Maximal Function Methods for Sobolev Spaces 1470465752, 9781470465759

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Mathematical Surveys and Monographs Volume 257

Maximal Function Methods for Sobolev Spaces

Juha Kinnunen Juha Lehrbäck Antti Vähäkangas

10.1090/surv/257

Maximal Function Methods for Sobolev Spaces

Mathematical Surveys and Monographs Volume 257

Maximal Function Methods for Sobolev Spaces

Juha Kinnunen Juha Lehrbäck Antti Vähäkangas

EDITORIAL COMMITTEE Ana Caraiani Robert Guralnick, Chair Bryna Kra

Natasa Sesum Constantin Teleman Anna-Karin Tornberg

2020 Mathematics Subject Classification. Primary 42B25, 46E35; Secondary 26D10, 28A12, 31B15, 35A23, 35J92, 42B37.

For additional information and updates on this book, visit www.ams.org/bookpages/surv-257

Library of Congress Cataloging-in-Publication Data Names: Kinnunen, Juha, author. | Lehrb¨ ack, Juha (Juha Tapio), 1979– author. | V¨ ah¨ akangas, Antti V., author. Title: Maximal function methods for Sobolev spaces / Juha Kinnunen, Juha Lehrb¨ ack, Antti V¨ ah¨ akangas. Description: Providence, Rhode Island: American Mathematical Society, [2021] | Series: Mathematical surveys and monographs, 0076-5376; volume 257 | Includes bibliographical references and index. Identifiers: LCCN 2021013318 | ISBN 9781470465759 (paperback) | ISBN 9781470466602 (ebook) Subjects: LCSH: Sobolev spaces. | Maximal functions. | Inequalities (Mathematics) | AMS: Harmonic analysis on Euclidean spaces – Harmonic analysis in several variables – Maximal functions, Littlewood-Paley theory. | Functional analysis – Linear function spaces and their duals – Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems. | Real functions – Inequalities – Inequalities involving derivatives and differential and integral operators. | Measure and integration – Classical measure theory – Contents, measures, outer measures, capacities. | Potential theory – Higher-dimensional theory – Potentials and capacities, extremal length and related notions in higher dimensions. | Partial differential equations – General topics in partial differential equations – Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals. | Partial differential equations – Elliptic equations and systems – Quasilinear elliptic equations with p-Laplacian. | Harmonic analysis on Euclidean spaces – Harmonic analysis in several variables – Harmonic analysis and PDE. Classification: LCC QA323 .K56 2021 | DDC 515/.782–dc23 LC record available at https://lccn.loc.gov/2021013318

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established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

26 25 24 23 22 21

Contents Preface

ix

Notation

xi

Chapter 1. Maximal Functions 1.1. Hardy–Littlewood maximal function 1.2. Hardy–Littlewood–Wiener maximal function theorem 1.3. Lebesgue differentiation theorem 1.4. A theorem of Stein 1.5. Restricted maximal function 1.6. Riesz potential 1.7. Fractional maximal function 1.8. Notes

1 1 6 10 13 15 17 20 23

Chapter 2. Lipschitz and Sobolev Functions 2.1. Lipschitz functions 2.2. Sobolev spaces 2.3. Approximation and calculus in Sobolev spaces 2.4. Sobolev spaces with zero boundary values 2.5. Weak convergence and Sobolev spaces 2.6. Difference quotients 2.7. Notes

25 25 29 31 35 37 43 46

Chapter 3. Sobolev and Poincar´e Inequalities 3.1. Pointwise estimates for Lipschitz functions 3.2. Sobolev–Gagliardo–Nirenberg inequality 3.3. Sobolev–Poincar´e inequalities 3.4. Poincar´e inequalities for zero boundary values 3.5. Morrey’s inequality 3.6. Notes

47 47 50 52 57 58 62

Chapter 4. Pointwise Inequalities for Sobolev Functions 4.1. Pointwise characterization of Sobolev spaces 4.2. Lipschitz truncation of Sobolev functions 4.3. Campanato and Morrey approaches to Sobolev spaces 4.4. Maximal operator on Sobolev spaces 4.5. Maximal function with respect to an open set 4.6. Fractional maximal operator on Sobolev spaces 4.7. Notes

63 63 66 69 73 75 79 82

v

vi

CONTENTS

Chapter 5. Capacities and Fine Properties of Sobolev Functions 5.1. Sobolev capacity 5.2. Estimates for capacity 5.3. Quasicontinuity and fine properties of capacity 5.4. Lebesgue points of Sobolev functions 5.5. Sobolev spaces with zero boundary values revisited 5.6. Variational capacity 5.7. Capacity and Hausdorff content 5.8. Lipschitz test functions for variational capacity 5.9. Mazya’s inequality 5.10. Notes

85 85 88 90 94 97 101 105 108 110 113

Chapter 6. Hardy’s Inequalities 6.1. Introduction to Hardy’s inequalities 6.2. Measure density and Hardy’s inequality 6.3. Self-improvement of Hardy’s inequality 6.4. Capacity density and pointwise Hardy inequalities 6.5. Wannebo’s approach 6.6. Stability of Sobolev spaces with zero boundary values 6.7. Notes

115 115 118 121 124 129 132 135

Chapter 7. Density Conditions 7.1. Hausdorff content density 7.2. Ahlfors–David regular sets 7.3. Lower dimension and capacity density 7.4. Density conditions and Hardy’s inequality in the borderline case 7.5. Self-improvement of the capacity density condition 7.6. Truncation and absorption 7.7. Local Hardy inequality 7.8. Concluding argument 7.9. Notes

137 137 138 140 144 147 148 151 155 158

Chapter 8. Muckenhoupt Weights 8.1. Doubling weights 8.2. Dyadic cubes and the Calder´ on–Zygmund lemma 8.3. Self-improvement of weighted norm inequalities 8.4. Muckenhoupt Ap weights for 1 < p < ∞ 8.5. Reverse H¨ older inequalities for Muckenhoupt weights 8.6. A1 weights and Coifman–Rochberg lemma 8.7. Self-improvement of reverse H¨older inequalities 8.8. General self-improvement result for reverse H¨ older inequalities 8.9. Notes

161 161 162 165 168 173 179 183 188 193

Chapter 9. Weighted Maximal and Poincar´e Inequalities 9.1. Poincar´e inequalities on cubes 9.2. Single weight maximal and Poincar´e inequalities 9.3. Weighted local Fefferman–Stein inequalities 9.4. Two weight maximal inequalities 9.5. Two weight Poincar´e inequalities 9.6. Local-to-global inequalities on open sets

195 195 198 201 205 209 211

CONTENTS

9.7. BMO and John–Nirenberg inequality 9.8. Maximal functions and BMO 9.9. Notes

vii

216 221 223

Chapter 10.1. 10.2. 10.3. 10.4. 10.5. 10.6. 10.7. 10.8.

10. Distance Weights and Hardy–Sobolev Inequalities Aikawa condition Ap properties of distance functions Assouad dimension Distance weighted Poincar´e inequalities Hardy–Sobolev inequalities Necessary conditions for Hardy–Sobolev inequalities Testing conditions Notes

225 225 228 232 238 240 243 247 252

Chapter 11.1. 11.2. 11.3. 11.4. 11.5. 11.6. 11.7. 11.8.

11. The p-Laplace Equation Weak solutions A variational approach Weak super- and subsolutions Energy estimates Local boundedness of weak solutions Harnack’s inequality Local H¨older continuity Notes

255 255 260 264 266 272 278 282 286

Chapter 12.1. 12.2. 12.3. 12.4.

12. Stability Results for the p-Laplace Equation Higher integrability of the gradient Stability with respect to the exponent Very weak solutions Notes

287 287 295 306 314

Bibliography

317

Index

335

Preface This book is about maximal functions and their applications in Sobolev spaces. There are many good texts on the use of maximal functions in harmonic analysis, but we feel that there is room for a source book gathering advances in maximal function methods related to Poincar´e and Sobolev inequalities, pointwise estimates and approximation for Sobolev functions, Hardy’s inequalities, and partial differential equations. A recurring theme throughout the book is self-improvement of uniform quantitative conditions. Our approach is partially motivated by the theory of analysis on metric measure spaces, but in order to avoid extra complications we restrict our attention only to prototypes in Euclidean space. Nevertheless, the methods applied in analysis on metric measure spaces are useful already in the Euclidean context. Besides maximal functions and Sobolev spaces, we discuss several related concepts. Capacities are needed for the study of fine properties of Sobolev functions and characterization of Sobolev spaces with zero boundary values. The capacity density condition is applied for Hardy’s inequalities and in partial differential equations. In addition to the Hausdorff dimension, we use the Assouad dimension and the lower dimension to characterize density conditions and to give sufficient and necessary conditions for Hardy’s inequalities and their generalizations. The distance function appears in Hardy’s inequality and also has applications in Sobolev spaces. We study the Muckenhoupt weight properties of distance functions and combine these with general weighted norm inequalities. At the end of the book we discuss the theory of weak solutions to the p-Laplace equation and show how maximal function techniques can be used in this context. The choice of topics is exclusive and reflects the research interests of the authors. Our style is concise and brief. Instead of lengthy motivations, we give detailed proofs in order to make the arguments flexible and transparent. For this reason some of the proofs are relatively long. We demonstrate interesting techniques, with the idea that the methods will have a wider range of applications beyond the topics covered by this book. This is not always the most direct approach and causes some overlap. Part of the material is rather standard, while some results appear for the first time in book form. One of our goals was to gather material that has been scattered in research papers and make it accessible to a wider audience. Standard references and related research papers are mentioned in the notes at the end of each chapter. We hope that the list of references, which is long but not complete, will make it easier for the reader to further investigate the literature. We have primarily aimed the book for graduate students, but we also believe that it can be used as a reference by researchers. Some parts of the material have been used by the authors in graduate courses. Most of the book is selfcontained, only assuming knowledge of measure and integration theory, in particular ix

x

PREFACE

the Lebesgue measure and Lp spaces, as well as some functional analysis. We have made an effort to arrange the material so that the chapters can be read relatively independently. We would like to thank many colleagues for discussions and comments, in particular, David Cruz-Uribe, Sylvester Eriksson-Bique, Jonathan Fraser, Napoleon Freitas Paajanen, Tero Kilpel¨ainen, Peter Lindqvist, Juan Manfredi, Derek Robinson, Nageswari Shanmugalingam and Xiao Zhong. The authors have been supported by the Academy of Finland during the writing of this book. Juha Kinnunen Juha Lehrb¨ack Antti V¨ah¨akangas

Notation We use the following notation for basic concepts throughout the text. • N = {1, 2, 3, . . .}, N0 = N ∪ {0}, Z = {. . . , −2, −1, 0, 1, 2, . . .}, Q is the set of rational numbers, R is the set of real numbers and [−∞, ∞] = R ∪ {−∞, ∞} is the set of extended real numbers. 1

• |x| = (x21 +x22 +· · ·+x2n ) 2 is the Euclidean norm and x·y = x1 y1 +x2 y2 +· · ·+xn yn is the inner product of x = (x1 , x2 , . . . , xn ) ∈ Rn and y = (y1 , y2 , . . . , yn ) ∈ Rn . • ej = (0, . . . , 0, 1, 0, . . . , 0), j = 1, 2, . . . , n, is the jth coordinate unit vector in Rn . • diam(A) = sup{|a − b| : a, b ∈ A} is the diameter of A ⊂ Rn . • d (A, B) = inf{|a − b| : a ∈ A and b ∈ B} is the distance between A, B ⊂ Rn and d (x, A) = d ({x}, A) is the distance between x ∈ Rn and A ⊂ Rn . • ∂A is the boundary, A = A ∪ ∂A is the closure and Ac = Rn \ A is the complement of A ⊂ Rn . • A is compactly contained in Ω ⊂ Rn , denoted A  Ω, if A is a compact subset of Ω. • |A| is the n-dimensional Lebesgue outer measure of A ⊂ Rn . • B(x, r) = {y ∈ Rn : |y − x| < r} is an open ball with center x ∈ Rn and radius r > 0. The corresponding closed ball is B(x, r) = {y ∈ Rn : |y − x| ≤ r}. • ωn = |B(0, 1)|, where B(0, 1) ⊂ Rn . Then |B(x, r)| = ωn r n for every x ∈ Rn and r > 0. • σ(∂B(0, 1)) = nωn is the (n − 1)-dimensional surface measure of ∂B(0, 1) ⊂ Rn . Here σ is a normalized Hausdorff measure; see Remark 3.1. • Q(x, r) = {y ∈ Rn : −r ≤ yj − xj < r, j = 1, . . . , n} is a half-open cube with center x ∈ Rn and side length 2r with r > 0. •

χA is the characteristic function of a set A ⊂ Rn , defined by χA (x) = 1 for x ∈ A and χA (x) = 0 for x ∈ Ac .

• For a function f : A → R and B ⊂ A, the restriction of f to B is f |B : B → R, f |B (x) = f (x) for every x ∈ B. • f+ (x) = max{f (x), 0} is the positive part and f− (x) = max{−f (x), 0} is the negative part of a function f : A → R. • supp f = {x ∈ A : f (x) = 0} is the support of a function f : A → R. If supp f is a compact subset of B ⊂ A, then f is compactly supported in B. • Ω ⊂ Rn is an open set. • C(Ω) = C 0 (Ω) = {u : Ω → R : u is continuous}. xi

xii

NOTATION

• C k (Ω) = {u : Ω → R : u is k times continuously differentiable}, k ∈ N. • C0 (Ω) = {u ∈ C(Ω) : u is compactly supported in Ω}. • C0k (Ω) = {u ∈ C k (Ω) : u is compactly supported in Ω}, k ∈ N.  ∞ k ∞ k • C ∞ (Ω) = ∞ k=0 C (Ω), C0 (Ω) = k=0 C0 (Ω). • The integral mean value of an integrable function f over a Lebesgue measurable set A ⊂ Rn with 0 < |A| < ∞ is  1 f (x) dx. fA =  f (x) dx = |A| A A

10.1090/surv/257/01

CHAPTER 1

Maximal Functions Maximal functions are standard tools in real and harmonic analysis, but they are also useful in connection with Sobolev spaces and partial differential equations. This chapter discusses definitions and basic properties of maximal functions. The Hardy–Littlewood–Wiener maximal function theorem is considered in Section 1.2, the Lebesgue differentiation theorem for locally integrable functions is proved in Section 1.3 and local integrability of the maximal function is discussed in Section 1.4. Bounds for the Riesz potentials in Section 1.6 are applied in the proofs of Sobolev–Poincar´e inequalities in Chapter 3. 1.1. Hardy–Littlewood maximal function The Hardy–Littlewood maximal function gives the maximal integral average of the absolute value of the function on balls centered at a point. Definition 1.1. Assume that f ∈ L1loc (Rn ), that is, f is Lebesgue measur able and K |f (x)| dx < ∞ for every compact set K ⊂ Rn . The centered Hardy– Littlewood maximal function M f : Rn → [0, ∞] is defined by M f (x) = sup  r>0

|f (y)| dy B(x,r)

for every x ∈ Rn . It is possible to consider the maximal function without the absolute values, but the definition above will suffice for our purposes. It follows from the Lebesgue differentiation theorem that the absolute value of a locally integrable function is bounded from above by the maximal function almost everywhere, see Corollary 1.23. This explains the terminology. It is enough to assume that f : Rn → [−∞, ∞] is a measurable function in the definition above, but the assumption f ∈ L1loc (Rn ) guarantees that the integral averages are finite. If the function f is not locally integrable, then the maximal function is infinite everywhere: if there is a compact set K where f is not integrable, then at every point x there is a ball B(x, r) that contains K, so M f (x) = ∞. Observe that M f (x) is defined at every point x ∈ Rn even if f is only defined up to a set of measure zero. Moreover, if f (x) = g(x) for almost every x ∈ Rn , then M f (x) = M g(x) for every x ∈ Rn . Remark 1.2. There are several alternative definitions for maximal functions. The noncentered maximal function of f ∈ L1loc (Rn ) at x ∈ Rn is M ∗ f (x) = sup  |f (y)| dy, Bx 1

B

2

1. MAXIMAL FUNCTIONS

where the supremum is taken over all open balls B with x ∈ B. Then M f (x) ≤ M ∗ f (x) for every x ∈ Rn . On the other hand, if B = B(z, r) x, then B(z, r) ⊂ B(x, 2r) and |B(x, 2r)| |f (y)| dy = 2n  |f (y)| dy ≤ 2n M f (x).  |f (y)| dy ≤ |B(z, r)|  B B(x,2r) B(x,2r) This implies M ∗ f (x) ≤ 2n M f (x) and thus M f (x) ≤ M ∗ f (x) ≤ 2n M f (x) for every x ∈ Rn . Instead of balls we will sometimes use cubes in the definition of the maximal function. Such maximal function is again comparable to M f with two-sided estimates as above. Maximal functions defined on dyadic cubes will be discussed in Chapter 8. Recall that a function g : Rn → [−∞, ∞] is lower semicontinuous if the distribution set {x ∈ Rn : g(x) > t} is open for every t ∈ R. Next we show that M f is always lower semicontinuous and thus measurable. This also indicates that the maximal operator improves the regularity of the function. Lemma 1.3. Assume that f ∈ L1loc (Rn ). Then M f is lower semicontinuous. Proof. Let Et = {x ∈ Rn : M f (x) > t} with t ∈ R. For a fixed x ∈ Et , there exists r > 0 such that 

|f (y)| dy > t.

B(x,r)

By the properties of the Lebesgue measure, we have  1 |f (y)| dy = lim |f (y)| dy,  R→r |B(x, R)| B(x,r) B(x,r) from which we conclude that there exists R > r with  1 |f (y)| dy > t. |B(x, R)| B(x,r) If z ∈ Rn with |x − z| < R − r, then |y − z| ≤ |y − x| + |x − z| < r + (R − r) = R for every y ∈ B(x, r), and thus B(x, r) ⊂ B(z, R). This implies   1 1 |f (y)| dy ≤ |f (y)| dy t< |B(x, R)| B(x,r) |B(x, R)| B(z,R)  1 = |f (y)| dy ≤ M f (z), |B(z, R)| B(z,R) whenever |x−z| < R−r. This shows that B(x, R−r) ⊂ Et and thus Et is open.



From the following example we see that the maximal function can be discontinuous. However, this cannot happen if the function is continuous, see Lemma 1.7 below.

1.1. HARDY–LITTLEWOOD MAXIMAL FUNCTION

3

Example 1.4. Let R > 0 and define f : Rn → R, f (x) = χB(0,R) (x) for every x ∈ Rn . Then M f (x) = 1 for every x ∈ B(0, R) and M f (x) < 1 for every x ∈ ∂B(0, R). The first claim holds since B(0, R) is open, and the second follows from the fact that there exists a ball B(y, r2 ) ⊂ B(x, r) \ B(0, R) if x ∈ ∂B(0, R) and r > 0. This shows that M f is not continuous at the points x ∈ ∂B(0, R). A few elementary properties of the Hardy–Littlewood maximal operator are collected below. Lemma 1.5. Assume that f, g ∈ L1loc (Rn ) and let x ∈ Rn . The following assertions hold: (a) (b) (c) (d) (e) (f)

M f (x) ≥ 0, M (f + g)(x) ≤ M f (x) + M g(x), |M f (x) − M g(x)| ≤ M (f − g)(x) if M f (x) < ∞ and M g(x) < ∞, M (af )(x) = |a|M f (x), for every a ∈ R, M (τh f )(x) = (τh M f )(x), where τh f (x) = f (x + h) with h ∈ Rn , M (δa f )(x) = (δa M f )(x), where δa f (x) = f (ax) with a > 0.

Proof. Assertion (a) is clear. To prove assertion (b), we observe that |f (y) + g(y)| dy

M (f + g)(x) = sup  r>0

B(x,r)





≤ sup  r>0

≤ sup  r>0

B(x,r)

B(x,r)

|f (y)| dy + 

|g(y)| dy

B(x,r)

|f (y)| dy + sup  r>0

|g(y)| dy

B(x,r)

= M f (x) + M g(x). For the proof of (c), let x ∈ Rn be such that M f (x) < ∞ and M g(x) < ∞. By property (b), we have M f (x) = M (f − g + g)(x) ≤ M (f − g)(x) + M g(x). By switching the roles of f and g above, we conclude |M f (x) − M g(x)| ≤ M (f − g)(x). Clearly M (af )(x) = sup  r>0

B(x,r)

|af (y)| dy = |a| sup  r>0

|f (y)| dy = |a|M f (x),

B(x,r)

for every a ∈ R. This proves (d), with the convention 0 · ∞ = 0. Assertion (e) follows from a change of variables  1 |f (y + h)| dy = sup |f (z)| dz M (τh f )(x) = sup  r>0 B(x,r) r>0 |B(x, r)| B(x+h,r) = sup  r>0

B(x+h,r)

|f (z)| dz = (τh M f )(x).

4

1. MAXIMAL FUNCTIONS

Another change of variables gives 1 |f (ay)| dy = sup M (δa f )(x) = sup  |B(x, r)| r>0 B(x,r) r>0

|f (z)|a−n dz

B(ax,ar)

|f (z)| dz = (δa M f )(x),

= sup  r>0



B(ax,ar)

where in the final step we used the fact that {ar : r > 0} = {r > 0}. This implies (f).  Example 1.6. Let 0 < α < n and define f : Rn \ {0} → R, f (x) = |x|−α . Let x . By Lemma 1.5 (f) and (d), we have x ∈ Rn \ {0} and write z = |x| M f (x) = M f (|x|z) = (δ|x| M f )(z) = M (δ|x| f )(z) = |x|−α M f (z). Thus M f (x) = M f (z)|x|−α for every x ∈ Rn \ {0}, where z ∈ ∂B(0, 1). Since f is radial, the value M f (z) is independent of z ∈ ∂B(0, 1). Moreover, M f (z) = sup  r>0

≤

B(z,r)

 1 −α 2

|f (y)| dy ≤ sup  1

B(z,r)

0 0. Then there exists a ball B(x, r) such that ε |τh f (y) − f (y)| dy + . 2 B(x,r)

M (τh f − f )(x) ≤ 

Since f is uniformly continuous in B(x, r + 1), there is 0 < δ < 1 such that ε |τh f (y) − f (y)| = |f (y + h) − f (y)| < 2 whenever y ∈ B(x, r) and |h| < δ. This implies |M (τh f )(x) − M f (x)| ≤ M (τh f − f )(x) ≤ 

B(x,r)

|τh f (y) − f (y)| dy +

ε 0. We will show that

C(n)Rn Rn ≤ M χB(0,R) (x) ≤ (1.1) n (|x| + R) (|x| + R)n for every x ∈ Rn . In order to prove (1.1), let x ∈ Rn . Lemma 1.5 (f) implies





M χB(0,R) (x) = M δR−1 χB(0,1) (x) = δR−1 M χB(0,1) (x). Therefore, it suffices to prove (1.1) for R = 1. First assume that |x| > 2 and let y ∈ B(0, 1). Then |y − x| ≤ |y| + |x| < 1 + |x|, and consequently B(0, 1) ⊂ B(x, |x| + 1). Moreover, if |x − y| < |x| − 1, then |y| ≥ |x| − |x − y| ≥ 1 and B(x, |x| − 1) ∩ B(0, 1) = ∅. From this we conclude that

|B(0, 1)| χB(0,1) (y) dy ≤ M χB(0,1) (x) = |B(x, |x| + 1)| B(x,|x|+1) ≤

sup 

ρ>|x|−1

B(x,ρ)

χB(0,1) (y) dy +

1 = sup |B(x, ρ)| ρ>|x|−1 ≤



sup |x|−1≥ρ>0



B(x,ρ)

χB(0,1) (y) dy

χB(0,1) (y) dy B(x,ρ)

|B(0, 1)| |B(0, 1)| = . |B(x, |x| − 1)| ρ>|x|−1 |B(x, ρ)| sup

Since |x| > 2, we have |x| + 1 = |x| + 4 − 3 < 3|x| − 3 = 3(|x| − 1), which implies |B(x, |x| + 1)| = |B(x, |x| − 1)|



|x| + 1 |x| − 1

n ≤ 3n .

Thus

|B(0, 1)| |B(0, 1)| |B(0, 1)| ≤ M χB(0,1) (x) ≤ ≤ 3n , |B(x, |x| + 1)| |B(x, |x| − 1)| |B(x, |x| + 1)| and this proves (1.1) in the case |x| > 2. On the other hand, if |x| ≤ 2, then |B(0, 1)| |B(0, 1)| ≥ = 3−n . |B(x, |x| + 1)| |B(x, 3)|

6

1. MAXIMAL FUNCTIONS

Thus by Lemma 1.10

|B(0, 1)| |B(0, 1)| ≤ M χB(0,1) (x) ≤ 1 ≤ 3n , |B(x, |x| + 1)| |B(x, |x| + 1)| and this completes the proof of (1.1) with C(n) = 3n . 1.2. Hardy–Littlewood–Wiener maximal function theorem This section discusses bounds for the maximal function on Lp (Rn ). Recall that for a Lebesgue measurable set A ⊂ Rn and 1 ≤ p ≤ ∞, the space Lp (A) consists of Lebesgue measurable functions f : A → [−∞, ∞] with f Lp (A) < ∞, where   p1 p f p = f Lp (A) = |f (x)| dx , for 1 ≤ p < ∞, A

and f ∞ = f L∞ (A) = ess sup |f (x)|. x∈A

The space Lp (A) becomes a Banach space when we identify two functions that differ on a set of measure zero. In addition, Lploc (A) is the space of functions f : A → [−∞, ∞] with f ∈ Lp (K) for every compact set K ⊂ A. First we show that the maximal function is essentially bounded if the function is essentially bounded. Intuitively this is clear since the integral averages in the maximal function cannot be larger than the essential supremum of the function. Lemma 1.10. Assume that f ∈ L∞ (Rn ). Then M f ∈ L∞ (Rn ) and M f (x) ≤ f L∞ (Rn ) for every x ∈ R . In particular, n

M f L∞ (Rn ) ≤ f L∞ (Rn ) . Proof. For every x ∈ Rn and r > 0, we have 

|f (y)| dy ≤ f L∞ (Rn ) . B(x,r)

Thus M f (x) ≤ f L∞ (Rn ) for every x ∈ Rn .



Lemma 1.10 asserts that M : L∞ (Rn ) → L∞ (Rn ), f → M f , is a bounded operator. In contrast, f ∈ L1 (R) does not imply M f ∈ L1 (R). Example 1.11. Example 1.9, with R = 1, shows that

1 M χB(0,1) (x) ≥ (|x| + 1)n

for every x ∈ Rn . This implies M χB(0,1) ∈ L1 (Rn ). Thus the Hardy–Littlewood maximal operator is not bounded on L1 (Rn ). Remark 1.12. In fact, the maximal function is in L1 (Rn ) if and only if the function is zero almost everywhere. In other words, M f ∈ / L1 (Rn ) for every nonzero 1 n f ∈ Lloc (R ). To see this, assume first that f is not zero almost everywhere and choose R > 0 to be large enough that  |f (y)| dy > 0. B(0,R)

1.2. HARDY–LITTLEWOOD–WIENER MAXIMAL FUNCTION THEOREM

7

For x ∈ Rn with |x| ≥ R we have 1 |f (y)| dy ≥ M f (x) ≥  |B(x, 2|x|)| B(x,2|x|)  C(n) |f (y)| dy, = |x|n B(0,R)

 |f (y)| dy B(0,R)

and thus M f ∈ / L1 (Rn ). On the other hand, if f (x) = 0 for almost every x ∈ Rn , then M f (x) = 0 for every x ∈ Rn and thus M f ∈ L1 (Rn ). The following Vitali-type 5r-covering lemma will be used several times throughout this book, for instance in the proof of the maximal function theorem, Theorem 1.15. For a proof, see Mattila [315, p. 23]. Lemma 1.13. Let B be a collection of balls B(x, rx ) ⊂ Rn , with the radius rx > 0 possibly depending on x. Assume that the balls in B have uniformly bounded radii. Then there exists a countable subcollection of pairwise disjoint balls B(xi , ri ) ∈ B, i ∈ N, such that 

B⊂

∞ 

B(xi , 5ri ).

i=1

B∈B

We remark that in Lemma 1.13 and in many corresponding statements the countable subcollection may be finite. We discuss briefly how the 5r-covering lemma is applied in practice. Let B be a collection of balls B(x, rx ) with 0 < rx < R. We would like to extract a countable subcollection of pairwise disjoint balls that covers the union of the original balls. In general, this is not possible without expanding the balls as in Lemma 1.13. Let B(xi , ri ) ∈ B, i ∈ N, be a countable subcollection of pairwise disjoint balls given by Lemma 1.13. Then ∞ ∞   ≤ ≤ B B(x , 5r ) |B(xi , 5ri )| i i i=1 i=1 B∈B ∞ ∞  

n n n |B(xi , ri )| = 5 B(xi , ri ) ≤ 5 B . =5 i=1

i=1

B∈B

  ∞ Thus the measures of B∈B B, ∞ i=1 B(xi , 5ri ) and i=1 B(xi , ri ) are comparable, and if one of these measures is infinite, then all of them are infinite. Moreover, if x ∈ B ∈ B, then there exists i ∈ N such that x ∈ B(xi , 5ri ). Also the following Cavalieri’s principle and its proof will be applied several times in the book, see Hewitt and Stromberg [191, Theorem 21.71]. Lemma 1.14. Let 0 < p < ∞. Assume that A ⊂ Rn is a measurable set and that f : A → [−∞, ∞] is a measurable function. Then 

 |f (x)|p dx = p A

0

∞

tp−1 |{x ∈ A : |f (x)| > t}| dt.

8

1. MAXIMAL FUNCTIONS

Proof. By Fubini’s theorem   |f (x)|  p χ |f (x)| dx = tp−1 dt dx A (x)p A Rn 0   ∞ χA (x)χ[0,|f (x)|) (t)tp−1 dt dx =p n R∞ 0 χA (x)χ[0,|f (x)|) (t)tp−1 dx dt =p Rn 0  ∞  χA (x)χ{x∈Rn :|f (x)|>t} (x) dx dt =p tp−1 Rn 0 ∞ =p tp−1 |{x ∈ A : |f (x)| > t}| dt.



0

Next we state the Hardy–Littlewood–Wiener maximal function theorem, which asserts that M : Lp (Rn ) → Lp (Rn ), f → M f , is a bounded operator for 1 < p ≤ ∞. The corresponding result does not hold for p = 1, but in this case we have a weak type estimate. The proof contains similar elements as the proof of the Marcinkiewicz interpolation theorem. Theorem 1.15. Let 1 ≤ p ≤ ∞ and assume that f ∈ Lp (Rn ). (a) Then M f is finite almost everywhere. (b) If p = 1, there exists a constant C = C(n) such that  C |{x ∈ Rn : M f (x) > t}| ≤ |f (x)| dx t Rn for every t > 0. (c) If 1 < p ≤ ∞, then M f ∈ Lp (Rn ) and there exists a constant C = C(n, p) such that M f Lp (Rn ) ≤ Cf Lp (Rn ) . Proof. Assume that f ∈ L1 (Rn ) and let Et = {x ∈ Rn : M f (x) > t} for t > 0. By the definition of M f , for every x ∈ Et there exists rx > 0 such that  (1.2) |f (y)| dy > t|B(x, rx )| = tωn rxn . B(x,rx )

This implies rxn

1 ≤ ωn t



1 |f (y)| dy ≤ ωn t B(x,rx )

 Rn

|f (y)| dy < ∞.

Thus B = {B(x, rx ) : x ∈ Et } is a collection of balls with uniformly bounded radii  and Et ⊂ B∈B B. The 5r-covering lemma, Lemma 1.13, gives pairwise disjoint balls B(xi , ri ) ∈ B, i ∈ N, such that |Et | ≤

∞

|B(xi , 5ri )| = 5

i=1

Thus |Et | ≤ 5n

∞

i=1

|B(xi , ri )| ≤

n

∞

|B(xi , ri )|.

i=1

 ∞  5n 5n |f (y)| dy ≤ |f (y)| dy. t i=1 B(xi ,ri ) t Rn

1.2. HARDY–LITTLEWOOD–WIENER MAXIMAL FUNCTION THEOREM

9

This proves assertion (b) with C(n) = 5n , and assertion (a) for p = 1. The latter claim follows by taking t → ∞. Next we prove assertion (a) and assertion (c) for 1 < p ≤ ∞. The case p = ∞ follows from Lemma 1.10, and thus we may assume that 1 < p < ∞. Let f ∈ Lp (Rn ) and t > 0, and define  f (x), if |f (x)| ≥ 2t , f(x) = 0, otherwise. Then |f | ≤ |f| + 2t , which implies by Lemma 1.5 (b) that M f (x) ≤ M f(x) + every x ∈ Rn . Consequently,  t {x ∈ Rn : M f (x) > t} ⊂ x ∈ Rn : M f(x) > . 2 Observe that  t 1−p |f (x)|p , |f(x)| = χ{y∈Rn : f(y) =0} (x)|f(x)|1−p |f(x)|p ≤ 2 which implies f ∈ L1 (Rn ). By (b) we have

t 2

for

2 · 5n  f L1 (Rn ) t  2 · 5n = |f (x)| dx. t {x∈Rn : |f (x)|≥ 2t }

|{x ∈ Rn : M f (x) > t}| ≤ (1.3)

By Lemma 1.14 and its proof, we obtain   ∞ (M f (x))p dx = p tp−1 {x ∈ Rn : M f (x) > t} dt Rn 0  ∞  ≤ p2 · 5n tp−2 |f (x)| dx dt 

(1.4)

0

{x∈Rn : |f (x)|≥ 2t }



2|f (x)|

= p2 · 5n

|f (x)| tp−2 dt dx Rn 0  2p−1 |f (x)||f (x)|p−1 dt dx. = p2 · 5n p − 1 Rn This proves assertion (c) with 1 < p < ∞ and  n  p1 5 p C(n, p) = 2 · . p−1 Finally, assertion (a) for 1 < p < ∞ holds since M f ∈ Lp (Rn ).



Remark 1.16. The centered maximal function M is a nonlinear operator. In general, the boundedness of a nonlinear operator does not imply its continuity. However, since M is a sublinear operator which is bounded on Lp (Rn ), it is also continuous on Lp (Rn ) for 1 < p ≤ ∞. To see this, let f, g ∈ Lp (Rn ). By Lemma 1.5 (c) and Theorem 1.15 (a), we have |M f (x) − M g(x)| ≤ M (f − g)(x) for almost every x ∈ R . Theorem 1.15 (c) implies n

M f − M gLp (Rn ) ≤ M (f − g)Lp (Rn ) ≤ C(n, p)f − gLp (Rn ) .

10

1. MAXIMAL FUNCTIONS

The continuity of M on Lp (Rn ) is a consequence of this estimate. When p = 1, the argument above shows that {x ∈ Rn : |M f (x) − M g(x)| > t} ≤ {x ∈ Rn : M (f − g)(x) > t}  C(n) ≤ |f (x) − g(x)| dx t Rn whenever t > 0 and f, g ∈ L1 (Rn ). In particular, it follows that if (fi )i∈N converges to f in L1 (Rn ), then M fi converges to M f in measure as i → ∞. It is clear that f (x) = 0 for almost every x ∈ Rn if and only if there exists a point x0 ∈ Rn such that M f (x0 ) = 0. Consequently, M f (x) = 0 for every x ∈ Rn if M f (x0 ) = 0 for some x0 ∈ Rn . Next we show that if the maximal function of a locally integrable function is finite at one point, then it is finite almost everywhere. Lemma 1.17. Assume that f ∈ L1loc (Rn ). If there exists a point x0 ∈ Rn such that M f (x0 ) < ∞, then M f (x) < ∞ for almost every x ∈ Rn . Proof. Let fk = f χB(0,k) for k ∈ N. Then fk ∈ L1 (Rn ), and it follows from Theorem 1.15 (a) that for every k ∈ N we have M fk (x) < ∞ for almost every x ∈ Rn . Hence if ∞  E= {x ∈ Rn : M fk (x) = ∞}, k=1

then |E| = 0. We claim that M f (x) < ∞ for every x ∈ Rn \ E. Let x ∈ Rn \ E and choose k ∈ N large enough that B(x, 1) ⊂ B(0, k). If 0 < r ≤ 1, then 

B(x,r)

|f (y)| dy = 

|fk (y)| dy ≤ M fk (x) < ∞.

B(x,r)

Then assume that r > 1. Since B(x, r) ⊂ B(x0 , r0 ) with r0 = r + |x − x0 |, we have 

|B(x0 , r0 )| |B(x0 , r0 )| M f (x0 ) |f (y)| dy ≤  |B(x, r)| B(x0 ,r0 ) |B(x, r)|  n r + |x − x0 | M f (x0 ) ≤ (1 + |x − x0 |)n M f (x0 ) < ∞. = r

|f (y)| dy ≤

B(x,r)

This implies M f (x) = sup  r>0

|f (y)| dy ≤ M fk (x) + (1 + |x − x0 |)n M f (x0 ) < ∞, B(x,r)

which shows that M f (x) < ∞ for every x ∈ Rn \ E.



Remark 1.18. Assume that f is a continuous function in Rn . If there exists a point x0 ∈ Rn with M f (x0 ) < ∞, then the proof of Lemma 1.17 shows that M f (x) < ∞ for every x ∈ Rn . Thus M f is a continuous function in Rn , by Lemma 1.7. 1.3. Lebesgue differentiation theorem We discuss the Lebesgue differentiation theorem as an application of the maximal function theorem. The core of the argument is that the weak type estimate implies almost everywhere convergence of integral averages. This is based on the fact that such convergence holds for a dense class of continuous functions.

1.3. LEBESGUE DIFFERENTIATION THEOREM

11

Theorem 1.19. Let 1 ≤ p < ∞ and assume that f ∈ Lploc (Rn ). Then (1.5)

|f (y) − f (x)|p dy = 0

lim 

r→0

B(x,r)

for almost every x ∈ Rn . Proof. By considering the functions f χB(0,k) , for k ∈ N, we may assume that f ∈ Lp (Rn ). Let   n p p Et = x ∈ R : lim sup  |f (y) − f (x)| dy > 2 · 3 t , r→0

for t > 0. Then  x ∈ Rn : lim sup  r→0

B(x,r)

B(x,r)

  ∞ |f (y) − f (x)|p dy > 0 = E 1j . j=1

Hence, it suffices to show that |Et | = 0 for every t > 0. Let t > 0, x ∈ Et and ε > 0, and let g : Rn → R be a continuous function satisfying f − gLp (Rn ) < ε. Since g is continuous, we have |g(y) − g(x)|p dy = 0,

lim sup  r→0

B(x,r)

and thus 2 · 3p t < lim sup  r→0

|f (y) − f (x)|p dy

B(x,r)



 ≤ 3p lim sup  r→0 p

B(x,r)

|f (y) − g(y)|p dy + 

|g(y) − g(x)|p dy

B(x,r)

+ 3 |g(x) − f (x)|p = 3p lim sup  r→0

|f (y) − g(y)|p dy + 3p |f (x) − g(x)|p B(x,r)

≤ 3p M (|f − g|p )(x) + 3p |f (x) − g(x)|p . It follows that if x ∈ Et , then M (|f − g|p )(x) > t

or

|f (x) − g(x)|p > t.

By Theorem 1.15 (b) and Chebyshev’s inequality, we obtain |Et | ≤ |{x ∈ Rn : M (|f − g|p )(x) > t}| + |{x ∈ Rn : |f (x) − g(x)|p > t}|   C(n) 1 C(n) + 1 p p ≤ ε . |f (x) − g(x)| dx + |f (x) − g(x)|p dx ≤ t t t n n R R Letting ε → 0 shows that |Et | = 0.



Definition 1.20. We say that x ∈ Rn is a Lebesgue point of f ∈ L1loc (Rn ) if the limit (1.6) f ∗ (x) = lim  f (y) dy r→0

B(x,r)

exists as a real number and lim 

r→0

|f (y) − f ∗ (x)| dy = 0.

B(x,r)

The set of Lebesgue points of f is denoted by Leb(f ).

12

1. MAXIMAL FUNCTIONS

In (1.6) one might be tempted to write f (x) instead of f ∗ (x), but the equivalence class of f ∈ L1loc (Rn ) is only defined up to a set of measure zero. If f = g almost everywhere, then f and g have the same set of Lebesgue points and f ∗ (x) = g ∗ (x) for every Lebesgue point x of f and g. Thus the notion of Lebesgue point is independent of the representative in the equivalence class in L1loc (Rn ). Whether x is a Lebesgue point of f is completely independent of the value f (x), and the function f does not even need to be defined at x. The following Lebesgue differentiation theorem asserts that a locally integrable function has Lebesgue points almost everywhere. It allows us to pass from information on averages to pointwise information outside a set of measure zero. Theorem 1.21. Assume that f ∈ L1loc (Rn ). Then |Rn \Leb(f )| = 0. Moreover, there exists a set E ⊂ Leb(f ) such that |Rn \ E| = 0 and f ∗ (x) = f (x) for every x ∈ E. Proof. Let

 |f (y) − f (x)| dy = 0 .

 E=

x ∈ Rn : lim  r→0

B(x,r)

Theorem 1.19 implies |Rn \ E| = 0. If x ∈ E, then 0 ≤  f (y) dy − f (x) =  (f (y) − f (x)) dy B(x,r)

≤

B(x,r)

r→0

|f (y) − f (x)| dy −−−→ 0.

B(x,r)

This implies f ∗ (x) = lim  r→0

f (y) dy = f (x)

B(x,r)

and 

B(x,r)

|f (y) − f ∗ (x)| dy = 

r→0

|f (y) − f (x)| dy −−−→ 0,

B(x,r)

for every x ∈ E. It follows that E ⊂ Leb(f ), and thus |Rn \ Leb(f )| ≤ |Rn \ E| = 0. This completes the proof.  If f is a pointwise defined function in an equivalence class in L1loc (Rn ), then by the Lebesgue differentiation theorem the limit f ∗ (x) = lim  r→0

f (y) dy

B(x,r)

equals f (x) for almost every x ∈ Rn . It follows that f ∗ belongs to the same equivalence class as f in L1loc (Rn ). In arguments related to Lebesgue points we use for f ∈ L1loc (Rn ) the representative f ∗ , unless otherwise specified. Example 1.22. Let f : R → R be the Heaviside function ⎧ ⎪ ⎨1, x > 0, f (x) = 12 , x = 0, ⎪ ⎩ 0, x < 0. Then 1 lim r→0 2r



x+r

f (y) dy = f (x) x−r

1.4. A THEOREM OF STEIN

13

for every x ∈ R, but x = 0 is not a Lebesgue point of f . Indeed, we have  r  0  r 1 1 1 1 1 |f (y) − a| dy = |a| dy + |1 − a| dy = |a| + |1 − a| = 0 2r −r 2r −r 2r 0 2 2 for every a ∈ R and r > 0. As a simple corollary of the Lebesgue differentiation theorem we obtain pointwise estimate for a function by its maximal function. Corollary 1.23. Assume that f ∈ L1loc (Rn ). Then |f (x)| ≤ M f (x) for almost every x ∈ Rn . Proof. Theorem 1.21 implies ∗ |f (x)| = |f (x)| = lim  f (y) dy ≤ sup  r→0

r>0

B(x,r)

|f (y)| dy = M f (x),

B(x,r)

for almost every x ∈ Rn .



Another consequence of Theorem 1.21 is the following Lebesgue density theorem. Theorem 1.24. Assume that A ⊂ Rn is a measurable set. Then  1, for a.e. x ∈ A, |A ∩ B(x, r)| lim = r→0 |B(x, r)| 0, for a.e. x ∈ / A. Proof. Consider the function f (x) = χA (x) for every x ∈ Rn , and let E ⊂ Rn be the corresponding set from Theorem 1.21. Then we have, for every x ∈ A ∩ E, 1 = f (x) = f ∗ (x) = lim  r→0

B(x,r)

f (y) dy = lim

r→0

|A ∩ B(x, r)| . |B(x, r)|

Similar reasoning shows that the limit is 0 for every x ∈ Ac ∩ E, and this proves the claim.  1.4. A theorem of Stein The following lemma gives a reverse weak type estimate, which is a converse of Theorem 1.15 (b). Reverse weak type estimates play a central role in selfimprovement of reverse H¨ older inequalities in Chapter 8. Lemma 1.25. Assume that f ∈ L1 (Rn ). There exists a constant C = C(n) such that  1 |f (x)| dx ≤ C|{x ∈ Rn : M f (x) > t}| t {x∈Rn :M f (x)>t} for every t > 0. Proof. Let t > 0. By Lemma 1.3, the set Et = {x ∈ Rn : M f (x) > t} is open. Theorem 1.15 (b) implies that |Et | < ∞, and thus Rn \ Et = ∅. Let K ⊂ Et be a compact set and define rx = d (x, ∂Et ) > 0 for x ∈ K. Then B = {B(x, rx ) : x ∈ K} is a cover of K. For every x ∈ K we have B(x, rx ) ⊂ Et and 0 < rx ≤ diam(K) + d (K, ∂Et ) < ∞.

14

1. MAXIMAL FUNCTIONS

Let x ∈ K. Since B(x, 2rx )∩(Rn \Et ) = ∅, there exists a point z ∈ B(x, 2rx )\Et with M f (z) ≤ t. Thus 

|f (y)| dy ≤ t B(z,r)

for every r > 0 and, in particular, |f (y)| dy ≤ t.



B(z,7rx )

Since B(x, 5rx ) ⊂ B(z, 7rx ), we have 

B(x,5rx )

|f (y)| dy ≤ C(n) 

|f (y)| dy ≤ C(n)t

B(z,7rx )

)n . Lemma 1.13 gives pairwise disjoint balls B(xi , ri ) ∈ B, i ∈ N, with C(n) = ( 57 satisfying K ⊂ ∞ i=1 B(xi , 5ri ). This implies  |f (y)| dy ≤ K

∞ 

i=1

|f (y)| dy ≤ C(n)t

B(xi ,5ri )

= C(n)t

∞

∞

|B(xi , 5ri )|

i=1

|B(xi , ri )| ≤ C(n)t|Et |.

i=1

The claim follows since 

 |f (y)| dy = sup K⊂Et

Et

|f (y)| dy ≤ C(n)t|Et | K

with C(n) = 7n , where the supremum is taken over all compact subsets K of Et .  By Remark 1.12, M f ∈ L1 (Rn ) only if f ∈ L1loc (Rn ) is zero almost everywhere. However, the maximal function M f can be locally integrable even if f is not zero almost everywhere. Next we characterize the local integrability of M f for compactly supported functions f ∈ L1 (Rn ) by Zygmund’s class L log+ L(Rn ). We say that a measurable function f : Rn → [−∞, ∞] belongs to L log+ L(Rn ) if  Rn

|f (x)| log+ |f (x)| dx < ∞.

Here log+ |f | = χ{x∈Rn :|f (x)|>1} log|f |. Theorem 1.26. Assume that f ∈ L1 (Rn ) is a compactly supported function. Then M f ∈ L1loc (Rn ) if and only if f ∈ L log+ L(Rn ).

1.5. RESTRICTED MAXIMAL FUNCTION

15

Proof. First assume that f ∈ L log+ L(Rn ) and let B ⊂ Rn be a ball. Lemma 1.14 and the estimate in (1.3) give  ∞  M f (x) dx = 2 |{x ∈ B : M f (x) > 2t}| dt B 0  ∞ ≤ 2|B| + 2 |{x ∈ B : M f (x) > 2t}| dt 1  ∞  1 |f (x)| dx dt ≤ 2|B| + C(n) t {x∈Rn :|f (x)|≥t} 1  max{|f (x)|,1}  1 dt dx |f (x)| ≤ 2|B| + C(n) t n 1 R = 2|B| + C(n) |f (x)| log+ |f (x)| dx < ∞. Rn

Since B is an arbitrary ball and M f ∈ L1 (B), we conclude that M f ∈ L1loc (Rn ). Then assume that M f ∈ L1loc (Rn ). Since the support of f is compact and f ∈ L1 (Rn ), the maximal function M f (x) decays to zero as |x| → ∞ and thus E = {x ∈ Rn : M f (x) > 1} is a bounded set. Corollary 1.23 and Lemma 1.25 imply    max{|f (x)|,1} 1 |f (x)| log+ |f (x)| dx = |f (x)| dt dx t n n R R 1  ∞  1 |f (x)| dx dt ≤ t {x∈Rn :|f (x)|>t} 1  ∞  1 ≤ |f (x)| dx dt t {x∈Rn :M f (x)>t} 1  ∞ ≤ C(n) |{x ∈ Rn : M f (x) > t}| dt 1  ∞ |{x ∈ E : M f (x) > t}| dt = C(n) 1  ≤ C(n) M f (x) dx < ∞, E

since M f is locally integrable. This shows that f ∈ L log+ L(Rn ).



1.5. Restricted maximal function We apply maximal functions with a restricted collection of radii in many arguments. Definition 1.27. Let R : Rn → [0, ∞] be a function and assume that f ∈ The restricted centered maximal function MR f : Rn → [0, ∞] is  |f (x)|, if R(x) = 0, (1.7) MR f (x) = if R(x) > 0, sup B(x,r) |f (y)| dy,

L1loc (Rn ).

where the supremum is taken over all radii r with 0 < r < R(x). If R ≥ 0 is a real number, we use the convention that R(x) = R for every x ∈ Rn .

16

1. MAXIMAL FUNCTIONS

Let f ∈ L1loc (Rn ). As in Corollary 1.23, we have |f (x)| ≤ MR f (x) ≤ M f (x) for almost every x ∈ Rn . Thus the maximal function theorem, see Theorem 1.15, also holds for the restricted maximal function, assuming that MR f is measurable whenever f ∈ L1loc (Rn ). If R : Rn → [0, ∞) is continuous, then the proof of Lemma 1.3 can be adapted to show that MR f is measurable for every f ∈ L1loc (Rn ). The following result is a variant of Lemma 1.7. Under the assumptions of Lemma 1.28, we have MR f (x) < ∞ for every x ∈ Rn . Lemma 1.28. Assume that R : Rn → [0, ∞) and f : Rn → R are continuous. Then MR f is a continuous function in Rn . Proof. We observe that F (x, r) =

 |f (x)|,

if r = 0, B(x,r) |f (y)| dy, if r > 0,

is a continuous function in Rn × [0, ∞). Let x ∈ Rn and ε > 0. By the uniform continuity of F in B(x, 1) × [0, R(x) + 1], there exists 0 < η < 1 such that |F (y, s) − F (x, t)| < ε whenever |y − x| + |s − t| < η and 0 ≤ s, t ≤ R(x) + 1. By the continuity of R at x, there exists 0 < δ < η2 such that |R(x) − R(y)|
0, and thus n



1 MR f (x) ≤ MR |f |p (x) p for every x ∈ Rn . The following maximal function with respect to an open set will be studied in Section 4.5.

1.6. RIESZ POTENTIAL

17

Definition 1.30. Let Ω  Rn be an open set and assume that f ∈ L1loc (Ω). The maximal function MΩ f : Ω → [0, ∞] is defined by |f (y)| dy,

MΩ f (x) = sup 

(1.8)

B(x,r)

where the supremum is taken over all radii r with 0 < r < d (x, ∂Ω). We remark that MΩ f is measurable if f ∈ L1loc (Ω). The maximal function theorem, see Theorem 1.15 (c), implies that the maximal operator MΩ is bounded on Lp (Ω) for 1 < p ≤ ∞, that is (1.9) MΩ f Lp (Ω) ≤ M (f χΩ )Lp (Rn ) ≤ C(n, p)f χΩ Lp (Rn ) = C(n, p)f Lp (Ω) , for every f ∈ Lp (Ω). The first inequality above follows since B(x, r) ⊂ Ω if x ∈ Ω and 0 < r < d(x, ∂Ω). Lemma 1.31. Let Ω  Rn be an open set and assume that f : Ω → R has a continuous extension to Rn . Then MΩ f is continuous in Ω. Proof. Let f: Rn → R be a continuous function such that f(x) = f (x) for every x ∈ Ω. Write R(x) = d (x, ∂Ω) for every x ∈ Rn . Then MΩ f (x) = MΩ (f χ )(x) = MΩ (fχ )(x) = MR (fχ )(x) = MR f(x) Ω

Ω

Ω

for every x ∈ Ω. The claim follows from Lemma 1.28.



1.6. Riesz potential Standard applications of Riesz potentials are in potential theory and harmonic analysis. We apply them in connection with Sobolev–Poincar´e type inequalities in Chapter 3. Definition 1.32. Let 0 < α < n and assume that f ∈ L1loc (Rn ) is a nonnegative function. The Riesz potential Iα f : Rn → [0, ∞] is defined by  f (y) Iα f (x) = dy, |x − y|n−α n R for every x ∈ Rn . It is possible to consider Riesz potentials also for signed functions, but nonnegative functions suffice for our purposes. We begin with a bound for the Riesz potential of a characteristic function of a measurable set with finite measure. Lemma 1.33. Assume that E ⊂ Rn is a measurable set with |E| < ∞. There exists a constant C = C(n) such that  1 1 dy ≤ C|E| n I1 χE (x) = n−1 E |x − y| for every x ∈ Rn . Proof. The claim is trivial if |E| = 0, and hence we may assume that |E| > 0. Let x ∈ Rn and let B = B(x, r) be a ball with |B| = |E|. Then |E \ B| = |E| − |E ∩ B| = |B| − |B ∩ E| = |B \ E| and thus

 E\B

1 |E \ B| |B \ E| dy ≤ n−1 = n−1 ≤ |x − y|n−1 r r

 B\E

1 dy. |x − y|n−1

18

1. MAXIMAL FUNCTIONS

This implies  E



 1 1 dy + dy n−1 |x − y| |x − y|n−1 E\B E∩B   1 1 ≤ dy + dy n−1 |x − y| |x − y|n−1 B\E E∩B  1 1 1 = dy = C(n)r = C(n)|B| n = C(n)|E| n . n−1 |x − y| B

1 dy = |x − y|n−1



We also have a norm estimate for the Riesz potential. Lemma 1.34. Let 1 ≤ p < ∞. Assume that Ω ⊂ Rn is an open set with |Ω| < ∞ and let f ∈ Lp (Ω). There exists a constant C = C(n, p) such that   1 I1 (|f |χ ) p n Ω L (Ω) ≤ C|Ω| f Lp (Ω) . p Proof. Let x ∈ Ω. If 1 < p < ∞ and p = p−1 , then H¨ older’s inequality and Lemma 1.33 give   |f (y)| |f (y)| 1 dy = dy 1 1 n−1 (n−1) (n−1) |x − y| p Ω Ω |x − y| |x − y| p   p1   1 p |f (y)|p 1 ≤ dy dy n−1 n−1 Ω |x − y| Ω |x − y|  p1  p 1 |f (y)|  ≤ C(n, p)|Ω| np dy n−1 Ω |x − y|  p1  p−1 |f (y)|p np = C(n, p)|Ω| dy . n−1 Ω |x − y|

Note that the inequality above also holds for p = 1. By Fubini’s theorem and Lemma 1.33, we obtain    p−1 |f (y)|p |I1 (|f |χΩ )(x)|p dx ≤ C(n, p)|Ω| n dy dx n−1 Ω Ω Ω |x − y|  p−1 1 |f (y)|p dy.  ≤ C(n, p)|Ω| n |Ω| n Ω

Next we consider a pointwise upper bound for the Riesz potential in terms of the maximal function. Lemma 1.35. Let 0 < α < n and assume that f ∈ L1loc (Rn ) is a nonnegative function. There exists a constant C = C(n, α) such that  B(x,r)

for every x ∈ Rn and r > 0.

f (y) dy ≤ Cr α M f (x) |x − y|n−α

1.6. RIESZ POTENTIAL

19

Proof. Let Bj = B(x, 2−j r) and Aj = Bj \ Bj+1 for j ∈ N0 . Since |x − y| ≥ r for every y ∈ Aj and |Bj | = ωn (2−j r)n , we obtain  ∞ 

f (y) f (y) dy = dy n−α |x − y| |x − y|n−α B(x,r) j=0 Aj  ∞

(2−j−1 r)α−n f (y) dy ≤

−j−1

2

Aj

j=0

≤2

n−α

∞

−j

(2

−j

α

r) (2

r)

−n

 f (y) dy Bj

j=0

= ωn 2n−α r α

∞

2−αj 

f (y) dy

Bj

j=0

≤ C(n, α)r α M f (x)

∞

2−αj

j=0

≤ C(n, α)r α M f (x).



The following Hardy–Littlewood–Sobolev theorem for the Riesz potential will be applied in Chapter 3. The exponent p∗ below is called the Sobolev conjugate of p. Observe that p1 − p1∗ = α n. np Theorem 1.36. Let 1 < p < ∞, 0 < α < np and p∗ = n−αp . Assume that p n f ∈ L (R ) is a nonnegative function. There exists a constant C = C(n, p, α) such that

Iα f Lp∗ (Rn ) ≤ Cf Lp (Rn ) . Proof. If f = 0 almost everywhere, there is nothing to prove, and thus we may assume that f > 0 on a set of positive measure. This implies M f (x) > 0 for p . By H¨older’s inequality every x ∈ Rn . Let x ∈ Rn , r > 0 and p = p−1  Rn \B(x,r)

f (y) dy ≤ |x − y|n−α

 p

 p1 

f (y) dy Rn \B(x,r)

≤ C(n, p, α)r

α− n p

Rn \B(x,r)

|x − y|

(α−n)p

 1 p

dy

f Lp (Rn ) .

Lemma 1.35 implies    f (y) f (y) f (y) Iα f (x) = dy = dy + dy n−α n−α n−α Rn |x − y| B(x,r) |x − y| Rn \B(x,r) |x − y|

n ≤ C(n, p, α) r α M f (x) + r α− p f Lp (Rn ) . By choosing

 r=

M f (x) f Lp (Rn )

− np > 0,

we obtain (1.10)

αp

αp

Iα f (x) ≤ C(n, p, α)M f (x)1− n f Lnp (Rn )

20

1. MAXIMAL FUNCTIONS

for every x ∈ Rn . Theorem 1.15 (c) gives   αp ∗ p∗ n p |Iα f (x)| dy ≤ C(n, p, α)f Lp (Rn ) Rn

≤

(M f (x))p dx

Rn αp ∗ p C(n, p, α)f Lnp (Rn ) f pLp (Rn ) ,

which shows that αp

p ∗

+

Iα f Lp∗ (Rn ) ≤ C(n, p, α)f Lnp (Rnp ) = C(n, p, α)f Lp (Rn ) .



From the proof of the previous theorem we also obtain a weak type estimate for p = 1. Lemma 1.37. Let 0 < α < n and assume that f ∈ L1 (Rn ) is a nonnegative function. There exists a constant C = C(n, α) such that n

|{x ∈ Rn : Iα f (x) > t}| ≤ Ct− n−α f Ln−α 1 (Rn ) n

for every t > 0. Proof. We may assume that f L1 (Rn ) > 0 since otherwise Iα f (x) = 0 for every x ∈ Rn . By a straightforward modification of (1.10) with p = 1, we have α

α

Iα f (x) ≤ C(n, α)M f (x)1− n f Ln1 (Rn ) for every x ∈ Rn . Theorem 1.15 (b) implies   α n−α |{x ∈ Rn : Iα f (x) > t}| ≤ x ∈ Rn : C(n, α)M f (x) n f Ln1 (Rn ) > t   n −α n ≤ x ∈ Rn : M f (x) > C(n, α)t n−α f L1n(Rn−α n) α

≤ C(n, α)t− n−α f Ln−α 1 (Rn ) f L1 (Rn ) n

n

= C(n, α)t− n−α f Ln−α 1 (Rn ) n



for every t > 0. 1.7. Fractional maximal function

The fractional maximal function has many applications in potential theory and partial differential equations. Definition 1.38. Let 0 ≤ α ≤ n and R > 0, and assume that f ∈ L1loc (Rn ). The centered fractional maximal function Mα,R f : Rn → [0, ∞] is defined by Mα,R f (x) = sup r α  0 n, we have Mα f (x) = ∞ for every x ∈ Rn if f = 0 on a set of positive measure. When 0 < α < n, there is a close connection between the fractional maximal function and the Riesz potential. Lemma 1.40. Let 0 < α < n and assume that f ∈ L1loc (Rn ). There exists a constant C = C(n) such that Mα f (x) ≤ CIα |f |(x)

(1.11) for every x ∈ R . n

Proof. Let x ∈ Rn and r > 0. Then  r α−n α r  |f (y)| dy = |f (y)| dy ωn B(x,r) B(x,r)   |f (y)| |f (y)| 1 1 ≤ dy ≤ dy. n−α ωn B(x,r) |x − y| ωn Rn |x − y|n−α Taking supremum over radii r > 0 proves the claim with C(n) =

1 ωn .



Remark 1.41. It is not possible to obtain a pointwise inequality in the reverse direction in (1.11). However, such an inequality holds in average by a result of Muckenhoupt and Wheeden. That is, for every 1 < p < ∞ there exists a constant C = C(n, p, α) such that   C −1 Mα f Lp (Rn ) ≤ Iα |f | p n ≤ CMα f Lp (Rn ) L (R )

for every f ∈ see Muckenhoupt and Wheeden [335] and Adams and Hedberg [4, Theorem 3.6.1]. L1loc (Rn ),

The following version of the Hardy–Littlewood–Sobolev theorem, see Theorem 1.36 and Lemma 1.37, will be useful in Section 4.6. Theorem 1.42. Let 1 ≤ p < ∞, 0 < α
1, there exists a constant C = C(n, p, α) such that Mα f Lp∗ (Rn ) ≤ Cf Lp (Rn ) for every f ∈ L (Rn ). (b) If p = 1, there exists a constant C = C(n, α) such that p

n

|{x ∈ Rn : Mα f (x) > t}| ≤ Ct− n−α f Ln−α 1 (Rn ) n

for every f ∈ L1 (Rn ) and t > 0. Proof. If f ∈ Lp (Rn ), p > 1 and 0 < α < np , then (1.11) and Theorem 1.36 imply   Mα f Lp∗ (Rn ) ≤ C(n)Iα |f |Lp∗ (Rn ) ≤ C(n, p, α)f Lp (Rn ) . On the other hand, if f ∈ L1 (Rn ) and 0 < α < n, then (1.11) and Lemma 1.37 give |{x ∈ Rn : Mα f (x) > t}| ≤ |{x ∈ Rn : C(n)Iα |f |(x) > t}| n

≤ C(n, α)t− n−α f Ln−α 1 (Rn ) n

for every t > 0.



22

1. MAXIMAL FUNCTIONS

The size of the distribution set {x ∈ Rn : Mα f (x) > t} can be estimated using the Hausdorff content. We recall here also the definition of Hausdorff measures, which are natural lower dimensional measures on Rn . Definition 1.43. Let E ⊂ Rn and λ ≥ 0. For 0 < δ ≤ ∞, the λ-dimensional Hausdorff δ-content of E is ∞  ∞

 λ λ ri : E ⊂ B(xi , ri ), 0 < ri ≤ δ . Hδ (E) = inf i=1

i=1

In the case λ = 0 we also allow finite summations. The (spherical) λ-dimensional Hausdorff measure of E is Hλ (E) = lim Hδλ (E) = sup Hδλ (E). δ→0

δ>0

Both Hδλ and Hλ are outer measures and Borel sets in Rn are Hλ -measurable. For every E ⊂ Rn , we have λ (E) ≤ Hδλ (E) ≤ Hλ (E) H∞

whenever 0 < δ ≤ ∞. On the other hand, Hλ (E) = 0 if and only if Hδλ (E) = 0 for every (or some) 0 < δ ≤ ∞. Definition 1.44. The Hausdorff dimension of E ⊂ Rn is     λ (E) = 0 . dimH (E) = inf λ ≥ 0 : Hλ (E) = 0 = inf λ ≥ 0 : H∞ We refer to Evans and Gariepy [122, Chapter 2] and Mattila [315, Chapter 4] for these and other basic properties of Hausdorff measures and Hausdorff dimension. A standard covering argument, as in the proof of Theorem 1.15, gives the following weak type inequality for the fractional maximal function. Lemma 1.45. Let 0 ≤ α < n and assume that f ∈ L1 (Rn ). There exists a constant C = C(n, α) such that  C n−α n |f (x)| dx H∞ ({x ∈ R : Mα f (x) > t}) ≤ t Rn for every t > 0. Proof. Let Et = {x ∈ Rn : Mα f (x) > t} for t > 0. By the definition of Mα f , for every x ∈ Et there exists rx > 0 such that rxα  This implies rxn−α

1 ≤ ωn t

|f (y)| dy > t. B(x,rx )



1 |f (y)| dy ≤ ω nt B(x,rx )

 Rn

|f (y)| dy < ∞.

Thus B =  {B(x, rx ) : x ∈ Et } is a collection of balls with uniformly bounded radii and Et ⊂ B∈B B. Lemma 1.13 gives pairwise disjoint balls B(xi , ri ) ∈ B, i ∈ N,  such that Et ⊂ ∞ i=1 B(xi , 5ri ). It follows that  ∞ ∞ 

5n−α 5n−α n−α H∞ (Et ) ≤ (5ri )n−α ≤ |f (y)| dy ≤ |f (y)| dy, ωn t i=1 B(xi ,ri ) ωn t Rn i=1 and this proves the claim with C(n, α) =

5n−α wn .



1.8. NOTES

23

1.8. Notes The maximal function was introduced by Hardy and Littlewood [179] in the one-dimensional case and by Wiener [397] in the higher dimensional case. References for maximal functions include Bennet and Sharpley [28], DeVore and Sharpley [100], Duoandikoetxea [107], Garc´ıa-Cuerva and Rubio de Francia [139], Grafakos [156, 157], de Guzm´ an [98, 99], Journ´e [216], Lu, Ding and Yan [286], Sadosky [356], Stein [369, 371] and Torchinsky [383]. The maximal function theorem, Theorem 1.15, holds for the centered maximal function with respect to more general measures than the Lebesgue measure by the Besicovitch covering theorem, see Mattila [315, p. 40]. Sj¨ ogren [362] showed that the noncentered maximal function for more general measures than the Lebesgue measure may fail to satisfy the weak type estimate. Theorem 1.17 is by Fiorenza and Krbec [129]. See also Wik [399]. For Hausdorff content estimates, see Orobitg and Verdera [341]. Boundedness of maximal operators on weighted Lp (Rn ) spaces is discussed in Chapter 8 and Chapter 9. Stein [370] showed that the Lp (Rn ) norm of the centered Hardy–Littlewood maximal operator with 1 < p ≤ ∞ is bounded from above by a dimension-free constant, see also Stein and Str¨omberg [372]. For the noncentered maximal function the constant depends on the dimension, see Grafakos and Montgomery-Smith [159]. The best constant in the weak type estimate for the centered Hardy–Littlewood maximal operator was obtained by Melas [323]. In the one-dimensional case best constants have been investigated, for example, by Aldaz [10], Bernal [29], Grafakos and Kinnunen [158] and Grafakos and Montgomery-Smith [159]. The class L log+ L(Rn ) in Theorem 1.26 was introduced by Zygmund and the necessity of this condition in Theorem 1.26 was observed by Stein [368]. Reverse weak type estimates as in Theorem 1.25 with more general measures have been investigated by Andersen and Young [15, 16]. For the Riesz potentials we refer to Adams and Hedberg [4, Chapter 3], Aikawa and Ess´en [8], Landkof [254, Chapter 1], Mizuta [329], Riesz [355], Stein [369, Section V.1] and Ziemer [407, Section 2.6]. The proof of Theorem 1.36 is from Hedberg [181]. Maximal function estimates for potentials are discussed, for example, in Adams and Hedberg [4, Chapter 3], Mal´ y and Ziemer [303, Section 1.2], Mazya [321, Chapter 12], Turesson [388, Chapter 3] and Ziemer [407, Section 2.8]. For the Riesz potentials and the fractional maximal function, see also Adams [3].

10.1090/surv/257/02

CHAPTER 2

Lipschitz and Sobolev Functions Sobolev spaces are natural function spaces in the theory of partial differential equations and variational problems. The definition of Sobolev function involves weak derivatives, but Sobolev spaces are also obtained by completion from smooth functions. In addition, we consider the flexible class of Lipschitz continuous functions. The definition and properties of Sobolev spaces are reviewed in Section 2.2 and Section 2.3. Sobolev spaces with zero boundary values are considered in Section 2.4 and the topic will be developed further in Section 5.5. This is the natural function space associated with Hardy’s inequalities in Chapter 6 and it enables us to assign and compare boundary values of functions in Chapter 11. Weak convergence methods in Sobolev spaces are considered in Section 2.5 and they are applied in Section 4.4, Section 4.5 and Section 12.2. 2.1. Lipschitz functions We begin by recalling the definition of a Lipschitz continuous function. Definition 2.1. Let A ⊂ Rn and 0 ≤ L < ∞. A function u : A → R is L-Lipschitz if |u(x) − u(y)| ≤ L|x − y| for every x, y ∈ A. The number L is called a Lipschitz constant of u. We say that u : A → R is Lipschitz in A if it is L-Lipschitz for some L. The space of all Lipschitz functions in A is denoted by Lip(A). A function u : Ω → R is locally Lipschitz in an open set Ω ⊂ Rn if u ∈ Lip(K) for every compact K ⊂ Ω. The space of all locally Lipschitz functions in Ω is denoted by Liploc (Ω). Observe that the Lipschitz constant of a function in Liploc (Ω) may depend on the compact subset K ⊂ Ω. Example 2.2. Let y ∈ Rn and let E ⊂ Rn with E = ∅. The triangle inequality implies that the distance functions x → |x − y| and x → d (x, E) are 1-Lipschitz functions in Rn . Note that these functions are not, in general, differentiable everywhere. However, by Rademacher’s theorem, see Theorem 2.9, they are differentiable almost everywhere. Example 2.3. Consider the slit domain Ω = {x ∈ R2 : |x| < 1} \ {(x1 , 0) ∈ R2 : x1 ≥ 0}. Let u : Ω → R, u(x) = θ, where θ is the argument of x in polar coordinates with 0 < θ < 2π. It is clear that u is not Lipschitz in Ω. However, this function u is locally Lipschitz in Ω. 25

26

2. LIPSCHITZ AND SOBOLEV FUNCTIONS

Functions with zero boundary values will be of particular importance for us. For Lipschitz functions, it turns out to be convenient to make the further distinction between compactly supported functions and functions that vanish in the complement and have a compact support. Definition 2.4. Let A ⊂ Rn . (a) The space of all compactly supported Lipschitz functions in A is Lipc (A) = {u ∈ Lip(Rn ) : supp u is a compact subset of A}. (b) The space of all compactly supported Lipschitz functions that vanish in the complement of A is Lip0 (A) = {u ∈ Lipc (Rn ) : u = 0 in Rn \ A}. Remark 2.5. It is clear that Lipc (A) ⊂ Lip0 (A). For a bounded set A ⊂ Rn , we have Lip0 (A) = {u ∈ Lip(Rn ) : u = 0 in Rn \ A}. Moreover, if Ω ⊂ Rn is an open set, then C0∞ (Ω) ⊂ C0k (Ω) ⊂ Lipc (Ω) for every k ∈ N. Indeed, by considering the zero extension of u ∈ C01 (Ω), we may assume that u ∈ C01 (Rn ). Let x, y ∈ Rn , x = y. By the mean value theorem, there exists z in the line-segment between x and y such that |u(x) − u(y)| = |∇u(z) · (x − y)| ≤ ∇uL∞ (Rn ) |x − y|. This shows that u is L-Lipschitz with L = ∇uL∞ (Rn ) . One useful property of Lipschitz functions, not shared by smooth functions, is that the pointwise minimum and maximum of L-Lipschitz functions are L-Lipschitz. This is also true for pointwise infimum and supremum of L-Lipschitz functions if the infimum and supremum are finite at one point. In particular, if u is an L-Lipschitz function in A ⊂ Rn and t ∈ R, then the truncations max{u, t} and min{u, t} are L-Lipschitz in A. Example 2.6. Let x ∈ Rn and r > 0. Define   u(y) = max 0, 1 − 1r d (y, B(x, r)) , for y ∈ Rn . The function u is 1r -Lipschitz in Rn , u = 1 in B(x, r), and u = 0 in Rn \ B(x, 2r). This kind of function is used as a cutoff to localize estimates. The following result gives a method to extend L-Lipschitz functions from a subset A ⊂ Rn to L-Lipschitz functions in the entire Rn . Theorem 2.7. Let A ⊂ Rn and assume that u : A → R is an L-Lipschitz function with 0 ≤ L < ∞. There exists an L-Lipschitz function u  : Rn → R such that u (x) = u(x) for every x ∈ A. Proof. Define (2.1)

u (x) = inf {u(a) + L|x − a|}, a∈A

for x ∈ Rn . We claim that u (x) = u(x) for every x ∈ A. To see this we fix x ∈ A and observe that, for every a ∈ A, u(x) − u(a) ≤ |u(x) − u(a)| ≤ L|x − a|,

2.1. LIPSCHITZ FUNCTIONS

27

which implies u(x) ≤ u(a) + L|x − a| for every a ∈ A. By taking infimum over a ∈ A, we obtain u(x) ≤ u (x). On the other hand, by the definition u (x) ≤ u(x), and thus u (x) = u(x) for every x ∈ A. Then we show that u  is L-Lipschitz in Rn . In particular, the proof below also shows that u  is real-valued since u  = u in A. Let x, y ∈ Rn . Then u (x) = inf {u(a) + L|x − a|} ≤ inf {u(a) + L|y − a| + L|x − y|} a∈A

a∈A

= inf {u(a) + L|y − a|} + L|x − y| = u (y) + L|x − y|. a∈A

By switching the roles of x and y, we obtain u (y) ≤ u (x) + L|x − y|. Thus −L|x − y| ≤ u (x) − u (y) ≤ L|x − y|, 

which shows that u  is L-Lipschitz in Rn .

