Gaussian capacity analysis 9783319950396, 9783319950402

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Lecture Notes in Mathematics  2225

Liguang Liu Jie Xiao Dachun Yang   Wen Yuan

Gaussian Capacity Analysis

Lecture Notes in Mathematics Editors-in-Chief: Jean-Michel Morel, Cachan Bernard Teissier, Paris Advisory Board: Michel Brion, Grenoble Camillo De Lellis, Princeton Alessio Figalli, Zurich Davar Khoshnevisan, Salt Lake City Ioannis Kontoyiannis, Athens Gábor Lugosi, Barcelona Mark Podolskij, Aarhus Sylvia Serfaty, New York AnnaWienhard, Heidelberg

2225

More information about this series at http://www.springer.com/series/304

Liguang Liu • Jie Xiao • Dachun Yang • Wen Yuan

Gaussian Capacity Analysis

123

Liguang Liu School of Mathematics Renmin University of China Beijing, China Dachun Yang Laboratory of Mathematics and Complex Systems (Ministry of Education of China) School of Mathematical Sciences Beijing Normal University Beijing, China

Jie Xiao Department Mathematics & Statistics Memorial University St. John’s, Newfoundland and Labrador Canada Wen Yuan Laboratory of Mathematics and Complex Systems (Ministry of Education of China) School of Mathematical Sciences Beijing Normal University Beijing, China

ISSN 0075-8434 ISSN 1617-9692 (electronic) Lecture Notes in Mathematics ISBN 978-3-319-95039-6 ISBN 978-3-319-95040-2 (eBook) https://doi.org/10.1007/978-3-319-95040-2 Library of Congress Control Number: 2018951056 Mathematics Subject Classification: 31B15, 31B35, 42B35, 52A38, 53A07, 53C65, 60D05 © Springer Nature Switzerland AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface This monograph Gaussian Capacity Analysis documents the functional p-capacity and its extreme BV -capacity in Gauss space with applications to tracing the Gaussian Sobolev p-space and its extreme BV -space as well as the induced geometric structure. Recall that the Gauss space Gn is the 1 ≤ n-dimensional Euclidean space Rn equipped with the standard Euclidean metric  n  12    |x − y| = (x j − y j )2 ∀ x = (x 1 , . . . , x n ), y = (y1 , . . . , yn ) ∈ Rn × Rn j=1

and the canonical Gaussian measure density n

γ(x) := (2π)− 2 e −

|x | 2 2

∀ x ∈ Rn .

Denote by dV and dVγ = γdV the Lebesgue volume element and the Gaussian volume element respectively. Note that Vγ is a canonical probability measure, i.e., ∫ Vγ (Rn ) = γ(x) dV (x) = 1. Rn

The Gauss space arises from probability theory, quantum mechanics, and differential geometry; see [8, 43, 44, 45, 26, 42], and the references therein. If Cc0 (Rn ) represents the class of continuous functions with compact support in n R , then for any integer k ≥ 1 the class Cck (Rn ) consists of all k-order continuously differentiable functions with compact support in Rn . Given p ∈ [1, ∞). It was proved by Pisier [47, Corollary 2.4] that p ∫

p1 ∫

p1 ∫ p f dVγ dVγ ≤ Cp,n |∇f | dVγ ∀ f ∈ Cc1 (Rn ), () f − Rn

Rn

Rn

where Cp,n is a positive constant depending only on n and p; see also Ledoux [34, pp. 212–213]. As is well known, this last inequality can be regarded as the Gaussian Poincaré p-inequality . Here, it is worth mentioning that ()-like inequalities over Gn were investigated by many authors from different perspectives such as probability, partial differential equation, complex analysis, and convex geometry. In particular, when p = 1, via a probability method Ledoux [34, p. 279] obtained () with a sharp con stant Cp,n = π2 . When p = 2, the Gaussian Poincaré 2-inequality is closely related to the first nontrivial eigenvalue of the following problem (see [12]): 



 2  2 −div exp(− |x2| ) ∇u(x) = λ exp(− |x2| ) u(x) in Ω; ∂u ∂ν

=0

on ∂Ω, v

vi

Preface

where Ω is a bounded convex domain of Rn and ν stands for the outward normal to ∂Ω. When p ∈ [2, ∞), () was also derived by Zeng [58, Proposition 2.1] with √ Cp,n = C p. Observe that an equivalent statement of () is as follows: there exists a positive constant Cp,n such that ∫ ()

Rn

p

| f | dVγ

p1

 ∫ ≤ Cp,n

Rn

p

|∇f | dVγ

p1

∫ +

Rn

 f dVγ ∀ f ∈ Cc1 (Rn ).

So, it is natural to ask the following restriction/trace question: Given q ∈ (0, ∞), for which nonnegative Borel measure μ on Rn does the inequality ∫ (  )

Rn

q

| f | dμ

q1

 ∫ ≤C

Rn

p

|∇f | dVγ

p1

∫ +

Rn

 f dVγ

hold uniformly over suitable functions f with a positive constant C independent of f ? In order to answer this question, we introduce and study the so-called Gaussian p-capacity, thereby completing this monograph of five chapters.  After exploring the basic structure of a Gaussian Sobolev p-space in Chapter 1, it is easy to observe that the right sides of () and () are equivalent to the Gaussian Sobolev norm  · W 1,p (Gn ) induced by the right side of ().  Meanwhile, the left side of () (coupled with [36, 37]) leads to Chapter 2 an investigation of the Gaussian Campanato class and its relationship to the Morrey or John-Nirenberg or Lipschitz class on Gn .  By the phenomena appeared in Chapters 1–2, it comes quite natural in Chapter 3 that the Gaussian p-capacity of a compact set K can be defined in the following way:   p inf  f W 1,p (Gn ) : f ∈ Cc1 (Rn ) with f ≥ 1 on K . Accordingly, the fundamental properties of this new capacity will be presented over there.  Via this new capacity, we shall later in Chapter 4 characterize such a nonnegative Radon measure μ on Rn obeying (  ).  As in [12, 27, 53], the Gaussian 1-capacity of fundamental importance can be characterized by the Gaussian Minkowski content of the boundary of a set consequently - not only an equivalence between the Cheeger’s isoperimetric inequality on Gn and the Gaussian 1-Poincaré inequality can be discovered

Preface

vii

to reprove () under p = 1, but also Ehrhard’s inequality and its immediate product - the Gaussian isoperimetry can be induced; and moreover the Gaussian ∞-capacity will be discussed as a dual form of the Gaussian 1-capacity – see Chapter 5.  As the weakest formulation of the Gaussian 1-capacity, the Gaussian BV capacity is addressed in Chapter 6 which is from a suitable modification of [53] and its main reference [27]. Briefly and historically speaking, the concept of a functional capacity is originated from electrostatics. The study of an electrical capacitance was started by Wiener in 1924 and has achieved a great development from then on; see, e.g., Pólya [48], Choquet [14], and Shubin [50]. The functional capacities are of fundamental importance in various branches of mathematics such as analysis, geometry, mathematical physics, partial differential equations, and probability theory; see, e.g., [2, 5, 10, 20, 41, 18, 59, 15, 16, 27, 30, 31, 54, 55, 21, 22, 38, 7, 29, 19]. Needless to say, this monograph is a new development of the geometric potential analysis based on the Gauss space and will be definitely useful for researchers working in the above-mentioned fields and their relatives. Liguang Liu was partially supported by NNSF of China Grant # 11771446 & #11761131002; Jie Xiao was partially supported by NSERC of Canada Grant # 20171864; Dachun Yang was partially supported by NNSF of China Grant # 11571039 & #11671185 & #11761131002 & #11726621; Wen Yuan was partially supported by NNSF of China Grant # 11471042 & #11871100 & #11761131002 & #11726621. Last but not least, we would like to point out that the following conventions will be used throughout this monograph.  N := {1, 2, 3, . . .} & Z = {0, ±1, ±2, . . .} ;  For any set E ⊂ Rn denote by E ◦ , E c = Rn \ E , and 1E the interior, the complement, and indicator of E respectively;  For k + 1 ∈ N ∪ {∞} the symbol Cck (Rn ) is the class of all functions f : Rn → R with k continuous partial derivatives (written as f ∈ C k (Rn ) ) and compact support, and ⎧ Cck (Rn ; Rm ) := Cck (Rn ) × · · · × Cck (Rn ); ⎪ ⎪ ⎪  ⎪ ⎪ ⎨ ⎪ m copies

⎪ C k (Rn ; Rm ) := C k (Rn ) × · · · × C k (Rn ); ⎪ ⎪ ⎪  ⎪ ⎪ m copies ⎩  C represents a positive constant depending only on the main parameters involved. Occasionally, C α,β, ... indicates that C depends on the parameters α, β, . . . ;

viii

Preface

 Given any two nonnegative quantities X and Y , the symbol X Y means X ≤ CY (denoted by X  Y ) and C −1X ≥ Y (denoted by X  Y ) for a positive constant C. Beijing, China St. John’s, Canada Beijing, China Beijing, China December 2017–March 2018

Liguang Liu Jie Xiao Dachun Yang Wen Yuan

Contents

1 Gaussian Sobolev p-Space 1.1 Definition and Approximation of W 1,p (Gn ) . . . 1.2 Approximating W 1,p (Gn )-Function with Cancellation . . . . . . . . . . . . . . . . . . . . 1.3 Compactness for W 1,p (Gn ) . . . . . . . . . . . . 1.4 Poincaré or Log-Sobolev Inequality for W 1, 2 (Gn )

. . . . . . . . . . .

1 1

. . . . . . . . . . . 8 . . . . . . . . . . . 12 . . . . . . . . . . . 14

2 Gaussian Campanato (p, κ)-Class 19 2.1 Location of Cp,κ (Gn ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 p 2.2 Another Look at Cp,κ (Gn ) for − n ≤ κ < 0 . . . . . . . . . . . . . . . 28 3 Gaussian p-Capacity 37 3.1 Gaussian p-Capacity for 1 ≤ p < ∞ . . . . . . . . . . . . . . . . . . 37 3.2 Alternative of Gaussian p-Capacity for 1 ≤ p < ∞ . . . . . . . . . . 48 4 Restriction of Gaussian Sobolev p-Space 4.1 Gaussian p-Capacitary-Strong-Type Inequality . . . . . . . . . . . . 4.2 Trace Inequality for W 1,p (Gn ) Under 1≤p ≤q N.

5

6

1 Gaussian Sobolev p-space

Consequently, when j > N we have φj ∗ f − f Lp (Gn ) ≤ φj ∗ f − φj ∗ дLp (Gn ) + φj ∗ д − дLp (Gn ) + д − f Lp (Gn ) 1

≤ (C Rp + 2)δ. Letting first j → ∞ and then δ → 0 yields lim φj ∗ f − f Lp (Gn ) = 0.

(1.6)

j→∞

Further, observing ∂(φj ∗ f )(x) ∂f = φj ∗ (x) ∀ ∂x i ∂x i

(x, j, i) ∈ Rn × N × {1, 2, . . . , n},

we apply (1.6) to derive     (1.7) lim  |∇(φj ∗ f ) − ∇f | Lp (Gn ) = lim  |φj ∗ ∇f − ∇f | Lp (Gn ) = 0. j→∞

j→∞

Combining (1.6) and (1.7) yields that (1.3) holds for f ∈ W 1,p (Gn ) with compact support. Finally, we show (1.4). By the above proof, we know that the following functions

 (1.8) fk, j (x) := φj ∗ η(2−k ·)f (x) ∀ x ∈ Rn satisfy

(1.9)

lim

k→∞

 lim  fk, j − f W 1,p (Gn ) = 0,

j→∞

Assume further that K is a nonempty compact subset of Rn and K ⊆ {x ∈ Rn : f (x) ≥ 1}◦ . Choose R > 0 and k 0 ∈ N such that K ⊆ B(0, 2−1 R) &

2k0 > R + 1.

