Geometric Potential Analysis 9783110741711, 9783110741674

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Mario Milman, Jie Xiao, and Boguslaw Zegarlinski (Eds.) Geometric Potential Analysis

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 6

Geometric Potential Analysis |

Edited by Mario Milman, Jie Xiao, and Boguslaw Zegarlinski

Editors Prof. Mario Milman Instituto Argentino de Matematica Buenos Aires Argentina [email protected] Prof. Jie Xiao Memorial University Department of Mathematics & Statistics 230 Elizabeth Ave. St. John’s, NL A1C 5S7 Canada [email protected]

Prof. Boguslaw Zegarlinski Imperial College London Department of Mathematics South Kensington Campus Huxley Building 180 Exhibition Road London SW7 2AZ UK [email protected]

ISBN 978-3-11-074167-4 e-ISBN (PDF) 978-3-11-074171-1 e-ISBN (EPUB) 978-3-11-074189-6 ISSN 2511-0438 Library of Congress Control Number: 2022932951 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. © 2022 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Foreword The stated goal of the Mathematical Congress of the Americas (MCA) is to highlight the excellence of mathematical achievements in the Americas, and to foster international collaborations of mathematicians at all levels. Due to the current pandemic, the 2021 edition that was scheduled to take place in Buenos Aires (Argentina) was conducted virtually. The program featured colloquium talks by renowned mathematicians and special sessions where prominent researchers presented the latest advances in their specialities. The editors of this volume organized a special session on “Geometric Potential Analysis”. The special session was run for three days (July 15–16 & 19, 2021) and was very well received.

At the request of many of the researchers and students who took part in our session, and taking into account the high quality of the event, we decided to edit a special volume in Advances in Analysis and Geometry – De Gruyter. The volume features papers by some of the speakers of our session, as well as a few specialists that were in attendance. It is our hope that the volume will be a valuable resource for graduate students and researchers working in such areas as Function Spaces and Potential Theory with applications to Analysis, Geometry, Probability, Differential Equations, Image Processing, and Mathematical Physics. Mario Milman ([email protected]) Jie Xiao ([email protected]) Boguslaw Zegarlinski ([email protected]) https://doi.org/10.1515/9783110741711-201

Contents Foreword | V Xinfeng Wu, Der-Chen Chang, and Yongsheng Han Hardy spaces associated to the Kohn Laplacian on a family of model domains in Cn+1 | 1 Julian Haddad and Jie Xiao Affine mixed Rayleigh quotients | 29 Liguang Liu and Suqing Wu Riesz potentials and Hardy–Morrey spaces | 49 Laurent Saloff-Coste A conjecture regarding compact Lie groups and its consequences | 79 Sheng-Ya Feng and Der-Chen Chang The singular Fredholm integral operators and related integral equations of Chandrasekhar type | 89 Lijia Zhang and Wengu Chen Extended multichannel weighted Schatten p-norm minimization for color image denoising | 105 Guoliang Li and Junfeng Li The boundedness of a bilinear oscillatory integral | 123 Sergey G. Bobkov and Cyril Roberto On sharp Sobolev-type inequalities for multidimensional Cauchy measures | 135 Guixiang Hong Banach lattice-valued q-variation and convexity | 153 Yanping Chen and Xueting Han Weighted boundedness of Marcinkiewicz integral operators and related singular integral operators | 167 Alina Stancu A spectral comparison note | 195

VIII | Contents Yuhua Sun and Fanheng Xu On existence and nonexistence of nonnegative solutions to heat equation on Riemannian manifolds | 207 Dorina Mitrea, Irina Mitrea, and Marius Mitrea The Neumann problem for the Laplacian in half-spaces | 219 Chunyan Liu and Ning Zhang An inequality on the mass of image of rectifiable chain under Lipschitz map | 253 Eduard Navas, Ebner Pineda, and Wilfredo O. Urbina Boundedness of general alternative Gaussian singular integrals on Gaussian variable Lebesgue spaces | 261 Yang Li and Ewelina Zatorska Remarks on weak–strong uniqueness for two-fluid model | 281 Esther Bou Dagher, Yifu Wang, and Boguslaw Zegarlinski Coercive inequalities on Carnot groups and applications | 291 Shihshu Walter Wei On exponential Yang–Mills fields and p-Yang–Mills fields | 317 Index | 359

Xinfeng Wu, Der-Chen Chang, and Yongsheng Han

Hardy spaces associated to the Kohn Laplacian on a family of model domains in Cn+1 Abstract: Consider a family of hypersurfaces in Cn+1 : n

2

m

Ωm = {(z1 , z2 , . . . , zn , zn+1 ) : Im(zn+1 ) = ( ∑ |zk | ) }, k=1

m ∈ N.

In this paper, we establish a Hardy space theory on Mm (the boundary manifold of Ωm ) via a new discrete square function constructed from the heat kernel. We prove that a class of singular integral operators are bounded on the Hardy spaces H p (Mm ), and are 2m+2n bounded from H p (Mm ) to Lp (Mm ) for 2m+2n+ϑ < p ≤ 1 with 0 < ϑ < 1. As an application, sharp estimates for the fundamental solution of the Kohn Laplacian n

Δλ,k = ∑(Zj Z̄ j + Z̄ j Zj ) + λ[Z̄ j , Zj ] j=1

on H p (Mm ) are derived. Keywords: Hardy spaces, Kohn Laplacian, model domains, Calderón’s reproducing formula, heat kernel, NIS operators MSC 2010: Primary 42B25, Secondary 42B20

Acknowledgement: Chang is partially supported by NSF grant DMS-1408839 and a McDevitt Endowment Fund at Georgetown University. Wu is supported by National Nature Science Foundation in China (Nos. 11671397, 12071473). This paper is based on a lecture that was delivered by the second author at the Geometric Potential Analysis session, Mathematical Congress of the Americas 2021. The authors would like to thank Professor Jie Xiao for his invitation. Xinfeng Wu, Department of Mathematics, China University of Mining & Technology, Beijing 100083, P.R. China, e-mail: [email protected] Der-Chen Chang, Department of Mathematics and Statistics, Georgetown University, Washington, DC 20057, USA; and Graduate Institute of Business Administration, College of Management, Fu Jen Catholic University, New Taipei City 242, Taiwan, ROC, e-mail: [email protected] Yongsheng Han, Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA, e-mail: [email protected] https://doi.org/10.1515/9783110741711-001

2 | X. Wu et al.

1 Subelliptic operators In this paper, we introduce a new method to prove estimates for the fundamental solution of a subelliptic operator (see, e. g., Hörmander [33]) ̄ + λ[Z,̄ Z] Δλ = Z Z̄ + ZZ

(1.1)

where Z=

𝜕 𝜕 + i𝒫 ′ (z)𝒫 (z) . 𝜕z 𝜕t

Here 𝒫 (z) is a holomorphic polynomial in the complex plane. As usual, set z = x +iy := (x, y) and 1 Z = (X − iY) 2 𝜕 𝜕 1 𝜕 𝜕 = {( − 2Im[𝒫 ′ (z)𝒫 (z)] ) − i( − 2Re[𝒫 ′ (z)𝒫 (z)] )}. 2 𝜕x 𝜕t 𝜕y 𝜕t With this notation, (1.1) takes the form 1 Δλ = (X 2 + Y 2 − iλ[X, Y]). 2

(1.2)

The fundamental solution Kλ (z, t; z0 , t0 ) = Kλ (x, y, t; x0 , y0 , t0 ) of the operator Δλ is a distribution in (z, t) which satisfies Δλ Kλ (z, t; z0 , t0 ) = δ(z − z0 )δ(t − t0 ). Since the operator Δλ is translation-invariant along the t-direction, we may assume that t0 = 0. Inspired by the works [3–5, 7, 9], we shall look for a Kλ of the form +∞

+∞

−∞

−∞

E(z, z0 , τ) V (z, z0 , τ) dτ = − ∫ Vλ d(log g), Kλ (z, t; z0 ) = ∫ g(z, t; z0 , τ) λ

(1.3)

where the function g satisfies the Hamilton–Jacobi equation for a certain complex Hamiltonian, namely 𝜕g + [(Xg)2 + (Yg)2 ] = 0, 𝜕τ and E = − 𝜕g is the associated total energy, the first invariant of motion. The function 𝜕τ Vλ is the volume element which is a solution of a second-order transport equation Δλ (EVλ ) +

𝜕 (T(Vλ ) + (Δλ g)Vλ ) = 0, 𝜕τ

(1.4)

Hardy spaces associated to the Kohn Laplacian on a family of model domains in Cn+1

| 3

where T=

𝜕 + (Xg)X + (Yg)Y 𝜕τ

is differentiation along the bicharacteristic, i. e., T is the usual first-order transport operator. More precisely, let 1 1 2 2 H(x, y, t; ξ , θ) = (ξ − 2Im[𝒫 ′ (z)𝒫 (z)]θ) + (ζ − 2Re[𝒫 ′ (z)𝒫 (z)]θ) 2 2

(1.5)

be the Hamilton function with (ξ , ζ , θ) being the variables dual to (x, y, t). The bicharacteristic curves are complex, which is a consequence of nonstandard boundary conditions. It is known that g is given by an action integral in terms of these bicharacteristics. The form (1.3) has a simple geometric interpretation. The operator Δλ in (1.1) has a characteristic variety in the cotangent bundle given by H(x, y, t, ξ , ζ , θ) = 0. Over every point (x, y, t) = (z, t), it is a line parametrized by the variable τ, ξ = 2Im[𝒫 ′ (z)𝒫 (z)]τ,

ζ = 2Re[𝒫 ′ (z)𝒫 (z)]τ,

θ = τ,

with τ ∈ R. Intuitively, Kλ (z, t; z0 , t0 ) may be considered as the (action) the characteristic variety over (z0 , t0 ) with respect to measure EVλ .

−1

integrated on

Remark 1. Before we go further, let us give an explanation of the geometric meaning of the formula (1.3). Recall that the fundamental singularity, i. e., “good” approximation to the fundamental solution, of the classical Laplace–Beltrami operator − 21 ∑nj=1 Xj∗ Xj n−2

is the Newtonian potential Cn (x0 )(g(x, x0 ))− 2 . Here {X1 , X2 , . . . , Xn } is an orthonormal basis of the tangent bundle TMn which yields a Riemannian distance function d(x, x0 ). Here Xj∗ denotes the formal adjoint of Xj with respect to the measure induced by this Riemannian metric and g(x, x0 ) = d2 (x, x0 ) is the classical action. Since the Laplace– Beltrami operator has trivial characteristic variety, the above interpretation of (1.3) formally subsumes the Newtonian potential over a zero-section; the power of the action to the denominator is immaterial, it is obtained from dimensionality considerations. The integral (1.3) can be calculated by a residue calculation. Let us start with the simplest case 𝒫 (z) = z. In this case, X=

𝜕 𝜕 + 2y , 𝜕x 𝜕t

Y=

𝜕 𝜕 − 2x . 𝜕y 𝜕t

It is easy to see that the operator Δλ is left invariant with respect to the Heisenberg translation (see, e. g., Stein [36]): (z, t) ∘ (z0 , t0 ) = (z + z0 , t + t0 + 2Im(z z0̄ )) = (z + z0 , t + t0 + 2[yx0 − xy0 ]).

(1.6)

4 | X. Wu et al. This yields Kλ (z, t; z0 , t0 ) = Kλ ((z0 , t0 )−1 ∘ (z, t); 0, 0). Explicitly, we have +∞

e−2λτ 2 csch(2τ) 1 dτ, Kλ (z, t; 0, 0) = − 2 ∫ 4π |z|2 coth(2τ) − it

(1.7)

−∞

see [4, 5, 9, 25]. The formula (1.7) is a special case of (1.3), where g = |z|2 coth(2τ) − it,

E=−

𝜕g 2|z|2 = , 𝜕τ sinh2 (2τ)

Vλ = −

1 −2λτ sinh(2τ) e . 4π 2 |z|2

Denote g± = lim g = ±|z|2 − it := ±r 2 − it. τ→±∞

It follows that +∞

g+

−∞

g−

V g V V Kλ (z, t; 0, 0) = − ∫ τ λ dτ = − ∫ λ dg = ∫ λ dg, g g g

(1.8)

𝒞−

where we made a branch cut 𝒞 = (−∞ − it, g− ] ∪ [g+ , +∞ − it), V

with upper and lower directed edges denoted by 𝒞+ and 𝒞− , respectively. Integrals of gλ vanish on upper and lower semicircles as their radii increase without bound. Therefore we can integrate on a “dumbbell” with waist at g± (see Figure 1.1).

Figure 1.1: Integration on a “dumbbell” with waist at g± .

Hardy spaces associated to the Kohn Laplacian on a family of model domains in Cn+1

Inside this domain,

Vλ g

| 5

has a simple pole at g = 0. Hence we have

iπ(1−λ)

1+λ V e 2 1 − 1−λ − g+ 2 (−g− )− 2 = Resg=0 λ = − 2 g 2πi 4π

∫ 𝒞− ∪(−𝒞+ )

Vλ dg g λ

=−

1 + e−iπλ Vλ ie−iπ 2 π dg = cos( λ)Kλ , ∫ 2πi g π 2 𝒞−

and thus we obtain Kλ (z, t; 0, 0) = −

)Γ( 1−λ ) Γ( 1+λ 2 2 4π 2

− 1−λ 2

(|z|2 − it)

(|z|2 + it)

− 1+λ 2

,

(1.9)

where |z| = ∑nj=1 |zj |2 . This agrees with the result of Folland–Stein [23] and Greiner– Stein [27, 28]. We note that the change of variable τ → g is reminiscent of classical calculations in action-angle coordinates. For a general holomorphic f (z), Δλ is not translation invariant with respect to any underlying group structure, and we must calculate Kλ (z, t; z0 , t0 ) for an arbitrary (z0 , t0 ). When (z0 , t0 ) ≠ (0, 0), we obtain a second invariant of motion, the angular momentum Ω(z, z0 , τ). The evaluation of formula (1.3) by a residue calculation shows that g± play a crucial role in the final formula, although g drops out, as it should. These observations help us construct new coordinates from g± and Ω± = limτ→±∞ Ω, which reduce the second-order transport equation (1.4) to a hypergeometric differential equation in two variables; this was already solved by Appell–Picard in the 1880s (see [1]). The main purpose of this paper is to study sharp estimates in Lebesgue and Hardy spaces for the fundamental solution of the operator Δλ on some model domains in C2 (which can be generalized to a family of model domains in Cn+1 ). This problem has been studied by many mathematicians, readers can find some of the results in, e. g., [5, 14, 20, 25, 26], etc. A complete solution for the case 𝒫 (z) = z m with |Re(λ)| < 1 can be found in a forthcoming paper [10]. In this case, 𝜕 𝜕 𝜕 𝜕 − imz m z̄m−1 = − imz|z|2(m−1) . Z̄ = 𝜕z̄ 𝜕t 𝜕z̄ 𝜕t

(1.10)

As we mentioned in the beginning of this section, the operator is closely related to the 𝜕̄b complex on a family of domains in C2 . Before we go further, let us give a brief introduction to the connection of function 𝒫 (z) = z m and complex geometry. When m = 1, the hypersurface {(z, w) ∈ C2 : Im(w) = |𝒫 (z)|2 = |z|2 } can be considered at the boundary of the Siegel domain in C2 , Ω1 = {(z, w) ∈ C2 : Im(w) > |z|2 },

6 | X. Wu et al. and the boundary 𝜕Ω1 can be considered as an orbit of the origin (0, 0) translated under the Heisenberg group (1.5). Hence, we may identify 𝜕Ω1 as the 1-dimensional Heisenberg group. See, e. g., [23, 27, 28]. In general, the nonisotropic Heisenberg group can be considered as the boundary of the strongly pseudoconvex domain n

Ω1,n = {(z1 , . . . , zn , zn+1 )) ∈ Cn+1 : Im(zn+1 ) > ∑ aj |zj |2 }, j=1

and Abelian extensions [6]. In this case, we have n “good” antiholomorphic vector fields, 𝜕 𝜕 − iaj zj , Z̄ j = 𝜕zj̄ 𝜕t with one “missing direction”

j = 1, 2, . . . , n,

𝜕 . 𝜕t

Remark 2. Unfortunately, the simple formula (1.9) is just a lucky coincidence of too much symmetry, and on general nonisotropic Heisenberg groups the fundamental solutions are of the form (1.3). The reason is that the fundamental solution must include all the distances, which necessitates the use of g, and the summation over all the distances means integration on τ. Note that the change of variables τ → g is reminiscent of classical calculations in action-angle coordinates. The fundamental solution for the Kohn Laplacian Δλ⃗ = ∑nj=1 (Z̄ j Zj + Zj Z̄ j ) + λj [Z̄ j , Zj ] on the boundary of Ω1,n has the following form: 2e−λj τ aj csch(2aj τ) (n − 1)! n Kλ⃗ (z, t) = − dτ, ∏ ∫ n (2π)n+1 j=1 (∑j=1 aj |zj |2 coth(2aj τ) − it)n +∞

−∞

where λ⃗ = (λ1 , . . . , λn ) with λj = ∑nk=1 ak − 2aj . This operator is invariant under left translations of the nonisotropic Heisenberg group: n

n

n

j=1

j=1

j=1

(z, t) ⋅ (w, s) = (z + w, t + s + 2Im ∑ aj zj wj ) = (∑ zj + wj , t + s + 2Im ∑ aj zj wj ). When aj = 1 for j = 1, . . . , n, the integral can be calculated explicitly and recovers (1.9), the famous Folland–Stein formula in [23], namely Kλ (z, t) =

Γ( n+λ )Γ( n−λ ) 1 2 2 . n−λ 2 2π (|z|2 + it) 2 (|z|2 − it) n+λ 2

In particular, if λ = 0, the fundamental solution will be K0 (z, t) =

2

2n−1 n −n Γ( ) [|z|4 + t 2 ] 2 . n+1 2 π

Hardy spaces associated to the Kohn Laplacian on a family of model domains in Cn+1

| 7

This is the fundamental solution for the “sub-Laplacian” on the Heisenberg group which was first studied by Folland [22]. To simplify our calculation, here we just consider the case for n = 1, i. e., C2 case. For 𝒫 (z) = z m with m ≥ 2, we may consider the Kohn Laplacian Δλ as a differential operator defined on the hypersurfaces 󵄨2 󵄨 Mm = 𝜕Ωm = {(z, w) ∈ C2 : Im(w) = 󵄨󵄨󵄨𝒫 (z)󵄨󵄨󵄨 = |z|2m } ⊂ C2 ,

m = 2, 3, . . . .

(1.11)

Since {Z, Z}̄ satisfies the bracket generating property of step 2m, by Chow’s theorem [8, 13], we known that, given A, B ∈ Mm , there exists a piecewise 𝒞 1 curve γ : [0, 1] → Mm which is a so-called horizontal curve such that γ(0) = A and γ(1) = B, and such that for s ∈ [0, 1], ̄ ̇ = α(s)Z(γ(s)) + β(s)Z(γ(s)). γ(s) In other words, the manifold Mm is topologically connected by horizontal curves. 1 Thus, the length of ℓ(γ) = ∫0 √|α(s)|2 + |β(s)|2 ds. The infimum of ℓ(γ) is the Carnot– Carathéodory distance dcc (A, B) between A and B. Hence Mm is a sub-Riemannian manifold. By a theorem of Fefferman–Phong [21], there exists a constant cm > 0 such that 1

BE (x, ρ) ⊆ Bℒ (x, cm ρ 2m ) for all x ∈ Mm , 0 < ρ < 1.

(1.12)

In fact, (1.12) is equivalent to saying that Δλ satisfies the subelliptic estimate 󵄩󵄩 m1 󵄩󵄩 󵄩󵄩|∇| u󵄩󵄩L2 ≤ cm {‖Δλ u‖L2 + c̃m ‖u‖L2 },

for all x ∈ Mm

1

(1.13)

and for all u ∈ C ∞ (Mm ). As usual, |∇| m is a pseudodifferential operator with symbol 1 |ξ | m . Thus, the operator Δλ is subelliptic and hence hypoelliptic (which recovers a theorem of Hörmander [33]). In the case of (1.9), Mm lacks a group structure when m ≥ 2 and the complex bicharacteristics run between 2 arbitrary points (z, t) and (w, s). We obtain 2 invariants of the motion, the energy E and the angular momentum Ω. It is impossible to calculate them explicitly, but we know their analytic properties, and g and Vλ may be found in terms of E and Ω. For example, g = −i(t − s) + (1 − +

1 )Eτ m

1 2 1 2 1 sgn(τ)[(2E|z|2 + W(|z|2 |) ) 2 − (2E|w|2 + W(|w|2 ) ) 2 ], 2m

where one uses the principal branch of the square roots, and W(v) = 2mvm − Ω, We have the following theorem.

Ω = Ω(z, t; w, s; τ).

8 | X. Wu et al. Theorem 1.1. Let 𝒫 (z) = z m , so that Z=

𝜕 𝜕 + imz̄ ⋅ |z|2(m−1) . 𝜕z 𝜕t

Then, for |Re(λ)| < 1, the fundamental solution Kλ (z, t; w, s) for the operator Δλ = Z Z̄ + Z̄ Z + λ[Z,̄ Z] has the following invariant representation: Kλ (z, t; w, s) = ∫ R

EVλ dτ, g

(1.14)

where the second-order transport equation for Vλ may be reduced to an Euler–Poisson– Darbeau equation and solved explicitly as a function of E and Ω. Namely, Vλ (z, t; w, s) =

1−λ 1+λ Cλ (A − g)− 2 (A− + g)− 2 Fλ (η+ , η− ), m +

(1.15)

where −eiπ

Cλ =

1−λ 2

)Γ( 1+λ ) 4π 2 Γ( 1−λ 2 2

and A± = |z|2m + |w|2m ± i(t − s) =

Ω± + g± , m

Ω± = lim Ω,

g± = lim g,

τ→±∞

τ→±∞

and Fλ (η+ , η− ) is a hypergeometric function of 2 variables, Γ(

1+λ 1−λ )Γ( )Fλ (p+ , p− ) 2 2 1 1

η = ∫ ∫{( + ) 1 − η+

1−λ 2

0 0

×

η ( − ) 1 − η−

1+λ 2

1

1 m

1 − p+ p− (η+ η− ) m 1 m

(1 − p+ η+ )(1 − p− η− )(1 −

(p+ p− )m η+ η− )

}

dη+ dη− η+ η−

with 1

p+ = p− =

2 m z w̄ 1

A+m

− m1

mg+ = (1 + ) Ω+

.

A fundamental solution for the case m = 2 was first given by Greiner [25] in various forms, one of which is Kλ (z, t; w, s) =

i −p − p̄ − i|1 − p2 | log[ ], 2 2π σ 1 + |p|2

(1.16)

Hardy spaces associated to the Kohn Laplacian on a family of model domains in Cn+1

| 9

where z = x + iy, w = u + iv, and 2 󵄨 󵄨4 σ 2 = 󵄨󵄨󵄨z 2 − w2 󵄨󵄨󵄨 + (t − s + 2Im[z 2 w̄ 2 ]) , 4 ̄ p̄ = 2(z w)(|z| + |w|4 − i(t − s)) . −1

After more detailed calculation, we also have the following theorem. Theorem 1.2. The function Kλ of Theorem 1.1 has a meromorphic extension in the variable λ with simple poles at points in the following exceptional set: ℰλ = {λ = ±(2k + 1 +

2j ), m

k ∈ Z+ , j = 0, 1, 2, . . . , m − 1}.

(1.17)

For λ ∈ ̸ ℰλ , Kλ is real-analytic at (z, t) = (w, t0 ) and is a fundamental solution for the operator Δλ . Moreover, Kλ (z, t; w, s) = K−λ (w, s; z, t),

Kλ (z, t; w, s) = K−λ̄ (z, t; w, s),

and Kλ (z, t; w, s) = Kλ̄ (w, s; z, t). Furthermore, Δλ is real-analytic hypoelliptic whenever λ is not among the poles (1.17). Moreover, from Theorem 1.2, we can also derive the following result. Theorem 1.3. The function Kλ of Theorem 1.1 has residues of the following form. At ±λ = 2j 2k + 1 + m , k ∈ Z+ , j = 0, 1, 2, . . . , m − 1, these residues have the form − 1−λ 2

A+

− 1+λ 2

A−

± ̄ ⋅ 𝒫k,j (p, p),

(1.18)

− − (p, p) ̄ are polynomials in two variables. where 𝒫k,j (p, p)̄ = 𝒫k,j At λ = 2k + 1, k ∈ Z+ , the residues are of the form

Ak+ A−k−1 ⋅ 𝒫k (p, p,̄ (1 − p)̄ −1 ), −

(1.19)

and at λ = −2k − 1, k ∈ Z+ , the residues are of the form A−k−1 Ak− ⋅ 𝒫k (p,̄ p, (1 − p)−1 ), + where 𝒫k , k ∈ Z+ , are polynomials in three variables.

(1.20)

10 | X. Wu et al. The construction of the fundamental solution (1.14) and the proofs of Theorems 1.2 and 1.3 are long and tedious, we are going to skip all the details here. We refer the interested readers to papers [2, 10, 26], etc. As we have seen in [7, 27, 28] for the case m = 1, the residue K0 (λ0 ) of Kλ at a point λ0 ∈ ℰλ yields a projection onto the kernel of Δλ0 , while the next term K1 (λ0 ) in the following Laurent series expansion is the kernel of a partial inverse. More precisely, Kλ = (λ − λ0 )−1 K0 (λ0 ) + K1 (λ0 ) + (λ − λ0 )K2 (λ0 ) + ⋅ ⋅ ⋅ . We may obtain similar results for the case m ≥ 2. In particular, when k = 0 and j = 0, we have λ = −1. We make use of the projection onto the kernel of Δ−1 = 2Z Z̄ to obtain the orthogonal projections onto the kernel of Z̄ which is the Cauchy–Szegő projection. After a long calculation (which we will omit here), one has the following theorem. Theorem 1.4. The Cauchy–Szegő kernel on the boundary of Ω̄ 2 = {(z1 , z2 ) ∈ C2 : Im(z2 ) ≥ |z1 |2m } is 1

𝒞 (z, t; w, s) =

=

2m

m+1

2π 2 A+m (1 − p+ )2

1

2π 2 (|z|2m + |w|2m − i(t − s))

m−1 m

2m

1

1

(|z|2m + |w|2m − i(t − s)) m − 2 m z w)̄ 2

. (1.21)

In particular, when m = 1, one has 𝒞 (z, t; w, s) =

π 2 (|z

−

1 . − i(t − s))2

w|2

2 Heat kernel and the discrete Calderón’s reproducing formula 2.1 Preliminaries Recall that a quasimetric d on Mm satisfies the following conditions: (i) d(x, y) = 0 iff x = y; (ii) d(x, y) = d(y, x); (iii) d(x, z) ≤ A[d(x, y) + d(y, z)] for some A ≥ 1. We shall use a pseudometric d (equivalent to the control metric dcc ) which has the optimal smoothness, i. e., d(x, y) is 𝒞 ∞ on {Mm × Mm \diagonal}, and for x ≠ y, 󵄨󵄨 K L 󵄨 1−K−L . 󵄨󵄨𝜕X 𝜕Y d(x, y)󵄨󵄨󵄨 ≲ d(x, y)

(2.1)

Hardy spaces associated to the Kohn Laplacian on a family of model domains in Cn+1

| 11

(Here 𝜕XK is a product of K of the vector fields {X1 , X2 } acting as derivatives on the x-variable, and 𝜕YL are corresponding L vector fields acting on the y-variable.) For the existence of such a pseudometric, see Theorems 3.3.1 and 4.4.6 in [34], where d is denoted by ρ.̃ It follows from (2.1) that for any 0 < ϑ < 1, 1−ϑ 󵄨 󵄨󵄨 ′ ϑ ′ ′ 󵄨󵄨d(x, y) − d(x , y)󵄨󵄨󵄨 ≤ C0 d(x, x ) [d(x, y) + d(x , y)]

for all x, x′ , y ∈ Mm . Throughout this section, we shall fix ϑ ∈ (0, 1). Following [35], we may identify Mm with C × R, and the measure we shall use is just the Lebesgue measure, which will be denoted by dx. We denote the measure of a set E by |E|. We define balls B(x, δ) = {y ∈ Mm : d(x, y) < r} with 0 < δ < ∞. We have the following formula for the volume |B(x, δ)|: 2m

󵄨󵄨 󵄨 k 2 󵄨󵄨B(x, δ)󵄨󵄨󵄨 ≈ ( ∑ Λk (x)δ )δ . k=2

Here Λk are the appropriate Levi invariants, and are continuous, nonnegative functions on Mm (see [35]). It is then easy to see that for s ≥ 1, 󵄨󵄨 󵄨 󵄨 2m+2 󵄨󵄨 󵄨󵄨B(x, sδ)󵄨󵄨󵄨 ≲ s 󵄨󵄨B(x, δ)󵄨󵄨󵄨,

x ∈ M, δ > 0.

We shall call 2m + 2 the “dimension” of Mm . The balls have the doubling property 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨B(x, 2δ)󵄨󵄨󵄨 ≤ C 󵄨󵄨󵄨B(x, δ)󵄨󵄨󵄨,

for all δ > 0.

We also use the volume functions 󵄨 󵄨 Vδ (x) = 󵄨󵄨󵄨B(x, δ)󵄨󵄨󵄨, 󵄨 󵄨 V(x, y) = 󵄨󵄨󵄨B(x, d(x, y))󵄨󵄨󵄨. It was observed in [35] that V(x, y) ≈ V(y, x). The following is the definition of test functions on Mm . Definition 2.1 ([31]). Fix 0 < γ, β < ϑ. A function f defined on Mm is said to be a test function of type (x0 , r, β, γ), x0 ∈ Mm , and r > 0, if f satisfies the following conditions: C (i) |f (x)| ≤ V (x )+V(x ( r )γ for all x ∈ Mm ; ,x) r+d(x ,x) r

0

0

0

d(x,y) β 1 (ii) |f (x)−f (y)| ≤ C( r+d(x ) V (x )+V(x ( r )γ for all x, y ∈ Mm such that d(x, y) ≤ 0 ,x) r 0 0 ,x) r+d(x0 ,x) (r + d(x0 , x))/2A.

12 | X. Wu et al. If f is a test function of type (x0 , r, β, γ), we write f ∈ ℳ(x0 , r, β, γ), and the norm of f in ℳ(x0 , r, β, γ) is defined by ‖f ‖ℳ(x0 ,r,β,γ) = inf{C : (i) and (ii) hold}. Now fix x0 ∈ Mm and denote ℳ(β, γ) = ℳ(x0 , 1, β, γ). It is easy to see that ℳ(x, r, β, γ) = ℳ(β, γ) with equivalent norms for all x ∈ Mm and r > 0. Furthermore, it is also easy to check that ℳ(β, γ) is a Banach space with respect to the norm in ℳ(β, γ). Denote by ℳ0 (β, γ) the collection of functions f ∈ ℳ(β, γ) with ∫M f (x)dx = 0. Let m

ℳϑ0 (β, γ) be the completion of ℳ0 (ϑ, ϑ) in ℳ(β, γ) when 0 < β, γ < ϑ. If f ∈ ℳϑ0 (β, γ),

then we define

‖f ‖ℳϑ (β,γ) = ‖f ‖ℳ(β,γ) . 0

We then define the distribution space (ℳϑ0 (β, γ))′ to be the set of all linear functionals L on ℳϑ0 (β, γ) with the property that there exists a constant C > 0 such that 󵄨󵄨 󵄨 󵄨󵄨L(f )󵄨󵄨󵄨 ≤ C‖f ‖ℳϑ (β,γ) , 0

∀ f ∈ ℳϑ0 (β, γ).

We denote by Cb (Mm ) the space of continuous functions on Mm with bounded support. For 0 < η ≤ 1, we set η

Cb (Mm ) = {f ∈ Cb (Mm ) :

sup

x,y∈Mm ,x =y̸

|f (x) − f (y)| < ∞}. d(x, y)η

η

η

We endow Cb (Mm ) with the natural topology and denote its dual by (Cb (Mm ))′ . Assume that T is a bounded linear operator on L2 (Mm ), or, more generally, T is a continuous η η linear mapping from Cb (Mm ) to (Cb (Mm ))′ for certain η ∈ (0, 1]. The operator T is said to have a distributional kernel K which is locally integrable away from the diagonal of η Mm × Mm if for any f , g ∈ Cb (Mm ) with supp f ∩ supp g = ⌀, ⟨Tf , g⟩ =

∫

g(x)K(x, y)f (y)dydx.

(2.2)

Mm ×Mm

The following result is a variant of [19, Theorem 2.4] (see also [31, Lemma 2.9]). Theorem 2.2. Let T be a bounded linear operator on L2 (Mm ) with distributional kernel K(x, y). Moreover, suppose that there exists a constant C > 0 such that the following four properties hold: (i) For all x, y ∈ Mm with x ≠ y, C 󵄨󵄨 󵄨 ; 󵄨󵄨K(x, y)󵄨󵄨󵄨 ≤ V(x, y)

Hardy spaces associated to the Kohn Laplacian on a family of model domains in Cn+1

| 13

(ii) For all x, y, y′ ∈ Mm with d(y, y′ ) ≤ d(x, y)/2A and x ≠ y, Cd(y, y′ )ϑ 󵄨󵄨 ′ 󵄨 , 󵄨󵄨K(x, y) − K(x, y )󵄨󵄨󵄨 ≤ V(x, y)d(x, y)ϑ

(2.3)

which also holds when x and y are interchanged; (iii) For all x, x′ , y, y′ ∈ Mm with d(x, x′ ), d(y, y′ ) ≤ d(x, y)/3A, Cd(x, x ′ )ϑ d(y, y′ )ϑ 󵄨󵄨 ′ ′ ′ ′ 󵄨 ; 󵄨󵄨K(x, y) − K(x , y) − K(x, y ) + K(x , y )󵄨󵄨󵄨 ≤ V(x, y)d(x, y)2ϑ (iv) T1 = 0. Denote by ‖T‖ the smallest constant so that the above (i)–(iii) hold. Then T is bounded from ℳ0 (β, γ) to ℳ(β, γ) for 0 < β, γ < ϑ. Moreover, 󵄩󵄩 󵄩 󵄩󵄩T(f )󵄩󵄩󵄩ℳ(β,γ) ≤ C(‖T‖ + ‖T‖L2 (Mm )→L2 (Mm ) )‖f ‖ℳ(β,γ) .

(2.4)

2.2 Heat kernels and the Calderón reproducing formula Consider the initial value problem for the heat equation: 𝜕u (x, s) + Δλ,2 u(x, s) = 0, 𝜕s

with u(x, 0) = f (x).

The solution is given by u(x, s) = Hs (f )(x), where Hs is the operator given via the spectral theorem by Hs = e−sΔλ,2 , and an appropriate self-adjoint extension of the nonnegative operator Δλ,2 initially defined on C0∞ (Mm ). Lemma 2.3 ([35, Proposition 2.3.1]). For f ∈ L2 (Mm ), we have Hs (f )(x) = ∫ H(s, x, y)f (y)dy, Mm

and H(s, x, y) satisfies the following properties: (1) For every integer J ≥ 0, 󵄨󵄨 j L K 󵄨 󵄨󵄨𝜕s 𝜕X 𝜕Y H(s, x, y)󵄨󵄨󵄨

J

2 √s 1 1 )( )( ) . ≲( V(x, y) + V√s (x) + V√s (y) d(x, y) + √s (d(x, y) + √s)2j+K+L

14 | X. Wu et al. (2) For each integer L ≥ 0, there exist an integer NL and a constant CL such that if φ ∈ C0∞ (B(x0 , δ)), then for all s ∈ (0, ∞), 󵄨 󵄨 󵄨󵄨 L −L |J| 󵄨 J 󵄨󵄨𝜕X Hs [φ](x0 )󵄨󵄨󵄨 ≤ CL δ sup ∑ δ 󵄨󵄨󵄨𝜕X φ(x)󵄨󵄨󵄨. x |J|≤NL

(3) For all s ∈ (0, ∞), ∫ H(s, x, y)dy = ∫ H(s, x, y)dx = 1. Mm

Mm

The aim of this section is to construct a discrete Calderón’s reproducing formula via the heat kernel. Set St (x, y) = H(t 2 , x, y) and Ψt (x, y) = √2t

𝜕 S (x, y). 𝜕t t

Denote Ψt f (x) = ∫ Ψt (x, y)f (y)dy. M

Lemma 2.4. For any N > 0, there exists a constant CN > 0 such that for all t > 0 and all x, x′ , y, y′ ∈ Mm with d(x, x′ ) ≤ (t + d(x, y))/2, t N (i) |Ψt (x, y)| ≤ ( V (x)+V C(y)+V(x,y) )( t+d(x,y) )N ; t

t

d(x,x ) t N (ii) |Ψt (x, y) − Ψt (x′ , y)| ≤ ( t+d(x,y) )( V (x)+V C(y)+V(x,y) )( t+d(x,y) )N ; t t (iii) Property (ii) holds with the roles of x and y interchanged; (iv) ∫M Ψt (x, y)dx = ∫M Ψt (x, y)dy = 0. m

′

m

The following Calderón reproducing formula was established by Nagel and Stein [35]. Proposition 2.5 ([35, Proposition 2.4.1]). For f ∈ L2 (Mm ), ∞

∫ Ψ2s (f ) 0

ds = f, s

f ∈ L2 (Mm ), 1/ϵ

where the integral on the left is defined as limϵ→0 ∫ϵ L2 -norm.

(2.5)

Ψ2s (f ) ds , with the limit taken in the s

Based on Proposition 2.5, we shall establish a discrete version of (2.5). To this end, we need the following grid of dyadic cubes of Christ [15].

Hardy spaces associated to the Kohn Laplacian on a family of model domains in Cn+1

| 15

Lemma 2.6 ( [15]). Let X be a space of homogeneous type (see [16, 17]). Then there exist a collection {Qkβ ⊂ X : k ∈ Z, β ∈ Ik } of open subsets, where Ik is some index set, and constants δ ∈ (0, 1) and C2 , C3 > 0 such that (i) μ(X\ ∪β Qkβ ) = 0 for each fixed k, and Qkβ ∩ Qkγ = ⌀ if β ≠ γ; (ii) for any β, γ, k, l with l ≥ k, either Qlγ ⊂ Qkβ or Qlγ ∩ Qkβ = ⌀;

(iii) for each (k, β) and each l < k there is a unique γ such that Qkβ ⊂ Qlγ ; (iv) diam (Qkβ ) ≤ C2 δk ;

(v) each Qkβ contains some ball B(zβk , C3 δk ), where zβk ∈ X.

We can think of Qkα as being a dyadic cube of side length ℓ(Qkα ) = 2−k centered at zαk . We shall use 𝒬j to denote the collection of dyadic cubes of side length 2−j−N for a large fixed integer N, whose precise value will be given by Theorem 2.7 below. Let 𝒬 denote the grid of dyadic cubes in Lemma 2.6. Our main result of this section is the following discrete Calderón’s reproducing formula, constructed via the heat kernel associated to the Kohn Laplacian. ̃ Q }Q∈𝒬 , each Theorem 2.7. Let 0 < β, γ < ϑ. There exists a collection of test functions {Ψ in ℳϑ0 (β, γ), such that f (x) =

̃ Q (x), ∑ ∑ |Q| ⋅ Ψj f (xQ )Ψ

(2.6)

j Q∈𝒬j

where xQ denotes any fixed point in Q, Ψj = Ψ2−αj , and the series converges in ℳϑ0 (β, γ), (ℳϑ0 (β, γ))′ , and Lp (Mm ), 1 < p < ∞. Moreover, ̃ Q ‖Lp (M ) ≲ ‖ΨQ ‖Lp (M ) , ‖Ψ m m

1 < p < ∞,

̃ Q ‖ℳ(β,γ) ≲ ‖ΨQ ‖ℳ(β,γ) . ‖Ψ

(2.7)

Proof. We borrow an idea from [29] to discretize (2.5): ∞

f (x) = ∫ ∫ Ψ2s (x, y)f (y)dy 0 Mm

2−αj

=∑ ∫ j∈Z

ds s

∫ Ψ2s (x, y)f (y)dy

2−α(j+1) Mm

=∑ ∫ j∈Z M

m

Ψ2j (x, y)f (y)dy

ds s 2−αj

+∑ ∫ j∈Z

∫ [Ψ2s (x, y) − Ψ2j (x, y)]f (y)dy

2−α(j+1) Mm

ds s

16 | X. Wu et al. = Tα f (x) + Rα f (x), where Ψj = Ψ2−αj . For Tα , we write ̃ j (x) + Rα,N f (x) Tα f (x) = ∑ ∑ |Q| ⋅ Ψj f (xQ )Ψ j Q∈𝒬j

= Tα,N f (x) + Rα,N f (x), where Tα,N f (x) = ∑ ∑ |Q| ⋅ Ψj f (xQ )ΨQ (x), j Q∈𝒬j

ΨQ (x) =

1 ∫ Ψj (x, x′ )dx′ , |Q|

for each Q ∈ 𝒬j ,

Q

Rα,N f (x) = ∑ ∑ ∫ Ψj (x, x ′ )[Ψj f (x′ ) − Ψj f (xQ )]dx′ . j Q∈𝒬j Q

Altogether we have f (x) = Tα,N f (x) + Rα f (x) + Rα,N f (x). The following theorem is an analog of [32, Theorem 1.19] for the operators Rα , Rα,N . Theorem 2.8. Let ϑ ∈ (0, 1] and 0 < β, γ < ϑ. If N is sufficiently large and α is sufficiently small, then 1 ‖Rα f ‖Lp (Mm ) + ‖Rα,N f ‖Lp (Mm ) ≤ ‖f ‖Lp (Mm ) , 2 1 ‖Rα f ‖ℳ(β,γ) + ‖Rα,N f ‖ℳ(β,γ) ≤ ‖f ‖ℳ(β,γ) , 2

f ∈ Lp (Mm ), 1 < p < ∞, f ∈ ℳϑ0 (β, γ).

Assume Theorem 2.8 holds for the moment. Then we obtain that Tα,N = 1 − Rα − Rα,N is a bounded and invertible operator on ℳϑ0 (β, γ) and on Lp (Mm ), 1 < p < ∞. Set ̃ Q (x) = T −1 (ΨQ )(x), Ψ α,N

for each Q ∈ 𝒬.

Hardy spaces associated to the Kohn Laplacian on a family of model domains in Cn+1

| 17

We have −1 −1 f = Tα,N (Tα,N (f )) = ∑ ∑ |Q| ⋅ Ψj f (xQ )Tα,N (ΨQ )(x) j Q∈𝒬j

̃ Q (x), = ∑ ∑ |Q| ⋅ Ψj f (xQ )Ψ j Q∈𝒬j

where the series converges in Lp (Mm ), 1 < p < ∞ and ℳϑ0 (β, γ), and, moreover, (2.7) holds. It thus remains to prove Theorem 2.8. We only treat Rα,N as the estimate for Rα is similar. By definition, Rα,N is given by the kernel Rα,N (x, y) := ∑ ∑ ∫ Ψj (x, x′ )[Ψj (x ′ , y) − Ψj (xQ , y)]dx ′ . j Q∈𝒬j Q

Note that if T satisfies the assumption in Theorem 2.2, then by a result from [35], T is bounded on Lp (Mm ), 1 < p < ∞. By Theorem 2.2, it then suffices to show that Rα,N (x, y) satisfies the assumption on K in Theorem 2.2 with small bounds. We first observe that Ψj (x′ , y) − Ψj (xQ , y) ∼ 2−N Ψj (x′ , y) in the sense that the left-hand side satisfies the same cancellation, size, and regularity conditions as the right-hand side. Thus we have ∑ ∫ Ψj (x, x′ )[Ψj (x ′ , y) − Ψj (xQ , y)]dx ′ ∼ 2−N ∑ ∫ Ψj (x, x ′ )Ψj (x ′ , y)dx′

Q∈𝒬j Q

Q∈𝒬j Q

∼2

−N

Ψj (x, y).

So we obtain Rα,N (x, y) ∼ 2−N ∑ Ψj (x, y) ∼ 2−N K(x, y), j

which gives the desired estimates for Rα,N . The proof of Theorem 2.8 is concluded.

3 H p -boundedness via discrete Calderón reproducing formula The following square function was introduced by Nagel and Stein [35].

18 | X. Wu et al. Definition 3.1. The Littlewood–Paley square function (associated to the heat kernel) of f is defined by 1 2

∞

󵄨2 dt 󵄨 g(f )(x) = { ∫ 󵄨󵄨󵄨Ψt (f )(x)󵄨󵄨󵄨 } . t 0

Theorem 2.7 leads us to define a new discrete Littlewood–Paley square function. Let 0 < β, γ < ϑ. For f ∈ ℳϑ0 (β, γ)′ , we define the following discrete Littlewood– Paley square function: 1/2

󵄨 󵄨2 gd (f )(x) = {∑ ∑ 󵄨󵄨󵄨Ψj f (xQ )󵄨󵄨󵄨 χQ (x)} , j Q∈𝒬j

where xQ denotes any fixed point in Q. The Hardy spaces H p (Mm ) can be defined via the above square function in a nat2m+2 ural way: For 0 < β, γ < ϑ and 2m+2+ϑ < p < ∞, define ′ 󵄩 󵄩 H p (Mm ) = {f ∈ (ℳϑ0 (β, γ)) : 󵄩󵄩󵄩gd (f )󵄩󵄩󵄩Lp (M

m)

< ∞}.

If f ∈ H p (Mm ), its (quasi)norm is given by ‖gd (f )‖Lp (Mm ) . To see that the Hardy spaces H p (Mm ) are well defined, we need to show that they do not depend on the choice of points xQ ∈ Q for every Q. This is given by the following. Theorem 3.2. Let 0 < β, γ < ϑ and f ∈ (ℳϑ0 (β, γ))′ ,

2m+2 2m+2+ϑ

< p < ∞. For a fixed large integer N and all

1/2 󵄩 1/2 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄨 󵄨2 󵄩 󵄨 󵄨2 ≈ 󵄩󵄩󵄩{ ∑ inf 󵄨󵄨󵄨Ψj (f )(z)󵄨󵄨󵄨 χQ } 󵄩󵄩󵄩 . 󵄩󵄩{ ∑ sup󵄨󵄨󵄨Ψj (f )(z)󵄨󵄨󵄨 χQ } 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩Lp (Mm ) 󵄩󵄩 󵄩󵄩Lp (Mm ) z∈Q j∈Z, z∈Q j∈Z, Q∈𝒬j

Q∈𝒬j

To prove Theorem 3.2, we need a Cotlar–Stein lemma, which is the following orthogonality estimate. Lemma 3.3 ([30, 31]). For any 0 < ϵ < ϑ, we have 2−|j−j |ϵ 󵄨󵄨 ̃ ′ 󵄨 󵄨󵄨Ψj Ψj (x, y)󵄨󵄨󵄨 ≲ ( V(x, y) + V ′ (x) + V 2−(j∧j

)

2−(j∧j

′)

(y)

)(

ϵ

2−(j∧j ) ) . ′ 2−(j∧j ) + d(x, y) ′

′

We also need the following maximal function estimate on Mm , which is an analog of a result of [24] in the Euclidean space. See also [30]. 2m+2 Lemma 3.4 ([24, 30]). Let ϵ > 0, k, k ′ ∈ Z, and xQ be any fixed point in Q. If 2m+2+ϵ < r < p ≤ 1, then there exists a constant C > 0 depending only on r such that for all fQ ∈ C

| 19

Hardy spaces associated to the Kohn Laplacian on a family of model domains in Cn+1

and all x ∈ Mm , 2−(k∧k )ϵ |Q| )( )|f | V2−(k∧k′ ) (x) + V(x, xQ ) (2−(k∧k′ ) + d(x, xQ ))ϵ Q ′

∑ (

Q∈𝒬k

1/r

≤ C2[(k∧k )−k](2m+2)(1−1/r) {ℳHL ( ∑ |fQ |r χQ (⋅))(x)} , ′

Q∈𝒬k

where ℳHL is the Hardy–Littlewood maximal function on Mm . Proof of Theorem 3.2. Applying the discrete Calderón reproducing formula in Theorem 2.7, we get 󵄨 󵄨 ̃ ′ (z)dz. Ψj f (xQ ) = ∑ ∑ 󵄨󵄨󵄨Q′ 󵄨󵄨󵄨Ψj′ f (xQ′ ) ∫ Ψj (xQ , z)Ψ Q j′ Q′ ∈𝒬j′

Mm

̃ ′ (x) ≈ Ψ(x, x ′ ), which means that Here xQ′ denotes any fixed point of Q′ . Note that Ψ Q Q they satisfy the same smoothness and cancellation conditions. By Lemmas 3.3, 3.4, and the Cauchy–Schwarz inequality, |Q′ ||Ψj′ f (xQ′ )| 󵄨󵄨 󵄨󵄨 −|j−j′ |ϵ Ψ f (x ) ≲ 2 ∑ ∑ 󵄨󵄨 j Q 󵄨󵄨 V(xQ , xQ′ ) + V2−(j∧j′ ) (xQ ) + V2−(j∧j′ ) (xQ′ ) j′ Q′ ∈𝒬 j′

ϵ

2−(j∧j ) ×( ) ′ 2−(j∧j ) + d(xQ , xQ′ ) ′

′

1/r

󵄨 󵄨r inf ′ 󵄨󵄨󵄨Ψj′ f (xQ′ )󵄨󵄨󵄨 χQ′ )(x)}

′

≲ ∑ 2−|j−j |ϵ {ℳHL ( ∑ j′

Q′ ∈𝒬j′

xQ′ ∈Q

2

≲ {∑ 2 j′

−|j−j′ |ϵ′

1

r 2 󵄨 󵄨r {ℳHL ( ∑ inf ′ 󵄨󵄨󵄨Ψj′ f (xQ′ )󵄨󵄨󵄨 χQ′ )(x)} } , x ∈Q

Q′ ∈𝒬j′

Q′

where ϵ′ = ϵ − (2m + 2)(1 − 1r ). It follows that ′ 󵄨 󵄨2 ∑ sup 󵄨󵄨󵄨Ψj f (xQ )󵄨󵄨󵄨 χQ (x) ≲ ∑ ∑ 2−|j−j |(ϵ−(2m+2)(1−1/r)) ( ∑ χQ (x))

j∈Z xQ ∈Q Q∈𝒬j

j

Q∈𝒬j

j′

2

r 󵄨 󵄨r × {ℳHL ( ∑ inf ′ 󵄨󵄨󵄨Ψj′ f (xQ′ )󵄨󵄨󵄨 χQ′ )(x)} x ∈Q

Q′ ∈𝒬j′

Q′

2

r 󵄨 󵄨r ≲ ∑{ℳHL ( ∑ inf ′ 󵄨󵄨󵄨Ψj′ f (xQ′ )󵄨󵄨󵄨 χQ′ )(x)} . x ∈Q

j′

Q′ ∈𝒬j′

Q′

20 | X. Wu et al. Applying the Fefferman–Stein vector-valued maximal inequality, we obtain 1

󵄩󵄩 2󵄩 󵄩󵄩 󵄨2 󵄨 󵄩󵄩 󵄩󵄩{∑ ∑ sup 󵄨󵄨󵄨Ψj f (xQ )󵄨󵄨󵄨 χQ } 󵄩󵄩󵄩 󵄩󵄩Lp (Mm ) 󵄩󵄩 j Q∈𝒬j xQ ∈Q 1

2/r 2 󵄩 󵄩󵄩 󵄩󵄩 󵄨r 󵄨 󵄩 ≲ 󵄩󵄩󵄩{∑{ℳHL ( ∑ inf ′ 󵄨󵄨󵄨Ψj′ f (xQ′ )󵄨󵄨󵄨 χQ′ (⋅))} } 󵄩󵄩󵄩 󵄩󵄩 ′ 󵄩󵄩Lp (Mm ) ∈Q x j Q′ ∈𝒬 ′ Q′ j

1

󵄩󵄩 2󵄩 󵄩󵄩 󵄩 󵄨 󵄨2 ≲ 󵄩󵄩󵄩{∑ ∑ inf ′ 󵄨󵄨󵄨Ψj′ f (xQ′ )󵄨󵄨󵄨 χQ′ } 󵄩󵄩󵄩 , 󵄩󵄩 ′ ′ 󵄩󵄩Lp (Mm ) x ∈Q j Q ∈𝒬 ′ Q′ j

concluding the proof. We consider on Mm a class of singular integrals of NIS type (nonisotropic smoothing operators of order 0). These operators occur naturally on the boundary of various domains in Cn (see [11]). They may be viewed as Calderón–Zygmund operators whose kernels are C ∞ away from the diagonal, and whose cancellation conditions are given quite simply in terms of their action on smooth bump functions. Let us recall from [35] the properties of the kernels of NIS operators. Suppose the operator T has a distribution kernel K(x, y) which is C ∞ away from the diagonal of Mm × Mm , and we suppose the following four properties hold: (I) If φ, ψ ∈ C0∞ (Mm ) have disjoint supports, then ⟨Tφ, ψ⟩ =

∫

K(x, y)φ(y)ψ(x) dxdy.

Mm ×Mm a (II) If φ is a normalized bump function associated to a ball of radius r, then |𝜕X,Y Tφ| ≲ −a r . More precisely, for each integer a ≥ 0, there are another integer b ≥ 0 and a constant Ca,b so that, whenever φ is a C ∞ function supported in a ball B(x0 , r),

󵄨 a 󵄨 󵄨 c 󵄨 sup r a 󵄨󵄨󵄨(𝜕X,Y Tφ)(x)󵄨󵄨󵄨 ≤ Ca,b sup sup r c 󵄨󵄨󵄨𝜕X,Y (φ)(x)󵄨󵄨󵄨.

x∈Mm

c≤b x∈B(x0 ,r)

Here X, Y are “good” vector fields. (III) If x ≠ y, then for every a ≥ 0, 󵄨󵄨 a 󵄨 −a −1 󵄨󵄨𝜕X,Y K(x, y)󵄨󵄨󵄨 ≲ d(x, y) V(x, y) . (IV) Properties (I) through (III) also hold with x and y interchanged. That is, these properties also hold for the adjoint operator T t defined by ⟨T t φ, ψ⟩ = ⟨Tψ, φ⟩.

| 21

Hardy spaces associated to the Kohn Laplacian on a family of model domains in Cn+1

The following are the main results from [35] and [31]. We will give a new proof using the Calderón reproducing formula given in Theorem 2.7. Theorem 3.5. Each singular integral operator T satisfying the above properties (I)–(IV) extends to a bounded operator on Lp (Mm ), 1 < p < ∞, and a bounded operator on 2m+2 H p (Mm ) to Lp (Mm ) with 2m+2+ϑ < p < ∞. Theorem 3.6. Each singular integral operator T satisfying the above properties (I)–(IV) 2m+2 extends to a bounded operator from H p (Mm ) to H p (Mm ) for 2m+2+ϑ < p ≤ 1. Let Kλ (z, w, t − s) be the fundamental solution to the Kohn Laplacian Δk,2 . Then the 2 second derivatives 𝜕X,Y Kλ (z, w, t − s) give rise to an NIS operator of order 0: 2 Tλ (f )(z, t) = p. v. ∫ 𝜕X,Y Kλ (z, w, t − s)f (w, s)dwds. Mm

As a consequence, we have Corollary 3.7. Each Tλ is bounded from H p (Mm ) to Lp (Mm ) for 2m+2 bounded from H p (Mm ) to itself for 2m+2+ϑ < p ≤ 1.

2m+2 2m+2+ϑ

< p < ∞, and

To prove Theorem 3.5, we need the following orthogonality estimate. Proposition 3.8. For each singular integral T satisfying (I) through (IV), we have 2−|j−j |ϵ 󵄨󵄨 ̃ ′ (x, y)󵄨󵄨󵄨 ≲ ( 󵄨󵄨Ψj T Ψ j 󵄨 V(x, y) + V ′ (x) + V 2−(j∧j

)

ϵ

2−(j∧j ) ) ′ 2−(j∧j ) + d(x, y) ′

′

2−(j∧j

′)

(y)

)(

for any 0 < ϵ < ϑ. Proof of Theorem 3.5. We prove first the H p (Mm ) boundedness of T. For f ∈ H p (Mm ), we apply the discrete Calderón reproducing formula in Theorem 2.7 to get 󵄨 󵄨 ̃ ′ )(z)dz. Ψj Tf (xQ ) = ∑ ∑ 󵄨󵄨󵄨Q′ 󵄨󵄨󵄨Ψj′ f (xQ′ ) ∫ Ψj (xQ , z)(T Ψ Q j′ Q′ ∈𝒬j′

Mm

̃ ′ (x) ≈ Ψ(x, x ′ ), applying Proposition 3.8, we get Recalling that Ψ Q Q 󵄨󵄨 ̃ ′ (xQ )󵄨󵄨󵄨 󵄨󵄨Ψj T Ψ Q 󵄨

ϵ

2−(j∧j ) 2−|j−j |ϵ ) . )( ′ V(xQ , xQ′ ) + V2−(j∧j′ ) (xQ ) + V2−(j∧j′ ) (xQ′ ) 2−(j∧j ) + d(xQ , xQ′ ) ′

≲(

′

By arguments similar to those in the proof of Theorem 3.2, we get 1

‖Tf ‖H p (Mm )

󵄩󵄩 2󵄩 󵄩󵄩 󵄩 󵄨 󵄨2 ≈ 󵄩󵄩󵄩{∑ ∑ sup 󵄨󵄨󵄨Ψj Tf (xQ )󵄨󵄨󵄨 χQ (x)} 󵄩󵄩󵄩 󵄩󵄩Lp (Mm ) 󵄩󵄩 j Q∈𝒬j xQ ∈Q

22 | X. Wu et al. 1

2/r 2 󵄩 󵄩󵄩 󵄩󵄩 󵄨r 󵄨 󵄩 ≲ 󵄩󵄩󵄩{∑{ℳHL ( ∑ inf ′ 󵄨󵄨󵄨Ψj′ f (xQ′ )󵄨󵄨󵄨 χQ′ (⋅))} } 󵄩󵄩󵄩 󵄩󵄩 ′ 󵄩󵄩Lp (Mm ) x ∈Q j Q′ ∈𝒬 ′ Q′ j

1

󵄩󵄩 2󵄩 󵄩󵄩 󵄨2 󵄨 󵄩 ≲ 󵄩󵄩󵄩{∑ ∑ inf ′ 󵄨󵄨󵄨Ψj′ f (xQ′ )󵄨󵄨󵄨 χQ′ } 󵄩󵄩󵄩 󵄩󵄩 ′ ′ 󵄩󵄩Lp (Mm ) x ∈Q j Q ∈𝒬 ′ Q′ j

≈ ‖f ‖H p (Mm ) . This completes the proof of Theorem 3.5.

4 Calderón’s identity and the proof of H p → Lp boundedness We shall use the following definition of approximation to the identity, whose existence follows from Coifman’s construction which first appeared in [18] on a space of homogeneous type. Definition 4.1. A sequence {Sk }k∈Z of operators is said to be an approximation to the identity on Mm if there exists a constant C > 0 such that for all k ∈ Z and all x, x ′ , y and y′ ∈ Mm , Sk (x, y), the kernel of Sk satisfies the following conditions: (i) Sk (x, y) = 0 if d(x, y) ≥ C2−k and |Sk (x, y)| ≤ C(V2−k (x) + V2−k (y))−1 ; (ii) |Sk (x, y) − Sk (x′ , y)| ≤ C2kϑ d(x, x′ )ϑ (V2−k (x) + V2−k (y))−1 ; (iii) Property (ii) holds with x and y interchanged; (iv) |[Sk (x, y) − Sk (x, y′ )] − [Sk (x′ , y) − Sk (x′ , y′ )]| ≤ C22kϑ d(x, x ′ )ϑ d(y, y′ )ϑ (V2−k (x) + V2−k (y))−1 ; (v) ∫M Sk (x, y)dx = ∫M Sk (x, y)dy = 1. m

m

The following Calderón-type identity is crucial for our purpose here, whose proof can be found in [31]. Lemma 4.2 ([31]). Let {Sk }k∈Z be an approximation to the identity on Mm and Dk = Sk − ̃ Sk−1 , k ∈ Z. Then there exist families of functions {̃ Dk } and {̃ Dk } such that for any yk+N ∈ Qk+N all f ∈ (ℳϑ0 (β, γ))′ with 0 < β, γ < ϑ, τ

τ

󵄨 󵄨󵄨̃ k+N f (x) = ∑ ∑󵄨󵄨󵄨Qk+N )Dk f (yτk+N ) τ 󵄨󵄨Dk (x, yτ k∈Z τ

󵄨 󵄨󵄨 k+N ̃ = ∑ ∑󵄨󵄨󵄨Qk+N )̃ Dk f (yτk+N ), τ 󵄨󵄨Dk (x, yτ

(4.1)

k∈Z τ

where the series converges in both ℳϑ0 (β, γ) and Lp (Mm ), 1 < p < ∞. Moreover, ̃ Dk (x, y) ̃ ̃ ̃ ̃ ̃ and Dk (x, y), the kernels of Dk and Dk , satisfy similar properties, with x and y interchanged: For any fixed 0 < ϵ < ϑ,

Hardy spaces associated to the Kohn Laplacian on a family of model domains in Cn+1

2 )( (2−k +d(x,y)) ϵ ),

C

(i) |̃ Dk (x, y)| ≤ ( V

| 23

−kϵ

(x)+V2−k (y)+V(x,y) 2−k ′ −k

)2 ϵ (ii) |̃ Dk (x, y) − ̃ Dk (x′ , y)| ≤ ( 2d(x,x −k +d(x,y) ) ( V

2−k

C ) (x)+V2−k (y)+V(x,y)

whenever d(x, x ′ ) ≤ (2−k +

d(x, y))(2A)−1 ; (iii) ∫M ̃ Dk (x, y)dx = ∫M ̃ Dk (x, y)dy = 0 for all k. m

m

Lemma 4.3 ([19, 31]). For

2m+2 2m+2+ϑ

< p < ∞, L2 (Mm ) ∩ H p (Mm ) is dense in H p (Mm ).