Remark 2.8. The function u  in (2.1) is called the McShane extension of u. A similar argument as used in the proof of Theorem 2.7 shows that the function sup {u(a) − L|x − a|},

(2.2)

a∈A

for x ∈ Rn , is also an L-Lipschitz extension of u. The McShane extension u  in (2.1) is the largest L-Lipschitz extension of u, in the sense that if v : Rn → R is LLipschitz and v|A = u, then v ≤ u . Correspondingly, the function defined by (2.2) is the smallest L-Lipschitz extension of u. Let Ω ⊂ Rn be an open set. A function u : Ω → R is differentiable at x ∈ Ω if there exists a linear map Du : Rn → R, called the derivative of u, such that |u(y) − u(x) − Du(y − x)| = 0. (2.3) lim y→x |y − x| If the derivative Du exists, it is unique and satisfies Du(y − x) = ∇u(x) · (y − x) for every y ∈ R , where n

 ∇u(x) =

∂u ∂u (x), . . . , (x) ∂x1 ∂xn



is the pointwise gradient of u at x. We recall the classical Rademacher theorem. See Evans and Gariepy [122, Section 3.1.2] for a proof. Theorem 2.9. Let Ω ⊂ Rn be an open set and assume that u ∈ Lip(Ω). Then u is differentiable almost everywhere in Ω. As a corollary of Theorem 2.9, we obtain differentiation rules for Lipschitz functions. Corollary 2.10. Let Ω ⊂ Rn be an open set. If u, v ∈ Lip(Ω), then the Leibniz rule (2.4)

∇(uv)(x) = ∇u(x)v(x) + u(x)∇v(x)

holds for almost every x ∈ Ω. Moreover, if f ∈ C 1 (R) and u ∈ Lip(Ω), then the chain rule (2.5)

∇(f ◦ u)(x) = f  (u(x))∇u(x)

is valid for almost every x ∈ Ω.

28

2. LIPSCHITZ AND SOBOLEV FUNCTIONS

Instead of the gradient ∇u, we are often interested in |∇u|, for which we have the following representation. Lemma 2.11. Let Ω ⊂ Rn be an open set and assume that u ∈ Lip(Ω). Then |∇u(x)| = lim

(2.6)

sup

r→0 y∈B(x,r)

|u(y) − u(x)| r

for almost every x ∈ Ω. Proof. By Theorem 2.7, we may assume that u ∈ Lip(Rn ), and by Theorem 2.9 u is differentiable at almost every x ∈ Ω. Let x ∈ Ω be such a point and let r > 0. Then |∇u(x) · (y − x)| |∇u(x)||y − x| ≤ sup ≤ |∇u(x)|. sup r r y∈B(x,r) y∈B(x,r) ∇u(x) On the other hand, we choose z = x+r |∇u(x)| if ∇u(x) = 0, and z = x if ∇u(x) = 0.

Since z ∈ B(x, r), we have

|∇u(x) · (y − x)| |∇u(x) · (y − x)| = sup r r y∈B(x,r) y∈B(x,r) sup

|∇u(x) · (z − x)| = |∇u(x)|. r Combining the estimates above we obtain |∇u(x) · (y − x)| (2.7) |∇u(x)| = sup , r y∈B(x,r) ≥

for every r > 0. Since u is differentiable at x, it follows from (2.3) that (2.8)

lim

sup

r→0 y∈B(x,r)

|u(y) − u(x) − ∇u(x) · (y − x)| = 0. r

Let r > 0 and y ∈ B(x, r), and write ∇u(x) · (y − x) , r u(y) − u(x) − ∇u(x) · (y − x) , b(y, r) = r u(y) − u(x) c(y, r) = . r Then c(y, r) = a(y, r) + b(y, r) and hence a(y, r) =

|a(y, r)| −

sup |b(z, r)| ≤ |a(y, r)| − |b(y, r)| ≤ |c(y, r)| z∈B(x,r)

≤ |a(y, r)| + |b(y, r)| ≤ |a(y, r)| +

sup |b(z, r)|. z∈B(x,r)

By taking supremums over all y ∈ B(x, r), we obtain sup |a(y, r)| − y∈B(x,r)

sup |b(y, r)| ≤ y∈B(x,r)

sup |c(y, r)| y∈B(x,r)

≤

sup |a(y, r)| + y∈B(x,r)

sup |b(y, r)|. y∈B(x,r)

The claim follows by taking r → 0 and using (2.7) and (2.8).



2.2. SOBOLEV SPACES

29

Remark 2.12. From (2.6) we see that if u ∈ Lip(Ω) is an L-Lipschitz function, then |∇u(x)| ≤ L for almost every x ∈ Ω. The following locality property is a useful consequence of (2.6). Lemma 2.13. Let Ω ⊂ Rn be an open set and assume that u ∈ Lip(Ω) and t ∈ R. Then ∇u = 0 almost everywhere in the set {x ∈ Ω : u(x) = t}. Proof. By Theorem 2.7, we may assume that Ω = Rn and u ∈ Lip(Rn ). Let A = {x ∈ Ω : u(x) = t} and let x ∈ A be such that (2.6) holds. By the Lebesgue density theorem, see Theorem 1.24, we may assume that (2.9)

|A ∩ B(x, r)| = 1. |B(x, r)|

lim

r→0

Define d(r) =

sup

d(y, A) + r 2 > 0

y∈B(x,r)

for r > 0. By (2.9) we have d(r) = 0. r Let r > 0 and y ∈ B(x, r). There exists a point z ∈ A ∩ B(y, d(r)). Since u ∈ Lip(Rn ) is an L-Lipschitz function for some constant L > 0, we have lim

r→0

|u(y) − u(x)| |u(y) − u(z)| L|y − z| Ld(r) = ≤ ≤ . r r r r This implies |u(y) − u(x)| Ld(r) ≤ lim = 0, r→0 y∈B(x,r) r→0 r r

|∇u(x)| = lim

sup



and the proof is complete. 2.2. Sobolev spaces

Let Ω ⊂ R be an open set and let u ∈ Liploc (Ω). Then u is absolutely continuous on all closed line segments that are contained in Ω. Using integration by parts on lines parallel to the coordinate axes and Fubini’s theorem, we obtain   ∂u ∂ϕ (2.10) u(x) (x) dx = − (x)ϕ(x) dx, ∂x ∂x i i Ω Ω n

for every test function ϕ ∈ C0∞ (Ω) and i = 1, . . . , n. Weak derivatives are defined as functions that satisfy the integration by parts formula (2.10). Definition 2.14. Let Ω ⊂ Rn be an open set and assume that u ∈ L1loc (Ω). A function v ∈ L1loc (Ω) is the ith weak partial derivative of u in Ω, for i = 1, . . . , n, if   ∂ϕ u(x) (x) dx = − v(x)ϕ(x) dx ∂xi Ω Ω for every ϕ ∈ C0∞ (Ω). We write i = 1, . . . , n, exist, then ∇u(x) =

∂u ∂xi



= v. If all weak partial derivatives

 ∂u ∂u (x), . . . , (x) , ∂x1 ∂xn

for x ∈ Ω, is the weak gradient of u in Ω.

∂u ∂xi ,

with

30

2. LIPSCHITZ AND SOBOLEV FUNCTIONS

Remark 2.15. A weak gradient is unique up to a set of measure zero. To see this, let f and g be weak gradients of u ∈ L1loc (Ω) in Ω. Then    f (x)ϕ(x) dx = − u(x)∇ϕ(x) dx = g(x)ϕ(x) dx, Ω

Ω

Ω

C0∞ (Ω),

for every ϕ ∈ where the integrals are taken componentwise. From this it follows that f = g almost everywhere in Ω. Moreover, we have f (x) = g(x) at every point x ∈ Ω that is a Lebesgue point of all the components of both f and g. It is also clear that changing a function on a set of measure zero does not affect its weak derivatives. Remark 2.16. Let u ∈ Liploc (Ω) and i = 1, . . . , n. Then the ith classical ∂u partial derivative ∂x exists almost everywhere in Ω by Rademacher’s theorem, i Theorem 2.9. By (2.10), the ith classical partial derivative coincides almost every∂u where in Ω with the ith weak partial derivative ∂x . We use the same notation for i the weak and pointwise derivatives of functions, and the interpretation will be clear from the context. Definition 2.17. Let 1 ≤ p ≤ ∞ and let Ω ⊂ Rn be an open set. The Sobolev ∂u , space W 1,p (Ω) consists of functions u ∈ Lp (Ω) whose weak partial derivatives ∂x i p 1,p i = 1, . . . , n, exist and belong to L (Ω). For 1 ≤ p < ∞, the space W (Ω) is equipped with the norm

1 uW 1,p (Ω) = upLp (Ω) + ∇upLp (Ω) p ,

(2.11)

and the space W 1,∞ (Ω) is equipped with the norm (2.12) where ∇uLp (Ω)

uW 1,∞ (Ω) = uL∞ (Ω) + ∇uL∞ (Ω) ,   = |∇u|Lp (Ω) for every 1 ≤ p ≤ ∞. A function u in Ω belongs

1,p to the local Sobolev space Wloc (Ω) if u ∈ W 1,p (Ω ) for every open set Ω  Ω.

Remark 2.18. In some cases, it is convenient to use different norms on the Sobolev space. (a) For 1 ≤ p < ∞, the norms  p1  n  

 ∂u p p , uLp (Ω) +   ∂xi Lp (Ω) i=1

uLp (Ω) +

n  

 ∂u  ,   ∂xi Lp (Ω) i=1

and uLp (Ω) + ∇uLp (Ω) are all equivalent to (2.11). (b) For p = ∞, the norm    ∂u   ∂u      ,..., max uL∞ (Ω) ,    ∂x1 L∞ (Ω) ∂xn L∞ (Ω) is equivalent to (2.12). The spaces W 1,p (Ω), 1 ≤ p ≤ ∞, are Banach spaces with the above norms. Moreover, the spaces W 1,p (Ω) are reflexive when 1 < p < ∞. As in Lp (Ω) spaces we identify functions in W 1,p (Ω) that are equal almost everywhere.

2.3. APPROXIMATION AND CALCULUS IN SOBOLEV SPACES

31

2.3. Approximation and calculus in Sobolev spaces In this section we review some basic properties of the first-order Sobolev spaces. Instead of discussing all details, we give standard references. We illustrate the wellknown density properties of smooth functions in Sobolev spaces by considering the case of W 1,p (Rn ) with 1 < p < ∞, see Evans and Gariepy [122, Section 4.2.1]. The argument is based on convolution approximation and maximal functions. Recall that for measurable functions f, g : Rn → [−∞, ∞] the convolution f ∗ g is defined by  (f ∗ g)(x) =

Rn

f (y)g(x − y) dy

for every x ∈ Rn for which the integral is defined. If 1 ≤ p ≤ ∞, f ∈ Lp (Rn ) and g ∈ L1 (Rn ), then (f ∗ g)(x) exists for almost every x ∈ Rn and f ∗ gLp (Rn ) ≤ f Lp (Rn ) gL1 (Rn ) ,

(2.13)

see Grafakos [156, Theorem 1.2.12]. This is Young’s convolution inequality. Let ϕ : Rn → R,   ⎧ 1 ⎨a exp , x ∈ B(0, 1), |x|2 − 1 ϕ(x) = ⎩ 0, x ∈ Rn \ B(0, 1),  where the constant a is chosen so that Rn ϕ(x) dx = 1. For x ∈ Rn , ε > 0 and f ∈ L1loc (Rn ), we define 1 x . (2.14) ϕε (x) = n ϕ ε ε The convolution  (ϕε ∗ f )(x) =

Rn

ϕε (y)f (x − y) dy

can be considered as a weighted mean value of the function f in B(x, ε) and it is clearly independent of the representative of f . Let u ∈ W 1,p (Rn ) and ε > 0. Since ϕ ∈ C0∞ (Rn ), the dominated convergence theorem implies that ϕε ∗ u ∈ C ∞ (Rn ) ∩ W 1,p (Rn ) and ∂u ∂ϕε ∂(ϕε ∗ u) = ϕε ∗ = ∗u ∂xi ∂xi ∂xi

(2.15)

for every i = 1, . . . , n. The Hardy–Littlewood maximal function gives uniform pointwise control for the convolution approximations. Lemma 2.19. Assume that f ∈ L1loc (Rn ). Then sup |(ϕε ∗ f )(x)| ≤ M f (x) ε>0

for every x ∈ R . n

Proof. First consider a function ψ of the form ψ(x) =

N

j=1

aj χB(0,rj ) (x),

32

2. LIPSCHITZ AND SOBOLEV FUNCTIONS

where x ∈ Rn , N ∈ N and aj > 0 for every j = 1, . . . , N . Then ψL1 (Rn ) =

N

aj |B(0, rj )|.

j=1

Let x ∈ Rn and ε > 0. A change of variables z =

y ε

gives

 y  1 f (x − y) dy ψε (y)f (x − y) dy = n ψ ε Rn ε Rn   N = ψ(z)f (x − εz) dz = aj f (x − εz) dz n R B(0,rj )

 |(ψε ∗ f )(x)| =

j=1

≤

N

aj |B(0, rj )| 

|f (x − εz)| dz.

B(0,rj )

j=1

With y = x − εz we have 

 1 |f (y)| dy εn |B(0, rj )| B(x,εrj )  1 = |f (y)| dy ≤ M f (x). |B(x, εrj )| B(x,εrj )

|f (x − εz)| dz =

B(0,rj )

Thus |(ψε ∗ f )(x)| ≤

N

aj |B(0, rj )|M f (x) = ψL1 (Rn ) M f (x)

j=1

for every x ∈ Rn . To conclude the proof of the lemma, consider an increasing sequence of functions ψk as above such that ψk (x) → ϕ(x) for every x ∈ Rn as k → ∞. This is possible since ϕ is nonnegative, radial and radially decreasing. By the monotone convergence theorem we have   ϕε (y)|f (x − y)| dy = lim (ψk )ε (y)|f (x − y)| dy |(ϕε ∗ f )(x)| ≤ Rn Rn k→∞  = lim (ψk )ε (y)|f (x − y)| dy = lim ((ψk )ε ∗ |f |)(x) k→∞

Rn

k→∞

≤ lim ψk L1 (Rn ) M f (x) = ϕL1 (Rn ) M f (x) = M f (x) k→∞

for every ε > 0 and x ∈ Rn , and the claim follows.



Lemma 2.20. Let 1 < p < ∞ and assume that u ∈ W 1,p (Rn ). Then ϕε ∗ u → u in W 1,p (Rn ) as ε → 0. In particular, C ∞ (Rn ) ∩ W 1,p (Rn ) is dense in W 1,p (Rn ). Proof. By (2.15), it suffices to show that if f ∈ Lp (Rn ), then ϕε ∗ f → f in L (Rn ) as ε → 0. To this end, let ε > 0 and f ∈ Lp (Rn ) with f = f ∗ , and let x be a Lebesgue point of f . By the Lebesgue differentiation theorem, Theorem 1.21, we p

2.3. APPROXIMATION AND CALCULUS IN SOBOLEV SPACES

obtain

33

 |(ϕε ∗ f )(x) − f (x)| =

ϕε (y)(f (x − y) − f (x)) dy Rn  ϕL∞ (Rn ) ≤ |f (x − y) − f (x)| dy εn B(0,ε) = C(n)ϕL∞ (Rn ) 

ε→0+

|f (y) − f (x)| dy −−−−→ 0.

B(x,ε)

By Lemma 2.19, we have |(ϕε ∗ f )(x) − f (x)| ≤ |(ϕε ∗ f )(x)| + |f (x)| ≤ M f (x) + |f (x)|. Theorem 1.21 implies that almost every point x ∈ Rn is a Lebesgue point of f . Since M f + |f | ∈ Lp (Rn ) by Theorem 1.15 (c), we can apply the dominated convergence theorem to obtain   |(ϕε ∗ f )(x) − f (x)|p dx = lim |(ϕε ∗ f )(x) − f (x)|p dx = 0, lim ε→0

Rn ε→0

Rn



and the proof is complete.

Lemma 2.20 also holds for p = 1, see Evans and Gariepy [122, Section 4.2.1]. Note that the proof of Lemma 2.20 shows that C ∞ (Rn ) is dense in Lp (Rn ), for every 1 < p < ∞. As another application of convolution approximation, we identify the space of bounded Lipschitz continuous functions with the space W 1,∞ (Rn ). Theorem 2.21. Assume that u ∈ L∞ (Rn ). (a) Then u ∈ W 1,∞ (Rn ) if and only if u has a representative in Lip(Rn ). (b) If u ∈ W 1,∞ (Rn ), then there exists a set E ⊂ Rn such that |E| = 0 and |u(x) − u(y)| ≤ ∇uL∞ (Rn ) |x − y| for every x, y ∈ Rn \ E. (c) If u is L-Lipschitz in Rn , then ∇uL∞ (Rn ) ≤ L. Proof. Assume first that u ∈ W 1,∞ (Rn ) and define uε = ϕε ∗ u for every ε > 0. Let ε > 0. By Young’s convolution inequality, we have   ∇uε L∞ (Rn ) = ϕε ∗ ∇uL∞ (Rn ) ≤ ϕε ∗ |∇u| ∞ n ≤ ∇uL∞ (Rn ) , L

(R )

where the first convolution is taken componentwise. Since uε ∈ C ∞ (Rn ), we have  1 |uε (x) − uε (y)| = ∇uε (y + t(x − y)) · (x − y) dt 0

≤ ∇uε L∞ (Rn ) |x − y| ≤ ∇uL∞ (Rn ) |x − y| for every x, y ∈ Rn . By the proof of Theorem 2.20, the convolutions uε converge to u pointwise almost everywhere in Rn as ε → 0. Hence, there is a set E ⊂ Rn such that |E| = 0 and |u(x) − u(y)| ≤ ∇uL∞ (Rn ) |x − y|

(2.16)

for every x, y ∈ R \ E. This proves (b). By uniform continuity, u can be redefined on the set E of measure zero in such a way that (2.16) holds for every x, y ∈ Rn . In particular, u has a representative in Lip(Rn ). n

34

2. LIPSCHITZ AND SOBOLEV FUNCTIONS

Then assume that u ∈ L∞ (Rn ) is L-Lipschitz in Rn . Using integration by parts as in connection with (2.10), we find that the classical gradient of u, which exists almost everywhere, is the weak gradient of u. Moreover, by Theorem 2.9 and (2.6) we see that |∇u(x)| ≤ L for almost every x ∈ Rn . Thus u ∈ W 1,∞ (Rn )  and ∇u∞ ≤ L, and this implies the remaining assertions. Remark 2.22. In addition to Lemma 2.20, we have the following density properties for smooth functions in Sobolev spaces. (a) The subspace C0∞ (Rn ) is dense in W 1,p (Rn ) with 1 ≤ p < ∞. For p > 1, this follows from Lemma 2.20 and a truncation argument. Since C0∞ (Rn ) functions are Lipschitz continuous, we conclude that Lipc (Rn ) is dense in W 1,p (Rn ) with 1 ≤ p < ∞. (b) If 1 ≤ p < ∞ and Ω ⊂ Rn is an open set, then W 1,p (Ω) ∩ C ∞ (Ω) is dense in W 1,p (Ω). This follows from a standard partition of unity and convolution argument, see Evans and Gariepy [122, Section 4.2.1]. Moreover, if u ∈ W 1,p (Ω) is nonnegative, the approximating functions can be taken to be nonnegative. However, Example 2.3 shows that Lipschitz continuous functions are not dense in W 1,p (Ω) for every open set Ω ⊂ Rn . The density properties above allow us to transfer results for classical derivatives to the weak derivatives, see Evans and Gariepy [122, Section 4.2.2] and Heinonen, Kilpel¨ ainen and Martio [187, Theorem 1.24]. Lemma 2.23. Let 1 ≤ p < ∞ and write p = 

p p−1 ,

for 1 < p < ∞, and

1,p (Ω) and p = ∞, for p = 1. Assume that Ω ⊂ R is an open set and that u ∈ Wloc 1,p v ∈ Wloc (Ω). Then n

(2.17)

∇(uv)(x) = ∇u(x)v(x) + u(x)∇v(x)

for almost every x ∈ Ω. Another useful result is the chain rule for weak derivatives, see Evans and Gariepy [122, Section 4.2.2] and Heinonen, Kilpel¨ainen and Martio [187, Theorem 1.18]. Lemma 2.24. Let Ω ⊂ Rn be an open set. Assume that u ∈ W 1,p (Ω) and f ∈ C 1 (R) with f  ∈ L∞ (R) and f (0) = 0. Then f ◦ u ∈ W 1,p (Ω) and ∇(f ◦ u)(x) = f  (u(x))∇u(x), for almost every x ∈ Ω. The space of Lipschitz functions is closed under taking absolute values, minima and maxima. These useful properties are inherited by the Sobolev spaces W 1,p (Ω) with 1 ≤ p < ∞, see Evans and Gariepy [122, 4.2.2] and Heinonen, Kilpel¨ainen and Martio [187, Theorem 1.20]. Theorem 2.25. Let 1 ≤ p < ∞ and let Ω ⊂ Rn be an open set. (a) If u ∈ W 1,p (Ω), then |u| ∈ W 1,p (Ω) and ⎧ ⎪ a.e. in {x ∈ Ω : u(x) > 0}, ⎨∇u, ∇|u| = 0, a.e. in {x ∈ Ω : u(x) = 0}, ⎪ ⎩ −∇u, a.e. in {x ∈ Ω : u(x) < 0}.

2.4. SOBOLEV SPACES WITH ZERO BOUNDARY VALUES

35

(b) If u ∈ W 1,p (Ω) and t ∈ R, then ∇u = 0 almost everywhere in the set {x ∈ Ω : u(x) = t}. (c) If u, v ∈ W Moreover,

1,p

(Ω), then max{u, v} ∈ W 1,p (Ω) and min{u, v} ∈ W 1,p (Ω). 

∇ max{u, v} = and

 ∇ min{u, v} =

∇u, ∇v,

a.e. in {x ∈ Ω : u(x) ≥ v(x)}, a.e. in {x ∈ Ω : u(x) ≤ v(x)},

∇u, ∇v,

a.e. in {x ∈ Ω : u(x) ≤ v(x)}, a.e. in {x ∈ Ω : u(x) ≥ v(x)}.

Corresponding claims as in (a)–(c) also hold for functions in the local Sobolev space 1,p (Ω). Wloc Remark 2.26. Theorem 2.25 implies that we can cut a Sobolev function at a certain level and the truncated function belongs to the same Sobolev space. More precisely, let 1 ≤ p < ∞ and t ≥ 0, and assume that u ∈ W 1,p (Ω). Then min{u, t} ∈ W 1,p (Ω) and  ∇u, a.e. in {x ∈ Ω : u(x) < t}, ∇ min{u, t} = 0, a.e. in {x ∈ Ω : u(x) ≥ t}. A similar claim also holds for max{u, −t} for t ≥ 0. The restriction t ≥ 0 is necessary if Ω is unbounded, since neither min{u, t} nor max{u, −t} belongs to W 1,p (Ω) if t < 0 and Ω is unbounded. Moreover, the truncated functions ⎧ ⎪ in {x ∈ Ω : u(x) ≥ t}, ⎨t, ut = max{−t, min{u, t}} = u, in {x ∈ Ω : −t < u(x) < t}, ⎪ ⎩ −t, in {x ∈ Ω : u(x) ≤ −t}, converge to u ∈ W 1,p (Ω) in W 1,p (Ω) as t → ∞. Thus bounded W 1,p (Ω) functions are dense in W 1,p (Ω) when 1 ≤ p < ∞. 2.4. Sobolev spaces with zero boundary values Roughly speaking, a Sobolev function belongs to the space W01,p (Ω) if it vanishes on the boundary ∂Ω. However, the meaning of boundary values is delicate since Sobolev functions are only defined up to Lebesgue measure zero. We discuss a pointwise approach to this question in Section 5.5. Definition 2.27. Let 1 ≤ p < ∞ and let Ω ⊂ Rn be an open set. The Sobolev space with zero boundary values W01,p (Ω) is the completion of C0∞ (Ω) with respect to the Sobolev norm (2.11). The space W01,p (Ω) is equipped with the norm (2.11). We say that functions u, v ∈ W 1,p (Ω) have the same boundary values in the Sobolev sense if u − v ∈ W01,p (Ω). By definition, u ∈ W01,p (Ω) if and only if there exist functions ui ∈ C0∞ (Ω), i ∈ N, such that ui → u in W 1,p (Ω) as i → ∞. The only difference compared to W 1,p (Ω) is that functions in W01,p (Ω) can be approximated by C0∞ (Ω) functions instead of more general C ∞ (Ω) functions. That is, for 1 ≤ p < ∞, we have W 1,p (Ω) = W 1,p (Ω) ∩ C ∞ (Ω) and

W01,p (Ω) = C0∞ (Ω),

36

2. LIPSCHITZ AND SOBOLEV FUNCTIONS

where the completions are taken with respect to the Sobolev norm. It follows that W01,p (Ω) is a closed subspace of W 1,p (Ω). Moreover, C0∞ (Ω) ⊂ Lipc (Ω) ⊂ Lip0 (Ω) ⊂ W01,p (Ω) ⊂ W 1,p (Ω) ⊂ Lp (Ω) for every open set Ω ⊂ Rn . The third inclusion above is proved in Lemma 2.29 and the other inclusions are clear. Remark 2.28. Let 1 ≤ p < ∞ and let Ω ⊂ Rn be an open set. (a) If u ∈ W 1,p (Ω) has a compact support in Ω, then u ∈ W01,p (Ω). (b) If u ∈ W 1,p (Ω) and ϕ ∈ Lipc (Ω), then uϕ ∈ W01,p (Ω). The proofs can be found in Heinonen, Kilpel¨ainen and Martio [187, Theorem 1.24 and Lemma 1.25]. Lemma 2.29. Let 1 ≤ p < ∞ and let Ω ⊂ Rn be an open set. Then Lip0 (Ω) ⊂

W01,p (Ω).

Proof. By Remark 2.28 (b), Lipc (Ω) ⊂ W01,p (Ω). Hence, it suffices to show that functions in Lip0 (Ω) can be approximated in W 1,p (Ω) by functions in Lipc (Ω). Let u ∈ Lip0 (Ω). Define v = u+ ∈ Lip0 (Ω) and vε (x) = max{v(x) − ε, 0}, for x ∈ Rn and 0 < ε < 1. Since v(x) = 0 for every x ∈ Rn \ Ω and the support of v is bounded, it follows from the Lipschitz continuity of v that vε ∈ Lipc (Ω) for every 0 < ε < 1. By Theorem 2.25 (c) we have  ∇v, a.e. in {x ∈ Ω : v(x) ≥ ε}, ∇vε = 0, a.e. in {x ∈ Ω : v(x) ≤ ε}, and hence

  ∇v − ∇vε Lp (Ω) ≤ χ{0≤v≤ε} ∇v Lp (Ω) .

By the dominated convergence theorem and Theorem 2.25 (b), we obtain   p1    p   χ{0≤v≤ε} (x)|∇v(x)| dx lim χ{0≤v≤ε} ∇v Lp (Ω) = lim ε→0

ε→0

 =

Ω

lim χ{0≤v≤ε} (x)|∇v(x)| dx p

Ω ε→0

 =

Ω

χ{v=0} (x)|∇v(x)|p dx

 p1

 p1 = 0.

Here χ{0≤v≤ε} |∇v|p ≤ |∇v|p for every 0 < ε < 1, and so |∇v|p ∈ L1 (Ω) may be used as an integrable majorant. Since 0 ≤ v − vε ≤ ε and supp(v − vε ) ⊂ supp v, we conclude that v − vε W 1,p (Ω) ≤ v − vε Lp (Ω) + ∇v − ∇vε Lp (Ω)      p ≤ εχ + χ ∇v  supp v L (Ω)

{0≤v≤ε}

Lp (Ω)

−→ 0,

as ε → 0. The function u− ∈ Lip0 (Ω) can be approximated with Lipc (Ω) functions  in a similar manner, and the claim follows since u = u+ − u− .

2.5. WEAK CONVERGENCE AND SOBOLEV SPACES

37

We have the following version of Theorem 2.25 for W01,p (Ω), see Heinonen, Kilpel¨ainen and Martio [187, Lemma 1.23]. Theorem 2.30. Let 1 ≤ p < ∞ and let Ω ⊂ Rn be an open set. If u, v ∈ then |u| ∈ W01,p (Ω), max{u, v} ∈ W01,p (Ω), and min{u, v} ∈ W01,p (Ω).

W01,p (Ω),

It is easy to extend functions in Sobolev spaces with zero boundary values. Theorem 2.31. Let 1 ≤ p < ∞ and let Ω ⊂ Rn be an open set. Assume that u ∈ W01,p (Ω) and let u0 be the zero extension of u, that is,  u(x), x ∈ Ω, u0 (x) = 0, x ∈ Rn \ Ω. Then u0 ∈ W 1,p (Rn ) and ∇u0 =



∇u, 0,

a.e. in Ω, a.e. in Rn \ Ω.

In particular, u0 W 1,p (Rn ) = uW 1,p (Ω) . Proof. Let ∇u be the weak gradient of u with respect to Ω, and let g be the componentwise zero extension of ∇u. By definition, there exist functions ui ∈ C0∞ (Ω), i ∈ N, such that ui → u in W 1,p (Ω) as i → ∞. If ϕ ∈ C0∞ (Rn ), then    u0 (x)∇ϕ(x) dx = − u(x)∇ϕ(x) dx = − lim ui (x)∇ϕ(x) dx. − Rn

i→∞

Ω

Ω

Since the weak gradient of ϕ coincides with the classical gradient of ϕ in Ω, the last term can be written as    lim ∇ui (x)ϕ(x) dx = ∇u(x)ϕ(x) dx = g(x)ϕ(x) dx. i→∞

Ω

Rn

Ω

By the uniqueness of weak gradients, see Remark 2.15, we conclude that the weak gradient ∇u0 of u0 with respect to Rn coincides almost everywhere with g. The remaining conclusions follow from this.  2.5. Weak convergence and Sobolev spaces Let 1 ≤ p < ∞ and m ∈ N, and let A ⊂ Rn be a measurable set. Then L (A; Rm ) is the space of functions f = (f1 , . . . , fm ) : A → Rm with fi ∈ Lp (A) for every i = 1, . . . , m. Integration of such functions is done componentwise. The space Lp (A; Rm ) is equipped with the norm   p1 p f Lp (A;Rm ) = |f (x)| dx , p

A

where |f (x)| is the Euclidean norm of f (x) ∈ Rm . We will often abbreviate Lp (A; Rm ) = Lp (A), but sometimes use Lp (A; Rm ) to emphasize the difference. The space Lp (A; Rm ) is a Banach space if we identify two functions that differ on a set of measure zero. For 1 < p < ∞, the dual space of Lp (A; Rm ) can be  p identified with Lp (A; Rm ), where p = p−1 is the conjugate exponent of p. The 

duality between f ∈ Lp (A; Rm ) and g ∈ Lp (A; Rm ) is given by  f (x) · g(x) dx. g, f  = A

In particular, the space Lp (A; Rm ) is reflexive for 1 < p < ∞.

38

2. LIPSCHITZ AND SOBOLEV FUNCTIONS

In this section we discuss weak convergence techniques for Lp (Ω; Rm ), where Ω ⊂ Rn is an open set. Most of the results hold for more general Banach spaces as well. Definition 2.32. Let 1 < p < ∞ and m ∈ N, and let Ω ⊂ Rn be an open set. A sequence (fi )i∈N of functions in Lp (Ω; Rm ) converges weakly in Lp (Ω; Rm ) to a function f ∈ Lp (Ω; Rm ), if   lim fi (x) · g(x) dx = f (x) · g(x) dx i→∞



Ω

Ω 

for every g ∈ L (Ω; R ) with p = p

m

p p−1 .

We remark that a weak limit is unique in Lp (Ω; Rm ) and hence almost everywhere. By the Cauchy–Schwarz and H¨ older inequalities, convergent sequences in Lp (Ω; Rm ) are weakly convergent, but the converse is not true. Example 2.33. Let 1 < p < ∞ and xi = (i, 0, . . . , 0), for i ∈ N. We claim that the functions fi = χB(xi ,1) converge weakly to zero in Lp (Rn ) = Lp (Rn ; R1 ) as i → ∞. Note that these functions do not converge to zero in Lp (Rn ) since fi Lp (Rn ) = f1 Lp (Rn ) > 0 for every i ∈ N.  In order to show the weak convergence to zero, let g ∈ Lp (Rn ) and ε > 0, and  choose ϕ ∈ Lp (Rn ) with compact support satisfying g − ϕLp (Rn ) ≤ ε. Then   lim sup fi (x)g(x) dx = lim sup fi (x)(g(x) − ϕ(x)) dx i→∞

Rn

i→∞

Rn

≤ lim sup fi Lp (Rn ) g − ϕLp (Rn ) ≤ εf1 Lp (Rn ) . i→∞

Since this holds for every ε > 0, we have  lim fi (x)g(x) dx = 0, i→∞

Rn

and thus the sequence (fi )i∈N converges weakly to zero in Lp (Rn ). Next we show that weakly convergent sequences are bounded and that the Lp norm is lower semicontinuous with respect to the weak convergence. Lemma 2.34. Let 1 < p < ∞ and m ∈ N, and let Ω ⊂ Rn be an open set. If a sequence (fi )i∈N converges to f weakly in Lp (Ω; Rm ), then (fi )i∈N is bounded in Lp (Ω; Rm ). Moreover, we have (2.18)

f Lp (Ω;Rm ) ≤ lim inf fi Lp (Ω;Rm ) . i→∞

Proof. The claim sup fi Lp (Ω;Rm ) < ∞ i

follows from the uniform boundedness principle in Wojtaszczyk [400, I.A.7] or the closed graph theorem in Wojtaszczyk [400, II.A.3]. In order to prove (2.18), let  g ∈ Lp (Ω; Rm ) with gLp (Ω;Rm ) = 1 and  f (x) · g(x) dx. f Lp (Ω;Rm ) = Ω

2.5. WEAK CONVERGENCE AND SOBOLEV SPACES

39

The Cauchy–Schwarz and H¨ older inequalities imply   f (x) · g(x) dx = lim fi (x) · g(x) dx Ω

i→∞

Ω

≤ lim inf fi Lp (Ω;Rm ) gLp (Ω;Rm ) = lim inf fi Lp (Ω;Rm ) , i→∞

i→∞



and the desired inequality follows. Pointwise uniform bounds are preserved under weak convergence.

Theorem 2.35. Let 1 < p < ∞ and m ∈ N, and let Ω ⊂ Rn be an open set. Assume that sequences (fi )i∈N and (gi )i∈N are such that fi converges to f weakly in Lp (Ω; Rm ) and gi converges to g weakly in Lp (Ω) as i → ∞. If |fi (x)| ≤ gi (x) for almost every x ∈ Ω, then |f (x)| ≤ g(x) for almost every x ∈ Ω. Proof. Let x ∈ Ω be a Lebesgue point of g and all components of f . Let  0 < r < d(x, ∂Ω) and assume that B(x,r) f (y) dy = 0. Define −1 f (y) dy  f (y) dy ∈ Rm z =  B(x,r)

B(x,r)

and



ψ = z|B(x, r)|−1 χB(x,r) ∈ Lp (Ω; Rm ). By the Cauchy–Schwarz inequality and the assumptions, we have  =z· f (y) dy f (y) dy = f (y) · ψ(y) dy   B(x,r) B(x,r) Ω   = lim fi (y) · ψ(y) dy ≤ lim inf |fi (y)||ψ(y)| dy i→∞ Ω i→∞ Ω  ≤ lim inf gi (y)|ψ(y)| dy =  g(y) dy. i→∞

Hence,



B(x,r)

Ω

f (y) dy ≤ 

B(x,r)

g(y) dy, B(x,r)

 which clearly also holds if B(x,r) f (y) dy = 0. The claim follows by taking r → 0 on both sides since almost every point x ∈ Ω is a Lebesgue point of g and all components of f , by Theorem 1.21.  A bounded sequence in Lp (Ω; Rm ) need not have a convergent subsequence. However, the following Theorem 2.36 shows that it always has a weakly convergent subsequence if 1 < p < ∞. This will be important in our applications of weak convergence. Since Lp (Ω; Rm ) is reflexive and separable when 1 < p < ∞, Theorem 2.36 follows from Wojtaszczyk [400, II.A.14 and II.C.3]. This theorem does not hold when p = 1, which can be seen by considering the sequence (ϕ k1 )k∈N of functions from (2.14) approximating Dirac’s delta. Theorem 2.36. Let 1 < p < ∞ and m ∈ N, and let Ω ⊂ Rn be an open set. Assume that (fi )i∈N is a bounded sequence in Lp (Ω; Rm ). Then there exists a subsequence (fik )k∈N and a function f ∈ Lp (Ω; Rm ) such that fik → f weakly in Lp (Ω; Rm ) as k → ∞.

40

2. LIPSCHITZ AND SOBOLEV FUNCTIONS

Weak convergence is often too weak a mode of convergence and we need tools to upgrade it to stronger modes of convergence. We begin with the following result, which is related to Lemma 2.34. Since Lp (Ω; Rm ) is a uniformly convex Banach space, see Smith and Turett [363], the next lemma follows from Brezis [56, Proposition 3.32] or Hewitt and Stromberg [191, 15.17]. Lemma 2.37. Let 1 < p < ∞ and m ∈ N, and let Ω ⊂ Rn be an open set. Assume that a sequence (fi )i∈N converges to f weakly in Lp (Ω; Rm ) and lim sup fi Lp (Ω;Rm ) ≤ f Lp (Ω;Rm ) .

(2.19)

i→∞

Then fi → f in L (Ω; R ), as i → ∞. p

m

Under the assumptions in Lemma 2.37 we have, by (2.18) and (2.19), f Lp (Ω;Rm ) ≤ lim inf fi Lp (Ω;Rm ) ≤ lim sup fi Lp (Ω;Rm ) ≤ f Lp (Ω;Rm ) , i→∞

i→∞

which implies lim fi Lp (Ω;Rm ) = f Lp (Ω;Rm ) .

i→∞

This means that the limit exists with an equality in (2.19). For a proof of the following version of Mazur’s lemma, see Yosida [401, pp. 120–121] and Bj¨ orn and Bj¨orn [31, Section 6.1]. Theorem 2.38. Assume that X is a normed linear space and that (xi )i∈N is a sequence in X such that xi → x weakly in X asi → ∞. Then for all i ∈ N i there exists mi > i and a convex combination yi = m j=i ai,j xj , with ai,j ≥ 0 and mi j=i ai,j = 1, such that yi → x in the norm of X as i → ∞. Mazur’s lemma asserts that for every weakly converging sequence, there is a sequence of convex combinations that converges strongly. Thus weak convergence is upgraded to strong convergence for a sequence of convex combinations. Observe that the convex combination is essentially for a subsequence since some of the coefficients ai,j may be zero. The feature that the convex combinations only involve the tail of the original sequence is applied to identify the weak limit in Theorem 2.40. Theorem 2.39. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. Assume that (ui )i∈N is a bounded sequence in W 1,p (Ω). Then there exists a subsequence (uik )k∈N and u ∈ W 1,p (Ω) such that (uik , ∇uik ) → (u, ∇u) weakly in Lp (Ω; Rn+1 ) as k → ∞. Moreover, if ui ∈ W01,p (Ω) for every i ∈ N, then u ∈ W01,p (Ω). Proof. Let fi = (ui , ∇ui ) ∈ Lp (Ω; Rn+1 ) for every i ∈ N. Then (fi )i∈N is a bounded sequence in Lp (Ω; Rn+1 ). By Theorem 2.36, there exists a subsequence (fik )k∈N that converges weakly to some f in Lp (Ω; Rn+1 ) as k → ∞. Write f = (u, v) with u ∈ Lp (Ω) and v ∈ Lp (Ω; Rn ). We show that u ∈ W 1,p (Ω) and that (uik , ∇uik ) converges to (u, ∇u) weakly in Lp (Ω; Rn+1 ) as k → ∞. It suffices to prove that v is the weak gradient of u in Ω. By using test functions of the form (g1 , 0, . . . , 0) or (0, g2 , . . . , gn+1 ), we conclude that uik → u weakly in Lp (Ω) and ∇uik → v weakly in Lp (Ω; Rn ) as k → ∞. Let ϕ ∈ C0∞ (Ω) and j = 1, . . . , n. Then    ∂uik ∂ϕ ∂ϕ k→∞ (x)ϕ(x) dx = − uik (x) (x) dx −−−−→ − u(x) (x) dx. ∂xj ∂xj Ω ∂xj Ω Ω

2.5. WEAK CONVERGENCE AND SOBOLEV SPACES

41

On the other hand, since ∇uik → v weakly in Lp (Ω; Rn ), by testing with function  (0, . . . , ϕ, . . . , 0) ∈ Lp (Ω; Rn ), where ϕ is in the jth position, we see that   ∂uik k→∞ (x)ϕ(x) dx −−−−→ vj (x)ϕ(x) dx. Ω ∂xj Ω This implies

 ∂ϕ u(x) (x) dx = − vj (x)ϕ(x) dx ∂xj Ω Ω for every ϕ ∈ C0∞ (Ω), and thus v = ∇u in Ω. For the last claim, we assume that ui ∈ W01,p (Ω) for every i ∈ N and that the sequence (fik )k∈N = ((uik , ∇uik ))k∈N converges weakly to f = (u, ∇u) in Lp (Ω; Rn+1 ). By Theorem 2.38, there exists a sequence of convex combinations m m  mk k k

hk = ak,j fij = ak,j uij , ∇ ak,j uij 

j=k

j=k

j=k

) as k → ∞. In particular, the first that converges to f = (u, ∇u) in L (Ω; R component mk

hk,1 = ak,j uij ∈ W01,p (Ω) p

n+1

j=k

converges to u in W (Ω) as k → ∞. Since W01,p (Ω) is a closed subspace of  W 1,p (Ω), it follows that u ∈ W01,p (Ω), and the proof is complete. 1,p

The following variant of Theorem 2.39 will be useful. It incorporates an additional mechanism for identifying the weak limit and also ensures that the original sequence converges weakly to this limit. We remark that Theorems 2.39 and 2.40 do not hold when p = 1. Theorem 2.40. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. Assume that (ui )i∈N is a bounded sequence in W 1,p (Ω) such that either ui → u weakly in Lp (Ω), as i → ∞, or ui → u almost everywhere in Ω as i → ∞. (a) Then u ∈ W 1,p (Ω), ui → u weakly in Lp (Ω), and ∇ui → ∇u weakly in Lp (Ω; Rn ) as i → ∞. (b) If ui ∈ W01,p (Ω) for every i ∈ N, then u ∈ W01,p (Ω). Proof. Assertion (b) follows from (a) and Theorem 2.39. For (a) it suffices to show that u ∈ W 1,p (Ω) and (ui , ∇ui ) → (u, ∇u) weakly in Lp (Ω; Rn+1 ) as i → ∞. We prove the latter claim by showing that each subsequence (uik )k∈N has a further subsequence, also denoted by (uik )k∈N , such that (uik , ∇uik ) → (u, ∇u)

(2.20)



weakly in Lp (Ω; Rn+1 ) as k → ∞. Indeed, if g ∈ Lp (Ω; Rn+1 ), then   i→∞ ai = (ui (x), ∇ui (x)) · g(x) dx −−−→ (u(x), ∇u(x)) · g(x) dx = a, Ω

Ω

since otherwise the definition of convergent real-valued sequences implies that (ai )i∈N has a subsequence all of whose subsequences fail to converge to a. This is a contradiction with respect to (2.20) when tested with g.

42

2. LIPSCHITZ AND SOBOLEV FUNCTIONS

Let (uik )k∈N be a subsequence of (ui )i∈N . By Theorem 2.39, there exists a subsequence, also denoted by (uik )k∈N , and a function v ∈ W 1,p (Ω) such that (uik , ∇uik ) → (v, ∇v) weakly in Lp (Ω; Rn+1 ) as k → ∞. It suffices to show that u = v almost everywhere. If ui → u weakly in Lp (Ω), then in particular uik → u weakly in Lp (Ω) and u = v almost everywhere by the uniqueness of weak limits. Hence we may assume that ui → u almost everywhere in Ω as i → ∞. By Theorem 2.38, there exists a sequence of convex combinations hk =

mk

ak,j (uij , ∇uij )

j=k p n+1 ) as k → ∞. In particular, the first compothat converges mtok (v, ∇v) in L (Ω; R nent hk,1 = j=k ak,j uij converges to v in Lp (Ω), and therefore some subsequence of (hk,1 )k∈N converges to v almost everywhere in Ω. On the other hand, since ui → u almost everywhere in Ω as i → ∞, we have

(2.21)

hk,1 =

mk

k→∞

ak,j uij −−−− →u

j=k

almost everywhere in Ω. Hence u = v almost everywhere in Ω, from which we conclude that u ∈ W 1,p (Ω) and that (2.20) holds.  Remark 2.41. The fact that ui → u almost everywhere in Ω, as i → ∞, does not imply for general convex combinations that mk

k→∞

ak,j uij −−−− →u

j=1

almost everywhere in Ω. However, for the tail convex combinations in (2.21) the convergence holds. In order to demonstrate weak convergence techniques, we prove continuity of the operator u → |u| on Sobolev spaces. This operator is easily shown to be bounded on Sobolev spaces, but continuity is not a consequence of boundedness since the operator is not linear. Theorem 2.42. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. Assume that (ui )i∈N is a sequence in W 1,p (Ω) that converges to u in W 1,p (Ω). Then |ui | → |u| in W 1,p (Ω) as i → ∞. Proof. Since ui → u in Lp (Ω) and |ui (x)| − |u(x)| ≤ |ui (x) − u(x)| for every x ∈ Ω, we see that |ui | → |u| in Lp (Ω) as i → ∞. By Theorem 2.25 (a) and (b), ∇|ui |(x) = |∇ui (x)| and ∇|u|(x) = |∇u(x)| (2.22) for almost every x ∈ Ω. Since (ui )i∈N converges in W 1,p (Ω), we have by (2.22) that (|ui |)i∈N is a bounded sequence in W 1,p (Ω). Since |ui | → |u| weakly in Lp (Ω) as i → ∞, Theorem 2.40 (a) implies ∇|ui | → ∇|u| weakly in Lp (Ω; Rn ) as i → ∞.

2.6. DIFFERENCE QUOTIENTS

43

From (2.22) we obtain   ∇|ui |(x) p dx = lim lim |∇ui (x)|p dx i→∞ Ω i→∞ Ω   (2.23) p p = |∇u(x)| dx = ∇|u|(x) dx. Ω

Ω

Lemma 2.37 implies that ∇|ui | converges to ∇|u| in Lp (Ω; Rn ), as i → ∞. This  shows that |ui | → |u| in W 1,p (Ω) as i → ∞. 2.6. Difference quotients This section gives a characterization of Sobolev spaces in terms of difference quotients. The proof is based on weak convergence methods from Section 2.5. Definition 2.43. Let Ω ⊂ Rn be an open set. Assume that u ∈ L1loc (Ω) and let Ω  Ω be an open set. The jth difference quotient, with j = 1, . . . , n, of u at x ∈ Ω is u(x + hej ) − u(x) Djh u(x) = , h for h ∈ R with 0 < |h| < d (Ω , ∂Ω). Here ej is the jth coordinate unit vector in Rn . We also write Dh u = (D1h u, . . . , Dnh u). Remark 2.44. The definition of difference quotients makes sense at every x ∈ Ω whenever 0 < |h| < d (x, ∂Ω). If Ω = Rn , then the definition makes sense for every h = 0. We interpret d (Ω , ∂Rn ) = ∞ in Definition 2.43 and Theorem 2.45. Pointwise derivatives are defined as limits of difference quotients. Theorem 2.45 shows that Sobolev spaces can be characterized by integrated difference quotients, bypassing this limiting process. We remark that assertions (b) and (c) in Theorem 2.45 do not hold for p = 1. Theorem 2.45. Let Ω ⊂ Rn and Ω  Ω be open sets. (a) Let 1 ≤ p < ∞ and assume that u ∈ W 1,p (Ω). There exists a constant C = C(n) such that (2.24)

Dh uLp (Ω ;Rn ) ≤ C∇uLp (Ω;Rn )

whenever 0 < |h| < d (Ω , ∂Ω). (b) Let 1 < p < ∞. Assume that u ∈ Lp (Ω) and that there exists a constant C such that Dh uLp (Ω ;Rn ) ≤ C whenever 0 < |h| < d(Ω , ∂Ω). Then the weak gradient ∇u exists in Ω , u ∈ W 1,p (Ω ) and ∇uLp (Ω ;Rn ) ≤ C. (c) Let 1 < p < ∞. Assume that u ∈ Lp (Rn ) and that there exists a constant C such that (2.25)

Dh uLp (Rn ;Rn ) ≤ C for every h = 0. Then the weak gradient ∇u with respect to Rn exists, u ∈ W 1,p (Rn ) and ∇uLp (Rn ;Rn ) ≤ C.

44

2. LIPSCHITZ AND SOBOLEV FUNCTIONS

Proof. To prove part (a), we first assume that u ∈ C ∞ (Ω) ∩ W 1,p (Ω). Let x ∈ Ω and let h ∈ R be such that 0 < |h| < d (Ω , ∂Ω). Then, for every j = 1, . . . , n,  h  h ∂ (u(x + tej )) dt = ∇u(x + tej ) · ej dt u(x + hej ) − u(x) = 0 ∂t 0  h ∂u (x + tej ) dt. = ∂x j 0 By H¨ older’s inequality  |h| ∂u u(x + hej ) − u(x) ≤ 1 dt (x + te ) |Djh u(x)| = j h |h| −|h| ∂xj p  p1  |h| ∂u 1 1 ≤ (x + te ) |2h|1− p , j dt |h| −|h| ∂xj which implies |Djh u(x)|p ≤

2p−1 |h|



|h| −|h|

p ∂u (x + te ) j dt. ∂xj



Next we integrate over Ω and switch the order of integration by Fubini’s theorem as follows p    |h| ∂u 2p−1 h p |Dj u(x)| dx ≤ ∂xj (x + tej ) dt dx |h|   Ω Ω −|h| p p    ∂u ∂u 2p−1 |h| p = (x + tej ) dx dt ≤ 2 (x) dx. |h| −|h| Ω ∂xj Ω ∂xj The final step above follows from the assumption 0 < |h| < d (Ω , ∂Ω), since if |t| ≤ |h|, then p p   ∂u ∂u (x + tej ) dx ≤ (x) dx. Ω ∂xj Ω ∂xj α Using the elementary inequality (a1 + · · · + an )α ≤ nα (aα 1 + · · · + an ), where ai ≥ 0, α > 0, we obtain  p2     n n p h p h 2 |D u(x)| dx = |Dj u(x)| dx ≤ n 2 |Djh u(x)|p dx Ω

Ω

=n

p 2

j=1 n 

j=1 p

Ω

|Djh u(x)|p

dx ≤ 2 n p

p 2

Ω j=1 n 

j=1



Ω

p ∂u ∂xj (x) dx

|∇u(x)|p dx.

≤ 2p n1+ 2

Ω

The case u ∈ C ∞ (Ω) ∩ W 1,p (Ω) of inequality (2.24) follows with C = C(n) = 2n 2 . Then we consider the general case u ∈ W 1,p (Ω). We choose functions ui ∈ ∞ C (Ω) ∩ W 1,p (Ω), i ∈ N, such that ui → u in W 1,p (Ω) as i → ∞. By passing to a subsequence, if necessary, we may also assume that ui → u pointwise almost everywhere in Ω as i → ∞. Assume that 0 < |h| < d (Ω , ∂Ω). Then Dh ui (x) → Dh u(x) for almost every x ∈ Ω as i → ∞. By Fatou’s lemma and inequality (2.24), 3

2.6. DIFFERENCE QUOTIENTS

we obtain  Ω

45

 |Dh u(x)|p dx ≤ lim inf

|Dh ui (x)|p dx   p ≤ C(n) lim inf |∇ui (x)| dx = C(n) |∇u(x)|p dx, i→∞

Ω

i→∞

Ω

Ω

and this proves part (a). For the proof of (b), we let ϕ ∈ C0∞ (Ω ) and j = 1, . . . , n, and assume that 0 < |h| < d (supp ϕ, ∂Ω ). A change of variables gives   u(x − hej ) − u(x) ϕ(x + hej ) − ϕ(x) dx = − ϕ(x) dx. u(x) h −h   Ω Ω This shows that   h (2.26) u(x)Dj ϕ(x) dx = − Dj−h u(x)ϕ(x) dx Ω

Ω



whenever 0 < |h| < d (supp ϕ, ∂Ω ). On the other hand, by assumption sup 0 t} B B

≤ {x ∈ Rn : I1 |∇ui | − |∇u| χB (x) > t} n

≤ C(n)t− n−1 ∇ui − ∇uLn−1 1 (B) n

for every t > 0. Thus, H¨ older’s inequality shows that I1 (|∇ui |χB ) → I1 (|∇u|χB ) in measure as i → ∞. By passing again to a subsequence, if necessary, we obtain a set E2 ⊂ B with |E2 | = 0 such that lim I1 (|∇ui |χB )(x) = I1 (|∇u|χB )(x) < ∞

i→∞

for every x ∈ B \ E2 .

50

´ INEQUALITIES 3. SOBOLEV AND POINCARE

Moreover, it follows from H¨ older’s inequality and Lp convergence that   p1 i→∞ p (3.4) |(ui )B − uB | ≤  |ui (x) − u(x)| dx ≤  |ui (x) − u(x)| dx −−−→ 0, B

B

and thus (ui )B → uB as i → ∞. By applying (3.2) and arguing as in the proof of Theorem 3.4, we obtain |u(x) − uB | = lim |ui (x) − (ui )B | ≤ C(n) lim I1 (|∇ui |χB )(x) i→∞

i→∞

= C(n)I1 (|∇u|χB )(x) ≤ C(n)rM (|∇u|χB )(x), for every x ∈ B \ (E1 ∪ E2 ), where |E1 ∪ E2 | = 0.



3.2. Sobolev–Gagliardo–Nirenberg inequality Pointwise estimates in Section 3.1 imply the Sobolev–Gagliardo–Nirenberg inequality. The proof of the case p > 1 is a consequence of Lemma 3.3 and the Hardy–Littlewood–Sobolev inequality for the Riesz potential in Theorem 1.36. The argument for p = 1 is based on a truncation method and the weak type estimate for the Riesz potential in Lemma 1.37. np Theorem 3.6. Let 1 ≤ p < n and p∗ = n−p , and assume that u ∈ W 1,p (Rn ). There exists a constant C = C(n, p) such that

uLp∗ (Rn ) ≤ C∇uLp (Rn ) . Proof. Assume first that u ∈ Lipc (Rn ). By Lemma 3.3, |u(x)| ≤

2n I1 |∇u|(x), nωn

for every x ∈ Rn . If 1 < p < n, then Theorem 1.36 implies   uLp∗ (Rn ) ≤ C(n)I1 |∇u|Lp∗ (Rn ) ≤ C(n, p)∇uLp (Rn ) , and this proves the claim for u ∈ Lipc (Rn ) when 1 < p < n. Then we consider the case p = 1 for u ∈ Lipc (Rn ). Let   Aj = x ∈ Rn : 2j < |u(x)| ≤ 2j+1 ,   with j ∈ Z. Define ϕ(t) = max 0, min{t, 1} , for t ∈ R, and ⎧ ⎪0, when |u(x)| ≤ 2j−1 , 1−j

⎨ 1−j uj (x) = ϕ 2 |u(x)| − 1 = 2 |u(x)| − 1, when 2j−1 < |u(x)| ≤ 2j , ⎪ ⎩ 1, when |u(x)| > 2j , for x ∈ Rn and j ∈ Z. Then uj ∈ Lipc (Rn ) for every j ∈ Z and by Lemma 2.13 we have ∇uj = 0 almost everywhere in Rn \ Aj−1 . Lemma 3.3 and Lemma 1.37 imply     |Aj | ≤ x ∈ Rn : |u(x)| > 2j = x ∈ Rn : uj (x) = 1 n  n−1   nωn  n |∇uj (x)| dx ≤ x ∈ R : I1 |∇uj |(x) > n+1 ≤ C(n) 2 Aj−1 n n   n−1  n−1  n −j n−1 1−j ≤ C(n) 2 |∇u(x)| dx ≤ C(n)2 |∇u(x)| dx , Aj−1

Aj−1

3.2. SOBOLEV–GAGLIARDO–NIRENBERG INEQUALITY

51

for every j ∈ Z. Since the sets Aj , with j ∈ Z, are pairwise disjoint and their union is {x ∈ Rn : u(x) = 0}, by summing over all j ∈ Z we obtain  Rn

|u(x)|

n n−1

∞ 

dx =

j=−∞

≤ C(n)

Aj

|∇u(x)| dx

n  n−1

∞ 

j=−∞

|∇u(x)| dx Aj−1

 = C(n)

Rn

n  n−1

Aj−1

j=−∞

≤ C(n)

n

2(j+1) n−1 |Aj |

j=−∞

∞ 



∞

n

|u(x)| n−1 dx ≤

|∇u(x)| dx

n  n−1

.

This proves the case p = 1 for u ∈ Lipc (Rn ). Assume then that u ∈ W 1,p (Rn ). By Remark 2.22 (a), there exist functions ui ∈ Lipc (Rn ), i ∈ N, such that ui → u in W 1,p (Rn ) as i → ∞. In particular, ui → u in Lp (Rn ) as i → ∞, and hence, by passing to a subsequence if necessary, we may in addition assume that ui → u almost everywhere in Rn . Fatou’s lemma and the claim for Lipschitz functions ui , i ∈ N, imply  Rn

|u(x)|

np n−p

 n−p np dx

 ≤ lim inf i→∞

Rn

|ui (x)|

np n−p

 n−p np dx



≤ C(n, p) lim inf i→∞



|∇ui (x)| dx p

Rn

|∇u(x)| dx p

= C(n, p) Rn

 p1

 p1 ,

where the final equality holds since ∇ui Lp (Rn ) → ∇uLp (Rn ) as i → ∞.



The proof above shows that in this case a weak type estimate implies a strong type estimate for p = 1. Observe from Theorem 1.15 (b) and Example 1.11 that this does not hold in general for p = 1. The reason why this is possible in the proof of Theorem 3.6 is that we consider gradients, which vanish almost everywhere on the set where the function is constant. From the Sobolev–Gagliardo–Nirenberg inequality we obtain the following estimate for Sobolev functions with zero boundary values. Corollary 3.7. Let 1 ≤ p < ∞, let Ω ⊂ Rn be an open set with |Ω| < ∞, and np for 1 ≤ p < n, and 1 ≤ q < ∞ assume that u ∈ W01,p (Ω). Let 1 ≤ q ≤ p∗ = n−p for n ≤ p < ∞. There exists a constant C = C(n, p, q) such that  |u(x)| dx q

(3.5) Ω

 1q

≤ C|Ω|

1 1 1 n−p+q

 |∇u(x)| dx p

 p1 .

Ω

Proof. Extend u as zero outside Ω. By Theorem 2.31 we then have ∇u(x) = 0 for almost every x ∈ Rn \ Ω. Assume first that 1 ≤ p < n. H¨ older’s inequality and

´ INEQUALITIES 3. SOBOLEV AND POINCARE

52

Theorem 3.6 imply    1q  n−p np np 1 1 −p + q1 q n n−p |u(x)| dx ≤ |Ω| |u(x)| dx Ω

Ω

≤ C(n, p)|Ω|

1 1 1 n−p+q

 |∇u(x)| dx p

 p1 .

Ω

Assume then that n ≤ p < ∞. If q > p, choose 1 < s < n satisfying q = By the first part of the proof and H¨ older’s inequality, we obtain  1q  1s   1 1 1 |u(x)|q dx ≤ C(n, p, q)|Ω| n − s + q |∇u(x)|s dx Ω

Ω

≤ C(n, p, q)|Ω|

1 1 1 n−p+q



|∇u(x)| dx p

ns n−s .

 p1 .

Ω

Finally, for q ≤ p, the claim follows by applying H¨ older’s inequality and the previous case with some t > q on the left-hand side.  Remark 3.8. Let 1 ≤ p < n and let Ω ⊂ Rn be an open set. The proof of Corollary 3.7 shows that the Sobolev inequality   n−p   p1 np np (3.6) |u(x)| n−p dx ≤ C(n, p) |∇u(x)|p dx Ω

Ω

holds for every u ∈ W01,p (Ω). 3.3. Sobolev–Poincar´ e inequalities Sobolev–Poincar´e inequalities for 1 < p < ∞ are straightforward consequences of the pointwise estimates in Section 3.1 and the Hardy–Littlewood–Wiener maximal function theorem and the Hardy–Littlewood–Sobolev inequality for the Riesz potential, see Theorem 1.15 and Theorem 1.36 respectively. The argument for p = 1 is based on a Lipschitz truncation method and weak type estimates, as in the proof of Theorem 3.6. We begin with a p-Poincar´e inequality for Lipschitz functions in the range 1 < p < ∞. Lemma 3.9. Let 1 < p < ∞ and assume that u ∈ Lip(Rn ). There exists a constant C = C(n, p) such that   |u(x) − uB |p dx ≤ Cr p |∇u(x)|p dx, B

B

for every ball B = B(x0 , r) ⊂ Rn . Proof. By Theorem 3.4, we have



p |u(x) − uB |p ≤ C(n, p)r p M (|∇u|χB )(x)

for every x ∈ B. Theorem 1.15 (c) implies  

p p p M (|∇u|χB )(x) dx |u(x) − uB | dx ≤ C(n, p)r n B R

p p |∇u(x)|χB (x) dx ≤ C(n, p)r n R = C(n, p)r p |∇u(x)|p dx. B



´ INEQUALITIES 3.3. SOBOLEV–POINCARE

53

In the range 1 < p < n we have a Sobolev–Poincar´e inequality for Lipschitz functions. np , and assume that u ∈ Lip(Rn ). Lemma 3.10. Let 1 < p < n and p∗ = n−p There exists a constant C = C(n, p) such that

 |u(x) − uB |

p∗

 p1∗ dx

 ≤C

|∇u(x)| dx p

B

 p1

B

for every ball B = B(x0 , r) ⊂ Rn . Proof. By Lemma 3.2, we have |u(x) − uB | ≤ C(n)I1 (|∇u|χB )(x) for every x ∈ B. Theorem 1.36, with α = 1, implies  p1∗   p1∗ 

p∗ ∗ I1 (|∇u|χB )(x) dx |u(x) − uB |p dx ≤ C(n) Rn

B

 ≤ C(n, p)

Rn



p |∇u(x)|χB (x) dx



|∇u(x)| dx p

= C(n, p)

 p1

 p1 .



B

Poincar´e and Sobolev–Poincar´e inequalities also hold for p = 1, and by approximation they can be extended for Sobolev functions, as well; see Theorem 3.14. The following two auxiliary results will be needed in the proof of the case p = 1 in Theorem 3.13. Lemma 3.11. Assume that E ⊂ Rn is a measurable set and that v ≥ 0 is a measurable function on E with {x ∈ E : v(x) = 0} ≥ |E| . 2 Then (3.7)

  {x ∈ E : v(x) > t} ≤ x ∈ E : |v(x) − a| > t 2

for every a ∈ R and t > 0. Proof. First assume that |a| ≤ 2t . If x ∈ E satisfies v(x) > t, then |v(x) − a| ≥ v(x) − |a| > and thus

t , 2

  {x ∈ E : v(x) > t} ≤ x ∈ E : |v(x) − a| > t . 2 Then assume that |a| > 2t . If x ∈ E and v(x) = 0, then |v(x) − a| = |a| >

t . 2

54

´ INEQUALITIES 3. SOBOLEV AND POINCARE

Hence, in the case |E| = ∞ inequality (3.7) follows readily from the assumptions. On the other hand, if |E| < ∞, then the assumptions imply {x ∈ E : v(x) > t} ≤ |E| − {x ∈ E : v(x) = 0} ≤ {x ∈ E : v(x) = 0}  t  ≤ x ∈ E : |v(x) − a| > , 2 and this proves the claim.  Lemma 3.12. Assume that u ∈ Lip(Rn ) b ∈ R such that {x ∈ B : u(x) ≥ b} ≥ |B| and 2

and let B ⊂ Rn be a ball. There exists {x ∈ B : u(x) ≤ b} ≥ |B| . 2

Proof. Let Et = {x ∈ B : u(x) ≥ t} for t ∈ R, and let   |B| b = sup t ∈ R : |Et | ≥ . 2 Then |b| ≤ uL∞ (B) < ∞ and there exists an increasing sequence (bj )j∈N such that bj → b as j → ∞ and |B| |Ebj | ≥ 2  for every j ∈ N. Since Eb = ∞ E and the sequence (Ebj )j∈N is decreasing, we b j j=1 have {x ∈ B : u(x) ≥ b} = |Eb | = lim |Ebj | ≥ |B| . j→∞ 2 A similar argument shows that also the second inequality holds for b since {x ∈ B : u(x) ≤ t} ≥ |B| − |Et | > |B| 2 whenever t > b.  We are ready to prove the Sobolev–Poincar´e inequality for p = 1. Theorem 3.13. Assume that u ∈ Lip(Rn ). There exists a constant C = C(n) such that  n−1   n n n−1 |u(x) − uB | dx ≤ C |∇u(x)| dx B

B

for every ball B = B(x0 , r) ⊂ Rn . Proof. By Lemma 3.12, there is a number b ∈ R satisfying {x ∈ B : u(x) ≥ b} ≥ |B| and {x ∈ B : u(x) ≤ b} ≥ |B| . 2 2 Define v+ = max{u − b, 0} and v− = − min{u − b, 0}. Both of these functions belong to Lip(Rn ). In the sequel v denotes either v+ or v− ; all statements are valid in both cases. Let Aj = {x ∈ B : 2j < v(x) ≤ 2j+1 },   with j ∈ Z. Define ϕ(t) = max 0, min{t, 1} , for every t ∈ R, and

vj (x) = ϕ 21−j v(x) − 1 ,

´ INEQUALITIES 3.3. SOBOLEV–POINCARE

55

for every x ∈ Rn and j ∈ Z. Then vj ∈ Lip(Rn ) for every j ∈ Z. Let j ∈ Z. By Lemma 3.2, we have |vj (x) − (vj )B | ≤ C(n)I1 (|∇vj |χB )(x) for every x ∈ B. Hence we obtain from Lemma 3.11, with t = 12 and a = (vj )B , that    1  |Aj | ≤ x ∈ B : 2j < v(x) ≤ x ∈ B : vj (x) > 2     1 ≤ x ∈ B : |vj (x) − (vj )B | > ≤ x ∈ Rn : I1 (|∇vj |χB )(x) > C(n) . 4 The right-hand side can be estimated using the weak type estimate for the Riesz potential in Lemma 1.37 and the fact that |∇vj (x)| = 21−j |∇v(x)|χAj−1 (x) for almost every x ∈ B, see Lemma 2.13. It follows that     n |Aj | ≤ x ∈ Rn : I1 (|∇vj |χB )(x) > C(n) ≤ C(n)|∇vj |χB Ln−1 1 (Rn ) n  jn  ≤ C(n)2− n−1 |∇v|χAj−1 ∩B Ln−1 1 (Rn ) . The sets Aj , for j ∈ Z, are pairwise disjoint, and thus we obtain, using the estimate above, 



(j+1)n n n n−1 v(x) dx = v(x) n−1 dx ≤ 2 n−1 |Aj | B

j∈Z

Aj

≤ C(n)

j∈Z

2

(j+1)n n−1

  n |∇v|χAj−1 ∩B Ln−1 1 (Rn )

jn − n−1 

2

j∈Z

n  n−1      ≤ C(n) |∇v|χAj−1 ∩B  1

L (Rn )

j∈Z

  n ≤ C(n)|∇u|χB Ln−1 1 (Rn ) .

Since |u − b| = v+ + v− , we arrive at  n−1   n−1  n n n n |u(x) − uB | n−1 dx ≤2 |u(x) − b| n−1 dx B

B

  n−1   n−1 n n n n ≤2 v+ (x) n−1 dx +2 v− (x) n−1 dx B B   ≤ C(n)|∇u|χB L1 (Rn ) = C(n) |∇u(x)| dx.  B

The next theorem gives a general Sobolev–Poincar´e inequality for Sobolev functions. In particular, this shows that Lemma 3.9, Lemma 3.10 and Theorem 3.13 do not only hold for Lipschitz functions but also for functions in the corresponding Sobolev space. We refer to (3.8) below as the (q, p)-Poincar´e inequality. Theorem 3.14. Let 1 ≤ p < ∞, let Ω ⊂ Rn be an open set, and assume that 1,p np (Ω). Let 1 ≤ q ≤ p∗ = n−p for 1 ≤ p < n, and 1 ≤ q < ∞ for n ≤ p < ∞. u ∈ Wloc There exists a constant C = C(n, p, q) such that  q1   p1  q p ≤ Cr  |∇u(x)| dx (3.8)  |u(x) − uB | dx B

for every ball B = B(x0 , r)  Ω.

B

´ INEQUALITIES 3. SOBOLEV AND POINCARE

56

Proof. Assume first that u ∈ Lip(Rn ). By Lemma 3.10, for 1 < p < n, and Theorem 3.13, for p = 1, we obtain 

np

 |u(x) − uB | n−p dx

 n−p np

= C(n, p)r −

n−p p

B

≤ C(n, p)r

(3.9)

1− n p



np

|u(x) − uB | n−p dx



B

|∇u(x)|p dx B



= C(n, p)r  |∇u(x)| dx p

 n−p np

 p1

 p1 .

