≥ 21−k , implying y ∉ Sk . Let S be a set. Set ∞

FS (x) ≡ ∑ Ψk ∗ (χ(Vk,r \Vk,r+1 )∩Sk ⋅ Ψ̃ k ∗ f )(x) k=−∞

(x ∈ ℝn ).

If S is bounded, then by the Fubini theorem, we see that FS satisfies the same moment condition as Ψk . Lemma 3.8. For any set S and r ∈ ℤ, ‖FS ‖L∞ ≲ 2r . Proof. Let x ∈ S and k ∈ ℤ. Then B(x, 2−k ) ⊂ Sk , and hence (Vk,r \ Vk,r+1 ) ∩ Sk ∩ B(x, 2−k ) = (Vk,r \ Vk,r+1 ) ∩ B(x, 2−k ). Thus ∞

FS (x) = ∑ Ψk ∗ (χVk,r \Vk,r+1 ⋅ Ψ̃ k ∗ f )(x) = O(2r ). k=−∞

Atomic decomposition in Hardy spaces

� 167

Suppose x ∈ Sl \ Sl+1 for some l ∈ ℤ. Then thanks to Lemmas 3.3, 3.6 and 3.7, l−1

FS (x) = ∑ Ψk ∗ (χVk,r \Vk,r+1 ⋅ Ψ̃ k ∗ f )(x) k=−∞ l+1

+ ∑ Ψk ∗ (χ(Vk,r \Vk,r+1 )∩Sk ⋅ Ψ̃ k ∗ f )(x) k=l

= O(2r ). We slightly generalize Lemma 3.8. Let S be a set and κ ∈ ℝ. Set ∞

FS,κ (x) ≡ ∑ χ(κ,∞) (k)Ψk ∗ (χ(Vk,r \Vk,r+1 )∩Sk ⋅ Ψ̃ k ∗ f )(x)

(x ∈ ℝn ).

k=−∞

Lemma 3.9. For any set S, κ ∈ ℝ and r ∈ ℤ, ‖FS,κ ‖L∞ ≲ 2r . We do not prove Lemma 3.9 since it is similar to Lemma 3.8. Form the Whitney decomposition of Ωr = {x ∈ ℝn : ℳN f (x) > 2r }

for each r ∈ ℤ. ρ

For x ∈ ℝn and r ∈ ℤ, we let ρr (x) ≡ dist(x, 𝜕Ωr ). We let {B(ξj,r , 5j,r )}∞ j=1 be a maximal disjoint family, where ρj,r ≡ ρr (ξj,r ) for j ∈ ℕ and r ∈ ℤ. Then we have the following properties: ∞

(1) Ωr = ⋃ B(ξj,r , 2−1 ρj,r ). j=1

(2) Let j ∈ ℕ and r ∈ ℤ. Set 3 4

3 4

𝒥j,r ≡ {ν ∈ ℕ ∩ (j, ∞) : B(ξj,r , ρj,r ) ∩ B(ξν,r , ρν,r ) ≠ 0}.

Then due to Lemma 2.2 ♯𝒥j,r ≤ 300n

and

7−1 ρν,r ≤ ρj,r ≤ 7ρν,r

for each ν ∈ 𝒥j,r .

Let j ∈ ℕ and k, r ∈ ℤ. We define Ej,k,r ≡ B(ξj,r , 2−1 ρj,r + 21−k ) ∩ (Vk,r \ Vk,r+1 ) whenever B(ξj,r , 2−1 ρj,r ) ∩ (Vk,r \ Vk,r+1 ) ≠ 0. If

168 � Y. Sawano and K. Kobayashi B(ξj,r , 2−1 ρj,r ) ∩ (Vk,r \ Vk,r+1 ) = 0, then we define Ej,k,r ≡ 0. Due to Lemma 2.2(1) ∞

⋃ Ej,k,r = Vk,r \ Vk,r+1 j=1

(k, r ∈ ℤ).

We set ∞

Rj,k,r ≡ Ej,k,r \ ⋃ Eν,k,r ν=j+1

(j ∈ ℕ, k, r ∈ ℤ).

We write Fj,k,r ≡ Ψk ∗ (χRj,k,r ⋅ Ψ̃ k ∗ f ) and ∞

Fj,r ≡ ∑ Fj,l,r l=−∞

for j ∈ ℕ and k, r ∈ ℤ.

As before, we can check that the sum defining Fj,r converges absolutely in 𝒮 ′ (ℝn ). The next lemma shows that the sum Fj,r belongs to L∞ (ℝn ). Also, observe that f =

∑

(k,r)∈ℤ2

fk,r =

∑

(j,k,r)∈ℕ×ℤ2

Fj,k,r =

∑

(j,r)∈ℕ×ℤ

Fj,r

in the sense of absolute convergence. Lemma 3.10. For all j ∈ ℕ and r ∈ ℤ, |Fj,r | ≲ 2r χB(ξj,r ,5ρj,r ) . Proof. The proof consists of two steps. – Let us verify that Fj,r vanishes outside B(ξj,r , 5ρj,r ). Let k ∈ ℤ satisfy Rj,k,r ≠ 0. Then B(ξj,r , 2−1 ρj,r ) ∩ (Vk,r \ Vk,r+1 ) ≠ 0. Let z ∈ B(ξj,r , 2−1 ρj,r ) ∩ (Vk,r \ Vk,r+1 ). Then 3 ρ ≥ |ξj,r − z| + dist(ξj,r , 𝜕Ωr ) ≥ dist(z, 𝜕Ωr ) ≥ 21−k , 2 j,r so that ρj,r ≥

4 3

⋅ 2−k . Thus

Atomic decomposition in Hardy spaces

� 169

B(ξj,r , 2−1 ρj,r + 21−k ) ⊂ B(ξj,r , 2ρj,r ). Since 1 supp(Fj,k,r ) ⊂ B(ξj,r , ρj,r + 21−k + 2−k ) ⊂ B(ξj,r , 5ρj,r ), 2 –

we obtain the desired result. Let us obtain the L∞ -bound of Fj,r . If k ∈ ℤ satisfies 2−k ≥ 2ρj,r , then from the definition of ρj,r , sup

z∈B(ξj,r ,2−1 ρj,r )

3 dist(z, 𝜕Ωr ) = ρj,r ≤ 2−k , 2

and hence B(ξj,r , 2−1 ρj,r ) ∩ (Vk,r \ Vk,r+1 ) = 0. Namely, if k ≤ − log2 ρj,r − 1, then B(ξj,r , 2−1 ρj,r ) ∩ (Vk,r \ Vk,r+1 ) = 0. A direct consequence of this equality is that the term for k ≤ −1 − log2 ρj,r in the sum defining Fj,r vanishes. From the definition of 𝒥j,r , 3 B(ξν,r , 2−1 ρν,r + 21−k ) ⊂ B(ξν,r , ρν,r ) 4 for all k ≥ 10 − log2 ρj,r and ν ∈ 𝒥j,r . Let S ≡ ⋃ B(ξν,r , 2−1 ρν,r ) and ν∈𝒥j,r

S̃ ≡ S ∪ B(ξj,r , 2−1 ρj,r ).

Then we have Sk = ⋃ B(ξν,r , 2−1 ρν,r + 21−k ), ν∈𝒥j,r

(S)̃ k = Sk ∪ B(ξj,r , 2−1 ρj,r + 21−k )

and Rj,k,r = {(S)̃ k ∩ (Vk,r \ Vk,r+1 )} \ {Sk ∩ (Vk,r \ Vk,r+1 )}. Thus

170 � Y. Sawano and K. Kobayashi Fj,r = FS,10−log ̃

2

ρj,r

− FS,10−log2 ρj,r +

∑

− log2 ρj,r ≤k≤− log2 ρj,r +10

Fj,k,r .

It remains to use Lemma 3.9. We conclude the proof of Proposition 3.1. Equality (3.1) is a consequence of Lemma 3.4. Thanks to Lemma 3.10, fk,r satisfies (3.2). It remains to prove (3.3). Using Lemma 3.10 again and the definition of Ωr , we estimate ∑

u

(j,r)∈ℕ×ℤ

(‖Fj,r ‖L∞ χB(ξj,r ,2−1 ρj,r ) ) ≲

∑

(j,r)∈ℕ×ℤ

2ur χB(ξj,r ,2−1 ρj,r )

∞

≲ ∑ 2ur χΩr r=−∞ ∞

= ∑ 2ur χ(2r ,∞] (ℳN f ) r=−∞

≲ (ℳN f )u , as required.

4 Applications to Hardy spaces based on other ball Banach spaces Here, we modify the proof especially (3.7) to obtain the decomposition results for distributions in Hardy spaces based on other ball Banach spaces. As we saw in Section 3, it matters that the distribution vanishes weakly at infinity and that the distribution satisfies (3.7). Section 4.1 considers the weighted Hardy space H p (w) with 0 < p < ∞ and w ∈ A1 . As an application of Section 4.1, we consider Hardy spaces based on ball Banach function spaces. We can locate Sections 4.3, 4.4, and 4.5 as further examples of Section 4.2. Hardy spaces with weight in A∞ , variable Hardy spaces, and Hardy–Morrey spaces are considered in Sections 4.3, 4.4, and 4.5, respectively. We will give a precise condition on L in Sections 4.3, 4.4, and 4.5. We need to define the above spaces by way of ℳN . It is known in [9] that the function spaces we are going to handle in this section do not depend on the choice of N as long as N ≫ 1. This condition L is used to obtain the boundedness of operators. However, as we mentioned, the condition on L can be tightened since we are considering the decompositions of distributions. So, although we present some concrete conditions on L in Sections 4.3, 4.4, and 4.5, we still may assume that L is large enough. We will make use of the Hardy–Littlewood maximal operator M. The space L0 (ℝn ) denotes the set of all complex/[0, ∞]-valued measurable functions considered modulo the difference on the set of measure zero. For f ∈ L0 (ℝn ), define a function Mf by

Atomic decomposition in Hardy spaces

(x ∈ ℝn ).

Mf (x) ≡ sup χB (x)mB (|f |) B∈ℬ

� 171

(4.1)

Here, mB (f ) stands for the average of a locally integrable or nonnegative function f over B. The mapping M : f 󳨃→ Mf is called the Hardy–Littlewood maximal operator. We also use the powered Hardy–Littlewood maximal operator M (η) defined by 1

M (η) f (x) ≡ sup(χB (x)mB (|f |η )) η , B∈ℬ

where 0 < η < ∞ and f ∈ L0 (ℝn ). Together with the Hardy–Littlewood maximal operator, we need to recall the notion of weights as well as their fundamental properties, which will be done in Sections 4.1 and 4.3. See [6] for more details on weights. We remark that the same idea can be used for Hardy spaces based on other function spaces such as the ones considered in [7, 8, 21–23].

4.1 Weighted Hardy space H p (w) with w ∈ A1 As the starting point, we seek to change Lp (ℝn ) by Lp (w) for some good class of weights. Although we work in a rather special setting, this setting will be a core of our argument. By a weight, we mean a function w ∈ L0 (ℝn ), which satisfies 0 < w(x) < ∞ for almost all x ∈ ℝn . We write w(A) ≡ ∫ w(x)dx A

if A is a measurable set of ℝn . The space Lp (w) is the set of all f ∈ L0 (ℝn ) for which (cf. [1]) 1

󵄩 󵄩 ‖f ‖Lp (w) ≡ 󵄩󵄩󵄩fw p 󵄩󵄩󵄩Lp < ∞. To proceed further, we compare the weights w and 1. Here, we introduce a general definition following the book [6, p. 402]. A weight w1 is comparable to a weight w2 if there exist α, β < 1 such that w1 (A) ≤ βw1 (B) for any measurable set A and any B ∈ ℬ satisfying A ⊂ B and w2 (A) ≤ αw2 (B). It is important that comparability is symmetric; w1 is comparable to w2 if and only if w2 is comparable to w1 . In this case, there exists δ > 0 such that δ

w (A) w1 (A) ≲( 2 ) w1 (B) w2 (B)

(4.2)

and that δ

w2 (A) w (A) ≲( 1 ) w2 (B) w1 (B)

(4.3)

172 � Y. Sawano and K. Kobayashi for any measurable set A and any B ∈ ℬ satisfying A ⊂ B. Let 0 < p < ∞, w be a weight and f ∈ 𝒮 ′ (ℝn ). Define ‖f ‖H p (w) ≡ ‖ℳN f ‖Lp (w) . The weighted Hardy space H p (w) is the set of all f ∈ 𝒮 ′ (ℝn ) for which the quantity ‖f ‖H p (w) is finite. In the present paper, as long as N ≫ 1, the definition of H p (w) does not depend on the choice of N. As a preliminary and important step, we consider A1 -weights among other classes of weights. Recall that a locally integrable weight w is said be an A1 -weight, if there exists C0 > 0 such that Mw(x) ≤ C0 w(x)

(4.4)

for a. e. x ∈ ℝn . The infimum of C0 satisfying (4.4) is called the A1 -norm. The class A1 of weights consists of all A1 -weights. Let Γ ∈ 𝒮 (ℝn ) and k ∈ ℤ. We estimate 󵄨󵄨 k 󵄨 󵄨 k 󵄨 󵄨󵄨Γ ∗ f (x)󵄨󵄨󵄨 ≤ A inf 󵄨󵄨󵄨Γ ∗ f (y)󵄨󵄨󵄨 ≤ y∈B(x,2−k )

ApN (Γ)

1

w(B(x, 2−k )) p

‖f ‖H p (w)

using Lemma 2.1. It follows from (4.2) and (4.3) that δ

w(B(x, 1)) |B(x, 1)| ≲( ) = 2knδ w(B(x, 2−k )) |B(x, 2−k )| for all x ∈ ℝn and k ∈ ℤ \ ℕ and that δ

w(B(x, 2−k )) |B(x, 2−k )| ≳( ) = 2−knδ w(B(x, 1)) |B(x, 1)| for all x ∈ ℝn and k ∈ ℕ. Also, it follows from (4.4) that ⟨x⟩−n w(B(1)) ≲ w(B(x, 1)) ≲ ⟨x⟩n w(B(1)). Therefore, 󵄨󵄨 k 󵄨 󵄨󵄨Γ ∗ f (x)󵄨󵄨󵄨 ≲

2

knδ p

w(B(x, 1))

1 p

‖f ‖H p (w) ≲ 2

knδ p

n

⟨x⟩ p ‖f ‖H p (w) .

(4.5)

Recall that Γ ∈ 𝒮 (ℝn ) is arbitrary. By letting Γ = Ψ,̃ we learn that a counterpart to (3.7) still holds. Estimate (4.5) also shows that f vanishes weakly at infinity. As in [19], A1 ∩ L1 (ℝn ) = 0. Thus, Ωr , the level set of ℳN f at 2r , cannot coincide with ℝn , allowing us to use Lemma 2.2. Therefore, the same conclusion with L ≫ 1 as Theorem 1.2 holds.

Atomic decomposition in Hardy spaces

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Theorem 4.1. Let 0 < p < ∞, f ∈ H p (w) with w ∈ A1 and let L ≫ 1. Then there exists ∞ a countable collection {fj }∞ j=1 of Lc -functions having moment of order L and a countable collection {Bj }∞ j=1 ⊂ ℬ satisfying (1.1), (1.2), and (1.7).

4.2 Hardy spaces based on ball Banach function spaces Based on Section 4.1, we establish a general theory of the decomposition of distributions in Hardy spaces based on ball Banach function spaces. Definition 4.2 (Ball Banach function space). A mapping ‖ ⋅ ‖Y → [0, ∞] is said to be a ball Banach function norm and the couple (Y (ℝn ), ‖ ⋅ ‖Y ) is said to be a ball Banach function space if (Y (ℝn ), ‖ ⋅ ‖Y ) satisfies (1)–(7) for all f , g, fj ⊂ L0 (ℝn ), j ∈ ℕ, and λ ∈ ℂ. (1) (Y (ℝn ), ‖ ⋅ ‖Y ) is a Banach space with the following property: f ∈ Y (ℝn ) if and only if ‖f ‖Y < ∞. (2) (Norm property): (A1) (Positivity): ‖f ‖Y ≥ 0. (A2) (Strict positivity): ‖f ‖Y = 0 if and only if f = 0 a. e. (B) (Homogeneity): ‖λf ‖Y = |λ| ⋅ ‖f ‖Y . (C) (Triangle inequality): ‖f + g‖Y ≤ ‖f ‖Y + ‖g‖Y . (3) (Symmetry): ‖f ‖Y = ‖|f |‖Y . (4) (Lattice property): If 0 ≤ g ≤ f a. e., then ‖g‖Y ≤ ‖f ‖Y . (5) (Fatou property): If 0 ≤ f1 ≤ f2 ≤ ⋅ ⋅ ⋅ and lim fj = f , then lim ‖fj ‖Y = ‖f ‖Y . j→∞

j→∞

(6) For B ∈ ℬ, χB ∈ Y (ℝn ). (7) If B ∈ ℬ and f ∈ Y (ℝn ), then χB f ∈ L1 (ℝn ).

For a ball Banach function space Y (ℝn ), we let Y ′ (ℝn ) ≡ {f ∈ L0 (ℝn ) : ‖f ‖Y ′ ≡

sup

g∈Y ,‖g‖Y =1

‖f ⋅ g‖L1 < ∞}.

The space Y ′ (ℝn ) is called the Köthe dual of Y (ℝn ) and it is known that Y ′ (ℝn ) is a ball Banach space if Y (ℝn ) is a ball Banach space; see [9, Proposition 2.3]. Assume that Y (ℝn ) is a ball Banach function space such that M is bounded on Y (ℝn ) and Y ′ (ℝn ). Then there exists η > 1 such that M (η) is also bounded on Y ′ (ℝn ) according to [15, Corollary 6.1]. Thus, for all f ∈ Y (ℝn ), 󵄩 󵄩 ‖f ‖L1 (M (η) χB(1) ) ≤ ‖f ‖Y 󵄩󵄩󵄩M (η) χB(1) 󵄩󵄩󵄩Y ′ ≲ ‖f ‖Y ‖χB(1) ‖Y ′ ∼ ‖f ‖Y .

(4.6)

We can develop the theory of the decomposition of Hardy spaces based on Y (ℝn ). But we can extend the class of linear spaces to some extent. Consider the power of Y (ℝn ): For 0 < p < ∞, we define

174 � Y. Sawano and K. Kobayashi 1

󵄩 󵄩 ‖f ‖Y (p) ≡ (󵄩󵄩󵄩|f |p 󵄩󵄩󵄩Y ) p for all f ∈ L0 (ℝn ). The p-convexification Y (p) (ℝn ) of Y (ℝn ) is the set of all f ∈ L0 (ℝn ) for which ‖f ‖Y (p) < ∞. For example, (Lp ) (ℝn ) = Lpu (ℝn ) for all 0 < u < ∞ (u)

and

1 ≤ p ≤ ∞.

Let Y (ℝn ) be as above and let X(ℝn ) ≡ Y (p) (ℝn ) for some 0 < p < ∞. The X-based Hardy space HX(ℝn ) collects all f ∈ 𝒮 ′ (ℝn ) for which ‖f ‖HX ≡ ‖ℳN f ‖X is finite. The number N will do as long as N ≫ 1. As is seen from (4.6), HX(ℝn ) is embedded into H p (w) for some w ∈ A1 . Therefore, the space HX(ℝn ) falls within the scope of Theorem 4.1. Theorem 4.3. Let Y (ℝn ) be a ball Banach function space such that M is bounded on Y (ℝn ) and Y ′ (ℝn ). Let 0 < p < ∞ and define X(ℝn ) ≡ Y (p) (ℝn ). Then for any f ∈ HX(ℝn ) and ∞ L ≫ 1, there exist a countable collection {fj }∞ j=1 of Lc -functions having moment of order L and a countable collection {Bj }∞ j=1 ⊂ ℬ satisfying (1.1), (1.2), and (1.7).

4.3 A∞ -weighted Hardy spaces We expand Section 4.1 using Section 4.2. A locally integrable weight w is said to be an A∞ -weight, if [w]A∞ ≡ sup mB (w) exp(−mB (log w)) < ∞. B∈ℬ

The quantity [w]A∞ is referred to as the A∞ -constant. The class A∞ of weights consists of all A∞ -weights. An important property of the class A∞ is that any weight in A∞ belongs to Ap for some 1 < p < ∞. Let 1 < p < ∞. A locally integrable weight w is an Ap -weight, if [w]Ap ≡ sup mB (w)(mB (w B∈ℬ

1 − p−1

p−1

))

< ∞.

The class Ap of weights consists of all Ap -weights. It is remarkable that w ∈ Ap if and only if M is bounded on Lp (w). A direct consequence of the definition is that w ∈ Ap if and only if σ ∈ Ap′ , where σ ≡ w

1 − p−1

. Remark also that {Ap }p∈[1,∞] is nested:

A1 ⊂ Ap ⊂ Aq ⊂ A∞

if 1 ≤ p ≤ q ≤ ∞.

Let w ∈ A∞ and 0 < p < ∞. Based on Section 4.1, we consider H p (w). Let w ∈ A∞ , so that w ∈ Au for some 1 < u < ∞. Then as we saw, M is bounded on Y (ℝn ) ≡ Lu (w) 1 ′ and on Y ′ (ℝn ) = Lu (σ), where σ ≡ w− u−1 . Since Y (p) (ℝn ) = Lpu (w) for all 0 < p < ∞, the space Lp (w) with 0 < p < ∞ and w ∈ A∞ falls within the scope of Theorem 4.3. In

Atomic decomposition in Hardy spaces

� 175

particular, Theorem 4.4 below can be used for another proof of the decomposition result in [20]. Theorem 4.4. The same conclusion as Theorem 4.1 holds if we assume merely w ∈ A∞ in Theorem 4.1.

4.4 Variable Hardy spaces For a measurable function p(⋅) : ℝn → (0, ∞), the variable Lebesgue space Lp(⋅) (ℝn ) with variable exponent p(⋅) is defined by Lp(⋅) (ℝn ) ≡ ⋃ {f ∈ L0 (ℝn ) : ρp (λ−1 f ) < ∞} where ρp (f ) ≡ ‖|f |p(⋅) ‖L1 . λ>0

Moreover, for f ∈ Lp(⋅) (ℝn ) we define the variable Lebesgue norm ‖ ⋅ ‖Lp(⋅) by ‖f ‖Lp(⋅) ≡ inf({λ > 0 : ρp (λ−1 f ) ≤ 1} ∪ {∞}). Here, we postulate the following conditions with some positive constants c∗ , c∗ and p∞ independent of x and y: – Local log-Hölder continuity condition: c∗ 󵄨 󵄨󵄨 󵄨󵄨p(x) − p(y)󵄨󵄨󵄨 ≤ log(|x − y|−1 ) –

1 for x, y ∈ ℝn satisfying |x − y| ≤ , 2

(4.7)

log-Hölder-type decay condition at infinity: c∗ 󵄨󵄨 󵄨 󵄨󵄨p(x) − p∞ 󵄨󵄨󵄨 ≤ log(e + |x|)

for x ∈ ℝn .

(4.8)

Assuming (4.7) and (4.8) as well as 0 < p− ≡ inf p(⋅) ≤ p+ ≡ sup p(⋅) < ∞, we can define variable Hardy space H p(⋅) (ℝn ) as the set of all f ∈ 𝒮 ′ (ℝn ) for which ℳN f ∈ Lp(⋅) (ℝn ). The number N will do as long as N ≫ 1. Theorem 1.2 did not use the structure of the underlying space Lp (ℝn ) heavily except in (3.7) and in the proof of the fact that the distribution vanishes weakly at infinity. Modify slightly the proof of Theorem 1.2, in particular (3.6), to have the following short proof of the key estimates of the decomposition theorems in [3, 14]. Theorem 4.5. Assume that the exponent p(⋅) satisfies the above conditions. Let f ∈ H p(⋅) (ℝn ) and L ∈ ℤ ∩ [[σp− ], ∞). Then there exist a countable collection {fj }∞ j=1 of ∞ L∞ -functions having moment of order L and a countable collection {B } ⊂ ℬ satisj c j=1 fying (1.1), (1.2), and (1.7).

176 � Y. Sawano and K. Kobayashi We may use Theorem 4.3 for another proof of Theorem 4.5, since M is bounded on ′ Lp(⋅) (ℝn ) and on Lp (⋅) (ℝn ) as long as p(⋅) satisfies (4.7) and (4.8) as well as 1 < p− ≤ p(⋅) p+ < ∞. Here, p′ (⋅) = p(⋅)−1 stands for the dual exponent.

4.5 Hardy–Morrey spaces p

First of all, let us recall the Morrey space ℳq (ℝn ) with 0 < q ≤ p < ∞. Define the Morrey norm ‖ ⋅ ‖ℳpq by 1

‖f ‖ℳpq ≡ sup{|B| p

− q1

‖f ‖Lq (B) : B ∈ ℬ} p

for f ∈ L0 (ℝn ). See [10] for example. The Morrey space ℳq (ℝn ) is the set of all f ∈ L0 (ℝn ) p for which ‖f ‖ℳpq is finite. The Hardy–Morrey space H ℳq (ℝn ) is the set of all f ∈ 𝒮 ′ (ℝn ) for which ‖f ‖Hℳpq ≡ ‖ℳN f ‖ℳpq is finite. The number N will do as long as N ≫ 1. We recall the following facts: p (1) Thanks to [4], M is bounded on ℳq (ℝn ) if 1 < q ≤ p < ∞. p n (2) In [17], the Köthe dual of ℳq (ℝ ) is specified if 1 < q ≤ p < ∞. p (3) Thanks to [16], M is bounded on the Köthe dual of ℳq (ℝn ) if 1 < q ≤ p < ∞. p

Let 0 < q ≤ p < ∞ again. Then from the above observation the space ℳq (ℝn ) falls within the scope of Theorem 4.3. p

Theorem 4.6. Let 0 < q ≤ p < ∞. Let f ∈ H ℳq (ℝn ) and L ∈ ℤ ∩ [[σq ], ∞). Then ∞ there exist a countable collection {fj }∞ j=1 of Lc -functions having moment of order L and a ∞ countable collection {Bj }j=1 ⊂ ℬ satisfying (1.1), (1.2), and (1.7). Theorem 4.6 recovers the results in [2, 11, 12]. It is noteworthy that in the present paper we did not depend on the diagonal argument in [2, 9]. As we did for variable Hardy spaces, we may also reexamine the proof of Theorem 1.2 to prove Theorem 4.6.

Bibliography [1] [2] [3] [4] [5]

D. R. Adams, Weighted nonlinear potential theory. Trans. Am. Math. Soc. 279 (1986), 73–94. A. Akbulut, V. S. Guliyev, T. Noi and Y. Sawano, Generalized Hardy-Morrey spaces. Z. Anal. Anwend. 36 (2017), 129–149. D. Cruz-Uribe and D. L. Wang, Variable Hardy spaces. Indiana Univ. Math. J. 63 (2014), 447–493. F. Chiarenza and M. Frasca, Morrey spaces and Hardy–Littlewood maximal function. Rend. Mat. 7 (1987), 273–279. S. Dekel, G. Kerkyacharian, G. Kyriazis and P. Petrushev, A New Proof of the Atomic Decomposition of Hardy Spaces. In: K. Ivanov, G. Nikolov and R. Uluchev (eds.) Constructive Theory of Functions, Sozopol 2016, 59–73. Prof. Marin Drinov Academic Publishing House, Sofia, 2018.

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[22] [23]

� 177

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Bernhard Ruf

A Bliss–Adams inequality Dedicated to the memory of David R. Adams

Abstract: The Trudinger–Moser inequality is the borderline inequality for the Sobolev inequalities in the case p = N. The famous Adams inequality concerns the generalization of the Trudinger–Moser inequality to mth-order derivatives in the case mp = N. Recently, in [11] a borderline inequality was derived for the Bliss-inequalities [5]. In this note, we give a second-order generalization of this inequality, which may be called of Bliss–Adams type. Keywords: Integral inequality, higher order, Adams inequality MSC 2020: Primary 35J20, 35J25, Secondary 35J50

1 On a personal note I met David R. Adams on the occasion of the II Workshop em Equações diferenciais não lineares in Campinas, Brazil, in 1998. I gave a talk presenting recent joint work with Djairo de Figueiredo, Olympio Miyagaki, and Joao Marcos do Ó [8–10, 19–21] on the application of the Trudinger–Moser inquality to partial differential equations. I remember that the hotel was in the center of Campinas, and we had to take a bus from the hotel to the conference venue on campus. The day after my talk, Professor Adams took a seat beside me on the bus. After some discussion on the Trudinger–Moser and related inequalities, he pulled a hardcover reprint from the Annals of Mathematics from his backpack and handed it to me—it contained his now famous work on the higher-order versions of the Trudinger–Moser inequality. Of course I was impressed—I have conserved this precious reprint up to this day. The influence of the article A sharp inequality of J. Moser for higher-order derivatives [1] by D. Adams cannot be overrated: there has been an enormous production of new results based on this inequality to this day and will without doubt continue so in the future.

2 The Trudinger–Moser inequality The well-known Sobolev embeddings say that H01 (Ω) ⊂ Lp (Ω), for 1 ≤ p ≤ 2∗ ; that is, in the special case of functions whose weak derivatives are square integrable in a given Acknowledgement: The author was supported in part by RIMS (Research Institute for Mathematical Sciences), and International Joint Usage/Research Center located in Kyoto University. Bernhard Ruf, Accademia di Scienze e Lettere - Istituto Lombardo, Milano, Italy, e-mail: [email protected] https://doi.org/10.1515/9783110792720-008

180 � B. Ruf

bounded domain Ω ⊂ ℝN , there exist constants Cp (N) such that sup

u∈H01 (Ω),∫Ω

|∇u(x)|2 dx=1

󵄨 󵄨p ∫󵄨󵄨󵄨u(x)󵄨󵄨󵄨 dx = Cp (N) < ∞,

1 ≤ p ≤ 2∗ =

Ω

2N . N −2

The case N = 2 is a borderline situation: the inequalities hold for any 1 ≤ p < ∞, but e not for p = ∞ as the following example shows: for Ω = B1 (0), let u(x) = log(log( |x| ));

then u ∈ H01 (B1 ), but u ∉ L∞ (Ω). It is then natural to ask if there exists some (maximal) function f (s) which grows faster than any polynomial such that u ∈ H01 (Ω) 󳨐⇒ ∫ f (u(x))dx < ∞. Ω

A positive answer to this question was given independently by V. I. Yudovic [25] (1961), S. Pohozaev [18] (1965), and N. Trudinger [24] (1967): they showed that the function of maximal growth is of exponential type, more precisely, they showed that there exists some α0 > 0 such that sup

u∈H01 (Ω),∫Ω |∇u(x)|2 dx=1

2

∫ eα|u(x)| dx = Cα < ∞,

for 0 < α < α0 .