To prove Theorem 3.6, we first establish the following 2m+2 Theorem 4.4. Let 2m+2+ϑ < p ≤ 1. If f ∈ L2 (Mm ) ∩ H p (Mm ), then f ∈ Lp (Mm ) and there exists a constant Cm > 0 which is independent of the L2 (Mm )-norm of f and the degree m of 𝒫 such that

‖f ‖H p (Mm ) ≤ Cm ‖f ‖H p (Mm ) . Proof. Assume f ∈ L2 (Mm ) ∩ H p (Mm ). By Lemma 4.2, ̃ f = ∑ ∑ |Q| ⋅ Dk (⋅, xQ ) ̃ Dk (f )(xQ ), k Q∈𝒬k

where the series converges in L2 (Mm ). Since Sk (x, y) are supported in {(x, y) ∈ Mm ×Mm : d(x, y) < C2−k }, so is Dk (x, xQ ) = Sk (x, xQ ) − Sk−1 (x, xQ ). Moreover, H p (Mm ) can be characterized by (cf. [19, 31]) 1/2 󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄨̃ 󵄨2 ‖f ‖H p (Mm ) ≈ 󵄩󵄩󵄩{∑ ∑ 󵄨󵄨󵄨̃ Dk (f )󵄨󵄨󵄨 χQ } 󵄩󵄩󵄩 . 󵄩󵄩 󵄩󵄩Lp (Mm ) k Q∈𝒬k

For each i ∈ Z, set 󵄨̃ 󵄨2 Ωi = {x ∈ Mm : {∑ ∑ 󵄨󵄨󵄨̃ Dk (f )(x)󵄨󵄨󵄨 χQ (x)} k Q∈𝒬k

1/2

> 2i }

and Bi = {(k, Q) : Q ∈ 𝒬k , |Q ∩ Ωi | >

|Q| 1 |Q|, |Q ∩ Ωi+1 | ≤ }. 2A 2A

We claim 󵄩󵄩 󵄩󵄩󵄩p 󵄩󵄩 Dk (f )(xQ )󵄩󵄩󵄩 ≤ C2ip |Ωi |. 󵄩󵄩 ∑ |Q| ⋅ Dk (⋅, xQ )̃ 󵄩󵄩 󵄩󵄩Lp (Mm ) (k,Q)∈Bi

(4.2)

24 | X. Wu et al. Assume that the claim holds for the moment. This, together with the fact that p

(∑ |ai |) ≤ ∑ |ai |p , i

0 < p ≤ 1,

i

yields ‖f ‖pLp (M

m)

󵄩󵄩 󵄩󵄩p 󵄩 󵄩 = 󵄩󵄩󵄩∑ ∑ |Q| ⋅ Dk (⋅, xQ )̃ Dk (f )(xQ )󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩Lp (Mm ) i∈Z (k,Q)∈Bi 󵄩󵄩p 󵄩󵄩 󵄩 󵄩 ≤ ∑󵄩󵄩󵄩 ∑ |Q| ⋅ Dk (⋅, xQ )̃ Dk (f )(xQ )󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩Lp (Mm ) i (k,Q)∈Bi ≲ ∑ 2ip |Ωi | ≲ ‖f ‖pH p (M ) . m

i

To finish the proof of Theorem 4.4, it thus remains to verify claim (4.2). Note that if (k, Q) ∈ Bi , then Dk (⋅, xQ ) is supported in ̃ = {x ∈ M : ℳ (χ )(x) > 1 }. Ω i m HL Ωi 2A Therefore, by Hölder’s inequality, 󵄩󵄩 󵄩󵄩󵄩p 󵄩󵄩 ̃ Dk (f )(xQ )󵄩󵄩󵄩 󵄩󵄩 ∑ |Q| ⋅ Dk (⋅, xQ )̃ 󵄩󵄩Lp (Mm ) 󵄩󵄩 (k,Q)∈Bi

󵄩󵄩p p󵄩 󵄩 󵄩󵄩 ̃ |1− 2 󵄩󵄩󵄩 ∑ |Q| ⋅ D (⋅, x )̃ ̃ ≤ |Ω . i k Q Dk (f )(xQ )󵄩 󵄩󵄩 2 󵄩󵄩 󵄩L (Mm ) 󵄩(k,Q)∈Bi

We now estimate the last L2 (Mm )-norm by the duality argument. For all g ∈ L2 (Mm ) with ‖g‖L2 (Mm ) ≤ 1, 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 ̃ Dk (f )(xQ ), g⟩󵄨󵄨󵄨 󵄨󵄨⟨ ∑ |Q| ⋅ Dk (⋅, xQ )̃ 󵄨󵄨 󵄨󵄨 (k,Q)∈Bi 󵄨󵄨 󵄨󵄨󵄨 󵄨 ̃ = 󵄨󵄨󵄨 ∑ |Q| ⋅ D∗k (g)(xQ )̃ Dk (f )(xQ )󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 (k,Q)∈Bi 1

1

󵄨 󵄨2 2 󵄨̃ 󵄨2 2 ≤ ( ∑ |Q|󵄨󵄨󵄨D∗k (g)(xQ )󵄨󵄨󵄨 ) ( ∑ |Q|󵄨󵄨󵄨̃ Dk (f )(xQ )󵄨󵄨󵄨 ) , (k,Q)∈Bi

(k,Q)∈Bi

where D∗k is an operator defined by D∗k (g)(x) = ∫ Dk (y, xQ )g(y)dy. Mm

Hardy spaces associated to the Kohn Laplacian on a family of model domains in Cn+1

| 25

Recall that Dk (y, xQ ) satisfies the same properties as Dk (xQ , y) for each k ∈ Z. By the Fefferman–Stein vector-valued maximal inequality and the Littlewood–Paley inequality, we have 1/2

󵄨 󵄨2 ( ∑ |Q|󵄨󵄨󵄨D∗k (g)(xQ )󵄨󵄨󵄨 ) (k,Q)∈Bi

󵄨󵄨 󵄨󵄨2 1/2 ≤ ( ∑ |Q|󵄨󵄨󵄨 inf ℳHL (D∗k (g))(u)󵄨󵄨󵄨 ) 󵄨u∈Q 󵄨 (k,Q)∈Bi

1/2 󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄨 󵄨2 ≤ 󵄩󵄩󵄩(∑󵄨󵄨󵄨ℳHL (D∗k (g))󵄨󵄨󵄨 ) 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩L2 (Mm ) k

≲ ‖g‖L2 (Mm ) ≤ 1. Altogether we obtain 1

󵄩󵄩 󵄩󵄩 󵄩 󵄨̃ 󵄨2 2 󵄩󵄩 ̃ Dk (f )(xQ )󵄩󵄩󵄩 ≲ ( ∑ |Q|󵄨󵄨󵄨̃ Dk (f )(xQ )󵄨󵄨󵄨 ) . 󵄩󵄩 ∑ |Q|Dk (⋅, xQ )̃ 󵄩󵄩L2 (Mm ) 󵄩󵄩 (k,Q)∈Bi (k,Q)∈Bi Note also that 󵄨̃ ̃k (f )(x)󵄨󵄨󵄨2 χQ (x)dx 22i |Ωi | ≳ ∫ ∑ ∑ 󵄨󵄨󵄨D 󵄨 ̃i \Ωi+1 k Q∈𝒬k Ω

≳∫

󵄨̃ 󵄨󵄨2 Dk (f )(xQ )ℳHL (χQ∩(Ω ∑ 󵄨󵄨󵄨̃ ̃i \Ωi+1 ) )(x)󵄨󵄨 dx

Mm (k,Q)∈Bi

󵄨2 󵄨̃ ≳ ∑ |Q|󵄨󵄨󵄨̃ Dk (f )(xQ )󵄨󵄨󵄨 , (k,Q)∈Bi

where in the last inequality we used the inequality ℳHL (χQ∩(Ω ̃ \Ω i

i+1 )

)(x) ≥

1 , 2A

̃ \Ω )| ≥ 1 |Q| whenever (k, Q) ∈ B . Putting together the above estimates, since |Q∩(Ω i i+1 i 2A we derive claim (4.2) and hence Theorem 4.4 follows. Now we may combine Theorems 3.5 and 4.4 to complete the proof of Theorem 3.6. We will skip the details here. Final remark. In this paper, we just concentrate on the calculations of the family of model domains in C2 : Ωm = Ωm,1 = {(z1 , z2 ) ∈ C2 : Im(z2 ) = |z1 |2m , m ∈ N}.

26 | X. Wu et al. In fact, techniques for proving our results can be generalized to a family of model domains in Cn+1 : n

2

m

Ωm = Ωm,n = {(z1 , z2 , . . . , zn , zn+1 ) : Im(zn+1 ) = ( ∑ |zk | ) , m ∈ N}. k=1

Due to the limitation of the page numbers, we will provide detailed constructions and proofs in forthcoming papers [10] and [12].

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Hardy spaces associated to the Kohn Laplacian on a family of model domains in Cn+1

| 27

[18] G. David, J. L. Journé, and S. Semmes, Opérateurs de Calderón–Zygmund, fonctions para-accrétives et interpolation, Rev. Mat. Iberoam. 1 (1985), 1–56. [19] D. G. Deng and Y. S. Han, Harmonic analysis on spaces of homogeneous type, Lecture notes in math., vol. 1966, Springer, Berlin, 2009. [20] K. P. Diaz, The Szegő kernel as a singular integral kernel on a family of weakly pseudoconvex domains, Trans. Am. Math. Soc. 304 (1987), 147–170. [21] C. Fefferman and D. H. Phong, Subelliptic eigenvalue problems, Proc. conf. harmonic analysis, Wadsworth math. series, 1981, pp. 590–606. [22] G. B. Folland, A fundamental solution for a subelliptic operator, Bull. Am. Math. Soc. 79 (1973), 373–376. [23] G. B. Folland and E. M. Stein, Estimates for the 𝜕̄b complex and analysis on the Heisenberg group, Commun. Pure Appl. Math. 27 (1974), 429–522. [24] M. Frazier and B. Jawerth, A discrete transform and decomposition of distribution spaces, J. Funct. Anal. 93 (1990), 34–170. [25] P. Greiner, A fundamental solution for a non-elliptic partial differential operator, Can. J. Math. 31 (1979), 1107–1120. [26] P. Greiner and Y. T. Li, A fundamental solution for a nonelliptic partial differential operator (II), Anal. Appl. 16 (2018), no. 3, 407–433. ̄ [27] P. Greiner and E. M. Stein, Estimates for the 𝜕-Neumann problem, Math. notes, vol. 19, Princeton University Press, Princeton, NJ, 1977. [28] P. Greiner and E. M. Stein, On the solvability of some differential operators of type ◻b , Proceedings of international conf. Cortona 1976–1977, Ann. Sc. Norm. Super. Pisa, Cl. Sci. 4 (1978), 106–165. [29] Y. Han, G. Lu, and E. Sawyer, Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group, Anal. PDE 7 (2014), no. 7, 1465–1534. [30] Y. Han, D. Müller, and D. Yang, Littlewood–Paley–Stein characterizations for Hardy spaces on spaces of homogeneous type, Math. Nachr. 279 (2006), 1505–1537. [31] Y. Han, D. Müller, and D. Yang, A theory of Besov and Triebel–Lizorkin spaces on metric measure spaces modeled on Carnot–Carathéodory spaces, Abstr. Appl. Anal. 2008 (2008), Article ID 893409. [32] Y. S. Han, Calderón-type reproducing formula and the Tb theorem, Rev. Mat. Iberoam. 10 (1994), no. 1, 51–91. [33] L. Hörmander, Hypoelliptic second-order differential equations, Acta Math. 119 (1967), 147–171. [34] A. Nagel and E. M. Stein, The ◻b -heat equation on pseudoconvex manifolds of finite type in ℂ2 , Math. Z. 238 (2001), no. 1, 37–88. [35] A. Nagel and E. M. Stein, On the product theory of singular integrals, Rev. Mat. Iberoam. 20 (2004), no. 2, 531–561. [36] E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton math. series, vol. 43, Princeton University Press, Princeton, NJ, 1993.

Julian Haddad and Jie Xiao

Affine mixed Rayleigh quotients Abstract: This note concerns the affine (q, p)-Rayleigh quotients and their infimum over the Sobolev space W01,p (Ω) \ {0} on a given bounded smooth domain Ω and possible minimizers, also presenting an affine version of Talenti’s inequality for the first eigenfunction of the affine p-Laplacian. Keywords: Affine (q, p)-Rayleigh quitient, affine p-Laplacian, affine p-Talenti inequality MSC 2010: Primary 35P30, Secondary 52A40, 35B09

1 Introduction From now on, assume that ℝn ⊃ Ω is open with boundary 𝜕Ω and denote by W01,p (Ω) with 1 ≤ p < ∞ the Sobolev space – the completion of all C ∞ -functions f with compact support in Ω with respect to the following norm (cf. [6, p. 245]): ‖f ‖W 1,p (Ω) := ‖f ‖p,Ω + ‖|∇f |‖p,Ω < ∞; { { { 1 ‖f ‖p,Ω := (∫Ω |f (x)|p dx) p ; { { 1 { p {‖|∇f ‖|p,Ω := (∫Ω |∇f (x)| dx) p . The affine energy ℰp , introduced in two papers within the theory of sharp affine inequalities by Zhang [15] and Lutwak–Yang–Zhang [9], is defined for f ∈ W01,p (ℝn ) as ℰp f = cn,p ( ∫

− n1

‖∇ξ f ‖−n p,ℝn dξ )

𝕊n−1

Acknowledgement: The first author was partially supported by CNPq (PQ 301203/2017-2) and Fapemig (APQ-01454-15). The second author was supported in part by NSERC of Canada (# 202979). Julian Haddad, Departamento de Matemática, ICEx, Universidade Federal de MinasGerais, 30.123-970, Belo Horizonte, Brazil, e-mail: [email protected] Jie Xiao, Department of Mathematics and Statistics, Memorial University, St. John’s, NL A1C5S7, Canada, e-mail: [email protected] https://doi.org/10.1515/9783110741711-002

30 | J. Haddad and J. Xiao with nω ω

1

1

cn,p := (nωn ) n ( 2ωn p−1 ) p ; { { n+p−2 { γ { π2 { {ωγ := Γ(1+ γ2 ) for γ ∈ (0, ∞); { { {∇ξ f (x) := ∇f (x) ⋅ ξ . It is proven in [9, 15] that the affine energy ℰp enjoys the following Sobolev-like inequality for f ∈ W01,p (ℝn ) with 1 ≤ p < n: ‖f ‖ np ,ℝn ≤ an,p ℰp f ,

(1.1)

n−p

where (cf. [7, (50)]) an,p

1

Γ(n)

− p1 p−1 p−1 ( n−p ) p

{( n n )n n = { ωn Γ( p )Γ(n+1− p ) an,p̂ ̂ {lim1 1 if and only if 󵄨 󵄨 p 1− p f (x) = a(1 + b󵄨󵄨󵄨A(x − x0 )󵄨󵄨󵄨 p−1 ) n for some (a, b, x0 , A) ∈ ℝ × (0, ∞) × ℝn × GLn (ℝ), where GLn (ℝ) denotes the set of invertible n × n-matrices. For p = 1, the equality in (1.1) is attained for the multiples of characteristic functions of ellipsoids in ℝn . As shown in [9], the affine energy satisfies ℰp f ≤ ‖|∇f |‖p,ℝn ,

(1.2)

thus (1.1) implies the classical Lp Sobolev inequality ‖f ‖ np ,ℝn ≤ an,p ‖|∇f |‖p,ℝn n−p

proven by Aubin [2] and Talenti [11]. In addition, the sharp L1 Sobolev inequality is equivalent to the classical isoperimetric inequality which is also the geometric core of the sharp Lp Sobolev inequality for 1 < p < n. When Ω is bounded and (1.1) is restricted to W01,p (Ω), we utilize Hölder’s inequality to obtain the following inequality: 󵄨 󵄨p κ(p, Ω) ∫󵄨󵄨󵄨f (x)󵄨󵄨󵄨 dx ≤ (ℰp f )p Ω

∀ f ∈ W01,p (Ω),

(1.3)

Affine mixed Rayleigh quotients | 31

where κ(p, Ω) is a positive constant depending only on {p, Ω}. In other words, as a consequence of the sharp affine Sobolev inequality (1.1), we deduce that the affine Poincaré inequality holds on W01,p (Ω) for any 1 ≤ p < n. In [8] this inequality is studied in detail, showing the existence of extremizers of inequality (1.3). Then the affine p-Laplace operator Δ𝒜 p is introduced (see Section 2 for the definition) as the secondorder differential operator governing the variational equation of the corresponding minimization problem μ𝒜 p,p (Ω) :=

ℰp f

inf

0=f̸ ∈W01,p (Ω)

‖f ‖p,Ω

.

Clearly, the existence of a positive constant κ(p, Ω) satisfying (1.3) is equivalent to μ𝒜 p,p (Ω) > 0. In this note we address three issues stemming from [8, 14]. The first one is to explore the equivalent forms of the infimum over W01,p (Ω) \ {0} of the affine (q, p)-Rayleigh quotient ℰp f

𝒜

ℛq,p (f ) :=

‖f ‖q,Ω

,

thereby discovering Theorem 1.1. Let Ω be bounded; { { { { { (q, p) ∈ [1, ∞) × [1, ∞); { { { { ‖|∇f |‖p,Ω { {μq,p (Ω) := inf ; { ‖f ‖q,Ω { { 0=f̸ ∈W01,p (Ω) { { { 𝒜 inf ℛ𝒜 q,p (f ); {μq,p (Ω) := 1,p { 0 =f ̸ ∈W { 0 (Ω) { { 1 { { 𝒜 (vol(K)) q { αq,p (Ω) := sup { 1 ; { 𝒜 { compact K⊂Ω (capp (K,Ω)) p { { p { { (vol(O)) q −1 { 𝒜 {βq,p (Ω) := sup , 𝒜 (μp,p (O)p bounded open O⊂Ω { where p

1,p cap𝒜 p (K, Ω) := inf{(ℰp (f )) : f ∈ W0 (Ω) ∩ C(Ω) and f ≥ 1K }

is the affine p-capacity of K with respect to Ω, C(Ω) is the class of all continuous functions on Ω and 1K is the characteristic function of K. (i) If p ≥ q, then there is a positive constant κ1 (Ω, n, q, p) depending on {Ω, n, q, p} such that 1

κ1 (Ω, n, q, p)(μq,p (Ω)) n ≤ μ𝒜 q,p (Ω) ≤ μq,p (Ω).

(1.4)

32 | J. Haddad and J. Xiao (ii) If p ≤ q, then there are two positive constants κj=2,3 (q, p) depending on {q, p} such that 𝒜 𝒜 −1 κ2 (q, p)(αq,p (Ω))−1 ≤ μ𝒜 q,p (Ω) ≤ (αq,p (Ω))

{

𝒜 κ3 (q, p)(βq,p (Ω))−1

≤

μ𝒜 q,p (Ω)

≤

𝒜 (βq,p (Ω))−1

for p ≥ 1;

for p > 1.

(1.5)

𝒜 𝒜 Here, both αq,p (Ω) and βq,p (Ω) are allowed to be ∞ whenever μ𝒜 q,p (Ω) = 0.

Observe that if 1 ≤ p < n; { { { { pn { {p∗ := n−p < q < ∞; { 1,p { f ∈ W0 (Ω) \ {0}; { { { { n {fr (x) = f (rx) ∀ (r, x) ∈ (0, ∞) × ℝ , then ℛ𝒜 q,p (fr ) ℛ𝒜 q,p (f )

=r

n( q1 − p1∗ )

→0

as r → ∞.

The second is to determine the existence of a minimizer for ℛ𝒜 q,p , thereby proving Theorem 1.2. If Ω is bounded; { { { 1 < p < n; { { { {1 ≤ q ≤ p, then there exists f† ∈ W01,p (Ω) \ {0} enjoying ℛq,p (f† ) = μq,p (Ω), 𝒜

(1.6)

𝒜

and hence f† is a weak solution to the Dirichelet problem q−p q−2 q Δ𝒜 f = (μ𝒜 f q,p (Ω)) ‖f ‖q,Ω |f | { p f =0

in Ω; on 𝜕Ω.

(1.7)

The third is to establish an affine version of Talenti’s inequality for the first (1, n) ∋ p-eigenfunction. To do so, for any f ∈ W01,p (Ω) on a bounded Ω and t > 0, we write Ωt (f ) := {x ∈ Ω : |f (x)| > t};

{

ν(t) := vol(Ωt (f )),

Affine mixed Rayleigh quotients | 33

for the t-level set of f and its distribution function, respectively. In [12] Talenti proved a useful differential inequality satisfied by the distribution function of the first eigenfunction of the Laplace operator. This inequality was exploited by many authors to better understand the behavior of the eigenfunctions of Ω, such as [4]. It is also called the p-Talenti inequality and represented in [4, (2.5)] as follows: If f ∈ W01,p (Ω) \ {0} is the first (1, n) ∋ p-eigenfunction of Δp f = λ1 (Ω)|f |p−2 f

{

f =0

in Ω; on 𝜕Ω,

or, equivalently, if f is a minimizer in W01,p (Ω) \ {0} for the quotient ℛp,p (f ) :=

‖|∇f |‖p,Ω ‖f ‖p,Ω

,

then ν satisfies 1 n

(nωn (ν(t))

n−1 n

)

p p−1

∞

ν(τ) ν(t) ≤ −ν (t)(λ1 (Ω)( 1−p + (p − 1) ∫ 2−p dτ)) t τ

1 p−1

′

t

.

This leads to an affine Talenti’s inequality for p

λ1𝒜 (Ω) = (μ𝒜 p,p (Ω)) as described below.

Theorem 1.3. Let 1 < p < n and Ω be bounded. If f‡ ∈ W01,p (Ω) \ {0} is the first eigenfunction of 𝒜 p−2 Δ𝒜 f p f = λ1 (Ω)|f |

{

f =0

in Ω; on 𝜕Ω,

or, equivalently, f‡ is a minimizer in W01,p (Ω) \ {0} for ℛ𝒜 p,p , then its distribution function (0, ∞) ∋ t 󳨃→ ν‡ (t) satisfies 1 n

(nωn (ν‡ (t))

n−1 n

)

p p−1

∞

ν‡ (t) ν‡ (τ) ≤ −ν‡′ (t)(λ1𝒜 (Ω)( 1−p + (p − 1) ∫ 2−p dτ)) t τ t

1 p−1

.

(1.8)

Theorem 1.3 will be a consequence of the Busemann–Petty centroid inequality stated in Section 2. This note is organized as follows. In Section 2, we prepare some necessary concepts from convex geometry and its associated partial differential equation. In Sec-

34 | J. Haddad and J. Xiao tions 3 and 4 we prove Theorems 1.1 and 1.2, respectively. In Section 5 we show Theorem 1.3.

2 Preliminaries First of all, we recall that a convex body K ⊂ ℝn is a convex compact subset of ℝn with nonempty interior. The support function hK is defined as hK (y) = max{⟨y, z⟩ : z ∈ K} which describes the (signed) distance of supporting hyperplanes of K to the origin and uniquely characterizes K. We also have the gauge g and radial r functions of K defined respectively as g(K, y) := inf{λ > 0 : y ∈ λK} ∀ y ∈ ℝn \ {0};

{

r(K, y) := max{λ > 0 : λy ∈ K} ∀ y ∈ ℝn \ {0} .

On the one hand, we have g(K, y) = (r(K, y)) , −1

and we recall that g(K, ⋅) is actually a norm when the convex body K is centrally symmetric and the unit ball with respect to g(K, ⋅) is just K. On the other hand, a general norm on ℝn is uniquely determined by its unit ball, which is a centrally symmetric convex body. Next, for a convex body K ⊂ ℝn we define the polar body, denoted by K ∘ , by K ∘ := {y ∈ ℝn : ⟨y, z⟩ ≤ 1 ∀ z ∈ K} . It is evident that ∘ h−1 K (⋅) = r(K , ⋅)

and (λK)∘ = λ−1 K ∘

∀ λ > 0.

A simple computation with polar coordinates reveals n

vol(K) = n−1 ∫ (r(K, ξ )) dξ = n−1 ∫ (g(K, ξ )) dξ . −n

(2.1)

.𝕊n−1

.𝕊n−1

Moreover, Lutwak–Zhang introduced in [10] the Lp centroid body Γp K which is defined by hpΓ K (y) := p

ω2 ωn ωp−1 󵄨 󵄨p ∫󵄨󵄨⟨y, z⟩󵄨󵄨󵄨 dz ωn+p vol(K) 󵄨 K

∀ y ∈ ℝn .

Affine mixed Rayleigh quotients | 35

There are some other normalizations of the Lp centroid body in the literature and the previous one is made so that Γp 𝔹n2 = 𝔹n2

for the origin-centered Euclidean unit ball 𝔹n2 in ℝn .

The definition of Γp K can also be written as hpΓ K (y) = p

ω2 ωp−1 ωn−2

nωp+n−2 vol(K)

n+p 󵄨

󵄨󵄨⟨y, ξ ⟩󵄨󵄨󵄨p dξ 󵄨 󵄨

∫ (r(K, ξ )) .𝕊n−1

∀ y ∈ ℝn .

Also, in [9], Lutwak–Yang–Zhang proved the Lp Busemann–Petty centroid inequality vol(Γp K) ≥ vol(K), see also [5] for an alternative proof. Now, for a function f ∈ W01,p (Ω) the formula ‖ξ ‖f ,p = ‖∇ξ f ‖p,Ω defines a norm in ℝn . If Lf ,p denotes the unit ball under this last norm, then formula (2.1) gives the following relation: − n1

ℰp f = cn,p (n vol(Lf ,p ))

.

Up to a minus sign, we utilize Δp f := −div(|∇f |p−2 ∇f ) to represent the p-Laplacian. Consequently, for a convex body K containing the origin as interior point, the Wulff p-Laplacian is defined as (cf. [3]) Δp,K f (x) = − div(∇(p−1 hpK (∇f (x))) = − div(hp−1 K (∇f (x))∇hK (∇f (x))). Furthermore, as shown in [8], the affine p-Laplacian (cf. [13] for an alternative) can be determined via {Δ𝒜 p f = Δp,Gf (f ); {G = ( nωn ) n1 Γ L , p f ,p vol(Lf ,p ) { f where it can be seen from [8] that Gf is a symmetric convex body with smooth boundary. Finally, according to [1, Proposition 2.3], if E is a set of finite perimeter PH (E, ℝn ) := sup{∫ div σ(x) dx : σ ∈ C01 (ℝn ; ℝn ) and H ∘ (σ) ≤ 1} E

36 | J. Haddad and J. Xiao with respect to a given norm H whose polar function is H ∘ (x) = sup ξ =0 ̸

⟨x, ξ ⟩ , H(ξ )

then 1

nκnn (vol(E))

n−1 n

≤ PH (E, ℝn )

(2.2)

holds, where κn is the volume of the polar of the unit ball with the norm H. As the H-isoperimetric inequality, (2.2) generalizes Minkowsky’s first inequality for the mixed volumes. Importantly, for any nonnegative function u of bounded variation there is the coarea formula ∞

∫ H(∇u(x)) dx = ∫ PH ({x ∈ Ω : u(x) > s}, ℝn ) ds.

(2.3)

0

Ω

3 Argument for Theorem 1.1 (i) Suppose q ≤ p. According to (1.2), [8, Theorem 9], and Hölder inequality for p/q ≥ 1, there are two positive constants c(Ω, n, p) and c(Ω, n, q, p) such that ‖|∇f |‖p,Ω ≥ ℰp f

1

n−1

n n ‖|∇f |‖p,Ω ≥ c(Ω, n, p)‖f ‖p,Ω n−1

1

n n ≥ c(Ω, n, q, p)‖f ‖q,Ω ‖|∇f |‖p,Ω

= c(Ω, n, q, p)‖f ‖q,Ω (

‖|∇f |‖p,Ω ‖f ‖q,Ω

1 n

) .

So, (1.4) holds with κ1 (Ω, n, q, p) = c(Ω, n, q, p). (ii) Suppose q ≥ p. Then (1.5) follows from the proof of [14, Theorems 3.3–3.5]. Firstly, according to the definition of cap𝒜 p (K, Ω) of a given compact K ⊂ Ω, we have 1K ≤ f ∈ W01,p (Ω) ∩ C(Ω) ⇒

1

‖f ‖q,Ω ≥ (vol(K)) q

Affine mixed Rayleigh quotients | 37 1

⇒

p (cap𝒜 p (K, Ω))

(vol(K))

1 q

ℰp (f )

inf

≥

1K ≤f ∈W01,p (Ω)∩C(Ω)

‖f ‖q,Ω

≥ μ𝒜 q,p (Ω),

thereby reaching −1

𝒜 (αq,p (Ω))

≥ μ𝒜 q,p (Ω).

Secondly, we utilize the capacitary strong inequality in [14, Theorem 3.1] to derive that if f ∈ W01,p (Ω) and

󵄨 󵄨 Ωt (f ) = {x ∈ Ω : 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 > t} ∀ t > 0,

then there is a constant τ > 0 such that ∞

= ∫ vol(Ωt (f )) dt q

‖f ‖qq

0

≤

q 𝒜 (αq,p (Ω))

∞

q

q p ∫ (cap𝒜 p (Ωt (f ), Ω)) dt 0

∞

q

𝒜 p ≤ (αq,p (Ω)) ( ∫ cap𝒜 p (Ωt (f ), Ω) dt )

≤

q p

0 q q 𝒜 τ(αq,p (Ω)) (ℰp (f )) ,

and hence 𝒜 −1 μ𝒜 q,p (Ω) ≥ κ2 (Ω, n, q, p)(αq,p (Ω)) ;

{

κ2 (q, p) = τ

− q1

.

Thirdly, from Hölder’s inequality it follows that for any open O ⊂ Ω, each f ∈ W01,p (O) and p ≤ q, p

−1 p (∫ |f (x)|q dx) q ((μ𝒜 q,p (Ω)) ℰp (f )) 󵄨 󵄨p ≤ , ∫󵄨󵄨󵄨f (x)󵄨󵄨󵄨 dx ≤ O p p −1 −1 (vol(O)) q (vol(O)) q O

and hence (ℰ (f ))p

1− pq

𝒜 {μq,p (Ω) ≤ ( ∫ |f p(x)|p dx )(vol(O)) O { 𝒜 𝒜 −1 μ (Ω) ≤ (β q,p (Ω)) . { q,p

;

38 | J. Haddad and J. Xiao Fourthly, for t > 0 and f ∈ W01,p>1 (Ω), we have 󵄨p 󵄨 󵄨p 󵄨 ∫󵄨󵄨󵄨f (x)󵄨󵄨󵄨 dx ≤ ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 dx + t p−1

Ω

Ωt (f )

󵄨 󵄨 ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 dx

Ω\Ωt (f )

p p−1 ≤ (μ𝒜 p,p (Ωt (f ))) (ℰp f ) + t

󵄨 󵄨 ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 dx

−p

1− pq

𝒜 ≤ βq,p (Ω)(vol(Ωt (f )))

Ω\Ωt (f )

(ℰp f )p + t p−1 1− pq

󵄨 󵄨 𝒜 ≤ βq,p (Ω)(t −1 ∫󵄨󵄨󵄨f (x)󵄨󵄨󵄨 dx)

󵄨 󵄨 ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 dx

Ω\Ωt (f )

󵄨 󵄨 (ℰp f )p + t p−1 ∫󵄨󵄨󵄨f (x)󵄨󵄨󵄨 dx. Ω

Ω

Upon taking t=(

q p(q−1)

(ℰp f )p

p

(∫Ω |f (x)| dx) q

)

,

we produce a constant 1

𝒜 κ† = (βq,p (Ω)) p > 0

such that q−p

q−1 p q(p−1) 󵄨 󵄨 󵄨p 󵄨 ∫󵄨󵄨󵄨f (x)󵄨󵄨󵄨 dx ≤ 2((κ† ℰp (f )) ) p(q−1) (∫󵄨󵄨󵄨f (x)󵄨󵄨󵄨 dx) .

Ω

(3.1)

Ω

If f in (3.1) is replaced by fk = min{max{f − 2k , 0}, 2k } ∀ k ∈ ℤ = {0, ±1, ±2, . . . }, then p(q−1)

p(q−p)

q(p−1) q(p−1) p(q−1) 󵄨 󵄨p 󵄨 󵄨 (∫󵄨󵄨󵄨fk (x)󵄨󵄨󵄨 dx) ≤ 2 q(p−1) (κ† ℰp fk )p (∫󵄨󵄨󵄨fk (x)󵄨󵄨󵄨 dx) ,

Ω

Ω

and hence p(q−1)

p(q−1)

p(q−p)

(2kp vol(Ω2k+1 (f ))) q(p−1) ≤ 2 q(p−1) (κ† ℰp fk )p (2k vol(Ω2k (f ))) q(p−1) .

(3.2)

Affine mixed Rayleigh quotients | 39

Let ak = 2kq vol(Ω2k (f )); { { { bk = (κ† ℰp fk )p ; { { { q(p−1) {θ = p(q−1) . Then, a combination of (3.2) and Hölder and Minkowski inequalities induces a positive constant pair {κ‡,1 , κ‡,2 } such that ∑ ak ≤ κ‡,1 ∑ bθk ap(1−θ) k

k∈ℤ

k∈ℤ

θ

1−θ

≤ κ‡,1 ( ∑ bk ) ( ∑ apk ) k∈ℤ

k∈ℤ

p(1−θ)

≤ κ‡,2 (κ† ℰp f )pθ ( ∑ ak )

,

k∈ℤ

thereby finding a constant κ3 (q, p) > 0 such that 𝒜 μ𝒜 q,p (Ω) ≥ κ3 (q, p)(βq,p (Ω)) . −1

4 Argument for Theorem 1.2 In what follows, we always assume that (p, q) ∈ (1, n) × [1, p] and Ω is a bounded subdomain of ℝn . Note that μ𝒜 q,p (Ω) =

inf

0≡f̸ ∈W01,p (Ω)

𝒜

ℛq,p (f ) =

inf

0≡f̸ ∈W01,p (Ω)

𝒜

ℰp (f /‖f ‖q,Ω ).

So, we may select a sequence (fk ) ⊂ W01,p (Ω) \ {0} such that ‖fk ‖q,Ω = 1

𝒜 and ℛ𝒜 q,p (fk ) → μq,p (Ω).

There is a constant κ(Ω, n, q, p) > 0 depending on {Ω, n, q, p} such that the Sobolev inequality 1 = ‖fk ‖q ≤ κ(Ω, n, q, p)‖|∇fk |‖p,Ω

(4.1)

40 | J. Haddad and J. Xiao is available; see also [6, p. 265, Theorem 3]. However, (4.1) and the argument for Theorem 1.1(i) imply 1

c(Ω,n,q,p) 𝒜 k p,Ω { 1 ; {ℛq,p (fk ) ≥ c(Ω, n, q, p)( ‖fk ‖q,Ω ) q ≥ (κ(Ω,n,q,p)) q 𝒜 { { 1+μq,p (Ω) ≥ ‖|∇fk |‖p,Ω ≥ (κ(Ω, n, q, p))−1 > 0. { c(Ω,n,q,p) ‖|∇f |‖

Consequently, there exist a function f† ∈ W01,p (Ω) and a subsequence (fkj ) such that

fkj → f† weakly in W01,p (Ω) and fkj → f† in Lq (Ω). This, along with (4.1) for f† , implies { (κ(Ω, n, q, p))−1 ≤ ‖|∇f† |‖p,Ω ≤ lim infk→∞ ‖|∇fk |‖p,Ω ≤ { { { { {f† ≢ 0; { { 𝒜 {μq,p (Ω) ≤ ℛq,p (f† ).

1+μ𝒜 q,p (Ω) ; c(Ω,n,q,p)

So, it remains to prove ℛq,p (f† ) ≤ lim inf ℛq,p (fkj ) = μq,p (Ω), 𝒜

𝒜

𝒜

j→∞

(4.2)

which in turn proves (1.6). For each fixed ξ ∈ 𝕊n−1 , we achieve that ∇ξ fkj → ∇ξ f† weakly in Lp (Ω), whence lim inf ‖∇ξ fkj ‖p,Ω ≥ ‖∇ξ f† ‖p,Ω . k→∞

By using ‖fkj ‖q,Ω = ‖f† ‖q,Ω = 1

∀ j ≥ 1,

(4.3)

we produce a positive constant c̃ such that ‖∇ξ fkj ‖p,Ω ≥ ĉ

∀ j ≥ 1.

Accordingly, Fatou lemma gives −n −n ∫ ‖∇ξ f† ‖−n p,Ω dξ ≥ ∫ lim sup ‖∇ξ fkj ‖p,Ω dξ ≥ lim sup ∫ ‖∇ξ fkj ‖p,Ω dξ . 𝕊n−1

𝕊n−1

k→∞

j→∞

𝕊n−1

This, along with (4.3), validates (4.2). To verify the but-also part of Theorem 1.2, we recall that the affine p-Laplacian Δ𝒜 p on W01,p (Ω) as a nonlocal quasilinear operator in divergence form is given by p−1 Δ𝒜 p f := −div(Hf (∇f )∇Hf (∇f )),

Affine mixed Rayleigh quotients | 41

where Hf (v) :=

p+n −n (cn,p (ℰp (f ))

󵄨󵄨p 󵄨󵄨 ‖∇ξ f ‖−n−p p,Ω 󵄨󵄨⟨ξ , v⟩󵄨󵄨 dξ )

∫ 𝕊n−1

1 p

for v ∈ ℝn ,

thereby finding via [8, Theorem 10] and [13, Theorem 2.3] that it is enough to show that any minimizer f for μ𝒜 p,q (Ω) enjoys the equation q (μ𝒜 q,p (Ω))

=

∫Ω ⟨Hfp−1 (∇f (x))∇Hf (∇f (x)), ∇ψ(x)⟩ dx

∀ ψ ∈ W01,p (Ω).

q−2 f (x)ψ(x) dx ‖f ‖q−p q,Ω ∫Ω |f (x)|

As a matter of fact, if f ∈ W01,p (Ω) is a minimizer for μ𝒜 q,p (Ω), then n 𝒜 −n ∫ ‖∇ξ g‖−n p dξ ≤ cn,p (μq,p (Ω)) ‖g‖q,Ω −n

∀ g ∈ W01,p (Ω)

𝕊n−1

with equality if g = f . Upon taking any ψ ∈ W01,p (Ω) and computing the derivative on the left-hand side with respect to ψ, which amounts to replacing g by f + εψ and taking the derivative with respect to ε at ε = 0, we obtain that if {x}p := |x|p sign(x) then ∫ n p|∇ξ f (x)|p−1 sign(∇ξ f )∇ξ ψ(x) dx n 𝜕 ( ∫ ‖∇ξ f ‖−n dξ ∫ ℝ n p dξ ) = − +1 𝜕ψ p (∫ n |∇ f (x)|p dx) p 𝕊n−1

𝕊n−1

= −n ∫

ξ ℝ p−1

∫ℝn {∇ξ f (x)}

𝕊n−1

∇ξ ψ(x) dx n

(∫ℝn |∇ξ f (x)|p dx) p

+1

dξ .

Moreover, an application of the Fubini theorem yields 𝜕 ( ∫ ‖∇ξ f ‖−n p dξ ) 𝜕ψ 𝕊n−1

= −n ∫ ∫ ‖∇ξ f ‖−n−p p,Ω {∇ξ f (x)}

p−1

⟨∇ψ(x), ξ ⟩dξdx

ℝn 𝕊n−1

= −n ∫ ⟨∇ψ(x), ∫ ‖∇ξ f ‖−n−p p,Ω {∇ξ f (x)} ℝn

𝕊n−1

p−1

ξ dξ ⟩ dx.

42 | J. Haddad and J. Xiao Suppose now that Kf is the convex body with support function hKf (z) = ( ∫ 𝕊n−1

󵄨󵄨p 󵄨󵄨 ‖∇ξ f ‖−n−p p,Ω 󵄨󵄨⟨z, ξ ⟩󵄨󵄨 dξ )

1 p

∀ z ∈ ℝn .

Then a straightforward computation gives p−1

∇(p−1 hpK )(z) = ∫ ‖∇ξ f ‖−n−p p,Ω {⟨z, ξ ⟩} f

ξ dξ

∀ z ∈ ℝn ,

𝕊n−1

then take z = ∇f (x) to produce 𝜕 −1 p ( ∫ ‖∇ξ f ‖−n p dξ ) = −n ∫ ⟨∇ψ(x), ∇(p hKf )(∇f (x))⟩ dx. 𝜕ψ ℝn

𝕊n−1

Note that − qn

𝜕 󵄨 󵄨q (( ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 dx) 𝜕ψ ℝn

− qn −1

n 󵄨 󵄨q ) = − ( ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 dx) q ℝn

q−1

= −n‖f ‖−n−q q,Ω ∫ {f (x)}

q−1

∫ q{f (x)}

ψ(x) dx

ℝn

ψ(x) dx.

ℝn

So we derive the weak formulation of (1.7) for f ∈ W01,p (Ω) \ {0}, namely we have ∫ ⟨∇ψ(x), ∇(p−1 hpK )(∇f (x))⟩ dx f

ℝn

=

n cn,p ∫ℝn {f (x)}q−1 ψ(x) dx n+q n (μ𝒜 q,p (Ω)) ‖f ‖q,Ω

∀ ψ ∈ W01,p (Ω).

(4.4)

Here it is worth pointing out that such a weak solution can be a classical one. On the one hand, if f ∈ C 2 (Ω) is a minimizer, then an integration-by-parts yields 𝜕 −1 p ( ∫ ‖∇ξ f ‖−n p dξ ) = −n ∫ ψ(x)div(∇(p hKf (∇f (x)))) dx 𝜕ψ 𝕊n−1

ℝn

= n ∫ ψ(x)Δp,Kf f (x) dx ℝn

and hence we employ (4.4) to obtain the classical equation −n 󵄨󵄨q−2 −n−q 󵄨󵄨 n −Δp,Kf f (x) + cn,p (μ𝒜 f (x) = 0. q,p (Ω)) ‖f ‖q,Ω 󵄨󵄨f (x)󵄨󵄨

Affine mixed Rayleigh quotients | 43

Through multiplying the last equation by n+p

−n cn,p (ℰp (f ))

f,

and using the equality μ𝒜 q,p (Ω) =

ℰp f

‖f ‖q,Ω

,

we get q 󵄨󵄨q−2 p−q 󵄨󵄨 𝒜 −Δ𝒜 f (x) = 0 p f (x) + (μq,p (Ω)) ‖f ‖q,Ω 󵄨󵄨f (x)󵄨󵄨

in Ω.

On the other hand, if f ∈ C 2 (Ω) is a weak solution of (1.7), then, taking ψ = f as a test function, we obtain ∫ ⟨∇f (x), ∇(p−1 Hfp )(∇f (x))⟩ dx

ℝn

p+n

−n = cn,p (ℰp (f ))

p+n

−n = cn,p (ℰp (f ))

p−1

∫ ⟨∇f (x), ∫ ‖∇ξ f ‖−n−p p,Ω {∇ξ f (x)}

ℝn

p−1

∫ ‖∇ξ f ‖−n−p p,Ω ( ∫ {∇ξ f (x)}

𝕊n−1 p+n

−n = cn,p (ℰp (f ))

p+n

−n = cn,p (ℰp (f ))

= (ℰp (f ))

p

ξ dξ ⟩ dx

𝕊n−1

⟨∇f (x), ξ ⟩ dx) dξ

ℝn

󵄨󵄨 󵄨󵄨p ∫ ‖∇ξ f ‖−n−p p,Ω ( ∫ 󵄨󵄨∇ξ f (x)󵄨󵄨 dx) dξ

𝕊n−1

ℝn

∫ ‖∇ξ f ‖−n p,Ω dξ 𝕊n−1

and q 2 q 󵄨󵄨 󵄨󵄨q−2 p−q p (μ𝒜 (f (x)) dx = (μ𝒜 q,p (Ω)) ‖f ‖q,Ω ∫ 󵄨󵄨f (x)󵄨󵄨 q,p (Ω)) ‖f ‖q,Ω . ℝn

Therefore, f is a minimizer for μ𝒜 q,p (Ω).

44 | J. Haddad and J. Xiao

5 Argument for Theorem 1.3 Note that if p′ =

p p−1

1 n

(nωn ν‡ (t)

and t > 0 then (1.8) amounts to

n−1 n

∞

ν‡ (s) ν‡ (t) ) ≤ −ν‡′ (t)((μ𝒜 p,p (Ω)) ( 1−p + (p − 1) ∫ 2−p ds)) t s p′

p

t

1 p−1

.

(5.1)

For the first eigenfunction f‡ ∈ W01,p (Ω) \ {0} consider the set K = Gf‡ defined in

Section 2. Recall that K is a convex body with C 2 -boundary. By Jensen’s inequality, for h > 0 we get (

∫t 0, j = 1, 2. Now the compatibility condition ω(x, y)ω(y, x) = ω(x, y) = ω 2 (x,y) 1 is equivalent to ω1 (x, y)ω1 (y, x) = ω2 (x, y)ω2 (y, x). In particular, ω1 (x, y) = ω2 (y, x) (or, equivalently, ω1 (y, x) = ω2 (x, y)) fulfills the compatibility condition. Denoting ω1 = Ω, Ω(x,y) one hence puts ω(x, y) = Ω(y,x) . Generally, the Lp -norm of the operator K cannot reach k0 (p). In fact, if it does hold true, Ω(x, y) will need to be controlled by a single-variable function. Specifically, there exist some nonnegative Φj ∈ Lp (0, ∞) (j = 1, 2) and a −

γ

−

γ

constant γ ⩾ 1 such that Ω(x, y) p ⩽ Φ1 (x), Ω(y, x) p ⩾ Φ2 (y), and ‖Φ1 ‖Lp ⩽ ‖Φ2 ‖Lp , where Φj ’s (j = 1, 2) are independent of γ. Thus, Φ1 = Φ2 almost everywhere in (0, ∞) and, consequently, the convergence factor takes the form ω(x, y) = σ(x) with some σ(y) measurable function σ > 0.

For the particular case, we will be able to further exploit the properties for the norm of the integral operator as follows. Corollary 2.3. Suppose that the convergence factor is ω(x, y) =

σ(x) σ(y) +

with σ > 0. If kε (p) =

kε (p, x) is independent of x > 0, kε (p) = k0 (p) + o(1) (ε → 0 ), k0 (p) = k0 (p′ ), and

Singular Fredholm integral equations of Chandrasekhar type

| 93

γ

σ p ∈ Lp (0, ∞) holds for any γ > 1, then K is a continuous linear operator mapping ′ Lp (0, ∞) (and Lp (0, ∞)) into itself with ‖K‖Lp →Lp = k0 (p). −

−1

Corollary 2.4. Suppose that the conditions in Corollary 2.3 are fulfilled, and σ p ∈ Lp (0, ∞), then there exist certain functions f ∈ Lp (0, ∞) keeping the equality ‖Kf ‖Lp = k0 (p)‖f ‖Lp valid. Remark 2.5. The norm of the integral operator K in Corollary 2.3 or 2.4 is attainable, which simplifies the technical conditions in [11–13]. The second assertion contains the critical case σ(x) = x, and the third corresponds to the supercritical case σ(x) = (1+x)α with α > 1. The symmetry of the kernel function implies that 1

1

‖K‖Lp →Lp = ‖K‖Lp′ →Lp′ = (k0 (p)) p′ (k0 (p′ )) p . By the same conditions of Lemma 2.1, for any f ∈ Lp (0, ∞), g ∈ Lp (0, ∞), it is equivalent to claim that ′

1

1

∞ ∞ 󵄨󵄨 ∞ ∞ 󵄨󵄨 p p′ ′ 󵄨󵄨 󵄨 󵄨󵄨 ∫ ∫ k(x, y)f (x)g(y)dxdy󵄨󵄨󵄨 ⩽ k0 (p){ ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨p dx} { ∫ 󵄨󵄨󵄨g(y)󵄨󵄨󵄨p dy} . 󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨 0 0 0 0

Moreover, the conditions of Corollary 2.4 guarantee that there exist certain functions ‖f ‖Lp = ‖g‖Lp = 1 providing the equality. Let us now consider the linear Fredholm integral equation (2.1). One would put it in the previous notations as φ = ψ + λKφ. Define an operator T from Lp (0, ∞) into itself by Tφ = ψ + λKφ. Noticing that 󵄩 󵄩 ‖Tφ1 − Tφ2 ‖Lp = |λ|󵄩󵄩󵄩K(φ1 − φ2 )󵄩󵄩󵄩Lp ⩽ |λ|k0 (p)‖φ1 − φ2 ‖Lp , one claims that T is a contraction operator if |λ|k0 (p) < 1 and thus, by means of contraction mapping principle, one reaches the following statement. Theorem 2.6. For the linear Fredholm integral equation ∞

φ(y) = ψ(y) + λ ∫ k(x, y)φ(x)dx, 0

0 < y < ∞,

(2.2)

94 | S.-Y. Feng and D.-C. Chang if the kernel k(x, y) is symmetric and nonnegative almost everywhere in (0, ∞) × (0, ∞) and fulfills the condition in Lemma 2.1, i. e., kε (p) = kε (r, x) (r = p or p′ ) is independent of x > 0 and kε (p) = k0 (p) + o(1) (ε → 0+ ), then the linear Fredholm integral equation (2.2) has the unique solution φ ∈ Lp (0, ∞) as long as |λ| < k 1(p) . 0

Under the conditions of Theorem 2.6, one actually gets the optimal range of the parameter λ in the equation.

3 Structure of Lp solutions to parametric equation In [7], the authors provide results on the existence and uniqueness of the classical solution and the L2 solution to the linearized Chandrasekar equation defined in a bounded interval b

φ(t) = ψ(t) + λ ∫ k(t, s)φ(s)ds, a

0 < a ⩽ t ⩽ b < +∞.

(3.1)

Among all the solutions of this equation, the classical and L2 solutions have independent mathematical and physical meanings. In a recent work, the authors generalize the previous results in two directions [14]. On the one hand, we consider equations defined on unbounded intervals. On the other hand, we consider all Lp (1 ⩽ p ⩽ ∞) solutions of the equation. In general, the L∞ solution to the equation is usually an effective substitute for the classical solution. Specifically, we will study the following equations: ∞

φ(t) = ψ(t) + λ ∫ k(t, s)φ(s)ds,

0 < t < ∞.

(3.2)

0

Making use of the notation in Section 3, one rewrites (3.2) as φ = Tφ = ψ + λKφ,

(3.3)

where the integral operators K and T from Lp (0, ∞) into itself are respectively given ∞ by (Kφ)(t) := ∫0 k(t, s)φ(s)ds and (Tφ)(t) := ψ(t) + λ(Kφ)(t).

3.1 Successive approximation method We construct the following approximation sequence. Let φ0 (t) = ψ(t),

(3.4)

Singular Fredholm integral equations of Chandrasekhar type

| 95

∞

φn (t) = ψ(t) + λ ∫ k(t, s)φn−1 (s)ds 0 n

j

∞

= ψ(t) + ∑ λ ∫ k (j) (t, s)ψ(s)ds, j=1

(3.5)

0

where k (1) (t, s) = k(t, s), ∞

∞

(j)

k (t, s) = ∫ ⋅ ⋅ ⋅ ∫ k(t, s1 )k(s1 , s2 ) ⋅ ⋅ ⋅ k(sj−1 , s)ds1 ⋅ ⋅ ⋅ dsj−1 , 0 0 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

j ⩾ 2.

(j−1) times

We expect that the solutions of (3.2) have the following form: ∞

j

∞

φ(t) = lim φn (t) = ψ(t) + ∑ λ ∫ k (j) (t, s)ψ(s)ds. n→∞

j=1

(3.6)

0

3.2 Existence of Lp (1 ⩽ p ⩽ ∞) solutions Denote M1 =

sup

(t,s)∈R+ ×R+

󵄨󵄨 󵄨 󵄨󵄨k(t, s)󵄨󵄨󵄨, ∞

󵄨 󵄨 1 := sup ∫ 󵄨󵄨k(t, s)󵄨󵄨dt, M2 = ‖K‖L∞ 󵄨 󵄨 s (Lt ) s∈R+

0 ∞

(3.7)

󵄨 󵄨 1 := sup ∫ 󵄨󵄨k(t, s)󵄨󵄨ds, M3 = ‖K‖L∞ 󵄨 󵄨 t (Ls ) t∈R+

0

∞∞

󵄨 󵄨 M4 = ‖K‖L1t (L1s ) = ‖K‖L1s (L1t ) := ∫ ∫ 󵄨󵄨󵄨k(t, s)󵄨󵄨󵄨dtds. 0 0

For 1 ⩽ p ⩽ ∞, let p′ be the dual index such that

1 p

+

1 p′

t

󵄨 󵄨 := { ∫ ( ∫ 󵄨󵄨󵄨k(t, s)󵄨󵄨󵄨ds) dt} , 0

0

∞ ∞

M6 = ‖K‖

′

Lpt (Lps )

1 p

p

∞ ∞

M5 = ‖K‖Lp (L1s )

= 1 and define

p p′

1 p

󵄨 󵄨p := { ∫ ( ∫ 󵄨󵄨󵄨k(t, s)󵄨󵄨󵄨 ds) dt} . 0

0

′

(3.8)

96 | S.-Y. Feng and D.-C. Chang ∞

In particular, M4 = M5 = ‖K‖L1t (L1s ) and M6 = ‖K‖L1t (L∞ = ∫0 sups∈R+ |k(t, s)|dt if p = 1, s ) 1 if p = ∞. while M3 = M5 = M6 = ‖K‖L∞ t (Ls )

It is transparent from (3.3) that the solution to (3.2) coincides with the fixed point

of operator T. Therefore, Tf1 − Tf2 = λK(f1 − f2 ) and hence

󵄨󵄨 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 |Tf1 − Tf2 |(t) = |λ|󵄨󵄨󵄨 ∫ k(t, s)[f1 (s) − f2 (s)]ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨0 ∞

󵄨 󵄨󵄨 󵄨 ⩽ |λ| ∫ 󵄨󵄨󵄨k(t, s)󵄨󵄨󵄨󵄨󵄨󵄨f1 (s) − f2 (s)󵄨󵄨󵄨ds. 0

Making use of the above norm conditions and pointwise estimates, one is ready

to get a uniform estimate in the L∞ space.

Proposition 3.1. Suppose that 0 < M3 < ∞. If |λ| < a unique fixed point in L (0, ∞). ∞

1 , then the integral operator T M3

has

As p < ∞, Hölder’s inequality brings us the result in the Lp spaces. Proposition 3.2. Suppose that 0 < M6 < ∞. If |λ| < has a unique fixed point in Lp (0, ∞) (1 ⩽ p < ∞).

1 , M6

then the integral operator T

In order for the series solution defined by (3.6) to converge, one needs to obtain the

uniform and integrable convergence of the series solution via the pointwise estimation of the iterative kernel. The first result below is for the L∞ case.

Proposition 3.3. Suppose that 0 < M1 < ∞, 0 < M2 < ∞, 0 < M4 < ∞, and ψ ∈ L∞ (0, ∞). If |λ| < L∞ (0, ∞).

1 , M2

∞

j (j) then the series ∑∞ j=1 λ ∫0 k (t, s)ψ(s)ds converges in

As p < ∞, one applies Hölder’s inequality and the corresponding norm conditions

to obtain the convergence of the series solution in the Lp spaces.

Proposition 3.4. Suppose that 0 < M1 < ∞, 0 < M3 < ∞, 0 < M5 < ∞, and ψ ∈ Lp (0, ∞) (1 ⩽ p < ∞). If |λ| < verges in Lp (0, ∞) (1 ⩽ p < ∞).

1 , M3

∞

j (j) then the series ∑∞ j=1 λ ∫0 k (t, s)ψ(s)ds con-

According to Propositions 3.1 and 3.3, it follows that Theorem 3.5. For the parameterized Fredholm equation φ = ψ + λKφ with the integral ∞

operator (Kφ)(t) := ∫0 k(t, s)φ(s)ds, suppose that ψ ∈ L∞ (0, ∞) and Mj ’s defined in (3.7) are such that 0 < Mj < ∞ (j = 1, 2, 3, 4). If |λ| < min{ M1 , M1 }, then there exists ∞

2

3

j (j) a series solution to the integral equation ψ(t) + ∑∞ j=1 λ ∫0 k (t, s)ψ(s)ds converging in

Singular Fredholm integral equations of Chandrasekhar type

| 97

L∞ (0, ∞), where the iterated kernels are given by k (1) (t, s) = k(t, s), ∞

∞

k (j) (t, s) = ∫ ⋅ ⋅ ⋅ ∫ k(t, s1 )k(s1 , s2 ) ⋅ ⋅ ⋅ k(sj−1 , s)ds1 ⋅ ⋅ ⋅ dsj−1 , 0 0 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

j ⩾ 2.

(j−1) times

Likewise, combining Propositions 3.2 and 3.4, one obtains Theorem 3.6. For the parameterized Fredholm equation φ = ψ + λKφ with the integral ∞ operator (Kφ)(t) := ∫0 k(t, s)φ(s)ds, suppose that ψ ∈ L∞ (0, ∞) and Mj ’s defined in (3.7) and (3.8) are such that 0 < Mj < ∞ (j = 1, 3, 5, 6). If |λ| < min{ M1 , M1 }, then ∞

3

6

(j) j there exists a series solution to the integral equation ψ(t) + ∑∞ j=1 λ ∫0 k (t, s)ψ(s)ds p converging in L (0, ∞) with 1 ⩽ p < ∞, where the iterative kernels are also given by

k (1) (t, s) = k(t, s), ∞

∞

(j)

k (t, s) = ∫ ⋅ ⋅ ⋅ ∫ k(t, s1 )k(s1 , s2 ) ⋅ ⋅ ⋅ k(sj−1 , s)ds1 ⋅ ⋅ ⋅ dsj−1 , 0 0 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

j ⩾ 2.

(j−1) times

4 Lp approximate solutions to nonparametric equations As the second part of this article, this section will study the approximate solutions to the integral equations of Chandrasekhar type. It needs to be pointed out that the kernel function of the integral operator under consideration does not have to be symmetric, and the equation need no longer contain parameters to ensure the existence of the solution of the equation. In the recent research on Chandrasekhar integral equation [12, 13], the authors discuss the convergence of approximate solution in the L2 space, while in this section we will supplement the latest advance in the Lp framework [15].

4.1 Truncation operator to the linearized Chandrasekhar equation For integral operators in infinite intervals, the main effort will be devoted to overcoming the noncompactness of integral operators. Anselone [3] utilizes the truncation operator in the finite interval to approximate the original operator. It turns out to be an effective method. Readers may consult [10, 12, 13, 15] and the references therein. Accordingly, we will first exploit the properties of the truncation operator in the Lp [0, ∞) space.

98 | S.-Y. Feng and D.-C. Chang To be exact, we consider the approximate solutions to the second type of Fredholm integral equation ∞

φ(x) = ψ(x) + ∫ k(x, y)φ(y)dy,

0 < x < ∞,

(4.1)

0

where φ is the solution of the equation and ψ is the initial data in the Lp space. For simplicity, it is shortly rewritten by φ = ψ + Kφ,

(4.2)

where ∞

Kφ(x) := ∫ k(x, y)φ(y)dy 0

with k(x, y) being a measurable real-valued function on [0, ∞) × [0, ∞). Apparently, if the kernel k(x, y) is symmetric, (4.1) is in the exact form of (2.2). One introduces a bilateral truncation operator KT = M[0,T] KM[0,T] , where MD is defined by MD φ(x) = χD (x)φ(x) for any φ ∈ Lp [0, ∞), and χD is the characteristic function of a subset D. The integral equation associated to the truncation operator KT is then φ = ψ + KT φ.

(4.3)

If (I − K)−1 and (I − KT )−1 exist for any T > 0, the solutions to (4.2) and (4.3) are respectively denoted by φ̂ = (I − K)−1 ψ,

φ̂ T = (I − KT )−1 ψ.

For k(x, y) = k1 (x, y)k2 (x, y), setting ∞

Kj φ(x) := ∫ kj (x, y)φ(y)dy 0

(4.4)

Singular Fredholm integral equations of Chandrasekhar type

| 99

and 1 p′

∞

A1 := ‖K1 ‖

′

p L∞ x (Ly )

󵄨p′ 󵄨 = sup{ ∫ 󵄨󵄨󵄨k1 (x, y)󵄨󵄨󵄨 dy} , x⩾0

0

∞

p A2 := ‖K2 ‖L∞ y (Lx )

1 p

󵄨p 󵄨 = sup{ ∫ 󵄨󵄨󵄨k2 (x, y)󵄨󵄨󵄨 dx} , y⩾0

0

󵄨 󵄨 ∞ = A3 := ‖K2 ‖L∞ sup 󵄨󵄨󵄨k2 (x, y)󵄨󵄨󵄨 y (Lx ) x⩾0,y⩾0

with 1 ⩽ p ⩽ ∞, one needs to extend the properties of the truncation operator in [10] from the L2 space to the Lp spaces as follows. Theorem 4.1. Suppose that 1 ⩽ p < ∞, Aj < ∞ (j = 1, 2, 3), and that the integral operators, K in (4.2) and KT in (4.3), respectively have solutions φ̂ and φ̂ T given by (4.4), then the following properties hold: (1) K is bounded in Lp [0, ∞). (2) For any T > 0, KT is a compact operator in Lp [0, ∞). (3) If (I − KT )−1 exists and is uniformly bounded with respect to T, then (I − K)−1 exists and φ̂ T → φ̂ in the Lp sense as T → ∞. Remark 4.2. For p = 1, the bounds Aj ’s (j = 1, 2, 3) take the form 󵄨 󵄨 ∞ = A1 := ‖K1 ‖L∞ sup 󵄨󵄨󵄨k1 (x, y)󵄨󵄨󵄨 < ∞, x (Ly ) x⩾0,y⩾0

∞

󵄨 󵄨 1 = sup{ ∫ 󵄨󵄨k2 (x, y)󵄨󵄨dx} < ∞, A2 := ‖K2 ‖L∞ 󵄨 󵄨 y (Lx ) y⩾0

0

󵄨 󵄨 ∞ = A3 := ‖K2 ‖L∞ sup 󵄨󵄨󵄨k2 (x, y)󵄨󵄨󵄨 < ∞. y (Lx ) x⩾0,y⩾0

The case p = ∞ is delicate. The first assertion stays valid, while the second becomes ∞ invalid unless an additional condition ‖K1 ‖L∞ := supx⩾0,y⩾0 |k1 (x, y)| < ∞ is prox (Ly ) vided. Due to the construction of the truncation operator, the third assertion generally does not hold in the L∞ sense.

4.2 Approximating solutions of polynomial decay In [11], the authors consider the polynomial decay estimate of the L2 solution to the integral equation as the characteristic part of the kernel function and the right-hand side term are dominated by a polynomial upper bound. This subsection will directly extend this result to the Lp situation, and get a complete picture of the physical parameter settings [15].

100 | S.-Y. Feng and D.-C. Chang Lemma 4.3. For 1 ⩽ p < ∞, suppose that K and L are linear bounded operators in Lp [0, ∞), (I − K)−1 exists, and Λ := ‖(I − K)−1 (L − K)L‖ < 1, then (I − L)−1 exists in Lp [0, ∞) with norm 1 + ‖(I − K)−1 ‖‖L‖ 󵄩󵄩 −1 󵄩 󵄩󵄩(I − L) 󵄩󵄩󵄩 ⩽ 1−Λ and, for any f ∈ Lp [0, ∞), it follows that ‖(I − K)−1 ‖‖Lf − Kf ‖ + Λ‖(I − K)−1 f ‖ 󵄩󵄩 −1 −1 󵄩 . 󵄩󵄩(I − L) f − (I − K) f 󵄩󵄩󵄩 ⩽ 1−Λ Set 1 p

∞T

󵄨 󵄨p Ω(T) := { ∫ ∫󵄨󵄨󵄨k2 (x, y)󵄨󵄨󵄨 dydx} , T 0

and, for any f ∈ Lp [0, ∞), let 1 p

∞T

󵄨 󵄨p Ωf (T) := { ∫ ∫󵄨󵄨󵄨k2 (x, y)f (y)󵄨󵄨󵄨 dydx} , T 0 ∞

1 p

󵄨 󵄨p ωf (T) := { ∫ 󵄨󵄨󵄨f (y)󵄨󵄨󵄨 dy} . T

The following technical lemma attributes the Lp error estimate to the decay of the characteristic part of the kernel function and the right-hand side term at infinity, which improves the result in the L2 space [10]. Lemma 4.4. Suppose that 1 ⩽ p < ∞, Aj < ∞ (j = 1, 2, 3), and that the integral operators, K in (4.2) and KT in (4.3), respectively have solutions φ̂ and φ̂ T given by (4.4). If 1 p Ω(T) < A2 A ‖(I−K) −1 ‖ , then one has the following L estimate: 1

3

‖φ̂ − φ̂ T ‖Lp ⩽

1−

A1 ‖(I − K)−1 ‖ [Ωψ (T) 2 A1 A3 Ω(T)‖(I − K)−1 ‖

̂ Lp ]. + A2 ωψ (T) + A1 A3 Ω(T)‖φ‖

(4.5)

We are now at the stage to formulate the main result as the characteristic part of the kernel function k2 and initial data ψ are both polynomially dominated. The notation X ≲ Y means that there exists some constant C > 0 such that X ⩽ CY. Theorem 4.5. Suppose that 1 ⩽ p < ∞, Aj < ∞ (j = 1, 2, 3), and that the integral operators, K in (4.2) and KT in (4.3), respectively have solutions φ̂ and φ̂ T given by (4.4). 1 + + If |k2 (x, y)|p ≲ (1+x)α1(1+y)β and |ψ(y)|p ≲ (1+y) γ for all (x, y) ∈ R × R with α > 1, β ⩾ 0,

Singular Fredholm integral equations of Chandrasekhar type |

101

α + β > 2, and γ > 1, then if the truncating endpoint T is sufficiently large, the following polynomial error estimates hold: – For β > 1, 1

‖φ̂ − φ̂ T ‖Lp ≲

A1 ‖(I − K)−1 ‖Lp →Lp [(α − 1)(β − 1)(1 + T)α−1 ] p 1

[(α − 1)(β − 1)(1 + T)α−1 ] p − A21 A3 ‖(I − K)−1 ‖Lp →Lp ×{

–

1

A2

+

1

[(γ − 1)(1 + T)γ−1 ] p

1

̂ Lp (β − 1) p + A1 A3 (β + γ − 1) p ‖φ‖

1

[(α − 1)(β − 1)(β + γ − 1)(1 + T)α−1 ] p

For β = 1,

}. (4.6)

1

‖φ̂ − φ̂ T ‖Lp ≲

A1 ‖(I − K)−1 ‖Lp →Lp [(α − 1)(1 + T)α−1 ] p 1

×{ –

1

[(α − 1)(1 + T)α−1 ] p − A21 A3 ‖(I − K)−1 ‖Lp →Lp [ln(1 + T)] p 1

A2

+

1

[(γ − 1)(1 + T)γ−1 ] p

(4.7)

1

̂ Lp [ln(1 + T)] p 1 + A1 A3 γ p ‖φ‖ 1

[(α − 1)γ(1 + T)α−1 ] p

}.