Choose j 0 ∈ N such that   2−j0 < dist ∂K, ∂({x ∈ Rn : f (x) ≥ 1}◦ ) . Then, for any

(x, y) ∈ K × B(x, 2−j0 ),

1.1 Definition and Approximation of W 1,p (Gn )

we observe f (y) ≥ 1 &

7

η(2−k0 y) = 1.

Thus, when x ∈ K we have ∫ ∫ fk0, j0 (x) = φj0 (x − y)η(2−k0 y)f (y) dV (y) ≥ Rn

Rn

φj0 (x − y) dV (y) = 1.

This proves (1.4), thereby completing the argument for Proposition 1.1.3.



From Proposition 1.1.3, we deduce the following generalizations of () and Remark 1.1.2. Corollary 1.1.4. Let p ∈ (1, ∞). Then there exists a positive constant Cp,n such that ∫ (1.10)

Rn

∫ f −

Rn

p

p1   f dVγ dVγ ≤ Cp,n  |∇f | Lp (Gn )

Consequently, for all

∀

f ∈ W 1,p (Gn ).

f ∈ W 1,p (Gn )

one has (1.11)

   f W 1,p (Gn )  |∇f | Lp (Gn ) +  f L1 (Gn ) ∫

Proof. Let

Rn

|∇f |p dVγ

p1

∫ +

Rn

f dVγ .

f ∈ W 1,p (Gn ).

For any > 0, by Proposition 1.1.3, there exists a function f ∈ Cc1 (Rn ) such that  f − f W 1,p (Gn ) < . Since

f ∈ Cc1 (Rn ),

it is known from () that ∫ Rn

∫ f −

Rn

p

p1 f dVγ dVγ ≤ Cp,n ∇f Lp (Gn ) ,

8

1 Gaussian Sobolev p-space

which, together with the Minkowski inequality and the Hölder inequality, implies ∫ f −

∫ Rn

Rn

∫ ≤

Rn

p

p1 ∫ (f − f ) − (f − f ) dVγ dVγ n

∫ +

p

p1 f dVγ dVγ

Rn

R

∫ f −

Rn

p

p1 f dVγ dVγ

≤  f − f Lp (Gn ) +

∫ Rn

  | f − f | dVγ + Cp,n  |∇f | Lp (Gn )

    ≤ 2 f − f Lp (Gn ) + Cp,n  |∇f − ∇f | Lp (Gn ) + Cp,n  |∇f | Lp (Gn )   ≤ (2 + Cp,n ) + Cp,n  |∇f | Lp (Gn ) . Letting → 0 in the above inequality yields (1.10). Notice that (1.11) is a consequence of (1.10) and the Hölder inequality. This finishes the proof of the corollary. 

1.2 Approximating W 1,p (Gn )-Function with Cancellation In the following proposition, we prove that if p ∈ (1, ∞), then   ∫ 1 n f dVγ = 0 f ∈ Cc (R ) : Rn

is dense in

Proposition 1.2.1. Let



f ∈ W 1,p (Gn ) :

∫ Rn

 f dVγ = 0 .

⎧ ⎪ p ∈ (1, ∞); ⎪ ⎪ ⎪ ⎨ ⎪ f ∈ W 1,p (Gn ); ⎪ ⎪ ∫ ⎪ ⎪ ⎪ n f dVγ = 0. ⎩ R

Then there exists a sequence of functions { f j }j ∈N ⊆ Cc1 (Rn )

1.2 Approximating W 1,p (Gn )-Function with Cancellation

such that (1.12)

9

∫ Rn

f j dVγ = 0 ∀

j ∈ N,

and (1.13)

lim  f j − f W 1,p (Gn ) = 0.

j→∞

Moreover, if K is a nonempty compact subset of Rn obeying  ◦ K ⊆ x ∈ Rn : f (x) ≥ 1 , then the above functions f j enjoy fj ≥ 1 Proof. Given any

on K

∀

j ∈ N.

f ∈ W 1,p (Gn ),

we shall construct a sequence of functions satisfying (1.12) and (1.13) by using fk, j as in (1.8). Fix ∈ (0, 14 ). For any R > 0, we can construct an even function τ ∈ Cc1 (R) such that

⎧ ⎪ 0≤τ≤1 ⎪ ⎪ ⎪ ⎪ ⎪ τ = 1 on [0, R]; ⎪ ⎨ ⎪ τ = 0 on [R + 1, ∞); ⎪ ⎪ ⎪ ⎪ τ is decreasing on [R, R + 1]; ⎪ ⎪ ⎪ ⎪ |τ(t)| ≤ 2 when t ∈ (R, R + 1). ⎩

When R > 0 is sufficiently large, the above construction of τ gives that ∫ τ(|x |) dVγ (x) > 2−2 3. Rn

In this case, for some 0 ∈ (0, 14 ), we write ∫ τ(|x |) dVγ (x) = 1 − 0 . (1.14) Rn

In the sequel, we denote by τR, 0 such a function τ(| · |). By (1.9), there exist large integers k and j such that (1.15)

 fk , j − f W 1,p (Gn ) < .

10

1 Gaussian Sobolev p-space

Without loss of generality, we may also choose the above k large enough to verify 2k > R + 1. Choose λ ∈ (0, 1) such that (1 + λ)(1 − 0 − ) = 1. Define (1.16)

 F := (1 + λ) (1 − 0 )fk , j −

∫ Rn



fk , j dVγ τR, 0 .

By (1.14) we get ∫ (1.17)

Rn

By

F (x) dVγ (x) = 0.

∫ Rn

f dVγ = 0,

the Hölder inequality and (1.15), we deduce ∫ ∫  fk , j dVγ = fk , j − f ) dVγ Rn

(1.18)

∫ ≤

Rn

Rn

| fk , j − f | dVγ

≤  fk , j − f Lp (Gn ) < . By



⎧ ⎪ (1 + λ)(1 − 0 − ) = 1; ⎪ ⎨ ⎪ ∈ (0, 2−2 ); ⎪ ⎪ ⎪ 0 ∈ (0, 2−2 ), ⎩

we observe 0 < (1 + λ)(1 − 0 ) − 1 = (1 + λ) = 1 − 0 − < 2 . From this, (1.16) and (1.18), it follows that if Λ = (1 + λ)(1 − 0 ),

1.2 Approximating W 1,p (Gn )-Function with Cancellation

then |F − f | ≤ Λ fk , j − ≤ Λ f k , j −

11

∫  f + (1 + λ)(1 − 0 ) − 1) | f | + (1 + λ)τR, 0 fk , j dVγ Rn f + 2 | f | + 2 ,

which, together with (1.15), further gives (1.19)

F − f Lp (Gn ) ≤ Λ fk , j − f Lp (Gn ) + 2  f Lp (Gn ) + 2   ≤ 2 2 +  f Lp (Gn ) .

To estimate ∇F − ∇f Lp (Gn ) , we have  |∇τR, 0 | |∇F − ∇f | ≤ Λ ∇fk , j − ∇f + 1 − Λ−1 |∇f | + 1 − 0 ≤ Λ ∇fk , j − ∇f + 2 |∇f | + 4 , where we have used

   |∇τ| 

L ∞ (Rn )

∫

Rn



fk , j dVγ

≤ 2,

(1.16) and (1.18). Consequently,    |∇F − ∇f |  p n L (G )     (1.20) ≤ Λ |∇fk , j − ∇f | Lp (Gn ) + 2  |∇f | Lp (Gn ) + 4 

  ≤ 2 3 +  |∇f | Lp (Gn ) . Summarizing the above arguments, we derive that for any ∈ (0, 2−2 ), there exists a function F satisfying (1.17), (1.19), and (1.20). This finishes the first part of the proposition. Assume further that K is a nonempty compact subset of Rn and  ◦ K ⊆ x ∈ Rn : f (x) ≥ 1 . For this case, we choose R such that K ⊆ B(0, 2−1R). We may as well assume that the above chosen j also satisfies   2−j < dist ∂K, ∂({x ∈ Rn : f (x) ≥ 1}◦ ) , which implies (1.21)

f (y) ≥ 1

∀ (x, y) ∈ K × B(x, 2−j ).

Since k was chosen to satisfy 2k > R + 1,

12

1 Gaussian Sobolev p-space

by the definition of η in the proof of Proposition 1.1.3 and especially the fact that η(x) = 1 ∀

|x | < 1,

we observe 2−k |y| ≤ 2−k (2−1 R + 1) < 1

∀ (x, y) ∈ K × B(x, 2−j ),

so that η(2−k y) = 1 ∀

(1.22)

(x, y) ∈ K × B(x, 2−j ).

Based on these, using (1.9), τ = 1 on K, (1.18), (1.21), and (1.22), we obtain that if x ∈ K, then 

∫ F (x) = (1 + λ) (1 − 0 )fk , j (x) − τR, 0 (x) fk , j dVγ Rn 

∫ ≥ (1 + λ) (1 − 0 ) φj (x − y)η(2−k y)f (y) dV (y) − Rn 

∫ ≥ (1 + λ) (1 − 0 ) φj (x − y) dV (y) − Rn

= (1 + λ)(1 − 0 − ) = 1. Thus, we complete the proof of Proposition 1.2.1.



1.3 Compactness for W 1,p (Gn ) For any two functions f and д, define their Gaussian inner product: ∫ f д dVγ . f , дγ = Rn

For any p ∈ (1, ∞], the symbol Lp (Gn ; Rn ) represents the space of vector-valued functions f = (f 1 , . . . , fn ) enjoying

fi ∈ Lp (Gn )

∀ i ∈ {1, 2, . . . , n}.

Proposition 1.3.1. Let p ∈ (1, ∞). Assume that the sequence { fk }k ∈N satisfies (1.23)

sup  fk W 1,p (Gn ) < ∞. k ∈N

1.3 Compactness for W 1,p (Gn )

13

Then there exist a subsequence { fki }i ∈N ⊆ { fk }k ∈N and a function

f ∈ W 1,p (Gn )

such that



(fki , ∇fki )

 i ∈N

converges to (f , ∇f ) weakly in Lp (Gn ) × Lp (Gn ; Rn ), that is, (1.24)



∀ φ ∈ Lp (Gn )

lim fki , φ γ = f , φ γ

i→∞

and (1.25) where p  (1.26)

lim ∇fki , Φ γ = ∇f , Φ γ

i→∞



∀ Φ ∈ Lp (Gn ; Rn ).

is the conjugate index of p. Moreover, ∫ ∫ fki dVγ = f dVγ . lim i→∞

Rn

Rn

Proof. (1.23) induces a subsequence { fki }i ∈N such that   (fki , ∇fki ) i ∈N tends to some (f , F ) weakly in Lp (Gn ) × Lp (Gn ; Rn ). This indicates that not only (1.24) holds, but also (1.25) holds with ∇f there replaced by F . To get (1.25) fully, we need to verify F = ∇f . Toward this end, recall  Mazur’s  Theorem in [57, p. 120, Theorem 2] - if {xj } in a normed linear space X,  ·  converges weakly to x∞ , then for any > 0 there is a convex combination of {xi }: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪

j

i=1 αi xi ;

0 ≤ αi ≤ 1; ⎪ ⎪ ⎪ ⎪ j α = 1, ⎩ i=1 i

14

1 Gaussian Sobolev p-space

such that

j      αi xi  < . x∞ − i=1

Then we obtain a convex combination m  i=1

λm,i (fki , ∇fki )

converging to (f , F ) strongly in Lp (Gn ) × Lp (Gn ; Rn ), where

 λm,i ∈ 0, 1]

&

m 

λm,i = 1.

i=1

In particular, we have lim

m 

m→∞

and lim

m 

m→∞

thereby getting

i=1

i=1

λm,i fki = f

λm,i ∇fki = F

in Lp (Gn )

in Lp (Gn ; Rn ),

F = ∇f a.e. on Rn .