B

For 1 ≤ p < n, inequality (3.8) follows from (3.9) and H¨older’s inequality on the left-hand side. In the case p ≥ n we proceed as in the proof of Corollary 3.7. For q > p ≥ n, ns , and (3.8) follows from (3.9) with exponent there exists 1 < s < n such that q = n−s s and an application of H¨ older’s inequality on the right-hand side. For q ≤ p, the claim follows from the previous case and H¨ older’s inequality on the left-hand side. 1,p (Ω), we obtain (3.8) by approximation. By considering R > r such For u ∈ Wloc that B ⊂ B(x0 , R)  Ω and applying the McShane extension, see Theorem 2.7, we find functions ui ∈ Lip(Rn ), i ∈ N, satisfying ui → u in W 1,p (B) as i → ∞; compare to the proof of Theorem 3.5. By passing to a subsequence, if necessary, we may in addition assume that ui → u almost everywhere in B. As in (3.4), we have (ui )B → uB as i → ∞. Fatou’s lemma and inequality (3.8) for the Lipschitz functions ui give   |u(x) − uB | dx q

B

 q1

 1q



≤ lim inf  |ui (x) − (ui )B | dx i→∞ B   p1 p ≤ lim inf C(n, p, q)r  |∇ui (x)| dx q

i→∞

B



≤ C(n, p, q)r  |∇u(x)| dx p

 p1 ,

B



and the proof is complete.

The following consequence of the Poincar´e inequality will be applied in Chapter 11 and Chapter 12. Lemma 3.15. Let 1 ≤ p < ∞ and let Ω ⊂ Rn be an open set. Assume that u ∈ W01,p (Ω) is such that |∇u| = 0 almost everywhere in Ω. Then u = 0 almost everywhere in Ω. Proof. Extend u as zero outside Ω. By Theorem 2.31 we then have |∇u| = 0 almost everywhere in Rn . Define Bj = B(0, j) ⊂ Rn for j ∈ N. By the (1, p)Poincar´e inequality in Theorem 3.14 we obtain  

Bj

|u(x) − uBj | dx ≤ C(n, p)r 

Bj

|∇u(x)| dx p

 p1 = 0,

´ INEQUALITIES FOR ZERO BOUNDARY VALUES 3.4. POINCARE

57

for every j ∈ N. Hence, for every j ∈ N there is a set Ej ⊂ Bj with |Ej | = 0 such that u(x) = uBj for every x ∈ Bj \ Ej . H¨older’s inequality implies   p1 1 j→∞ p ≤ |Bj |− p uLp (Rn ) −−−→ 0. (3.10) |uBj | ≤  |u(x)| dx Bj

Thus u(x) = limj→∞ uBj χBj (x) = 0 for every x ∈ Rn \ E, where E = zero measure.

∞ j=1

Ej has 

Remark 3.16. The Sobolev–Poincar´e inequality in Theorem 3.14 gives an alternative proof for the Sobolev–Gagliardo–Nirenberg inequality in Theorem 3.6. In order to show this, let 1 ≤ p < n, assume that u ∈ W 1,p (Rn ), and define Bj = B(0, j), for j ∈ N. By (3.10) we have uBj → 0 as j → ∞, and thus np

np

|u(x)| n−p = lim inf |u(x) − uBj | n−p χBj (x) j→∞

for every x ∈ Rn . Using Fatou’s lemma and Theorem 3.14, with q = p∗ , we obtain  n−p  n−p   np np np np |u(x)| n−p dx ≤ lim inf |u(x) − uBj | n−p dx j→∞

Rn

Bj



≤ C(n, p) lim inf j→∞

 ≤ C(n, p)

|∇u(x)| dx p

Bj

|∇u(x)| dx p

Rn

 p1

 p1 .

3.4. Poincar´ e inequalities for zero boundary values If a Lipschitz or a Sobolev function u is zero outside a ball B ⊂ Rn , then we can remove the mean value uB from the left-hand side of the (q, p)-Poincar´e inequality (3.8). This result follows from Corollary 3.7, but we give an alternative proof which is based on the (q, p)-Poincar´e inequality. This technique can be adapted to many other situations as well, which is illustrated in Remark 3.18. np for 1 ≤ p < n, and 1 ≤ q < ∞ for Theorem 3.17. Let 1 ≤ q ≤ p∗ = n−p n ≤ p < ∞. There exists a constant C = C(n, p, q) such that   q1   p1 q p (3.11) |u(x)| dx ≤ Cr |∇u(x)| dx   B

B

for every ball B = B(x0 , r) ⊂ Rn and every u ∈ W01,p (B). Proof. We may assume that q > 1 since the claim for q = 1 follows from H¨older’s inequality. Consider first the case u ∈ Lip0 (B). Then u = 0 in B(x0 , 2r) \ B(x0 , r). By H¨ older’s inequality |uB(x0 ,2r) | ≤ 

B(x0 ,2r)



(3.12)

|u(x)|χB(x0 ,r) (x) dx

|B(x0 , r)| |B(x0 , 2r)|  −n 1− q1 = (2 ) 

≤

1− q1 

B(x0 ,2r)

|u(x)|q dx



B(x0 ,2r)

|u(x)| dx q

 q1 .

 1q

´ INEQUALITIES 3. SOBOLEV AND POINCARE

58

Using Minkowski’s inequality, the (q, p)-Poincar´e inequality from Theorem 3.14 for B(x0 , 2r), and estimate (3.12), we obtain  1q   q1  q q |u(x)| dx ≤  |u(x) − uB(x0 ,2r) | dx + |uB(x0 ,2r) |  B(x0 ,2r)

B(x0 ,2r)



≤ C(n, p, q)r 

|∇u(x)| dx p

 p1

−n 1− 1q

+ (2

)

B(x0 ,2r)

 |u(x)| dx q



 q1

< ∞.

B(x0 ,2r)

1

Since (2−n )1− q < 1, the second term on the right-hand side can be absorbed to the left-hand side, and thus  1q   p1  q p |u(x)| dx ≤ C(n, p, q)r |∇u(x)| dx .   B(x0 ,2r)

B(x0 ,2r)

Finally, the mean value integrals on both sides can be taken with respect to B(x0 , r) since u = 0 and ∇u = 0 in B(x0 , 2r) \ B(x0 , r). In the general case u ∈ W01,p (B) the claim follows by approximation since Lip0 (B) is dense in W01,p (B). The details are essentially same as in the proof of Theorem 3.6.  Remark 3.18. With minor modifications to the proof above, we can generalize the statement of Theorem 3.17 as follows. Let n, p and q > 1 be as in Theorem 3.17. Assume that u ∈ Lip(Rn ) is such that u = 0 in a set E ⊂ B = B(x0 , r) satisfying |E| ≥ γ|B| with 0 < γ ≤ 1. As in (3.12), we obtain   q1 1− 1q  |B \ E| q χ |uB | ≤  |u(x)| B\E (x) dx ≤  |u(x)| dx |B| B B  q1  1 ≤ (1 − γ)1− q  |u(x)|q dx . B

Since 0 ≤ (1 − γ) such that

1− q1

< 1, we conclude that there exists a constant C = C(n, p, q, γ)

  |u(x)| dx q

B

 1q

 ≤ Cr  |∇u(x)| dx p

 p1 .

B

These ideas will be taken one step further in Section 5.9 with the concept of variational capacity; compare to Theorem 5.47 and also to Theorem 6.22. 3.5. Morrey’s inequality For n < p < ∞, we obtain Morrey’s inequality, which can be regarded as a pointwise version of the Sobolev–Poincar´e inequality in Section 3.3. The limiting case p = n is studied in Section 9.7 together with the space of bounded mean oscillation. Theorem 3.19. Let n < p < ∞ and assume that u ∈ Lip(Rn ). There exists a constant C = C(n, p) such that  p1  (3.13) |u(x) − u(y)| ≤ Cr  |∇u(z)|p dz B

for every ball B = B(x0 , r) ⊂ Rn and every x, y ∈ B.

3.5. MORREY’S INEQUALITY

59

Proof. Let B0 = B = B(x0 , r) and choose a ball B1 = B(x1 , 2−1 r) such that x ∈ B1 ⊂ B0 . For j = 2, 3, . . . , choose recursively a ball Bj = B(xj , 2−j r) such that x ∈ Bj ⊂ Bj−1 . We apply a telescoping argument with the balls Bj . First, using the continuity of u, the triangle inequality, and the facts that x ∈ Bj ⊂ Bj−1 and |Bj−1 | = 2n |Bj | for every j ∈ N, we obtain |u(x) − uB0 | = lim |uBj − uB0 | ≤ j→∞

≤

∞

|uBj − uBj−1 | ≤

|u(z) − uBj−1 | dz



j=1 Bj ∞

j=1

∞

|Bj−1 | j=1

∞

|u(z) − uBj−1 | dz = 2n  |u(z) − uBj | dz. |Bj | Bj−1 j=0 Bj

The (1, p)-Poincar´e inequality, see Theorem 3.14, for each Bj , implies   p1 ∞

|u(x) − uB0 | ≤ C(n, p) 2−j r  |∇u(z)|p dz Bj

j=0

= C(n, p)

∞



− n 2−j r 2−j r p

 |∇u(z)|p dz

 p1

Bj

j=0 n



≤ C(n, p)r 1− p

|∇u(z)|p dz B0

 p1 ∞ −j 1− np 2 . j=0

The geometric series converges since p > n, and so  p1   1− n p |∇u(z)| dz = C(n, p)r  |u(x) − uB0 | ≤ C(n, p)r p B0

|∇u(z)| dz p

 p1 .

B0

The same estimate holds for y ∈ B as well, and thus



|u(x) − u(y)| ≤ |u(x) − uB | + |uB − u(y)| ≤ C(n, p)r  |∇u(z)| dz p

 p1

B

for every x, y ∈ B.



Remark 3.20. We consider an alternative proof of Morrey’s inequality (3.13) through pointwise estimates and the Riesz potential. Let n < p < ∞, assume that u ∈ Lip(Rn ), and let x, y ∈ B = B(x0 , r). By Lemma 3.2, we have |u(x) − u(y)| ≤ |u(x) − uB | + |uB − u(y)|   |∇u(z)| |∇u(z)| ≤ C(n) dz + C(n) dz. n−1 |x − z| |y − z|n−1 B B H¨older’s inequality gives   p1   1  p |∇u(z)| p (1−n)p dz ≤ |∇u(z)| dz |x − z| dz , n−1 B |x − z| B B where p = 

p p−1


0. Let x ∈ Rn and B = B(x, R), where R > 0 is to be chosen later. By Theorem 3.19, we have n

|u(x) − uB | ≤  |u(x) − u(y)| dy ≤ C(n, p)∇uLp (Rn ) R1− p . B

This implies |u(x)| ≤ |u(x) − uB | + |uB | ≤ C(n, p)∇uLp (Rn ) R1− p + |B|− p uLp (Rn ) n

1

= C(n, p)R1− p ∇uLp (Rn ) + C(n, p)R− p uLp (Rn ) . n

n

Choosing R = uLp (Rn ) ∇u−1 Lp (Rn ) > 0 concludes the proof of (3.16). As another application of Morrey’s inequality we show that all functions in 1,p (Rn ), with p > n, are differentiable almost everywhere. Wloc

´ INEQUALITIES 3. SOBOLEV AND POINCARE

62

1,p Theorem 3.26. Let n < p ≤ ∞ and assume that u ∈ Wloc (Rn ). Then u has a continuous representative which is differentiable almost everywhere. Moreover, the pointwise derivative of this representative equals the weak derivative of u almost everywhere. 1,∞ 1,p (Rn ) ⊂ Wloc (Rn ), we may assume n < p < ∞. Let Proof. Since Wloc n ϕj ∈ Lipc (R ) be such that ϕj (x) = 1 for every x ∈ B(0, j), with j ∈ N. Applying Theorem 3.23 to the functions uϕj ∈ W 1,p (Rn ) and taking j → ∞, we see that u has a continuous representative in Rn . From now on, we consider this continuous representative, still denoted by u. Moreover, all the derivatives below are weak, unless otherwise specified. By Theorem 1.19, we have

(3.17)

lim 

r→0

|∇u(z) − ∇u(x)|p dz = 0

B(x,r)

for almost every x ∈ Rn . Let x ∈ Rn be such a point and define v(y) = u(y) − u(x) − ∇u(x) · (y − x), 1,p (Rn ) ∩ C(Rn ). By Morrey’s inequality in Corolfor every y ∈ R . Then v ∈ Wloc lary 3.21, we have  p1  p |∇v(z)| dz , |v(y) − v(x)| ≤ C(n, p)|y − x|  n

B(x,2|y−x|)

for every y ∈ R . Here we also use the fact that v is continuous. Since v(x) = 0 and ∇v(z) = ∇u(z) − ∇u(x) for almost every z ∈ Rn , we obtain   p1 |u(y) − u(x) − ∇u(x) · (y − x)| ≤ C(n, p)  |∇u(z) − ∇u(x)|p dz . |y − x| B(x,2|y−x|) n

By (3.17) the right-hand side tends to zero, as y → x. This shows that u is differentiable at x and that the weak and pointwise derivatives coincide at x.  3.6. Notes Sobolev–Poincar´e inequalities are discussed in Adams and Hedberg [4], Adams and Fournier [5], Bj¨orn and Bj¨orn [31], Edmunds and Evans [116], Evans [121], Evans and Gariepy [122], Gilbarg and Trudinger [152], Hajlasz and Koskela [171], Heinonen [185], Heinonen, Koskela, Shanmugalingam and Tyson [190], Mazya [318, 321] and Ziemer [407]. The proof of Lemma 3.2 is similar to Edmunds and Evans [116, Theorem 3.18], see also Evans and Gariepy [122, 4.5.2] and Gilbarg and Trudinger [152, Lemma 7.15]. A version of Theorem 3.4 can be found in Bennett and Sharpley [28, Theorem V.5.6]. Theorem 3.6 was proved by Sobolev for 1 < p < n and by Gagliardo and Nirenberg for p = 1. For Lemma 3.3 and Theorem 3.6, for 1 < p < n, we refer to Ziemer [407, Remark 2.8.6]. For the truncation method applied in the proof of Theorem 3.6, we refer to Mazya [318, 2.3.1, p. 110], [321, 2.3.1, p. 155]. See also Hajlasz [167] and Hajlasz and Koskela [171]. The proofs of Theorem 3.17 and Remark 3.18 are similar to Evans and Gariepy [122, 5.6.1]. Lemma 3.11 can be found in Hajlasz, [167, Lemma 5]. Theorem 3.26 is from Evans and Gariepy [122, 6.2] and Mal´ y and Ziemer [303, Theorem 1.71].

10.1090/surv/257/04

CHAPTER 4

Pointwise Inequalities for Sobolev Functions This chapter continues discussion on pointwise inequalities for Sobolev functions by the maximal function of the gradient. A Lipschitz truncation method in Section 4.2 gives an approximation of a Sobolev function with a Lipschitz continuous function which coincides with the original function outside a set of arbitrarily small measure. This result is applied in Section 12.3. A pointwise characterization for Sobolev functions is provided in Section 4.3. Regularity properties of maximal operators, with pointwise estimates for the gradient, are discussed in Section 4.4, Section 4.5 and Section 4.6. 4.1. Pointwise characterization of Sobolev spaces The result below is closely related to Theorem 3.4. For the definition of the restricted maximal function, see Definition 1.27. Theorem 4.1. Assume that u ∈ Lip(Rn ). There exists a constant C = C(n) such that

|u(x) − u(y)| ≤ C|x − y| M2|x−y| |∇u|(x) + M2|x−y| |∇u|(y) for every x, y ∈ Rn . Proof. Let x, y ∈ Rn . The claim is clear if x = y, and thus we may assume that x = y. Let B = B(x0 , |x − y|) be a ball with x, y ∈ B. Then B ⊂ B(x, 2|x − y|) ∩ B(y, 2|x − y|), and Theorem 3.4 with R = 2|x − y| gives |u(x) − u(y)| ≤ |u(x) − uB | + |uB − u(y)|

≤ C(n)|x − y| M (|∇u|χB )(x) + M (|∇u|χB )(y)

≤ C(n)|x − y| M2|x−y| |∇u|(x) + M2|x−y| |∇u|(y) . The final step above follows from the general inequality (4.1)

M (f χB )(z) ≤ M (f χB(z,R) )(z) ≤ MR f (z),

where R > 0 and B ⊂ Rn is a ball with z ∈ B ⊂ B(z, R) and f ∈ L1 (B(z, R)).  In (4.1) we also use the fact that f χB(z,R) vanishes in Rn \ B(z, R). Theorem 4.1 has a counterpart for Sobolev functions. This generalization is particularly interesting since it can be used to give a characterization for Sobolev functions, see Theorem 4.6. 63

64

4. POINTWISE INEQUALITIES FOR SOBOLEV FUNCTIONS

Theorem 4.2. Let 1 ≤ p < ∞ and assume that u ∈ W 1,p (Rn ). There exist a constant C = C(n) and a set E ⊂ Rn with |E| = 0 such that

|u(x) − u(y)| ≤ C|x − y| M |∇u|(x) + M |∇u|(y) for every x, y ∈ Rn \ E. Proof. Since compactly supported Lipschitz functions are dense in W 1,p (Rn ), see Remark 2.22 (a), there exist functions ui ∈ Lipc (Rn ), i ∈ N, such that ui → u in W 1,p (Rn ) as i → ∞. By the maximal function theorem, see Theorem 1.15, and Remark 1.16, (M |∇ui |)i∈N converges to M |∇u| in Lp (Rn ) if 1 < p < ∞, and in measure if p = 1. By passing twice to a subsequence, if necessary, we obtain a set E ⊂ Rn with |E| = 0 such that lim ui (x) = u(x) < ∞

i→∞

and lim M |∇ui |(x) = M |∇u|(x) < ∞

i→∞

for every x ∈ Rn \ E. Theorem 4.1 implies |u(x) − u(y)| = lim |ui (x) − ui (y)| i→∞



≤ C(n) lim sup |x − y| M |∇ui |(x) + M |∇ui |(y) i→∞

= C(n)|x − y| M |∇u|(x) + M |∇u|(y)

for every x, y ∈ Rn \ E.

 1,p Wloc (Rn ),

Remark 4.3. Let 1 ≤ p < ∞, assume that u ∈ and let B = B(x0 , R) ⊂ Rn be a ball. By Theorem 3.5, inequality (4.1) and Remark 1.29, there exists a set EB ⊂ B with |EB | = 0 such that (4.2)

|u(x) − uB | ≤ C(n)RM (|∇u|χB )(x) ≤ C(n)RM2R |∇u|(x)

1 ≤ C(n)R M2R |∇u|p (x) p

for every x ∈ B \ EB . Let E = B EB , where the union is taken over all balls with center in Qn and rational radii. Since the union is countable, we have |E| = 0. Moreover, for every x, y ∈ Rn , x = y, there exists a ball B = B(x0 , r) from the union with x, y ∈ B and r ≤ |x − y|. By arguing as in the proof of Theorem 4.1 and applying (4.2), we obtain 

1

1  |u(x) − u(y)| ≤ C(n)|x − y| M2|x−y| |∇u|p (x) p + M2|x−y| |∇u|p (y) p for every x, y ∈ Rn \ E. Remark 4.4. Assume that u ∈ W 1,∞ (Rn ). Lemma 1.10 implies M |∇u|(x) ≤ ∇uL∞ (Rn ) 1,1 for every x ∈ Rn . Since u ∈ W 1,∞ (Rn ) ⊂ Wloc (Rn ), by Remark 4.3 there exists a n set E ⊂ R with |E| = 0 such that

|u(x) − u(y)| ≤ C(n)∇uL∞ (Rn ) |x − y| for every x, y ∈ R \ E. This gives another proof for the fact that every function u ∈ W 1,∞ (Rn ) has a Lipschitz continuous representative; compare to Theorem 2.21. n

4.1. POINTWISE CHARACTERIZATION OF SOBOLEV SPACES

65

Motivated by Theorem 4.2, we introduce the notion of a maximal gradient and the Hajlasz–Sobolev space. Definition 4.5. Let 1 < p < ∞ and assume that u ∈ Lp (Rn ). We write g ∈ D(u) for a measurable function g : Rn → [0, ∞] if there exists a set E ⊂ Rn with |E| = 0 such that

(4.3) |u(x) − u(y)| ≤ |x − y| g(x) + g(y) for every x, y ∈ Rn \ E. The Hajlasz–Sobolev space M 1,p (Rn ) consists of functions u ∈ Lp (Rn ) with D(u) ∩ Lp (Rn ) = ∅. This space is equipped with the norm uM 1,p (Rn ) = uLp (Rn ) + inf gLp (Rn ) . g∈D(u)

Theorem 4.6. Let 1 < p < ∞. Then M 1,p (Rn ) = W 1,p (Rn ) and the associated norms are equivalent, that is, there exists a constant C = C(n, p) ≥ 1 such that C −1 uW 1,p (Rn ) ≤ uM 1,p (Rn ) ≤ CuW 1,p (Rn ) for every u ∈ W 1,p (Rn ). Proof. First we discuss the inclusion W 1,p (Rn ) ⊂ M 1,p (Rn ). Let u ∈ W 1,p (Rn ) and let C(n) be the constant in Theorem 4.2. Theorem 1.15 (c) implies C(n)M |∇u| ∈ D(u) ∩ Lp (Rn ) and

  uM 1,p (Rn ) ≤ uLp (Rn ) + C(n)M |∇u|Lp (Rn ) ≤ C(n, p)uW 1,p (Rn ) .

Assume then that u ∈ M 1,p (Rn ) with g ∈ D(u) ∩ Lp (Rn ). Let h ∈ R \ {0} and j = 1, . . . , n. By the assumption we have

|u(x + hej ) − u(x)| ≤ |h| g(x + hej ) + g(x) for almost every x ∈ Rn , from which it follows that  

p p p g(x + hej ) + g(x) dx ≤ 2p gpLp (Rn ) |h|p . |u(x + hej ) − u(x)| dx ≤ |h| Rn

Rn

By the characterization of the Sobolev space with difference quotients, see Theorem 2.45 (c), we conclude that u ∈ W 1,p (Rn ) and uW 1,p (Rn ) ≤ uLp (Rn ) + C(n, p)gLp (Rn ) < ∞. The inequality uW 1,p (Rn ) ≤ C(n, p)uM 1,p (Rn ) follows by taking infimum over all g ∈ D(u) ∩ Lp (Rn ).



Remark 4.7. The pointwise characterization in Theorem 4.6 can be used to give alternative proofs for many properties of Sobolev spaces. For example, if 1 < p < ∞, u ∈ M 1,p (Rn ) and g ∈ D(u) ∩ Lp (Rn ), then by the triangle inequality

|u(x)| − |u(y)| ≤ |u(x) − u(y)| ≤ |x − y| g(x) + g(y) whenever x, y ∈ Rn \ E with |E| = 0. Thus g ∈ D(|u|) ∩ Lp (Rn ) and consequently |u| ∈ M 1,p (Rn ). By Theorem 4.6 we conclude that if u ∈ W 1,p (Rn ), then also |u| ∈ W 1,p (Rn ); compare to Theorem 2.25 (a). The next result is similar to Theorem 2.40, but the proof is based on pointwise inequalities.

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4. POINTWISE INEQUALITIES FOR SOBOLEV FUNCTIONS

Theorem 4.8. Let 1 < p < ∞ and assume that u ∈ Lp (Rn ). Then u ∈ W (Rn ) if and only if there exist functions ui ∈ Lp (Rn ) and gi ∈ D(ui ) ∩ Lp (Rn ), i ∈ N, such that ui → u almost everywhere and gi → g ∈ Lp (Rn ) almost everywhere, as i → ∞. 1,p

Proof. For u ∈ W 1,p (Rn ) the claim follows from Theorem 4.6. To prove the reverse claim, assume that ui , g ∈ Lp (Rn ), gi ∈ D(ui ) ∩ Lp (Rn ), and ui → u almost everywhere and gi → g almost everywhere, as i → ∞. Then for every i ∈ N there exists a set Ei ⊂ Rn with |Ei | = 0 such that

(4.4) |ui (x) − ui (y)| ≤ |x − y| gi (x) + gi (y) for every x, y ∈ Rn \ Ei . Let A ⊂ Rn with |A| = 0 be such that lim ui (x) = u(x) < ∞ and

i→∞

lim gi (x) = g(x) < ∞

i→∞

 for every x ∈ Rn \ A. Define E = A ∪ ∞ i=1 Ei and observe that |E| = 0. Let x, y ∈ Rn \ E with x = y. By taking the limits in (4.4) we obtain

|u(x) − u(y)| = lim |ui (x) − ui (y)| ≤ lim sup |x − y| gi (x) + gi (y) i→∞ i→∞

= |x − y| g(x) + g(y) , and thus g ∈ D(u) ∩ Lp (Rn ). This implies u ∈ M 1,p (Rn ) and u ∈ W 1,p (Rn ) by Theorem 4.6.  4.2. Lipschitz truncation of Sobolev functions Theorem 4.2 implies that a Sobolev function is Lipschitz continuous in the set where the Hardy–Littlewood maximal function of the gradient is bounded. The size of the set where the maximal function of the gradient is large can be estimated by maximal function arguments. In contrast to the standard convolution approximation, see Remark 2.22, this approach gives an approximation of u ∈ W 1,p (Rn ) with a Lipschitz continuous function which coincides with u outside a set of arbitrarily small measure. Theorem 4.9. Let 1 ≤ p < ∞ and assume that u ∈ W 1,p (Rn ). For every ε > 0 there exists ϕε ∈ Lip(Rn ) such that u − ϕε W 1,p (Rn ) < ε and {x ∈ Rn : u(x) = ϕε (x)} < ε. Proof. By Theorem 4.2 there exists E ⊂ Rn with |E| = 0 such that

|u(x) − u(y)| ≤ C(n)|x − y| M |∇u|(x)| + M |∇u|(y) for every x, y ∈ Rn \ E. Let   Ft = x ∈ Rn \ E : M |∇u|(x) ≤ t and |u(x)| ≤ t for t > 0. Since |E| = 0, we have

|Rn \ Ft | = |(Rn \ E) \ Ft | = {x ∈ Rn \ E : M |∇u|(x) > t or |u(x)| > t} .

If x, y ∈ Ft , then



|u(x) − u(y)| ≤ C(n)|x − y| M |∇u|(x) + M |∇u|(y) ≤ 2C(n)t|x − y|,

4.2. LIPSCHITZ TRUNCATION OF SOBOLEV FUNCTIONS

67

and thus the restriction u|Ft is a 2C(n)t-Lipschitz function. Theorem 2.7 allows us to extend u|Ft from Ft to a 2C(n)t-Lipschitz function vt : Rn → R. By truncation we obtain a 2C(n)t-Lipschitz function ut = max{−t, min{vt , t}}. By inequality (1.3)  {x ∈ Rn : M |∇u|(x) > t} ≤ C(n) tp−1 |∇u(x)| dx t · tp−1 {x∈Rn :|∇u(x)|≥ 2t }  C(n, p) ≤ |∇u(x)|p dx. tp {x∈Rn :|∇u(x)|≥ 2t } Hence, by Chebyshev’s inequality, we obtain

tp |Rn \ Ft | ≤ tp {x ∈ Rn : M |∇u|(x) > t} + {x ∈ Rn : |u(x)| > t}   ≤ C(n, p) |∇u(x)|p dx + |u(x)|p dx. {x∈Rn :|∇u(x)|≥ 2t }

{x∈Rn :|u(x)|>t}

The dominated convergence theorem implies lim tp |Rn \ Ft | = 0

(4.5)

t→∞

and consequently |Rn \ Ft | → 0 as t → ∞. Observe that (4.6)

lim |{x ∈ Rn : u(x) = ut (x)}| ≤ lim |Rn \ Ft | = 0,

t→∞

t→∞

since u = ut in Ft for every t > 0. Then we consider an estimate for u − ut W 1,p (Rn ) . Since u = ut in Ft and |ut | ≤ t, the measure estimate in (4.5) implies  p ut − uLp (Rn ) = |ut (x) − u(x)|p dx Rn \Ft



(4.7)

≤2

p



 |ut (x)| dx + p

Rn \Ft

|u(x)| dx p

Rn \Ft

  ≤ 2p tp |Rn \ Ft | +

Rn \Ft

|u(x)|p dx

 t→∞

−−−→ 0.

By Theorem 2.25 (b), we obtain ∇(ut − u)(x) = χRn \Ft (x)∇(ut − u)(x) = χRn \Ft (x)∇ut (x) − χRn \Ft (x)∇u(x) for almost every x ∈ Rn . Since ut is 2C(n)t-Lipschitz in Rn , we have |∇ut | ≤ 2C(n)t almost everywhere in Rn , and by (4.5),    p p p p |∇ut (x)| dx + |∇u(x)| dx ∇(ut − u)Lp (Rn ) ≤ 2 Rn \Ft

  p ≤ 2 (2C(n)t)p |Rn \ Ft | +

Rn \Ft

Rn \Ft



|∇u(x)| dx p

t→∞

−−−→ 0.

Together with (4.7) this implies lim u − ut W 1,p (Rn ) = 0.

t→∞

By (4.6) we conclude that, for a given ε > 0, we may choose ϕε = ut for a sufficiently large t > 0. 

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4. POINTWISE INEQUALITIES FOR SOBOLEV FUNCTIONS

There are close connections between Sobolev spaces with zero boundary values and inequalities involving the distance to the boundary. Similar phenomena appear in connection with Hardy’s inequalities, see Section 6.6. Lemma 4.10. Let 1 < p < ∞, let Ω  Rn be an open set, and assume that u ∈ W 1,p (Ω) satisfies  |u(x)|p (4.8) dx < ∞. p Ω d (x, ∂Ω) Then u ∈ W01,p (Ω). Proof. By considering the positive part u+ and the negative part u− separately, we may assume that u ≥ 0. For every t > 0 we define ut : Ω → [0, ∞) by   ut (x) = max min{u(x), td (x, ∂Ω)} − 1t , 0 . We show that {ut : t > 0} is a bounded collection of functions in W01,p (Ω). Clearly 0 ≤ ut ≤ u in Ω and thus   ut (x)p dx ≤ u(x)p dx < ∞. Ω

Ω

To estimate the gradient we define Ft = {x ∈ Ω : u(x) > td (x, ∂Ω)}, for t > 0. Theorem 2.25 (c) implies    |∇ut (x)|p dx ≤ |∇u(x)|p dx + tp |∇d (x, ∂Ω)|p dx Ω Ω\F Ft  t ≤ |∇u(x)|p dx + tp |Ft |, Ω

where we also used the fact that |∇d (x, ∂Ω)| ≤ 1 for almost every x ∈ Ω. Since  u(x)p p t |Ft | ≤ dx < ∞, p Ω d (x, ∂Ω) for every t > 0, and ut (x) = 0 if d(x, ∂Ω) < t−2 , the estimates above give the uniform boundedness for the functions ut in W01,p (Ω). Since ut → u almost everywhere  in Ω, as t → ∞, we conclude from Theorem 2.40 (b) that u ∈ W01,p (Ω). Theorem 4.9, for 1 < p < ∞, and Lemma 4.10 are special cases of the following general result. The key point is that the distance function gives an effective control for the zero extension of a Lipschitz function. This idea will be also used in many subsequent arguments. Theorem 4.11. Let 1 < p < ∞, let Ω  Rn be an open set, and assume that u ∈ W 1,p (Ω) satisfies  |u(x)|p dx < ∞. (4.9) p Ω d (x, ∂Ω) Then u ∈ W01,p (Ω) and for every ε > 0 there exists ϕε ∈ Lip(Rn ) such that c n (a) ϕ ε (x) = 0 for every x ∈ Ω = R \ Ω, (b) {x ∈ Ω : u(x) = ϕε (x)} < ε, (c) u − ϕε W 1,p (Ω) < ε.

4.3. CAMPANATO AND MORREY APPROACHES TO SOBOLEV SPACES

69

Proof. Lemma 4.10 implies u ∈ W01,p (Ω). By Theorem 2.31, we may assume that u ∈ W 1,p (Rn ) and u(x) = 0 for every x ∈ Ωc . By Theorem 4.6 we have u ∈ M 1,p (Rn ), and thus there exist g ∈ D(u) ∩ Lp (Rn ) and E ⊂ Rn with |E| = 0 such that

(4.10) |u(x) − u(y)| ≤ |x − y| g(x) + g(y) for every x, y ∈ Rn \ E. Let   Ft = x ∈ Ω \ E : |u(x)| ≤ t, |u(x)|d (x, ∂Ω)−1 ≤ t and g(x) ≤ t for t > 0. Since |E| = 0, we have |Ω \ Ft | ≤ {x ∈ Ω \ E : |u(x)| > t} + {x ∈ Ω \ E : |u(x)|d (x, ∂Ω)−1 > t} + {x ∈ Ω \ E : g(x) > t} . Chebyshev’s inequality implies  tp {x ∈ Ω \ E : |u(x)| > t} ≤

t→∞

{x∈Ω\E:|u(x)|>t}

|u(x)|p dx −−−→ 0.

Similar estimates for |u|d (x, ∂Ω)−1 ∈ Lp (Ω) and g ∈ Lp (Rn ) show that lim tp |Ω \ Ft | = 0.

(4.11)

t→∞

By (4.10) we have



|u(x) − u(y)| ≤ |x − y| g(x) + g(y) ≤ 2t|x − y|

for every x, y ∈ Ft . On the other hand, if x ∈ Ft and y ∈ Ωc , then |u(x) − u(y)| = |u(x)| ≤ td (x, ∂Ω) = td (x, Ωc ) ≤ t|x − y|. These estimates and the fact that u = 0 in Ωc show that the restriction u|Ft ∪Ωc is a 2t-Lipschitz function. Theorem 2.7 allows us to extend u|Ft ∪Ωc to a 2t-Lipschitz function vt : Rn → R. By truncation we obtain a 2t-Lipschitz function ut = max{−t, min{vt , t}}. Clearly ut = 0 in Ω . Since u = vt and |u| ≤ t in Ft , we have ut = vt = u in Ft . By (4.11) we obtain c

(4.12)

lim |{x ∈ Ω : u(x) = ut (x)}| ≤ lim |Ω \ Ft | = 0.

t→∞

t→∞

Proceeding as in the proof of Theorem 4.9, we conclude that lim ut − uW 1,p (Ω) = 0.

t→∞

For a given ε > 0, we can choose ϕε = ut for a sufficiently large t > 0, and this completes the proof.  4.3. Campanato and Morrey approaches to Sobolev spaces We study a Campanato-type approach based on the fractional sharp maximal function, which controls the oscillation of a locally integrable function. Definition 4.12. Let 0 ≤ β < ∞ and R > 0, and assume that f ∈ L1loc (Rn ).  The restricted fractional sharp maximal function Mβ,R f : Rn → [0, ∞] is defined as  f (x) = sup r −β  Mβ,R 0 0 and f is H¨ older continuous in Rn , with exponent β and constant C, then f (z) dz ≤  |f (y) − f (z)| dz ≤ C(2r)β |f (y) − fB(x,r) | = f (y) −  B(x,r)

B(x,r)

4.3. CAMPANATO AND MORREY APPROACHES TO SOBOLEV SPACES

71

for every x ∈ Rn and y ∈ B(x, r). Thus Mβ f (x) = sup r −β  r>0

|f (y) − fB(x,r) | dy ≤ 2β C < ∞

B(x,r)

for every x ∈ Rn . This shows that Mβ f is bounded in Rn by 2β C. Since Mβ f is invariant under redefinition of f on a set of measure zero, we conclude that Mβ f ∈ L∞ (Rn ) if and only if f ∈ L1loc (Rn ) can be redefined on a set of measure zero in such a way that f becomes H¨ older continuous in Rn with exponent β. Remark 4.15. In the limiting case β = 0 we obtain the space of bounded mean oscillation BMO(Rn ), which consists of functions f ∈ L1loc (Rn ) satisfying M0 f ∈ L∞ (Rn ). Functions of bounded mean oscillation are discussed in Section 9.7. The following characterization of Sobolev spaces extends Theorem 4.6. Theorem 4.16. Let 1 < p < ∞. The following conditions are equivalent. (a) u ∈ W 1,p (Rn ). (b) u ∈ Lp (Rn ) and there exist a nonnegative function g ∈ Lp (Rn ) and a set E ⊂ Rn with |E| = 0 such that

|u(x) − u(y)| ≤ |x − y| g(x) + g(y) for every x, y ∈ Rn \ E. (c) u ∈ Lp (Rn ) and there exists a nonnegative function g ∈ Lp (Rn ) such that the Poincar´e inequality 

B(x,r)

|u(y) − uB(x,r) | dy ≤ r 

g(y) dy

B(x,r)

holds for every x ∈ Rn and r > 0. (d) u ∈ Lp (Rn ) and M1 u ∈ Lp (Rn ). Proof. Theorem 4.6 gives the equivalence of (a) and (b). Then we prove that (b) implies (c). Let y ∈ B(x, r) \ E. We integrate the pointwise inequality in (b) twice over the ball B(x, r). This gives u(z) dz ≤  |u(y) − u(z)| dz |u(y) − uB(x,r) | = u(y) −  B(x,r) B(x,r)   g(z) dz ≤ 2r g(y) +  B(x,r)

and

 

B(x,r)

|u(y) − uB(x,r) | dy ≤ 2r  ≤ r

 B(x,r)

g(y) dy + 

g(z) dz B(x,r)

4g(y) dy.

B(x,r)

Thus (c) holds with the function 4g ∈ Lp (Rn ). To show that (c) implies (d), we observe that 0 ≤ M1 u(x) = sup r>0

1 |u(y) − uB(x,r) | dy ≤ sup  g(y) dy = M g(x) r B(x,r) r>0 B(x,r)

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4. POINTWISE INEQUALITIES FOR SOBOLEV FUNCTIONS

for every x ∈ Rn . By Theorem 1.15 (c), we have M g ∈ Lp (Rn ), and thus M1 u ∈ Lp (Rn ). Finally, we show that (d) implies (b). By Lemma 4.13, with β = 1, there exists E ⊂ Rn such that |E| = 0 and

|u(x) − u(y)| ≤ C(n)|x − y| M1 u(x) + M1 u(y) for every x, y ∈ Rn \ E. This gives (b) with g = C(n)M1 u ∈ Lp (Rn ).



Next we consider a Morrey-type approach to pointwise inequalities. Compared to the Campanato approach above, here the sharp maximal function is replaced with a fractional maximal function of the gradient, see Definition 1.38. 1,1 (Rn ). There exist a Theorem 4.17. Let 0 ≤ α < 1 and assume that u ∈ Wloc n constant C = C(n, α) and a set E ⊂ R with |E| = 0 such that

|u(x) − u(y)| ≤ C|x − y|1−α Mα,4|x−y| |∇u|(x) + Mα,4|x−y| |∇u|(y)

for every x, y ∈ Rn \ E. Proof. Let x ∈ Rn , R > 0 and 0 < r < R. By the Poincar´e inequality in Theorem 3.14, we have 

B(x,r)

|u(y) − uB(x,r) | dy ≤ C(n)r 

|∇u(y)| dy.

B(x,r)

This implies r α−1 

B(x,r)

|u(y) − uB(x,r) | dy ≤ C(n)r α 

|∇u(y)| dy, B(x,r)

and consequently  M1−α,R u(x) ≤ C(n)Mα,R |∇u|(x). The claim follows from Lemma 4.13 with β = 1 − α > 0.



Remark 4.18. Theorem 4.17 gives a Morrey-type condition for H¨older continuity; compare to Remark 4.14, where H¨ older continuity was characterized by 1,1 (Rn ) and a Campanato approach. Let 0 ≤ α < 1 and assume that u ∈ Wloc ∞ n n Mα |∇u| ∈ L (R ). By Theorem 4.17 there exists E ⊂ R with |E| = 0 such that

(4.14) |u(x) − u(y)| ≤ C(n, α)|x − y|1−α Mα |∇u|(x) + Mα |∇u|(y) for every x, y ∈ Rn \ E. By (4.14) we have   |u(x) − u(y)| ≤ C(n, α)Mα |∇u|

L∞ (Rn )

|x − y|1−α ,

for every x, y ∈ Rn \ (E ∪ F ), where F is an exceptional set for essential supremum with |F | = 0. This shows that u is H¨older continuous with the exponent 1 − α, after a possible redefinition on a set of measure zero. Remark 4.19. From (4.14) we recover Theorem 3.23, which was obtained as a consequence of Morrey’s inequality in Section 3.5. To see this, assume that older’s inequality we have u ∈ W 1,p (Rn ) with n < p < ∞. By H¨

1 M np |∇u|(x) ≤ C(n) Mn |∇u|p (x) p ≤ C(n)∇uLp (Rn ) < ∞, for every x ∈ Rn . Thus (4.14), with α =

n p,

implies n

|u(x) − u(y)| ≤ C(n, p)∇uLp (Rn ) |x − y|1− p

4.4. MAXIMAL OPERATOR ON SOBOLEV SPACES

73

for every x, y ∈ Rn \ E. This shows that u is H¨older continuous with the exponent 1 − np after a possible redefinition on a set of measure zero. 4.4. Maximal operator on Sobolev spaces We show that the Hardy–Littlewood maximal operator is not only bounded on Lp (Rn ) but also on W 1,p (Rn ) for 1 < p ≤ ∞. We begin with the limiting case p = ∞. Recall that W 1,∞ (Rn ) essentially coincides with the space of bounded Lipschitz continuous functions, see Theorem 2.21. Theorem 4.20. Assume that u ∈ W 1,∞ (Rn ). Then M u ∈ W 1,∞ (Rn ) and M uW 1,∞ (Rn ) ≤ uW 1,∞ (Rn ) . Proof. Theorem 2.21 implies that the function u ∈ W 1,∞ (Rn ), perhaps after redefinition on a set of measure zero, is bounded and Lipschitz continuous with constant ∇uL∞ (Rn ) . Hence |τh u(y) − u(y)| = |u(y + h) − u(y)| ≤ ∇uL∞ (Rn ) |h| for every y, h ∈ Rn , where τh u(y) = u(y + h). On the other hand, it follows from Lemma 1.10 that M u(y) ≤ uL∞ (Rn ) < ∞ for every y ∈ Rn . Since the maximal function commutes with translations and the maximal operator is sublinear, see Lemma 1.5, we have |M u(x + h) − M u(x)| = |(τh M u)(x) − M u(x)| = |M (τh u)(x) − M u(x)| ≤ M (τh u − u)(x) = sup  r>0

|τh u(y) − u(y)| dy

B(x,r)

≤ ∇uL∞ (Rn ) |h|, for every x, h ∈ R . This shows that M u is Lipschitz continuous in Rn with constant ∇uL∞ (Rn ) . Since M u ∈ L∞ (Rn ), by Theorem 2.21 (c) we conclude that M u ∈ W 1,∞ (Rn ) and n

M uW 1,∞ (Rn ) = M uL∞ (Rn ) + ∇M uL∞ (Rn ) ≤ uL∞ (Rn ) + ∇uL∞ (Rn ) = uW 1,∞ (Rn ) .



The proof above shows that if u ∈ Lip(Rn ) and M u(x0 ) < ∞ for some x0 ∈ Rn , then M u ∈ Lip(Rn ) with the same Lipschitz constant as u, see Remark 1.18. Theorem 2.9 implies that then M u is differentiable almost everywhere. In general, the question of differentiability for maximal functions is more delicate and the maximal function of a positive differentiable function is not necessarily differentiable. Nevertheless, certain weak differentiability properties are preserved. Theorem 4.21. Let 1 < p < ∞ and assume that u ∈ W 1,p (Rn ). Then M u ∈ W (Rn ) and there exists a constant C = C(n, p) such that 1,p

(4.15)

M uW 1,p (Rn ) ≤ CuW 1,p (Rn ) .

Proof. The proof is based on the characterization of W 1,p (Rn ) by difference quotients, see Theorem 2.45. By the maximal function theorem with 1 < p < ∞, see Theorem 1.15 (c), we have M u ∈ Lp (Rn ). Using Lemma 1.5, Theorem 1.15 (c), and Theorem 2.45 (a), we obtain (τhej M u) − M uLp (Rn ) = M (τhej u) − M uLp (Rn ) ≤ M (τhej u − u)Lp (Rn ) ≤ C(n, p)τhej u − uLp (Rn ) ≤ C(n, p)∇uLp (Rn ) |h|

74

4. POINTWISE INEQUALITIES FOR SOBOLEV FUNCTIONS

for every h ∈ R \ {0} and j = 1, . . . , n. Theorem 2.45 (c) implies that M u ∈ W 1,p (Rn ) with ∇M uLp (Rn ) ≤ C(n, p)∇uLp (Rn ) . Thus, again by Theorem 1.15 (c), M uW 1,p (Rn ) ≤ M uLp (Rn ) + ∇M uLp (Rn )

≤ C(n, p) uLp (Rn ) + ∇uLp (Rn ) ≤ C(n, p)uW 1,p (Rn ) .



A more careful analysis gives a pointwise estimate for the gradient. Theorem 4.22. Let 1 < p < ∞ and assume that u ∈ W 1,p (Rn ). Then M u ∈ W (Rn ) and 1,p

|∇M u(x)| ≤ M |∇u|(x)

(4.16) for almost every x ∈ R . n

Proof. For r > 0 and x ∈ Rn , we define

χr (x) =

(4.17) where

χB(0,r) (x) |B(0, r)|

,

χB(0,r) is the characteristic function of B(0, r). Then 

B(x,r)

|u(y)| dy = (|u| ∗ χr )(x),

for every x ∈ Rn . For almost every x ∈ Rn we have ∇(|u| ∗ χr )(x) = (∇|u| ∗ χr )(x), where the convolution on the right-hand side is taken componentwise. Young’s convolution inequality (2.13) and Theorem 2.25 (a) give |u| ∗ χr ∈ W 1,p (Rn ). Let (rj )j∈N be an enumeration of positive rational numbers. Then

M u(x) = sup |u| ∗ χrj (x) j∈N

for every x ∈ R . We define functions vk : Rn → R, with k ∈ N, by

vk (x) = max |u| ∗ χrj (x). n

1≤j≤k

By applying Theorem 2.25 (c) inductively, we find that (vk )k∈N is a pointwise increasing sequence of functions in W 1,p (Rn ) with lim vk (x) = M u(x)

k→∞

for every x ∈ Rn . Moreover, for every k ∈ N, we have



|∇vk (x)| ≤ max ∇ |u| ∗ χrj (x) = max ∇|u| ∗ χrj (x) ≤ M |∇u|(x) 1≤j≤k

1≤j≤k

for almost every x ∈ R . Theorem 1.15 (c) gives n

  supvk W 1,p (Rn ) ≤ M uLp (Rn ) + M |∇u|Lp (Rn ) k∈N

≤ C(n, p) uLp (Rn ) + ∇uLp (Rn ) < ∞,

and thus the sequence (vk )k∈N is bounded in W 1,p (Rn ). Since vk → M u pointwise, as k → ∞, Theorem 2.40 (a) implies that M u ∈ W 1,p (Rn ), vk → M u weakly in

4.5. MAXIMAL FUNCTION WITH RESPECT TO AN OPEN SET

75

Lp (Rn ), and ∇vk → ∇M u weakly in Lp (Rn ; Rn ) as k → ∞. Theorem 2.35, with  fk = ∇vk and gk = M |∇u| for every k ∈ N, implies (4.16). Remark 4.23. Theorem 4.22 gives an alternative proof for Theorem 4.21. To see this, let 1 < p < ∞ and assume that u ∈ W 1,p (Rn ). By Theorem 1.15 (c) and (4.16), we have M uW 1,p (Rn ) ≤ M uLp (Rn ) + ∇M uLp (Rn )  

≤ C(n, p) uLp (Rn ) + M |∇u|Lp (Rn )

≤ C(n, p) uLp (Rn ) + ∇uLp (Rn ) ≤ C(n, p)uW 1,p (Rn ) . 4.5. Maximal function with respect to an open set We study boundedness of the maximal operator with respect to an open set, see Definition 1.30, on Sobolev spaces. The arguments in the previous section do not apply directly, mainly because this maximal operator does not commute with translations. Results in this section hold for every nontrivial open set in Rn . We assume throughout this section that Ω  Rn is an open set, and thus d (x, ∂Ω) is finite and positive for every x ∈ Ω. For u ∈ L1loc (Ω), we define integral average functions ut : Ω → [0, ∞), 0 < t < 1, by ut (x) = 

|u(y)| dy,

B(x,tδ(x))

where δ(x) = d (x, ∂Ω) ∈ (0, ∞) for every x ∈ Ω  Rn . Then MΩ u(x) = sup ut (x) 0 0 such that 1 |f (y)| dy ≥ Mα f (x + hk ) − . j B(x+hk ,rj )

rjα 

Since x + hk ∈ K, we may assume that r0 ≤ rj ≤ R0 for every j ∈ N. Let j ∈ N to be chosen later. Since B(x + hk , rj ) ⊂ B(x, rj + |hk |), we have Mα f (x) ≥ (rj + |hk |)α 

B(x,rj +|hk |)

≥ ωn−1 (rj + |hk |)α−n

|f (y)| dy



|f (y)| dy. B(x+hk ,rj )

From the estimates above we obtain 

+ −1 α−n α−n |hk |D Mα f (x) ≤ ωn rj − (rj + |hk |)

|f (y)| dy +

B(x+hk ,rj )

1 |hk | + . j k

By the mean value theorem there exists ρj,k , with rj < ρj,k < rj + |hk |, such that rjα−n − (rj + |hk |)α−n ≤ (n − α)ρα−n−1 |hk |, j,k and hence



|hk |D+ Mα f (x) ≤ C(n, α)ρα−n−1 |hk | j,k

|f (y)| dy + 

≤ C(n, α)rjα−n−1 |hk |  ≤ C(n, α)

rj rj + |hk |

B(x+hk ,rj )

B(x,rj +|hk |)

1 |hk | + j k

|f (y)| dy +

α−n−1

|hk |Mα−1 f (x) +

We now choose j ∈ N to be so large that 1j ≤ |hkk | . Then  α−n−1 rj + D Mα f (x) ≤ C(n, α) Mα−1 f (x) + rj + |hk |  α−n−1 r0 ≤ C(n, α) Mα−1 f (x) + r0 + |hk |

1 |hk | + j k 1 |hk | + . j k

2 k 2 , k

and by letting k → ∞ we conclude that (4.26)

D+ Mα f (x) ≤ C(n, α)Mα−1 f (x)

for every x ∈ Rn . Clearly, this estimate also holds if f is identically zero. Then assume that f ∈ Lp (Rn ), and let fj ∈ C0∞ (Rn ), j ∈ N, be such that fj → f in Lp (Rn ) as j → ∞. Let j ∈ N. Since Mα fj is Lipschitz continuous in compact subsets of Rn , it is differentiable almost everywhere in Rn by Theorem 2.9. If x ∈ Rn is a point of differentiability, then inequality (4.26) implies ∂Mα fj + ∂xi (x) ≤ D Mα fj (x) ≤ C(n, α)Mα−1 fj (x) for every i = 1, . . . , n. Thus (4.27)

|∇Mα fj (x)| ≤ C(n, α)Mα−1 fj (x)

82

4. POINTWISE INEQUALITIES FOR SOBOLEV FUNCTIONS

for almost every x ∈ Rn . Theorem 1.42, or the maximal function theorem if α = 1, implies ∇Mα fj Lq (Rn ;Rn ) ≤ C(n, α)Mα−1 fj Lq (Rn ) ≤ C(n, p, α)fj Lp (Rn ) ,

and so ∇Mα fj j∈N is a bounded sequence in Lq (Rn ; Rn ). Using Theorem 1.42 we obtain Mα fj − Mα f Lp∗ (Rn ) ≤ Mα (fj − f )Lp∗ (Rn ) j→∞

≤ C(n, p, α)fj − f Lp (Rn ) −−−→ 0, ∗

and therefore Mα fj → Mα f in Lp (Rn ) as j → ∞. A similar argument shows q n there exists that Mα−1 fj → M

L (R ) as j → ∞. By Theorem 2.36, α−1 f in a subsequence of ∇Mα fj j∈N , which converges to some v ∈ Lq (Rn ; Rn ). Since ∗

Mα fj → Mα f in Lp (Rn ), it follows from a straightforward adaptation of the proof of Theorem 2.39 that ∇Mα f = v ∈ Lq (Rn ; Rn ) and, by passing to a subsequence, that ∇Mα fj → ∇Mα f weakly in Lq (Rn ; Rn ) as j → ∞. By Theorem 2.35 we also see that the estimate in (4.27) is preserved up to the weak limit and hence |∇Mα f (x)| ≤ C(n, α)Mα−1 f (x) for almost every x ∈ Rn . This concludes the proof of part (a). Part (b) follows immediately from part (a) and Theorem 1.42, or the maximal function theorem if α = 1, since Mα f Lp∗ (Rn ) ≤ C(n, p, α)f Lp (Rn ) and ∇Mα f Lq (Rn ) ≤ C(n, α)Mα−1 f Lq (Rn ) ≤ C(n, p, α)f Lp (Rn ) .



4.7. Notes The estimate in Theorem 4.1 has been studied by Calder´ on in [69] and Calder´on and Scott [70]. Such pointwise inequalities are also discussed in Bennett and Sharpley [28, Section V.5] and DeVore and Sharpley [100]. Definition 4.5 on metric measure spaces is from Hajlasz [165], see also Varopoulos [391] and Vodopyanov [392] for special cases. For Theorem 4.6 we refer to Bennett and Sharpley [28, Theorem V.5.6, Corollary V.5.9], Calder´ on [69] and Hajlasz [165]. See also Hajlasz [164], Hajlasz and Martio [175] and Vodopyanov [392]. The equivalence fails for p = 1, see Hajlasz [164]. For p = 1 there is a characterization for the Hardy–Sobolev space in Koskela and Saksman [245]. See also Miyachi [328]. A counterpart of Theorem 4.6 also holds for open subsets of Rn with a sufficiently regular boundary, but not in general. However, we have M 1,p (Ω) ⊂ W 1,p (Ω) for every open set Ω ⊂ Rn and 1 ≤ p < ∞, see [164, Proposition 1] and [175, Lemma 6]. For Lemma 4.13, we refer to Calder´on and Scott [70], DeVore and Sharpley [100, Theorem 2.5] and Mac´ıas and Segovia [296]. See also Bojarski [48] and Hajlasz and Kinnunen [170, Lemma 3.6]. The equivalence of (a) and (c) in Theorem 4.16 is by Calder´ on [69], see also DeVore and Sharpley [100, Theorem 6.2]. For a general version of Theorem 4.16, see Hajlasz and Kinnunen [170, Theorem 3.4]. Franchi, Hajlasz and Koskela [132] showed that the equivalence of (a) and (c) in Theorem 4.16 also holds for p = 1. For a version of Theorem 4.16 in the range 1 ≤ p < ∞ we refer to Hajlasz [169, Theorem 2.1]. See also Franchi, Lu

4.7. NOTES

83

and Wheeden [133–135], Hajlasz [168], Hajlasz and Koskela [171], Heinonen and Koskela [188] and Koskela and MacManus [244]. The approximation property in Theorem 4.9 has been studied by Calder´ on and Zygmund [71]. Liu [284] showed the approximation in the Sobolev norm. See also Michael and Ziemer [326]. For extensions of Theorem 4.9, see Bojarski and Hajlasz [50], Bojarski, Hajlasz and Strzelecki [51], Hajlasz [165], Hajlasz and Kinnunen [170, Theorem 5.3] and Mal´ y [299]. See also Bennett and Sharpley [28, Theorem V.5.10], Mal´ y and Ziemer [303, Theorem 1.69], Mazya [321] and Ziemer [407]. The Lipschitz truncation method in Theorem 4.9 will be applied in Section 12.3. See also the notes at the end of Chapter 12. The argument in the proof of Lemma 4.10 is from Zhong [405, Theorem 1.9]. See also Kinnunen and Martio [235, Theorem 3.13] and Kilpel¨ ainen, Kinnunen and Martio [219, Theorem 5.1]. Theorem 4.22 is from Kinnunen [224]. A result of Luiro [287] shows that M : W 1,p (Rn ) → W 1,p (Rn ) is a continuous operator for 1 < p < ∞. Kurka [251] and Tanaka [378] study boundedness of maximal operators for u ∈ W 1,1 (R). For extensions of Tanaka’s result to functions of bounded variation in the onedimensional case we refer to Aldaz [10], Aldaz and P´erez L´ azaro [12, 13], Bernal [29] and Liu, Chen and Wu [283]. For extensions of Theorem 4.22, see Aldaz, Colzani and P´erez L´ azaro [11], Buckley [62], Carneiro and Moreira [77], Carneiro and Sveiter [78], Korry [239, 240], Hajlasz and Liu [173], Hajlasz and Mal´ y [174] and P´erez, Picon, Saari and Sousa [346]. Theorem 4.25 has been proved by Kinnunen and Lindqvist [232]. Continuity of the maximal operator with respect to an open set is discussed by Luiro [288–290]. For extensions see Aalto and Kinnunen [1], Hajlasz and Liu [172], Hajlasz and Onninen [176] and Luiro and V¨ ah¨ akangas [291, 292]. Theorem 4.31 is proved by Kinnunen and Saksman [237]. For a corresponding result on an open set, we refer to Heikkinen, Kinnunen, Korvenp¨a¨ a and Tuominen [183]. See also Beltran, Ramos and Saari [26], Carneiro, Finder and Sousa [74], Carneiro and Madrid [75], Carneiro, Madrid and Pierce [76] and Heikkinen and Tuominen [184].

10.1090/surv/257/05

CHAPTER 5

Capacities and Fine Properties of Sobolev Functions Capacity is an appropriate outer measure in the study of pointwise behavior of Sobolev functions. We begin with the definition, basic properties and estimates for Sobolev capacity. Section 5.3 studies quasicontinuity of Sobolev functions. A capacitary version of the Lebesgue differentiation theorem for Sobolev functions is proved in Section 5.4, and Section 5.5 discusses pointwise characterizations of Sobolev spaces with zero boundary values. At the end of the chapter we consider a capacity relative to an open set, which is applied, for example, in the capacity density condition in Definition 6.17 and in Mazya’s inequality in Theorem 5.47. 5.1. Sobolev capacity The Sobolev capacity is induced by the Sobolev norm. Definition 5.1. Let 1 < p < ∞. The Sobolev p-capacity of a set E ⊂ Rn is defined by

upLp (Rn ) + ∇upLp (Rn ) , Capp (E) = inf upW 1,p (Rn ) = inf u∈Ap (E)

u∈Ap (E)

where Ap (E) consists of u ∈ W (R ) with u ≥ 1 almost everywhere in a neighborhood of E. Functions in Ap (E) are called p-admissible functions for E. If Ap (E) = ∅, we set Capp (E) = ∞. 1,p

n

It is not enough to require u ≥ 1 on E instead of u ≥ 1 almost everywhere in an open set containing E in the definition of Ap (E). The reason for this is that Sobolev functions are defined almost everywhere, and we are often interested in capacities of sets of Lebesgue measure zero. If |E| = 0 and we only require u ≥ 1 on E, we could choose u ∈ W 1,p (Rn ) to be zero in Rn \ E, and thus all sets of measure zero would also be of capacity zero. We may restrict the class of admissible functions in the definition of capacity. Lemma 5.2. Let 1 < p < ∞. For E ⊂ Rn , we have Capp (E) =

inf

u∈Ap (E)

upW 1,p (Rn ) ,

where Ap (E) consists of u ∈ W 1,p (Rn ) with 0 ≤ u ≤ 1 almost everywhere in Rn and u = 1 in a neighborhood of E. Proof. Since Ap (E) ⊂ Ap (E), we have Capp (E) ≤

inf

u∈Ap (E) 85

upW 1,p (Rn ) .

86

5. CAPACITIES AND FINE PROPERTIES OF SOBOLEV FUNCTIONS

For the converse inequality, we may assume that Ap (E) = ∅. Let ε > 0 and let uε ∈ Ap (E) be such that uε pW 1,p (Rn ) ≤ Capp (E) + ε. By Theorem 2.25, we have v = min{|uε |, 1} ∈ Ap (E), |v| ≤ |uε | and |∇v| ≤ |∇uε | almost everywhere in Rn . This implies inf

u∈Ap (E)

upW 1,p (Rn ) ≤ vpW 1,p (Rn ) ≤ uε pW 1,p (Rn ) ≤ Capp (E) + ε,

and by letting ε → 0 we obtain inf

u∈Ap (E)

upW 1,p (Rn ) ≤ Capp (E).



The Sobolev capacity is an outer measure. Theorem 5.3. Let 1 < p < ∞. (a) Capp (∅) = 0. (b) For E1 ⊂ E2 ⊂ Rn , we have Capp (E1 ) ≤ Capp (E2 ). (c) For Ei ⊂ Rn , i ∈ N, we have Capp

∞ 

 Ei

i=1

≤

∞

Capp (Ei ).

i=1

Proof. Clearly Capp (∅) = 0. If E1 ⊂ E2 ⊂ Rn , then Ap (E2 ) ⊂ Ap (E1 ) and thus Capp (E1 ) ≤ Capp (E2 ). This proves assertion (b). ∞In order to prove the countable subadditivity in (c), we may assume that i=1 Capp (Ei ) < ∞ since otherwise there is nothing to prove. Let ε > 0. For every i ∈ N, let ui ∈ Ap (Ei ) = ∅ with ui pW 1,p (Rn ) ≤ Capp (Ei ) + 2−i ε.

(5.1)

∞ We claim that v = supi∈N ui is p-admissible for i=1 Ei . First we show that v ∈ W 1,p (Rn ). For k ∈ N, define vk (x) = max1≤i≤k ui (x) for every x ∈ Rn . The sequence (vk )k∈N is increasing and vk → v pointwise almost everywhere as k → ∞. Moreover, 0 ≤ vk (x) = max ui (x) ≤ sup ui (x) = v(x), 1≤i≤k

i∈N

and, by Theorem 2.25, |∇vk (x)| ≤ max |∇ui (x)| ≤ sup|∇ui (x)|, 1≤i≤k

i∈N

5.1. SOBOLEV CAPACITY

87

for almost every x ∈ Rn and every k ∈ N. By (5.1), we obtain   p p |vk (x)| dx + |∇vk (x)|p dx vk W 1,p (Rn ) = Rn Rn   ≤ sup |ui (x)|p dx + sup |∇ui (x)|p dx  ≤

(5.2)

= ≤

Rn i∈N ∞

Rn i=1 ∞

 i=1 ∞

 |ui (x)|p dx +

Rn

Rn i∈N ∞

Rn i=1

|∇ui (x)|p dx 

 |ui (x)|p dx +

Rn

(Capp (Ei ) + 2−i ε) =

i=1

|∇ui (x)|p dx

∞

Capp (Ei ) + ε < ∞,

i=1

for every k ∈ N. This shows that (vk )k∈N is a a bounded sequence in W 1,p (Rn ). Theorem 2.40 implies that v ∈ W 1,p (Rn ), vk → v weakly in Lp (Rn ), and ∇vk → ∇v weakly in Lp (Rn ; Rn ) as k → ∞. Let i ∈ N. Since ui ∈ Ap (Ei ), there exists an open set Gi ⊃ Ei such that ui (x) = 1 for almost every x ∈ Gi and 0 ≤ ui (x) ≤ 1 foralmost every x∈ Rn . It ∞ ∞ ui (x) = 1 for almost every x ∈ i=1 Gi . Since  follows that v(x) = supi∈N  i=1 Gi is ∞ ∞ an open set that contains i=1 Ei , we conclude that v is p-admissible for i=1 Ei . ∞  Thus  Capp Ei ≤ vpW 1,p (Rn ) = vpLp (Rn ) + ∇vpLp (Rn ) i=1

≤ lim inf vk pLp (Rn ) + lim inf ∇vk pLp (Rn ) k→∞

k→∞ ∞

≤ lim inf vk pW 1,p (Rn ) ≤ k→∞

Capp (Ei ) + ε,

i=1

where we applied the lower semicontinuity of Lp norms with respect to weak limits, see inequality (2.18), and the estimate in (5.2). Property (c) follows by letting ε → 0.  The Sobolev capacity is outer regular in the sense that capacity of an arbitrary set can be approximated by capacity of a larger open set. Lemma 5.4. Let 1 < p < ∞. For every E ⊂ Rn , we have   Capp (E) = inf Capp (G) : E ⊂ G, G is open . Proof. Theorem 5.3 (b) implies   Capp (E) ≤ inf Capp (G) : E ⊂ G, G is open . In order to prove the converse inequality, we may assume that Ap (E) = ∅. Let ε > 0 and let u ∈ Ap (E) with upW 1,p (Rn ) ≤ Capp (E) + ε. Since u ∈ Ap (E), there exists an open set Gε ⊃ E such that u(x) ≥ 1 for almost every x ∈ Gε . Thus   inf Capp (G) : E ⊂ G, G is open ≤ Capp (Gε ) ≤ upW 1,p (Rn ) ≤ Capp (E) + ε, and the claim follows by letting ε → 0.



88

5. CAPACITIES AND FINE PROPERTIES OF SOBOLEV FUNCTIONS

5.2. Estimates for capacity We collect basic estimates for Sobolev capacity in this section. More refined estimates are considered in Section 5.7. We are mainly interested in sets of zero capacity since they are negligible in the theory of Sobolev spaces. Sets of capacity zero are of measure zero and thus capacity is a finer outer measure than Lebesgue outer measure. Lemma 5.5. Let 1 < p < ∞. For every E ⊂ Rn , we have |E| ≤ Capp (E). Proof. If Capp (E) = ∞, there is nothing to prove, and thus we may assume that Capp (E) < ∞. Let ε > 0 and let u ∈ Ap (E) with upW 1,p (Rn ) ≤ Capp (E) + ε. There exists an open set G such that E ⊂ G and u(x) ≥ 1 for almost every x ∈ G. This implies  |u(x)|p dx ≤ upLp (Rn ) ≤ upW 1,p (Rn ) ≤ Capp (E) + ε, |E| ≤ |G| ≤ G

and the claim follows by letting ε → 0.



Lemma 5.5 shows that Capp (B(x, r)) > 0 for every x ∈ Rn and r > 0. This implies that capacity is nontrivial in the sense that every nonempty open set has positive capacity. On the other hand, every bounded set has finite Sobolev capacity. Lemma 5.6. Let 1 < p < ∞ and let B(x, r) ⊂ Rn be a ball. There exists a constant C = C(n) such that  Cr n−p , if 0 < r ≤ 1, Capp (B(x, r)) ≤ n if 1 ≤ r < ∞. Cr , Proof. Let 

1 u(y) = max 0, 1 − d y, B(x, r) r



⎧ ⎪ ⎨1, = 2− ⎪ ⎩ 0,

|y−x| r ,

y ∈ B(x, r), y ∈ B(x, 2r) \ B(x, r), y ∈ Rn \ B(x, 2r).

Then u ∈ Lip0 (B(x, 2r)) with 0 ≤ u(y) ≤ χB(x,2r) (y) for every y ∈ Rn and u is 1 1 r -Lipschitz. This implies u ∈ Ap (B(x, r)). Moreover, |∇u(y)| ≤ r for almost every n y ∈ R . Hence we obtain, for 0 < r ≤ 1,   p |u(y)| dy + |∇u(y)|p dy ≤ (1 + r −p )|B(x, 2r)| Capp (B(x, r)) ≤ B(x,2r)

B(x,2r)

≤ (r −p + r −p )|B(x, 2r)| = 2r −p |B(x, 2r)| = C(n)r n−p . For r ≥ 1 we apply above the estimate 1 + r −p ≤ 2, and the claim follows.



For 1 < p < n, there is a corresponding estimate for the capacity of a ball from below. Lemma 5.7. Let 1 < p < n and let B(x, r) ⊂ Rn be a ball. There exists a constant C = C(n, p) such that  Cr n−p , if 0 < r ≤ 1, Capp (B(x, r)) ≥ n if 1 ≤ r < ∞. Cr ,

5.2. ESTIMATES FOR CAPACITY

89

Proof. In the case 1 ≤ r < ∞ the claim follows from Lemma 5.5 since Capp (B(x, r)) ≥ |B(x, r)| = C(n)r n . Assume then that r > 0 and let u ∈ Ap (B(x, r)). Since u(y) ≥ 1 for almost every y ∈ B(x, r), by Theorem 3.6 we obtain  n−p  n−p   np np np p−n np n−p np n−p |u(y)| dy ≤ |B(x, r)| |u(y)| dy 1≤  B(x,r)

≤ C(n, p)r

p−n p

 Rn

|∇u(y)|p dy

Rn

 p1 .

This implies C(n, p)r n−p ≤ ∇upLp (Rn ) ≤ upW 1,p (Rn ) , and consequently Capp (B(x, r)) ≥ C(n, p)r n−p for every r > 0.



Remark 5.8. Let x ∈ Rn . For 1 < p < n and 0 < r ≤ 1, Lemma 5.6 implies Capp ({x}) ≤ Capp (B(x, r)) ≤ C(n)r n−p . On the other hand, for p = n and 0 < r ≤ 12 , we apply the admissible function ⎧ −1 ⎪ 1 1 ⎪ log |y−x| , y ∈ B(x, 1) \ B(x, r), ⎨ log r u(y) = 1, y ∈ B(x, r), ⎪ ⎪ ⎩ 0, y ∈ Rn \ B(x, 1), and obtain

 1 1−n Capn ({x}) ≤ Capn (B(x, r)) ≤ C(n) log . r By letting r → 0, we conclude that Capp ({x}) = 0 for 1 < p ≤ n. Countable subadditivity of capacity implies that all countable sets E ⊂ Rn have zero capacity for 1 < p ≤ n. All points have positive capacity for p > n, and thus there are no nontrivial sets of capacity zero when p > n. In practice this means that capacity is useful only for 1 < p ≤ n. Lemma 5.9. Let n < p < ∞. For every x ∈ Rn , we have Capp ({x}) > 0. Proof. Let z ∈ Rn . By Lemma 5.6 and monotonicity, Capp ({z}) ≤ Capp (B(z, 1)) ≤ C(n) < ∞. Let u ∈ Ap ({z}). There exists 0 < r ≤ 1 such that u(x) ≥ 1 for almost every x ∈ B(z, r). By redefining u on a set of measure zero, if necessary, we may assume that u(x) ≥ 1 for every x ∈ B(z, r). Let 

 ψ(x) = max 0, 1 − d x, B(z, 1) for every x ∈ Rn . Then ψ ∈ Lip0 (B(z, 2)), 0 ≤ ψ ≤ 1, ψ(x) = 1 for every x ∈ B(z, r), and |∇ψ(x)| ≤ 1 for almost every x ∈ B(z, 2). Let v(x) = ψ(x)u(x) for every x ∈ Rn . By (2.17), we have v ∈ W 1,p (Rn ). By Morrey’s inequality, see Theorem 3.23, there exists a set E ⊂ Rn such that |E| = 0 and n

|v(x) − v(y)| ≤ C(n, p)∇vLp (Rn ) |x − y|1− p

90

5. CAPACITIES AND FINE PROPERTIES OF SOBOLEV FUNCTIONS

for every x, y ∈ Rn \ E. Choose x ∈ B(z, r) and y ∈ B(z, 4) \ B(z, 2) such that x, y ∈ / E. Then v(x) ≥ 1 and v(y) = 0, and thus  |∇v(x)|p dx = ∇vpLp (Rn ) ≥ C(n, p)|x − y|n−p |v(x) − v(y)|p B(z,2)

≥ C(n, p)5n−p > 0. On the other hand, we have  

p p |∇ψ(x)u(x)| + |ψ(x)∇u(x)| dx |∇v(x)| dx ≤ B(z,2)

B(z,2)





 |∇ψ(x)| |u(x)| dx +

≤2

p

p

B(z,2)

 ≤ 2p

|ψ(x)| |∇u(x)| dx

p

B(z,2)

 |u(x)|p dx +

B(z,2)

p

p



|∇u(x)|p dx B(z,2)

≤ 2p upW 1,p (Rn ) . This shows that there exists a constant C = C(n, p) such that upW 1,p (Rn ) ≥ C > 0 for every u ∈ Ap ({z}), and consequently Capp ({z}) ≥ C > 0.  5.3. Quasicontinuity and fine properties of capacity We discuss properties of Sobolev functions outside a set of small or vanishing capacity. Definition 5.10. Let 1 < p < ∞ and G ⊂ Rn . We say that a property holds p-quasieverywhere in G, or for p-quasievery x ∈ G, if there is a set E ⊂ Rn with Capp (E) = 0 such that the property holds for every x ∈ G \ E. If G = Rn , we say that the property holds p-quasieverywhere. We consider a Sobolev space version of the result which asserts that for every Cauchy sequence in Lp (Rn ) there is a subsequence that converges pointwise almost everywhere. The part concerning the uniform convergence is a Sobolev space version of Egorov’s theorem. Theorem 5.11. Let 1 < p < ∞. Assume that ui ∈ W 1,p (Rn ) ∩ C(Rn ), for every i ∈ N, and that (ui )i∈N is a Cauchy sequence in W 1,p (Rn ). There is a subsequence of (ui )i∈N that converges pointwise p-quasieverywhere in Rn . Moreover, the convergence is uniform outside a set of arbitrarily small Sobolev p-capacity. Proof. By induction we obtain a subsequence, which we still denote by (ui )i∈N , such that ∞

(5.3) 2ip ui − ui+1 pW 1,p (Rn ) < ∞. i=1

  Let Ei = x ∈ Rn : |ui (x) − ui+1 (x)| > 2−i for every i ∈ N. Since ui ∈ C(Rn ) for every i ∈ N, we have 2i |ui − ui+1 | ∈ Ap (Ei ) and thus Capp (Ei ) ≤ 2ip ui − ui+1 pW 1,p (Rn ) .