For β < 1, 1

‖φ̂ − φ̂ T ‖Lp ≲

A1 ‖(I − K)−1 ‖Lp →Lp [(α − 1)(1 − β)(1 + T)α+β−2 ] p 1

[(α − 1)(1 − β)(1 + T)α+β−2 ] p − A21 A3 ‖(I − K)−1 ‖Lp →Lp ×{

A2

1

1

[(γ − 1)(1 + T)γ−1 ] p

+

(1 − β) p (1 + T)

β−1 p

1

̂ Lp + A1 A3 (β + γ − 1) p ‖φ‖

}. 1 [(α − 1)(1 − β)(β + γ − 1)(1 + T)α+β−2 ] p (4.8)

Remark 4.6. First of all, the range of parameters in the theorem is necessary. In fact, A3 < ∞ implies α ⩾ 0 and β ⩾ 0, and A2 < ∞ further requires α > 1, while γ > 1 guarantees ωψ (T) < ∞. Secondly, as the kernel function has the form k(x, y) = k1 (x, y)k2 (x, y), k1 (x, y) can only depend on the y variable, and k2 (x, y) can only depend on the x variable. These components are regarded as the symbolic and characteristic functions in the case of separation of kernel function variables. This specific form of the kernel function corresponds to the case of β = 0, which has an independent significance in the theory of celestial radiation.

4.3 Approximating solutions of exponential decay We study the large-time error estimation of the approximate solution when the characteristic part of the kernel function and the right-hand side term have exponential decay in this section. Taking account that there are no parameters in the integral equation, one may use iterative kernel techniques to approximate the solution of the equation.

102 | S.-Y. Feng and D.-C. Chang In fact, this successive approximation method has a wide range of applicability, it is suitable in both the finite and infinite interval cases [7, 10, 13, 15]. Since the truncation operator is the main analysis tool, one needs to structure the truncated solution of the equation. Denoting by φ̄ T the solution to the truncation equation φ − KT φ = ψχ[0,T] , one easily checks that the approximating solution to (4.3) is represented by φ̄ T (x), φ̂ T (x) = { ψ(x),

x ∈ [0, T],

(4.9)

x ∈ (T, ∞).

Similar to Lemma 4.4 in the polynomial case, one needs an error estimate compatible with the upper bound of exponential decay. Lemma 4.7. Suppose that 1 ⩽ p < ∞, Aj < ∞ (j = 1, 2, 3), and that the integral operators, K in (4.2) and KT in (4.3), respectively have solutions φ̂ and φ̂ T given by (4.4). If (I − KT )−1 exists and is uniformly bounded with respect to T, then one has the following Lp estimate: 󵄩 󵄩 ‖φ̂ − φ̂ T ‖Lp ⩽ 2A1 󵄩󵄩󵄩(I − K)−1 󵄩󵄩󵄩Lp →Lp [Ωφ̂ T (T) + A2 ωψ (T)].

(4.10)

Now we extend the L2 case in [13] to the Lp case. Compared with [10], Feng and Chang [13] remove the displacement dominance restriction on the characteristic part of the kernel function and increase the choice of the parameters. Theorem 4.8. Suppose that 1 ⩽ p < ∞, Aj < ∞ (j = 1, 2, 3) with A1 A2 < 1, (I − KT )−1 exists in Lp [0, ∞) and is uniformly bounded in T, and that the integral operators, K in (4.2) and KT in (4.3), respectively have solutions φ̂ and φ̂ T given by (4.4). If 󵄨󵄨 󵄨p −αx−βy , 󵄨󵄨k2 (x, y)󵄨󵄨󵄨 ⩽ e for some α > 0, β > 0, and γ > 0, λ := follows that

󵄨󵄨 󵄨p −γy 󵄨󵄨ψ(y)󵄨󵄨󵄨 ⩽ e , A1

1

(β+γ) p

x ⩾ 0, y ⩾ 0

(4.11)

< 1, then for sufficiently large T > 0, it α

‖φ̂ − φ̂ T ‖Lp ⩽

− T A −γT 2A1 e p [ 12 e p + ]. 1 1 1 − A1 A2 γ p (1 − λ)α p (α + β) p

(4.12)

Remark 4.9. The convergence results in this subsection show that, since the truncation operator is defined pointwise, under the most natural conditions (the conditions proposed in the theorem), the approximate solution of the integral equation generally does not converge in the L∞ sense, unless other conditions are provided to ensure uniform convergence of the operator.

Singular Fredholm integral equations of Chandrasekhar type

| 103

5 Final remarks This article gives a relatively complete picture of the Lp solutions to the singular Fredholm integral equations, but our approach is different from other existing literature. In short, we have avoided the discussion of the H-function. Obviously, how to match the results of this article with the properties of the H-function will become a natural and interesting question. We leave the details to upcoming papers.

Bibliography [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

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104 | S.-Y. Feng and D.-C. Chang

[18] N. Karamollahi, G. B. Loghmani, and M. Heydari, Dual solutions of the nonlinear problem of heat transfer in a straight fin with temperature-dependent heat transfer coefficient, Int. J. Numer. Methods Heat Fluid Flow (2020). https://doi.org/10.1108/HFF-04-2020-0201. [19] G. Mastroianni and G. Monegato, Truncated quadrature rules over and Nyström-type methods, SIAM J. Numer. Anal. 41 (2004), 1870–1892. [20] T. W. Mullikin, Chandrasekhar’s X and Y equations, Trans. Am. Math. Soc. 113 (1964), 316–332. [21] M. S. Muthuvalu and J. Sullaiman, Half-sweep geometric mean iterative method for the repeated Simpson solution of second kind linear Fredholm integral equations, Proyecciones 31 (2012), 65–79. [22] N. K. Nichols, On the convergence of two-stage iterative process for solving linear equations, SIAM J. Numer. Anal. 10 (1973), 460–469. [23] G. M. Phillips, Explicit forms for certain Hermite approximations, BIT Numer. Math. 13 (1973), 177–180. [24] V. Ruggiero and E. Galligani, An iterative method for large sparse systems on a vector computer, Comput. Math. Appl. 20 (1990), 25–28. [25] J. Sulaiman, M. Othman, and M. K. Hasan, A new half-sweep arithmetic mean (HSAM) algorithm for two-point boundary value problems, Proceedings of the international conference on statistics and mathematics and its application in the development of science and technology, 2004, pp. 169–173. [26] J. Sulaiman, M. Othman, N. Yaacob, and M. K. Hasan, Half-sweep geometric mean (HSGM) method using fourth-order finite difference scheme for two-point boundary problems, Proceedings of the first international conference on mathematics and statistics, 2006, pp. 25–33. [27] B. Yang, On the norm of an integral operator and applications, J. Math. Anal. Appl. 321 (2006), 182–192.

Lijia Zhang and Wengu Chen

Extended multichannel weighted Schatten p-norm minimization for color image denoising Abstract: Most existing image denoising methods assume that the noise is additive white Gaussian noise (AWGN). In this paper, considering that the realistic noise may be of different kinds or combinations in practical applications, we propose an extended multichannel weighted Schatten p-norm minimization model based on noise prior for RGB color image denoising. This model uses the Lq (1 ≤ q ≤ 2) norm of the matrix as the data fidelity term instead of the Frobenius norm. To solve the model effectively, we utilize the alternating direction multiplier method (ADMM) via incorporating the generalized soft threshold algorithm and the proximity operator. Numerical experiments show that the proposed mode with L1 -norm data fitting term can result in better performance compared with the Frobenius one in impulsive noise. Keywords: Image denoising, alternating direction multiplier method, generalized soft threshold algorithm MSC 2010: Primary 65F22, 68U10, Secondary 65F10, 68Q25

1 Introduction As a classic inverse problem in computer vision, image denoising aims to recover a clean image x from its noisy observation y = x + n, where n is the noise. Most of the existing imaging denoising methods focus on additive white Gaussian noise (AWGN), which can be categorized into filter [1], sparse coding [12], low-rank approximation [16], effective priors [32], and deep learning [31] based methods. In this paper, we focus on the low-rank matrix approximation (LRMA) methods. LRMA, which aims to recover the underlying low-rank matrix from its degraded observation, has a wide range of applications in computer vision. LRMA methods can be generally categorized into two categories: the low-rank matrix factorization (LRMF) methods [2, 13, 19, 22] and the nuclear norm minimization (NNM) methods [3, 4, 14, 20, 21]. Given a matrix Y ∈ ℝm×n , LRMF aims to find a matrix X ∈ ℝm×n of rank r (r ≪ min(m, n)), which is as close to Y as possible under certain data fitting Acknowledgement: This work was supported by the NSF of China (No. 11871109), CAEP Foundation (Grant No. CX20200027), Key Laboratory of Computational Physics Foundation (Grant No. 6142A05210502). Lijia Zhang, Wengu Chen, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China, e-mails: [email protected], [email protected] https://doi.org/10.1515/9783110741711-006

106 | L. Zhang and W. Chen functions. While X is able to be factorized into the product of two low-rank matrices, namely X = ABT , where A ∈ ℝm×r and B ∈ ℝn×r are two smaller matrix factors. A variety of LRMF methods have been proposed, ranging from the classical singular value decomposition (SVD) under ℓ2 -norm loss to the robust LRMF algorithms [2, 13, 19, 22] under ℓ1 -norm loss. The LRMA methods based on the above mentioned LRMF generally solve a nonconvex optimization problem. The rank minimization methods represent another main branch along this line of research, which reconstruct the data matrix by imposing additional rank constraints on the estimated matrix. However, the direct rank minimization is NP-hard, so it can alternatively be relaxed to minimize the nuclear norm of the estimated matrix to approximate the observation data matrix. This method is called nuclear norm minimization (NNM). The nuclear norm of a matrix X, denoted by ‖X‖∗ , is defined as the sum of its singular values, i. e., ‖X‖∗ = ∑ σi (X), i

where σi (X) denotes the ith singular value of X. NNM aims to approximate Y by X, while minimizing the nuclear norm of X. One significant advantage of NNM lies in that it is the tightest convex relaxation of the nonconvex LRMF problem with a certain data fitting term. Hence, the problem has attracted great interest and attention of researchers in recent years. Candès and Recht [6] have proved that most low-rank matrices can be recovered perfectly by solving an NNM problem. Furthermore, Cai et al. [3] proved that the NNM-based LRMA problem with Frobenius norm data fidelity can be efficiently solved by a soft-threshold operation on the singular values of the observation data matrix. The model can be expressed as X̂ = arg min ‖Y − X‖2F + λ‖X‖∗ , X

(1.1)

where λ is a positive constant, and the solution of (1.1) can be obtained by X̂ = U𝒮λ (Σ)VT ,

(1.2)

where Y = UΣVT is the SVD of Y and 𝒮λ (Σ) is the soft-thresholding function on the diagonal matrix Σ with parameter λ. For each diagonal element Σii in Σ, 𝒮λ (Σ)ii = max{Σii − λ, 0}.

(1.3)

The above-mentioned singular value soft-threshold shrinkage algorithm has been widely used in solving NNM based low-rank matrix problems, such as matrix completion [3], robust principal component analysis (RPCA) [4], and low-rank representation for subspace clustering [11].

Extended MC-WSNM for color image denoising

| 107

Although NNM has been widely used in LRMA, but still flawed. The traditional NNM treats each singular value equally, which results in that the soft-thresholding operator in (1.3) shrinks each singular value by the same amount λ. This, however, ignores the prior knowledge that singular values of different sizes are of different importance. For instance, the larger singular values generally deliver the major edge and texture information. To solve this problem, Zhang et al. proposed a truncated nuclear norm regularization (TNNR) method [30], and Sun et al. introduced the concept of capped nuclear function [25]. Considering that larger singular values should be shrunk less than the smaller ones while recovering an image from its corrupted one, Gu proposed the weighted nuclear norm and studied its minimization [15]. The weighted nuclear norm of a matrix X is defined as ‖X‖w,∗ = ∑ wi σi (X), i

(1.4)

where w = [w1 , . . . , wn ]⊤

(wi ⩾ 0)

is the weight vector assigned to σi (X), which is the corresponding shrinkage threshold. Therefore, the weighted nuclear norm minimization (WNNM) problem is proposed as X̂ = arg min ‖Y − X‖2F + ‖X‖w,∗ . X

(1.5)

Most of the existing images are RGB color images in addition to grayscale images. Xu et al. [28] extended WNNM to RGB color images by using the low-rank prior knowledge of nonlocally similar blocks of the image and introducing a weight matrix to adjust the different degrees of the low-rank constraints of the three channels under different noise levels. The multichannel weighted nuclear norm minimization (MC-WNNM) model is described as 󵄩 󵄩2 X̂ = arg min󵄩󵄩󵄩W(Y − X)󵄩󵄩󵄩F + ‖X‖w,∗ , X

(1.6)

where σr−1 I W=( 0 0

0 σg−1 I 0

0 0 ), σb−1 I

(1.7)

σr , σg , σb correspond to the standard deviations of the three channels, and I is the identity matrix. In the denoising algorithm based on the NNM mentioned above, NNM tends to excessively shrink the singular values and treat different rank components equally, which limits its flexibility in practical applications. WNNM considers the importance

108 | L. Zhang and W. Chen of different sizes of singular values, and weighted singular values are used to shrink them to different degrees. However, the problem of excessive scaling is still unavoidable. The Schatten-p norm is the lp -norm defined on the singular value, which can prevent the singular value from being contracted excessively. Thanks to this, Xie [27] proposed a minimization method based on the weighted Schatten-p norm with 0 < p ≤ 1, which is defined as min(m,n)

‖X‖w,Sp = ( ∑ i=1

1 p

wi σip ) ,

(1.8)

where w = [w1 , . . . , wmin(m,n) ]⊤ is a nonnegative weight vector, and σi is the ith singular value of X, and all singular values are arranged in nonascending order. Therefore, the weighted Schatten-p norm minimization (WSNM) model has the following objective function: X̂ = arg min ‖Y − X‖2F + ‖X‖pw,S . X

p

(1.9)

Applied to the problem of grayscale image denoising, the model has achieved better denoising effect than WNNM when comparing related indicators. Considering that there is not only Gaussian noise but also non-Gaussian noises in practical applications, this paper proposes an extended MC-WSNM model based on some prior information on noises. This model uses the Lq (1 ≤ q ≤ 2) norm of the matrix as the data fidelity term instead of the Frobenius norm. To find the optimal solution of the model effectively, the alternating direction multiplier method is used to optimize it, and then the optimal solution of the model can be obtained effectively through the generalized soft-threshold algorithm and the proximity operator. It is worth noting that the MC-WSNM model is a special form of the extended model for removing Gaussian noise when q = 2, which can also be used to remove impulse noise by setting q = 1. The experimental results show that the L1 -norm data fitting model can result in better performance compared with the Frobenius one in the impulsive noise case. Therefore, the extended multichannel weighted Schatten-p norm minimization model is more applicable.

2 Multichannel weighted Schatten p-norm minimization As a generalization to the multichannel weighted nuclear norm minimization (MC-WNNM) model [28] and Schatten-p norm minimization [27], the multichannel

Extended MC-WSNM for color image denoising

| 109

weighted Schatten-p norm minimization (MC-WSNM) model [24] is described as 󵄩2 󵄩 X̂ = arg min󵄩󵄩󵄩W(Y − X)󵄩󵄩󵄩F + ‖X‖pw,S , p

X

(2.1)

where σr−1 I W=( 0 0

0 σg−1 I 0

0 0 ), σb−1 I

(2.2)

σr , σg , σb correspond to the standard deviations of the three channels, and I is the identity matrix. Also ‖X‖w,Sp is defined as in (1.8). Since larger singular values tend to encode more important information than smaller singular values, the weight parameter wi is set to be wi = C/(σi1/p + ϵ), where C > 0 is a constant and ϵ is a small positive number which can prevent the denominator from being zero. Through this setting, the original information components related to the larger singular value will be less affected. Given a noisy color image Yc , suppose that we have extracted N local patches {yj }Nj=1 and patches similar to them. We then arrange the nonlocally similar blocks corresponding to each yj (including yj itself) column by column to form N noise block matrices {Yj }Nj=1 . Then clean matrices {Xj }Nj=1 can be estimated from noisy patch matrices {Yj }Nj=1 by MC-WSNM. The patches in matrices {Xj }Nj=1 are aggregated to form the denoised image X̂ c .

The MC-WSNM model has achieved better experimental results than other advanced algorithms. However, this algorithm only considers the Gaussian noise and ignores the prior knowledge of other types of noise in practical applications. In response to this shortcoming, both low-rank and noise priors are used to improve the model in this paper, and an extended multichannel weighted Schatten-p norm minimization model is proposed based on the Lq (1 ≤ q ≤ 2) norm data fidelity term and low-rank regular constraints, and applied to color image noise removal.

3 Extended multichannel weighted Schatten-p norm minimization 3.1 The proposed color image denoising model The Frobenius norm data-fitting has been widely studied and applied to (1.1), (1.5), (1.6), (1.9), (2.1) and many other variants since it models well Gaussian noise in the

110 | L. Zhang and W. Chen maximum likelihood sense. However, the measurement noise may be of different kinds or a mixture in many practical applications, among which impulsive noise is a typical case. Impulsive noise affects the original image by replacing part of the image pixel values with random noise values while keeping the rest unchanged. It can model large errors in observations and has been extensively studied in robust statistics. The impulsive corruption in measurements may come from missing data in the measurement process, transmission problems [5, 7, 23], faulty memory locations [8], buffer overflow [17], and so on. When the image and video is corrupted by impulsive noise, the Frobenius norm data-fitting model is inefficient as it is well known that the model based on least-squares estimators is highly sensitive to outliers in the observation. Even an excellent denoising algorithm like WSNM cannot effectively work. Yang and Zhang [29] obtained a robust model by using L1 -norm as a data-fitting item, and proved that L1 -norm loss function can significantly improve the denoising performance compared to Frobenius one when there is a large error or impulsive noise in the measurement. Impulsive noise can be well modeled as a symmetric stable (SαS) process [26], which can be described by a characteristic function φ(ω) = exp(jaω − γ α |ω|α ), where 0 < α ≤ 2 is the characteristic index for measuring the tail thickness of the distribution. The smaller the α, the thicker the tail of the SαS distribution, and the stronger the impulsive noise. Further, a ∈ ℝ is the position parameter while γ > 0 is the dispersion coefficient about the deviation of the distribution sample from the mean. The probability density function (PDF) of the SαS distribution has no closedform expression except for a few special cases. Therefore, when the impulsive noise is modeled as the SαS distribution, it is difficult to obtain the maximum posterior estimate of X. The generalized Gaussian distribution (GGD) can also be used to model impulsive noise. The probability density function (PDF) of the zero-mean generalized Gaussian distribution variable x is P(x) =

v

2βΓ( v1 )

exp(−

|x|v ), βv

(3.1)

where v > 0 is the shape parameter that controls the shape of the distribution; β is the scale parameter; Γ(⋅) is the gamma function; β = σ√

Γ(1/v) , Γ(3/v)

where σ is the standard deviation.

The flexible parametric form of GGD (3.1) is suitable for a large class of symmetric distributions from super-Gaussian (v < 2) to sub-Gaussian (v > 2), including various specific distributions, such as Laplace distribution (v = 1) and Gaussian distribution

Extended MC-WSNM for color image denoising |

111

(v = 2). When v < 2, the generalized Gaussian distribution has a heavy tail, so it is more suitable for modeling impulsive noise. In this paper, we use the Lq (1 ≤ q ≤ 2) norm as the loss function of the data-fitting term, and propose the following denoising model (Ex-MC-WSNM): 󵄩q 󵄩 X̂ = arg min󵄩󵄩󵄩W(Y − X)󵄩󵄩󵄩q + ‖X‖pw,S , p

X

(3.2)

where ‖X‖q represents the Lq -norm of the matrix X, namely m n

q

1 q

‖X‖q = (∑ ∑ |xij | ) . i=1 j=1

The weight matrix W can be determined under the maximum a posteriori (MAP) estimation framework: X̂ = arg max ln P(X|Y, w) X

P(X, Y, w) P(Y, w) P(Y|X, w)P(X|w)P(w) = arg max ln P(Y|w)P(w) X P(Y|X)P(Y|w)P(X|w) = arg max ln P(Y|w) X = arg max ln X

= arg max ln P(Y|X) + ln P(X|w). X

(3.3)

Assuming that impulsive noise is independent of RGB channels and independently and identically distributed (i. i. d.) in each channel with the generalized Gaussian distribution (GGD) and standard deviations {σr , σg , σb }. Assume every column of Y is 3s2 vector. Then v

3s2

{P(Y|X) = ∏c∈{r,g,b} [ 2β Γ( 1 ) ] c v { Γ(1/v) √ β = σ . c Γ(3/v) { c

exp(− β1v ‖Yc − Xc ‖vv ), c

(3.4)

In addition, P(X|w) ∝ exp(−‖X‖pw,S ). p

Putting (3.4) and (3.5) into (3.3), we have X̂ = arg min X

1 ‖Yc − Xc ‖vv + ‖X‖pw,S v p β c∈{r,g,b} c ∑

󵄩 󵄩v = arg min󵄩󵄩󵄩W(Y − X)󵄩󵄩󵄩v + ‖X‖pw,S , X

p

(3.5)

112 | L. Zhang and W. Chen with βr−1 I W=( 0 0

0 βg−1 I 0

0 0 ). −1 βb I

It is noteworthy that, when q = 2, the extended model (3.2) degenerates to the MC-WSNM model (2.1), which can be used to remove Gaussian noise; when q = 1, the model given in (3.2) can be used to remove impulsive noise. Furthermore, when the weight matrix W is an identity matrix and q = 1, the extended model (3.2) degenerates to a grayscale image impulsive denoising model [9], and the WSNM model is also a special form of the model (3.2) for q = 2. Hence, we can conclude that the proposed model (3.2) is a more generalized model, which includes both the denoising of Gaussian noise and the processing of impulsive noise.

3.2 Model optimization We have known that the Ex-MC-WSNM model (3.2) can be reduced to MC-WSNM model (2.1) when q = 2. When 1 ≤ q < 2, the following Theorem 3.2 will be used to help solve the extended model (3.2). Theorem 3.1 ([18]). Let A, B ∈ ℝm×n , r = min{m, n}, and σ1 (A) ≥ ⋅ ⋅ ⋅ ≥ σr (A),

σ1 (B) ≥ ⋅ ⋅ ⋅ ≥ σr (B)

are singular values of matrices A and B, respectively, then σi (A⊤ B) ≤ σi (A)‖B‖,

(3.6)

where ‖B‖ = σ1 (B), 1 ≤ i ≤ r. Theorem 3.2. The Ex-MC-WSNM model (3.2) is equivalent to the following problem: 󵄩 󵄩q X̂ = arg min󵄩󵄩󵄩W(Y − X)󵄩󵄩󵄩q + ‖WX‖pw′ ,S , p

X

(3.7)

where wi′ = wi ‖W‖−p . Proof. By Theorem 3.1, we have σi (WX) ≤ σi (W)‖X‖,

1 ≤ i ≤ r.

Note that the singular value and the spectral norm are unchanged under the matrix transposition. So σi (WX) = σi (X⊤ W) ≤ σi (X⊤ )‖W‖ = ‖W‖σi (X).

(3.8)

Extended MC-WSNM for color image denoising |

113

Accordingly, we get 󵄩 󵄩 σi (X) = σ(W−1 WX) ≤ 󵄩󵄩󵄩W−1 󵄩󵄩󵄩σi (WX) = ‖W‖−1 σi (WX) and ‖W‖σi (X) ≤ σi (WX).

(3.9)

It can be derived from (3.8) and (3.9) that ‖W‖σi (X) = σi (WX), { σi (X) = ‖W‖−1 σi (WX). Therefore, r

r

i=1

i=1

‖X‖pw,S = ∑ wi σip (X) = ∑ wi ‖W‖−p σip (WX). p

Set w′ = [w1′ , . . . , wr′ ] , ⊤

where wi′ = wi ‖W‖−p .

Then r

‖X‖pw,S = ∑ wi′ σip (WX) = ‖WX‖pw′ ,S . p

p

i=1

Altogether, we can conclude that 󵄩󵄩 󵄩q 󵄩 󵄩q p p 󵄩󵄩W(Y − X)󵄩󵄩󵄩q + ‖X‖w,Sp = 󵄩󵄩󵄩W(Y − X)󵄩󵄩󵄩q + ‖WX‖w′ ,S . p To solve problem (3.7), let WY = E and WX = L. Then (3.7) can be transformed into L̂ = arg min ‖E − L‖qq + ‖L‖pw′ ,S . L

p

(3.10)

Hence, we only need to get the solution of problem (3.10), and the solution of problem (3.7) as X̂ = W−1 L̂ can be further obtained. For the optimization problem (3.10), the alternating direction multiplier method (ADMM) is used to solve it.

114 | L. Zhang and W. Chen First of all, we introduce the augmented variable Q so that the problem can be transformed into the following equivalent linear equality constraint problem: min ‖Q‖qq + ‖L‖pw′ ,S L,Q

such that

p

E − L = Q,

(3.11)

and minimize its Lagrangian function q

p

ℒ(L, Q, A, ρ) = ‖Q‖q + ‖L‖w′ ,S + ⟨A, E − L − Q⟩ + p

ρ ‖E − L − Q‖2F , 2

(3.12)

where A is the Lagrangian multiplier, and ρ > 0 is the penalty parameter. Next, we update the variables iteratively. Specifically, in the (k + 1)th iteration, the minimization equation (3.12) can be decomposed into the following subproblems: (1) For the fixed variables Q and A, update L as L(k+1) = arg min ‖L‖pw′ ,S + L

p

2 ρ(k) 󵄩󵄩󵄩󵄩 1 (k) 󵄩󵄩󵄩 (k) 󵄩󵄩E − L − Q + (k) A 󵄩󵄩󵄩 . 󵄩󵄩F 2 󵄩󵄩 ρ

(3.13)

(2) For the fixed variables L and A, update Q as Q(k+1) = arg min ‖Q‖qq + Q

󵄩󵄩2 ρ(k) 󵄩󵄩󵄩󵄩 1 󵄩 (k+1) − Q + (k) A(k) 󵄩󵄩󵄩 . 󵄩󵄩E − L 󵄩󵄩F 2 󵄩󵄩 ρ

(3.14)

(3) For the fixed variables L and Q, update A as A(k+1) = A(k) + ρ(k) (E − L(k+1) − Q(k+1) ). (4) Update ρ as ρ(k+1) = μρ(k)

(μ > 1).

Finally, we repeat the above iterative update steps until the convergence condition is met or the number of iterations exceeds the preset threshold. It can be seen that the subproblem (3.13) is similar to the problem (1.9) in [27]. Therefore, it can be similarly transformed into several unconstrained independent subproblems with lp -norm minimization, and the generalized soft-threshold (GST) algorithm can be used for optimizing and solving. In order to effectively solve the subproblem (3.14), the proximity operator has been introduced, which is defined as in [10]: proxg,η (t) = arg min g(x) + x

Then, the following proposition has been given.

η ‖x − t‖22 . 2

Extended MC-WSNM for color image denoising |

115

Proposition 3.3 ([26]). When 1 < q < 2, the proximity operator for the Lq -norm function satisfies prox 1 |z|q ,1 (b) = sign(b) z∗ , ρ

where z∗ is the solution of the equation z − |b| +

q q−1 z = 0, ρ

z ≥ 0.

Next, the subproblem (3.14) is optimized and solved. Specifically, let 1

B(k) = E − L(k+1) +

ρ(k)

A(k) .

The subproblem (3.14) can be transformed into Q(k+1) = arg min ‖Q‖qq + Q

ρ(k) 󵄩󵄩 (k) 󵄩2 󵄩B − Q󵄩󵄩󵄩F . 2 󵄩

(3.15)

When q = 1, the solution of the problem (3.15) is Q(k+1) = prox‖Q‖1 ,ρ(k) (B(k) ) = S1/ρ(k) (B(k) ), where S1/ρ(k) (⋅) is a widely used soft-threshold operator defined by 󵄨󵄨 (k) 󵄨󵄨 (k) S1/ρ(k) (B(k) )ij = sign(B(k) ij ) max{󵄨󵄨Bij 󵄨󵄨 − 1/ρ , 0}. When 1 < q < 2, Q(k+1) = arg min ‖Q‖qq + Q

ρ(k) 󵄩󵄩 (k) 󵄩2 󵄩󵄩B − Q󵄩󵄩󵄩F 2

1󵄩 1 󵄩2 = arg min ρ(k) ( 󵄩󵄩󵄩B(k) − Q󵄩󵄩󵄩F + (k) ‖Q‖qq ) 2 ρ Q 1 1 2 = arg min ∑(B(k) − Qij ) + (k) ∑ |Qij |q . ij 2 i,j ρ i,j Q It can be known from Proposition 3.3 that the solution Q(k+1) of the subproblem (3.14) satisfies the equation ∗ Q(k+1) = sign(B(k) ij ij ) Qij ,

where Q∗ij satisfies the equation q q−1 󵄨 󵄨󵄨 h(m) = m − 󵄨󵄨󵄨B(k) = 0, ij 󵄨󵄨 + (k) m ρ

m ≥ 0.

116 | L. Zhang and W. Chen Note that q(q−1) q−2 ′ > 0, {h (m) = 1 + ρ(k) m { ′′ q(q−1)(q−2) q−3 h (m) = m < 0. ρ(k) {

So h(m) is a monotonically increasing and concave function. When B(k) ≠ 0, we have ij {h(0) = −|Bij | < 0, q−1 { h(|B(k) |) = ρq(k) |B(k) | > 0. ij ij { (k)

Consequently, Q∗ij satisfies 0 < Q∗ij < |B(k) | and can be calculated by Newton’s method. ij The initial point can be chosen to be a positive lower bound of the solution as 1

m

(0)

ϕ q−1 ,

={ ϕ,

ϕ < 1, ϕ ≥ 1,

with ϕ =

|B(k) | ij

q ρ(k)

+1

.

1

In practical applications, ϕ q−1 may be very small when ϕ < 1 and q → 1+ . In this case, a very small constant δ > 0 will be preset. If h(δ) ≤ 0, the initial point m(0) = δ will be selected. Otherwise, we directly set Q∗ij = 0 since the true solution is very small and less than δ when h(δ) > 0. As a result, the minimum solution of the Ex-MC-WSNM model (3.2) is obtained.

3.3 Numerical experiments In this section, we present the experimental results of impulse noise removal of the proposed Ex-MC-WSNM model. The 24 color images in the Kodak PhotoCD dataset (http://r0k.us/graphics/kodak/) are used in our experiments as test images, and saltand-pepper impulse noise with the noise density percentages of 5 %, 15 %, and 20 % are respectively added to these images to produce the noisy images. We have compared the denoising performance of the proposed Ex-MC-WSNM model with the MC-WSNM model and MC-WNNM model, using peak signal-to-noise ratio (PSNR) and intuitive visual effects as measurement indicators. For the fairness and comprehensiveness of the comparison, the necessary parameters in the algorithm proposed in this paper are consistent with the parameter settings in the comparison algorithms. Among them, we have set the size of the image block to s = 6, the number of nonlocally similar blocks to M = 70, the window size for searching for nonlocally similar blocks is 40 × 40, the initial penalty parameter ρ(0) = 3, update parameter μ = 1.001. The numerical experimental results of impulse noise removal under different noise density percentage settings is carried out first. The PSNR values obtained by the denoising algorithms in this paper and comparison algorithms mentioned are shown

Extended MC-WSNM for color image denoising |

117

Table 6.1: Denoising results (PSNR) by different methods for impulse noise density of 5%. Images kodim01 kodim02 kodim03 kodim04 kodim05 kodim06 kodim07 kodim08 kodim09 kodim10 kodim11 kodim12 kodim13 kodim14 kodim15 kodim16 kodim17 kodim18 kodim19 kodim20 kodim21 kodim22 kodim23 kodim24 Average

MC-WNNM

MC-WSNM

5% Ex-MC-WSNM

25.11 24.45 26.25 26.38 24.15 24.63 27.54 25.02 27.66 27.33 25.48 25.80 23.02 24.60 24.66 25.91 25.56 23.46 25.94 23.99 25.64 25.19 26.16 24.87 25.37

25.05 23.88 24.86 25.15 23.99 23.86 26.71 24.20 25.49 25.28 23.88 23.57 22.81 23.92 22.94 24.52 23.83 22.50 25.02 22.29 25.33 24.34 24.48 24.23 24.26

24.63 27.39 28.85 27.88 23.25 25.32 27.87 23.51 28.63 28.27 25.91 28.47 21.80 24.81 27.28 27.60 26.89 24.04 27.12 26.84 25.58 26.39 28.41 24.40 26.30

in Tables 6.1–6.3. The optimal PSNR value of each color image is marked in bold. It can be seen from Table 6.1, when the impulse noise density is 5 %, the algorithm proposed in this paper obtains the optimal PSNR value in 18 of the 24 color test pictures, and the average PSNR is 0.93 and 2.04 dB higher than that of MC-WNNM and MC-WSNM algorithms, respectively. When the impulse noise density is 15 % as presented in Table 6.2, the proposed algorithm obtains a higher PSNR value in 20 images than other comparison algorithms. On average, PSNR value of the proposed algorithm is 2.28 and 0.84 dB higher than that of the MC-WNNM and MC-WSNM algorithms, respectively. It can be seen from Table 6.3 with the impulse noise density 20 % that the algorithm proposed in this paper obtains the highest average PSNR value among the 24 images, which is 2.11 and 1.84 dB higher than the MC-WNNM and MC-WSNM algorithms. In the following, a comparison chart of visual denoising effects under different impulse noise densities has been plotted, as shown in Figs. 6.1–6.3, which show the impulse noise removal effect when the impulse noise density is 5, 15, and 20, respectively. As can be seen in the three figures, for the impulsive noise color images with

118 | L. Zhang and W. Chen Table 6.2: Denoising results (PSNR) by different methods for impulse noise density 15%. Images kodim01 kodim02 kodim03 kodim04 kodim05 kodim06 kodim07 kodim08 kodim09 kodim10 kodim11 kodim12 kodim13 kodim14 kodim15 kodim16 kodim17 kodim18 kodim19 kodim20 kodim21 kodim22 kodim23 kodim24 Average

MC-WNNM

MC-WSNM

15 % Ex-MC-WSNM

21.74 22.32 22.47 22.53 20.28 21.26 22.74 20.73 23.59 23.22 21.66 22.48 20.12 20.94 20.72 22.18 21.09 20.22 22.26 19.52 22.13 22.05 22.04 21.20 21.65

22.88 24.26 24.18 24.32 21.22 22.60 24.31 21.80 25.25 25.02 22.78 24.26 21.02 22.07 22.51 23.78 22.40 21.47 23.74 21.26 23.41 23.56 23.91 22.17 23.09

22.48 24.92 26.29 25.27 20.91 23.45 24.63 21.05 25.94 25.60 23.73 26.42 20.34 22.56 24.57 25.39 24.22 22.06 24.67 24.07 23.55 24.31 25.52 22.43 23.93

Figure 6.1: Visual comparison of different denoising methods on the image kidom07.

Extended MC-WSNM for color image denoising |

Table 6.3: Denoising results (PSNR) by different methods for impulse noise density 20%. Images kodim01 kodim02 kodim03 kodim04 kodim05 kodim06 kodim07 kodim08 kodim09 kodim10 kodim11 kodim12 kodim13 kodim14 kodim15 kodim16 kodim17 kodim18 kodim19 kodim20 kodim21 kodim22 kodim23 kodim24 Average

MC-WNNM

MC-WSNM

20 % Ex-MC-WSNM

20.79 21.62 21.58 21.74 19.17 20.43 21.64 19.55 22.56 22.28 20.47 21.55 19.30 20.00 19.88 21.46 20.06 19.43 21.29 18.60 21.15 21.25 21.20 20.09 20.71

20.92 22.34 21.90 22.06 19.41 20.64 21.96 19.67 22.91 22.61 20.75 21.92 19.40 20.16 20.37 21.60 20.27 19.53 21.53 18.96 21.39 21.53 21.48 20.26 20.98

21.53 23.83 25.02 24.05 19.87 22.61 23.40 19.95 24.81 24.43 22.68 25.13 19.65 21.53 23.01 24.46 22.90 21.11 23.64 22.41 22.66 23.35 24.08 21.55 22.82

Figure 6.2: Visual comparison of different denoising methods on the image kidom23.

119

120 | L. Zhang and W. Chen

Figure 6.3: Visual comparison of different denoising methods on the image kidom01.

different noise densities, different degrees of noise still remain in the image after denoising by models MC-WNNM and MC-WSNM, the image is not clear enough, and the effect of removing impulse noise is not ideal. However, the denoising algorithm in this paper has a significant improvement in noise suppression, and achieved a better denoising effect. In summary, this paper proposed the Ex-MC-WSNM model without the knowledge of the noise type of the degraded images. Numerical experiments present better performance on removing impulse noise in terms of the PSNR value or the visual effect compared with the state-of-the-art MC-WNNM and MC-WSNM models.

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Guoliang Li and Junfeng Li

The boundedness of a bilinear oscillatory integral Abstract: In this paper, the bilinear oscillatory integral 1

Tβ (f , g)(x) = ∫ f (x − t)g(x + t)ei|t|

−β

−1

dt , |t|

is proved to be Lp (ℝ) × Lq (ℝ) → Lr (ℝ) bounded for all 1 < p, q < ∞.

1 p

β>0

+

1 q

= 1r ,

1 2

< r < ∞, and

Keywords: Bilinear oscillatory integral, bilinear Hilbert transform MSC 2010: 42B20, 42B25

1 Introduction In this short note, we investigate the boundedness of the following bilinear oscillatory operator: 1

Tβ (f , g)(x) = ∫ f (x − t)g(x + t)ei|t| −1

−β

dt , |t|

where β > 0.

The main result is Theorem 1.1. Let 0 < β < ∞ and f , g ∈ 𝒮 (ℝ1 ). Then 󵄩󵄩 󵄩 󵄩󵄩Tβ (f , g)󵄩󵄩󵄩r ≤ Cβ,r,p,q ‖f ‖p ‖g‖q

Acknowledgement: Junfeng Li∗ (the corresponding author) is supported by NSFC (# 12071052) and the Fundamental Research Funds for the Central Universities. Guoliang Li, Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China, e-mail: [email protected] Junfeng Li, School of Mathematical Sciences, Dalian University of Technology, Dalian, LN, 116024, China, e-mail: [email protected] https://doi.org/10.1515/9783110741711-007

124 | G. Li and J. Li holds for any triple (p, q, r) obeying 1 1 1 = + , r p q

p, q > 1,

1 < r < ∞, 2

(1.1)

and the constant Cβ,r,p,q < ∞ depending only on β, p, q, and r. If we remove the oscillatory term ei|t| , we arrive at the bilinear Hilbert transform −β

1

H(f , g) = p. v. ∫ f (x − t)g(x + t) −1

dt . t

Here p. v. stands for the principal value. This operator was introduced by Calderón in the 1960s in connection with the first Calderón commutator. There was a long-standing open problem to establish the L2 × L2 → L1 boundedness of H. This problem was settled by Lacey–Thiele in the late 1990s. They proved the Lp × Lq → Lr boundedness of H by a newly-developed method which is called the time–frequency analysis. This method was introduced by Carleson [2] and developed by Fefferman [4]. Below is Lacey–Thiele’s result. Theorem 1.2 (Lacey–Thiele [10, 11]). The bilinear Hilbert transform H is Lp × Lq → Lr bounded whenever 1 1 1 = + , r p q

p, q > 1,

2 < r < ∞. 3

Consequently, Lacey [9] obtained the same boundedness of the corresponding maximal function. For the case 21 < r ≤ 32 , he showed that the model operator is unbounded. But the boundedness problem for the standard bilinear Hilbert transform H is still open. These results were extended uniformly by Grafakos–Li [6] and Li [14] to the parameterized bilinear Hilbert transform Hα,β (f , g)(x) := p. v. ∫ f (x − αt)g(x − βt)

dt , t

α, β ∈ ℝ.

Recently, Benea–Bernicot–Lie–Vitturi [1] introduced the bilinear Hilbert–Carleson operator 󵄨󵄨 a dt 󵄨󵄨 󵄨 󵄨󵄨 BCa (f , g)(x) := sup󵄨󵄨󵄨∫ f (x − t)g(x + t)eiλt 󵄨 t 󵄨󵄨󵄨 λ∈ℝ 󵄨󵄨 and showed that in the nonresonant case a ∈ (0, ∞) \ {1, 2} the operator BCa has the same boundedness.

The boundedness of a bilinear oscillatory integral | 125

The obstacle to push the index to r > 21 in the above results is that the operators are invariant under modulations. For example, the bilinear Hilbert transform satisfies 󵄨 󵄨 󵄨 󵄨󵄨 ic 󵄨󵄨H(Mc f , Mc g)󵄨󵄨󵄨 = 󵄨󵄨󵄨T(f , g)󵄨󵄨󵄨 where Mc f (x) := e f (x). This invariance makes the analysis around the resonance set very difficult. Here the resonance set is the diagonal line of the first and third quadrants in the frequency space. Thus the bilinear Hilbert transform must be split into scales and the frequency behavior of each scale should be understood. Now the time–frequency analysis comes in. However, if the operator has extra curvature which breaks the invariance, then the boundedness is better. For instance, consider the bilinear Hilbert transform along the parabola ∞

Ht 2 (f , g)(x) := p. v. ∫ f (x − t)g(x − t 2 ) −∞

dt . t

It was proved to be Lp (ℝ) × Lq (ℝ) → Lr (ℝ) bounded whenever r > 21 by Li [15] and Li– Xiao [16], respectively. A similar discussion on the bilinear Hilbert transform along a general curve can be found in [3, 7, 13] and the references therein. Recently the authors of this paper also considered the bilinear oscillatory integral along the parabola β

Tβ,t 2 (f , g)(x) := p. v. ∫ f (x − t)g(x − t 2 )ei|t| ℝ

dt , t

thereby obtaining the following result. Theorem 1.3 (Li–Li [12]). Let −∞ < β < 0 or 1 < β < ∞. Then 󵄩󵄩 󵄩 󵄩󵄩Tβ,t 2 (f , g)󵄩󵄩󵄩r ≤ Cβ,r,p,q ‖f ‖p ‖g‖q holds for any triple (p, q, r) obeying 1 1 1 = + , r p q

p, q > 1,

1 < r < ∞, 2

and the constant Cβ,r,p,q < ∞ depending only on β, p, q, and r. This result settles a problem posted in [5]. The main effort within the argument is β to handle the conflict between the curvature term t 2 and the oscillatory term ei|t| . In −β this paper, we consider only the oscillatory term ei|t| . Even for a bilinear oscillatory operator, Tβ is also modulation invariant, but the extra oscillatory term helps us gain r > 21 . Via Theorem 1.1, we can only deal with the case |t| < 1 since the oscillatory term satisfies ei|t| ∼ 1 −β

when |t| ≫ 1.

126 | G. Li and J. Li Unfortunately, the method used here is not good enough to push the result to r > the latter case.

1 2

for

2 Strategy of the proof In this section, let us explain the strategy of the proof. There exists a nonnegative smooth bump function ρ supported on [−3, −1] ∪ [1, 3] such that the following decomposition is true: 1 = ∑ ρ (t) |t| j∈ℤ j

∀ t ≠ 0

where ρj (t) = 2j ρ(2j t).

(2.1)

Under this consideration and for natural number j ≥ 1 and β > 0, let βj

Tj (f , g)(x) = ∫ f (x − 2−j t)g(x + 2−j t)ei2

|t|−β

ρ(t)dt.

ℝ

We first obtain the following uniform annual estimate. Theorem 2.1. Let 0 < β < ∞,

1 ≤ r < ∞, 2

p ≥ 1, q ≥ 1,

1 1 1 + = . p q r

Then there is a constant C independent of j ∈ {1, 2, 3, . . . } such that 󵄩󵄩 󵄩 󵄩󵄩Tj (f , g)󵄩󵄩󵄩r ≤ C‖f ‖p ‖g‖q .

(2.2)

Proof. We denote βj

T̃ j (f , g)(x) = ∫ f (x − t)g(x + t)ei2

|t|−β

ρ(t)dt.

ℝ

By rescaling, (2.2) is equivalent to 󵄩󵄩 ̃ 󵄩 󵄩󵄩Tj (f , g)󵄩󵄩󵄩r ≤ C‖f ‖p ‖g‖q ,

that is,

󵄩󵄩 ̃ 󵄩 󵄩󵄩Tj (f , g)󵄩󵄩󵄩r ≲ ‖f ‖p ‖g‖q .

(2.3)

Since ρ is integrable, when 1 ≤ r < ∞, (2.3) follows directly from Hölder’s inequality. For the case 21 ≤ r < 1, by using the finite support of ρ, we may assume that f and g are locally supported in some intervals with length comparable to 1. Thus, we get 󵄩󵄩 ̃ 󵄩 󵄩 󵄩 󵄩󵄩Tj (f , g)󵄩󵄩󵄩r ≲ 󵄩󵄩󵄩T̃ j (f , g)󵄩󵄩󵄩1 ≲ ‖f ‖1 ‖g‖1 ≲ ‖f ‖p ‖g‖q , thereby finishing the proof.

The boundedness of a bilinear oscillatory integral | 127

From (2.1), we have the following decomposition: Tβ (f , g)(x) = ∑ Tj (f , g)(x). j≥0

To obtain Theorem 1.1, it is sufficient to show that for β > 0, p, q, and r satisfying (1.1), there exists ϵ > 0 obeying 󵄩 󵄩󵄩 −ϵj 󵄩󵄩Tj (f , g)󵄩󵄩󵄩r ≲ϵ 2 ‖f ‖p ‖g‖q . In the above and below the notation A ≲ϵ B means that there is some constant Cϵ depending on ϵ but not on A and B such that A ≤ Cϵ B. By rescaling, this last estimate is equivalent to 󵄩󵄩 ̃ 󵄩 −ϵj 󵄩󵄩Tj (f , g)󵄩󵄩󵄩r ≲ϵ 2 ‖f ‖p ‖g‖q . Upon noticing the uniform estimate (2.3), we are required to show that there exists ϵ > 0 such that 󵄩󵄩 ̃ 󵄩 −ϵj 󵄩󵄩Tj (f , g)󵄩󵄩󵄩1 ≲ϵ 2 ‖f ‖2 ‖g‖2 holds for sufficiently large j > 0. Again, noting that ρ has finite support, our task can be reduced further to validating Theorem 2.2. For 0 < β < ∞, there is a constant ϵ > 0 independent of j ∈ {1, 2, 3, . . . } such that 󵄩󵄩 ̃ 󵄩 −ϵj 󵄩󵄩Tj (f , g)󵄩󵄩󵄩2 ≲ϵ 2 ‖f ‖2 ‖g‖2 . As a matter of fact, applying the Fourier transform, we have ̂ σ̃ j (ξ , η)eix(ξ +η) dξdη T̃ j (f , g)(x) = ∬ f ̂(ξ )g(η) ℝ2

with σ̃ j (ξ , η) = ∫ ρ(t)ei(tξ −tη+2

βj

|t|−β )

dt.

ℝ

The main idea is to obtain a suitable decay estimate for the bilinear multiplier σ̃ j (ξ , η). Here, the critical points of the phase function ϕj,ξ ,η (t) = 2βj (2−βj ξt − 2−βj ηt + |t|−β ) in the support of ρ play the crucial role. Therefore, the following decomposition in terms of 2−βj ξ and 2−βj η will be natural.

128 | G. Li and J. Li ̂ is a nonnegative bump function Let Φ : ℝ → ℝ be a Schwartz function such that Φ supported on [−3, −1] ∪ [1, 3] and satisfies ̂ ∑ Φ( k

ξ )=1 2k

for all ξ ∈ ℝ \ {0}.

For a function f ∈ 𝒮 (ℝ), we define ̂ ξ )eixξ dξ . Pk f (x) := fk (x) := ∫ ̂f (ξ )Φ( 2k ℝ

Moreover, we denote ̂ x ))eixξ dξ . P≤k f (x) := f≤k (x) := ∫ ̂f (ξ )( ∑ Φ( ′ 2k k ′ ≤k ℝ

With these notations and for j ≥ 1, we write T̃ j (f , g)(x) =

∑ T̃ j,m,m′ (f , g)(x)

m,m′ ∈ℤ

with ξ η i(ξ +η)x ̂ ̂ ̃ ̂ Φ( )g(η) dξdη. T̃ j,m,m′ (f , g)(x) = ∬ f ̂(ξ )Φ( ′ )σj (ξ , η)e 2βj+m 2βj+m ℝ2

By symmetry, we can also assume m′ ≤ m, and renew our aim to establish 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 −ϵj 󵄩󵄩 ∑ T̃ j,m,m′ (f , g)󵄩󵄩󵄩 ≲ϵ 2 ‖f ‖2 ‖g‖2 . 󵄩󵄩 ′ 󵄩󵄩2 m ≤m

(2.4)

m,m′ ∈ℤ

By the Plancherel theorem, (2.4) could be estimated via two cases below. For m ∈ ℤ, we have 󵄩󵄩 ̃ 󵄩 −ϵj 󵄩󵄩Tj (fm+βj , g≤m+βj−10 )󵄩󵄩󵄩2 ≲ϵ 2 ‖fm+βj ‖2 ‖g≤m+βj−10 ‖2 , and 󵄩 󵄩󵄩 ̃ −ϵj 󵄩󵄩Tj (fm+βj , gm+βj )󵄩󵄩󵄩2 ≲ϵ 2 ‖fm+βj ‖2 ‖gm+βj ‖2 . By rescaling again, the argument for Theorem 2.2 will be completed via proving m+βj 󵄩󵄩 󵄩 −ϵj − 󵄩󵄩Tj,m (f0 , g≤−10 )󵄩󵄩󵄩2 ≲ϵ 2 2 2 ‖f0 ‖2 ‖g≤−10 ‖2

(2.5)

The boundedness of a bilinear oscillatory integral | 129

and m+βj 󵄩 󵄩󵄩 −ϵj − 󵄩󵄩Tj,m (f0 , g0 )󵄩󵄩󵄩2 ≲ϵ 2 2 2 ‖f0 ‖2 ‖g0 ‖2 ,

(2.6)

where ix(ξ +η) ̂ Tj,m (f , g)(x) := ∫ℝ2 f ̂(ξ )g(η)σ dξdη, j,m (ξ , η)e

{

βj

σj,m (ξ , η) := ∫ ρ(t)ei2

ϕj,m (t,ξ ,η)

dt and ϕj,m (t, ξ , η) = t2m (ξ − η) + |t|−β .

3 The noncritical cases In this section, we settle some elementary estimates for the multiplier which, of course, depend on the critical points of ϕj,m . Without loss of generality, we may consider the case t > 0 only. The case t < 0 can be handled similarly. Let v = 2m (ξ − η) and ϕj,v (t) = tv + t −β ,

t > 0.

Then ϕ′j,v (t) = v − βt −(β+1) . Its possible critical point would be 1 − β+1

t0 = (v/β)

.

At this point we have β

ϕj,v (t0 ) = βv β+1 , and we always have −(β+2) ϕ′′ . j,v (t) = β(β + 1)t

We call it the noncritical case if t0 is not in the support of ρ. This means |t0 | ≁ 1. By the van der Corput lemma, we can obtain the following so-called noncritical case estimates: (i) For problem (2.5), we have 2−βj , 󵄨󵄨 󵄨 󵄨󵄨σj,m (ξ , η)󵄨󵄨󵄨 ≲ { −(m+βj) 2 ,

m < −Cβ , m > Cβ .

Here Cβ denotes a constant only depending on β.

130 | G. Li and J. Li (ii) For problem (2.6), we have 󵄨 󵄨󵄨 −βj 󵄨󵄨σj,m (ξ , η)󵄨󵄨󵄨 ≲ 2 ,

m < −Cβ .

The noncritical case for either (2.5) or (2.6) follows directly.

4 The critical cases In this section, we will establish the critical cases, |m| < Cβ for (2.5) and m > −Cβ for (2.6). It suffices to verify the case m > −Cβ for (2.6) since the case |m| < cβ for (2.5) can be handled in the same way. We now face showing that 󵄨󵄨 󵄨󵄨 m+βj 󵄨󵄨 ̂ ̂ ̂ 󵄨 −ϵj − 󵄨󵄨∬ f (ξ )g (η)h(ξ + η)σj,m (ξ , η)dξdη󵄨󵄨󵄨 ≲ 2 2 2 ‖f ‖2 ‖g‖2 ‖h‖2 󵄨󵄨 󵄨󵄨 2

(4.1)

ℝ

holds for any h ∈ L2 (ℝ). The left-hand side of (4.1) can be decomposed into three parts I–II–III as defined below: ̂ + η)dξdη|, ̂ |ξ −η| { I := | ∬ℝ2 ̂f (ξ )ĝ (η) ∑k≤−m−Cβ Φ( )σj,m (ξ , η)h(ξ { 2k { { |ξ −η| ̂ + η)dξdη|, ̂ k )σj,m (ξ , η)h(ξ II := | ∬ℝ2 ̂f (ξ )ĝ (η) ∑|k+m|−m+Cβ 󵄨 2 ∑

ℝ

By the van der Corput lemma, for any fixed k > −m + Cβ , we have 󵄨󵄨 󵄨 󵄨 󵄨󵄨 󵄨 󵄨󵄨 ̂ |ξ − η| −k−βj 󵄨󵄨󵄨 ̂ |ξ − η| 󵄨󵄨󵄨 󵄨󵄨Φ( k )󵄨󵄨. 󵄨󵄨Φ( k )σj,m (ξ , η)󵄨󵄨󵄨 ≲ 2 󵄨󵄨 󵄨󵄨 󵄨󵄨 2 2 󵄨󵄨

The boundedness of a bilinear oscillatory integral | 131

̂ we get Thus, by Hölder’s inequality and for the support of Φ, III ≲

3m

k

∑ k≥−m+Cβ

2− 2 −m−βj ‖f ‖2 ‖g‖2 ‖h‖2 ≲ 2− 2 2−βj ‖f ‖2 ‖g‖2 ‖h‖2 .

Now, we consider II. In fact, we are required to verify 󵄨󵄨 󵄨 m+βj 󵄨󵄨 ̂ ̂ ̂ m ̂ + η)dξdη󵄨󵄨󵄨󵄨 ≲ 2−ϵj 2− 2 ‖f ‖ ‖g‖ ‖h‖ . 󵄨󵄨∬ f (ξ )g (η)Φ(2 |ξ − η|)σj,m (ξ , η)h(ξ 2 2 2 󵄨󵄨 󵄨󵄨 󵄨 2 ℝ

Notice that the possible critical point of the phase function ϕj,m is 2m |ξ − η|. So, by the stationary phase principle, we need to show that if 0 0, let Q(u, v) := |u − 2v|s ,

{

Qτ (u, v) := (|u − 2v|s − |u − 2v + τ|s ).

Since 󵄨󵄨 𝜕3 Q 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 ≳ 1, 󵄨󵄨 𝜕u𝜕2 v 󵄨󵄨󵄨 for the fixed τ > 0 we have 󵄨󵄨 𝜕2 Q 󵄨󵄨 󵄨󵄨 τ 󵄨󵄨 󵄨󵄨 󵄨󵄨 ≳ τ 󵄨󵄨 𝜕u𝜕v 󵄨󵄨

The boundedness of a bilinear oscillatory integral | 133

and 󵄩󵄩 𝜕2 Q 󵄩󵄩 󵄩󵄩 τ󵄩 󵄩󵄩 ≲ τ. 󵄩󵄩 󵄩󵄩 𝜕u𝜕v 󵄩󵄩󵄩C2 By Lemma 4.1, we bound the second part of (4.3) by 2

βj

−2

− 21

(τ0 )

ℝ βj

≲2

−2

1

1

2 2 󵄨2 󵄨2 󵄨̂ ̂ 󵄨 h(u − τ)󵄨󵄨󵄨 du] [∫󵄨󵄨󵄨ĝ (v)ĝ (v − τ)󵄨󵄨󵄨 dv] dτ ∫[∫󵄨󵄨󵄨h(u)

(τ0 )

− 21

ℝ

‖h‖22 ‖g‖22 .

βj

By choosing τ0 = 2− 3 , we obtain βj

L ≤ 2− 3 ‖h‖22 ‖g‖22 , thereby finishing the proof of Theorem 2.2.

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Sergey G. Bobkov and Cyril Roberto

On sharp Sobolev-type inequalities for multidimensional Cauchy measures Abstract: We are discussing some Sobolev-type inequalities for Cauchy measures and their information-theoretic counterparts. Keywords: Sobolev-type inequalities, Cauchy measures, Rényi entropy, Fisher information MSC 2010: Primary 60Exx, 60Fxx

1 Introduction One of the classical Sobolev inequalities on the Euclidean space ℝn is the relation ( ∫ |f |

2n n−2

)

n−2 2n

2

1 2

≤ Sn ( ∫ |∇f | ) ,

ℝn

n ≥ 3,

(1.1)

ℝn

which holds true for all smooth functions f on ℝn vanishing at infinity. The best constant Sn =

1

√πn(n − 2)

1

(

Γ(n) n ) Γ(n/2)

was determined in the 1970s by Aubin [1] and Talenti [13], see also [3, 10]. In the sequel, the integrals are always understood with respect to the Lebesgue measure on ℝn , if the measure is not indicated explicitly. Information-theoretic aspects of (1.1) are recently discussed in [6]. After the change of functions p = f 2 / ∫ f 2 , this inequality enters the family of entropic isoperiAcknowledgement: The first author was partially supported by the NSF grant DMS-2154001. The second author was supported by the Labex MME-DII funded by ANR, reference ANR-11-LBX-0023-01 and ANR-15-CE40-0020-03 – LSD – Large Stochastic Dynamics, and the grant of the Simone and Cino Del Duca Foundation, France. Sergey G. Bobkov, School of Mathematics, University of Minnesota, Minneapolis, MN, USA, e-mail: [email protected] Cyril Roberto, MODAL’X, UMR CNRS 9023, UPL, FP2M, CNRS FR 2036, Université Paris Nanterre, Nanterre, France, e-mail: [email protected] https://doi.org/10.1515/9783110741711-008

136 | S. G. Bobkov and C. Roberto metric inequalities Nα (X) I(X) ≥ Cn (α) with a particular index α =

n . n−2

(1.2)

Here, 2 − n(α−1)

α

Nα (X) = Nα (p) = (∫ p ) is the Rényi entropy power, and I(X) = I(p) = ∫

|∇p|2 p

is the Fisher information hidden in the distribution of the random vector X in ℝn with a smooth density p. Since the function α 󳨃→ Nα is nonincreasing, the inequality (1.2) is getting stronger for the growing index α. With some constants Cn (α) > 0 independent of p, (1.2) holds true for all α ∈ [0, ∞] n in dimension n = 1, and for all α ∈ [0, ∞) in dimension n = 2. However, α = n−2 is the maximal possible value in (1.2) in the case n ≥ 3. When α = 1, the Rényi entropy power is reduced to the Shannon entropy power 2 N1 (X) = N(X) = exp{− ∫ p log p}. n In this case, being written with an optimal constant, (1.2) becomes a well-known relation due to Stam [12], N(X) I(X) ≥ 2πe n,

(1.3)

in which the standard Gaussian measure plays an extremal role (for any n ≥ 1). Costa and Cover [9] pointed out a remarkable analogy between (1.3) and the isoperimetric inequality relating the surface of an arbitrary body in ℝn to its volume. The terminology isoperimetric inequality for entropies goes back to Dembo, Costa, and Thomas [11]. Rather than describing the best constant, it should be emphasized that an equality in (1.1) is always attained, and only for the functions of the form f (x) =

c (1 + b |x − x0

|2 )

n−2 2

,

c ∈ ℝ, b > 0, x0 ∈ ℝn

(sometimes called the Barenblatt profiles). Up to numerical factors, they serve as densities of the generalized multidimensional Cauchy measures, also called Student’s distributions. So, choosing b = 1 and x0 = 0, we put dms (x) = qs (x) = cs φs (x), dx

φs (x) =

1 , (1 + |x|2 )s

x ∈ ℝn .

On sharp Sobolev-type inequalities for multidimensional Cauchy measures | 137

The function φs is integrable, if and only if s > n2 , and then cs = c(s, n) is defined as a normalizing constant so that ms (ℝn ) = 1. The probability distribution ms will be called the Cauchy measure on ℝn with parameter s. Thus, φs with s = n−2 represents an extremizer in (1.1), which leads to the ex2 tremizer qn−2 = cn−2 φn−2 in the entropic isoperimetric inequality (1.2). It is indeed a probability density as long as n ≥ 5. However, φn−2 is not integrable in dimensions n = 3 and n = 4. As a consequence, in this case there is no extremizer in (1.2) in the class of all (smooth) probability densities on ℝn . We refer an interested reader to [6] for details. One of the aims in this note is to show the relationship of (1.1)–(1.2) with a weighted Poincaré-type inequality for the Cauchy measure mn with parameter s = n. It is well-defined for all n ≥ 1 and has density cn dmn (x) = , dx (1 + |x|2 )n

cn =

n

Γ(n)

π 2 Γ(n/2)

(1.4)

.

In particular, as a consequence of (1.1), we prove: Theorem 1.1. For any C 1 -smooth function g: ℝn → ℝ, n ≥ 3, Varmn (g) ≤

1 󵄨󵄨 2 󵄨2 ∫󵄨∇g(x)󵄨󵄨󵄨 (1 + |x|2 ) dmn (x). 4n 󵄨

The constant 1/(4n) is optimal, and the equality in (1.5) is attained for g(x) =

(1.5) 1 . 1+|x|2

As usual, 2

Varmn (g) = ∫ g 2 dmn − (∫ g dmn )

stands for the variance of g under mn . As we will see, inequality (1.5) expresses the fact that p = qn−2 is a “point” of local n minimum to the functional Nα (p) I(p) for α = n−2 . Equivalently, f = φn−2 is a “point” of local minimum to the functional ‖∇f ‖2 /‖f ‖ 2n . n−2

Weighted Poincaré-type inequalities such as (1.5) have been studied quite intensively, although for a different weight function. In particular, it was shown in [4] that, for all s ≥ n, Varms (g) ≤

As 󵄨 󵄨2 ∫󵄨󵄨∇g(x)󵄨󵄨󵄨 (1 + |x|2 ) dms (x) 2(s − 1) 󵄨

(1.6)

138 | S. G. Bobkov and C. Roberto with As = (√1 +

2

2 2 +√ ). s−1 s−1

Up to a universal factor, (1.6) is stronger than (1.5), however, it does not contain information about extremizers. Similar weighted Poincaré-type and isoperimetric inequalities of Cheeger-type remain to hold for general convex measures, cf. [2, 5, 7]. Let us also mention that after rescaling of the space variable, (1.5) implies in the limit as n → ∞ the Poincaré-type inequality with respect to the standard Gaussian measure γk on ℝk (which is also true about the inequality (1.6) with s → ∞ and fixed n). We provide details in the end of these notes (Section 5), while Theorem 1.1 is proved in Section 4. In this connection it is worthwhile to note that the Stam entropic inequality (1.3) may be used to derive the Gross logarithmic Sobolev inequality in the Gauss space (ℝk , γk ), which is stronger than the Poincaré-type inequality. Hence, one may wonder whether or not a similar derivation is applicable to mn on the basis of the entropic isoperimetric inequality (1.2). We propose one variant of a log-Sobolev-type inequality in the “Cauchy space” (ℝn , mn ) in Section 3. In Section 2 we address a closely related question: Is it true that the Cauchy measures ms play an extremal role when minimizing the Fisher information I(X) subject to certain moment-type constraints? This might give another analogy with a well-known assertion that I(X) is minimized for the normal distribution under a second moment assumption.