This completely proves (1.25). Finally, we take φ = 1 in (1.24) and then obtain (1.26).



1.4 Poincaré or Log-Sobolev Inequality for W 1,2 (Gn ) As shown in [25], the space W 1, 2 (Gn ) deserves special treatment. Due to ∇γ(x) = −x · γ(x) ∀ x ∈ Rn , we can utilize the chain rule to obtain that if f , д ∈ Cc1 (Rn ) and i ∈ {1, . . . , n}, then    ∫ ∫  ∂ д(x)γ(x) ∂ f (x) f (x) dx д(x) dVγ (x) = − ∂x i ∂x i Rn Rn   ∫ ∂д(x) =− f (x) − x i д(x) dVγ (x). ∂x i Rn

1.4 Poincaré or Log-Sobolev Inequality for W 1, 2 (Gn )

15

Upon introducing L д(x) := −∇д(x) + xд(x), we see that L exists as the adjoint operator of ∇ and enjoys the integration-byparts formula of Malliavin calculus ∫ ∫ (1.27) д∇f dVγ = f L д dVγ ∀ (f , д) ∈ W 1, 2 (Gn ) × Cc1 (Rn ) Rn

Rn

thanks to Proposition 1.1.3. Moreover, if L f (x) := (∇ · ∇)f (x) − x · ∇f (x) = Δf (x) − x · ∇f (x) ∀ (f , д) ∈ Cc2 (Rn ), then ∫

∫

Rn

дL f dVγ =

∫

Rn

∫

=− =− ∫ =

n

∫R

Rn

Rn

  д(x)x · ∇f (x) dVγ (x) Rn ∫   ∇ д(x)γ(x) · ∇f (x) dx − д(x)x · ∇f (x) dVγ (x)

дΔf dVγ −

Rn

∇д · ∇f dVγ

f Lд dVγ ,

and hence L is symmetric in L2 (Gn ). Proposition 1.4.1. For a bounded and continuous function f on Rn , denoted by f ∈ C 0 (Rn ), let

∫ Pt f (x) =

Rn

√   f e −t x + 1 − e −2t y dVγ (y) ∀

(x, t) ∈ Rn × [0, ∞).

Then the Ornstein-Uhlenbeck semigroup (Pt )t ∈[0,∞) satisfies the following properties. (i) P0 f = f ∀ f ∈ C 0 (Rn ). (ii) t → Pt f is continuous from [0, ∞) to L2 (Gn ). (iii) Pt ◦ Ps = Ps+t ∀ s, t ∈ [0, ∞). (iv) Pt 1 = 1 & Pt f ≥ 0 ∀ t ≥ 0 & 0 ≤ f ∈ C 0 (Rn ). (v) Pt f L∞ (Rn ) ≤  f L∞ (Rn ) ∀ f ∈ C 0 (Rn ). (vi) ∂t Pt f = L(Pt f ) = Pt (L f ) ∀

(t, f ) ∈ [0, ∞) × Cc2 (Rn ).

16

1 Gaussian Sobolev p-space

∫ (vii) limt →∞ Pt f (x) = Rn f dVγ ∀ (x, f ) ∈ Rn × C 0 (Rn ). ∫ (viii) Rn L f dVγ = 0 ∀ f ∈ Cc2 (Rn ). (ix) |∇Pt f (x)| ≤ e −t Pt |∇f |(x) ∀ (t, f , x) ∈ [0, ∞) × Cc1 (Rn ) × Rn . Proof. It suffices to verify (vi) and (ix). For (vi) we use f ∈ Cc1 (Rn ), the definition of Pt f and (1.27) to compute

∫  −2t √   e y ∂t Pt f (x) = − e −t x · ∇f e −t x + 1 − e −2t y dVγ (y) √ Rn 1 − e −2t 

∫ − 2t √   e = √ y · ∇f e −t x + 1 − e −2t y dVγ (y) − 2 t n R 1−e∫ √   −t −e x · ∇f e −t x + 1 − e −2t y dVγ (y) Rn

= ΔPt f (x) − x · ∇Pt f (x) = L(Pt f )(x). For (ix) we calculate ∇Pt f (x) = e −t

∫

√   ∇f e −t x + 1 − e −2t y dVγ (y),

Rn



thereby reaching the pointwise inequality. Theorem 1.4.2. For

f ∈ Cc1 (Rn )

let

∫ Varγ (f ) :=

and Entγ (f 2 ) :=

Rn

∫ Rn



2

∫ f −

Rn

f dVγ



f2

f 2 ln ∫ Rn

dVγ 

f 2 dVγ

dVγ

be the Gauss variance of f and the Gauss entropy of f 2 respectively. (i) The Poincaré inequality

∫ Varγ (f ) ≤

Rn

|∇f | 2 dVγ ,

equivalently, Varγ (Pt f ) ≤ e −2t Varγ (f )

∀ t ∈ [0, ∞),

holds. Moreover, both inequalities are optimal and extremal functions are determined by ∇f = c for some constant c ∈ Rn .

1.4 Poincaré or Log-Sobolev Inequality for W 1, 2 (Gn )

17

(ii) The logarithmic Sobolev inequality ∫

Entγ (f 2 ) ≤ 2

Rn

|∇f | 2 dVγ ,

equivalently,   Entγ Pt (f 2 ) ≤ e 2t Entγ (f 2 )

∀ t ∈ [0, ∞),

holds. Moreover, both inequalities are optimal and extremal functions are determined by ∇f = c f for some constant c ∈ Rn . Proof. (i) According to Proposition 1.4.1 and the Cauchy-Schwarz inequality, we obtain

∫ ∞ ∫ 2 Varγ (f ) = − ∂t (Pt f ) dVγ dt 0 Rn ∫ ∞∫ = −2 LPt f Pt f dVγdt n 0 ∫ R ∫ ∞ |∇Pt f | 2 dVγdt =2 n ∫0 ∞ ∫R   e −2t Pt |∇f | 2 dVγdt ≤2 n ∫0 ∞ ∫R   e −2t Pt |∇f | 2 dVγdt ≤2 n ∫0 ∞ ∫R e −2t |∇f | 2 dVγdt =2 0 Rn ∫ |∇f | 2 dVγ . = Rn

This, along with

∫ ∂t Varγ (Pt f ) = −2

Rn

|∇Pt f | 2 dVγ

and the Grönwall lemma, implies Varγ (Pt f ) ≤ e −2t Varγ (f )

∀ t ∈ [0, ∞).

Conversely, if the last inequality holds, then its derivation at time t = 0 derives ∫ Varγ (f ) ≤ |∇f | 2 dVγ . Rn

While checking all stages of the argument for this last inequality, we can see that smooth functions obeying ∇f = c are the unique function ensuring the equalities in the two inequalities of (i).

18

1 Gaussian Sobolev p-space

(ii) For simplicity, let

д = f 2 ∈ Cc1 (R).

Just like the argument for (i) we have

∫ ∞ ∫ Endγ (д) = − ∂t (Pt д) ln(Pt д) dVγ dt 0 Rn ∫ ∞∫ =− LPt д ln(Pt д) dVγdt n ∫ ∞0 ∫ R  (∇Pt д) · (∇ ln(Pt д) dVγdt = n 0 R ∫ ∞∫ |∇Pt д| 2 dVγdt = Pt д 0 Rn  2 ∫ ∞∫ −2t P t |∇д| dVγdt . e ≤ Pt д 0 Rn Now, the Cauchy-Schwarz inequality or the convexity of the map (0, ∞) × (0, ∞)  (s, t) → s 2t −1 , derives



Pt |∇д| Pt д

2

 ≤ Pt

|∇д| 2 , д

whence ∫ Entγ (д) ≤

0

∞∫ Rn

e

−2t

 Pt

∫ |∇д| 2 |∇д| 2 −1 dVγ . dVγdt = 2 д д Rn

In fact, this last inequality amounts to Entγ (Pt д) ≤ e 2t Entγ (д) ∀ t ∈ [0, ∞), due to the following variation formula ∫ ∂t Entγ (Pt д) = −

Rn

|∇Pt д| 2 d vγ . Pt д

Similarly, we can obtain the extremal functions for (ii) via examining the equalities for (i), i.e., the functions satisfying ∇(f 2 ) = c f 2 for a constant c ∈ Rn . 

Chapter 2

Gaussian Campanato (p, κ)-Class In this chapter we are motivated by the left side of () to investigate the Campanato (p, κ)-class on Gn and its relationship with the Morrey space, John-Nirenberg space, and Lipschitz space on Gn .

2.1 Location of Cp,κ (Gn ) In what follows, define m(x) := min{1, |x | −1 }

∀ x ∈ Rn .

For any a ∈ (0, ∞) denote by Ba the set of all balls B in Rn such that r B ≤ a m(c B ), where c B and r B denote the center and the radius of B respectively. Balls in Ba are called admissible balls with scale a. If (B, x) ∈ Ba × B, then (2.1)

(a + 1)−1m(x) ≤ m(c B ) ≤ (a + 1)m(x)

and (2.2)

e−

a 2 +2a

≤e

2

|c B | 2 −|x | 2 2

≤ ea ;

see [40, Proposition 2.1]. Consequently, for all B ∈ Ba , (2.3)

e−

a 2 +2a 2



Vγ (B) e−

|c B | 2 2

r Bn

 ea .

This inequality implies that dVγ is doubling on the admissible class Ba , that is, for all balls B ∈ Ba one has (2.4)

Vγ (2B)  Vγ (B)

(locally doubling property).

© Springer Nature Switzerland AG 2018 L. Liu et al., Gaussian Capacity Analysis, Lecture Notes in Mathematics 2225, https://doi.org/10.1007/978-3-319-95040-2_2

19

2 Gaussian Campanato (p, κ)-class

20

A locally integrable function f is said to be of bounded mean oscillation on Gn , denoted by BMO(Gn ), provided that ∫ 1 n  f BMO(G ) := sup | f (x) − f B,γ | dVγ (x) < ∞. B ∈Ba Vγ (B) B In the above and below, (2.5)

f B,γ

1 := Vγ (B)

∫ B

f dVγ .