(2.1)

Ω

This is now commonly called the Trudinger inequality. The proofs by Pohozaev [18] and Trudinger [24] use a Taylor series expansion

A Bliss–Adams inequality

� 181

1 󵄨2 k 󵄨 ∫(α󵄨󵄨󵄨u(x)󵄨󵄨󵄨 ) dx k! i=0 ∞

2

∫ eα|u(x)| dx = ∑

Ω

Ω

αk 󵄨󵄨 󵄨2k =∑ ∫󵄨󵄨u(x)󵄨󵄨󵄨 dx. k! k=0 ∞

Ω

By the Sobolev embeddings mentioned above, we can estimate k

󵄨 󵄨2k 󵄨 󵄨2 ∫󵄨󵄨󵄨u(x)󵄨󵄨󵄨 dx ≤ Ck (∫󵄨󵄨󵄨∇u(x)󵄨󵄨󵄨 dx) Ω Ω ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ =1

and one has an estimate for the Sobolev embedding constants Ck of the form Ck ≤ c(|Ω|)k(kC)k Hence, we can estimate ∞

2

∫ eα|u(x)| dx ≤ c(|Ω|) ∑ k(αC)k k=0

Ω

∞

= c(|Ω|) ∑ (αC)k k=0

kk k!

kk . (k − 1)!

The series converges by the ratio test for α < eC1 =: α0 . This is the Trudinger inequality. The question about the best possible exponent α0 was asked and answered by J. Moser in [15]. Indeed, J. Moser showed (1971) that the best exponent in (2.1) is α0 = 4π To obtain this result, J. Moser used symmetrization to reduce inequality (2.1) to a one-dimensional integral inequality. To a function u, one associates a radially symmetric function u∗ such that sublevelsets of u∗ are balls with the same volume as the corresponding sublevel-sets of u, i. e., 󵄨󵄨 󵄨 󵄨 󵄨 N ∗ 󵄨󵄨{x ∈ ℝ : u (x) < d}󵄨󵄨󵄨 = 󵄨󵄨󵄨{x ∈ Ω : u(x) < d}󵄨󵄨󵄨, where |A| denotes the Lebesgue measure of the set A. Then u∗ is a positive and nonincreasing function defined on BR (0) with |BR | = |Ω|. The symmetrized function u∗ has the following properties: – let f ∈ C(ℝ), then ∫B f (u∗ (x))dx = ∫Ω f (u(x))dx (by construction); –

R

∫B |∇u∗ (x)|2 dx ≤ ∫Ω |∇u(x)|2 dx (Pólya–Szegö inequality). R

From this, one clearly deduces that 2

sup ∫ eα|u(x)| dx ≤ sup ∫ eα|u

‖∇u‖2 ≤1

Ω

‖∇u∗ ‖2 ≤1

BR

∗

(|x|)|2

dx,

182 � B. Ruf and hence it is sufficient to consider the radial case. Next, performing the following change of variables: r = |x| = Re−t/2

and

w(t) = √4πu∗ (r),

one checks that ∞

󵄨 󵄨2 󵄨 󵄨2 ∫ 󵄨󵄨󵄨∇u∗ (x)󵄨󵄨󵄨 dx = ∫ 󵄨󵄨󵄨w′ (t)󵄨󵄨󵄨 dt

BR

0

and ∫e

α|u∗ (x)|2

∞

α

2

dx = |BR | ∫ e 4π |w(t)| −t dt.

BR

0

So, to prove Trudinger’s inequality it is sufficient to show that the following onedimensional integral inequality holds: Moser’s inequality

∞ {∫0

∞

2

∫ eβu (t)−t dt < ∞ ⇐⇒ β :=

sup

|u′ (t)|2 dt=1,u(0)=0} 0

α ≤ 1. 4π

(2.2)

Moser showed in particular, and somewhat surprisingly, that inequality (2.2) holds α also for the limiting exponent β = 4π = 1. To prove optimality, Moser introduced the following sequence of functions (the socalled Moser sequence): wn (t) = {

1 t, √n

√n,

0 ≤ t ≤ n;

(2.3)

t ≥ n.

Indeed, one checks that ∫0 |wn′ (t)|2 dt = 1 and ∞

∞

2

∞

∫ eβwn −t dt ≥ ∫ eβn−t dt = e(β−1)n → ∞, 0

n

for β > 1.

Note that the Moser sequence is a weakly convergent sequence in the space H01 (0, ∞), ∞ the closure of the space E with respect to the norm (∫0 |u′ (t)|2 dt)1/2 ; indeed, wn ⇀ 0. One also shows that wn is a noncompactness sequence for the functional

A Bliss–Adams inequality

∞

� 183

2

J(u) = ∫ eu (t)−t dt, 0

namely, lim inf J(wn ) ≥ 2 > 1 = J(0). We recall that it was shown, surprisingly, by L. Carleson and S.-Y. A. Chang [6] that the best exponent β = 1 is attained. This is in sharp contrast to the case of the critical Sobolev embeddings where it is known that the best constant is never attained on bounded domains.

3 The Bliss inequalities In an article in 1930, Hardy–Littlewood [12] published the following inequality (now known as “Hardy inequality”): ∞ 󵄨󵄨 u(x) 󵄨󵄨2 󵄨 ′ 󵄨2 󵄨󵄨 󵄨󵄨 ∫ 󵄨󵄨 󵄨 dx ≤ C1 ∫ 󵄨󵄨󵄨u (x)󵄨󵄨󵄨 dx 󵄨󵄨 x 󵄨󵄨󵄨 ∞

(3.1)

0

0

for all u ∈ E := {u ∈ C 1 (0, +∞), u(0) = 0}. In fact, Hardy and Littlewood wrote in their article that they believed that also the following inequalities should be true: k

k

|u(x)|2 󵄨 󵄨2 ∫ ( 1+1/k ) dx ≤ Ck ( ∫ 󵄨󵄨󵄨u′ (x)󵄨󵄨󵄨 dx) x ∞

∞

0

(3.2)

0

for all u ∈ E, for all k ∈ ℕ. They say they were not able to prove this, and so they told G. A. Bliss about it, who was a well-known specialist of the Calculus of Variations; see [5], working at the University of Chicago. Hardy–Littlewood went on to say that Bliss [4] came back within in one week with the full proof of the inequalities (3.2). Bliss did not only show the validity of (3.2), but he also showed that the sharp constants Ck in (3.2) are explicitly given by Ck =

1 (k − 1)Γ(2 + [ k Γ( 1 )Γ(2 + k−1

2 ) k−1 k−1 ] , 1 ) k−1

and he gave the family of functions which attain them:

(3.3)

184 � B. Ruf uλ (x) = ak

λx

1

[(λ2 x)k−1 + 1] k−1

(3.4)

.

We remark that the Bliss inequalities (3.2) are similar in form to the (radial) Sobolev inequalities, which were proved by Sobolev [22] in 1936, that is 6 years after the publication of the paper of Bliss. Furthermore, also the form of the attaining functions (3.4) are quite similar to the family of attaining functions in the Sobolev case, which were found by Th. Aubin [2] and G. Talenti [23] independently in 1976, that is 46 years (!) after the paper of Bliss.

4 A Bliss–Moser inquality We remark that the inequalities (3.2) also hold on the bounded interval (0, 1), i. e., considering E = {u ∈ C 1 (0, 1), u(0) = 0}. The optimal constants (3.3) remain the same (see [3, 17]), but the functions (3.4) do not attain the values (3.3), but they produce an optimizing sequence of functions for 1 1

sup

∫(

{∫0 |u′ (x)|2 dx=1,u(0)=0} 0

|u(x)|2 1

x 1+ k

k

) dx,

k = 1, 2, 3, . . .

Note that the situation of the inequalities (3.2) is similar to the Sobolev case in two dimensions: there is an inequality for every k ≥ 1. So, the question arises if there is a “limiting” inequality for the inequalities (3.2), just as in the 2d-Sobolev case. The form of 1 a possible limiting inequality is not clear because of the exponent x 1+ k in the denominator of the terms on the left. But considering that for all k ≥ 1, there exists a constant dk such that 1 e log( ) ≤ dk x − k , x

for x ∈ (0, 1),

we can try to estimate the following expression: 1

1

sup

e

∫ eα log( x )

{u∈E,∫0 |u′ (x)|2 dx=1} 0

|u(x)|2 x

dx

(4.1)

by mimicking the Trudinger case, i. e., estimating the integral by a power series of integrals of Bliss type. Indeed, we have the following. Theorem 4.1. The supremum in (4.1) is finite for 0 ≤ α < 1, and infinite for α > 1. Proof. We begin with an observation: the constants Ck in the Bliss inequalities (3.2) have the following asymptotics.

A Bliss–Adams inequality

� 185

Proposition 4.2. kCk = [

(k − 1)Γ(2 + 1 Γ( k−1 )Γ(2 +

2 ) k−1 k−1 ] 1 ) k−1

󳨀→ e,

as k → ∞.

Proof. Writing m = k − 1, we get [

mΓ(2 +

Γ( m1 )Γ(2

2 ) m m ] + m1 )

(4.2)

By the functional equation for the Gamma function, Γ(x + 1) = xΓ(x), we get Γ(2 +

2 2 2 2 ) = Γ(1 + + 1) = (1 + )Γ(1 + ), m m m m

and Γ(

1 1 1 ) = Γ( + 1). m m m

Hence, (4.2) becomes [

mΓ(1 + (1 +

2 )) m

mΓ( m1 + 1)Γ(1 + (1 +

m 1 )) m

] =[ =(

(1 +

2 )Γ(1 m

Γ( m1 + 1)Γ(1 + 1+

1+

= (1 +

+

2 ) m

m

]

1 )(1 + m1 ) m 2 m Γ(1 + m2 ) m m ) [ ] 1 (Γ(1 + m1 ))2 m m Γ(1 + m2 ) m 1 ) [ ] . m+1 (Γ(1 + m1 ))2

The first factor converges to e, while for the second factor we get by Taylor expansion Γ(1 +

2 2 1 ) = Γ(1) + Γ′ (1) + O( 2 ), m m m

and similarly for Γ(1 +

1 ), m

and hence [

Γ(1 + (Γ(1 +

2 ) m m ] 1 2 )) m

=[

Γ(1) + Γ′ (1) m2 + O( m12 )

(Γ(1) + Γ′ (1) m1 + O( m12 ))2

= [(1 + Γ′ (1) = [1 + O(

m

] 2 m

1 1 1 2 + O( 2 ))(1 − Γ′ (1) + O( 2 )) ] m m m m m

1 )] → 1, m2

as m → ∞.

186 � B. Ruf Next, we give the proof of Theorem 4.1. Claim 1: If α ∈ (0, 1), then 1

∫e

2

α log( xe ) u x(x)

1

󵄨2 󵄨 for all u ∈ E with ∫󵄨󵄨󵄨u′ (x)󵄨󵄨󵄨 dx ≤ 1.

dx ≤ C,

0

(4.3)

0

Proof. Write 1

e

∫ eα log( x )

u2 (x) x

dx

0

as a Taylor series: 1

∫e

2

α log( xe ) u x(x)

1

dx = ∫ ∑ 0 k

0

1

k

1 e u2 (x) [α log( ) ] dx k! x x k

= ∫∑ 0 k

1 e u2 (x) [α log( )x 1/k 1+1/k ] dx k! x x

Now note that e k max{log( )x 1/k , x > 0} = 1−1/k . x e

(4.4)

Hence, we can estimate the above sum by using (4.4) and the Bliss inequalities (3.2) 1

k

∫∑ 0 k

1 αk u2 (x) [ 1−1/k 1+1/k ] dx k! e x k 1

k

1 u2 (x) αk ≤ ∑ [ 1−1/k ] ∫[ 1+1/k ] dx k! e x k 0

k

1

1 αk 󵄨 󵄨2 ≤ e ∑ [ ] Ck (∫󵄨󵄨󵄨u′ (x)󵄨󵄨󵄨 dx ) k! e k 0 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ≤ e∑ k

k

=1

1 α [ ] k k−1 ⏟⏟kC ⏟⏟⏟⏟k⏟ k! e →e k

α k k−1 = e2 ∑( ) (1 + o(1)). e k! k Applying the ratio test, we get

k

A Bliss–Adams inequality k

( αe )k+1 (k+1) (k+1)! ( αe )k

k k−1 k!

� 187

=

α (k + 1)k k! e k k−1 (k + 1)!

(4.5)

=

α (k + 1)k−1 e k k−1

(4.6)

k−1

=

α 1 (1 + ) e k

=

α 1 1 (1 + ) e k 1+

(4.7)

k

1 k

→ α.

(4.8)

Hence, if α < 1, then the series converges. This proves Claim 1. Claim 2: For α > 1, the supremum (4.1) is infinite. Proof. Note that the Moser sequence (2.3) blows up at +∞, and converges pointwise to zero in every t ≥ 0. We can alternatively define an infinitesimal Moser sequence zn (t) = {

√nt, 1 , √n

0 ≤ t ≤ n1 ; t ≥ n1 .

The sequence satisfies again ∞

󵄨 󵄨2 ∫ 󵄨󵄨󵄨z′n (t)󵄨󵄨󵄨 dt = 1, 0

and it converges uniformly to 0 on [0, 1], while z′ (t) = √n blows up near zero. We use this sequence to demonstrate the sharpness of the Bliss–Moser inequality (4.1) with respect to α = 1. Indeed, setting α = 1 + δ we have 1

∫e

(1+δ) log( et )

|zn (t)|2 t

1/n

e

dt ≥ ∫ e(1+δ) log( t )nt dt

0

0 1

1 ≥ ∫ e(1+δ) log(en)s ds n 0

=

1 1 e(1+δ) log(en) n (1 + δ) log(en)

= e1+δ

nδ → ∞, (1 + δ) log(en)

as n → ∞.

188 � B. Ruf Remark 4.3. (1) We remark that in the Trudinger inequality, the series argument did not allow to reach the optimal exponent α = 4π, while in the Bliss–Moser inequality we obtain that α = 1 is the limiting exponent. We believe that this is so because in the Bliss inequalities we have the sharp embedding constants Ck , while for the (subcritical) Sobolev embeddings no explicit sharp embedding constants are available. (2) It has been shown recently by do Ó, Ubilla, and the author [11], using other methods, that for α = 1 the inequality (4.1) continues to hold. Indeed, it was proved that even a lower-order log(log(x)) term can be added in the exponent; one has 1

1

sup

e

e

∫ e(log( x )+β log(log x ))

|u(x)|2 x

dx < ∞ ⇐⇒ β ≤ 1.

{u∈E,∫0 |u′ (x)|2 dx=1} 0

(4.9)

5 The Adams inequality In the article [1], A sharp inequality of J. Moser for higher-order derivatives, D. R. Adams extended the Trudinger–Moser inequality to the case of higher-order derivatives. N Adams treats the general case of Lp -integrable mth-order derivatives, with p = m . N For this exposition, let us consider the special case m = 2, and N = 4, so that p = 2 = 2. For the Sobolev space W02,2 (Ω) = H02 (Ω), Ω ⊂ ℝ4 bounded, we have again that H02 (Ω) ⊂ Lq (Ω),

1 ≤ q < ∞,

but H02 (Ω) ⊄ L∞ (Ω).

By power series approximation, one can again show that there exists β0 > 0 such that sup

2

{u∈H02 (Ω),∫Ω |Δu|2 ≤1}

∫ eβ|u(x)| dx ≤ c|Ω|,

for 0 < β < β0 .

(5.1)

Ω

The best possible β0 was determined explicitly by D. R. Adams, namely β0 = 32π 2 and he proved that (5.1) holds for β ≤ β0 (i. e., in particular also for β = β0 ), while the supremum (5.1) becomes infinite for β > β0 . As D. R. Adams pointed out in [1], the symmetrization argument of J. Moser cannot be applied in the case of higher-order derivatives, since no estimates are available by which the L2 -norm of Δu can be dominated by the L2 -norm of the Laplacian of the symmetrized function. The key idea in Adams’ proof is to express u as the Riesz potential of its Laplacian, and then to apply the following sharp version of an estimate by Hedberg [13] for Riesz potentials. Indeed, writing I2 ∗ f (x) = ∫ Ω

f (y) dy, |x − y|2

A Bliss–Adams inequality

� 189

for f ∈ L2 (ℝ4 ) with support contained in Ω, Adams proved the following sharp form of Hedberg’s estimate. Theorem. There exists a constant c0 > 0 such that 2

∫ e π2

|

I2 ∗f (x) 2 | ‖f ‖2

dx ≤ c0

Ω

for all f ∈ L2 (ℝ4 ) with support contained in Ω ⊂ ℝ4 , and the constant

2 π2

is best possible.

To prove this theorem, Adams relied on an inequality by O’Neil [16] on nonincreasing rearrangements for convolution integrals: set f ∗ (t) = inf{s > 0 : λ(s) ≤ t} where 󵄨 󵄨 λ(s) = 󵄨󵄨󵄨{x ∈ ℝN : f (x) > s}󵄨󵄨󵄨 is the distribution function, and t

f

∗∗

1 (t) = ∫ f ∗ (s)ds. t 0

Then for h := g ∗ f one has the inequality h (t) ≤ tg ∗∗

+∞

∗∗

(t)f

∗∗

(t) + ∫ g ∗ (s)f ∗ (s)ds. t

Thus, for g(x) = |x|−2 we have 1

1 2π 2 2 g (t) = ( ) t 4

and g ∗∗ (t) = 2g ∗ (t).

∗

Then O’Neil’s inequality yields that for u = g ∗ f , we get u∗ (t) ≤ u∗∗ (t) ≤ tf

∞

∗∗

(t)g 1 2

=(

∗∗

(t) + ∫ f ∗ (s)g ∗ (s)ds t

0

2 1 π2 ) ( 1 ∫ f ∗ (s)ds + ∫ f ∗ (s) 1 ds). 2 t2 0 s2 t ∞

190 � B. Ruf Next, Adams performed the following change of variables: 1

s

ϕ(s) := |Ω| 2 f ∗ (|Ω|e−s )e− 2 with which he reduced the problem to showing that there exists a fixed constant c0 such that ∞

∫ ϕ2 (s)ds ≤ 1 0

implies ∞

∫e

∞

[∫−∞ a(s,t)ϕ(s)ds]2 −t

∞

dt =: ∫ e−F(t) dt ≤ c0 0

0

where 1, { { a(s, t) := { 2e(t−s)/2 , { { 0,

for 0 < s < t; for t < s < ∞; for − ∞ < s ≤ 0.

Note that this reduces inequality (5.1) again to a one-dimensional integral inequality. Indeed, Adams proves the following lemma. Lemma. Let a(s, t) be a nonnegative, measurable function on (−∞, ∞) × [0, ∞) such that a(s, t) ≤ 1 when 0 < s < t; 0

∞

2

1/2

sup( ∫ + ∫ a(s, t) ds) t>0

−∞

t

= b < ∞,

then there exists a constant c0 such that for ϕ ≥ 0 one has

∞ ∫−∞

sup

ϕ(s)2 ds≤1

∞

∫ e−F(t) dt ≤ c0 . 0

It is interesting to note that this lemma contains Moser’s inequality, taking a(s, t) = 1 when 0 < s < t and zero otherwise.

6 Second-order Bliss inequalities In this section, we derive the second-order Bliss inequalities. More precisely, we show the following.

A Bliss–Adams inequality

� 191

Theorem 6.1. There exist constants Dk such that 1

∫( 0

u2 (t) t

3+ k1

1

k

󵄨2 󵄨 ) dt ≤ Dk ∫󵄨󵄨󵄨u′′ (t)󵄨󵄨󵄨 dt

(6.1)

0

for all u ∈ F := {u ∈ C 2 (0, 1) | u(0) = u′ (0) = 0},

k = 1, 2, . . . .

The optimal constants Dk satisfy Dk ≤

4k C . 9 k

Proof. The inequalities (6.1) are known, and follow, e. g., from the extension of the Caffarelli–Kohn–Nirenberg inequalities [7] to higher-order derivatives by C. S. Lin [14]. We derive the inequalities together with the estimates for Dk from an iteration of the Bliss inequalities (3.2) with the Hardy–Littlewood inequality (3.1). Indeed, applying (3.1) to the functions {u′ ∈ C 1 (0, 1) | u′ (0) = 0} we obtain that 1

1

0

0

|u′ (t)|2 󵄨 󵄨2 dt ≤ 4 ∫󵄨󵄨󵄨u′′ (t)󵄨󵄨󵄨 dt. ∫ 2 t Next, we show that 1

∫( 0

|u(t)|2 1

t 3+ k

k

1

) ≤ Bk (∫ 0

k

|u′ (t)|2 dt) . t2

This can be obtained by a change of variables from the Bliss inequalities (3.2). Let t = x 1/3 and v(t) = u(x 1/3 ). Then 1

1

0

0

1 |v′ (t)|2 󵄨 󵄨2 dt ∫󵄨󵄨󵄨u′ (x)󵄨󵄨󵄨 dx = ∫ 3 t2 and 1

∫( 0

and thus we have

|u(x)|2 1

x 1+ k

k

1

) dx = 3 ∫( 0

|v(t)|2 1

t 3+ k

k

) dt,

192 � B. Ruf 1

∫( 0

|v(t)|2 t

3+ k1

1

k

k

1 |u(x)|2 ) dt = ∫( ) dx 1 3 x 1+ k 0

≤

k

1

Ck 󵄨2 󵄨 (∫󵄨󵄨󵄨u′ (x)󵄨󵄨󵄨 dx) 3 0

k

1

C |v′ (t)|2 = k (∫ dt) 9 t2 0

k

1

k Ck

󵄨 󵄨2 ≤4 (∫󵄨󵄨󵄨v′′ (t)󵄨󵄨󵄨 dt) , 9 0

and consequently we have shown inequality (6.1), with 4k C . 9 k

Dk ≤

7 A Bliss–Adams inequality In this section, we derive an exponential inequality for square integrable second-order derivatives. We again start from the inequality (4.4), e k 1 log( ) ≤ ( 1 ) 1 , 1− k x e xk

x ∈ (0, 1).

Since we want to apply the second-order Bliss inequalities, we are led to consider 1 1

sup

∫e

2

β log( xe ) |u(x)| 3 x

{u∈F|∫0 |u′′ (x)|2 dx≤1} 0

dx.

(7.1)

We have the following. Theorem 7.1. The supremum in (7.1) is finite for β < 41 , and it is infinite for β > 4 + π 2 . Proof. We write the integral as a power series of the form: 1

∫e 0

2

β log( xe ) |u(x)| 3 x

1

k

k

1 e |u(x)|2 dx = ∑ ) dx ∫(β log( )) ( k! x x3 k∈ℕ 0

k

k

≤ ∑ β ( k∈ℕ

e

1− k1

k 1

) ∫( 0

|u(x)|2 x

3+ k1

k

) dx

A Bliss–Adams inequality

k

≤ ∑ β ( k∈ℕ

≤ ∑ βk ( k∈ℕ

k

1

e1− k k

e

󵄨2 󵄨 ) Dk (∫󵄨󵄨󵄨u′′ (x)󵄨󵄨󵄨 dx ) 0 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ =1

k

1− k1

k

1

k

) 4k

Ck 9

k

1 4β e ∑ ( ) k k−1 kCk 9 k∈ℕ k! e

= Now we use that

kCk → e ,

by Proposition 4.2

and that k

1 4β ( ) k k−1 k! e k∈ℕ ∑

converges by (4.8) if 4β < 1. Next, consider the sequence of functions: 1

n 2 x2, { { { { { 1 3 sin(nπ(x − un (x) = a ⋅ { { πn 2 { { { (1 + 1) 1 , 3 { π 4 n2

1 )) 2n

+

1

1 ; 2n 1 ≤ n;

0≤x≤ 3

4n 2

,

1 2n 1 n

≤x

≤ x ≤ 1.

Then un (

1 a )= 3 2n n2

and un′ (

1 a ) = 12 . 2n n

Furthermore, we calculate 1

1

1

2n n 2 󵄨󵄨2 1 2 (nπ)2 󵄨󵄨󵄨󵄨 1 󵄨󵄨 ′′ 󵄨󵄨2 󵄨 2 2 ) 󵄨󵄨sin(nπ(x − ))󵄨󵄨󵄨 dx ∫󵄨󵄨un (x)󵄨󵄨 dx = a ∫ (2n ) dx + ∫ a2 ( 3 󵄨 󵄨󵄨 2n 󵄨 πn 2 1 0 0 2n

= a2 (2 + nπ 2 = a2 (2 + 1

2 2 Finally, set a = ( 4+π 2 ) . Then

1 ) 2n

π2 ) 2

� 193

194 � B. Ruf 1

2

un ∈ H (0, 1) and

u(0) = 0,

󵄨2 󵄨 with ∫󵄨󵄨󵄨un′′ (x)󵄨󵄨󵄨 dx = 1.

u (0) = 0 ′

0

Finally, we estimate 1

∫e

β log( xe )

|un |2 x

3

1 2n

dx ≥ ∫ e

0

4

β log(2ne)a2 nx3 x

dx

0 1 2n

2

= ∫ eβ log(2ne)a nx dx 0

=

1 2 1 eβ log(2ne)a nx |02n 2 β log(2ne)a n βa2

(2ne) 2 − 1 . = 2 βa n log(2ne) Now, by assumption β > 1

e

∫ eβ log( x )

2 , say β a2 |un |2 x

3

dx ≥

0

=

(1) (2) (3) (4)

=

2 (1 + δ), for some δ a2

(2ne)1+δ

βa2 n log(2ne)

> 0. Then

βa2 2

= 1 + δ, and hence

− o(1)

2e(2ne)δ − o(1) → ∞, βa2 log(2ne)

as n → ∞.

We end this note with some open problems: Determine the best constants in the second-order Bliss inequalities, and find functions attaining them. Prove Bliss inequalities for mth-order derivatives u(m) , with best constants and optimizing functions. Find the best exponent β0 in the Bliss–Adams inequality (conjecture β0 = 1). Is the best exponent attained? Prove mth-order Bliss–Adams inequalities.

Bibliography [1] [2] [3] [4] [5]

D. R. Adams, A sharp inequality of J. Moser for higher order derivatives. Ann. Math. 128 (1988), 385–398. Th. Aubin, Problèmes isopérimétriques et espaces de Sobolev. J. Differ. Geom. 11 (1976), 573–598. M. Biswas, On the best constant of the one-dimensional Bliss inequality. Houst. J. Math. 46 (2020), 189–200. G. A. Bliss, An integral inequality. J. Lond. Math. Soc. 5 (1930), 40–46. G. A. Bliss, Lectures on the Calculus of Variations. Chicago University Press, 1947.

A Bliss–Adams inequality

[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

[19] [20] [21]

[22] [23] [24] [25]

� 195

L. Carleson and S.-Y. A. Chang, On the existence of an extremal function for an inequality of J. Moser. Bull. Sci. Math. 110 (1986), 113–127. L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights. Composition Matematica 53 (1984), 259–275. D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in R2 with nonlinearities in the critical growth range. Calc. Var. Partial Differ. Equ. 3 (1995), 139–153. D. G. de Figueiredo, J. M. do Ó and B. Ruf, On an inequality by N. Trudinger and J. Moser and related elliptic equations. Commun. Pure Appl. Math. 55 (2002), 135–152. D. G. de Figueiredo and B. Ruf, Existence and non-existence of radial solutions for elliptic equations with critical exponent in ℝ2 . Commun. Pure Appl. Math. 48 (1995), 639–655. J. M. do Ó, B. Ruf and P. Ubilla, A critical Moser type inequality with loss of compactness due to infinitesimal shocks. Calc. Var. Partial Differ. Equ., 62, 008 (2023). G. H. Hardy and J. E. Littlewood, Notes on the theory of series ({XII}): On Certain Inequalities Connected with the Calculus of Variations. J. Lond. Math. Soc. 5 (1930), 34–39. L. I. Hedberg, On certain convolution inequalities. Proc. Am. Math. Soc. 36 (1972), 505–510. C. S. Lin, Interpolation inequalities with weights. Commun. Partial Differ. Equ. 11 (1986), 1515–1538. J. Moser, A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20 (1971), 1077–1092. R. O’Neil, Convolution operators and L(p, q) spaces. Duke Math. J. 30 (1963), 129–142. B. Opic and A. Kufner, Hardy-type inequalities. Pitman Research Notes in Matheamtics Series, 219. Longman Scientific & Technical, Harlow, 1990. S. I. Pohozaev, The Sobolev embedding in the case pl = n. In: Proceedings of the Technical Scientific Conference on Advances of Scientific Research 1964–1965. Mathematics Section, 158–170. Moskov. Ènerget. Inst., Moscow, 1965. B. Ruf, On a result by Carleson-Chang concerning the Trudinger-Moser inequality. Nonlinear Anal. 47 (2001), 6041–6051. B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in R2 . J. Funct. Anal. 219 (2005), 340–367. B. Ruf, Superlinear elliptic equations and systems. In: Handbook of differential equations: stationary partial differential equations. Vol. V, 211–276. Handb. Differ. Equ.. Elsevier/North-Holland, Amsterdam, 2008. S. L. Sobolev, On some estimates relating to families of functions having derivatives that are square integrable. Dokl. Adad. Nauk SSSR 1 (1936), 267–270 (in Russian). G. Talenti, Best constants in Sobolev inequality. Ann. Pura Appl. 110 (1976), 353–372. N. Trudinger, On the embedding into Orlicz spaces and some applications. J. Math. Mech. 17 (1967), 473–483. V. I. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations. Dokl. Acad. Nauk SSSR 138 (1961), 805–808. English translation: Sov. Math. Dokl. 2 (1961), 746–749.

Cristina Tarsi

Trudinger-type inequalities in ℝN with radial increasing mass-weight Dedicated to the memory of David R. Adams

Abstract: We prove Trudinger-type inequalities with radial increasing mass-weights in the whole ℝN , in the setting of mass-weighted Sobolev spaces Ww1,N (ℝN ). Due to the presence of increasing weights, we will not apply symmetrization tools: the proofs of our inequalities mainly rely on a proper transformation of variables, which allows us to reduce the weighted case to the unweighted classical one. Keywords: Weighted Sobolev spaces, Trudinger–Moser inequalities, exponential growth MSC 2020: Primary 35A23, Secondary 46E35

1 Introduction Let Ω ⊂ ℝN be an open domain with Lipschitz boundary and finite measure. It is well known that W01,N (Ω) 󳨅→ Lp (Ω) for p ∈ [1, ∞);

{

W01,N (Ω) 󳨅→ ̸ L∞

for p = ∞.