2 Minimizing the Fisher information subject to moment conditions Recall that, given two random variables X and Z with (smooth) densities p and q, respectively, the relative Fisher information is defined by I(X ‖ Z) =

󵄨󵄨 ∇p ∇q 󵄨󵄨2 󵄨 󵄨󵄨 − ∫ 󵄨󵄨󵄨 󵄨 p. 󵄨󵄨 p q 󵄨󵄨󵄨

{p,q>0}

To avoid technical issues, suppose that q is everywhere positive and well-behaving. Then, if p = fq with f smooth and bounded, I(X ‖ Z) = ∫

|∇f |2 q. f

On sharp Sobolev-type inequalities for multidimensional Cauchy measures | 139

Using an integration by parts formula, we get I(X) − I(Z) = I(X ‖ Z) + 2 ∫⟨∇f , ∇q⟩ + ∫ f

|∇q|2 |∇q|2 −∫ q q

|∇q|2 |∇q|2 ]−∫ , q q

= I(X ‖ Z) + ∫ f [−2Δq +

(2.1)

where n

Δg = ∑ 𝜕2 g/𝜕x2 i=1

i

stands for the Laplacian operator. As a classical example, one may consider the standard Gaussian random variable Z with density q(x) = (2π)−n/2 exp{−|x|2 /2}. In this case, ∇q = −xq and Δq = −nq + |x|2 q, so that the above identity amounts to I(X) − I(Z) = I(X ‖ Z) + n − ∫ |x|2 f (x)q(x) = I(X ‖ Z) + 𝔼 |X|2 − 𝔼 |Z|2 . Since the relative Fisher information is nonnegative, this implies in particular that among all random vectors X in ℝn with the second moment 𝔼 |X|2 = 𝔼 |Z|2 = n, the Fisher information I(X) is minimized for the standard Gaussian distribution. Here, we obtain a similar comparison for the Cauchy measures ms with densities qs (x) =

cs , (1 + |x|2 )s

x ∈ ℝn ,

(2.2)

where cs is the normalizing constant so that ∫ qs = 1. Let us denote by Zs a random vector in ℝn with distribution ms , s > n2 . As a direct analogue of the above result for the Gaussian measure, we prove: Theorem 2.1. Let X be a random vector in ℝn with a smooth density and finite Fisher information. If 𝔼

n − (s − n + 2) |Zs |2 n − (s − n + 2) |X|2 = 𝔼 , (1 + |X|2 )2 (1 + |Zs |2 )2

(2.3)

then I(X ‖ Zs ) = I(X) − I(Zs ). In particular, I(X) ≥ I(Zs ).

(2.4)

140 | S. G. Bobkov and C. Roberto The expectation in (2.3) may easily be evaluated explicitly, so that one may rewrite this moment condition as 𝔼

n − (s − n + 2) |X|2 = (1 + |X|2 )2

n 2

(s − n2 ) s+1

.

For example, for s = n − 2 with n ≥ 5, (2.3) is simplified to 𝔼

1 1 n−4 =𝔼 = , (1 + |X|2 )2 (1 + |Zs |2 )2 4(n − 1)

(2.5)

and then we get (2.4). For the proof of Theorem 2.1, we need a few calculus lemmas. Lemma 2.2. For any s > n/2 and x ∈ ℝn , |∇qs (x)|2 n − (s − n + 2) |x|2 = 4s qs (x), qs (x) (1 + |x|2 )2

−2Δqs (x) +

|∇qs (x)|2 |x|2 = 4s2 q (x). qs (x) (1 + |x|2 )2 s This is verified directly. Putting φs (x) = (1 + |x|2 )

−s

as before, we have 𝜕xi φs (x) = −2s

xi , (1 + |x|2 )s+1

|x| 󵄨󵄨 󵄨 , 󵄨󵄨∇φs (x)󵄨󵄨󵄨 = 2s (1 + |x|2 )s+1

so that |∇φs (x)|2 |x|2 = 4s2 . φs (x) (1 + |x|2 )s+2 This is the same as (2.7). Further differentiation gives 𝜕x22 φs (x) = −2s [ i

Δφs (x) = −2s [

xi2 1 − 2(s + 1) ], (1 + |x|2 )s+1 (1 + |x|2 )s+2 n |x|2 − 2(s + 1) ], (1 + |x|2 )s+1 (1 + |x|2 )s+2

and thus |∇φs (x)|2 n − (s − n + 2) |x|2 − 2Δφs (x) = 4s . φs (x) (1 + |x|2 )s+2 Hence, we arrive at (2.6).

(2.6) (2.7)

On sharp Sobolev-type inequalities for multidimensional Cauchy measures | 141

For example, for s = n − 2, |∇φn−2 (x)|2 1 , − 2Δφn−2 (x) = 4n(n − 2) φn−2 (x) (1 + |x|2 )n

(2.8)

which is a multiple of φn . In order to compute the constant cα in (2.2), we need a technical lemma. As usual, 1

B(x, y) = ∫(1 − t)x−1 t y−1 dt = 0

Γ(x) Γ(y) , Γ(x + y)

x, y > 0,

stands for the beta function. In the sequel, ωn denotes the volume of the unit ball of ℝn . Lemma 2.3. For any s > n/2, nωn 1 1 n n = =∫ B( , s − ). 2 s cs 2 2 2 (1 + |x| ) n

(2.9)

ℝ

As a consequence, we get Lemma 2.4. For any s > n/2, (s − n2 )(s + 1 − n2 ) 1 dm (x) = , s s(s + 1) (1 + |x|2 )2 n (s − n2 ) |x|2 2 . dm (x) = ∫ s s(s + 1) (1 + |x|2 )2

∫

Proof. Using polar coordinates, we have ∞

1 r n−1 = nω dr dσn−1 ∫ ∫ ∫ n (1 + |x|2 )s (1 + r 2 )s n 𝕊n−1 0 ∞

ℝ

= nωn ∫ 0

Changing variable u =

1 , 1+r 2

r n−1 dr. (1 + r 2 )α

we get

∞

1

0

0

r n−1 1 1 dr = ∫( − 1) ∫ 2 u (1 + r 2 )s 1

n−2 2

us−2 du

n−2 n−2 1 = ∫(1 − u) 2 us−2− 2 du. 2

0

This leads to the first desired conclusion (2.9).

(2.10) (2.11)

142 | S. G. Bobkov and C. Roberto Applying this identity, we see that the integral in (2.10) is equal to B( n2 , s + 2 − n2 ) cs = cs+2 B( n2 , s − n2 ) =

Γ(s + 2 − n2 )

=

(s − n2 )(s + 1 − n2 )

Γ(s + 2)

Γ(s) Γ(s − n2 )

s(s + 1)

.

Applying (2.9) once more, the integral in (2.11) may be written as cs [∫

1 1 −1 −1 −∫ ] = cs (cs+1 − cs+2 ) (1 + |x|2 )s+1 (1 + |x|2 )s+2 B( n2 , s + 1 − n2 ) − B( n2 , s + 2 − n2 ) = B( n2 , s − n2 ) = =

s−

n 2

−

s (s − n2 ) n2 s(s + 1)

(s − n2 )(s + 1 − n2 ) s(s + 1)

.

Proof of Theorem 2.1. Applying (2.6)–(2.7) in (2.1), we see that the random variable X with density p = fqs satisfies I(X) − I(Zs ) = I(X ‖ Zs ) + ∫ 4s −∫

n − (s − n + 2) |x|2 f (x)qs (x) (1 + |x|2 )2

4s2 |x|2 q (x). (1 + |x|2 )2 s

(2.12)

In particular, if ∫

|x|2 n − (s − n + 2) |x|2 p(x) = s ∫ q (x), 2 2 (1 + |x| ) (1 + |x|2 )2 s

(2.13)

we get I(X) − I(Zs ) = I(X ‖ Zs ), that is, the desired relation (2.4). Moreover, choosing f = 1 in (2.12), we see that the two integrals therein must coincide, that is, ∫

n − (s − n + 2) |x|2 |x|2 q (x) = s q (x). ∫ s (1 + |x|2 )2 (1 + |x|2 )2 s

On sharp Sobolev-type inequalities for multidimensional Cauchy measures | 143

This may also be verified on the basis of Lemma 2.3. Indeed, by (2.10)–(2.11), the above first integral is equal to n

(s − n2 )(s + 1 − n2 ) s(s + 1)

− (s − n + 2)

n 2

(s − n2 )

s(s + 1)

=

n 2

(s − n2 ) s+1

,

which is exactly the second integral, according to (2.11). Thus, the moment condition (2.13) coincides with the condition (2.3).

3 Log-Sobolev-type inequality In analogy with the equivalence between the Stam isoperimetric inequality for entropies (that is, in the case α = 1 as in (1.3)) and the logarithmic Sobolev inequality for the standard Gaussian measure, we derive in this section one inequality involving, as a reference measure, the Cauchy measure (2.2) with parameter s = n − 2, that is, with density q(x) =

c , (1 + |x|2 )n−2

x ∈ ℝn , n ≥ 5,

(3.1)

where c−1 =

nωn n n B( , − 2) 2 2 2

is a normalizing constant (cf. Lemma 2.3). Recall that this function is an extremizer in the isoperimetric inequality for enn , so that, for all smooth densities p on ℝn , tropies (1.2) of order α = n−2 (∫ p

n n−2

)

− n−2 n

− n−2 n

n |∇p|2 ≥ (∫ q n−2 ) ∫ p

∫

|∇q|2 . q

(3.2)

Here the expression on the right-hand side represents the constant 2

n n Cn = 4π n(n − 2)(Γ( )/Γ(n)) . 2

(3.3)

Let X be random vector in ℝn with density p = fq, and, as before, denote by Z a random vector with density q. Taking the logarithm in (3.2) leads to −

n n n n−2 n−2 log ∫ f n−2 q n−2 + log I(X) ≥ − log ∫ q n−2 + log I(Z). n n

144 | S. G. Bobkov and C. Roberto Therefore, if f satisfies the moment condition (2.5), then Theorem 2.1 is applicable, and hence from (2.4) we obtain n

n

n (log I(X) − log I(Z)) n−2 n I(X ‖ Z) = log(1 + ). n−2 I(Z)

n

log ∫ f n−2 q n−2 − log ∫ q n−2 ≤

Using log(1 + x) ≤ x, this is simplified to n

n

n

log ∫ f n−2 q n−2 − log ∫ q n−2 ≤

n 1 |∇f |2 q ∫ n − 2 I(Z) f

= Bn ∫

|∇f |2 q f

for some constant Bn that can be made explicit. Namely, since ∇q = −2(n − 2)

x q, 1 + |x|2

as in the relation (2.7) from Lemma 2.2 with s = n − 2, we may apply Lemma 2.4 to get I(Z) = 4(n − 2)2 ∫

|x|2 n(n − 2)(n − 4) . q= n−1 (1 + |x|2 )2

Thus, Bn =

n−1 . (n − 2)2 (n − 4)

As a summary, we proved the following statement. Theorem 3.1. Let q be the density of the Cauchy distribution on ℝn with parameter α = n − 2, n ≥ 5, as in (3.1). For any smooth function f : ℝn → ℝ+ satisfying ∫ fq = 1

and

∫

fq n−4 , = 2 2 4(n − 1) (1 + |x| )

(3.4)

we have n

n

n

log ∫ f n−2 q n−2 − log ∫ q n−2 ≤

n−1 |∇f |2 q. ∫ f (n − 2)2 (n − 4)

(3.5)

Turning back to the previous computations, let us note that we have actually proved a stronger inequality n

n

n

log ∫ f n−2 q n−2 − log ∫ q n−2 ≤

n n−1 |∇f |2 log(1 + q). ∫ n−2 n(n − 2)(n − 4) f

On sharp Sobolev-type inequalities for multidimensional Cauchy measures | 145

Similarly to the usual log-Sobolev inequality, the last integral in (3.5) describes the relative Fisher information I(X ‖ Z) of the random vector X in ℝn with density p = fq with respect to the random vector Z with density q. Let us show that the left-hand side of (3.5), which replaces the relative entropy D(X ‖ Z) in the usual log-Sobolev inequality, is always nonnegative. We claim that, under the moment condition (3.4), we have n

n

n

∫ q n−2 ≤ ∫ f n−2 q n−2 .

(3.6)

Indeed, by Hölder’s inequality with exponents n/(n − 2) and n/2, n n fq ≤ (∫ f n−2 q n−2 ) ∫ 2 2 (1 + |x| )

n−2 n

2

n 1 (∫ ) , 2 n (1 + |x| )

so that ∫f

n n−2

q

n n−2

n

2 − n−2

n−2 1 fq ) (∫ ) ≥ (∫ (1 + |x|2 )2 (1 + |x|2 )n

.

Therefore, (3.6) would follow from 2

n

n−2 n−2 n 1 fq n−2 ≤ (∫ (∫ q . ) ) ∫ (1 + |x|2 )n (1 + |x|2 )2

By (3.4) and (2.5), this is equivalent to 2

n n 1 (∫ ) (∫ q n−2 ) (1 + |x|2 )n

n−2 n

≤∫

1 q. (1 + |x|2 )2

But, since q is proportional to (1 + |x|2 )−(n−2) , the above inequality is actually an equality.

4 Proof of Theorem 1.1 In the proof of Theorem 1.1, we follow ideas from [8]. Recall that the function φ(x) =

1 , (1 + |x|2 )n−2

x ∈ ℝn ,

is an extremizer in the isoperimetric inequality for entropies with order α = (∫ p

n n−2

)

− n−2 n

− n−2 n

n |∇p|2 ≥ (∫ φ n−2 ) ∫ p

∫

|∇φ|2 . p

n , n−2

(4.1)

146 | S. G. Bobkov and C. Roberto If n ≥ 5 (and only then), φ is integrable, and then after normalization it represents the density of the Cauchy probability measure mn−2 . But, applying (4.1) to p = f / ∫ f , one realizes that the inequality holds for any f ≥ 0 smooth enough, not necessarily a density, like in the Sobolev inequality (1.1). Our aim is to apply (4.1) to p = (1 + εg)φ and to expand in the limit ε → 0. We may assume that g is smooth enough and compactly supported so that all approximations are uniform in space. We also assume that ε is small enough so that p(x) > 0 for all n x ∈ ℝn . Set α = n−2 . On the one hand, we have ∫ pα = ∫ φα (1 + εg)α = ∫ φα + εα ∫ gφα + ε2

α(α − 1) ∫ g 2 φα + o(ε2 ). 2

Therefore, −1/α

(∫ pα )

−1/α

= (∫ φα )

(1 + εα

∫ gφα ∫ φα

+ ε2

−1/α 2 α α(α − 1) ∫ g φ 2 + o(ε )) . 2 ∫ φα

In terms of the probability measure mn on ℝn with density mn (dx) = φ(x)α / ∫ φα , dx the latter expression may be written as −1/α

(∫ φα )

(1 − ε ∫ g dmn + ε2 (−

2

α+1 α−1 (∫ g dmn ) )) ∫ g 2 dmn + 2 2

with error of order o(ε2 ). On the other hand, ∫

|∇p|2 p

(1 + εg)2 |∇φ|2 + 2ε(1 + εg) φ ∇φ ⋅ ∇g + ε2 φ2 |∇g|2 (1 + εg) φ 1 󵄨 󵄨2 = ∫ (|∇φ|2 + 2ε (g |∇φ|2 + φ∇φ ⋅ ∇g) + ε2 󵄨󵄨󵄨∇(gφ)󵄨󵄨󵄨 )(1 − εg + ε2 g 2 + o(ε2 )) φ

=∫

=∫

|∇φ|2 |∇φ|2 + ε ∫[ g + 2 ∇φ ⋅ ∇g ] φ φ

+ ε2 ∫[ =∫

|∇(gφ)|2 |∇φ|2 − 2g ∇φ ⋅ ∇g − g 2 ] + o(ε2 ) φ φ

|∇φ|2 |∇φ|2 + ε ∫ g[ − 2Δφ] + ε2 ∫ φ|∇g|2 + o(ε2 ), φ φ

On sharp Sobolev-type inequalities for multidimensional Cauchy measures | 147

where in the last line we used an integration by parts to ensure that ∫ ∇g ⋅ ∇φ = − ∫ gΔφ. Multiplying the two expressions, it follows that −1/α

(∫ pα )

∫

|∇p|2 p

−1/α

= (∫ φα )

∫

|∇φ|2 φ

−1/α

+ ε (∫ φα )

[∫ g[

−1/α

+ ε2 (∫ φα )

|∇φ|2 |∇φ|2 − 2Δφ] − ∫ g dmn ∫ ] φ φ

[∫ φ|∇g|2 − ∫ g dmn ∫ g[

|∇φ|2 − 2Δφ] φ 2

+∫

|∇φ|2 α−1 α+1 (− (∫ g dmn ) )]. ∫ g 2 dmn + φ 2 2

(4.2)

By (4.1), and since ε may be both positive and negative, the coefficient in front of ε in (4.2) must vanish, that is, ∫ g[

|∇φ|2 |∇φ|2 − 2Δφ] = ∫ g dmn ∫ . φ φ

(4.3)

Of course, this may be verified directly on the basis of Lemma 2.2 with s = n − 2, from which we know that |∇φ(x)|2 4(n − 2)2 |x|2 = φ(x) (1 + |x|2 )n

(4.4)

|∇φ(x)|2 4n(n − 2) − 2Δφ(x) = . φ(x) (1 + |x|2 )n

(4.5)

and

Up to a normalizing constant, the right-hand side is the density of the probability measure mn . Therefore, ∫ g[

|∇φ|2 − 2Δφ] = 4n(n − 2) ∫ g dmn ∫ φα . φ

148 | S. G. Bobkov and C. Roberto To obtain (4.3), it remains to check that 4n(n − 2) ∫ φα = ∫

|∇φ|2 , φ

which follows by Lemmas 2.3–2.4 (in view of (4.4)). Thus, the linear term in (4.2) vanishes. As a consequence, the coefficient in front of ε2 must be nonnegative, that is, ∫ φ |∇g|2 ≥ ∫ g dmn ∫ g[ +(

|∇φ|2 − 2Δφ] φ

2

|∇φ|2 α+1 α−1 (∫ g dmn ) ) ∫ . ∫ g 2 dmn − 2 2 φ

(4.6)

Recalling (4.4)–(4.5), up to the factor 4n(n − 2) ∫ φα , the above right-hand side represents just the normalized variance 2

(∫ g dmn ) +

2

α−1 α+1 α−1 (∫ g dmn ) = Varmn (g). ∫ g 2 dmn − 2 2 2

As a result, (4.6) is simplified to ∫ φ |∇g|2 ≥ 4n Varmα (g) ∫ φα . Since φ(x) 2 = (1 + |x|2 ) , φ(x)α we arrive at the weighted Poincaré-type inequality Varmn (g) ≤

1 󵄨󵄨 2 󵄨2 ∫󵄨󵄨∇g(x)󵄨󵄨󵄨 (1 + |x|2 ) dmn (x). 4n

At this step, the assumption that g is compactly supported may be dropped. Note also that, since the volume of the unit ball is n

ωn = 2π 2 /(nΓ(n/2)), the density of mn is φα (x) 2 1 Γ(n) 1 = = n . nωn B( n2 , n2 ) (1 + |x|2 )n π 2 Γ(n/2) (1 + |x|2 )n ∫ φα

(4.7)

On sharp Sobolev-type inequalities for multidimensional Cauchy measures | 149

We end this section by proving that the factor 1/(4n) is optimal in (4.7). In fact, let us check that g(x) =

1 1 + |x|2

is an extremizer. To this aim, we will repeatedly use the following identities: 1 n n n n B( , + 1) = B( , ) 2 2 2 2 2

n n 1 n+2 n n and B( , + 2) = B( , ), 2 2 4 n+1 2 2

that are consequences of (x + y) B(x, y + 1) = yB(x, y). We have ∫ g dmn = Zn ∫

nω n n 1 dx = Zn n B( , + 1) = 2 2 2 2 (1 + |x|2 )n+1

and ∫ g 2 dmn = Zn ∫

nω dx n n n+2 = Zn n B( , + 2) = . 2 2 2 4(n + 1) (1 + |x|2 )n+2

Therefore, 2

Varmn (g) = ∫ g 2 dmα − (∫ g dmn ) =

1 1 n+2 ( − 1) = . 4 n+1 4(n + 1)

On the other hand, |x|2 2 󵄨 󵄨2 ∫󵄨󵄨󵄨∇g(x)󵄨󵄨󵄨 (1 + |x|2 ) dmn (x) = 4Zn ∫ (1 + |x|2 )n+2

1 1 −∫ ) 2 n+1 (1 + |x| ) (1 + |x|2 )n+2 nω n n n n = 4Zn n (B( , + 1) − B( , + 2)) 2 2 2 2 2 n n n n B( , + 1) − B( 2 , 2 + 2) =4 2 2 B( n2 , n2 )

= 4Zn (∫

1 n+2 = 4( − ) 2 4(n + 1) n = . n+1

150 | S. G. Bobkov and C. Roberto Therefore, the smallest constant C such that 2 󵄨2 󵄨 Varmn (g) ≤ C ∫󵄨󵄨󵄨∇g(x)󵄨󵄨󵄨 (1 + |x|2 ) dmn (x)

holds for any g must satisfy 1 Cn ≤ , 4(n + 1) n + 1 from which we deduce that C ≥ 1/(4n) and hence that C = 1/(4n) is indeed the optimal constant in the weighted Poincaré inequality (4.7).

5 Relationship with the Gaussian Poincaré-type inequality Let us explain how (1.5) implies the Poincaré-type inequality Varγk (g) ≤ ∫ |∇g|2 dγk

(5.1)

with respect to the standard Gaussian measure γk on ℝk . As is well known, the Cauchy measure mn may be characterized as the distribution of the random vector X=

Z √ξ12 + ⋅ ⋅ ⋅ + ξn2

,

where ξi ’s are independent random variables with a standard normal distribution on the real line, that are independent of a random vector Z = (Z1 , . . . , Zn ) having a standard normal distribution on ℝn . Rescaling the space variable, (1.5) may be rewritten in terms of the random vector Y = √n X = (Y1 , . . . , Yn ) as Var(g(Y)) ≤

2

1 󵄨󵄨 1 󵄨2 𝔼 󵄨∇g(Y)󵄨󵄨󵄨 (1 + |Y|2 ) . 4 󵄨 n

If g = g(y1 , . . . , yk ) depends on the first k variables (k < n), and Zk,n = (Y1 , . . . , Yk )

On sharp Sobolev-type inequalities for multidimensional Cauchy measures | 151

is the k-dimensional projection of Y, we obtain that Var(g(Zk,n )) ≤

2

1 1 1 󵄨󵄨 󵄨2 𝔼 󵄨∇g(Zk,n )󵄨󵄨󵄨 (1 + |Zk,n |2 + |Vk,n |2 ) , 4 󵄨 n n

(5.2)

where Vk,n = (Yk+1 , . . . , Yn ). As n → ∞, we have Zk,n ⇒ γk weakly in distribution, so that the variance in (5.2) is convergent to Varγk (g) as long as the function g is bounded and continuous on ℝk . Moreover, assuming that the gradient ∇g is bounded and continuous as well, the asymptotic behavior of the right-hand side in (5.2) is easily explored. First, putting χn = √ξ12 + ⋅ ⋅ ⋅ + ξn2 , we have 𝔼 |Zk,n |2 = 𝔼 √Z12 + ⋅ ⋅ ⋅ + Zk2 𝔼

√n √n 2kn = 𝔼 χk 𝔼 χn ≤ , χn n−1 n−1

which is bounded in n. Since Zk,n and Vk,n are asymptotically independent, we conclude that the limit of the right-hand side in (5.2) is equal to the integral in (5.1). Now, one may use the identity 2

𝔼 (1 +

2 1 2 |Z| ) = 4 + . n n

Thus, in the limit (5.2) leads to (5.1).

Bibliography [1] [2] [3] [4] [5] [6]

T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differ. Geom. 11 (1976), 573–598. S. G. Bobkov, Large deviations and isoperimetry over convex probability measures with heavy tails, Electron. J. Probab. 12 (2007), 1072–1100. S. G. Bobkov and M. Ledoux, From Brunn–Minkowski to sharp Sobolev inequalities, Ann. Mat. Pura Appl. (4) 187 (2008), 369–384. S. G. Bobkov and M. Ledoux, Weighted Poincaré-type inequalities for Cauchy and other convex measures, Ann. Probab. 37 (2009), 403–427. S. G. Bobkov and M. Ledoux, On weighted isoperimetric and Poincaré-type inequalities, IMS collections. High dimensional probability V: the luminy volume, vol. 5, 2009, pp. 1–29. S. G. Bobkov and C. Roberto, Entropic isoperimetric inequalities, Submitted to: High dimensional probability proceedings, Progress in probability, vol. 9, Springer, 2021.

152 | S. G. Bobkov and C. Roberto

[7] [8] [9] [10] [11] [12] [13]

P. Cattiaux, N. Gozlan, A. Guillin, and C. Roberto, Functional inequalities for heavy tailed distributions and application to isoperimetry, Electron. J. Probab. 15 (2010), 346–385. D. Cordero-Eurasquin and C. Roberto, Private communication, 2008. M. H. M. Costa and T. M. Cover, On the similarity of the entropy power inequality and the Brunn–Minkowski inequality, IEEE Trans. Inf. Theory 30 (1984), 837–839. M. Del Pino and J. Dolbeault, Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl. (9) 81 (2002), 847–875. A. Dembo, T. M. Cover, and J. Thomas, Information theoretic inequalities, IEEE Trans. Inf. Theory 37 (1991), 1501–1518. A. J. Stam, Some inequalities satisfied by the quantities of information of Fisher and Shannon, Inf. Control 2 (1959), 101–112. G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. 110 (1976), 353–372.

Guixiang Hong

Banach lattice-valued q-variation and convexity Abstract: In this paper, we show that the q-variation for a differential operator is not bounded in Lp (ℝ; L∞ (ℝ)) for any 1 < p < ∞. As a consequence, the q-variation operator cannot be used to characterize the Hardy–Littlewood property of the underlying Banach lattice. Moreover, for Köthe function spaces X with X ∗ norming such that X is r-convex for some large r, and X is not s-convex for any s, r < s < ∞, we obtain a lower bound of the (Lp (ℝ; X), Lp (ℝ; X))-bound of the q-variation operator, which tends to ∞, as r tends to ∞. Keywords: Variational inequalities, Hardy–Littlewood property, behavior in L∞ (ℝ) MSC 2010: Primary 42B25, 46B20, Secondary 46B99

1 Introduction In recent years, many research papers in probability, ergodic theory, and harmonic analysis (see, e. g., the references appearing in the Introduction of [5]) have been devoted to the study of the boundedness of the q-variation operators, with 2 < q < ∞, acting on scalar-valued functions. The q-variation operators can be viewed as “better” operators than the maximal operators in the sense that they immediately imply the pointwise convergence of the underlying family of operators without using the Banach principle via the corresponding maximal inequality, and they can be used to measure the speed of convergence of the family. Very recently, vector-valued q-variations for differential operators and semigroups, i. e., q-variation operators acting on vector-valued functions, have also been considered in [5, 6, 8, 9]. Let X be a Banach lattice. Given a locally integrable function f : ℝ → X, for any t > 0, the differential average At is defined as t

At f (x) =

1 ∫ f (x − y)dy. t −t

Acknowledgement: The author was supported in part by NSFC Grant #12071355. The author would like to express his great appreciation to Professor J. L. Torrea for the discussions related to the topic of this paper. In particular, Theorem 1.2 was suggested by him to the author. Guixiang Hong, School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China, e-mail: [email protected] https://doi.org/10.1515/9783110741711-009

154 | G. Hong Let J be a finite subset of ℝ+ . Then the Hardy–Littlewood operator ℳJ on X-valued functions f is defined as ℳJ f (x) = sup󵄨󵄨󵄨At f (x)󵄨󵄨󵄨, t∈J

󵄨

󵄨

where the supremum is in the lattice X. This accounts for the need to take a finite collection of radii J. When X is a Köthe function space, we can take the supremum on ℝ+ . We refer the readers to the nice book by Lindenstrauss and Tzafriri [7] for more information on Banach lattices and function spaces. Moreover, the q-variation operator 𝒱q,J (A) on any X-valued function f is defined as 󵄨q

1/q

𝒱q,J (A)f (x) = sup (∑󵄨󵄨󵄨Ati f (x) − Ati+1 f (x)󵄨󵄨󵄨 )

󵄨

{ti }⊂J

i

where the supremum is taken over all decreasing sequences {ti } in J. It is obvious that the q-variation operator is bigger than the maximal operator, thus if X is a Banach lattice such that 𝒱q,J (A) is bounded in Lp (ℝ; X), then so does ℳJ . Now, a natural question is whether or not the reverse implication could remain true. In this paper, we give a negative answer. Precisely, we will show the following result. Theorem 1.1. Let 2 < q < ∞, then 𝒱q,J (A) is not uniformly (with respect to J’s) bounded in Lp (ℝ; L∞ (ℝ)) for any 1 < p < ∞. This result provides a negative answer of the above question due to the wellknown fact that ℳJ ’s are uniformly bounded on Lp (ℝ; L∞ (ℝ)). The idea of the proof is to construct an L∞ (ℝ)-valued function such that we can compute the lower bound, which is partially inspired by the arguments for exploring the behavior of q-variation for the heat semigroup in L∞ (ℝ) developed in [1] by Betancor et al. Next, we would like to give a quantitative description of Theorem 1.1 at the Banach function space level. To do so, given a Köthe function space X, we define the following quantity for fixed 1 < p < ∞ and 2 < q < ∞: CX (A) =

sup sup

f ∈Lp (ℝ;X)

J

‖𝒱q,J (A)f ‖Lp (X) ‖f ‖Lp (X)

,

(1.1)

which may equal infinity if 𝒱q,J ’s are not uniformly bounded on Lp (ℝ; X). However, by the result in [6] that if X is a UMD lattice then 𝒱q,J ’s are indeed uniformly bounded on Lp (ℝ; X) and hence CX (A) is finite for a UMD lattice X. Below is the quantitative result. Theorem 1.2. Let 1 < p < ∞ and 2 < q < ∞. Let X be a Köthe function space with X ∗ norming such that not only X is r-convex for some large r > 0 but also X is not s-convex for all s ∈ (r, ∞). Then there exists a constant C > 0 such that CX (A) ≥ Cr 1/q /cr , where cr is the constant of r-convexity of X.

Banach lattice-valued q-variation and convexity | 155

In particular, there holds CLr (ℝ) (A) ≥ Cr 1/q

and

lim CLr (ℝ) (A) = ∞.

r→∞

Note that if q = ∞ then the variation operator is reduced to the maximal operator and hence Theorem 1.2 gives no new information. The formulation and the proof of this result is partially motivated by the work [4] by Harboure et al. Recall that a Banach lattice X is said to have the Hardy–Littlewood (H. L.) property if there exists some p ∈ (1, ∞) such that the operators ℳ′J s are uniformly bounded in Lp (ℝ; X). See [2] and the references therein for more information on this property. Theorem 1.1 implies that the H. L. property does not ensure the uniform Lp (X)boundedness of q-variation. While the UMD property of a Banach lattice X is sufficient for the uniform Lp (X)-boundedness of q-variation with 2 < q < ∞ thanks to the previously-mentioned result in [5], it is interesting to know whether the UMD property of X is also necessary for the uniform Lp (X)-boundedness of q-variation for all 2 < q < ∞. At the time of writing the paper, we have no idea about this. We will show Theorem 1.1 in Section 3. Theorem 1.2 will be proved in Section 4. One intermediate step will be shown in Section 2. Throughout this paper, C always denotes a positive constant that may vary from line to line.

2 Reduction In this section, we will reduce the statements in Theorems 1.1 and 1.2 for the q-variation associated with the differential averages to similar statements for that associated with the heat semigroup on the real line. Let {e−tΔ }t>0 be the heat semigroup on ℝ and {Ht }t>0 be the associated kernels. It is well known that Ht (x) = t −1/2 H(x/√t)

2

with H(x) = (4π)−1/2 e−|x| /4 .

Let H = {Ht }t>0 and denote the corresponding q-variation by 𝒱q,J (H). In the sequel, X is assumed to be a Köthe function space on a measure space (Ω, ν). Thus any X-valued measurable function on ℝ can be viewed as a measurable function on ℝ × X. Accordingly, as the definition of CX (A), for fixed 1 < p < ∞ and 2 < q < ∞, we define CX (H). As a variant of Lemma 2.4 in [3], the main result of this section is formulated below. Lemma 2.1. Let X be a Köthe function space on a measure space (Ω, ν) obeying Fatou property. Then there is a positive constant C such that CX (H) ≤ CCX (A).

156 | G. Hong Proof of Lemma 2.1. If h(|x|) = H(x), then h(t) is trivially differentiable and ∞

󵄨 󵄨 ∫ 󵄨󵄨󵄨h′ (t)󵄨󵄨󵄨t dt = C < ∞. 0

Note that ∞

∞

h(s) = − ∫ h (t)dt = − ∫ χ[s,∞) (t)h′ (t)dt. ′

s

0

So we have ∞

H(x) = − ∫ χ[s,∞) (|x|)h′ (t)dt. 0

Also, for any X-valued function f on ℝ, At f can be viewed as a function on ℝ × Ω. If At (x) = t −1 χ[−t,t] (x), then At f can be rewritten as At f (x, ω) = At ∗ f (x, ω), and hence ∞

1 |x| 1 |x| Hs (x) = H( ) = − ∫ χ[0,t] ( )h′ (t)t dt √s √s √s t√s 0 ∞

= − ∫ At√s (x)h′ (t)t dt. 0

Consequently, ∞

Hs f (x, ω) = − ∫ At√s f (x, ω)h′ (t)t dt. 0

Upon fixing a finite subset J ⊂ ℝ+ , we have 𝒱q,J (H)f (x, ω)

1/q

󵄨 󵄨q = sup (∑󵄨󵄨󵄨Hsj f (x, ω) − Hsj+1 f (x, ω)󵄨󵄨󵄨 ) {sj }⊂J

j

Banach lattice-valued q-variation and convexity | 157

󵄨󵄨 ∞ 󵄨󵄨q 1/q 󵄨󵄨 󵄨󵄨 ′ 󵄨 ≤ sup (∑󵄨󵄨 ∫ (At √sj f (x, ω) − At √sj+1 f (x, ω))h (t)t dt 󵄨󵄨󵄨 ) 󵄨 󵄨󵄨 {sj }⊂J j 󵄨󵄨 0 󵄨 1/q

∞

󵄨 󵄨q ≤ sup ∫ (∑󵄨󵄨󵄨At √sj f (x, ω) − At √sj+1 f (x, ω)󵄨󵄨󵄨 ) {sj }⊂J

0

j

1/q

∞

󵄨 󵄨q ≤ ∫ sup (∑󵄨󵄨󵄨At √sj f (x, ω) − At √sj+1 f (x, ω)󵄨󵄨󵄨 ) {sj }⊂J

0

j

󵄨󵄨 ′ 󵄨󵄨 󵄨󵄨h (t)󵄨󵄨t dt 󵄨󵄨 ′ 󵄨󵄨 󵄨󵄨h (t)󵄨󵄨t dt

∞

󵄨 󵄨 ≤ sup 𝒱q,J (A)f (x, ω) ⋅ ∫ 󵄨󵄨󵄨h′ (t)󵄨󵄨󵄨t dt J

0

≤ C sup 𝒱q,J (A)f (x, ω). J

Using this pointwise estimate, we obtain the desired result. By Lemma 2.1, in order to prove Theorem 1.1, it suffices to prove CL∞ (ℝ) (H) ≥ M

∀ M > 0.

(2.1)

Indeed, suppose that Theorem 1.1 were not true, i. e., CL∞ (ℝ) (A) < ∞, then Lemma 2.1 yields CL∞ (ℝ) (H) < ∞, which contradicts (2.1).

3 Proof of Theorem 1.1 This section is devoted to the proof of (2.1), and consequently the proof of Theorem 1.1 is finished. Given a > 1, define the function −1

G = ∑ (−1)k+1 χ[ak ,ak+1 ) . k=−∞

We need the following estimate, which can be calculated directly (see, e. g., (2.2) in [1]). Lemma 3.1. There is some a > 1 such that there exist C > 0 and j0 ∈ ℕ satisfying 󵄨󵄨 󵄨 󵄨󵄨Ha−2j G(0) − Ha−2(j+1) G(0)󵄨󵄨󵄨 ≥ C

∀ j ≥ j0 .

Now we are ready to prove Theorem 1.1. Proof Theorem 1.1. We define a function on ℝ × ℝ by ̃ y) = G(x − y)χ G(x, [−1,1] (y).

(3.1)

158 | G. Hong It is easy to check that G̃ ≠ 0 only if −1 < x < 2, and hence 󵄩 󵄩 ̃ y)󵄩󵄩󵄩 ≤ 31/p . ̃ p ∞ = 󵄩󵄩󵄩 sup G(⋅, ‖G‖ L (L ) 󵄩󵄩p 󵄩󵄩y∈[−1,1] As a consequence, for any j1 greater than j0 which has appeared in Lemma 3.1, we have 1/q 󵄩 󵄩󵄩 j1 󵄩󵄩 󵄩 q . 󵄩󵄩( ∑ |Ha−2j G̃ − Ha−2(j+1) G|̃ ) 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 p ∞ 󵄩 j=j0 󵄩L (L )

󵄩󵄩 −1/p 󵄩

CL∞ (ℝ) (H) ≥ 3 It is easy to verify that

̃ y) = ∫ H −2j (x − z)G(z, ̃ y) dz Ha−2j G(x, a ℝ

= ∫ Ha−2j (x − z)G(z − y)χ[−1,1] (y) dz ℝ

= ∫ Ha−2j (x − y − z)G(z)χ[−1,1] (y) dz ℝ

= Ha−2j G(x − y)χ[−1,1] (y). Note that whenever x ∈ (0, 1), the interval (x − 1, x + 1) contains the interval [0, 1]. Thus CL∞ (ℝ) (H) is not less than 1

3

−1/p

p/q

j1

󵄨 󵄨q (∫ sup ( ∑ 󵄨󵄨󵄨Ha−2j G(x − y) − Ha−2(j+1) G(x − y)󵄨󵄨󵄨 ) 0

y∈(−1,1) j=j 0 1

= 3−1/p (∫ 0

y∈(x−1,x+1) j=j 0

1

≥3

−1/p

j1

󵄨 󵄨q ( ∑ 󵄨󵄨󵄨Ha−2j G(y) − Ha−2(j+1) G(y)󵄨󵄨󵄨 )

sup

j1

󵄨 󵄨q (∫ sup ( ∑ 󵄨󵄨󵄨Ha−2j G(y) − Ha−2(j+1) G(y)󵄨󵄨󵄨 )

p/q

y∈[0,1] j=j 0 0

1/p

dx) p/q

1/p

dx) 1/p

dx)

.

On the other hand, after changing the variable, for every z ∈ ℝ, we have 󵄨󵄨 󵄨 󵄨󵄨Ha−2j G(y) − Ha−2(j+1) G(y)󵄨󵄨󵄨 󵄨󵄨 2 −2(j+1) 1 󵄨󵄨󵄨󵄨 1 1 󵄨 −|y−z|2 /4a−2j G(z) dz − −(j+1) ∫ e−|y−z| /4a G(z) dz 󵄨󵄨󵄨 = 󵄨󵄨 −j ∫ e 󵄨󵄨 √4π 󵄨󵄨 a a ℝ

ℝ

1 󵄨󵄨󵄨󵄨 j −u2 /4 = g(u + a−j y)χ[0,a−j ) (u + a−j y) du 󵄨(−1) ∫ e √4π 󵄨󵄨󵄨 ℝ

Banach lattice-valued q-variation and convexity | 159

󵄨󵄨 2 󵄨 − (−1)j+1 ∫ e−u /4 g(u + a−(j+1) y)χ[0,a−(j+1) ) (u + a−(j+1) y) du󵄨󵄨󵄨. 󵄨󵄨 ℝ

Observe that when t tends to 0, 2

2

∫ e−u /4 g(u + t)χ[0,B) (u + t) du → ∫ e−u /4 g(u)χ[0,B) (u) du ℝ

ℝ

uniformly in B ∈ (0, ∞). Hence, for all j0 ≤ j ≤ j1 , y → 0 implies 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨Ha−2j G(y) − Ha−2(j+1) G(y)󵄨󵄨󵄨 → 󵄨󵄨󵄨Ha−2j G(0) − Ha−2(j+1) G(0)󵄨󵄨󵄨. Thus, there exists δ > 0 such that for all |y| < δa−(j1 +1)

and j0 ≤ j ≤ j1 ,

we have 󵄨󵄨 󵄨 󵄨 −1 󵄨 󵄨󵄨Ha−2j G(y) − Ha−2(j+1) G(y)󵄨󵄨󵄨 ≥ 2 󵄨󵄨󵄨Ha−2j G(0) − Ha−2(j+1) G(0)󵄨󵄨󵄨.

(3.2)

Finally, by Lemma 3.1, we obtain CL∞ (ℝ) (H) ≥ 3

−1/p

j1

󵄨 󵄨q sup ( ∑ 󵄨󵄨󵄨Ha−2j G(y) − Ha−2(j+1) G(y)󵄨󵄨󵄨 )

1/q

y∈[0,1] j=j 0

≥3

2 C(j1 − j0 )

≥M

∀ M > 0,

−1/p −1

provided that j1 is sufficiently large. This completes the proof of (2.1), hence of Theorem 1.1.

4 Proof of Theorem 1.2 The starting point of the proof is the following proposition, which is Theorem 1.2 in the case X = Lr (ℝ). Proposition 4.1. Let 1 < p < ∞ and 2 < q < ∞. Then there exists a constant C > 0 such that CLr (ℝ) (A) ≥ Cr 1/q .

160 | G. Hong The proof is postponed to the next section. The proof of Theorem 1.2 is divided into two steps. The first step is to deduce Cℓr (A) ≥ Cr 1/q from CLr (ℝ) (A) ≥ Cr 1/q, which is Proposition 4.1. The second step is to show CX (A) ≥ Cr 1/q /cr from Cℓr (A) ≥ Cr 1/q through Proposition 3.11 of [4]. We state this proposition here as the following lemma. Lemma 4.2. Let X be a Köthe function space with X ∗ norming such that X is r-convex for some r ∈ (1, ∞), and X is not s-convex for any s ∈ (r, ∞). Then given ε ∈ (0, 1) and a positive integer m, there exists a sequence {ei }m i=1 of pairwise disjoint elements of X such that m

(1 −

ε) ∑ bri i=1

󵄩󵄩 m 󵄩󵄩r m 󵄩󵄩 󵄩󵄩 󵄩 ≤ 󵄩󵄩∑ bi ei 󵄩󵄩󵄩 ≤ crr ∑ bri 󵄩󵄩 󵄩󵄩 i=1 󵄩i=1 󵄩X

(4.1)

holds for any sequence {bi }m i=1 of nonnegative scalars and cr is the constant of r-convexity of X. Proof of Theorem 1.2. Two situations are handled below. (i) Cℓr (A) ≥ Cr 1/q . Since the set m

n

k=1

j=1

S = { ∑ (∑ ajk χFj (x))χEk (y) : ajk ∈ ℂ, Fj ⊂ ℝ , Ek ⊂ ℝ disjoint} is dense in Lp (ℝ; Lr (ℝ)), by the definition of CLr (ℝ) (A), we can find some J and f ∈ S such that 󵄩󵄩 󵄩 1/q 󵄩󵄩𝒱q,J (A)f 󵄩󵄩󵄩Lp (Lr ) ≥ Cr ‖f ‖Lp (Lr ) .

Banach lattice-valued q-variation and convexity | 161

For f ∈ S, we define an ℓr -valued function f ̃ on ℝ as m

n

f ̃(x) = {|Ek |1/r ∑ ajk χFj (x)} j=1

k=1

.

Then the desired estimate follows from the two identities ‖f ̃‖Lp (ℓr ) = ‖f ‖Lp (Lr ) and 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩𝒱q,J (A)f ̃󵄩󵄩󵄩Lp (ℓr ) = 󵄩󵄩󵄩𝒱q,J (A)f 󵄩󵄩󵄩Lp (Lr ) . Let us just verify the second equality, since the first can be proved in a similar way. Denote ∑nj=1 ajk χFj (x) by Fjk (x). Given x ∈ ℝ, from the fact that Ek ’s are disjoint it follows that r

󵄩󵄩 󵄩r 󵄨 󵄨q q 󵄩󵄩𝒱q,J (A)f (x, ⋅)󵄩󵄩󵄩Lr = ∫ sup (∑󵄨󵄨󵄨Ati f (x, y) − Ati+1 f (x, y)󵄨󵄨󵄨 ) dy ℝ

{ti }⊂J

i

r

󵄨󵄨 m 󵄨󵄨q q 󵄨󵄨 󵄨󵄨 = ∫ sup (∑󵄨󵄨󵄨 ∑ (Ati − Ati+1 )Fjk (x)χEk (y)󵄨󵄨󵄨 ) dy 󵄨 󵄨󵄨 {t }⊂J i 󵄨󵄨k=1 󵄨 ℝ i r

m

󵄨 󵄨q q = ∑ ∫ sup (∑󵄨󵄨󵄨(Ati − Ati+1 )Fjk (x)χEk (y)󵄨󵄨󵄨 ) dy i

k=1 ℝ {ti }⊂J

r

m

󵄨 󵄨q q = ∑ |Ek | sup (∑󵄨󵄨󵄨(Ati − Ati+1 )Fjk (x)󵄨󵄨󵄨 ) k=1

{ti }⊂J

i

1

m 󵄩r 󵄩󵄩 1 󵄩󵄩 󵄩 󵄨q q 󵄨 = 󵄩󵄩󵄩{sup (∑󵄨󵄨󵄨|Ek | r (Ati − Ati+1 )Fjk (x)󵄨󵄨󵄨 ) } 󵄩󵄩󵄩 󵄩󵄩 {ti }⊂J 󵄩ℓr k=1 󵄩 i 󵄩󵄩 󵄩̃󵄩r = 󵄩󵄩𝒱q,J (A)f 󵄩󵄩ℓr .

(ii) CX (A) ≥ Cr 1/q /cr . Similarly, by the density argument, we can find some J and f ∈ Lp (ℝ; ℓr ) of the form m

f (x) = {fk (x)}k=1 with fk ∈ Lp (ℝ) such that 󵄩󵄩 󵄩 1/q 󵄩󵄩𝒱q,J (A)f 󵄩󵄩󵄩Lp (ℓr ) ≥ Cr ‖f ‖Lp (ℓr ) .

162 | G. Hong Upon fixing x ∈ ℝ, taking ε = 1/2, and using Lemma 4.2, we get a sequence {ek }m k=1 of pairwise disjoint elements of X such that 󵄩󵄩 m 󵄩󵄩r m 󵄩󵄩 󵄩 󵄩󵄩 ∑ fk (x)ek 󵄩󵄩󵄩 ≤ cr ∑ f r (x) = cr 󵄩󵄩󵄩f (x)󵄩󵄩󵄩r r r k r󵄩 󵄩󵄩 󵄩󵄩 󵄩ℓ 󵄩󵄩k=1 󵄩󵄩X k=1 and 󵄩󵄩 m 󵄩󵄩r m 󵄩󵄩 󵄩󵄩 r 󵄩󵄩r −1 2 󵄩󵄩𝒱q,J (A)f (x)󵄩󵄩ℓr = 2 ∑ (𝒱q,J (A)fk (x)) ≤ 󵄩󵄩󵄩 ∑ (𝒱q,J (A)fk (x))ek 󵄩󵄩󵄩 . 󵄩󵄩 󵄩󵄩 k=1 󵄩k=1 󵄩X −1 󵄩 󵄩

Now, via defining the X-valued function m

f ̃(x) = ∑ fk (x)ek , k=1

and using the disjoint property of ek ’s, we easily obtain 󵄩󵄩 m 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩𝒱q,J (A)f ̃(x)󵄩󵄩󵄩X = 󵄩󵄩󵄩 ∑ (𝒱q,J (A)fk (x))ek 󵄩󵄩󵄩 . 󵄩󵄩 󵄩󵄩 󵄩k=1 󵄩X To conclude, using the result obtained in first step, we deduce that 󵄩󵄩 󵄩 󵄩 −1/r 󵄩 󵄩󵄩𝒱q,J (A)f ̃󵄩󵄩󵄩Lp (X) ≥ 2 󵄩󵄩󵄩𝒱q,J (A)f 󵄩󵄩󵄩Lp (ℓr ) ≥ 2−1/r Cr 1/q ‖f ‖Lp (ℓr ) ≥ Cr 1/q cr−1 ‖f ̃‖Lp (X) , which implies the desired estimate.

5 Proof of Proposition 4.1 As in Section 2, by Lemma 2.1, in order to prove Proposition 4.1, it suffices to prove CLr (ℝ) (H) ≥ Cr 1/q .

(5.1)

Now we adapt the previous argument for the proof of (2.1) to the proof of (5.1). Proof of Proposition 4.1. If the function G̃ is taken as in the proof of Theorem 1.1, then it is easy to check ̃ p r ≤ 21/r 31/p . ‖G‖ L (L )

Banach lattice-valued q-variation and convexity | 163

As a consequence, for any j1 greater than j0 which has appeared in Lemma 3.1, we have 1/q 󵄩 󵄩󵄩 j1 󵄩󵄩 󵄩󵄩 󵄩 . CLr (ℝ) (H) ≥ 2−1/r 3−1/p 󵄩󵄩󵄩( ∑ |Ha−2j G̃ − Ha−2(j+1) G|̃ q ) 󵄩󵄩󵄩 󵄩󵄩 p r 󵄩󵄩 󵄩L (L ) 󵄩 j=j0

As in the proof of Theorem 1.1, we arrive at the step that CLr (ℝ) (H) is not less than 1

1

j1

󵄨 󵄨q 2−1/r 3−1/p (∫(∫( ∑ 󵄨󵄨󵄨Ha−2j G(y) − Ha−2(j+1) G(y)󵄨󵄨󵄨 ) 0

0

r/q

j=j0

p/r

dy)

1/p

dx)

.

Now using the estimate (3.2) for all |y| < δa−(j1 +1)

and j0 ≤ j ≤ j1 ,

we get 1

j1

󵄨 󵄨q CLr (ℝ) (H) ≥ 2−1/r 3−1/p (∫( ∑ 󵄨󵄨󵄨Ha−2j G(y) − Ha−2(j+1) G(y)󵄨󵄨󵄨 )

r/q

j=j0

0

δ a1+j1

j1

󵄨 󵄨q ≥ 2−1/r 3−1/p ( ∫ ( ∑ 󵄨󵄨󵄨Ha−2j G(y) − Ha−2(j+1) G(y)󵄨󵄨󵄨 ) 0

j=j0

1

1/r

dy) r/q

1/r

dy)

1/r

≥ C2−1/r−1 3−1/p (j1 − j0 ) q (δa−(j1 +1) ) = C2−1 3−1/p (

1/r

δ ) a−j1 /r (j1 − j0 )1/q . 2a

Upon taking j1 = [r]j0 where [r] is the integer part of r, and noting (δ/2a)1/r → 1

and a−j1 /r → a−j0

as r → ∞,

we obtain the desired result, i. e., for large r, we have CLr (ℝ) (H) ≥ Cr 1/q . Proposition 4.1 should be compared to the following behavior of Hilbert transform. Proposition 5.1. For any 1 < p < ∞, there exists a positive constant C such that ‖H‖Lp (ℝ;Lr (ℝ))→Lp (ℝ;Lr (ℝ)) ≥ Cr

∀ r ≫ 1.

(5.2)

164 | G. Hong Meanwhile, it is trivial that the Hardy–Littlewood maximal operator satisfies ‖M‖Lp (ℝ;Lr (ℝ))→Lp (ℝ;Lr (ℝ)) ≥ C. Thus, the q variation operator can be regarded as an operator between singular integral operators and the maximal operator. Proposition 5.1 should have been known somewhere, but we have not found it in the literature. Hence we give a proof here. Proof of Proposition 5.1. We will consider a function on ℝ × ℝ given by F(x, y) = f (x − y)χ[−1,1] (y) with f = χ[0,1) defined on ℝ. Then it is easy to check that F ≠ 0 only if −1 < x < 2, and ‖F‖Lp (Lr ) ≤ 21/r 31/p . Hence 󵄩 󵄩 ‖H‖Lp (Lr )→Lp (Lr ) ≥ 2−1/r 3−1/p 󵄩󵄩󵄩H(F)󵄩󵄩󵄩Lp (Lr ) . Moreover, a simple calculation gives HF(x, y) = Hf (x − y)χ[−1,1] (y). Therefore, via using the inclusion (0, 1) ⊂ (x − 1, x + 1)

∀ x ∈ (0, 1),

we get ‖HF‖Lp (Lr )

1

1

0

−1

󵄨 󵄨r ≥ (∫( ∫ 󵄨󵄨󵄨Hf (x − y)󵄨󵄨󵄨 dy) 1

x+1

󵄨 󵄨r = (∫( ∫ 󵄨󵄨󵄨Hf (y)󵄨󵄨󵄨 dy) 0

1

x−1 1

󵄨 󵄨r ≥ (∫(∫󵄨󵄨󵄨Hf (y)󵄨󵄨󵄨 dy) 0

p/r

1/p

dx)

p/r

p/r

1/p

dx) 1/p

dx)

0

= ‖Hf ‖Lr ([0,1]) . Note that Hilbert transform is a principle value, i. e., Hf (x) = lim

ε→0

∫ |x−y|>ε

f (y) dy x−y

∀ x ∈ (−∞, ∞).

Banach lattice-valued q-variation and convexity | 165

So, it is easy to conclude by the cancellation condition of the kernel 1/x that 1

Hf (x) = ∫ 2x

x dy = ln x−y 1−x

∀ x ∈ (0, 1/2).

Consequently, for large M > 0 and 0 < x ≤ e−M , we have 1−x 󵄨󵄨 󵄨 ≥ 2−1 ln x −1 ≥ 2−1 M, 󵄨󵄨Hf (x)󵄨󵄨󵄨 = ln x whence ‖Hf ‖Lr ([0,1]) ≥ ‖Hf ‖Lr ([0,e−M ]) ≥ 2−1 Me−M/r . Taking M = r, we conclude that ‖H‖Lp (Lr )→Lp (Lr ) ≥ 2−1/r 3−1/p (2e)−1 r which implies the desired estimate (5.2).

Bibliography [1] J. J. Betancor, R. Crescimbeni, and J. L. Torrea, The ρ-variation of the heat semigroup in the Hermite setting: behaviour in L∞ , Proc. Edinb. Math. Soc. 54 (2011), 1–17. [2] J. J. Betancor, J. C. Fariña, and L. Rodríguez-Mesa, Hardy–Littlewood and UMD Banach lattices via Bessel convolution operators, J. Oper. Theory 67 (2012), 349–368. [3] J. T. Campbell, R. L. Jones, K. Reinhold, and M. Wierdl, Oscillation and variation for singular integrals in higher dimensions, Trans. Am. Math. Soc. 355 (2003), 2115–2137. [4] E. Harboure, R. A. Macías, C. Segovia, and J. L. Torrea, Some estimates for maximal functions on Köthe function spaces, Isr. J. Math. 90 (1995), 349–371. [5] G. Hong and T. Ma, Vector-valued q-variation for differential operators and semigroups I, Math. Z. 286 (2017), 89–120. [6] G. Hong and T. Ma, Vector-valued q-variation for ergodic averages and analytic semigroups, J. Math. Anal. Appl. 437 (2016), 1084–1100. [7] J. Lindenstrauss and L. Tzafriri, Classical Banach space II, Springer, Berlin–Heidelberg–New York, 1979. [8] T. Ma, J. Torrea, and Q. Xu, Weighted variation inequalities for differential operators and singular integrals, J. Funct. Anal. 268 (2015), 376–416. [9] T. Ma, J. Torrea, and Q. Xu, Weighted variation inequalities for differential operators and singular integrals in higher dimensions, Sci. China Math. 60 (2017), 1419–1442.

Yanping Chen and Xueting Han

Weighted boundedness of Marcinkiewicz integral operators and related singular integral operators Abstract: When α ≥ 0 and Ω ∈ Lq (𝕊n−1 ) (q > 1), μΩ,α and τΩ,α can be defined by 1/2

󵄨 󵄨2 μΩ,α (f )(x) = (∫ 2−2tα 󵄨󵄨󵄨FΩ,t (f )(x)󵄨󵄨󵄨 dt) ℝ

and τΩ,α (f )(x) = ∫ 2−tα FΩ,t (f )(x)dt, ℝ

where 2t

FΩ,t (f )(x) = 2−t ∫ b(s) ∫ Ω(y′ )f (x − sy′ )dσ(y′ )ds. 0

𝕊n−1

This paper gives the boundedness of μΩ,α and τΩ,α from the homogeneous weighted Sobolev space Lpα (w) to Lp (w), where p, q, and w satisfy certain conditions. Furthermore, this paper shows that the Marcinkiewicz integral operators μΩ,α,S and μ∗Ω,α,λ are also bounded from Lpα (w) to Lp (w). Keywords: Marcinkiewicz integral, Sobolev space, weighted boundedness, rough kernel MSC 2010: Primary 42B20, 42B99, Secondary 47G10

Acknowledgement: The first author was supported by the National Natural Science Foundation of China grant #11871096 and grant #11471033. Yanping Chen, Xueting Han, Department of Applied Mathematics, University of Science and Technology Beijing, Beijing, 100083, China, e-mails: [email protected], [email protected] https://doi.org/10.1515/9783110741711-010

168 | Y. Chen and X. Han

1 Introduction For a nonnegative and locally integrable function w on the Euclidean space ℝn , we say that w is an Ap weight [7], if and only if Mw(x) ≤ Cw(x),

a. e. x ∈ ℝn ,

p = 1;

and sup( Q

p−1

1 1 − 1 ∫ w(x)dx)( ∫ w(x) p−1 dx) |Q| |Q| Q

< ∞,

1 < p < ∞,

Q

for any cube Q ⊂ ℝn . Besides, for any weight w, we consider f ∈ Lp (w) (1 < p < ∞) if 1/p

󵄨 󵄨p ‖f ‖Lp (w) := ( ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 w(x)dx)

< ∞.

ℝn

Let α ∈ ℝ and 1 < p < ∞. The homogeneous Sobolev space Lpα (ℝn ) [7] is made up of all f ∈ S ′ (ℝn )/P such that ∨ (| ⋅ |α f ̂)

exists and (| ⋅ |α f ̂)∨ belongs to Lp (ℝn ). For f ∈ Lpα (ℝn ), we define ∨󵄩 󵄩 ‖f ‖Lpα (ℝn ) = 󵄩󵄩󵄩(| ⋅ |α f ̂) 󵄩󵄩󵄩Lp (ℝn ) .

Also, for the multiindex k = (k1 , . . . , kn ) with ki ∈ ℕ (i = 1, . . . , n), we set |k| = k1 + ⋅ ⋅ ⋅ + kn ∈ ℕ. When α is a nonnegative integer, it is known that ‖(| ⋅ |α f ̂)∨ ‖Lp (ℝn ) is comparable to ∑|k|=α ‖𝜕k f ‖Lp (ℝn ) (see [7]), that is, 󵄩󵄩 α 󵄩󵄩 󵄩 󵄩 k 󵄩 α ∨󵄩 󵄩󵄩D f 󵄩󵄩Lp (ℝn ) = 󵄩󵄩󵄩(| ⋅ | f ̂) 󵄩󵄩󵄩Lp (ℝn ) ≅ ∑ 󵄩󵄩󵄩𝜕 f 󵄩󵄩󵄩Lp (ℝn ) ,

(1.1)

|k|=α

α f = | ⋅ |α f ̂. Since any test function f belongs to the homogeneous weighted ̂ where D Sobolev space Lpα (w) with α ≥ 0, 1 < p < ∞, and w ∈ Ap , the norm of f in the space Lpα (w) is defined by 1/p

󵄨 󵄨p ‖f ‖Lpα (w) = ( ∫ 󵄨󵄨󵄨Dα f (x)󵄨󵄨󵄨 w(x)dx)

.

ℝn

Note that D0 f = f . So Lp0 (w) = Lp (w). Of course, (1.1) also holds on Lp (w).

Weighted boundedness of Marcinkiewicz integral operators |

169

Suppose that Ω ∈ Lq (𝕊n−1 ) (q > 1) is homogeneous of degree zero and satisfies ∫ Ω(y′ )Ym (y′ )dσ(y′ ) = 0,

(1.2)

𝕊n−1

for all spherical harmonic polynomials Ym whose degrees m ≤ [α] with α ≥ 0. Here 𝕊n−1 = {x ∈ ℝn : |x| = 1} denotes the unit sphere of ℝn (n ≥ 2); dσ(x′ ) is the Lebesgue measure for each x′ ∈ 𝕊n−1 . Let the test function f ∈ 𝒮 (ℝn ) and b be a bounded function on ℝ+ . For all t ∈ ℝ and x ∈ ℝn , we denote 2t

FΩ,t (f )(x) = 2 ∫ b(s) ∫ Ω(y′ )f (x − sy′ )dσ(y′ )ds. −t

0

𝕊n−1

The Marcinkiewicz integral operator μΩ,α can be defined by 1/2

󵄨 󵄨2 μΩ,α (f )(x) = (∫ 2−2tα 󵄨󵄨󵄨FΩ,t (f )(x)󵄨󵄨󵄨 dt) ,

α ≥ 0.

ℝ

The boundedness of μΩ,α has been extensively studied. When α = 0, in 1990, Torchinsky and Wang [10] proved that if Ω ∈ Lipβ (𝕊n−1 ) (0 < β ≤ 1) and w ∈ Ap (1 < p < ∞), then μΩ,0 is bounded on Lp (w). Later, in 1999, Ding, Fan, and Pan showed that μΩ,0 is bounded on Lp (w) (1 < p < ∞) when Ω ∈ Lq (𝕊n−1 ) (q > 1) in [3]. When α ≥ 0, suppose Ω ∈ H r (𝕊n−1 ) with r = (n − 1)/(n − 1 + α) and 1 < p < ∞, then Chen, Ding, and Fan [2] proved that μΩ,α is bounded from Lpα (ℝn ) to Lp (ℝn ) in 2008. In this paper, we prove the boundedness of μΩ,α from Lpα (w) to Lp (w) with the kernel Ω ∈ Lq (𝕊n−1 ) (q > 1) and 1 < p < ∞, α ≥ 0. Below is our main result. Theorem 1.1. Suppose that Ω ∈ Lq (𝕊n−1 ) (q > 1) satisfies (1.2) and b(s) ∈ L∞ (ℝ+ ). If α ≥ 0, p, q > 1, and w satisfy one of the following conditions: (a) q′ < p < ∞ and w ∈ Ap/q′ ; (b) 1 < p < q and w1−p ∈ Ap′ /q′ ; ′

(c) 1 < p < ∞ and wq ∈ Ap , ′

then there exists a constant C > 0, independent of f , such that 󵄩󵄩 󵄩 󵄩󵄩μΩ,α (f )󵄩󵄩󵄩Lp (w) ≤ C‖f ‖Lpα (w) .

(1.3)

On the other hand, for α ≥ 0, the singular integral operator τΩ,α is related closely to the Marcinkiewicz integral operator μΩ,α . The form of τΩ,α can be expressed by τΩ,α (f )(x) = ∫ 2−tα FΩ,t (f )(x)dt. ℝ

170 | Y. Chen and X. Han It is known that τΩ,α is bounded from Lpα (ℝn ) to Lp (ℝn ) (1 < p < ∞) when Ω ∈ H (𝕊n−1 ) with r = (n − 1)/(n − 1 + α) (see [2]). By the method of [5], from Theorem 1.1 and Lemma 2.5 (in Section 2), we can get the next conclusion immediately. r

Corollary 1.2. Under the conditions of Theorem 1.1, there exists a constant C > 0, independent of f , such that 󵄩 󵄩󵄩 󵄩󵄩τΩ,α (f )󵄩󵄩󵄩Lp (w) ≤ C‖f ‖Lpα (w) .

(1.4)

The Marcinkiewicz integral operators μΩ,α,S and μ∗Ω,α,λ are defined by μΩ,α,S (f )(x) = (∫

1/2

∫

ℝ |x−y| 1,

which are related to the area integral S and gλ∗ function, respectively. In fact, Ding, Fan, and Pan [4] got the Lp -boundedness of μΩ,0,S and μ∗Ω,0,λ provided Ω ∈ Lq (𝕊n−1 ) (q > 1). In their article [3], they proved that these two operators are bounded on Lp (w). With Ω ∈ Lq (𝕊n−1 ) (q > 1), we here give the boundedness of μΩ,α,S and μ∗Ω,α,λ from Lpα (w) to Lp (w). Theorem 1.3. Suppose that Ω ∈ Lq (𝕊n−1 ) (q > 1) satisfies (1.2) and b(s) ∈ L∞ (ℝ+ ). If α ≥ 0, λ > 1, p, q > 1, and w satisfy one of the following conditions: (ã) max{q′ , 2} = p0 < p < ∞ and w ∈ Ap/p0 ; ′ (b̃) 2 < p < q and w1−(p/2) ∈ Ap′ /q′ ; (c̃) 2 ≤ p < ∞ and wq ∈ Ap/2 , ′

then there exists a constant C > 0, independent of f , such that 󵄩󵄩 ∗ 󵄩 󵄩󵄩μΩ,α,λ (f )󵄩󵄩󵄩Lp (w) ≤ C‖f ‖Lpα (w)

(1.5)

󵄩󵄩 󵄩 󵄩󵄩μΩ,α,S (f )󵄩󵄩󵄩Lp (w) ≤ C‖f ‖Lpα (w) .