It was proved in [40] that, for all p ∈ (1, ∞), f ∈ L1 (Gn ) ∩ BMO(Gn )

p1  ∫ 1 ⇔  f L1 (Gn ) + sup | f (x) − f B,γ |p dVγ (x) < ∞ B ∈Ba Vγ (B) B with implicit constants depending only on a, p, and n. Definition 2.1.1. Let a ∈ (0, ∞), p ∈ [1, ∞), and κ ∈ (−∞, 1]. (i) Any locally integrable function f is said to be in the Gaussian Campanato p,κ (p, κ)-class CBa (Gn ) provided 

f 

p, κ CBa (Gn )

1 := sup 1−κ V (B) γ B ∈Ba

∫

p

B

| f (x) − f B,γ | dVγ (x)

p1

< ∞,

where f B,γ is as in (2.5). Moreover, p,κ p,κ L Ba (Gn ) := L1 (Gn ) ∩ CBa (Gn );  f  Lp, κ (Gn ) :=  f L1 (Gn ) +  f  Cp, κ (Gn ) . Ba

Ba

When a = 1, we write p,κ

p,κ

CBa (Gn ) & L Ba (Gn ) as Cp,κ (Gn ) & L p,κ (Gn ). (ii) Qa consists of all admissible cubes Q with sides parallel to the axes, the center c Q , and the side-length Q ≤ a m(c Q ). Then



p,κ

p,κ

CQ (Gn ) & L Q (Gn ); a

a

 ·  Cp, κ (Gn ) &  ·  Lp, κ (Gn ) , Qa

Qa

are defined via replacing B ∈ Ba with Q ∈ Qa in (i). Simply, we write p,κ p,κ CBa (Gn ) or CQ (Gn ) as Cp,κ (Gn ); p,κ

a

p,κ

L Ba (Gn ) or L Q (Gn ) as L p,κ (Gn ), a

if there exists no confusion.

2.1 Location of Cp,κ (Gn )

21

p

Lemma 2.1.2. For any cube Q ⊆ Rn and p ∈ [1, ∞], denote by L 0 (Q; Gn ) the family of all functions ∫ p n φ ∈ L (G ) satisfying supp φ ⊆ Q & φ dVγ = 0. Q

(i) If

⎧ ⎪ 0 < b < a < ∞; ⎪ ⎨ ⎪ p ∈ [1, ∞); ⎪ ⎪ ⎪ κ ∈ (−∞, 1], ⎩

then  f  Cp, κ (Gn )  f  Cp, κ (Gn )  f  Cp, κ (Gn )  f  Cp, κ (Gn ) Ba

Qa

Bb

Qb

with implicit constants depending only on n, a, b, p, and κ. Consequently, p,κ

p,κ

p,κ

p,κ

L Ba (Gn ) = L Q (Gn ) = L Q (Gn ) = L Bb (Gn ). a

(ii) If

b

⎧ ⎪ p ∈ [1, ∞); ⎪ ⎨ ⎪ κ ∈ (−∞, 0]; ⎪ ⎪ ⎪ f ∈ Cp,κ (Gn ), ⎩

then for all (B, B ) ∈ B1 × B1 with c B = c B  & r B < r B  , it holds that

 ⎧ r B ⎪ ⎪ ⎨ 1 + ln r B  f  Cp, κ (Gn ) ⎪ (2.6) | f B,γ − f B ,γ |  r  nκ p f C p, κ (Gn ) B ⎪ ⎪ κ ⎪ r B Vγ (B) p ⎩

as κ = 0; as κ ∈ (−∞, 0),

with implicit constants depending only on p, κ, and n. Proof. (i) By a reexamination of the proof of [40, Lemma 2.3], we can get a nonnegative integer N (depending only on n, p, a, and b) such that for any cube Q ∈ Qa and each function p φ ∈ L 0 (Q; Gn ), there exist at most N subcubes {Q 1 , . . . , Q N } in the admissible class Qb and N functions p {φ1 , . . . , φN } ∈ L 0 (Q; Gn ) such that

⎧ supp φj ⊆ Q j ∀ j ∈ {1, . . . , N }; ⎪ ⎪ ⎪ ⎪ ⎨ φ = N ⎪ j=1 φ j ; ⎪ φj Lp (Gn )  φLp (Gn ) ; ⎪ ⎪ ⎪ ⎪V (Q ) V (Q) ∀ j ∈ {1, . . . , N }. γ ⎩ γ j

2 Gaussian Campanato (p, κ)-class

22

The rest of the argument is completely parallel to that for [40, Proposition 2.4], so the details are omitted. (ii) We show (2.6) by considering the following two cases: r B ≥ 2−1r B 

& r B < 2−1r B  .

Case r B ≥ 2−1r B  . In this case, we have B ⊆ B  ⊆ 2B, and hence

Vγ (B) γ(B )

by (2.4). From this and Hölder’s inequality, we deduce (2.7)

| f B,γ − f B ,γ | ≤

1 Vγ (B)

∫ B

| f (x) − f B ,γ | dVγ (x) 

 f  Cp, κ (Gn ) κ

Vγ (B) p

,

thereby reaching (2.6). Case r B < 2−1r B  . Under this situation we choose j 0 ≥ 1 such that 2 j 0 r B ≤ r B  < 2 j 0 +1 r B . Via setting

B j := B(c B , 2j r B )

∀

j ∈ {1, . . . , j 0 },

we have | f B,γ − f B ,γ | ≤

j 0 −1 j=0

| f B j ,γ − f B j+1,γ | + | f B j0 ,γ − f B ,γ |.

For any j ∈ {1, . . . , j 0 − 1}, we use

r B j = 2−1r B j+1

and (2.7) to conclude (2.8)

| f B j ,γ − f B j+1,γ | 

 f  Cp, κ (Gn ) κ

Vγ (B j ) p

.

Similarly, | f B j0 − f B ,γ | has the same upper bound as in (2.8). Accordingly, via summing the inequality (2.8), we know | f B,γ − f B ,γ | 

j0   f  Cp, κ (Gn ) j=0

Vγ (B j )

κ p



j0  f  Cp, κ (Gn ) 

Vγ (B)

κ p

j=0

2−

jnκ p

,

2.1 Location of Cp,κ (Gn )

23

where we have used the fact that γ(B j ) 2jnVγ (B). Then, applying 2j0

r B , rB

we see that (2.6) holds for the case r B < 2−1r B  .



Using Lemma 2.1.2, we can locate each Gaussian Campanato class. Proposition 2.1.3. Let p ∈ [1, ∞). (i) If κ ∈ (0, 1), then

L p,κ (Gn ) = M p,κ (Gn ),

the Morrey space of all functions f obeying f 

M p, κ (Gn )



p1 ∫ κ−1 p :=  f L1 (Gn ) + sup Vγ (B) | f | dVγ < ∞. B ∈B1

(ii) If κ = 0, then

B

Cp,κ (Gn ) = BMO(Gn ).

p

(iii) If κ ∈ [− n , 0), then

Cp,κ (Gn ) = Lip− pκ (Gn ),

the Lipschitz space of all locally integrable functions f obeying  f Lip− κ (Gn ) := p

sup

B ∈B1 & x,y ∈B

κ

Vγ (B) p | f (x) − f (y)| < ∞.

p

(iv) If κ ∈ (−∞, − n ), then Cp,κ (Gn ) consists of only functions which are constant almost everywhere. Proof. (i) Clearly, if

f ∈ M p,κ (Gn ),

then an application of the Hölder inequality gives  f  Lp, κ (Gn )   f  M p, κ (Gn ) . and so

M p,κ (Gn ) ⊆ L p,κ (Gn ).

Conversely, we need only to show that if (f , B) ∈ L p,κ (Gn ) × Ba ,

24

2 Gaussian Campanato (p, κ)-class

then 

p1 ∫ κ−1 p Vγ (B) | f | dVγ   f  Lp, κ (Gn ) .

(2.9)

Ba

B

We prove (2.9) by distinguishing the following two cases. Case |c B | ≤ 1 + a. Under this situation we have 

p1 ∫ κ−1 p Vγ (B) | f | dVγ B



p1 ∫ κ κ−1 p ≤ Vγ (B) | f − f B,γ | dVγ + Vγ (B) p | f B,γ | B ∫ κ κ ≤  f  Cp, κ (Gn ) + Vγ (B) p −1 | f − f B(0, 1),γ | dVγ + Vγ (B) p | f B(0, 1),γ | Ba

B

  f  Cp, κ (Gn ) +  f L1 (Gn ) . Ba

Case |c B | > 1 + a. Under this situation, we recall the notion of maximal admissible balls and the mother of a maximal admissible ball introduced in [40]. Given a ∈ (0, ∞). A ball B ∈ Ba is said to be maximal if r B = a m(c B ). For each maximal ball B ∈ Ba containing no the origin, we denote by M(B) the maximal ball in Ba centered at a point in the segment   [0, c B ] = tc B : t ∈ [0, 1] such that the boundary of M(B) contains c B , and naturally we call M(B) the mother of B. In other words, the relation between B and its mother M(B) is as follows: ⎧ r M (B) = a m(c M (B) ); ⎪ ⎪ ⎨ ⎪ |c M (B) | + r M (B) = |c B |; ⎪ ⎪ ⎪c M (B) = |c M (B) | c B . |c B | ⎩ For notational convenience put M 0 (B) := B. If M(B) does not contain the origin, then we may consider the mother of M(B), which is written as M 2 (B). Therefore, for any maximal ball B ∈ Ba , we may find a chain of maximal balls, B, M(B), M 2 (B), . . . , M k (B), with the property that M j (B) is the mother of M j−1 (B) ∀ j ∈ {1, . . . , k } and M k (B) contains the origin.

2.1 Location of Cp,κ (Gn )

25

Now, we may assume that 

M k (B) : 0 ≤ k ≤ k 0



is a chain of maximal balls in Ba , with the property that M k (B) is the mother of M k−1 (B) ∀ k ∈ {1, . . . , k 0 }, where k 0 is the smallest number such that c k0 ≤ 1 + a. M (B) For the notational convenience let B 0 := B & Bk := M k (B) ∀ k ∈ {1, . . . , k 0 }. Then



p1 ∫ κ−1 p | f | dVγ ≤ I1 + I2 + I3 , Vγ (B) B

where

 p1 ∫ κ ⎧ ⎪ ⎪ I1 := Vγ (B) p Vγ1(B) B | f − f B,γ |p dVγ ; ⎪ ⎪ ⎨ ⎪ κ  0 |f − f Bk ,γ |; I2 := Vγ (B) p kk= ⎪ 1 B k −1,γ ⎪ ⎪ κ ⎪ ⎪ I := V (B) p | f γ B k0 ,γ |. ⎩ 3

Obviously, I1 ≤  f  Cp, κ (Gn ) . Ba

For term I3 , we use to get

c k0 ≤ 1 + a M (B)   Vγ (Bk0 ) = Vγ M k0 (B) 1,

which, combined with Vγ (B) < 1, yields κ

Vγ (B) p I3 ≤ Vγ (Bk0 )

∫ Bk0

| f | dVγ   f L1 (Gn ) .

To estimate I2 , for 1 ≤ k ≤ k 0 , since Bk is the mother of Bk−1 , we use [36, Lemma 3.2] to get ⎧ ⎪ ⎨ Bk−1 ⊆ (a + 2) Bk ; ⎪

  ⎪ ⎪Vγ (Bk−1 ) Vγ (Bk ) Vγ (a + 2)Bk , ⎩

2 Gaussian Campanato (p, κ)-class

26

which, together with the Hölder inequality, implies κ

I2 ≤ Vγ (B) p

k0 

| f Bk −1,γ − f (a+2)Bk ,γ | + | f (a+2)Bk ,γ − f Bk ,γ |



k=1

≤ Vγ (B)

κ p

k0   k=1

≤  f  Cp, κ (Gn ) Ba

1 Vγ (Bk−1 )

k0  k=1



∫

1 | f − f (a+2)Bk ,γ | dVγ + Vγ (Bk )

B k −1

 pκ 

Vγ (B)

  Vγ (a + 2)Bk



∫ Bk

···

    Vγ (a + 2)Bk Vγ (a + 2)Bk   + Vγ (Bk ) Vγ Bk−1

k0   Vγ (B) p . Vγ (Bk ) κ

 f 

p, κ CBa (Gn )

k=1

Note that (cf. [36, Lemma 3.3(iv)]) Vγ (B)  e −ak Vγ (Bk )

∀ k ∈ {1, . . . , k 0 }.