A counterexample is given by the function 󵄨 󵄨 u(x) = (− log󵄨󵄨󵄨log |x|󵄨󵄨󵄨)+ , when Ω is the unit ball. The maximal degree of summability for functions in W01,N (Ω) was established independently by Pohožaev [15] and Trudinger [19] (see also [20]) and is of exponential type: more precisely, they proved that there exists α > 0 such that sup{∫ eα|u| Ω

N N−1

|u ∈ W01,N (Ω), ‖∇u‖N ≤ 1} < ∞

(1.1)

Several years later, Moser [12] was able to simplify Trudinger’s proof, and to determine the optimal exponent αN such that (1.1) holds for every 0 ≤ α ≤ αN , and fails for α > αN . 1

N−1 , where ωN−1 is the (N − 1)-dimensional This optimal exponent is given by αN = NωN−1 N surface of the unit sphere in ℝ . In particular, for N = 2, α2 = 4π. While the proof of

Cristina Tarsi, Dipartimento di Matematica, Università degli Studi di Milano, Milano 20133, Italy, e-mail: [email protected] https://doi.org/10.1515/9783110792720-009

198 � C. Tarsi the validity of (1.1) for α < αN is not difficult, the very delicate point is to prove that it holds also for α = αN . This is done by showing, after reducing by symmetrization to the case where is the unit ball, that is, {vk } is a maximizing sequence, it cannot be “too far” from the so-called Moser sequence. The same sequence is also used to prove the failure of (1.1) for α > αN . Subsequently, Adams [2] was able to extend the result to functions u k,p in the higher-order Sobolev spaces W0 (Ω) with kp = N, by writing u as Riesz potential of its gradient of order k. If |Ω| = ∞, then the inequality (1.1) is obviously false, and the exponential needs to be suitably regularized when u is near 0. The standard way to do this is to consider the reduced exponential functions et − 1

expN (t) := {

t

e −

tk ∑N−2 k=0 k!

if N = 2;

if N > 2,

and recast [19] into N

sup ∫ expN (α|u| N−1 )dx, ℝN

where, for simplicity, we have considered Ω = ℝN . Then, by restricting the class of functions in the supremum and considering smooth functions such that ‖∇u‖2 ≤ 1 and ‖u‖2 ≤ K, K > 0, as developed by Cao [5] one has sup

2

‖∇u‖2 ≤1, ‖u‖2 ≤K

∫ (eαu − 1) dx ≤ C(α, K) < ∞

if α ≤ 4π(1 − δ),

(1.2)

ℝ2

where δ ∈ (0, 1). A further result in this direction was obtained by Adachi–Tanaka in [1] which reads as follows: for all u ∈ H 1 (ℝ2 ) \ {0}, one has ∫ (e

α

u2 ‖∇u‖2 2

− 1) dx ≤ C(α)

ℝ2

‖u‖22 , ‖∇u‖22

(1.3)

where C(α) < ∞ as long as α < 4π Note that the inequalities (1.2) and (1.3) involve only values of the parameter α < 4π, so that they are usually named as subcritical TM inequalities. The critical Moser case in which α = 4π remained uncovered until Ruf in [16] established the following inequality: sup

‖∇u‖22 + ‖u‖22 ≤1

2

∫ (eαu − 1) dx < ∞ if α ≤ 4π, ℝ2

Trudinger-type inequalities in ℝN with radial increasing mass-weight

� 199

which is sharp, namely the supremum becomes infinity as α > 4π. Starting from these pioneering results, a very large amount of literature has been developed, which is still very active and motivated either by theoretical aspects or by applications to PDEs (partial differential equations). Note that, from the point of view of functional embeddings, Moser and Trudinger-type inequalities have the same role, since they both identify the optimal target space (in the Orlicz framework) where the Sobolev spaces embed in the limiting cases; roughly speaking, they both allow to hanN dle with functionals having the same exponential growth (≈ exp(t N−1 )), except for the exponent appearing in front of t. They differ exactly in identifying (or not) the sharp exponent in front of t necessary to gain uniform integral bounds. For this reason, subcritical Trudinger-type inequalities are enough to handle with a large amount of PDEs involving optimal growth, in the limiting case of Sobolev embeddings, even if the delicate analysis of critical phenomena often requires the availability of a sharp Moser-type version. In this paper, we are interested in developing a new class of Trudinger-type inequality, settled in the whole space ℝ2 or ℝN , whose main feature is the presence of an increasing, radial (continuous) weight. We are motivated by some very recent developments in the study of Schrödinger–Poisson and Choquard-type equations in dimension two and, more in general, in dimension N (see [4, 6, 7] and references therein): both of them are characterized by the presence of free energy functionals involving logarithmic kernels, which arises in the reduction of planar Schrödinger–Poisson systems to a single integrodifferential equation. In [4, 6], new log-weighted Trudinger-type inequalities have been proved, to allow the variational approach up to the maximal optimal growth; see also [8, 18] for related functional discussions. Note that the presence of an increasing radial weight prevents the application of standard symmetrization tools, which are one of the key steps when proving Trudinger–Moser-type inequalities: for this reason, up to now weighted inequalities with increasing weights have been proved only in the framework of Sobolev (mass weighted) radial spaces (i. e., the weight appears only in the mass component of the norm); see, for instance, [3, 11, 13, 14]. The main novelty in [4, 6, 18] has been the introduction of a proper change of variable that allows to relate a function in the mass-weighted Sobolev space to another function in W 1,N (ℝN ), inspired by [9, 14]. More precisely, let w(|x|) be an increasing, 𝒞 1 positive radial weight, and let Hw1 (ℝ2 ) := {u ∈ H 1 (ℝ2 ) : ∫ |u|2 w(|x|)dx < ∞}, ℝ2

equipped with the norm ‖u‖2w := ‖∇u‖22 + ‖u‖22,w = ∫ |∇u|2 dx + ∫ u2 w(|x|)dx. ℝ2

ℝ2

200 � C. Tarsi With the notation of [10], Hw1 is nothing but W 1,2 (ℝ2 , S), where S is the set of weights given by S = {w(|x|), 1}. Since the weight w(|x|) satisfies the condition w−1 ∈ L1loc (ℝ2 ), it turns out that Hw1 (ℝ2 ) is a Banach space [10, Theorem 1.11], and further, it is a Hilbert space, endowed with the inner product ⟨u, v⟩ = ∫ ∇u ⋅ ∇vdx + ∫ uvw(|x|)dx ℝ2

ℝ2

Further, its dual can be characterized thanks to the Hahn–Banach theorem as Hw−1 (ℝ2 ) = (H 1 (ℝ2 ) ∩ L2w (ℝ2 )) = H −1 (ℝ2 )|Hw1 +(L2w ) (ℝ2 )|Hw1 ′

′

(see Theorem 14.9 in [17]). If w(x) = |x|β , then we will have ‖u‖2w := ‖∇u‖2 + ‖u‖22,w = ∫ |∇u|2 dx + ∫ u2 |x|β dx ℝ2

ℝ2

For the sake of clarity, we first address the easier, prototype case of Trudinger-type inequalities in the whole plane with monomial radial increasing weights, extending the (sharp) result proved in [14] to a the nonradial setting (with loss of sharpness), as follows. Theorem 1.1. For any α ≤

8π , 2+β

one has 2

sup ∫ (eαu − 1)|x|β dx < ∞.

‖u‖2w ≤1

(1.4)

ℝ2

Furthermore, 2

∫ (eαu − 1)|x|β dx < ∞ ℝ2

holds for any u ∈ Hw1 (ℝ2 ) and any α > 0. Then we consider the case of a general increasing weight w in the whole plane. Theorem 1.2. Let w(|x|) be a radial 𝒞 1 increasing weight, such that w′ (r) γ ≤ w(r) r for some γ > 0, where r = |x|. Then, for any α ≤ 2

∀r > 0 8π 2+γ

one has

sup ∫ (eαu − 1)w(|x|)dx < ∞

‖u‖2w ≤1

ℝ2

(1.5)

(1.6)

Trudinger-type inequalities in ℝN with radial increasing mass-weight

� 201

Furthermore, 2

∫ (eαu − 1)w(|x|)dx < ∞ ℝ2

holds for any u ∈ Hw1 (ℝ2 ) and any α > 0. We end with the Trudinger-type inequalities in dimension N ≥ 3, discussing directly the case of a general increasing radial weight. We denote the space of measurable functions by Ww1,N (ℝN ) := {u ∈ W 1,N (ℝN ) : ∫ |u|N w(|x|)dx < ∞}, ℝN

equipped with the norm ‖u‖Nw := ‖∇u‖NN + ‖u‖NN,w = ∫ |∇u|N dx + ∫ |u|N w(|x|)dx. ℝN

ℝN

Further, let us recall the definition of the reduced exponential function N−2 j

expN (t) = et − ∑

j=0

t . j!

We have then the following. Theorem 1.3. Let w(|x|) be a radial 𝒞 1 increasing weight such that w′ (r) γ ≤ w(r) r

∀r > 0 1

N N−1 ) one has for some γ > 0, where r = |x|. Then, for any α ≤ αN ( N+γ N

sup ∫ expN (α|u| N−1 )w(|x|)dx < ∞,

‖u‖Nw ≤1

(1.7)

ℝN

where 1

N−1 αN = NωN−1

is the sharp Moser exponent, and ωN−1 is the (N −1)-dimensional surface of the unit sphere in ℝN . Furthermore, N

∫ expN (α|u| N−1 )w(|x|)dx < ∞ ℝN

202 � C. Tarsi holds for any u ∈ Ww1,N (ℝN ) and any α > 0. Remark 1.4. If we consider weights w(r) such that for any γ > 0 r → +∞, then we easily obtain, in the case N = 2,

w′ w

≤

γ r

definitively as

2

sup ∫ (e(4π−ε)u − 1) log(e + |x|)dx = Cε < ∞

‖u‖2w ≤1

ℝ2

for any ε > 0, and similarly for N ≥ 3. So, the Moser exponent 4π (and αN in higher dimensions) could be again the optimal one, when considering slowly increasing massweights, such as w(|x|) = log(e + |x|). The question is addressed in a forthcoming paper [18]. It is still open to verify the sharpness for more general weights.

2 Inequalities in ℝ2 In this section, we address the Trudinger-type inequalities in ℝ2 , proving Theorem 1.1 and Theorem 1.2. As specified in the Introduction, the main tool will be a transformation, which relates functions in Hw1 to functions in H 1 , inspired by [9, 14]. Due to the radial nature of the weight, we will write the Sobolev norms in term of polar coordinates: this trick will allow us to bound the weighted norm ‖u‖w with the classical one ‖u‖, up to a (nonsharp) constant cw depending on the weight. Proof of Theorem 1.1. We will follow the ideas introduced in [6]. To do so, let us perform the change of variable β

y = (y1 , y2 ) = |x|1+ 2 (cos θ, sin θ), where {x = (x1 , x2 ) = |x|(cos θ, sin θ); { 2 2 {|x| = √x1 + x2 . Equivalently, y x = , |y| |x|

T(|x|) = |y|,

β

|y| = |x|1+ 2 .

Since T acts only on the radial part of any point in ℝ2 , to shorten the notation we will write β

ρ = T(r) = r 1+ 2 with inverse map

where r = |x|, ρ = |y|

Trudinger-type inequalities in ℝN with radial increasing mass-weight

� 203

2

T −1 (ρ) = ρ 2+β . Let us define v(y) := u(x),

2

2

that is, v(y) = u(|y| 2+β cos θ, |y| 2+β sin θ)

or, equivalently u(r cos θ, r sin θ) = v(T(r) cos θ, T(r) sin θ). Then, by a direct calculation, if ̃z(ρ, θ) := v(ρ cos θ, ρ sin θ),

z(r, θ) := u(r cos θ, r sin θ),

z(r, θ) = ̃z(T(r), θ),

then we have zr (r, θ) = ̃zρ (T(r), θ)T ′ (r),

zθ (T(r), θ) = ̃zθ (T(r), θ),

whence getting 2π ∞

∫ |∇v|2 dy1 dy2 = ∫ ∫ [̃z2ρ + 0 0

ℝ2

̃z2θ ]ρdρdθ ρ2

2π ∞

= ∫ ∫ [̃z2ρ (T(r), θ) + 0 0

2π ∞

= ∫ ∫ [z2r (r, θ) ⋅ 0 0

̃z2θ (T(r), θ) ′ ]T (r)T(r)drdθ T 2 (r)

z2θ (r, θ) 1 r2 + ⋅ ]T ′ (r)T(r)drdθ [T ′ (r)]2 r2 T 2 (r)

Thanks to 1

[T ′ (r)]2

=

4 r −β , (2 + β)2

r2

T 2 (r)

= r −β ,

we obtain 2π ∞

z2θ r 2 T ′ (r) 4 2 [z + ] drdθ ≤ ∫ |∇v|2 dy1 dy2 ∫ ∫ r (2 + β)2 r 2 T(r) 0 0

ℝ2

2π ∞

≤ ∫ ∫ [z2r + 0 0

Upon noting

z2θ r 2 T ′ (r) ] drdθ. r 2 T(r)

204 � C. Tarsi r 2 T ′ (r) β + 2 = r, T(r) 2 we have, at the end, β+2 2 ∫ |∇u|2 dx1 dx2 < ∫ |∇v|2 dy1 dy2 < ∫ |∇u|2 dx1 dx2 . 2+β 2 ℝ2

ℝ2

ℝ2

On the other hand, since 2π ∞

∫ v2 dy = ∫ ∫ ̃z2 (ρ, θ)ρdρdθ 0 0

ℝ2

2π ∞

= ∫ ∫ ̃z2 (T(r), θ)T ′ (r)T(r)drdθ 0 0

2π ∞

= ∫ ∫ z2 (r, θ)T ′ (r)T(r)drdθ 0 0

2π ∞

β+2 = ∫ ∫ z2 (r, θ)r 1+β drdθ, 2 0 0

it follows that ∫ v2 dy = ℝ2

β+2 ∫ u2 |x|β dx. 2 ℝ2

Finally, we achieve β+2 2 2 ‖u‖2w < ‖v‖2 = ‖∇v‖22 + ‖v‖22 < ‖u‖w . 2+β 2

(2.1)

We have then proved that the map 1

2

1

2

𝒯 : Hw (ℝ ) → H0 (ℝ )

u 󳨃→ v

is an invertible, continuous map, with a continuous inverse map, too. Then [16] derives ∫ (e ℝ2

αu2

β

2π ∞

2

− 1)|x| dx = ∫ ∫ (eαu (r cos θ,r sin θ) − 1)r 1+β drdθ 0 0 2π ∞

2 2 = ∫ ∫ (eαv (ρ cos θ,ρ sin θ) − 1)ρdρdθ 2+β

0 0

Trudinger-type inequalities in ℝN with radial increasing mass-weight

=

� 205

2 2 ∫ (eαv − 1)dx < ∞ 2+β

ℝ2

for any α > 0. The uniform bound (1.4) follows directly, once we consider the norm 2 estimate (2.1): for any u ∈ Hw1 and α ≤ 4π 2+β , there holds 2

2

∫ (eαu /‖u‖w − 1)|x|β ℝ2

≤ ∫ (e

2 4π 2+β u2 /‖u‖2w

− 1)|x|β dx

ℝ2

2π ∞

= ∫ ∫ (e 0 0

2 4π 2+β u2 (r cos θ,r sin θ)/‖u‖2w

− 1)r 1+β drdθ

2π ∞

2 4π 2 v2 (ρ cos θ,ρ sin θ)/‖u‖2w = − 1)ρdρdθ ∫ ∫ (e 2+β 2+β 0 0

2π ∞

≤

2 2 2 ∫ ∫ (e4πv (ρ cos θ,ρ sin θ)/‖v‖ − 1)ρdρdθ 2+β

0 0

2 2 2 = ∫ (e4πv /‖v‖ − 1)dx < ∞ 2+β

ℝ2

by Ruf’s inequality in [16]. Remark 2.1. We remark that the transformation T is the same one introduced in [14]. The estimate (2.1) was (partially) observed there: the authors pointed out that, since β > 0, the inequality 2 ‖u‖2w < ‖v‖2 = ‖∇v‖22 + ‖v‖22 2+β holds, whereas it is reversed if the weight is decreasing, that is, if β < 0. For this reason, they proved a sharp weighted Moser-type inequality in the framework of radial Sobolev spaces. Taking into account polar coordinates, instead, allows us to obtain the upper bound as in (2.1), thereby yielding the Trudinger inequality in the nonradial setting. Proof of Theorem 1.2. As in the previous proof, let us perform the change of variable y = (y1 , y2 ) = |x|√w(|x|)(cos θ, sin θ), that is, T(|x|) = |y|,

y x = , |y| |x|

|y| = |x|√w(|x|).

206 � C. Tarsi We shorten the notation as follows: ρ = T(r) = r √w(r)

where r = |x|, ρ = |y|.

Note that T ′ (r) =

2w(r) + rw′ (r) > 0, 2√w(r)

T(0) = 0,

lim T(r) = ∞

r→∞

so that T is invertible on ℝ2 , even if its inverse map is not explicit. Let us define v(y) := u(x),

that is, v(y) = u(T −1 (|y|) cos θ, T −1 (|y|) sin θ),

or, equivalently u(r cos θ, r sin θ) = v(T(r) cos θ, T(r) sin θ). Consequently, if z(r, θ) := u(r cos θ, r sin θ)̃z(ρ, θ) := v(ρ cos θ, ρ sin θ),

z(r, θ) = ̃z(T(r), θ),

then zr (r, θ) = ̃zρ (T(r), θ)T ′ (r),

zθ (T(r), θ) = ̃zθ (T(r), θ),

and hence ∫ |∇v|2 dy1 dy2 ℝ2

2π ∞

= ∫ ∫ [̃z2ρ (T(r), θ) + 0 0

2π ∞

= ∫ ∫ [z2r (r, θ) ⋅ 0 0

̃z2θ (T(r), θ) ′ ]T (r)T(r)drdθ T 2 (r)

z2θ (r, θ) 1 r2 + ⋅ ]T ′ (r)T(r)drdθ. [T ′ (r)]2 r2 T 2 (r)

Now, a combination of 1

[T ′ (r)]2

=

w(r) , (w(r) + rw′ (r)/2)2

r2

T 2 (r)

=

1 w(r)

and the assumption (1.5) derives r2 1 > , T 2 (r) [T ′ (r)]2

T(r) r 2 = > r. T ′ (r) w + r2 w′ 2 + γ

Trudinger-type inequalities in ℝN with radial increasing mass-weight

Then we reach 2π ∞

z2 2 ∫ ∫ [z2r + θ2 ]rdrdθ ≤ ∫ |∇v|2 dy1 dy2 2+γ r 0 0

ℝ2

2π ∞

≤ ∫ ∫ [z2r + 0 0

z2θ r 2 T ′ (r) ] drdθ. r 2 T(r)

Via noting r w′ r 2 T ′ (r) 2 + γ r 2 T ′ (r) = r(1 + ) 󳨐⇒ r ≤ ≤ r, T(r) 2 w T(r) 2 we have, at the end, 2+γ 2 ∫ |∇u|2 dx1 dx2 < ∫ |∇v|2 dy1 dy2 < ∫ |∇u|2 dx1 dx2 . 2+γ 2 ℝ2

ℝ2

ℝ2

On the other hand, since 2+γ T ′ (r)T(r) r 2 T ′ (r) = 󳨐⇒ rw(r) ≤ T ′ (r)T(r) ≤ rw(r), w(r) T(r) 2 we conclude that 2π ∞

∫ v dy = ∫ ∫ ̃z2 (ρ, θ)ρdρdθ 2

0 0

ℝ2

2π ∞

= ∫ ∫ ̃z2 (T(r), θ)T ′ (r)T(r)drdθ 0 0

2π ∞

= ∫ ∫ z2 (r, θ)T ′ (r)T(r)drdθ 0 0

≤

2π ∞

γ+2 ∫ ∫ z2 (r, θ)rw(r)drdθ 2 0 0

so that ∫ u2 w(|x|)dx ≤ ∫ v2 dy ≤ ℝ2

ℝ2

γ+2 ∫ u2 w(|x|)dx. 2 ℝ2

� 207

208 � C. Tarsi Finally, we reach γ+2 2 2 ‖u‖2 < ‖v‖2 = ‖∇v‖22 + ‖v‖22 < ‖u‖w . 2+γ w 2

(2.2)

As before, 1

2

1

2

𝒯 : Hw (ℝ ) → H0 (ℝ )

u 󳨃→ v is an invertible, continuous map, with continuous inverse map. Then [16] is used to imply that

∫ (e

αu2

2π ∞

2

− 1)w(|x|)dx = ∫ ∫ (eαu (r cos θ,r sin θ) − 1)rw(r)drdθ 0 0

ℝ2

2π ∞

2

≤ ∫ ∫ (eαv (ρ cos θ,ρ sin θ) − 1)ρdρdθ 0 0 2

= ∫ (eαv − 1)dx < ∞ ℝ2

holds for any α > 0. The uniform bound (1.6) follows directly, once we consider the norm 8π , there holds estimate (2.2): for any u ∈ Hw1 and α ≤ 2+γ 2

2

∫ (eαu /‖u‖w − 1)w(|x|) ≤ ∫ (e ℝ2

2 4π 2+γ u2 /‖u‖2w

− 1)w(|x|)dx

ℝ2

2π ∞

= ∫ ∫ (e

2 4π 2+γ u2 (r cos θ,r sin θ)/‖u‖2w

− 1)rw(r)drdθ

0 0

2π ∞

≤ ∫ ∫ (e

2 2 v (ρ cos θ,ρ sin θ)/‖u‖2w 4π 2+γ

− 1)ρdρdθ

0 0

2π ∞

2

2

≤ ∫ ∫ (e4πv (ρ cos θ,ρ sin θ)/‖v‖ − 1)ρdρdθ 0 0 2

2

= ∫ (e4πv /‖v‖ − 1)dx < ∞ ℝ2

by Ruf’s inequality in [16].

Trudinger-type inequalities in ℝN with radial increasing mass-weight

� 209

3 Inequalities in ℝN In this section, we address the Trudinger-type inequalities in the whole ℝN with N ≥ 3, proving Theorem 1.3. The main tool will be as in the plane, a transformation, which relates functions in Ww1,N to functions in W 1,N , combined with the evaluation of the Sobolev norms via spherical coordinates. We follow the same ideas introduced in [4]. Proof of Theorem 1.3. As before, we perform a change of variables to pass from Ww1,N (ℝN ) to the functions in W 1,N (ℝN ). We use the hyperspherical coordinates in ℝN below: x1 = |x| sin θ1 sin θ2 ⋅ ⋅ ⋅ sin θN−2 sin θN−1 ; { { { { { { {x2 = |x| sin θ1 sin θ2 ⋅ ⋅ ⋅ sin θN−2 cos θN−1 ; { { x = {x3 = |x| sin θ1 sin θ2 ⋅ ⋅ ⋅ cos θN−2 ; { { { { ... { { { { {xN = |x| cos θ1 , where θ1 , . . . , θN−2 ∈ [0, π]; { { { θN−1 ∈ [0, 2π); { { { 2 2 2 {|x| = x1 + ⋅ ⋅ ⋅ + xN . Let us apply the following change of variables, by acting only on the radial component of a point in ℝN : T(|x|) = |y|,

y x = , |y| |x|

|y| = |x| √w(|x|). N

We set r = |x| and s = |y|, whence N w(r). s = T(r) = r √

We obtain T ′ (r) =

Nw(r) + rw′ (r) Nw

N−1 N

> 0,

T(0) = 0,

lim T(r) = ∞.

r→∞

Accordingly, T is invertible on ℝN (though the inverse map is not explicitly known). Let v(y) := u(x), or, equivalently u(r sin θ1 ⋅ ⋅ ⋅ sin θN−1 , . . . , r cos θ1 ) = v(T(r) sin θ1 ⋅ ⋅ ⋅ sin θN−1 , . . . , T(r) cos θ1 ). Then, upon denoting

210 � C. Tarsi θ = (θ1 , . . . , θN−1 ); { { { { { {z(r, θ) := u(r sin θ1 ⋅ ⋅ ⋅ sin θN−1 , . . . , r cos θ1 ); { {̃z(s, θ) := v(s sin θ1 ⋅ ⋅ ⋅ sin θN−1 , . . . , s cos θ1 ); { { { { {z(r, θ) = ̃z(T(r), θ), we compute zr (r, θ) = ̃zs (T(r), θ)T ′ (r),

{

zθi (T(r), θ) = ̃zθi (T(r), θ)

∀i ∈ {1, . . . , N − 1}.

Therefore, we obtain ∫ |∇v|N dy ℝN

2π π

π

= ∫ ∫ sin θN−2 ⋅ ⋅ ⋅ ∫ sin 0 0

+

N−2

0

̃z2θ (s, θ) N−1

s2 sin2 θ1 . . . sin2 θN−2

+∞

θ1 ∫ [̃z2s (s, θ) + N 2

̃z2θ (s, θ) 1 s2

0

] sN−1 ds dθ1 . . . dθN−2 dθN−1

2π π

π

+∞

0 0

0

0

= ∫ ∫ sin θN−2 ⋅ ⋅ ⋅ ∫ sinN−2 θ1 ∫ [̃z2s (T(r), θ) +

+

̃z2θ (T(r), θ) N−1

T 2 (r) sin2 θ1 . . . sin2 θN−2

2π π

π

0 0

0

+

r2

T 2 (r)

+ ⋅⋅⋅

N 2

+∞ 0

r

̃z2θ (T(r), θ) 1

] T N−1 (r)T ′ (r)dr dθ1 . . . dθN−2 dθN−1

= ∫ ∫ sin θN−2 ⋅ ⋅ ⋅ ∫ sinN−2 θ1 ∫ [ z2θN−1 (r, θ)

+ ⋅⋅⋅

2 z2r (r, θ) zθ1 (r, θ) r 2 + + ⋅⋅⋅ [T ′ (r)]2 r 2 T 2 (r) N 2

2

T 2 (r) sin2 θ1 . . . sin2 θN−2

] T N−1 (r)T ′ (r)dr dθ1 . . . dθN−2 dθN−1 .

Now, since 2(N−1)

1 [w(r)] N = , ′ 2 [T (r)] [w(r) + Nr w′ (r)]2

r2 1 = , 2 T (r) [w(r)]2/N

we get N2 r2 1 r2 < < , (N + γ)2 T 2 (r) [T ′ (r)]2 T 2 (r) thereby finding

Trudinger-type inequalities in ℝN with radial increasing mass-weight 2π π

π

0 0

0

� 211

N z2θ1 (r, θ) N N−2 2 ( ) ∫ ∫ sin θN−2 ⋅ ⋅ ⋅ ∫ sin θ1 ∫ [zr (r, θ) + + ⋅⋅⋅ N +γ r2

+

∞

N 2 z2θN−1 (r, θ) ] 2 2 2 r sin θ1 . . . sin θN−2

0

N

r T (r) dr dθ1 . . . dθN−2 dθN−1 T(r) ′

≤ ∫ |∇v|N dy1 . . . dyN ℝN

2π π

π

N−2

≤ ∫ ∫ sin θN−2 ⋅ ⋅ ⋅ ∫ sin 0 0

+

r 2 sin2 θ

1 . . . sin

2

θ1 ∫ [z2r (r, θ) + 0

0

z2θN−1 (r, θ)

∞

N 2

θN−2

]

z2θ1 (r, θ) r2

+ ⋅⋅⋅

r N T ′ (r) dr dθ1 . . . dθN−2 dθN−1 . T(r)

Since r N T ′ (r) r w′ (r) r N T ′ (r) N + γ N−1 = r N−1 [1 + ] 󳨐⇒ r N−1 ≤ ≤ r , T(r) N w(r) T(r) N we gain N

(

N +γ N ) ∫ |∇u|N dx < ∫ |∇v|N dy < ∫ |∇u|N dx. N +γ N ℝN

ℝN

ℝN

On the other hand, we have 2π π

π

∞

󵄨 󵄨N ∫ |v|N dy = ∫ ∫ sin θN−2 ⋅ ⋅ ⋅ ∫ sinN−2 θ1 ( ∫ 󵄨󵄨󵄨̃z(s, θ)󵄨󵄨󵄨 sN−1 ds)dθ1 ⋅ ⋅ ⋅ dθN−2 dθN−1

ℝN

0 0

0

0

2π π

π

∞

0 0

0

= ∫ ∫ sin θN−2 ⋅ ⋅ ⋅ ∫ sinN−2 θ1 ( ∫ 2π π

0

π

∞

0

0

= ∫ ∫ sin θN−2 ⋅ ⋅ ⋅ ∫ sinN−2 θ1 ( ∫ 0 0

|̃z(T(r), θ)|N dr) dθ1 ⋅ ⋅ ⋅ dθN−2 dθN−1 (T ′ (r)T N−1 (r))−1 |z(r, θ)|N dr)dθ1 ⋅ ⋅ ⋅ dθN−2 dθN−1 . (T ′ (r)T N−1 (r))−1

Meanwhile, note that N + γ N−1 T ′ (r)T N−1 (r) r N T ′ (r) = 󳨐⇒ r N−1 w(r) ≤ T ′ (r)T N−1 (r) ≤ r w(r). w(r) T(r) N So, from

212 � C. Tarsi 2π π

N

π

∫ |v| dy = ∫ ∫ sin θN−2 ⋅ ⋅ ⋅ ∫ sin 0 0

ℝN

N−2

∞

θ1 ( ∫

0

0

r N T ′ (r) w(r)dr)dθ1 ⋅ ⋅ ⋅ dθN−2 dθN−1 |z(r, θ)|−N T(r)

it follows that ∫ |u|N w(|x|)dx ≤ ∫ |v|N dy ≤ ℝN

ℝN

N +γ ∫ |u|N w(|x|)dx. N ℝN

Finally, N

(

N +γ N N ) ‖u‖Nw < ‖v‖N < ‖u‖w . N +γ N

(3.1)

We have hence proved that the map 1,N

N

1,N

N

𝒯 : Ww (ℝ ) → W0 (ℝ )

u 󳨃→ v

is an invertible, continuous, and with a continuous inverse map. Then, as before, 󵄨 󵄨N ∫ expN (α󵄨󵄨󵄨u(x)󵄨󵄨󵄨 N−1 )w(|x|)dx

ℝN

2π

π

0

0

∞

󵄨 󵄨N = ∫ ⋅ ⋅ ⋅ ∫ sinN−2 θ1 ∫ expN (α󵄨󵄨󵄨z(r, θ)󵄨󵄨󵄨 N−1 )w(r)r N−1 dr dθ1 ⋅ ⋅ ⋅ dθN−1 2π

0

π

∞

󵄨 󵄨N ≤ ∫ ⋅ ⋅ ⋅ ∫ sinN−2 θ1 ∫ expN (α󵄨󵄨󵄨̃z(ρ, θ)󵄨󵄨󵄨 N−1 )ρN−1 dρ dθ1 ⋅ ⋅ ⋅ dθN−1 0

0

0

= ∫ expN (α|v|

N N−1

)dx < ∞

ℝN

for any α > 0. The uniform bound (1.7) follows directly from (3.1), given that for any u ∈ Ww1,N (ℝN )

and α ≤ αN (

1/(N−1)

N ) N +γ

there holds N

∫ expN (α(|u|/‖u‖w ) N−1 )w(|x|)dx ℝN

2π

π

= ∫ ⋅ ⋅ ⋅ ∫ sin 0

0

N−2

∞

N 󵄨 󵄨 θ1 ∫ expN (α(󵄨󵄨󵄨z(r, θ)󵄨󵄨󵄨/‖u‖w ) N−1 )w(r)r N−1 dr dθ1 ⋅ ⋅ ⋅ dθN−1

0

Trudinger-type inequalities in ℝN with radial increasing mass-weight 2π

π

= ∫ ⋅ ⋅ ⋅ ∫ sin 0

2π

0

N−2

� 213

∞

N 󵄨 󵄨 θ1 ∫ expN (α(󵄨󵄨󵄨̃z(T(r), θ)󵄨󵄨󵄨/‖u‖w ) N−1 )wr N−1 dr dθ1 ⋅ ⋅ ⋅ dθN−1

0

π

∞

0

0

N 󵄨 󵄨 ≤ ∫ ⋅ ⋅ ⋅ ∫ sinN−2 θ1 ∫ expN (αN (󵄨󵄨󵄨̃z(ρ, θ)󵄨󵄨󵄨/‖v‖) N−1 )ρN−1 dρ dθ1 ⋅ ⋅ ⋅ dθN−1

0

N

= ∫ expN (αN (|v|/‖v‖) N−1 )dx < ∞. ℝN

Bibliography [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

S. Adachi and K. Tanaka, Trudinger type inequalities in ℝN and their best exponents. Proc. Am. Math. Soc. 128 (2000), 2051–2057. D. R. Adams, A sharp inequality of J. Moser for higher order derivatives. Ann. Math. 128 (1988), 385–398. F. S. B. Albuquerque, On a weighted Adachi-Tanaka type Trudinger-Moser inequality in nonradial Sobolev spaces. Z. Anal. Anwend. 40 (2021), 209–216. C. Bucur, D. Cassani and C. Tarsi, Quasilinear logarithmic Choquard equations with exponential growth in ℝN . J. Differ. Equ. 328 (2022), 261–294. D. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in ℝ2 . Commun. Partial Differ. Equ. 17 (1992), 407–435. D. Cassani and C. Tarsi, Schrödinger-Newton equations in dimension two via a Pohozaev-Trudinger log-weighted inequality. Calc. Var. Partial Differ. Equ., 60, 197 (2021). S. Cingolani and T. Weth, On the planar Schrödinger–Poisson system. Ann. Inst. Henri Poincaré C, Anal. Non Linéaire 33 (2016), 169–197. S. Cingolani and T. Weth, Trudinger-Moser-type inequality with logarithmic convolution potentials. J. Lond. Math. Soc. 105 (2022), 1897–1935. M. Dong and G. Lu, Best constants and existence of maximizers for weighted Trudinger-Moser inequalities. Calc. Var. Partial Differ. Equ., 55, 88 (2016). A. Kufner and B. Opic, How to define reasonably weighted Sobolev spaces. Comment. Math. Univ. Carol. 25 (1984), 537–554. M. Ishiwata, M. Nakamura and H. Wadade, On the sharp constant for the weighted Trudinger-Moser type inequality of the scaling invariant form. Ann. Inst. Henri Poincaré C, Anal. Non Linéaire 31 (2014), 297–314. J. Moser, A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20 (1970/71), 1077–1092. V. H. Nguyen, The weighted Moser-Trudinger inequalities of Adimurthi-Druet type in ℝN . Nonlinear Anal., 195, 111723 (2020). V. H. Nguyen and F. Takahashi, On a weighted Trudinger-Moser type inequality on the whole space and related maximizing problem. Differ. Integral Equ. 31 (2018), 785–806. S. I. Pohožaev, The Sobolev embedding in the case pl = n. In: Proc. Tech. Sci. Conf. on Adv. Sci., Research 1964–1965, Mathematics Section, 158–170. Moskov. Ènerget. Inst, Moscow, 1965. B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in ℝ2 . J. Funct. Anal. 219 (2005), 340–367. J. Simon, Banach, Fréchet, Hilbert and Neumann spaces. In: Analysis for PDEs set. Mathematics and Statistics Series, 1. John Wiley & Sons, Inc., Hoboken, NJ, 2017. C. Tarsi, Moser type inequalities on the plane with radial increasing weight. Preprint 2022. N. S. Trudinger, On embeddings into Orlicz spaces and some applications. J. Math. Mech. 17 (1967), 473–483. V. I. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations. Dokl. Akad. Nauk SSSR 138 (1961), 805–808.