Let j ∈ N and Fj =

∞

i=j

Capp (Fj ) ≤

Ei . Using Theorem 5.3 (c), we obtain ∞

i=j

Capp (Ei ) ≤

∞

i=j

2ip ui − ui+1 pW 1,p (Rn ) .

5.3. QUASICONTINUITY AND FINE PROPERTIES OF CAPACITY

By monotonicity and (5.3), we have ∞  Capp

≤ lim inf Capp (Fj ) ≤ lim inf

Fj

j→∞

j=1

j→∞

∞

91

2ip ui − ui+1 pW 1,p (Rn ) = 0,

i=j

since the tail of a convergent series converges to zero.  ∞ Observe that (ui )i∈N converges pointwise in Rn \ j=1 Fj . Moreover, |um (x) − uk (x)| ≤

k−1

i=m

|ui (x) − ui+1 (x)| ≤

k−1

2−i ≤ 21−m

i=m

for every x ∈ R \ Fj and every k > m > j. Cauchy’s criterion for uniform convergence in Rn \Fj is satisfied, and thus (ui )i∈N convergences uniformly in Rn \Fj for each j ∈ N. Since limj→∞ Capp (Fj ) = 0, we conclude that the convergence is uniform outside a set of arbitrarily small Sobolev p-capacity.  n

Definition 5.12. Let 1 < p < ∞. A function u : Rn → [−∞, ∞] is pquasicontinuous in Rn if for every ε > 0 there is a set E ⊂ Rn such that Capp (E) < ε and the restriction of u to Rn \ E, denoted by u|Rn \E , is a continuous real-valued function. By outer regularity of capacity, see Lemma 5.4, we may assume that E is open and Lebesgue measurable in the definition above. Remark 5.13. Let 1 < p < ∞ and assume that u is p-quasicontinuous in Rn . If we redefine or modify u on a set of zero Sobolev p-capacity, the function remains p-quasicontinuous. Moreover, by definition

Capp {x ∈ Rn : |u(x)| = ∞} = 0. If v is p-quasicontinuous in Rn and ϕ : R → R is continuous, then it is possible to redefine u and v on a set of p-capacity zero in such a way that uv, u + v, and ϕ ◦ u are well-defined p-quasicontinuous functions in Rn . Every Sobolev function has a quasicontinuous representative. Theorem 5.14. Let 1 < p < ∞ and assume that u ∈ W 1,p (Rn ). There exists a p-quasicontinuous v ∈ W 1,p (Rn ) such that u = v almost everywhere in Rn . Proof. By Lemma 2.20, there are functions ui ∈ W 1,p (Rn ) ∩ C(Rn ), i ∈ N, such that ui → u in W 1,p (Rn ) and almost everywhere in Rn as i → ∞. From Theorem 5.11 we obtain a subsequence of (ui )i∈N that converges p-quasieverywhere to a function v : Rn → [−∞, ∞], and the convergence is uniform outside a set of arbitrarily small Sobolev p-capacity. In particular u = v almost everywhere in Rn , by Lemma 5.5, and so also v ∈ W 1,p (Rn ). Uniform convergence implies continuity of the limit function, and thus v is a continuous real-valued function outside a set of arbitrarily small Sobolev p-capacity. This implies that v is p-quasicontinuous.  Theorem 5.16 below shows that the quasicontinuous representative given by Theorem 5.14 is unique. In the proof of Theorem 5.16 we apply the following useful observation. Lemma 5.15. Let 1 < p < ∞. For every open set G ⊂ Rn and every E ⊂ Rn with |E| = 0, we have Capp (G) = Capp (G \ E).

92

5. CAPACITIES AND FINE PROPERTIES OF SOBOLEV FUNCTIONS

Proof. By monotonicity Capp (G) ≥ Capp (G \ E). To prove the reverse inequality, we may assume that Ap (G \ E) = ∅. Let ε > 0 and let u ∈ Ap (G \ E) with upW 1,p (Rn ) ≤ Capp (G \ E) + ε. There is an open set U ⊂ Rn such that G \ E ⊂ U and u(x) ≥ 1 for almost every x ∈ U . Then U ∪ G is open and G ⊂ U ∪ G. Since |E| = 0 and u ≥ 1 almost everywhere in U ∪ (G \ E), we have u ≥ 1 almost everywhere in U ∪ G. This implies u ∈ Ap (U ∪ G) and Capp (G) ≤ Capp (U ∪ G) ≤ upW 1,p (Rn ) ≤ Capp (G \ E) + ε. The claim follows by letting ε → 0.



Theorem 5.16. Let 1 < p < ∞. Assume that u and v are p-quasicontinuous functions in Rn and u = v almost everywhere. Then u = v holds p-quasieverywhere in Rn . Proof. Let ε > 0 and let V ⊂ Rn be an open set such that Capp (V ) < ε and the restrictions of u and v to Rn \ V are continuous. The set {x ∈ Rn \ V : u(x) = v(x)} is open in the relative topology of Rn \ V , that is, there exists an open set U ⊂ Rn with U \ V = U ∩ (Rn \ V ) = {x ∈ Rn \ V : u(x) = v(x)}. Moreover, {x ∈ Rn : u(x) = v(x)} ⊂ V ∪ {x ∈ Rn \ V : u(x) = v(x)} = U ∪ V, and, by assumption, we have |U \ V | = |{x ∈ Rn \ V : u(x) = v(x)}| = 0. Lemma 5.15, with G = U ∪ V and E = U \ V , implies

Capp {x ∈ Rn : u(x) = v(x)} ≤ Capp (U ∪ V ) = Capp (V ) < ε. The claim follows by letting ε → 0.



Remark 5.17. Let 1 < p < ∞. If u and v are p-quasicontinuous in R and u ≥ v almost everywhere in Rn , then max{v − u, 0} = 0 almost everywhere in Rn and max{v − u, 0} is p-quasicontinuous; see also Remark 5.13. Theorem 5.16 implies that max{v − u, 0} = 0 p-quasieverywhere in Rn , and consequently u ≥ v p-quasieverywhere in Rn . n

The reasoning above can be generalized to open subsets of Rn . Lemma 5.18. Let 1 < p < ∞ and let G ⊂ Rn be an open set. Assume that u and v are p-quasicontinuous functions in Rn and u ≥ v almost everywhere in G. Then u ≥ v holds p-quasieverywhere in G. Proof. If G = Rn , the claim follows from Remark 5.17, and thus we may assume that G  Rn . Let  1 Gk = x ∈ G : d(x, Rn \ G) > k for k ∈ N, and define   uk (x) = u(x) min 1, kd(x, Rn \ G)

5.3. QUASICONTINUITY AND FINE PROPERTIES OF CAPACITY

93

and

  vk (x) = v(x) min 1, kd(x, Rn \ G) ∞ for every k ∈ N and x ∈ Rn . Then G = k=1 Gk and uk (x) = u(x) and vk (x) = v(x) for every k ∈ N and x ∈ Gk . Let k ∈ N. Since the functions uk and vk are p-quasicontinuous in Rn and uk ≥ vk almost everywhere in Rn , by Theorem 5.16 uk ≥ vk holds p-quasieverywhere in Rn . This implies Capp (Ek ) = 0 for every k ∈ N, where Ek = {x ∈ Rn : uk (x) < vk (x)}. It follows that ∞  ∞ 



Ek ≤ Capp (Ek ) = 0, Capp {x ∈ G : u(x) < v(x)} ≤ Capp k=1

k=1

and hence u ≥ v p-quasieverywhere in G, as claimed.



Sobolev capacity can be characterized in terms of quasicontinuous functions. Definition 5.19. Let 1 < p < ∞. For E ⊂ Rn , we define Cap∗p (E) =

inf

u∈A∗ p (E)

upW 1,p (Rn ) ,

where A∗p (E) consists of u ∈ W 1,p (Rn ) such that u is p-quasicontinuous and u ≥ 1 p-quasieverywhere in E. Here we use the convention Cap∗p (E) = ∞ if A∗p (E) = ∅. Theorem 5.20. Let 1 < p < ∞. For E ⊂ Rn we have Cap∗p (E) = Capp (E). Proof. For inequality Cap∗p (E) ≤ Capp (E) we may assume that Capp (E) < ∞. Let u ∈ Ap (E). Then u ∈ W 1,p (Rn ) and u ≥ 1 almost everywhere in a neighborhood G of E. Let v be the p-quasicontinuous representative of u, given by Theorem 5.14. Since u = v almost everywhere, we have v ≥ 1 almost everywhere in G. Lemma 5.18 implies that v ≥ 1 p-quasieverywhere in G, and hence v ≥ 1 p-quasieverywhere in E. Thus v ∈ A∗p (E) and it follows that Cap∗p (E) ≤ Capp (E). For the reverse inequality, we may assume that Cap∗p (E) < ∞. Let v ∈ A∗p (E). By considering the p-quasicontinuous function min{|v|, 1} and arguing as in the proof of Lemma 5.2, we may assume that 0 ≤ v ≤ 1 everywhere and v = 1 p-quasieverywhere in E. Let 0 < ε < 1 and let V be an open set such that Capp (V ) < εp , v = 1 in E \ V , and the restriction v|Rn \V is continuous. By relative topology, there exists an open set U ⊂ Rn satisfying   x ∈ Rn \ V : v(x) > 1 − ε = U ∩ (Rn \ V ) = U \ V. Since v = 1 in E \ V , we have E \ V ⊂ U \ V . Let w ∈ Ap (V ) with wW 1,p (Rn ) < ε, and let v(x) + w(x) 1−ε for every x ∈ Rn . Then u ≥ 1 almost everywhere in (U \ V ) ∪ V = U ∪ V , which is an open neighborhood of E. This implies u ∈ Ap (E) and u(x) =

1

Capp (E) p ≤ uW 1,p (Rn ) ≤

1 vW 1,p (Rn ) + wW 1,p (Rn ) 1−ε

1 vW 1,p (Rn ) + ε. 1−ε Since ε > 0 and v ∈ A∗p (E) were arbitrary, and v is independent of ε, we obtain  Capp (E) ≤ Cap∗p (E). ≤

94

5. CAPACITIES AND FINE PROPERTIES OF SOBOLEV FUNCTIONS

5.4. Lebesgue points of Sobolev functions The Lebesgue differentiation theorem, see Theorem 1.21, states that a locally integrable function has Lebesgue points almost everywhere. In this section we consider a capacitary version of this result for Sobolev functions, see Theorem 5.23. In the proof we apply the following capacitary weak type estimate for Sobolev functions. Theorem 5.21. Let 1 < p < ∞ and assume that u ∈ W 1,p (Rn ). There exists a constant C = C(n, p) such that

C Capp {x ∈ Rn : M u(x) > t} ≤ p upW 1,p (Rn ) t for every t > 0. Proof. Let t > 0 and let Et = {x ∈ Rn : M u(x) > t}. The set Et is open by Lemma 1.3. Theorem 4.21 implies M u ∈ W 1,p (Rn ), and thus Mu ∈ Ap (Et ). t Since the maximal operator M is bounded on W 1,p (Rn ), see inequality (4.15), we conclude that Capp (Et ) ≤

1 C(n, p) M upW 1,p (Rn ) ≤ upW 1,p (Rn ) . p t tp



Also the following measure theoretic result will be needed in the proof of Theorem 5.23. Roughly speaking, this lemma asserts that the set where an Lp function blows up rapidly is of capacity zero. Lemma 5.22. Let 1 < p < ∞. Assume that f ∈ Lp (Rn ) and define   E = x ∈ Rn : lim sup r p  |f (y)|p dy > 0 . r→0

B(x,r)

Then Capp (E) = 0. Proof. Let

 Eε =

x ∈ Rn : lim sup r p  r→0

 |f (y)|p dy > ε , B(x,r)

for ε > 0 We show that Capp (Eε ) = 0 for every ε > 0, and the claim follows by subadditivity of capacity. Let ε > 0 and 0 < δ < 15 . For every x ∈ Eε there exists rx , with 0 < rx ≤ δ, such that rxp 

|f (y)|p dy > ε.

B(x,rx )

Then B = {B(x, rx ) : x ∈ Eε } is a cover of the set Eε . By Lemma 1.13, there exists a countable collection of pairwise disjoint balls B(xi , ri ) ∈ B, i ∈ N, such ∞ that Eε ⊂ i=1 B(xi , 5ri ). Using countable subadditivity of capacity, the estimate in Lemma 5.6, the choice of the radii ri , and the pairwise disjointness of the balls

5.4. LEBESGUE POINTS OF SOBOLEV FUNCTIONS

95

B(xi , ri ), we obtain Capp (Eε ) ≤

∞

Capp (B(xi , 5ri )) ≤ C(n)

i=1 ∞

C(n) ≤ ε i=1



∞

rin−p

i=1

C(n) |f (y)| dy = ε B(xi ,ri )



p

|f (y)|p dy.

∞

i=1 B(xi ,ri )

The pairwise disjointness of the balls B(xi , ri ), i ∈ N, also gives ∞  ∞ ∞ 

rip |B(xi , ri )| ≤ |f (y)|p dy B(xi , ri ) = ε B(x ,r ) i i i=1 i=1 i=1  δp δ→0 ≤ |f (y)|p dy −−−→ 0. ε Rn By absolute continuity of integral, we obtain  C(n) δ→0 Capp (Eε ) ≤ |f (y)|p dy −−−→ 0. ∞ ε i=1 B(xi ,ri ) This implies Capp (Eε ) = 0 for every ε > 0, and the claim follows.



We are ready to state and prove a refined version of the Lebesgue differentiation theorem for Sobolev functions. This result gives an expression for the quasicontinuous representative in Theorem 5.14. Theorem 5.23. Let 1 < p < ∞ and assume that u ∈ W 1,p (Rn ). There exists a set E ⊂ Rn with Capp (E) = 0 such that the limit (5.4)

lim 

r→0

u(y) dy = u∗ (x)

B(x,r)

exists for every x ∈ Rn \ E and any extension of u∗ : Rn \ E → R to entire Rn is a p-quasicontinuous representative of u. Moreover, there exists a set E  ⊃ E such that Capp (E  ) = 0 and (5.5)

lim 

r→0

|u(y) − u∗ (x)| dy = 0

B(x,r)

for every x ∈ Rn \ E  . Proof. By Lemma 2.20, there exist functions ui ∈ C(Rn ) ∩ W 1,p (Rn ) such that u − ui pW 1,p (Rn ) ≤ 2−i(p+1)   for every i ∈ N. Let Fi = x ∈ Rn : M (u − ui )(x) > 2−i for i ∈ N. Theorem 5.21 gives (5.6)

Capp (Fi ) ≤ C(n, p)2ip u − ui pW 1,p (Rn ) ≤ C(n, p)2−i ,

for every i ∈ N. Let i ∈ N and note that |ui (x) − uB(x,r) | ≤ 

|ui (x) − u(y)| dy

B(x,r)

≤

B(x,r)

|ui (x) − ui (y)| dy + 

|ui (y) − u(y)| dy, B(x,r)

96

5. CAPACITIES AND FINE PROPERTIES OF SOBOLEV FUNCTIONS

for every x ∈ Rn . Since ui is continuous, we have |ui (x) − ui (y)| dy = 0,

lim sup  r→0

B(x,r)

and thus (5.7)

lim sup |ui (x) − uB(x,r) | ≤ lim sup  r→0

r→0

|ui (y) − u(y)| dy

B(x,r)

≤ M (ui − u)(x) ≤ 2−i for every x ∈ Rn \ Fi .  Let k ∈ N and define Ek = ∞ m=k Fm . By countable subadditivity of capacity and inequality (5.6), we obtain (5.8)

Capp (Ek ) ≤

∞

∞

Capp (Fm ) ≤ C(n, p)

m=k

2−m = C(n, p)21−k .

m=k

If i > j ≥ k and x ∈ R \ Ek , then (5.7) implies n

|ui (x) − uj (x)| ≤ lim sup |ui (x) − uB(x,r) | + lim sup |uB(x,r) − uj (x)| r→0 −i

≤2

r→0

−j

+2

≤2

1−j

.

Hence (ui )i∈N satisfies the Cauchy criterion for uniform convergence in Rn \ Ek . Since the functions ui are continuous, the sequence (ui )i∈N converges uniformly in Rn \ Ek to a continuous function vk : Rn \ Ek → R. Furthermore, if x ∈ Rn \ Ek and i ≥ k, then lim sup |vk (x) − uB(x,r) | ≤ |vk (x) − ui (x)| + lim sup |ui (x) − uB(x,r) | r→0

r→0

≤ |vk (x) − ui (x)| + 2−i . The right-hand side of the previous inequality tends to zero as i → ∞. Thus lim sup |vk (x) − uB(x,r) | = 0, r→0

and consequently vk (x) = lim  r→0

u(y) dy = u∗ (x)

B(x,r)

for every x ∈  Rn \ E k . n Let E = ∞ k=1 Ek . Then the limit in (5.4) exists for every x ∈ R \ E, and by monotonicity of capacity and (5.8), we have Capp (E) ≤ lim inf Capp (Ek ) ≤ C(n, p) lim inf 21−k = 0. k→∞

k→∞

Let ε > 0 and choose k ∈ N to be so large that Capp (Ek ) < 2ε . Since u∗ |Rn \Ek = vk is continuous, any extension of u∗ : Rn \ E → R to entire Rn is p-quasicontinuous in Rn . Lemma 5.5 implies |E| = 0 and by Theorem 1.21 we have u = u∗ almost everywhere in Rn \ E. Thus any extension of u∗ : Rn \ E is a p-quasicontinuous representative of u. Finally, let   n p p |∇u(y)| dy > 0 . F = x ∈ R : lim sup r  r→0

B(x,r)

5.5. SOBOLEV SPACES WITH ZERO BOUNDARY VALUES REVISITED

97

Lemma 5.22 shows that Capp (F ) = 0. The p-Poincar´e inequality, see Theorem 3.14, implies  p1  p lim sup  |u(y) − uB(x,r) | dy r→0

(5.9)

B(x,r)



≤ C(n, p) lim sup r  r→0

|∇u(y)| dy p

 p1 =0

B(x,r)

for every x ∈ Rn \ F . Choose E  = E ∪ F . Then Capp (E  ) ≤ Capp (E) + Capp (F ) = 0. Since E ⊂ E  , the limit in (5.4) exists for every x ∈ Rn \ E  ⊂ Rn \ E. By (5.9), we have  p1  ∗ ∗ p |u(y) − u (x)| dy ≤ lim sup  |u(y) − u (x)| dy lim sup  r→0

B(x,r)

r→0



≤ lim sup  r→0

|u(y) − uB(x,r) |p dy

B(x,r)

+ lim sup |uB(x,r) − u∗ (x)| = 0, r→0

for every x ∈ R \ E ⊂ R \ F , and thus (5.5) holds for every x ∈ Rn \ E  . n



B(x,r)

 p1

n



Remark 5.24. Let n < p < ∞. Lemma 5.9 implies that Capp ({x}) > 0 for every x ∈ Rn , and Theorem 5.23 holds for every u ∈ W 1,p (Rn ) with E = ∅. Consequently, every function u ∈ W 1,p (Rn ) has a continuous representative and every point x ∈ Rn is a Lebesgue point of u. Recall that this fact follows more directly from Morrey’s inequality, see Theorem 3.23, which is also applied in the proof of Lemma 5.9. Remark 5.25. Let 1 < p < ∞ and assume that u ∈ W 1,p (Rn ). Let 1 ≤ q ≤ for 1 < p < n, and 1 ≤ q < ∞ for n ≤ p < ∞. By applying the (q, p)Poincar´e inequality, see Theorem 3.14, instead of the p-Poincar´e inequality in (5.9), we conclude that there exists a set E ⊂ Rn such that Capp (E) = 0 and  q1  ∗ q lim  |u(y) − u (x)| dy =0 np n−p

r→0

B(x,r)

for every x ∈ R \ E. n

5.5. Sobolev spaces with zero boundary values revisited We return to Sobolev spaces with zero boundary values, see Definition 2.27. Applying pointwise properties of Sobolev functions, we obtain characterizations for these spaces. The first result is a version of Theorem 5.14. Theorem 5.26. Let 1 < p < ∞, let Ω ⊂ Rn be an open set, and assume that u ∈ W01,p (Ω). There exists a p-quasicontinuous function v ∈ W 1,p (Rn ) such that v = u almost everywhere in Ω and v = 0 holds p-quasieverywhere in Rn \ Ω. Proof. Since u ∈ W01,p (Ω), there exist ui ∈ C0∞ (Ω), i ∈ N, such that ui → u in W 1,p (Ω) as i → ∞. By taking a subsequence, if necessary, we may assume that ui → u pointwise almost everywhere in Ω, as i → ∞. Since (ui )i∈N is a Cauchy sequence in W 1,p (Rn ), by Theorem 5.11 it has a subsequence that converges pointwise p-quasieverywhere in Rn to a function v. By Lemma 5.5 and Theorem 2.40,

98

5. CAPACITIES AND FINE PROPERTIES OF SOBOLEV FUNCTIONS

we have v ∈ W 1,p (Rn ) and v = u almost everywhere in Ω. Moreover, the convergence is uniform outside a set of arbitrarily small Sobolev p-capacity and, as in Theorem 5.14, the limit function v is p-quasicontinuous. Observe that v = 0  p-quasieverywhere in Rn \ Ω since ui = 0 in Rn \ Ω for every i ∈ N. Next we consider a converse of Theorem 5.26. Theorem 5.27. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. Assume that u ∈ W 1,p (Rn ) is p-quasicontinuous and u = 0 holds p-quasieverywhere in Rn \ Ω. Then u ∈ W01,p (Ω). Proof. Since W 1,p (Rn ) = W01,p (Rn ), see Remark 2.22, we may assume that Ω  Rn . By Remark 2.28, see also Lemma 4.10, and the fact that W01,p (Ω) is a closed subspace of W 1,p (Ω), it suffices to show that u can be approximated in W 1,p (Rn ) by functions having a compact support in Ω. If we can construct such a sequence for u+ = max{u, 0}, then we can do it for u− = − min{u, 0} as well, and we obtain the result for u = u+ − u− . Thus we may assume that u ≥ 0 everywhere in Rn . Moreover, we may also assume that u has a compact support in Rn , and, by considering the truncations min{u, t}, for t > 0, that u is bounded. Let δ > 0 and let G ⊂ Rn be an open set such that Capp (G ) < δ and the restriction of u to Rn \ G is continuous. Define E = {x ∈ Rn \ Ω : u(x) = 0}. By assumption Capp (E) = 0. Since Capp (G ∪ E) ≤ Capp (G ) + Capp (E) < δ, there exists a function v = vδ ∈ Ap (G ∪ E) with 0 ≤ v ≤ 1 almost everywhere, v = 1 almost everywhere in a neighborhood of G ∪ E and vpW 1,p (Rn ) < δ, see Lemma 5.2. By modifying v on a set of measure zero, if necessary, we may assume that v = 1 in an open set G containing G ∪ E. Let 0 < ε < 1 and define uε (x) = max{u(x) − ε, 0}, for every x ∈ R . Let x ∈ (Rn \ Ω) \ G. Since u(x) = 0 and the restriction of u to Rn \ G is continuous, there exists rx > 0 such that uε = 0 in B(x, rx ) \ G. Thus (1 − v)uε = 0 in B(x, rx ) for every x ∈ (Rn \ Ω) \ G, and so (1 − v)uε is zero in the neighborhood  G∪ B(x, rx ) n

x∈(Rn \Ω)\G

of R \ Ω. Since u, and therefore also (1 − v)uε , are compactly supported in Rn , we conclude that (1 − v)uε is compactly supported in Ω for every 0 < ε < 1. Moreover, since both (1 − v) and u are bounded, we have (1 − v)uε ∈ W 1,p (Rn ) and (1 − v)uε ∈ W01,p (Ω) by Lemma 4.10. To conclude that u ∈ W01,p (Ω), it suffices to show that the functions (1 − v)uε converge to u in W 1,p (Rn ). Observe that n

(5.10)

u − (1 − v)uε W 1,p (Rn ) ≤ u − uε W 1,p (Rn ) + vuε W 1,p (Rn ) .

Hence it is enough to prove that each of the terms on the right hand side can be made arbitrarily small by choosing δ and ε to be small enough; recall here that v = vδ depends on δ. Since  u − ε in {x ∈ Rn : u(x) ≥ ε}, uε = 0 in {x ∈ Rn : u(x) ≤ ε},

5.5. SOBOLEV SPACES WITH ZERO BOUNDARY VALUES REVISITED

by Theorem 2.25 (c) we have  ∇u, ∇uε = 0, Thus

99

a.e. in {x ∈ Rn : u(x) ≥ ε}, a.e. in {x ∈ Rn : u(x) ≤ ε}.

  ∇u − ∇uε Lp (Rn ) ≤ χ{0≤u≤ε} ∇uLp (Rn ) .

By the dominated convergence theorem and Theorem 2.25 (b), we obtain   p1    p   χ χ lim {0≤u≤ε} ∇u Lp (Rn ) = lim {0≤u≤ε} (x)|∇u(x)| dx ε→0

ε→0

 =

Rn

lim χ{0≤u≤ε} (x)|∇u(x)|p dx

Rn ε→0

 =

Rn

χ{u=0} (x)|∇u(x)|p dx

 p1

 p1 = 0.

Here χ{0≤u≤ε} |∇u|p ≤ |∇u|p for every 0 < ε < 1, and so |∇u|p ∈ L1 (Rn ) may be used as an integrable majorant. Since 0 ≤ u − uε ≤ ε and supp(u − uε ) ⊂ supp u, we have u − uε W 1,p (Rn ) ≤ u − uε Lp (Rn ) + ∇u − ∇uε Lp (Rn )      p n + χ ≤ εχ ∇u p supp u L (R )

{0≤u≤ε}

L (Rn )

,

and it follows that lim u − uε W 1,p (Rn ) = 0.

(5.11)

ε→0

Then we consider the term vuε W 1,p (Rn ) in (5.10). Since |uε (x)| ≤ |u(x)| ≤ uL∞ (Rn ) for almost every x ∈ Rn , we have vuε Lp (Rn ) ≤ uL∞ (Rn ) vLp (Rn ) ≤ uL∞ (Rn ) vW 1,p (Rn ) . On the other hand, |∇uε | ≤ |∇u| almost everywhere, and using (2.17) we obtain ∇(vuε )Lp (Rn ) ≤ uε ∇vLp (Rn ) + v∇uε Lp (Rn ) ≤ u∇vLp (Rn ) + v∇uε Lp (Rn ) ≤ uL∞ (Rn ) ∇vLp (Rn ) + v∇uLp (Rn ) ≤ uL∞ (Rn ) vW 1,p (Rn ) + v∇uLp (Rn ) . Recalling that

vpW 1,p (Rn )

< δ, we arrive at the estimate

vuε W 1,p (Rn ) ≤ vuε Lp (Rn ) + ∇(vuε )Lp (Rn ) (5.12)

≤ 2uL∞ (Rn ) vW 1,p (Rn ) + v∇uLp (Rn ) 1

≤ 2δ p uL∞ (Rn ) + v∇uLp (Rn ) . Let δi > 0, i ∈ N, be such that δi → 0 as i → ∞, and let vi = vδi be the corresponding functions as above, with vi pW 1,p (Rn ) < δi . Then vi → 0 in Lp (Rn ).

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5. CAPACITIES AND FINE PROPERTIES OF SOBOLEV FUNCTIONS

By passing to a subsequence, we may also assume that vi → 0 almost everywhere in Rn as i → ∞. By the dominated convergence theorem, we have   p1  |vi (x)|p |∇u(x)|p dx lim sup vi ∇uLp (Rn ) ≤ lim i→∞

i→∞



Rn

lim |vi (x)| |∇u(x)| dx p

=

Rn i→∞

p

 p1 = 0.

Here |vi |p |∇u|p ≤ |∇u|p , and so |∇u|p ∈ L1 (Rn ) may be used as an integrable majorant. Hence we obtain from (5.12), with with v = vi , that  1  (5.13) lim sup vi uε W 1,p (Rn ) ≤ lim sup 2δip uL∞ (Rn ) + vi ∇uLp (Rn ) = 0. i→∞

i→∞

By combining (5.10) and (5.11), with v = vi , and (5.13), we conclude that u − (1 − vi )uε W 1,p (Rn ) −→ 0 as ε → 0 and i → ∞, and the claim u ∈ W01,p (Ω) follows.



Remark 5.28. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. If u ∈ W 1,p (Rn ) is continuous and zero everywhere in Rn \ Ω, then u ∈ W01,p (Ω) by Theorem 5.27. By combining Theorem 5.26 and Theorem 5.27, we obtain a Havin and Bagby type characterization of Sobolev spaces with zero boundary values on an arbitrary open set. Corollary 5.29. Let 1 < p < ∞, let Ω ⊂ Rn be an open set, and assume that u ∈ W 1,p (Ω). Then u ∈ W01,p (Ω) if and only if there exists a p-quasicontinuous function v ∈ W 1,p (Rn ) such that v = u almost everywhere in Ω and v = 0 holds p-quasieverywhere in Rn \ Ω. It is also possible to characterize Sobolev spaces with zero boundary values in terms of Lebesgue points of Sobolev functions. Theorem 5.30. Let 1 < p < ∞, let Ω ⊂ Rn be an open set, and assume that u ∈ W 1,p (Ω). Then u ∈ W01,p (Ω) if and only if there exists v ∈ W 1,p (Rn ) such that u = v almost everywhere in Ω and lim 

r→0

v(y) dy = 0

B(x,r)

for p-quasievery x ∈ Rn \ Ω. Proof. Assume first that u ∈ W01,p (Ω). By applying Theorem 5.26 to u, we obtain a p-quasicontinuous function v ∈ W 1,p (Rn ) satisfying v = u almost everywhere in Ω and v = 0 p-quasieverywhere in Rn \ Ω. On the other hand, Theorem 5.23 shows that the limit lim 

r→0

v(y) dy = v ∗ (x)

B(x,r)

exists for p-quasievery x ∈ Rn and that the zero extension v ∗ : Rn → R is a pquasicontinuous representative of v. Since v ∗ = v almost everywhere in Rn and

5.6. VARIATIONAL CAPACITY

101

both v ∗ and v are p-quasicontinuous, Theorem 5.16 implies that v ∗ = v holds p-quasieverywhere in Rn . From this we conclude that lim 

r→0

v(y) dy = v ∗ (x) = v(x) = 0

B(x,r)

for p-quasievery x ∈ Rn \ Ω, as desired. Conversely, let u ∈ W 1,p (Ω) and v ∈ W 1,p (Rn ) be such that u = v almost everywhere in Ω and (5.14)

lim 

r→0

v(y) dy = 0

B(x,r)

for p-quasievery x ∈ Rn \ Ω. As above, by Theorem 5.23 the limit (5.15)

v ∗ (x) = lim  r→0

v(y) dy

B(x,r)

exists for p-quasievery x ∈ Rn and the zero extension v ∗ : Rn → R is a p-quasicontinuous representative of v. Hence v ∗ ∈ W 1,p (Rn ). From (5.14) and (5.15) we obtain v ∗ (x) = 0 for p-quasievery x ∈ Rn \ Ω. Since v ∗ = v = u almost everywhere in Ω, Corollary 5.29 implies that u ∈ W01,p (Ω).  Example 5.31. Let Ω = B(0, 1) \ {0} and consider u : Ω → R, u(x) = 1 − |x|, for every x ∈ Ω. Then u ∈ W01,p (Ω) for 1 < p ≤ n and u ∈ / W01,p (Ω) for p > n. This is a consequence of Remark 5.8, Lemma 5.9 and Theorem 5.30. Hence, a function that belongs to a Sobolev space with zero boundary values does not have to be zero at every Lebesgue point in the complement. 5.6. Variational capacity Variational capacity is defined relative to an open subset of Rn with admissible functions having zero boundary values. Moreover, only the integral of the gradient is taken into account instead of the full Sobolev norm. This notion of capacity will be useful, for example, in connection with the capacity density condition in Definition 6.17. Definition 5.32. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. The variational p-capacity of a subset E  Ω is  capp (E, Ω) = inf |∇u(x)|p dx, Ω

where the infimum is taken over all u ∈ a neighborhood of E.

W01,p (Ω)

with u ≥ 1 almost everywhere in

Let Ω ⊂ Rn be an open set and let E  Ω. Since d(E, ∂Ω) > 0, there exists a neighborhood G of E with G  Ω. Then u : Rn → R,   d(x, G) u(x) = max 0, 1 − , d(G, Ωc ) satisfies u = 1 in G and u ∈ Lip0 (Ω) ⊂ W01,p (Ω), see Lemma 2.29. It follows that u is an admissible test function for E  Ω and thus capp (E, Ω) < ∞. Variational capacity can be defined for all E ⊂ Ω, for instance, by using the characterization in Theorem 5.46 below, but then capp (E, Ω) = ∞ may occur.

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5. CAPACITIES AND FINE PROPERTIES OF SOBOLEV FUNCTIONS

Remark 5.33. As in Lemma 5.2, we may assume 0 ≤ u ≤ 1 in Definition 5.32. This follows by considering the function v = min{|u|, 1} since |∇v| ≤ |∇u| almost everywhere and v ∈ W01,p (Ω) by Corollary 5.29. Variational capacity has similar basic properties as the Sobolev capacity. Theorem 5.34. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. (a) capp (∅, Ω) = 0. (b) For E1 ⊂ E2  Ω, we have capp (E1 , Ω) ≤ capp (E2 , Ω). (c) For E  Ω ⊂ Ω ⊂ Rn , where Ω is an open set, we have capp (E, Ω ) ≤ capp (E, Ω). (d) For unbounded Ω and E  Ω, we have capp (E, Ω) = lim capp (E, Ω ∩ B(0, j)). ∞

j→∞

(e) For bounded Ω and i=1 Ei  Ω, we have ∞  ∞ 

Ei , Ω ≤ capp (Ei , Ω). capp i=1

i=1

Proof. Properties (a) and (b) are immediate from the definition. Property (c) follows easily from Theorem 2.31. For the proof of (d), let u ∈ W01,p (Ω) with u ≥ 1 almost everywhere in a neighborhood of E. Let ϕ ∈ C0∞ (B(0, 1)) be such that ϕ = 1 in B(0, 12 ) and let vj (x) = u(x)ϕ( xj ) for every x ∈ Ω. Then vj ∈ W01,p (Ω ∩ B(0, j)) and vj ≥ 1 almost everywhere in a neighborhood of E for j large enough. Observe also that



∇vj (x) = ∇u(x)ϕ xj + 1j u(x)∇ϕ xj for almost every x ∈ Ω. By the dominated convergence theorem   lim sup capp (E, Ω ∩ B(0, j)) ≤ lim sup |∇vj (x)|p dx = |∇u(x)|p dx. j→∞

j→∞

Ω∩B(0,j)

Ω

Inequality lim sup capp (E, Ω ∩ B(0, j)) ≤ capp (E, Ω) j→∞

follows by taking infimum over all functions u as above. The reverse inequality capp (E, Ω) ≤ lim inf capp (E, Ω ∩ B(0, j)) j→∞

is a consequence of (c). This proves (d). The proof of the subadditivity in (e) is similar to the proof of the corresponding property for the Sobolev capacity in Theorem 5.3 (c). We may assume that ∞ 1,p i=1 capp (Ei , Ω) < ∞. Let ε > 0 and let ui ∈ W0 (Ω) be an admissible test function for Ei , with 0 ≤ ui ≤ 1 almost everywhere in Ω and  (5.16) |∇ui (x)|p dx ≤ capp (Ei , Ω) + 2−i ε Ω

for every i ∈ N. Since ui ∈ W01,p (Ω) and Ω is bounded, we have by Corollary 3.7,   |ui (x)|p dx ≤ C(n, p, Ω) |∇ui (x)|p dx Ω

Ω

5.6. VARIATIONAL CAPACITY

103

for every i ∈ N. Define v = supi∈N ui . It can be shown, as in the proof of Theorem 5.3, that v ∈ W 1,p (Ω). Moreover, it follows from Theorem 2.30 and Theo rem 2.40 (b) that v ∈ W01,p (Ω). There is a neighborhood Gof ∞ i=1 Ei such that ∞ v = 1 almost everywhere in G, and thus v is admissible for i=1 Ei . This implies ∞   ∞  ∞ 

capp Ei , Ω ≤ |∇v(x)|p dx ≤ |∇ui (x)|p dx ≤ capp (Ei , Ω) + ε, Ω

i=1

i=1

Ω

i=1

and we obtain property (e) for bounded Ω by letting ε → 0.



We have the following measure estimates for variational capacity. Lemma 5.35. Let 1 < p < ∞, let B(x, r) ⊂ Rn be a ball, and assume that E ⊂ B(x, r). There exist constants C1 = C(n, p) and C2 = C(n) such that C1 In particular,

|E| |B(x, r)| ≤ capp (E, B(x, 2r)) ≤ C2 . p r rp



C(n, p)r n−p ≤ capp B(x, r), B(x, 2r) ≤ C(n)r n−p .

Proof. First we prove the upper bound. Let r < ρ < 2r and define   1 d (y, B(x, ρ)) u(y) = max 0, 1 − 2r − ρ for every y ∈ Rn . Then u ∈ Lip0 (B(x, 2r)) ⊂ W01,p (B(x, 2r)) and u = 1 in the open neighborhood G = B(x, ρ) of E. Thus u is an admissible test function for 1 for almost every y ∈ B(x, 2r), we conclude capp (E, B(x, 2r)). Since |∇u(y)| ≤ 2r−ρ that  |B(x, 2r)| |B(x, r)| capp (E, B(x, 2r)) ≤ |∇u(y)|p dy ≤ = C(n) . p (2r − ρ) (2r − ρ)p B(x,2r) The upper bound follows by letting ρ → r from above. For the lower bound we apply the Poincar´e inequality in Theorem 3.17. Let u ∈ W01,p (B(x, 2r)) be such that u ≥ 1 almost everywhere in a neighborhood G ⊂ B(x, 2r) of E. By (3.11) with q = p, we have    p p p |E| ≤ |G| ≤ |u(y)| dy ≤ |u(y)| dy ≤ C(n, p)r |∇u(y)|p dy. G

B(x,2r)

B(x,2r)

The lower bound follows by taking infimum over all admissible test functions. The  final claim holds by choosing E = B(x, r). We review capacity estimates for concentric balls. For detailed formulas, see also Heinonen, Kilpel¨ainen and Martio [187, Example 2.12] Lemma 5.36. Let x ∈ Rn and assume that 0 < r ≤

R 2.

(a) For 1 < p < n, there exists a constant C = C(n, p) such that

C −1 r n−p ≤ capp B(x, r), B(x, R) ≤ Cr n−p . (b) For n < p < ∞, there exists a constant C = C(n, p) such that

C −1 Rn−p ≤ capp B(x, r), B(x, R) ≤ CRn−p .

104

5. CAPACITIES AND FINE PROPERTIES OF SOBOLEV FUNCTIONS

(c) There exists a constant C = C(n) such that  

R 1−n R 1−n C −1 log ≤ capn B(x, r), B(x, R) ≤ C log . r r Proof. First assume that 1 < p < n. The upper bound in (a) follows from Theorem 5.34 (c) and Lemma 5.35 since



capp B(x, r), B(x, R) ≤ capp B(x, r), B(x, 2r) ≤ C(n)r n−p . Here we applied the assumption 2r ≤ R. To prove the lower bound in (a), let u ∈ W01,p (B(x, R)) be such that u ≥ 1 almost everywhere in an open set G satisfying B(x, r) ⊂ G. Then G ⊂ B(x, R), and we obtain from the Sobolev inequality, see Remark 3.8, that r

n−p p

n−p

n−p

= C(n, p)|B(x, r)| np ≤ C(n, p)|G| np   n−p  np np n−p ≤ C(n, p) |u(y)| dy ≤ C(n, p) B(x,R)

|∇u(y)| dy p

 p1 .

B(x,R)

The claim follows by taking infimum over all functions u as above. Then we consider the case n < p < ∞. The upper bound in (b) follows from Theorem 5.34 (b) and Lemma 5.35 since



capp B(x, r), B(x, R) ≤ capp B(x, R2 ), B(x, R) ≤ C(n)Rn−p . For the lower bound we use Morrey’s inequality. Let u ∈ W01,p (B(x, R)) be an admissible test function for capp B(x, r), B(x, R) and extend u as zero so that u ∈ W 1,p (Rn ), see Theorem 2.31. By Theorem 3.23 we may assume that u is continuous, and thus u = 1 in B(x, r) and u = 0 in Rn \ B(x, R). Let z ∈ ∂B(x, R). Then u(x) = 1 and u(z) = 0, and by Corollary 3.21 and continuity of u we have  p1  p |∇u(y)| dy 1 = |u(x) − u(z)| ≤ C(n, p)2R  ≤ C(n, p)R

1− n p

B(x,2R)



|∇u(y)| dy p

 p1 .

B(x,R)

Hence

 Rn−p ≤ C(n, p)

|∇u(y)|p dy, B(x,R)

and the lower bound in (b) follows. The upper bound in (c) follows by testing capacity with the Lipschitz function ⎧ ⎪ 1, if y ∈ B(x, r), ⎪ ⎨ −1 R R u(y) = log r log |y−x| , if y ∈ B(x, R) \ B(x, r), ⎪ ⎪ ⎩ 0, if y ∈ / B(x, R), compare to Remark 5.8. We omit the details concerning the lower bound in (c), but this can be obtained with a direct computation, as in Heinonen, Kilpel¨ainen and Martio [187, Example 2.12], or using a telescoping argument, as in Bj¨ orn, Bj¨ orn and Lehrb¨ack [32, Proposition 7.1]. The latter approach also applies in more general metric spaces. 

5.7. CAPACITY AND HAUSDORFF CONTENT

105

Remark 5.37. Let 1 < p < n. Assume that Ω ⊂ Rn is an open set and B(x, 2r) ⊂ Ω. By replacing B(x, R) in the proof of Lemma 5.36 (a) by Ω, we obtain

C −1 r n−p ≤ capp B(x, r), Ω ≤ Cr n−p , where C = C(n, p). Next we make comparisons between the Sobolev p-capacity and the variational p-capacity. Theorem 5.38. Let 1 < p < ∞ and assume that E ⊂ B(x, r) ⊂ Rn . There exists a constant C = C(n, p) such that  1 1 p 1 + Capp (E). Cap (E) ≤ cap (E, B(x, 2r)) ≤ 2 (5.17) p p 1 + Cr p rp Proof. Let u ∈ W01,p (B(x, 2r)) be admissible for capp (E, B(x, 2r)). We extend u as zero outside B(x, 2r). Then u ∈ Ap (E), where Ap (E) is as in Definition 5.1. Using the p-Poincar´e inequality for Sobolev functions with zero boundary values, see Theorem 3.17, we obtain   p p |u(y)| dy + |∇u(y)|p dy Capp (E) ≤ uW 1,p (Rn ) = B(x,2r) B(x,2r)  p ≤ (C(n, p)r + 1) |∇u(y)|p dy. B(x,2r)

The first inequality in (5.17) follows by taking infimum over all such u. To obtain the second inequality in (5.17), we let u ∈ Ap (E) and r < ρ < 2r, and consider 

 1 d y, B(x, 2−1 (r + ρ)) , ψ(y) = max 0, 1 − 2r − ρ for every y ∈ Rn . Then 0 ≤ ψ(y) ≤ 1 and |∇ψ(y)| ≤

1 2r−ρ

for almost every

W01,p (B(x, 2r))

y ∈ B(x, 2r), and ψu ∈ is admissible for capp (E, B(x, 2r)). By Leibniz’s rule (2.17),  capp (E, B(x, 2r)) ≤ |∇(ψu)(y)|p dy B(x,2r) 

p ≤ |∇ψ(y)||u(y)| + ψ(y)|∇u(y)| dy B(x,2r)



≤ 2p

(2r − ρ)−p |u(y)|p dy +

B(x,2r)



≤ 2p (2r − ρ)−p + 1 upW 1,p (Rn ) .



 |∇u(y)|p dy B(x,2r)

Taking ρ → r and then infimum over u ∈ Ap (E) concludes the proof.



5.7. Capacity and Hausdorff content There are close connections between capacities and Hausdorff contents, see Definition 1.43. We begin with an upper bound. In general, upper bounds for capacity are easier to obtain than lower bounds. Theorem 5.39. Let 1 < p ≤ n and assume that E ⊂ B(x, r) ⊂ Rn . There exists a constant C = C(n) such that n−p (E). capp (E, B(x, 2r)) ≤ CH∞

106

5. CAPACITIES AND FINE PROPERTIES OF SOBOLEV FUNCTIONS

Proof. Let {B(xi , ri ) : i ∈ N} be a cover of E. Without loss of generality, we may assume that B(xi , ri ) ∩ E = ∅ for every i ∈ N. If rj ≥ r4 for some j ∈ N, we obtain from Lemma 5.35 that ∞

rin−p . capp (E, B(x, 2r)) ≤ C(n)r n−p ≤ 4n C(n)rjn−p ≤ 4n C(n) i=1

On the other hand, if 0 < ri < 4r for every i ∈ N, then B(xi , 2ri ) ⊂ B(x, 2r) for every i ∈ N, and Theorem 5.34 (c) and Lemma 5.35 imply



capp B(xi , ri ), B(x, 2r) ≤ capp B(xi , ri ), B(xi , 2ri ) ≤ C(n)rin−p ∞ for every i ∈ N. Observe also that i=1 B(xi , ri ) ⊂ B(x, 32 r)  B(x, 2r). By monotonicity and countable subadditivity, we have  ∞  B(xi , ri ), B(x, 2r) capp (E, B(x, 2r)) ≤ capp i=1

≤

∞

∞



capp B(xi , ri ), B(x, 2r) ≤ C(n) rin−p .

i=1

i=1



The claim follows by taking infimum over all such covers.

The reverse inequality does not hold in Theorem 5.39, see Remark 7.23. Theorem 5.41 gives a slightly weaker estimate to this direction. Before that, we prove a comparison result for the Hausdorff contents with different dimensions. Lemma 5.40. Let 0 ≤ λ1 < λ2 and assume that E ⊂ B(x, r) ⊂ Rn . Then (5.18)

λ2 λ1 H∞ (E) ≤ r λ2 −λ1 H∞ (E).

λ2 Proof. Since E ⊂ B(x, r + ε), for any ε > 0, we have H∞ (E) ≤ (r + ε)λ2 . λ2 λ2 By letting ε → 0 we obtain H∞ (E) ≤ r . Let {B(xi , ri : i ∈ N)} be a cover of E. Assume first that rj > r for some j ∈ N. Then λ2 (E) ≤ r λ2 = r λ2 −λ1 r λ1 ≤ r λ2 −λ1 rjλ1 ≤ r λ2 −λ1 H∞

∞

riλ1 .

i=1

r λ2 −λ1 riλ1

On the other hand, if ri ≤ r for every i ∈ N, then ≤ Hence, ∞ ∞

λ2 H∞ (E) ≤ riλ2 ≤ r λ2 −λ1 riλ1 . riλ2

i=1

for every i ∈ N.

i=1

The claim follows by taking infimum over all such covers of E in both cases.



Let E ⊂ B(x, r) and 1 < p ≤ n. By Lemma 5.40 we have λ n−p H∞ (E)r n−p−λ ≤ H∞ (E)

for λ2 = λ > n − p = λ1 ≥ 0. A combination of Theorem 5.39 and the following Theorem 5.41 implies λ n−p C(n, p, λ)H∞ (E)r n−p−λ ≤ capp (E, B(x, 2r)) ≤ C(n)H∞ (E).

Theorem 5.41. Let 1 < p ≤ n and λ > n − p, and assume that E ⊂ B(x, r) ⊂ Rn . There exists a constant C = C(n, p, λ) such that λ H∞ (E)r n−p−λ ≤ C capp (E, B(x, 2r)).

5.7. CAPACITY AND HAUSDORFF CONTENT

107

Proof. Let u ∈ W01,p (B(x, 2r)) be admissible for capp (E, B(x, 2r)). Extending u as zero outside B(x, 2r), we may assume that u ∈ W 1,p (Rn ), see Theorem 2.31. We may also assume that 0 ≤ u ≤ 1, u = 0 in Rn \ B(x, 2r), and that there exists a neighborhood G of E such that u(y) = 1 for every y ∈ G. Let z ∈ E and let rk = 22−k r for k ∈ N0 . Since B(x, 2r) ⊂ B(z, 4r), u = 0 in B(z, 4r) \ B(x, 2r), and 0 ≤ u ≤ 1, we have  (2r)n 1 |B(x, 2r)| = u(y) dy ≤ = 2−n < 1. |uB(z,4r) | = |B(z, 4r)| B(x,2r) |B(z, 4r)| (4r)n Since u = 1 in G, there exists j ∈ N with uB(z,rj ) = 1, and thus |uB(z,rj ) − uB(z,4r) | = 1 − 2−n > 0. Using this lower bound and a telescoping argument, as in the proof of Theorem 3.19 for the balls B(z, rk ), we obtain  p1  ∞

p (5.19) 1 ≤ C(n)|uB(z,rj ) − uB(z,4r) | ≤ C(n, p) rk  |∇u(y)| dy . On the other hand, for δ > 0, we have ∞

2−kδ ≤ C(n, p, δ)

k=0

∞

k=0

∞

k=0

 rk 

B(z,rk )

2−kδ = C(δ), and (5.19) gives  p1 p |∇u(y)| dy .

B(z,rk )

k=0

From this we conclude that there exist an index kz ∈ N0 such that  p1  (5.20) rk z  |∇u(y)|p dy ≥ C(n, p, δ)2−kz δ = C(n, p, δ)r −δ (rkz )δ , B(z,rkz )

where C(n, p, δ) > 0. We choose δ = λ−n+p > 0 and write rz = rkz for the p corresponding radius in (5.20). By raising both sides of (5.20) to power p, we obtain for every z ∈ E a ball B(z, rz ) with  (5.21) rzλ ≤ C(n, p, λ)r λ−n+p |∇u(y)|p dy. B(z,rz )

Observe that the radii rz with z ∈ E are uniformly bounded. We apply Lemma 1.13 zi ∈ E, i ∈ N, such that the balls for the balls B(z, rz ), z ∈ E, and obtain points  B(zi , rzi ) are pairwise disjoint and E ⊂ ∞ i=1 B(zi , 5rzi ). From (5.21), pairwise disjointness of the balls B(zi , rzi ), and the fact that ∇u = 0 almost everywhere in Rn \ B(x, 2r), we conclude that λ (E) ≤ H∞

∞

(5rzi )λ = 5λ

i=1

≤ C(n, p, λ)

∞

rzλi

i=1 ∞

 |∇u(y)|p dy

r λ−n+p

i=1



B(zi ,rzi )

|∇u(y)|p dy.

≤ C(n, p, λ)r λ−n+p B(x,2r)

The claim follows by taking infimum over all admissible test functions.



108

5. CAPACITIES AND FINE PROPERTIES OF SOBOLEV FUNCTIONS

The comparison results above give relations between zero Sobolev capacity and the Hausdorff dimension, see Definition 1.44. Theorem 5.42. Let 1 < p < n and let E ⊂ Rn . (a) If Capp (E) = 0, then dimH (E) ≤ n − p. (b) If dimH (E) < n − p, then Capp (E) = 0. Proof. First assume that Capp (E) = 0. Let λ > n − p and r > 0. Then Capp (E ∩ B(0, r)) = 0, and Theorem 5.41 and Theorem 5.38 imply λ (E ∩ B(0, r))r n−p−λ ≤ C(n, p, λ) capp (E ∩ B(0, r), B(0, 2r)) H∞  1 ≤ C(n, p, λ) 1 + p Capp (E ∩ B(0, r)) = 0. r Since this holds for every r > 0, by countable subadditivity we conclude that ∞

λ λ H∞ (E) ≤ H∞ (E ∩ B(0, j)) = 0 j=1

for every λ > n − p. Thus dimH (E) ≤ n − p, proving part (a). Then assume that dimH (E) < n − p and let r > 0. Since n−p n−p H∞ (E ∩ B(0, r)) ≤ H∞ (E) = 0,

we obtain from Theorem 5.38 and Theorem 5.39 that 1 Capp (E ∩ B(0, r)) ≤ capp (E ∩ B(0, r), B(0, 2r)) 1 + Cr p n−p ≤ C(n, p)H∞ (E ∩ B(0, r)) = 0. Countable subadditivity implies that Capp (E) = 0.



Remark 5.43. The claim in Theorem 5.42 (b) can be improved to the following: if Hn−p (E) < ∞, then Capp (E) = 0. We refer to Evans and Gariepy [122, Section 4.7.2] for the proof. 5.8. Lipschitz test functions for variational capacity For compact sets, variational capacity can be characterized in terms of Lipschitz functions with zero boundary values. Theorem 5.44. Let 1 < p < ∞, let Ω ⊂ Rn be an open set, and assume that E ⊂ Ω is compact. Then  (5.22) capp (E, Ω) = inf |∇u(x)|p dx, Ω

where the infimum is taken over all u ∈ Lip0 (Ω) with u = 1 in E. Proof. Assume first that u ∈ Lip0 (Ω) satisfies u(x) = 1 for every x ∈ E. Let 1 u(x), for every x ∈ Ω. Then the set G = {x ∈ Ω : 0 < ε < 1 and let v(x) = 1−ε u(x) > 1 − ε} is a neighborhood of E and the function v ∈ Lip0 (Ω) is admissible for the capacity capp (E, Ω), by Theorem 2.29. Thus   1 p  |∇u(x)|p dx. capp (E, Ω) ≤ |∇v(x)|p dx = 1−ε Ω Ω By letting ε → 0 and taking infimum over all u as above, we conclude that capp (E, Ω) is at most the infimum in (5.22).

5.8. LIPSCHITZ TEST FUNCTIONS FOR VARIATIONAL CAPACITY

109

For the proof of the reverse inequality, we may assume by Theorem 5.34 (d) that Ω is bounded. Let 0 < ε < 1 and let u ∈ W01,p (Ω) be such that u = 1 almost everywhere in a neighborhood G ⊂ Ω of E and  |∇u(x)|p dx ≤ capp (E, Ω) + ε . Ω

Let v ∈ W (R ) be a p-quasicontinuous function such that v = u almost everywhere in Ω and v = 0 p-quasieverywhere in Ωc , see Corollary 5.29. Since v = u almost everywhere in Ω, we have v = 1 almost everywhere in G . Lemma 5.18 implies that v = 1 p-quasieverywhere in G . By modifying v on a set of Sobolev p-capacity zero, if necessary, we may assume that v = 1 in the open set G and v = 0 in Ωc . Let 0 < ε < 1 and let G  G be an open set with E ⊂ G. Choose an open set V ⊂ Gc such that Capp (V ) < εp and the restriction v|Rn \V is continuous; here we need the fact that v = 1 in the open set G  G. Let w ∈ Ap (V ) be such that wW 1,p (Rn ) < ε. By modifying w on a set of measure zero, we may assume that w = 1 in V . By quasicontinuity and relative topology, there exists an open set U1 ⊂ Rn such that   Ωc \ V ⊂ x ∈ Rn \ V : v(x) < ε = U1 ∩ (Rn \ V ) = U1 \ V. 1,p

n

subset of Ω. This implies Ωc ⊂ U1 ∪ V . Since Ω is bounded, (U1 ∪ V )c is a compact Thus there exists an open set U2 with Ωc ⊂ U2 ⊂ U1 ∪ V and d U2 , (U1 ∪ V )c > 0. Let G1 and G2 be open sets satisfying E ⊂ G2  G1  G. This is possible since E is compact. Observe that d (V, G1 ) ≥ d (Gc , G1 ) = δ > 0 since V ⊂ Gc . Here it is important that δ is independent of ε. Let   1 ψ(x) = max 0, 1 − d (x, V ) , δ for every x ∈ Rn . Then ψ ∈ Lip(Rn ), 0 ≤ ψ ≤ 1, ψ = 1 in V , ψ = 0 in G1 , and |∇ψ| ≤ 1δ for almost every x ∈ Rn . Let   (v(x) − ε)+ , 1 − ψ(x)w(x) ϕ(x) = min 1−ε for every x ∈ Rn . Then ϕ ∈ W 1,p (Rn ), ϕ = 1 in G1 , and ϕ = 0 in U1 ∪ V . By Theorem 2.25 (c) and Leibniz’s rule (2.17) we have 1 1 ∇vLp (Rn ) + ∇wLp (Rn ) + wLp (Rn ) 1−ε δ 1 1 ∇uLp (Ω) + wW 1,p (Rn ) + wW 1,p (Rn ) ≤ 1−ε δ   1

1 1  p capp (E, Ω) + ε ≤ + 1+ wW 1,p (Rn ) . 1−ε δ

∇ϕLp (Rn ) ≤ (5.23)

Let r1 = d (U2 , (U1 ∪ V )c ) > 0 and r2 = d (G2 , Gc1 ) > 0. If z ∈ U2 and 0 < r < r1 , then 

|∇ϕ(x)| dx = 0. B(z,r)

110

5. CAPACITIES AND FINE PROPERTIES OF SOBOLEV FUNCTIONS

By H¨ older’s inequality, we obtain  M |∇ϕ|(z) = sup  r≥r1

(5.24) ≤

B(z,r)

|∇ϕ(x)| dx ≤ sup 

1 1

|B(z, r1 )| p

r≥r1

|∇ϕ(x)| dx p

 p1

B(z,r)

∇ϕLp (Rn ) = C1 < ∞

for every z ∈ U2 . Similarly, we have M |∇ϕ|(z) ≤

1 1

|B(z, r2 )| p

∇ϕLp (Rn ) = C2 < ∞

for every z ∈ G2 . For t > 0, let

  Ft = x ∈ Rn \ A : M |∇ϕ|(x) ≤ t and |ϕ(x)| ≤ t ,

as in the proof of Theorem 4.9, where A ⊂ Rn is an exceptional set with |A| = 0 from Theorem 4.2. Since 0 ≤ ϕ(z) ≤ 1 and M |∇ϕ|(z) ≤ max{C1 , C2 } for every z ∈ U2 ∪ G2 , we have (U2 ∪ G2 ) \ A ⊂ Ft for large t. By the proof of Theorem 4.9, we obtain t > 0 and a Lipschitz function ϕt such that ϕ − ϕt W 1,p (Rn ) < ε and ϕt = ϕ in Ft ⊃ (U2 ∪ G2 ) \ A. Since U2 ∪ G2 is open and ϕt is continuous, we conclude that ϕt = 0 in U2 and ϕt = 1 in G2 . Therefore ϕt ∈ Lip0 (Ω) and ϕt = 1 in E, and hence ϕt is an admissible function for (5.22). By (5.23) and the choice of w, we finally obtain  p1   |∇u(x)|p dx : u ∈ Lip0 (Ω), u = 1 in E ≤ ∇ϕt Lp (Rn ) inf Ω

≤ ∇ϕLp (Rn ) + ϕ − ϕt W 1,p (Rn )  

1 1 1 capp (E, Ω) + ε p + 1 + ≤ ε + ε. 1−ε δ Since δ is independent of ε, the claim follows by letting ε → 0 and then ε → 0.  5.9. Mazya’s inequality Variational capacity can be characterized using quasicontinuous functions; compare to Definition 5.12 and Corollary 5.29. Definition 5.45. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. A function u ∈ W01,p (Ω) is p-quasicontinuous if the zero extension v ∈ W 1,p (Rn ) of u to entire Rn is p-quasicontinuous in Rn . We have a version of Theorem 5.20 for variational capacity. This characterization is applied in Mazya’s inequality, see Theorem 5.47, which is a capacitary version of the Poincar´e inequality. Theorem 5.46. Let 1 < p < ∞, let Ω ⊂ Rn be an open set, and assume that E  Ω. Then  (5.25) capp (E, Ω) = inf |∇u(x)|p dx, Ω

where the infimum is taken over all p-quasicontinuous functions u ∈ W01,p (Ω) with u ≥ 1 p-quasieverywhere in E.

5.9. MAZYA’S INEQUALITY

111

Proof. Assume first that u ∈ W01,p (Ω) is admissible for capp (E, Ω), see Definition 5.32. Let v ∈ W 1,p (Rn ) be the p-quasicontinuous representative of u given by Corollary 5.29. Then v = u almost everywhere in Ω and v = 0 holds pquasieverywhere in Rn \ Ω. Since u = v almost everywhere in Ω and u ≥ 1 almost everywhere in a neighborhood G ⊂ Ω of E, also v ≥ 1 almost everywhere in G. Lemma 5.18 implies that v ≥ 1 holds p-quasieverywhere in G, and hence v ≥ 1 p-quasieverywhere in E. Thus v|Ω ∈ W01,p (Ω) is admissible for (5.25). Since ∇u(x) = ∇v(x) = ∇v|Ω (x) for almost every x ∈ Ω, capp (E, Ω) is at least the infimum in (5.25). For the converse inequality, assume that v ∈ W01,p (Ω) is p-quasicontinuous and v ≥ 1 p-quasieverywhere in E. We extend v as zero to Ωc . Then v ∈ W 1,p (Rn ) is p-quasicontinuous in Rn and, by truncation, we may assume that 0 ≤ v(x) ≤ 1 for every x ∈ Rn . Fix 0 < ε < 1 and choose an open set V such that Capp (V ) < εp , v = 1 in E \ V , and the restriction v|Rn \V is continuous. By relative topology, there exists an open set U ⊂ Rn satisfying   x ∈ Rn \ V : v(x) > 1 − ε = U ∩ (Rn \ V ) = U \ V. Then E \ V ⊂ U \ V since v = 1 on E \ V . As E  Ω, we have d (Ωc , E) = δ > 0. Define   2 ψ(x) = max 0, 1 − d (x, E) , δ n for every x ∈ R . Then ψ ∈ Lipc (Ω), 0 ≤ ψ ≤ 1, ψ = 1 in E, and |∇ψ| ≤ 2δ for almost every x ∈ Ω. Let G = {x ∈ Ω : ψ(x) > 12 }. Let w ∈ Ap (V ) with wW 1,p (Rn ) < ε and define u(x) =

v(x) + 2ψ(x)w(x) 1−ε

for every x ∈ Rn . Then u ∈ W01,p (Ω) and u ≥ 1 almost everywhere in ((U \ V ) ∪ V ) ∩ G = (U ∪ V ) ∩ G, which is a neighborhood of E. Hence u is an admissible test function for capp (E, Ω). By Leibniz’s rule (2.17) and the triangle inequality we have |∇v(x)| + 2|∇w(x)| + 2w(x)|∇ψ(x)|, 1−ε for almost every x ∈ Rn , and thus 1 1 4 ∇vLp (Ω) + 2∇wLp (Ω) + wLp (Ω) capp (E, Ω) p ≤ 1−ε δ  1 1 ∇vLp (Ω) + 4 1 + wW 1,p (Rn ) ≤ 1−ε δ  p1   1 1 p ε. ≤ |∇v(x)| dx +4 1+ 1−ε Ω δ |∇u(x)| ≤

The claim follows by letting ε → 0 and taking infimum over all functions v as above.  The following Mazya’s inequality for quasicontinuous representatives provides an important link between capacities and Poincar´e inequalities. Note that the claim 1,p (Rn ). is not true for an arbitrary function u ∈ Wloc

112

5. CAPACITIES AND FINE PROPERTIES OF SOBOLEV FUNCTIONS

Theorem 5.47. Let 1 < p < ∞, let B(z, r) ⊂ Rn be a ball, and assume that 1,p np (Rn ) is a p-quasicontinuous function. Let 1 ≤ q ≤ n−p for 1 < p < n, u ∈ Wloc and 1 ≤ q < ∞ for n ≤ p < ∞. There exists a constant C = C(n, p, q) such that  pq   C q

|u(x)| dx ≤ |∇u(x)|p dx.  capp {u = 0} ∩ B(z, r2 ), B(z, r) B(z,r) B(z,r) Here {u = 0} = {y ∈ Rn : u(y) = 0}. Proof. By considering |u| instead of u, we may assume that u ≥ 0 in B(z, r). Write B = B(z, r) and   q1 1 q = |B|− q uLq (B) . uB,q =  u(x) dx B

By the (q, p)-Poincar´e inequality in Theorem 3.14 and H¨ older’s inequality we have 1  q + uB uB,q ≤  |u(x) − uB |q dx B



≤ C(n, p, q)r  |∇u(x)| dx p

B

 p1

 p

+  u(x) dx

 p1

< ∞.

B

If uB,q = 0 the claim is true, and thus we may assume that uB,q > 0. Let 

 3 ψ(x) = max 0, 1 − d x, B(z, r2 ) , r for every x ∈ Rn . Then ψ ∈ Lipc (B(z, r)), 0 ≤ ψ ≤ 1, ψ = 1 in B(z, r2 ), and |∇ψ| ≤ 3r for almost every x ∈ B(z, r). Let   u(x) v(x) = ψ(x) 1 − , uB,q for every x ∈ Rn . Then v ∈ W01,p (B(z, r)) is p-quasicontinuous and v = 1 in {u = 0} ∩ B(z, r2 ). The Leibniz rule implies |∇v(x)| ≤

3 |uB,q − u(x)| |∇u(x)| + r uB,q uB,q

for almost every x ∈ B(z, r) and thus, by Theorem 5.46, 

capp {u = 0} ∩ B(z, 2r ), B(z, r) ≤ |∇v(x)|p dx B   (5.26) C(p) C(p) p ≤ p |u(x) − u | dx + |∇u(x)|p dx. B,q r (uB,q )p B (uB,q )p B We use Minkowski’s inequality and the p-Poincar´e inequality, see Theorem 3.14, to estimate the first term on the right-hand side of (5.26), and obtain   p1   p1 p p ≤  |u(x) − uB | dx + |uB,q − uB |  |u(x) − uB,q | dx B B (5.27)   p1 1 p ≤ C(n, p)r  |∇u(x)| dx + |B|− q uLq (B) − uB Lq (B) . B

5.10. NOTES

113

By the triangle inequality and the (q, p)-Poincar´e inequality (3.8), we have 1 1 |B|− q uLq (B) − uB Lq (B) ≤ |B|− q u − uB Lq (B)   q1 =  |u(x) − uB |q dx B

  p1 ≤ C(n, p, q)r  |∇u(x)|p dx . B

Together with (5.27) this gives  p1   p1  p p |u(x) − u | dx ≤ C(n, p, q)r |∇u(x)| dx , B,q   B

and thus

B



 |u(x) − uB,q |p dx ≤ C(n, p, q)r p

B

|∇u(x)|p dx. B

Substituting this to (5.26) and recalling that B = B(z, r), we arrive at 

C(n, p, q) capp {u = 0} ∩ B(z, r2 ), B(z, r) ≤ |∇u(x)|p dx. (uB,q )p B(z,r) The claim follows by reorganizing the terms.



Remark 5.48. If n ≤ p < ∞ and 1 ≤ q ≤ 2p, then the constant C in Theorem 5.47 can be chosen to be independent of q. This follows immediately from H¨older’s inequality on the left-hand side. 5.10. Notes For the capacity theory, we refer to Adams and Hedberg [4], Bj¨orn and Bj¨orn [31, Chapter 6], Evans and Gariepy [122, Section 4.7], Diening, Harjulehto, H¨ast¨ o and R˚ uˇziˇcka [104, Chapter 10], Heinonen, Kilpel¨ ainen and Martio [187, Chapter 2], Mal´ y and Ziemer [303, Section 2.1], Mazya [318, 321], Ponce [350], Reshetnyak [354], Turesson [388] and Ziemer [407]. See also Kinnunen and Martio [234]. The proof of Theorem 5.16 is from Kilpel¨ ainen [218], see also Kilpel¨ ainen, Kinnunen and Martio [219]. Theorem 5.23 is discussed in Adams and Hedberg [4, Section 6.2], Evans and Gariepy [122, Section 4.8], Federer and Ziemer [126], Mal´ y and Ziemer [303, 2.1.8] and Ziemer [407, Chapter 3]. See also Kinnunen and Latvala [229]. We considered capacity for p > 1. The case p = 1 has been studied in Federer and Ziemer [126], Hakkarainen and Kinnunen [177], Turesson [388] and Ziemer [407, Section 5.12]. Fleming proved in [130] that in this case capacity has the same zero sets as the Hausdorff measure of codimension one. Moreover, capacity is equivalent by two sided estimates to the Hausdorff content of codimension one, see Kinnunen, Korte, Shanmugalingam and Tuominen [227], Mal´ y [300, 301], Mal´ y, Swanson and Ziemer [302], M¨ak¨ al¨ainen [298], Turesson [388, Theorem 3.5.5] and Ziemer [407, Lemma 5.12.3]. Sobolev spaces with zero boundary values are discussed in Adams and Hedberg [4, Chapters 10–11], Diening, Harjulehto, H¨ ast¨ o and R˚ uˇziˇcka [104, Section 11.3], Heinonen, Kilpel¨ainen and Martio [187, Chapter 1] and Mazya [318, 321]. See also Kilpel¨ainen, Kinnunen and Martio [219]. For Corollary 5.29, we refer to Adams and Hedberg [4, Theorem 9.1.3]. See also Diening, Harjulehto, H¨ast¨ o and

114

5. CAPACITIES AND FINE PROPERTIES OF SOBOLEV FUNCTIONS

R˚ uˇziˇcka [104, Section 11.2], Heinonen, Kilpel¨ainen and Martio [187, Theorem 4.5] and Turesson [388, Section 4.4]. Theorem 5.41 is based on Heinonen and Koskela [188, Theorem 5.9]. For the connections between capacities and Hausdorff dimension, see also Adams and Hedberg [4, Section 5.1], Aikawa and Ess´en [8, Section I.6], Evans and Gariepy [122, Section 4.7.2], Heinonen, Kilpel¨ainen and Martio [187, Chapter 2], Mazya and Havin [322] and Reshetnyak [354, Section 3.4]. The proof of Theorem 5.44 is inspired by Bj¨ orn and Bj¨orn [31, Theorem 6.19 (x) and Lemma 5.4]. Theorem 5.47 can be found in Mazya [318, Chapter 10] and [321, Chapter 14]. See also Adams and Hedberg [4, Section 8.2], Kilpel¨ainen and Koskela [220] and Mikkonen [327, Lemma 8.11]. For a general version, see Bj¨ orn and Bj¨orn [31, Theorem 6.21].