(1.6)

and

Throughout this paper, the letter “C” always stands for a positive constant, which is independent of the essential variables and may change from line to line. In addition, we said the constants p and p′ are Hölder conjugates of each other if 1/p + 1/p′ = 1. This paper is organized as follows. In Section 2 we collect some useful lemmas. In Section 3 we give the proof of Theorem 1.1. In Section 4 we verify Theorem 1.3.

Weighted boundedness of Marcinkiewicz integral operators | 171

2 Five lemmas Firstly, there are some important properties of Ap weights with 1 < p < ∞. Lemma 2.1 ([6]). For any Ap (1 < p < ∞) weight, we have (i) Ap1 ⊂ Ap2 if 1 < p1 < p2 < ∞;

(ii) w ∈ Ap if and only if w1−p ∈ Ap′ ; (iii) If w ∈ Ap , then there is an ε > 0 such that p − ε > 1 and w ∈ Ap−ε ; (iv) If w ∈ Ap , then there is an ε > 0 such that w1+ε ∈ Ap ; (v) If w ∈ Ap , then for any 0 < ε < 1, wε ∈ Ap . ′

Secondly, we introduce the Stein–Weiss interpolation theorem with change of measures. Lemma 2.2 ([9]). Suppose that u0 , v0 , u1 , v1 are positive weight functions, 1 < p0 , p1 < ∞, and S is a sublinear operator enjoying ‖Sf ‖Lp0 (u0 ) ≤ C0 ‖f ‖Lp0 (v0 ) and ‖Sf ‖Lp1 (u1 ) ≤ C1 ‖f ‖Lp1 (v1 ) . Then ‖Sf ‖Lp (u) ≤ C‖f ‖Lp (v) holds for any 0 < θ < 1 and θ 1−θ 1 = + , p p0 p1 where pθ/p0 p(1−θ)/p1 u1 ,

u = u0

pθ/p0 p(1−θ)/p1 v1 ,

v = v0

and C ≤ C0θ C11−θ . Thirdly, for any f ∈ L1loc (ℝn ), the Hardy–Littlewood maximal function is given by Mf (x) = sup r>0

1 󵄨 󵄨 ∫ 󵄨󵄨󵄨f (y)󵄨󵄨󵄨dy, |B(x, r)|

∀x ∈ ℝn .

B(x,r)

Lemma 2.3 ([7]). If 1 < p < ∞, then the Hardy–Littlewood maximal operator M is bounded on Lp (w) if and only if w ∈ Ap .

172 | Y. Chen and X. Han Fourthly, the mixed norm can be defined by p/r

󵄨r 󵄨 ‖ft ‖Lp (Lr ,w) := ( ∫ (∫󵄨󵄨󵄨ft (x)󵄨󵄨󵄨 dt)

1/p

w(x)dx)

,

ℝn ℝ

for 1 < r, p < ∞. Upon writing Lp (Lr (ℝ), w, ℝn ) as the set of all functions f with their mixed norm ‖ft ‖Lp (Lr ,w) < ∞, we have the following important result. Lemma 2.4 ([1]). If M is the Hardy–Littlewood maximal operator, then for 1 < r, p < ∞ and w ∈ Ap , there is a constant C such that for any ft ∈ Lp (Lr (ℝ), w, ℝn ), ‖Mft ‖Lp (Lr ,w) ≤ C‖ft ‖Lp (Lr ,w) . Fifthly, suppose that ψ ≥ 0 is a bounded radial function satisfying supp(ψ) ⊂ {x ∈ ℝn : 1/2 ≤ |x| ≤ 2} and ∞

∫ 0

ψ(t) dt = log 2. t

̂ ) = ψ(ξ ), then for any f ∈ 𝒮 (ℝn ), ϕ ̂ (ξ ) = ψ(tξ ) and If ϕt (x) = t −n ϕ(x/t) and ϕ(ξ t f (x) = ∫ ϕ2t ∗ f (x)dt

(2.1)

ℝ

hold (see [3]). With g-function of f being defined by 1/2

󵄨 󵄨2 g(f )(x) = (∫󵄨󵄨󵄨ϕ2t ∗ f (x)󵄨󵄨󵄨 dt) , ℝ

we introduce the following lemma. Lemma 2.5 ([8]). For 1 < p < ∞ and w ∈ Ap , there are constants C1 and C2 such that 󵄩 󵄩 C1 ‖f ‖Lp (w) ≤ 󵄩󵄩󵄩g(f )󵄩󵄩󵄩Lp (w) ≤ C2 ‖f ‖Lp (w) . In fact, it is not hard to see that ‖g(f )‖Lp (w) = ‖ϕ2t ∗ f ‖Lp (L2 ,w) .

(2.2)

Weighted boundedness of Marcinkiewicz integral operators | 173

3 Proof of Theorem 1.1 The proof for α = 0 was given in [3]. In this section, we prove the theorem in three cases: (1) 0 < α < 1; (2) α ∈ ℕ; and (3) 1 < α ∉ ℕ.

3.1 Proof of Theorem 1.1 when 0 < α < 1 Note that we can write f = Gα ∗ Dα f where Gα (x) = |x|α−n

̂ (ξ ) ≅ |ξ |−α and G α

(0 < α < n).

By (1.2), for 2−tα FΩ,t (f )(x), we can write 2t

2

−tα

FΩ,t (f )(x) = 2

∫ b(s) ∫ Ω(y′ )f (x − sy′ )dσ(y′ )ds

−tα−t

0

𝕊n−1

t

2

= 2−tα−t ∫ b(s) ∫ Ω(y′ )Gα ∗ Dα f (x − sy′ )dσ(y′ )ds 0

α

𝕊n−1

=: H1,t ∗ D f (x). Here 2t

H1,t (z) = 2

−tα−t

∫ b(s) ∫ Ω(y′ )Gα (z − sy′ )dσ(y′ )ds. 0

𝕊n−1

Then by the Minkowski’s inequality and (2.1), we have 1/2

󵄨 󵄨2 μΩ,α (f )(x) = (∫󵄨󵄨󵄨2−tα FΩ,t (f )(x)󵄨󵄨󵄨 dt) ℝ

1/2

󵄨 󵄨2 = (∫󵄨󵄨󵄨H1,t ∗ Dα f (x)󵄨󵄨󵄨 dt) ℝ

1/2 󵄨󵄨 󵄨󵄨󵄨2 󵄨 = (∫󵄨󵄨󵄨∫ ϕ2v+t ∗ H1,t ∗ Dα f (x)dv󵄨󵄨󵄨 dt) 󵄨󵄨 󵄨󵄨 ℝ ℝ

1/2

󵄨 󵄨2 ≤ ∫(∫󵄨󵄨󵄨ϕ2v+t ∗ H1,t ∗ Dα f (x)󵄨󵄨󵄨 dt) dv ℝ

ℝ

=: ∫ J1,v Dα f (x)dv, ℝ

(3.1)

174 | Y. Chen and X. Han where 1/2

󵄨2 󵄨 J1,v Dα f (x) = (∫󵄨󵄨󵄨ϕ2v+t ∗ H1,t ∗ Dα f (x)󵄨󵄨󵄨 dt) . ℝ

If we can obtain the estimate 󵄩󵄩 −vθ α 󵄩 vη 󵄩 α 󵄩 󵄩󵄩J1,v D f 󵄩󵄩󵄩Lp (w) ≤ C min{2 1 , 2 1 }󵄩󵄩󵄩D f 󵄩󵄩󵄩Lp (w) ,

(3.2)

where 0 < θ1 , η1 < 1, then by (3.1) and (3.2), we can conclude (1.3) via 󵄩󵄩 󵄩 󵄩 α 󵄩 󵄩󵄩μΩ,α (f )󵄩󵄩󵄩Lp (w) ≤ ∫󵄩󵄩󵄩J1,v D f 󵄩󵄩󵄩Lp (w) dv ℝ

0

∞

−∞

0

󵄩 󵄩 󵄩 󵄩 ≤ C ∫ 2vη1 󵄩󵄩󵄩Dα f 󵄩󵄩󵄩Lp (w) dv + C ∫ 2−vθ1 󵄩󵄩󵄩Dα f 󵄩󵄩󵄩Lp (w) dv 󵄩 󵄩 ≤ C 󵄩󵄩󵄩Dα f 󵄩󵄩󵄩Lp (w)

= C‖f ‖Lpα (w) .

Now, it remains to prove (3.2). Here is the sketch of the proof. Under conditions (a), (b), and (c), we can obtain the following facts: 󵄩󵄩 󵄩 α 󵄩 −v(1−α) vα 󵄩 , 2 }󵄩󵄩󵄩Dα f 󵄩󵄩󵄩L2 󵄩󵄩J1,v D f 󵄩󵄩󵄩L2 ≤ C min{2

(3.3)

󵄩󵄩 󵄩 α 󵄩 α 󵄩 󵄩󵄩J1,v D f 󵄩󵄩󵄩Lp (w) ≤ C 󵄩󵄩󵄩D f 󵄩󵄩󵄩Lp (w) .

(3.4)

and

Then by conditions (a)–(c) and the properties of Ap weights, we can deduce that there exist 1 < p < ∞, 0 < δ < 1 such that w1+δ ∈ Ap and (3.4) also holds, that is, 󵄩󵄩 󵄩 α 󵄩 α 󵄩 󵄩󵄩J1,v D f 󵄩󵄩󵄩Lp (w1+δ ) ≤ C 󵄩󵄩󵄩D f 󵄩󵄩󵄩Lp (w1+δ ) .

(3.5)

After that, using the Stein–Weiss interpolation theorem with change of measures between (3.3) and (3.5), we can get (3.2). The process in more details is as follows. 3.1.1 Proof of (3.2) under condition (a) In order to estimate (3.3), denoting z − sy′ by x yields 2t

̂ H 1,t (ξ ) = ∫ 2

−tα−t

ℝn

∫ b(s) ∫ Ω(y′ )Gα (z − sy′ )dσ(y′ )dse−2πi⟨z,ξ ⟩ dz 0

𝕊n−1

Weighted boundedness of Marcinkiewicz integral operators | 175 2t

=2

−tα−t

′

∫ b(s) ∫ Ω(y′ ) ∫ Gα (x)e−2πi⟨x,ξ ⟩ dxe−2πis⟨y ,ξ ⟩ dσ(y′ )ds 0

ℝn

𝕊n−1

t

2

′

̂ (ξ ). = 2−tα−t ∫ b(s) ∫ Ω(y′ )e−2πis⟨y ,ξ ⟩ dσ(y′ )dsG α 0

𝕊n−1

̂ (ξ ) ≅ |ξ |−α , from (1.2) we get Since G α 2t

′ 󵄨󵄨 ̂ 󵄨󵄨 󵄨 󵄨 󵄨󵄨 󵄨 󵄨 ̂ 󵄨󵄨 −tα−t 󵄨󵄨 ∫󵄨󵄨b(s)󵄨󵄨󵄨 ∫ 󵄨󵄨󵄨Ω(y′ )󵄨󵄨󵄨󵄨󵄨󵄨e−2πis⟨y ,ξ ⟩ − 1󵄨󵄨󵄨dσ(y′ )ds󵄨󵄨󵄨G 󵄨󵄨H1,t (ξ )󵄨󵄨 ≤ 2 α (ξ )󵄨󵄨

0

𝕊n−1

t

2

≤ C2

−tα−t

󵄨 󵄨 󵄨 󵄨 ∫ ∫ 󵄨󵄨󵄨Ω(y′ )󵄨󵄨󵄨s󵄨󵄨󵄨y′ 󵄨󵄨󵄨|ξ |dσ(y′ )ds|ξ |−α 0 𝕊n−1

≤ C‖Ω‖L1 (𝕊n−1 ) 2

−tα−t

|ξ |

1−α

1−α

t

2t

∫ sds 0

≤ C‖Ω‖L1 (𝕊n−1 ) (2 |ξ |) and 2t

′ 󵄨󵄨 ̂ 󵄨󵄨 󵄨 󵄨 󵄨󵄨 󵄨 󵄨 ̂ 󵄨󵄨 −tα−t 󵄨󵄨 ∫󵄨󵄨b(s)󵄨󵄨󵄨 ∫ 󵄨󵄨󵄨Ω(y′ )󵄨󵄨󵄨󵄨󵄨󵄨e−2πis⟨y ,ξ ⟩ 󵄨󵄨󵄨dσ(y′ )ds󵄨󵄨󵄨G 󵄨󵄨H1,t (ξ )󵄨󵄨 ≤ 2 α (ξ )󵄨󵄨

0

t

𝕊n−1

2

󵄨 󵄨 ≤ C2−tα−t ∫ ds ∫ 󵄨󵄨󵄨Ω(y′ )󵄨󵄨󵄨dσ(y′ )|ξ |−α 0

𝕊n−1 t

≤ C‖Ω‖L1 (𝕊n−1 ) (2 |ξ |) . −α

It is easy to see that 1−α −α 󵄨󵄨 ̂ 󵄨󵄨 t t 󵄨󵄨H1,t (ξ )󵄨󵄨 ≤ C‖Ω‖L1 (𝕊n−1 ) min{(2 |ξ |) , (2 |ξ |) }.

From (3.6) and the Plancherel theorem, we obtain 󵄩󵄩 α 󵄩2 󵄩󵄩J1,v D f 󵄩󵄩󵄩L2 󵄨̂ 󵄨󵄨2 󵄨󵄨 ̂ 󵄨󵄨2 󵄨󵄨 ̂ 󵄨󵄨2 α = ∫ ∫ 󵄨󵄨󵄨ϕ 2v+t (ξ )󵄨󵄨 󵄨󵄨H1,t (ξ )󵄨󵄨 󵄨󵄨D f (ξ )󵄨󵄨 dξdt ℝ ℝn

≤ C∫

∫

ℝ 2−v−t−1 ≤|ξ |≤2−v−t+1

−2α 󵄨󵄨 ̂ 󵄨2 t−v−t+1 2(1−α) ) , (2t−v−t−1 ) }dξdt 󵄨󵄨Dα f (ξ )󵄨󵄨󵄨 min{(2

(3.6)

176 | Y. Chen and X. Han

≤ C min{2

−2v(1−α)

,2

2vα

2−v−t+1

}∫ ∫

󵄨̂ α f (rξ ′ )󵄨󵄨󵄨2 r n−1 drdσ(ξ ′ )dt ∫ 󵄨󵄨󵄨D 󵄨

ℝ 𝕊n−1 2−v−t−1 ∞

󵄨̂ α f (rξ ′ )󵄨󵄨󵄨2 r n−1 ≤ C min{2−2v(1−α) , 22vα } ∫ ∫ 󵄨󵄨󵄨D 󵄨 0 𝕊n−1

≤ C min{2

−2v(1−α)

,2

2vα

−v+1−log2 r

dtdσ(ξ ′ )dr

∫ −v−1−log2 r

󵄨̂ α f (ξ )󵄨󵄨󵄨2 dξ } ∫ 󵄨󵄨󵄨D 󵄨 ℝn

󵄩 󵄩2 = C min{2−2v(1−α) , 22vα }󵄩󵄩󵄩Dα f 󵄩󵄩󵄩L2 . This estimate implies (3.3). We proceed to estimate (3.4), we observe that 1/2

󵄨2 󵄨 J1,v Dα f (x) = (∫󵄨󵄨󵄨H1,t ∗ ϕ2v+t ∗ Dα f (x)󵄨󵄨󵄨 dt) . ℝ

Then, by (1.2), we have H1,t ∗ ϕ2v+t ∗ Dα f (x) 2t

= ∫2

−tα−t

∫ b(s) ∫ Ω(y′ )Gα (z − sy′ )dσ(y′ )dsϕ2v+t ∗ Dα f (x − z)dz 0

ℝn t

𝕊n−1

2

= ∫ ∫ b(s)Ω(y′ ) ∫ [Gα (z − sy′ ) − Gα (z)]ϕ2v+t ∗ Dα f (x − z) 0 𝕊n−1

|z|≥2t

t

2

+ ∫ ∫ b(s)Ω(y′ ) ∫ Gα (z − sy′ )ϕ2v+t ∗ Dα f (x − z) 0 𝕊n−1

|z| 0 and all x, y ∈ M. If the manifold has nonnegative Ricci curvature, by a famous result of Li and Yau [12], we have Pt (x, y) ≍

C d2 (x, y) exp(− ), ct Vx (√t)

where the sign ≍ means that both ⩽ and ⩾ are true but with different values of C and c. It follows that (H)-condition is satisfied if the manifold has nonnegative Ricci curvature. Theorem 1.2. The following statements are valid: (a) If +∞

∫

t dt = ∞, V(t)p−1

then equation (1.1) admits no nonnegative solutions.

Existence and nonexistence of nonnegative solutions to heat equation

| 211

(b) If +∞

∫

t dt < ∞ V(t)p−1

and u0 is smaller than a small Gaussian, then (1.1) admits positive solutions. In part (a) of Theorem 1.2, the “solution” can also be understood in a weak sense. Our proof is based on the method originating from [20] (cf. [19]), and also the idea used in the proof of Theorem 11.14 of [5] via the Laplacian of the distance function and a carefully chosen test function. The argument for part (b) is purely functional-analytic. 1,2 We denote by Wc1,2 (M × [0, ∞)) the subspace of Wloc (M × [0, ∞)) of functions with compact support. In the above and below, U ≲ stands for U ≤ cV for a constant c > 0; U ≍ V means both U ≲ V and V ≳ U. The symbols c, C, c0 , C0 , c1 , C1 , . . . denote positive constants whose values are unimportant and may vary at different occurrences.

2 Proof of Theorem 1.2 Proof of part (a) Suppose that u is a global positive solution to (1.1). Let 0 ≤ ψ ∈ Wc1,2 (M × [0, ∞)). Multiplying both sides of (1.1) by ψ and integrating by parts, we obtain ∞

p

∞

∞

0 M ∞

0 M ∞

0 M

0 M

∫ ∫ u ψdxdt = − ∫ ∫ uΔψdxdt − ∫ ∫ u𝜕t ψdxdt − ∫ u0 ψdx 0 M

⩽ − ∫ ∫ uΔψdxdt − ∫ ∫ u𝜕t ψdxdt.

M

(2.1)

Let us choose the following test function: ψ(x, t) = φq (r(x), t), where q = p/(p − 1) and r(x) = d(o, x) for the reference point o ∈ M. It follows that Δψ = qφq−1 Δφ + q(q − 1)φq−2 |∇φ|2 ,

{

𝜕t ψ = qφq−1 𝜕t φ.

212 | Y. Sun and F. Xu Substituting the above into (2.1), we obtain ∞

p q

∞

q−1

∫ ∫ u φ dxdt ≲ ∫ ∫ uφ 0 M

∞

(−Δφ)dxdt + ∫ ∫ uφq−1 (−𝜕t φ)dxdt.

0 M

(2.2)

0 M

For some constant C1 , let h ∈ C ∞ [0, ∞) be a function satisfying h(r) = 1,

r ∈ [0, 1];

h(r) = 0,

r ∈ [2, ∞);

−C1 ⩽ h′ ⩽ 0,

󵄨󵄨 ′′ 󵄨󵄨 󵄨󵄨h 󵄨󵄨 ⩽ C1 ,

r ∈ (1, 2).

Fix a finite strictly increasing sequence {rk } (k = 0, 1, . . . , i). Let Qk ≡ B(o, rk ) × [0, rk2 ). Define φ by 1, (x, t) ∈ Q0 , { { { { (rk −rk−1 )2 r(x) t φ(x, t) = {a V(r )p−1 h( r )h( r2 ) + Tk , (x, t) ∈ Qk \Qk−1 , k = 1, . . . , i, k−1 k { k−1 { { 0, (x, t) ∈ M\Qi , { where −1

i

(rk − rk−1 )2 ) V(rk )p−1 k=1

a = (∑ and i

2p

(rj −rj−1 ) { {a ∑ V(r )p−1 , k = 1, . . . , i − 1, j Tk = { j=k+1 { k = i. {0,

Clearly, φ ∈ Wc2 (M × [0, ∞)). Note that Δφ =

𝜕2 φ n − 1 𝜕 log g 1/2 𝜕φ +( + ) . 2 r 𝜕r 𝜕r 𝜕r

(2.3)

So, when (x, t) ∈ Qk \Qk−1 , letting {ri } be a geometric sequence satisfying rk = 2rk−1 , we have −Ca

(rk − rk−1 ) 𝜕φ ⩽ ⩽0 𝜕r V(rk )p−1

(2.4)

Existence and nonexistence of nonnegative solutions to heat equation

| 213

and 󵄨󵄨 𝜕2 φ 󵄨󵄨 Ca 󵄨󵄨 󵄨 . 󵄨󵄨 2 󵄨󵄨󵄨 ⩽ 󵄨󵄨 𝜕r 󵄨󵄨 V(rk )p−1

(2.5)

Combining (2.3)–(2.5) and (G)-condition, we have −Δφ ≲

a , V(rk )p−1

(x, t) ∈ Qk \Qk−1 .

(2.6)

(x, t) ∈ Qk \Qk−1 .

(2.7)

Also by the definition of φ, we have −𝜕t φ ≲

a , V(rk )p−1

Substituting (2.6) and (2.7) into (2.2), we have ∞

i

∞

p q

uφq−1 χ dxdt. V(rk )p−1 {Qk \Qk−1 } k=1

∫ ∫ u φ dxdt ≲ a ∫ ∫ ∑ 0 M

0 M

Using Hölder’s inequality, we obtain ∞

∫ ∫ up φq dxdt 0 M ∞

1/p

p q

≲ a( ∫ ∫ u φ dxdt) 0 M

( ∫ ∫( ∑

1/p

∞

≲ a( ∫ ∫ up φq dxdt) 0 M

i

∞

k=1

0 M

χ{Qk \Qk−1 } V(rk )p−1

q

1/q

) dxdt) 1/q

i

(∑ ∬ k=1 Q \Q k k−1

1 dxdt) V(rk )p

.

Accordingly, ∞

p q

(p−1)/p

( ∫ ∫ u φ dxdt) 0 M

i

≲ a( ∑ ∬

k=1 Q \Q k k−1

1/q

1 dxdt) V(rk )p 1/q

rk2 ≲ a( ∑ ) V(rk )p−1 k=1 i

i

1/q

(r − r )2 ≲ a( ∑ k k−1 ) V(rk )p−1 k=1 ≲ a(q−1)/q .

(2.8)

214 | Y. Sun and F. Xu On the other hand, i

(rk − rk−1 )2 V(rk )p−1 k=1

a−1 = ∑

2 rk2 − rk−1 V(rk )p−1 k=1 i

=C∑ i

rk

r dx V(r)p−1

⩾C∑ ∫

k=2 rk−1 ri

=C∫ 2r0

r dr. V(r)p−1

(2.9)

Since ∞

∫

r dx = ∞, V(r)p−1

for every r0 > 0, we use (2.9) to get that if i → +∞, then a → 0. Thus we can choose a sufficiently large i such that a ≤ r0 −1 . Substituting the above into (2.8), we have r0

∫ ∫ up dxdt ⩽ 0 B(r0 )

C

r0q−1

.

Letting r0 → +∞ yields ∞

∫ ∫ up dxdt = 0, 0 M

and thus u ≡ 0. But u0 ≥ 0 is not identically zero, so, by the Maximum Principle, we know that u is positive almost everywhere, which leads to a contradiction, and hence there exist no nonnegative solutions to problem (1.1).

Proof of part (b) Fix the point o ∈ M. Let us define the operator t

Tu(x, t) = ∫ Pt (x, y)u0 (y)dy + ∫ ∫ Pt−s (x, y)up (y, s)dx(y)ds, M

0 M

(2.10)

Existence and nonexistence of nonnegative solutions to heat equation

| 215

acting on the space SM = {u ∈ L∞ (M × [0, ∞)) | 0 ⩽ u(x, t) ⩽ λPt+δ (x, o)},

(2.11)

where λ > 0 is a constant to be chosen later, and δ > 1 is a fixed constant. It is easy to see that SM is a close set of L∞ (M × [0, ∞)). Let u0 satisfy 0 ⩽ u0 (x) ⩽

λ P (x, o). 2 δ

(2.12)

Now let us show TSM ⊂ SM . Using (1.5) and (2.12), we have ∫ Pt (x, y)u0 (y)dx(y) ⩽ M

λ λ ∫ Pt (x, y)Pδ (y, o)dx(y) = Pt+δ (x, o). 2 2

(2.13)

M

Using (1.6), (1.5), and (2.11), we have t

∫ ∫ Pt−s (x, y)up (y, s)dx(y)ds 0 M

t

p (y, o)dx(y)ds ⩽ λ ∫ ∫ Pt−s (x, y)Ps+δ p

0 M

t

⩽ λp C3p−1 ∫ 0

1

V(√s + δ)p−1 t

⩽ λp C3p−1 Pt+δ (x, o) ∫ 0

∫ Pt−s (x, y)Ps+δ (y, o)dx(y)ds M

1

V(√s + δ)p−1

ds.

(2.14)

By the volume condition ∞

∫

s ds < ∞, V(s)p−1

there exists a constant C4 such that t

∫ 0

1

V(√s + δ)p−1

t+δ

ds = ∫ δ

1 ds V(√s)p−1

√t+δ

=C ∫ √δ

s ds V(s)p−1

⩽ C4 < ∞.

(2.15)

216 | Y. Sun and F. Xu Substituting (2.15) into (2.14), we obtain that if λ is small enough then t

∫ ∫ Pt−s (x, y)up (y, s)dx(y)ds ⩽ λp C1p−1 C4 Pt+δ (x, o) ⩽ 0 M

λ P (x, o). 2 t+δ

(2.16)

Combining (2.10) and (2.13) with (2.16), we obtain 0 ⩽ Tu ⩽ λPt+δ (x, o)

and TSM ⊂ SM .

Now let us show that T is a contraction map. For u1 , u2 ∈ SM , we have t

󵄨 p 󵄨 󵄨󵄨 󵄨 p 󵄨󵄨Tu1 (x, t) − Tu2 (x, t)󵄨󵄨󵄨 ≤ ∫ ∫ Pt−s (x, y)󵄨󵄨󵄨u1 (y, s) − u2 (y, s)󵄨󵄨󵄨dx(y)ds. 0 M

Noticing 󵄨󵄨 p 󵄨 󵄨 󵄨 p p−1 p−1 󵄨󵄨u1 (y, s) − u2 (y, s)󵄨󵄨󵄨 ⩽ p max{u1 (y, s), u2 (y, s)}󵄨󵄨󵄨u1 (y, s) − u2 (y, s)󵄨󵄨󵄨, and combining with (1.6), (1.4), (2.11), and (2.15), we obtain that 󵄨󵄨 󵄨 󵄨󵄨Tu1 (x, t) − Tu2 (x, t)󵄨󵄨󵄨 ⩽ pλ

p−1

‖u1 − u2 ‖

t

L∞

p−1 (y, o)dyds ∫ ∫ Pt−s (x, y)Ps+δ 0 M

t

⩽ pλp−1 C1p−1 ‖u1 − u2 ‖L∞ ∫ 0

t

= pλp−1 C1p−1 ‖u1 − u2 ‖L∞ ∫ ⩽

pλp−1 C1p−1 C4 ‖u1

0

1

V(√s + δ)p−1

∫ Pt−s (x, y)dyds M

1 ds √ V( s + δ)p−1

− u2 ‖L∞ .

Readjusting λ to be small enough such that pλp−1 C1p−1 C4 < 1, we obtain that T is a contraction map. Applying the fixed point theorem, we derive that there exists a fixed point u ∈ SM satisfying t

u(x, t) = ∫ Pt (x, y)u0 (y)dy + ∫ ∫ Pt−s (x, y)up (y, s)dyds. M

0 M

(2.17)

Existence and nonexistence of nonnegative solutions to heat equation

| 217

Thanks to u0 ≩ 0, u is positive on M. Since u0 , u ∈ L2 (M), by a standard argument of regularity (see [5, Theorems 7.6–7.7]), we know that the two integrals in (2.17) are smooth on M × (0, ∞), whence u is a global positive solution of problem (1.1).

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Dorina Mitrea, Irina Mitrea, and Marius Mitrea

The Neumann problem for the Laplacian in half-spaces Abstract: Even in the case of the Laplacian considered in a half-space, there exist infinitely many Neumann problems and their well-posedness is affected by how the coefficient matrix interacts with the unit normal for the half-space in question. In this paper we study this interplay between algebra and geometry. Keywords: Laplacian, boundary value problem, Neumann problem, regularity problem, nontangential maximal operator, conormal derivative, single layer potential, Riesz transforms, Calderón–Zygmund theory, jump-formulas MSC 2010: Primary 35J25, 42B20, 42B37, 45E05, Secondary 35C15, 35J05, 42B20, 42B25

1 Introduction In this paper we are concerned with the Lp -Neumann problem for the Laplacian in half-spaces in ℝn . The latter are sets of the form H + := {x ∈ ℝn : ⟨x, h⟩ > 0},

(1.1)

for some unit vector h in ℝn . Whenever this is the case, we agree to abbreviate H := 𝜕H + = {x ∈ ℝn : ⟨x, h⟩ = 0},

(1.2)

and consider the measure induced by ℋn−1 , the (n−1)-dimensional Hausdorff measure in ℝn , as the surface measure on H = 𝜕H + . Having fixed an aperture parameter κ > 0, at each point x ∈ H = 𝜕H + we define the nontangential approach region with vertex at x as + √ 2 ΓH κ (x) := {y ∈ H : |x − y| < κ + 1 dist (y, H)}. +

(1.3)

Acknowledgement: The first named author has been supported in part by the Simons Foundation grant # 426669. The second named author has been supported by the NSF grant # 1900938. The third named author has been supported in part by the Simons Foundation grant # 637481. Dorina Mitrea, Marius Mitrea, Department of Mathematics, Baylor University, Sid Richardson Building, 1410 S 4th Street, Waco, TX 76706, USA, e-mails: [email protected], [email protected] Irina Mitrea, Department of Mathematics, Temple University, 1805 N. Broad Street, Philadelphia, PA 19122, USA, e-mail: [email protected] https://doi.org/10.1515/9783110741711-013

220 | D. Mitrea et al. The nontangential pointwise trace of a continuous (possibly vector-valued) function u defined in H + is then given by 󵄨κ−n.t. (u󵄨󵄨󵄨𝜕H + )(x) :=

lim

+ ΓHκ (x)∋y→x

u(y)

for all x ∈ H = 𝜕H + ,

whenever this limit is meaningful. We also consider the action of the nontangential + maximal operator 𝒩κH with aperture parameter κ in H + on any continuous (again, possibly vector-valued) function u defined in H + as + 󵄨 󵄨 (𝒩κH u)(x) := sup 󵄨󵄨󵄨u(y)󵄨󵄨󵄨 at each x ∈ H = 𝜕H + . +

y∈ΓHκ (x)

Given any p ∈ (1, ∞), the Lp -Neumann problem for the Laplacian in H + asks for finding a harmonic function in H + with the nontangential maximal function of its gradient belonging to Lp (H, ℋn−1 ) and whose conormal derivative is prescribed in Lp (H, ℋn−1 ). In stark contrast to the Dirichlet problem, there are infinitely many conormal derivatives associated with the Laplacian. Specifically, corresponding to each writing of the Laplacian as a divergence-form operator, i. e., each way of expressing Δ = divA∇ in ℝn ,

(1.4)

for some n × n matrix with complex entries A ∈ ℂn×n , we shall associate a conormal derivative operator, acting on each function u ∈ C ∞ (H + ),

Δu = 0

in H + ,

H+

p

𝒩κ (∇u) ∈ L (H, ℋ

n−1

),

according to 󵄨κ−n.t. 𝜕νA + u := ⟨−h, A(∇u)󵄨󵄨󵄨𝜕H + ⟩ at ℋn−1 -a. e. point on H. H

(1.5)

Then 𝜕νA + u is well-defined in Lp (H, ℋn−1 ). H

Indeed, this becomes a consequence of Calderón’s theorem (see [1], [8, Theorem 3, p. 201]) which, bearing in mind the fact that the Laplacian is rotation invariant and n n−1 (3.3)–(3.4), presently gives that the nontangential trace (∇u)|κ−n.t. 𝜕H + exists (in ℂ ) at ℋ a. e. point in H. For example, in the case when we pick h := en := (0, . . . , 0, 1) ∈ Sn−1 ,

The Neumann problem for the Laplacian in half-spaces | 221

it follows that H + = ℝn+ , and the conormal derivative introduced in (1.5) becomes 󵄨κ−n.t. 𝜕νA u := ⟨−en , A(∇u)󵄨󵄨󵄨𝜕ℝn ⟩ at ℒn−1 -a. e. point on ℝn−1 ≡ 𝜕ℝn+ . +

(1.6)

Here and elsewhere, ℒn−1 stands for the (n−1)-dimensional Lebesgue measure in ℝn−1 . In addition, corresponding to A := In×n , the n×n identity matrix (which is a valid choice in (1.4)), we obtain the following particular conormal derivative operator for the upper half-space: 󵄨κ−n.t. u = ⟨−en , (∇u)󵄨󵄨󵄨𝜕ℝn ⟩ 𝜕νIn×n ℝn +

+

󵄨κ−n.t. = −(𝜕n u)󵄨󵄨󵄨𝜕ℝn +

at ℒn−1 -a. e. point on ℝn−1 ≡ 𝜕ℝn+ ,

i. e., the familiar normal derivative operator for the upper half-space. Note that condition (1.4) is equivalent to having A = (ajk )1≤j,k≤n ∈ ℂn×n

ajk + akj = 2δjk

with

for all j, k ∈ {1, . . . , n},

(1.7)

or, equivalently, having A = In×n + B for some antisymmetric matrix B ∈ ℂn×n .

(1.8)

There are infinitely many such antisymmetric matrices B ∈ ℂn×n , and a moment’s reflection shows that they give rise to infinitely many conormal derivatives of the sort just described above, for A := In×n + B. Ultimately, this discussion shows that the formulation of the Lp -Neumann problem for the Laplacian in a half-space in ℝn uses, in addition to an aperture parameter κ > 0, the following ingredients:1 an integrability exponent p ∈ (1, ∞), a half-space H + := {x ∈ ℝn : ⟨x, h⟩ > 0} associated with some vector h ∈ Sn−1 , and some complex n × n matrix A ∈ ℂn×n whose entries satisfy the condition specified in the last line of (1.7). Given all these, the boundary value problem we wish to study reads u ∈ C ∞ (H + ), { { { { { {Δu = 0 in H + , NBVP(A, H + , p) { H + {𝒩 (∇u) ∈ Lp (H, ℋn−1 ), { κ { { { A p n−1 𝜕 { νH + u = f ∈ L (H, ℋ ).

(1.9)

1 It is well known that, even though the specific choice of the aperture parameter κ affects the pointwise values of the corresponding nontangential maximal operator of a given function, its membership to Lp is independent of said parameter (see, e. g., [9, p. 62]). This is the reason for which we shall not include the explicit dependence on κ in the symbol used in connection with the Neumann problem (1.9).

222 | D. Mitrea et al. The goal of this paper is to identify all Neumann problems NBVP(A, H + , p) which are well posed. As is apparent from the theorem below, one remarkable aspect is that the well-posedness of the Neumann problem depends on how A and H + relate to one another, that is, how the algebra underscoring the writing of the Laplacian as in (1.4) interacts with the geometry of the underlying domain H + . Theorem 1.1. Let A ∈ ℂn×n be such that Δ = div A∇ (a condition equivalent to (1.7), or (1.8)). Pick h ∈ Sn−1 and consider the half-space H + and its boundary H, defined as in (1.1)–(1.2), for this unit vector h. Then the Neumann problem NBVP(A, H + , p) from (1.9) is solvable uniquely modulo constants when p = 2 or, equivalently, for each p ∈ (1, ∞), if and only if ⟨h, Aξ ⟩ ≠ −i|ξ |

for all ξ ∈ H \ {0}.

(1.10)

In [4, 7] the Lp -Neumann problem has been solved in much more general classes of domains than half-spaces, in the larger category of homogeneous, second-order, constant (complex) coefficient, weakly elliptic systems, and for conormal derivatives associated with so-called distinguished coefficient tensors. It turns out that, in dimension n ≥ 3, any weakly elliptic scalar operator L = divA∇ possesses exactly one such distinguished coefficient tensor, namely 21 (A + A⊤ ), which is necessarily symmetric. In the current work, we deal with less general domains and (scalar) differential operators, but we allow a larger variety of conormal derivatives. Indeed, from Theorem 1.1 we see that the Neumann problem (1.9) is well posed for matrices A used in the writing of the Laplacian which are not necessarily symmetric, or even close to being symmetric. For example, any matrix A with real entries will do. This being said, we wish to stress that complex coefficients are allowed, as long as (1.10) is satisfied. As a corollary of Theorem 1.1, we see that if A ∈ ℂn×n is as in (1.7) (or, equivalently, (1.8)) then having i ∈ ̸ {⟨ω1 , Aω2 ⟩ : ω1 , ω2 ∈ Sn−1 with ⟨ω1 , ω2 ⟩ = 0} is equivalent to having the Neumann problem NBVP(A, H + , p) formulated in (1.9) solvable uniquely, modulo constants, for any half-space H + in ℝn and any integrability exponent p ∈ (1, ∞). We prove Theorem 1.1 using boundary layer potentials, making use of recent progress in this area described in [4, 7]. As such, our approach is robust, and may be adapted to other types of boundary value problems, such as those involving oblique derivatives, or transmission boundary conditions, and even to more general system than the Laplacian, like the Lamé system of elasticity. The layout of the paper is as follows. Basic notation and preliminary results are collected in Section 2. The proof of Theorem 1.1, presented in Section 5, requires that we first reduce matters to the standard upper half-space ℝn+ via a suitable rotation. As a preamble, in Section 3 we clarify how basic notions, like nontangential boundary

The Neumann problem for the Laplacian in half-spaces | 223

traces and nontangential maximal operators (which are employed in the formulation of (1.9)) transform under such rotations. A key step, carried out in Section 4, is the equivalent reduction of the Lp -Neumann problem to inverting in Lp a certain singular integral operator, which is a finite linear combination of classical Riesz transforms and the identity operator. Of course, the property of being invertible is algebraically tied up with the coefficients involved in said linear combination which, in turn, are traced back to the coefficient matrix A. This is how condition (1.10) comes into existence. Relying on the results established in Section 4, the proof of Theorem 1.1 is completed in Section 5. This is done first for the case of the upper half-space and then in general, by reducing matter to the latter case via a suitable rotation. In Section 6 we discuss a concrete example in which condition (1.10) is violated, and indicate how this gives rise to an Lp -Neumann problem which actually fails to be Fredholm solvable for each p ∈ (1, ∞). Finally, in Section 7 we present a duality result at the level of Neumann problems formulated in an arbitrary half-space and for a given pair of Hölder conjugate integrability exponents.

2 Preliminary results Throughout, fix n ∈ ℕ with n ≥ 2 (serving as the dimension of the ambient Euclidean space). The area of unit sphere Sn−1 := {x ∈ ℝn : |x| = 1} is denoted by ωn−1 . We agree to identify the boundary of the upper half-space ℝn+ := {x = (x ′ , xn ) ∈ ℝn = ℝn−1 × ℝ : xn > 0} with the horizontal hyperplane ℝn−1 via (x′ , 0) ≡ x ′ for any x′ ∈ ℝn−1 . The origin in ℝn−1 is denoted by 0′ , and we let 󵄨 󵄨 Bn−1 (x ′ , r) := {y ′ ∈ ℝn−1 : 󵄨󵄨󵄨y ′ − x ′ 󵄨󵄨󵄨 < r} stand for the (n−1)-dimensional ball of radius r ∈ (0, ∞) centered at x ′ ∈ ℝn−1 . Having fixed some background parameter κ > 0, at each point x ′ ∈ 𝜕ℝn+ we define the conical nontangential approach region with vertex at x′ as 󵄨 󵄨 Γκ (x′ ) := {y = (y′ , t) ∈ ℝn+ : 󵄨󵄨󵄨x ′ − y′ 󵄨󵄨󵄨 < κ t}.

(2.1)

Whenever meaningful, the nontangential pointwise trace of a continuous (possibly vector-valued) function u defined in ℝn+ is given by 󵄨κ−n.t. (u󵄨󵄨󵄨𝜕ℝn )(x′ ) := +

lim

Γκ (x′ )∋y→(x′ ,0)

u(y) for all x ′ ∈ ℝn−1 ≡ 𝜕ℝn+ ,

224 | D. Mitrea et al. and its nontangential maximal function is defined as 󵄨 󵄨 (𝒩κ u)(x′ ) := sup 󵄨󵄨󵄨u(y)󵄨󵄨󵄨 ′ y∈Γκ (x )

for all x ′ ∈ ℝn−1 ≡ 𝜕ℝn+ .

̂ of any We agree to adopt the following normalization of the Fourier transform ϕ n given complex-valued Schwartz function ϕ in ℝ : ̂ ) := ∫ e−i⟨x,ξ ⟩ ϕ(x) dx ϕ(ξ

for all ξ ∈ ℝn .

(2.2)

ℝn

The action of the Fourier transform then extends to tempered distributions via duality, in the usual fashion (cf., e. g., [6]). In particular, if for each j ∈ {1, . . . , n} we define Kj (x) :=

xj

1

ωn−1 |x|n

for all x ∈ ℝn \ {0},

then the tempered distribution canonically induced by Kj (via integration against Schwartz function in ℝn ) turns out to be of function type and is given by (see [6]) ̂j (ξ ) = −i K

ξj

|ξ |2

for all ξ ∈ ℝn \ {0}.

(2.3)

The theorem below is a consequence of the classical Calderón–Zygmund theory of singular integral operators (a result of this flavor in a much more general geometric setting may be found in [7]). Theorem 2.1. There exists a positive integer N = N(n) with the following significance. Consider a complex-valued function K ∈ C N (ℝn \ {0}) with K(−x) = −K(x) K(λ x) = λ

−(n−1)

K(x)

and n

for all λ > 0 and x ∈ ℝ \ {0},

and for each f ∈ L1 (ℝn−1 , 1+|xdx′ |n−1 ) define the integral operator ′

for all x ∈ ℝn+ ,

(𝒯 f )(x) := ∫ K(x − (y′ , 0))f (y′ ) dy′ ℝn−1

along with the maximal operator 󵄨 󵄨 T∗ f (x′ ) := sup󵄨󵄨󵄨Tε f (x′ )󵄨󵄨󵄨 for all x′ ∈ ℝn−1 ε>0

Tε f (x′ ) :=

∫ y′ ∈ℝn−1

|x′ −y′ |>ε

K(x′ − y′ , 0)f (y′ ) dy′

where

for all x′ ∈ ℝn−1 .

The Neumann problem for the Laplacian in half-spaces | 225

Then for each p ∈ (1, ∞) there exists a constant C ∈ (0, ∞) depending only on n and p such that ‖T∗ f ‖Lp (ℝn−1 ) ≤ C‖K|Sn−1 ‖C N ‖f ‖Lp (ℝn−1 ,ℒn−1 ) for every function f ∈ Lp (ℝn−1 , ℒn−1 ).

(2.4)

Furthermore, for each function f ∈ L1 (ℝn−1 ,

dx ′ ), 1 + |x ′ |n−1

the limit Tf (x′ ) := lim+ Tε f (x′ ) ε→0

(2.5)

exists for ℒn−1 -a. e. point x ′ ∈ ℝn−1 and the induced singular integral operator (of principal-value type) T : Lp (ℝn−1 , ℒn−1 ) 󳨀→ Lp (ℝn−1 , ℒn−1 ) with p ∈ (1, ∞),

(2.6)

is well defined, linear, and bounded. Moreover, as a consequence of the boundedness of the maximal operator (2.4), the definition in (2.5), and Lebesgue’s dominated convergence theorem, it follows that if p, p′ ∈ (1, ∞) are such that 1/p + 1/p′ = 1, then the (real) transpose of T in (2.6) is the operator −T : Lp (ℝn−1 , ℒn−1 ) 󳨀→ Lp (ℝn−1 , ℒn−1 ). ′

′

(2.7)

In addition, with ℳ denoting the Hardy–Littlewood maximal function in ℝn−1 , the following Cotlar inequality (𝒩κ (𝒯 f ))(x′ ) ≤ C(ℳ(T∗ f ))(x′ ) + C(ℳf )(x′ ) for all x′ ∈ ℝn−1 holds for each aperture parameter κ ∈ (0, ∞) and each given function f ∈ L1 (ℝn−1 ,

dx ′ ). 1 + |x ′ |n−1

In particular, (2.4) together with the boundedness of ℳ implies that for each integrability exponent p ∈ (1, ∞) there exists a finite constant C = C(K, n, p) > 0 such that 󵄩󵄩 󵄩 󵄩󵄩𝒩κ (𝒯 f )󵄩󵄩󵄩Lp (ℝn−1 ,ℒn−1 ) ≤ C‖K|Sn−1 ‖C N ‖f ‖Lp (ℝn−1 ,ℒn−1 )

226 | D. Mitrea et al. holds for each function f ∈ Lp (ℝn−1 , ℒn−1 ). Finally, with the “hat” denoting the Fourier transform in ℝn (cf. (2.2)) and with en := (0, . . . , 0, 1) ∈ Sn−1 , the jump-formula 󵄨κ−n.t. ((𝒯 f )󵄨󵄨󵄨𝜕ℝn )(x′ ) := +

(𝒯 f )(y) =

lim

Γκ (x′ )∋y→(x′ ,0)

1 ̂ K(−en )f (x′ ) + Tf (x′ ) 2i

(2.8)

is valid at ℒn−1 -a. e. point x ′ ∈ ℝn−1 , whenever f ∈ L1 (ℝn−1 ,

dx ′ ). 1 + |x ′ |n−1

Prominent examples of singular integral operators (of principal-value type) which fall under the scope of the above theorem are the Riesz transforms. Specifically, for each j ∈ {1, . . . , n − 1} introduce the operator Rj acting on functions f ∈ L1 (ℝn−1 ,

dx ′ ), 1 + |x ′ |n−1

at ℒn−1 -a. e. x′ ∈ ℝn−1 , by (Rj f )(x ′ ) := lim+ ε→0

2

ωn−1

xj − yj

∫ n−1

y ∈ℝ |x′ −y′ |>ε ′

|x ′ − y′ |n

f (y′ ) dy′ .

(2.9)

From (2.6) we then see that for every j ∈ {1, . . . , n − 1}, Rj : Lp (ℝn−1 , ℒn−1 ) → Lp (ℝn−1 , ℒn−1 )

is linear and bounded whenever p ∈ (1, ∞).

(2.10)

Recall next the standard fundamental solution for the Laplacian in ℝn , i. e., the function given at each x ∈ ℝn \ {0} by 1

1

E(x) := { ω1n−1 (2−n) |x| ln |x| 2π

n−2

if n ≥ 3, if n = 2.

(2.11)

The Neumann problem for the Laplacian in half-spaces | 227

In particular, in all dimensions we have (∇E)(x) =

1 x ωn−1 |x|n

for all x ∈ ℝn \ {0}.

(2.12)

Use the fundamental solution (2.11) to define the following modified version of the boundary-to-domain single layer operator in the upper half-space: Smod f (x) := ∫ {E(x − (y , 0)) − E∗ (−(y , 0))}f (y ) dy ′

ℝn−1

where

1

f ∈ L (ℝ

n−1

′

dx′ , ) 1 + |x′ |n−1

′

′

for all x ∈ ℝn+ ,

(2.13)

and E∗ := E ⋅ 1ℝn \B(0,1) ,

(here and elsewhere, 1E denotes the characteristic function of the set E). Then for each f ∈ L1 (ℝn−1 ,

dx ′ ) 1 + |x ′ |n−1

the function Smod f is well defined, Smod f ∈ C

∞

(ℝn+ ),

(2.14)

and for each multiindex α ∈ ℕn0 with |α| ≥ 1 one has 𝜕α (Smod f )(x) = ∫ (𝜕α E)(x − (y′ , 0))f (y) dy′ ℝn−1

for all x ∈ ℝn+ .

(2.15)

dx′ ), 1 + |x′ |n−1

(2.16)

In particular, (2.15) implies that Δ(Smod f ) = 0

in ℝn+ for each f ∈ L1 (ℝn−1 ,

while (2.15) and Theorem 2.1 guarantee that for each aperture parameter κ ∈ (0, ∞) and each p ∈ (1, ∞) there exists a constant C ∈ (0, ∞) such that 󵄩󵄩 󵄩 󵄩󵄩𝒩κ (∇Smod f )󵄩󵄩󵄩Lp (ℝn−1 ,ℒn−1 ) ≤ C‖f ‖Lp (ℝn−1 ,ℒn−1 ) ,

(2.17)

for each function f ∈ Lp (ℝn−1 , ℒn−1 ). From (2.15) and Theorem 2.1, we also see that for each index j ∈ {1, . . . , n}, each aperture parameter κ ∈ (0, ∞), and each function f ∈ L1 (ℝn−1 ,

dx ′ ), 1 + |x ′ |n−1

228 | D. Mitrea et al. the nontangential boundary trace 󵄨κ−n.t. 𝜕j (Smod f )󵄨󵄨󵄨𝜕ℝn +

exists at ℒn−1 -a. e. point on ℝn−1 ≡ 𝜕ℝn+ .

In fact, from (2.15), (2.12), (2.8), (2.3), and (2.9), we see that for each function f ∈ L1 (ℝn−1 ,

dx ′ ), 1 + |x ′ |n−1

we have 1

󵄨κ−n.t. 𝜕j (Smod f )󵄨󵄨󵄨𝜕ℝn = { 21 +

2

Rj f f

if j ∈ {1, . . . , n − 1}, if j = n,

(2.18)

at ℒn−1 -a. e. point on ℝn−1 ≡ 𝜕ℝn+ . Moving on, we introduce the Lp -based homogeneous Sobolev space of order one in the context of the entire (n−1)-dimensional Euclidean space (regarded as the boundary of the upper half-space in ℝn ). In this setting, a natural definition is to consider, for each p ∈ (1, ∞), L̇ p1 (ℝn−1 , ℒn−1 ) := {f ∈ L1loc (ℝn−1 , ℒn−1 ) : 𝜕j f ∈ Lp (ℝn−1 , ℒn−1 )

(2.19)

for each j ∈ {1, . . . , n − 1}}.

We shall equip this space with the seminorm n−1

L̇ p1 (ℝn−1 , ℒn−1 ) ∋ f 󳨃→ ‖f ‖L̇ p (ℝn−1 ,ℒn−1 ) := ∑ ‖𝜕j f ‖Lp (ℝn−1 ,ℒn−1 ) . 1

j=1

(2.20)

In this regard, we wish to remark that any function in L̇ p1 (ℝn−1 , ℒn−1 ) automatically enjoys a global integrability property, stemming from the inclusion L̇ p1 (ℝn−1 , ℒn−1 ) ⊆ L1 (ℝn−1 ,

dx′ ). 1 + |x′ |n

(2.21)

To justify this, we recall a result from [5] elaborating on the manner in which global integrability properties of a given function are related to the behavior at infinity of its mean oscillation function. In what follows, a barred integral indicates integral average. Lemma 2.2. Fix ε > 0 arbitrary. Then there exists a constant Cn,ε ∈ (0, ∞) such that for each function f ∈ L1loc (ℝn−1 , ℒn−1 )

The Neumann problem for the Laplacian in half-spaces | 229

and each radius r > 0 there holds |f (y′ ) − fBn−1 (0′ ,r) |

∫ ℝn−1

dy′

[r + |y′ |]n−1+ε ∞

Cn,ε ≤ ε ∫( r 1

󵄨󵄨 ′ 󵄨 ′ dλ 󵄨󵄨f (y ) − fBn−1 (0′ ,λr) 󵄨󵄨󵄨 dy ) 1+ε , λ

− ∫

(2.22)

Bn−1 (0′ ,λr)

where fBn−1 (0′ ,R) :=

f dℒn−1

− ∫

for each R ∈ (0, ∞).

Bn−1 (0′ ,R)

Turning to the proof of (2.21) in earnest, note that, by design, L̇ p1 (ℝn−1 , ℒn−1 ) is con1,1 tained in the (local) Sobolev space Wloc (ℝn−1 ). As such, for any function f ∈ L̇ p1 (ℝn−1 , ℒn−1 ) Poincaré’s inequality (cf., e. g., [2, Theorem 2, p. 141]) implies that for any λ ∈ (0, ∞) we have |f − fBn−1 (0′ ,λ) | dℒn−1

− ∫ Bn−1 (0′ ,λ)

1/p

≤ Cn λ(

− ∫

󵄨󵄨 ′ 󵄨󵄨p n−1 󵄨󵄨∇ f 󵄨󵄨 dℒ )

Bn−1 (0′ ,λ)

≤ Cn λ

1−(n−1)/p 󵄩 󵄩 ′

󵄩 󵄩󵄩∇ f 󵄩󵄩󵄩[Lp (ℝn−1 ,ℒn−1 )]n−1 ,

(2.23)

where ∇′ denotes the gradient in ℝn−1 . In concert with (2.23), (2.22) gives ∫ ℝn−1

|f (x ′ ) − fBn−1 (0′ ,1) | 1 + |x ′ |n

∞

dλ 󵄩 󵄩 dx ≤ Cn 󵄩󵄩󵄩∇′ f 󵄩󵄩󵄩[Lp (ℝn−1 ,ℒn−1 )]n−1 ∫ λ1−(n−1)/p 2 λ ′

1

󵄩 󵄩 ≤ Cn,p 󵄩󵄩󵄩∇′ f 󵄩󵄩󵄩[Lp (ℝn−1 ,ℒn−1 )]n−1 ,

(2.24)

for some constant Cn,p ∈ (0, ∞). The inclusion claimed in (2.21) now readily follows from (2.24). In light of (2.21), the definition given in (2.19) self-improves to L̇ p1 (ℝn−1 , ℒn−1 ) = {f ∈ L1 (ℝn−1 ,

dx′ ) : 𝜕j f ∈ Lp (ℝn−1 , ℒn−1 ) 1 + |x′ |n for each j ∈ {1, . . . , n − 1}}.

230 | D. Mitrea et al. For the purposes we have in mind, this space is natural since for any aperture parameter κ ∈ (0, ∞) and any integrability exponent p ∈ (1, ∞) we have: if u ∈ C 1 (ℝn+ ) and 𝒩κ (∇u) ∈ Lp (ℝn−1 , ℒn−1 ) then u|κ−n.t. exists ℒn−1 -a. e. on ℝn−1 ≡ 𝜕ℝn+ , { 𝜕ℝn+ { { κ−n.t. u|𝜕ℝn belongs to L̇ p1 (ℝn−1 , ℒn−1 ), and { + { { κ−n.t. {‖u|𝜕ℝn ‖L̇ p (ℝn−1 ,ℒn−1 ) ≤ C‖𝒩κ (∇u)‖Lp (ℝn−1 ,ℒn−1 ) ,

(2.25)

1

+

for some dimensional constant C ∈ (0, ∞). In analogy with (2.13), define the following modified version of the boundary-toboundary single layer operator at ℒn−1 -a. e. point x ′ ∈ ℝn−1 as Smod f (x′ ) := ∫ {E(x − (y′ , 0)) − E∗ (−(y′ , 0))}f (y′ ) dy′ ℝn−1 1

n−1

for each f ∈ L (ℝ

dx′ ), , 1 + |x′ |n−1

(2.26) where E∗ := E ⋅ 1ℝn \B(0,1) .

Then the operator defined in (2.26) is meaningful, via an absolutely convergent integral, at ℒn−1 -a. e. point in ℝn−1 , and for each p ∈ (1, ∞) the mapping induced in the context Smod : Lp (ℝn−1 , ℒn−1 ) → L̇ p1 (ℝn−1 , ℒn−1 ) is well defined, linear, and bounded, when the target space is endowed with the seminorm introduced in (2.20). As a consequence, if L̇ p1 (ℝn−1 , ℒn−1 )/ ∼ denotes the quotient space of classes [ ⋅ ] of equivalence modulo constants of functions in L̇ p1 (ℝn−1 , ℒn−1 ), equipped with the seminorm n−1

󵄩󵄩 󵄩󵄩 p 󵄩󵄩[f ]󵄩󵄩L̇ (ℝn−1 ,ℒn−1 )/∼ := ∑ ‖𝜕j f ‖Lp (ℝn−1 ,ℒn−1 ) 1

j=1

for each f ∈ L̇ p1 (ℝn−1 , ℒn−1 ), then the operator [Smod ] : Lp (ℝn−1 , ℒn−1 ) → L̇ p1 (ℝn−1 , ℒn−1 )/ ∼ [Smod ]f := [Smod f ] ∈

defined as

L̇ p1 (ℝn−1 , ℒn−1 )/

∼,

∀ f ∈ Lp (ℝn−1 , ℒn−1 )

The Neumann problem for the Laplacian in half-spaces | 231

is well defined, linear, and bounded. In fact, it has been shown in [4, 7] that [Smod ] : Lp (ℝn−1 , ℒn−1 ) → L̇ p1 (ℝn−1 , ℒn−1 )/ ∼

is an isomorphism for each p ∈ (1, ∞).

(2.27)

Also, with the modified boundary-to-domain single layer operator Smod as in (2.13), for each aperture parameter κ > 0 and each f ∈ L1 (ℝn−1 ,

dx′ ), 1 + |x′ |n−1

one has 󵄨κ−n.t. ((Smod f )󵄨󵄨󵄨𝜕ℝn )(x′ ) = (Smod f )(x ′ ) at ℒ

n−1

+

-a. e. point x′ ∈ ℝn−1 ≡ 𝜕ℝn+ .

(2.28)

See [7] for a more in-depth discussion on this topic. For each boundary datum f ∈ L̇ p1 (ℝn−1 , ℒn−1 ) with p ∈ (1, ∞), if [f ] ∈ L̇ p1 (ℝn−1 , ℒn−1 )/ ∼ denotes the equivalence class (modulo constants) of f , and if g ∈ Lp (ℝn−1 , ℒn−1 )

is selected so that [Smod ]g = [f ],

then from (2.14), (2.16), (2.17), and (2.28) we see that there exists a constant c ∈ ℂ with the property that the function u := Smod g + c

in ℝn+

(2.29)

is a solution of the homogeneous regularity problem for the Laplacian in ℝn+ with boundary datum f , i. e., u solves u ∈ C ∞ (ℝn+ ), { { { { { {Δu = 0 in ℝn+ , { { 𝒩κ (∇u) ∈ Lp (ℝn−1 , ℒn−1 ), { { { { κ−n.t. n−1 n−1 {u|𝜕ℝn+ = f at ℒ -a. e. point in ℝ .

(2.30)

Moreover, it has been shown in [4, 7] that the solution of the homogeneous regularity problem (2.30) is unique. When used in concert with (2.25) and (2.29), we obtain the following basic integral representation result:

232 | D. Mitrea et al. Theorem 2.3. Pick an integrability exponent p ∈ (1, ∞) and fix an aperture parameter κ > 0. Then any function u satisfying u ∈ C ∞ (ℝn+ ),

in ℝn+ ,

Δu = 0

p

n−1

𝒩κ (∇u) ∈ L (ℝ

, ℒn−1 ),

(2.31)

may be written uniquely in the form u = Smod g + c

(2.32)

for some g ∈ Lp (ℝn−1 , ℒn−1 ) and some constant c ∈ ℂ.

3 Rotating half-spaces Unraveling notation shows that (1.3) is in agreement with (2.1) in the case when H + is the upper half-space ℝn+ . Another reason for which (1.3) is natural is as follows. Let R : ℝn → ℝn be a rotation satisfying R(en ) = h; we then also have R(ℝn+ ) = H +

and R(𝜕ℝn+ ) = H.

Having fixed an aperture parameter κ > 0, we claim that ℝn

R(Γκ + (x)) = ΓH κ (Rx) +

for each x ∈ 𝜕ℝn+ ,

(3.1)

where the superscripts for the nontangential cones indicate the domain with respect to which the cones in question are considered. To prove (3.1), select some arbitrary point x = (x′ , 0) ∈ 𝜕ℝn+ . Then ℝn

y = (y′ , t) ∈ Γκ + (x)

󵄨 󵄨 if and only if 󵄨󵄨󵄨x ′ − y′ 󵄨󵄨󵄨 < κt

which, in turn, is equivalent to 󵄨󵄨 ′ ′ 󵄨2 2 2 2 󵄨󵄨x − y 󵄨󵄨󵄨 + t < (κ + 1)t , hence also to the condition 2

|x − y|2 < (κ2 + 1)[dist (y, 𝜕ℝn+ )] . This is further equivalent to |x − y| < √κ 2 + 1 dist (y, 𝜕ℝn+ ),

The Neumann problem for the Laplacian in half-spaces | 233

thus 󵄨 󵄨 |Rx − Ry| = 󵄨󵄨󵄨R(x − y)󵄨󵄨󵄨 = |x − y|

< √κ2 + 1 dist (y, 𝜕ℝn+ )

= √κ2 + 1 inf{|y − z| : z ∈ 𝜕ℝn+ } = √κ2 + 1 inf{|Ry − Rz| : Rz ∈ H} = √κ2 + 1 dist (Ry, 𝜕H + ).

(3.2)

Note that (3.2) is equivalent to having Ry ∈ ΓH κ (Rx). This shows that (3.1) holds. In view of (3.1), for each continuous function u : H + → ℝ and each point x ∈ 𝜕ℝn+ we may write +

󵄨κ−n.t. (u󵄨󵄨󵄨𝜕H + )(Rx) = = =

lim

ΓHκ (Rx)∋y→Rx +

u(y)

lim

R−1 (ΓHκ (Rx))∋z→x +

ℝn

lim

Γκ + (x)∋z→x

u(Rz)

(u ∘ R)(z)

󵄨κ−n.t. = ((u ∘ R)󵄨󵄨󵄨𝜕ℝn )(x),

(3.3)

+

and (again using superscripts to indicate the domain with respect to which the nontangential cones and the nontangential maximal operators are considered) (𝒩κH u)(Rx) = +

󵄨 󵄨 sup 󵄨󵄨󵄨u(y)󵄨󵄨󵄨 = +

y∈ΓHκ

(Rx)

sup

ℝn y∈R(Γκ + (x))

󵄨󵄨 󵄨 󵄨󵄨u(y)󵄨󵄨󵄨

ℝn 󵄨 󵄨 = sup 󵄨󵄨󵄨u(Rz)󵄨󵄨󵄨 = (𝒩κ + (u ∘ R))(x). n

(3.4)

z∈Γκ + (x) ℝ

4 Inverting linear combinations of Riesz transforms In the proof of Theorem 1.1, we shall first pay attention to the special case when h := en = (0, . . . , 0, 1) ∈ Sn−1 , a scenario in which H + = ℝn+ is the upper half-space, and H = 𝜕ℝn+ ≡ ℝn−1 . This offers the most natural context in which we can invert certain singular integral operators which are finite linear combinations of Riesz transforms and the identity operator. To explain the genesis of this type of singular integral operators, fix a complex n × n matrix A = (ajk )1≤j,k≤n ∈ ℂn×n

234 | D. Mitrea et al. whose entries satisfy the condition formulated in the last line of (1.7). For each function u as is (2.31), we then use (1.6) to write 󵄨κ−n.t. 𝜕νA u = ⟨−en , A(∇u)󵄨󵄨󵄨𝜕ℝn ⟩ +

n−1

󵄨κ−n.t. 󵄨κ−n.t. = − ∑ ank (𝜕k u)󵄨󵄨󵄨𝜕ℝn − (𝜕n u)󵄨󵄨󵄨𝜕ℝn . + +

(4.1)

k=1

Since from Theorem 2.3 we know that for each u as in (2.31) there exist some function g ∈ Lp (ℝn−1 , ℒn−1 ) and some constant c ∈ ℂ such that u is of the form (2.32), we may further use (4.1) and (2.18) to write in this case: n−1

󵄨κ−n.t. 󵄨κ−n.t. 𝜕νA u = − ∑ ank (𝜕k Smod g)󵄨󵄨󵄨𝜕ℝn − (𝜕n Smod g)󵄨󵄨󵄨𝜕ℝn +

k=1

1 n−1 1 = − ∑ ank Rk g − g. 2 k=1 2

+

(4.2)

The computation in (4.2) underscores the fact that the existence of a solution for the Lp -Neumann problem for the Laplacian in the upper half-space with boundary datum f ∈ Lp (ℝn−1 , ℒn−1 ), i. e., u ∈ C ∞ (ℝn+ ), { { { { { {Δu = 0 in ℝn+ , { { 𝒩κ (∇u) ∈ Lp (ℝn−1 , ℒn−1 ), { { { { A n−1 n−1 {𝜕ν u = f at ℒ -a. e. point on ℝ ,

(4.3)

becomes equivalent to determining g ∈ Lp (ℝn−1 , ℒn−1 ) such that n−1

(I + ∑ ank Rk )g = −2f k=1

in Lp (ℝn−1 , ℒn−1 ),

where I denotes the identity operator. This brings into focus the connection between the invertibility of the operator n−1

I + ∑ ank Rk k=1

on Lp (ℝn−1 , ℒn−1 )

and the solvability of the Lp -Neumann problem (4.3). This is made precise in the proposition below.