Liguang Liu and Jie Xiao

In response to David R. Adams’ October 12, 2001, letter Dedicated to the memory of David R. Adams

Abstract: This note is in response to the David R. Adams October 12, 2001, letter on BMOp and its predual space. Keywords: Adams, Hardy, John–Nirenberg, Lorentz, Sweezy spaces MSC 2020: Primary 42B25, Secondary 46E35

1 The Adams letter A tiny variant of the handwritten October 12, 2001, letter of David R. Adams

Acknowledgement: Liu was supported by NNSF of China No. 11771446; Xiao was supported by NSERC of Canada No. 202979 and MUN’s SBM-Fund #214311. Liguang Liu, School of Mathematics, Renmin University of China, Beijing 100872, P.R. China, e-mail: [email protected] Jie Xiao, Department of Mathematics and Statistics, Memorial University, St. John’s A1C 5S7, NL, Canada, e-mail: [email protected] https://doi.org/10.1515/9783110792720-010

216 � L. Liu and J. Xiao is texted below: Dear Prof. Xiao Suppose we define a space called BMOp (not the same use as in some papers!) as those p

functions u(x), x ∈ ℝn , such that supQ ∫–Q eb|u−uQ | dx < ∞ for some constant b = b(u) independent of Q, Q = cube with sides parallel to coordinate axes. uQ = p

1 ∫ u dx. |Q| Q

Set ‖u‖BMOp = inf{λ : supQ ∫–Q (eλ |u−uQ | − 1) dx ≤ 1}, 1 ≤ p < ∞. (cf. pg. 1057 of my paper with R. Bagby [4]). Note: BMO1 = BMO by John–Nirenberg. Also, BMOp ⊊ BMO for p > 1. The question is: what is the predual of BMOp , 1 < p < ∞? I have forgotten all the reasons why the following is a reasonable guess, it here goes! Let Ep = {f ∈ L1 : Rj f ∈ L(1,p) , j = 1, . . . , n}. Here, L(1,p) is the Lorentz space: 1 < p < ∞. Clearly, L(1,1) = L1 . Also, Rj f is the jth Riesz transform of f : −1

Rj f (x) =

Γ( n+1 ) 2 π

(p. v.) ∫

n+1 2

ℝn

xj − yj

|x − y|n+1

f (y) dy.

The idea is Ep∗ = BMOp (??) Notice that E1∗ = BMO

{

∗ E∞

=L

∞

(since E1 = H 1 );

(since E∞ = L1 ).

One of the reasons I had hoped to get such a result was to have a space closer to L1 than H 1 for which the Calderon–Zygmund (C-Z) singular integrals are bounded operators. Notice that if K is C-Z operator, then K : BMOp → BMOp . If something like this is true, then there are lots of possibilities for future work. First of all, is it true that K : Ep → Ep ? Let me know if you have any ideas along these lines—this is a problem I have put off working on. Best wishes, DRA.

2 The space BMOp Regarding Adams’ aforementioned space BMOp , after a normalization of the seminorm ‖ ⋅ ‖BMOp , we state its precise definition as below. Definition 2.1. Given p ∈ [1, ∞), the space BMOp consists of all locally integrable functions u on ℝn such that sup — ∫ eλ

−p

B⊂ℝn

B

|u(x)−uB |p

dx < ∞,

In response to David R. Adams’ October 12, 2001, letter

� 217

where the supremum is taken over all balls B ⊂ ℝn and ∫–B stands for the integral mean over B. For any u ∈ BMOp , let ‖u‖BMOp = inf{λ : sup — ∫ (eλ

−p

B⊂ℝn

|u(x)−uB |p

− 1) dx ≤ 1}.

B

Sweezy [15, Definition 3] introduced the following space BMO(p), which indeed appears in some form in an unpublished work of S. Janson [6]. Definition 2.2. Given p ∈ [1, ∞), the space BMO(p) consists of all locally integrable functions u on ℝn such that ‖u‖BMO(p) = sup [ 1≤r 0, by the definition of ‖ ⋅ ‖BMOp , we have sup — ∫ (e

B⊂ℝn

|u(x)−uB | p ) BMOp +ϵ

( ‖u‖

− 1) dx ≤ 1.

B

For any B ⊂ ℝn , by utilizing tk , k! k=1 ∞

et − 1 = ∑ we deduce

pk

|u(x) − uB | 1 ) ∫( — k! ‖u‖ BMOp + ϵ k=1 ∞

∑

dx ≤ 1.

B

Given any r ∈ [1, ∞), taking k ∈ ℕ such that p(k − 1) < r ≤ pk and applying the Hölder inequality, we derive r

− p1

1

1

r pk −1 󵄨 󵄨r 󵄨 󵄨pk (— ∫ 󵄨󵄨󵄨u(x) − uB 󵄨󵄨󵄨 dx) ≤ r p (— ∫ 󵄨󵄨󵄨u(x) − uB 󵄨󵄨󵄨 dx)

B

B

218 � L. Liu and J. Xiao 1

≤ (‖u‖BMOp + ϵ)(

k! pk ) . rk

Obviously, 1

1

k! pk 1 p k = 1 󳨐⇒ ( k ) = ( ) ≤ 1. r r When k ≥ 2, there is 1

1

1

1

1

pk pk pk 1 k! 1 p k k k! pk ) ≤( ) ( ) ≤( ) ≤ 2 2p , ( k) ≤( k k p k−1 k−1 r p (k − 1) 1

1 pk ) is decreasing with respect to k. Summarizing where the last step holds because (1+ k−1 all, we get

1

1

rp

1

r 1 󵄨 󵄨r (— ∫ 󵄨󵄨󵄨u(x) − uB 󵄨󵄨󵄨 dx) ≤ 2 2p (‖u‖BMOp + ϵ).

B

So, letting ϵ → 0 and taking supremum over all r ∈ [1, ∞) derive 1

‖u‖BMO(p) ≤ 2 2p ‖u‖BMOp . Next, suppose that u ∈ BMO(p) with seminorm 1. So, for any r ∈ [1, ∞) and any ball B ⊂ ℝn , there is 1

r 1 1 󵄨 󵄨r (— ∫ 󵄨󵄨󵄨u(x) − uB 󵄨󵄨󵄨 dx) ≤ r p ‖u‖BMO(p) = r p .

B

Consequently, for any B ⊂ ℝn , r ∈ [1, ∞), and α ∈ (0, ∞), r 󵄨󵄨 󵄨 󵄨 󵄨 −r 󵄨󵄨{x ∈ B : 󵄨󵄨󵄨u(x) − uB 󵄨󵄨󵄨 > α}󵄨󵄨󵄨 ≤ |B|α r p .

(2.3)

Under the condition α ≥ 2, taking r = ( α2 )p in (2.3) and using r

r

α−r r p = e−r ln α e p

ln r

αp

= e− 2p

ln α

αp

e 2p

ln( α2 )

ln 2

= e− 2p α

p

we derive from (2.3) that 󵄨󵄨 󵄨 󵄨 󵄨 − ln 2 αp 󵄨󵄨{x ∈ B : 󵄨󵄨󵄨u(x) − uB 󵄨󵄨󵄨 > α}󵄨󵄨󵄨 ≤ |B|e 2p . For λ ∈ (0, ∞), write

(2.4)

In response to David R. Adams’ October 12, 2001, letter

∫ (e —

|u(x)−uB | p ( ) λ

(

|u(x)−uB | p ) λ

− 1) dx = — ∫

B

B

=

� 219

et dt dx

∫ 0 ∞

1 1 󵄨󵄨 󵄨 󵄨 󵄨 ∫ 󵄨󵄨{x ∈ B : 󵄨󵄨󵄨u(x) − uB 󵄨󵄨󵄨 > λt p }󵄨󵄨󵄨et dt. |B|

0

Note that ( λ2 )p

( λ2 )p

0

0

∞

∞

( λ2 )p

( λ2 )p

1 2 p 󵄨 󵄨 󵄨 󵄨 ∫ 󵄨󵄨󵄨{x ∈ B : 󵄨󵄨󵄨u(x) − uB 󵄨󵄨󵄨 > λt p }󵄨󵄨󵄨et dt ≤ |B| ∫ et dt = |B|(e( λ ) − 1)

and, by (2.4), 1 ln 2 p 󵄨 󵄨 󵄨 󵄨 ∫ 󵄨󵄨󵄨{x ∈ B : 󵄨󵄨󵄨u(x) − uB 󵄨󵄨󵄨 > λt p }󵄨󵄨󵄨et dt ≤ |B| ∫ e− 2p λ t et dt

= = − p1

With λ = 2(ln √2) 3

|B| exp(−[( λ2 )p ln 2 − 1]( λ2 )p ) ( λ2 )p ln 2 − 1

|B| exp(( λ2 )p − ln 2) ( λ2 )p ln 2 − 1

.

, we then obtain ∫ (e —

(

|u(x)−uB | p ) λ

− 1) dx ≤ e

( λ2 )p

−1+

exp(( λ2 )p − ln 2) ( λ2 )p ln 2 − 1

B 3

= √2 − 1 + 3

= √2 − 1 + < 1,

2(

√3 2

ln 2 3 ln √2

− 1)

√3 2 4

which implies 3

− p1

‖u‖BMOp ≤ 2(ln √2)

‖u‖BMO(p) .

Remark 2.4. From (2.1) and Proposition 2.3, it is obvious that p 󳨃→ ‖⋅‖BMO(p) is increasing, which implies not only BMOp2 = BMO(p2 ) ⊊ BMO(p1 ) = BMOp1

when p1 < p2

220 � L. Liu and J. Xiao but also the existence of lim ‖u‖BMO(p) .

p→∞

In what follows, X ≲ Y stands for X ≤ cY for a constant c > 0; X ≈ Y represents X ≲ Y ≲ X. Proposition 2.5. For any locally integrable function u on ℝn , there is lim ‖u‖BMO(p) ≈

p→∞

sup

B(x0 ,r0 )⊂ℝn

‖u − uB(x0 ,r0 ) ‖L∞ .

(2.5)

Consequently, ⋂ BMO(p) = L∞ /ℝ = BMO∞ .

p∈[1,∞)

(2.6)

Proof. Equation (2.6) is a direct by-product of (2.5). So, it remains to prove (2.5). Note that lim ‖u‖BMO(p) ≲

p→∞

sup

B(x0 ,r0 )⊂ℝn

‖u − uB(x0 ,r0 ) ‖L∞

follows directly from choosing − p1 󵄩 󵄩

λ = (ln 2)

󵄩 󵄩󵄩u(x) − uB 󵄩󵄩󵄩L∞

to get − p1

‖u‖BMOp ≤ (ln 2)

sup

B(x0 ,r0

)⊂ℝn

󵄩󵄩 󵄩 󵄩󵄩u(x) − uB 󵄩󵄩󵄩L∞ .

Next, we show lim ‖u‖BMO(p) ≳

p→∞

sup

B(x0 ,r0 )⊂ℝn

‖u − uB(x0 ,r0 ) ‖L∞ .

For any B ⊂ ℝn and r ∈ [1, ∞), by the definition of BMO(p), we have 1

r 1 󵄨 󵄨r (— ∫ 󵄨󵄨󵄨u(x) − uB 󵄨󵄨󵄨 dx) ≤ r p ‖u‖BMO(p) .

B

In both sides of the above formula, letting p → ∞ yields 1

r 󵄨 󵄨r (— ∫ 󵄨󵄨󵄨u(x) − uB 󵄨󵄨󵄨 dx) ≤ lim ‖u‖BMO(p) . p→∞

B

(2.7)

In response to David R. Adams’ October 12, 2001, letter

� 221

Further, letting r → ∞ yields (see [11, p. 43, (6)]) 1

r 󵄩 󵄨 󵄨r 󵄩󵄩 ∫ 󵄨󵄨󵄨u(x) − uB 󵄨󵄨󵄨 dx) ≤ lim ‖u‖BMO(p) . 󵄩󵄩u(x) − uB 󵄩󵄩󵄩L∞ (B) = lim (— r→∞ p→∞

(2.8)

B

For an arbitrary ball B(x0 , r0 ), taking R ∈ (0, ∞) large enough such that B(x0 , r0 ) ⊂ B(0, R), we then have |uB(0,R) − uB(x0 ,r0 ) | ≤

󵄨 󵄨 ∫ 󵄨󵄨󵄨f (x) − fB(0,R) 󵄨󵄨󵄨 dx —

B(x0 ,r0 )

1

p 󵄨 󵄨p ≤( — ∫ 󵄨󵄨󵄨f (x) − fB(0,R) 󵄨󵄨󵄨 dx)

B(x0 ,r0 ) n

1

p R p 󵄨 󵄨p ≤( ) ( — ∫ 󵄨󵄨󵄨f (x) − fB(0,R) 󵄨󵄨󵄨 dx) r0 n

B(0,R)

R p 1 ≤ ( ) p p ‖u‖BMO(p) , r0 and hence letting p → ∞ gives |uB(0,R) − uB(x0 ,r0 ) | ≤ lim ‖u‖BMO(p) . p→∞

(2.9)

From (2.8)–(2.9), it follows that ‖u − uB(x0 ,r0 ) ‖L∞ (B(0,R)) ≤ ‖u − uB(0,R) ‖L∞ (B(0,R)) + |uB(0,R) − uB(x0 ,r0 ) | ≤ 2 lim ‖u‖BMO(p) . p→∞

This last estimate implies (2.7) by letting R → ∞.

3 The space Ep It was proved in [15, Theorem 1] that when 1 ≤ p < ∞ the predual space of BMOp is the following Xp , which may be called the Sweezy space. Definition 3.1. Let 1 ≤ p < ∞ and 1 < q ≤ 2. A function a on ℝn is called an (q, p)-atom if: (i) a is supported on a ball B ⊂ ℝn ; (ii) ∫B a(x) dx = 0; 1

− p1

(iii) (∫–B |a(x)|q dx) q ≤ (q′ )

|B|−1 , where q′ is the conjugate exponent of q with q1 + q1′ = 1.

222 � L. Liu and J. Xiao Define the space Xp to be the collection of all functions in L1 that can be decomposed into ∞

f = ∑ λi ai , i=1

(3.1)

where every ai is a (qi , p)-atom for some 1 < qi ≤ 2 and ∑∞ i=1 |λi | < ∞. Define ∞

‖f ‖Xp = inf ∑ |λi |, i=1

where the infimum is taken over all decompositions of f as in (3.1). When pushing p → ∞, we have (cf. [1]) X∞ = L10 = {f ∈ L1 : ∫ f (x) dx = 0}., ℝn

whose dual space is BMO∞ (cf. [3, Theorem 2]). Lemma 3.2. Let q ∈ [1, 2]. For any j ∈ {1, 2, . . . , n}, there is ‖Rj ‖Lq →L(q,∞) ≤ Cn ,

(3.2)

where Cn is a positive constant depending only on n. Proof. Let us apply the Calderón–Zygmund decomposition of a function f ∈ Lq . According to [10, p. 299, Exercise 4.3.8], if f ∈ Lq and α ∈ (0, ∞), then f can be decomposed into ∞

f = g + b = g + ∑ bk , k=1

where g and {bk }∞ k=1 satisfy the following: n

(i) ‖g‖Lq ≤ ‖f ‖Lq and ‖g‖L∞ ≤ 2 q α; (ii) each bk is supported on a cube Qk . Any two cubes Qk and Qℓ have disjoint interiors when k ≠ ℓ; q

(iii) ‖bk ‖Lq ≤ 2n+q αq |Qk |;

(iv) ∫Q bk dx = 0; k

q

−q (v) ∑∞ k=1 |Qk | ≤ α ‖f ‖Lq ;

(vi) ‖b‖Lq ≤ 2

n+q q

q

‖f ‖Lq and ‖b‖L1 ≤ 2α1−q ‖f ‖Lq .

For any cube Qk , denote by cQk and ℓQk its center and side length, respectively. Let Qk∗ = 4√nQk , that is, Qk∗ has center cQk and side-length 4√nℓQk . For α ∈ (0, ∞), write

In response to David R. Adams’ October 12, 2001, letter

� 223

󵄨 󵄨 󵄨󵄨 n 󵄨 󵄨󵄨{x ∈ ℝ : 󵄨󵄨󵄨Rj f (x)󵄨󵄨󵄨 > 2α}󵄨󵄨󵄨

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ∞ ∞ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 ∗ 󵄨󵄨 ∗ 󵄨󵄨󵄨 󵄨󵄨󵄨 n 󵄨󵄨 ≤ 󵄨󵄨{x ∈ ℝ : 󵄨󵄨Rj g(x)󵄨󵄨 > α}󵄨󵄨 + 󵄨󵄨 ⋃ Qk 󵄨󵄨 + 󵄨󵄨{x ∉ ⋃ Qk : 󵄨󵄨Rj b(x)󵄨󵄨 > α}󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 k=1 󵄨 󵄨 k=1 󵄨 󵄨 =: Z1 + Z2 + Z3 .

Recall that ‖Rj ‖L2 (ℝn )→L2 (ℝn ) ≤ 1. With this, we apply (i) and 1 ≤ q ≤ 2 to deduce 󵄨 󵄨2 Z1 ≤ α−2 ∫ 󵄨󵄨󵄨Rj g(x)󵄨󵄨󵄨 dx ≤α

ℝn

−2

󵄨 󵄨2 ∫ 󵄨󵄨󵄨g(x)󵄨󵄨󵄨 dx

ℝn

2−q 󵄨 󵄨q ≤ α ‖g‖L∞ ∫ 󵄨󵄨󵄨g(x)󵄨󵄨󵄨 dx −2

≤ ≤

ℝn n( q2 −1) −q q α ‖f ‖Lq 2 q n −q 2 α ‖f ‖Lq .

Applying (v) yields ∞

∞

k=1

k=1

q 󵄨 󵄨 Z2 ≤ ∑ 󵄨󵄨󵄨Qk∗ 󵄨󵄨󵄨 = (4√n)n ∑ |Qk | ≤ (4√n)n α−q ‖f ‖Lq .

Now, we deal with Z3 . Notice that x ∉ Qk∗ & y ∈ Qk 󳨐⇒ |x − cQk | ≥ 2√nℓQk & |y − cQk | < 2−1 √nℓQk 󳨐⇒ 2−1 |x − cQk | < |x − y| < 2|x − cQk |. So, for any x ∉ Qk∗ , by (ii), (iv), the mean value theorem, and the Hölder inequality, we have n+1 󵄨 󵄨󵄨 xj − (cQk )j xj − zj 󵄨󵄨 󵄨󵄨 󵄨 Γ( 2 ) 󵄨󵄨󵄨 − )b (z) dz ∫( 󵄨󵄨Rj bk (x)󵄨󵄨󵄨 = 󵄨󵄨 k n+1 󵄨󵄨 n+1 n+1 󵄨 󵄨󵄨 |x − z| |x − c | Qk π 2 󵄨Q k

≤

) (n + 2)Γ( n+1 2

≤( ≤(

π

n+1 2

∫ sup Qk

θ∈[0,1]

2n √n(n + 2)Γ( n+1 ) 2 n+1 2

π 2n √n(n + 2)Γ( n+1 ) 2 π

n+1 2

|z − cQk ||bk (z)|

|x − (θcQk + (1 − θ)z)|n+1

dz

)ℓQk |x − cQk |−n−1 ‖bk ‖L1 1− q1

)ℓQk |x − cQk |−n−1 |Qk |

‖bk ‖Lq

224 � L. Liu and J. Xiao

n

≤ (2 q

) + 2)Γ( n+1 2

n +1 2 √n(n

≤ an α(

π

ℓQk

|x − cQk |

n+1 2

)ℓQk |x − cQk |−n−1 |Qk |α

n+1

)

,

where 22n+1 √n(n + 2)Γ( n+1 ) 2

an =

π

n+1 2

.

Denote by νn the Lebesgue measure of the unit ball in ℝn . Then Z3 ≤ α−1

∫ ∗ x∉⋃∞ k=1 Qk

󵄨󵄨 󵄨 󵄨󵄨Rj b(x)󵄨󵄨󵄨 dx

∞

󵄨 󵄨 ≤ α−1 ∑ ∫ 󵄨󵄨󵄨Rj bk (x)󵄨󵄨󵄨 dx k=1 x∉Q∗ k

∞

≤ an ∑ ∫ ( k=1 x∉Q∗ k

n+1

ℓQk

|x − cQk |

∞ ∞

= an νn ∑ ∫ ( k=1 ℓ

Qk

ℓQk ρ

n+1

)

)

dx

ρn−1 dρ

∞

= an νn ∑ |Qk | k=1 −q

q

≤ an νn α ‖f ‖Lq . Combining the estimates of Z1 , Z2 , and Z3 , we arrive at q 󵄨󵄨 󵄨 󵄨 n −q n 󵄨 n 󵄨󵄨{x ∈ ℝ : 󵄨󵄨󵄨Rj f (x)󵄨󵄨󵄨 > 2α}󵄨󵄨󵄨 ≤ (2 + (4√n) + an νn )α ‖f ‖Lq .

Replacing 2α by t yields q 󵄨󵄨 󵄨 󵄨 n q −q n 󵄨 n 󵄨󵄨{x ∈ ℝ : 󵄨󵄨󵄨Rj f (x)󵄨󵄨󵄨 > t}󵄨󵄨󵄨 ≤ (2 + (4√n) + an νn )2 t ‖f ‖Lq .

In other words, 1

‖Rj ‖Lq →L(q,∞) ≤ 2(2n + (4√n)n + an νn ) q ≤ 2(2n + (4√n)n + an νn ), as desired. Lemma 3.3. Let 1 < p < ∞ and 1 < q ≤ 2. If a is a (q, p)-atom supported on a ball B = B(x0 , r), then

In response to David R. Adams’ October 12, 2001, letter

� 225

‖Rj a‖L(1,p) ≤ Cn,p ,

(3.3)

where Cn,p is a positive constant depending only on n and p, but independent of q. Proof. Write ∞

‖Rj a‖L(1,p) = ( ∫ t

1 p

󵄨p 󵄨 󵄨 󵄨󵄨{x ∈ ℝ : 󵄨󵄨󵄨Rj a(x)󵄨󵄨󵄨 > t}󵄨󵄨󵄨 dt) .

p−1 󵄨󵄨

0

n

Let Cn be the constant within (3.2). Then, by Lemma 3.2 and 1 < q ≤ 2, we have ∞

󵄨 󵄨 󵄨 󵄨p ∫ t p−1 󵄨󵄨󵄨{x ∈ ℝn : 󵄨󵄨󵄨Rj a(x)󵄨󵄨󵄨 > t}󵄨󵄨󵄨 dt |B|−1 ∞

≤ ∫ t

p−1

q

(

q

‖Rj ‖Lq →L(q,∞) ‖a‖Lq tq

|B|−1

≤ (Cn ‖a‖Lq )

qp

p

) dt

∞

∫ t −p(q−1)−1 dt |B|−1

≤ (Cn )qp =

(q′ )−q p(q − 1)

(Cn )qp (q − 1)q−1 pqq

≤ (Cn )2p . Note that

x ∉ 2B & ξ ∈ B 󳨐⇒ |x − ξ| ≥ 2−1 |x − x0 |. So, for any x ∉ 2B, by supp a ⊂ B & ∫ a(x) dx = 0, ℝn

the mean value theorem and the Hölder inequality, we have n+1 󵄨 󵄨󵄨 xj − (x0 )j xj − zj 󵄨󵄨 󵄨 Γ( 2 ) 󵄨󵄨󵄨 󵄨󵄨 − )a(z) dz ∫( 󵄨󵄨Rj a(x)󵄨󵄨󵄨 = 󵄨󵄨 n+1 󵄨󵄨 n+1 󵄨󵄨 |x − x0 |n+1 π 2 󵄨󵄨B |x − z|

≤

(n + 2)Γ( n+1 ) 2 π

n+1 2

󵄨 󵄨−n−1 󵄨󵄨 󵄨 ∫ |z − x0 | sup 󵄨󵄨󵄨x − (θx0 + (1 − θ)z)󵄨󵄨󵄨 󵄨󵄨a(z)󵄨󵄨󵄨 dz B

θ∈[0,1]

226 � L. Liu and J. Xiao ) 2n+1 (n + 2)Γ( n+1 2

)r|x − x0 |−n−1 ‖a‖L1 n+1 π 2 = bn r|x − x0 |−n−1 ‖a‖L1 ,

≤(

where bn =

) 2n+1 (n + 2)Γ( n+1 2 π

n+1 2

.

Since the definition of the (q, p)-atom implies ‖a‖L1 ≤ |B|

1− q1

1

‖a‖Lq

q−1 p ≤( ) ≤ 1, q

it follows that n

b r n+1 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 −n−1 > t}󵄨󵄨󵄨 ≤ νn ( n ) . 󵄨󵄨{x ∉ 2B : 󵄨󵄨󵄨Rj a(x)󵄨󵄨󵄨 > t}󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨{x ∉ 2B : bn r|x − x0 | t Consequently, |B|−1

(∫ t

󵄨 󵄨 󵄨p 󵄨󵄨{x ∈ ℝ : 󵄨󵄨󵄨Rj a(x)󵄨󵄨󵄨 > t}󵄨󵄨󵄨 dt)

p−1 󵄨󵄨

0

|B|−1

n

≤( ∫ t

p−1

1 p

p

|B|−1

|2B| dt) + ( ∫ t

0

0

n

1 p

n

1 p

1 p

󵄨 󵄨 󵄨p 󵄨󵄨{x ∉ 2B : 󵄨󵄨󵄨Rj a(x)󵄨󵄨󵄨 > t}󵄨󵄨󵄨 dt)

p−1 󵄨󵄨

|B|−1

p n 2 ≤ ( ) + νn (bn r) n+1 ( ∫ t n+1 −1 dt) p

=(

1 p

0

n 2 ) + (νn bn ) n+1 . p

Altogether, we get |B|−1

󵄨p

󵄨 󵄨 󵄨 ‖Rj a‖L(1,p) ≤ ( ∫ t p−1 󵄨󵄨󵄨{x ∈ ℝn : 󵄨󵄨󵄨Rj a(x)󵄨󵄨󵄨 > t}󵄨󵄨󵄨 dt)

1 p

0

∞

+ (∫ t

󵄨 󵄨 󵄨p 󵄨󵄨{x ∈ ℝ : 󵄨󵄨󵄨Rj a(x)󵄨󵄨󵄨 > t}󵄨󵄨󵄨 dt)

p−1 󵄨󵄨

0

≤ (Cn )2 + (

n

1

n 2n p ) + (νn bn ) n+1 . p

1 p

1 p

In response to David R. Adams’ October 12, 2001, letter

� 227

Given p ∈ [1, ∞], a distribution f on ℝn and a nonnegative radial decreasing smooth function ϕ ∈ C ∞ with supp ϕ ∈ B(0, 1); { { { { { {∫ℝn ϕ(x) dx = 1; { { {ϕt (x) = t −n ϕ(t −1 x) ∀(x, t) ∈ ℝn × (0, ∞); { { { ⋆ {f = supt∈(0,∞) |f ∗ ϕt |, let the Hardy–Lorentz space H 1,p be the collection of all distributions f obeying (cf. [2, 9]) 󵄩 󵄩 ‖f ‖H (1,p) = 󵄩󵄩󵄩f ⋆ 󵄩󵄩󵄩L(1,p) < ∞. Of course, H (1,1) is the classic Hardy space H 1 . Using the atomic decompositions of the Hardy space H 1 , we observe that every (q, p)-atom a within Definition 3.1 belongs to the Hardy space H 1 . However, the H 1 -norm of an (q, p)-atom a may depend on q. Proposition 3.4. The questions on the Adams space Ep stated in Section 1 are answered below: (i) If 1 ≤ p1 ≤ p2 ≤ ∞, then H 1 ⊆ Ep1 ⊆ Ep2 ⊆ L1 . (ii) Xp ⊆ Ep ⊆ H (1,p) ; { (1,p) ∗ [H ] ⊆ Ep∗ ⊆ BMOp , with “⊆” becoming “=” as p = 1. So, the Calderón-Zygmund operator (https:// encyclopediaofmath.org/ wiki/ Calderón-Zygmund_operator) K : Ep → Ep is valid for p = 1 but not for p = ∞. Proof. (i) This follows immediately from the fact that [1, ∞] ∋ p 󳨃→ L(1,p) is increasing (cf. [10, p. 49, Proposition 1.4.10]). (ii) For p ∈ (1, ∞), suppose that f ∈ Xp is not equal to zero. Then we may write ∞

f = ∑ λi ai , i=1

where every ai is a (qi , p)-atom for some 1 < qi ≤ 2, supp ai ⊂ Bi , and ∞

∑ |λi | ≤ 2‖f ‖Xp . i=1

228 � L. Liu and J. Xiao By the Hölder inequality, we have 1

qi −1 󵄨 󵄨q ‖ai ‖L1 ≤ |Bi |(— ∫ 󵄨󵄨󵄨ai (x)󵄨󵄨󵄨 i dx) ≤ (qi′ ) p ≤ 1,

Bi

thereby leading to ∞

∞

i=1

i=1

‖f ‖L1 ≤ ∑ |λi |‖ai ‖L1 ≤ ∑ |λi | ≤ 2‖f ‖Xp . By (3.3) in Lemma 3.3, we get ∞

‖Rk f ‖L(1,p) ≲ ∑ |λj |‖Rk aj ‖L(1,p) ≲ ∑ |λj | ≲ ‖f ‖Xp , j=1

j=1

thereby finding f ∈ Ep , which ensures Xp ⊆ Ep . Next, if f ∈ Ep , then f ∈ L1 ⊆ L(1,p) & Rj f ∈ L(1,p) . According to [18, Theorem 1.2], we have n

n

j=1

j=1

󵄩 󵄩 ‖f ∗ ϕt ‖L(1,p) + ∑󵄩󵄩󵄩(Rj f ) ∗ ϕt 󵄩󵄩󵄩L(1,p) ≲ ‖f ‖L(1,p) + ∑ ‖Rj f ‖L(1,p) < ∞ whence giving f ∈ H (1,p) & Ep ⊆ H (1,p) . Consequently, since 1 ≤ p1 ≤ p2 ≤ ∞ 󳨐⇒ H (1,p1 ) ⊆ H (1,p2 ) , the inclusion Xp ⊆ Ep ⊆ H (1,p) improves [3, Proposition 2.2] with X1 = E1 = H (1,1) = H 1 ; { X∞ ⊊ E∞ ⊊ H (1,∞) . Now, it is automatical to obtain

∀t ∈ (0, ∞),

In response to David R. Adams’ October 12, 2001, letter

� 229

BMOp ⊇ Ep∗ ⊇ [H (1,p) ] ; ∗

see also [17, Theorem 4.6] for a BMOp -type characterization of [H (1,p) ]∗ . Especially, we have [X1 ]∗ = [H 1 ]∗ = E1∗ = BMO;

{

∗ ∗ [H (1,∞) ]∗ ⊊ E∞ = [L1 ]∗ = L∞ ⊊ X∞ .