10.1090/surv/257/06

CHAPTER 6

Hardy’s Inequalities This chapter studies Hardy’s inequalities for functions with zero boundary values. A pointwise Hardy inequality captures local phenomena better than an integral Hardy inequality and, by the Hardy–Littlewood–Wiener maximal function theorem, it implies the integral Hardy inequality. Hardy’s inequalities hold in every open set for p > n, but the corresponding result is not true for 1 < p ≤ n. Several sufficient and necessary conditions for Hardy’s inequalities are obtained in Chapter 6, Chapter 7 and Chapter 10. 6.1. Introduction to Hardy’s inequalities We begin with an integral version of Hardy’s inequality. Definition 6.1. Let 1 ≤ p < ∞ and let Ω  Rn be an open set. We say that the p-Hardy inequality holds in Ω, with a constant C, if   |u(x)|p dx ≤ C |∇u(x)|p dx, (6.1) p d (x, ∂Ω) Ω Ω for every u ∈ Lip0 (Ω). It is essential that the constant in (6.1) is independent of u. This implies that we may consider (6.1) not only for u ∈ Lip0 (Ω) but also for some other classes of functions with zero boundary values. Lemma 6.2. Let 1 ≤ p < ∞ and 0 < C < ∞, and assume that Ω  Rn is an open set. The following conditions are equivalent. (a) The p-Hardy inequality holds in Ω with the constant C. (b) Inequality (6.1) holds for every u ∈ Lipc (Ω) with the constant C. (c) Inequality (6.1) holds for every u ∈ W01,p (Ω) with the constant C. Proof. Since Lipc (Ω) ⊂ W01,p (Ω), the implication from (c) to (b) is clear. Next we show that (b) implies (a). Let u ∈ Lip0 (Ω) and define   (6.2) uj (x) = max 0, |u(x)| − 1j for every x ∈ Ω and j ∈ N. The sequence (uj (x))j∈N converges to |u(x)| for every x ∈ Ω. Moreover, uj ∈ Lipc (Ω) and |∇uj (x)| ≤ |∇u(x)| for almost every x ∈ Ω and every j ∈ N. By Fatou’s lemma and (6.1) for uj , with j ∈ N, we obtain   |u(x)|p |uj (x)|p dx ≤ lim inf dx p p j→∞ Ω d (x, ∂Ω) Ω d (x, ∂Ω)   (6.3) ≤ C lim inf |∇uj (x)|p dx ≤ C |∇u(x)|p dx. j→∞

Ω 115

Ω

116

6. HARDY’S INEQUALITIES

To complete the proof, we show that (a) implies (c). To this end, let u ∈ and consider a sequence of functions uj ∈ C0∞ (Ω) ⊂ Lip0 (Ω), with j ∈ N, such that uj → u in W 1,p (Ω) as j → ∞. By passing to a subsequence, we may assume that uj (x) → u(x) for almost every x ∈ Ω. An application of Fatou’s lemma, as in (6.3), gives (6.1) for u with the constant C. 

W01,p (Ω)

Remark 6.3. The proof of Lemma 6.2 shows that the p-Hardy inequality holds in Ω if and only if (6.1) holds for every u ∈ C0∞ (Ω). Next we consider a pointwise variant of Hardy’s inequality, in terms of the restricted Hardy–Littlewood maximal operator from Definition 1.27. Definition 6.4. Let 1 ≤ p < ∞ and let Ω  Rn be an open set. We say that the pointwise p-Hardy inequality holds in Ω, with a constant C, if (6.4)



1 |u(x)| ≤ Cd (x, ∂Ω) M2d(x,∂Ω) |∇u|p (x) p ,

for every x ∈ Ω and every u ∈ Lip0 (Ω). As in (6.1) it is essential that the constant in (6.4) is independent of u. Remark 6.5. Let 1 ≤ q < p < ∞ and assume that the pointwise q-Hardy older’s inequality implies that also the inequality holds in an open set Ω  Rn . H¨ pointwise p-Hardy inequality holds in Ω. Remark 6.6. Let 1 ≤ q < p < ∞ and let Ω  Rn be an open set. It follows from the maximal function theorem that if the pointwise q-Hardy inequality holds in Ω, then also the p-Hardy inequality holds in Ω. To see this, let u ∈ Lip0 (Ω) and assume that (6.4) holds for u with a constant C1 and the exponent q. Since pq > 1, Theorem 1.15 (c) for the restricted maximal function implies  

p |u(x)|p p M2d(x,∂Ω) |∇u|q (x) q dx dx ≤ C 1 p Ω d (x, ∂Ω) Ω 

p (6.5) |∇u(x)|q q dx ≤ C(n, p, q, C1 ) Ω = C(n, p, q, C1 ) |∇u(x)|p dx. Ω

In (6.5) we used the fact that u vanishes in Ωc = Rn \ Ω and thus ∇u = 0 almost everywhere in Ωc . The reasoning in (6.5) does not apply for q = p since the maximal operator is not bounded on L1 (Rn ). However, it is true that the pointwise p-Hardy inequality implies the p-Hardy inequality, but the proof is more complicated. We discuss this question in Section 6.5, see Corollary 6.26. The pointwise p-Hardy inequality for p > n is an easy consequence of Morrey’s inequality, see Theorem 3.19. Theorem 6.7. Let n < p < ∞ and let Ω  Rn be an open set. The pointwise pHardy inequality and the p-Hardy inequality hold in Ω with a constant C = C(n, p).

6.1. INTRODUCTION TO HARDY’S INEQUALITIES

117

Proof. Let u ∈ Lip0 (Ω) and x ∈ Ω, and choose z ∈ ∂Ω with |x−z| = d (x, ∂Ω). Since u(z) = 0, Theorem 3.19 implies   p1 p |u(x)| = |u(x) − u(z)| ≤ C(n, p)2d (x, ∂Ω)  |∇u(y)| dy B(x,2d(x,∂Ω))



1 ≤ C(n, p)d (x, ∂Ω) M2d(x,∂Ω) |∇u|p (x) p .

This shows that the pointwise p-Hardy inequality holds in Ω with a constant C = C(n, p). In particular, the pointwise q-Hardy inequality holds in Ω for q = n+p 2 >n with a constant C = C(n, q) = C(n, p). Since q < p, also the p-Hardy inequality holds in Ω by Remark 6.6.  Pointwise Hardy inequalities also hold for Sobolev functions with zero boundary values. Lemma 6.8. Let 1 ≤ p < ∞. Assume that the pointwise p-Hardy inequality holds in an open set Ω  Rn with a constant C and let u ∈ W01,p (Ω). Then inequality (6.4) holds for the zero extension of u at almost every x ∈ Ω with the same constant C. Proof. By considering the zero extension of u to Ωc as in Theorem 2.31, we may assume that u ∈ W 1,p (Rn ). Since Lipc (Ω) is dense in W01,p (Ω), there exist functions uj ∈ Lipc (Ω) with j ∈ N such that uj → u in W 1,p (Rn ) as j → ∞. By Minkowski’s inequality

1

1 M2d(x,∂Ω) |∇uj |p (x) p = M2d(x,∂Ω) |∇u + (∇uj − ∇u)|p (x) p

1

1 ≤ M2d(x,∂Ω) |∇u|p (x) p + M2d(x,∂Ω) |∇uj − ∇u|p (x) p for every x ∈ Ω. Theorem 1.15 (b) implies    x ∈ Rn : M2d(x,∂Ω) |∇uj − ∇u|p (x) > εp ≤ C(n) |∇uj (x) − ∇u(x)|p dx εp Rn for every ε > 0. Thus

1 M2d(x,∂Ω) |∇uj − ∇u|p (x) p → 0 in measure as j → ∞. By passing twice to a subsequence, if necessary, we obtain a set E ⊂ Ω with |E| = 0 such that uj (x) → u(x) < ∞ as j → ∞ and

1 lim M2d(x,∂Ω) |∇uj − ∇u|p (x) p = 0 j→∞

for every x ∈ Ω \ E. Since Lipc (Ω) ⊂ Lip0 (Ω), the pointwise p-Hardy inequality and the estimates above give

1 |u(x)| = lim |uj (x)| ≤ C lim sup d (x, ∂Ω) M2d(x,∂Ω) |∇uj |p (x) p j→∞

j→∞



1 ≤ Cd (x, ∂Ω) M2d(x,∂Ω) |∇u|p (x) p for every x ∈ Ω \ E. Remark 6.9. Consider the Rayleigh quotient  |∇u(x)|p dx , (6.6) λ = λ(p, Ω) = inf Ω |u(x)|p dx Ω d(x,∂Ω)p



118

6. HARDY’S INEQUALITIES

where the infimum is taken over all u ∈ W01,p (Ω) such that u is not zero almost everywhere. This variational problem is related to the p-Hardy inequality since λ > 0 if and only if the p-Hardy inequality holds in Ω. Note that u ∈ W01,p (Ω) is a minimizer of (6.6) if and only if u is a weak solution to the nonlinear eigenvalue problem |u(x)|p−2 u(x) = 0. (6.7) div(|∇u(x)|p−2 ∇u(x)) + λ d (x, ∂Ω)p The p-Laplace operator div(|∇u(x)|p−2 ∇u(x)) will be discussed in Chapter 11 and Chapter 12. This approach to Hardy inequalities is considered, for example, by Barbatis, Filippas and Tertikas [23–25], Brezis, Marcus, Mizel, Pinchover and Shafrir [57,58,305,306], Davies [96], Matskewich and Sobolevskii [314], Pinchover and Tintarev [347–349] and Tidblom [380, 381]. Remark 6.10. The origins of Hardy’s inequalities are in the one-dimensional case. Let 1 < p < ∞, β = p − 1 and u ∈ Lip0 (0, ∞). Then  ∞  ∞ |u(x)|p β pp x dx ≤ |u (x)|p xβ dx, (6.8) xp |p − 1 − β|p 0 0 where the constant on the right-hand side is the best possible. The proof of this weighted one-dimensional Hardy inequality is based on integration by parts and H¨ older’s inequality. We refer to Hardy, Littlewood and P´olya [180, § 330] and Kufner [248, Theorem 5.2] for more details. The inequality in (6.8) implies the p-Hardy inequality for Ω = Rn \{0}; this will be applied in Example 6.13 below. Let 1 < p < ∞ with p = n, let u ∈ Lip0 (Rn \{0}), and write x ∈ Rn as x = rz, where 0 < r = |x| < ∞ and z ∈ ∂B(0, 1). By (3.1), Fubini’s theorem, and (6.8) with β = n − 1 = p − 1, we obtain  ∞  |u(x)|p |u(x)|p dx = dσ(x) dr |x|p rp Rn ∂B(0,r) 0   ∞ |u(rz)|p n−1 = r dr dσ(z) rp ∂B(0,1) 0 (6.9)   ∞ ∂ u(rz) p r n−1 dr dσ(z) ≤C ∂r ∂B(0,1) 0  ≤C |∇u(x)|p dx, Rn

where C = C(n, ∂ p). In the final step we used again Fubini’s theorem, (3.1), and the fact that ∂r u(rz) ≤ |∇u(rz)| = |∇u(x)| for every x = rz ∈ Rn . 6.2. Measure density and Hardy’s inequality In contrast with the case p > n in Theorem 6.7, for 1 ≤ p ≤ n there exist open sets where the p-Hardy inequality, or the pointwise p-Hardy inequality, does not hold, see Example 6.13. The following geometric property is a sufficient condition for the Hardy inequality. Definition 6.11. A set E ⊂ Rn satisfies the measure density condition if there exists a constant C, with 0 < C ≤ 1, such that (6.10) for every x ∈ E and r > 0.

|E ∩ B(x, r)| ≥ C|B(x, r)|

6.2. MEASURE DENSITY AND HARDY’S INEQUALITY

119

Theorem 6.12. Let 1 < p < ∞ and assume that Ω  Rn is an open set. If Ωc satisfies the measure density condition, with a constant C1 , then the pointwise p-Hardy inequality and the p-Hardy inequality hold in Ω with a constant C = C(n, p, C1 ). Proof. Let u ∈ Lip0 (Ω) and x ∈ Ω. Choose a point z ∈ ∂Ω with |x − z| = d (x, ∂Ω) and let R = d (x, ∂Ω). Then |u(x)| ≤ |u(x) − uB(x,R) | + |uB(x,R) − uB(z,R) | + |uB(z,R) |.

(6.11)

For the first term on the right-hand side we apply Theorem 3.4, inequality (4.1) and Remark 1.29, as in (4.2), and obtain

1 (6.12) |u(x) − uB(x,R) | ≤ C(n)R M2R |∇u|p (x) p . For the second term on the right-hand side of (6.11) we use the Poincar´e inequality in Theorem 3.14 and the inclusion B(x, R) ∪ B(z, R) ⊂ B(x, 2R). This gives |uB(x,R) − uB(z,R) | ≤ |uB(x,R) − uB(x,2R) | + |uB(z,R) − uB(x,2R) | ≤ C(n) 

|u(y) − uB(x,2R) | dy

B(x,2R)



≤ C(n, p)R 

|∇u(y)| dy p

 p1

B(x,2R)



1 ≤ C(n, p)R M2R |∇u|p (x) p . The measure density assumption is applied to estimate the last term on the right-hand side of (6.11). Here we employ the same idea as in the proof of Theorem 3.17, see also Remark 3.18. Let B = B(z, R). By Minkowski’s inequality, the p-Poincar´e inequality, and H¨older’s inequality, we obtain   p1   p1 p p |u(y)| dy ≤ |u(y) − u | dy + |uB | B   B



B

(6.13)

 p1

 1 |u(y)| dy |B| B\Ωc B  p1   p1  1− p1  |B \ Ωc | p + |u(y)| dy . ≤ C(n, p)R  |∇u(y)|p dy  |B| B B

≤ C(n, p)R  |∇u(y)| dy p

+

The measure density condition, with 0 < C1 ≤ 1, implies |B \ Ωc | = |B| − |Ωc ∩ B| ≤ (1 − C1 )|B|, and thus

1− p1 1 |B \ Ωc | ≤ (1 − C1 )1− p < 1. |B| The last term in (6.13) can be absorbed to the left-hand side. Since B = B(z, R) ⊂ B(x, 2R), we obtain   p1   p1 p p |uB | ≤  |u(y)| dy ≤ C(n, p, C1 )R  |∇u(y)| dy 

B

B



≤ C(n, p, C1 )R 

B(x,2R)

|∇u(y)|p dy

 p1



1 ≤ C(n, p, C1 )R M2R |∇u|p (x) p .

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6. HARDY’S INEQUALITIES

Combining the estimates above for the right-hand side of (6.11) we conclude that

1 |u(x)| ≤ C(n, p, C1 )d (x, ∂Ω) M2d(x,∂Ω) |∇u|p (x) p . This proves that the pointwise p-Hardy inequality holds in Ω with a constant C C(n, p, C1 ). By the argument above, the pointwise q-Hardy inequality holds in for q = p+1 2 ∈ (1, p) with a constant C = C(n, q, C1 ) = C(n, p, C1 ). Since q < also the p-Hardy inequality holds in Ω by Remark 6.6.

= Ω p, 

In Section 6.4 we obtain more sensitive sufficient conditions for Hardy’s inequalities based on a capacity density condition, which is a refined version of the measure density condition. Next example shows that there exist open sets for which Hardy’s inequality does not hold. Example 6.13. Let Ω = B(0, 1) \ {0} ⊂ Rn . We show that the p-Hardy inequality holds in Ω when 1 < p < n or p > n, but the n-Hardy inequality does not hold in Ω. For p > n, the p-Hardy inequality follows immediately from Theorem 6.7. Thus we may assume that 1 < p < n. Let u ∈ Lip0 (Ω). Since B(0, 1)c satisfies the measure density condition with C = 12 and u ∈ Lip0 (B(0, 1)), the p-Hardy inequality with respect to B(0, 1) holds for u with a constant C = C(n, p) by Theorem 6.12. On the other hand, Remark 6.10 implies that the pHardy inequality (6.9) holds for u ∈ Lip0 (Ω) ⊂ Lip0 (Rn \ {0}). Since d (x, ∂Ω) = min{|x|, 1 − |x|} for every x ∈ Ω, where 1 − |x| = d (x, ∂B(0, 1)), and |∇u(x)| = 0 for almost every x ∈ Ωc , we conclude that    |u(x)|p |u(x)|p |u(x)|p dx ≤ dx + dx p p |x|p Ω d (x, ∂Ω) B(0,1) (1 − |x|) Rn    p p |∇u(x)| dx + |∇u(x)| dx ≤ C(n, p) B(0,1) Rn  ≤ C(n, p) |∇u(x)|p dx. Ω

Next we show that the n-Hardy inequality does not hold in Ω. Let Ai = B(0, 2−i ) \ B(0, 2−i−1 ) with i ∈ N. For j ∈ N, consider uj ∈ Lip0 (Ω) defined as ⎧ j+1 ⎪ |x| ≤ 2−j−1 ⎨1 − 2 d (x, Aj ), uj (x) = 1, 2−j−1 < |x| < 12 , ⎪ ⎩ |x| ≥ 12 . max{0, 1 − 2d (x, A1 )}, Then  Ω

|uj (x)|n dx ≥ d (x, ∂Ω)n i=1 j

 Ai

j

1 j→∞ dx ≥ C(n) 2−in 2in = C(n)j −−−→ ∞, d (x, ∂Ω)n i=1

while the left-hand side of the n-Hardy inequality remains bounded since    n n |∇uj (x)| dx ≤ 2 dx + (2j+1 )n dx Ω

Ω

B(0,2−j−1 )

≤ C(n) + C(n)2−(j+1)n 2(j+1)n = C(n) < ∞ for every j ∈ N. Thus the n-Hardy inequality does not hold in Ω.

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121

Example 6.13 illustrates general features related to Hardy’s inequality that will be discussed more in the following sections and chapters. For instance, the p-Hardy inequality holds in Ω if the complement Ωc is sufficiently large or sufficiently small, or an appropriate combination of large and small parts. Hardy’s inequality in these three cases will be considered in Theorem 7.17, Corollary 10.36 and Theorem 10.40, respectively. On the other hand, if Ωc contains an isolated (n − p)-dimensional part, then the p-Hardy inequality does not hold in Ω. This occurs in Example 6.13 with the 0-dimensional part {0} ⊂ Ωc . A precise necessary condition is given in Theorem 10.46, and Example 10.49 generalizes the idea in Example 6.13. 6.3. Self-improvement of Hardy’s inequality If the p-Hardy inequality holds in an open set Ω  Rn for some 1 < p < ∞, then it is not obvious whether the q-Hardy inequality holds in Ω for some q = p. In this respect there is a difference between integral and pointwise Hardy inequalities since the pointwise p-Hardy inequality implies pointwise q-Hardy inequalities for every p < q < ∞, see Remark 6.5. On the other hand, Example 6.13 shows that the p-Hardy inequality does not necessarily imply the q-Hardy inequality for every p < q < ∞, nor for every 1 ≤ q < p. In this section we show that if the p-Hardy inequality holds in Ω, for some 1 < p < ∞, then there exists ε > 0 such that also the q-Hardy inequality holds in Ω for p − ε < q < p + ε, see Theorem 6.16. The proofs of the cases q > p and q < p are rather different and we consider them separately. Lemma 6.14. Let 1 < p < ∞ and assume that the p-Hardy inequality holds in an open set Ω  Rn with a constant C1 . There exists ε = ε(p, C1 ) > 0 such that the q-Hardy inequality holds in Ω whenever p < q < p + ε, with a constant C = C(p, C1 ). Proof. By Lemma 6.2 (b), we may assume that u ∈ Lipc (Ω). Let ε > 0 and define ε ε v(x) = |u(x)|1+ p d (x, ∂Ω)− p for every x ∈ Ω. Then v ∈ Lipc (Ω), and by the Leibniz rule (2.4) and the chain rule (2.5) we obtain  ε ε ε ε ε ε |u(x)| p |∇u(x)|d (x, ∂Ω)− p + |u(x)|1+ p d (x, ∂Ω)− p −1 |∇v(x)| ≤ 1 + p p for almost every x ∈ Ω. Here we also used the fact that |∇d (x, Ω)| ≤ 1 for almost every x ∈ Ω. The p-Hardy inequality for v and the estimate above for |∇v(x)| give p+ε p      |u(x)| |v(x)| dx = dx ≤ C1 |∇v(x)|p dx Ω d (x, ∂Ω) Ω d (x, ∂Ω) Ω  p  ε (6.14) ≤ 2p C1 1 + |u(x)|ε |∇u(x)|p d (x, ∂Ω)−ε dx p Ω  ε p  p + 2 C1 |u(x)|p+ε d (x, ∂Ω)−p−ε dx. p Ω −1

By choosing ε < p2 C1 p we have 2p C1 ( pε )p < 1, and we can absorb the last term in (6.14) to the left-hand side. Note that this term is finite for u ∈ Lipc (Ω). Let

122

6. HARDY’S INEQUALITIES

εα α = 1 + pε . Then αp = p + ε = α−1 , and H¨ older’s inequality gives   ε p    |u(x)| p+ε 1 − 2p C1 dx p d (x, ∂Ω)  Ω ≤ 2p C1 αp |∇u(x)|p |u(x)|ε d (x, ∂Ω)−ε dx Ω



(6.15) ≤ 2 C1 α p

|∇u(x)|

p

αp

 α1 

|∇u(x)|p+ε dx Ω

This implies   Ω

|u(x)| d (x, ∂Ω)

d (x, ∂Ω)

Ω

 = 2p C1 αp

|u(x)|

dx

Ω

εα α−1

 α1   Ω

p+ε

 dx ≤

2p C1 αp 1 − 2p C1 ( pε )p

|u(x)| d (x, ∂Ω)

−εα α−1

 α−1 α dx

1− α1

p+ε dx

.

α  |∇u(x)|p+ε dx. Ω

−1

We conclude that if 0 < ε < p2 C1 p , then the (p + ε)-Hardy inequality holds in Ω with the constant  p 1+ pε 2 C1 (1 + pε )p . (6.16) C(p, ε, C1 ) = 1 − 2p C1 ( pε )p −1

From this expression we see that for 0 < ε < p4 C1 p the constant in the (p+ε)-Hardy inequality can be chosen to be independent of ε.  Next we consider self-improvement for q < p. The proof is based on Lipschitz truncation and maximal function techniques, similar to the proof of Theorem 4.11, and integration over distribution sets. This approach is applied for self-improvement properties several times, see the proofs of Theorem 7.32, Theorem 8.11, Lemma 8.24, Theorem 8.43 and Theorem 12.23. Lemma 6.15. Let 1 < p < ∞ and assume that the p-Hardy inequality holds in an open set Ω  Rn with a constant C1 . There exists ε = ε(n, p, C1 ) > 0 such that the q-Hardy inequality holds in Ω whenever p − ε < q < p, with a constant C = C(n, p, C1 ). Proof. Let u ∈ Lip0 (Ω). We may assume that u is not identically zero. For t > 0, define Ft = Gt ∩ Ht , where Gt = {x ∈ Ω : M |∇u|(x) ≤ t} and

Ht = {x ∈ Ω : |u(x)| ≤ td (x, ∂Ω)}.

The pointwise estimate in Theorem 4.1 implies

|u(x) − u(y)| ≤ C(n)|x − y| M |∇u|(x) + M |∇u|(y) ≤ C(n)t|x − y| for every x, y ∈ Ft . For x ∈ Ft and y ∈ Ωc , we have |u(x) − u(y)| = |u(x)| ≤ td(x, ∂Ω) ≤ t|x − y|. Since u vanishes in Ωc , we conclude from the estimates above that the restriction u|Ft ∪Ωc is a C(n)t-Lipschitz function. We use Theorem 2.7 and extend u|Ft ∪Ωc to a C(n)t-Lipschitz function u  in Rn . Then u  = u = 0 in Ωc . Moreover, the support of u  is bounded. To see this, fix x0 ∈ supp u. There exists R = R(t, u) > 0 such that if x ∈ Ω \ B(x0 , R), then u(x) = 0 and x ∈ Ft . Here the inclusion x ∈ Gt is a

6.3. SELF-IMPROVEMENT OF HARDY’S INEQUALITY

123

consequence of the fact that |∇u| = |∇u|χsupp u ∈ L1 (Rn ). Thus u  = u = 0 outside . It follows that u  ∈ Lip0 (Ω). B(x0 , R), and this proves the boundedness of supp u Since the p-Hardy inequality holds for u  with the constant C1 and |∇ u(x)| ≤ |∇u(x)|χFt (x) + C(n)tχΩ\Ft (x) for almost every x ∈ Ω, we obtain p p     |u(x)| | u(x)| dx = dx Ft d (x, ∂Ω) Ft d (x, ∂Ω)     p | u(x)| ≤ dx ≤ C1 |∇ u(x)|p dx Ω d (x, ∂Ω) Ω   p ≤ C1 |∇u(x)| dx + C1 C(n, p) tp dx. Ft

Thus

 Ht



|u(x)| d (x, ∂Ω)

p

Ω\Ft



 dx ≤ C1

|∇u(x)|p dx + C1 C(n, p) Ft

 + 

Ht \Ft



|u(x)| d (x, ∂Ω)

tp dx Ω\Ft

p dx

 |∇u(x)|p dx + C1 C(n, p) tp dx Gt Ω\Ht  + (1 + C1 C(n, p)) tp dx,

(6.17)

≤ C1

Ω\Gt

where we have used the definition of Ht and the inclusions Ω\Ft ⊂ (Ω\Ht )∪(Ω\Gt ) and Ht \ Ft ⊂ Ω \ Gt . Next we multiply (6.17) by t−ε−1 , where 0 < ε < p − 1, and then integrate with respect to t over (0, ∞). After changing the order of integration on the left-hand side, this leads to p−ε   |u(x)| 1 dx ε Ω d (x, ∂Ω)  ∞  ∞  −ε−1 p (6.18) ≤ C1 t |∇u(x)| dx dt + C1 C(n, p) tp−ε−1 Ω \ Ht dt Gt 0 0  ∞ + (1 + C1 C(n, p)) tp−ε−1 Ω \ Gt dt. 0

Since |∇u(x)| ≤ M |∇u|(x) for almost every x ∈ Ω, see Corollary 1.23, the first integral on the right-hand side of (6.18) can be estimated as  ∞  ∞   t−ε−1 |∇u(x)|p dx dt ≤ t−ε−1 |∇u(x)|p dx dt Gt {x∈Ω:|∇u(x)|≤t} 0 0  1 = |∇u(x)|p−ε dx. ε Ω For the second integral on the right-hand side of (6.18), we have p−ε  ∞   1 |u(x)| p−ε−1 Ω \ Ht dt = t dx. p − ε Ω d (x, ∂Ω) 0

124

6. HARDY’S INEQUALITIES

Finally, we estimate the last integral on the right-hand side of (6.18) by using the definition of Gt and Theorem 1.15 (c) with p − ε > 1, and obtain  ∞  

p−ε C2 1 p−ε−1 M |∇u|(x) t dx ≤ |∇u(x)|p−ε dx, Ω \ Gt dt = p−ε Ω p−ε Ω 0 p−ε where C2 = C2 (n, p, ε) = 5n 2p−ε p−ε−1 . In the final step, we also used the fact that c n u vanishes in Ω = R \ Ω and thus ∇u = 0 almost everywhere in Ωc . ε ≤ 12 , it follows By choosing 0 < ε < p − 1 so small that (1 + C1 C(n, p)) p−ε from (6.18) and the previous estimates that  p−ε    |u(x)| ε dx ≤ 2 C1 + (1 + C1 C(n, p))C2 |∇u(x)|p−ε dx d (x, ∂Ω) p − ε Ω Ω 

p−ε ≤ 2C1 + C2 |∇u(x)| dx. Ω p−1 2 ,

we may assume that the constant C2 If we in addition require that ε ≤ above only depends on p and n. Thus there exists ε0 = ε0 (n, p, C1 ) > 0 such that if 0 < ε ≤ ε0 , then the (p − ε)-Hardy inequality holds in Ω with the constant  C = C(n, p, C1 ) = 2C1 + C2 . Combination of the previous lemmas gives two-sided improvement for Hardy’s inequality. Theorem 6.16. Let 1 < p < ∞ and assume that the p-Hardy inequality holds in an open set Ω  Rn with a constant C1 . Then there exists ε = ε(n, p, C1 ) > 0 such that the q-Hardy inequality holds in Ω whenever p − ε < q < p + ε, with a constant C = C(n, p, C1 ). Proof. Let ε0 = ε0 (n, p, C1 ) > 0 be as in Lemma 6.15 and choose   p − p1 ε = min C1 , ε0 > 0. 4 It follows from Lemma 6.14 and Lemma 6.15 that the q-Hardy inequality holds in  Ω whenever p − ε < q < p + ε, with a constant C = C(n, p, C1 ). We emphasize that the constant in the q-Hardy inequality in Theorem 6.16 is independent of q. 6.4. Capacity density and pointwise Hardy inequalities We characterize validity of the pointwise Hardy inequality with a capacity density condition for the complement of the open set. This is a generalization of the measure density condition in Definition 6.11. Definition 6.17. Let 1 ≤ p < ∞. A closed set E ⊂ Rn satisfies the p-capacity density condition if there exists a constant C > 0 such that



(6.19) capp E ∩ B(x, r), B(x, 2r) ≥ C capp B(x, r), B(x, 2r) for every x ∈ E and r > 0. From Lemma 5.35 we see that (6.19) is equivalent to the requirement that

capp E ∩ B(x, r), B(x, 2r) ≥ C1 r n−p for every x ∈ E and r > 0. We may also assume that 0 < C ≤ 1 in (6.19).

6.4. CAPACITY DENSITY AND POINTWISE HARDY INEQUALITIES

125

Example 6.18. If 1 ≤ p < ∞ and a closed set E ⊂ Rn satisfies the measure density condition (6.10) with a constant C1 , then E satisfies the p-capacity density condition, with a constant C = C(n, p, C1 ). This follows from the lower bound in Lemma 5.35 since

capp E ∩ B(x, r), B(x, 2r) ≥ C(n, p)|E ∩ B(x, r)|r −p ≥ C1 C(n, p)|B(x, r)|r −p = C(n, p, C1 )r n−p for every x ∈ E and r > 0. Lemma 6.19. Let n < p < ∞ and let E ⊂ Rn be a nonempty closed set. Then E satisfies the p-capacity density condition, with a constant C = C(n, p). Proof. Let x ∈ E and r > 0. Let u ∈ Lip0 (B(x, 2r)) with u = 1 in E ∩B(x, r), and let z ∈ ∂B(x, 2r). By Morrey’s inequality, see Theorem 3.19, we have  1 = |u(x)| = |u(x) − u(z)| ≤ C(n, p)r 

|∇u(y)| dy p

 p1 ,

B(x,2r)

and thus



|∇u(y)|p dy ≥ C(n, p)|B(x, 2r)|r −p = C(n, p)r n−p.

B(x,2r)

By taking infimum over all functions u as above, Theorem 5.44 gives

capp E ∩ B(x, r), B(x, 2r) ≥ C(n, p)r n−p .



In Chapter 7, we will encounter more examples of sets that satisfy the p-capacity density condition, see Theorem 7.9 and Example 7.10. Remark 6.20. If 1 ≤ p < q < ∞ and a closed set E ⊂ Rn satisfies the p-capacity density condition, with a constant C1 , then E satisfies the q-capacity density condition, with a constant C = C(n, p, q, C1 ). To see this, let x ∈ E and r > 0, and let u ∈ Lip0 (B(x, 2r)) with

u = 1 in E ∩ B(x, r). By Theorem 5.44, we can test capp E ∩ B(x, r), B(x, 2r) with u. Thus the p-capacity density condition and H¨ older’s inequality imply 

n−p ≤ C capp E ∩ B(x, r), B(x, 2r) ≤ C |∇u(y)|p dy r p

|∇u(y)|q dy

≤ C|B(x, 2r)|1− q = Cr

n− np q

B(x,2r)





B(x,2r)

|∇u(y)| dy q

 pq

 pq ,

B(x,2r)

where C = C(n, p, q, C1 ). A straightforward simplification gives  n−q r ≤ C(n, p, q, C1 ) |∇u(y)|q dy. B(x,2r)

The claim follows from Theorem 5.44 by taking infimum over all functions u as above.

126

6. HARDY’S INEQUALITIES

Remark 6.21. The capacity density condition has several applications in the theory of partial differential equation. For example, the p-capacity density condition for Ωc is stronger than the Wiener criterion

 1  1 capp Ωc ∩ B(y, r), B(y, 2r) p−1 dr

= ∞, r capp B(y, r), B(y, 2r) 0 which characterizes regular boundary points y ∈ ∂Ω for the Dirichlet problem for the p-Laplace equation. See Remark 11.43 and Remark 11.44 for more details, and Chapter 12 for other applications. The p-capacity density condition implies the following boundary Poincar´e inequality. Notice that if the complement of an open set satisfies the measure density condition in (6.10), then Theorem 6.22 follows from Remark 3.18, but under the more general capacity density condition we apply Mazya’s inequality from Theorem 5.47. By approximation, Theorem 6.22 holds for every u ∈ W01,p (Ω). Theorem 6.22. Let 1 < p < ∞ and assume that Ω  Rn is an open set such that the complement Ωc satisfies the p-capacity density condition with a constant np C1 . Let x ∈ Ωc and r > 0. Let 1 ≤ q ≤ n−p for 1 ≤ p < n, and 1 ≤ q < ∞ for n ≤ p < ∞. There exists a constant C = C(n, p, q, C1 ) such that  q1   p1  q p |u(y)| dy ≤ Cr |∇u(y)| dy   B(x,r)

B(x,r)

for every u ∈ Lip0 (Ω). Proof. Let u ∈ Lip0 (Ω). Then Ωc ⊂ {u = 0} = {y ∈ Rn : u(y) = 0}. Since Ω satisfies the p-capacity density condition, we obtain



capp {u = 0} ∩ B(x, r2 ), B(x, r) ≥ capp Ωc ∩ B(x, r2 ), B(x, r) c

≥ C(n, p, C1 )r n−p = C(n, p, C1 )|B(x, r)|r −p . Together with Theorem 5.47, this gives  pq   C(n, p, q) q

|u(y)| dy ≤ |∇u(y)|p dy  capp {u = 0} ∩ B(x, r2 ), B(x, r) B(x,r) B(x,r) ≤ C(n, p, q, C1 )r p 

|∇u(y)|p dy.



B(x,r)

The pointwise p-Hardy inequality holds in an open set if and only if the complement satisfies the p-capacity density condition. In addition, validity of the pointwise p-Hardy inequality in Ω is equivalent to the boundary Poincar´e inequality in Theorem 6.22 and to a mean value version of the pointwise Hardy inequality. Compare also to Theorem 6.12, where the pointwise Hardy inequality was obtained from the measure density condition instead of the capacity density condition. Theorem 6.23. Let 1 < p < ∞ and assume that Ω  Rn is an open set. The following conditions are equivalent. (a) Ωc satisfies the p-capacity density condition. (b) For every x ∈ Ωc and r > 0, and every u ∈ Lip0 (Ω), (6.20)



B(x,r)

|u(y)|p dy ≤ Cr p 

B(x,r)

|∇u(y)|p dy.

6.4. CAPACITY DENSITY AND POINTWISE HARDY INEQUALITIES

127

(c) For every x ∈ Ω and every u ∈ Lip0 (Ω), |uB(x,d(x,∂Ω)) |p ≤ Cd (x, ∂Ω)p 

|∇u(y)|p dy.

B(x,2d(x,∂Ω))

(d) The pointwise p-Hardy inequality holds in Ω. Moreover, the constants in each of the conditions only depend on each other, n and p. Proof. The implication from (a) to (b) follows from Theorem 6.22 with q = p. Next we show that (b) implies (c). Assume that (6.20) holds with a constant C1 . Let u ∈ Lip0 (Ω), x ∈ Ω and R = d (x, ∂Ω). Choose z ∈ ∂Ω with |x − z| = R. Then |uB(x,R) | ≤ |uB(x,R) − uB(z,R) | + |uB(z,R) |. By the (1, p)-Poincar´e inequality, see Theorem 3.14, and the inclusion B(x, R) ∪ B(z, R) ⊂ B(x, 2R), we have |uB(x,R) − uB(z,R) | ≤ |uB(x,R) − uB(x,2R) | + |uB(x,2R) − uB(z,R) | |u(y) − uB(x,2R) | dy

≤ C(n) 

B(x,2R)



≤ C(n, p)R 

|∇u(y)| dy p

 p1 .

B(x,2R)

On the other hand, H¨ older’s inequality and assumption (b) imply   p1   p1 1 |uB(z,R) | ≤  |u(y)|p dy ≤ C1p R  |∇u(y)|p dy B(z,R)

B(z,R)



≤ C(n, p, C1 )R 

|∇u(y)|p dy

 p1

.

B(x,2R)

Condition (c) follows by combining the estimates above. Then we show that (c) implies (d). Assume that (c) holds with a constant C1 . Let u ∈ Lip0 (Ω), x ∈ Ω and R = d (x, ∂Ω). By the estimate in (6.12) and assumption (c), we obtain

1 |u(x)| ≤ |u(x) − uB(x,R) | + |uB(x,R) | ≤ C(n)R M2R |∇u|p (x) p + |uB(x,R) |   p1 1

p1 p p p ≤ C(n)R M2R |∇u| (x) + C1 R  |∇u(y)| dy B(x,2R)

1

1

1 ≤ C(n)R M2R |∇u|p (x) p + C1p R M2R |∇u|p (x) p

1 ≤ C(n, p, C1 )d (x, ∂Ω) M2d(x,∂Ω) |∇u|p (x) p .

Finally, we establish the implication from (d) to (a). Assume that the pointwise p-Hardy inequality (6.4) holds in Ω with a constant C1 . Let x ∈ Ωc and R > 0. By Theorem 5.44 and Lemma 5.35, it suffices to find a constant C = C(n, p, C1 ) such that  n−p ≤C |∇v(y)|p dy (6.21) R B(x,2R)

128

6. HARDY’S INEQUALITIES

whenever v ∈ Lip0 (B(x, 2R)) and v = 1 in Ωc ∩ B(x, R). Moreover, by truncation we may assume that 0 ≤ v ≤ 1, compare to Remark 5.33. Let γ = 16 . If γn v(y) dy > ,  4 B(x,R) then the Sobolev–Poincar´e inequality (3.11) with q = 1, see Theorem 3.17, gives 1 < 4γ −n 

B(x,R)

v(y) dy ≤ C(n) 



≤ C(n, p)R 

|v(y)| dy

B(x,2R)

|∇v(y)| dy p

 p1

,

B(x,2R)

and (6.21) follows. We may hence assume that 

v(y) dy ≤

B(x,R)

γn . 4

Let F = {y ∈ B(x, γR) : v(y) < 12 }. Since v = 1 in Ωc ∩ B(x, R), we have F ⊂ Ω. By definition v ≥ 12 in B(x, γR) \ F , and thus   γn |B(x, R)|. v(y) dy ≤ 2 v(y) dy ≤ |B(x, γR) \ F | ≤ 2 2 B(x,γR)\F B(x,R) This implies |F | = |B(x, γR)| − |B(x, γR) \ F | γn γn ≥ γ n |B(x, R)| − |B(x, R)| = |B(x, R)|. 2 2

(6.22) Let

 ψ(y) = max 0, 1 −

2 Rd



 y, B(x, R2 )

for every y ∈ Rn . Then ψ ∈ Lip0 (B(x, R)) and ψ = 1 in B(x, R2 ). Define u(x) = ψ(x)(1 − v(x)) for every y ∈ R . Then u ∈ Lip0 (Ω) and u = 1 − v in B(x, R2 ). In particular u = 0 in B(x, R2 ) ∩ Ωc and |∇u| = |∇v| almost everywhere in B(x, R2 ). For every z ∈ F there exists a radius 0 < rz ≤ 2d (z, ∂Ω) such that n

M2d(z,∂Ω) |∇u|p (z) ≤ 2 

|∇u(y)|p dy.

B(z,rz )

By Lemma 1.13, there exist pairwise disjoint balls B(zi , ri ), where zi ∈ F and ∞ ri = rzi are as above, such that F ⊂ i=1 B(zi , 5ri ). It follows from (6.22) that (6.23)

|B(x, R)| ≤

∞

2 |F | ≤ C(n) |B(zi , ri )|. γn i=1

Let i ∈ N. Since zi ∈ F ∩ B(x, γR) ⊂ Ω ∩ B(x, γR) and x ∈ / Ω, we have d (zi , ∂Ω) < γR. If y ∈ B(zi , ri ), then |x − y| ≤ |x − zi | + |zi − y| ≤ γR + 2d (zi , ∂Ω) < γR(1 + 2) =

R , 2

6.5. WANNEBO’S APPROACH

129

and thus B(zi , ri ) ⊂ B(x, R2 ). This implies that |∇u| = |∇v| almost everywhere in B(zi , ri ). Since zi ∈ F , we have u(zi ) = 1 − v(zi ) > 12 . The pointwise p-Hardy inequality and the choice of the radius ri imply  1 p ≤ |u(zi )|p ≤ C1p d (zi , ∂Ω)p M2d(zi ,∂Ω) |∇u|p (zi ) 2 ≤ C1p γ p Rp 

|∇u(y)|p dy,

B(zi ,ri )

and consequently

 |B(zi , ri )| ≤ C(p, C1 )R

|∇v(y)|p dy

p B(zi ,ri )

for every i ∈ N. Substituting these estimates into (6.23) gives ∞ 

p |B(x, R)| ≤ C(n, p, C1 )R |∇v(y)|p dy i=1

B(zi ,ri )

|∇v(y)|p dy,

≤ C(n, p, C1 )Rp B(x,2R)

where we also used the fact that the balls B(zi , ri ) ⊂ B(x, 2R) are pairwise disjoint. This shows that (6.21) holds, and the proof is complete.  Theorem 6.23 settles the question of the validity of pointwise p-Hardy inequalities, for 1 < p < ∞. By Remark 6.6, this also implies sufficient conditions for q-Hardy inequalities for p < q. However, Theorem 6.23 does not characterize the validity of the p-Hardy inequality (6.1). For instance, let Ω = B(0, 1) \ {0} as in Example 6.13. Then the complement Ωc satisfies the p-capacity density condition only when p > n. Indeed, if 1 < p ≤ n and 0 < r < 12 , then by monotonicity and the upper bounds in parts (a) and (c) of Lemma 5.36, we obtain



capp Ωc ∩ B(0, 12 ), B(0, 1) = capp {0}, B(0, 1)

r→0 ≤ capp B(0, r), B(0, 1) −−−→ 0.

Since capp B(0, 12 ), B(0, 1) > 0, it follows that Ωc does not satisfy the p-capacity density condition when 1 < p ≤ n. Hence we conclude from Theorem 6.23 that the pointwise p-Hardy inequality holds in Ω if and only if p > n. On the other hand, by Example 6.13 the p-Hardy inequality holds in Ω also when 1 < p < n. In particular, this shows that the pointwise and integral versions of the p-Hardy inequality are not equivalent. 6.5. Wannebo’s approach If 1 ≤ q < p and the pointwise q-Hardy inequality holds in Ω ⊂ Rn , then the p-Hardy inequality holds in Ω, see Remark 6.6. In this section we give a direct proof for the fact that the p-capacity density condition for the complement implies the p-Hardy inequality with the same p > 1. By the equivalence in Theorem 6.23, this also shows that the pointwise p-Hardy inequality implies the p-Hardy inequality, see Corollary 6.26. We apply a quantitative version of a weighted Hardy inequality under the assumption that the complement satisfies the capacity density condition.

130

6. HARDY’S INEQUALITIES

Lemma 6.24. Let 1 < p < ∞ and β < 0. Assume that Ω  Rn is an open set and that Ωc satisfies the p-capacity density condition with a constant C1 . There exists a constant C = C(n, p, β, C1 ) such that   (6.24) Ω

|u(x)| d (x, ∂Ω)

p

 d (x, ∂Ω)β dx ≤ C

|∇u(x)|p d (x, ∂Ω)β dx Ω

for every u ∈ Lipc (Ω). Moreover, for − 12 < β < 0, we have C ≤ |β|−1 C(n, p, C1 ). Proof. For j ∈ Z, let Aj = {x ∈ Ω : 2−j−1 ≤ d (x, ∂Ω) < 2−j } and ∞ 

Ωj =

Ak = {x ∈ Ω : d (x, ∂Ω) < 2−j }.

k=j

Let j ∈ Z. By Lemma exists a collection {B(xi , 2−j ) : i ∈ N, xi ∈ Aj } ∞ 1.13, there −j such that Aj ⊂ i=1 B(xi , 2 ) and the balls B(xi , 15 2−j ), i ∈ N, are pairwise disjoint. For each B(xi , 2−j ) there exists B(yi , 2−j+1 ), with yi ∈ ∂Ω, such that B(xi , 2−j ) ⊂ B(yi , 2−j+1 ) and B(yi , 2−j+1 ) ∩ Ω ⊂ Ωj−1 . Since the measures of B(xi , 15 2−j ) and B(yi , 2−j+1 ) are comparable and the balls B(xi , 15 2−j ) are pairwise disjoint, the balls B(yi , 2−j+1 ), i ∈ N, have bounded overlap, that is, ∞

χB(yi ,2−j+1 ) (x) ≤ C(n) < ∞,

i=1

for every x ∈ Rn . Let u ∈ Lipc (Ω). Since Ωc satisfies the p-capacity density condition, it follows from the case q = p of the boundary Poincar´e inequality in Theorem 6.22 that 

 |u(x)| dx ≤ p

B(xi ,2−j )

|u(x)|p dx  ≤ C(n, p, C1 )2−jp B(yi ,2−j+1 )

B(yi ,2−j+1 )

|∇u(x)|p dx.

By summing up these inequalities over i ∈ N and using the bounded overlap of the corresponding balls and the fact that ∇u = 0 in Ωc , we obtain  |u(x)|p dx ≤ Aj

∞ 

i=1

B(xi ,2−j )

|u(x)|p dx

≤ C(n, p, C1 )2−jp ≤ C(n, p, C1 )2−jp −jp

= C(n, p, C1 )2

∞ 

i=1

B(yi

,2−j+1 )

|∇u(x)|p dx

|∇u(x)|p dx Ωj−1 ∞

k=j−1

 |∇u(x)|p dx, Ak

6.5. WANNEBO’S APPROACH

for every j ∈ Z. This implies   ∞

|u(x)|p d (x, ∂Ω)β−p dx ≤ 2p−β 2jp−jβ Ω

|u(x)|p dx

Aj

j=−∞

≤ C(n, p, C1 )

131

∞

2p−β 2−jβ

j=−∞

= C(n, p, C1 )2p

∞

∞ 

k=j−1

k+1

2−(j+1)β

k=−∞ j=−∞ ∞ −(k+2)β

|∇u(x)|p dx

Ak

 |∇u(x)|p dx Ak

 2 |∇u(x)|p dx ≤ C(n, p, C1 )2 1 − 2β Ak k=−∞ ∞  −2β

2 ≤ C(n, p, C1 ) |∇u(x)|p d (x, ∂Ω)β dx 1 − 2β k=−∞ Ak −2β  2 |∇u(x)|p d (x, ∂Ω)β dx. = C(n, p, C1 ) 1 − 2β Ω p

−2β

2 Thus (6.24) holds with the constant C = C(n, p, C1 ) 1−2 β. 1 −2β ≤ 2 and Finally, if − 2 < β < 0, then 2  0 log 2 β 1 − 2 = log 2 et log 2 dt ≥ log 2 · e− 2 |β|. β

Therefore

log 2 2 2−2β , ≤ C(n, p, C1 )e 2 1 − 2β |β| log 2 proving the claim concerning the constant for these values of β.

C = C(n, p, C1 )



Theorem 6.25. Let 1 < p < ∞. Assume that Ω  Rn is an open set and that Ω satisfies the p-capacity density condition, with a constant C1 . Then the p-Hardy inequality holds in Ω with a constant C(n, p, C1 ). c

Proof. By Lemma 6.2 (b) we may assume that u ∈ Lipc (Ω). Let − 12 < β < 0 and

β

v(x) = u(x)d (x, ∂Ω)− p , for every x ∈ Rn . Then v ∈ Lipc (Ω) and by the Leibniz rule (2.4) and the chain rule (2.5) we have β

|∇v(x)| ≤ |∇u(x)|d (x, ∂Ω)− p +

β |β| |u(x)|d (x, ∂Ω)− p −1 p

for almost every x ∈ Ω. By Lemma 6.24, the weighted Hardy inequality (6.24) holds for v ∈ Lipc (Ω), and we obtain   |u(x)|p |v(x)|p dx = d (x, ∂Ω)β dx p p Ω d (x, ∂Ω) Ω d (x, ∂Ω)  C(n, p, C1 ) ≤ |∇v(x)|p d (x, ∂Ω)β dx (6.25) |β| Ω   C(n, p, C1 ) |u(x)|p C(n, p, C1 ) |β|p ≤ |∇u(x)|p dx + dx. |β| |β| pp Ω d (x, ∂Ω)p Ω

132

6. HARDY’S INEQUALITIES

We choose − 12 < β < 0 to be so close to zero that C(n, p, C1 ) |β|p 1 = C(n, p, C1 )p−p |β|p−1 ≤ . |β| pp 2 Since u ∈ Lipc (Ω), the last integral on the right-hand side of (6.25) is finite. Thus the last term in (6.25) can be absorbed to the left-hand side, and we arrive at   |u(x)|p dx ≤ C(n, p, C1 ) |∇u(x)|p dx.  p Ω d (x, ∂Ω) Ω Corollary 6.26. Let 1 < p < ∞ and assume that the pointwise p-Hardy inequality holds in an open set Ω  Rn . Then the p-Hardy inequality holds in Ω. Proof. By Theorem 6.23, Ωc satisfies the p-capacity density condition. Thus Theorem 6.25 implies that the p-Hardy inequality holds in Ω.  6.6. Stability of Sobolev spaces with zero boundary values There is a close connection between Hardy’s inequalities and Sobolev spaces with zero boundary values. By Lemma 4.10, finiteness of the right-hand side of Hardy’s inequality implies that a function has zero boundary values in the Sobolev sense. This condition is necessary and sufficient if the complement satisfies the capacity density condition. Theorem 6.27. Let 1 < p < ∞ and let Ω  Rn be an open set. Assume that u ∈ W 1,p (Ω) and that the complement Ωc satisfies the p-capacity density condition. Then u ∈ W01,p (Ω) if and only if  |u(x)|p dx < ∞. (6.26) p Ω d (x, ∂Ω) Proof. The sufficiency of the condition in (6.26) was proved in Lemma 4.10. To prove the necessity, assume that u ∈ W01,p (Ω). Theorem 6.25 implies that the p-Hardy inequality holds in Ω, with a constant C, and by Lemma 6.2 (c) we have   |u(x)|p dx ≤ C |∇u(x)|p dx < ∞.  p Ω d (x, ∂Ω) Ω The main purpose of this section is to find when the Sobolev spaces with zero boundary values satisfy the stability property (6.27)

W01,q (Ω) = W01,p (Ω),

W 1,p (Ω) ∩ 1 0, by Theorem 5.42 (b) we have dimH (K) ≥ n−p. We show that (6.27) does not hold for Ω = B(0, 2) \ K. Let 1 < q < p. Since dimH (K) = n − p < n − q, Theorem 5.42 (b) implies Capq (K) = 0. Let   u(x) = max 0, 1 − d (x, B(0, 1)) for every x ∈ Rn . Then u ∈ Lipc (Rn ) ⊂ W 1,q (Rn ) and u = 0 q-quasieverywhere in Ωc since Capq (K) = 0 and u = 0 everywhere in Rn \ B(0, 2). Corollary 5.29 implies that u ∈ W01,q (Ω) for every 1 < q < p. As u ∈ W 1,p (Ω), u belongs to the left-hand side of (6.27). Assume, for a contradiction, that u ∈ W01,p (Ω). By Theorem 5.30 there exists v ∈ W 1,p (Rn ) such that v = u almost everywhere in Ω and v(y) dy = 0

lim 

r→0

B(x,r)

for p-quasievery x ∈ Ωc . On the other hand, since |K| = 0 and v = u almost everywhere in B(0, 1) \ K ⊂ Ω, by the definition of u we have lim 

r→0

B(x,r)

v(y) dy = lim  r→0

u(y) dy = 1

B(x,r)

for every x ∈ K ⊂ Ωc . As Capp (K) > 0, we arrive at a contradiction with the / W01,p (Ω). claim that this limit is zero for p-quasievery x ∈ Ωc . Hence u ∈ If p > n, then (6.27) holds for all bounded open sets Ω ⊂ Rn . Theorem 6.29. Let n < p < ∞ and let Ω ⊂ Rn be an open set. Assume that u ∈ W 1,p (Ω) and that u ∈ W01,q (Ω) for every 1 < q < p. Then u ∈ W01,p (Ω). Proof. Let n < q < p. Since u ∈ W01,q (Ω), the zero extension u0 of u belongs to W 1,q (Rn ) by Theorem 2.31. Furthermore, the weak gradient ∇u0 equals almost everywhere in Rn to the zero extension of the weak gradient ∇u. Thus u0 W 1,p (Rn ) = uW 1,p (Ω) < ∞, since u ∈ W 1,p (Ω). This shows that u0 ∈ W 1,p (Rn ). Theorem 3.23 implies that u0 ∈ W 1,p (Rn ) has a continuous representative u∗ , and by continuity u∗ (x) = lim  r→0

u0 (y) dy

B(x,r)

for every x ∈ Rn . Since all points in Rn have positive q-capacity by Lemma 5.9, we obtain from Theorem 5.26 a continuous function v ∈ W 1,q (Rn ) satisfying v = u almost everywhere in Ω and v = 0 everywhere in Rn \ Ω. Then u∗ = u0 = v almost everywhere in Rn , and by continuity u∗ = v everywhere in Rn . Thus u∗ ∈ W 1,p (Rn ) is continuous, u∗ = u almost everywhere in Ω, and u∗ = 0 in Rn \ Ω. The claim  u ∈ W01,p (Ω) follows from Corollary 5.29. As Example 6.28 indicates, in the case 1 < p < n some additional conditions must be imposed on Ω in order to obtain (6.27).

134

6. HARDY’S INEQUALITIES

Definition 6.30. Let 1 < p < ∞. The pointwise upper p-capacity density of a set E ⊂ Rn at a point x ∈ Rn is

capp E ∩ B(x, r), B(x, 2r) densp (E, x) = lim sup . r n−p r→0 A closed set E ⊂ Rn satisfies the pointwise p-capacity density condition, with exponents P ⊂ (1, p), if for p-quasievery x ∈ E there exists q ∈ P such that densq (E, x) > 0. The pointwise capacity density condition for the complement implies a stability result for Sobolev functions with zero boundary values. Theorem 6.31. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. Assume that Ω satisfies the pointwise p-capacity density condition, with exponents P ⊂ (1, p), and that u ∈ W 1,p (Ω) and u ∈ W01,q (Ω) for every q ∈ P . Then u ∈ W01,p (Ω). c

Proof. Let q ∈ P . As in the proof of Theorem 6.29, the zero extension u0 of u belongs to W 1,q (Rn ) ∩ W 1,p (Rn ). Define u∗ : Rn → R by  limr→0 B(x,r) u0 (y) dy, if x ∈ Leb(u0 ), ∗ u (x) = 0, if x ∈ / Leb(u0 ). Theorem 5.23 implies that u∗ is both a p-quasicontinuous and a q-quasicontinuous representative of u0 ∈ W 1,p (Rn ) ∩ W 1,q (Rn ). Thus u∗ = 0 holds q-quasieverywhere in Ωc for every q ∈ P , by Theorem 5.26 and Theorem 5.16. By the assumptions, p-quasievery point x ∈ Ωc satisfies densq (Ωc , x) > 0 for some q ∈ P . Fix such x ∈ Ωc and a corresponding q ∈ P , and choose δ = 1 c 2 densq (Ω , x) > 0. Then there is a sequence (ri )i∈N with limi→∞ ri = 0 and

(6.28) δrin−q ≤ capq Ωc ∩ B(x, ri ), B(x, 2ri ) for every i ∈ N. The q-quasicontinuity of u∗ and Theorem 5.47 give |u∗ (y)| dy



B(x,2ri )



≤

C(n, q)

capq {u∗ = 0} ∩ B(x, ri ), B(x, 2ri )



|∇u∗ (y)|q dy

 1q .

B(x,2ri )

Since u∗ = 0 q-quasieverywhere in Ωc , using the monotonicity and subadditivity of the variational capacity with Theorem 5.38, we obtain from (6.28)



capq {u∗ = 0} ∩ B(x, ri ), B(x, 2ri ) ≥ capq Ωc ∩ B(x, ri ), B(x, 2ri ) ≥ δrin−q for every i ∈ N. By H¨ older’s inequality  ∗ |u (y)| dy ≤ C r q   B(x,2ri )

∗

q

∗

p

|∇u (y)| dy

 1q

B(x,2ri )

 ≤ C rp 

|∇u (y)| dy

 p1 ,

B(x,2ri )

where C = C(n, q, δ). As u∗ = u0 and ∇u∗ = ∇u0 almost everywhere in Rn , we have   p1 p p (6.29) lim inf  |u0 (y)| dy ≤ C lim sup r  |∇u0 (y)| dy r→0

B(x,r)

r→0

B(x,r)

6.7. NOTES

135

for p-quasievery x ∈ Ωc , where the constant C = C(n, q, δ) may depend on x via q and δ. By Lemma 5.22, the right-hand side of (6.29) is zero p-quasieverywhere in Rn . Thus u0 (y) dy ≤ lim inf  |u0 (y)| dy = 0 |u∗ (x)| = lim  r→0 r→0 B(x,r)

B(x,r)

for p-quasievery x ∈ Ωc , by Theorem 5.23, and hence u∗ = 0 p-quasieverywhere in Ωc . Since u∗ ∈ W 1,p (Rn ) is p-quasicontinuous and u∗ = u almost everywhere in Ω, we conclude from Corollary 5.29 that u ∈ W01,p (Ω).  As a consequence of Theorem 6.31 we obtain the following characterization of W01,p (Ω) under the pointwise p-capacity density condition. Theorem 6.32. Let 1 < p < ∞ and let Ω ⊂ Rn be a bounded open set. Assume that u ∈ W 1,p (Ω) and that Ωc satisfies the pointwise p-capacity density condition, with exponents P ⊂ (1, p). Then u ∈ W01,p (Ω) if and only if u ∈ W01,q (Ω) for every q ∈ P. Proof. If u ∈ W01,p (Ω), then u ∈ W01,q (Ω) for every q ∈ P by H¨older’s inequality and the fact that W01,p (Ω) is the completion of C0∞ (Ω) in W 1,p (Ω). The assumption that Ω is bounded is needed here. The reverse implication follows directly from Theorem 6.31.  Remark 6.33. Let 1 < p < ∞ and assume that Ω ⊂ Rn is a bounded open set and that Ωc satisfies the p-capacity density condition. In Chapter 7 we will show that then there is 1 < q < p such that the q-capacity density condition holds for Ωc , see Theorem 7.21. Hence Ωc satisfies the pointwise p-capacity density condition with P = {q} ⊂ (1, p). Thus we obtain from Theorem 6.27 and Theorem 6.32 that under these assumptions the following conditions are equivalent for a function u ∈ W 1,p (Ω): (a) u ∈ W01,p (Ω), (b) u ∈ W01,q (Ω) for every 1 < q < p,  |u(x)|p (c) Ω d(x,∂Ω) p dx < ∞. 6.7. Notes Several sufficient Lipschitz and H¨ older type boundary conditions for Hardy’s inequality have been given by Balinsky, Evans and Lewis [22], Edmunds and Evans [115, 116], Kufner [248], Neˇcas [337] and Opic and Kufner [340]. Capacitary characterizations of Hardy’s inequality are given in Mazya [318, 321], see also Section 10.7. For details on the origins and history of one-dimensional Hardy’s inequalities, we refer to Hardy, Littlewood and P´olya [180, Chapter IX] and Kufner, Maligranda and Persson [249]. General theory of one-dimensional Hardy-type inequalities can be found in Kufner and Persson [250]. The measure density condition plays a role in harmonic analysis, function spaces and the regularity theory up to the boundary for partial differential equations, see Brudnyi and Brudnyi [59, 60], Giaquinta [147, 148], Giusti [153], Heinonen, Kilpel¨ainen and Martio [187], Jonsson and Wallin [215] and Shvartsman [361]. Theorem 6.25 for p = 2 is by Ancona [14] and for p ≥ 1 by Lewis [271] and Wannebo [393]. See also Mikkonen [327]. The proof of Theorem 6.25 in Section 6.5

136

6. HARDY’S INEQUALITIES

is based on Wannebo [393], see also Buckley and Koskela [64]. Hardy’s inequalities for functions vanishing only on a part of the boundary are considered by Egert, Haller-Dintelmann and Rehberg [118]. The pointwise Hardy inequality has been studied by Hajlasz [166] and Kinnunen and Martio [235]. Weighted pointwise Hardy inequalities are considered in Koskela and Lehrb¨ ack [243]. Self-improvement of the pointwise Hardy inequality is proved by Eriksson-Bique and V¨ ah¨ akangas [119]. Theorem 6.23 has been studied by Kinnunen and Korte [226], Korte, Lehrb¨ack and Tuominen [241] and Lehrb¨ ack [256]. More characterizations in the borderline case p = n are discussed in Theorem 7.20. Section 6.3 is from Koskela and Zhong [246], see also Lehrb¨ ack [257] for weighted results. Section 6.6 is based on Hedberg and Kilpel¨ainen [182]. Many arguments related to Hardy’s inequality are based on general principles which apply even on metric measure spaces, see Bj¨ orn, MacManus and Shanmugalingam [36], Kilpel¨ainen, Kinnunen and Martio [219], Korte, Lehrb¨ack and Tuominen [241], Korte and Shanmugalingam [242], Koskela and Zhong [246] and Lehrb¨ack [260, 261]. In [95] Danielli, Garofalo and Phuc studied Hardy’s inequalities in Carnot–Carath´eodory spaces. We refer to Buckley and Koskela [64] and Cianci [86] for results in Orlicz–Sobolev spaces. For other function spaces, see Edmunds, Hurri-Syrj¨ anen and V¨ ah¨ akangas [117], Ihnatsyeva, Lehrb¨ ack, Tuominen and V¨ ah¨ akangas [201] and Ihnatsyeva and V¨ah¨akangas [203, 204].

10.1090/surv/257/07

CHAPTER 7

Density Conditions Geometric characterizations of the capacity density condition in Definition 6.17 are given in terms of Hausdorff content, Ahlfors–David regular sets and the lower dimension in Section 7.1, Section 7.2 and Section 7.3, respectively. The case p > n is not of interest since by Lemma 6.19 every nonempty closed set satisfies the p-capacity density condition. On the other hand, the borderline case p = n is special for density conditions and also for Hardy’s inequalities. Several analytic, metric and geometric characterizations for the n-capacity density condition and the n-Hardy inequality are given in Section 7.4. These include a self-improvement property for the n-capacity density condition. Section 7.5 discusses a corresponding self-improvement property of the capacity density condition for 1 < p < n. The argument is completed in Section 7.6, Section 7.7 and Section 7.8. 7.1. Hausdorff content density A density condition for the Hausdorff content is analogous to the measure and capacity density conditions, see Definition 6.11 and Definition 6.17. Recall the definitions of the Hausdorff content and the Hausdorff measure from Definition 1.43. Definition 7.1. Let 0 ≤ λ ≤ n. A set E ⊂ Rn satisfies the λ-Hausdorff content density condition if there exists a constant C > 0 such that

λ (7.1) H∞ E ∩ B(x, r) ≥ Cr λ for every x ∈ E and r > 0. For λ = n, the n-dimensional Hausdorff content is equivalent to the n-dimensional Lebesgue outer measure, and hence n-Hausdorff content density condition is equivalent to the measure density condition. Observe that if a set ∅ = E ⊂ Rn satisfies the λ-Hausdorff content density condition with λ > 0, then E is unbounded. Remark 7.2. Let 0 ≤ λ1 < λ2 and assume that E ⊂ Rn satisfies the λ2 Hausdorff content density condition, with a constant C1 . Let x ∈ E and r > 0. Lemma 5.40 implies



λ2 λ1 E ∩ B(x, r) ≤ r λ2 −λ1 H∞ E ∩ B(x, r) , (7.2) C1 r λ2 ≤ H∞

λ1 and thus H∞ E ∩ B(x, r) ≥ C1 r λ1 . This shows that E satisfies the λ1 -Hausdorff content density condition, with the same constant C1 . Theorem 7.3. Let 1 < p < ∞ and 0 ≤ λ ≤ n, with λ > n − p, and assume that a closed set E ⊂ Rn satisfies the λ-Hausdorff content density condition, with a constant C1 . Then E satisfies the p-capacity density condition, with a constant C = C(n, p, λ, C1). 137

138

7. DENSITY CONDITIONS

Proof. For p > n, the claim holds by Lemma 6.19. Hence we may assume 1 < p ≤ n. Theorem 5.41 and (7.1) imply



λ E ∩ B(x, r) r n−p−λ capp E ∩ B(x, r), B(x, 2r) ≥ C(n, p, λ)H∞ ≥ C(n, p, λ, C1)r n−p for every x ∈ E and r > 0.



Remark 7.4. Theorem 7.3 has also a converse, see Theorem 7.22, but this is based on the self-improvement property of the capacity density condition discussed in Section 7.5. However, we observe that if 1 < p ≤ n and a closed set E ⊂ Rn satisfies the p-capacity density condition, with a constant C1 , then by Theorem 5.39 we have



n−p E ∩ B(x, r) r n−p ≤ C(n, p, C1 ) capp E ∩ B(x, r), B(x, 2r) ≤ C(n, p, C1 )H∞ for every x ∈ E and r > 0. Hence E satisfies the λ-Hausdorff content density condition with λ = n − p. The Hausdorff content density condition gives a sufficient condition for Hardy’s inequality without applying Theorem 6.25. Corollary 7.5. Let 1 < p < ∞ and 0 ≤ λ ≤ n, with λ > n − p. Assume that Ω  Rn is an open set and that Ωc satisfies the λ-Hausdorff content density condition. Then the pointwise p-Hardy inequality and the p-Hardy inequality hold in Ω. Proof. Theorem 7.3 implies that Ωc satisfies the q-capacity density condition with p > q > max{1, n − λ}. The pointwise q-Hardy inequality holds in Ω by Theorem 6.23 and the p-Hardy inequality follows as in Remark 6.6.  7.2. Ahlfors–David regular sets Ahlfors–David regularity implies the capacity density condition, see Theorem 7.9. Examples of Ahlfors–David regular sets include subspaces and compact submanifolds of Rn and many self-similar fractals. Definition 7.6. Let 0 < λ ≤ n. A closed set E ⊂ Rn is (Ahlfors–David) λ-regular, or a λ-set, if there exists a constant C ≥ 1 such that

(7.3) C −1 r λ ≤ Hλ E ∩ B(x, r) ≤ Cr λ for every x ∈ E and 0 < r ≤ diam(E). It is easy to see that if E is λ-regular, then dimH (E) = λ. Cantor type constructions show that there exist λ-regular sets for every λ with 0 < λ ≤ n. We refer to Falconer [125] and Mattila [315] for more details concerning fractal geometry and self-similar sets. The next remark is related to the definitions of the Hausdorff content and the Hausdorff measure, see Definition 1.43. Remark 7.7. In Definition 1.43 it is sometimes convenient to impose the additional restriction that the covering balls are centered in the set. This leads to quanλ and Hλ with two-sided bounds. Let 0 < δ ≤ ∞ tities which are comparable to H∞ and let {B(xi , Ri ) : i ∈ N} be a cover of E ⊂ Rn , with 0 < Ri ≤ δ2 for every i ∈ N. We may assume that for every i ∈ N there exists zi ∈ E ∩ B(xi , Ri ) since otherwise

7.2. AHLFORS–DAVID REGULAR SETS

139

B(xi , Ri ) is redundant in the cover. Then B(xi , Ri ) ⊂ B(zi , 2Ri ), and the balls B(zi , 2Ri ), i ∈ N, cover the set E. This implies  ∞ ∞ ∞ ∞



riλ : E ⊂ B(yi , ri ), yi ∈ E, 0 < ri ≤ δ ≤ (2Ri )λ = 2λ Riλ . inf i=1

i=1

i=1

i=1

By taking infimum over all such covers of E, we obtain ∞  ∞

 λ λ ri : E ⊂ B(yi , ri ), yi ∈ E, 0 < ri ≤ δ ≤ C(λ)Hλδ (E). Hδ (E) ≤ inf i=1

2

i=1

λ (E), and at the limit δ → 0 we obtain For δ = ∞ we have two-sided bounds by H∞ λ two-sided bounds by H (E).