The Neumann problem for the Laplacian in half-spaces | 235

Proposition 4.1. Fix an integrability exponent p ∈ (1, ∞) along with some aperture parameter κ ∈ (0, ∞). Suppose the matrix A = (ajk )1≤j,k≤n ∈ ℂn×n has entries satisfying the condition stipulated in the second line of (1.7). Then for each boundary datum f ∈ Lp (ℝn−1 , ℒn−1 ) the Lp -Neumann problem for the Laplacian in the upper half-space formulated in (4.3) has a solution if and only if the operator TA : Lp (ℝn−1 , ℒn−1 ) → Lp (ℝn−1 , ℒn−1 ), n−1

TA := I + ∑ ank Rk , k=1

(4.4)

is surjective. In addition, a solution for (4.3) is unique modulo constants if and only if the operator (4.4) is injective. As a consequence, the Lp -Neumann problem for the Laplacian in the upper halfspace formulated in (4.3) is solvable unique modulo constants if and only if the operator TA is an isomorphism in the context of (4.4). Proof. Fix some arbitrary function f ∈ Lp (ℝn−1 , ℒn−1 ). Then any solution u of (4.3) for this boundary datum f satisfies the conditions in (2.31), thus is of the form (2.32) for some g ∈ Lp (ℝn−1 , ℒn−1 )

and c ∈ ℂ (cf. Theorem 2.3).

As such, on account of (4.4) and (4.2) we then conclude that TA g = 𝜕νA u = −2f ∈ Lp (ℝn−1 , ℒn−1 ). This shows that TA is surjective. Conversely, if TA is surjective then given any f ∈ Lp (ℝn−1 , ℒn−1 ) one can find g ∈ Lp (ℝn−1 , ℒn−1 )

such that TA g = −2f .

236 | D. Mitrea et al. If we now define u := Smod g in ℝn+ , then from (2.14), (2.16), (2.17), (4.2), and (4.4) we see that u solves the Lp -Neumann problem formulated in (4.3) for the current boundary datum f . Altogether, this argument proves that (4.3) has a solution if and only if the operator (4.4) is surjective. In addition, this reasoning also shows that if TA is also injective, then the only solution for (4.3) corresponding to f = 0 is a constant. Suppose now that we have uniqueness for the boundary value problem (4.3), modulo constants. Let g ∈ Lp (ℝn−1 , ℒn−1 ) obey TA g = 0. In concert with (2.14), (2.16), (2.17), and (4.2), this shows that the function defined as u := Smod g in ℝn+ is a null solution for (4.3), hence u = c, some complex constant. Taking the trace of u to the boundary and availing ourselves of (2.28) we obtain 󵄨κ−n.t. c = u󵄨󵄨󵄨𝜕ℝn = Smod g +

at ℒn−1 -a. e. point on ℝn−1 ≡ 𝜕ℝn+ .

In turn, the latter implies [Smod g] = [0]

in Lp (ℝn−1 , ℒn−1 )/ ∼ .

Since [Smod ] is an isomorphism from Lp (ℝn−1 , ℒn−1 ) into L̇ p1 (ℝn−1 , ℒn−1 )/ ∼ (cf. (2.27)), we conclude that g=0

in Lp (ℝn−1 , ℒn−1 ).

This shows that TA must be injective. In light of Proposition 4.1, it is natural to take a closer look at operators of the form n−1

I + ∑ λk Rk k=1

on Lp (ℝn−1 , ℒn−1 ) where λk ∈ ℂ, k ∈ {1, . . . , n − 1},

to determine the conditions on the coefficients {λk }1≤k≤n−1 ensuring the invertibility of said operators on Lp (ℝn−1 , ℒn−1 ). This issue is addressed in Proposition 4.3 below. As a preamble, we establish the following useful result. Lemma 4.2. Let b : ℝn−1 → ℂ be some ℒn−1 measurable function, and consider the operator Mb of pointwise multiplication by b. Then Mb : L2 (ℝn−1 , ℒn−1 ) → L2 (ℝn−1 , ℒn−1 ) is bounded if and only if

b ∈ L∞ (ℝn−1 , ℒn−1 ).

(4.5)

In addition, Mb : L2 (ℝn−1 , ℒn−1 ) → L2 (ℝn−1 , ℒn−1 ) is an isomorphism

(4.6)

The Neumann problem for the Laplacian in half-spaces | 237

if and only if b ∈ L∞ (ℝn−1 , ℒn−1 ), { { { b ≠ 0 at ℒn−1 -a. e. point in ℝn−1 , { { { −1 ∞ n−1 n−1 {and b ∈ L (ℝ , ℒ ).

(4.7)

Proof. If b ∈ L∞ (ℝn−1 , ℒn−1 ), then Mb is bounded on L2 (ℝn−1 , ℒn−1 ) with operator norm ≤ ‖b‖L∞ (ℝn−1 ,ℒn−1 ) . Conversely, assume Mb : L2 (ℝn−1 , ℒn−1 ) → L2 (ℝn−1 , ℒn−1 ) is bounded. With C ∈ (0, ∞) denoting its operator norm, for every ball B in ℝn−1 we have 󵄩 󵄩2 ∫ |b|2 dℒn−1 = 󵄩󵄩󵄩Mb (1B )󵄩󵄩󵄩L2 (ℝn−1 ,ℒn−1 ) ≤ C 2 ‖1B ‖2L2 (ℝn−1 ,ℒn−1 ) B

= C 2 ℒn−1 (B).

This shows that |b|2 ∈ L1loc (ℝn−1 , ℒn−1 ), which, after invoking Lebesgue’s differentiation theorem, further implies that |b|2 ≤ C 2 at ℒn−1 -a. e. point in ℝn−1 . Consequently, b ∈ L∞ (ℝn−1 , ℒn−1 ). This proves (4.5). Suppose next that b satisfies (4.7). Then Mb−1 is a well-defined operator which is bounded on L2 (ℝn−1 , ℒn−1 ) and Mb−1 ∘ Mb = Mb ∘ Mb−1 = I

on L2 (ℝn−1 , ℒn−1 ),

hence (4.6) holds. If now we assume that (4.6) is true, by (4.5) we have b ∈ L∞ (ℝn−1 , ℒn−1 ),

238 | D. Mitrea et al. thus the first condition in (4.7) holds. In addition, there exists C ∈ (0, ∞) such that for each f ∈ L2 (ℝn−1 , ℒn−1 ) one may find g ∈ L2 (ℝn−1 , ℒn−1 ) with bg = f and ‖g‖L2 (ℝn−1 ,ℒn−1 ) ≤ C‖f ‖L2 (ℝn−1 ,ℒn−1 ) .

(4.8)

Indeed, for each f ∈ L2 (ℝn−1 , ℒn−1 ), one may take g := Mb−1 f and the statement in (4.8) follows. To proceed, define the set Ω := {x ∈ ℝn−1 : b(x) = 0}. Since the function b is ℒn−1 -measurable, it follows that Ω is ℒn−1 -measurable. In particular, if we introduce Ωj := Ωj ∩ Bn−1 (0′ , j) for each j ∈ ℕ, then Ωj ↗ Ω

as j → ∞ and ℒn−1 (Ωj ) ↗ ℒn−1 (Ω) as j → ∞.

In the case when ℒn−1 (Ω) > 0, it follows that there exists j0 ∈ ℕ such that ℒn−1 (Ωj0 ) > 0. Applying (4.8) with f := 1Ωj , one may find 0

g ∈ L2 (ℝn−1 , ℒn−1 ) such that bg = f = 1Ωj

n−1

0

ℒ

-a. e. in ℝn−1 .

(4.9)

However, when restricted to Ωj0 the equality in (4.9) leads to a contradiction since b = 0 in Ωj0 ⊂ Ω. Consequently, the assumption that ℒn−1 (Ω) > 0 is false and we obtain ℒn−1 (Ω) = 0. This proves the second condition in (4.7). In turn, the latter implies that b−1 is well defined and measurable. In addition, if f ∈ L2 (ℝn−1 , ℒn−1 ) is arbitrary and g is as in (4.8), then ‖Mb−1 f ‖L2 (ℝn−1 ,ℒn−1 ) = ‖g‖L2 (ℝn−1 ,ℒn−1 ) ≤ C‖f ‖L2 (ℝn−1 ,ℒn−1 ) , which ensures that Mb−1 is bounded on L2 (ℝn−1 , ℒn−1 ). Now the third condition in (4.7) follows by applying (4.5). Here is the proposition alluded to earlier, whose proof makes use of Lemma 4.2.

The Neumann problem for the Laplacian in half-spaces | 239

Proposition 4.3. Pick a finite family λ := {λk }1≤k≤n−1 ∈ ℂn−1 and, for each integrability exponent p ∈ (1, ∞), consider the operator Tλ : Lp (ℝn−1 , ℒn−1 ) → Lp (ℝn−1 , ℒn−1 ), n−1

Tλ := I + ∑ λk Rk , k=1

(4.10)

where I is the identity, and Rk is defined in (2.9) for each k ∈ {1, . . . , n − 1}. Then Tλ is invertible for p = 2 or, equivalently, for all p ∈ (1, ∞), if and only if n−1

ξk λk ≠ −i |ξ ′ | k=1 ∑

for all ξ ′ ∈ ℝn−1 \ {0′ }.

(4.11)

Proof. To set the stage, recall that for each k ∈ {1, . . . , n − 1} the operator Rk : L2 (ℝn−1 , ℒn−1 ) → L2 (ℝn−1 , ℒn−1 ) is bounded, and that for every f ∈ L2 (ℝn−1 , ℒn−1 ) we have ̂f (ξ ′ ) = − iξk ̂f (ξ ′ ) for all ξ ′ = (ξ , . . . , ξ ) ∈ ℝn−1 \ {0′ } R 1 n−1 k |ξ ′ | where the “hat” denotes the Fourier transform in ℝn−1 (cf. (2.2)). In concert with (4.10), this implies that for every function f ∈ L2 (ℝn−1 , ℒn−1 ) we have n−1 ̂f (ξ ′ ) = (1 − i ∑ ξk λk )̂f (ξ ′ ) for all ξ ′ ∈ ℝn−1 \ {0′ }. T λ |ξ ′ | k=1

(4.12)

Consequently, Tλ is a multiplier operator on L2 (ℝn−1 , ℒn−1 ) and its corresponding multiplier function m is given by n−1

ξk λk |ξ ′ | k=1

m(ξ ′ ) := 1 − i ∑

for all ξ ′ ∈ ℝn−1 \ {0′ }.

(4.13)

240 | D. Mitrea et al. Let us now assume that Tλ : L2 (ℝn−1 , ℒn−1 ) → L2 (ℝn−1 , ℒn−1 ) is invertible. Thanks to this assumption, (4.12)–(4.13), and the fact that Fourier transform is an isomorphism of L2 (ℝn−1 , ℒn−1 ), we conclude from Lemma 4.2 that for ℒn−1 -a. e. ξ ′ ∈ ℝn−1 \ {0′ },

m(ξ ′ ) ≠ 0

and m−1 ∈ L∞ (ℝn−1 , ℒn−1 ).

As a consequence, there exists C ∈ (0, ∞) such that 󵄨󵄨 ′ −1 󵄨 󵄨󵄨m(ξ ) 󵄨󵄨󵄨 ≤ C

for ℒn−1 -a. e. ξ ′ ∈ ℝn−1 \ {0′ }

or, equivalently, 󵄨 󵄨 C −1 ≤ 󵄨󵄨󵄨m(ξ ′ )󵄨󵄨󵄨

at ℒn−1 -a. e. point ξ ′ ∈ ℝn−1 \ {0′ }.

Since, as is apparent from (4.13), the function m is continuous in ℝn−1 \ {0′ } (and the complement of any null-set for the Lebesgue measure is dense in ℝn−1 ), the latter inequality self-improves to 󵄨 󵄨 C −1 ≤ 󵄨󵄨󵄨m(ξ ′ )󵄨󵄨󵄨 at each point ξ ′ ∈ ℝn−1 \ {0′ }. Obviously, this implies that the condition demanded in (4.11) holds. Next, assume that (4.11) is true, and fix an arbitrary p ∈ (1, ∞). The goal is to show that Tλ : Lp (ℝn−1 , ℒn−1 ) → Lp (ℝn−1 , ℒn−1 ) is an invertible operator. To this end, start by noting that n−1

−1

ξk λk ] |ξ ′ | k=1

m−1 (ξ ′ ) = [1 − i ∑

for all ξ ′ ∈ ℝn−1 \ {0′ }

(4.14)

is a well-defined bounded function. In particular, using ℱ , ℱ −1 as alternative notation for the Fourier transform and its inverse in ℝn−1 , the operator Qf := ℱ −1 (m−1 ℱ f ) for each f ∈ L2 (ℝn−1 , ℒn−1 ),

(4.15)

is linear and bounded from L2 (ℝn−1 , ℒn−1 ) into L2 (ℝn−1 , ℒn−1 ). Returning to (4.14), it is also clear that m−1 ∈ C ∞ (ℝn−1 \ {0′ })

The Neumann problem for the Laplacian in half-spaces | 241

and that m−1 is positive homogeneous of degree 0 in ℝn−1 \{0′ }. Consequently, for each multiindex α ∈ ℕn0 there exists a constant Cα ∈ (0, ∞) such that for all ξ ′ ∈ ℝn−1 \ {0′ }.

󵄨 ′ 󵄨−|α| 󵄨󵄨 α −1 ′ 󵄨󵄨 󵄨󵄨𝜕 m (ξ )󵄨󵄨 ≤ Cα 󵄨󵄨󵄨ξ 󵄨󵄨󵄨 If we now introduce C :=

sup

|α|≤[n/2]+1

Cα ∈ (0, ∞),

then 󵄨󵄨 ′ 󵄨󵄨|α| 󵄨󵄨 α −1 ′ 󵄨󵄨 n 󵄨󵄨ξ 󵄨󵄨 󵄨󵄨𝜕 m (ξ )󵄨󵄨 ≤ C for all α ∈ ℕ0 such that |α| ≤ [n/2] + 1, and for all ξ ′ ∈ ℝn−1 \ {0′ }.

(4.16)

By the Mikhlin–Hörmander multiplier theorem (cf., e. g., [3, Theorem 5.2.7, p. 362]), condition (4.16) implies that the operator Q, originally defined as in (4.15) on the space L2 (ℝn−1 , ℒn−1 ), extends to Q : Lp (ℝn−1 , ℒn−1 ) → Lp (ℝn−1 , ℒn−1 ) linear and bounded.

(4.17)

In addition, (4.15) implies ̂ = m−1 ̂f for all f ∈ L2 (ℝn−1 , ℒn−1 ). Qf

(4.18)

and we claim that QTλ = Tλ Q = I

on L2 (ℝn−1 , ℒn−1 ).

Indeed, for each f ∈ L2 (ℝn−1 , ℒn−1 ), using (4.12), (4.13), (4.18), and (4.14), we may write ̂) Tλ (Qf ) = ℱ −1 (ℱ (Tλ (Qf ))) = ℱ −1 (m Qf = ℱ −1 (m m−1 ̂f ) = f . Similarly, Q(Tλ f ) = f as wanted.

for each f ∈ L2 (ℝn−1 , ℒn−1 ),

(4.19)

242 | D. Mitrea et al. From (2.10) and the definition of the operator Tλ given in (4.10), we see that Tλ : Lp (ℝn−1 , ℒn−1 ) → Lp (ℝn−1 , ℒn−1 ) is bounded.

(4.20)

In concert, the identity in (4.19), the boundedness in (4.20) and (4.17), and the fact that the space L2 (ℝn−1 , ℒn−1 ) ∩ Lp (ℝn−1 , ℒn−1 ) is dense in Lp (ℝn−1 , ℒn−1 ) ultimately imply QTλ = Tλ Q = I

on Lp (ℝn−1 , ℒn−1 ).

This proves that Tλ is invertible on Lp (ℝn−1 , ℒn−1 ) for any given p ∈ (1, ∞), as wanted.

5 Proof of main result This section is devoted to presenting the proof of Theorem 1.1: Proof of Theorem 1.1. Fix a matrix A = (ajk )1≤j,k≤n ∈ ℂn×n satisfying (1.7). We will first prove the theorem in the case of the upper half-space, that is, when H + = ℝn+ ,

h = en ,

and H = ℝn−1 × {0} ≡ ℝn−1 .

In this setting, (1.10) becomes 󵄨 󵄨 ⟨en , A(ξ ′ , 0)⟩ ≠ −i󵄨󵄨󵄨ξ ′ 󵄨󵄨󵄨 for all (ξ ′ , 0) ∈ ℝn \ {0}

(5.1)

or, equivalently, n−1

󵄨 󵄨 ∑ ank ξk ≠ −i󵄨󵄨󵄨ξ ′ 󵄨󵄨󵄨 for all ξ ′ ∈ ℝn−1 \ {0′ },

k=1

(5.2)

while for each p ∈ (1, ∞) the Neumann problem NBVP(A, H + , p) from (1.9) takes the form (4.3). Invoking Proposition 4.3 (with λk := ank for k ∈ {1, . . . , n − 1}), we first obtain that (5.1) holds if and only if the operator n−1

TA := I + ∑ ank Rk k=1

is invertible from Lp (ℝn−1 , ℒn−1 ) into Lp (ℝn−1 , ℒn−1 ),

The Neumann problem for the Laplacian in half-spaces | 243

either for p = 2, or for every p ∈ (1, ∞). The latter, combined with Proposition 4.1, proves that (5.1) is true if and only if the Lp -Neumann problem (4.3) (considered either for every p ∈ (1, ∞), or just for p = 2) has a unique solution modulo constants.

(5.3)

This establishes Theorem 1.1 in the case of the upper half-space. To prove Theorem 1.1 in the general case, let R : ℝn → ℝn be a rotation that satisfies R(en ) = h. Thus, R(ℝn+ ) = H +

and R(𝜕ℝn+ ) = H.

Also, introduce the notation ̃ := R⊤ MR ∈ ℂn×n M

for each M ∈ ℂn×n .

(5.4)

Next we make a series of observations. The first observation is that (1.10) is equivalent to 󵄨 󵄨 ⟨Ren , A(R(ξ ′ , 0))⟩ ≠ −i󵄨󵄨󵄨R(ξ ′ , 0)󵄨󵄨󵄨

(5.5)

for all (ξ ′ , 0) ∈ 𝜕ℝn+ \ {0}.

Employing (5.4), from (5.5) and the fact that R is a rotation, we conclude that the condition in (1.10) is equivalent to ̃ ′ , 0)⟩ ≠ −i󵄨󵄨󵄨ξ ′ 󵄨󵄨󵄨 for all (ξ ′ , 0) ∈ 𝜕ℝn \ {0}. ⟨en , A(ξ + 󵄨 󵄨

(5.6)

Next, suppose that u : H + → ℂ and w : ℝn+ → ℂ are twice differentiable functions related by the identity w(x) = u(Rx)

for each x ∈ ℝn+ .

(5.7)

By the chain rule, we have (∇w)(x) = R⊤ (∇u)(Rx)

and

(Δw)(x) = (Δu)(Rx) for all x ∈ ℝn+ .

(5.8)

Consequently, w ∈ C ∞ (ℝn+ )

Δw = 0

in ℝn+

if and only if u ∈ C ∞ (H + ), if and only if

Δu = 0

in H + ,

(5.9)

244 | D. Mitrea et al. and p

n−1

𝒩κ (∇w) ∈ L (ℝ H+

, ℒn−1 ) if and only if

p

𝒩κ (∇u) ∈ L (H, ℋ

n−1

).

(5.10)

Moreover, 󵄨κ−n.t. (∇w)󵄨󵄨󵄨𝜕ℝn +

exists ℋn−1 -a. e. on 𝜕ℝn+

if and only if 󵄨κ−n.t. (∇u)󵄨󵄨󵄨𝜕H +

exists ℋn−1 -a. e. on H,

and when either (hence both) exists, from (5.8) and (3.3) we conclude that 󵄨κ−n.t. 󵄨κ−n.t. 󵄨κ−n.t. (∇w)󵄨󵄨󵄨𝜕ℝn = [R⊤ (∇u) ∘ R]󵄨󵄨󵄨𝜕ℝn = R⊤ [(∇u)󵄨󵄨󵄨𝜕H + ] ∘ R +

at ℋn−1 -a. e. point on 𝜕ℝn+ .

+

(5.11)

Thus, whenever (5.11) holds, we have 󵄨κ−n.t. (𝜕νA + u) ∘ R = ⟨−h, A(∇u)󵄨󵄨󵄨𝜕H + ∘ R⟩ H

󵄨κ−n.t. = ⟨−en , R⊤ A(∇u)󵄨󵄨󵄨𝜕H + ∘ R⟩ 󵄨κ−n.t. = ⟨−en , (R⊤ AR)R⊤ (∇u)󵄨󵄨󵄨𝜕H + ∘ R⟩ 󵄨󵄨κ−n.t. ̃ = ⟨−en , A(∇w) 󵄨󵄨𝜕ℝn+ ⟩ = 𝜕νA w ̃

at ℋn−1 -a. e. point on 𝜕ℝn+ .

(5.12)

The final observation we make is that if the matrix A is as in (1.7), i. e., if A = In×n + B for some B ∈ ℂn×n ̃ (defined in relation to A as in antisymmetric, then the matrix A (5.4)) has the same form.

(5.13)

Indeed, ̃ = R⊤ (In×n + B)R = In×n + B ̃ A and (using B⊤ = −B) we have ̃ ⊤ = R⊤ B⊤ R = −R⊤ BR = −B. ̃ (B)

(5.14)

The Neumann problem for the Laplacian in half-spaces | 245

From (5.9), (5.10), (5.12), and (5.13), we see that, given 1 < p < ∞, the function u solves the Neumann problem NBVP(A, H + , p), as in (1.9) for some boundary datum f ∈ Lp (H, ℋn−1 ), if and only if the function w (rẽ ℝn , p) formulated to u as in (5.7)) solves the Neumann problem NBVP(A, + ̃ in place of A and corresponding to the boundary lated as in (4.3) with A datum

(5.15)

f ∘ R ∈ Lp (ℝn−1 , ℒn−1 ). Finally, from (5.15), (5.3), and (5.6), all desired conclusions follow. The proof of Theorem 1.1 is therefore complete.

6 An example It is of interest to see directly, in a more transparent fashion, the failure of wellposedness for the Lp -Neumann problem (4.3) in the case when condition (5.1) is violated. In this section we shall present a concrete example in the two-dimensional setting, a scenario in which (5.1) simply becomes (see (5.2)) the demand that the 2 × 2 complex matrix A = (ajk )1≤j,k≤2 satisfies a21 ≠ ±i.

(6.1)

Specifically, in this section we shall consider the Laplacian Δ in ℝ2 ≡ ℂ, written as Δ = ∑2j,k=1 ajk 𝜕j 𝜕k , where the coefficient matrix A = (ajk )1≤j,k≤2 is 1 i

A := (

−i ). 1

(6.2)

Clearly, this violates (6.1), so Theorem 1.1 implies that the corresponding Lp -Neumann problem (4.3) cannot be well-posed for each p ∈ (1, ∞). Our goal here is to explore this issue and prove that, in fact, the Lp -Neumann problem (4.3) in the upper half-plane with A as in (6.2) fails to be well-posed, or even Fredholm solvable, for each p ∈ (1, ∞). To set the stage, fix an arbitrary exponent p ∈ (1, ∞) and pick an aperture parameter κ > 0. We shall first describe the space of admissible boundary data for the Lp -Neumann problem for the Laplacian in the upper half-plane where the prescribed conormal derivative is the one associated with the matrix A from (6.2). In this regard, we make the claim {𝜕νA u : u ∈ C ∞ (ℝ2+ ), Δu = 0 in ℝ2+ , 𝒩κ (∇u) ∈ Lp (ℝ, ℒ1 )} = {f ∈ Lp (ℝ, ℒ1 ) : Hf = −if },

(6.3)

246 | D. Mitrea et al. where H is the classical Hilbert transform on the real line, defined for each function dx ϕ ∈ L1 (ℝ, 1+|x| ) as 1 π

Hϕ(x) := lim+ ε→0

∫

y∈ℝ |x−y|>ε

ϕ(y) dy x−y

for ℒ1 -a. e. x ∈ ℝ.

Since the space on the right-hand side of (6.3) has infinite codimension in Lp (ℝ, ℒ1 ) (given that H 2 = −I on this space), the identification in (6.3) suits our current goals. To prove the left-to-right inclusion in (6.3), pick a complex-valued function u satisfying u ∈ C ∞ (ℝ2+ ),

in ℝ2+ ,

Δu = 0

p

1

𝒩κ (∇u) ∈ L (ℝ, ℒ ).

(6.4)

Thanks to the Fatou-type result established by A. P. Calderón (cf. [1], [8, Theorem 3, p. 201]), these properties guarantee that 󵄨κ−n.t. (∇u)󵄨󵄨󵄨𝜕ℝ2 +

exists at ℒ1 -a. e. point on 𝜕ℝ2+ .

In particular, f := 𝜕νA u is a well-defined function in Lp (ℝ, ℒ1 ). More specifically, bearing in mind (1.6), we see that 2

󵄨κ−n.t. 󵄨κ−n.t. 󵄨κ−n.t. f = 𝜕νA u = − ∑ a2k (𝜕k u)󵄨󵄨󵄨𝜕ℝ2 = −i(𝜕1 u)󵄨󵄨󵄨𝜕ℝ2 − (𝜕2 u)󵄨󵄨󵄨𝜕ℝ2 =

k=1 󵄨κ−n.t. (−i)((𝜕1 u)󵄨󵄨󵄨𝜕ℝ2 +

󵄨κ−n.t. = −2i(𝜕z u)󵄨󵄨󵄨𝜕ℝ2 +

+

+

+

󵄨κ−n.t. − i(𝜕2 u)󵄨󵄨󵄨𝜕ℝ2 ) +

1

at ℒ -a. e. point on 𝜕ℝ2+ ≡ ℝ,

where 1 𝜕z := (𝜕1 − i𝜕2 ) 2 is the complex conjugate of the Cauchy–Riemann operator 1 𝜕z̄ := (𝜕1 + i𝜕2 ). 2 Hence, if we define w := 2𝜕z u

in ℝ2+ ,

(6.5)

The Neumann problem for the Laplacian in half-spaces | 247

upon recalling that Δ = 4𝜕z̄ 𝜕z , the properties in (6.4) imply w ∈ C ∞ (ℝ2+ ),

𝜕z̄ w = 0

in ℝ2+ ,

p

1

𝒩κ w ∈ L (ℝ, ℒ ).

These properties simply amount to stating that w is a holomorphic function belonging to the classical Hardy space ℋp (ℝ2+ ) (consisting of holomorphic functions in the upper half-plane, whose nontangential maximal function is pth power integrable on the real line). In addition, (6.5) tells us that 󵄨κ−n.t. w󵄨󵄨󵄨𝜕ℝ2 = if

at ℒ1 -a. e. point on 𝜕ℝ2+ ≡ ℝ.

+

(6.6)

Together with Cauchy’s reproducing formula for holomorphic functions in the Hardy space associated with the upper half-plane, this gives 󵄨κ−n.t. w = C (w󵄨󵄨󵄨𝜕ℝ2 ) = i C f

in ℝ2+ ,

+

(6.7)

where C is the boundary-to-domain Cauchy integral operator, acting on any function dx ) according to ϕ ∈ L1 (ℝ, 1+|x| C ϕ(z) :=

ϕ(x) 1 dx ∫ 2πi x − z

for each z ∈ ℂ+ .

(6.8)

ℝ

After taking the nontangential trace to the boundary in (6.7) and using (6.6), we arrive at the conclusion that 1 1 if = i( I − H)f 2 2i

at ℒ1 -a. e. point in ℝ.

This ultimately implies that f must satisfy the compatibility condition Hf = −if

at ℒ1 -a. e. point in ℝ,

proving the left-to-right inclusion in (6.3). For the converse inclusion, consider f ∈ Lp (ℝ, ℒ1 ) satisfying Hf = −if

at ℒ1 -a. e. point in ℝ.

Bring Smod , the modified boundary-to-domain harmonic single layer potential operator associated with the Laplacian in the upper half-plane (cf. (2.13)), and define u := −Smod f

in ℝ2+ .

Then, according to (2.14), (2.16), and (2.17), this function belongs to C ∞ (ℝ2+ ), satisfies Δu = 0 in ℝ2+ , and also has p

1

𝒩κ (∇u) ∈ L (ℝ, ℒ ).

248 | D. Mitrea et al. In addition, (6.5), (2.15), (2.11), and (6.8) permit us to compute 󵄨κ−n.t. 󵄨κ−n.t. 𝜕νA u = −2i(𝜕z u)󵄨󵄨󵄨𝜕ℝ2 = 2i(𝜕z Smod f )󵄨󵄨󵄨𝜕ℝ2 +

+

1 1 1 󵄨κ−n.t. 1 = (C f )󵄨󵄨󵄨𝜕ℝ2 = f − Hf = f + f = f . + 2 2i 2 2

(6.9)

Clearly, (6.9) together with the properties of u imply the right-to-left inclusion in (6.3). The equality claimed in (6.3) is therefore established. As regards the space of null-solutions for the Lp -Neumann problem (4.3) in the case when n = 2 and A is as in (6.2), we claim that {u ∈ C ∞ (ℝ2+ ) : Δu = 0 in ℝ2+ , 𝒩κ (∇u) ∈ Lp (ℝ, ℒ1 ), 𝜕νA u = 0} = {w : w holomorphic in

ℝ2+ ,

p

(6.10)

1

with 𝒩κ (∇w) ∈ L (ℝ, ℒ )}.

To see that this is the case, pick an arbitrary function u belonging to the space in the left-hand side of (6.10). Then 𝜕z u is a holomorphic function in ℝ2+ (since 𝜕z̄ 𝜕z = 41 Δ), and satisfies p

1

𝒩κ (𝜕z u) ∈ L (ℝ, ℒ ).

These properties place 𝜕z u in ℋp (ℝ2+ ), the classical Hardy space in the upper half-plane for the Cauchy–Riemann operator. Since from (6.5) and current assumptions we have 󵄨κ−n.t. (𝜕z u)󵄨󵄨󵄨𝜕ℝ2 = 0 +

at ℒ1 -a. e. point on 𝜕ℝ2+ ≡ ℝ,

(6.11)

the property (6.11), along with Cauchy’s reproducing formula, forces 𝜕z u to vanish identically in ℝ2+ . The latter property may be interpreted simply as saying that w := u is holomorphic in ℝ2+ . This puts w (and, ultimately, u) in the space on the right-hand side of (6.10). Conversely, given any holomorphic function w in ℝ2+ satisfying p

1

p

1

𝒩κ (∇w) ∈ L (ℝ, ℒ ),

the function u := w is harmonic in ℝ2+ , has 𝒩κ (∇u) ∈ L (ℝ, ℒ )

and, much as in (6.5), we see that 󵄨κ−n.t. 󵄨κ−n.t. 𝜕νA u = −2i(𝜕z u)󵄨󵄨󵄨𝜕ℝ2 = −2i(𝜕z̄ u)󵄨󵄨󵄨𝜕ℝ2 +

󵄨κ−n.t. = −2i(𝜕z̄ w)󵄨󵄨󵄨𝜕ℝ2 = 0 +

+

1

at ℒ -a. e. point on 𝜕ℝ2+ ≡ ℝ,

since w is holomorphic in ℝ2+ ≡ ℂ+ . The proof of (6.10) is therefore complete. Note that the space on the right-hand side of (6.10) is infinite dimensional since, for example,

The Neumann problem for the Laplacian in half-spaces | 249

for each m ∈ ℕ the function ℂ+ ∋ z 󳨃→ (z̄ − i)−m ∈ ℂ belongs to said space. In view of this, we conclude that the space of null-solutions for the Lp -Neumann problem (4.3) is, in fact, infinite dimensional.

7 A duality result Let us agree to use the notation A# := 2In×n − A,

for each A ∈ ℂn×n ,

(7.1)

where the “bar” denotes complex conjugation. Then for each A ∈ ℂn×n we have #

(A# ) = A and

Δ = div A∇

if and only if

Δ = div A# ∇,

hence, the entries of A satisfy the condition formulated in the last line of (1.7) if and only if the entries of A# do so. A noteworthy consequence of Proposition 4.1 is the following duality result for the Neumann problem formulated in ℝn+ . Proposition 7.1. Assume p, p′ ∈ (1, ∞) are such that 1/p + 1/p′ = 1, and fix an aperture parameter κ ∈ (0, ∞). Also, let A ∈ ℂn×n be a matrix with entries satisfying the condition stipulated in the second line of (1.7). Then the Lp -Neumann problem for the Laplacian in the upper half-space formulated in (4.3) for the given matrix A is solvable uniquely ′ modulo constants if and only if the Lp -Neumann problem for the Laplacian in the upper half-space formulated as in (4.3), but for the matrix A# in place of A and with p′ in lieu of p, is solvable uniquely modulo constants. Proof. Associate with the given matrix A = (ajk )1≤j,k≤n ∈ ℂn×n the operator TA as in (4.4). Bearing in mind that for each k ∈ {1, . . . , n − 1} the transpose ′ of the Riesz transform Rk acting on Lp (ℝn−1 , ℒn−1 ) is −Rk acting on Lp (ℝn−1 , ℒn−1 ) (itself a consequence of (2.7)), we conclude that the Hermitian adjoint of TA in (4.4) is the operator (TA )∗ : Lp (ℝn−1 , ℒn−1 ) → Lp (ℝn−1 , ℒn−1 ), ′

′

n−1

(TA )∗ = I − ∑ ank Rk = TA# k=1

(7.2)

250 | D. Mitrea et al. where the last equality follows from the definition in (7.1) together with (4.4) written for A# in place of A and with p′ replacing p. Granted (7.2), the equivalence claimed in the statement becomes a direct consequence of Proposition 4.1. ̃ Moving on, we observe that the operations of taking (⋅)# as in (7.1) and taking (⋅) as in (5.4) commute. That is, given any rotation R in ℝn , we have ̃# ̃ #=A (A)

for each A ∈ ℂn×n .

(7.3)

Indeed, if A ∈ ℂn×n is arbitrary and R : ℝn → ℝn is a rotation, then #

̃ # = (R⊤ AR) = 2In×n − R⊤ AR = R⊤ (2In×n − A)R (A) ̃# = R⊤ (A# )R = A as wanted. This formalism allows for an extension of Proposition 7.1, from dealing exclusively with the upper half-space ℝn+ , to actually allowing any half-space H + in ℝn . Specifically, we have the following general duality result for the Neumann problem formulated in arbitrary half-spaces. Theorem 7.2. Assume p, p′ ∈ (1, ∞) are such that 1/p + 1/p′ = 1, and fix an aperture parameter κ ∈ (0, ∞). Pick h ∈ Sn−1 and consider the half-space H + and its boundary H, defined as in (1.1)–(1.2), for this unit vector h. Also, write Δ = div A∇ for some A ∈ ℂn×n (hence A is a matrix satisfying (1.7)). Then the Lp -Neumann problem in H + formulated as in (1.9) is solvable uniquely mod′ ulo constants if and only if the Lp -Neumann problem in H + formulated as in (1.9), but for the matrix A# in place of A and with p′ in place of p, is solvable uniquely modulo constants. Proof. Based on (5.15) (used twice), Proposition 7.1, and formula (7.3), we obtain the following chain of equivalences: the Neumann problem NBVP(A, H + , p) is well posed, ̃ ℝn , p) is well posed, if and only if the Neumann problem NBVP(A, +

̃ # , ℝn , p′ ) is well posed, if and only if the Neumann problem NBVP((A) + ̃# , ℝn , p′ ) is well posed, if and only if the Neumann problem NBVP(A +

if and only if the Neumann problem NBVP(A# , H + , p′ ) is well posed, as wanted.

The Neumann problem for the Laplacian in half-spaces | 251

Bibliography [1] A. P. Calderón, On the behavior of harmonic functions near the boundary, Trans. Amer. Math. Soc. 68 (1950), 47–54. [2] L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. [3] L. Grafakos, Classical and modern Fourier analysis, Pearson Education, Inc., New Jersey, 2004. [4] J. J. Marín, J. María Martell, D. Mitrea, I. Mitrea, and M. Mitrea, Singular integrals, quantitative flatness, and boundary problems, Progress in Mathematics, Birkhäuser, 2022, to appear. [5] J. M. Martell, D. Mitrea, I. Mitrea, and M. Mitrea, The BMO-Dirichlet problem for elliptic systems in the upper half-space and quantitative characterizations of VMO, Anal. PDE 12 (2019), 605–720. [6] D. Mitrea, Distributions, partial differential equations, and harmonic analysis, 2nd ed., Springer, Switzerland, 2018. [7] D. Mitrea, I. Mitrea, and M. Mitrea, Geometric harmonic analysis, Volumes I–V, Developments in mathematics series, Springer, Switzerland, 2022, to appear. [8] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton mathematical series, vol. 30, Princeton University Press, Princeton, NJ, 1970. [9] E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton mathematical series, vol. 43, Monographs in Harmonic Analysis, III, Princeton University Press, Princeton, NJ, 1993.

Chunyan Liu and Ning Zhang

An inequality on the mass of image of rectifiable chain under Lipschitz map Abstract: For a complete normed abelian group G, we show that the mass of image of a rectifiable G-chain S under chain map f♯ induced by Lipschitz map f is controlled by the integral of Jacobian of f restricted on the support of S with respect to the associated radon measure μS . Keywords: Mass, rectifiable chain, Lipschitz map MSC 2010: 52A20, 52A38

1 Introduction In our previous paper [4], we proved in Lemma 3.5 an inequality on the mass of the image of the rectifiable d-chain S under chain map f♯ , that is, M(f♯ S) ≤ ∫ ap Jd (f |M )dμS (x),

(1.1)

where f : ℝn → ℝn is a C 1 map, μS is the Radon measure associated to chain S, and M is a d-rectifiable set such that μS (ℝn \M) = 0. We further assumed that P is a polyhedral d-chain and the inequality M(f♯ P) ≤ ∫ ap Jd (f |spt P )(x)dμP (x) holds for any Lipschitz map f : ℝn → ℝn . The estimation of mass of the image of a flat chain with finite mass under the induced chain map plays an important role in dealing with the related problems of geometric measure theory. Given a flat d-chain S with finite mass, a consequence on the mass of chain f♯ S under the chain map induced by Lipschitz map f in [3] was proved to be M(f♯ S) ≤ ∫ φd dμS (x)

󵄩 󵄩 as 󵄩󵄩󵄩Df (x)󵄩󵄩󵄩 ≤ φ(x),

Acknowledgement: Both authors were supported in part by NSF of China (# 11901217 and # 11971005). We would like to express our gratitude to Dr. Yangqin Fang for fruitful discussions. Chunyan Liu, Ning Zhang, School of Mathematics and Statistics, Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan, Hubei 430074, China, e-mails: [email protected], [email protected] https://doi.org/10.1515/9783110741711-014

254 | C. Liu and N. Zhang where φ is continuous on ℝn and Df denotes the differential. The most commonly used form of this consequence is M(f♯ S) ≤ Lip(f )d M(S). The above inequality establishes the relationship between the mass of f♯ S under the chain map and the mass of the original chain S. Since this estimate of the mass of a flat d-chain f♯ S is too large for some problems, an improvement of inequality (1.1) is necessary. In this paper, our goal is to prove that inequality (1.1) is also true if we change any Lipschitz map by a C 1 map. The main theorem of the paper is the following. Theorem 1.1. Let f be a Lipschitz map of ℝn into ℝn , and let S ∈ Rd (ℝn ; G). If M = spt S, then M(f♯ S) ≤ ∫ ap Jd (f |M )(x)dμS (x). M

In order to prove this theorem, we collect some definitions and basic results that we use throughout the paper. For further facts, we refer the reader to the books by Royden [5], Whitney [6], Federer [2], and the article by Fleming [3]. Let d be a positive integer. For any set E ⊆ ℝn , the d-dimensional Hausdorff measure ℋd (E) is defined by d

d

n

ℋ (E) = lim inf{∑ diam (Ui ) : E ⊆ ⋃ Ui , Ui ⊆ ℝ , and diam (Ui ) ≤ δ}. δ→0

A normed abelian group is an abelian group G equipped with a norm | ⋅ | : G → [0, ∞) satisfying (1) | − g| = |g|, (2) |g + h| ≤ |g| + |h|, (3) |g| = 0 if and only if g = 0. A normed abelian group G is complete if it is complete with respect to the metric induced by the norm. The group of polyhedral chains of dimension d in ℝn , with coefficients in G, denoted by Pd (ℝn ; G), is a collection of elements ∑ni=1 gi Δi with gi ∈ G, and Δi being interior disjoint polyhedra of dimension d. The mass of P is defined by n

M(P) = ∑ |gi |ℋd (Δi ). i=1

The Whitney flat norm on Pd (ℝn ; G) is defined by W(P) = inf{M(Q) + M(R) : P = Q + 𝜕R, Q ∈ Pd (ℝn ; G), R ∈ Pd+1 (ℝn ; G)}.

An inequality on the mass of image of rectifiable chain under Lipschitz map

| 255

The group of flat chains of dimension d, Fd (ℝn ; G), is the completion of Pd (ℝn ; G) W

with respect to the Whitney flat norm W. For any S ∈ Fd (ℝn ; G), we denote by Pi 󳨀→ S a sequence of polyhedral chains {Pi } converging to S in Whitney flat norm W. Then the mass of S is defined by W

M(S) = inf{lim inf M(Pi ) : Pi 󳨀→ S, Pi ∈ Pd (ℝn ; G)}. i→∞

Associated to every finite mass d-chain S and Borel set E, there is a finite mass d-chain S E that is roughly the portion of S in E. Then E 󳨃→ M(S

E)

defines a Radon measure μS such that μS (E) = M(S

E).

A flat chain S is supported by a closed set X if for every open set U containing X, there is a sequence of polyhedral chains {Pi } tending to S with cells of each Pi contained in U. The support of S, denoted by spt S, is the smallest closed set X supporting S, if it exits. A Lipschitz map f : ℝn → ℝn induces a homomorphic chain map f♯ : Fd (ℝn ; G) → Fd (ℝn ; G). If P is a polyhedral d-chain, then f♯ P is a Lipschitz d-chain, defined in [6, p. 297] by approximating f by piecewise-linear functions. A flat chain S ∈ Fd (ℝn ; G) is called rectifiable if for each ε > 0 there exists a polyhedral d-chain P ∈ Pd (ℝn ; G) and a Lipschitz map f : ℝn → ℝn such that M(S − f♯ P) < ε. We denote by Rd (ℝn ; G) the collection of all rectifiable d-chains. For any Lipschitz mapping f : ℝn ⊇ X → ℝn , the approximate d-dimensional Jacobian of f is defined by the formula 󵄩 󵄩 ap Jd f (x) = 󵄩󵄩󵄩∧d ap Df (x)󵄩󵄩󵄩 whenever f is approximately differentiable at x.

2 Proof of Theorem 1.1 First, we will introduce the following approximation theorem in [1] by De Pauw.

256 | C. Liu and N. Zhang Theorem 2.1 (Approximation theorem). Let S ∈ Rd (ℝn ; G) and ε > 0. There exist Q ∈ Pd (ℝn ; G) and g : ℝn → ℝn (a diffeomorphism of class C 1 ) with the following properties: (1) max{Lip(g), Lip(g −1 )} ≤ 1 + ε, (2) |g(x) − x| ≤ ε for every x ∈ ℝn , (3) g(x) = x whenever dist(x, spt S) ≥ ε, (4) M(Q − g♯ S) ≤ ε. A signed measure μ on the measurable space (ℝn , ℳ) can be decomposed into two mutually singular measure μ+ and μ− on (ℝn , ℳ) such that μ = μ+ − μ− , where ℳ is a σ-algebra of subsets of ℝn . The measure |μ| is defined on ℳ by |μ|(E) = μ+ (E) + μ− (E)

for all E ∈ ℳ.

Then |μ|(ℝn ) is called the total variation of μ and denoted by ‖μ‖TV , that is, m

󵄨 󵄨 ‖μ‖TV = |μ|(ℝn ) = sup ∑ 󵄨󵄨󵄨μ(Ek )󵄨󵄨󵄨, k=1

where the supremum is taken over all finite disjoint collection {Ek }m k=1 of measurable n subsets of ℝ . We start with the proof of the following result on the total variation of a measure. Lemma 2.2. Let S ∈ Rd (ℝn ; G). For any 0 < ε < 1, there exist Qε ∈ Pd (ℝn ; G) and gε : ℝn → ℝn (a diffeomorphism of class C 1 ) such that 󵄩󵄩 −1 󵄩 d 󵄩󵄩gε ♯ μQε − μS 󵄩󵄩󵄩TV ≤ (1 + 2 M(S))ε. Proof. Since S ∈ Rd (ℝn ; G), by Lemma 2.1, we get that for any 0 < ε < 1, there exist Qε ∈ Pd (ℝn ; G) and gε : ℝn → ℝn (a diffeomorphism of class C 1 ) such that max{Lip(gε ), Lip(gε−1 )} ≤ 1 + ε,

M(Qε − gε♯ S) ≤ ε.

Thus for any measurable set E, we have M(gε♯ (S

{

M(S

E)) ≤ Lip(gε )d M(S

E) =

M(gε−1 ♯ (gε♯ (S

E) ≤ (1 + ε)d M(S

E);

d

E))) ≤ (1 + ε) M(gε♯ (S

E)),

that is, (1/(1 + ε)d − 1)M(S

E) ≤ M(gε♯ (S

d

E)) − M(S

≤ ((1 + ε) − 1)M(S

E).

E)

An inequality on the mass of image of rectifiable chain under Lipschitz map

| 257

Since 1 − 1/(1 + ε)d = ((1 + ε)d − 1)/(1 + ε)d < (1 + ε)d − 1, we get 󵄨󵄨 󵄨󵄨M(gε♯ (S

E)) − M(S

󵄨 E)󵄨󵄨󵄨 ≤ ((1 + ε)d − 1)M(S

E),

thereby finding 󵄩󵄩 −1 󵄩 󵄩󵄩gε ♯ μQε − μS 󵄩󵄩󵄩TV 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩gε−1 ♯ (μQε − μgε ♯ S )󵄩󵄩󵄩TV + 󵄩󵄩󵄩gε−1 ♯ μgε ♯ S − μS 󵄩󵄩󵄩TV 󵄨 󵄨 󵄨 󵄨 = 󵄨󵄨󵄨gε−1 ♯ (μQε − μgε ♯ S )󵄨󵄨󵄨(ℝn ) + 󵄨󵄨󵄨gε−1 ♯ μgε♯ S − μS 󵄨󵄨󵄨(ℝn ) m

m

󵄨 󵄨 󵄨 󵄨 = sup ∑ 󵄨󵄨󵄨gε−1 ♯ (μQε − μgε ♯ S )(Ek )󵄨󵄨󵄨 + sup ∑ 󵄨󵄨󵄨(gε−1 ♯ μgε ♯ S − μS )(Ek )󵄨󵄨󵄨 k=1 m

k=1

m

󵄨 󵄨 󵄨 󵄨 = sup ∑ 󵄨󵄨󵄨(μQε − μgε ♯ S )(Ek′ )󵄨󵄨󵄨 + sup ∑ 󵄨󵄨󵄨μgε ♯ S (Ek′ ) − μS (Ek )󵄨󵄨󵄨 k=1 m

󵄨 = sup ∑ 󵄨󵄨󵄨M(Qε k=1 m

k=1

Ek′ ) − M((gε♯ S)

≤ sup ∑ M((Qε − gε♯ S) k=1

m

󵄨 󵄨 Ek′ )󵄨󵄨󵄨 + sup ∑ 󵄨󵄨󵄨M(gε♯ (S m

k=1

Ek′ ) + sup ∑ ((1 + ε)d − 1)M(S

≤ M(Qε − gε♯ S) + ((1 + ε)d − 1)M(S)

Ek )) − M(S

k=1

󵄨 Ek )󵄨󵄨󵄨

Ek )

≤ ε + 2d M(S)ε,

where the supremum is taken over all finite disjoint collection {Ek }m k=1 of measurable subsets of ℝn , Ek′ = gε (Ek ). We are now in a position to prove the main theorem of this paper. Proof of Theorem 1.1. Let gε be a diffeomorphism of class C 1 and Qε be a polyhedral d-chain as in Lemma 2.2. Then f♯ S = f♯ (gε−1 ∘ gε )♯ S = f♯ ∘ gε−1 ♯ (gε♯ S), gε♯ S = Qε + gε♯ S − Qε . Then we obtain M(f♯ S) = M(f♯ ∘ gε−1 ♯ (gε♯ S)) = M(f♯ ∘ gε−1 ♯ (Qε + gε♯ S − Qε )) ≤ M(f♯ ∘ gε−1 ♯ (Qε )) + M(f♯ ∘ gε−1 ♯ (gε♯ S − Qε )).

258 | C. Liu and N. Zhang Since 󵄩d 󵄩󵄩 󵄩 󵄩 −1 d −1 d −1 󵄩󵄩∧d ap D(gε |spt Qε )(x)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩ap D(gε |spt Qε )(x)󵄩󵄩󵄩 ≤ Lip(gε ) ≤ (1 + ε) , we can estimate M(f♯ ∘ gε−1 ♯ (Qε )) ≤ ∫ ap Jd ((f ∘ gε−1 )|spt Qε )(x)dμQε (x) spt Qε

= ∫ ap Jd (f |gε−1 (spt Q) )(gε−1 (x)) ⋅ ap Jd (gε−1 |spt Qε )(x)dμQε (x) spt Qε

󵄩 󵄩 󵄩 󵄩 ≤ ∫ 󵄩󵄩󵄩∧d ap D(f |gε−1 (spt Q) )(gε−1 (x))󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩∧d ap D(gε−1 |spt Qε )(x)󵄩󵄩󵄩dμQε (x) spt Qε

󵄩 󵄩 ≤ (1 + ε)d ∫ 󵄩󵄩󵄩∧d ap D(f |gε−1 (spt Q) )(gε−1 (x))󵄩󵄩󵄩dμQε (x) spt Qε

d

= (1 + ε)

∫ gε−1 (spt Qε )

ap Jd (f |gε−1 (spt Qε ) )(x)dgε−1 ♯ μQε (x).

Set Mε = gε−1 (spt Qε ),

MS = spt S,

με = gε−1 ♯ μQε ,

then ∫ ap Jd (f |Mε )(x)dμε (x)

Mε

≤ ∫ ap Jd (f |Mε )d(με − μS )(x) + ∫ ap Jd (f |Mε )dμS (x) ℝn

ℝn

≤ ∫ ap Jd (f |Mε )d|με − μS |(x) + ∫ ap Jd (f |MS )dμS (x). ℝn

ℝn

By Lemma 2.2, we achieve ∫ ap Jd (f |Mε )d|με − μS |(x) ≤ Lip(f )d |με − μS |(ℝn )

ℝn

≤ Lip(f )d (1 + 2d M(S))ε.

An inequality on the mass of image of rectifiable chain under Lipschitz map

| 259

Hence letting ε → 0 yields M(f♯ ∘ gε−1 ♯ (Qε )) ≤ ∫ ap Jd (f |MS )dμS (x) = ∫ ap Jd (f |MS )dμS (x). MS

ℝn

Moreover, if ε → 0, then M(f♯ ∘ gε−1 ♯ (gε♯ S − Qε )) ≤ Lip(f )d (1 + ε)d M(gε♯ S − Qε ) ≤ Lip(f )d (1 + ε)d ε → 0, and hence M(f♯ S) ≤ ∫ ap Jd (f |MS )dμS (x). MS

Bibliography [1] T. De Pauw and I. Vasilyev, On the existence of mass minimizing rectifiable G chains in finite dimensional normed spaces. arXiv:1812.04520v1, 11 December 2018. [2] H. Federer, Geometric measure theory, Springer, New York, 1969. [3] W. Fleming, Flat chains over a finite coefficient group, Trans. Am. Math. Soc. 121 (1966), 160–186. [4] C. Liu, Y. Fang, and N. Zhang, The weak convergence of varifolds generated by rectifiable flat G-chains. arXiv:2008.13633v3, 21 May 2021. [5] H. Royden and P. Fitzpatrick, Real analysis, 4th ed., Pearson modern classics for advanced mathematics series, Pearson, 2010. [6] H. Whitney, Geometric integration theory, Princeton, 1957.

Eduard Navas, Ebner Pineda, and Wilfredo O. Urbina

Boundedness of general alternative Gaussian singular integrals on Gaussian variable Lebesgue spaces Abstract: In a previous paper [9], we introduced a new class of Gaussian singular integrals, that we called the general alternative Gaussian singular integrals and studied the boundedness of them on Lp (γd ), 1 < p < ∞. In this paper, we study the boundedness of those operators on Gaussian variable Lebesgue spaces under a certain additional condition of regularity on p(⋅) following [5]. Keywords: Gaussian harmonic analysis, variable Lebesgue spaces, Ornstein–Uhlenbeck semigroup, singular integrals MSC 2010: Primary 42B25, 42B35, Secondary 46E30, 47G10

1 Introduction and preliminaries The general Gaussian singular integrals, generalizing the Gaussian higher order Riesz transforms were initially introduced by W. Urbina in [11] and later extended by S. Pérez [10] to a much larger class. Definition 1.1. Given a C 1 -function F, satisfying the orthogonality condition ∫ F(x)γd (dx) = 0, ℝd

and such that for every ε > 0 there exist constants Cε and Cε′ such that 󵄨󵄨 󵄨 ϵ|x|2 󵄨󵄨F(x)󵄨󵄨󵄨 ≤ Cϵ e

2 󵄨 󵄨 and 󵄨󵄨󵄨∇F(x)󵄨󵄨󵄨 ≤ Cϵ′ eϵ|x| ,

Eduard Navas, Departamento de Matemáticas, Universidad Nacional Experimental Francisco de Miranda, Punto Fijo, Venezuela, e-mail: [email protected] Ebner Pineda, Departamento de Matemática, Facultad de Ciencias Naturales y Matemáticas, ESPOL, Guayaquil 09-01-5863, Ecuador, e-mail: [email protected] Wilfredo O. Urbina, Department of Mathematics and Actuarial Sciences, Roosevelt University, Chicago, IL, 60605, USA, e-mail: [email protected] https://doi.org/10.1515/9783110741711-015

262 | E. Navas et al. for each m ∈ ℕ, the generalized Gaussian singular integral is defined as 1

− log r ) TF,m f (x) = ∫ ∫( 1 − r2

m−2 2

ℝd 0

2

− |y−rx|

dr y − rx e 1−r2 r F( ) f (y)dy. √1 − r 2 (1 − r 2 )d/2+1 r m

The operators TF,m can be written as TF,m f (x) = ∫ 𝒦F,m (x, y)f (y)dy, ℝd

where we denote 1

− log r ) 𝒦F,m (x, y) = ∫( 1 − r2

m−2 2

2

r

m−1

0

− |y−rx|

e 1−r2 y − rx ) dr F( √1 − r 2 (1 − r 2 )d/2+1 2

1

− |y−rx|

y − rx e 1−r2 = ∫ φm (r)F( ) dr √1 − r 2 (1 − r 2 )d/2+1 0

=

1

1 y − √1 − t x e−u(t) ) d/2+1 dt, ∫ ψm (t)F( √t 2 t 0

and − log r φm (r) = ( ) 1 − r2

m−2 2

r m−1 ;

which, applying the change of variables t = 1 − r 2 , has been replaced by ψm (t) = φm (√1 − t)/√1 − t

and

u(t) =

|√1 − tx − y|2 . t

In [10], S. Pérez proved the boundedness of TF,m on Gaussian Lp spaces. Theorem 1.2. The operators TF,m are Lp (γd ) bounded for 1 < p < ∞, that is, there exists C > 0, depending only in p and the dimension, such that ‖TF,m f ‖p,γd ≤ C‖f ‖p,γd

for all f ∈ Lp (γd ).

Regarding the weak (1, 1)-boundedness with respect to the Gaussian variable, she proved a negative result. Theorem 1.3. Let Ωt = {z ∈ ℝd : min |zi | ≥ t} and 1≤i≤d

Θ(t) =

infΩt F(z) t2

.

Boundedness of general alternative Gaussian singular integrals | 263

If lim sup Θ(t) = ∞, t→∞

then the operators TF,m are not of weak type (1, 1) with respect to the Gaussian measure. Also, she obtained a positive result that is contained in the following theorem. So what is needed to get sufficient conditions on F for the weak type (1, 1) of TF,m , since it is known that the Gaussian Riesz transform ℛβ for |β| ≥ 3 is not weak (1, 1) with respect to the Gaussian measure. Thus, since the weak type property is not true, the natural question is what weights can be put in order to get a weak type inequality. She got that the weight should be of the form w(y) = 1 + |y||β|−2 . Moreover, for every 0 < ϵ < |β| − 2, there exists a function F ∈ L1 ((1 + | ⋅ |ϵ )γd )

such that TF,m f ∉ L1,∞ (γd ),

see [7]. The weights w that will be considered, in order to ensure that TF,m are bounded from L1 (wγd ) into L1,∞ (γd ), depend on the function Φ. Theorem 1.4. The operators TF,m map continuously L1 (wγd ) into L1,∞ (γd ) with w(y) = 1 ∨ max η(t) and 1≤t≤|y|

Φ(t)/t η(t) = { Φ(t)/t 2

if 1 ≤ m < 2, if m ≥ 2.

In a previous paper [9], we introduced a new class of Gaussian singular integrals, the general alternative Gaussian singular integrals as follows: Definition 1.5. Given a C 1 -function F, satisfying the orthogonality condition ∫ F(x) γd (dx) = 0, ℝd

and such that for every ε > 0 there exist constants Cε and Cε′ such that 󵄨󵄨 󵄨 ϵ|x|2 󵄨󵄨F(x)󵄨󵄨󵄨 ≤ Cϵ e

2 󵄨 󵄨 and 󵄨󵄨󵄨∇F(x)󵄨󵄨󵄨 ≤ Cϵ′ eϵ|x| ,

for each m ∈ ℕ, the general alternative Gaussian singular integrals are defined as 1

− log r T F,m f (x) = ∫ ∫( ) 1 − r2 ℝd 0

m−2 2

2

r

d−1

− |y−rx|

x − ry e 1−r2 F( ) drf (y)dy. √1 − r 2 (1 − r 2 )d/2+1

264 | E. Navas et al. Thus, T F,m can be written as T F,m f (x) = ∫ 𝒦F,m (x, y)f (y)dy, ℝd

where 1

− log r 𝒦F,m (x, y) = ∫( ) 1 − r2

m−2 2

0

2

r

d−1

− |y−rx|

x − ry e 1−r2 F( dr ) √1 − r 2 (1 − r 2 )d/2+1 2

1

− |y−rx|

x − ry e 1−r2 = ∫ φm (r) F( dr ) √1 − r 2 (1 − r 2 )d/2+1 0

=

1

1 x − √1 − t y e−u(t) ) d/2+1 dt, ∫ ψm (t) F( √t 2 t 0

with − log r φm (r) = ( ) 1 − r2

m−2 2

r d−1 ;

which, after a change of the variable t = 1 − r 2 , is replaced by ψm (t) = φm (√1 − t)/√1 − t

and

u(t) =

|y − √1 − tx|2 . t

Additionally, in [9] their boundedness in Lp (γd ) was proved for d > 1 and 1 < p < ∞, Theorem 1.6. For d > 1, the operators T F,m are Lp (γd ) bounded for 1 < p < ∞, that is, there exists C > 0, depending only in p and the dimension, such that ‖T F,m f ‖p,γd ≤ C‖f ‖p,γd for any f ∈ Lp (γd ). In [1], H. Aimar, L. Forzani, and R. Scotto obtained a surprising result: the alternative Riesz transforms ℛβ are weak type (1, 1) for all multiindices β, i. e., independently of their orders which is a contrasting fact with respect to the anomalous behavior of the higher order Riesz transforms ℛβ . For the general alternative Gaussian singular integrals T F,m , we also proved

Boundedness of general alternative Gaussian singular integrals | 265

Theorem 1.7. For d > 1, there exists a constant C depending only on d and m such that for all λ > 0 and f ∈ L1 (γd ), we have γd ({x ∈ ℝd : T F,m (x) > λ}) ≤

C 󵄨󵄨 󵄨 ∫ 󵄨f (y)󵄨󵄨󵄨γd (dy). λ 󵄨 ℝd

On the other hand, in [5] E. Dalmasso and R. Scotto proved the boundedness of the general Gaussian singular integrals TF,m , on Gaussian variable Lebesgue spaces under certain condition of regularity on p(⋅). In order to understand their result, we need to get more background on variable Lebesgue spaces with respect to a Borel measure in general, and the Gaussian measure in particular. As usual in what follows, C represents a constant that is not necessarily the same in each occurrence; also we will use the following notation: given two functions f , g, the symbols ≲ and ≳ denote that there is a constant c such that f ≤ cg and cf ≥ g, respectively. When both inequalities are satisfied, that is, f ≲ g ≲ f , we will denote f ≈ g. Any μ-measurable function p(⋅) : Ω ⊂ ℝd → [1, ∞] is an exponent function; the set of all the exponent functions will be denoted by 𝒫 (Ω, μ). For E ⊂ ℝd , we set p− (E) = ess inf p(x) x∈E

and p+ (E) = ess sup p(x). x∈E

We use the abbreviations p+ = p+ (ℝd ) and p− = p− (ℝd ). Ω∞ = {x ∈ Ω : p(x) = ∞}. Definition 1.8. Let E ⊂ ℝd . We say that α(⋅) : E → ℝ is locally log-Hölder continuous, and denote this by α(⋅) ∈ LH0 (E), if there exists a constant C1 > 0 such that 󵄨󵄨 󵄨 󵄨󵄨α(x) − α(y)󵄨󵄨󵄨 ≤

C1

log(e +

1 ) |x−y|

for all x, y ∈ E. We say that α(⋅) is log-Hölder continuous at infinity with base point at x0 ∈ ℝd , and denote this by α(⋅) ∈ LH∞ (E), if there exist constants α∞ ∈ ℝ and C2 > 0 such that C2 󵄨󵄨 󵄨 󵄨󵄨α(x) − α∞ 󵄨󵄨󵄨 ≤ log(e + |x − x0 |) for all x ∈ E. We say that α(⋅) is log-Hölder continuous, and denote this by α(⋅) ∈ LH(E), if both conditions are satisfied. The maximum, max{C1 , C2 }, is called the log-Hölder constant of α(⋅). Definition 1.9. We denote p(⋅) ∈ 𝒫dlog (ℝd ), if

by Clog (p) or Clog the log-Hölder constant of

1 p(⋅) 1 . p(⋅)

is log-Hölder continuous and denote

We will need the following technical result; for its proof, see Lemma 3.26 in [4].

266 | E. Navas et al. Lemma 1.10. Let ρ(⋅) : ℝd → [0, ∞) be such that ρ(⋅) ∈ LH∞ (ℝd ),

0 < ρ∞ < ∞,

and let −N

R(x) = (e + |x|)

,

N > d/ρ− .