Remark 3.5. Two comments are in order. (i) On the one hand, it is worth mentioning that 1 1 {f ∈ L 󳨐⇒ d(f , H ) = infg∈H 1 ‖f − g‖L1 = | ∫ℝn f (x) dx|; { 1 (1,p) & 1 < p < ∞ 󳨐⇒ d(f , H 1 ) = ∫ℝn f (x) dx = 0. {f ∈ L ∩ H

As a matter of fact, the first implication is due to [3, Theorem 5]. To verify the second implication (indicating that {Xp , Ep , L1 ∩ H (1,p) } between H 1 & L1 are sufficiently close), note that any function f ∈ H (1,p) can be written as (cf. [2, Theorem 2.1]) f = ∑ λj,k aj,k j,k

in L(1,p)

p 1 with aj,k being an H 1 atom and ∑∞ k=1 (∑j |λj,k |) < ∞. So, if f ∈ L , then a slight modification of the argument presented in [14, pp. 101–102] derives 1 {f = ∑j,k λj,k aj,k in L ; { {∫ℝn f (x) dx = ∑j,k λj,k ∫ℝn aj,k (x) dx = 0.

(ii) On the other hand, the kernels of the Riesz transforms R1 & R2 on the plane ℝ2 :

are applicable to the image processing (cf. [13, p. 42]).

230 � L. Liu and J. Xiao

4 The trace of Ep Thanks to a failure of the boundedness of Rj on L(1,p) , we are led to a consideration of the geometric nature of a given nonnegative Radon measure μ obeying ‖Rj f ‖L(1,p) ≲ ‖Rj f ‖L(1,p) ≴ ‖f ‖L(1,p) < ∞ ∀j ∈ {1, . . . , n}, μ

where q

∞

‖g‖L(p,q) μ

1

{(p ∫ [(t p μ({x ∈ ℝn : |g(x)| > t})] p dtt ) q ={ 0 1 n p {supt∈(0,∞) t[μ({x ∈ ℝ : |g(x)| > t})]

as q ∈ (0, ∞); as q = ∞.

Meanwhile, it is perhaps appropriate to mention three basic facts below. (i) From [8, p. 749] and μ being chosen as the n-dimensional Lebesgue measure on ℝn , it follows that: n

‖Rf ‖L(p,q) = ∑ ‖Rj f ‖L(p,q) ≈ ‖f ‖L(p,q) j=1

∀(p, q) ∈ (1, ∞) × (0, ∞].

(ii) From [7, Theorem 1.1], it can been seen that if 1 ≤ p0 < p < ∞; { { { 0 < q < ∞; { { { (p0 ,∞) ∩ BMO, {f ∈ L then there holds the interpolation inequality p0

1−

p0

‖f ‖L(p,q) ≲ ‖f ‖Lp(p0 ,∞) ‖f ‖BMOp . (iii) If ℝn ∋ x 󳨃→ Rj μ(x) =

Γ( n+1 ) 2 π

n+1 2

(p. v.) ∫ ℝn

xj − yj

|x − y|n+1

dμ(y)

is in Lp with p ∈ (1, ∞), then upon choosing any nonnegative function ϕ ∈ C0∞ such that its support is contained in a given ball B(x, 2r) with ‖ϕ‖L∞ ≲ 1 = ϕ|B(x,r) , we use the identity n

I = − ∑ R2j = −R ⋅ R j=1

In response to David R. Adams’ October 12, 2001, letter

� 231

p

and the Hölder inequality as well as the boundedness of Rj on L p−1 to derive (cf. [12, Lemma 4.1]) μ(B(x, r)) ≤ ∫ ϕ dμ ℝn

n

= − ∑ ∫ R2j ϕ dμ n

j=1 ℝn

= ∑ ∫ Rj ϕ(y)Rj μ(y) dy j=1 ℝn n

≲ ∑ ‖Rj ϕ‖

p

L p−1

j=1

‖Rj μ‖Lp

n

p L p−1

≲ ‖ϕ‖ ≲r

(p−1)n p

∑ ‖Rj μ‖Lp j=1

.

However, the above argument breaks down at p = ∞. Accordingly, we discover the following trace principle for Ep in order to resolve the foregoing geometric problem. Proposition 4.1. Let (a, p) ∈ (0, n) × [1, ∞]; { { { { { Γ( n−a ) { f (y) { Ia f (x) = a n 2 a ∫ℝn |x−y| be the a-Riesz potential; { n−a dy { { 2 π 2 Γ( 2 ) { Γ( n−a ) { f (y)(x−y) { Ra f (x) = a n 2 a ∫ℝn |x−y| be the vector-valued a-Riesz potential; { n+1−a dy { { 2 Γ( ) 2 π { 2 { { { μ(B(x,r)) n {μ be a nonnegative Radon measure on ℝ with |||μ|||n = supB(x,r)⊂ℝn |B(x,r)| . Then the following statements are equivalent: n

(i) Ia : Ep → Lμn−a (

,∞)

is bounded; n

n

(ii) Ra : E1 = H 1 → [Lμn−a ]n = [Lμn−a (iii) I = −R ⋅ R : L

(1,p)

→

(

(1,p) Lμ

n , n−a ) n

] is bounded;

is bounded, i. e., ‖f ‖L(1,p) ≲ ‖f ‖L(1,p) < ∞; μ

(iv) ‖Rf ‖[L(1,p) ]n ≲ ‖Rf ‖[L(1,p) ]n < ∞; μ

(v) |||μ|||n < ∞.

Proof. (i)⇐⇒(v) If (i) holds, then ‖Ia f ‖

( n ,∞) Lμ n−a

n

≲ ‖f ‖Ep = ‖f ‖L1 + ∑ ‖Rj f ‖L(1,p) j=1

∀f ∈ Ep ,

232 � L. Liu and J. Xiao and hence the inclusion pair L1 ⊆ L1,p & H 1 ⊆ Ep ensures ‖Ia f ‖

( n ,∞)

Lμ n−a

≲ ‖f ‖H 1

∀f ∈ H 1 .

(4.1)

Upon choosing f† = |B(x− , r)|−1 1B(x− ,r) − |B(x+ , r)|−1 1B(x+ ,r) ; { { { B(x− , r) ∩ B(x+ , r) = 0; { { { n {1B(x± ,r) = the indicator of the ball B(x± , r) ⊂ ℝ , we find {‖f† ‖H 1 ≈ 1; n−a −n {‖I f ‖ n n , a † ( n−a ,∞) ≳ (r μ(B(x+ , r))) Lμ { thereby using (4.1) for f† to reveal |||μ|||n < ∞. Conversely, if |||μ|||n < ∞, then for any Borel set S ⊆ ℝn and any sequence of balls ∞

{B(xj , rj )}j=1

satisfying S ⊆ ⋃ B(xj , rj ),

∞

j=1

we have ∞

∞

j=1

j=1

󵄨 󵄨 μ(S) ≤ ∑ μ(B(xj , rj )) ≤ |||μ|||n ∑󵄨󵄨󵄨B(xj , rj )󵄨󵄨󵄨, and hence taking the infimum over all possible ball coverings of S gives μ(S) ≤ |||μ|||n

inf ∞

S⊆⋃j=1

∞

󵄨 󵄨 ∑󵄨󵄨󵄨B(xj , rj )󵄨󵄨󵄨 = |||μ|||n |S|. B(x ,r ) j

j

j=1

This last inequality, plus Ep ⊆ L1 , yields ‖Ia f ‖

( n ,∞)

Lμ n−a

n ,∞) ≲ ‖f ‖ 1 ≲ ‖f ‖ , ≲ ‖Ia f ‖L( n−a Ep L

as desired in (i). (ii)⇐⇒(v) Suppose that (ii) is valid. Then ‖Ra f ‖

n

[Lμn−a ]n

≲ ‖f ‖H 1

∀f ∈ H 1 .

(4.2)

In response to David R. Adams’ October 12, 2001, letter

� 233

Note that Ra f = (1 + a − n)−1 Ia Rf ;

{

(4.3)

‖Rf ‖[H 1 ]n = ∑nj=1 ‖Rj f ‖H 1 ≈ ‖f ‖H 1 .

So, (4.3) is put into (4.2) to derive ‖Ia f ‖

≲ ‖Ia f ‖

( n ,∞)

Lμ n−a

n

Lμn−a

≲ ‖f ‖H 1

∀f ∈ H 1 ,

which implies |||μ|||n < ∞ via (4.1). Conversely, if |||μ|||n < ∞, then using (4.3) and [5, Theorem 3.12] produces ‖Ra f ‖

n

[Lμn−a ]n

≈ ‖Ia Rf ‖

n

[Lμn−a ]n

≲ ‖Rf ‖[H 1 ]n ≈ ‖f ‖H 1 .

In other words, (ii) holds. As an immediate by-product of (v)󳨐⇒(ii), we get that if R∗a stands for the adjoint operator of Ra then n

n

n ∗

n

|||μ|||n < ∞ 󳨐⇒ R∗a : ([Lμn−a ] ) = [Lμa ] → BMO = [H 1 ]

∗

being bounded.

(iii)⇐⇒(v) This follows from Lμ ’s definition and the indicator of a Borel set. (iv)⇐⇒(v) On the one hand, if |||μ|||n < ∞, then (1,p)

μ(S) ≲ |S| ∀ Borel set S ⊆ ℝn , and hence n

n

‖Rf ‖[L(1,p) ]n = ∑ ‖Rj f ‖L(1,p) ≲ |||μ|||n ∑ ‖Rj f ‖L(1,p) = |||μ|||n ‖Rf ‖[L(1,p) ]n . μ

μ

j=1

j=1

So, (iv) is valid. On the other hand, suppose that (iv) holds. Given any Borel set S ⊆ ℝn and its indicator 1S , let f‡ = − ∑nj=1 Rj 1S .;

{

(4.4)

Rf‡ = (1S , . . . , 1S ).

The second formula of (4.4) is placed in (iv) to induce n

n

‖Rf‡ ‖[L(1,p) ]n = ∑ ‖1S ‖L(1,p) ≲ ‖Rf‡ ‖[L(1,p) ]n = ∑ ‖1S ‖L(1,p) , μ

whence

j=1

μ

j=1

234 � L. Liu and J. Xiao {μ(S) ≈ ‖1S ‖L(1,p) ≲ ‖1S ‖L(1,p) ≈ |S|; μ { |||μ||| < ∞. n {

Bibliography [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

[14]

[15] [16] [17] [18]

W. Abut-Shammala and A. Torchinsky, The atomic decomposition in L1 (ℝn ). Proc. Am. Math. Soc. 135 (2007), 2839–2843. W. Abut-Shammala and A. Torchinsky, The Hardy-Lorentz spaces Hp,q (ℝn ). Stud. Math. 182 (2007), 283–294. W. Abut-Shammala and A. Torchinsky, Spaces between H 1 and L1 . Proc. Am. Math. Soc. 136 (2008), 1743–1748. D. R. Adams and R. J. Badby, Translation-dilation invariant estimates for Riesz potentials. Indiana Univ. Math. J. 23 (1974), 1051–1067. A. Bonami and R. Johnson, Tent spaces based on the Lorentz spaces. Math. Nachr. 132 (1987), 81–99. S-Y. A. Chang, J. M. Wilson and T. Wolff, Some Weighted norm inequalities concerning the Schrödinger Operators. Comment. Math. Helv. 60 (1985), 217–246. N. A. Dao, N. T. N. Hanh, T. M. Hieu and H. B. Nguyen, Interpolation inequalities between Lorentz space and BMO: the endpoint case (L1,∞ , BMO). Electron. J. Differ. Equ., 2019, 56 (2019). D. E. Edmunds and B. Opic, Equivalent quasi-norms on Lorentz spaces. Proc. Am. Math. Soc. 131 (2002), 745–754. R. Fefferman and F. Soria, The space Weak H1 . Stud. Math. 85 (1987), 1–16. L. Grafakos, Classical and Modern Fourier Analysis. Pearson Education, Inc., 2004. E. H. Lieb and M. Loss, Analysis, 2nd ed. Graduate Studies in Mathematics, 14. Amer. Math. Soc., 2001. J. Mateu, L. Prat and J. Verdera, The capacity associated to signed Riesz kernels, and Wolff potentials. J. Reine Angew. Math. 578 (2005), 201–223. M. Reinhardt, Applications of Riesz transforms and monogenic wavelet frames in imaging and image processing. Thesis of Technische Universität Bergakademie Freiberg (2019). https://d-nb.info/ 1226101542/34. E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. In: Monographs in Harmonic Analysis, III. Princeton Mathematical Series, 43. Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy, xiv+695 pp. G. Sweezy, Subspaces of L1 (ℝd ). Proc. Am. Math. Soc. 132 (2004), 3599–3606. G. Sweezy, The maximal function on spaces that lie between L∞ and BMO. In: Proceedings of the 10th WSEAS International Conferences on Applied Mathematics. Dallas, Texas, USA, Nov. 1–3, 2006. W. Wang and A. Wang, The dual spaces of variable anisotropic Hardy-Lorentz spaces and continuity of a class of linear operators. Turk. J. Math. 46 (2022), 2466–2484. L. Wu, D. Zhou, C. Zhuo and Y. Jiao, Riesz transform characterizations of variable Hardy-Lorentz spaces. Rev. Mat. Complut. 31 (2018), 747–780.

Augusto C. Ponce and Daniel Spector

Some remarks on capacitary integrals and measure theory Dedicated to the memory of David R. Adams

Abstract: We present results for Choquet integrals with minimal assumptions on the monotone set function through which they are defined. They include the equivalence of sublinearity and strong subadditivity independent of regularity assumptions on the capacity, as well as various forms of standard measure theoretic convergence theorems for these nonadditive integrals, e. g., Fatou’s lemma and Lebesgue’s dominated convergence theorem. Keywords: Monotonicty, strong or finite or countable subadditivity, evaescence, semifinite, locally finite, inner or outer or zero-capacity regularity MSC 2020: Primary 31C15, Secondary 28C99, 31C45, 35J85, 46E35, 46N20

1 Introduction The oeuvre of David R. Adams has had a profound influence on the study of Sobolev inequalities, including results early in his career on trace inequalities [1–3], numerous papers over the years concerning potentials [4, 5, 11, 14, 17–19], and of special interest in this paper, his body of work on capacities and Choquet integrals [6–10, 12, 13, 16, 20–23]. That one should be interested in the study of Choquet integration is clear from the consideration of strong forms of the Sobolev inequality, namely V. Maz’ya’s capacitary inequalities [39–41] and their various extensions [6, 10, 31, 43, 45, 46], as it is precisely in these improvements to typical Lebesgue or Lorentz inequalities that these integrals make an appearance. These inequalities give usual compactness results, though are strong enough even to provide information about fine properties of functions and, therefore, motivate the need for as robust as possible of a framework of Choquet integration, which contains Acknowledgement: D. Spector is supported by the National Science and Technology Council of Taiwan under research grant no. 110-2115-M-003-020-MY3 and the Taiwan Ministry of Education under the Yushan Fellow Program. Part of this work was undertaken while D. Spector was visiting IRMP Institute of the Université catholique de Louvain. He would like to thank the IRMP Institute for its support and A. C. Ponce for his warm hospitality during the visit. Augusto C. Ponce, Institut de recherche en mathématique et physique, Chemin du cyclotron 2, L7.01.02, Université catholique de Louvain, Louvain-la-Neuve 1348, Belgium, e-mail: [email protected] Daniel Spector, Department of Mathematics, National Taiwan Normal University, No. 88, Section 4, Tingzhou Road, Wenshan District, Taipei City 116, Taiwan, ROC, e-mail: [email protected] https://doi.org/10.1515/9783110792720-011

236 � A. C. Ponce and D. Spector these main capacitary inequalities as examples. The work we reference of D. R. Adams provides a number of results in this direction, most notably his survey [12]. The starting place of D. R. Adams is the treatise of G. Choquet [29], who developed a theory of integration with respect to monotone, countably subadditive set functions with additional regularity assumptions: We say that H : 𝒫 (ℝd ) → [0, ∞], defined on the class 𝒫 (ℝd ) of all subsets of ℝd , is a capacity in the sense of Choquet whenever it satisfies the conditions empty set: H(0) = 0; monotonicity: If E ⊂ F ⊂ ℝd , then H(E) ≤ H(F); countable subadditivity: For every sequence of sets En ⊂ ℝd , ∞

∞

n=0

n=0

H( ⋃ En ) ≤ ∑ H(En ); outer regularity: For every nonincreasing sequence of compact subsets Kn ⊂ ℝd , ∞

H( ⋂ Kn ) = lim H(Kn ); n→∞

n=0

inner regularity: For every nondecreasing sequence of sets En ⊂ ℝd , ∞

H( ⋃ En ) = lim H(En ). n→∞

n=0

Given a set function H : 𝒫 (ℝd ) → [0, ∞] that merely satisfies monotonicity, one can define the Choquet integral with respect to H of any function f : ℝd → [0, ∞] as ∞

∫ f dH := ∫ H({f > t}) dt,

(1.1)

0

where the right-hand side is understood as the Lebesgue integral of the non-increasing function t ∈ (0, ∞) 󳨃󳨀→ H({f > t}). Such an integral has a number of desirable properties, for example, (1.1) is positively 1-homogeneous and monotone. Moreover, one may replace the sets {f > t} with {f ≥ t} and obtain the same value for the integral. However, one consequence of the choice to integrate outside the framework of measure theory is that this integral need not be linear, and in fact may not even be sublinear. Indeed, G. Choquet [29, 54.2 on p. 289] established a necessary and sufficient condition on H that the integral be sublinear.

Some remarks on capacitary integrals and measure theory

� 237

Theorem 1.1. Let H be a capacity in the sense of Choquet. Then the Choquet integral (1.1) is sublinear if and only if H is strongly subadditive. Here, we recall the notion of strong subadditivity: For every sets E, F ⊂ ℝd , H(E ∩ F) + H(E ∪ F) ≤ H(E) + H(F). The sublinearity of the integral implies one has a triangle inequality, from which Hölder’s and Minkowski’s inequalities follow from usual convexity arguments. These inequalities in turn serve as a basis for the study of a family of Banach spaces of functions Lp (H), those suitably regular functions whose pth power has a finite Choquet integral. Here, typical questions have concerned the boundedness of maximal functions [10, 47, 49], characterizations of the topological duals [10, 48], and interpolation theory [25–27]. The assumption one has a capacity in the sense of Choquet ensures that even without the full strength of results from measure theory one has a number of useful tools, e. g., Fatou’s lemma (which follows from monotonicity and inner regularity, see [27, Theorem 1 on pp. 98–99]): ∫ lim inf fn dH ≤ lim inf ∫ fn dH, n→∞

n→∞

(1.2)

for every sequence of functions fn : ℝd → [0, ∞]. As a result, the Choquet integral built on a strongly subadditive capacity in the sense of Choquet enjoys countable sublinearity: For every sequence of functions fn : ℝd → [0, ∞], one has ∞

∞

n=0

n=0

∫ ∑ fn dH ≤ ∑ ∫ fn dH,

(1.3)

and a Fatou-type lemma that is often appealed to (see, e. g., the argument on p. 123 of [10]): If fn → f locally in L1 (ℝd ) (or pointwise almost everywhere), then ∫ ℳf dH ≤ lim inf ∫ ℳfn dH, n→∞

where ℳf (x) := sup r>0

1 󵄨 󵄨 ∫ 󵄨󵄨󵄨 f (y)󵄨󵄨󵄨 dy rd Br (x)

is the Hardy–Littlewood maximal function. These results are a small sample of the theory of Choquet integration developed and recorded for capacities in the sense of Choquet, and we refer the reader to [10, 27, 29] for further details. Unfortunately, in practice the capacities that arise in various inequalities [6, 10, 31, 39–41, 43] may fail to satisfy inner regularity or outer regularity, a notable example

238 � A. C. Ponce and D. Spector being the Hausdorff content or its dyadic version; see [34] and Example 1.4 below. It is natural then that one address the necessity of these regularity assumptions in the resulting theory of Choquet integration. This program was initiated by Adams, who typically did not require that the set functions under consideration be capacities in the sense of Choquet, often with the initial assumptions of only empty set, monotonicity, and countable subadditivity. He referred to these objects as capacities in the sense of N. Meyers, though the reader may also recognize these are the defining properties of an outer measure. To these, he then added a continuity assumption, outer regularity or inner regularity, either of which is sufficient to obtain Choquet’s characterization of sublinearity of the integral (see Anger’s paper [24] for the proof assuming outer regularity or Saito, Tanaka, and Watanabe’s paper [49, Proposition 3.2] for the proof assuming inner regularity). The first observation of this paper is that neither assumption is necessary, that one has the following characterization independent of regularity assumptions. Theorem 1.2. Suppose that H satisfies monotonicity. Then the Choquet integral (1.1) is sublinear if and only if H is strongly subadditive. Here is a simple example for which Theorem 1.2 applies but not Theorem 1.1. Example 1.3. Let H be the set function defined by 0

if A is finite,

1

if A is infinite.

H(A) := {

Then H is strongly subadditive and satisfies the empty set and monotonicity, but does not satisfy countable subadditivity, outer regularity, or inner regularity. A second pertinent example is the dyadic Hausdorff content for which Yang and Yuan [54] observed the following. β Example 1.4. Let 0 < β < d and let H = ℋ̃ ∞ be the set function defined by β

∞

β

∞

ℋ̃ ∞ (E) := inf{ ∑ ℓ(Qn ) : Qn is a dyadic cube and E ⊂ ⋃ Qn }, n=0

n=0

(1.4)

where ℓ(Qn ) denotes the side-length of Qn . Then H is strongly subadditive and satisfies empty set, monotonicity, countable subadditivity, and inner regularity, but not outer regularity for β ≤ d − 1. The initial impetus for this work was the question of the validity of Theorem 1.2, though after obtaining a proof we discovered in our broader literature review that this was known to the community of nonadditive measure theory [32, Chapter 6]. As it seems to have not been referenced in the results after Adams, we give the proof below for the convenience of the reader, the idea of which is as follows: First, one proves an algebraic result, which amounts to sublinearity for finite sums of characteristic functions (see,

Some remarks on capacitary integrals and measure theory

� 239

e. g., the argument at the top of p. 249 of [24], the argument on pp. 766–768 of [49], or Proposition 4.2 below); Second, one argues the general case by approximation. In the papers [24, 49], this is performed invoking either outer regularity or inner regularity to justify the limit, though as we show below it can be done by using only monotonicity and properties of the Lebesgue integral in (1.1). That the Choquet integral is sublinear without any regularity assumptions is perhaps surprising to the community working on capacitiary inequalities in the spirit of Adams, e. g. [28, 30, 33, 35–38, 42, 43, 45–53], and suggests that it should be interesting to understand what other aspects of the theory of Choquet integration relies on these regularity assumptions and in what areas it can be dispensed. In this paper, we take up this question as pertains to analogues of measure theoretic results, in particular, Fatou’s lemma and Lebesgue’s dominated convergence theorem, as well as functional analysis results concerning spaces of functions with finite capacitary integral. Our results show that while one does not need regularity of the capacity, in one way or another, regularity must make an appearance in order to obtain such results. In particular, if one assumes regularity on any of the capacity, the mode of convergence, or the functions involved, then it is possible to obtain analogues of (1.2) and (1.3); see Figure 1.

Figure 1: Trinity of assumptions.

The plan of the paper is to make precise the sketch presented in Figure 1, as well as to meticulously develop the assumptions that lead to various results of the measure theoretic and functional analytic aspects of Choquet integration. Our results are roughly organized in terms of increasing assumptions on the capacity as one proceeds through the paper. With this framework, in Section 2, we rely on a strong form of pointwise convergence with respect to H, namely quasiuniform convergence, which is sufficient to obtain versions of Fatou’s lemma and Lebesgue’s dominated convergence theorem for Choquet integrals with minimal assumptions on the capacity. Conversely, we show that convergence of a sequence of functions with respect to the Choquet integral implies that the sequence has this strong convergence property, the idea of which follows the typical completeness argument for the functional space L1 (H). In Section 3, we recall the notion of quasicontinuity. When one imposes certain additional conditions on H, we show that

240 � A. C. Ponce and D. Spector quasicontinuous functions admit approximation with respect to the Choquet integral by functions in Cc (ℝd ), the class of continuous functions that are compactly supported in ℝd . In Section 4, we prove Theorem 1.2 on the sublinearity of the Choquet integral. In Section 5, we show how one can relax the notion of convergence provided one works within the class of quasicontinuous functions. This relies on a classical application of the Hahn–Banach theorem that realizes the Choquet integral as a supremum over Lebesgue integrals with respect to locally finite measures underneath the capacity. In Section 6, we introduce the Banach space L1 (H) as the set of equivalence classes of quasicontinuous functions for which the Choquet integral is finite. The results in the preceding sections are then shown to imply standard results concerning this space such as completeness, density of Cc (ℝd ), after which we discuss Fatou’s lemma in this context, namely we provide closure properties that guarantee that the limit of a sequence of functions remains in L1 (H). For ease of reference, we provide proofs of all of our assertions. This paper is an homage to David R. Adams’ work, for which we are extremely grateful, and is dedicated to his memory.

2 Quasiuniform convergence In this section, we present some convergence properties for the Choquet integral with respect to a monotone set function H : 𝒫 (ℝd ) → [0, ∞] under quasiuniform convergence of the integrand. Definition 2.1. Let (fn )n∈ℕ be a sequence of functions fn : ℝd → [−∞, ∞]. We say that (fn )n∈ℕ converges quasiuniformly to f : ℝd → [−∞, ∞] whenever, for each ϵ > 0, there exists E ⊂ ℝd such that H(E) ≤ ϵ and (fn )n∈ℕ is finite and converges uniformly to f in ℝd \ E. We denote fn → f

q. u.

For simplicity, we omit the dependence of H. In the case where H is a measure, almost everywhere pointwise convergence implies quasiuniform convergence on sets of finite measure. However, this is not true for capacities in general. Example 2.2. Take a capacity H such that H(𝜕Br ) ≥ η for every 1 < r < 2 and some fixed η > 0, where Br := Br (0) is the ball of radius r centered at 0. Such is the case with β the Hausdorff content ℋ∞ with β ≤ d − 1 or the Sobolev capacity Cap1,p for 1 ≤ p < d; see [15, 44] for their definitions and further properties. If (fn )n∈ℕ∗ is any sequence of functions in ℝd such that fn ≥ 1 on 𝜕B1+2/n with fn supported in B1+3/n \ B1+1/n , then fn → 0 pointwise in ℝd , but not quasiuniformly since H(𝜕B1+2/n ) ≥ η for every n ∈ ℕ∗ . In this section, we rely mostly on subadditivity involving finitely many sets:

Some remarks on capacitary integrals and measure theory

� 241

finite subadditivity: For every E, F ⊂ ℝd , H(E ∪ F) ≤ H(E) + H(F). The following version of Fatou’s lemma for quasiuniform convergence then holds. Proposition 2.3. Suppose that H satisfies monotonicity and finite subadditivity. If (fn )n∈ℕ is a sequence of nonnegative functions in ℝd such that fn → f q. u., then ∫ f dH ≤ lim inf ∫ fn dH. n→∞

Proof. Given ϵ > 0, there exists E ⊂ ℝd such that H(E) ≤ ϵ and fn → f uniformly in ℝd \ E. Given η > 0, take N ∈ ℕ such that, for n ≥ N, | fn − f | ≤ η

in ℝd \ E.

We then have f ≤ fn + | fn − f | ≤ fn + η

in ℝd \ E.

Hence, for every t > 0, {f > t + η} \ E ⊂ {fn > t} which implies that {f > t + η} ⊂ {fn > t} ∪ E. By monotonicity and finite subadditivity of H, we get H({f > t + η}) ≤ H({fn > t}) + H(E). Fix k ∈ ℕ. Integrating with respect to t over the interval (0, k), k+η

k

k

∫ H({f > s}) ds = ∫ H({f > t + η}) dt ≤ ∫ H({fn > t}) dt + H(E)k η

0

0

≤ ∫ fn dH + ϵk. This estimate holds for every n ≥ N. Letting n → ∞, we get k+η

∫ H({f > s}) ds ≤ lim inf ∫ fn dH + ϵk. η

n→∞

242 � A. C. Ponce and D. Spector As ϵ → 0, we deduce that k+η

∫ H({f > s}) ds ≤ lim inf ∫ fn dH. n→∞

η

To conclude, we let k → ∞ and η → 0. We then get by Fatou’s lemma for the Lebesgue measure, ∞

∫ f dH = ∫ H({f > s}) ds ≤ lim inf ∫ fn dH. n→∞

0

We now show the following version of the dominated convergence theorem. Proposition 2.4. Suppose that H satisfies monotonicity and finite subadditivity. If (fn )n∈ℕ is a sequence of real-valued functions in ℝd such that fn → f q. u. and if there exists F : ℝd → [0, ∞] such that ∫ F dH < ∞ and |fn | ≤ F in ℝd for every n ∈ ℕ, then ∫ |f | dH < ∞ and lim ∫ | fn − f | dH = 0.

n→∞

Proof. Since |fn | → |f | q. u. and |fn | ≤ F in ℝd , by Proposition 2.3 and by monotonicity of the Choquet integral, we have ∫ | f | dH ≤ lim inf ∫ | fn | dH ≤ ∫ F dH < ∞. n→∞

Given ϵ > 0, take E ⊂ ℝd with H(E) ≤ ϵ such that (fn )n∈ℕ converges uniformly to f in ℝd \ E. Then, given η > 0, let N ∈ ℕ be such that, for every n ≥ N, |fn − f | ≤ η in ℝd \ E. Thus, for t > η and n ≥ N, we have {| fn − f | > t} ⊂ E, whence, by monotonicity of H, H({| fn − f | > t}) ≤ H(E) ≤ ϵ.