By the following lemma, unbounded λ-regular sets satisfy the λ-Hausdorff content density condition. Lemma 7.8. Let 0 < λ ≤ n and assume that a closed set E ⊂ Rn is λ-regular with a constant C1 . There exists a constant C = C(λ, C1 ) such that

λ E ∩ B(x, r) ≥ Cr λ H∞ for every x ∈ E and 0 < r ≤ diam(E). Proof. Fix x ∈ E and 0 < r ≤ diam(E). Let {B(xi , ri ) : i ∈ N} be a cover of E ∩ B(x, r), with xi ∈ E ∩ B(x, r) ⊂ E and ri > 0 for every i ∈ N. If rj > diam(E) for some j ∈ N, then ∞

riλ ≥ rjλ > diam(E)λ ≥ r λ . i=1

On the other hand, if 0 < ri ≤ diam(E) for every i ∈ N, we apply (7.3) and obtain ∞ ∞





r λ ≤ C1 Hλ E ∩ B(x, r) ≤ C1 Hλ E ∩ B(xi , ri ) ≤ C12 riλ . i=1

i=1

By taking infimum over all such covers, and recalling that by Remark 7.7 it suffices to consider balls centered at E ∩ B(x, r), we conclude that

λ E ∩ B(x, r) . r λ ≤ C(λ, C1 )H∞  Theorem 7.9. Let 1 < p < ∞ and 0 < λ ≤ n, with λ > n − p, and assume that E ⊂ Rn is a closed and unbounded λ-regular set. Then E satisfies the p-capacity density condition. Proof. Since E is unbounded, Lemma 7.8 implies that the λ-Hausdorff content density condition in (7.1) holds for every x ∈ E and r > 0. The claim follows from Theorem 7.3.  Example 7.10. Assume that Ω0 ⊂ Rn is a bounded open set whose complement satisfies the measure density condition. Let 0 < λ ≤ n, let K ⊂ Ω0 be a compact λ-regular set, with a constant C1 , and define Ω = Ω0 \ K. We claim that Ωc satisfies the λ-Hausdorff content density condition in (7.1). For x ∈ Ωc0 , this follows from Remark 7.2 since the measure density condition is equivalent to the n-Hausdorff content density condition and λ ≤ n. Assume then that x ∈ K. For 0 < r ≤ 2 diam(Ω0 ) the λ-Hausdorff content density condition, with a constant depending on C1 , λ, diam(K) and diam(Ω0 ), follows from the λ-regularity of K Ωc0

140

7. DENSITY CONDITIONS

and Theorem 7.8. On the other hand, if r > 2 diam(Ω0 ), then the measure density condition of Ωc0 implies the λ-Hausdorff content density condition. By Theorem 7.3, the complement Ωc satisfies the p-capacity density condition for every p > max{1, n − λ}. Consequently, the pointwise p-Hardy inequality holds in Ω for all such p. 7.3. Lower dimension and capacity density The λ-Hausdorff content density condition holds for a set E ⊂ Rn if and only if there is a constant C > 0 such that if {B(xi , ri ) : i ∈ N} is a cover of E ∩ B(x, R), where x ∈ E and R > 0, then ∞

(7.4) riλ ≥ CRλ . i=1

Next we consider this condition for finite covers of E ∩ B(x, R) with balls that have a fixed radius 0 < ri = r ≤ R. For such covers {B(xi , r) : i = 1, . . . , N }, where N ∈ N, the condition corresponding to (7.4) is  λ N

R λ λ r ≥ CR , or equivalently, N ≥ C . r i=1 This leads to the definition of the lower dimension. It turns out that this concept of dimension can be used to characterize the Hausdorff content density condition and thus also the capacity density condition, see Theorem 7.14 and Theorem 7.22. See also Section 10.3 for the dual concept, the Assouad dimension. Definition 7.11. The lower dimension dimL (E) of a set E ⊂ Rn is the supremum of the numbers λ ≥ 0 for which there exists a constant C > 0 such that  λ

R (7.5) N E ∩ B(x, R), r ≥ C r for every x ∈ E and 0 < r < R ≤ diam(E). Here N (·, r) denotes the minimal number of open balls of radius r that are needed to cover the set. In particular, (7.5) holds whenever 0 ≤ λ < dimL (E), and possibly also for λ = dimL (E). Observe that 0 ≤ dimL (E) ≤ n for every E ⊂ Rn . In the case E = {x0 }, x0 ∈ Rn , we remove the requirement R ≤ diam(E) from the definition above and thus dimL ({x0 }) = 0. It should be noted that unlike many other notions of dimension, the lower dimension is not monotone. For instance, if E = {0} ∪ [1, 2] ⊂ R, then dimL (E) = 0 due to the isolated point 0, but for the subset [1, 2] ⊂ E we have dimL ([1, 2]) = 1. The latter fact is a special case of the following theorem. Theorem 7.12. Let 0 < λ ≤ n and assume that a closed set E ⊂ Rn is λ-regular. Then dimL (E) = λ. Proof. Fix x ∈ E and 0 < r < R ≤ diam(E), and consider balls B(xi , r), i = 1, . . . , N , which cover E ∩ B(x, R). By Lemma 7.8 and the definition of the Hausdorff content, we obtain N



λ E ∩ B(x, R) ≤ C r λ = CN r λ . Rλ ≤ CH∞

Thus N ≥ C

R λ r

i=1

, which implies dimL (E) ≥ λ.

7.3. LOWER DIMENSION AND CAPACITY DENSITY

141

On the other hand, by Lemma 1.13 and boundedness, there exist N0 ∈ N and  0 points xi ∈ E ∩ B(x, R), i = 1, 2, . . . , N0 , such that E ∩ B(x, R) ⊂ N i=1 B(xi , r) r and the balls B(xi , 5 ), i = 1, . . . , N0 , are pairwise disjoint. If 2R ≤ diam(E), then by the λ-regularity we have N0 N0





Hλ E ∩ B xi , r5 ≥ C r λ = CN0 r λ . Rλ ≥ CHλ E ∩ B(x, 2R) ≥ C i=1

i=1

The same estimate also holds if 2R > diam(E) since in this case



Hλ E ∩ B(x, 2R) ≤ Hλ E ∩ B(x, diam(E)) ≤ C diam(E)λ ≤ CRλ . λ Thus N0 ≤ C Rr . Here the constant C is independent of x, r and R. It follows that the lower bound in (7.5) cannot hold for any exponent larger than λ. Hence  dimL (E) = λ, as desired. The following lemma gives a connection between the lower dimension and the Hausdorff content density condition. Roughly speaking, the lemma asserts that the estimate in (7.6), for covers using balls of fixed radii r, implies a corresponding estimate (7.7) for covers where balls of all radii are allowed. However, we obtain the latter only for dimensional parameters 0 ≤ λ < λ0 , and so the endpoint λ0 is lost. Lemma 7.13. Let E ⊂ Rn be a closed set. Assume that there exist 0 < λ0 ≤ n and a constant C1 > 0 such that  λ0

R (7.6) N E ∩ B(x, R), r ≥ C1 r for every x ∈ E and 0 < r < R ≤ diam(E). Then, for every 0 ≤ λ < λ0 , there exists a constant C = C(n, λ0 , λ, C1 ) > 0 such that

λ E ∩ B(x, R) ≥ CRλ (7.7) H∞ for every x ∈ E and 0 < R ≤ diam(E). Proof. Let 0 ≤ λ < λ0 , x ∈ E and 0 < R ≤ diam(E). We may assume C1 < 1. Choose K ∈ N to be so large that λ
11 m = 11 C1 gives the ratio of the radii used in the consecutive steps of the construction. Notice that log K λ< < λ0 . log m R 11 .

Let B 0 = B(x, R). In the first step of the construction we take r1 = Rm−1 < Using the assumption (7.6) and Lemma 1.13 with radius 11 5 r1 , we find closed

142

7. DENSITY CONDITIONS

balls B i = B(zi , r1 ), i = 1, 2, . . . , N0 , with zi ∈ E ∩ B 0 , such that the balls B(zi , 2r1 ) ⊂ B(x, 2R), i = 1, . . . , N0 , are pairwise disjoint, E ∩ B0 ⊂

N0 

B(zi , 11r1 ),

i=1

and N0 (11r1 )λ0 ≥ C1 Rλ0 . By the choices of r1 and m, we have N0 ≥ K. We proceed the construction with the balls B i , i = 1, . . . , K, and omit the rest of them. r1 . We find balls B i1 i2 = B(zi1 i2 , r2 ), In the second step we take r2 = Rm−2 < 11 i1 = 1, 2, . . . , K and i2 = 1, 2, . . . , Ni1 , with zi1 i2 ∈ E ∩ B i1 , such that, for every i1 = 1, 2, . . . , K, the balls B(zi1 i2 , 2r2 ) ⊂ B(zi1 , 2r1 ), i2 = 1 . . . , Ni1 , are pairwise disjoint, 

Ni1

E ∩ B i1 ⊂

B(zi1 i2 , 11r2 ),

i2 =1

and, by (7.6), we have Ni1 (11r2 )λ0 ≥ C1 r1λ0 . Again, Ni1 ≥ K for every i1 = 1, . . . , K. We continue with the balls B i1 i2 = B(zi1 i2 , r2 ), i1 , i2 = 1, 2, . . . , K. Since B(zi1 , 2r1 ) ∩ B(zj1 , 2r1 ) = ∅, if i1 = j1 , and B(zi1 i2 , 2r2 ) ⊂ B(zi1 , 2r1 ), for every i1 , i2 = 1, . . . , K, we conclude that balls B(zi1 i2 , 2r2 ), i1 , i2 = 1, . . . , K, are pairwise disjoint. Continuing recursively, in the kth step of the construction we obtain a collection of closed balls B i1 i2 ...ik , where ij = 1, . . . , K for j ∈ {1, . . . , k − 1} and ik = 1, . . . , Ni1 i2 ...ik−1 , with center points zi1 i2 ...ik ∈ E ∩ B i1 i2 ...ik−1 and radius rk = Rm−k , satisfying the following properties: B(zi1 i2 ...ik , 2rk ) ⊂ B(zi1 i2 ...ik−1 , 2rk−1 ) are pairwise disjoint, Ni1 i2 ...ik−1



E ∩ B i1 i2 ...ik−1 ⊂

B(zi1 i2 ...ik , 11rk ),

ik =1 λ0 λ0 . Since rk−1 = mλ0 rkλ0 , the defand, by (7.6), we have Ni1 i2 ...ik−1 (11rk )λ0 ≥ C1 rk−1 inition of m implies that Ni1 i2 ...ik−1 ≥ K for every i1 i2 . . . ik−1 , where ij = 1, . . . , K for j ∈ {1, . . . , k − 1}. The construction continues with the balls B i1 i2 ...ik , where i1 , i2 , . . . , ik = 1, 2, . . . , K. Observe that the corresponding balls B(zi1 i2 ...ik , 2rk ) are pairwise disjoint. We define a Cantor-type set ∞

F =

K 

B(zi1 i2 ...ik , 2rk ).

k=1 i1 ,...,ik =1

Observe that F ⊂ E ∩ B(x, 2R). Indeed, if z ∈ F , then z ∈ B(x, 2R), and since zi1 i2 ...ik ∈ E and rk → 0 as k → ∞, we have z ∈ E = E. Let μ be the equally distributed probability measure supported on F . That is, μ is constructed by distributing a unit mass to the set F via the balls B(zi1 i2 ...ik , 2rk ) in such a way that in each step all the balls B(zi1 i2 ...ik , 2rk ) obtain the same amount of mass. Then

(7.8) μ E ∩ B(x, 2R) = μ(F ) = 1,

7.3. LOWER DIMENSION AND CAPACITY DENSITY

and more generally (7.9)

143



μ E ∩ B(zi1 i2 ...ik , 2rk ) = K −k ,

for every 1 ≤ i1 , i2 , . . . , ik ≤ K. The existence of such a measure μ can be proved using the Carath´eodory construction with respect to the balls B(zi1 i2 ...ik , 2rk ). We refer to Falconer [125, pp. 13–14] and Mattila [315, pp. 54–55] for further details. Let y ∈ Rn and 0 < r < R, and choose k ∈ N with Rm−k ≤ r < Rm−k+1 . By the pairwise disjointness of the balls B(zi1 i2 ...ik , 2rk ), there exists a constant C2 = C2 (n, m) such that B(y, r) intersects at most C2 of the balls B(zi1 i2 ...ik , 2rk ) from the kth step of the construction. By (7.9) and the choice of k, we obtain  r λ

log K (7.10) μ(B(y, r)) ≤ μ F ∩ B(y, r) ≤ C2 K −k = C2 m−k log m ≤ C2 , R K where we also used the fact that λ < log log m . Let {B(xi , ri ) : i ∈ N} be a cover of E ∩ B(x, 2R). First assume that ri < R for every i ∈ N. Applying (7.8) and (7.10), we obtain ∞ ∞  r λ



i 1 = μ E ∩ B(x, 2R) ≤ μ(B(xi , ri )) ≤ C(n, λ0 , λ, C1 ) . R i=1 i=1

Thus (7.11)

(2R)λ ≤ C(n, λ0 , λ, C1 )

∞

riλ .

i=1

Observe that (7.11) also holds in the case that rj ≥ R for some j ∈ N. The claim (7.7) for 2R follows by taking infimum over all such covers.  Lemma 7.13 has several consequences. In the following theorem the Hausdorff content density condition is obtained for all radii up to the diameter of the set. Theorem 7.14. Let E ⊂ Rn be a closed set and assume that 0 ≤ λ < dimL (E). Then there exists a constant C = C(n, λ, E) such that

λ E ∩ B(x, R) ≥ CRλ H∞ for every x ∈ E and 0 < R ≤ diam(E). Moreover, dimL (E) is the supremum of the exponents λ ≥ 0 for which such a constant C exists. Proof. Let 0 ≤ λ < λ0 < dimL (E). The definition of the lower dimension implies that (7.6) holds with a constant C1 for every x ∈ E and 0 < r < R ≤ diam(E). Thus we obtain from Lemma 7.13 that

λ E ∩ B(x, R) ≥ C(n, λ, λ0 , C1 )Rλ H∞ for every x ∈ E and 0 < R ≤ diam(E). By choosing λ0 = 12 (dimL (E) + λ), the constant above only depends on n, λ and E. Assume then that

λ E ∩ B(x, R) ≥ CRλ H∞ for every x ∈ E and 0 < R ≤ diam(E). Let x ∈ E and 0 < r < R ≤ diam(E), and let {B(xi , r) : i = 1, . . . , N } be a cover of E ∩ B(x, R). Then N



λ E ∩ B(x, R) ≤ C Rλ ≤ CH∞ r λ = CN r λ , i=1

144

7. DENSITY CONDITIONS

and thus N ≥ C

R λ r

. Since this holds for all such covers, we have  λ

R N E ∩ B(x, R), r ≥ C . r

This implies that dimL (E) ≥ λ, and the proof is complete.



Theorem 7.14 gives a comparison between Hausdorff and lower dimensions of closed sets. n Corollary 7.15. Assume that E ⊂ R is a closed set and let x ∈ E and r > 0. Then dimL (E) ≤ dimH E ∩ B(x, r) .

Proof. We may clearly assume that dimL (E) > 0 and 0 < r ≤ diam(E). λ E ∩ B(x, r) > 0. Hence Let 0 ≤ λ < dimL (E). By Theorem 7.14, we have H∞  λ ≤ dimH (E ∩ B(x, r)), and the claim follows. Remark 7.16. The assumption that E is closed is necessary in Corollary 7.15. It is easy to see that dimL (E) = dimL (E) for all E ⊂ Rn , and hence for instance

dimL (Qn ) = dimL (Rn ) = n  0 = dimH Qn ∩ B(x, r) for every x ∈ Qn and r > 0. By combining Theorem 7.14 and Corollary 7.5, we obtain a sufficient condition for Hardy inequalities in terms of the lower dimension of the complement of the set. See also Theorem 7.22 for a converse implication and Corollary 10.36 for a dual pair of this statement for Hardy inequalities in terms of the Assouad dimension of the complement of the set. Theorem 7.17. Let 1 < p < ∞. Assume that Ω ⊂ Rn is an open set such that Ω is unbounded and dimL (Ωc ) > n − p. Then the pointwise p-Hardy inequality and the p-Hardy inequality hold in Ω. c

Proof. Let λ ≥ 0 be such that dimL (Ωc ) > λ > n − p. By Theorem 7.14 and the unboundedness of Ωc , the complement Ωc satisfies the λ-Hausdorff content density condition (7.1). The claim follows from Corollary 7.5.  7.4. Density conditions and Hardy’s inequality in the borderline case This section discusses density conditions and Hardy’s inequalities in the borderline case p = n. We begin with the following metric definition. Definition 7.18. A set E ⊂ Rn is uniformly perfect if E contains more than one point and there exists a constant C, with 0 < C < 1, such that

E ∩ B(x, R) \ B(x, CR) = ∅ for every x ∈ E and 0 < R ≤ diam(E). Uniform perfectness can be characterized in terms of the lower dimension. Lemma 7.19. A set E ⊂ Rn is uniformly perfect if and only if dimL (E) > 0. Proof. Assume first that E ⊂ Rn is uniformly perfect, with a constant 0 < C < 1. We show that there exists λ > 0 such that (7.5) holds for every x ∈ E and 0 < r < R ≤ diam(E). Fix x0 ∈ E and

0 < r < R ≤ diam(E). By uniform perfectness, there exists x1 ∈ E ∩ B x0 , R2 such that |x1 − x0 | ≥ C R2 , and thus

7.4. DENSITY CONDITIONS AND HARDY’S INEQUALITY

145

at least two disjoint balls B0 , B1 ⊂ B(x0 , R) of radius 8−1 CR are needed to cover E ∩ B(x0 , R). By applying a similar reasoning to the balls B(x0 , 8−1 CR) and B(x1 , 8−1 CR), which do not intersect, we obtain that at least four pairwise disjoint balls of radius (8−1 C)2 R are needed to cover E ∩ B(x0 , R). Continuing recursively we conclude that, for every N ∈ N0 , at least 2N balls of radius (8−1 C)N R are needed to cover E ∩ B(x0 , R). Let N ∈ N0 be such that r (8−1 C)N +1 ≤ < (8−1 C)N , R and let log 2 > 0. λ=− log(8−1 C) We have



N E ∩ B(x0 , R), r ≥ N E ∩ B(x0 , R), (8−1 C)N R  λ

−λ 1 1 1 R ≥ , ≥ 2N = 2N +1 = (8−1 C)N +1 2 2 2 r and thus dimL (E) ≥ λ > 0. Assume then that dimL (E) > 0 and let 0 < λ < dimL (E). There exists 0 < C1 < 1 such that for every x ∈ E and 0 < r < R ≤ diam(E) we have  λ

R N E ∩ B(x, R), r ≥ C1 . r

1 Choose C = C21 λ and r = CR. Then N E ∩ B(x, R), CR ≥ 2, and thus

B(x, CR) does not cover E ∩B(x, R). We conclude that E ∩ B(x, R)\B(x, CR) = ∅, and this proves that E is uniformly perfect.  The next theorem contains several equivalent statements, which are specific to the borderline case p = n. The proof below avoids the use of the improvement of Hardy inequalities (Lemma 6.15) and the fact that the p-capacity density condition implies the p-Hardy inequality (Theorem 6.25). However, the crux of the proof is the fact that uniform perfectness, through positivity of the lower dimension, implies the Hausdorff content density condition. Theorem 7.20. Assume that Ω  Rn is an open set. The following conditions are equivalent. (a) The n-Hardy inequality holds in Ω. (b) Ωc satisfies the n-capacity density condition. (c) Ωc is uniformly perfect and unbounded. (d) dimL (Ωc ) > 0 and Ωc is unbounded. (e) There exists λ > 0 such that Ωc satisfies the λ-Hausdorff content density condition. (f) Ωc satisfies the p-capacity density condition for some 1 < p < n. (g) The pointwise p-Hardy inequality holds in Ω for some 1 < p < n. (h) The pointwise n-Hardy inequality holds in Ω. Moreover, the constants in each of the conditions only depend on each other, n and p.

146

7. DENSITY CONDITIONS

Proof. First we show that (a) implies (c). Let y ∈ Ωc and R > 0. If

Ωc ∩ B(y, R) \ B(y, 18 R) = ∅, then the uniform perfectness condition holds for y and R with the constant 18 and the diameter of Ωc is at least 18 R. Hence it suffices to consider the cases where

Ωc ∩ B(y, R) \ B(y, 18 R) = ∅, which implies that Ωc ∩ B(y, R) ⊂ B(y, C1 R) for some 0 < C1 ≤ 18 . We show that there is a uniform lower bound for such C1 > 0, and this proves the claim. Let u : Rn → R, ⎧ 0, if |x − y| ≤ C1 R or |x − y| ≥ R, ⎪ ⎪ ⎪ ⎨ 1 |x − y| − 1, if C R < |x − y| < 2C R, 1 1 u(x) = C1 R R ⎪ 1, if 2C1 R ≤ |x − y| ≤ 2 , ⎪ ⎪ ⎩ if R2 < |x − y| < R. 2 − R2 |x − y|, Observe that u is supported in the annulus B(y, R) \ B(y, C1 R) ⊂ Ω and u ∈

Lip0 (Ω). Let A = B y, R2 \ B(y, 2C1 R). For every x ∈ A, we have u(x) = 1 and d (x, ∂Ω) ≤ |x − y|, and thus    |u(x)|n −n dx ≥ d (x, ∂Ω) dx ≥ |x − y|−n dx n d (x, ∂Ω) Ω A A   R = C(n) log − log(2C1 R) = −C(n) log(4C1 ). 2 The n-Hardy inequality implies   |u(x)|n −C(n) log(4C1 ) ≤ dx ≤ C |∇u(x)|n dx 2 n d (x, ∂Ω) Ω Ω   R −n  −n ≤ C2 |B(y, 2C1 R)|(C1 R) + |B(y, R)| ≤ C(n, C2 ), 2 where C2 is the constant in the assumed n-Hardy inequality. Hence − log(4C1 ) ≤

C(n, C2 ) , C(n)

and so C1 ≥

2) 1 − C(n,C e C(n) . 4

This shows that Ωc is uniformly perfect and unbounded. Then we show that (b) implies (c). Let y ∈ Ωc and R > 0, and assume that c Ω ∩ B(y, R) ⊂ B(y, C1 R) for some 0 < C1 < 1. The n-capacity density condition of Ωc , monotonicity of the capacity, and Lemma 5.36 (c) give

C(n, C2 ) ≤ capn Ωc ∩ B(y, R), B(y, 2R)  1−n

2 ≤ capn B(y, C1 R), B(y, 2R) ≤ C(n) log , C1 where C2 is the constant in the n-capacity density condition. Hence   1 C(n,C ) 1−n 1 2 C(n, C2 ) 1−n 2 − C(n) log ≤ , and so C1 ≥ 2e > 0. C1 C(n) This shows that Ωc is uniformly perfect and also unbounded. The assertion (c) implies (d) is the necessity part of Lemma 7.19.

7.5. SELF-IMPROVEMENT OF THE CAPACITY DENSITY CONDITION

147

The assertion (d) implies (e) follows from Theorem 7.14 for every 0 < λ < dimL (Ωc ). In particular, since Ωc is assumed to be unbounded, the density condition holds for every 0 < R < ∞. The assertion (e) implies (f) follows from Theorem 7.3. The assertion (f) implies (g) follows from the implication from (a) to (d) in Theorem 6.23. The assertion (g) implies (h) is a special case of Remark 6.5. The assertion (h) implies (b) is the implication from (d) to (a) in Theorem 6.23, for p = n. The assertion (g) implies (a) follows from Remark 6.6.



7.5. Self-improvement of the capacity density condition The implication from (b) to (f) in Theorem 7.20 gives a relatively elementary proof for the self-improvement of the capacity density condition for p = n. Our next goal is to show that a corresponding improvement holds for 1 < p < n. Theorem 7.21. Let 1 < p < ∞ and assume that a closed set E ⊂ Rn satisfies the p-capacity density condition, with a constant C0 . Then there exists 1 < q < p such that E satisfies the q-capacity density condition. Moreover, the exponent q and the constant in the q-capacity density condition depend on n, p and C0 in an explicit and quantitative fashion. The proof of Theorem 7.21, which will be completed in Section 7.6, Section 7.7 and Section 7.8, is based on quantitative estimates and absorption arguments, where it is crucial to track the dependencies of constants. Before proceeding to this somewhat lengthy proof, we record the following consequence of Theorem 7.21. Compare to Theorem 7.20 for p = n. Theorem 7.22. Let 1 < p ≤ n and assume that E ⊂ Rn is a closed set. The following conditions are equivalent. (a) E satisfies the p-capacity density condition. (b) There exists λ > n−p such that E satisfies the λ-Hausdorff content density condition. (c) dimL (E) > n − p and E is unbounded. (d) The pointwise p-Hardy inequality holds in the open set Ω = E c . Moreover, the constants in each of the conditions only depend on each other, n and p. Proof. First we show that (a) implies (b). By Theorem 7.21, there exists 1 < q < p such that E satisfies the q-capacity density condition, with a constant C1 . Let λ = n − q > n − p ≥ 0. As in Remark 7.4, we obtain

r λ = r n−q ≤ C(n, q, C1 ) capq E ∩ B(x, r), B(x, 2r)

λ E ∩ B(x, r) ≤ C(n, q, C1 )H∞ for every x ∈ E and r > 0, and this proves the claim. The assertion (b) implies (c) follows from Theorem 7.14. Note that if E is nonempty and bounded, then the λ-Hausdorff content density condition fails for large r > 0.

148

7. DENSITY CONDITIONS

The assertion (c) implies (d) is Theorem 7.17. The assertion (d) implies (a) is the implication from (d) to (a) in Theorem 6.23.  n−p (E) of Remark 7.23. Theorem 7.22 implies that capp (E, B(x, 2r)) and H∞ a set E ⊂ B(x, r) are not equivalent for 1 < p < n. To see this, let F ⊂ Rn be (n − p)-regular and unbounded, and assume for contradiction that there exists a constant C1 such that



n−p F ∩ B(x, r) ≤ C1 capp F ∩ B(x, r), B(x, 2r) H∞

n−p F ∩ B(x, r) ≥ C2 r n−p by Lemma 7.8, we for every x ∈ F and r > 0. Since H∞ conclude that F satisfies the p-capacity density condition. By Theorem 7.22 there exists λ > n − p such that F satisfies the λ-Hausdorff content density condition. Consequently dimH (F ) ≥ λ, which contradicts the fact that dimH (F ) = n − p < λ by the (n − p)-regularity.

To obtain the improvement of the p-capacity density condition in Theorem 7.21, it will be sufficient to show that the p-capacity density condition of E implies (6.20) for every z ∈ E and r > 0, but with an exponent q < p instead of p. This is the content of the next theorem. The proof of Theorem 7.21 is a combination of Theorem 6.23 and Theorem 7.24. Theorem 7.24. Let 1 < p < ∞ and assume that E ⊂ Rn is a closed set which satisfies the p-capacity density condition with a constant C0 . There exist ε = ε(n, p, C0 ), with 0 < ε < p − 1, and a constant C = C(n, p, C0 ) such that   |u(x)|p−ε dx ≤ Cr p−ε |∇u(x)|p−ε dx B(z,r)

B(z,r)

for every u ∈ Lip0 (E ), z ∈ E and r > 0. c

r ) be any Proof. Let z ∈ E, r > 0 and u ∈ Lip0 (E c ). Let F ⊂ E ∩ B(z, 12 closed set with z ∈ F . Since    |u(x)|p−ε |u(x)|p−ε |u(x)|p−ε dx = dx + dx p−ε p−ε r r r p−ε B(z,r) B(z,r)∩F B(z,r)\F  |u(x)|p−ε dx, ≤ p−ε B(z,r)\F d (x, F )

it suffices to find ε and C, and a closed set F as above, such that   |u(x)|p−ε (7.12) dx ≤ C |∇u(x)|p−ε dx. p−ε B(z,r)\F d (x, F ) B(z,r) We prove this improved local Hardy inequality in Theorem 7.32 below, and the claim follows.  7.6. Truncation and absorption In this section we establish some technical tools that will be used in the proofs of the local Hardy inequalities in Sections 7.7 and 7.8. The following truncation procedure provides us with the closed set F that is needed in the proof of Ther ) in the proof of orem 7.24. Here the ball B(z, r) corresponds to the ball B(z, 12 Theorem 7.24.

7.6. TRUNCATION AND ABSORPTION

149

Lemma 7.25. Assume that E ⊂ Rn is a closed set and let z ∈ E and r > 0. Let F0 = E ∩ B(z, r2 ) and define recursively  E ∩ B(x, 2−j−1 r) Fj = for every j ∈ N. Let F = for every j ∈ N.

x∈Fj−1

 j∈N0

Fj . Then z ∈ F ⊂ E ∩ B(z, r), and Fj−1 ⊂ Fj ⊂ F

Proof. The claim z ∈ Fis clear since z ∈ F0 ⊂ F , and the claim F ⊂ E follows since E is closed and j∈N0 Fj ⊂ E by definition. To verify F ⊂ B(z, r), let x ∈ Fj . If j = 0, then x ∈ B(z, r). If j ∈ N, then by induction we find a sequence xj , . . . , x0 such that xj = x, xk ∈ Fk for every k = 0, . . . , j, and xk ∈ E ∩ B(xk−1 , 2−k−1 r) for every k = 1, . . . , j. It follows that |x − z| ≤

j

|xk − xk−1 | + |x0 − z| ≤

k=1

j

2−k−1 r + 2−1 r < r,

k=1

and thus x ∈ B(z, r). We have shown that Fj ⊂ B(z, r) for every j ∈ N0 , and hence F ⊂ B(z, r). To prove Fj−1 ⊂ Fj ⊂ F , let j ∈ N and x ∈ Fj−1 . Since x ∈ E, we conclude that x ∈ E ∩ B(x, 2−j−1 r) ⊂ Fj .  Next we show that the truncated set F in Lemma 7.25 has the property that there always exist certain balls whose intersection with F contain big pieces of the original set E. Lemma 7.26. Assume that E, B(z, r) and F are as in Lemma 7.25. Let m ∈ N0 and x ∈ Rn be such that d (x, F ) < 2−m+1 r, and write rm = 2−m−1 r. There exists yx,m ∈ E such that B(yx,m , rm ) ⊂ B(x, 8rm ) and E ∩ B(yx,m , 12 rm ) = F ∩ B(yx,m , 12 rm ).

(7.13)

Proof. In this proof we will apply  Lemma 7.25 without further notice. Since d (x, F ) < 2−m+1 r, there exists y ∈ j∈N0 Fj ⊂ E satisfying |y − x| < 2−m+1 r. Let j ∈ N0 be such that y ∈ Fj . We consider two cases. First, assume that j > m ≥ 0. By induction, there are points yk ∈ Fk with k = m, . . . , j such that yj = y and yk ∈ E ∩ B(yk−1 , 2−k−1 r) for every k = m + 1, . . . , j. It follows that j j

|ym − y| = |yj − ym | ≤ |yk − yk−1 | ≤ 2−k−1 r < 2−m−1 r. k=m+1 

k=m+1 −m−1

Let yx,m = ym ∈ Fm ⊂ E. If y ∈ B(ym , 2 

r), then



|y − x| ≤ |y − ym | + |ym − y| + |y − x| < 2−m−1 r + 2−m−1 r + 2−m+1 r < 2−m+2 r, and thus B(ym , 2−m−1 r) ⊂ B(x, 2−m+2 r). Moreover, since ym ∈ Fm , we have  E ∩ B(ym , 2−m−2 r) ⊂ E ∩ B(y  , 2−m−2 r) = Fm+1 ⊂ F. y  ∈Fm

On the other hand F ⊂ E, and (7.13) follows. Then consider the case m ≥ j ≥ 0. Let yx,m = y ∈ E. We have |y  − x| ≤ |y  − y| + |y − x| < 2−m−1 r + 2−m+1 r < 2−m+2 r

150

7. DENSITY CONDITIONS

for every y  ∈ B(y, 2−m−1 r). This implies B(y, 2−m−1 r) ⊂ B(x, 2−m+2 r). Since y ∈ Fj ⊂ Fm , we can now repeat the final part of the argument above, with ym replaced by y, and (7.13) follows also in this case.  The main reason for truncating the set E is to obtain the following absorption result that is needed twice during the subsequent arguments. Dependencies of the constants below are delicate. Note that inequality (7.15) is useful only if C3 > 0. This can be ensured by requiring that C1 is sufficiently small, depending on n, p, σ and ς, but not depending on q. Lemma 7.27. Let 1 < p < ∞ and 1 ≤ q ≤ p, and assume that E, B(z, r) and F are as in Lemma 7.25. Moreover, let σ ≥ 1 and ς ≥ 2, and assume that u ∈ Lip0 (F c ) satisfies the inequality  B(z,ςr)\F

(7.14)

|u(x)|q dx d (x, F )q 

≤ C1 B(z,σςr)\F

|u(x)|q dx + C2 d (x, F )q

 |∇u(x)|q dx B(z,σςr)

with some constants C1 , C2 . Then  C3

(7.15)

B(z,σςr)\F

|u(x)|q dx ≤ C4 d (x, F )q

 |∇u(x)|q dx, B(z,σςr)

where C3 = 1 − C1 (1 + C(n, p, σ, ς)) and C4 = (1 + C2 )C(n, q, σ, ς). Proof. Since F ⊂ B(z, r) and ς ≥ 2, we have  (B(z,σςr)\F )\B(z,ςr)

≤ 3q r −q



 |u(x)|q −q dx ≤ r |u(x)|q dx d (x, F )q B(z,σςr) u(x) − uB(z,σςr) q dx

B(z,σςr)

 q q . + |B(z, σςr)| uB(z,σςr) − uB(z,ςr) + |B(z, σςr)| uB(z,ςr) Theorem 3.14 implies 

u(x) − uB(z,σςr) q dx + |B(z, σςr)| uB(z,σςr) − uB(z,ςr) q B(z,σςr)  u(x) − uB(z,σςr) q dx ≤ C(n, σ, q)r −q B(z,σςr)  ≤ C(n, q, σ, ς) |∇u(x)|q dx.

3q r −q

B(z,σςr)



7.7. LOCAL HARDY INEQUALITY

151

On the other hand, since q ≤ p and u = 0 in F , by (7.14) we obtain  q q −q q −q |u(x)|q dx 3 r |B(z, σςr)| uB(z,ςr) ≤ 3 C(n, σ)r B(z,ςr)\F

 ≤ 3q ς q C(n, σ) B(z,ςr)\F



|u(x)|q dx d (x, F )q

≤ 3p ς p C(n, σ)C1 B(z,σςr)\F



|u(x)|q dx d (x, F )q |∇u(x)|q dx.

+ 3q ς q C(n, σ)C2 B(z,σςr)

Combining the estimates above, we find that  |u(x)|q dx q B(z,σςr)\F d (x, F )   |u(x)|q |u(x)|q = dx + dx q q B(z,ςr)\F d (x, F ) (B(z,σςr)\F )\B(z,ςr) d (x, F )   |u(x)|q dx + C |∇u(x)|q dx, ≤ C1 (1 + C(n, p, σ, ς)) 4 q B(z,σςr)\F d (x, F ) B(z,σςr) where C4 = (1 + C2 )C(n, q, σ, ς). Let 0 ≤ L < ∞ be a Lipschitz constant of u ∈ Lip0 (F c ). Then sup |u(x)|q d(x, F )−q ≤ Lq , x∈F c

and therefore the first integral on the right-hand side is finite. Absorption of this integral to the left-hand side completes the proof.  7.7. Local Hardy inequality In this section we prove a local Hardy inequality with respect to the truncated set F , see Theorem 7.31. The strategy is to adapt Wannebo’s argument in Section 6.5, together with covering and absorption arguments, to the present context. For the rest of this section, we let 1 < p < ∞ and assume that a closed set E ⊂ Rn satisfies the p-capacity density condition with a constant C0 and that F is as in Lemma 7.25, with fixed z ∈ E and r > 0. We begin with Lemma 7.28 and Lemma 7.29, which provide information concerning the individual balls in the following collections Bm .   Let Am = x ∈ B(z, 8r) : 2−m r ≤ d (x, F ) < 2−m+1 r and Ωm =

∞ 

  Ak = x ∈ B(z, 8r) : 0 < d (x, F ) < 2−m+1 r

k=m

for m ∈ Z. By Lemma 1.13, for every Am there exists a cover   Bm = B(xi , 2−m+2 r) : xi ∈ Am , i = 1, . . . , Nm , such that the balls B(xi , 15 2−m+2 r), i = 1, . . . , Nm , are pairwise disjoint. The balls in Bm have uniformly bounded overlap, that is, (7.16)

Nm

i=1

for every x ∈ Rn and m ∈ N0 .

χB(xi ,2−m+2 r) (x) ≤ C(n),

152

7. DENSITY CONDITIONS

Lemma 7.28. Let m ∈ N0 and assume that B(xi , 2−m+2 r) ∈ Bm . Then B(xi , 2−m+2 r) \ F ⊂ Ωm−2 .

(7.17)

Proof. Let x ∈ B(xi , 2−m+2 r) \ F . Then d (x, F ) > 0, and d (x, F ) ≤ |x − xi | + d (xi , F ) < 2−m+2 r + 2−m+1 r < 2−(m−2)+1 r. Since m ∈ N0 , we have |x − z| ≤ d (x, B(z, r)) + r ≤ d (x, F ) + r < 2−m+2 r + 2−m+1 r + r < 8r, and thus x ∈ B(z, 8r). Estimates above imply that x ∈ Ωm−2 .



The following lemma is a substitute for the boundary Poincar´e inequality, see Theorem 6.22, to the present setting. Note that this is the only place in the proof of Theorem 7.21 where the p-capacity density condition, with the constant C0 , is applied. Lemma 7.29. Let m ∈ N0 and assume that B(xi , 2−m+2 r) ∈ Bm and v ∈ Lip0 (F c ). Then   p −mp p (7.18) |v(x)| dx ≤ C2 r |∇v(x)|p dx, B(xi ,2−m+2 r)

B(xi ,2−m+2 r)

with C = C(n, p, C0 ). Proof. Let Bi = B(xi , 2−m+2 r). Lemma 7.26, applied to m and xi , gives a closed ball B(y, rm ) ⊂ Bi , with y ∈ E and rm = 2−m−1 r. Note that   p p p p (7.19)  |v(x)| dx ≤ C(p)  |v(x)−vBi | dx+ vBi −vB(y,rm ) + vB(y,rm ) . Bi

Bi

By H¨ older’s inequality, p ≤ vB − v i B(y,rm )



p



|v(x) − vBi | dx

B(y,rm )

≤ C(n)  |v(x) − vBi |p dx, Bi

and by Theorem 3.14,

  p C(p)  |v(x) − vBi |p dx + vBi − vB(y,rm )

(7.20)

Bi

≤ C(n, p)2−mp r p  |∇v(x)|p dx. Bi

To estimate the last term in (7.19), we recall that by assumption v = 0 in F and, by Lemma 7.26, we have E ∩ B(y, 12 rm ) = F ∩ B(y, 12 rm ). By the p-capacity density condition and the fact that |Bi | = C(n)(2−m+2 r)n , we obtain



capp {v = 0} ∩ B(y, 12 rm ), B(y, rm ) ≥ capp F ∩ B(y, 12 rm ), B(y, rm )

= capp E ∩ B(y, 12 rm ), B(y, rm ) ≥ C(n, p, C0 )(2−m−2 r)n−p = C(n, p, C0 )|Bi |2mp r −p .

7.7. LOCAL HARDY INEQUALITY

153

Theorem 5.47 and the inclusions B(y, rm ) ⊂ B(y, rm ) ⊂ Bi imply v

B(y,rm )

p ≤ 

|v(x)|p dx

B(y,rm )

 C(n, p)

|∇v(x)|p dx capp {v = 0} ∩ B(y, 12 rm ), B(y, rm ) B(y,rm )  ≤ C(n, p, C0 )2−mp r p |Bi |−1 |∇v(x)|p dx

≤

Bi −mp p

≤ C(n, p, C0 )2

r  |∇v(x)|p dx. Bi

The desired estimate (7.18) follows by substituting (7.20) and the estimate above to (7.19) and multiplying both sides by |Bi |. 

We turn to the proof of the local Hardy inequality. As in Section 6.5, we first prove a weighted version of the inequality. This is then bootstrapped in Theorem 7.31 to the unweighted inequality by choosing a suitable test function. Lemma 7.30. Let − 12 < β < 0 and assume that v ∈ Lip0 (F c ). Then 

 |v(x)| d (x, F ) p

(7.21)

β−p

dx ≤ C

B(z,2r)\F

with C =

|∇v(x)|p d (x, F )β dx, B(z,8r)\F

1 |β| C(n, p, C0 ).

Proof. Since v ∈ Lip0 (F c ), we have ∇v(x) = 0 for almost every x ∈ F . By (7.18), (7.16) and (7.17), we obtain  |v(x)|p dx ≤ Am

Nm 

i=1

B(xi

,2−m+2 r)

|v(x)|p dx

≤ C(n, p, C0 )2−mp r p (7.22) ≤ C(n, p, C0 )2−mp r p −mp p

≤ C(n, p, C0 )2

r

Nm 

i=1

B(xi ,2−m+2 r)\F

|∇v(x)|p dx

|∇v(x)|p dx Ωm−2 ∞

k=m−2

 |∇v(x)|p dx Ak

for every m ∈ N0 . Since   B(z, 2r) \ F ⊂ Ω0 = x ∈ B(z, 8r) : 0 < d (x, F ) < 2r

154

7. DENSITY CONDITIONS

and d (x, F )β−p ≤ 2mp−mβ r β−p for every x ∈ Am , applying (7.22) we obtain   p β−p |v(x)| d (x, F ) dx ≤ |v(x)|p d (x, F )β−p dx B(z,2r)\F

Ω0

≤

∞

 |v(x)|p dx

mp−mβ β−p

2

r

Am

m=0

= C(n, p, C0 )

∞

m=0

= C(n, p, C0 )r β

 ∞

2−mβ r β

k=m−2

∞ k+2

2−mβ

k=−2 m=0

|∇v(x)|p dx

Ak

 |∇v(x)|p dx Ak

 ∞ C(n, p, C0 ) −kβ β ≤ 2 r |∇v(x)|p dx |β| Ak k=−2  C(n, p, C0 ) ≤ |∇v(x)|p d (x, F )β dx. |β| B(z,8r)\F In the penultimate step above, we used the estimate k+2

m=0

2−mβ ≤

2−(k+2)β ≤ C2−kβ |β|−1 , 1 − 2β

which holds for every − 12 < β < 0 with a constant C independent of k and β; compare to the end of the proof of Lemma 6.24. Inequality (7.21) follows.  Theorem 7.31. Let 1 < p < ∞ and assume that a closed set E ⊂ Rn satisfies the p-capacity density condition with a constant C0 . Let z ∈ E and r > 0, and let F be as in Lemma 7.25 for B(z, r). Assume that u ∈ Lipc (F c ). Then   |u(x)|p (7.23) dx ≤ C |∇u(x)|p dx, p B(z,8r)\F d (x, F ) B(z,8r) where C = C(n, p, C0 ). Proof. Let − 12 < β < 0 and define β

v(x) = u(x)d (x, F )− p , for every x ∈ Rn . Then v ∈ Lipc (F c ). By the Leibniz rule (2.4) and the chain rule (2.5), we have β

|∇v(x)| ≤ |∇u(x)|d (x, F )− p +

β |β| |u(x)|d (x, F )− p −1 , p

for almost every x ∈ B(z, 8r) \ F . By (7.21) for v, we obtain   |u(x)|p dx = |v(x)|p d (x, F )β−p dx p B(z,2r)\F d (x, F ) B(z,2r)\F  C(n, p, C0 ) |∇v(x)|p d (x, F )β dx ≤ |β| B(z,8r)\F   |u(x)|p C(n, p, C0 ) p p−1 ≤ |∇u(x)| dx + C(n, p, C0 )|β| dx. p |β| B(z,8r) B(z,8r)\F d (x, F )

7.8. CONCLUDING ARGUMENT

155

We apply Lemma 7.27 with ς = 2, σ = 4, q = p, C1 = C(n, p, C0 )|β|p−1 and 1 C(n, p, C0 ). We choose − 12 < β < 0, depending on n, p and C0 , to be such C2 = |β| that 1 C3 = 1 − C1 (1 + C(n, p, σ, ς)) ≥ . 2 Lemma 7.27 gives   |u(x)|p dx ≤ C(n, p, C ) |∇u(x)|p dx, 0 p B(z,8r)\F d (x, F ) B(z,8r) and this concludes the proof.



7.8. Concluding argument In the final step of the proof of Theorem 7.24 we improve the local Hardy inequality in Theorem 7.31. This is done by adapting the proof of Lemma 6.15 to the present setting and applying Lemma 7.27. Theorem 7.32. Let 1 < p < ∞ and assume that a closed set E ⊂ Rn satisfies the p-capacity density condition with a constant C0 . Let z ∈ E and r > 0, let F be as in the statement of Lemma 7.25 for B(z, r), and let C2 = C2 (n, p, C0 ) be the constant in (7.23), see Theorem 7.31. Then there exist 0 < ε = ε(n, p, C2 ) < p − 1 and C = C(n, p, C2 ) such that the inequality   |u(x)|p−ε (7.24) dx ≤ C |∇u(x)|p−ε dx p−ε B(z,12r)\F d (x, F ) B(z,12r) holds for every u ∈ Lip0 (E c ). Proof. Without loss of generality, we may assume that C2 ≥ 1 in (7.23). We shall first prove (7.24) under the additional assumption that u ∈ Lipc (E c ). Let U = Rn \ supp u. Then u = 0 in the open set U and d (F, U c ) ≥ d (E, U c ) > 0. Let t > 0 and define Ft = Gt ∩ Ht , where   Gt = x ∈ B(z, 8r) : M 12 d(x,F ) |∇u|(x) ≤ t and Ht = {x ∈ B(z, 8r) : |u(x)| ≤ td (x, F )}. We show that the restriction of u to Ft is C(n)t-Lipschitz. To this end, fix x, y ∈ Ft , x = y, such that d (y, F ) ≤ d (x, F ). If d (x, F ) ≥ 5|x − y|, then d(y, F ) ≥ d (x, F ) − |x − y| ≥ 4|x − y|. Hence, by the pointwise estimate in Theorem 4.1, we obtain

|u(x) − u(y)| ≤ C(n) |x − y| M2|x−y| |∇u|(x) + M2|x−y| |∇u|(y)

≤ C(n) |x − y| M 12 d(x,F ) |∇u|(x) + M 12 d(y,F ) |∇u|(y) ≤ C(n)t|x − y|. On the other hand, if d (y, F ) ≤ d (x, F ) ≤ 5|x − y|, then

|u(x) − u(y)| ≤ |u(x)| + |u(y)| ≤ t d (x, F ) + d (y, F ) ≤ 10t|x − y|. Thus u is C(n)t-Lipschitz in Ft . We apply Theorem 2.7 and extend the restriction u|Ft to a C(n)t-Lipschitz  ⊂ U with d (F, Rn \ U  ) > 0. Indeed, function u  in Rn . Then u  = 0 in an open set U if  = {x ∈ B(z, 8r) : d (x, F ) < 1 d (F, U c )}, x∈U 2

156

7. DENSITY CONDITIONS

then u(x) = 0. Since |∇u| = 0 in U , we have x ∈ Ft , and so u (x) = 0. Define   v(x) = u (x) max 0, 1 − d (x, B(z, 8r)) , for x ∈ Rn . Then v ∈ Lipc (F c ) and |∇v(x)| = |∇ u(x)| ≤ χFt (x)|∇u(x)| + C(n)tχFtc (x) for almost every x ∈ B(z, 8r). By applying Theorem 7.31 to the function v ∈ Lipc (F c ), we obtain   |u(x)|p |v(x)|p dx ≤ dx p p (B(z,8r)\F )∩Ft d (x, F ) B(z,8r)\F d (x, F )  ≤ C2 |∇u(x)|p dx + C2 C(n)p tp |B(z, 8r) \ Ft |. Ft

Since Ht = Ft ∪ (Ht \ Gt ) and C2 ≥ 1, it follows that  |u(x)|p dx p (B(z,8r)\F )∩Ht d (x, F )  ≤ C2 |∇u(x)|p dx + C2 C(n)p tp |B(z, 8r) \ Ft | Ft  |u(x)|p + dx (7.25) p (Ht \F )\Gt d (x, F ) 

≤ C2 |∇u(x)|p dx + C2 C(n, p)tp |B(z, 8r) \ Ft | + |Ht \ Gt | Gt

|∇u(x)|p dx + C2 C(n, p)tp |B(z, 8r) \ Ht | + |B(z, 8r) \ Gt | . ≤ C2 Gt

Here t > 0 was arbitrary, and thus we conclude that (7.25) holds for every t > 0. Next we multiply (7.25) by t−1−ε , where 0 < ε < p − 1, and integrate with respect to t over the set (0, ∞). With a change of the order of integration on the left-hand side, this gives  p−ε   ∞  |u(x)| 1 dx ≤ C2 t−1−ε |∇u(x)|p dx dt ε B(z,8r)\F d (x, F ) Gt 0  ∞

+ C2 C(n, p) tp−1−ε |B(z, 8r) \ Ht | + |B(z, 8r) \ Gt | dt. 0

Since |∇u(x)| ≤ M 12 d(x,F ) |∇u|(x) for almost every x ∈ B(z, 8r), see Corollary 1.23, we find by the definition of Gt that the first term on the right-hand side is dominated by  C2 |∇u(x)|p−ε dx. ε B(z,8r) Using the definitions of Ht and Gt , the second term on the right-hand side can be estimated from above by    p−ε 

p−ε |u(x)| C2 C(n, p) M 12 d(x,F ) |∇u|(x) dx + dx . p−ε B(z,8r)\F d (x, F ) B(z,8r) Fix x ∈ B(z, 8r) for a moment. Since 12 d (x, F ) ≤ 4r, we have B(x, ρ) ⊂ B(z, 12r) for every 0 < ρ ≤ 12 d (x, F ). This implies



M 12 d(x,F ) |∇u|(x) = M 12 d(x,F ) χB(z,12r) |∇u| (x) ≤ M χB(z,12r) |∇u| (x)

7.8. CONCLUDING ARGUMENT

157

for almost every x ∈ B(z, 8r). By Theorem 1.15 (c), we have  

p−ε

p−ε M 12 d(x,F ) |∇u|(x) M (χB(z,12r) |∇u|)(x) dx ≤ dx B(z,8r) Rn  ≤ C(n, p, ε) |∇u(x)|p−ε dx. B(z,12r)

By combining the estimates above, we obtain  p−ε  |u(x)| dx B(z,8r)\F d (x, F )  p−ε   |u(x)| dx + C4 |∇u(x)|p−ε dx, ≤ C3 B(z,12r)\F d (x, F ) B(z,12r)

ε ε where C3 = C2 C(n, p) p−ε and C4 = C2 1 + C(n, p) p−ε C(n, p, ε) . In order to apply Lemma 7.27, let ς = 8, σ = 32 and q = p − ε. We choose 0 < ε < p − 1 depending on n, p and C2 to be so small that C5 = 1 − C3 (1 + C(n, p, σ, ς)) ≥ 12 . Lemma 7.27 implies   |u(x)|p−ε dx ≤ C(n, p, C2 ) |∇u(x)|p−ε dx, p−ε B(z,12r)\F d (x, F ) B(z,12r) and this proves inequality (7.24) when u ∈ Lipc (E c ). To prove the general case, let u ∈ Lip0 (E c ). Then u has a bounded support. Let   uj (x) = max |u(x)| − 1j , 0 , for every x ∈ Rn and every j ∈ N. Then |∇uj | ≤ |∇u| almost everywhere and uj vanishes in the open neighborhood Uj = {|u| < 1j } of E. In particular, uj ∈ Lipc (E c ) for every j ∈ N. By Fatou’s lemma and the special case of inequality (7.24) that was established above, we conclude that   |u(x)|p−ε |uj (x)|p−ε dx ≤ lim inf dx p−ε p−ε j→∞ B(z,12r)\F d (x, F ) B(z,12r)\F d (x, F )  ≤ C(n, p, C2 ) lim inf |∇uj (x)|p−ε dx j→∞ B(z,12r)  ≤ C(n, p, C2 ) |∇u(x)|p−ε dx. B(z,12r)

This proofs the claim in the general case u ∈ Lip0 (E c ).



We close this chapter by showing that the nonquantitative capacity condition

(7.26) capp E ∩ B(x, r), B(x, 2r) > 0, for every x ∈ E and r > 0, is not self-improving. Example 7.33. Let 1 < p < 2. We construct a set E ⊂ R2 which satisfies (7.26) with p, but for every 1 < q < p the corresponding condition does not hold since there exist a point x ∈ E and a radius r > 0, depending on q, such that capq E ∩

B(x, r), B(x, 2r) = 0. We consider here only the planar case n = 2, but it is easy to generalize this example also to the higher dimensions.

158

7. DENSITY CONDITIONS

For i ∈ N, let λi = 2 − p + 1i and let Ei ⊂ [2−i , 2−i+1 ] × [0, 2−i ] be a λi -regular compact set. For instance, Ei can be taken to be a Cantor-type fractal set. Let E=

∞ 

Ei ∪ {0} ∪ B(0, 2)c

i=1

where 0 = (0, 0) ∈ R . Let x ∈ E and r > 0. If x ∈ B(0, 2)c or r > 2, then |E ∩ B(x, r)| > 0, and Lemma 5.35 implies that capp E ∩ B(x, r), B(x, 2r) > 0. Thus we may assume that x = 0 or x ∈ Ei for some i ∈ N, and that 0 < r ≤ 2. If x ∈ Ei , then by Lemma 7.8



λi λi E ∩ B(x, r) ≥ H∞ Ei ∩ B(x, r) ≥ C(λi , Ei ) min{r, diam(Ei )}λi > 0. H∞ 2

Theorem 5.41 implies



λi E ∩ B(x, r) r 2−p−λi > 0. capp E ∩ B(x, r), B(x, 2r) ≥ C(p, λi , Ei )H∞ On the other hand, if x = 0, then we choose i ∈ N such that Ei ⊂ B(0, r), and so



λi Ei ∩ B(0, r) r 2−p−λi > 0. capp E ∩ B(0, r), B(0, 2r) ≥ C(p, λi , Ei )H∞ We conclude that (7.26), with p, holds for the set E. This p is also the smallest exponent for which (7.26) is valid. Indeed, if 1 < q < p, we choose k ∈ N to be so large that q < p − k1 , and consider x = 0 and r < 2−k . Then, using Theorem 5.39 and countable subadditivity, we obtain



2−q E ∩ B(0, r) capq E ∩ B(0, r), B(0, 2r) ≤ CH∞   ∞

2−q 2−q ≤ C H∞ ({0}) + H∞ (Ei ) . i=k

Observe that 2 − q > 2 − p + k1 ≥ 2 − p + 1i = λi for every i ≥ k, and thus 2−q 2−q H∞ (Ei ) = 0 for every i ≥ k. Since H∞ ({0}) = 0, we conclude that

capq E ∩ B(0, r), B(0, 2r) = 0. This shows that (7.26) does not hold for any 1 < q < p. 7.9. Notes References for Ahlfors–David regular sets include Brudnyi and Brudnyi [59,60], Ihnatsyeva and V¨ ah¨ akangas [202], Jonsson and Wallin [215] and Triebel [385]. Lemma 7.8 has been considered in Egert, Haller-Dintelmann and Rehberg [118, Lemma 4.5]. For more information concerning the lower dimension we refer to Fraser [136]. This concept has also been called the lower Assouad dimension and denoted by dimA (E), see for instance [217]. Essentially the same dimension was studied already by Larman [255]. The proof of Lemma 7.13 is inspired by J¨arvi and Vuorinen [211, Theorem 4.1], see also Lehrb¨ ack [259, Lemma 4.1]. Corollary 7.15 was proved by Larman [255]. Lemma 7.19 has been considered by K¨aenm¨ aki, Lehrb¨ack and Vuorinen [217], and the necessity part is essentially by Rajala and Vilppolainen [351, Remark 3.6(iii)].

7.9. NOTES

159

Many of the implications in Theorem 7.20 can be found in J¨arvi and Vuorinen [211, Theorem 4.1] and Korte and Shanmugalingam [242, Theorem 1.1]. Ancona [14] proved that the capacity density condition is also necessary for the validity of Hardy’s inequality when p = n = 2 and Lewis [271] generalized this for p = n ≥ 2. See also Buckley and Koskela [64] for corresponding results in Orlicz–Sobolev spaces. Sugawa [377] proved that Hardy’s inequality is equivalent to uniform perfectness of the complement for p = n = 2. Sugawa’s paper also contains a survey on properties of uniformly perfect sets. Theorem 7.21 has been proved by Lewis [271] for the Riesz capacities. The corresponding result for the variational capacity in a Euclidean case with weights is by Mikkonen [327]. This approach has been generalized to metric measure spaces by Bj¨ orn, MacManus and Shanmugalingam [36]. These arguments are based on nonlinear potential theory. The direct proof of Theorem 7.21 in Section 7.6, Section 7.7 and Section 7.8 is an adaptation of the metric space proof in Lehrb¨ack, Tuominen and V¨ ah¨ akangas [264]. It is also possible to consider restricted versions of the p-capacity density condition and the λ-Hausdorff content density condition. Then, instead of every r > 0, the conditions are required to hold for every 0 < r < r0 , with some fixed r0 > 0. It is straightforward to modify our results for these restricted conditions.

10.1090/surv/257/08

CHAPTER 8

Muckenhoupt Weights This chapter discusses Muckenhoupt weights and related weighted norm inequalities for maximal functions. Self-improving phenomena are studied for weighted norm inequalities in Section 8.3, for Muckenhoupt weights in Section 8.4 and Section 8.5, and for reverse H¨older inequalities in Section 8.7 and Section 8.8. Instead of aiming at the most direct approach, we give several independent points of view to the self-improving phenomena. This causes some overlap, but these methods and results are applied in Chapter 9, Chapter 10 and Chapter 12. 8.1. Doubling weights It is often more convenient to use cubes instead of balls. Unless specified otherwise, we apply half-open cubes whose sides are parallel to the coordinate axis. Definition 8.1. A cube in Rn is a set of the form Q = [a1 , b1 ) × . . . × [an , bn ), with b1 − a1 = · · · = bn − an . The side length of Q is l(Q) = b1 − a1 . For x ∈ Rn and r > 0,   Q(x, r) = y ∈ Rn : −r ≤ yj − xj < r for all j = 1, . . . , n is the cube with center x and side length 2r. Clearly, |Q(x, r)| = (2r)n

and

√ diam(Q(x, r)) = (2 n)r.

Definition 8.2. A function w ∈ L1loc (Rn ), with w(x) > 0 for almost every x ∈ Rn , is called a weight in Rn . For a weight w and a measurable set E ⊂ Rn , we write  w(E) = w(x) dx. E

Note that the measure w dx induced by a weight and the Lebesgue measure are mutually absolutely continuous. For 1 ≤ p < ∞, the space Lp (E; w dx) consists of measurable functions f on E satisfying   p1 f Lp (E;w dx) = |f (x)|p w(x) dx < ∞. E

∈ Lploc (E; w dx) if p p n

f ∈ Lp (K; w dx) for every compact set K ⊂ E. In addition, f We also write L (w dx) = L (R ; w dx). Definition 8.3. A weight w in Rn is doubling if there exists a constant C such that w(B(x, 2r)) ≤ Cw(B(x, r)) for every x ∈ Rn and r > 0. 161

162

8. MUCKENHOUPT WEIGHTS

It is immediate that the doubling condition in Definition 8.3 can be equivalently stated in the form that there exists a constant C such that w(Q(x, 2r)) ≤ Cw(Q(x, r)) for every x ∈ Rn and r > 0. We collect definitions and properties of weighted maximal functions in the remark below. Remark 8.4. Let w be a doubling weight in Rn , with a constant C1 . (a) For a measurable function f ∈ L1loc (w dx) the weighted maximal function M w f at x ∈ Rn is  1 M w f (x) = sup |f (y)|w(y) dy. r>0 w(B(x, r)) B(x,r) Let 1 < p ≤ ∞. There exists a constant C = C(n, p, C1 ) such that M w f Lp (w dx) ≤ Cf Lp (w dx) for every f ∈ Lp (w dx). More precisely, we have C = 1 for p = ∞ and  3  p1 C1 p C =2· p−1 for 1 < p < ∞. The proof of this statement is a straightforward adaptation of the proof of Theorem 1.15 (c). (b) For a measurable function f ∈ L1loc (w dx), we define  1 c,w |f (y)|w(y) dy, M f (x) = sup Qx w(Q) Q where the supremum is taken over all cubes Q ⊂ Rn with x ∈ Q. Let 1 < p ≤ ∞. There exists a constant C = C(n, p, C1 ) such that M c,w f Lp (w dx) ≤ Cf Lp (w dx) for every f ∈ Lp (w dx). This follows from (a) since M c,w and M w are pointwise comparable by the doubling property of w, compare to Remark 1.2. For w = 1, we write M c f = M c,w f . (c) Let f ∈ L1loc (w dx). There exists a measurable set E with w(E) = |E| = 0 such that  1 lim |f (y) − f (x)|w(y) dy = 0 r→0 w(B(x, r)) B(x,r) for every x ∈ Rn \ E; compare to Theorem 1.19. As a consequence, if x ∈ Rn \ E and (Qj )j∈N is a sequence of cubes such that x ∈ Qj for every j ∈ N and l(Qj ) → 0 as j → ∞, then  1 f (y)w(y) dy. f (x) = lim j→∞ w(Qj ) Q j 8.2. Dyadic cubes and the Calder´ on–Zygmund lemma Let Q0 ⊂ Rn be a cube. The dyadic decomposition D(Q0 ) of Q0 is defined as D(Q0 ) =

∞  j=0

Dj (Q0 ),

´ 8.2. DYADIC CUBES AND THE CALDERON–ZYGMUND LEMMA

163

where each Dj (Q0 ) consists of 2jn pairwise disjoint half-open cubes Q, with side length l(Q) = 2−j l(Q0 ), such that  Q0 = Q Q∈Dj (Q0 )

for every j ∈ N0 . If j ≥ 1 and Q ∈ Dj (Q0 ), there exists a unique cube πQ ∈ Dj−1 (Q0 ), satisfying Q ⊂ πQ. The cube πQ is called the dyadic parent of Q, and Q is a dyadic child of πQ. The dyadic maximal function will be a useful tool for us. Definition 8.5. Let Q0 ⊂ Rn be a cube and let f ∈ L1 (Q0 ). The dyadic d f is defined as (noncentered) maximal function MQ 0 (8.1)

d f (x) = sup  |f (y)| dy, MQ 0 Qx

Q

where the supremum is taken over all dyadic cubes Q ∈ D(Q0 ) with x ∈ Q. Let f, g ∈ L1 (Q0 ) and x ∈ Q0 . Using the definition it is easy to show that ≥ 0, d d d (f + g)(x) ≤ MQ f (x) + MQ g(x), MQ 0 0 0

d f (x) MQ 0

and d d (af )(x) = |a|MQ f (x) MQ 0 0 for every a ∈ R, compare to Lemma 1.5. The Calder´ on–Zygmund lemma and its variants will be used several times in the sequel.

Lemma 8.6. Let Q0 ⊂ Rn be a cube. Assume that f ∈ L1 (Q0 ) is a nonnegative function and let (8.2)

d f (x) > t} Et = {x ∈ Q0 : MQ 0

for t > 0. For every t ≥ fQ0 , there exists a collection Dt of pairwise disjoint dyadic cubes Q ∈ D(Q0 ) such that  (8.3) Et = Q Q∈Dt

and (8.4)

t <  f (x) dx ≤ 2n t Q

for every Q ∈ Dt . Moreover, f (x) ≤ t for almost every x ∈ Q0 \ Et . Proof. Let t > 0. If Et = ∅, then the claim holds with Dt = ∅. In this case the union in (8.3) is the empty set. Hence we may assume Et = ∅. For every x ∈ Et there exists a dyadic cube Q ∈ D(Q0 ) with x ∈ Q and fQ > t. It follows that there exists a unique maximal dyadic cube Qx ∈ D(Q0 ) with x ∈ Qx and fQx > t. Maximality means that if Qx  Q ∈ D(Q0 ), then fQ ≤ t. The collection of all such maximal cubes is Dt = {Qx ∈ D(Q0 ) : x ∈ Et }. If Qx = Q0 for some x ∈ Q0 , then Dt = {Q0 } and there are no cubes Q ∈ D(Q0 ) with Qx  Q. This happens if and only if t < fQ0 , which contradicts the assumption t ≥ fQ0 . Hence Dt = {Q0 }.

164

8. MUCKENHOUPT WEIGHTS

Since either Q ⊂ R or R ⊂ Q for Q, R ∈ D(Q0 ) with Q ∩ R = ∅, by maximality the cubes in Dt are pairwise disjoint. Moreover, if x ∈ Et , then x ∈ Qx ⊂ Et , and thus (8.3) holds. Assume then that Q ∈ Dt ∩ Dj (Q0 ), for some j ∈ N, and let πQ ∈ Dj−1 (Q0 ) be the dyadic parent of Q. Then Q  πQ ⊂ Q0 and |πQ| = 2n |Q|. By maximality of Q, we have fπQ ≤ t. Thus   1 2n t < fQ = f (x) dx ≤ f (x) dx ≤ 2n t. |Q| Q |πQ| πQ d Finally, by the Lebesgue differentiation theorem f (x) ≤ MQ f (x) ≤ t for almost 0 every x ∈ Q0 \ Et . 

The Calder´ on–Zygmund lemma gives simple proofs for norm estimates for the dyadic maximal function. The next result is a weak type estimate, compare to Theorem 1.15 (b). Lemma 8.7. Let Q0 ⊂ Rn be a cube. Assume that f ∈ L1 (Q0 ) is a nonnegative d function and let Et = {x ∈ Q0 : MQ f (x) > t}. Then 0  1 f (x) dx |Et | ≤ t Et for every t > 0. Proof. Let t > 0. If t < fQ0 , then Et = Q0 and thus   1 1 f (x) dx = f (x) dx. |Et | = |Q0 | ≤ t Q0 t Et Then assume that t ≥ fQ0 . Let Dt be the collection of dyadic subcubes of Q0 given by Lemma 8.6. By using (8.3) and the facts that the cubes Q ∈ Dt are pairwise disjoint and t < fQ for every Q ∈ Dt , we obtain   

1 1 |Et | = Q = |Q| ≤ f (x) dx = f (x) dx.  t t Et Q Q∈Dt

Q∈Dt

Q∈Dt

There exists a reverse weak type estimate for the dyadic maximal function, compare to Lemma 1.25. Lemma 8.8. Let Q0 ⊂ Rn be a cube. Assume that f ∈ L1 (Q0 ) is a nonnegative d function and let Et = {x ∈ Q0 : MQ f (x) > t}. Then 0  f (x) dx ≤ 2n t|Et | Et

for every t ≥ fQ0 . Proof. Without loss of generality, we may assume that t > 0. Let Dt be the collection of dyadic subcubes of Q0 given by Lemma 8.6. Using (8.3) and the facts that the cubes Q ∈ Dt are pairwise disjoint and fQ ≤ 2n t for every Q ∈ Dt , we obtain 



f (x) dx = f (x) dx ≤ 2n t |Q| = 2n t|Et |.  Et

Q∈Dt

Q

Q∈Dt

8.3. SELF-IMPROVEMENT OF WEIGHTED NORM INEQUALITIES

165

8.3. Self-improvement of weighted norm inequalities We consider maximal function inequalities not only for the Lebesgue measure but also for weighted measures. The main result of this section, Theorem 8.11, gives a self-improvement property for weighted norm inequalities related to the Hardy–Littlewood–Wiener maximal function theorem. We begin with a Whitney covering theorem. For the proof we refer to Stein [369, Chapter 6], see also Bennett and Sharpley [28, Section V.5] and Grafakos [156, Appendix J]. Lemma 8.9. Assume that Ω  Rn is a nonempty open set.There exist pairwise disjoint half-open dyadic cubes Qi ⊂ Ω, i ∈ N, such that Ω = ∞ i=1 Qi and diam(Qi ) ≤ d (Qi , ∂Ω) ≤ 4 diam(Qi )

(8.5)

for every i ∈ N. Moreover, let Qi = Q(xi , ri ) and Q∗i = Q(xi , 98 ri ), for every i ∈ N. There exists a constant C = C(n) such that ∞

χQ∗i (x) ≤ C

i=1

for every x ∈ Ω, and 3 diam(Qi ) ≤ d (x, ∂Ω) ≤ 6 diam(Qi ) 4

(8.6) for every x ∈ Q∗i .

A collection {Qi : i ∈ N} of cubes Qi ⊂ Ω satisfying the properties in Lemma 8.9 is called a Whitney decomposition of Ω. Let f ∈ L∞ (Rn ) be a nonnegative function with a compact support. Let t > 0 and Et = {x ∈ Rn : M ∗ f (x) > t}, where M ∗ f is the noncentered Hardy–Littlewood maximal function of f as in Remark 1.2. Then Et  Rn is an open set. Assume that Et = ∅ and let {Qi : i ∈ N} be a Whitney decomposition of Et , see Lemma 8.9. Define  f (x), x ∈ Rn \ E t , (8.7) ft (x) = x ∈ Qi for some i ∈ N. Qi f (y) dy, If Et = ∅, then ft (x) = f (x) for every x ∈ Rn . Since f ∈ L∞ (Rn ) is a nonnegative function and the support of f is compact, also ft is nonnegative and has a compact support. Lemma 8.10. Assume that f ∈ L∞ (Rn ) is a nonnegative function with a compact support. Let t > 0 and let Et and ft be as in (8.7). Then the following assertions hold. (a) ft (x) ≤ C(n)t for almost every x ∈ Rn . (b) M ∗ f (x) ≤ C(n)M ∗ ft (x) for every x ∈ Rn \ Et . Proof. By Corollary 1.23 we have ft (x) = f (x) ≤ M ∗ f (x) ≤ t for almost every x ∈ Rn \ Et . On the other hand, if x ∈ Et , then x ∈ Qi for some i ∈ N. By (8.5), there exists a ball B ⊂ Rn that intersects Rn \ Et and satisfies

166

8. MUCKENHOUPT WEIGHTS

Qi ⊂ B and |B| ≤ C(n)|Qi |. For z ∈ B \ Et we have ft (x) = 

Qi

f (y) dy ≤ C(n)  f (y) dy ≤ C(n)M ∗ f (z) ≤ C(n)t. B

Thus (a) holds for almost every x ∈ R . To prove part (b), let x ∈ Rn \ Et . Let z ∈ Rn and r > 0 be such that x ∈ B(z, r). Then    f (y) dy = f (y) dy + f (y) dy n

B(z,r)

B(z,r)\Et

B(z,r)∩Et

 =

f (y) dy + B(z,r)\Et

∞ 

i=1

f (y) dy.

B(z,r)∩Qi

Consider a Whitney cube Qi of Et satisfying B(z, r)∩Qi = ∅. Since x ∈ B(z, r)\Et , by the lower bound in (8.5) we have Qi ⊂ B(z, 4r). Thus (8.7) implies    f (y) dy ≤ f (y) dy = f (y) dy B(z,r)∩Qi B(z,4r)∩Qi Qi   = ft (y) dy = ft (y) dy. Qi

B(z,4r)∩Qi

By collecting the estimates above, we obtain   ∞ 

f (y) dy ≤ ft (y) dy + B(z,r)

B(z,4r)\Et

i=1

 ft (y) dy =

B(z,4r)∩Qi

ft (y) dy, B(z,4r)

and consequently 

B(z,r)

f (y) dy ≤ C(n) 

ft (y) dy ≤ C(n)M ∗ ft (x).

B(z,4r)

By taking supremum over all balls B(z, r) as above, we conclude that (b) holds.



Theorem 8.11. Let 1 < p < ∞ and let w be a weight in Rn . Assume that there exists a constant C1 such that   ∗

p (8.8) M f (x) w(x) dx ≤ C1 f (x)p w(x) dx Rn

Rn

for all nonnegative functions f ∈ L∞ (Rn ) with compact support. Then there exist ε = ε(n, p, C1 ), with 0 < ε < p − 1, and C = C(n, p, C1 ) such that   ∗

p−ε M f (x) w(x) dx ≤ C f (x)p−ε w(x) dx Rn

Rn

for all nonnegative f ∈ L∞ (Rn ) with compact support. Proof. Fix a nonnegative function f ∈ L∞ (Rn ) with a compact support. Let t > 0, and let Et and ft be as in (8.7) for f . By the assumption   ∗

p M ft (x) w(x) dx ≤ C1 ft (x)p w(x) dx. Rn

Rn

8.3. SELF-IMPROVEMENT OF WEIGHTED NORM INEQUALITIES

167

By Lemma 8.10, we obtain   ∗ ∗

p

p M f (x) w(x) dx ≤ C(n, p) M ft (x) w(x) dx n n R \Et R \Et  ∗

p ≤ C(n, p) M ft (x) w(x) dx (8.9) n R   p p ≤ C(n, p, C1 ) f (x) w(x) dx + t w(Et ) . Rn \Et

This implies, for a fixed t0 > 0,   ∞ ∗

p −1−ε M f (x) w(x) dx dt t Rn \Et

t0



≤ C(n, p, C1 )

(8.10)

∞

t0

t−1−ε 

∞

+ C(n, p, C1 )

 f (x)p w(x) dx dt Rn \Et

tp−1−ε w(Et ) dt.

t0

By Fubini’s theorem, we have  ∞  ∗

p M f (x) w(x) dx dt t−1−ε t0

Rn \Et



∗

p M f (x)

= Rn

 =

∗

p M f (x)

1 ε

∞

t0 ∞





χRn \Et (x)t−1−ε dt w(x) dx t

−1−ε



dt w(x) dx

max{t0 ,M ∗ f (x)}

Rn

=





Rn

∗ 

p  −ε M f (x) max t0 , M ∗ f (x) w(x) dx.