So we continuously deduce I2   f  Cp, κ (Gn ) Ba

k0 

e−

ak κ p

k=1

  f  Cp, κ (Gn ) . Ba

Via combining the estimates for I1 , I2 and I3 , we achieve the desired inequality (2.9). (ii) This follows from the definition. p (iii) For κ ∈ [− n , 0), it is easy to see that Lip− pκ (Gn ) ⊆ Cp,κ (Gn ) p

by their definitions. Also, when κ ∈ (−∞, − n ), it is obvious that almost everywhere constant functions belong to Cp,κ (Gn ). Let f ∈ Cp,κ (Gn ) & κ ∈ (−∞, 0). For any B ∈ B1 and almost every x, y ∈ B, by the differential theorem of integrals, we obtain | f (x) − f (y)| = lim f B(x, 2−j r ),γ − f B(y, 2−j r ),γ j→∞

(2.10)

B

B

≤ lim | f B(x, 2−j r B ),γ − f B(x,r B ),γ | + f B(x,r B ),γ − f B(y,r B ),γ | j→∞

+ lim | f B(y,r B ),γ − f B(y, 2−j r B ),γ |. j→∞

2.1 Location of Cp,κ (Gn )

27

From Lemma 2.1.2 and Vγ (B(x, 2−j r B )) 2−jnVγ (B), we deduce (2.11)

f B(x, 2−j r

jnκ

2− p  f  Cp, κ (Gn )  f  Cp, κ (Gn ) − f B(x,r B ),γ  . κ  κ B ),γ −j p Vγ (B(x, 2 r B )) Vγ (B) p

Likewise, for almost every y ∈ B, we have (2.12)

f B(y, 2−j r

B ),γ

 f  Cp, κ (Gn ) − f B(y,r B ),γ  . κ Vγ (B) p

For all x, y ∈ B, we observe B(x, r B ) ⊆ 2B & B(y, r B ) ⊆ 2B whence, by (2.2) and (2.3),     Vγ (B) Vγ B(x, r B ) Vγ B(y, r B ) . From these and Lemma 2.1.2, it follows that f B(x,r ),γ − f B(y,r ),γ B B ≤ f B(x,r B ),γ − f 2B,γ + f 2B,γ − f B(y,r B ),γ (2.13) 

 f  Cp, κ (Gn ) κ

Vγ (B) p

.

Inserting (2.11), (2.12), and (2.13) into (2.10), and using (2.3), we see that for almost every (x, y) ∈ B × B ∈ Ba × Ba there holds  f  Cp, κ (Gn )

(2.14)

| f (x) − f (y)| 

whence finding

p  f ∈ Lip− pκ (Gn ) ∀ κ ∈ − , 0 . n p

κ

Vγ (B) p

,

(iv) When κ ∈ (−∞, − n ), from (2.14) one deduces that the derivative of f is 0 for almost every x ∈ B, and hence f is a constant function almost everywhere. 

28

2 Gaussian Campanato (p, κ)-class

p

2.2 Another Look at Cp,κ (Gn ) for − n ≤ κ < 0 As well known (cf. [51]), the Ornstein-Uhlenbeck-Poisson semigroup {Pt }t > 0 is determined by ∫ Pt f (x) = Pt (x, y)f (y) dV (y) Rn

where Pt (x, y) = is valid for any

∫

1 2π

n+1 2

0

∞



t 3

s2



2

e

− t4s

|y−e −s x | 2

− ! e 1−e −2s $ " n % ds (1 − e −2s ) 2 # &

(t, x, y) ∈ (0, ∞) × Rn × Rn .

In accordance with [23, 24, 46], we say that for 0 < α < 1 a function f on Rn is said to be in Lip ,α (Gn ) provided  f Lip , α (Gn ) :=  f L∞ (Rn ) + sup t 1−α ∂t Pt f L∞ (Rn ) < ∞. t ∈(0,∞)

Lemma 2.2.1. Let α ∈ (0, 1). Then f ∈ Lip ,α (Gn ) if and only if ' 

α |x − yx | 2 α  α + |yx | (2.15) | f (x) − f (y)|  min |x − y| , ∀ x, y ∈ Rn 1 + |x | after correction of f on a null set. Here and hereafter, for any x, y ∈ Rn with x  0, we decompose y as y = yx + yx , where yx is parallel to x and yx orthogonal to x; if x = 0, then yx = y and yx = 0, and this also holds for all x in case n = 1. Moreover,    f Lip , α (Gn )  f L∞ (Rn ) + inf C > 0 : constants C in (2.15)  

| f (0)| + inf C > 0 : constants C in (2.15) and the implicit equivalent constants are independent of f . Proof. Note that this lemma means that the combined Lipschitz condition applies in the radial direction, but in the orthogonal direction the exponent is always α. So this lemma follows from [35, Theorem 1.1] and the fact that (2.15) amounts to α ' 

2α  |x − y| sin θ | f (x) − f (y)|  min |x − y| α , +  ∀ x, y ∈ Rn  −1 1 + |x | + |y| |x | + |y| after correction of f on a null set; where θ is the angle between x and y and vanishes whenever x or y is the origin, along with a combination of the ordinary Lipschitz continuity conditions (cf. [52, Section V.4.2]), some sharp estimates for the  Ornstein-Uhlenbeck-Poisson semigroup (Pt )t ≥0 and some of its derivatives.

p

2.2 Another Look at Cp,κ (Gn ) for − n ≤ κ < 0

29

From the above characterization of the Gaussian Lipschitz space, we know that any f ∈ Lip ,α (Gn ) with α ∈ (0, 1) behaves like the classical Lipschitz function when x is close to y (namely, |x − y| is smaller than a fixed constant multiple of (1 + |x |)−1 ); and the combined Lipschitz condition plays a role when |x − y| is large but not “quite” large - in other words |x − y| is smaller than a fixed constant; for the remaining case the fact f ∈ L∞ (Rn ) will be important. Based on these, for any 

1 ,1 (x, r ) ∈ R × 1 + |x |

(

n

we define the Gaussian cylinder   R(x, r ) := y ∈ Rn : |yx − x | <  = 2−1r 2 (1 + |x |) & |yx | < r . Notice that R(x, r ) is a cylinder centered at x, with bottom radius r and length 2, but it goes along the direction x. Definition 2.2.2. For α ∈ (0, 1) and p ∈ [1, ∞) let Lipα,p (Gn ) be the collection of all locally integrable functions f on Rn such that  f Lipα,p (Gn ) :=

sup

1 (x,r )∈Rn ×( 1+|x | , 1]

r

−α



1 V (R(x, r ))

∫ R(x,r )

f − f R(x,r ) p dV

p1

p1 ∫ p 1 f − f B(x,r ) dV + sup r V (B(x, r )) B(x,r ) (x,r )∈Rn ×(0, 1]

p1  ∫ 1 + sup r −α | f |p dV V (B(x, r )) B(x,r ) (x,r )∈Rn ×(1,∞) −α



is finite. Here and hereafter in this section, for any set E ⊆ Rn we make a convention ∫ 1 fE = f dV . V (E) E Another perspective of the formula )

p Cp,κ (Gn ) = Lip− pκ (Gn ) ∀ k ∈ − , 0 n is the following assertion.



2 Gaussian Campanato (p, κ)-class

30

Theorem 2.2.3. Let α ∈ (0, 1) and p ∈ [1, ∞). Then Lipα,p (Gn ) = Lip ,α (Gn ) holds with equivalent norms. Proof. The required identification follows from a consideration of two parts. Part 1 : f ∈ Lipα,p (Gn ) ⇐ f ∈ Lip ,α (Gn ). Let  f Lip , α (Gn ) = 1. By Lemma 2.2.1, after correction of f on a null set, one has that if x, z ∈ Rn then | f (x) − f (z)|  |x − z| α

(2.16) and



(2.17) For any

|x − z x | | f (x) − f (z)|  1 + |x |

2α

+ |z x | α .

( 1 , 1 & z ∈ R(x, r ), (x, r ) ∈ Rn × 1 + |x | 

we have

|z x | < r & |z x − x | < 2−1r 2 (1 + |x |),

so that (2.17) implies

| f (z) − f (x)|  r α .

Consequently, 

p1 ∫ 1 f − f R(x,r ) p dV V (R(x, r )) R(x,r ) 

p1 ∫ 1 p ≤2 | f − f (x)| dV V (R(x, r )) R(x,r )  r α.

Similarly, for any

(x, r ) ∈ Rn × (0, 1] & z ∈ B(x, r ),

the estimation (2.16) also implies | f (z) − f (x)|  r α , whence leading to 

1 V (B(x, r ))

∫ B(x,r )

f − f B(x,r ) p dV

p1

p

2.2 Another Look at Cp,κ (Gn ) for − n ≤ κ < 0

 ≤2

1 V (B(x, r ))

∫ B(x,r )

| f − f (x)|p dV

31

p1

 r α. Also, for any

(x, r ) ∈ Rn × (1, ∞)

we utilize

f ∈ L∞ (Rn )

to estimate r

−α



1 V (B(x, r ))

∫

p

B(x,r )

| f | dV

p1

  f L∞ (Rn ) .

Combining the last three formulae gives  f Lipα,p (Gn )  1  f Lip , α (Gn ) . Part 2 : f ∈ Lipα,p (Gn ) ⇒ f ∈ Lip ,α (Gn ). Let  f Lipα,p (Gn ) = 1. We are about to show  f Lip , α (Gn )  1. From the definition of Lipα,p (Gn ) it follows that sup

(x,r )∈Rn ×(0,∞)

r

−α



1 V (B(x, r ))

∫ B(x,r )

f − f B(x,r ) p dV

p1

 1,

which, together with a standard classical argument, implies that for almost all x, y ∈ Rn one has (2.18)

| f (x) − f (y)|  |x − y| α .

It remains to show that for almost all x, y ∈ Rn one has 

(2.19)

|x − yx | | f (x) − f (y)|  1 + |x |

2α

+ |yx | α .

Via writing | f (x) − f (y)| ≤ | f (x) − f (yx )| + | f (yx ) − f (y)| and noticing

| f (yx ) − f (y)|  |yx − y| α |yx | α

by means of (2.18), we observe that the proof of (2.19) can be reduced to proving  | f (x) − f (yx )| 

|x − yx | 1 + |x |

2α

.

2 Gaussian Campanato (p, κ)-class

32

In other words, we only need to prove that if (x, y) ∈ Rn × Rn with x = τy for some τ ∈ (−∞, ∞) then  | f (x) − f (y)| 

(2.20)

|x − y| 1 + |x |

2α

.

The demonstration of (2.20) can be split into the following three cases. 10 ⎧ |x − y| ≤ 110 ⎪ + |x | ⎪ ⎪ ⎨ 10 ⎪ Case i = 110 + |x | < |x − y| < ⎪ ⎪ ⎪ ⎪ |x − y| ≥ 1+ |x | ⎩ 1010

For Case 1 we use |x − y| ≤

as i = 1; 1+ |x | 1010

as i = 2; as i = 3.