(2.1)

{| fn − f | > t} ⊂ {F + | f | ≥ t}.

(2.2)

Also, for any t > 0,

It follows from (2.1) and (2.2) that, for any k > η and n ≥ N, η

k

∞

∫ | fn − f | dH = ∫ + ∫ + ∫ H({| fn − f | > t}) dt 0

η

k

Some remarks on capacitary integrals and measure theory η

� 243

∞

≤ ∫ H({F + | f | > t}) dt + kϵ + ∫ H({F + | f | > t}) dt. 0

k

As n → ∞, η

∞

lim sup ∫ | fn − f | dH ≤ ∫ H({F + | f | > t}) dt + kϵ + ∫ H({F + | f | > t}) dt. n→∞

0

(2.3)

k

By finite subadditivity of H, H({F + | f | > t}) ≤ H({F ≥ t/2}) + H({| f | ≥ t/2}). Since both F and |f | have finite Choquet integrals, we have the conclusion by letting ϵ, η → 0 and then k → ∞ in (2.3). The proof of the partial converse of the dominated convergence theorem involves countable subadditivity. Proposition 2.5. Suppose that H satisfies monotonicity and countable subadditivity. Let (fn )n∈ℕ be a sequence of real-valued functions in ℝd such that ∫ | fn − f | dH ≤

1 4n

for every n ∈ ℕ,

where f : ℝd → ℝ satisfies ∫ |f | dH < ∞. Then fn → f q. u. and there exists F : ℝd → [0, ∞] such that ∫ F dH < ∞ and |fn | ≤ F in ℝd for every n ∈ ℕ. Proof. By monotonicity of H, one has an analogue of Chebyshev’s inequality, 1 1 H({|fn − f | > 1/2n }) ≤ ∫ | fn − f | dH ≤ n . 2n 4

(2.4)

For each k ∈ ℕ, denoting ∞

Ak := ⋃ {|fn − f | > 1/2n }, n=k

then, by countable subadditivity of H and (2.4), ∞

∞

n=k

n=k

H(Ak ) ≤ ∑ H({|fn − f | > 1/2n }) ≤ ∑

1 1 ≤ . 2n 2k−1

(2.5)

Since the sequence (1/2n )n∈ℕ is summable, by the Weierstrass M-test the series d ∑∞ n=k |fn − f | converges uniformly in ℝ \ Ak for every k ∈ ℕ, and then so does the sequence (fn )n∈ℕ . We deduce from (2.5) that (fn )n∈ℕ converges quasiuniformly to f in ℝd .

244 � A. C. Ponce and D. Spector To conclude, it suffices to verify that F := |f |+∑∞ n=0 |fn − f | has finite Choquet integral. We first observe that, by quasi-sublinearity of the Choquet integral, one has that for every two functions g and h, ∫ |g + h| dH ≤ 2 ∫ |g| dH + 2 ∫ |h| dH.

(2.6)

Iterating this inequality, for every j ∈ ℕ one gets j

j

n=0

n=0

j

1 ≤ 4. n−1 2 n=0

∫ ∑ | fn − f | dH ≤ ∑ 2n+1 ∫ | fn − f | dH ≤ ∑

By quasi-sublinearity of the Choquet integral and quasiuniform convergence of the series ∑∞ n=0 |fn − f |, we deduce from Proposition 2.3 that ∫ F dH ≤ 2 ∫ | f | dH + 8 < ∞. The need for countable subadditivity in the statement of Proposition 2.5 can be seen in the following. Example 2.6. Take a sequence (an )n∈ℕ of distinct points in ℝd and, for each n ∈ ℕ, let fn (an ) = 1 and fn (x) = 0 for x ≠ an . If H is the capacity given by Example 1.3, then ∫ |fn − 0| dH = H({an }) = 0 for each n but (fn )n∈ℕ does not converge q. u. to 0 since, for every j ∈ ℕ, we have H(⋃n≥j {an }) = 1.

3 Approximation of quasicontinuous functions We investigate in this section the question of approximation of quasicontinuous functions with finite Choquet integral by sequences of continuous functions. Definition 3.1. A function f : ℝd → [−∞, ∞] is quasicontinuous whenever, for each ϵ > 0, there exists an open set ω ⊂ ℝd such that H(ω) ≤ ϵ and f |ℝd \ω is finite and continuous in ℝd \ ω. For later use, we observe that, for every t ∈ ℝ, {f > t} ∪ ω is an open set in ℝd ,

(3.1)

even though {f > t} itself need not be open. Indeed, by continuity of f |ℝd \ω the set {f > t} \ ω is open in ℝd \ ω with respect to the relative topology. Hence, there exists an open set U in ℝd such that {f > t} \ ω = U \ ω and then {f > t} ∪ ω = U ∪ ω

Some remarks on capacitary integrals and measure theory

� 245

is open in ℝd as claimed. We begin by approximating quasicontinuous functions by sequences of bounded continuous functions. Proposition 3.2. Suppose that H satisfies monotonicity and finite subadditivity. If f : ℝd → [−∞, ∞] is quasicontinuous and ∫ |f | dH < ∞, then there exists a sequence (fn )n∈ℕ of bounded continuous functions in ℝd such that lim ∫ | fn − f | dH = 0.

n→∞

Proof. Suppose that f is quasicontinuous and has finite Choquet integral. We compose f with the truncation function Tk : [−∞, ∞] → ℝ at height k > 0 defined by k { { { Tk (t) = {t { { {−k

if t > k,

(3.2)

if −k ≤ t ≤ k, if t ≤ −k.

Then the composition Tk (f ) satisfies ∞

∞

∞

0

0

k

󵄨 󵄨 󵄨 󵄨 ∫󵄨󵄨󵄨Tk (f ) − f 󵄨󵄨󵄨 dH = ∫ H({󵄨󵄨󵄨Tk (f ) − f 󵄨󵄨󵄨 > t}) dt = ∫ H({| f | > k + t}) dt = ∫ H({| f | > s}) ds. Since ∫ |f | dH < ∞, the integral in the right-hand side tends to zero as k → ∞. Next, by quasicontinuity of f , for every ϵ > 0 we may find an open set ω ⊂ ℝd such that H(ω) ≤ ϵ and f |ℝd \ω is continuous on the closed set ℝd \ ω. Then, by composition, Tk (f )|ℝd \ω is also continuous. Thus, the Tietze extension theorem allows us to extend Tk (f )|ℝd \ω as a bounded continuous function gk,ϵ : ℝd → ℝ with |gk,ϵ | ≤ k. We can therefore estimate using the quasi-sublinearity (2.6) of the Choquet integral, 󵄨 󵄨 󵄨 󵄨 ∫ |gk,ϵ − f | dH ≤ 2 ∫󵄨󵄨󵄨gk,ϵ − Tk (f )󵄨󵄨󵄨 dH + 2 ∫󵄨󵄨󵄨Tk (f ) − f 󵄨󵄨󵄨 dH. Taking n ∈ ℕ, we can find k = kn > 0 such that the second term is bounded by 1/(n + 1). For the first term, since |gkn ,ϵ − Tkn (f )| is bounded by 2kn and vanishes on ℝd \ ω, we have 2kn

2kn

󵄨 󵄨 󵄨 󵄨 ∫󵄨󵄨󵄨gkn ,ϵ − Tk (f )󵄨󵄨󵄨 dH = ∫ H({󵄨󵄨󵄨gkn ,ϵ − Tkn (f )󵄨󵄨󵄨 > t}) dt ≤ ∫ H(ω) dt ≤ 2kn ϵ. 0

0

Thus ∫ |gkn ,ϵ − f | dH ≤ 4kn ϵ +

2 ϵ. n+1

246 � A. C. Ponce and D. Spector Choosing ϵ = ϵn > 0 so that 4kn ϵn ≤ 1/(n + 1), the right-hand side is less than or equal to 3/(n + 1) and we have the conclusion with fn := gkn ,ϵn . To obtain the approximation by continuous functions with compact support, one needs a vanishing property of the capacity at infinity. evanescence: For every open subset U ⊂ ℝd with H(U) < ∞ and every closed subset F ⊂ U, lim H(F \ Br ) = 0.

r→∞

Under this additional assumption, we prove the following. Proposition 3.3. Suppose that H satisfies monotonicity, finite subadditivity, and evanescence. If f : ℝd → [−∞, ∞] is quasicontinuous and ∫ |f | dH < ∞, then there exists a sequence (φn )n∈ℕ in Cc (ℝd ) such that lim ∫ |φn − f | dH = 0.

n→∞

Proof. By Proposition 3.2 and quasi-sublinearity (2.6) of the Choquet integral, we may assume that f is bounded and continuous. Given r > 0, take ψr ∈ Cc (ℝd ) such that 0 ≤ ψr ≤ 1 in ℝd and ψr = 1 in Br . Since |fψr − f | ≤ |f | in ℝd and |fψr − f | = 0 in Br , for every t > 0 we have {| fψr − f | > t} ⊂ {| f | ≥ t} \ Br .

(3.3)

By boundedness of f , there exists M ≥ 0 such that |f | ≤ M. For any 0 < η ≤ M, we then have η

M

M

∫ | fψr − f | dH = ∫ H({| fψr − f | > t}) dt = ∫ + ∫ H({| fψr − f | > t}) dt. 0

0

η

Using the monotonicity of H and (3.3), we estimate η

η

∫ H({| fψr − f | > t}) dt ≤ ∫ H({| f | > t}) dt 0

0

and M

∫ H({| fψr − f | > t}) dt ≤ MH({| fψr − f | > η}) ≤ MH({| f | ≥ η} \ Br ). η

Thus

Some remarks on capacitary integrals and measure theory

� 247

η

∫ | fψr − f | dH ≤ ∫ H({| f | > t}) dt + MH({| f | ≥ η} \ Br ).

(3.4)

0

Observe that {|f | ≥ η} is closed and contained in the open set {|f | > η}, which satisfies H({|f | > η}) < ∞ by the Chebyshev inequality. Thus, by evanescence, the second term in the right-hand side of (3.4) converges to zero as r → ∞. Since ∫ |f | dH < ∞, the first term in the right-hand side of (3.4) also tends to zero as η → 0. Therefore, if we let r → ∞ and then η → 0 in (3.4), we obtain lim ∫ | fψr − f | dH = 0,

r→∞

which implies the desired conclusion since fψr ∈ Cc (ℝd ). The assumption evanescence is necessary for the density of functions with compact support. Indeed, we have the following. Proposition 3.4. Suppose that H satisfies monotonicity. If every quasicontinuous function f : ℝd → ℝ with ∫ |f | dH < ∞ can be approximated in terms of the Choquet integral by a sequence in Cc (ℝd ), then H satisfies evanescence. Proof. Given an open set U ⊂ ℝd with H(U) < ∞ and a closed subset F ⊂ U, let f : ℝd → ℝ be a continuous function supported in U such that f = 1 on F and 0 ≤ f ≤ 1 in ℝd . Then, by monotonicity of H, ∫ | f | dH ≤ H(U) < ∞. By assumption, there exists a sequence (fn )n∈ℕ in Cc (ℝd ) such that ∫ |fn − f | dH → 0. Given k ∈ ℕ to be chosen below, let R > 0 be such that supp fk ⊂ BR . By monotonicity of H, we have ∫ | fk − f | dH ≥ ∫ | fk − f |χF\BR dH = ∫ χF\BR dH = H(F \ BR ). Given ϵ > 0, take k ∈ ℕ so that the integral in the left-hand side is less than ϵ. Then, for every r ≥ R, we have by monotonicity of H, H(F \ Br ) ≤ H(F \ BR ) ≤ ∫ | fk − f | dH ≤ ϵ.

4 Sublinearity of the Choquet integral The proof of the sublinearity of the Choquet integral in the discrete case relies on the following minimization property.

248 � A. C. Ponce and D. Spector Lemma 4.1. Suppose that H satisfies strong subadditivity. Then, for every n ∈ ℕ∗ and C1 , . . . , Cn ⊂ ℝd , there exist D1 , . . . , Dn−1 ⊂ Dn such that n

n

i=1

i=1

∑ χDi = ∑ χCi

and

n

n

i=1

i=1

∑ H(Di ) ≤ ∑ H(Ci ).

From the identity between the characteristic functions, we have ⋃ni=1 Di = ⋃ni=1 Ci and, since Dn contains all sets Di , it then follows that Dn = ⋃ni=1 Ci . Proof of Lemma 4.1. We proceed by induction on n. For n = 1, it suffices to take D1 = C1 . We now suppose the statement is true for some n ∈ ℕ∗ . Assume we are given sets C1 , . . . , Cn+1 ⊂ ℝd , and apply the induction assumption to the first n sets C1 , . . . , Cn to get ̃ 1, . . . , D ̃ n−1 ⊂ D ̃ n that satisfy the conclusion. By strong subadditivity of H, we have D ̃ n ∩ Cn+1 ) + H(D ̃ n ∪ Cn+1 ) ≤ H(D ̃ n ) + H(Cn+1 ). H(D ̃ i for i ∈ {1, . . . , n − 1}, Dn := D ̃ n ∩ Cn+1 and Dn+1 := D ̃ n ∪ Cn+1 . Then Let Di := D n+1

n−1

̃ i ) + H(D ̃ n ∩ Cn+1 ) + H(D ̃ n ∪ Cn+1 ) ∑ H(Di ) = ∑ H(D i=1

i=1

n−1

n

i=1

i=1

̃ i ) + H(D ̃ n ) + H(Cn+1 ) = ∑ H(D ̃ i ) + H(Cn+1 ). ≤ ∑ H(D By an application of the induction hypothesis, we then have n+1

n

n+1

i=1

i=1

i=1

∑ H(Di ) ≤ ∑ H(Ci ) + H(Cn+1 ) = ∑ H(Ci ).

We also observe that χD̃

n ∩Cn+1

+ χD̃

n ∪Cn+1

= χD̃ + χCn+1 . n

Since ∑ni=1 χD̃ = ∑ni=1 χCi , one sees that i

n+1

n−1

i=1

i=1

∑ χDi = ∑ χD̃ + χD̃ i

n ∩Cn+1

n

+ χD̃

n ∪Cn+1

n+1

= ∑ χD̃ + χCn+1 = ∑ χCi , i=1

i

i=1

as claimed. Finally, it remains to verify that Di ⊂ Dn+1 for i ∈ {1, . . . , n}. This follows ̃ n ∪ Cn+1 . Indeed, the first n − 1 sets D1 , . . . , Dn−1 are contained from the choice of Dn+1 = D ̃ ̃ n ∩ Cn+1 , which is also in Dn , which is a subset of Dn+1 . To conclude, one notes that Dn = D contained in Dn+1 .

Some remarks on capacitary integrals and measure theory

� 249

The following proposition is claimed true by an induction argument at the top of page 249 of [24], while a detailed proof can be found on pages 766–768 of [49]. We rely on a different organization of the proof based on Lemma 4.1. Proposition 4.2. Suppose that H satisfies strong subadditivity. Then, for every C1 , . . . , Cn ⊂ ℝd , the nested family of sets A1 ⊂ . . . ⊂ An ⊂ ℝd such that n

n

i=1

i=1

∑ χAi = ∑ χCi satisfies n

n

i=1

i=1

∑ H(Ai ) ≤ ∑ H(Ci ). For every i ∈ {1, . . . , n}, the set Ai is the collection of points that belong to at least n − i + 1 sets among C1 , . . . , Cn . Note that Ai can be also identified as the superlevel set {a ≥ i} of the function a := ∑ni=1 χCi . Proof of Proposition 4.2. We proceed by induction on the number of sets C1 , . . . , Cn . The conclusion is trivially true for n = 1 since in this case A1 = C1 . We now assume that n ≥ 2 and that the statement holds for any family of n − 1 sets. Then, given C1 , . . . , Cn ⊂ ℝd , take D1 , . . . , Dn−1 ⊂ Dn

(4.1)

given by Lemma 4.1, so that n

n

i=1

i=1

∑ χDi = ∑ χCi

and

n

n

i=1

i=1

∑ H(Di ) ≤ ∑ H(Ci ).

(4.2)

By the induction assumption applied to D1 , . . . , Dn−1 , we have a family of nested sets A1 ⊂ . . . ⊂ An−1 ⊂ ℝd such that n−1

n−1

i=1

i=1

∑ χAi = ∑ χDi

and

n−1

n−1

i=1

i=1

∑ H(Ai ) ≤ ∑ H(Di ).

(4.3)

In particular, An−1 = ⋃n−1 i=1 Di . Define An := Dn . From (4.1), we deduce that A1 ⊂ . . . ⊂ An−1 ⊂ An are nested. Moreover, by (4.2) and (4.3), they satisfy n

n

n

i=1

i=1

i=1

∑ χAi = ∑ χDi = ∑ χCi and also

250 � A. C. Ponce and D. Spector n

n

n

i=1

i=1

i=1

∑ H(Ai ) ≤ ∑ H(Di ) ≤ ∑ H(Ci ). By induction, the conclusion then follows. From this result, sublinearity of the Choquet integral where the functions in consideration have values in a discrete set follows easily. Corollary 4.3. Suppose that H satisfies monotonicity and strong subadditivity. If f , g : ℝd → ℕ/k for some k ∈ ℕ∗ , then ∫(f + g) dH ≤ ∫ f dH + ∫ g dH.

(4.4)

Proof. By positive 1-homogeneity of the Choquet integral we may assume that f and g take values in ℕ. Let us first assume they are both bounded from above by n, in which case n

f = ∑ χ{f ≥i} i=1

n

and

∫ f dH = ∑ H({f ≥ i}), i=1

with analogous identities for g and f + g; note that this last function is bounded from above by 2n. Since 2n

n

n

i=1

i=1

i=1

∑ χ{f +g≥i} = f + g = ∑ χ{f ≥i} + ∑ χ{g≥i} , and the sets {f + g ≥ i} for i ∈ {1, . . . , 2n} are nested, we may apply Proposition 4.2 with sets {f ≥ i} and {g ≥ i} for i ∈ {1, . . . , n} to get 2n

n

n

i=1

i=1

i=1

∑ H({f + g ≥ i}) ≤ ∑ H({f ≥ i}) + ∑ H({g ≥ i}). From the identities verified by the Choquet integrals of f + g, f , and g, this inequality is precisely (4.4). To handle the case where f or g is unbounded, we compose them with the truncation function Tn at height n ∈ ℕ defined by (3.2). As Tn (f ) and Tn (g) are both bounded and have values in ℕ, we can apply the sublinearity of the Choquet integral we already proved in this setting to deduce that ∫(Tn (f ) + Tn (g)) dH ≤ ∫ Tn (f ) dH + ∫ Tn (g) dH. Since Tn (f ) ≤ f and Tn (g) ≤ g, by monotonicity of the Choquet integral we get ∫(Tn (f ) + Tn (g)) dH ≤ ∫ f dH + ∫ g dH.

Some remarks on capacitary integrals and measure theory

� 251

We also note that Tn (f + g) ≤ Tn (f ) + Tn (g), which again by monotonicity of the Choquet integral gives ∫ Tn (f + g) dH ≤ ∫ f dH + ∫ g dH.

(4.5)

Since n

∫ H({f + g > t}) dt = ∫ Tn (f + g) dH, 0

it follows from Fatou’s lemma for the Lebesgue integral that ∞

∫(f + g) dH = ∫ H({f + g > t}) dt ≤ lim inf ∫ Tn (f + g) dH. n→∞

0

(4.6)

It now suffices to combine (4.5) and (4.6). Proof of Theorem 1.2. Given E, F ⊂ ℝd , we have H(E ∩ F) + H(E ∪ F) = ∫(χE + χF ) dH. Thus, if the Choquet integral is sublinear, then we have H(E ∩ F) + H(E ∪ F) ≤ ∫ χE dH + ∫ χF dH = H(E) + H(F). Hence, H is strongly subadditive. To prove the converse, we now assume that H is strongly subadditive. Denote by ⌊α⌋ the integer part of the real number α. Let f , g : ℝd → [0, ∞] and k ∈ ℕ∗ . An application of Corollary 4.3 to the ℕ/k-valued functions ⌊kf ⌋/k and ⌊kg⌋/k gives ∫(

⌊kf ⌋ ⌊kg⌋ ⌊kf ⌋ ⌊kg⌋ + ) dH ≤ ∫ dH + ∫ dH. k k k k

Since ⌊kf ⌋/k ≤ f and ⌊kg⌋/k ≤ g, by monotonicity of the Choquet integral, we then have ∫(

⌊kf ⌋ ⌊kg⌋ + ) dH ≤ ∫ f dH + ∫ g dH. k k

Since kα − 1 ≤ ⌊kα⌋ for every α ≥ 0, we have f +g− which implies

2 ⌊kf ⌋ ⌊kg⌋ ≤ + , k k k

(4.7)

252 � A. C. Ponce and D. Spector

(f + g −

⌊kf ⌋ ⌊kg⌋ 2 ) ≤ + . k k k +

By definition and monotonicity of the Choquet integral, ∞

∫ H({f + g > t}) dt = ∫(f + g − 2/k

⌊kf ⌋ ⌊kg⌋ 2 ) dH ≤ ∫( + ) dH. k k k +

(4.8)

Hence, by (4.7) and (4.8), ∞

∫ H({f + g > t}) dt ≤ ∫ f dH + ∫ g dH. 2/k

An application of Fatou’s lemma for the Lebesgue integral then yields ∞

∫(f + g) dH ≤ lim inf ∫ H({f + g > t}) dt ≤ ∫ f dH + ∫ g dH. k→∞

2/k

As a consequence of Proposition 2.3, one gets countable sublinearity for the Choquet integral for series of functions that converge quasiuniformly. Note that countable subadditivity of H is not necessary in this case. Corollary 4.4. Suppose that H satisfies monotonicity and strong subadditivity. If (fn )n∈ℕ is a sequence of real-valued functions in ℝd such that (∑kn=0 fn )k∈ℕ converges q. u. to F : ℝd → ℝ, then ∞

∫ |F| dH ≤ ∑ ∫ | fn | dH. n=0

Proof. By monotonicity and sublinearity of the Choquet integral, for every k ∈ ℕ we have k k 󵄨󵄨 k 󵄨󵄨 󵄨 󵄨 ∫󵄨󵄨󵄨 ∑ fn 󵄨󵄨󵄨 dH ≤ ∫ ∑ | fn | dH ≤ ∑ ∫ | fn | dH. 󵄨󵄨 󵄨 n=0 󵄨 n=0 n=0

By Proposition 2.3, we have the conclusion as k → ∞. Remark 4.5. The statements above and their proofs in this section apply without change to a set function H defined on an algebra 𝒳 associated to a set X, that is, 𝒳 is a family of subsets of X such that 0 ∈ 𝒳 and A ∩ B, A ∪ B ∈ 𝒳

for every A, B ∈ 𝒳 .

More generally, one expects that results concerning the Choquet integral stated for capacities satisfying finite subadditivity hold in the setting of an algebra, provided one

Some remarks on capacitary integrals and measure theory

� 253

works with the class of H-capacitable functions, i. e., those for which their upper-level sets are elements in this algebra.

5 Lower semicontinuity of the Choquet integral To obtain a Fatou’s lemma without quasiuniform convergence, we restrict ourselves to the setting of quasicontinuous functions. The main tool is based on the following standard consequence of the Hahn–Banach theorem. Proposition 5.1. Suppose that H satisfies monotonicity, strong subadditivity, and evanescence. If f : ℝd → [0, ∞] is a quasicontinuous function such that H({f > t}) < ∞

for every t > 0,

(5.1)

then 󵄨󵄨 μ ≥ 0 is a locally finite Borel measure, 󵄨󵄨 }. ∫ f dH = sup {∫ f dμ 󵄨󵄨󵄨 󵄨󵄨 μ ≤ H on open subsets of ℝd 󵄨 We handle separately the inequality “≥” in the following. Lemma 5.2. Suppose that H satisfies monotonicity and finite subadditivity. If f : ℝd → [0, ∞] is quasicontinuous then, for every locally finite nonnegative Borel measure μ in ℝd such that μ ≤ H on open subsets of ℝd , we have ∫ f dμ ≤ ∫ f dH. Proof of Lemma 5.2. Since f is quasicontinuous, for every ϵ > 0 there exists an open set ω ⊂ ℝd such that H(ω) ≤ ϵ and f |ℝd \ω is continuous. Then, by (3.1), the set {f > t} ∪ ω is open in ℝd for every t > 0. Since μ is monotone, μ ≤ H on open sets, and H is finitely subadditive, we get μ({f > t}) ≤ μ({f > t} ∪ ω) ≤ H({f > t} ∪ ω) ≤ H({f > t}) + H(ω). Given k > 0, we integrate both members with respect to t ∈ (0, k) to get k

k

∫ μ({f > t}) dt ≤ ∫ H({f > t}) dt + H(ω)k ≤ ∫ f dH + ϵk. 0

0

Letting ϵ → 0 and then k → ∞, we have the conclusion using Cavalieri’s principle for the Lebesgue integral.

254 � A. C. Ponce and D. Spector Proof of Proposition 5.1. Inequality “≥” in the statement readily follows from Lemma 5.2 for any nonnegative quasicontinuous function. We thus focus on the proof of the reverse inequality “≤.” We first show it for a nonnegative function f ∈ Cc (ℝd ), assuming in addition that H is finite. Since H is strongly subadditive, the function P : ψ ∈ Cc (ℝd ) 󳨃󳨀→ ∫ ψ+ dH ∈ ℝ+ is well-defined and sublinear. By the Hahn–Banach theorem, the linear functional tf 󳨃󳨀→ t ∫ f dH defined on the one-dimensional vector subspace {tf : t ∈ ℝ} has a linear extension F : Cc (ℝd ) → ℝ such that F ≤ P on Cc (ℝd ). Observe that if ψ ∈ Cc (ℝd ) is nonpositive, we have F(ψ) ≤ P(ψ) = 0. Thus F is positive and then, for every compact subset S ⊂ ℝd , its restriction to C(S) is a continuous linear functional. Hence, by the Riesz representation theorem, there exists a locally, finite, nonnegative Borel measure μ in ℝd such that F(ψ) = ∫ ψ dμ for every ψ ∈ Cc (ℝd ). In particular, since f is nonnegative, ∫ f dμ = F(f ) = P(f ) = ∫ f dH. It remains to observe that μ ≤ H on open subsets of ℝd . To this end, take a nonempty, open set U ⊂ ℝd and a compact subset K ⊂ U. There exists ψ ∈ Cc (ℝd ) such that 0 ≤ ψ ≤ 1 in ℝd , ψ = 1 on K and supp ψ ⊂ U. We then have by monotonicity of μ and H, μ(K) ≤ ∫ ψ dμ = F(ψ) ≤ P(ψ) = ∫ ψ dH ≤ H(U). Since this inequality holds for every compact subset K ⊂ U, by inner regularity of μ we conclude that μ(U) ≤ H(U). We next prove the inequality “≤” for a nonnegative function f ∈ Cc (ℝd ) that satisfies (5.1). Take η > 0 to be chosen below and consider the set function Hη defined by contraction for every A ⊂ ℝd as Hη (A) := H(A ∩ {f > η}). Observe that Hη is also monotone and strongly subadditive. Moreover, for every n ∈ ℕ, ∞

∞

0

η

∫ f dHη = ∫ H({f > max {t, η}}) dt ≥ ∫ H({f > t}) dt.

(5.2)

Some remarks on capacitary integrals and measure theory

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We may apply the previous case with finite set function Hη to get ∫ f dHη ≤ sup ∫ f dμ ≤ sup ∫ f dμ. μ≤Hη

μ≤H

(5.3)

Combining (5.2) and (5.3), we then obtain ∞

∫ H({f > t}) dt ≤ sup ∫ f dμ. μ≤H

η

As η → 0, it follows from Fatou’s lemma for the Lebesgue integral that ∫ f dH ≤ sup ∫ f dμ. μ≤H

(5.4)

To conclude, we now prove (5.4) for an arbitrary nonnegative quasicontinuous function f for which (5.1) holds. To this end, we apply Proposition 3.2 that ensures the existence of a sequence of nonnegative functions (φn )n∈ℕ in Cc (ℝd ) such that ∫ |φn −f | dH → 0. For every n ∈ ℕ, we have by sublinearity of the Choquet integral and by Lemma 5.2 applied to the quasicontinuous function |φn − f |, ∫ φn dμ ≤ ∫ f dμ + ∫ |φn − f | dμ ≤ ∫ f dμ + ∫ |φn − f | dH. Taking the supremum on both sides with respect to μ ≤ H, we have by the first part of the proof, ∫ φn dH ≤ sup ∫ f dμ + ∫ |φn − f | dH. μ≤H

As n → ∞, by convergence of the sequence (φn )n∈ℕ with respect to the Choquet integral we get (5.4), which completes the proof. Assumption (5.1) can be removed under inner regularity of open sets with infinite capacity. semifinite: For every open set U ⊂ ℝd such that H(U) = ∞ and every M ≥ 0, there exists a subset E ⊂ U with M ≤ H(E) < ∞. We then get the counterpart of Proposition 5.1 for all nonnegative quasicontinuous functions with infinite Choquet integral. Proposition 5.3. Suppose that H satisfies monotonicity, strong subadditivity, evanescence, and semifinite. If f : ℝd → [0, ∞] is a quasicontinuous function such that H({f > t}) = ∞ for some t > 0,

(5.5)

256 � A. C. Ponce and D. Spector then 󵄨󵄨 μ ≥ 0 is a locally finite Borel measure, 󵄨󵄨 }. ∫ f dH = ∞ = sup {∫ f dμ 󵄨󵄨󵄨 󵄨󵄨 μ ≤ H on open subsets of ℝd 󵄨 Proof. From (5.5), we have ∫ f dH = ∞. Next, by quasicontinuity of f , there exists an open set ω ⊂ ℝd such that H(ω) ≤ 1 and f |ℝd \ω is continuous. In particular, by (3.1), the set {f > t} ∪ ω is open in ℝd . By monotonicity of H and (5.5), H({f > t} ∪ ω) ≥ H({f > t}) = ∞. Applying semifinite, for every n ∈ ℕ there exists a subset En ⊂ {f > t} ∪ ω with n ≤ H(En ) < ∞. Let Hn be the set function defined for every A ⊂ ℝd by Hn (A) := H(A ∩ En ). Note that t

tH({f > t} ∩ En ) ≤ ∫ Hn ({f > s}) ds ≤ ∫ f dHn . 0

Since En ⊂ {f > t} ∪ ω, by monotonicity and finite subadditivity of H, n ≤ H(En ) ≤ H({f > t} ∩ En ) + H(ω) ≤ H({f > t} ∩ En ) + 1 Thus t(n − 1) ≤ ∫ f dHn .