In a similar way, we obtain   ∞    −ε 1 t−1−ε f (x)p w(x) dx dt = f (x)p max t0 , M ∗ f (x) w(x) dx ε Rn Rn \Et t0  1 ≤ f (x)p−ε w(x) dx, ε Rn   where the inequality holds since f (x) ≤ max t0 , M ∗ f (x) for almost every x ∈ Rn . The last term in (8.10) can be estimated as   ∞   ∞ p−1−ε p−1−ε χEt (x)t t w(Et ) dt = dt w(x) dx t0



Rn

t0



M ∗ f (x)

= Et0

1 ≤ p−ε 1 ≤ p−ε



 tp−1−ε dt w(x) dx

t0

∗

p−ε M f (x) w(x) dx

Et0



Rn

∗ 

p  −ε M f (x) max t0 , M ∗ f (x) w(x) dx.

168

8. MUCKENHOUPT WEIGHTS

Multiplication of the obtained inequalities by ε > 0, together with (8.10), gives   ∗ 

p  −ε ∗ M f (x) max t0 , M f (x) w(x) dx ≤ C(n, p, C1 ) f (x)p−ε w(x) dx Rn Rn  ∗ 

p  −ε ε M f (x) max t0 , M ∗ f (x) + C(n, p, C1 ) w(x) dx. p − ε Rn Here the last term is finite by (8.8) and the fact that t0 > 0. We choose 0 < ε < p−1 to be so small that 1 ε < . C(n, p, C1 ) p−ε 2 This allows us to absorb the last term to the left-hand side. Finally, by Fatou’s lemma,   ∗ ∗ 

p−ε

p  −ε M f (x) M f (x) max t0 , M ∗ f (x) w(x) dx ≤ lim inf w(x) dx t0 →0+ Rn Rn  f (x)p−ε w(x) dx, ≤ C(n, p, C1 ) Rn



and this concludes the proof. 8.4. Muckenhoupt Ap weights for 1 < p < ∞

Muckenhoupt weights are related to weighted norm inequalities for maximal functions and they are applied in harmonic analysis and partial differential equations. Definition 8.12. A weight w in Rn belongs to the Muckenhoupt class Ap , for 1 < p < ∞, if there exists a constant C such that  p−1 1 1−p (8.11) dx ≤C  w(x) dx  w(x) Q

Q

for every cube Q ⊂ R . The smallest possible constant C in (8.11) is called the Ap constant of w, and it is denoted by [w]Ap . n

The following basic properties of Muckenhoupt weights follow rather directly from the definitions and H¨older’s inequality. The first assertion shows that the Muckenhoupt classes are nested and the second assertion is a duality statement. Lemma 8.13. The following assertions hold. (a) For 1 < p < q < ∞, we have Ap ⊂ Aq .  (b) For 1 < p < ∞, we have w ∈ Ap if and only if w1−p ∈ Ap , where p p = p−1 . Proof. To prove assertion (a), let 1 < p < q < ∞, let Q ⊂ Rn be a cube, and older’s inequality assume that w ∈ Ap . By H¨ q−1  (q−1) p−1  p−1  q−1 1 1 q−1 1 p−1 1−q 1−q 1−p dx ≤  w(x) dx =  w(x) dx .  w(x) Q

Q

Q

This implies

 q−1  p−1 1 1 1−q dx 1−p dx w(x) dx w(x) ≤ w(x) dx w(x) ≤ [w]Ap ,     Q

Q

and assertion (a) follows.

Q

Q

8.4. MUCKENHOUPT Ap WEIGHTS FOR 1 < p < ∞

169

To prove assertion (b), let 1 < p < ∞ and assume that w ∈ Ap . Then  p −1

1 1−p 1−p 1−p dx  w(x) dx  w(x) Q

Q

1   p−1 1  =  w(x)1−p dx  w(x) dx ≤ [w]Ap−1 p

Q

Q





for every cube Q ⊂ Rn . This shows that w1−p ∈ Ap . Conversely, if w1−p ∈ Ap  then w = (w1−p )1−p ∈ Ap by the argument above.  Next, we consider a weighted H¨ older inequality for integral averages. Lemma 8.14. Let 1 < p < ∞ and assume that w ∈ Ap . Let Q ⊂ Rn be a cube and f ∈ L1 (Q) be a nonnegative function. Then  p1   1 1 p p f (x) w(x) dx .  f (x) dx ≤ [w]Ap w(Q) Q Q Proof. H¨ older’s inequality and (8.11) imply  p−1  p1   p 1 1 p 1−p w(x) dx f (x) w(x) dx  f (x) dx ≤ |Q| Q Q Q   p−1  p1  1   p 1 w(Q) p 1 p = f (x) w(x) dx  w(x) 1−p dx |Q| w(Q) Q Q  p1   1 1 ≤ [w]Ap p f (x)p w(x) dx . w(Q) Q



As an application of Lemma 8.14 we obtain that a Muckenhoupt weight is doubling, see Definition 8.3. Theorem 8.15. Let 1 < p < ∞ and assume that w ∈ Ap . Then w is doubling, that is, there exists a constant C = C(n, p, w) such that w(B(x, 2r)) ≤ Cw(B(x, r)) for every x ∈ Rn and r > 0. Proof. Let x ∈ Rn and r > 0. Since B(x, 2r) ⊂ Q(x, 2r), Lemma 8.14 with f = χB(x,r) implies 1 |B(x, r)| ≤ C(n)  = f (y) dy 2n |B(x, 2r)| Q(x,2r)  p1   1 1 p p ≤ C(n)[w]Ap f (y) w(y) dy w(Q(x, 2r)) Q(x,2r)  1 1 w(B(x, r)) p p ≤ C(n)[w]Ap . w(B(x, 2r)) The claim follows by raising both sides to power p and reorganizing the terms.



The class of Muckenhoupt weights is closed under maximum and minimum. Theorem 8.16. Let 1 < p < ∞ and assume that v, w ∈ Ap . Then max{v, w} ∈ Ap and min{v, w} ∈ Ap .

170

8. MUCKENHOUPT WEIGHTS

Proof. The function max{v, w} is a weight since max{v(x), w(x)} > 0 for almost every x ∈ Rn and max{v, w} ∈ L1loc (Rn ). By Definition 8.12 we have  max{v(x), w(x)} dx ≤  v(x) dx +  w(x) dx Q Q Q 1−p 1−p   1 1 1−p 1−p dx + [w]Ap  w(x) dx ≤ [v]Ap  v(x) Q

≤ [v]Ap

Q

1−p 

1 1−p + [w]Ap  max{v(x), w(x)} dx Q

for every cube Q ⊂ R . Thus max{v, w} ∈ Ap . Consider σ(x) = min{v(x), w(x)}, for x ∈ Rn . Clearly σ(x) > 0 for almost every x ∈ Rn and σ ∈ L1loc (Rn ), and so σ is a weight. Lemma 8.13 (b) implies 1 1 p . By the first part of the proof we have v 1−p , w 1−p ∈ A p−1  1 1 1 1  p , σ 1−p = (min{v, w}) 1−p = max v 1−p , w 1−p ∈ A p−1 n

and min{v, w} = σ ∈ Ap follows from Lemma 8.13 (b).



A weighted norm inequality for the dyadic maximal function is related to Muckenhoupt weights. Lemma 8.17. Let 1 < p < ∞ and assume that w is a weight such that   d

p MQ f (x) w(x) dx ≤ C1 f (x)p w(x) dx Rn

Q

for every nonnegative f ∈ L∞ (Rn ) with compact support and for every cube Q ⊂ Rn . Then w ∈ Ap and [w]Ap ≤ C1 . 1

Proof. Fix a cube Q ⊂ Rn . Let ε > 0 and f = (ε + w) 1−p χQ . Clearly 1

d  (ε + w(y)) 1−p dy ≤ MQ f (x) Q

for every x ∈ Q, and thus  p d

p 1 1−p dy w(x) dx (ε + w(y)) ≤  MQ f (x) w(x) dx.   Q

Q

Q

By the assumption, we have  d

p 1 M f (x) w(x) dx ≤ C 1 Q  |Q|

f (x)p w(x) dx

Rn

Q

p

≤ C1  (ε + w(x)) 1−p (ε + w(x)) dx Q

1

= C1  (ε + w(x)) 1−p dx. Q

This gives

  w(x) dx  (ε + w(y)) Q

Q

1 1−p

p−1 dy

≤ C1

8.4. MUCKENHOUPT Ap WEIGHTS FOR 1 < p < ∞

171

for every ε > 0. Fatou’s lemma implies  p−1 1 1−p dx w(x) dx w(x)   Q

Q



≤ lim inf  w(x) dx  (ε + w(x)) ε→0+ Q

1 1−p

p−1 dx

≤ C1 ,

Q



and the claim follows.

Theorem 8.19 gives a converse of Theorem 8.17, see also Theorem 9.9 and Theorem 8.28. First, we discuss a partial converse that provides a weighted weak type estimate for the dyadic maximal operator. The proof is based on the Calder´ on– Zygmund lemma and the argument is similar to the proof of Lemma 8.7. Theorem 8.18. Let Q0 ⊂ Rn be a cube, f ∈ L∞ (Q0 ) be a nonnegative function d f (x) > t} for t > 0. Let 1 < p < ∞ and assume that and Et = {x ∈ Q0 : MQ 0 w ∈ Ap . Then  [w]Ap w(Et ) ≤ f (x)p w(x) dx tp Q0 for every t > 0. Proof. Let t > 0 and let Dt be the associated collection of maximal dyadic subcubes of Q0 as in the proof of Lemma 8.6. Recall that in the case 0 < t < fQ0 we have Dt = {Q0 }. Since t < fQ for every Q ∈ Dt , by (8.3) we have   

Q ≤ w(Q) w(Et ) = w Q∈Dt

Q∈Dt

 p 1 1 p w(Q)t ≤ p w(Q)  f (x) dx . = p t t Q Q∈Dt

Q∈Dt

The integrals on the right-hand side can be estimated with Lemma 8.14. This, together with the pairwise disjointness of the cubes Q ∈ Dt , implies   [w]Ap [w]Ap p f (x) w(x) dx = f (x)p w(x) dx w(Et ) ≤ p tp t Et Q∈Dt Q  [w]Ap ≤ f (x)p w(x) dx.  tp Q0 The following result shows that the Hardy–Littlewood maximal operator is bounded on Lp (w dx) for 1 < p < ∞ and w ∈ Ap . In particular, we obtain a converse of Lemma 8.17. Theorem 8.19. Let 1 < p < ∞ and assume that w ∈ Ap . There exists a constant C = C(n, p, [w]Ap ) such that   ∗

p M f (x) w(x) dx ≤ C |f (x)|p w(x) dx Rn

for every f ∈ L (w dx). p

Rn

172

8. MUCKENHOUPT WEIGHTS 1

Proof. Let σ = w 1−p . Lemma 8.13 (b) implies that σ ∈ Ap , where p = Let f ∈ Lp (w dx), and fix x ∈ Rn and a ball B = B(x0 , r) with x ∈ B. Then 1  p−1  p−1   |B| 1 |f (y)| dy ,  |f (y)| dy = A w(B(x , 3r)) σ(B(x , 3r)) 0 0 B B where

p p−1 .



 1 w(B(x0 , 3r))σ(B(x0 , 3r))p−1 p−1 A= |B|p   1 w(Q(x0 , 3r))σ(Q(x0 , 3r))p−1 p−1 ≤ C(n, p) |Q(x0 , 3r)|p 1

≤ C(n, p)[w]Ap−1 . p For z ∈ B, we have 1 σ(B(x0 , 3r))

 |f (y)| dy ≤ B

1 σ(B(z, 2r))



|f (y)σ(y)−1 |σ(y) dy B(z,2r)

≤ M σ (f σ −1 )(z). Hence,  |f (y)| dy B

1  p−1  σ

p−1 1 −1 −1 M (f σ )(z) w(z) w(z) dz ≤ C(n, p)[w]Ap w(B(x0 , 3r)) B 1   p−1  σ

p−1 1 −1 −1 M (f σ )(z) ≤ C(n, p, [w]Ap ) w(z) w(z) dz w(B(x, 2r)) B(x,2r)

1 ≤ C(n, p, [w]Ap ) M w M σ (f σ −1 )p−1 w−1 (x) p−1 . 1 p−1



By taking supremum over all balls B as above, it follows that

1 M ∗ f (x) ≤ C(n, p, [w]Ap ) M w M σ (f σ −1 )p−1 w−1 (x) p−1 for all x ∈ Rn . Observe that w ∈ Ap and σ ∈ Ap are doubling weights by Theorem 8.15, with constants depending on n, p and [w]Ap . Remark 8.4 (a) implies  ∗

p M f (x) w(x) dx n R  w σ

p M M (f σ −1 )p−1 w−1 (x) p−1 w(x) dx ≤ C(n, p, [w]Ap ) n R σ

p M (f σ −1 )(x) σ(x) dx ≤ C(n, p, [w]Ap ) n R ≤ C(n, p, [w]Ap ) |f (x)|p w(x) dx, Rn

where the facts that σ = w

1 1−p

and w = σ 1−p are used.



Theorem 8.11 and Theorem 8.19 together imply the following Muckenhoupt Ap weighted norm inequality on Lp−ε (w dx).

¨ 8.5. REVERSE HOLDER INEQUALITIES FOR MUCKENHOUPT WEIGHTS

173

Corollary 8.20. Let 1 < p < ∞ and assume that w ∈ Ap . There exist ε = ε(n, p, [w]Ap ), with 0 < ε < p − 1, and C = C(n, p, [w]Ap ) such that   ∗

p−ε M f (x) w(x) dx ≤ C f (x)p−ε w(x) dx Rn

Rn

∞

for all nonnegative f ∈ L (R ) with compact support. n

Next we obtain a self-improvement result for Muckenhoupt Ap weights for 1 < p < ∞. Corollary 8.21. Let 1 < p < ∞ and assume that w ∈ Ap . There exist ε = ε(n, p, [w]Ap ) > 0 and C = C(n, p, [w]Ap ) such that w ∈ Ap−ε and [w]Ap−ε ≤ C[w]Ap . Proof. Let Q ⊂ Rn be a cube. By Corollary 8.20 there exists ε = ε(n, p, [w]Ap ), with 0 < ε < p − 1, such that   d ∗

p−ε

p−ε MQ f (x) M f (x) w(x) dx ≤ C(n, p) w(x) dx Q Rn  ≤ C(n, p, [w]Ap ) f (x)p−ε w(x) dx Rn

∞

for all nonnegative f ∈ L (R ) with compact support. Lemma 8.17 implies that  w ∈ Ap−ε and [w]Ap−ε ≤ C(n, p, [w]Ap ). n

8.5. Reverse H¨ older inequalities for Muckenhoupt weights We prove a reverse H¨ older inequality for Muckenhoupt Ap weights, with 1 < p ≤ ∞. For this purpose, we first discuss the limiting Muckenhoupt condition for p = ∞, which is the weakest Ap condition. Definition 8.22. A weight w in Rn belongs to the Muckenhoupt class A∞ if there exist constants C, δ > 0 such that  δ |E| w(E) ≤C (8.12) w(Q) |Q| whenever Q ⊂ Rn is a cube and E ⊂ Q is a measurable set. An A∞ weight is doubling, see Definition 8.3. Theorem 8.23. Assume that w ∈ A∞ . Then w is doubling, that is, there exists a constant C = C(n, w) such that w(B(x, 2r)) ≤ Cw(B(x, r)) for every x ∈ Rn and r > 0. Proof. Let C1 and δ be the constants in the A∞ condition (8.12) for w. Choose 0 < β = β(w) < 1 satisfying 0 < C1 β δ < 1 and define α = 1 − C1 β δ > 0. First assume that Q ⊂ Rn is a cube and E ⊂ Q is a measurable set with |E| ≥ (1 − β)|Q|. Then |Q \ E| ≤ β|Q| and (8.12) implies  δ |Q \ E| w(Q \ E) ≤ C1 w(Q) ≤ C1 β δ w(Q). |Q| Consequently, (8.13)

w(E) = w(Q) − w(Q \ E) ≥ (1 − C1 β δ )w(Q) = αw(Q).

174

8. MUCKENHOUPT WEIGHTS 1

Fix x ∈ Rn and r > 0. Define r1 = 2r and rj+1 = (1 − β) n rj for every j = 1, . . . , k − 1, where k = k(n, β) is the smallest integer such that Q(x, rk ) ⊂ B(x, r). Then Q(x, rj+1 ) ⊂ Q(x, rj ) and |Q(x, rj+1 )| ≥ (1 − β)|Q(x, rj )|, for every j = 1, . . . , k − 1. Hence we may apply (8.13) iteratively for these cubes, and obtain w(B(x, 2r)) ≤ w(Q(x, 2r)) = w(Q(x, r1 )) ≤ α−1 w(Q(x, r2 )) ≤ · · · ≤ α1−k w(Q(x, rk )) ≤ α1−k w(B(x, r)). This proves the claim with the constant C(n, w) = α1−k .



Let Q ⊂ Rn be a cube, σ be a weight and f ∈ L1 (Q; σ dx) be a nonnegative function. The σ-weighted average of f on Q is defined as  1 f (x)σ(x) dx. (8.14) fQ;σ = σ(Q) Q In the next lemma a distributional inequality implies a reverse H¨older inequality. Lemma 8.24. Let σ ∈ A∞ and let w ≥ 0 be a measurable function such that wσ ∈ L1loc (Rn ). Assume that there exist α > 0 and 0 < β < 1 such that σ({x ∈ Q : w(x) > βwQ;σ }) ≥ ασ(Q) for every cube Q ⊂ Rn . Then there exist constants q = q(n, σ, α, β) > 1 and C = C(n, σ, α, β) such that   1q   1 C w(x)q σ(x) dx ≤ w(x)σ(x) dx σ(Q0 ) Q0 σ(Q0 ) Q0 for every cube Q0 ⊂ Rn . Proof. Let Q0 ⊂ Rn be a cube. For t > wQ0 ;σ , let Dt ⊂ D(Q0 ) be the collection of maximal dyadic subcubes Q ⊂ Q0 such that wQ;σ > t, compare the proof of Lemma 8.6. Observe that Q  Q0 for every Q ∈ Dt since t > wQ0 ;σ . Let Q ∈ Dt and let πQ ∈ D(Q0 ) be the dyadic parent cube of Q, which satisfies the conditions Q ⊂ πQ and 2n |Q| = |πQ|. Theorem 8.23 implies σ(πQ) ≤ C(n, σ). σ(Q) By the maximality of Q ∈ Dt , we have wπQ;σ ≤ t and   1 1 σ(πQ) · w(x)σ(x) dx ≤ w(x)σ(x) dx ≤ C(n, σ)t. t < wQ;σ = σ(Q) Q σ(Q) σ(πQ) πQ  On the other hand, if x ∈ Q0 \ Q∈Dt Q, then wQ ;σ ≤ t whenever Q ∈ D(Q0 ) with x ∈ Q ; otherwise Q ⊂ Q for some Q ∈ Dt , and this would contradict the choice of x. Since there are arbitrarily small such cubes Q containing x, the Lebesgue differentiation theorem for the doublingmeasure σ dx, see Remark 8.4 (c), shows that w(x) ≤ t for almost every x ∈ Q0 \ Q∈Dt Q with respect to the measure σ dx.

¨ 8.5. REVERSE HOLDER INEQUALITIES FOR MUCKENHOUPT WEIGHTS

175

Let dμ = (σw) dx. Using the absolute continuity of μ with respect to σ dx and the assumption, we obtain

μ(Q) ≤ C(n, σ)t σ(Q) μ({x ∈ Q0 : w(x) > t}) ≤ Q∈Dt

Q∈Dt

C(n, σ)t σ({x ∈ Q : w(x) > βwQ;σ }) ≤ α Q∈Dt

(8.15)

C(n, σ)t σ({x ∈ Q : w(x) > βt}) ≤ α Q∈Dt

C(n, σ)t σ({x ∈ Q0 : w(x) > βt}), ≤ α where the final step holds since the cubes in Dt ⊂ D(Q0 ) are pairwise disjoint. Next we multiply (8.15) by tq−2 , where 1 < q < 2 is to be specified later, and then integrate the resulting estimate from wQ0 ;σ to a fixed t0 > wQ0 ;σ . This gives  t0 tq−2 μ({x ∈ Q0 : w(x) > t}) dt wQ0 ;σ

 C(n, σ) t0 dt ≤ (βt)q σ({x ∈ Q0 : w(x) > βt}) q αβ t 0  C(n, σ) t0 dt = (βt)q σ({x ∈ Q0 : min{t0 , w(x)} > βt}) q αβ t 0 C(n, σ) w(x) min{t0 , w(x)}q−1 σ(x) dx < ∞. ≤ qαβ q Q0

By Fubini’s theorem,  t0 tq−2 μ({x ∈ Q0 : w(x) > t}) dt wQ0 ;σ





min{t0 ,w(x)}

w(x)

= {x∈Q0 :w(x)>wQ0 ;σ }



 tq−2 dt σ(x) dx

wQ0 ;σ

q−1  wQ min{t0 , w(x)}q−1 0 ;σ − w(x) = σ(x) dx q − 1 q − 1 {x∈Q0 :w(x)>wQ0 ;σ }  2 1 wq σ(Q0 ). w(x) min{t0 , w(x)}q−1 σ(x) dx − ≥ q − 1 Q0 q − 1 Q0 ;σ



Combining the estimates above, we arrive at   1q 1   1 1 C(n, σ) q q−1 − w(x) min{t0 , w(x)} σ(x) dx q−1 qαβ q σ(Q0 ) Q0   1q 2 ≤ wQ0 ;σ q−1 for every t0 > wQ0 ;σ . The claim follows by first choosing 1 < q < 2 to be so small that C(n, σ) 1 − ≥ 1, q−1 qαβ q and then letting t0 → ∞ inside the final integral and applying Fatou’s lemma. 

176

8. MUCKENHOUPT WEIGHTS

The next result shows that A∞ =

 1 0 and δ = δ(σ) > 0 be the constants in the A∞ condition (8.12) for σ. Choose 0 < β = β(σ) < 1 satisfying C(σ)β δ < 1 and define α = 1 − C(σ)β δ > 0. Let w = σ −1 , let Q ⊂ Rn be a cube and define E = {x ∈ Q : w(x) > βwQ;σ }. Since w(x) ≤ βwQ;σ for every x ∈ Q \ E, we obtain  |Q \ E| 1 β |Q| = σ(Q \ E) ≤ β. w(x)σ(x) dx ≤ |Q| |Q| Q\E |Q| σ(Q) By (8.12), we have

 δ |Q \ E| σ(E) + σ(Q \ E) σ(E) σ(E) 1= ≤ + C(σ) + C(σ)β δ , ≤ σ(Q) σ(Q) |Q| σ(Q)

(8.16) and thus

ασ(Q) = 1 − C(σ)β δ σ(Q) ≤ σ(E). From Lemma 8.24 we obtain constants q = q(n, σ, α, β) = q(n, σ) > 1 and C = C(n, σ, α, β) = C(n, σ) such that  q1   q1   1 1 1−q q σ(x) dx = w(x) σ(x) dx 1 σ(Q) Q Q σ(Q) q  C ≤ w(x)σ(x) dx σ(Q) Q 1

q−1

|Q| q |Q| q |Q| =C =C q−1 , 1 σ(Q) σ(Q) q σ(Q) q where we also used the fact that w(x)σ(x) = 1 for almost every x ∈ Q. Reorganization of the terms gives  q1  q−1 1 − q−1 − 1−q σ(x) dx ≤ C(n, σ). |Q| q σ(Q) q |Q| q Q

For p =

q q−1 ,

the estimate above implies  p−1 1 1−p dx σ(x) dx σ(x) ≤ C.   Q

Q

Hence σ ∈ Ap , with p = p(n, σ) and the Ap constant C = C(n, σ).

¨ 8.5. REVERSE HOLDER INEQUALITIES FOR MUCKENHOUPT WEIGHTS

177

Next we show that (b) implies (c). Let 1 < p < ∞ and assume that w ∈ Ap . 1 Choose 0 < β = β(p, w) < 1 satisfying (β[w]Ap ) p−1 < 1 and define α = 1 − 1 (β[w]Ap ) p−1 > 0. Let Q ⊂ Rn be a cube and E = {x ∈ Q : w(x) > βwQ }. By (8.11), we have p−1   p−1  1 1 1 |Q \ E| 1−p = wQ (βwQ ) dx β |Q| |Q| Q\E p−1  1 ≤ wQ  w(x) 1−p dx ≤ [w]Ap . Q

As in (8.16), we obtain

1 |E| ≥ 1 − (β[w]Ap p−1 )|Q| = α|Q|. Lemma 8.24, with σ(x) = 1 for every x ∈ Rn , gives constants q = q(n, α, β) = q(n, p, w) > 1 and C = C(n, α, β) = C(n, p, w) > 0 such that 

 q1

q

 w(x) dx Q

≤ C  w(x) dx Q

for every cube Q ⊂ Rn , and thus assertion (c) holds. Finally we show that (c) implies (a). Let w be a weight and assume that there are constants C > 0 and q > 1 such that the reverse H¨older condition (c) holds. Let Q ⊂ Rn be a cube and E ⊂ Q be a measurable set. Using first H¨older’s inequality and then the reverse H¨older inequality, we obtain w(E) ≤ |E|

q−1 q

 q

 q1

w(x) dx

≤ C|E|

Q

q−1 q



|E| |Q|  w(x) dx = C |Q| Q 1 q

Thus w satisfies the A∞ condition (8.12) with δ =

q−1 q

 q−1 q

> 0, and (a) holds.

w(Q). 

Remark 8.26. If 1 < p < ∞ and w ∈ Ap , then C and q > 1 in Theorem 8.25 (c) only depend on w through [w]Ap . This follows from the proof of (b) implies (c) since α and β in the proof only depend on [w]Ap and p. From Theorem 8.25 we obtain a second proof for the self-improvement property of Muckenhoupt weights, compare to Corollary 8.21. Yet another proof will be based on Theorem 8.39. Theorem 8.27. Let 1 < p < ∞ and assume that w ∈ Ap . There exists a constant ε = ε(n, p, [w]Ap ) > 0 such that w ∈ Ap−ε , with [w]Ap−ε only depending on n, p and [w]Ap . 1

p Proof. By Lemma 8.13 (b), we have w 1−p ∈ Ap with p = p−1 , and [w]Ap only depends on p and [w]Ap . Theorem 8.25 and Remark 8.26 imply that there exist C = C(n, p, [w]Ap ) and q = q(n, p, [w]Ap ) > 1 such that

 (8.17)

 w(x) Q

q 1−p

 1q dx

1

≤ C  w(x) 1−p dx Q

178

8. MUCKENHOUPT WEIGHTS

for every cube Q ⊂ Rn . Let ε = ε(p, q) = ε(n, p, [w]Ap ), with 0 < ε < p − 1, be q 1 determined by p−1 = p−ε−1 . By (8.17) and (8.11), we have 

1

 w(x) 1−(p−ε) dx Q

p−ε−1



 p−1 q q 1−p dx w(x)  Q  p−1 1 ≤ C(n, p, [w]Ap )  w(x) 1−p dx =



Q

−1

≤ C(n, p, [w]Ap )  w(x) dx Q

for every cube Q ⊂ Rn . This shows that w ∈ Ap−ε , with a constant C(n, p, [w]Ap ).  By using the self-improvement of Muckenhoupt weights, we obtain a second proof for the fact that the Hardy–Littlewood maximal operator is bounded on Lp (w dx) for 1 < p < ∞ and w ∈ Ap , compare to Theorem 8.19. Theorem 8.28. Let 1 < p < ∞ and assume that w ∈ Ap . There exists a constant C = C(n, p, [w]Ap ) such that   M f (x)p w(x) dx ≤ C |f (x)|p w(x) dx Rn

Rn

for every f ∈ L1loc (Rn ). Proof. Without loss of generality, we may assume that f is a nonnegative function for which the right-hand side is finite. By Theorem 8.27, there exists 1 < q = q(n, p, [w]Ap ) < p such that w ∈ Aq . Let x ∈ Rn , r > 0 and Q = Q(x, r). Lemma 8.14 implies   q1  1 q f (y) dy ≤ C(n)  f (y) dy ≤ C(n, q, w) f (y) w(y) dy  w(Q) Q B(x,r) Q

1 ≤ C(n, q, w) M c,w (f q )(x) q . By taking supremum over all r > 0, we obtain

1 M f (x) ≤ C(n, q, w) M c,w (f q )(x) q . Using Remark 8.4 (b) we arrive at   c,w q

p M (f )(x) q w(x) dx M f (x)p w(x) dx ≤ C(n, p, q, w) n Rn  R |f (x)|p w(x) dx. ≤ C(n, p, w) Rn

Finally, the Aq constant of w depends on w only through [w]Ap , see Theorem 8.27. The constant in Lemma 8.14 only depends on [w]Aq and q. The constant in Remark 8.4 (b) depends on n, pq and the doubling constant of w, which in turn only depends on [w]Aq , n and q, see the proof of Theorem 8.15. Hence we conclude  that the claim holds with C = C(n, p, w) = C(n, p, [w]Ap ).

8.6. A1 WEIGHTS AND COIFMAN–ROCHBERG LEMMA

179

8.6. A1 weights and Coifman–Rochberg lemma This section discusses the Muckenhoupt condition with p = 1, which is the strongest Ap condition. Definition 8.29. A weight w in Rn belongs to the Muckenhoupt class A1 if there exists a constant C such that 1 (8.18)  w(x) dx ess sup w(x) ≤ C, x∈Q Q for every cube Q ⊂ Rn . The smallest possible constant C in (8.18) is called the A1 constant of u, and it is denoted by [w]A1 . Assume that w ∈ A1 and let 1 < p < ∞. Then  p−1 1 1 |Q| − p−1 ≤ [w]A1 · w(x) dx ≤ ess sup  w(Q) x∈Q w(x) Q for every cube Q ⊂ Rn , and thus w ∈ Ap with [w]Ap ≤ [w]A1 . Together with Theorem 8.25 this implies A1 ⊂ Ap ⊂ A∞ ,

(8.19) for 1 < p < ∞.

Remark 8.30. If w and σ are A1 weights, then it is straightforward to show that wσ 1−p ∈ Ap for every 1 < p < ∞. Jones [213] proved that the converse claim holds as well, see also Coifman, Jones and Rubio de Francia [88] and Cruz-Uribe, Martell and P´erez [92]. In particular, this gives a characterization of Ap weights in terms of A1 weights. The Muckenhoupt A1 condition in (8.18) can also be written as inf w(x),  w(x) dx ≤ C ess x∈Q

(8.20)

Q

for every cube Q ⊂ R . This is useful in the proof of the following characterization. n

Lemma 8.31. A weight w in Rn belongs to A1 if and only if there exists a constant C1 such that (8.21)

M c w(x) ≤ C1 w(x)

for almost every x ∈ Rn .

Furthermore, if w ∈ A1 , then (8.21) holds with C1 = [w]A1 . Proof. It is clear that (8.21) implies (8.20), with C = C1 , since c  w(y) dy ≤ M w(x) ≤ C1 w(x) Q

for almost every x ∈ Q whenever Q ⊂ Rn is a cube. Assume then that w ∈ A1 . Define E = {x ∈ Rn : M c w(x) > [w]A1 w(x)} and let x ∈ E. There exists a cube Q ⊂ Rn containing x such that (8.22)

 w(y) dy > [w]A1 w(x). Q

180

8. MUCKENHOUPT WEIGHTS

 whose corners have rational coordinates, with For every ε > 0 there is a cube Q,  and |Q  \ Q| < ε. Then |Q|  = |Q| + |Q  \ Q| < |Q| + ε, and by choosing ε > 0 Q⊂Q small enough, we have   1 1 w(y) dy > [w]A1 w(x).   w(y) dy =   w(y) dy ≥ |Q| + ε |Q| Q Q Q Hence we may assume that the corner points of the cube Q satisfying (8.22) are rational. The conditions in (8.20) and (8.22) imply [w]A1 w(x) <  w(y) dy ≤ [w]A1 ess inf w(y). y∈Q

Q

Since [w]A1 > 0, we conclude that w(x) < ess inf w(y). y∈Q

Let (Qi )i∈N be an enumeration of cubes in Rn with rational corners and define   Ei = x ∈ Qi : w(x) < ess inf w(y) y∈Qi

for i  ∈ N. Then |Ei | = 0 for every i ∈ N, and the argument above shows that  E⊂ ∞ i=1 Ei . Thus |E| = 0, and the claim in (8.21) follows, with C1 = [w]A1 . The following lemma provides a weighted weak type estimate for the dyadic maximal function, compare to Theorem 8.18 and Lemma 8.7. Lemma 8.32. Let Q0 ⊂ Rn be a cube, f ∈ L∞ (Q0 ) be a nonnegative function d f (x) > t} for t > 0. Assume that w ∈ A1 . Then and Et = {x ∈ Q0 : MQ 0  [w]A1 w(Et ) ≤ f (x)w(x) dx t Q0 for every t > 0. Proof. Fix t > 0 and let Dt be the associated collection of maximal dyadic subcubes of Q0 as in the proof of Lemma 8.6. Recall that in the case 0 < t < fQ0 we have Dt = {Q0 }. Since t < fQ for every Q ∈ Dt , by (8.3) we have 

1 w(Q) w(Et ) = w(Q) ≤ f (x) dx t |Q| Q Q∈Dt Q∈Dt     w(Q) 1 1 [w]A1 · f (x)w(x) dx.  = f (x)w(x) dx ≤ t |Q| w(x) t Q Q0 Q∈Dt

A lemma of Coifman and Rochberg gives a maximal function characterization for A1 weights. Moreover, this result gives a method for constructing A1 weights. Recall that M c f is the noncentered Hardy–Littlewood maximal function of f ∈ L1loc (Rn ) with respect to cubes, see Remark 8.4 (b). Theorem 8.33. The following assertions hold. (a) Let 0 < δ < 1 and assume that f ∈ L1loc (Rn ) satisfies 0 < M c f (x) < ∞ for almost every x ∈ Rn . Then w = (M c f )δ is an A1 weight and [w]A1 ≤ C(n) 1−δ . (b) Conversely, assume that w ∈ A1 . There exist f ∈ L1loc (Rn ), 0 < δ < 1, and a function g, with g, g1 ∈ L∞ (Rn ), such that w(x) = g(x)(M c f (x))δ for almost every x ∈ Rn .

8.6. A1 WEIGHTS AND COIFMAN–ROCHBERG LEMMA

181

Proof. To prove part (a), it suffices to show that C(n) c δ c δ  (M f (y)) dy ≤ 1 − δ (M f (x)) Q for every cube Q ⊂ Rn and for every x ∈ Q. Let Q = Q(x0 , r) and write f = f1 +f2 , where f1 = f χQ(x0 ,2r) and f2 = f χRn \Q(x0 ,2r) . We may consider (M c f1 )δ and (M c f2 )δ separately, since M c f ≤ M c f1 + M c f2 and thus (M c f )δ ≤ (M c f1 )δ + (M c f2 )δ for 0 < δ < 1. Hence it is enough to show that (8.23)

C(n) c δ c δ  (M fi (y)) dy ≤ 1 − δ (M f (x)) , Q

for every x ∈ Q with i = 1, 2. Moreover, we may assume that fi L1 (Rn ) > 0 for i = 1, 2, since otherwise M c fi is identically zero and (8.23) follows immediately. Let t0 > 0 to be chosen later. By Theorem 1.15 (b), we have  ∞ δ c δ tδ−1 {y ∈ Q : M c f1 (y) > t} dt  (M f1 (y)) dy = |Q| Q 0  ∞   δ C(n) ≤ f1 L1 (Rn ) dt tδ−1 min |Q|, |Q| 0 t  t0  ∞    δ C(n) f1 L1 (Rn ) dt = + tδ−1 min |Q|, |Q| 0 t t0    ∞  t0 δ ≤ tδ−1 dt + C(n)f1 L1 (Rn ) tδ−2 dt |Q| |Q| 0 t0   1 (Rn )  f δ 1 L = tδ0 1 + C(n) t−1 . |Q| 1−δ 0 By choosing t0 =

f1 L1 (Rn ) |Q|

> 0, we obtain



   f1 L1 (Rn ) δ f1 L1 (Rn ) δ |Q| 1 + C(n) |Q| |Q| 1 − δ f1 L1 (Rn )  δ   1 C(n)δ |f (y)| dy = 1+ 1−δ |Q| Q(x0 ,2r)  δ   1 C(n)δ2nδ nδ = 2 + |f (y)| dy 1−δ |Q(x0 , 2r)| Q(x0 ,2r)

c δ  (M f1 (y)) dy ≤ Q

≤

C(n) (M c f (x))δ , 1−δ

for every x ∈ Q. This proves (8.23) for (M c f1 )δ . To obtain an estimate for (M c f2 )δ , we derive a pointwise estimate in Q. Let y ∈ Q and let Q = Q(y  , r  ) be any cube containing y. If Q ⊂ Q(x0 , 2r), then  1 |f2 (z)| dz = 0. |Q | Q

182

8. MUCKENHOUPT WEIGHTS

Next we assume that Q intersects the complement of Q(x0 , 2r). Then 2r  ≥ r. Thus Q ⊂ Q(y  , 5r  ), and   1 5n |f (z)| dz ≤ |f2 (z)| dz 2 |Q | Q |Q(y  , 5r  )| Q(y ,5r )  5n ≤ |f (z)| dz ≤ 5n M c f (x) |Q(y  , 5r  )| Q(y ,5r ) for every x ∈ Q. The estimates above show that M c f2 (y) ≤ 5n M c f (x) for every x, y ∈ Q, and this implies (8.23) for (M c f2 )δ . The proof of (a) is complete. We turn to part (b). Since w ∈ A1 ⊂ A2 , by Theorem 8.25 there exist C and q > 1 such that   q1 q ≤ C  w(y) dy  w(y) dy Q

Q

for every cube Q ⊂ R . Together with Lemma 8.31, this implies   1q c q

q1 q M (w )(x) = sup  w(y) dy ≤ C sup  w(y) dy n

Qx c

Qx

Q

Q

= CM w(x) ≤ C[w]A1 w(x) for almost every x ∈ R . On the other hand, by H¨older’s inequality   q1

1 = M c (wq )(x) q w(x) ≤ M c w(x) = sup  w(y) dy ≤ sup  w(y)q dy n

Qx

Qx

Q

Q

for almost every x ∈ R . Thus we have shown that n

w(x) ≤ (M c f (x))δ ≤ Cw(x) for almost every x ∈ Rn , where f = wq ∈ L1loc (Rn ) and δ = 1q . The claim in (b) follows by choosing w(x) g(x) = , (M c f (x))δ 

and the proof is complete.

Remark 8.34. The claim in Theorem 8.33 (a) does not hold for δ = 1. In fact, if f is a compactly supported measurable function with 0 < f L∞ (Rn ) < ∞, then the function M c f cannot be an A1 weight, or even an A2 weight. To see this, assume that M c f ∈ A2 ⊃ A1 . Then also w = (M c f )−1 is an A2 weight by Lemma 8.13 (b) for p = 2. By the boundedness of the maximal function on L2 (w dx), see Theorem 8.28, and the fact that w(x) ≤ |f (x)|−1 almost everywhere, we obtain    c

2 M f (x) w(x) dx ≤ C M c f (x) dx = |f (x)|2 w(x) dx n n n R R R  |f (x)| dx < ∞. ≤C Rn

However, it can be shown that the left-hand side is finite only if f L∞ (Rn ) = 0, compare to Remark 1.12.

¨ 8.7. SELF-IMPROVEMENT OF REVERSE HOLDER INEQUALITIES

183

Remark 8.35. The original result of Coifman and Rochberg in [89] is more general. Assume that μ is a positive Borel measure on Rn and define the maximal function  1 μ(Q) 1 dμ = sup M μ(x) = sup |Q| Qx Qx |Q| Q for every x ∈ Rn . If 0 < M μ(x) < ∞ for almost every x ∈ Rn and 0 < δ < 1, then w = (M μ)δ is an A1 weight and the A1 constant only depends on n and δ. The claim (a) in Theorem 8.33 follows from this result by considering the measure dμ = |f (x)| dx. In particular, for μ = δ, where δ is the Dirac measure at the origin, we have M δ(x) = C(n)|x|−n . Thus |x|−α ∈ A1 for 0 ≤ α < n. By Remark 8.30, we conclude that |x|−α ∈ Ap whenever 1 < p < ∞ and n(1 − p) < α < n. In Chapter 10, we examine more carefully the Ap properties of distance weights w(x) = d (x, E)−α , for nonempty sets E ⊂ Rn . In particular, we recover the above facts concerning the special case E = {0} ⊂ Rn from our more general results; see Example 10.27. 8.7. Self-improvement of reverse H¨ older inequalities A Muckenhoupt weight satisfies a reverse H¨ older inequality by Theorem 8.25. The main goal in this section is to show that reverse H¨older inequalities are selfimproving, compare to Theorem 8.27 for the corresponding result for Muckenhoupt weights. For a nonnegative function f on a set E ⊂ Rn and t ≥ 0, we write {f > t} = {x ∈ E : f (x) > t}.

(8.24)

Lemma 8.36. Let μ be a measure in Rn . Let E ⊂ Rn be a μ-measurable set with μ(E) < ∞ and let f be a nonnegative μ-measurable function on E. For 0 < q < ∞ and 0 ≤ t0 ≤ t1 < ∞, we have  f (x)q dμ(x) {t0 t}) dt + tq0 μ({f > t0 }) − tq1 μ({f > t1 }).

t0

Proof. Lemma 1.14 for the measure μ implies   ∞ f (x)q dμ(x) = q tq−1 μ({t0 < f ≤ t1 } ∩ {f > t}) dt, {t0 t}) dt   t0 tq−1 μ({t0 < f ≤ t1 }) dt + = 0

tq = 0 μ({t0 < f ≤ t1 }) + q



t1

tq−1 μ({t < f ≤ t1 }) dt

t0 t1

t0

tq−1 μ({t < f ≤ t1 }) dt.

184

8. MUCKENHOUPT WEIGHTS

Since {t < f ≤ t1 } = {f > t} \ {f > t1 } and the measures of the sets are finite by the assumption μ(E) < ∞, we obtain μ({t < f ≤ t1 }) = μ({f > t}) − μ({f > t1 }) for every t0 ≤ t ≤ t1 . Consequently  t1  t1  tq−1 μ({t < f ≤ t1 }) dt = tq−1 μ({f > t}) dt − μ({f > t1 }) t0



t0 t1

= t0

t1

tq−1 dt

t0

tq − tq0 μ({f > t1 }). tq−1 μ({f > t}) dt − 1 q 

The claim follows by combining the equations above.

The next lemma is a core of the self-improvement result for reverse H¨ older inequalities. Lemma 8.37. Let 1 < p < ∞ and let Q0 ⊂ Rn be a cube. Assume that f, g ∈ Lp (Q0 ) are nonnegative functions and that there exist t0 ≥ 0 and C1 > 1 such that      p p−1 p (8.26) f (x) dx ≤ C1 t f (x) dx + g(x) dx {f >t}

{f >t}

for every t0 ≤ t < ∞. Let q > p with C = C(p, q, C1 ) such that    q−p q f (x) dx ≤ C t0 Q0

{g>t}

< 1. Then there exists a constant

q

f (x) dx +

g(x) dx . Q0

 q

f (x)q dx

f (x) dx + {f ≤t0 }

Q0



 p

Q0

Proof. Clearly   q f (x) dx = (8.27)

C1 q−p q−1

{f >t0 }





≤ tq−p 0

f (x)p dx +

f (x)q dx.

{f ≤t0 }

{f >t0 }

It suffices to estimate the second integral on the right-hand side. Let t1 > t0 . Using equation (8.25) with the exponent q − p > 0 and the measure dμ = f p dx, we obtain   q f (x) dx = f (x)q−p dμ(x) {t0 t}  f (x)p dx − tq−p 1

f (x)p dx.

{f >t1 }

Assumption (8.26) implies   t1 tq−p−1 f (x)p dx dt t0

 ≤ C1

{f >t} t1 q−2



t

t0

{f >t}





t1

f (x) dx dt + C1

t t0

q−p−1

g(x)p dx dt. {g>t}

¨ 8.7. SELF-IMPROVEMENT OF REVERSE HOLDER INEQUALITIES

185

By (8.25), with the exponent q − 1 > 0 and the measure dμ = f dx, we obtain      t1 1 tq−2 f (x) dx dt ≤ f (x)q dx + tq−1 f (x) dx . 1 q − 1 {t0 t} {f >t1 } On the other hand, with dμ = g p dx, we have  ∞   t1 tq−p−1 g(x)p dx dt ≤ tq−p−1 μ({g > t}) dt t0 {g>t} 0   1 1 q−p = g(x) dμ(x) = g(x)q dx. q − p Q0 q − p Q0 Consequently  (8.28)

{t0 t1 }

Q0

where we also applied the estimate  p−1    f (x) f (x) dx ≤ f (x) dx = t1−p f (x)p dx. 1 t1 {f >t1 } {f >t1 } {f >t1 } 

Since

{t0 t1 } Q0 Here 0 < 1 − C1 q−p q−1 < 1, and thus  f (x)q dx {t0 t0 } {f >t1 } Q0   q−p ≤ Ct0 f (x)p dx + C g(x)q dx, {f >t0 }

Q0

with C = C(p, q, C1 ) ≥ 1. This upper bound does not depend on t1 , and by letting t1 → ∞ and using Fatou’s lemma, we obtain    q−p q p f (x) dx ≤ Ct0 f (x) dx + C g(x)q dx. {f >t0 }

{f >t0 }

Finally, by (8.27), we arrive at    f (x)q dx ≤ C(p, q, C1 ) tq−p 0 Q0

Q0

 g(x)q dx ,

 f (x)p dx +

Q0

which is the required estimate. We are ready for the main result of this section.

Q0



186

8. MUCKENHOUPT WEIGHTS

Theorem 8.38. Let 1 < p < ∞ and let Q0 ⊂ Rn be a cube. Assume that f ∈ Lp (Q0 ) is a nonnegative function and that there exists a constant C1 such that  p1  p ≤ C1  f (x) dx (8.29)  f (x) dx Q

Q

for every cube Q ⊂ Q0 . Then there exist an exponent q = q(n, p, C1 ) > p and a constant C = C(n, p, C1 ) such that   q1 q (8.30) ≤ C  f (x) dx  f (x) dx Q

Q

for every cube Q ⊂ Q0 . Proof. It suffices to prove that (8.30) holds for Q = Q0 . We may clearly d f be the dyadic maximal function as in (8.1) and assume that fQ0 > 0. Let MQ 0 define   p1 p (8.31) t0 =  f (x) dx . Q0

d Let t ≥ t0 ≥ fQ0 and Et = {x ∈ Q0 : MQ f (x) > t}. By Lemma 8.6, there exists a 0  collection Dt of pairwise disjoint dyadic cubes Q ∈ D(Q0 ) such that Et = Q∈Dt Q and t < fQ ≤ 2n t for every Q ∈ Dt . Let Ft = {x ∈ Q0 : f (x) > t}. Since d f (x) ≤ MQ f (x) for almost every x ∈ Q0 and f satisfies (8.29), we obtain 0  p  



f (x)p dx ≤ f (x)p dx = f (x)p dx ≤ C1p |Q|  f (x) dx Ft

Et

≤ C1p (2n t)p

Q

Q∈Dt

Q

Q∈Dt

|Q| = C(n, p, C1 )tp |Et |.

Q∈Dt



Let f(x) = Then f (x) ≤ f(x) + have

t 2

if f (x) > 2t , otherwise.

f (x), 0,

for every x ∈ Q0 and, as in the proof of Theorem 1.15, we d d  f (x) ≤ MQ f (x) + MQ 0 0

t 2

for every x ∈ Q0 . Lemma 8.32 implies d d  |Et | = {x ∈ Q0 : MQ f (x) > t} ≤ {x ∈ Q0 : MQ f (x) > 2t } 0 0   2 2  ≤ f (x) dx, f (x) dx = t Q0 t Ft 2

and thus



 f (x)p dx ≤ C(n, p, C1 )tp |Et | ≤ C(n, p, C1 )tp−1

f (x) dx.

Ft

Ft 2

On the other hand, we have   p f (x) dx = F t \Ft 2

F t \Ft 2

 f (x)

p−1

f (x) dx ≤ t

p−1

f (x) dx, Ft 2

¨ 8.7. SELF-IMPROVEMENT OF REVERSE HOLDER INEQUALITIES

187

and combination of the estimates above shows that   t p−1  f (x)p dx ≤ C(n, p, C1 ) f (x) dx 2 Ft Ft 2

2

for t ≥ t0 . We apply Lemma 8.37, with g = 0, and obtain q = q(n, p, C1 ) > p and a constant C = C(n, p, C1 ) such that   f (x)q dx ≤ Ctq−p f (x)p dx. 0 Q0

Q0

Finally (8.31) and (8.29) give   f (x)q dx ≤ C  Q0

Q0

 ≤C 

(8.32)

f (x)p dx

 =C 

f (x)p dx Q0

f (x)p dx

 pq −1   pq  pq

Q0

f (x)p dx Q0



−1 

|Q0 |

f (x)p dx Q0

f (x)p dx Q0



q

|Q0 | ≤ C|Q0 | 

f (x) dx

,

Q0



proving (8.30) for Q0 .

The next theorem gives an alternative way to show that every Ap weight satisfies a reverse H¨older inequality, compare to Theorem 8.25. Following the proof of Theorem 8.27, we obtain a third proof for the self-improvement of Muckenhoupt Ap weights. Theorem 8.39. Let 1 ≤ p < ∞ and assume that w ∈ Ap . There exist δ = δ(n, p, [w]Ap ) > 0 and C = C(n, p, [w]Ap ) such that 1  1+δ  1+δ w(x) dx ≤ C  w(x) dx  Q

Q

for every cube Q ⊂ Rn . Proof. By Theorem 8.13 (a) and (8.19) we have Ap ⊂ Aq if 1 ≤ p < q < ∞. older’s inequality, for Thus we may assume that p > 2. Fix a cube Q ⊂ Rn . By H¨ every measurable function f > 0, we have 1≤

Q

Inserting f = w

1 p−1

1 dx  f (x) dx. f (x) Q

gives 1

1

1 ≤  w(x) 1−p dx  w(x) p−1 dx. Q

Q

Since w ∈ Ap , we obtain  p−1 1 1−p dx ≤ [w]Ap = [w]Ap · 1p−1  w(x) dx  w(x) Q

Q

 ≤ [w]Ap

 w(x) Q

1 1−p

p−1  dx

 w(x) Q

1 p−1

p−1 dx

.

188

8. MUCKENHOUPT WEIGHTS

By dividing both sides by

 0
p − 1 and C = C(n, p, [w]Ap ) such that 1  q1   p−1  q 1 ≤ C  w(x) p−1 dx ≤ C  w(x) dx  w(x) p−1 dx n

Q

Q

Q

for every cube Q ⊂ R . The last inequality follows from H¨older’s inequality. Thus we arrive at   p−1 q q p−1 dx ≤ C  w(x) dx  w(x) n

Q

Q

for every cube Q ⊂ Rn . The claim follows by choosing δ > 0 such that 1 + δ = q  p−1 . 8.8. General self-improvement result for reverse H¨ older inequalities There exists a more general version of Theorem 8.38, which is relevant in the regularity theory for nonlinear partial differential equations in Chapter 12. First we introduce an appropriate maximal function on cubes. Definition 8.40. Let 1 ≤ p < ∞, let Q0 ⊂ Rn be a cube, and assume that p f ∈ Lp (Q0 ) is a nonnegative function. The noncentered maximal function MQ f 0 on Q0 is defined as  p1  p p MQ0 f (x) = sup  f (y) dy , Qx

Q

where the supremum is taken over all cubes Q ⊂ Q0 with x ∈ Q. We begin with a reverse weak type inequality for this maximal operator. Lemma 8.41. Let 1 ≤ p < ∞ and let Q0 ⊂ Rn be a cube. Assume that f ∈ Lp (Q0 ) is a nonnegative function. There exists a constant C = C(n) such that  p f (x)p dx ≤ Ctp {x ∈ Q0 : MQ f (x) > t} (8.33) 0 {x∈Q0 :f (x)>t}

whenever t ≥ (f p )Q0 . p

Proof. We apply Lemma 8.6 for f p at the level tp and obtain a collection Dtp of pairwise disjoint dyadic cubes Q ∈ D(Q0 ) such that tp < (f p )Q ≤ 2n tp for every Q ∈ Dtp and  d p p {x ∈ Q0 : MQ f (x) > t } = Q. 0 Q∈Dtp

8.8. GENERAL SELF-IMPROVEMENT RESULT

189

d Since f (x)p ≤ MQ f p (x) for almost every x ∈ Q0 , we have 0  

 f (x)p dx ≤ f (x)p dx = f (x)p dx {x∈Q0 :f (x)>t}

d f p (x)>tp } {x∈Q0 :MQ

=

Q∈Dtp

0

|Q|  f (x)p dx ≤ 2n tp Q

Q∈Dtp

Q

|Q|

Q∈Dtp

d p ≤ 2n tp {x ∈ Q0 : MQ f (x) > tp } .

d p p1 p Since MQ f (x) ≤ MQ f (x) for every x ∈ Q0 , inequality (8.33) follows. 0 0 The next result is a weak that M f is the usual centered



p . Recall type estimate for the maximal operator MQ 0 Hardy–Littlewood maximal function of f ∈ L1loc (Rn ).

Lemma 8.42. Let 1 ≤ p < ∞ and let Q0 ⊂ Rn be a cube. Assume that f ∈ Lp (Q0 ) is a nonnegative function. There exist constants C1 = C1 (n) and C2 = C2 (n) such that  {x ∈ Q0 : M p f (x) > t} ≤ C1 (8.34) f (x)p dx Q0 tp {x∈Q0 :f (x)> Ct } 2

for every t > 0. Proof. Let g = f p χQ0 ∈ L1 (Rn ). We have

1 p p MQ f (x) = MQ (f χQ0 )(x) ≤ C2 (n)M g(x) p for every x ∈ Q0 , where C2 (n) ≥ 1. The proof of Theorem 1.15 shows that   {x ∈ Q0 : M p f (x) > t}| ≤ x ∈ Rn : M g(x) > tp Q0 C2 (n)  C1 (n) ≤ g(x) dx p tp {x∈Rn :g(x)≥ 2Ct (n) } 2  C1 (n) ≤ f (x)p dx. t tp {x∈Q0 :f (x)≥ 2C (n) } 2

1 p

In the final step we used the inequality (2C2 (n)) ≤ 2C2 (n).



Theorem 8.43. Let 1 < p < ∞ and C1 > 0, and let Ω ⊂ Rn be an open set. There exist θ = θ(n, p) > 0, q = q(n, p, C1 ) > p and C = C(n, p, C1 ) ≥ 1 such that if f, g ∈ Lploc (Ω) are nonnegative functions satisfying !  p1   p1 "  p p f (x) dx ≤ C1  f (x) dx +  g(x) dx  (8.35)

Q(z,r)

Q(z,2r)

Q(z,2r)

 +θ 

f (x)p dx

 p1 ,

Q(z,2r)

for every cube Q(z, r) with Q(z, 2r) ⊂ Ω, then  q1   p1   q p f (x) dx ≤C  f (x) dx +C   Q(z,r)

Q(z,2r)

for every cube Q(z, r) with Q(z, 2r) ⊂ Ω.

Q(z,2r)

q

g(x) dx

 q1

190

8. MUCKENHOUPT WEIGHTS

Proof. Let Q0 = Q(x0 , r0 ) be a cube with Q0 ⊂ Ω. We begin by constructing a specific Whitney type decomposition W of Q0 . For i ∈ N let

Qi = Q x0 , (1 − 2−i )r0 . We divide each Qi into (2i+1 − 2)n pairwise disjoint Q0 -dyadic half-open cubes, with common side length 2−i r0 , which cover Qi . Denote this collection by Li . We define recursively a collection Wi of pairwise disjoint cubes by setting W1 = L1 and    = ∅ for every Q  ∈ Wi Wi+1 = Q ∈ Li+1 : Q ∩ Q  for every i ∈ N. Let W = ∞ i=1 Wi . The cubes in W are pairwise disjoint and they cover the interior of Q0 , and hence they cover Q0 up to measure zero. Moreover, if Q = Q(z, r) ∈ W, then the doubled cube Q(z, 2r) is a subset of Q0 . Let f, g ∈ Lploc (Ω) be nonnegative functions such that (8.35) holds for some 0 < θ < 1, to be specified later, and define   p1 p < ∞. (8.36) t0 =  f (x) dx Q0

Without loss of generality, we may assume that t0 > 0. Let t ≥ t0 . For Q ∈ W, we have  1 |Q0 | p f (x) dx ≤ f (x)p dx = f (x)p dx ≤ aQ tp , (8.37)  |Q| Q0 |Q| Q0 Q where aQ =

|Q0 | |Q|

> 1. Define functions f and g in the interior of Q0 by setting − f(x) = aQ p f (x) 1

and

−1

g(x) = aQ p g(x),

for every x ∈ Q ∈ W. Clearly 0 ≤ f ≤ f and 0 ≤ g ≤ g almost everywhere in Q0 , and thus f, g ∈ Lp (Q0 ). Let Q ∈ W. By (8.37), we have p  p  f (x) dx ≤ t , Q

and Lemma 8.41 gives  (8.38)

{x∈Q:f(x)>t}

p  f(x)p dx ≤ C(n)tp {x ∈ Q : MQ f (x) > t} .

To estimate the right-hand side of (8.38), let x ∈ Q and let Qx = Q(zx , rx ) ⊂ Q be a subcube of Q containing x. Then the construction above guarantees that the doubled cube Q(zx , 2rx ) is contained in Q0 ⊂ Ω, and (8.35) implies  p1  p1   1 −p p p  = aQ  f (y) dy  f (y) dy Qx

(8.39)

Qx

!

1 −p

≤ C 1 aQ



 Q(zx ,2rx )

 −1 + θaQ p 

g(y)p dy

f (y) dy + 

 p1 "

Q(zx ,2rx )

f (y)p dy

 p1

.

Q(zx ,2rx )

It is easy to see that the cube Q(zx , 2rx ) intersects at most those cubes in W which touch Q, that is, those Whitney type cubes whose closures intersect the closure

8.8. GENERAL SELF-IMPROVEMENT RESULT

191

of Q. In particular, there exists a cube Q = Q(x , r  ) ∈ W which touches Q and satisfies 1 1 f(y) ≥ (aQ )− p f (y) and g(y) ≥ (aQ )− p g(y) for almost every y ∈ Q(zx , 2rx ). Moreover, by the construction of the cubes it holds that Q ⊂ Q(x , 5r  ). This implies |Q| ≤ |Q(x , 5r  )| = 5n |Q | and aQ ≤ 5n aQ . Concluding from above, we obtain f(y) dy.

1



Q(zx ,2rx )

f (y) dy ≤ (5n aQ ) p 

Q(zx ,2rx )

A similar reasoning shows that   p1  1 p n p g(y) dy ≤ (5 aQ )   Q(zx ,2rx )

and

 p1

Q(zx ,2rx )

 f (y)p dy



g(y) dy p

Q(zx ,2rx )

 p1

 1 ≤ (5n aQ ) p 

f(y)p dy

 p1 .

Q(zx ,2rx )

By substituting the estimates above to (8.39) and taking supremum over all cubes Qx as above, we have

M p f(x) ≤ C(n, C1 ) MQ f(x) + M p g(x) + C(n)θM p f(x) Q

0

Q0

Q0

for every x ∈ Q. This implies p  f (x) > t} ⊂ Q ∩ (E ∪ F ∪ G), {x ∈ Q : MQ

where

 t , 3C(n, C1 )   t p F = x ∈ Q0 : MQ , g(x) > 0 3C(n, C1 )   t p  G = x ∈ Q0 : MQ . f (x) > 0 3C(n)θ

 E = x ∈ Q0 : MQ0 f(x) >

By (8.38) we have  (8.40)

{x∈Q:f(x)>t}

f(x)p dx ≤ C(n)tp |Q ∩ (E ∪ F ∪ G)|.

By summing (8.40) over all Whitney cubes Q ∈ W and applying Lemma 8.42, we obtain  f(x)p dx ≤ C(n)tp |E ∪ F ∪ G| ≤ C(n)tp (|E| + |F | + |G|) {f>t}   ≤ C(n, C1 )tp−1 g(x)p dx f(x) dx + C(n, p, C1 ) (8.41) {f>τ t} { g >τ t}  + C(n, p)θ p f(x)p dx, {f>τ t}

where 0 < τ = τ (n, C1 ) < 1, and notation as in (8.24) is used for the set Q0 . On the other hand, we have   p p−1  f (x) dx ≤ t f(x) dx. (8.42) {τ tτ t}

192

8. MUCKENHOUPT WEIGHTS

By adding (8.41) and (8.42) and reorganizing terms, we arrive at 

1 − C(n, p)θ p f(x)p dx {f>τ t}

  p−1 ≤ C(n, p, C1 ) (τ t)

{f>τ t}

f(x) dx +





g(x) dx . p

{ g >τ t}

Recall that here t ≥ t0 was arbitrary. Also note that the term that is absorbed into the left-hand side is finite since f ∈ Lp (Q0 ). Let 0 < θ = θ(n, p) < 1 be so small that 1 − C(n, p)θ p ≥ 12 . Lemma 8.37, applied for the functions f, g ∈ Lp (Q0 ), and the trivial estimates f ≤ f , g ≤ g imply the existence of q = q(n, p, C1 ) > p such that      q q−p p q  f (x) dx + g(x) dx . f (x) dx ≤ C(n, p, C1 ) (τ t0 ) Q0

Q0

Q0

By applying (8.36) and estimating as in (8.32), we obtain   q1   p1  q p  dx ≤ C f (x) dx + C f (x)    Q0

Q0

q

 q1

g(x) dx

Q0

with C = C(n, p, C1 ). Finally, it follows from the construction of W that Q1 = Q(x0 , r20 ) is divided 0| n to 2n cubes Q ∈ W, each having side length r20 . Since aQ = |Q |Q| = 4 for these n cubes, it holds by the definition of f that f(x) = 4− p f (x) for every x ∈ Q1 . Hence we conclude that  q1   p1   q1  q p q f (x) dx ≤ C f (x) dx + C g(x) dx    Q1

Q0

Q0

with C = C(n, p, C1 ). This is the desired inequality for the cube Q(z, r) = Q1 , and the proof is complete.  Remark 8.44. By covering cubes in (8.35) with balls by Lemma 1.13, and covering balls in the final inequality by dyadic cubes, we obtain the following variant of Theorem 8.43. Let 1 < p < ∞, C1 > 0, 1 < τ1 , τ2 < ∞, and let Ω ⊂ Rn be an open set. There exist θ = θ(n, p, τ1 ) > 0, q = q(n, p, C1 , τ1 ) > p and C = C(n, p, C1 , τ1 , τ2 ) ≥ 1 such that if f, g ∈ Lploc (Ω) are nonnegative functions satisfying ! 1  1 "  

f (x)p dx B(z,r)

p

≤ C1 

B(z,τ1 r)

p

B(z,τ1 r)



+θ 

g(x)p dx

f (x) dx +  f (x)p dx

 p1

,

B(z,τ1 r)

for every ball B(z, r), with B(z, τ1 r) ⊂ Ω, then  1q   p1   q p f (x) dx ≤C  f (x) dx +C   B(z,r)

B(z,τ2 r)

for every ball B(z, r) with B(z, τ2 r) ⊂ Ω.

q

g(x) dx B(z,τ2 r)

 1q

8.9. NOTES

193

Remark 8.45. By tracking the constants in the proof of Theorem 8.43 and using covering arguments, we also obtain the following variant of Theorem 8.43 for balls. Let 1 < p < ∞, C1 > 0, 1 < τ1 , τ2 < ∞, and let Ω ⊂ Rn be an open set. There exist θ = θ(n, p, τ1 ) > 0, ε0 = ε0 (n, p, C1 , τ1 ) > 0 and C = C(n, p, C1 , τ1 , τ2 ) ≥ 1 such that the following claim holds for every ε with 0 < ε < ε0 : For nonnegative functions f, g ∈ Lp−ε loc (Ω) satisfying ! 1 1 "  p−ε   p−ε  p−ε p−ε f (x) dx ≤ C1  f (x) dx +  g(x) dx  B(z,r)

B(z,τ1 r)

B(z,τ1 r)



+θ 

f (x)p−ε dx

1  p−ε

B(z,τ1 r)

for every ball B(z, r) with B(z, τ1 r) ⊂ Ω, we have 1  p1   p−ε   p p−ε f (x) dx ≤C  f (x) dx +C   B(z,r)

B(z,τ2 r)

p

 p1

g(x) dx B(z,τ2 r)

for every ball B(z, r) with B(z, τ2 r) ⊂ Ω. 8.9. Notes For the dyadic maximal function in Section 8.2 we refer to Journ´e [216, Chapter 1]. The proof of Theorem 8.11 is based on an unpublished note on Muckenhoupt weights by Stephen Keith and Xiao Zhong. Weighted norm inequalities have been studied by Muckenhoupt [333, 334] and by Coifman and Fefferman [87]. Other references for Muckenhoupt weights include Cruz-Uribe, Martell and P´erez [92], Duoandikoetxea [107, Chapter 7], [108], Garc´ıa-Cuerva and Rubio de Francia [139, Chapter 4], Garnett [142, Chapter 6], Genebashvili, Gogatishvili, Kokilashvili and Krbec [145], Grafakos [157, Chapter 9], Heinonen, Kilpel¨ainen and Martio [187, Chapter 15], Journ´e [216, Chapter 2], Lu, Ding and Yan [286, Section 1.4], Stein [371, Chapter 5], Str¨ omberg and Torchinsky [376, Chapter 1], Torchinsky [383, Chapter 9] and Turesson [388, Section 1.2]. Theorem 8.16 can be found in Kilpel¨ainen, Koskela and Masaoka [221, Proposition 4.3]. The elementary proof of Theorem 8.19 is from Lerner [267]. Theorem 8.25 is in Coifman and Fefferman [87], see also Turesson [388, Section 1.2.4]. Theorem 8.33 is from Coifman and Rochberg [89]. Theorem 8.38 has been studied by Gehring [144] and Theorem 8.43 by Meyers and Elcrat [325]. The proof of Theorem 8.38 is based on Bojarski and Iwaniec [52], Garnett [142, Chapter 6] and Kinnunen [223]. See also Duoandikoetxea [108, Section 4.2]. For the proof of Theorem 8.43 we refer to Bj¨orn and Bj¨orn [31, Section 3.5], Giaquinta [147, Chapter 5], [148, Section 6.2], Giaquinta and Martinazzi [149], Giaquinta and Modica [150], Giusti [153, Section 6.4], Iwaniec and Martin [207, Chapter 14], Mart´ın and Milman [309], Stredulinsky [373, 374], Str¨omberg and Torchinsky [376, Chapter 1] and Zatorska-Goldstein [402].

10.1090/surv/257/09

CHAPTER 9

Weighted Maximal and Poincar´ e Inequalities This chapter begins with Poincar´e inequalities on cubes and, more generally, on open sets satisfying a chaining condition. In Section 9.2, we study weighted norm inequalities for dyadic maximal operators, and as a consequence obtain a weighted Poincar´e inequality on cubes. Two weight Poincar´e inequalities are discussed in Section 9.3, Section 9.4 and Section 9.5. Weighted Poincar´e inequalities are proved using the sparse domination method in Lemma 9.13 and Lemma 9.19. In Section 9.6 we investigate two weight Poincar´e inequalities on open sets satisfying a chaining condition. The sparse domination method is applied in Section 9.7 to prove the John–Nirenberg inequality for functions of bounded mean oscillation (BMO), and at the end of the section we obtain weighted Sobolev inequalities for p = n as a consequence of the John–Nirenberg inequality. Finally, in Section 9.8 we examine connections between Muckenhoupt weights and BMO functions and prove a boundedness result for a maximal operator on BMO. 9.1. Poincar´ e inequalities on cubes We consider Poincar´e inequalities for more general open sets than balls, compare to Section 3.3. The proof is based on a chaining argument and a local-to-global inequality, which allows us to reduce the problem to Poincar´e inequalities on balls. As in Lemma 8.9, we consider a Whitney decomposition W(Ω) of an open set Ω and write Q∗ = Q(x, 98 r) for Q = Q(x, r) ∈ W(Ω). Definition 9.1. Let Ω ⊂ Rn be a connected open set and let W(Ω) be a Whitney decomposition of Ω. We say that C(Q) = (Q0 , . . . , Qk ) ⊂ W(Ω) is chain in Ω joining Q0 to Q = Qk , if Qi = Qj whenever i = j, and for each j ∈ {1, . . . , k} there exists a cube Q(z, r) ⊂ Q∗j ∩ Q∗j−1 satisfying (9.1)

r ≥ C(n) max{l(Q∗j ), l(Q∗j−1 )}.

For a fixed cube Q0 ∈ W(Ω), the collection {C(Q) : Q ∈ W(Ω)} is called a chain decomposition of Ω. The shadow of a Whitney cube Q ∈ W(Ω) is the set S(Q) = {Q ∈ W(Ω) : Q ∈ C(Q )}. Notice the duality of the concepts of chain and shadow: Q ∈ C(Q ) if and only if Q ∈ S(Q). 

Definition 9.2. An open set Ω ⊂ Rn satisfies the Boman chain condition if Ω is connected and there exist τ > 1 and a chain decomposition {C(Q) : Q ∈ W(Ω)} of Ω, with a fixed cube Q0 ∈ W(Ω), such that if Q = Q(x, r) ∈ W(Ω) and Q ∈ S(Q), then (9.2)

Q ⊂ Q(x, τ r). 195

196

´ INEQUALITIES 9. WEIGHTED MAXIMAL AND POINCARE

Note that if an open set Ω ⊂ Rn satisfies the Boman chain condition, then Ω is bounded and connected. Remark 9.3. All open cubes and balls satisfy the Boman chain condition, with τ = τ (n) independent of the cube. More generally, if Ω ⊂ Rn is a so-called John domain, then Ω satisfies the Boman chain condition, see Buckley, Koskela and Lu [65]. The class of John domains includes bounded and connected open sets with a smooth or a Lipschitz boundary. The following lemma provides a special case of a local-to-global inequality for open sets satisfying the Boman chain condition, including all cubes. See also Theorem 9.28 for a more general local-to-global inequality. Lemma 9.4. Assume that Ω ⊂ Rn is an open set satisfying the Boman chain condition, with τ > 1. There exists a constant C = C(n, τ ) such that  |f (x) − fΩ | dx ≤ C Ω

 Q∗

Q∈W(Ω)

|f (x) − fQ∗ | dx

for every f ∈ L1 (Ω). Proof. Observe that    |f (x) − fΩ | dx ≤ |f (x) − fQ∗0 | dx + |fΩ − fQ∗0 | dx Ω Ω Ω  ≤ 2 |f (x) − fQ∗0 | dx (9.3) Ω



≤2 |f (x) − fQ∗ | dx + 2 |Q||fQ∗ − fQ∗0 |. Q∗

Q∈W(Ω)

Q∈W(Ω)

The first term on the right-hand side of (9.3) is already of the desired form. To estimate the second term, we use the chain C(Q) = (Q0 , . . . , Qk ) joining the cube Q0 to Qk = Q ∈ W(Ω), and obtain

|Q||fQ∗ − fQ∗0 | ≤

Q∈W(Ω)

|Q|

Q∈W(Ω)

k

|fQ∗i − fQ∗i−1 |.

i=1

By (9.1), we have fQ∗ − fQ∗ ≤ fQ∗ − fQ∗ ∩Q∗ + fQ∗ ∩Q∗ − fQ∗ i i−1 i i i−1 i i−1 i−1 ≤

∗ Q∗ i ∩Qi−1

≤ C(n)

|f (x) − fQ∗i | dx + 

i

j=i−1

∗ Q∗ i ∩Qi−1



Q∗ j

|f (x) − fQ∗j | dx

|f (x) − fQ∗i−1 | dx

´ INEQUALITIES ON CUBES 9.1. POINCARE

197

for every i ∈ {1, . . . , k}. Using the estimates above, the assumption Qi1 = Qi2 for i1 = i2 , and the duality between chains and shadows, we arrive at



|Q||fQ∗ − fQ∗0 | ≤ C(n)

Q∈W(Ω)

|Q|

≤ C(n)

≤ C(n)

|Q|

Q∈W(Ω)



R∈C(Q)



R∗



R∈W(Ω)

R∗

R∗

|Q| 

R∈W(Ω) Q∈S(R)

≤ C(n, τ )



i=1 j=i−1

Q∈W(Ω)

k i

Q∗ j

|f (x) − fQ∗j | dx

|f (x) − fR∗ | dx |f (x) − fR∗ | dx

|f (x) − fR∗ | dx, 

where the final inequality follows from (9.2).