1010 1 + |x |

and (2.18) to get the required inequality 

|x − y| | f (x) − f (y)|  |x − y|  1 + |x | α

2α

.

For Case 2 we have 1010 1 + |x | < |x − y| < , 1 + |x | 1010 whence

|x | ≥ 109 & 2−1 |y| < |x | < 2|y|.

For each j + 1 ∈ N let * 2−j ⎧ | ⎪ ⎨r j = 2 1+|x|x−y ⎪ & R j := R(x, r j ); * −j | 2 |x −y | ⎪  ⎪r j = & R j := R(y, r j), 1+ |y | ⎩ and observe

V (R j ) r jn+1 (1 + |x |) V (R j+1 );  ). V (R j ) (r j)n+1 (1 + |y|) V (R j+ 1

From the last two estimates and the equivalence 1 + |x | 1 + |y| it follows that

V (R 1 ) V (R 0 ).

p

2.2 Another Look at Cp,κ (Gn ) for − n ≤ κ < 0

33

Now, we show R 1 ⊆ R 0 . To this end, let z ∈ R 1 . Then |zy | < r & |zy − y| < 2−1 (r 1 )2 (1 + |y|) = 2−2 |x − y|. Using the facts 2−1 |y| < |x | < 2|y| and x = τy for some τ ∈ (−∞, ∞), we obtain

|z x | = |zy | < r

and |z x − x | =

|zy − y||x | |x − y||x | < < 2|x − y| = 2−1r 02 (1 + |x |), |y| 4|y|

which implies z ∈ R 0 and proves R 1 ⊆ R 0 . By the Lebesgue differential theorem of integrals we obtain (2.21)

| f (x) − f (y)| ≤

∞ ∞    f R j − f R j−1 + f R0 − f R1 + f R j − f R j +1 . j=1

j=1

Upon using V (R j ) V (R j−1 ) ∀ j ∈ N, we gain fR − fR ≤ j j−1

 

1 V (R j )

∫

f − f R p dV j −1

Rj

1  V (R j−1 )

∫ R j−1

 p1

f − f R p dV j −1

 p1

α

 (r j−1 )  −j

α 2 |x − y| 2

. 1 + |x | Similarly,  −j

2α  −j

α 2 |x − y| 2 f R  − f R   (r )α 2 |x − y|

. j j j+1 1 + |y| 1 + |x | Applying

R 1 ⊆ R 0 & V (R 1 ) V (R 0 ),

we derive fR − fR ≤ 0 1



1 V (R 1 )

∫ R 1

f − f R p dV 0

 p1

34

2 Gaussian Campanato (p, κ)-class

 

1 V (R 0 )

∫ R0

f − f R p dV 0

p1

 (r 0 )α 

α |x − y| 2

. 1 + |x | Inserting the last three formulae in (2.21) leads to (2.20). For Case 3 we utilize   |x − y| ≥ 1 + |x | 10−10 to get that if then

z ∈ B(x, 10−10 )   max{|z x − x |, |z x |} ≤ |z − x | < 10−10 < 1 + |x | 10−10

and hence | f (z) − f (x)| ≤ | f (z x ) − f (x)| + | f (z x ) − f (z)| 

α |z x − x | 2  α  |z x | + 1 + |x |  1, where we have used the already-established (2.18) to estimate | f (z x ) − f (z)|, and the already-proved results in Case 1 and Case 2 to control | f (z x ) − f (x)|. Thus f (x) − f B(x, 10−10 ) ≤



1 V (B(x, 10−10 ))

∫

p

B(x, 10−10 )

| f − f (x)| dV

Meanwhile, the definition of Lipα,p implies f B(x, 10−10 ) 



1 V (B(x, 1))

∫ B(x, 1)

| f |p dV

p1

 1.

Combining the last two formulae implies | f (x)| ≤ f (x) − f B(x, 10−10 ) + f B(x, 10−10 )  1.

p1

 1.

p

2.2 Another Look at Cp,κ (Gn ) for − n ≤ κ < 0

35

Analogously, we have ⎧ ⎪ ⎨ | f (y)|  1; ⎪ ⎪ ⎪ | f (x) − f (y)|  1  ⎩

|x −yx | 1+ |x |

 2α

,

thereby reaching (2.20) and completing the proof of Part 2.  Corollary 2.2.4. Let α ∈ (0, 1) and p ∈ [1, ∞). Then one has the Gaussian Campanato estimation  f Lip , α (Gn )  f L∞ (Rn ) +

sup

(x,r )∈Rn ×

+

1 ,1 1+|x |

sup

(x,r )∈Rn ×(0, 1]

−α +r

−α

r





1 V (R(x, r ))

1 V (B(x, r ))

∫ R(x,r )

∫ B(x,r )

f − f R(x,r ) p dV

f − f B(x,r ) p dV

p1

p1

,

thereby finding the Gaussian Morrey inequality  f Lip , α (Gn )   f L∞ (Rn ) +

sup

(x,r )∈Rn ×

+

sup

1

,1 1+|x |

(x,r )∈Rn ×(0, 1]

r

+r

−α



−α



1 V (R(x, r ))

1 V (B(x, r ))

∫

p

R(x,r )

∫ B(x,r )

|∇f | dV p

|∇f | dV

p1

p1

.

Proof. This follows from the argument for Theorem 2.2.3 and the Poincaré inequality:     | f − f E |p E  |∇f |p E as E = R(x, r ) or E = B(x, r ). 

Chapter 3

Gaussian p-Capacity In this chapter we introduce the notion of the Gaussian p-capacity which is important at least in describing the appropriate tracing of the Gaussian Sobolev p-space.

3.1 Gaussian p-Capacity for 1 ≤ p < ∞ Definition 3.1.1. For p ∈ [1, ∞) and E ⊆ Rn let   Ap (E) := f ∈ W 1,p (Gn ) : E ⊆ {x ∈ Rn : f (x) ≥ 1}◦ . Define the Gaussian p-capacity of E as:   p (3.1) Capp (E; Gn ) := inf  f W 1,p (Gn ) : f ∈ Ap (E) . Recall that Cc1 (Rn ; Rn ) is the collection of all vector-valued functions Φ = (φ1 , . . . , φn )

with φi ∈ Cc1 (Rn ).

Similarly, Cc0 (Rn ; Rn ) is the collection of all vector-valued functions Φ = (φ1 , . . . , φn )

with φi ∈ Cc0 (Rn ).

Lemma 3.1.2. Given p ∈ [1, ∞). For a sequence of functions { fk }k ∈N ⊆ W 1,p (Gn ) let д = sup fk k ∈N

If both д and h are in L

1

(Gn ),

and h = sup |∇fk |. k ∈N

then |∇д(x)| ≤ h(x)

holds for almost all x ∈ Rn . Proof. Thanks to

д ∈ L1 (Gn ) ⊆ L1loc (Rn ),

© Springer Nature Switzerland AG 2018 L. Liu et al., Gaussian Capacity Analysis, Lecture Notes in Mathematics 2225, https://doi.org/10.1007/978-3-319-95040-2_3

37

38

3 Gaussian p-Capacity

it makes sense to consider ∇д. According to (1.1), we have ∫ ∫ ∇д · Φ dV = − д divΦ dV ∀ Φ ∈ Cc1 (Rn ; Rn ). Rn

Rn

For any l ∈ N, define

дl = sup fk . 1 ≤k ≤l

Due to

fk ∈ W 1,p (Gn ),

one has

дl ∈ Lp (Gn ) ⊆ L1loc (Rn ).

An application of [18, p. 148, Lemma 2(iii)] yields |∇дl | ≤ sup |∇fk |

(3.2)

1 ≤k ≤l

a. e. on Rn .

Of course, this follows by induction and from verifying the case l = 2:

 ∇ max{ f 1 , f 2 }(x) = 2−1 ∇f 1 (x) + ∇f 2 (x) + ∇| f 1 (x) − f 2 (x)| ∇f 1 (x) for almost all x ∈ Rn with f 1 (x) ≥ f 2 (x); = for almost all x ∈ Rn with f 1 (x) ≤ f 2 (x). ∇f 2 (x) According to (1.1) and the Lebesgue dominated convergence theorem, for any Φ ∈ Cc1 (Rn ; Rn ), we have

∫ д divΦ dV = lim дl divΦ dV l →∞ Rn Rn ∫ = lim ∇дl · Φ dV l →∞ Rn ∫ ≤ |Φ|h dV .

∫

Rn

Based on this and Rudin [49, p.58, Theorem 3.3], the linear functional L defined by ∫ L(Φ) := д divΦ dV ∀ Φ ∈ Cc1 (Rn ; Rn ) Rn

extends to a linear functional L¯ on Cc0 (Rn ; Rn ) such that ∫ ¯ |L(Φ)| ≤ |Φ|h dV ∀ Φ ∈ Cc0 (Rn ; Rn ). Rn

3.1 Gaussian p-Capacity for 1 ≤ p < ∞

39

In particular, for each compact set K ⊆ Rn ,   ¯ sup L(Φ) : Φ ∈ Cc0 (Rn ; Rn ), |Φ| ≤ 1, supp Φ ⊆ K ∫ ≤ h dV K

≤ C(K, n)hL1 (Gn ) < ∞. By this and [18, p. 49, Theorem 1], there exist a Radon measure μ on Rn and a μ-measurable function σ : Rn → Rn such that ∫ ¯ L(Φ) = Φ · σ dμ ∀ Φ ∈ Cc0 (Rn ; Rn ), Rn

where |σ(x)| = 1 for almost all x ∈ Rn . Moreover, the construction of μ (see [18, p. 49, Definition]) gives us that for any Lebesgue measurable set A ⊆ Rn , μ(A) = inf {μ(O) : open O ⊇ A}   ¯ = inf sup L(Φ) : Φ ∈ Cc0 (Rn ; Rn ), |Φ| ≤ 1, supp(Φ) ⊆ O open O ⊇A ∫ h dV ≤ inf open O ⊇A O ∫ = h dV . A

Therefore, μ is absolutely continuous with respect to the Lebesgue measure, so that dμ = udV for some function u satisfying that |u(x)| ≤ h(x) for almost all x ∈ Rn . Hence, for all Φ ∈ Cc1 (Rn ; Rn ), we have

∫

∫ Rn

¯ д divΦ dV = L(Φ) = L(Φ) =

Rn

Φ · σu dV ,

which implies ∇д = σu and

|∇д(x)| ≤ h(x) for almost all x ∈ Rn .

This completes the argument. Remark 3.1.3. Three comments are in order.



40

3 Gaussian p-Capacity

(i) Let p ∈ [1, ∞). Because of Remark 1.1.2 and Corollary 1.1.4, we have p     Capp (E; Gn ) inf  |∇f | Lp (Gn ) +  f L1 (Gn ) : f ∈ Ap (E) p ∫    

inf  |∇f | Lp (Gn ) + f dVγ : f ∈ Ap (E) . Rn

(ii) Given any function f ∈ W 1,p (Gn ) with p ∈ [1, ∞), since | f | = max{ f , −f }, it follows from Lemma 3.1.2 that |∇(| f |)(x)| ≤ |∇f (x)| for almost all x ∈ Rn , which consequently implies | f |W 1,p (Gn ) ≤  f W 1,p (Gn ) ∀ f ∈ W 1,p (Gn ). Observe that f ∈ Ap (E) ⇒ | f | ∈ Ap (E). So

  p Capp (E; Gn ) = inf  f W 1,p (Gn ) : f ≥ 0, f ∈ Ap (E) .