(5.6)

Since Hn is finite and Hn ≤ H, by Proposition 5.1 we have ∫ f dHn ≤ sup ∫ f dμ ≤ sup ∫ f dμ. μ≤Hn

μ≤H

(5.7)

Combining (5.6) and (5.7), we get t(n − 1) ≤ sup ∫ f dμ. μ≤H

This implies the conclusion since n can be chosen arbitrarily large. From Propositions 5.1 and 5.3, we deduce the following analogue of Fatou’s lemma for quasicontinuous functions. Corollary 5.4. Suppose that H satisfies monotonicity, strong subadditivity, evanescence, and semifinite. If (fn )n∈ℕ is a sequence of nonnegative quasicontinuous functions in ℝd then, for every quasicontinuous function f : ℝd → [0, ∞] such that f ≤ lim infn→∞ fn in ℝd , we have

Some remarks on capacitary integrals and measure theory

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∫ f dH ≤ lim inf ∫ fn dH. n→∞

Proof. We may assume that each fn has a finite Choquet integral and that lim inf ∫ fn dH < ∞. n→∞

For every nonnegative locally finite Borel measure μ in ℝd such that μ ≤ H on open subsets of ℝd , we have by monotonicity and Fatou’s lemma for the Lebesgue integral, ∫ f dμ ≤ ∫ lim inf fn dμ ≤ lim inf ∫ fn dμ. n→∞

n→∞

Thus, by Lemma 5.2, ∫ f dμ ≤ lim inf ∫ fn dH. n→∞

(5.8)

Since f is quasicontinuous and the right-hand side is finite, by Proposition 5.3 we must have H({f > t}) < ∞ for every t > 0. The assumptions of Proposition 5.1 are then satisfied by f and it thus suffices to take the supremum with respect to μ in the left-hand side of (5.8). The assumption semifinite avoids a gap between sets of finite and infinite capacities, which is illustrated in the following example where the conclusion of Corollary 5.4 fails. Example 5.5. Let H be defined for every nonempty subset A ⊂ ℝd as 1

if A is bounded,

∞

if A is unbounded.

H(A) = {

For each n ∈ ℕ, take fn ∈ Cc (ℝd ) such that 0 ≤ fn ≤ 1 in ℝd , fn = 1 in Bn and supp fn ⊂ Bn+1 . Then ∫ fn dH = 1 and fn → 1 pointwise in ℝd as n → ∞, but ∫ 1 dH = ∞. One can fulfill assumption semifinite by replacing H with a more regular set func̃ : 𝒫 (ℝd ) → [0, ∞] defined for every A ⊂ ℝd by tion H ̃ H(A) := sup {H(D) : D ⊂ A, H(D) < ∞}. ̃ is monotone regardless of H. Under monotonicity of H itself, these set Observe that H functions coincide on sets of finite H capacity. Proposition 5.6. If H satisfies monotonicity, then ̃ H(A) = H(A)

for every A ⊂ ℝd with H(A) < ∞.

258 � A. C. Ponce and D. Spector ̃ also satisfies semifinite. Hence, H ̃ Proof. By the monotonicity of H, the supremum in the definition of H(A) is achieved by the set A itself whenever H(A) < ∞. That semifinite holds, then follows from the fact ̃ as that we may then reformulate the definition of H ̃ ̃ ̃ H(A) = sup {H(D) : D ⊂ A, H(D) < ∞}. ̃ inherits several properties of H. Note that H Proposition 5.7. Suppose that H satisfies monotonicity. If H also satisfies any of the assumptions finite subadditivity, countable subadditivity, strong subadditivity, or ẽ vanescence, then so does H. ̃ verifies evanescence whenever H does follows from the fact that H ̃=H Proof. That H on sets where H is finite. We now assume that H satisfies countable subadditivity. Given a sequence of sets (En )n∈ℕ and D ⊂ ⋃∞ n=0 En with H(D) < ∞, by monotonicity of H for every n ∈ ℕ, we have H(En ∩ D) < ∞. Hence, by countable subadditivity of H and ̃ definition of H, ∞

∞

∞

n=0

n=0

n=0

̃ n ). H(D) = H( ⋃ (En ∩ D)) ≤ ∑ H(En ∩ D) ≤ ∑ H(E It now suffices to take the supremum with respect to D in the left-hand side. Similarly, ̃ satisfies finite subadditivity whenever H does. one verifies that H We now assume that H verifies strong subadditivity. Given E, F ⊂ ℝd , take C ⊂ E∩F and D ⊂ E ∪ F with H(C) < ∞ and H(D) < ∞. By finite subadditivity of H, H(C ∪ D) < ∞, and then, by monotonicity, H is finite on every subset of C ∪ D. Note that, since C ⊂ E ∩ F and D ⊂ E ∪ F, C ⊂ ((C ∪ D) ∩ E) ∩ ((C ∪ D) ∩ F) and

D ⊂ ((C ∪ D) ∩ E) ∪ ((C ∪ D) ∩ F).

By monotonicity and strong subadditivity of H, we then have H(C) + H(D) ≤ H(((C ∪ D) ∩ E) ∩ ((C ∪ D) ∩ F)) + H(((C ∪ D) ∩ E) ∪ ((C ∪ D) ∩ F)) ≤ H((C ∪ D) ∩ E) + H((C ∪ D) ∩ F).

Since (C ∪ D) ∩ E and (C ∪ D) ∩ F are subsets of E and F where H is finite, we get by ̃ definition of H, ̃ ̃ H(C) + H(D) ≤ H(E) + H(F). It now suffices to take the supremum in the left-hand side with respect to C and D.

Some remarks on capacitary integrals and measure theory

� 259

6 The space L1 (H) We assume that H satisfies monotonicity and strong subadditivity. We introduce an equivalence relation ∼ among elements in the vector space of real-valued quasicontinuous functions in ℝd by denoting f ∼ g whenever f = g quasi-everywhere (q. e.), that is, there exists E ⊂ ℝd such that H(E) = 0 and f = g in ℝd \ E. Then [f ] is the equivalence class that contains f . We let L1 (H) := {[f ] : f : ℝd → ℝ is quasicontinuous and ∫ | f | dH < ∞}. We may naturally equip this set with addition and multiplication by scalar: For every quasicontinuous functions f , g and λ ∈ ℝ, let [f ] + [g] := [f + g] and

λ[f ] := [λf ].

Observe that the function 󵄩󵄩 󵄩󵄩 󵄩󵄩[f ]󵄩󵄩L1 (H) := ∫ | f | dH is well-defined in L1 (H) and, by sublinearity and 1-homogeneity of the Choquet integral, is a norm in this space. Proposition 6.1. Suppose that H satisfies monotonicity, strong subadditivity, and countable subadditivity. Then L1 (H) is a Banach space. The proof is standard, e. g., [42, Proposition 2.1 and 2.2], though as stated in the Introduction we include it for completeness. Proof. Let ([fn ])n∈ℕ be a Cauchy sequence in L1 (H), where each fn : ℝd → ℝ is quasicontinuous. We may find positive integers n1 < n2 < . . . < nj such that 1 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩[fm − fn ]󵄩󵄩󵄩L1 (H) = 󵄩󵄩󵄩[fm ] − [fn ]󵄩󵄩󵄩L1 (H) ≤ j 4

for every m, n ≥ nj .

This implies 1/2j

∫ H({|fnj − fnj+1 | > 1/2j }) dt ≤ 0

and, therefore, H({|fnj − fnj+1 | > 1/2j }) ≤

1 . 2j

1 4j

(6.1)

260 � A. C. Ponce and D. Spector Since |fnj −fnj+1 | is quasicontinuous, there exists an open set ωj ⊂ ℝd such that H(ωj ) ≤ 1/2j

and |fnj − fnj+1 | is continuous in ℝd \ ωj . By (3.1), the set

Gj := {|fnj − fnj+1 | > 2−j } ∪ ωj is open and, by finite subadditivity of H, H(Gj ) ≤ 1/2j−1 . d Let Fm := ⋃∞ j=m Gj . For any x ∈ ℝ \ Fm , we have ∞ ∞ 1 󵄨 󵄨 ∑ 󵄨󵄨󵄨 fnl (x) − fnl+1 (x)󵄨󵄨󵄨 ≤ ∑ l < +∞. 2 l=m l=m

Therefore, if for x ∈ ℝd \ Fm one defines k

f (x) := lim fnj (x) = fn1 (x) + lim ∑(fnl+1 (x) − fnl (x)), j→∞

k→∞

l=1

then by the Weierstrass M-test, (fnj )j∈ℕ converges uniformly to f in ℝd \ Fm , whence f

is continuous in ℝd \ Fm . As (Fm )m≥1 is nonincreasing, the function f is well-defined on ⋃m≥1 (ℝd \ Fm ). Finally, let f (x) = 0 for x ∈ ⋂m≥1 Fm . Since, by countable subadditivity of H, ∞

H(Fm ) ≤ ∑ H(Gj ) ≤ j=m

1

2m−2

→0

as m → ∞, we deduce that f is quasicontinuous and fnj → f q. u.

Since ([fn ])n∈ℕ is a Cauchy sequence in L1 (H), to prove its convergence to [f ] in L (H) it suffices to prove that the subsequence ([fnj ])j≥1 converges q. u. to [f ]. For every i ≥ j ≥ 1, by (6.1) we have 1

∫ | fnj − fni | dH ≤

1 . 4j

As (|fnj − fni |)i≥1 converges to |fnj − f | when i → ∞, it follows from Proposition 2.3 that ∫ | fnj − f | dH ≤ lim inf ∫ | fnj − fni | dH ≤ i→∞

1 , 4j

from which the conclusion follows. Every function in Cc (ℝd ) has finite Choquet integral provided that H satisfies locally finite: H(U) < +∞ for every bounded open set U ⊂ ℝd . Then the quotient space Cc (ℝd )/∼ is contained in L1 (H). Moreover, we have the following.

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Corollary 6.2. Suppose that H satisfies monotonicity, strong subadditivity, countable subadditivity, evanescence, and locally finite. Then L1 (H) is the completion of Cc (ℝd )/∼ with respect to the L1 (H) norm. Proof. From Proposition 3.3, we have that Cc (ℝd )/∼ is dense in L1 (H). By Proposition 6.1, L1 (H) is complete. We conclude this section with a closure property for bounded sequences in L1 (H). First, concerning quasiuniform convergence, we have the following Corollary 6.3. Suppose that H satisfies monotonicity and strong subadditivity. If ([fn ])n∈ℕ is a bounded sequence in L1 (H) such that (fn )n∈ℕ converges q. u. to a quasicontinuous function f , then [f ] ∈ L1 (H) and 󵄩 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩[f ]󵄩󵄩L1 (H) ≤ lim inf 󵄩󵄩󵄩[fn ]󵄩󵄩󵄩L1 (H) . n→∞ Proof. It suffices to apply Proposition 2.3, which implies that ∫ |f | dH < ∞ and then [f ] ∈ L1 (H) by quasicontinuity of f . Observe that if f , g : ℝd → ℝ are such that f = g q. e. and f is quasicontinuous, it need not be true that g is quasicontinuous. To make sure that such a property holds, one may require some regularity on sets where H vanishes. zero-capacity regularity: For every E ⊂ ℝd with H(E) = 0 and every ϵ > 0, there exists an open set ω ⊃ E such that H(ω) ≤ ϵ. Corollary 6.4. Suppose that H satisfies monotonicity, strong subadditivity, evanescence, semifinite, and zero-capacity regularity. If ([fn ])n∈ℕ is a bounded sequence in L1 (H) such that (fn )n∈ℕ converges q. e. to a quasicontinuous function f , then [f ] ∈ L1 (H) and 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩[f ]󵄩󵄩L1 (H) ≤ lim inf 󵄩󵄩󵄩[fn ]󵄩󵄩󵄩L1 (H) . n→∞ Proof. We observe that (|fn |)n∈ℕ converges q. e. to |f | and that |f | is quasicontinuous. It then follows from zero-capacity regularity that the function lim infn→∞ |fn |, which equals |f | q. e., is also quasicontinuous. Moreover, by Corollary 5.4, we deduce that ∫ | f | dH = ∫ lim inf | fn | dH ≤ lim inf ∫ | fn | dH, n→∞

from which the conclusion follows.

n→∞

262 � A. C. Ponce and D. Spector

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Javier Martínez Perales and Carlos Pérez

Remarks on vector-valued Gagliardo and Poincaré–Sobolev-type inequalities with weights Dedicated to the memory of David R. Adams

Abstract: In this paper, we review certain extensions of the Gagliardo and Poincaré– Sobolev-type inequalities to later explore the possibility of extending them to the vectorvalued setting. We restrict ourselves to the most classical case of ℓq -valued functions, where already some difficulties arise, due to the lack of a vector-valued variant of the truncation method, both on the classical and the fractional case. We think that these difficulties may be overcome in the future, and we pose some conjectures in this direction. Keywords: Poincaré–Sobolev, Gagliardo, maximal operator, weight, Muckenhoupt, Wheeden MSC 2020: Primary 54C40, 14E20, Secondary 46E25, 20C20

1 Introduction 1.1 A personal note As the second author of this paper, Carlos has the following personal story to share: “I met David in the 1991–1992 academic year, in the Department of Mathematics at the University of Kentucky where I was his post-doc. It was my last year in North America where I had been since 1985. My stay in Lexington was fantastic because the environment in the department was very pleasant but especially because of David’s hospitality, kindness, and generosity. In my beginning, David’s Lectures on Lp -Potential Theory, Vol. 2, Department of Mathematics, University of Umea, Umea, Sweden, 1981, were very influential. Also, David’s paper Weighted nonlinear potential theory, Trans. Amer. Math. Soc. 297 (1986), 73–94, was very important to me as one can see in my doctoral thesis. Other papers of him, besides his book [1] with L. Hedberg, have continued to be very influential in my later career. I will always be very grateful to David.” Acknowledgement: The second author is supported by grant PID2020-113156GB-I00, Spanish Government; by the Basque Government through grant IT1247-19 and the BERC 2014-2017 program and by BCAM Severo Ochoa accreditation CEX2021-001142-S, Spanish Government. He is also very grateful to the Mittag-Leffler Institute where part of this research was carried out. Javier Martínez Perales, Department of Mathematical Analysis, University of Málaga, Málaga, Spain, e-mail: [email protected] Carlos Pérez, Department of Mathematics, University of the Basque Country, Ikerbasque and BCAM, Bilbao, Spain, e-mail: [email protected] https://doi.org/10.1515/9783110792720-012

266 � J. M. Perales and C. Pérez

1.2 A historical backgroud In 1958, E. Gagliardo [12] proved for n ≥ 2 the existence of a dimensional constant C(n) > 0 such that 1 ′

n 󵄨n′ 󵄨 󵄨 󵄨 ( ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 dx) ≤ C(n) ∫ 󵄨󵄨󵄨∇f (x)󵄨󵄨󵄨 dx

ℝn

(1.1)

ℝn

for any smooth function f with compact support in the Euclidean space ℝn . This is a very relevant inequality since it is equivalent to the celebrated isoperimetric inequality (see, for instance, [5] for a survey). The best constant C(n) in the inequality above was obtained by Maz’ya [18] and, independently, by H. Federer and W. H. Fleming [8] in 1960, and its value is β(n) := inf{C(n) : (1.1) holds for n} =

1

1

nωnn

(1.2)

,

where ωn denotes the volume of the Euclidean unit ball in ℝn . It is well known that Gagliardo’s inequality can be stated in a local form, that is, there exists a constant C(n) > 0 such that the inequality 1 ′

n 󵄨 󵄨 󵄨 󵄨n ′ (∫󵄨󵄨󵄨f (x) − fQ 󵄨󵄨󵄨 dx) ≤ C(n) ∫󵄨󵄨󵄨∇f (x)󵄨󵄨󵄨 dx,

(1.3)

Q

Q

holds for any Q ∈ 𝒬, where 𝒬 denotes the family of open cubes Q of ℝn . As usual, a cube of ℝn will be a Cartesian product of n intervals of the same length ℓ(Q) and fQ := ∫ − f (x) dx := 1/|Q| ∫ |f | dx Q

Q

is the average of f over the cube Q. Note that the above local Gagliardo inequality (1.3) can be seen as the limiting case p = 1 of the local Poincaré–Sobolev inequality, which, for any given 1 < p < n, states the existence of a constant C(n, p) > 0 such that, for any sufficiently good function f defined over ℝn , the inequality 1/p∗

󵄨 󵄨p∗ (∫󵄨󵄨󵄨f (x) − fQ 󵄨󵄨󵄨 dx) Q

1/p

󵄨 󵄨p ≤ C(n, p)(∫󵄨󵄨󵄨∇f (x)󵄨󵄨󵄨 dx)

,

(1.4)

Q

holds for every cube Q ∈ 𝒬, with p∗ being the so-called Sobolev conjugate of p, defined by the equation

Remarks on vector-valued Gagliardo and Poincaré–Sobolev-type inequalities with weights

� 267

1 1 1 − ∗ = , p p n that is, p∗ =

np . n−p

This inequality (1.4) can be proved in different ways, a classical one is the use of the ∗ strong Lp − Lp boundedness properties of the Riesz potentials if p > 1. In the limiting case p = 1, the strong L1 boundedness of the Riesz potentials is well known to fail, but one can still use the weak L1 boundedness of the Riesz potential I1 to prove a weak local Gagliardo inequality, which is then used to get the strong one by the so-called truncation method.

1.3 An outline of the paper In this paper, we explore several extensions of the local Gagliardo inequality (1.3) to the vector-valued setting, where the truncation method is currently not known to work. This will have some consequences on the results we will be able to obtain. We use Section 2 to introduce two already known extensions of the Gagliardo inequality in the scalar case. Later, we present in the subsequent Subsection 2.3 the vector-valued extensions of these inequalities that will be proved in this work. Section 3 contains some preliminary and classical results that will be used in the proofs of the main results, together with some additional extensions of Poincaré–Sobolev-type inequalities to the vector-valued setting. Finally, in Section 4 we prove the main results of this paper.

2 Scalar results and vector-valued extensions 2.1 Scalar weighted Gagliardo inequality In the recent paper [22], Ezequiel Rela and the second author addressed the following result, which was previously studied in [11] and [21] improving results from [6] and [24]. We will refer to the inequalities in this theorem as generalized isoperimetric inequalities. Theorem 2.1. Let μ be a Radon measure in ℝn , n ≥ 2, then there exists a dimensional constant C1 (n) > 0 such that for any cube Q ∈ 𝒬 and any function f ∈ C 1 (Q), 1 󵄨 󵄨 ‖f − fQ ‖Ln′ ,∞ (Q,dμ) ≤ C1 (n) ∫󵄨󵄨󵄨∇f (x)󵄨󵄨󵄨M c ( χQ μ)(x) n′ dx,

(2.1)

Q

where M c denotes the centered Hardy–Littlewood maximal operator defined with balls of ℝn (see Section 3 for a definition).

268 � J. M. Perales and C. Pérez As a consequence, we have 1

󵄨 󵄨 ‖f − fQ ‖Ln′ (Q,dμ) ≤ C2 (n) ∫󵄨󵄨󵄨∇f (x)󵄨󵄨󵄨M c ( χQ μ)(x) n′ dx.

(2.2)

Q

Hence, the global inequality 1 ′

n 1 󵄨 󵄨n ′ 󵄨 󵄨 ( ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 dμ(x)) ≤ C2 (n) ∫ 󵄨󵄨󵄨∇f (x)󵄨󵄨󵄨M c μ(x) n′ dx,

(2.3)

ℝn

ℝn

holds for any f ∈ C 1 (ℝn ) with compact support. 1

The presence of the weight M c μ(x) n′ at the right-hand side of the general inequalities in Theorem 2.1 naturally inspires the definition of the A1,n′ class of weights. This class is defined as that of the weights w ∈ L1loc (ℝn ), for which there exists a constant C > 0 such that M(wn )(x) ≤ Cwn (x), ′

′

for almost every x ∈ ℝn . In fact, A1,n′ is part of a larger family of weights denoted by Ap,p∗ .1 The above result applied to weights in this class has been used in [22] to get the following corollary using extrapolation theory from harmonic analysis, more precisely the sharp version of the Harboure–Macías–Segovia extrapolation theorem [15] as obtained in [14]. This is a sharp weighted local Poincaré–Sobolev inequality, which extends the classical inequality (1.4). Corollary 2.2. Let 1 ≤ p < n and let w ∈ Ap,p∗ . There exists a constant C(n, p) > 0 such that for any cube Q ∈ 𝒬 and any Lipschitz function f , 1

1

p ′ 󵄩󵄩 󵄩 p 󵄩󵄩w( f − fQ )󵄩󵄩󵄩Lp∗ (Q,dx) ≤ C(n, p)[w]An p,p∗ (∫ |w∇f | dx) .

Q

As a consequence, we have the global estimate 1 ′

‖wf ‖Lp∗ (ℝn ) ≤ C(n, p)[w]An

p,p∗

‖w∇f ‖Lp (ℝn )

for any Lipschitz function f with compact support.

1 The Ap,p∗ condition was introduced by B. Muckenhoupt and R. Wheeden in [20]. We remit to Section 3 for more information about this class of weights.

Remarks on vector-valued Gagliardo and Poincaré–Sobolev-type inequalities with weights

� 269

We remark that Corollary 2.2 was obtained by B. Muckenhoupt and R. Wheeden in [20] with a proof based on the good-λ method, which is not sharp enough to derive the 1 ′

precise constant [w]An

p,p∗

at the right-hand side, which is indeed the sharpest possible.

This is the main novelty of this theorem.

2.2 Scalar weighted fractional Gagliardo inequality A fractional variant of Theorem 2.1 was obtained by the authors together with R. HurrySyrjänen and A. Vahakängäs in [16], motivated by the influential work by J. Bourgain, H. Brezis, and P. Mironescu in [3, Theorem 2.10]. In [22], an improvement of this result was obtained, enlarging the validity of the interval of the parameter δ from [ 21 , 1) to (0, 1), which is the natural domain. Theorem 2.3 ([16] and [22]). Let μ be a Radon measure in ℝn and let 0 < δ < 1. Then there exists a dimensional constant C(n) > 0 such that, for any Q ∈ 𝒬, and any function f ∈ C 1 (Q), ‖f − fQ ‖

n L n−δ

(Q,dμ)

≤ C(n)

1 |f (x) − f (y)| 1−δ c (n/δ)′ dx, dyM (χ μ)(x) ∫∫ Q δ |x − y|n+δ

(2.4)

Q Q

and hence ‖f ‖

n L n−δ

(ℝn ,dμ)

≤ C(n)

1 |f (x) − f (y)| 1−δ dyM c ( χQ μ)(x) (n/δ)′ dx, ∫∫ n+δ δ |x − y| n n

ℝ ℝ

for any C 1 (ℝn ) function with compact support. These results are sort of fractional isoperimetric inequalities. Here, the presence of the constant (1 − δ) in the right-hand side of the inequality is key. This result improves the one in [16] since it is valid for general measures μ instead of (general) weights and the smaller centered maximal function can be written at the right-hand side, instead of the noncentered one. Observe that there is no problem with the definition of the inte1

gral at the right-hand side, as the measure dν(x, y) := dyM c ( χQ μ)(x) (n/δ)′ dx is absolutely continuous with respect to the Lebesgue measure dy dx. Actually it is an interesting fact 1

that M c ( χQ μ) (n/δ)′ is an A1 weight. The same is true for the global case.

2.3 Main results: extensions to the vector-valued setting The main contribution of this paper is the investigation of extensions of the results mentioned above to the vector-valued setting. We restrict ourselves to the case of ℓq -valued

270 � J. M. Perales and C. Pérez functions, for q ≥ 1. This is enough to realize that some difficulties appear in the vectorvalued setting. About the classical scalar weighted Gagliardo inequality of Subsection 2.1, the following result is obtained. Theorem A. Let n ≥ 2 and let q ≥ 1. There exists a dimensional constant C(n) such that for any Radon measure μ, any cube Q ∈ 𝒬 and any vector-valued function f : ℝn → ℓq with C 1 (Q) components, 1 󵄩 󵄩󵄩 󵄩 󵄩 c 󵄩󵄩‖f − fQ ‖ℓq 󵄩󵄩󵄩Ln′ ,∞ (Q,dμ) ≤ C(n) ∫󵄩󵄩󵄩∇f (x)󵄩󵄩󵄩ℓq M ( χQ μ)(x) n′ dx.

(2.5)

Q

The lack of a truncation method in the vector-valued setting is the main reason why the above inequality is a weak inequality. Nevertheless, for a value of q below n′ , we are able to prove a strong version of the above inequality. Theorem B. Let n ≥ 2 and let 1 ≤ q ≤ n′ . Then for any Radon measure μ, any cube Q ∈ 𝒬 and any vector-valued function f : ℝn → ℓq with C 1 (Q) components, 1 󵄩󵄩 󵄩 󵄩 󵄩 c 󵄩󵄩‖f − fQ ‖ℓq 󵄩󵄩󵄩Ln′ (Q,dμ) ≤ γ(n) ∫󵄩󵄩󵄩∇f (x)󵄩󵄩󵄩ℓq M ( χQ μ)(x) n′ dx,

(2.6)

Q

where γ(n) is the best possible constant in the weighted local strong Gagliardo inequality (2.2). Consequently, if f : ℝn → ℓq is a vector-valued function with C 1 (ℝn ) components with common compact support, 1 󵄩󵄩 󵄩 󵄩 󵄩 c 󵄩󵄩‖f ‖ℓq 󵄩󵄩󵄩Ln′ (ℝn ,dμ) ≤ γ(n) ∫ 󵄩󵄩󵄩∇f (x)󵄩󵄩󵄩ℓ M ( χQ μ)(x) n′ dx. q

ℝn

In the special case of dμ = dx, we have the best possible estimate 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩‖f ‖ℓq 󵄩󵄩󵄩Ln′ (ℝn ,dx) ≤ β(n) ∫ 󵄩󵄩󵄩∇f (x)󵄩󵄩󵄩ℓq dx,

(2.7)

ℝn

where β(n) the isoperimetric constant (1.2). It seems natural, in view of the above two results, to pose the following conjectures. Conjecture A. Let n ≥ 2 and let q ≥ 1. There exists a dimensional constant C(n) > 0 such that for any Radon measure μ, any cube Q ∈ 𝒬 and any vector-valued function f : ℝn → ℓq with C 1 (Q) components, 1 󵄩 󵄩󵄩 󵄩 󵄩 c 󵄩󵄩‖f − fQ ‖ℓq 󵄩󵄩󵄩Ln′ (Q,dμ) ≤ C(n) ∫󵄩󵄩󵄩∇f (x)󵄩󵄩󵄩ℓq M ( χQ μ)(x) n′ dx.

Q

Remarks on vector-valued Gagliardo and Poincaré–Sobolev-type inequalities with weights

� 271

Conjecture B. Let n ≥ 2 and let q ≥ 1. For any vector-valued function f : ℝn → ℓq with C 1 components with compact support, 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩‖f ‖ℓq 󵄩󵄩󵄩Ln′ (ℝn ,dx) ≤ β(n) ∫ 󵄩󵄩󵄩∇f (x)󵄩󵄩󵄩ℓq dx, ℝn

where β(n) the isoperimetric constant (1.2). In the fractional context, our results are the following. Theorem C. Let n ≥ 2, let 0 < δ < 1, and let 1 ≤ q < ∞. There exists a constant C(n) > 0 such that, for any Radon measure μ, any cube Q ∈ 𝒬 and any vector-valued function f : ℝn → ℓq with C 1 (Q) components, ‖f (x) − f (y)‖ℓq 1 1−δ 󵄩󵄩 󵄩 dyM c μ(x) (n/δ)′ dx. ∫∫ 󵄩󵄩‖f − fQ ‖ℓq 󵄩󵄩󵄩L(n/δ)′ ,∞ (Q,dμ) ≤ C(n) n+δ δ |x − y| Q Q

Again, not having a truncation method in the vector-valued setting is the main reason why the above inequality is a weak inequality. For a value of q below (n/δ)′ , we are able to prove a strong version of the above inequality. n . Then, for any Radon measure μ, Theorem D. Let n ≥ 2, let 0 < δ < 1, and let 1 ≤ q ≤ n−δ n any cube Q ∈ 𝒬 and any vector-valued function f : ℝ → ℓq with C 1 (Q) components,

‖f (x) − f (y)‖ℓq 1 1−δ 󵄩󵄩 󵄩 dyM c μ(x) (n/δ)′ dx, ∫∫ 󵄩󵄩‖f − fQ ‖ℓq 󵄩󵄩󵄩L(n/δ)′ (Q,dμ) ≤ γ(n) n+δ δ |x − y| Q Q

where γ(n) is the best possible constant in the weighted, local, strong, Gagliardo inequality (2.4). Consequently, if f : ℝn → ℓq is a vector-valued function with C 1 (ℝn ) components with common compact support, ‖f (x) − f (y)‖ℓq 1 1−δ 󵄩󵄩 󵄩 dyM c μ(x) (n/δ)′ dx. ∫∫ 󵄩󵄩‖f ‖ℓq 󵄩󵄩󵄩L(n/δ)′ (ℝn ,dμ) ≤ γ(n) n+δ δ |x − y| n n ℝ ℝ

The conjecture in this case is then the following. Conjecture C. Let n ≥ 2, let 0 < δ < 1, and let 1 ≤ q < ∞. There exists a constant C(n) > 0 such that, for any Radon measure μ, any cube Q ∈ 𝒬 and any vector-valued function f : ℝn → ℓq with C 1 (Q) components, ‖f (x) − f (y)‖ℓq 1 1−δ 󵄩󵄩 󵄩 dyM c μ(x) (n/δ)′ dx. ∫∫ 󵄩󵄩‖f − fQ ‖ℓq 󵄩󵄩󵄩L(n/δ)′ (Q,dμ) ≤ C(n) n+δ δ |x − y| Q Q

272 � J. M. Perales and C. Pérez

3 Representation formulas and applications Representation formulas (sometimes called subrepresentation) are a very powerful tool in applications. This section is dedicated to the statement and proof of some of these results, which will be applied for proving Theorems A and C. In passing, we also derive some applications related to weighted Poincaré–Sobolev-type inequalities.

3.1 Riesz potentials and representation formulas As mentioned above, a very powerful tool in the study of Poincaré–Sobolev-type inequalities is the so-called representation formula, which is a pointwise estimate for the pointwise oscillation of a function in a cube. This formula involves the classical Riesz potential I1 , which is a particular case of the fractional integral operators (or Riesz potentials) Iα , defined for 0 < α < n and a measure ν by Iα (ν)(x) := ∫ ℝn

dν(y) , |x − y|n−α

x ∈ ℝn .

(3.1)

More precisely, the aforementioned representation formula establishes the existence of a constant C(n) > 0 such that, for any Q ∈ 𝒬 and any f ∈ C 1 (Q), 󵄨󵄨 󵄨 󵄨󵄨f (x) − fQ 󵄨󵄨󵄨 ≤ C(n)I1 ( χQ |∇f |)(x),

x ∈ Q.