Then there exists a constant C depending on d, N, and the LH∞ constant of r(⋅) such that given any set E and any function F with 0 ≤ F(y) ≤ 1 for all y ∈ E, one has ∫ F ρ(y) (y)dy ≤ C ∫ F(y)ρ∞ dy + ∫ Rρ− (y)dy, E

E

∫ F ρ∞ (y)dy ≤ C ∫ F ρ(y) (y)dy + ∫ Rρ (y)dy. −

E

E

(1.1)

E

(1.2)

E

Definition 1.11. For a μ-measurable function f : ℝd → ℝ, we define the modular ρp(⋅),μ (f ) =

󵄨 󵄨p(x) ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 μ(dx) + ‖f ‖L∞ (Ω∞ ,μ) . ℝd \Ω∞

Definition 1.12. The variable exponent Lebesgue space on ℝd , Lp(⋅) (ℝd , μ), consists on those μ-measurable functions f for which there exists λ > 0 such that f ρp(⋅),μ ( ) < ∞, λ i. e., f Lp(⋅) (ℝd , μ) = {f : ℝd → ℝ : f measurable, ρp(⋅),μ ( ) < ∞ for some λ > 0}, λ and the norm ‖f ‖Lp(⋅) (ℝd ,μ) = inf{λ > 0 : ρp(⋅),μ (f /λ) ≤ 1}. It is well known that if p(⋅) ∈ LH(ℝd ) with 1 < p− ≤ p+ < ∞

Boundedness of general alternative Gaussian singular integrals | 267

then the classical Hardy–Littlewood maximal function ℳ is bounded on the variable Lebesgue space Lp(⋅) , see [3]. However, it is known that even though these are the sharpest possible pointwise conditions, they are not necessary. In [6] a necessary and sufficient condition is given for the Lp(⋅) -boundedness of ℳ, but it is not an easy to work condition. The class LH(ℝd ) is also sufficient for the boundedness on Lp(⋅) -spaces of classical singular integrals of Calderón–Zygmund type, see [4, Theorem 5.39]. If ℬ is a family of balls (or cubes) in ℝd , we say that ℬ is N-finite if it has bounded overlappings for N, that is, ∑ χB (x) ≤ N

B∈ℬ

for all x ∈ ℝd .

In other words, there exist at most N balls (resp. cubes) that intersect at the same time. The following definition was introduced for the first time by Berezhnoǐ in [2], defined for family of disjoint balls or cubes. In the context of variable spaces, it has been considered in [6], allowing the family to have bounded overlappings. Definition 1.13. Given an exponent p(⋅) ∈ 𝒫 (ℝd ), we will say that p(⋅) ∈ 𝒢 if for every family of balls (or cubes) ℬ which is N-finite, ∑ ‖fχB ‖p(⋅) ‖gχB ‖p′ (⋅) ≲ ‖f ‖p(⋅) ‖g‖p′ (⋅)

B∈ℬ

for all functions f ∈ Lp(⋅) (ℝd ) and g ∈ Lp (⋅) (ℝd ). ′

The constant only depends on N. Lemma 1.14 ([6, Teorema 7.3.22]). If p(⋅) ∈ LH(ℝd ), then p(⋅) ∈ 𝒢 . We will consider only variable Lebesgue spaces with respect to the Gaussian measure γd , Lp(⋅) (ℝd , γd ). The next condition was introduced by E. Dalmasso and R. Scotto in [5]. Definition 1.15. Let p(⋅) ∈ 𝒫 (ℝd , γd ). We say that p(⋅) ∈ 𝒫γ∞ (ℝd ) if there exist constants d Cγd > 0 and p∞ ≥ 1 such that 󵄨󵄨 󵄨 Cγ 󵄨󵄨p(x) − p∞ 󵄨󵄨󵄨 ≤ d2 , |x| for x ∈ ℝd \ {(0, 0, . . . , 0)}. Observation 1.16. If p(⋅) ∈ 𝒫γ∞ (ℝd ), then p(⋅) ∈ LH∞ (ℝd ). d

268 | E. Navas et al. Lemma 1.17. If 1 < p− ≤ p+ < ∞, the following statements are equivalent: (i) p(⋅) ∈ 𝒫γ∞ (ℝd ). d (ii) There exists p∞ > 1 such that 2

C1−1 ≤ e−|x| (p(x)/p∞ −1) ≤ C1

and

2

′

′

C2−1 ≤ e−|x| (p (x)/p∞ −1) ≤ C2

for all x ∈ ℝd ,

where C1 = eCγd /p∞

and

C2 = eCγd (p− ) /p∞ . ′

Definition 1.15 with Observation 1.16 and Lemma 1.17 end up strengthening the regularity conditions on the exponent functions p(⋅) to obtain the boundedness of the Ornstein–Uhlenbeck semigroup {Tt }, see [8]. As a consequence of Lemma 1.14, we have Corollary 1.18. If p(⋅) ∈ 𝒫γ∞ (ℝd ) ∩ LH0 (ℝd ), d then p(⋅) ∈ 𝒢 . As it has been mentioned already, in [5] E. Dalmasso and R. Scotto proved the boundedness of TF,m on Gaussian variable Lebesgue spaces under the additional condition of regularity p(⋅) ∈ 𝒫γ∞ (ℝd ). d Theorem 1.19. Let p(⋅) ∈ 𝒫γ∞ (ℝd ) ∩ LH0 (ℝd ) with 1 < p− ≤ p+ < ∞. d Then there exists a constant C > 0, depending only in p and the dimension, such that ‖TF,m f ‖p(⋅),γd ≤ C‖f ‖p(⋅),γd

for all f ∈ Lp(⋅) (γd ).

The main result in this paper is the proof, following the arguments of Dalmasso and Scotto [5], that the general alternative Gaussian singular integrals T F,m are also bounded on Gaussian variable Lebesgue spaces under the same condition of regularity on p(⋅) considered by Dalmasso and Scotto. Theorem 1.20. Let d > 1 and

p(⋅) ∈ 𝒫γ∞ (ℝd ) ∩ LH0 (ℝd ) with 1 < p− ≤ p+ < ∞. d

Then there exists a constant C > 0, depending only in p and the dimension, such that ‖T F,m f ‖p(⋅),γd ≤ C‖f ‖p(⋅),γd

for all f ∈ Lp(⋅) (γd ).

Boundedness of general alternative Gaussian singular integrals | 269

2 Proof of the main result We are ready for the proof of our main result, Theorem 1.20. As usual we split each operator T F,m into local and global parts: T F,m f (x) =

𝒦F,m (x, y)f (y)dy +

∫ |x−y| 1, p′ (⋅) ∈ Pγ∞d (ℝd ). Thus, from Lemma 1.17, for every x ∈ ℝd ,

2

e−|x| (p(x)/p∞ −1) ≤ C1

2

′

′

and e−|x| (p (x)/p∞ −1) ≤ C2 .

(2.4)

Moreover, since the values of the Gaussian measure γd are all equivalent on each ball B,̂ we have ∫( B̂

p(y)

|f (y)| 2

e|cB | /p∞ ‖fχB̂ ‖p(⋅),γd

)

p(y)

≲ ∫( B̂

p(y)

≲ ∫( B̂

|f (y)| ) ‖fχB̂ ‖p(⋅),γd |f (y)| ) ‖fχB̂ ‖p(⋅),γd

dy 2

e−|y| (p(y)/p∞ −1) γd (dy) γd (dy) ≲ 1,

which yields 2

e−|cB | /p∞ ‖fχB̂ ‖p(⋅) ≲ ‖fχB̂ ‖p(⋅),γd . Similarly, by applying the second inequality of (2.4), we get 2

′

e−|cB | /p∞ ‖gχB̂ ‖p′ (⋅) ≲ ‖gχB̂ ‖p′ (⋅),γd .

272 | E. Navas et al. Replacing both estimates in (2.3), we obtain 󵄨 󵄨󵄨 󵄨 ∫ 󵄨󵄨󵄨T F,m (fχBh (⋅) )(x)󵄨󵄨󵄨󵄨󵄨󵄨g(x)󵄨󵄨󵄨γd (dx) ℝd

≲ ∑ ‖fχB̂ ‖p(⋅),γd ‖gχB̂ ‖p′ (⋅),γd B∈ℱ

2 ′ 2 󵄩 󵄩 󵄩 󵄩 = ∑ 󵄩󵄩󵄩fχB̂ e−|⋅| /p(⋅)| 󵄩󵄩󵄩p(⋅) 󵄩󵄩󵄩gχB̂ e−|⋅| /p (⋅) 󵄩󵄩󵄩p′ (⋅) .

B∈ℱ

Since the family of balls ℱ̂ has bounded overlaps, from Corollary 1.18 applied to 2

2

fe−|⋅| /p(⋅) ∈ Lp(⋅) (ℝd )

and ge−|⋅| /p (⋅) ∈ Lp (⋅) (ℝd ), ′

′

it follows that 󵄨 󵄨󵄨 󵄨 ∫ 󵄨󵄨󵄨T F,m (fχBh (⋅) )(x)󵄨󵄨󵄨󵄨󵄨󵄨g(x)󵄨󵄨󵄨γd (dx) ⩽ ‖f ‖p(⋅),γd ‖g‖p′ (⋅),γd . ℝd

Taking supremum over all functions g with ‖g‖p′ (⋅),γd ≤ 1, from (2.2) we finally get 󵄩 󵄩 ‖T F,m,L f ‖p(⋅),γd = 󵄩󵄩󵄩T F,m (fχBh (⋅) )󵄩󵄩󵄩p(⋅),γ ≤ C‖f ‖p(⋅),γd . d

(ii) For the global part, to handle the kernel 𝒦, we need the following result (see [9]): Lemma 2.1. Consider the kernel 𝒦F,m (x, y) in the global region, i. e., y ∈ Bch (x). If a = |x|2 + |y|2

and

b = 2⟨x, y⟩,

we have the following inequalities: (i) If b ≤ 0, for each 0 < ϵ < 1, there exists Cϵ > 0 such that 󵄨󵄨 󵄨 −|y|2 +ϵ|x|2 . 󵄨󵄨𝒦F,m (x, y)󵄨󵄨󵄨 ≤ Cϵ e (ii) If b > 0, for each 0 < ϵ < d1 , there exists Cϵ > 0 such that 󵄨󵄨 󵄨 ϵ(|x|2 −|y|2 ) e 󵄨󵄨𝒦F,m (x, y)󵄨󵄨󵄨 ≤ Cϵ e

−(1−ϵ)u(t0 )

t0d/2

,

where t0 = 2

√a2 − b2 a + √a2 − b2

and

1 u0 = (|y|2 − |x|2 + |x + y||x − y|). 2

Boundedness of general alternative Gaussian singular integrals | 273

Now, for the study of the global part T F,m,G , let us look at the set Ex = {y : ⟨x, y⟩ > 0} and consider two cases: – Case b = 2⟨x, y⟩ ≤ 0. Let 0 < ϵ < we have, by Lemma 2.1(i),

1 , p+

then for f ∈ Lp(⋅) (ℝd , γd ) with ‖f ‖p(⋅),γd = 1 p(x)

∫(

󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨𝒦F,m (x, y)󵄨󵄨󵄨󵄨󵄨󵄨f (y)󵄨󵄨󵄨dy)

∫

γd (dx)

c c ℝd Bh (⋅)∩Ex

≤ ∫( ℝd

p(x)

2 2 󵄨 󵄨 e−|y| +ϵ|x| 󵄨󵄨󵄨f (y)󵄨󵄨󵄨dy)

∫

γd (dx)

Bch (⋅)∩Exc p− p(x) p

󵄨 󵄨 ≤ ∫ ( ∫ 󵄨󵄨󵄨f (y)󵄨󵄨󵄨γd (dy))

2

2

eϵp(x)|x| −|x| dx

−

ℝd ℝd p(x)

p− 2 󵄨 󵄨p ≤ ∫ ( ∫ 󵄨󵄨󵄨f (y)󵄨󵄨󵄨 − γd (dy)) e(ϵp+ −1)|x| dx.

ℝd ℝd

Since by hypothesis 󵄨 󵄨p(x) ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 γd (dx) ≤ 1, ℝd

we get, 󵄨 󵄨p 󵄨 󵄨p(y) ∫ 󵄨󵄨󵄨f (y)󵄨󵄨󵄨 − γd (dy) ≤ ∫ 󵄨󵄨󵄨f (y)󵄨󵄨󵄨 γd (dy) + ∫ γd (dy) ℝd

|f |>1

|f |≤1

≤ 1 + Cd , we obtain p(x)

∫(

∫

󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨𝒦F,m (x, y)󵄨󵄨󵄨󵄨󵄨󵄨f (y)󵄨󵄨󵄨dy)

γd (dx)

c c ℝd Bh (⋅)∩Ex p(x)

2

≤ ∫ (1 + Cd ) p− e(ϵp+ −1)|x| dx ℝd p+

2

≤ (1 + Cd ) p− ∫ e(ϵp+ −1)|x| dx = Cp,d . ℝd

274 | E. Navas et al. In other words, for all f ∈ Lp(⋅) (ℝd , γd ) with ‖f ‖p(⋅),γd = 1.

󵄩 󵄩󵄩 󵄩󵄩T F,m (fχBch (⋅)∩Exc )󵄩󵄩󵄩p(⋅),γd ≤ Cp,d –

Case b = 2⟨x, y⟩ > 0. For f ∈ Lp(⋅) (ℝd , γd ) with ‖f ‖p(⋅),γd = 1, we have p(x)

󵄨 󵄨 󵄨󵄨 (I) = ∫ ( ∫ 󵄨󵄨󵄨𝒦F,m (x, y)󵄨󵄨󵄨󵄨󵄨󵄨f (y)󵄨󵄨󵄨dy) ℝd

γd (dx)

Bch (⋅)∩Ex 2

2

eϵ(|x| −|y| )

≤ C ∫( ∫ c ℝd Bh (⋅)∩Ex

p(x)

e−(1−ϵ)u(t0 ) 󵄨󵄨 󵄨 󵄨󵄨f (y)󵄨󵄨󵄨dy) t0d/2

2

e−|x| (dx) p(x)

2

|x| 󵄨 −ϵ(|y|2 −|x|2 ) 󵄨󵄨 −(1−ϵ)u(t0 )− p(x) t0−d/2 e 󵄨󵄨f (y)󵄨󵄨󵄨 dy)

= C ∫( ∫ c ℝd Bh (⋅)∩Ex

Now, using the inequality 󵄨󵄨 2 2󵄨 󵄨󵄨|y| − |x| 󵄨󵄨󵄨 ≤ |x + y||x − y| and that on the global region |x + y||x − y| > d, as b > 0, we have e−(1−ϵ)u(t0 ) t0d/2

≤C

e

−|x|2

2

2

2

2

−(1−ϵ)u(t0 )+ |y|p−|x| −ϵ(|y|2 −|x|2 )

C

=

t0d/2

=

t0d/2

C

|y|2

e p(y) e p(x) e−ϵ(|y| −|x| ) ∞

t0d/2

e

− (1−ϵ) (|y|2 −|x|2 +|x+y||x−y|)+( p 1 −ϵ)(|y|2 −|x|2 ) 2

e

(− (1−ϵ) + p 1 −ϵ)(|y|2 −|x|2 ) − (1−ϵ) (|x+y||x−y|) 2 2 ∞

∞

e

≤ C|x + y|d e−α∞ |x+y||x−y| where α∞ =

(1 − ϵ) 󵄨󵄨󵄨󵄨 1 (1 − ϵ) 󵄨󵄨󵄨󵄨 − 󵄨󵄨 −ϵ− 󵄨>0 󵄨󵄨 p∞ 2 2 󵄨󵄨󵄨

if ϵ
21 }.

Therefore, Bch (x) ⊂ Ax ∪ Cx . Define 2

J1 = ∫ P(x, y)f (y)e−|y| /p(y) dy Ax ∩Ex

and 2

J2 = ∫ P(x, y)f (y)e−|y| /p(y) dy. Cx ∩Ex

Let us estimate J1 first. Observe that if y ∈ Ax

and

3 5 |x| ≤ |y| ≤ |x| 4 4

then |x| ≈ |y| and |x| ≈ |x + y|, and hence J1 ≲

∫

2

|x|d e−α∞ |x||x−y| f (y)e−|y| /p(y) dy

d 0,

where q is the finite index conjugate to p and 1

N ≡ (|x|4 + 16|z|2 ) 4 . The authors of [9] extend, giving a simpler proof, the result of J. Inglis by obtaining a q-Poincaré inequality in the setting of step-two Carnot groups for the measures dμ = with

g ′ (N) N2

e−g(N) dλ, Z

an increasing function, q ≥ 2, and where 1

N ≡ (|x|4 + a|z|2 ) 4 . The route to prove the q-Poincaré inequality is through the following U-bound theorem: Theorem 2.2 ([9]). Let 1

N = (|x|4 + a|z|2 ) 4

with a ∈ (0, ∞),

and let g : [0, ∞) → [0, ∞) be a differentiable increasing function such that g ′′ (N) ≤ g ′ (N)3 N 3

on {N ≥ 1}.

Coercive inequalities on Carnot groups and applications | 297

Let dμ =

e−g(N) dλ Z

be a probability measure, where Z is the normalization constant. Then, given q ≥ 2, ∫

g ′ (N) q |f | dμ ≤ C ∫ |∇f |q dμ + D ∫ |f |q dμ N2

holds for all locally Lipschitz functions f , supported outside the unit ball {N < 1}, with C and D positive constants independent of f . The authors of [9] expect that this U-bound can be used to extend those coercive inequalities to (nonproduct) measures in an infinite-dimensional setting [28]. The role ′ of 𝒰 in Theorem 2.1 is played by g N(N) from the U-bound of Theorem 2.2. Hence, we get 2 the following corollaries: Corollary 2.3 ([9]). The Poincaré inequality for q ≥ 2 holds for the measure dμ =

exp(− cosh(N k )) dλ, Z

where λ is the Lebesgue measure, and k ≥ 1 in the setting of the step-two Carnot group. Corollary 2.4 ([9]). The Poincaré inequality for q ≥ 2 holds for the measure dμ =

exp(−N k ) dλ, Z

where λ is the Lebesgue measure, and k ≥ 4 in the setting of the step-two Carnot group. Corollary 2.5 ([9]). The Poincaré inequality for q ≥ 2 holds for the measure dμ =

exp(−N k log(N + 1)) dλ, Z

where λ is the Lebesgue measure, and k ≥ 3 in the setting of the step-two Carnot group. After proving the q-Poincaré inequality for measures as a function of the homogeneous norm 1

N = (|x|4 + a|z|2 ) 4 , a natural question would be if one could obtain other coercive inequalities. J. Inglis et al.’s Theorem 2.1 [20] proved that, for the measure dμ =

e−U dλ , Z

298 | E. Bou Dagher et al. provided we have the U-bound μ(|f |(|U|β + |∇U|)) ≤ Aμ|∇f | + Bμ(|f |), one obtains β 󵄨󵄨 |f | 󵄨󵄨󵄨󵄨 󵄨 μ(|f |󵄨󵄨󵄨log 󵄨󵄨 ) ≤ Cμ(|∇f |) + Bμ(|f |). 󵄨󵄨 μ(|f |) 󵄨󵄨

In [9], an extension to their theorem is provided: Theorem 2.6 ([9]). Let U be a locally Lipschitz function on ℝN which is bounded below such that Z = ∫ e−U dλ < ∞ and

dμ =

e−U dλ. Z

Let ϕ : [0, ∞) → ℝ+ be a nonnegative, nondecreasing, concave function such that ϕ(0) > 0 and ϕ′ (0) > 0. Assume that the following classical Sobolev inequality is satisfied: (∫ |f |

q+ϵ

dλ)

q q+ϵ

≤ a ∫ |∇f |q dλ + b ∫ |f |q dλ

for some a, b ∈ [0, ∞) and ϵ > 0. Moreover, if for some A, B ∈ [0, ∞) we have μ(|f |q (ϕ(U) + |∇U|q )) ≤ Aμ(|∇f |q ) + Bμ(|f |q ), then there exist constants C, D ∈ [0, ∞) such that 󵄨󵄨 |f |q 󵄨󵄨󵄨󵄨 󵄨 q q μ(|f |q ϕ(󵄨󵄨󵄨log 󵄨)) ≤ Cμ(|∇f | ) + Dμ(|f | ) 󵄨󵄨 μ(|f |q ) 󵄨󵄨󵄨 holds for all locally Lipschitz functions f on ℝN . Theorem 2.2 is used to get more general coercive inequalities: Corollary 2.7 ([9]). Let 𝔾 be a step-two Carnot group and 1

N = (|x|4 + a|z|2 ) 4

with a ∈ (0, ∞).

Let the probability measure be p

e−βN dμ = dλ, Z

Coercive inequalities on Carnot groups and applications | 299

where Z is the normalization constant. Then, for p ≥ 4,

q ≥ 2,

0 0, p ≥ 1, let p

dμ =

e−αN dλ, Z

where Z is the normalization constant. The measure μ satisfies no logβ -Sobolev inequal2pβ ity (0 < β ≤ 1) for 1 < q < p−1 . We note that W. Hebisch and B. Zegarliński showed in [18] that there is no Logarithmic Sobolev inequality for measures as a function of a homogeneous norm. In [10], the Logarithmic Sobolev inequality, for measures as a function of the Kaplan norm, is forced by introducing a multiplicative or an additive singularity. The aim of [8] is to obtain similar results, that are stated in this section, in the setting of the higher-dimensional anisotropic Heisenberg group with the measure as a function of the fundamental solution. Before we give its definition, we introduce polarizable groups. In [3], Z. Balogh and J. Tyson introduced the concept of polarizable Carnot groups defined by the condition that N is ∞-harmonic in 𝔾\{0}, i. e., if ∇ :=(Xi )1≤i≤n , then 1 Δ∞ N := ⟨∇(|∇N|2 ), ∇N⟩ = 0 2

in 𝔾\{0}.

(2.1)

They showed that, using the ∞-harmonicity of N, one can provide a procedure to construct polar coordinates of special type where the curves passing through the points on the unit sphere {N = 1} are horizontal. Moreover, they showed in [3] that the fundamental solution of the p-subLaplacian can be expressed as the fundamental solution N of the sub-Laplacian, proved capacity formulas, and produced sharp constants for the Moser–Trudinger

300 | E. Bou Dagher et al. inequality. For the time being, there is no classification of polarizable Carnot groups, and the only examples till now are Euclidean spaces and Heisenberg-type groups. In addition, the concept of a polarizable Carnot group is a delicate one in the sense that under a small perturbation of the Lie algebra, the group is no longer polarizable. Z. Balogh and J. Tyson provided in [3] the anisotropic Heisenberg group in ℝ4 as a counterexample with the following generators of the Lie algebra: 𝜕 + 2ay ⋅ 𝜕t𝜕 , X = 𝜕x { { { { 𝜕 { { Y = 𝜕y − 2ax ⋅ 𝜕t𝜕 , { { { 𝜕 Z = 𝜕z − 2w ⋅ 𝜕t𝜕 , { { { { 𝜕 { − 2z ⋅ 𝜕t𝜕 , W = 𝜕w { { { { 1 {a = 2 .

Note that, if a = 1, we have the polarizable Heisenberg group. To show (2.1) does not hold true for the anisotropic Heisenberg group, they computed explicitly the fundamental solution of the sub-Laplacian using R. Beals, B. Gaveau, and P. Greiner’s [5] explicit integral representation for the fundamental solution in the setting of general step-two Carnot groups. The goal of [8] was to study coercive inequalities such as the q-Poincaré inequality and the β-logarithmic Sobolev inequality in the setting of the anisotropic Heisenberg group with respect to measures as a function of the explicit fundamental solution. The authors of [8] show the computations, which were partially omitted in [3] in the ℝ5 setting, and use them to get an explicit fundamental solution for higher dimensions. Let θ > 0, and consider a generalization of the anisotropic Heisenberg group ℍ2n ( θ2 , θ), as introduced in [2], on ℝ2n+1 with the dilation δλ (x1 , x2 , . . . , x2n , t) = (λx1 , λx2 , . . . , λx2n , λ2 t) and the composition law (x1 , x2 , . . . , x2n , t) ∘ (η1 , η2 , . . . , η2n , τ) = (y1 , y2 , . . . , y2n , s) with y1 = x1 + η1 , { { { { { { {y2 = x2 + η2 , { { {. .. { { { { { { y2n = x2n + η2n , { { { { θx1 ηn+1 −θxn+1 η1 + θ ∑nj=2 (xj ηj+n − ηj xj+n ). 2 {s = t + τ +

Coercive inequalities on Carnot groups and applications | 301

Clearly, ℍ2n ( θ2 , θ) is a homogeneous Carnot group of step two with generators θx

for j = 1,

𝜕x1 − 2n+1 𝜕t { { { { { {𝜕x + θx2 1 𝜕t Xj = { n+1 {𝜕x − θxj+n 𝜕t { { j { { 𝜕 { xj + θxj−n 𝜕t

for j = n + 1, for j = 2, 3, . . . , n, for j = n + 2, n + 3, . . . , 2n.

Theorem 2.9 ([8]). The fundamental solution for the group ℍ2n ( θ2 , θ) is given by the following homogeneous norm: 1

1

1

(B2 + t 2 ) 4n (AB + t 2 + A√B2 + t 2 ) 2 − 4n , N(x, t) = 1 (B + √B2 + t 2 ) 2

(2.2)

where {A = { {B =

θx12 2 θx12 4

+ +

2 θxn+1 2 2 θxn+1 4

+ θ2 ∑2n xj2 , j=2,j=n+1 ̸

+ θ2 ∑2n xj2 . j=2,j=n+1 ̸

A similar theorem is obtained to get the U-bound inequality: Theorem 2.10 ([8]). Let N −2n be the fundamental solution in the setting of the anisotropic Heisenberg group ℝ2n+1 with n > 5. Let g : [0, ∞) → [0, ∞) be a differentiable increasing function such that g ′′ (N) ≤ g ′ (N)2

on {N ≥ 1}.

Let dμ =

e−g(N) dλ Z

be a probability measure and Z the normalization constant. Then, for q ≥ 2, ∫

g ′ (N) q |f | dμ ≤ C ∫ |∇f |q dμ + D ∫ |f |q dμ N2

holds outside the unit ball {N < 1} with C and D positive constants independent of a function f for which the right-hand side is well defined. The role of 𝒰 in Theorem 2.1 is played by Hence, we get the following corollaries:

g ′ (N) N2

from the U-bound of Theorem 2.10.

Corollary 2.11 ([8]). The Poincaré inequality for q ≥ 2 holds for the measure dμ =

exp(− cosh(N k )) dλ, Z

302 | E. Bou Dagher et al. where λ is the Lebesgue measure, and k ≥ 1 in the setting of the anisotropic Heisenberg group ℝ2n+1 with n > 5. Corollary 2.12 ([8]). The Poincaré inequality for q ≥ 1 holds for the measure dμ =

exp(−N k ) dλ, Z

where λ is the Lebesgue measure, and k ≥ 4 in the setting of the anisotropic Heisenberg group ℝ2n+1 with n > 5. Corollary 2.13 ([8]). The Poincaré inequality for q ≥ 2 holds for the measure dμ =

exp(−N k log(N + 1)) dλ, Z

where λ is the Lebesgue measure, and k ≥ 3 in the setting of the anisotropic Heisenberg group ℝ2n+1 with n > 5. In addition, similarly to the work in [9], a β-logarithmic Sobolev inequality for p

dμ =

e−αN dλ, Z

is proved for the triple p ≥ 4,

q ≥ 2,

0 0 such that 󵄨 󵄨r 󵄨 󵄨q ∫󵄨󵄨󵄨f − Mk,r,μ (f )󵄨󵄨󵄨 dμ ≤ C ∫󵄨󵄨󵄨∇k f 󵄨󵄨󵄨 dμ for all f ∈ W k,q . By (3.2), the Lebesgue measure λ satisfies Poincaré inequality of order 1.

(3.4)

304 | E. Bou Dagher et al. Remark 3.4. Lemma 3.1 only confirms the existence of the minimizer for q ∈ (1, ∞]. For q = 1, we interpret the left-hand side of (3.4) as the infimum inf ∫ |f − w|dμ

w∈ℙk

in case the minimizer does not exist. We also have two different approaches to establish Poincaré inequalities of higher order. One is an abstract approach via functional analysis, see [29] (originally from [24]). The other is the downhill induction. Theorem 3.5 ([27]). Let q ∈ (1, ∞). Suppose that we have Poincaré inequalities of the first order: 󵄨 󵄨q 󵄨 󵄨q μ󵄨󵄨󵄨f − m1,q (f )󵄨󵄨󵄨 ≤ C1,q ∑ μ󵄨󵄨󵄨∇α f 󵄨󵄨󵄨 , |α|=1

with a constant m1,q (f ) and a constant C1,q ∈ (0, ∞). Then we have the following inequality: 󵄨 󵄨q 󵄨 󵄨q μ󵄨󵄨󵄨f − mk,q (f )󵄨󵄨󵄨 ≤ Ck,q ∑ μ󵄨󵄨󵄨∇α f 󵄨󵄨󵄨 ,

(PIk,q )

|α|=k

with some polynomial mk,q (f ) of order k − 1 and a constant Ck,q ∈ (0, ∞). The idea is mainly from [25]. The constant in the above result grows exponentially and is probably not optimal, and different from the optimal constant in the (PIk,q ) with the minimizing polynomial Mk,q (f ). But it still provides us with at least two results: (1) It inspires us to study the constant of inequalities of the first order, so that an upper bound can be obtained for constants of higher order. (2) It provides us with an alternative approach to obtain Poincaré inequality of higher order from that of the first order, so that the abstract analysis in [29] can be curved under some particular circumstances.

4 Adams’ regularity and coercive inequalities In 1979, R. A. Adams introduced the Adams’ inequalities in [1]. We will introduce his main result. We denote by log∗ t = max{1, log t}, and set log∗k t = log∗ (log∗ (⋅ ⋅ ⋅ log∗ t)) for k iterations of log∗ . We assume that U(x) : ℝn → ℝ

in L1 (ℝn ) ∩ C 2 (ℝn )

Coercive inequalities on Carnot groups and applications | 305

satisfies the coercive condition lim U(x) = +∞.

(4.1)

|x|→+∞

Consider the probability measure dμ =

1 −U e dλ, Z

where Z is the normalizer chosen in such a way that μ is a probability measure. Under this assumption, it is easy to check that e−U ∈ Lp (ℝn , dλ) for all p ≥ 1. Also letting k be an integer, p ≥ 1 as usual, we can define the weighted Sobolev spaces under the probability measure dμ similar as what we did for the Lebesgue measure, and they will be endowed with the norm 1

‖f ‖W k,p (Ω,dμ) = ‖f ‖k,p,Ω,μ

p 󵄨 󵄨p = { ∑ ∫󵄨󵄨󵄨∇α f (x)󵄨󵄨󵄨 dμ} ,

(4.2)

|α|≤k Ω

where α is a multiindex. Next we define the following Adams’ regularity condition: Definition 4.1. Let the potential function U be defined as in (4.1). Then U is said to satisfy the Adams’ regularity condition if ∃ ε, C > 0

such that

2−ε 󵄨 󵄨 ∑ 󵄨󵄨󵄨∇α U 󵄨󵄨󵄨 ≤ C(1 + |∇U|)

for a. e. x ∈ ℝn .

(4.3)

|α|=2

Assuming this condition, Adams showed in [1] the following result. Theorem 4.2. There exists a constant C > 0 such that 󵄨 󵄨p 󵄨 󵄨 mp ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 (1 + 󵄨󵄨󵄨∇U(x)󵄨󵄨󵄨) dμ(x) ⩽ C‖f ‖pm,p,μ,ℝn

ℝn

for all f ∈ C0∞ (ℝn ).

The Adams’ regularity condition is not tight at all; in fact, in ℝ1 , functions including, but not limited to, polynomials U(x) = xα , where α is a multiindex, U(x) = xα log∗ (x), and e−x all satisfy (4.3). When it comes to the higher-dimensional case, Adams presented a series of interesting examples in [1]: Example 4.3. Let d : ℝn → ℝ be a smooth, positive function such that d(x) = |x| for all |x| ≥ 1,

306 | E. Bou Dagher et al. and similarly log∗∗ (t) be a smooth, positive function such that log∗∗ (t) = log t

for t

such that

log t ≥ 1.

Notice that by applying the continuous extension method such d and log∗∗ defined above exist. For j ∈ ℕ, denote ∗∗ ∗∗ log∗∗ j (t) = log (logj−1 (t)).

Then for σ > 0, all functions U0 (x) = dσ (x), { { { { { {U1 (x) = eU0 (x) , { { Uj (x) = eUj−1 (x) , { { { { ∗∗ σ {U−j (x) = d(x)(logj d(x)) satisfy the Adams’ regularity condition.

5 Generalized regularity condition and tight coercive inequalities Theorem 5.1 ([27]). Under the assumption of Adams’ regularity condition (4.3), there exists γ > 0 such that if p ∈ [2 − γ, ∞) then ∃K >0

such that

2m−1 p

∫(1 + |∇U|)

|f |p dμ ≤ K‖f ‖pm,p

for all f ∈ Wm,p (μ),

where ‖f ‖m,p ≡ ‖f ‖m,p,μ is the weighted Sobolev norm in the space Wm,p (μ). If we generalize the Adams’ regularity condition to the following: Definition 5.2. If there exists a constant K ∈ (0, ∞) such that ∃ K ∈ (0, ∞) obeying |∇2 U| ≤ K(1 + |∇U|)2−ϵ , { for all 3 ≤ k ≤ m ∃ Kk ∈ (0, ∞) obeying |∇k U| ≤ Kk (1 + |∇U|)k , then we say that U satisfies assumption Am . Then we have Theorem 5.3 ([27]). Let m ≥ 2 be an integer, and let ϵ > 0. Suppose the assumption Am holds. Then, there exist constants K > 0 and γ > 0 such that for all p ∈ [2 − γ, ∞), the

Coercive inequalities on Carnot groups and applications | 307

following inequality holds true: 󵄩p 󵄩 󵄨p 󵄨 ∫󵄨󵄨󵄨∇k U 󵄨󵄨󵄨 |f |p dμ ≤ ϵ󵄩󵄩󵄩∇k f 󵄩󵄩󵄩p + K‖f ‖pp , where f ∈ Wk,p (μ) and k ≤ m. In the current setup we can discuss the boundedness of the Riesz-type transforms for certain probability measures. Let dμ = e−U dλ, where U is a smooth function in ℝn as previous discussion. Define the adjoint of the gradient in the corresponding space by ∇∗ = −∇ + ∇U. Consider the Friedrichs extension of the following Dirichlet operator: L ≡ ∇∗ ⋅ ∇ = − ∑ ∇j2 + ∑ ∇j (U)∇j ≡ −Δ + ∇U ⋅ ∇ j=1,...,n

j=1,...,n

defined initially on the dense set of smooth compactly supported functions. Then the extension, denoted later on by the same symbol L, is a positive, self-adjoint operator 1 for which L 2 is a well-defined, positive, and self-adjoint linear operator. In this section we study the equivalence of norms defined in terms of higher-order derivatives and powers of the operator L, respectively, given as follows with k ∈ ℕ and p ∈ [1, ∞): 󵄩 k 󵄩 ‖f ‖L,k,p ≡ ‖f ‖p + 󵄩󵄩󵄩L 2 f 󵄩󵄩󵄩p

(5.1)

󵄩 󵄩 ‖f ‖∼k,p ≡ ‖f ‖p + 󵄩󵄩󵄩∇k f 󵄩󵄩󵄩p .

(5.2)

and

We have Theorem 5.4 ([27]). The norms (5.1) and (5.2) are equivalent for p ≥ 2. Next we present a few results of applications of Adams’ regularity condition to Orlicz spaces. Let Φ be an Orlicz function and let ‖⋅‖Φ be the corresponding Luxemburg norm. Define a functional MΦ,μ (f ) similar as in Definition 3.1: 󵄩󵄩 󵄩 󵄩󵄩f − MΦ,μ (f )󵄩󵄩󵄩Φ ≡ inf ‖f − a‖Φ . a∈ℝ By the definition of MΦ,μ (f ), we have 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩f − MΦ,μ (f )󵄩󵄩󵄩Φ ≤ 󵄩󵄩󵄩f − μ(f )󵄩󵄩󵄩Φ . The functional is well defined because one has the following property.

308 | E. Bou Dagher et al. Theorem 5.5 ([27]). Suppose that the norm ‖ ⋅ ‖Φ is strictly convex. Then MΦ,μ (f ) is unique, and we have MΦ,μ (sf ) = |s|MΦ,μ (f ),

{

‖(sf + tg) − MΦ,μ (sf + tg)‖Φ ≤ |s|‖f − MΦ,μ (f )‖Φ + |t|‖g − MΦ,μ (g)‖Φ .

Let Φ be an Orlicz function. Then |μ(f ) − MΦ,μ (f )| ≤ Φ−1 (1)‖f − MΦ,μ (f )‖Φ ≤ Φ−1 (1)‖f − μ(f )‖Φ , { { { ‖f − μ(f )‖Φ ≤ (1 + Φ−1 (1) ⋅ ‖𝕀‖Φ )‖f − MΦ,μ (f )‖Φ , { { { {|‖f − MΦ,μ (f )‖Φ − ‖g − MΦ,μ (g)‖Φ | ≤ ‖g − f ‖Φ . By perturbing the measure, we have the following simple estimate for Orlicz norms. Consider a probability measure dν = e−V dμ, defined with ‖V‖∞ < ∞. Then we have einf(V) ‖f ‖Φ,ν ≤ ‖f ‖Φ,μ ≤ esup(V) ‖f ‖Φ,ν . Similar as what we have done with the Poincaré inequalities, we are going to generalize the Orlicz–Sobolev inequalities to higher order. Let k

ℋk ≡ {g ∈ 𝕃Φ (μ) : ∇ g = 0}.

For k ∈ ℕ, define a functional of order k by 󵄩󵄩 󵄩 󵄩󵄩f − Mk,Φ,μ (f )󵄩󵄩󵄩Φ ≡ inf ‖f − g‖Φ,μ . g∈ℋ k

Definition 5.6. We say that a probability measure ν satisfies (Φ − Ψ) Orlicz–Sobolev inequality of order k ∈ ℕ if and only if 󵄩 󵄩 󵄩󵄨 󵄨󵄩 ∃ Ck ∈ (0, ∞) such that 󵄩󵄩󵄩f − Mk,Φ,μ (f )󵄩󵄩󵄩Φ,μ ≤ Ck 󵄩󵄩󵄩󵄨󵄨󵄨∇k f 󵄨󵄨󵄨󵄩󵄩󵄩Ψ,μ holds for any f for which the right-hand side is finite. The merit of choosing the minimizer Mk,Φ,μ (f ) lies in the fact that then we can apply the perturbation of Orlicz norm property mentioned above to the obtain corresponding perturbation property for (Φ − Ψ) Orlicz–Sobolev inequality. We remark that, for the log-Sobolev inequality, the perturbation property was first proved using a very special property of the relative entropy (Rothaus lemma, see, e. g., [17]) which

Coercive inequalities on Carnot groups and applications | 309

is consequence of a convexity inequality for 𝕃p -norms. Later this was generalized in [4, 6, 23] to a few other cases. However, such methods are difficult to generalize to other entropy functions. Next, using Adams’ inequalities mentioned above, together with Poincaré inequality of higher order, we can obtain the following tight Orlicz–Sobolev inequality of higher order. Theorem 5.7 ([27]). Let Φ(t) = t ∏nj=1 (logj (γj + |t|))pj ≡ tΘ(t),

{

ΦA,p (t) = |t|p ∏nj=1 (log∗j (|t|))pj ≡ |t|p A(log∗ t).

Suppose that the following Adams inequality holds: 󵄨 󵄨p μ(ΦA,p (f )) ≤ C̃ A μ󵄨󵄨󵄨∇k f 󵄨󵄨󵄨 + D̃ A ΦA,p (‖f ‖pp ) with some C̃ A , D̃ A ∈ (0, ∞) independent of f . Then 󵄩󵄩 p 󵄩󵄩 ′ 󵄨 k 󵄨p ′ p 󵄩󵄩 |f | 󵄩󵄩Φ ≤ C μ󵄨󵄨󵄨∇ f 󵄨󵄨󵄨 + D ‖f ‖p with some C ′ , D′ ∈ (0, ∞) independent of f . Moreover, if the following (p, k)-Poincaré inequality holds: 󵄩󵄩 󵄩p 󵄨 k 󵄨p 󵄩󵄩(f − Mp,k (f ))󵄩󵄩󵄩p ≤ cp,k μ󵄨󵄨󵄨∇ f 󵄨󵄨󵄨 with some cp,k ∈ (0, ∞) for all f for which the right-hand side is well defined, then the following tight (Φ, p, k)-inequality holds: 󵄩󵄩 󵄨󵄨 󵄨p 󵄩 󵄨 k 󵄨p 󵄩󵄩 󵄨󵄨f − Mp,k (f )󵄨󵄨󵄨 󵄩󵄩󵄩Φ ≤ C μ󵄨󵄨󵄨∇ f 󵄨󵄨󵄨 .

(OSI)

Conversely, (OSI) implies (Φ, p, k)-inequality with a constant cp,k = CΦ−1 (1).

6 Examples to Adam’s regularity condition Recall that Adams’ regularity condition states that ∃ ϵ, C ∈ (0, ∞) such that

2−ϵ 󵄨 󵄨 ∑ 󵄨󵄨󵄨∇α U 󵄨󵄨󵄨 ≤ C(1 + |∇U|) . |α|=2

Before we discuss the higher-dimensional case, recall that on ℝ a necessary and sufficient condition for the log-Sobolev inequality in terms of U was provided in

310 | E. Bou Dagher et al. [6, Theorem 5.5], which essentially amounts to the Adams’ regularity condition. In Example 5.7 there, a very nonconvex function U was provided, for which the log-Sobolev inequality still holds. We would also like to mention an example given in [18], in which Adams’ regularity condition is not satisfied and even Poincaré inequality fails. Let U = β|x|p (1 + ε cos |x|q ) with 0 < |ε| < 1. We have 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 |∇U|2 = 󵄨󵄨󵄨U ′ 󵄨󵄨󵄨 󵄨󵄨󵄨∇|x|󵄨󵄨󵄨 = 󵄨󵄨󵄨U ′ 󵄨󵄨󵄨 = β2 󵄨󵄨󵄨p|x|p−1 (1 + ε cos |x|q ) − q|x|p+q−1 ε sin |x|q 󵄨󵄨󵄨 and ΔU = div(U ′ ∇|x|) = U ′′ + U ′ Δ|x| = β(

2pq|x|p+q−2 ε εq|x|p+2(q−1) p(p − 1)|x|p−1 − − ) + U ′ Δ|x|. q −1 (1 + ε cos |x| ) (sin |x|q )−1 (cos |x|q )−1

For example, if p ≥ 2 or q > 1 and p + 2(q − 1) > 2p then Poincaré inequality fails and Adams’ regularity condition cannot be satisfied as well, no matter how small |ε| > 0 is. Another family of examples of functions not satisfying Adams’ regularity condition in dimension one is as follows. Example 6.1. Consider x

f (x) = xn ∫ 1

sin(t m ) dt √t

for all x > 1.

By Abel–Dirichlet test, we have ∞

∫ 1

sin(t m ) dt < ∞ √t

whence lim f (x) = +∞ x→∞

as n > 0.

Then f (x) = nx ′

n−1

x

∫ 1

1 sin(t m ) dt + x n− 2 sin(x m ) √t

Coercive inequalities on Carnot groups and applications | 311

is bounded when n
n + 21 , then f does not satisfy Adams’ regularity condition. To get an example in a k-dimensional space, we can now simply replace x by the Euclidean distance r = √x12 + ⋅ ⋅ ⋅ + xk2 .

6.1 Examples to Adam’s regularity condition in Carnot groups Here, as studied in [7], we discuss some examples in the context of Carnot groups. In particular, we will demonstrate that the relation of Adams’ regularity with coercive inequalities is rather complicated. First of all, we consider the Heisenberg group ℍ1 ≅ ℝ3 with a group action given by 1 (x, z) ∘ (x′ , z ′ ) = (x + x′ , y + y′ , z + z ′ + ⟨Λx, x′ ⟩) 2 with a matrix 0 1

Λ=(

−1 ). 0

Then we consider the following subgradient and the corresponding sub-Laplacian: ∇ ≡ (X, Y)

and

Δ ≡ X2 + Y 2,

where 1 X = 𝜕x − y𝜕z 2

1 and Y = 𝜕y + x𝜕z . 2

Let 1

N ≡ (|x|4 + 16z 2 ) 4 . Then one has the following well known formulas (see, e. g., [19]) outside the origin: |∇N| =

|x| N

and

ΔN = 3

|x|2 , N3

312 | E. Bou Dagher et al. as well as XN = YN = X2N = Y 2N = XYN = YXN =

1 ((x2 + y2 )x − 4yz), N3 1 ((x2 + y2 )y + 4xz), N3 3 2 2 (x y + 4xz + y3 ) , N7 3 2 (xy2 − 4yz + x3 ) , N7 1 2 (−8z(−x2 y2 − 2y4 + x4 ) − 3xy(x2 + y2 ) + 48xyz 2 + 64z 3 ), 7 N 1 2 (−8z(x2 y2 + 2x4 − y4 ) − 3xy(x 2 + y2 ) + 48xyz 2 − 64z 3 ). N7

Hence we have 2−ε

󵄨󵄨 󵄨2−ε 󵄨 ′ 󵄨2−ε 󵄨 ′ 󵄨2−ε |x| 2−ε 󵄨󵄨∇U(N)󵄨󵄨󵄨 = 󵄨󵄨󵄨U (N)󵄨󵄨󵄨 ⋅ |∇N| = 󵄨󵄨󵄨U (N)󵄨󵄨󵄨 N 2−ε and

󵄨 󵄨 󵄨 󵄨 ∑ 󵄨󵄨󵄨∇α U(N)󵄨󵄨󵄨 = ∑ 󵄨󵄨󵄨U ′ (N)∇α N + U ′′ (N)∇α1 N ⋅ ∇α1 N 󵄨󵄨󵄨. |α|=2

|α|=2

In particular, when x = 0, we get 1 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ∑ 󵄨󵄨󵄨∇α U(N)󵄨󵄨󵄨 = 󵄨󵄨󵄨U ′ (N)󵄨󵄨󵄨 ⋅ (|XYN| + |YXN|) = const ⋅ 󵄨󵄨󵄨U ′ (N)󵄨󵄨󵄨|z|− 2 ,

|α|=2

so we may have a problem with regularity only locally. Otherwise, homogeneity considerations tell us that we can have regularity condition satisfied provided that 󵄨󵄨 ′′ 󵄨󵄨 󵄨 ′ 󵄨2 󵄨󵄨U (N)󵄨󵄨 ≤ C 󵄨󵄨󵄨U (N)󵄨󵄨󵄨 holds for a constant C > 0. As we discussed at the beginning of this paper, within this class one can achieve Poincaré inequality. However, as shown in [18], the log-Sobolev inequality fails in such the case (in fact, for all smooth homogeneous norms). If in the present setup, if we consider the function U(d) defined by a smooth real function U of the control distance d, the situations changes radically. In this case Adams’ regularity condition fails because, by definition, of the control distance is |∇d| = 1 while (as shown in [18] for Heisenberg group) |Δd| diverges to infinity as one approaches the z axis. However, as one shows [18], in this case the log-Sobolev inequality holds (see also [8, 9, 19]).

Coercive inequalities on Carnot groups and applications | 313

7 Multiparticle decay 7.1 Higher-order estimates for Gaussian semigroup Let L=Δ−x⋅∇

and Pt = etL .

Then we have Theorem 7.1. 󵄨󵄨 α 󵄨󵄨2 −2|α| 󵄨󵄨 α 󵄨󵄨2 Pt 󵄨󵄨∇ f 󵄨󵄨 . 󵄨󵄨∇ Pt f 󵄨󵄨 ≤ e Hence with the corresponding invariant Gaussian measure γ, we have 󵄨 󵄨2 󵄨 󵄨2 ∫󵄨󵄨󵄨∇α Pt f 󵄨󵄨󵄨 dγ ≤ e−2|α| ∫󵄨󵄨󵄨∇α f 󵄨󵄨󵄨 dγ. Proof. In the case of interest to us, it is well known that the kernel is smooth. Then, following an idea of Bakry–Emery, we have not only 󵄨 󵄨2 󵄨 󵄨2 𝜕s Pt−s 󵄨󵄨󵄨∇α Ps f 󵄨󵄨󵄨 = Pt−s (−L󵄨󵄨󵄨∇α Ps f 󵄨󵄨󵄨 + 2∇α LPs f ⋅ ∇α Ps f ), but also 󵄨 󵄨2 𝜕s Pt−s 󵄨󵄨󵄨∇α Ps f 󵄨󵄨󵄨 󵄨 󵄨2 = Pt−s ((−L󵄨󵄨󵄨∇α Ps f 󵄨󵄨󵄨 + 2∇α Ps f ⋅ L∇α Ps f ) + 2[∇α , L]Ps f ⋅ ∇α Ps f ). Since [∇, L] = −∇, using the following inductive formula: [∇n , L] ≡ ∇[∇n−1 , L] + [∇, L]∇n−1 , we have [∇n , L] = −n∇n . Hence for a component, we get [∇α , L] = −|α|∇α .

314 | E. Bou Dagher et al. Using the fact that the Markovian form of L satisfies L(g 2 ) − 2gLg = 2|∇g|2 ≥ 0, we obtain the following differential equality: 󵄨2 󵄨 󵄨2 󵄨 𝜕s Pt−s 󵄨󵄨󵄨∇α Ps f 󵄨󵄨󵄨 = −2|α|Pt−s (󵄨󵄨󵄨∇α Ps f 󵄨󵄨󵄨 ), thereby concluding 󵄨󵄨 α 󵄨󵄨2 −2|α| 󵄨󵄨 α 󵄨󵄨2 Pt 󵄨󵄨∇ f 󵄨󵄨 . 󵄨󵄨∇ Pt f 󵄨󵄨 ≤ e Accordingly, with the corresponding invariant Gaussian measure γ, we have 󵄨 󵄨2 󵄨 󵄨2 ∫󵄨󵄨󵄨∇α Pt f 󵄨󵄨󵄨 dγ ≤ e−2|α| ∫󵄨󵄨󵄨∇α f 󵄨󵄨󵄨 dγ.

7.2 Optimal higher-order Poincaré constants Note that for Hermite polynomials Hk , k ∈ ℕ, in one dimension, we have ∇Hk (t) = √kHk−1 (t). and hence ∇n Hk (t) = √k ⋅ ⋅ ⋅ (k − n)Hk−1 (t)

for all k ≥ n.

Thus, using representation f = ∑ fk Hk k∈ℕ

with respect to the orthonormal basis of Hermite polynomials, we compute 󵄨 󵄨2 󵄨 󵄨2 ∫󵄨󵄨󵄨∇n f 󵄨󵄨󵄨 dγ = ∑ k ⋅ ⋅ ⋅ (k − n)|fk−n |2 ≥ n! ∫󵄨󵄨󵄨f − M2,n,γ (f )󵄨󵄨󵄨 dγ, k≥n+1

where in 𝕃2 (γ) we have M2,n,γ (f ) = ∑ fk Hh . k≤n

Thus, we conclude with the following result.

Coercive inequalities on Carnot groups and applications | 315

Theorem 7.2 (The optimal n-th order Poincare inequality). 󵄨2 󵄨 󵄨2 󵄨 n! ∫󵄨󵄨󵄨f − M2,n,γ (f )󵄨󵄨󵄨 dγ ≤ ∫󵄨󵄨󵄨∇n f 󵄨󵄨󵄨 dγ. Comparing this with the downhill induction where we got a bound on a gap given by mn0 only, we see that optimal constants can in general be much better.

7.3 Exponential decay for other drifts Next suppose that we consider a generator of the form L = Δ − v∇. Repeating our standard calculations, with fs ≡ Ps , we have 󵄨 󵄨2 󵄨 󵄨2 𝜕s Pt−s 󵄨󵄨󵄨∇n fs 󵄨󵄨󵄨 = Pt−s (−2󵄨󵄨󵄨∇∇n fs 󵄨󵄨󵄨 − 2∇α fs ⋅ [∇n , v]∇fs ). Under some natural conditions on v, one can still show pointwise exponential decay as follows: 󵄨󵄨 n 󵄨󵄨2 −mt 󵄨 n 󵄨2 󵄨󵄨∇ Pt f 󵄨󵄨 ≤ e Pt 󵄨󵄨󵄨∇ f 󵄨󵄨󵄨 with a constant m ∈ (0, ∞) independent of n ∈ ℕ. However, for the moment the characterization of cases where a better type decay with exponent growing with n is an open question.

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Shihshu Walter Wei

On exponential Yang–Mills fields and p-Yang–Mills fields Abstract: We introduce normalized exponential Yang–Mills energy functional 𝒴ℳ0e , stress–energy tensor Se,𝒴ℳ0 associated with the normalized exponential Yang–Mills energy functional 𝒴ℳ0e , e-conservation law. We also introduce the notion of the e-degree de which connects two separate parts in the associated normalized exponential stress–energy tensor Se,𝒴ℳ0 (cf. (3.10) and (4.15)), derive monotonicity formula for exponential Yang–Mills fields, and prove a vanishing theorem for exponential Yang–Mills fields. These monotonicity formula and vanishing theorem for exponential Yang–Mills fields augment and extend the monotonicity formula and vanishing theorem for F-Yang–Mills fields in [18] and [68, 9.2]. We also discuss an average principle (cf. Proposition 8.1), isoperimetric and Sobolev inequalities, convexity and Jensen’s inequality, p-Yang–Mills fields, an extrinsic average variational method in the calculus of variation and Φ(3) -harmonic maps, from varied, coupled, generalized viewpoints and perspectives (cf. Theorems 6.1, 7.1, 9.1, 9.2, 10.1, 10.2, 11.13, 11.14, and 11.15). Keywords: Normalized exponential Yang–Mills energy functional, stress–energy tensor, e-conservation law, exponential Yang–Mills connection, monotonicity formula, vanishing theorem, exponential Yang–Mills field, p-Yang–Mills field MSC 2010: Primary 58E20, 53C21, 81T13, Secondary 26D15, 53C20

1 Introduction The Yang–Mills functional, brought to mathematics by physics, is broadly analogous to functionals such as the length functional in geodesic theory, the area functional in minimal surface, or minimal submanifold, theory, the energy (resp. p-energy) functional in harmonic (resp. p-harmonic) map theory, or the mass functional in stationary or minimal current, geometric measure theory (cf., e. g., [24, 37, 40]). A critical point of the Yang–Mills functional with respect to any compactly supported variation in the space of smooth connections ∇ on the adjoint bundle is called aYang–Mills connection. Its associated curvature field R∇ is known asYang–Mills field and is “harmonic,” i. e., a harmonic 2-form with values in the vector bundle. The Euler–Lagrange equation for the Yang–Mills functional isYang–Mills equation. Whereas Hodge theory Shihshu Walter Wei, Department of Mathematics, University of Oklahoma, 601 Elm Avenue, # 423, Norman, OK 73072, USA, e-mail: [email protected] https://doi.org/10.1515/9783110741711-018

318 | S. W. Wei of harmonic forms is motivated in part byMaxwell’s equations of unifying magnetism with electricity in the physics world, and harmonic forms are privileged representatives in a de Rham cohomology class picked out by the Hodge Laplacian, harmonic maps can be viewed as a nonlinear generalization of a harmonic 1-form, and Yang– Mills field can be viewed as a nonlinear generalization of harmonic 2-form. On the other hand, Yang–Mills equation, which can be viewed as a nonabelian generalization of Maxwell’s equations, has had wide-ranging consequences, and influenced developments in other fields such as low-dimensional topology, particularly the topology of smooth 4-manifolds. For example, M. Freedman and R. Kirby first observed the startling fact that there exists an exotic ℝ4 , i. e., a manifold homeomorphic to, but not diffeomorphic to, ℝ4 (cf. [39, p. 95], [16, 29, 31]). This is in stunning contrast to a phenomenal theorem of J. Milnor in compact high-dimensional topology which shows that there exist exotic seven-spheres S7 , i. e., manifolds that are homeomorphic to, but not diffeomorphic to, the standard Euclidean S7 (cf. [46]). In [18], we unify the concept of minimal hypersurfaces in Euclidean space ℝn+1 , maximal spacelike hypersurfaces in Minkowski space ℝn,1 , harmonic maps, pharmonic maps, F-harmonic maps, Yang–Mills fields, and introduce F-Yang–Mills fields, F-degree, and generalized Yang–Mills–Born–Infeld fields (with the plus or minus sign) on manifolds, where F : [0, ∞) → [0, ∞) is a strictly increasing C 2 function with F(0) = 0.

(1.1)

When p

F(t) = t, p−1 (2t) 2 , √1 + 2t − 1, and 1 − √1 − 2t, F-Yang–Mills field becomes an ordinary Yang–Mills field, p-Yang-Mills field, a generalized Yang–Mills–Born–Infeld field with the plus sign, and a generalized Yang– Mills–Born–Infeld field with the minus sign on a manifold, respectively (cf. [8–11, 41– 43, 55, 69, 77]). When p

F(t) = t, et , p−1 (2t) 2 , √1 + 2t − 1, and 1 − √1 − 2t , F-harmonic map or the graph of F-harmonic map becomes an ordinary harmonic map, exponentially harmonic map, p-harmonic map, minimal hypersurface in Euclidean space ℝn+1 , and maximal spacelike hypersurface in Minkowski space ℝn,1 , respectively (cf. [3, 21, 22, 69, 72, 73]). We use ideas from physics – stress–energy tensors and conservation laws to simplify and unify various properties in F-Yang–Mills fields, F-harmonic maps, and, more generally, differential k-forms, k ≥ 0, with values in vector bundles. In this paper, we introduce normalized exponential Yang–Mills energy functional 𝒴ℳ0e (resp. exponential Yang–Mills energy functional 𝒴ℳe ), stress–energy tensor Se,𝒴ℳ0 associated with the normalized exponential Yang–Mills energy functional 𝒴ℳ0e

On exponential Yang–Mills fields and p-Yang–Mills fields | 319

(resp. stress–energy tensor Se,𝒴ℳ associated with the exponential Yang–Mills energy functional 𝒴ℳe ) (A critical point of 𝒴ℳ0e , i. e., a normalized exponential Yang–Mills connection, and its associated normalized exponential Yang–Mills field are just the same as the Yang–Mills connection and its associated exponential Yang–Mills field.). We also introduce the notion of the e-degree de which connects two separate parts in the associated normalized exponential stress–energy tensors Se,𝒴ℳ0 (cf. (4.15)). These stress–energy tensors arise from calculating the rate of change of various functionals when the metric of the domain or base manifold is changed, and are naturally linked to various conservation laws. For example, we prove that every normalized exponential Yang–Mills field or every exponential Yang–Mills field R∇ satisfies an e-conservation law (cf. Theorem 3.11). Every normalized exponential Yang–Mills connection or exponential Yang–Mills connection satisfies the exponential Yang–Mills equation (cf. Corollary 3.7). We then prove monotonicity formulae, via the coarea formula and comparison theorems in Riemannian geometry (cf. [18, 32, 33, 68]). Whereas a “microscopic” approach to some of these monotonicity formulae leads to celebrated blow-up techniques due to E. de-Giorgi [15] and W. L. Fleming [28], and regularity theory in geometric measure theory (cf. [1, 2, 24, 35, 44, 51, 52], for example, the regularity results of Allard [1] depend on the monotonicity formulae for varifolds. Monotonicity properties are also dealt with by Price and Simon [51], Price [50] for Yang–Mills fields, and by Hardt–Lin [35] and Luckhaus [44] for p-harmonic maps), a “macroscopic” version of these monotonicity formulae enables us to derive some vanishing theorems under suitable growth conditions on Cartan–Hadamard manifolds or manifolds which possess a pole with appropriate curvature assumptions. In particular, we have Theorem 5.1, the monotonicity formula for exponential Yang–Mills fields, and Theorem 6.1, the vanishing theorem for exponential Yang–Mills fields. These monotonicity formula and vanishing theorem for exponential Yang–Mills fields augment and extend vanishing theorems for F-Yang–Mills fields in [18] and [68]. We note that even when F(t) = et

or F(t) = et − 1

for t =

‖R∇ ‖2 , 2

F-Yang–Mills field becomes exponential Yang–Mills field, the following vanishing theorem for F-Yang–Mills fields is not applicable to exponential Yang–Mills fields. This is due to the fact that for F(t) = et , the degree of F is dF := sup t≥0

tF ′ (t) = ∞, F(t)

and the F-Yang–Mills energy functional growth condition (1.3) is not satisfied for λ = −∞ in (1.4). To overcome this difficulty in getting estimates, we introduce the notion of e-degree de , for a given curvature tensor R∇ (cf. (4.15)).

320 | S. W. Wei Theorem A (Vanishing theorem for F-Yang–Mills fields [18, 68]). Suppose that the radial curvature K(r) of M satisfies one of the seven conditions: (i) −α2 ≤ K(r) ≤ −β2 , with α > 0, β > 0, and (n − 1)β − 4αdF ≥ 0;

(ii) K(r) = 0, with n − 4dF > 0;

B A ≤ K(r) ≤ , with ϵ > 0 , A ≥ 0 , 0 < B < 2ϵ, and (1 + r 2 )1+ϵ (1 + r 2 )1+ϵ A B n − (n − 1) − 4e 2ϵ dF > 0; 2ϵ A A (iv) − 2 ≤ K(r) ≤ − 21 , with 0 ≤ A1 ≤ A, and r r 1 + √1 + 4A1 1 + (n − 1) − 2(1 + √1 + 4A)dF > 0; 2 A (A − 1) A(A − 1) ≤ K(r) ≤ − 1 12 , with A ≥ A1 ≥ 1, and (v) − r2 r 1 + (n − 1)A1 − 4AdF > 0;

(iii) −

(vi)

(vii)

(1.2)

B1 (1 − B1 ) B(1 − B) ≤ K(r) ≤ , with 0 ≤ B, B1 ≤ 1, and r2 r2 󵄨󵄨 1 󵄨󵄨󵄨 1 󵄨 1 + (n − 1)(󵄨󵄨󵄨B − 󵄨󵄨󵄨 + ) − 2(1 + √1 + 4B1 (1 − B1 ))dF > 0; 󵄨󵄨 2 󵄨󵄨 2 B1 B 1 ≤ K(r) ≤ 2 , with 0 ≤ B1 ≤ B ≤ , and 4 r2 r 1 + √1 − 4B 󵄩 󵄩2 − (1 + √1 + 4B1 )󵄩󵄩󵄩R∇ 󵄩󵄩󵄩∞ > 0. 1 + (n − 1) 2

If R∇ ∈ A2 (Ad(P)) is an F-Yang–Mills field satisfying ∫ F( Bρ (x0 )

‖R∇ ‖2 ) dv = o(ρλ ) as ρ → ∞, 2

(1.3)

where λ is such that n − 4 αβ dF { { { { { { n − 4dF { { { A { {n − (n − 1) B − 4e 2ϵ d { F { 2ϵ { { 1+√1+4A1 − 2(1 + √1 + 4A)dF λ ≤ {1 + (n − 1) 2 { { { { 1 + (n − 1)A1 − 4AdF { { { { n−1 { { {1 + (|B− 1 |+2−1 )−1 − 2(1 + √1 + 4B1 (1 − B1 ))dF { 2 { { 1+√1−4B − 2(1 + √1 + 4B1 )dF {1 + (n − 1) 2

if K(r) obeys (i), if K(r) obeys (ii), if K(r) obeys (iii), if K(r) obeys (iv),

(1.4)

if K(r) obeys (v), if K(r) obeys (vi), if K(r) obeys (vii),

then R∇ ≡ 0 on M . In particular, every F-Yang–Mills field R∇ with finite F-Yang–Mills energy functional vanishes on M.