(iii) When f ∈ Ap (E) with p ∈ [1, ∞), one easily deduces from [18, p. 130, Theorem 4(iii)] that |∇ min{ f , 1}| ≤ |∇f | a. e. on Rn , so that  min{ f , 1}W 1,p (Gn ) ≤  f W 1,p (Gn ) and then min{ f , 1} ∈ Ap (E). Thus, we can equivalently write   p Capp (E; Gn ) = inf  f W 1,p (Gn ) : 0 ≤ f ≤ 1, f ∈ Ap (E) . Proposition 3.1.4. Let p ∈ [1, ∞). The set-function Capp (· ; Gn ) enjoys the following properties. (i) Capp (∅ ; Gn ) = 0 and Capp (Rn ; Gn ) ≤ 1.

3.1 Gaussian p-Capacity for 1 ≤ p < ∞

41

(ii) If E 1 ⊆ E 2 ⊆ Rn , then Capp (E 1 ; Gn ) ≤ Capp (E 2 ; Gn ). n (iii) For any sequence {E j }∞ j=1 of subsets of R , ∞

  n ≤ Capp ∪∞ E ; G Capp (E j ; Gn ). j j=1

(3.3)

j=1

(iv) For any 1 ≤ p < q < ∞ and any set E ⊆ Rn , 1  1  1 1 2− p Capp (E; Gn ) p ≤ 2− q Capq (E; Gn ) q .

n (v) For any sequence {K j }∞ j=1 of compact subsets of R such that



K1 ⊇ K2 ⊇ · · · ;

   n . limj→∞ Capp K j ; Gn = Capp ∩∞ j=1 K j ; G

Proof. (i) It is easy to see that Capp (∅ ; Gn ) = 0 holds by using Definition 3.1.1. To see the inequality Capp (Rn ; Gn ) ≤ 1, we can take the test function f ≡ 1 in Definition 3.1.1. (ii) For any sets E 1 ⊆ E 2 ⊆ Rn , we have Ap (E 2 ) ⊆ Ap (E 1 ), so that Capp (E 1 ; Gn ) =

inf

f ∈Ap (E 1 )

p

 f W 1,p (Gn ) ≤

inf

f ∈Ap (E 2 )

p

 f W 1,p (Gn ) = Capp (E 2 ; Gn ).

(iii) Without loss of generality, we may assume ∞  j=1

Capp (E j ; Gn ) < ∞

- otherwise - (3.3) holds trivially. For any > 0 and j ∈ N, by (3.1), we find a function f j, ∈ Ap (E j ) such that p

Capp (E j ; Gn ) ≤  f j, W 1,p (Gn ) ≤ Capp (E j ; Gn ) + 2−j .

42

3 Gaussian p-Capacity

Let f = sup f j, . j ∈N

Clearly,

 ◦ n ∪∞ j=1 E j ⊆ x ∈ R : f (x) ≥ 1 .

Upon observing  p  p         + sup f j,  sup |∇f j, |   j ∈N  p n  j ∈N  p n L (G ) L (G )   p   f j, W 1,p (Gn ) ≤

(3.4)

j ∈N

≤

 j ∈N

 Capp (E j ; Gn ) + 2−j ,

we apply Lemma 3.1.2 to deduce that ∇f exists a. e. on Rn and |∇f | ≤ sup |∇f j, | j ∈N

a. e. on Rn .

Notice that (3.4) also implies f ∈ W 1,p (Gn ). Thus,

f ∈ Ap (∪∞ j=1 E j ),

and (3.4) again gives

  p n ≤  f Capp ∪∞ E ; G  ≤ Capp (E j ; Gn ) + . j j=1 W 1,p (Gn ) j ∈N

Letting → 0 yields (3.3). This proves (iii). (iv) When 1 ≤ p < q < ∞, using the Hölder inequality and the elementary inequality (3.5)

1

1

a + b ≤ 21− κ (a κ + b κ ) κ

∀

(κ, a, b) ∈ [1, ∞) × (0, ∞) × (0, ∞),

q

we have that if κ = p , then

(3.6)

  p1 p 1 1 p 2− p  f W 1,p (Gn ) = 2− p  |∇f | Lp (Gn ) +  f Lp (Gn )

  p1 p 1 p ≤ 2− p  |∇f | Lq (Gn ) +  f Lq (Gn ) 1

≤ 2− q  f W 1,q (Gn ) .

3.1 Gaussian p-Capacity for 1 ≤ p < ∞

43

For any set E ⊆ Rn , notice that Aq (E) ⊆ Ap (E). So, we use (3.6) and (3.1) to obtain 1  1 2− p Capp (E; Gn ) p = ≤ ≤

1

inf

2− p  f W 1,p (Gn )

inf

2− q  f W 1,q (Gn )

inf

2− q  f W 1,q (Gn )

f ∈Ap (E)

1

f ∈Ap (E)

1

f ∈Aq (E)

1  1 = 2− q Capq (E; Gn ) q .

(v) By (ii), it is trivial to find that

   n . lim Capp K j ; Gn ≥ Capp ∩∞ j=1 K j ; G

j→∞

It remains to prove the converse of this last inequality. To this end, we let K = ∩∞ j=1 K j , which is also compact. For any ∈ (0, 1), choose f ∈ Ap (K) such that

p

 f W 1,p (Gn ) ≤ Capp (K; Gn ) + .

Notice that the compact set K is contained in the open set ◦  x ∈ Rn : f (x) ≥ 1 . Since Kj  K

as

j → ∞,

it is a basic fact that there exists some j 0 ∈ N such that  ◦ K j0 ⊆ x ∈ Rn : f (x) ≥ 1 . Thus, f ∈ Ap (K j0 ) and     lim Capp K j ; Gn ≤ Capp K j0 ; Gn

j→∞

p

≤  f W 1,p (Gn ) ≤ Capp (K; Gn ) + . Letting → 0 in the last formulae yields the desired result of (v).



44

3 Gaussian p-Capacity

Proposition 3.1.5. Let p ∈ (1, ∞). Then for any sequence {E j }∞ j=1 with E j ⊆ E j+1 ⊆ Rn ∀ j ∈ N, one has

   n . E ; G lim Capp E j ; Gn = Capp ∪∞ j=1 j

j→∞

Proof. We adopt the idea used in Costea [15, Theorem 3.1(iv)]. Let ∞ E = ∪i= 1 Ei .

By Proposition 3.1.4(ii), we obtain lim Capp (Ei ; Gn ) ≤ Capp (E; Gn ) .

i→∞

Thus, we only need to prove the converse of the above inequality. Without loss of generality, we may as well assume lim Capp (Ei ; Gn ) < ∞.

i→∞

Fix ∈ (0, 1). For any i ∈ N, choose ui ∈ Ap (Ei ) such that p

ui W 1,p (Gn ) ≤ Capp (Ei ; Gn ) + . ∞ increases and Since {Ei }i= 1

lim Capp (Ei ; Gn ) < ∞,

i→∞

it follows that sup ui W 1,p (Gn ) < ∞. i ∈N

Applying Proposition 1.3.1, we find a subsequence, which we denote again by {ui }i ∈N , and a function u ∈ W 1,p (Gn ) such that {(ui , ∇ui )}i ∈N converges to (u, ∇u) weakly in Lp (Gn ) × Lp (Gn ; Rn ). Upon fixing i 0 ∈ N and using Mazur’s Theorem, for the sequence {ui }i ≥i 0 we find a finite convex combination of {ui }i ≥i 0 , denoted by vi 0 , such that (3.7)

vi 0 − u W 1,p (Gn ) < 2−i 0 .

3.1 Gaussian p-Capacity for 1 ≤ p < ∞

45

Since every ui with i ≥ i 0 satisfies  ◦ Ei 0 ⊆ Ei ⊆ x ∈ Rn : ui (x) ≥ 1 , it follows that

 ◦ Ei 0 ⊆ x ∈ Rn : vi 0 (x) ≥ 1 .

In this way, we obtain a sequence {vi }i ∈N with each vi being a finite convex combination of {uk }k ≥i such that vi → u strongly in W 1,p (Gn ) as i → ∞, and so that vi ∈ Ap (Ei ). Passing to a subsequence if necessary, we may even assume that for any i ∈ N, vi+1 − vi W 1,p (Gn ) < 2−i .

(3.8) Next, for any j ∈ N, define

wj = sup vi . i ≥j

It is easy to verify that for all j ∈ N and x ∈ Rn , |wj (x) − vj (x)| ≤

(3.9)

∞ 

|vi+1 (x) − vi (x)|.

i=j

and (3.10)

 ∞ sup |∇vi (x)| − |∇vj (x)| ≤ |∇vi+1 (x) − ∇vi (x)|. i ≥j

i=j

By (3.8) and (3.9), we have wj Lp (Gn ) ≤ vj Lp (Gn ) + ≤ vj Lp (Gn ) +

(3.11)

≤ vj 

Lp (Gn )

∞  i=j ∞ 

vi+1 − vi Lp (Gn ) 2−i

i=j 1−j

+2

.

Similarly, by (3.9) and (3.10), we obtain (3.12)

    sup |∇vi |   i ≥j 

Lp (Gn )

∞       |∇vi+1 − ∇vi |  p n ≤  |∇vj | Lp (Gn ) + L (G ) i=j

  ≤  |∇vj | Lp (Gn ) + 21−j .

46

3 Gaussian p-Capacity

Notice that (3.11)-(3.12) and Lemma 3.1.2 imply that ∇wj exists a. e. on Rn and |∇wj (x)| ≤ sup |∇vi (x)| a.e. x ∈ Rn .

(3.13)

i ≥j

By (3.13)-(3.12)-(3.11), we see wj ∈ W 1,p (Gn ). Now, we calculate the  · W 1,p (Gn ) norm of wj . To this end, we observe that the mean value theorem implies the following inequality: (3.14)

(a + b)p ≤ ap + p(M + 1)p−1b

for (M, a, b) ∈ (0, ∞) × [0, M] × [0, 1].

Notice that (3.7) implies   max vj Lp (Gn ) , |∇vj |Lp (Gn ) ≤ u W 1,p (Gn ) + 1. Below we will apply (3.14) with M = u W 1,p (Gn ) + 1. Consequently, we deduce from (3.11), (3.12), and (3.13) that  p   p p wj W 1,p (Gn ) ≤ wj Lp (Gn ) + sup |∇vi |  i ≥j Lp (Gn )

    p (3.15) ≤ vj Lp (Gn ) + 21−j +  |∇vj |  p

L (Gn )

+ 21−j

p

p

≤ vj W 1,p (Gn ) + Cu,p 2−j , where

Cu,p := 4p(u W 1,p (Gn ) + 2)p−1 .

Recalling that vi ∈ Ap (Ei ) and {Ei }i increases to E, we have ◦  E ⊆ x ∈ Rn : wj (x) ≥ 1 . Thus, wj ∈ Ap (E). Moreover, (3.15) implies p

p

Capp (E; Gn ) ≤ wj W 1,p (Gn ) ≤ vj W 1,p (Gn ) + Cu,p 2−j

∀

j ∈ N.