(3.2)

The importance of this lemma relies on the fact that the Riesz potential I1 is a bounded ∗ operator from Lp (ℝn ) into Lp (ℝn ) for any 1 < p < n (see [1] and the recent monograph [17]). This fact is crucial in the proof of the weighted Poincaré–Sobolev inequality (1.4). Different techniques for proving Poincaré–Sobolev-type inequalities are known; see [21] for instance. n = n′ . In Note that the above boundedness result is false when p = 1 and 1∗ = n−1

this case, the Riesz potential I1 is just weakly bounded from L1 (ℝn ) into Ln (ℝn ). Nevertheless, the Poincaré–Sobolev inequality (1.4) can be proved for p = 1 by combining the representation formula (3.2), the weak boundedness of I1 and the so-called truncation method of Maz’ya [18]. This is a classical way to prove the classical Gagliardo inequality. An extension of the boundedness result we mentioned above to the weighted setting was obtained by B. Muckenhoupt and R. Wheeden [20]. They showed in that paper that I1 satisfies the weighted bound ′

‖wI1 f ‖Lp∗ (ℝn ) ≤ C‖wf ‖Lp (ℝn ) if and only if w satisfies the Ap,p∗ condition defined by

(3.3)

Remarks on vector-valued Gagliardo and Poincaré–Sobolev-type inequalities with weights

[w]Ap,p∗ = sup (− ∫ w(x)p dx)(− ∫ w(x)−p dx) ∗

Q∈𝒬

′

Q

p∗ p′

� 273

< ∞.

Q

At the endpoint p = 1, only the weak boundedness holds, and Muckenhoupt and Wheeden proved that the weak inequality ‖I1 f ‖Ln′ ,∞ (wn′ ) ≤ C‖wf ‖L1 (ℝn ) holds if and only if w satisfies the A1,n′ condition defined by (− ∫ wn (x) dx) ≤ C ess inf wn (x), ′

Q

′

Q

Q∈𝒬

where the smallest constant C is denoted as [w]A ′ . 1,n Using a different proof, (3.3) was improved in [14] where, for any 1 < p < n, the following optimal weighted bound for I1 was proved: 1

p′

′ max{1, p∗ } 󵄩󵄩 󵄩 ‖wf ‖Lp (ℝn ) . 󵄩󵄩wI1 ( f )󵄩󵄩󵄩Lp∗ (ℝn ) ≤ C(p)[w]An p,p∗

(3.4)

This allows to prove the existence of a constant C(n, p) > 0 such that, for any 1 < p < n and any sufficiently good function f , 1

1

p′

p ′ max{1, p∗ } 󵄩󵄩 󵄩 (∫ |w∇f |p dx) . 󵄩󵄩w( f − fQ )󵄩󵄩󵄩Lp∗ (Q,dx) ≤ C(n, p)[w]An p,p∗

(3.5)

Q

However, a better result has been derived in [22], as stated in Corollary 2.2. Observe that the case p = 1 is also covered by the result in [22]. Since this work is dedicated to the study of vector-valued extensions of these inequalities, the following lemma, which is a vector-valued variant of the representation formula (3.2), will be of great help in the sequel. Lemma 3.1. Let q ≥ 1. There is a dimensional constant C(n) > 0 such that, any cube Q of ℝn and for any vector-valued function f : ℝn → ℓq with components in C 1 (Q), the representation formula 󵄩󵄩 󵄩 󵄩󵄩f (x) − fQ 󵄩󵄩󵄩ℓq ≤ C(n)I1 (χQ ‖∇f ‖ℓq )(x) holds for every x ∈ Q.

274 � J. M. Perales and C. Pérez Proof. By the classical representation formula (3.2), we have, for every x ∈ Q, ∞

󵄨q

󵄩 󵄨 󵄩󵄩 󵄩󵄩f (x) − fQ 󵄩󵄩󵄩ℓq = [∑󵄨󵄨󵄨fj (x) − ( fj )Q 󵄨󵄨󵄨 ]

1 q

j=1

∞

≤ C(n)[∑(∫ j=1 Q

|∇fj (y)|

|x − y|n−1

q

dy) ]

1 q

1 q

∞

dy 󵄨 󵄨q ≤ C(n) ∫[∑󵄨󵄨󵄨∇fj (y)󵄨󵄨󵄨 ] |x − y|n−1 j=1 Q

= C(n)I1 (χQ ‖∇f ‖ℓq )(x), by Minkowski’s integral inequality. Combining the representation formula in Lemma 3.1 together with the precise boundedness properties of the fractional integral operator I1 stated in (3.4), we immediately get the following vector-valued version of the weighted Poincaré–Sobolev inequality. Corollary 3.2. Let 1 < p < n. Let w be a weight with w ∈ Ap,p∗ . There is a positive constant C = C(p, n) such that, for any cube Q of ℝn and for any vector-valued function f : ℝn → ℓq with components in C 1 (Q), we have 1 ∗

p 󵄩 󵄩 p∗ (∫(w(x)󵄩󵄩󵄩f (x) − fQ 󵄩󵄩󵄩ℓ ) dx) q

Q

1 n′

≤ C[w]A

p′

max{1, p∗ }

p,p∗

1

p 󵄩 󵄩 p (∫(w(x)󵄩󵄩󵄩∇f (x)󵄩󵄩󵄩ℓ ) dx) . q

Q

This result should be compared with Corollary 2.2 in the scalar situation. The exponent obtained here is much worse since, again, the truncation method is not at hand. On the fractional setting, we will be using the following representation lemma, which is based on the ideas in [10] and was used in [16] in its rough version. Here, we use the following variant for continuous functions, which provides a representation formula that holds for every point in the cube. The notation we use has been already defined in (3.1). Lemma 3.3. Let Q0 be a cube in ℝn . Assume that 0 < α < n and consider 0 < η < n−α. Let f ∈ C 1 (Q0 ) and let g be a nonnegative measurable function on Q0 such that for a positive constant κ,

Remarks on vector-valued Gagliardo and Poincaré–Sobolev-type inequalities with weights

� 275

󵄨 󵄨 − g(x) dx − 󵄨󵄨󵄨f (x) − fQ 󵄨󵄨󵄨 dx ≤ κℓ(Q)α ∫ ∫

(3.6)

Q

Q

for every cube Q ⊂ Q0 . Then there exists a dimensional constant C(n) > 0 such that κ 󵄨 󵄨󵄨 󵄨󵄨f (x) − fQ0 󵄨󵄨󵄨 ≤ C(n) Iα (gχQ0 )(x) η for every x ∈ Q0 . In the particular case that α < 1, we can choose η = 1, so we get 󵄨󵄨 󵄨 󵄨󵄨f (x) − fQ0 󵄨󵄨󵄨 ≤ C(n)κIα (gχQ0 )(x). For each 0 < δ < 1, we can use the fractional Poincaré-type inequality, |f (x) − f (y)| 󵄨 󵄨 dy dx, − 󵄨󵄨󵄨f (x) − fQ 󵄨󵄨󵄨 dx ≤ cn (1 − δ)ℓ(Q)δ ∫ −∫ ∫ |x − y|n+δ

Q

Q Q

which can be found in [3], as the starting inequality to get the representation formula 󵄨 󵄨󵄨 󵄨󵄨f (x) − fQ 󵄨󵄨󵄨 ≤ C(n)(1 − δ)Iδ (gf ,Q )(x)

(3.7)

for every x ∈ Q, after an application of Lemma 3.3 with α = δ and κ = C(n)(1 − δ) to the functions f and

gf ,Q (x) := ∫ Q

|f (x) − f (y)| dyχQ (x). |x − y|n+δ

To get our vector-valued result, we need a version of Lemma 3.1 within this frac-

tional context.

Lemma 3.4. Let q ≥ 1. There is a dimensional constant C(n) > 0 such that, for any cube

Q of ℝn and any vector-valued function f : ℝn → ℓq with components in C 1 (Q), the representation formula

󵄩󵄩 󵄩 󵄩󵄩f (x) − fQ 󵄩󵄩󵄩ℓq ≤ C(n)(1 − δ)Iδ (‖g fj ,Q ‖ℓq )(x), holds for every x ∈ Q. Proof. By using the scalar representation formula (3.7), we have that, for every x ∈ Q,

276 � J. M. Perales and C. Pérez

∞

󵄨q 󵄩 󵄨 󵄩󵄩 󵄩󵄩f (x) − fQ 󵄩󵄩󵄩ℓq = [∑󵄨󵄨󵄨fj (x) − ( fj )Q 󵄨󵄨󵄨 ]

1 q

j=1

q

∞

1 ≤ C(n)(1 − δ)[∑(∫ gf ,Q (y)dy) ] |x − y|n−δ j j=1

1 q

Q

= C(n)(1 − δ) ∫ Q

1 󵄩 󵄩󵄩 󵄩g f ,Q (y)󵄩󵄩󵄩ℓq dy, |x − y|n−δ 󵄩 j

by Minkowski’s integral inequality.

3.2 Riesz potentials and maximal operators Another very useful tool from harmonic analysis, which we will be using through Hedberg’s lemma, is the Hardy–Littlewood maximal operator. This operator can be defined in several ways, but we will just give two of them. The first one is the usual one, using cubes of ℝn : for a measure ν, we define M(ν)(x) = sup

𝒬∋Q∋x

ν(Q) , |Q|

x ∈ ℝn .

(3.8)

The second one is the one in which balls centered at the point of interest are used. For a measure ν, we define M c (ν)(x) = sup r>0

ν(B(y, r)) , |B(y, r)|

x ∈ ℝn .

(3.9)

As in the classical situation, the above representation formula using Riesz potentials combined with the following Hedberg-type lemma will be very useful. The proof is included for the convenience of the reader. Lemma 3.5. There is a constant a dimensional constant C(n) > 0 such that, for any finite measure ν and any 0 < α < n, Iα (ν)(x) ≤

α n−α C(n) ν(ℝn ) n M c (ν)(x) n , α

x ∈ ℝn .

(3.10)

Hence, if Q ∈ 𝒬, the following more classical inequality is recovered: Iα (χQ ν)(x) ≤

C(n) ℓ(Q)α M c ( χQ ν)(x), α

x ∈ Q.2

(3.11)

2 Here, νχQ means the measure ν restricted to Q. That is, the measure defined by νχQ (A) := ν(A ∩ Q) for any measurable set A.

Remarks on vector-valued Gagliardo and Poincaré–Sobolev-type inequalities with weights

� 277

Estimate (3.11) is really what is usually called Hedberg’s inequality; however, (3.10) is more relevant since it is the key in proving the generalized isoperimetric inequalities from Theorem 2.1 and their fractional counterpart in Theorem 2.3. Proof. Let us assume first that (3.10) holds and pick a cube Q of ℝn . By applying inequality (3.10) to the measure χQ ν, we get Iα ( χQ ν)(x) ≤

α n−α C(n) ν(Q) n M c (χQ ν)(x) n α α

n−α C(n) αn ν(Q) n c |Q| ( ) M ( χQ ν)(x) n = α |Q| α n−α C(n) αn c ≤ |Q| M (χQ ν)(x) n M c (χQ ν)(x) n α C(n) = ℓ(Q)α M c ( χQ ν)(x) α

for every x ∈ Q. This is the second part of the lemma. For the first one, let x ∈ ℝn and t > 0. By the layer-cake formula, ∞

∫ ℝn

dν(y) 1 = ∫ ν({y ∈ ℝn : > t}) dt |x − y|n−α |x − y|n−α 0 ∞

1

= ∫ ν({y ∈ ℝn : |x − y| < t − n−α }) dt, 0 1

and we recognize in the last line the measure of the ball B(x, t − n−α ). Then we can continue the above with ∞

∫ ℝn

1 dν(y) ≤ ∫ min{ν(ℝn ), ν(B(x, t − n−α ))} dt n−α |x − y|

0 ∞

1

ν(B(x, t − n−α )) 󵄨󵄨 − 1 󵄨 = ∫ min{ν(ℝ ), 󵄨󵄨B(x, t n−α )󵄨󵄨󵄨} dt 1 − n−α |B(x, t )| 0 n

∞

1

󵄨 󵄨 ≤ ∫ min{ν(ℝn ), M c ν(x)󵄨󵄨󵄨B(x, t − n−α )󵄨󵄨󵄨} dt 0

∞

n

≤ C(n) ∫ min{ν(ℝn ), M c ν(x)t − n−α } dt 0 c

( Mν(ℝν(x) n) )

= C(n)

∫ 0

n−α n

ν(ℝn ) dt

278 � J. M. Perales and C. Pérez ∞

+ C(n) c

( Mν(ℝν(x) n) )

=

n

M c ν(x)t − n−α dt

∫ n−α n

α n−α C(n) ν(ℝn ) n M c ν(x) n . α

3.3 Poincaré and Poincaré–Sobolev-type inequalities related to degenerate elliptic PDE The combination of the representation formula (3.2) with Hedberg’s inequality (3.11) is very effective. For instance, it yields the following result, which improves a well-known inequality by Fabes, Kenig, and Serapioni [7]. It is possible to prove the existence of a dimensional constant C(n) > 0 such that, for any p ≥ 1, any cube Q ∈ 𝒬 and any f ∈ C 1 (Q), we have 1

1

p p 1/p 󵄨 󵄨p 󵄨 󵄨p (∫󵄨󵄨󵄨f (x) − fQ 󵄨󵄨󵄨 u(x) dx) ≤ C(n)[u, v]A ℓ(Q)(∫󵄨󵄨󵄨∇f (x)󵄨󵄨󵄨 v(x) dx) , p

Q

(3.12)

Q

where recall that, for a given pair of weights u, v ∈ L1loc (ℝn ), we say that (u, v) ∈ Ap if p−1

[u, v]Ap := sup(− ∫ u(x) dx)(− ∫ v1−p (x) dx) ′

Q

Q

< ∞.

(3.13)

Q

Indeed, combining (3.2) and (3.11), we get that, for any Q ∈ 𝒬 and any f ∈ C 1 (Q), the inequality 󵄨󵄨 󵄨 󵄨󵄨f (x) − fQ 󵄨󵄨󵄨 ≤ C(n)ℓ(Q)M( χQ ∇f )(x) holds for any x ∈ Q. Then, if u ∈ L1loc (ℝn ) is a weight, we have the inequality 󵄩 󵄩 ‖f − fQ ‖Lp,∞ (du) ≤ C(n)ℓ(Q)󵄩󵄩󵄩M( χQ ∇f )󵄩󵄩󵄩Lp,∞ (du) for any cube Q ∈ 𝒬 and any f ∈ C 1 (Q). If now v ∈ L1loc (ℝn ) is such that (u, v) ∈ Ap , then it is well known that 1

‖M‖Lp (dv)→Lp,∞ (du) ≈ [u, v]Ap , p

and hence we get inequality

Remarks on vector-valued Gagliardo and Poincaré–Sobolev-type inequalities with weights

� 279

1

1

p 󵄨p 󵄨 ‖f − fQ ‖Lp,∞ (du) ≤ C(n)ℓ(Q)[u, v]Ap (∫󵄨󵄨󵄨∇f (x)󵄨󵄨󵄨 dv(x)) . p

Q

This yields (3.12) after an application of the truncation method. As a consequence of the above, and since (w, Mw) ∈ Ap with constant 1, we have 1

1

p p 󵄨p 󵄨 󵄨p 󵄨 (∫󵄨󵄨󵄨f (x) − fQ 󵄨󵄨󵄨 w(x) dx) ≤ C(n)ℓ(Q)(∫󵄨󵄨󵄨∇f (x)󵄨󵄨󵄨 Mw(x) dx) ,

Q

Q

and in particular, if w ∈ A1 , 1

1

p p 󵄨 󵄨p 󵄨 󵄨p (∫󵄨󵄨󵄨f (x) − fQ 󵄨󵄨󵄨 w(x) dx) ≤ C(n)[w]A1 ℓ(Q)(∫󵄨󵄨󵄨∇f (x)󵄨󵄨󵄨 w(x) dx) .

Q

Q

These estimates are central objects in the regularity theory of PDE [7, 13], mainly due to the presence of the fractional factor ℓ(Q). This allows to self-improve the above Poincaré-type inequality to a Poincaré–Sobolev-type inequality, in which the exponent of the left-hand side norm is larger than the one at the right-hand side. More precisely, it is possible to find a constant Cn,p and a positive number λ such that, for any f ∈ C 1 (Q) and any Q ∈ 𝒬, 1

p+λ 1 󵄨 󵄨p+λ ( ∫󵄨󵄨󵄨f (x) − fQ 󵄨󵄨󵄨 w(x) dx) w(Q)

Q

2 1 + p n

≤ Cn,p [w]A

1

1

p 1 󵄨 󵄨p ℓ(Q)( ∫󵄨󵄨󵄨∇f (x)󵄨󵄨󵄨 w(x) dx) . w(Q)

(3.14)

Q

We remit the reader to [21] for sharp results in this direction (see also [4]). To derive a vector-valued result in the spirit of (3.12), we cannot use the weighted bound of the maximal function for Ap -weights, since an ulterior application of the truncation method is not possible at the moment. However, we can still combine the vectorvalued representation formula in Lemma 3.1 together with Hedberg’s estimate (3.11) in Lemma 3.5 to derive Poincaré-type estimates. Indeed, it is well known (see [23] and [19]) that the strong two-weight boundedness of the maximal function is given in terms of the Sp condition, that is, given a pair of weights u, v ∈ L1loc(ℝn ) with (u, v) ∈ Sp , it holds that ‖M‖Lp (dv)→Lp (du) ≈n p′ [u, v]Sp , where the constant [u, v]Sp is defined by

(3.15)

280 � J. M. Perales and C. Pérez

[u, v]Sp := (

1/p

1 ∫ M(σχQ )p udx) σ(Q)

< ∞,

Q

with σ := v1−p . Then the following result is true: there is a dimensional constant C(n) > 0 such that, for any cube Q ∈ 𝒬, any vector-valued function f : ℝn → ℓq with C 1 (Q) components and any pair (u, v) ∈ Sp satisfying Sawyer’s condition, we have that ′

1

1

p p 󵄩 󵄩p 󵄩 󵄩p (∫󵄩󵄩󵄩f (x) − fQ 󵄩󵄩󵄩ℓ u(x)dx) ≤ C(n)p′ [u, v]Sp ℓ(Q)(∫󵄩󵄩󵄩∇f (x)󵄩󵄩󵄩ℓ v(x)dx) q q

Q

Q

if p > 1. We remark that this result should be compared with the scalar case (3.12) where the constant that appears is 1

‖M‖Lp (u)→Lp,∞ (v) ≈n [u, v]Ap ≤ [u, v]Sp . p

We believe that this constant should be here the same as well. When p = 1, we can prove the existence of a constant C(n) > 0 such that, for any cube Q ∈ 𝒬, any vector-valued function f : ℝn → ℓq with C 1 (Q) components and any weight w ∈ L1loc(ℝn ) , the inequality 󵄩 󵄩 󵄩 󵄩 ∫󵄩󵄩󵄩f (x) − fQ 󵄩󵄩󵄩ℓ w(x)dx ≤ C(n)ℓ(Q) ∫󵄩󵄩󵄩∇f (x)󵄩󵄩󵄩ℓ Mw(x)dx q q

Q

Q

holds. The argument is as follows: by Lemma 3.1, there is a constant C(n) > 0 such that, for any cube Q ∈ 𝒬, any vector-valued function f : ℝn → ℓq with C 1 (Q) components and any weight w ∈ L1loc(ℝn ) , 󵄩 󵄩 ∫󵄩󵄩󵄩f (x) − fQ 󵄩󵄩󵄩ℓ w(x)dx ≤ C(n) ∫ I1 (χQ ‖∇f ‖ℓq )w(x)dx q

Q

Q

󵄩 󵄩 = C(n) ∫󵄩󵄩󵄩∇f (x)󵄩󵄩󵄩ℓ I1 ( χQ w)(x)dx q

Q

󵄩 󵄩 ≤ C(n)ℓ(Q) ∫󵄩󵄩󵄩∇f (x)󵄩󵄩󵄩ℓ M( χQ w)(x)dx, q

Q

where the self-adjointness of I1 and inequality (3.11) were used. Then, in the case of A1 weights, the obtained inequality is 󵄩 󵄩 󵄩 󵄩 ∫󵄩󵄩󵄩f (x) − fQ 󵄩󵄩󵄩ℓ w(x)dx ≤ C(n)[w]A1 ℓ(Q) ∫󵄩󵄩󵄩∇f (x)󵄩󵄩󵄩ℓ w(x)dx. q q

Q

Q

Remarks on vector-valued Gagliardo and Poincaré–Sobolev-type inequalities with weights

� 281

If we want to derive vector-valued Poincaré–Sobolev-type estimates in the spirit of (3.14) we cannot use the tools in [21] but we can use the following result for the fractional operator I1 obtained in [2]. We just consider the special case, which best fits our interests, but the result there is a more general result. Theorem 3.6. Let 1 < p < n and let w ∈ Ap . Then there exists a constant C(n, p) > 0 such that, for all Q ∈ 𝒬 and for all f ≥ 0, 1

pn′ ′ 1 ( ∫ I1 ( f )pn w(x)dx) w(Q)

Q

1 (p−1)(np)′

≤ C(n, p)[w]A

p

1

p 1 ℓ(Q)( ∫ f p w(x)dx) . w(Q)

(3.16)

Q

Combining this result with the representation formula in Lemma 3.1, we get the following vector-valued Poincaré–Sobolev-type estimate. Theorem E. Let p and w as in the previous theorem and let q ≥ 1. There exists a constant C(n, p) > 0 such that, for all Q ∈ 𝒬 and each vector-valued function f : ℝn → ℓq with C 1 (Q) components, 1

(

pn′ 1 󵄩 󵄩pn′ ∫󵄩󵄩󵄩f (x) − fQ 󵄩󵄩󵄩ℓ w(x)dx) q w(Q)

Q

1 (p−1)(np)′

≤ C(n, p)[w]A

p

1

p 1 󵄩 󵄩p ℓ(Q)( ∫󵄩󵄩󵄩∇f (x)󵄩󵄩󵄩ℓ w(x)dx) . q w(Q)

Q

Observe that n′ p > p, so this is a weighted Poincaré–Sobolev type estimate.

4 Verification of Theorems A, B, C, and D In this section, we investigate the validity of vector-valued extensions of the so-called Gagliardo inequality (2.3). As already mentioned, in the scalar case, the way to prove this inequality is to first prove a weak inequality (using the weak boundedness of I1 in the case p = 1), and then to use the so-called truncation method, which allows us to get a strong inequality from a weak one. This tool is not available in principle for vectorvalued functions, and then we shall not be able to prove a general strong inequality. Nevertheless, it is possible to prove a strong inequality by restricting the range of possibilities for the ℓq norm.

282 � J. M. Perales and C. Pérez

4.1 Proof of Theorems A and C The arguments for both results are essentially the same, and they come basically from an application of the corresponding representation formula together with the following lemma, which is kind of a Fefferman–Stein inequality [9] for fractional integrals. Lemma 4.1. Let 0 < δ ≤ 1. Let Q ∈ 𝒬 be a cube of ℝn . There exists a dimensional constant C(n) > 0 such that 1 C(n) 󵄩󵄩 󵄩 ∫ h(x)(M c μ(x)) (n/δ)′ dx, 󵄩󵄩Iδ (hχQ )󵄩󵄩󵄩L(n/δ)′ ,∞ (Q,dμ) ≤ δ

(4.1)

Q

for any h ≥ 0 and any Radon measure μ. Proof. Let g ≥ 0 and consider a cube Q in ℝn . Let 0 < δ ≤ 1. Consider the level set EQ := {x ∈ Q : Iδ (gχQ )(x) > 1}, and assume μ(EQ ) > 0. Since Iδ is self-adjoint, μ(EQ ) ≤ ∫ Iδ (gχQ )(x) dμ(x) = ∫ g(x)Iδ (μχEQ )(x) dx. EQ

Q

Thus, applying Lemma 3.5 with ν = μχEQ and α = δ, we get μ(EQ ) ≤

1 δ C(n) μ(EQ ) n ∫ g(x)M c (μχQ )(x) (n/δ)′ dx. δ

Q

Hence, since μ(EQ ) > 0 1

μ(EQ ) (n/δ)′ ≤

1 C(n) ∫ g(x)M c (μχQ )(x) (n/δ)′ dx. δ

Q

Therefore, if we take a function h ≥ 0 and a cube Q ∈ 𝒬, then for any t > 0 such that μ({x ∈ Q : Iδ (hχQ )(x) > t}) > 0, we have that tμ({x ∈ Q : Iδ (hχQ )(x) > t})

1 (n/δ)′

1

(n/δ)′ h = tμ({x ∈ Q : Iδ ( χQ )(x) > 1}) t 1 C(n) h(x) c ≤ t∫ M (μχQ )(x) (n/δ)′ dx δ t

Q

Remarks on vector-valued Gagliardo and Poincaré–Sobolev-type inequalities with weights

=

� 283

1 C(n) ∫ h(x)M c (μχQ )(x) (n/δ)′ dx. δ

Q

This yields (4.1) by taking supremum on t > 0. Proof of Theorems A and C. Just apply Lemmas 3.1, 3.4, and 4.1. In the fractional case, apply also Minkowski’s integral inequality.

4.2 Proof of Theorems B and D Nevertheless, for vector-valued functions with values in ℓq with 1 ≤ q ≤ (n/δ)′ , 0 < δ ≤ 1, we are able to prove a strong inequality. Proof of Theorems B and D. We give the proof of both theorems at once since the key points are the same. This can be done by considering 1 ≤ q ≤ ( nδ )′ . This corresponds to Theorem B in the case δ = 1 and to Theorem D for 0 < δ < 1. The case q = ( nδ )′ is immediate since it reduces to the scalar situation. For the case 1 ≤ q < ( nδ )′ , let γ(n) be the optimal constant in (2.2) or (2.4), respectively and let r = ( nδ )′ q

> 1. Then

1 n ′

r

∞

( ) δ 󵄨 󵄨q 󵄩 󵄩( n )′ = [∫[∑󵄨󵄨󵄨fj (x) − ( fj )Q 󵄨󵄨󵄨 ] dμ(x)] (∫󵄩󵄩󵄩f (x) − fQ 󵄩󵄩󵄩ℓ δ dμ(x)) q

Q

Q

1 1 r q

j=1

∞

󵄨 󵄨q = [∫ ∑󵄨󵄨󵄨fj (x) − ( fj )Q 󵄨󵄨󵄨 h(x) dμ(x)] Q j=1 ∞

1 q

1 q

󵄨q

󵄨 = [∑ ∫󵄨󵄨󵄨fj (x) − ( fj )Q 󵄨󵄨󵄨 h(x) dμ(x)] , j=1 Q

where h ≥ 0 is a real-valued function such that r′

‖h‖Lr′ (Q,dμ) = (∫ h(x) dμ(x))

1 r′

= 1.

Q

Now, we note that, by Hölder’s inequality, the scalar inequality (2.2) in the classical case, the scalar inequality (2.4) in the fractional case, and Minkowski’s integral inequality (where we use q ≥ 1), we can continue with ∞

1 q

∞

q ′

n 󵄨 󵄨q 󵄨 󵄨n′ [∑ ∫󵄨󵄨󵄨fj (x) − ( fj )Q 󵄨󵄨󵄨 h(x) dx] = [∑[∫󵄨󵄨󵄨fj (x) − ( fj )Q 󵄨󵄨󵄨 dμ(x)] ]

j=1 Q

j=1 Q

1 q

284 � J. M. Perales and C. Pérez q

∞

1 󵄨 󵄨 ≤ γ(n)[∑[∫󵄨󵄨󵄨∇fj (x)󵄨󵄨󵄨M c μ(x) n′ dx] ]

1 q

j=1 Q

1 q

∞

1 󵄨q 󵄨 ≤ γ(n) ∫(∑󵄨󵄨󵄨∇fj (x)󵄨󵄨󵄨 ) M c μ(x) n′ dx,

j=1

Q

in the classical case, and with q n ′

∞

( ) δ 󵄨( n )′ 󵄨 [∑[∫󵄨󵄨󵄨fj (x) − ( fj )Q 󵄨󵄨󵄨 δ dμ(x)] ]

1 q

j=1 Q

q

1 |fj (x) − fj (y)| 1−δ ∞ c (n/δ)′ dx] ] [∑[∫ ∫ dyM μ(x) ≤ γ(n) δ |x − y|n+δ j=1

1 q

Q Q

q

1 q

∞ 1 |fj (x) − fj (y)| 1−δ dy) ] M c μ(x) (n/δ)′ dx, ≤ γ(n) ∫[∑(∫ n+δ δ |x − y| j=1 Q

Q

in the fractional case. The fractional inequality follows by applying once again Minkowski’s integral inequality. The global results follow by standard methods from the local one. We are only left with the special case of (2.7). The idea is the same being the case ′ q = n′ is immediate. So, assume that 1 ≤ q < n′ and let r = nq > 1, then 1 ′

r

∞

n 󵄨 󵄨q 󵄩 󵄩n ′ ( ∫ 󵄩󵄩󵄩f (x)󵄩󵄩󵄩ℓ dx) = [ ∫ [∑󵄨󵄨󵄨fj (x)󵄨󵄨󵄨 ] dx] q

ℝn

ℝn

1 1 r q

j=1

1 q

∞

󵄨 󵄨q = [∑ ∫ 󵄨󵄨󵄨fj (x)󵄨󵄨󵄨 h(x) dx] , j=1 ℝn

where h is a real-valued function such that ‖h‖Lr′ (ℝn ) = 1. Now, we can continue using

Hölder’s inequality with exponents r and r ′ , combined with the Gagliardo inequality (1.1) with optimal constant β(n) to get 1 ′

q ′

∞

1 q

∞

q

n n 󵄩 󵄩n ′ 󵄨 󵄨n ′ 󵄨 󵄨 ( ∫ 󵄩󵄩󵄩f (x)󵄩󵄩󵄩ℓ dx) ≤ [∑( ∫ 󵄨󵄨󵄨fj (x)󵄨󵄨󵄨 dx) ] ≤ β(n)[∑( ∫ 󵄨󵄨󵄨∇fj (x)󵄨󵄨󵄨 dx) ] q

ℝn

j=1 ℝn

j=1 ℝn

󵄩 󵄩 ≤ β(n) ∫ 󵄩󵄩󵄩∇f (x)󵄩󵄩󵄩ℓ dx. ℝn

q

1 q

Remarks on vector-valued Gagliardo and Poincaré–Sobolev-type inequalities with weights

� 285

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Index ℱN 158 Adams inequality 179 Adams/Hardy/John–Nirenberg/Lorentz/Sweezy spaces 215 Ahlfors regular 23 atomic decomposition 157 ball Banach function space 173 ball quasi-Banach function space 173 boundary value problems 49 conformal change in metric 23 CR volume doubling property 1 critical point setting 77 Euclidean logarithmic Sobolev inequality 77 evaescence 235 exponential growth 197 Exponential integrability 127 Gagliardo 265 Generalized ultraspherical polynomials 1 Gromov hyperbolic filling 23 Hardy spaces 157 Hausdorff content 127 higher order 179 inner or outer or zero-capacity regularity 235 Integral inequality 179 Liquid crystal droplets 103 locally finite 235 M-uniform domains 103 maximal operator 265

https://doi.org/10.1515/9783110792720-013

metric space 23 mixed-norm Besov spaces 49 Monotonicty 235 Muckenhoupt 265 non-smooth domain 127 optimal stability 77 outer minimal sets 103 Poincaré inequality 127 Poincaré–Sobolev 265 point-wise estimate 127 polynomial growth harmonic functions 1 quasisymmetry 23 Riesz potential 127 semifinite 235 singular and degenerate coefficients 49 Sobolev Inequality and mean value inequality 1 strong or finite or countable subadditivity 235 Trace theorems 49 Trudinger–Moser inequalities 197 uniformization 23 uniformly perfect 23 variable exponent 127, 157 weight 265 weighted mixed-norm Sobolev spaces 49 Weighted Sobolev spaces 197 Wheeden 265

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