On exponential Yang–Mills fields and p-Yang–Mills fields | 321

We also discuss an average principle (cf. Proposition 8.1) and Jensen’s inequality from varied, generalized viewpoints and perspectives of exponential Yang–Mills fields, p-Yang–Mills fields, and Yang–Mills fields (see Theorems 7.1, 9.1, 9.2, 10.1, and 10.2). In the context of harmonic maps, the stress–energy tensor was introduced and studied in detail by Baird and Eells [6]. Following Baird–Eells [6], Sealey [54] introduced the stress–energy tensor for vector-bundle-valued p-forms and established some vanishing theorems for L2 harmonic p-forms (cf. [17, 53, 76]). In a more general framework, Dong and Wei use a unified method to study the stress–energy tensors and obtain monotonicity inequalities and vanishing theorems for vector-bundle-valued p-forms [18]. The idea and methods can be extended and unified in a σ2 -version of harmonic maps, namely Φ-Harmonic maps (cf. [34]). These are the second elementary symmetric functions of a pullback tensor, whereas harmonic maps are the first elementary symmetric functions of a pullback tensor. More recently, Feng– Han–Li–Wei use stress–energy tensors to unify properties for ΦS -harmonic maps (cf. [26]), Feng–Han–Wei extend and unify results for ΦS,p -harmonic maps (cf. [27]), and Feng–Han–Jiang–Wei further extend and unify results for Φ(3) -harmonic maps (cf. [25]). Whereas we can view harmonic maps as Φ(1) -harmonic maps (involving σ1 ) and Φ-harmonic maps as Φ(2) -harmonic maps (involving σ2 ), Φ(3) -harmonic maps involve σ3 , the third elementary symmetric function of the pullback tensor. In fact, an extrinsic average variational method in the calculus of variations can be carried over to more general settings, so we introduce a notion of an Φ(3) -harmonic map and find a large class of manifolds, Φ(3) -superstrongly unstable (Φ(3) -SSU) manifolds, introduce the notions of a stable Φ(3) -harmonic map and Φ(3) -strongly unstable (Φ(3) -SU) manifolds (cf. Theorems 11.8, 11.9, 11.10, and 11.11). By an extrinsic average variational method in the calculus of variations proposed in [60], we find multiple large classes of manifolds with geometric and topological Table 1: An extrinsic average variational method. Mappings

Functionals

New manifolds found Geometry Topology

harmonic map or energy functional E or Φ(1) -harmonic map EΦ(1)

SSU manifolds or Φ(1) -SSU manifolds

SU or Φ(1) -SU

π1 = π2 = 0 π1 = π2 = 0

p-harmonic map

p-SSU manifolds

p-SU

π1 = ⋅ ⋅ ⋅ = π[p] = 0

Φ-harmonic map or Φ-energy functional EΦ or Φ(2) -harmonic map EΦ(2)

Φ-SSU manifolds or Φ(2) -SSU manifolds

Φ-SU or Φ(2) -SU

π1 = ⋅ ⋅ ⋅ = π4 = 0 π1 = ⋅ ⋅ ⋅ = π4 = 0

ΦS -harmonic map EΦS

ΦS -SSU manifolds

ΦS -SU

π1 = ⋅ ⋅ ⋅ = π4 = 0

ΦS,p -harmonic map EΦS,p

ΦS,p -SSU manifolds

ΦS,p -SU

π1 = ⋅ ⋅ ⋅ = π[2p] = 0

Φ(3) -harmonic map Φ(3) -energy functional EΦ(3) Φ(3) -SSU manifolds

Φ(3) -SU

π1 = ⋅ ⋅ ⋅ = π6 = 0

Ep

322 | S. W. Wei properties in the setting of varied, coupled, generalized type of harmonic maps, and summarize some of the results in Table 1. For some details, related ideas, techniques, we refer to [14, 58–70].

2 Fundamentals of vector bundles and principal G-bundle This section is devoted to a brief discussion of the fundamental notions of vector bundles and principal G-bundle. Definition 2.1. A (differentiable) vector bundle of rank n consists of a total space E, a base M, and a projection π : E → M , where E and M are differentiable manifolds, π is differentiable, each fiber Ex := π −1 (x) for x ∈ M, carries the structure of an n-dimensional (real) vector space, with the following local triviality: For each x ∈ M, there exist a neighborhood U and a diffeomorphism φ : π −1 (U) → U × ℝn such that for every y ∈ U, φy := φ|Ey : Ey → {y} × ℝn is a vector space isomorphism. Such a pair (φ, U) is called a bundle chart. Note that local trivializations φα , φβ with Uα ∩ Uβ ≠ 0 determine transition maps φβα : Uα ∩ Uβ → Gl(n, ℝ) by n φβ ∘ φ−1 α (x, v) = (x, φβα (x)v) for x ∈ M, v ∈ ℝ ,

where Gl(n, ℝ) is the general linear group of bijective linear self-maps of ℝn . As direct consequences, the transition maps satisfy: φαα (x) = idℝn

φαβ (x)φβα (x) = idℝn

φαγ (x)φγβ (x)φβα (x) = idℝn

for x ∈ Uα ;

for x ∈ Uα ∩ Uβ ; for x ∈ Uα ∩ Uβ ∩ Uγ

(cf. [38]). A vector bundle can be reconstructed from its transition maps E = ∐ Uα × ℝn / ∼, α

On exponential Yang–Mills fields and p-Yang–Mills fields | 323

where ∐ denotes disjoint union, and the equivalence relation ∼ is defined by (x, v) ∼ (y, w)

:⇐⇒

x=y

and w = φβα (x)v (x ∈ Uα , y ∈ Uβ , v, w ∈ ℝn ) .

(2.1)

Definition 2.2. Let G be a subgroup of Gl(n, ℝ), for example, the orthogonal group O(n) or special orthogonal group SO(n). By saying that a vector bundle has the structure group G, we mean that there exists an atlas of bundle charts for which all transition maps have their values in G . Definition 2.3. Let G be a Lie group. A principal G-bundle consists of a base M, the total space P of the bundle, and a differentiable projection π : P → M , where P and M are differentiable manifolds, with an action of G on P satisfying: (i) G acts freely on P from the right, that is, (q, p) ∈ P × G is mapped to qp ∈ P , and qp ≠ q for q ≠ e . The G action then defines an equivalence relation on P : p ∼ q :⇐⇒ ∃g ∈ G such that p = qg. (ii) M is the quotient of P by this equivalence relation, and π : P → M maps q ∈ M to its equivalence class. By (i), each fiber π −1 (x) can then be identified with G. (iii) P is locally trivial in the following sense: For each x ∈ M, there exists a neighborhood U of x and a diffeomorphism φ : π −1 (U) → U × G of the form φ(p) = (π(p), ψ(g)) which is G-equivariant, i. e., φ(pg) = (π(p), ψ(p)g) for all g ∈ G. Example 2.4. We have the following results. (i) The projection Sn → P n (ℝ) of the n-sphere to the real projective space is a principal bundle with group G = O(1) = Z2 . (ii) The Hopf map S2n+1 → P n (ℂ) of the (2n + 1)-sphere to the complex projective space is a principal bundle with group G = U(1) = S1 . (iii) The Hopf map S4n+1 → P n (ℚ) of the (4n + 1)-sphere to the quaternionic projective space is a principal bundle with group G = Sp(1) = S3 . (iv) Hopf fibrations S1 → S1 , S3 → S2 , S7 → S4 , and S15 → S8 . For k = 1, 2, 4, 8 , the Hopf construction is defined by (z, w) 󳨃→ u(z, w) = (|z|2 − |w|2 , 2z ⋅ w) : ℝk × ℝk → ℝk+1 . In fact, Hopf fibrations are p-harmonic maps and p-harmonic morphisms for every p > 1 (cf., e. g., [12, 66]).

324 | S. W. Wei We recall that a C 2 map u : M → N is said to be a p-harmonic morphism if for any p-harmonic function f defined on an open set V of N, the composition f ∘ u is p-harmonic on u−1 (V). Example 2.5. If E → M is a vector bundle with fiber V, the bundle of bases of E, B(E) → M is a principle bundle with group Gl(V) .

2.1 Reversibility of principal and vector bundles (󳨐⇒) Given a principal G-bundle P → M and a vector space V on which G acts from the left, we construct the associated vector bundle E → M with fiber V as follows: We have a free action of G on P × V from the right: P × V × G → P × V, (p, v) ⋅ g = (p ⋅ g, g −1 v) . If we divide out this G-action, i. e., identify (p, v) and (p, v) ⋅ g, the fibers of (P × V)/G → P/G become vector spaces isomorphic to V, and E := P ×G V := (P × V)/G → M is a vector bundle with fiber G ×G V := (G × V)/G = V and structure group G . The transition functions for P also give transition functions for E via the left action of G on V. (⇐󳨐) Conversely, given a vector bundle E with structure group G, we construct a principal G-bundle as ∐ Uα × G/ ∼ α

with (xα , gα ) ∼ (xβ , gβ )

:⇐⇒

xα = xβ ∈ Uα ∩ Uβ

and gβ = φβα (x)gα ,

where {Uα } is a local trivialization of E with transition functions φβα as in (2.1). Example 2.6. We have the following assertions: (i) The canonical line bundles (real, complex, and quaternionic) over the projective spaces P n (ℝ) , P n (ℂ), and P n (ℚ) are the associated bundles of the principal bundles in Example 2.4 (i)–(iii) via the canonical actions of O(1), U(1), and Sp(1) on ℝ, ℂ, and ℚ, respectively. (ii) Let E → M be a bundle with fiber F and structure group G, and f : N → M be a map between manifolds N and M. Then the pullback of E → M is a bundle f −1 E → M

On exponential Yang–Mills fields and p-Yang–Mills fields | 325

with fiber F, structure group G, and bundle charts (φ ∘ f , f −1 (U)), where φ(U) are bundle charts of E . The pullback f −1 E → M is called the pullback bundle.

3 Normalized exponential Yang–Mills functionals and e-conservation laws Our basic setup is the following: We consider a Riemannian manifold M, and a principal bundle P with compact structure Lie group G over M. Let Ad(P) be the adjoint bundle Ad(P) = P ×Ad 𝒢 ,

(3.1)

where 𝒢 is the Lie algebra of G. Every connection ρ on P induces a connection ∇ on Ad(P). A connection ∇ on the vector bundle Ad(P) is a rule that allows us to take derivatives of smooth cross-sections of Ad(P). We also have the Riemannian connection ∇M on the tangent bundle TM, and the induced connection on the tensor product Λ2 T ∗ M ⊗ Ad(P), where Λ2 T ∗ M is the second exterior power of the cotangent bundle T ∗ M. An AdG invariant inner product on 𝒢 induces a fiber metric on Ad(P) and makes Ad(P) and Λ2 T ∗ M ⊗ Ad(P) into Riemannian vector bundles. Denote by Γ(Λ2 T ∗ M ⊗ Ad(P)) the (infinite-dimensional) vector space of smooth sections of Λ2 T ∗ M ⊗ Ad(P) . For k ≥ 0, let Ak (Ad(P)) = Γ(Λk T ∗ M ⊗ Ad(P)) be the space of smooth k-forms on M with values in the vector bundle Ad(P). Although ρ is not a section of A1 (Ad(P)), via its induced connection ∇, the associated curvature tensor R∇ , given by R∇X,Y = [∇X , ∇Y ] − ∇[X,Y] , is in A2 (Ad(P)). Let 𝒞 be the space of smooth connections ∇ on Ad(P), and dv be the volume element of M . Recall that the Yang–Mills functional is the mapping 𝒴ℳ : 𝒞 → ℝ+ given by 1 󵄩 ∇ 󵄩2 𝒴ℳ(∇) = ∫ 󵄩󵄩󵄩R 󵄩󵄩󵄩 dv, M

2

(3.2)

326 | S. W. Wei the p-Yang–Mills functional, for p ≥ 2 (resp. the F-Yang–Mills functional) is the mapping 𝒴ℳp : 𝒞 → ℝ+ given by 𝒴ℳp (∇) = ∫ M

1 󵄩󵄩 ∇ 󵄩󵄩p 󵄩R 󵄩 dv p󵄩 󵄩

1 󵄩 󵄩2 (resp. 𝒴ℳF (∇) = ∫ F( 󵄩󵄩󵄩R∇ 󵄩󵄩󵄩 ) dv ), 2

(3.3)

M

where the norm is defined in terms of the Riemannian metric on M and a fixed AdG invariant inner product on the Lie algebra 𝒢 of G . That is, at each point x ∈ M, its norm is 󵄩󵄩 ∇ 󵄩󵄩2 󵄩 ∇ 󵄩2 󵄩󵄩R 󵄩󵄩x = ∑󵄩󵄩󵄩Rei ,ej 󵄩󵄩󵄩x , i 0, β > 0

(i [32])

β coth(βr)(g − dr ⊗ dr) ≤ Hess (r) ≤ α coth(αr)(g − dr ⊗ dr);

(4.1) (ii [32])

1 (g − dr ⊗ dr) = Hess (r); (4.2) r A B − ≤ K(r) ≤ with ϵ > 0, A ≥ 0, and 0 ≤ B < 2ϵ (1 + r 2 )1+ϵ (1 + r 2 )1+ϵ (iii [32], [18, Lemma 4.1.(iii)]) ⇒

⇒

1−

r

B 2ϵ

A

(g − dr ⊗ dr) ≤ Hess (r) ≤

e 2ϵ (g − dr ⊗ dr); r

(4.3)

A A ≤ K(r) ≤ − 21 with 0 ≤ A1 ≤ A (iv [33], [68, Theorem A]) 2 r r 1 + √1 + 4A1 1 + √1 + 4A ⇒ (g − dr ⊗ dr) ≤ Hess (r) ≤ (g − dr ⊗ dr); (4.4) 2r 2r A (A − 1) A(A − 1) − ≤ K(r) ≤ − 1 12 with A ≥ A1 ≥ 1 (v [68, Corollary 3.1]) r2 r A1 A (g − dr ⊗ dr) ≤ Hess (r) ≤ (g − dr ⊗ dr); (4.5) ⇒ r r B1 (1 − B1 ) B(1 − B) ≤ K(r) ≤ , with 0 ≤ B, B1 ≤ 1 (vi [68, Corollary 3.5]) r2 r2 |B − 21 | + 21 1 + √1 + 4B1 (1 − B1 ) ⇒ (g − dr ⊗ dr) ≤ Hess (r) ≤ (g − dr ⊗ dr); (4.6) r 2r B1 B 1 (vii [68, Theorem 3.5]) ≤ K(r) ≤ 2 with 0 ≤ B1 ≤ B ≤ 4 r2 r 1 + √1 + 4B1 1 + √1 − 4B ⇒ (g − dr ⊗ dr) ≤ Hess (r) ≤ (g − dr ⊗ dr); (4.7) 2r 2r 2q 2q − Ar ≤ K(r) ≤ −Br with A ≥ B > 0 , q > 0 (viii [32]) −

⇒

B0 r q (g − dr ⊗ dr) ≤ Hess (r) ≤ (√A coth √A)r q (g − dr ⊗ dr),

for r ≥ 1 , (4.8)

where 2

1

2 q+1 q+1 B0 = min{1, − + (B + ( ) ) }. 2 2

(4.9)

On exponential Yang–Mills fields and p-Yang–Mills fields | 333

Proof. Claims (i), (ii) and (viii) are treated in Section 2 of [32], (iii) is proved in [18], (iv) is derived in [33, 68], (v)–(vii) are proved in [68]. We note there are many applications of this Theorem (cf., e. g., [71]), (iv) extends the asymptotic comparison theorem in [32], [49, p. 39], and (vii) generalizes [23, Lemma 1.2 (b)]. Let ♭ denote the bundle isomorphism that identifies the vector field X with the differential one-form X ♭ , and let ∇ be the Riemannian connection of M. Then the covariant derivative ∇X ♭ of X ♭ is a (0, 2)-type tensor, given by ∇X ♭ (Y, Z) = ∇Y X ♭ Z = ⟨∇Y X, Z⟩,

∀ X, Y ∈ Γ(M).

(4.10)

If X is conservative, then X = ∇f ,

X ♭ = df ,

and ∇X ♭ = Hess(f )

(4.11)

for some scalar potential f (cf. [13, p. 1527]). A direct computation yields (cf., e. g., [18]) div(iX Se,𝒴ℳ0 ) = ⟨Se,𝒴ℳ0 , ∇X ♭ ⟩ + (divSe,𝒴ℳ0 )(X) ,

∀ X ∈ Γ(M).

(4.12)

By Theorem 3.11, every normalized exponential Yang–Mills field R∇ satisfies an e-conservation law. It follows from the divergence theorem that for every bounded domain D in M with C 1 boundary 𝜕D , ∫ Se,𝒴ℳ0 (X, ν)dsg = ∫⟨Se,𝒴ℳ0 , ∇X ♭ ⟩dvg ,

(4.13)

D

𝜕D

where ν is unit outward normal vector field along 𝜕D with (n − 1)-dimensional volume element dsg . When we choose scalar potential f (x) = 21 r 2 (x), (4.11) becomes X = r∇r,

X ♭ = rdr,

1 and ∇X ♭ = Hess( r 2 ) = dr ⊗ dr + rHess (r). 2

(4.14)

The conservative vector field X and e-conservation law will illuminate that the curvature of the base manifold M via Hessian comparison Theorem 4.1 influences the behavior of the stress–energy tensor Se,𝒴ℳ0 and the behavior of the underlying criticality– curvature field R∇ ∈ A2 (Ad(P)) with the help from the following concept (4.15) and estimate (4.20). Analogous to F-degree, we introduce Definition 4.2. For a given curvature field R∇ , the e-degree de is the quantity given by ∇ 2

de = sup x∈M

exp( ‖R2 ‖ (x)) ∇ 2

exp( ‖R2 ‖ (x)) − 1

.

(4.15)

334 | S. W. Wei The e-degree de will play a role in connecting two separated parts of the normalt t ized stress–energy tensor Se,𝒴ℳ0 . Since ete−1 is a decreasing function, with 1 ≤ ete−1 ≤ ∞, we have Proposition 4.3. Suppose ‖R∇ ‖2 (x) ≤ c 2

∀ x ∈ M,

(4.16)

where c > 0 is a constant. Then de ≥

ec . −1

(4.17)

ec

Lemma 4.4. Let M be a complete n-manifold with a pole x0 . Assume that there exist two positive functions h1 (r) and h2 (r) such that h1 (r)(g − dr ⊗ dr) ≤ Hess (r) ≤ h2 (r)(g − dr ⊗ dr)

(4.18)

on M\{x0 }. If h2 (r) satisfies rh2 (r) ≥ 1,

(4.19)

and ‖R∇ ‖ > 0 on M, then ∇ 2

‖R ‖ 󵄩 󵄩2 ) − 1), ⟨Se,𝒴ℳ0 , ∇X ♭ ⟩ ≥ (1 + (n − 1)rh1 (r) − 2de 󵄩󵄩󵄩R∇ 󵄩󵄩󵄩∞ rh2 (r))(exp( 2

(4.20)

where X = r∇r. Proof. Choose an orthonormal frame {ei , 𝜕r𝜕 }i=1,...,n−1 around x ∈ M\{x0 }. Take X = r∇r. Then ∇𝜕 X = 𝜕r

𝜕 , 𝜕r

∇ei X = r∇ei

(4.21) 𝜕 = r Hess (r)(ei , ej )ej . 𝜕r

(4.22)

Using (3.10), (4.14) (or (4.21), (4.22)), we have ⟨Se,𝒴ℳ0 , ∇X ♭ ⟩ = (exp( n−1

n−1 ‖R∇ ‖2 ) − 1)(1 + ∑ rHess (r)(ei , ei )) 2 i=1

− ∑ exp( i,j=1

− exp(

‖R∇ ‖2 )⟨iei R∇ , iej R∇ ⟩rHess (r)(ei , ej ) 2

‖R∇ ‖2 )⟨i 𝜕 R∇ , i 𝜕 R∇ ⟩. 𝜕r 𝜕r 2

(4.23)

On exponential Yang–Mills fields and p-Yang–Mills fields | 335

By (4.18) and (4.15), (4.23) implies that ⟨Se,𝒴ℳ0 , ∇X ♭ ⟩ ≥ (exp(

‖R∇ ‖2 ) − 1)(1 + (n − 1)rh1 (r)) 2

∇ 2

n−1 exp( ‖R2 ‖ ) ‖R∇ ‖2 − (exp( ) − 1) ∑ ⟨iei R∇ , iei R∇ ⟩rh2 (r) ∇ 2 2 exp( ‖R ‖ ) − 1 i=1 ∇ 2

− (exp(

≥ (exp(

‖R ‖ )− 2

∇ 2 exp( ‖R2 ‖ ) ∇ ∇ 1)⟨i 𝜕 R , i 𝜕 R ⟩ ∇ 2 𝜕r 𝜕r exp( ‖R2 ‖ ) −

2

(4.24)

1 ∇ 2

exp( ‖R2 ‖ ) ‖R∇ ‖2 󵄩 󵄩2 ) − 1)(1 + (n − 1)rh1 (r) − 2󵄩󵄩󵄩R∇ 󵄩󵄩󵄩 rh2 (r) ) ∇ 2 2 exp( ‖R ‖ ) − 1 ∇ 2

+ (exp(

‖R ‖ ) − 1)(rh2 (r) − 2

2 ‖R∇ ‖2 exp( 2 ) 1)⟨i 𝜕 R∇ , i 𝜕 R∇ ⟩ ∇ 2 𝜕r 𝜕r exp( ‖R2 ‖ ) − 1 ∇ 2

‖R ‖ 󵄩 󵄩2 ) − 1). ≥ (1 + (n − 1)rh1 (r) − 2󵄩󵄩󵄩R∇ 󵄩󵄩󵄩 rh2 (r)de )(exp( 2 The last two steps follow from (4.19) and the fact that n

n−1

∑ ⟨iei R∇ , iei R∇ ⟩ + ⟨i 𝜕 R∇ , i 𝜕 R∇ ⟩ = ∑ ∑⟨R∇ (ei , ej1 ), R∇ (ei , ej1 )⟩ i=1

𝜕r

𝜕r

1≤j1 ≤n i=1

󵄩 󵄩2 = 2󵄩󵄩󵄩R∇ 󵄩󵄩󵄩 , where en =

𝜕 𝜕r

(4.25)

. Now the lemma follows immediately from (4.24) and (3.7).

5 Monotonicity formulae In this section, we will establish monotonicity formulae on complete manifolds with a pole. Theorem 5.1 (Monotonicity formulae). Let (M, g) be an n-dimensional complete Riemannian manifold with a pole x0 , Ad(P) be the adjoint bundle, and the curvature tensor R∇ ∈ A2 (Ad(P)) be an exponential Yang–Mills field. Assume that the radial curvature K(r) of M and the curvature tensor R∇ satisfy one of the following seven conditions: 󵄩 󵄩2 (i) −α2 ≤ K(r) ≤ −β2 with α > 0, β > 0, and (n − 1)β − 2de α󵄩󵄩󵄩R∇ 󵄩󵄩󵄩∞ ≥ 0; 󵄩 󵄩2 (ii) K(r) = 0, with n − 2de 󵄩󵄩󵄩R∇ 󵄩󵄩󵄩∞ > 0;

336 | S. W. Wei B A ≤ K(r) ≤ , with ϵ > 0 , A ≥ 0 , 0 < B < 2ϵ, and 2 1+ϵ (1 + r ) (1 + r 2 )1+ϵ A B 󵄩 󵄩2 n − (n − 1) − 2de e 2ϵ 󵄩󵄩󵄩R∇ 󵄩󵄩󵄩∞ > 0; 2ϵ A A − 2 ≤ K(r) ≤ − 21 , with 0 ≤ A1 ≤ A , and r r 1 + √1 + 4A1 󵄩 󵄩2 1 + (n − 1) − de (1 + √1 + 4A)󵄩󵄩󵄩R∇ 󵄩󵄩󵄩∞ > 0; 2 A (A − 1) A(A − 1) − ≤ K(r) ≤ − 1 12 , with A ≥ A1 ≥ 1 and r2 r 󵄩 󵄩2 1 + (n − 1)A1 − 2de A󵄩󵄩󵄩R∇ 󵄩󵄩󵄩∞ > 0; B1 (1 − B1 ) B(1 − B) ≤ K(r) ≤ , with 0 ≤ B, B1 ≤ 1, and r2 r2 󵄨󵄨 1 󵄨󵄨󵄨 1 󵄨 󵄩 󵄩2 1 + (n − 1)(󵄨󵄨󵄨B − 󵄨󵄨󵄨 + ) − de (1 + √1 + 4B1 (1 − B1 ))󵄩󵄩󵄩R∇ 󵄩󵄩󵄩∞ > 0; 󵄨󵄨 2 󵄨󵄨 2 B1 B 1 ≤ K(r) ≤ 2 , with 0 ≤ B1 ≤ B ≤ and 2 4 r r 1 + √1 − 4B 󵄩 󵄩2 1 + (n − 1) − de (1 + √1 + 4B1 )󵄩󵄩󵄩R∇ 󵄩󵄩󵄩∞ > 0. 2

(iii) −

(iv)

(v)

(vi)

(vii)

(5.1)

Then 1 ρλ1

∫ (exp( Bρ1 (x0 )

‖R∇ ‖2 1 ) − 1) dv ≤ λ 2 ρ2

∫ (exp( Bρ2 (x0 )

‖R∇ ‖2 ) − 1) dv, 2

(5.2)

for any 0 < ρ1 ≤ ρ2 , where { n − 2de αβ ‖R∇ ‖2∞ { { { { { n − 2de ‖R∇ ‖2∞ { { { A { { {n − (n − 1) B − 2de e 2ϵ ‖R∇ ‖2∞ { 2ϵ { { { 1+√1+4A1 − de (1 + √1 + 4A)‖R∇ ‖2∞ λ ≤ {1 + (n − 1) 2 { { { { 1 + (n − 1)A1 − 2de A‖R∇ ‖2∞ { { { { n−1 ∇ 2 { { {1 + (|B− 21 |+ 21 )−1 − de (1 + √1 + 4B1 (1 − B1 ))‖R ‖∞ { { { { √ 1 + (n − 1) 1+ 21−4B − de (1 + √1 + 4B1 )‖R∇ ‖2∞ {

if K(r) obeys (i), if K(r) obeys (ii), if K(r) obeys (iii), if K(r) obeys (iv), if K(r) obeys (v),

(5.3)

if K(r) obeys (vi), if K(r) obeys (vii).

Proof. Take a smooth vector field X = r∇r on M. If K(r) satisfies (i), then by Theorem 4.1 and since the increasing function αr coth(αr) → 1 as r → 0, (4.19) holds. Now Lemma 4.1 is applicable and, by (4.20), we have on Bρ (x0 )\{x0 } , for every ρ > 0, ⟨Se,𝒴ℳ0 , ∇X ♭ ⟩

∇ 2

‖R ‖ 󵄩 󵄩2 ≥ (1 + (n − 1)βr coth(βr) − 2de αr coth(αr)󵄩󵄩󵄩R∇ 󵄩󵄩󵄩∞ )(exp( ) − 1) 2

On exponential Yang–Mills fields and p-Yang–Mills fields | 337

= (1 + βr coth(βr)(n − 1 − 2de

αr coth(αr) 󵄩󵄩 ∇ 󵄩󵄩2 ‖R∇ ‖2 ) − 1) 󵄩󵄩R 󵄩󵄩∞ ))(exp( βr coth(βr) 2

(5.4)

∇ 2

α 󵄩 󵄩2 ‖R ‖ > (1 + 1 ⋅ (n − 1 − 2de 󵄩󵄩󵄩R∇ 󵄩󵄩󵄩∞ ))(exp( ) − 1) β 2 ≥ λ(exp(

‖R∇ ‖2 ) − 1), 2

provided that α 󵄩 󵄩2 n − 1 − 2de 󵄩󵄩󵄩R∇ 󵄩󵄩󵄩∞ ≥ 0, β since βr coth(βr) > 1

for r > 0,

coth(αr) 0, ⟨Se,𝒴ℳ0 , ∇X ♭ ⟩ ≥ λ(exp(

‖R∇ ‖2 ) − 1) in Bρ (x0 ), 2

‖R∇ ‖2 𝜕 ρ (exp( ) − 1) ≥ Se,𝒴ℳ0 (X, ) 2 𝜕r

on 𝜕Bρ (x0 ) .

(5.5)

It follows from (4.13) and (5.5) that ρ

∫ (exp( 𝜕Bρ (x0 )

‖R∇ ‖2 ‖R∇ ‖2 ) − 1) ds ≥ λ ∫ (exp( ) − 1) dv. 2 2

(5.6)

Bρ (x0 )

Hence, we get from (5.6) the following inequality: ∇ 2

∫𝜕B

ρ (x0 )

(exp( ‖R2 ‖ ) − 1) ds

∇ 2 ∫B (x ) (exp( ‖R2 ‖ ) ρ 0

− 1) dv

≥

λ . ρ

(5.7)

The coarea formula implies that d ‖R∇ ‖2 ) − 1) dv = ∫ (exp( dρ 2 Bρ (x0 )

∫ (exp( 𝜕Bρ (x0 )

‖R∇ ‖2 ) − 1) ds . 2

(5.8)

338 | S. W. Wei Thus we have ∇ 2 d exp( ‖R2 ‖ ) ∫ dρ Bρ (x0 )

∫B

ρ (x0 )

exp(

‖R∇ ‖2 2

− 1 dv

) − 1 dv

≥

λ ρ

(5.9)

for a. e. ρ > 0. Integrating (5.9) over [ρ1 , ρ2 ], we have ln ∫ (exp( Bρ2 (x0 )

‖R∇ ‖2 ‖R∇ ‖2 ) − 1) dv − ln ∫ (exp( ) − 1) dv ≥ ln ρλ2 − ln ρλ1 . (5.10) 2 2 Bρ1 (x0 )

This proves (5.2). Corollary 5.2. Suppose that M has constant sectional curvature −α2 ≤ 0 and n − 1 − 2de ‖R∇ ‖2∞ ≥ 0 if α ≠ 0; { n − 2de ‖R∇ ‖2∞ > 0 if α = 0. Let R∇ ∈ A2 (Ad(P)) be an exponential Yang–Mills field. Then 1

n−2de ‖R∇ ‖2∞

ρ1

≤

1

∫ (exp( Bρ1 (x0 )

n−2de ‖R∇ ‖2∞

ρ2

‖R∇ ‖2 ) − 1) dv 2

‖R∇ ‖2 ) − 1) dv, ∫ (exp( 2

(5.11)

Bρ2 (x0 )

for any x0 ∈ M and 0 < ρ1 ≤ ρ2 . Proof. In Theorem 5.1, if we take α = β ≠ 0 for the case (i) or α = 0 for the case (ii), this corollary follows immediately.

Proposition 5.3. Let (M, g) be an n-dimensional complete Riemannian manifold whose radial curvature satisfies

(viii) −Ar 2q ≤ K(r) ≤ −Br 2q ,

with A ≥ B > 0 and q > 0.

(5.12)

Let R∇ be an exponential Yang–Mills field and 󵄩 󵄩2 δ := (n − 1)B0 − 2de 󵄩󵄩󵄩R∇ 󵄩󵄩󵄩∞ √A coth √A ≥ 0,

(5.13)

On exponential Yang–Mills fields and p-Yang–Mills fields | 339

where B0 is as in (4.9). Suppose that (5.18) holds. Then 1

ρ1+δ 1 ≤

(exp(

∫ Bρ1 (x0 )−B1 (x0 )

1

ρ1+δ 2

∫

‖R∇ ‖2 ) − 1) dv 2

(exp(

Bρ2 (x0 )−B1 (x0 )

‖R∇ ‖2 ) − 1) dv, 2

(5.14)

for any 1 ≤ ρ1 ≤ ρ2 . Proof. Take X = r∇r. Applying Theorem 4.1, (4.19), and (4.20), we have ⟨Se,𝒴ℳ0 , ∇X ♭ ⟩ ≥ (exp(

‖R∇ ‖2 ) − 1)(1 + δr q+1 ) 2

(5.15)

and Se,𝒴ℳ0 (X,

𝜕 ‖R∇ ‖2 ) = exp( )(1 − ⟨i 𝜕 R∇ , i 𝜕 R∇ ⟩) − 1 𝜕r 𝜕r 𝜕r 2

on 𝜕B1 (x0 ),

𝜕 ‖R∇ ‖2 Se,𝒴ℳ0 (X, ) = ρ exp( )(1 − ⟨i 𝜕 R∇ , i 𝜕 R∇ ⟩) − ρ 𝜕r 𝜕r 𝜕r 2

on 𝜕Bρ (x0 ).

(5.16)

It follows from (4.13) that ρ

exp(

∫ 𝜕Bρ (x0 )

−

‖R∇ ‖2 )(1 − ⟨i 𝜕 R∇ , i 𝜕 R∇ ⟩) − 1 ds 𝜕r 𝜕r 2

∫ exp( 𝜕B1 (x0 )

≥

‖R∇ ‖2 )(1 − ⟨i 𝜕 R∇ , i 𝜕 R∇ ⟩) − 1 ds 𝜕r 𝜕r 2

(1 + δr q+1 )(exp(

∫ Bρ (x0 )−B1 (x0 )

(5.17)

‖R∇ ‖2 ) − 1). 2

Whence, if ∫ exp( 𝜕B1 (x0 )

‖R∇ ‖2 )(1 − ⟨i 𝜕 R∇ , i 𝜕 R∇ ⟩) − 1 ds ≥ 0, 𝜕r 𝜕r 2

(5.18)

then ρ

∫ (exp( 𝜕Bρ (x0 )

‖R∇ ‖2 ) − 1) ds ≥ (1 + δ) 2

∫ Bρ (x0 )−B1 (x0 )

(exp(

‖R∇ ‖2 ) − 1) dv, 2

(5.19)

340 | S. W. Wei for any ρ > 1. The coarea formula then implies ∇ 2

d ∫B

ρ (x0 )−B1 (x0 )

(exp( ‖R2 ‖ ) − 1) dv

∇ 2 ∫B (x )−B (x ) (exp( ‖R2 ‖ ) ρ 0 1 0

− 1) dv

≥

1+δ dρ ρ

(5.20)

for a. e. ρ ≥ 1. Integrating (5.20) over [ρ1 , ρ2 ], we get ln(

∫

(exp(

‖R∇ ‖2 ) − 1) dv) 2

∫

(exp(

Bρ2 (x0 )−B1 (x0 )

− ln(

Bρ1 (x0 )−B1 (x0 )

‖R∇ ‖2 ) − 1) dv) 2

(5.21)

≥ (1 + δ) ln ρ2 − (1 + δ) ln ρ1 . Hence we have proved the proposition. Corollary 5.4. Let K(r) and δ be as in Proposition 5.3, satisfying (5.12) and (5.13), respectively, and let R∇ be an exponential Yang–Mills field. Suppose exp(

‖R∇ ‖2 )(1 − ⟨i 𝜕 R∇ , i 𝜕 R∇ ⟩) ≥ 1 𝜕r 𝜕r 2

(5.22)

on 𝜕B1 . Then (5.14) holds. Proof. The assumption (5.22) implies that (5.18) holds, and the assertion follows from Proposition 5.3.

6 Vanishing theorems for exponential Yang–Mills fields Theorem 6.1 (Vanishing theorem). Suppose that the radial curvature K(r) of M satisfies one of the seven growth conditions in (5.1) (i)–(vii) of Theorem 5.1. Let R∇ be an exponential Yang–Mills field satisfying the 𝒴ℳ0e -energy functional growth condition ∫ ig(exp( Bρ (x0 )

‖R∇ ‖2 ) − 1) dv = o(ρλ ) as ρ → ∞, 2 ∇ 2

(6.1)

where λ is given by (5.3). Then exp( ‖R2 ‖ ) ≡ 1 , and hence R∇ ≡ 0. In particular, every exponential Yang–Mills field R∇ with finite normalized exponential Yang–Mills 𝒴ℳ0e -energy functional vanishes on M.

On exponential Yang–Mills fields and p-Yang–Mills fields | 341

Proof. This follows at once from Theorem 5.1. Proposition 6.2. Let (M, g) be an n-dimensional complete Riemannian manifold whose radial curvature satisfies (5.12) (viii) of Proposition 5.1. Let δ be as in (5.13) in which B0 is as in (4.9). Suppose (5.18) holds. Then every exponential Yang–Mills field R∇ , with the growth condition (exp(

∫ Bρ (x0 )−B1 (x0 )

‖R∇ ‖2 ) − 1) dv = o(ρ1+δ ) 2

as ρ → ∞,

(6.2)

vanishes on M − B1 (x0 ). In particular, if R∇ has finite normalized exponential Yang–Mills energy on M − B1 (x0 ), then R∇ ≡ 0 on M − B1 (x0 ). Proof. This follows at once from Proposition 5.3.

7 Vanishing theorems from exponential Yang–Mills fields to F -Yang–Mills fields Theorem 7.1. Suppose that the radial curvature K(r) of M satisfies one of the seven growth conditions in (1.1) (i)–(vii) of Theorem A, in which dF = 1. Let R∇ be an exponential Yang–Mills field with ‖R∇ ‖ = constant and Volume(Bρ (x0 )) = o(ρλ ) as ρ → ∞ ,

(7.1)

where λ is given by (1.4), in which dF = 1. Then R∇ ≡ 0. In particular, every exponential Yang–Mills field R∇ with constant ‖R∇ ‖ over manifold which has finite volume, Volume(M) < ∞, vanishes. Proof. By Corollary 3.8, this exponential Yang–Mills field R∇ is a Yang–Mills field which is a special case of F-Yang–Mills field, where F is the identity map. Thus the F-degree of the identity map dF = 1 . Now we apply F-Yang–Mills vanishing Theorem A in which F(t) = t, dF = 1, the F-Yang–Mills functional 𝒴ℳF growth condition (1.3) is transformed to the volume of the base manifold growth condition (7.1), and the conclusion R∇ ≡ 0 follows.

8 An average principle, isoperimetric and Sobolev inequalities In this section, we state, interpret, and apply an average principle in a simple discrete version, then extend it to a dual (or continuous) version:

342 | S. W. Wei Proposition 8.1 (An average principle of concavity (resp. convexity, linearity)). Let f be a concave function (resp. convex function, linear function). Then f (average) ≥ average (f ),

(resp. f (average) ≤ average (f ),

(8.1)

f (average) = average (f )).

Applying (8.1), where a convex function f = exp and “average” is taken over two positive numbers with respect to the sum, yields one of the simplest inequalities that has far-reaching implications “f (average)”↙

√a ⋅ b = exp( A + B ) 2

(Average principle)

≤

“average(f )”↙

exp A + exp B a + b = . 2 2

(8.2)

That is, we get Example 8.2 (GM ≤ AM). The geometric mean is no greater than the arithmetic mean: √a ⋅ b ≤ a + b , for a, b > 0, 2 with “ = ” holding if and only if a = b .

(8.3)

Indeed, let a = exp(A) and b = exp(B) . Then applying the average principle of convexity (8.1), where f = exp yields A geometric interpretation of this inequality Among all rectangles on the Euclidean plane with a given perimeter ℒ, the square has the largest area 𝒜. By duality, this means in parallel that Among all rectangles on the Euclidean plane with a given area 𝒜, the square has the least perimeter ℒ. Indeed, 16 𝒜 = 16 a ⋅ b ≤ (2a + 2b)2 = ℒ2 . Equality holds if and only if the rectangles are squares, i. e., a = b.

(8.4)

On exponential Yang–Mills fields and p-Yang–Mills fields | 343

A dual approach from discreteness to continuity yields A sharp isoperimetric inequality for plane curves Among all simple closed smooth curves on the Euclidean plane with a given length L, the circle encloses the largest area A: 4πA ≤ L2 .

(8.5)

Equality holds if and only if the curve encloses a disk. This is equivalent to The Sobolev inequality on ℝ2 with optimal constant If u ∈ W 1,1 (ℝ2 ), then 2

4π ∫ |u|2 dx ≤ (∫ |∇u| dx) . ℝ2

(8.6)

ℝ2

Similarly, applying (8.1), where f = exp and “average” is averaging the sum of n positive numbers, n ≥ 2, yields “f (average)”↙ n −1

n √a 1 ⋅ ⋅ ⋅ an = exp(n ∑ Aj )

(Average principle)

j=1

≤

“average(f )”↙ n −1

n

n ∑ exp Aj = n−1 ∑ aj . j=1

j=1

(8.7)

That is, we obtain Example 8.3 (The geometric mean of the numbers is no greater than the arithmetic mean of n positive numbers). a1 + ⋅ ⋅ ⋅ + an , for a1 , . . . , an > 0 , n with “ = ” holding if and only if a1 = ⋅ ⋅ ⋅ = an .

n √a 1 ⋅ ⋅ ⋅ an ≤

(8.8)

For a dual version, consider a concave function f = log. Then the average principle of Proposition 8.1 yields Example 8.4. Let g be a nonnegative measurable function on [0, 1]. Then 1

1

log ∫ g(t) dt ≥ ∫ log(g(t)) dt, 0

whenever the right-hand side is defined.

0

(8.9)

344 | S. W. Wei Isoperimetric and Sobolev inequalities can be generalized to higher dimensional Euclidean spaces. As in dimension two, the n-dimensional sharp isoperimetric inequality is equivalent (for sufficiently smooth domains) to: The Sobolev inequality on ℝn with optimal constant If u ∈ W 1,1 (ℝn ) and ωn is the volume of the unit ball in ℝn , then ( ∫ |u|

n n−1

dx)

n−1 n

≤

1 1 ∫ |∇u| dx. n √n ωn

(8.10)

ℝn

ℝn

Isoperimetric and Sobolev inequalities are extended to Riemannian manifolds M with sharp constants and applications to optimal sphere theorems (cf., e. g., [74]). Theorem 8.5 (A sharp isoperimetric inequality [19, 74]). For every domain Ω (in M), there exists a constant C(M) depending on M such that 2

P n ≥ nn ωn V n−1 (1 − C(M)V n ),

(8.11)

where P = vol(𝜕Ω), V = vol(Ω), and ωn is the volume of the unit ball in Rn . Furthermore, on simply connected Riemannian manifolds of dimension n with Ricci curvature bounded from below by n−1, the best C(M) one can take in the above inequality (8.10) is greater than or equal to C0 =

n(n − 1)

2

2(n + 2)ωnn

(8.12)

.

It is then by a standard technique, via the coarea formula and Cavalieri’s principle, that (8.11) is equivalent to the following: Theorem 8.6 (A sharp Sobolev inequality [74]). There exists a constant A = A(M) such that ∀φ ∈ W 1,1 (M), (∫ |φ| M

n n−1

dv)

n−1 n

≤ K(n, 1) ∫ |∇φ|dv + A(M)(∫ |φ| M

n n+1

dv)

n+1 n

,

(8.13)

M

where K(n, 1) = lim K(n, p) = p→+1

1

1

nωnn

.

This isoperimetric inequality (8.11) certainly has its roots in global analysis and partial differential equations (see, e. g., [4]). Furthermore, the optimal constants in (8.11) will have some geometric and even topological applications. An immediate example is that a sharp estimate on C(M) recaptures

On exponential Yang–Mills fields and p-Yang–Mills fields | 345

Theorem 8.7 (Bernstein isoperimetric inequality [7]). On the 2-sphere S2 , L2 ≥ 4πA(1 −

1 A) 4π

“ = ” holds if and only if the domain in question is a disk.

(8.14)

Remark 8.8. For a generalization of isoperimetric inequality to n-dimensional integer multiplicity rectifiable current in ℝn+k , which follows from the deformation theorem in geometric measure theory, we refer to Federer and Fleming [24].

9 Convexity and Jensen’s inequalities We note that, by Proposition 8.1, every convex function f enjoys the average principle of convexity and Jensen’s inequality in an average sense. From the duality between discreteness and continuity, we consider Jensen’s inequality involving normalized exponential Yang–Mills energy functional 𝒴ℳ0e . Let M be a compact manifold and E be a vector bundle over M. Denote by ℒp1 (E) the Sobolev space of connections of E which are p-integrable and so are their first derivatives. Set p

0

𝒲 (E) = ⋂ ℒ1 (E) ∩ {∇ : 𝒴ℳe (∇) < ∞} . p≥1

(9.1)

Theorem 9.1 (Jensen’s inequality involving normalized exponential Yang–Mills energy functional 𝒴ℳ0e ). Let ∇ be a connection in 𝒲 (E). Then applying (8.1) yields exp(

‖R∇ ‖2 1 ‖R∇ ‖2 1 dv) − 1 ≤ ) − 1) dv, ∫ ∫(exp( Volume(M) 2 Volume(M) 2 M

(9.2)

M

that is, exp(

1 1 0 𝒴ℳ(∇)) − 1 ≤ 𝒴ℳe (∇) . Volume(M) Volume(M)

(9.3)

Equality is valid if and only if ‖R∇ ‖ is constant almost everywhere. Proof. This is a form of Jensen’s inequality for the convex function et −1 (cf. [47, p. 21]). Theorem 9.2. Let ∇ be a minimizer in 𝒲 (E) of the Yang–Mills functional ℳ𝒴 , and the norm ‖R∇ ‖ be constant almost everywhere. Then the same connection ∇ is a minimizer of the normalized exponential Yang–Mills functional 𝒴ℳ0e , and for any minimizer ∇̃ of the

346 | S. W. Wei normalized exponential Yang–Mills functional 𝒴ℳ0e in 𝒲 (E), the norm ‖R∇ ‖ is almost everywhere constant. ̃

Proof. By the definition of minimizer ∇, the monotonicity of t 󳨃→ et − 1, and Jensen’s inequality (9.3), we have for each ∇̃ in 𝒲 (E), exp(

1 1 ̃ −1 𝒴ℳ(∇)) − 1 ≤ exp( 𝒴ℳ(∇)) Volume(M) Volume(M) 1 0 ≤ 𝒴ℳe (∇)̃ . Volume(M)

(9.4)

1 1 0 𝒴ℳ(∇)) − 1 ≤ inf 𝒴ℳe (∇)̃ . ̃ Volume(M) Volume(M) ∇∈𝒲(E)

(9.5)

so that exp(

On the other hand, since ‖R∇ ‖ = constant a. e., 1 ‖R∇ ‖2 1 0 𝒴ℳe (∇) = exp( ) − 1 = exp( 𝒴ℳ(∇)) − 1, Volume(M) 2 Volume(M)

(9.6)

so that ∇ is also a minimizer of the normalized exponential Yang–Mills functional 𝒴ℳ0e . Now we assume that ∇̃ is any minimizer of the normalized exponential Yang–Mills functional 𝒴ℳ0e in 𝒲 (E). Then 1 1 0 0 𝒴ℳe (∇)̃ ≤ 𝒴ℳe (∇) Volume(M) Volume(M)

(9.7)

and combining (9.7), (9.6), and (9.4) allows us to improve all inequalities in (9.4) to ̃ equalities, so that we are ready to apply Theorem 9.1 and conclude that ‖R∇ ‖ is constant almost everywhere.

10 p-Yang–Mills fields Similarly, we set p

p

2

𝒲 (E) = ℒ1 (E) ∩ ℒ1 (E),

and obtain via (8.1)

p≥2

On exponential Yang–Mills fields and p-Yang–Mills fields | 347

Theorem 10.1 (Jensen’s inequality involving p-Yang–Mills energy functional 𝒴ℳp , p ≥ 2). Let ∇ be a connection in 𝒲 p (E). Then p

2 1 2 ‖R∇ ‖2 1 ‖R∇ ‖p ( dv) ≤ ) dv, ∫ ∫( p Volume(M) 2 Volume(M) p

M

(10.1)

M

that is, p

2 1 1 2 ( 𝒴ℳ(∇)) ≤ 𝒴ℳp (∇). p Volume(M) Volume(M)

(10.2)

Equality is valid if and only if ‖R∇ ‖ is constant almost everywhere. p

Proof. This is a form of Jensen’s inequality for the convex function t 󳨃→ p1 (2t) 2 , p ≥ 2 (cf. [47, p. 21]). Theorem 10.2. Let ∇ be a minimizer in 𝒲 p (E) of the Yang–Mills functional ℳ𝒴 , and the norm ‖R∇ ‖ be constant almost everywhere. Then the same connection ∇ is a minimizer of the p-Yang–Mills functional 𝒴ℳp , and for any minimizer ∇̃ of the p-Yang–Mills functional 𝒴ℳp in 𝒲 p (E), the norm ‖R∇ ‖ is almost everywhere constant. ̃

Proof. By the definition of minimizer ∇, and Jensen’s inequality (10.2), we have for each ∇̃ in 𝒲 p (E), p

p

2 2 1 2 1 2 ̃ ( 𝒴ℳ(∇)) ≤ ( 𝒴ℳ(∇)) p Volume(M) p Volume(M) 1 ̃ 𝒴ℳp (∇), ≤ Volume(M)

(10.3)

so that p

2 1 2 1 ̃ ( 𝒴ℳ(∇)) ≤ inf 𝒴ℳp (∇). p (E) Volume(M) ̃ p Volume(M) ∇∈𝒲

(10.4)

On the other hand, since ‖R∇ ‖ = constant a. e., p

2 1 ‖R∇ ‖p 1 2 𝒴ℳp (∇) = = ( 𝒴ℳ(∇)) , Volume(M) p p Volume(M)

(10.5)

so that ∇ is also a minimizer of the p-Yang–Mills functional 𝒴ℳp . Now we assume ∇̃ is any minimizer of the p-Yang–Mills functional 𝒴ℳp in 𝒲 p (E). Then 1 1 𝒴ℳp (∇)̃ ≤ 𝒴ℳp (∇) Volume(M) Volume(M)

(10.6)

348 | S. W. Wei and combining (10.6), (10.5), and (10.3) allows us to improve all inequalities in (10.3) to ̃ equalities, so that we are ready to apply Theorem 9.1 and conclude that ‖R∇ ‖ is constant almost everywhere. Remark 10.3. J. Eells and L. Lemaire first derived Jensen’s inequality and established its optimality in the setting of exponentially harmonic maps [21]. F. Matsuura and H. Urakawa showed exp(

𝒴ℳe (∇) 𝒴ℳ(∇) )≤ Volume(M) Volume(M)

for any ∇ ∈ 𝒲 (E) ,

and the validity of equality [45].

11 An extrinsic average variation method and Φ(3) -harmonic maps We propose an extrinsic, average variational method as an approach to confront and resolve problems in global nonlinear analysis and geometry (cf. [58, 60]). In contrast to an average method in PDE that we applied in [14] to obtain sharp growth estimates for warping functions in multiply warped product manifolds, we employ an extrinsic average variational method in the calculus of variations [60], find a large class of manifolds of positive Ricci curvature that enjoy rich properties, and introduce the notions of superstrongly unstable (SSU) manifolds and p-superstrongly unstable (p-SSU) manifolds [58, 61, 63, 73]. Definition 11.1. A Riemannian manifold M with its Riemannian metric ⟨⋅, ⋅⟩M is said to be superstrongly unstable (SSU), if there exists an isometric immersion of M in (ℝq , ⟨ ⋅ ⟩ℝq ) with its second fundamental form B such that for every unit tangent vector v to M at every point x ∈ M, the following symmetric linear operator QM x is negative definite: m

⟨QM x (v), v⟩M = ∑(2⟨B(v, ei ), B(v, ei )⟩ℝq − ⟨B(v, v), B(ei , ei )⟩ℝq ) i=1

(11.1)

and M is said to be p-superstrongly unstable (p-SSU) for p ≥ 2 if the following functional is negative valued: Fp,x (v) = (p − 2)⟨B(v, v), B(v, v)⟩ℝq + ⟨QM x (v), v⟩M ,

(11.2)

where {e1 , . . . , em } is a local orthonormal frame on M. We prove, in particular, that every compact SSU manifold must be strongly unstable (SU), i. e., (a) a compact SSU manifold cannot be the target of any nonconstant

On exponential Yang–Mills fields and p-Yang–Mills fields | 349

stable harmonic maps from any manifold, (b) the homotopic class of any map from any manifold into a compact SSU manifold contains elements of arbitrarily small energy E, (c) a compact SSU manifold cannot be the domain of any nonconstant stable harmonic map into any manifold, and (d) the homotopic class of any map from a compact SSU manifold into any manifold contains elements of arbitrarily small energy E (cf. [36, Theorem 2.2, p. 321]).

11.1 Harmonic maps and p-harmonic maps, from a viewpoint of the first elementary symmetric function σ1 We recall at any fixed point x0 ∈ M, a symmetric 2-covariant tensor field α on (M, g) in general, or the pullback metric u∗ in particular, has the eigenvalues λ relative to the metric g of M, i. e., the m real roots of the equation det(gij λ − αij ) = 0

where gij = g(ei , ej ), αij = α(ei , ej ),

and {e1 , . . . , em } is a basis for Tx0 (M) (cf., e. g., [34]). A harmonic map u : (M, g) → (N, h) can be viewed as a critical point of the energy functional, given by the integral of a half of the first elementary symmetric function σ1 , of eigenvalues relative to the metric g, or the trace of the pullback metric tensor u∗ h, with respect to g, where {e1 , . . . , em } is a local orthonormal frame field on M, that is, E(u) = ∫ M

1 1 m ∑ h(du(ei ), du(ei )) dv = ∫ (σ1 (u∗ )) dv. 2 i=1 2

(11.3)

M

A p-harmonic map can be viewed as a critical point of the p-energy functional Ep (u), given by the integral of p1 times σ1 or the trace of the pullback metric tensor to the power p2 , i. e., m

p 2

p 1 1 E(u) = ∫ (∑ h(du(ei ), du(ei ))) dv = ∫ (σ1 (u∗ )) 2 dv. p i=1 p

M

(11.4)

M

For the study of the stability of harmonic maps (resp. p-harmonic maps), Howard and Wei [36] (resp. Wei and Yau [73]) introduce the following notions: Definition 11.2. A Riemannian manifold M is said to be strongly unstable (SU) (resp. p-strongly unstable (p-SU)) if M is neither the domain nor the target of any nonconstant smooth stable harmonic map (resp. stable p-harmonic map), and the homotopic class of maps from or into M contains a map of arbitrarily small energy E (resp. p-energy Ep ).

350 | S. W. Wei This definition leads to Theorem 11.3. Every compact superstrongly unstable (SSU)-manifold (resp. p-superstrongly unstable (p-SSU)) manifold is strongly unstable (SU) (resp. p-strongly unstable (p-SU)). And, we make the following classification. Theorem 11.4 ([37, 48]). Let M be a compact irreducible symmetric space. The following statements are equivalent: (1) M is SSU. (2) M is SU. (3) M is U, i. e., IdM is an unstable harmonic map. (4) M is one of the following: (i) the simply connected simple Lie groups (Al )l≥1 , B2 = C2 , and (Cl )l≥3 ;

(ii) SU(2n)/Sp(n), n ≥ 3;

(iii) spheres Sk , k > 2; (iv) (v) (vi)

quaternionic Grassmannians Sp(m + n)/Sp(m) × Sp(n), m ≥ n ≥ 1;

(11.5)

E6 /F4 ;

Cayley plane F4 /Spin(9).

Theorem 11.5 (Topological vanishing theorem). Suppose that M is a compact SSU (resp. p-SSU) manifold. Then M is SU and π1 (M) = π2 (M) = 0,

(resp. π1 (M) = ⋅ ⋅ ⋅ = π[p] = 0).

(11.6)

Furthermore, the following three statements are equivalent: (a) π1 (M) = π2 (M) = 0.

(b) the infimum of the energy E is 0 among maps homotopic to the identity on M. (c) the infimum of the energy E is 0 among maps homotopic to a map from M.

That is, π1 (M) = π2 (M) = 0

[75]

⇐⇒ [20]

⇐⇒

inf{E(u′ ) : u′ is homotopic to Id on M}, inf{E(u ) : u is homotopic to u : M → ∙}.

(Cf. [60, the diagram on p. 58].)

′

′

(11.7)

(11.8)

On exponential Yang–Mills fields and p-Yang–Mills fields | 351

11.2 Φ-harmonic maps, from a viewpoint of the second elementary symmetric function σ2 [34] We introduce the notion of a Φ-harmonic map which is the second symmetric function σ2 of the pullback metric tensor u∗ h, an analogue of σ1 in the above Section 11.1. In [34], Han and Wei show that the extrinsic average variational method in the calculus of variations employed in the study of harmonic maps, p-harmonic maps, F-harmonic maps, and Yang–Mills fields can be extended to the study of Φ-harmonic maps. In fact, we find a large class of manifolds with rich properties, Φ-superstrongly unstable (Φ-SSU) manifolds, establish their links to p-SSU manifolds and topology, and apply the theory of p-harmonic maps, minimal varieties, and Yang–Mills fields to study such manifolds. With the same notations as above, we introduce the following notions: Definition 11.6. A Riemannian manifold (M m , g) with a Riemannian metric g is said to be Φ-superstrongly unstable (Φ-SSU) if there exists an isometric immersion ℝq such that, for all unit tangent vectors v to M m at every point x ∈ M m , the following functional is always negative: m

FΦx (v) = ∑(4⟨B(v, ei ), B(v, ei )⟩ − ⟨B(v, v), B(ei , ei )⟩), i=1

(11.9)

where B is the second fundamental form of M m in ℝq , and {e1 , . . . , em } is a local orthonormal frame on M near x. Definition 11.7. A Riemannian manifold M is Φ-strongly unstable (Φ-SU) if it is neither the domain nor the target of any nonconstant smooth Φ-stable stationary map, and the homotopic class of maps from or into M contains a map of arbitrarily small energy. Theorem 11.8. Every compact Φ-superstrongly unstable (Φ-SSU) manifold is Φstrongly unstable (Φ-SU).

11.3 ΦS -harmonic maps, from a viewpoint of an extended second symmetric function σ2 [26] We introduce the notion of a ΦS -harmonic map, which is a σ2 version of the stress– energy tensor S. In [26], Feng, Han, Li, and Wei show that the extrinsic average variational method in the calculus of variations employed in the study of σ1 and σ2 versions of the pullback metric u∗ h on M can be extended to the study of a σ2 version of the stress–energy tensor S. In fact, we find a large class of manifolds, ΦS -superstrongly unstable (ΦS -SSU)

352 | S. W. Wei manifolds, introduce the notions of a stable ΦS harmonic map, ΦS -strongly unstable (ΦS -SU) manifolds, and prove Theorem 11.9. Every compact ΦS -superstrongly unstable (ΦS -SSU) manifold is ΦS strongly unstable (ΦS -SU).

11.4 ΦS,p -harmonic maps, from a viewpoint of a combined extended second symmetric function σ2 [27] We introduce the notion of a ΦS,p -harmonic map, which is a combined generalized σ2 version of the stress–energy tensor S, and a σ1 version of the pullback u∗ . In [26], Feng, Han, Li, and Wei show that the extrinsic average variational method in the calculus of variations employed in the study of σ1 and σ2 versions of the pullback metric u∗ h on M and stress–energy tensor can be extended to the study of a combined extended second symmetric function σ2 version. In fact, we find a large class of manifolds, ΦS,p -superstrongly unstable (ΦS,p -SSU) manifolds, introduce the notions of a stable ΦS,p -harmonic map, ΦS,p -strongly unstable (ΦS,p -SU) manifolds, and prove Theorem 11.10. Every compact ΦS,p -superstrongly unstable (ΦS,p -SSU) manifold is ΦS,p -strongly unstable (ΦS,p -SU) .

11.5 Φ(3) -harmonic maps, from a viewpoint of the third elementary symmetric function σ3 [25] We introduce the notion of a Φ(3) -harmonic map, which is a σ3 version of the pullback u∗ . In fact, Feng, Han, Jiang, and Wei show that the extrinsic average variational method in the calculus of variations employed in the study of σ1 and σ2 versions of the pullback metric u∗ h on M can be extended to the study of the third symmetric function σ3 version. Whereas we can view harmonic maps as Φ(1) -harmonic maps (involving σ1 ) and Φ-harmonic maps as Φ(2) -harmonic maps (involving σ2 ), we introduce the notion of a Φ(3) -harmonic map and find a large class of manifolds, Φ(3) -superstrongly unstable (Φ(3) -SSU) manifolds, introduce the notions of a stable Φ(3) -harmonic map, Φ(3) -strongly unstable (Φ(3) -SU) manifolds, and prove Theorem 11.11 ([25]). Every compact Φ(3) -superstrongly unstable (Φ(3) -SSU) manifold is Φ(3) -strongly unstable (Φ(3) -SU). Definition 11.12 ([25]). A Riemannian manifold M m is said to be Φ(3) -superstrongly unstable (Φ(3) -SSU) if there exists an isometric immersion of M m in ℝq with its second fundamental form B such that for all unit tangent vectors v to M m at every point

On exponential Yang–Mills fields and p-Yang–Mills fields | 353

x ∈ M m , the following functional is negative valued: m

FΦ(3) (v) = ∑(6⟨B(v, ei ), B(v, ei )⟩ℝq − ⟨B(v, v), B(ei , ei )⟩ℝq ), x

i=1

(11.10)

where {e1 , . . . , em } is a local orthonormal frame field on M m near x. Theorem 11.13. Every Φ(3) -SSU manifold M is p-SSU for any 2 ≤ p ≤ 6. Proof. By Definition 11.12, Φ(3) -SSU manifold enjoys m

FΦ(3) (v) = ∑(6⟨B(v, ei ), B(v, ei )⟩ℝq − ⟨B(v, v), B(ei , ei )⟩ℝq ) < 0 x

i=1

(11.11)

for all unit tangent vector v ∈ Tx (M). It follows from (11.2) and (11.10) that Fp,x (v) = (p − 2)⟨B(v, v), B(v, v)⟩ℝq + ⟨QM x (v), v⟩M n

≤ (p − 2) ∑(2⟨B(v, ei ), B(v, ei )⟩ℝq ) n

i=1

+ ∑(2⟨B(v, ei ), B(v, ei )⟩ℝq − ⟨B(v, v), B(ei , ei )⟩ℝq ) n

i=1

(11.12)

≤ ∑(p⟨B(v, ei ), B(v, ei )⟩ − ⟨B(v, v), B(ei , ei )⟩) i=1 n

≤ ∑(6⟨B(v, ei ), B(v, ei )⟩ − ⟨B(v, v), B(ei , ei )⟩) < 0, i=1

for 2 ≤ p ≤ 6. So, by Definition 11.1, M is p-SSU for any 2 ≤ p ≤ 6. Theorem 11.14. Every compact Φ(3) -SSU manifold M is 6-connected, i. e., π1 (M) = ⋅ ⋅ ⋅ = π6 (M) = 0.

(11.13)

Proof. Since every compact p-SSU is [p]-connected (cf. [63, Theorem 3.10, p. 645]) and p = 6, by the previous theorem, the result follows. Theorem 11.15 (Sphere theorem). Every compact Φ(3) -SSU manifold M of dimension m ≤ 13 is homeomorphic to an m-sphere. Proof. In view of Theorem 11.13, M is 6-connected. By the Hurewicz isomorphism theorem, the 6-connectedness of M implies that homology groups satisfy H1 (M) = ⋅ ⋅ ⋅ = H6 (M) = 0. It follows from Poincaré duality theorem and the Hurewicz isomorphism theorem [57] again that Hm−6 (M) = ⋅ ⋅ ⋅ = Hm−1 (M) = 0, Hm (M) ≠ 0, and M is (m − 1)connected. Hence M is a homotopy m-sphere. Since M is Φ(2) -SSU, m ≥ 7. Conse-

354 | S. W. Wei quently, a homotopy m-sphere M for m ≥ 5 is homeomorphic to an m-sphere by a theorem of Smale [56]. We summarize some of new manifolds found and these results obtained by an extrinsic average method in Table 1 of Section 1.

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Index (0, n) ∋ αth order Riesz potential operator 49

hyperbolic space 196

Adams’ regularity conditions 305 affine p-Laplacian 35 alternating direction multiplier method 108 approximate solutions 89

Kohn Laplacian 1

bilinear Hilbert transform 124 boundary value problem 221 Calderón–Zygmund theory 224 Calderón’s reproducing formula 14 Cauchy measures 136 Chandrasekhar kernels 91 coercive inequalities 291 conormal derivatives 220 conservation laws 318 Dirichlet Laplacian 195 doubling 81 Fisher information 136 Fredholm integral equation 91 Gaussian variable Lebesgue spaces 265 general Gaussian singular integrals 261 generalized soft threshold algorithm 108 H-function 89 Hardy spaces 50 Hardy spaces Hp (Mm ) 1 Hardy–Littlewood property 153 heat equation 210 heat kernel 14 higher order Poincaré inequalities 292

https://doi.org/10.1515/9783110741711-019

Lipschitz map 253 Marcinkiewicz integral operators 167 model domains 5 Morrey space 50 NIS operators 20 noncompact complete connected Riemannian manifolds 207 nontangential maximal operator 220 Ornstein–Uhlenbeck semigroup 268 p-Talenti inequality 33 principal eigenvalue 195 proximity operator 108 Rényi entropy power 136 Riesz transforms 223 singular integral operators 224 Sobolev space 168 Sobolev-type inequalities 135 stress–energy tensors 318 two-fluid model 281 volume growth 207 weak–strong uniqueness 283

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