3.1 Gaussian p-Capacity for 1 ≤ p < ∞

47

According to the construction of vj , we may assume vj =

Nj 

λj,k uk ,

k=j

where ⎧ ⎪ j ≤ N j ∈ N; ⎪ ⎨ ⎪ λj,k ∈ [0, 1]; ⎪ ⎪ ⎪ N j λj,k = 1. ⎩ k=j Consequently, by the Minkowski inequality, the Hölder inequality, and Proposition 3.1.4 (ii), we achieve  p p p vj W 1,p (Gn ) = vj Lp (Gn ) +  |∇vj | Lp (Gn ) p

p

Nj Nj   ! ! $ $ ≤ " λj,k uk Lp (Gn ) % + " λj,k  |∇uk | Lp (Gn ) % # k=j # k=j & & N N j j    p p ≤ λj,k uk Lp (Gn ) + λj,k  |∇uk | Lp (Gn ) k=j

=

k=j

Nj  k=j

≤

Nj 

p

λj,k uk W 1,p (Gn )

 λj,k Capp (Ek ; Gn ) +

k=j

 ≤ Capp E N j ; Gn + . We then deduce that for any j ∈ N,

 Capp (E; Gn ) ≤ Capp E N j ; Gn + + Cu,p 2−j . Letting j → ∞ and → 0 yields

   Capp (E; Gn ) ≤ lim Capp E N j ; Gn = lim Capp E j ; Gn , j→∞

thereby completing the argument.

j→∞



Remark 3.1.6. The limiting case p → 1 of Proposition 3.1.5 will be presented through Theorem 6.2.1(v) of Chapter 6 for the Gaussian BV-capacity.

48

3 Gaussian p-Capacity

3.2 Alternative of Gaussian p-Capacity for 1 ≤ p < ∞ Definition 3.2.1. Let p ∈ [1, ∞) and K ⊆ Rn be a compact set. Define   A(K) := f ∈ Cc1 (Rn ) : f ≥ 1 on K and

  p Cap0,p (K; Gn ) := inf  f W 1,p (Gn ) : f ∈ A(K) .

(3.16)

If O ⊆ Rn is an open set, then (3.17)

  Cap0,p (O; Gn ) := sup Cap0,p (K; Gn ) : compact K ⊆ O .

Remark 3.2.2. As in Remark 3.1.3(ii)-(iii), by    | f |  1,p n ≤  f W 1,p (Gn ) ∀ W (G )

f ∈ Cc1 (Rn ),

we also have

  p Cap0,p (K; Gn ) = inf  f W 1,p (Gn ) : 0 ≤ f ∈ A(K)   p = inf  f W 1,p (Gn ) : 0 ≤ f ≤ 1, f ∈ A(K) .

Lemma 3.2.3. Let p ∈ [1, ∞). Then one has the following properties. (i) For compact sets K 1 and K 2 satisfying that K 1 ⊆ K 2 ⊆ Rn , Cap0,p (K 1 ; Gn ) ≤ Cap0,p (K 2 ; Gn ). (ii) For compact set K and open set O satisfying O ⊆ K ⊆ Rn , Cap0,p (O; Gn ) ≤ Cap0,p (K; Gn ). (iii) For open sets O 1 and O 2 satisfying O 1 ⊆ O 2 ⊆ Rn , Cap0,p (O 1 ; Gn ) ≤ Cap0,p (O 2 ; Gn ). Proof. First of all, if K 1 and K 2 are compact subsets of Rn satisfying K 1 ⊆ K 2 , then A(K 2 ) ⊆ A(K 1 ), and hence

Capp (K 1 ; Gn ) ≤ Capp (K 2 ; Gn ),

which proves (i). Next, (ii) follows from (i) and (3.17). Finally, (iii) follows directly from (ii) and (3.17).



3.2 Alternative of Gaussian p-Capacity for 1 ≤ p < ∞

49

Lemma 3.2.4. Let p ∈ [1, ∞) and K be a compact subset of Rn . Then   (3.18) Cap0,p (K; Gn ) = inf Cap0,p (O; Gn ) : open O ⊇ K . Proof. By (3.17), we see that for any open set O ⊇ K, Cap0,p (K; Gn ) ≤ Cap0,p (O; Gn ). Taking the infimum over all such sets O yields   Cap0,p (K; Gn ) ≤ inf Cap0,p (O; Gn ) : open O ⊇ K . To prove the converse of this inequality, it suffices to verify that, for any ∈ (0, ∞), there exists an open set O ⊇ K such that

 (3.19) Cap0,p (O; Gn ) ≤ (1 + )p Cap0,p (K; Gn ) + . By (3.16), there exists a function f ∈ Cc1 (Rn ) such that f ≥ 1 on K and

p

 f W 1,p (Gn ) < Cap0,p (K; Gn ) + .

Define

Since

f := (1 + )f ;   K = x ∈ Rn : f (x) ≥ 1 .

f ∈ Cc1 (Rn ) and f > 1 on K,

it follows that K is a compact set with K ⊆ K ◦ ⊆ K . This, along with Lemma 3.2.3(ii), implies Cap0,p (K ◦ ; Gn ) ≤ Cap0,p (K ; Gn ). Upon noticing f ∈ A(K ), we have

 p Cap0,p (K ; Gn ) ≤  f W 1,p (Gn ) < (1 + )p Cap0,p (K; Gn ) + .

Via combining the last two inequalities, we find an open set O = K ◦ ⊇ K such that (3.19) holds, thereby completing the argument for (3.18).



50

3 Gaussian p-Capacity

Due to Lemma 3.2.4, we can extend the definition of Cap0,p (·; Gn ) from a compact set to any set. Definition 3.2.5. Let p ∈ [1, ∞) and E be an arbitrary subset of Rn . Define   Cap0,p (E; Gn ) := inf Cap0,p (O; Gn ) : open O ⊇ E . (3.20) Applying Proposition 1.1.3, we give the following equivalent characterization of the Gaussian p-capacity. Theorem 3.2.6. If p ∈ (1, ∞) and E is a subset of Rn , then Cap0,p (E; Gn ) = Capp (E; Gn )

(3.21)

Also, if K is a compact subset of Rn , then (3.22)

Cap0, 1 (K; Gn ) = Cap1 (K; Gn ).

Proof. We will prove (3.21) and (3.22) according to the following three steps. Step 1. E = K is compact. For any > 0, by (3.16) and the definition of A(K), there exists f ∈ Cc1 (Rn ) such that f ≥ 1 on K and

p

Cap0,p (K; Gn ) ≤  f W 1,p (Gn ) ≤ Cap0,p (K; Gn ) + .

For any λ ∈ (0, ∞) define

f λ := (1 + λ)f .

Since f is continuous, it follows that   ◦  K ⊆ x ∈ Rn : f λ (x) > 1 ⊆ x ∈ Rn : f λ (x) ≥ 1 ; f λ ∈ Ap (K). Accordingly, p

Capp (K; Gn ) ≤  f λ W 1,p (Gn )

p

= (1 + λ)p  f W 1,p (Gn )   ≤ (1 + λ)p Cap0,p (K; Gn ) + . In the above inequality, letting → 0 and λ → 0 yields (3.23)

Capp (K; Gn ) ≤ Cap0,p (K; Gn ).

Starting from the definition of Capp (K; Gn ), we see that, for any ∈ (0, ∞), there exists a function p

f ∈ Ap (K) such that Capp (K; Gn ) ≤  f W 1,p (Gn ) ≤ Capp (K; Gn ) + .

3.2 Alternative of Gaussian p-Capacity for 1 ≤ p < ∞

51

It follows from the definition of Ap (K) that  ◦ f ∈ W 1,p (Gn ) & K ⊆ x ∈ Rn : f (x) ≥ 1 . By Proposition 1.1.3, there exists a sequence of functions { f j }j ∈N ⊆ Cc1 (Rn ) such that lim  f j − f W 1,p (Gn ) = 0

j→∞

and each f j satisfies

f j ≥ 1 on K .

Therefore, we obtain: f j ∈ A(K); p p Cap0,p (K; Gn ) ≤ limj→∞  f j W 1,p (Gn ) =  f W 1,p (Gn ) ≤ Capp (K; Gn ) + . Letting → 0 in the above inequality implies Cap0,p (K; Gn ) ≤ Capp (K; Gn ).

(3.24)

Combining (3.23) and (3.24) implies that (3.21) and (3.22) hold when E = K is a compact subset of Rn . Step 2. E = O is open. For each compact set K ⊆ O, it is easy to verify Ap (O) ⊆ Ap (K). By this, (3.17) and Step 1, we get that if p ∈ [1, ∞) then Cap0,p (O; Gn ) = = = ≤

sup

Cap0,p (K; Gn )

sup

Capp (K; Gn )

compact K ⊆O compact K ⊆O

sup

inf

compact K ⊆O f ∈Ap (K ) p inf  f W 1,p (Gn ) f ∈A (O )

p

 f W 1,p (Gn )

p

= Capp (O; Gn ). It remains to prove Capp (O; Gn ) ≤ Cap0,p (O; Gn ) as p ∈ (1, ∞). Since O is an open set of Rn , there exists an increasing sequence of compact sets {K j }j ∈N such that ∪∞ j=1 K j = O.

52

3 Gaussian p-Capacity

Applying (vi) of Proposition 3.1.4, the already-proved result in Step 1 and (3.17), we conclude Capp (O; Gn ) = lim Capp (K j ; Gn ) j→∞

= lim Cap0,p (K j ; Gn ) j→∞

≤

sup

compact K ⊆O

Cap0,p (K; Gn )

= Cap0,p (O; Gn ). Hence, we obtain (3.21) when E = O is open. Step 3. E is an arbitrary set. According to (3.20), for any > 0 there exists an open set O ⊇ E such that Cap0,p (O; Gn ) ≤ Cap0,p (E; Gn ) + . By the definition of Capp (O; Gn ) and Step 2, we can find an f ∈ Ap (O) such that p

 f W 1,p (Gn ) ≤ Capp (O; Gn ) + = Cap0,p (O; Gn ) + . Since O ⊇ E ⇒ Ap (O) ⊆ Ap (E), the above f also belongs to Ap (E), which, along with the last two formulae, yields p

Capp (E; Gn ) ≤  f W 1,p (Gn ) ≤ Cap0,p (O; Gn ) + ≤ Cap0,p (E; Gn ) + 2 . Then letting → 0 gives Capp (E; Gn ) ≤ Cap0,p (E; Gn ). Now we prove the converse of this last inequality. For any > 0, by the definition of Capp (E; Gn ), there exists p

f ∈ Ap (E) such that  f W 1,p (Gn ) ≤ Capp (E; Gn ) + . The definition of Ap (E) implies f ∈ W 1,p (Gn )

& E ⊆ {x ∈ Rn : f (x) ≥ 1}◦ =: U .

Observe that U is open with Ap (U ) ⊆ Ap (E) &

f ∈ Ap (U ).

So, from the definition of Cap0,p (E; Gn ) and Step 2, it follows that Cap0,p (E; Gn ) ≤ Cap0,p (U ; Gn )

3.2 Alternative of Gaussian p-Capacity for 1 ≤ p < ∞

53

= Capp (U ; Gn ) p

≤  f W 1,p (Gn ) ≤ Capp (E; Gn ) + . Letting → 0 yields Cap0,p (E; Gn ) ≤ Capp (E; Gn ). Thus, we derive that (3.21) holds for any general set E.



Note that Definition 3.2.5 and Theorem 3.2.6 reveal that Cap1