Theorem 9.5. Assume that Ω ⊂ Rn is an open set satisfying the Boman chain condition, with τ > 1. There exists a constant C = C(n, τ ) such that   |u(x) − uΩ | dx ≤ C diam(Ω) |∇u(x)| dx Ω

for every u ∈ W

1,1

Ω

(Ω).

Proof. Let W(Ω) = {Qi : i√∈ N} be a Whitney decomposition of Ω, where

Qi = Q(xi , ri ). Let Bi = B xi , 9 8 n ri for every i ∈ N. Then Bi  Ω, the interior of Q∗i is contained in Bi , and ∞

χBi ≤ C(n)χΩ . (9.4) i=1

Lemma 9.4 and the 1-Poincar´e inequality for balls, see Theorem 3.14, imply  ∞ 

|u(x) − uΩ | dx ≤ C(n, τ ) |u(x) − uQ∗i | dx Ω

≤ C(n, τ ) ≤ C(n, τ ) ≤ C(n, τ )

∗ i=1 Qi ∞ 

∗ i=1 Qi  ∞

|u(x) − uBi | dx

|u(x) − uBi | dx

i=1 ∞

Bi

 |∇u(x)| dx

ri Bi

i=1

 |∇u(x)| dx,

≤ C(n, τ ) diam(Ω) Ω



where the final inequality follows from (9.4).

Corollary 9.6. Let Q ⊂ R be a cube. There exists a constant C = C(n) such that   |u(x) − uQ | dx ≤ Cl(Q) |∇u(x)| dx n

Q

for every u ∈ W 1,1 (Q).

Q

198

´ INEQUALITIES 9. WEIGHTED MAXIMAL AND POINCARE

Proof. The interior of the cube Q satisfies the Boman chain condition with τ = τ (n) > 1, and hence the claimed inequality, with a constant C = C(n), follows from Theorem 9.5.  9.2. Single weight maximal and Poincar´ e inequalities We prove weighted Poincar´e inequalities by applying weighted norm inequalities for dyadic maximal functions. We begin by defining a weighted dyadic maximal d function. The corresponding unweighted dyadic maximal function MQ f was de0 ∞ fined in (8.1). For simplicity, we only consider functions f ∈ L (Q0 ). Definition 9.7. Let w be a weight in Rn and let Q0 ⊂ Rn be a cube. For d,w f is defined at x ∈ Q0 f ∈ L∞ (Q0 ), the weighted dyadic maximal function MQ 0 by  1 d,w f (x) = sup |f (y)|w(y) dy, MQ 0 Qx w(Q) Q where the supremum is taken over all dyadic cubes Q ∈ D(Q0 ) with x ∈ Q. The proof of the following weighted norm inequality is a variant of the proof of Theorem 1.15; compare also to the proof of Lemma 8.32. Observe that the result holds for an arbitrary weight. In particular, the weight is not assumed to be doubling nor to belong to a Muckenhoupt class. Theorem 9.8. Let 1 < p < ∞ and let Q0 ⊂ Rn be a cube. Assume that w is a weight in Rn . Then   d,w

p p2p MQ0 f (x) w(x) dx ≤ |f (x)|p w(x) dx p − 1 Q0 Q0 for every f ∈ L∞ (Q0 ). Proof. Without loss of generality, we may assume that f ∈ L∞ (Q0 ) is nond,w negative. For t > 0, let Et = {x ∈ Q0 : MQ f (x) > t} and let Dt be the collection 0 of maximal dyadic cubes Q ∈ D(Q0 ) satisfying  1 f (x)w(x) dx > t, w(Q) Q  as in the proof of Lemma 8.6. Then Et = Q∈Dt Q and the cubes Q ∈ Dt are pairwise disjoint. Thus we obtain the weak type inequality  

1 1 (9.5) w(Et ) = w(Q) ≤ f (x)w(x) dx ≤ f (x)w(x) dx, t t Q0 Q Q∈Dt

Q∈Dt

for every t > 0. Let

 f(x) =

if f (x) > 2t , otherwise,

f (x), 0,

for every x ∈ Q0 . Then f (x) ≤ f(x) +

t 2

for every x ∈ Q0 and we obtain

d,w d,w  MQ f (x) ≤ MQ f (x) + 0 0

t 2

´ INEQUALITIES 9.2. SINGLE WEIGHT MAXIMAL AND POINCARE

199

for every x ∈ Q0 . The weak type estimate in (9.5) implies d,w d,w  w({x ∈ Q0 : MQ f (x) > t}) ≤ w({x ∈ Q0 : MQ f (x) > 2t }) 0 0  2 ≤ f(x)w(x) dx t Q0  2 f (x)w(x) dx. = t {x∈Q0 :f (x)> 2t }

As in the proof of Theorem 1.15, we obtain   ∞ d,w

p d,w MQ0 f (x) w(x) dx = p tp−1 w({x ∈ Q0 : MQ f (x) > t}) dt 0 Q0 0    ∞ ≤ 2p tp−2 f (x)w(x) dx dt 0



 = 2p

f (x) Q0

{x∈Q0 :f (x)> 2t }  ∞ tp−2 {x∈Q0 :f (x)> t } (x) dt w(x) dx 2 0

χ

 2f (x)  f (x) tp−2 dt w(x) dx Q0 0  p f (x)p w(x) dx. = 2p p − 1 Q0 

= 2p



The following result is a strong type version of Theorem 8.18 for the dyadic maximal function. The proof is based on the self-improvement of Muckenhoupt weights. Theorem 9.9. Let 1 < p < ∞, let Q0 ⊂ Rn be a cube, and assume that w ∈ Ap . There exists a constant C = C(n, p, w) such that   d

p MQ0 f (x) w(x) dx ≤ C |f (x)|p w(x) dx Q0

Q0

for every f ∈ L∞ (Q0 ). Proof. Without loss of generality, we may assume that f ∈ L∞ (Q0 ) is nonnegative. By Theorem 8.27, there exists 0 < ε = ε(n, p, w) < p − 1 such that w ∈ Ap−ε . Let x ∈ Q0 and let Q ∈ D(Q0 ) with x ∈ Q. Theorem 8.14 implies 1   p−ε  1 p−ε f (y) w(y) dy .  f (y) dy ≤ C(n, p, w, ε) w(Q) Q Q By taking supremums, we obtain

1 d,w p−ε p−ε d f (x) ≤ C(n, p, w, ε) MQ f (x) . MQ 0 0

p Theorem 9.8 with the exponent p−ε > 1 implies   p d d,w p−ε p−ε

p MQ0 f (x) w(x) dx ≤ C(n, p, w, ε) MQ0 f (x) w(x) dx Q0 Q  0 p (f (x)p−ε ) p−ε w(x) dx ≤ C(n, p, w, ε) Q  0 = C(n, p, w, ε) f (x)p w(x) dx. Q0



200

´ INEQUALITIES 9. WEIGHTED MAXIMAL AND POINCARE

We continue with a pointwise estimate, which is a counterpart of Theorem 3.4 for cubes. Lemma 9.10. Let Q ⊂ Rn be a cube and assume that u ∈ Lip(Q). There exists a constant C = C(n) such that d |∇u|(x) |u(x) − uQ | ≤ C l(Q)MQ

for every x ∈ Q. Proof. Let Q0 = Q and x ∈ Q0 . For each j ∈ N, we choose recursively the dyadic child Qj ⊂ Qj−1 with x ∈ Qj and 2n |Qj | = |Qj−1 |. Since u is continuous at x, we have ∞ ∞ |u(x) − uQ | = uQj − uQj−1 ≤ |uQj − uQj−1 | j=1

≤

∞

j=1

j=1



|u(y) − uQj−1 | dy ≤

Qj

∞

2n 

|u(y) − uQj−1 | dy.

Qj−1

j=1

By Corollary 9.6, we obtain |u(x) − uQ | ≤

∞



C(n) 2−j+1 l(Q) 

j=1 d |∇u|(x) ≤ C(n)l(Q)MQ 0

|∇u(y)| dy

Qj−1

∞

d 2−j+1 = C(n)l(Q)MQ |∇u|(x). 0



j=1

Weighted Poincar´e inequalities for cubes follow as in Lemma 3.9. Theorem 9.11. Let 1 < p < ∞ and assume that w ∈ Ap . Let Q ⊂ Rn be a cube and let u ∈ Lip(Q). There exists a constant C = C(n, p, w) such that   p p (9.6) |u(x) − uQ | w(x) dx ≤ Cl(Q) |∇u(x)|p w(x) dx. Q

Q

Proof. By Lemma 9.10 and Theorem 9.9, we have   d

p p p MQ |∇u|(x) w(x) dx |u(x) − uQ | w(x) dx ≤ C(n, p)l(Q) Q Q  p |∇u(x)|p w(x) dx. ≤ C(n, p, w)l(Q)



Q

Remark 9.12. For n = 1, the validity of the weighted Poincar´e inequality (9.6) for all cubes in R implies that w ∈ Ap (R), see Bj¨ orn, Buckley and Keith [35]. For n ≥ 2, this implication is not true. For instance, let α > −n and let w(x) = |x|α , for every x ∈ Rn \ {0}. Then the Poincar´e inequality in (9.6) with respect to w holds for any 1 < p < ∞, see Heinonen, Kilpel¨ ainen and Martio [187, p. 10]. However, w is an Ap weight if and only if p > 1 + α n or α = 0, see Remark 8.35 and Theorem 10.26. See also Gong [155] for absolute continuity of doubling measures supporting a Poincar´e inequality.

9.3. WEIGHTED LOCAL FEFFERMAN–STEIN INEQUALITIES

201

9.3. Weighted local Fefferman–Stein inequalities The following sparse domination lemma is a fundamental tool here and also in Section 9.7. It can be regarded as a refined version of Lemma 9.10 and Lemma 4.13. This technique allows to change simple nested sequences of cubes, as used for instance in the proof of Lemma 9.10, to an appropriate sparse collection of dyadic subcubes, with a controlled overlap property. Notice that the sparse collection of cubes depends on the function in question. Lemma 9.13. Assume that w ∈ A∞ . Let Q0 ⊂ Rn be a cube and let f ∈ L1 (Q0 ). There exists a collection S ⊂ D(Q0 ) of dyadic subcubes of Q0 satisfying the following conditions. (a) There exists a constant C = C(n, w) such that

χQ (x)  |f (y) − fQ | dy, |f (x) − fQ0 | ≤ C Q

Q∈S

for every Lebesgue point x ∈ Q0 of f . (b) There exist a constant 0 < κ = κ(w) < 1 and a collection {EQ : Q ∈ S} of pairwise disjoint measurable sets EQ ⊂ Q with w(EQ ) ≥ κw(Q) for every Q ∈ S. Moreover, if R, S ∈ S and R  S, then R ⊂ S \ ES . Proof. We may assume that 

|f (y) − fQ0 | dy > 0, Q0

since otherwise there is nothing to prove. Let C(w) > 0 and δ(w) > 0 be the constants in the A∞ condition (8.12) for w, and choose ρ > 1 such that C(w)ρ−δ(w) < 1. The collection S is constructed recursively. Let Q0 ∈ S. For S ∈ S, we add to S the maximal dyadic cubes S   S that satisfy the stopping condition   |f (y) − fS | dy > ρ  |f (y) − fS | dy. S

S

By iterating this process, we obtain a collection S ⊂ D(Q0 ) satisfying the following properties. (S1) For Q ∈ D(Q0 ), let πS Q be the minimal cube in S with Q ⊂ πS Q. If πS (Q) = S ∈ S, then  |f (y) − fS | dy ≤ ρ  |f (y) − fS | dy. Q

S

(S2) For S ∈ S, let chS (S) denote the collection of maximal cubes in S that are strictly contained in S; these are the children of S with respect to S. For every S  ∈ chS (S) we have   1 ρ |f (y) − fS | dy > |f (y) − fS | dy. |S  | S  |S| S Moreover, the sets S  ∈ chS (S) are pairwise disjoint, and thus  

|S| |S| S  ∈chS (S) S  |f (y) − fS | dy   . ≤ |S | ≤ ρ ρ |f (y) − f | dy S  S S ∈chS (S)

´ INEQUALITIES 9. WEIGHTED MAXIMAL AND POINCARE

202

(S3) Let S ∈ S. From the pairwise disjointness of the sets S  ∈ chS (S), condition (S2), the A∞ property of w, and the choice of ρ, we obtain 1 w(S)

S  ∈ch



1 w(S ) ≤ C(w) |S| 

S (S)

S  ∈ch

For every S ∈ S, let



|S |

≤ C(w)ρ−δ(w) < 1.

S (S)



ES = S \

δ(w)

S  ∈ch

S .

S (S)

If S, R ∈ S and S = R, then ES ∩ ER = ∅. This is clear if S ∩ R = ∅. On the other hand, if R  S, there exists S  ∈ chS (S) with R ⊂ S  , and thus ER ∩ ES ⊂ R ∩ ES ⊂ S  ∩ ES = ∅. If S  R, then ER ∩ ES = ∅, and we conclude that the sets ES , for S ∈ S, are pairwise disjoint. Moreover, we see that R ⊂ S \ ES if R, S ∈ S and R  S. Let S ∈ S. To complete the proof of assertion (b), it suffices to show that w(ES ) ≥ κw(S). To this end, condition (S3) implies

w(S  ) ≤ C(w)ρ−δ(w) w(S), S  ∈chS (S)

where C(w)ρ−δ(w) < 1. Hence w(ES ) = w(S) −



w(S  ) ≥ 1 − C(w)ρ−δ(w) w(S),

S  ∈chS (S)

and we conclude that (b) holds with 0 < κ = 1 − C(w)ρ−δ(w) < 1. In the proof of assertion (a) we apply the operators

χQ (x)(fQ − fQ ), ΔQ f (x) = Q ∈ch(Q)

where x ∈ Q0 and Q ∈ D(Q0 ) is a dyadic cube and ch(Q) is the collection of its 2n dyadic children. For S ∈ S and Q ∈ D(Q0 ), we write Q ∈ C(S) if πS Q = S. Let x ∈ Q0 be a Lebesgue point of f . Then |f (x) − fQ0 | = ΔQ f (x) = ΔQ f (x) S∈S Q∈C(S) Q∈D(Q0 ) (9.7)

ΔQ f (x) . ≤ S∈S Q∈C(S)

Let S ∈ S and assume that x ∈ S. Then either x ∈ S \ ES or x ∈ ES . First we consider the case x ∈ S \ ES . By the definition of ES there exists S  ∈ chS (S) with x ∈ S  , and by telescoping we obtain

Q∈C(S)

ΔQ f (x) = fS  − fS =  (f (y) − fS ) dy.  S

9.3. WEIGHTED LOCAL FEFFERMAN–STEIN INEQUALITIES

203

Let πS  ∈ D(Q0 ) be the dyadic parent of S  . Then πS (πS  ) = S ∈ S, and condition (S1) implies ≤ = Δ f (x) (f (y) − f ) dy  Q S   |f (y) − fS | dy S S Q∈C(S)

≤ 2n 

πS 

|f (y) − fS | dy ≤ ρ2n  |f (y) − fS | dy. S

Assume then that x ∈ ES . For k ∈ N0 , let Qk ⊂ S be the unique dyadic cube Qk ∈ D(Q0 ) with x ∈ Qk and l(S) = 2k l(Qk ). Then Qk ∈ C(S) for every k ∈ N0 since otherwise x ∈ S  for some S  ∈ chS (S), but this is not possible since x ∈ ES . By telescoping, we obtain

ΔQ f (x) = lim (fQk − fS ). k→∞

Q∈C(S)

We remark that the limit exists, and is equal to f (x) − fS , since x is assumed to be a Lebesgue point of f . Since x ∈ ES , we have πS (Qk ) = S for every k ∈ N0 . By property (S1), |fQk − fS | =  (f (y) − fS ) dy ≤  |f (y) − fS | dy ≤ ρ  |f (y) − fS | dy, Qk

and thus

Qk

S

ΔQ f (x) ≤ ρ  |f (y) − fS | dy ≤ ρ2n  |f (y) − fS | dy S S Q∈C(S)

for every x ∈ ES .  By combining the estimates above, and noticing that Q∈C(S) ΔQ f (x) = 0 if x ∈ S, we obtain ΔQ f (x) ≤ ρ2n χS (x)  |f (y) − fS | dy S Q∈C(S)

for every Lebesgue point x ∈ Q0 of f . This, together with (9.7), concludes the proof of assertion (a).  The next result can be viewed as a weighted local Fefferman–Stein inequality. It is not clear how to modify the proofs of Lemma 9.10 or Lemma 4.13 to obtain a more direct proof of this theorem. Instead, we make use of the sparse domination lemma, Lp -duality, and boundedness results for weighted maximal operators. We need the following dyadic sharp maximal function, compare to Definition 4.12. Definition 9.14. Let Q0 ⊂ Rn be a cube. For f ∈ L1 (Q0 ), the dyadic sharp d, maximal function MQ f is defined at x ∈ Q0 by 0 d, MQ f (x) = sup  |f (y) − fQ | dy, 0 Qx

Q

where the supremum is taken over all dyadic cubes Q ∈ D(Q0 ) with x ∈ Q.

´ INEQUALITIES 9. WEIGHTED MAXIMAL AND POINCARE

204

Theorem 9.15. Let 1 < p < ∞ and assume that w ∈ A∞ . Let Q0 ⊂ Rn be a cube and let f ∈ L1 (Q0 ). There exists a constant C = C(n, p, w) such that   d,

p p MQ0 f (x) w(x) dx. |f (x) − fQ0 | w(x) dx ≤ C (9.8) Q0

Q0

Proof. Without loss of generality, we may assume that the right-hand side of (9.8) is finite. Let S ⊂ D(Q0 ) be the sparse collection of cubes associated to function f given by Lemma 9.13, and define

χQ (x)  |f (y) − fQ | dy, F (x) = Q

Q∈S

for every x ∈ Q0 . Lemma 9.13 implies   p |f (x) − fQ0 | w(x) dx ≤ C(n, p, w) Q0

F (x)p w(x) dx.

Q0

We estimate the integral on the right-hand side by the Lp -duality  p1   p F (x) w(x) dx = sup F (x)g(x)w(x) dx, g

Q0

Q0

where the supremum is taken over all nonnegative g ∈ L∞ (Q0 ) ∩ Lq (Q0 ; w dx) with p and gLq (Q0 ;w dx) ≤ 1; this fact follows from density of simple functions q = p−1 q (Q ; in L 0 w dx). Let g be a function as above. By the definition of F , we have  

F (x)g(x)w(x) dx = |f (y) − f | dy g(x)w(x) dx. Q  (9.9) Q0

Q∈S

Q

Q

From Lemma 9.13 we obtain for every Q ∈ S a subset EQ ⊂ Q such that      1 −1 g(x)w(x) dx ≤ κ g(x)w(x) dx w(z) dz Q EQ w(Q) Q  d,w ≤ κ−1 MQ g(z)w(z) dz, 0 EQ

d,w MQ g 0

where is as in Definition 9.7 and κ = κ(w). From (9.9) we obtain, by pairwise disjointness of the sets EQ and H¨older’s inequality, that  

  d,w F (x)g(x)w(x) dx ≤ κ−1 |f (y) − f | dy MQ g(z)w(z) dz Q  0 Q0

Q∈S

≤ κ−1

Q∈S

≤ κ−1

EQ





Q0

EQ

Q

d, d,w MQ f (z)MQ g(z)w(z) dz 0 0

d, d,w MQ f (z)MQ g(z)w(z) dz 0 0

 d,  ≤ κ−1 MQ f Lp (Q 0

0 ;w dx)

 d,w  M g  Q0

Lq (Q0 ;w dx)

.

Finally, Theorem 9.8 implies  d,w  M g  q ≤ C(p)gLq (Q0 ;w dx) ≤ C(p), Q0 L (Q0 ;w dx) and the claim follows by taking supremum over all functions g as above.



9.4. TWO WEIGHT MAXIMAL INEQUALITIES

205

9.4. Two weight maximal inequalities In this section we consider two weight inequalities for the following dyadic fractional maximal function, compare to Definition 1.38. Definition 9.16. Let Q0 ⊂ Rn be a cube and let 0 ≤ α < n. For f ∈ L1 (Q0 ), d the dyadic fractional maximal function Mα,Q f at x ∈ Q0 is 0  1 d Mα,Q0 f (x) = sup |f (y)| dy, 1− α n Qx |Q| Q where the supremum is taken over all dyadic cubes Q ∈ D(Q0 ) with x ∈ Q. From the Poincar´e inequality on cubes we obtain pointwise estimates in terms of the dyadic fractional maximal function, compare to Section 4.3. Lemma 9.17. Let Q0 ⊂ Rn be a cube and let u ∈ Lip(Q0 ). Then d, d MQ u(x) ≤ C(n)M1,Q |∇u|(x) 0 0

for every x ∈ Q0 . Proof. Let x ∈ Q0 and let Q ∈ D(Q0 ) with x ∈ Q. By Corollary 9.6, we have  |u(y) − uQ | dy ≤ C(n)l(Q)  |∇u(y)| dy Q Q  C(n) d = |∇u(y)| dy ≤ C(n)M1,Q |∇u|(x). 1 0 |Q|1− n Q The claim follows by taking supremum over all cubes Q as above.



The next lemma gives a necessary condition for a weighted Hardy–Littlewood– Sobolev theorem for the dyadic fractional maximal function, compare to Theorem 1.42. This result is also a generalization of Lemma 8.17. The corresponding sufficient condition will be discussed in Theorem 9.20. Lemma 9.18. Let 1 < p ≤ q < ∞ and 0 ≤ α < n, and let Q0 ⊂ Rn be a cube. Assume that (w, v) is a pair of weights such that   q1  p1  d

q p Mα,Q0 f (x) w(x) dx ≤ C1 |f (x)| v(x) dx Q0

Q0

1 1−p

for every f ∈ L∞ (Q0 ), and write σ = v . Then  p p 1 (9.10) w(Q) q σ(Q)p−1 ≤ C1p α 1− n |Q| for every dyadic cube Q ∈ D(Q0 ). Proof. Let ε > 0 and Q ∈ D(Q0 ), and define 1

f (x) = (ε + v(x)) 1−p χQ (x) for every x ∈ Q0 . Then 1 α |Q|1− n

 Q

1

d (ε + v(y)) 1−p dy ≤ Mα,Q f (x) 0

´ INEQUALITIES 9. WEIGHTED MAXIMAL AND POINCARE

206

for every x ∈ Q, and thus  q    d

q 1 1 1−p dy Mα,Q0 f (x) w(x) dx. w(x) dx (ε + v(y)) ≤ α 1− |Q| n Q Q Q Since Q ⊂ Q0 and f ∈ L∞ (Q0 ), we have   d

q Mα,Q0 f (x) w(x) dx ≤ C1q Q

 pq

p

f (x) v(x) dx

Q0

 ≤

C1q

(ε + v(x))

p 1−p

 pq (ε + v(x)) dx

Q

 =

C1q

(ε + v(x))

1 1−p

 pq dx

,

Q

and thus



1 α |Q|1− n

q

 w(Q)

(ε + v(y))

1 1−p

q− pq dy

≤ C1q

Q

for every ε > 0. By Fatou’s lemma, we conclude that  p p 1 w(Q) q σ(Q)p−1 α 1− |Q| n p−1  p  p 1 1 q 1−p ≤ lim inf w(Q) (ε + v(y)) dy ≤ C1p , α ε→0+ |Q|1− n Q 

and the claim follows.

For the converse of Lemma 9.18 we need another variant of the sparse domination lemma. Here we write  1 fα,Q = |f (y)| dy α |Q|1− n Q for Q ∈ D(Q0 ) and f ∈ L1 (Q0 ). Lemma 9.19. Let 0 ≤ α < n and assume that σ ∈ A∞ . Let Q0 ⊂ Rn be a cube and let f ∈ L∞ (Q0 ). There exists a collection S ⊂ D(Q0 ) of dyadic subcubes of Q0 satisfying the following conditions. (a) For 1 ≤ p < ∞, there exists a constant C = C(n, p, σ) such that

d

p Mα,Q0 f (x) ≤ C (fα,Q )p χQ (x) Q∈S

for every x ∈ Q0 . (b) There exist a constant κ = κ(n, σ) > 0 and a collection {EQ : Q ∈ S} of pairwise disjoint measurable sets EQ ⊂ Q with σ(EQ ) ≥ κσ(Q) for every Q ∈ S. Proof. Let C(σ) > 0 and δ(σ) > 0 be the constants in the A∞ condition (8.12) for w. Choose a > 2n such that  n δ(σ) 2 ρ = C(σ) < 1. a

9.4. TWO WEIGHT MAXIMAL INEQUALITIES

207

Without loss of generality, we may assume that fα,Q0 > 0. Let k0 be the smallest integer satisfying ak0 ≥ fα,Q0 , and let d f (x) ≤ ak+1 } Dk = {x ∈ Q0 : ak < Mα,Q 0

and d Sk = {x ∈ Q0 : ak < Mα,Q f (x)} 0

for every integer k > k0 . Let Sk0 = {Q0 }. For k > k0 , let Sk be the collection of pairwise disjoint maximal dyadic cubes Q ∈ D(Q0 ) satisfying  1 (9.11) ak < |f (y)| dy. α |Q|1− n Q Since  k > k0 , each of these maximal cubes is strictly contained in Q0 . Observe that Sk = Q∈Sk Q for every integer k > k0 . For every k ≥ k0 and every R ∈ Sk+1 , there exists a unique Q ∈ Sk with R ⊂ Q. In particular, the sets   R =Q\ R Ek,Q = Q \ R∈Sk+1

R∈Sk+1 R⊂Q

in the collection {Ek,Q : k ≥ k0 and Q ∈ Sk } are pairwise disjoint. We claim that σ(Ek,Q ) ≥ (1 − ρ)σ(Q)

(9.12)

for every k ≥ k0 and Q ∈ Sk . In order to prove this inequality, we first apply (9.11) and obtain α 

|R| αn 

|Q| n 2n |Q|, |R| ≤ |f (y)| dy ≤ |f (y)| dy ≤ ak+1 R ak+1 Q a R∈Sk+1 R⊂Q

R∈Sk+1 R⊂Q

whenever Q ∈ Sk ; recall here that α ≥ 0. The final step above follows from the maximality of the collection Sk if k > k0 and from inequality ak0 ≥ fα,Q0 if k = k0 . Since σ ∈ A∞ , we have    δ(σ)   n δ(σ) σ R∈Sk+1 :R⊂Q R 1  2 ≤ C(σ) R ≤ C(σ) = ρ. σ(Q) |Q| R∈Sk+1 :R⊂Q a From this we deduce that σ(Ek,Q ) ≥ σ(Q) − ρσ(Q) = (1 − ρ)σ(Q), and (9.12) follows. Moreover, since Ek,Q = ∅ for every k ≥ k0 and Q ∈ Sk , we have Sk ∩ Sj = ∅ fork = j. We conclude from above that condition (b) holds for the collection S = k≥k0 Sk , with κ = 1 − ρ > 0. Observe that α

α

d d Mα,Q f (x) ≤ |Q0 | n MQ f (x) ≤ |Q0 | n f L∞ (Q0 ) < ∞ 0 0

for every x ∈ Q0 , and thus d f (x) ≤ ak0 +1 } ∪ Q0 = {x ∈ Q0 : MQ 0

 k>k0

Dk .

´ INEQUALITIES 9. WEIGHTED MAXIMAL AND POINCARE

208

Since ak0 −1 < fα,Q0 , we obtain

d

p

p d Mα,Q Mα,Q0 f (x) ≤ a2p (fα,Q0 )p χQ0 (x) + f (x) χDk (x) 0 k>k0

≤ a (fα,Q0 ) χQ0 (x) + ap 2p

p

akp χSk (x)

k>k0

≤ a (fα,Q0 ) 2p

≤a

2p

p

χQ0 (x) + a





p

(fα,Q )p χQ (x)

k>k0 Q∈Sk

(fα,Q )

p

χQ (x)

k≥k0 Q∈Sk

= a2p

(fα,Q )p χQ (x)

Q∈S

for every x ∈ Q0 , and this proves condition (a).



The following result is an extension of Theorem 9.9 for two weights and two exponents. Indeed, if α = 0, p = q and w = v, then (9.13) reduces to a localized Ap condition for w. Theorem 9.20. Let 1 < p ≤ q < ∞ and 0 ≤ α < n, and assume that (w, v) 1 is a pair of weights such that σ = v 1−p ∈ A∞ . Let Q0 ⊂ Rn be a cube and assume that there exists a constant C1 > 0 such that  p p 1 w(Q) q σ(Q)p−1 ≤ C1 (9.13) α |Q|1− n for all dyadic cubes Q ∈ D(Q0 ). Then there exists a constant C = C(n, p, q, σ, C1 ) such that  Q0

d

q Mα,Q0 f (x) w(x) dx

 1q

 ≤C

|f (x)|p v(x) dx

 p1

Q0

for every u ∈ L∞ (Q0 ). Proof. Let f ∈ L∞ (Q0 ) and let S be the associated sparse collection of cubes, given by Lemma 9.19 with σ ∈ A∞ and exponent p. Then  Q0

d

q Mα,Q0 f (x) w(x) dx

 pq

 d

p  =  Mα,Q f  pq 0 L (Q0 ;w dx)     pχ  ≤ C (f ) α,Q Q q 

L p (Q0 ;w dx)

Q∈S

  (fα,Q )p χ  ≤C Q ≤C



Q∈S

q

L p (Q0 ;w dx)

Q∈S

1 α |Q|1− n

p

 |f (y)| dy Q

p

w(Q) q ,

´ INEQUALITIES 9.5. TWO WEIGHT POINCARE

209

where C = C(n, p, q, σ). On the other hand, by (9.13), p 

 1 p |f (y)| dy w(Q) q 1− α n |Q| Q Q∈S p  p  

p 1 1 p−1 −1 q = w(Q) σ(Q) |f (y)|σ (y)σ(y) dy σ(Q) α σ(Q) Q |Q|1− n Q∈S p

 1  |f (y)|σ −1 (y)σ(y) dy σ(Q). ≤ C1 σ(Q) Q Q∈S

We use the properties of the collection S and apply Lemma 9.8 to bound the final term by p   1 C1 |f (y)| σ −1 (y)σ(y) dy σ(EQ ) κ σ(Q) Q Q∈S p    1 C1 −1 |f (y)| σ (y)σ(y) dy σ(x) dx = κ σ(Q) Q Q∈S EQ  d,σ

p C1 ≤ MQ0 (f σ −1 )(x) σ(x) dx κ Q0  C(p)C1 ≤ |f (y)σ −1 (y)|p σ(y) dy. κ Q0 This concludes the proof since κ = κ(n, σ) and v = σ 1−p .



9.5. Two weight Poincar´ e inequalities The following result is a generalization of Theorem 9.11 and it gives a sufficient condition for the validity of a two weight Poincar´e inequality. Theorem 9.21. Let 1 < p ≤ q < ∞ and let (w, v) be a pair of weights with 1 w ∈ A∞ and σ = v 1−p ∈ A∞ . Let Q0 ⊂ Rn be a cube and assume that there exists a constant C1 such that  p p 1 w(Q) q σ(Q)p−1 ≤ C1 (9.14) 1 1− |Q| n for all dyadic cubes Q ∈ D(Q0 ). Then there exists a constant C = C(n, p, q, w, σ, C1 ) such that   q1   p1 q p |u(x) − uQ0 | w(x) dx ≤C |∇u(x)| v(x) dx Q0

Q0

for every u ∈ Lip(Q0 ). Proof. Let u ∈ Lip(Q0 ). Theorem 9.15 and Lemma 9.17 imply   d,

q q MQ0 u(x) w(x) dx |u(x) − uQ0 | w(x) dx ≤ C(n, q, w) Q0 Q  0 d

q M1,Q0 |∇u|(x) w(x) dx. ≤ C(n, q, w) Q0

´ INEQUALITIES 9. WEIGHTED MAXIMAL AND POINCARE

210

By Theorem 9.20, we obtain   d

q M1,Q0 |∇u|(x) w(x) dx ≤ C(n, p, q, σ, C1) Q0

|∇u(x)| v(x) dx p

 pq ,

Q0



and the claim follows.

Remark 9.22. We show that Theorem 9.21 is a generalization of Theorem 9.11. Let 1 < p < ∞ and let Q0 ⊂ Rn be a cube. Assume that w ∈ Ap ⊂ A∞ and let 1 v = l(Q0 )p w and σ = v 1−p . Lemma 8.13 (b) implies σ ∈ Ap ⊂ A∞ . For q = p and every Q ∈ D(Q0 ), we have  p−1  p  p p 1 1 q σ(Q)p−1 = l(Q)p |Q|−p w(Q) l(Q )− p−1 1−p dx w(Q) w(x) 0 1 |Q|1− n Q   p−1 p 1 l(Q) 1−p dx = w(x) dx w(x)  l(Q0 ) Q Q ≤ [w]Ap , where the final step holds since l(Q) ≤ l(Q0 ). This shows that the assumption of Theorem 9.21 is satisfied, and the claim of Theorem 9.11 follows from Theorem 9.21. np Remark 9.23. Let 1 < p < n and q = p∗ = n−p , and let w(x) = v(x) = 1 for n n every x ∈ R . Then σ(x) = 1 for every x ∈ R and p  p p p 1 w(Q) q σ(Q)p−1 = |Q| n −p |Q|1− n |Q|p−1 = 1 1 1− n |Q|

for every cube Q ⊂ Rn . By Theorem 9.21, we find that the Sobolev–Poincar´e inequality  p1∗   p1  ∗ |u(x) − uQ |p dx ≤ C(n, p) |∇u(x)|p dx (9.15) Q

Q

holds for all cubes Q ⊂ R and every u ∈ Lip(Q). Note that the 1-Poincar´e inequality for cubes is needed in the proof of Theorem 9.21 via the proof of Lemma 9.17. n

The following Poincar´e inequality for cubes follows from Remark 9.23 by adapting the proof of Theorem 3.14. We remark that inequalities (9.15) and (9.16) are obtained as special cases of Theorem 9.21, but there are also direct proofs, for instance by following the lines of the proofs of the corresponding inequalities for balls in Chapter 3. 1,p (Rn ). Let 1 ≤ Corollary 9.24. Let 1 < p < ∞ and assume that u ∈ Wloc np q ≤ n−p for 1 < p < n, and 1 ≤ q < ∞ for n ≤ p < ∞. There exists a constant C = C(n, p, q) such that   1q   p1 q p (9.16) |u(x) − u | dx ≤ Cl(Q) |∇u(x)| dx Q   Q

Q

for every cube Q ⊂ R . n

For compactly supported Lipschitz functions we obtain the following two weight Sobolev–Gagliardo–Nirenberg inequality; compare to Theorem 3.6 and Remark 3.16.

9.6. LOCAL-TO-GLOBAL INEQUALITIES ON OPEN SETS

211

Corollary 9.25. Let 1 < p ≤ q < ∞ and let (w, v) be a pair of weights with 1 w ∈ A∞ and σ = v 1−p ∈ A∞ . Assume that there exists a constant C1 such that p  p 1 w(Q) q σ(Q)p−1 ≤ C1 1 1− |Q| n for all cubes Q ⊂ Rn . Then there exists a constant C = C(n, p, q, w, σ, C1 ) such that  q1   p1  q p |u(x)| w(x) dx ≤C |∇u(x)| v(x) dx Rn

Rn

for every u ∈ Lipc (Rn ).

Proof. Let u ∈ Lipc (Rn ) and let Qk = Q(0, k) for every k ∈ N. Since u ∈ L (Rn ), we have uL∞ (Rn ) = 0. lim |uQk | ≤ lim k→∞ k→∞ |Qk | Fatou’s lemma and Theorem 9.21 imply   q1  q1  q q |u(x)| w(x) dx ≤ lim inf |u(x) − uQk | w(x) dx ∞

k→∞

Rn

Qk

 ≤ C lim inf k→∞



|∇u(x)| v(x) dx p

Qk

|∇u(x)| v(x) dx p

=C Rn

 p1

 p1 , 

where C = C(n, p, q, w, σ, C1). 9.6. Local-to-global inequalities on open sets The covering lemma below is proved by maximal function arguments.

Lemma 9.26. Let 1 ≤ p < ∞ and 1 ≤ τ < ∞ and assume that w is a doubling weight in Rn . Let Ω ⊂ Rn be an open set with a Whitney decomposition W(Ω) = {Qi : i ∈ N}, where Qi = Q(xi , ri ). There exists a constant C = C(n, p, τ, w) such that ∞   ∞         ai χQ(xi ,τ ri )  ≤ C ai χQi     p  p  i=1

L (w dx)

i=1

L (w dx)

whenever ai ≥ 0 for every i ∈ N. Proof. The case p = 1 follows from the fact that w is a doubling weight. Hence, we may assume that 1 < p < ∞. By duality and density of compactly  p supported continuous functions in Lp (w dx), with p = p−1 , it suffices to verify that there exists a constant C = C(n, p, τ, w) such that   ∞ ∞  

  (9.17) ai χQ(xi ,τ ri ) (y)g(y)w(y) dy ≤ C  ai χQi   p Rn  i=1

i=1

L (w dx)

for every g ∈ C0 (R ) with gLp (w dx) ≤ 1. Let i ∈ N and assume that g is as above. Then  (9.18) |g(y)|w(y) dy ≤ w(Q(xi , τ ri ))M c,w g(x) n

Q(xi ,τ ri )

´ INEQUALITIES 9. WEIGHTED MAXIMAL AND POINCARE

212

for every x ∈ Q(xi , τ ri ). Here M c,w is the weighted maximal operator as in Remark 8.4 (b). By integrating (9.18) over the cube Qi ⊂ Q(xi , τ ri ) and using the fact that w is doubling, we obtain 

 w(Q(xi , τ ri ) M c,w g(x)w(x) dx w(Qi ) Qi  ≤ C(w, τ ) M c,w g(x)w(x) dx.

|g(y)|w(y) dy ≤ Q(xi ,τ ri )

Qi

By the triangle inequality we conclude   ∞ ∞

ai χQ(xi ,τ ri ) (y)g(y)w(y) dy ≤ ai |g(y)|w(y) dy Rn Q(xi ,τ ri ) i=1 i=1  ∞

ai M c,w g(x)w(x) dx ≤ C(w, τ ) Qi

i=1

 = C(w, τ )

∞

Rn i=1

ai χQi (x)M c,w g(x)w(x) dx.

The required estimate (9.17) follows by applying H¨ older’s inequality to the right  hand side and using the boundedness of M c,w on Lp (w dx), see Remark 8.4 (b).  For a weight w in Rn and a measurable set E ⊂ Rn with 0 < w(E) < ∞, we write  1 f (x)w(x) dx fE;w = w(E) E for every f ∈ L1 (E; w dx); compare to (8.14). Lemma 9.27. Assume that w is a doubling weight in Rn and that Ω ⊂ Rn is an open set with a Whitney decomposition W(Ω). Let Q0 , Q ∈ W(Ω) and let C(Q) be a chain in Ω joining Q0 to Q. Then 

1 |f (x) − fR∗ ;w |w(x) dx |fQ∗ ;w − fQ∗0 ;w | ≤ C(n, w) w(R∗ ) R∗ R∈C(Q)

for every f ∈ L1loc (Ω; w dx). Proof. Let C(Q) = (Q0 , . . . , Qk ), with Q = Qk , and let f ∈ L1loc (Ω; w dx). Then k fQ∗ ;w − fQ∗ ;w . fQ∗ ;w − fQ∗ ;w ≤ 0 i i−1 i=1

Let i ∈ {1, . . . , k}. By (9.1) and the assumption that w is doubling, we obtain max{w(Q∗i ), w(Q∗i−1 )} ≤ C(n, w)w(Q∗i ∩ Q∗i−1 ).

9.6. LOCAL-TO-GLOBAL INEQUALITIES ON OPEN SETS

213

Thus fQ∗ ;w − fQ∗

i−1 ;w

i

= fQ∗ ;w − fQ∗ ∩Q∗ ;w + fQ∗ ∩Q∗ ;w − fQ∗ ;w i i i−1 i i−1 i−1  1 ≤ |f (x) − fQ∗i ;w |w(x) dx w(Q∗i ∩ Q∗i−1 ) Q∗i ∩Q∗i−1  1 |f (x) − fQ∗i−1 ;w |w(x) dx + w(Q∗i ∩ Q∗i−1 ) Q∗i ∩Q∗i−1  i

1 ≤ C(n, w) |f (x) − fQ∗j ;w |w(x) dx. w(Q∗j ) Q∗j j=i−1

Since Qi1 = Qi2 for i1 = i2 , we conclude |fQ∗ ;w − f

Q∗ 0 ;w

| ≤ 2C(n, w)

R∈C(Q)

1 w(R∗ )

 R∗

|f (x) − fR∗ ;w |w(x) dx, 

and the proof is complete. We are ready for a weighted local-to-global inequality.

Theorem 9.28. Let 1 ≤ p < ∞ and let w be a doubling weight in Rn . Assume that Ω ⊂ Rn is an open set satisfying the Boman chain condition, with τ > 1. There exists a constant C = C(n, p, w, τ ) such that 

 p inf |f (x) − c| w(x) dx ≤ C |f (x) − fQ∗ ;w |p w(x) dx (9.19) c∈R

Ω

Q∈W(Ω)

Q∗

for every f ∈ L1loc (Ω; w dx). Proof. Throughout this proof C is a constant that depends at most on the parameters p, n, w and τ . Let Q0 be the fixed Whitney cube in the chain decomposition of Ω, with τ > 1, see Definition 9.2. By the triangle inequality,

f (x) − fQ∗ ;w ≤ f (x) − ∗ ;w χ (x) + ∗ ;w χ (x) − fQ∗ ;w f f Q Q Q Q 0 0 Q∈W(Ω)

Q∈W(Ω)

= g1 (x) + g2 (x), for every x ∈ Ω. This implies    |f (x) − fQ∗0 ;w |p w(x) dx ≤ 2p g1 (x)p w(x) dx + 2p g2 (x)p w(x) dx. Ω

Ω

Ω

We estimate the integrals on the right-hand side separately. By the properties of the Whitney cubes we have 



 g1 (x)p w(x) dx = g1 (x)p w(x) dx = |f (x) − fQ∗ ;w |p w(x) dx Ω

Q∈W(Ω)

≤

Q∈W(Ω)

Q



Q∗

Q∈W(Ω)

|f (x) − fQ∗ ;w |p w(x) dx.

Q

´ INEQUALITIES 9. WEIGHTED MAXIMAL AND POINCARE

214

The second integral is more difficult to estimate. We begin with p   g2 (x)p w(x) dx = (fQ∗ ;w − fQ∗0 ;w )χQ (x) w(x) dx Ω Ω Q∈W(Ω) p   ≤ |fQ∗ ;w − fQ∗0 ;w |χQ (x) w(x) dx. Rn

Define

 aR =

1 w(R∗ )

Q∈W(Ω)

 R∗

|f (x) − fR∗ ;w |p w(x) dx

 p1

≥0

for cubes R = Q(xR , rR ) ∈ W(Q). By Lemma 9.27 and H¨ older’s inequality, we have

aR χ Q |fQ∗ ;w − fQ∗0 ;w |χQ ≤ C R∈C(Q)

for every Q ∈ W(Ω). By summing these estimates over all Whitney cubes and applying (9.2), we obtain



χQ |fQ∗ ;w − fQ∗0 ;w |χQ ≤ C aR χQ = C aR Q∈W(Ω)

Q∈W(Ω) R∈C(Q)

≤C

R∈W(Ω)

Q∈S(R)

aR χQ(xR ,τ rR ) .

R∈W(Ω)

An application of Lemma 9.26 shows that    p g2 (x) w(x) dx ≤ C Rn

Ω

 ≤C ≤C

aR χQ(xR ,τ rR ) (x)

R∈W(Ω)



Rn

p

w(x) dx

p aR χR (x)

w(x) dx

R∈W(Ω)



apR

R∈W(Ω)

Rn

χR (x)w(x) dx

 w(R) |f (x) − fR∗ ;w |p w(x) dx w(R∗ ) R∗ R∈W(Ω)

 ≤C |f (x) − fR∗ ;w |p w(x) dx,

≤C

R∈W(Ω)

R∗



and the proof is complete.

Remark 9.29. Theorem 9.28 applies for f ∈ L1loc (Ω; w dx) even if fΩ;w is not defined. However, if f ∈ L1 (Ω; w dx) and 1 ≤ p < ∞, then   |f (x) − fΩ;w |p w(x) dx ≤ 2p inf |f (x) − c|p w(x) dx. (9.20) Ω

c∈R

Ω

9.6. LOCAL-TO-GLOBAL INEQUALITIES ON OPEN SETS

215

This follows, since   p1   p1 1 p p |f (x) − fΩ;w | w(x) dx ≤ |f (x) − c| w(x) dx +w(Ω) p |fΩ;w − c| Ω

Ω

 1 = |f (x) − c| w(x) dx +w(Ω) (f (x) − c)w(x) dx w(Ω) Ω Ω  p1  p1    1 1 p p p |f (x) − c| w(x) dx +w(Ω) |f (x) − c| w(x) dx ≤ w(Ω) Ω Ω 

p

 p1

1 p

for every c ∈ R. Recall that the dilated cubes Q∗ appearing in Theorem 9.30 have a bounded overlap property. Theorem 9.30. Let 1 ≤ p < ∞ and w ∈ A∞ . Assume that Ω ⊂ Rn is an open set satisfying the Boman chain condition, with τ > 1. There exists a constant C = C(n, p, w, τ ) such that 

 d,

p MQ∗ f (x) w(x) dx |f (x) − c|p w(x) dx ≤ C inf c∈R

Ω

Q∈W(Ω)

Q∗

for every f ∈ Liploc (Ω). Proof. Theorem 8.23 implies that w is a doubling weight. By Theorem 9.28, inequality (9.20), and Theorem 9.15, we obtain 

 p inf |f (x) − c| w(x) dx ≤ C |f (x) − fQ∗ ;w |p w(x) dx c∈R

Ω

Q∗

Q∈W(Ω)

≤ 2p C

(9.21)



|f (x) − fQ∗ |p w(x) dx

Q∈W(Ω)

≤ 2p C

Q∈W(Ω)

Q∗



Q∗

d,

p MQ∗ f (x) w(x) dx, 

where C = C(n, p, w, τ ).

The following two-weight Poincar´e inequality holds for open sets satisfying the Boman chain condition. Theorem 9.31. Let 1 < p ≤ q < ∞ and let (w, v) be a pair of weights with 1 w ∈ A∞ and σ = v 1−p ∈ A∞ . Assume that Ω ⊂ Rn is an open set satisfying the Boman chain condition, with τ > 1, and that there exists a constant C1 such that p  p 1 w(Q ) q σ(Q )p−1 ≤ C1 1 1−  n |Q | for all cubes Q satisfying Q ⊂ Q∗ for some Whitney cube Q ∈ W(Ω). Then there exists a constant C = C(n, p, q, w, σ, τ, C1) such that  q1   p1  |u(x) − c|q w(x) dx ≤C |∇u(x)|p v(x) dx inf c∈R

Ω

for every u ∈ Liploc (Ω).

Ω

´ INEQUALITIES 9. WEIGHTED MAXIMAL AND POINCARE

216

Proof. Theorem 8.23 implies that w ∈ A∞ is a doubling weight. As in (9.21), we have 

 inf |u(x) − c|q w(x) dx ≤ 2p C(n, p, w, τ ) |u(x) − uQ∗ |q w(x) dx. c∈R

Ω

Q∈W(Ω) Q∗

Theorem 9.21 and the assumption q ≥ p imply 

 q inf |u(x) − c| w(x) dx ≤ C c∈R

Ω

Q∈W(Ω)



≤C

|∇u(x)| v(x) dx p

Q∗

 pq

 Q∗

Q∈W(Ω)

|∇u(x)|p v(x) dx

 ≤C

 pq

|∇u(x)| v(x) dx p

 pq ,

Ω



where C = C(n, p, q, w, σ, τ, C1).

As a consequence of Theorem 9.31, we obtain a Sobolev–Poincar´e inequality on open sets satisfying the Boman chain condition. np Theorem 9.32. Let 1 < p < n and p∗ = n−p , and assume that Ω ⊂ Rn is an open set satisfying the Boman chain condition, with τ > 1. There exists a constant C = C(n, p, τ ) such that  p1∗   p1  p∗ p |u(x) − uΩ | dx ≤C |∇u(x)| dx , Ω

for every u ∈ W

1,p

Ω

(Ω).

Proof. Assume first that u ∈ W 1,p (Ω) ∩ C ∞ (Ω) ⊂ Liploc (Ω). Then  p1∗   p1  p∗ p |u(x) − c| dx ≤C |∇u(x)| dx inf c∈R

Ω

Ω

by Remark 9.23 and Theorem 9.31. Since W (Ω) ⊂ L1 (Ω), in this case we obtain the claimed inequality from inequality (9.20). In the general case u ∈ W 1,p (Ω) the claim follows by approximation in W 1,p (Ω) and Fatou’s lemma, compare to the proof of Theorem 3.14.  1,p

9.7. BMO and John–Nirenberg inequality As another application of the sparse domination technique in Lemma 9.13, we consider functions of bounded mean oscillation. The central result is the John– Nirenberg inequality, Theorem 9.39, which implies a self-improvement property for BMO functions. Definition 9.33. A function f ∈ L1loc (Rn ) belongs to the space of bounded mean oscillation BMO(Rn ) if f BMO(Rn ) = sup  |f (x) − fQ | dx < ∞, Q

Q

where the supremum is taken over all cubes Q ⊂ Rn .

9.7. BMO AND JOHN–NIRENBERG INEQUALITY

217

Remark 9.34. If f ∈ BMO(Rn ), then |f | ∈ BMO(Rn ) and   |f | (9.22) n ≤ 2f BMO(Rn ) . BMO(R )

Indeed, if Q ⊂ R is a cube, then by (9.20) and the triangle inequality  |f (x)| − |f |Q dx ≤ 2  |f (x)| − |fQ | dx n

Q

Q

≤ 2  |f (x) − fQ | dx ≤ 2f BMO(Rn ) . Q

Inequality (9.22) follows by taking supremum over all cubes Q ⊂ Rn . Remark 9.35. We note that L∞ (Rn ) ⊂ BMO(Rn ) and f BMO(Rn ) ≤ 2f L∞ (Rn ) . This follows, since  |f (x) − fQ | dx ≤  Q

Q



|f (x)| + |fQ | dx =  |f (x)| dx + |fQ | Q

≤  |f (x)| dx + |f |Q ≤ 2  |f (x)| dx ≤ 2f L∞ (Rn ) Q

Q

for every cube Q ⊂ Rn . The following example gives an unbounded BMO function. Thus L∞ (Rn ) is a proper subset of BMO(Rn ). Example 9.36. We show that log|x| ∈ BMO(Rn ). By Remark 9.29 it is enough to show that for every cube Q(x, r), with x ∈ Rn and r > 0, there exists a constant cQ(x,r) ∈ R such that log|y| − cQ(x,r) dy < ∞. sup  Q(x,r)

Q(x,r)

We consider two cases. √ First assume that |x| < 2r n. In this case, we choose cQ(x,r) = log(2r). By a change of variables y = 2rz, dy = (2r)n dz, we obtain   1 log|y| − cQ(x,r) dy = log(2r|z|) − log(2r) dz n x 1 (2r) Q(x,r) Q( 2r ,2)   log|z| dz ≤ log|z| dz = C(n) < ∞. = √ x 1 Q( 2r ,2)

B(0,2 n)

x 1 The final estimate holds since if z ∈ Q( 2r , 2 ), then √ √ √ n √ n |x| |z| ≤ + < n+ ≤ 2 n, 2r 2 2 √ x 1 , 2 ) ⊂ B(0, 2 n). and thus Q( 2r √ Assume then that |x| ≥ 2r n. In this case, we choose cQ(x,r) = log|x|. A change of variables, as above, gives   1 log|y| − cQ(x,r) dy = log|2rz| − log|x| dz n (2r) Q(x,r) Q( x , 1 )  2r 2 (9.23) log |2rz| dz. = x 1 |x| Q( 2r , 2 )

´ INEQUALITIES 9. WEIGHTED MAXIMAL AND POINCARE

218

x 1 Let z ∈ Q( 2r , 2 ). Then

Thus

√ √ |x| |x| n n − ≤ |z| ≤ + . 2r 2 2r 2 √ |2rz| 2r n 1 1 ≥1− ≥1− = |x| |x| 2 2 2

and

√ |2rz| 2r n 1 ≤1+ ≤ 1 + < 2. |x| |x| 2 2 These estimates show that 1 |2rz| − log 2 = log ≤ log ≤ log 2 2 |x| and consequently

log |2rz| ≤ log 2 |x|

x 1 for every z ∈ Q( 2r , 2 ). This, together with (9.23), gives   1 log |2rz| dz ≤ log 2 < ∞. log|y| − cQ(x,r) dy ≤ n x 1 (2r) Q(x,r) |x| Q( 2r ,2)

We conclude that the mean oscillation of log|x| is uniformly bounded and thus log|x| ∈ BMO(Rn ). Observe however that log|x| ∈ / L∞ (Rn ). We also need a local dyadic variant of BMO. Definition 9.37. Let Q0 ⊂ Rn be a cube. A function f ∈ L1 (Q0 ) belongs to the space BMOd (Q0 ) of dyadic bounded mean oscillation if f BMOd (Q0 ) =

sup Q∈D(Q0 )

 |f (y) − fQ | dy < ∞, Q

where the supremum is taken over all dyadic subcubes of Q0 . d, f be the dyadic sharp maximal function of f ∈ L1 (Q0 ), see DefiniLet MQ 0 tion 9.14. Then  d,  f L∞ (Q ) , f BMOd (Q0 ) = sup  |f (y) − fQ | dy = MQ 0 0 Q∈D(Q0 )

Q

d, f ∈ L∞ (Q0 ). and thus f ∈ BMOd (Q0 ) if and only if MQ 0 1,n Functions in the Sobolev space W are examples of BMO functions.

Example 9.38. Let Q0 ⊂ Rn be a cube and let u ∈ W 1,n (Q0 ). Then by H¨ older’s inequality u ∈ W 1,1 (Q0 ). The 1-Poincar´e inequality on cubes, see Corollary 9.6, and H¨ older’s inequality give   n1 n  |u(x) − uQ | dx ≤ C(n)l(Q)  |∇u(x)| dx Q

Q

1 −n

 |∇u(x)| dx n

= C(n)l(Q)|Q|

Q

 n1

9.7. BMO AND JOHN–NIRENBERG INEQUALITY

219

for every Q ∈ D(Q0 ). Since l(Q)|Q|− n = 1 and Q ⊂ Q0 if Q ∈ D(Q0 ), we have 1

uBMOd (Q0 ) =

 |u(x) − uQ | dx

sup Q∈D(Q0 )



Q

≤ C(n)

|∇u(x)| dx n

 n1

≤ C(n)uW 1,n (Q0 ) .

Q0

Thus W 1,n (Q0 ) ⊂ BMOd (Q0 ), and the inclusion is a bounded linear operator. In a similar fashion, it can be shown that W 1,n (Rn ) ⊂ BMO(Rn ). We prove a weighted John–Nirenberg inequality for the dyadic BMO. Theorem 9.39. Assume that w ∈ A∞ . Let Q0 ⊂ Rn be a cube and assume that f ∈ BMOd (Q0 ). There exists a constant C = C(n, w) such that   w({x ∈ Q0 : |f (x) − fQ0 | > t}) t ≤ C exp − w(Q0 ) Cf BMOd (Q0 ) for every t > 0. Proof. We may assume that f BMOd (Q0 ) > 0. Let S ⊂ D(Q0 ) be the sparse collection of cubes given by Lemma 9.13, associated to the function f ∈ BMOd (Q0 ). For j ∈ N0 , let Sj denote the collection of those pairwise disjoint cubes S ∈ S for which there is a sequence S = Sj  · · ·  S0 = Q0 such that Si ∈ chS (Si−1 ) for every i = 1, . . . , j. Observe that  S1 = chS (Q0 ), and Sj = chS (Q) S0 = {Q0 }, Q∈Sj−1

for j ∈ N. Let x ∈ Q0 be a Lebesgue point of f . If |f (x) − fQ0 | > t, then by Lemma 9.13 (a),

χQ (x)  |f (y) − fQ | dy ≤ C1 f BMOd (Q0 ) χQ (x), t < |f (x) − fQ0 | ≤ C1 Q∈S

Q

Q∈S

where C1 = C1 (n, w) > 0. Let j ∈ N0 be such that t ≤ j + 1. j< C1 f BMOd (Q0 )  Then j < Q∈S χQ (x), and thus x belongs to at least j + 1 cubes in S. It follows that x ∈ S for some S ∈ Sj , and therefore   

w({x ∈ Q0 : |f (x) − fQ0 | > t}) ≤ w S = w(S). S∈Sj

S∈Sj

Here we also used the fact that w dx-almost every point in Q0 is a Lebesgue point of f . If j > 0 then, by Lemma 9.13 (b),



w(S) = w(S) ≤ w(Q \ EQ ) ≤ (1 − κ) w(Q). S∈Sj

Q∈Sj−1 S∈chS (Q)

Q∈Sj−1

Q∈Sj−1

By iterating this estimate, and using the fact that S0 = {Q0 }, we obtain

1

exp −(j + 1) log 1−κ w(Q0 ). w(S) ≤ (1 − κ)j w(Q0 ) = 1−κ S∈Sj

220

´ INEQUALITIES 9. WEIGHTED MAXIMAL AND POINCARE

 The desired conclusion, with C = max

C1 1 1 1−κ , log 1−κ

 , follows by combining the

estimates above.



The John–Nirenberg inequality implies a self-improvement property for the dyadic BMO. This result holds for all A∞ -weights. We remark that inequality (9.24) also follows from Theorem 9.15. Theorem 9.40. Let 1 < p < ∞ and assume that w ∈ A∞ . Let Q0 ⊂ Rn be a cube and let f ∈ BMOd (Q0 ). There exists a constant C = C(n, p, w) such that  1 (9.24) |f (x) − fQ0 |p w(x)dx ≤ Cf pBMOd (Q0 ) . w(Q0 ) Q0 Proof. Let C1 = C1 (n, w) > 0 be the constant as in Theorem 9.39. By Theorem 9.39 and Fubini’s theorem,  1 |f (x) − fQ0 |p w(x) dx w(Q0 ) Q0  ∞

p tp−1 w {x ∈ Q0 : |f (x) − fQ0 | > t} dt ≤ w(Q0 ) 0    ∞ t p−1 ≤ pC1 t exp − dt C1 f BMOd (Q0 ) 0  ∞ tp−1 e−t dt, ≤ pC11+p f pBMOd (Q0 ) 0



and the claim follows.

We also obtain a weighted exponential integrability result for dyadic BMO functions. Theorem 9.41. Assume that w ∈ A∞ . Let Q0 be a cube and let f ∈ BMOd (Q0 ). There exists a constant C = C(n, w) such that    |f (x) − fQ0 | 1 exp w(x) dx ≤ C. w(Q0 ) Q0 Cf BMOd (Q0 ) Proof. Let C1 = C1 (n, w) > 0 be the constant in Theorem 9.39 and define g(x) =

|f (x) − fQ0 | 2C1 f BMOd (Q0 )

for every x ∈ Q0 . By Fubini’s theorem and Theorem 9.39,   1 1 g(x) e w(x) dx = 1 + (eg(x) − 1)w(x) dx w(Q0 ) Q0 w(Q0 ) Q0  ∞ 1 =1+ et w({x ∈ Q0 : g(x) > t}) dt w(Q0 ) 0  ∞ e−t dt = C2 < ∞. ≤ 1 + C1 0

The proof is complete, with C = max{2C1 , C2 }.



By Example 9.38 we immediately obtain the following corollary of Theorem 9.40 and Theorem 9.41. Part (b) of Corollary 9.42 is a version of Trudinger’s inequality.

9.8. MAXIMAL FUNCTIONS AND BMO

221

Corollary 9.42. Assume that w ∈ A∞ . Let Q0 be a cube and let u ∈ W 1,n (Q0 ). (a) Let 1 < p < ∞. There exists a constant C = C(n, p, w) such that  p1   n1   1 |u(x) − uQ0 |p w(x)dx ≤C |∇u(x)|n dx . w(Q0 ) Q0 Q0 (b) There exists a constant C = C(n, w) such that    |u(x) − uQ0 | 1 exp w(x) dx ≤ C. w(Q0 ) Q0 C∇uLn (Q0 ) 9.8. Maximal functions and BMO The main result of this section is a boundedness result for a maximal operator on BMO(Rn ), see Theorem 9.46. For this purpose, we first study the connection between Muckenhoupt weights and BMO functions. Lemma 9.43. Assume that w ∈ A∞ . Then log w ∈ BMO(Rn ). Moreover, if w ∈ A2 , then log wBMO(Rn ) ≤ log(2[w]A2 ). Proof. By Theorem 8.25, we have w ∈ Ap for some 1 < p < ∞. By using 1 Lemma 8.13 and replacing w with w 1−p ∈ Ap if p > 2, we may assume that w ∈ A2 . n Let Q ⊂ R be a cube and let f = log w. Then exp f = w ∈ A2 , and thus  exp(f (x)) dx  exp(−f (x)) dx ≤ [w]A2 . Q

Q

Multiplying this inequality by exp(−fQ ) exp(fQ ) = 1 gives  exp(f (x) − fQ ) dx  exp(fQ − f (x)) dx ≤ [w]A2 . Q

Q

By Jensen’s inequality

  1 = exp(0) = exp  (fQ − f (x)) dx ≤  exp(fQ − f (x)) dx, Q

and hence

Q

−1



 exp(f (x) − fQ ) dx ≤ [w]A2  exp(fQ − f (x)) dx Q

≤ [w]A2 .

Q

In a similar way we obtain  exp(fQ − f (x)) dx ≤ [w]A2 . Q

Thus,    exp|f (x) − fQ | dx =  max exp(f (x) − fQ ), exp(fQ − f (x)) dx Q

Q

≤  exp(f (x) − fQ ) dx +  exp(fQ − f (x)) dx ≤ 2[w]A2 , Q

Q

and Jensen’s inequality implies   exp  |f (x) − fQ | dx ≤  exp|f (x) − fQ | dx ≤ 2[w]A2 . Q

Q

222

´ INEQUALITIES 9. WEIGHTED MAXIMAL AND POINCARE

After taking logarithms and then supremum over all cubes Q ⊂ Rn , we conclude  that f = log w ∈ BMO(Rn ) with f BMO(Rn ) ≤ log(2[w]A2 ). The following result is a converse of Lemma 9.43. The proof is based on the John–Nirenberg inequality. Theorem 9.44. Assume that f ∈ BMO(Rn ). There exists δ = δ(n, f ) > 0 such that exp(δf ) ∈ A2 . Proof. We may assume that f BMO(Rn ) > 0. Let C = C(n) be the constant in Theorem 9.41 with w = 1 in Rn . Let Q ⊂ Rn be a cube and define g = δf with 1 . δ= Cf BMO(Rn ) Since f BMOd (Q) ≤ f BMO(Rn ) , Theorem 9.41 implies  exp(|g(x) − gQ |) dx =  exp(δ|f (x) − fQ |) dx ≤ C. Q

Q

Thus 2  exp(g(x) − gQ ) dx  exp(gQ − g(x)) dx ≤ C , Q

Q

and multiplying this inequality by exp(gQ ) exp(−gQ ) = 1 gives 2  exp(g(x)) dx  exp(−g(x)) dx ≤ C . Q

Q

This is the A2 condition for exp g on the cube Q. The claim follows by taking  supremum over all cubes Q ⊂ Rn . Remark 9.45. The parameter δ > 0 is necessary in Theorem 9.44. Indeed, we have −n log |x| ∈ BMO(Rn ) but |x|−n = exp(−n log |x|) ∈ A2 . Using Lemma 9.43, Theorem 9.41, which was based on the John–Nirenberg inequality, and the Coifman–Rochberg lemma, see Theorem 8.33, we obtain the following boundedness result for the maximal operator on BMO(Rn ). Theorem 9.46. Let f ∈ BMO(Rn ) and assume that M c f is finite almost everywhere in Rn . Then M c f ∈ BMO(Rn ) and there exists a constant C = C(n) such that M c f BMO(Rn ) ≤ Cf BMO(Rn ) . Proof. We may assume that f BMO(Rn ) > 0, and by Remark 9.34, we may also assume that f ≥ 0. Let C1 = C1 (n) be the constant in Theorem 9.41 with w = 1 in Rn . Let Q ⊂ Rn be a cube and define g = δf with 1 . δ= C1 f BMO(Rn ) Theorem 9.41 implies  exp(g(x) − gQ ) dx ≤  exp(|g(x) − gQ |) dx =  exp(δ|f (x) − fQ |) dx ≤ C1 . Q

Q

Q

By Jensen’s inequality, exp(gQ ) ≤  exp(g(x)) dx ≤ C1 exp(gQ ). Q

9.9. NOTES

223

This holds for all cubes Q ⊂ Rn , and thus (9.25)

exp(M c g(x)) ≤ M c (exp(g))(x) ≤ C1 exp(M c g(x))

for every x ∈ Rn . Since exp(g) ∈ L1loc (Rn ) and M c g(x) = δM c f (x) < ∞ for almost every x ∈ Rn , Theorem 8.33 (a) shows that

1 w = M c (exp(g)) 2 is an A1 weight with [w]A1 ≤ C2 = C2 (n). Let



1 v = (exp(M c g)) 2 = exp 12 M c g .

The inequalities in (9.25) imply     1 1 1 2 v(x) dx ess sup w(x) dx ess sup ≤ C  1  v(x) w(x) x∈Q x∈Q Q Q 1

≤ C12 [w]A1 ≤ C(C1 , C2 ), for every cube Q ⊂ Rn . Thus v is an A1 weight and [v]A1 ≤ C(C1 , C2 ). By Lemma 9.43 and the fact that v ∈ A2 with [v]A2 ≤ [v]A1 , we obtain M c gBMO(Rn ) = log vBMO(Rn ) ≤ log(2[v]A2 ) ≤ log(2[v]A1 ) ≤ C(C1 , C2 ). 2 Hence, M c gBMO(Rn ) 2C(C1 , C2 ) ≤ = C(n)f BMO(Rn ) .  M c f BMO(Rn ) = δ δ 9.9. Notes Sobolev–Poincar´e type inequalities on open subsets of Rn have been studied by Bojarski [49], Goldshte˘ın and Reshetnyak [154], Martio [310] and Reshetnyak [353]. See also Mazya [318, 321]. Chaining arguments can be found in Bojarski [49], Boman [53], Hajlasz [167], Hajlasz and Koskela [171], Hurri-Syrj¨ anen [194, 195], Iwaniec and Nolder [208], Jones [212, 214], Ohtsuka [338], Smith and Stegenga [364] and Staples [366, 367]. Section 9.1 is based on Chua [85] and Iwaniec and Nolder [208]. Theorem 9.11 has been studied by Fabes, Kenig and Serapioni [124]. The argument in Section 9.2 is based on Chiarenza and Frasca [83]. See also Heinonen, Kilpel¨ainen and Martio [187, 15.20]. Characterization of localto-global inequality in Theorem 9.32 on domains satisfying a separation property has been given by Buckley and Koskela [63], see also Dyda, Ihnatsyeva and V¨ ah¨akangas [112]. A nonweighted version of Lemma 9.13 can be found in Lerner, Ombrosi and Rivera-R´ıos [269]. Theorem 9.15 is a version of an inequality of Fefferman and Stein [128]. See also Duoandikoetxea [107, Lemma 6.9], Garc´ıa-Cuerva and Rubio de Francia [139, Theorem 3.6], Grafakos [157, Theorem 7.4.5], Journ´e [216, p. 41], Stein [371, Corollary 1, p. 154]. For the corresponding result with a good lambda inequality, we refer to Str¨ omberg and Torchinsky [376, Chapter 3]. A local version of the Fefferman–Stein inequality has been studied in Iwaniec [206], Lerner [268] and Wik [399]. We follow Kurki and V¨ ah¨ akangas [252]. The necessity in Lemma 9.18 is discussed in Cruz-Uribe [91] and Muckenhoupt and Wheeden [335]. The proof of Theorem 9.20 is adapted from P´erez [344]. The A∞ -condition in Theorem 9.20 can be relaxed, see P´erez [345] and Cruz-Uribe and Moen [93]. Section 9.5 and Section 9.6 are based on Kurki and V¨ah¨ akangas [252];

224

´ INEQUALITIES 9. WEIGHTED MAXIMAL AND POINCARE

see also Chanillo, Str¨ omberg and Wheeden [80], Chanillo and Wheeden [81], Chua [85] and Cruz-Uribe, Martell and P´erez [92]. For applications of weighted norm inequalities in the theory of Schr¨odinger equations, see Chang, Wilson and Wolff [79] and Fefferman [127]. For Lemma 9.26 we refer to Str¨omberg and Torchinsky [375], [376, Lemma 4, p. 115]. See also Bojarski [49, Lemma 4.2], Boman [53], Chua [85], Iwaniec and Nolder [208, Lemma 4] and Staples [367]. For related local-to-global results we refer to Hurri-Syrj¨ anen [196], Hurri-Syrj¨ anen, Marola and V¨ ah¨akangas [197], Maasalo [294], Marola and Saari [308], Reimann and Rychener [352], Smith and Stegenga [364] and Staples [366, 367]. The John–Nirenberg lemma can be found in Bennet and Sharpley [28], Duoandikoetxea [107], Garc´ıa-Cuerva and Rubio de Francia [139], Garnett [142], Grafakos [157], Journ´e [216], Lu, Ding and Yan [286], Stein [371] and Torchinsky [383]. The weighted John–Nirenberg theorem has been studied by Muckenhoupt and Wheeden [336]. See also Ombrosi, P´erez, Rivera-R´ıos and Rela [339] and Canto and P´erez [72]. Corollary 9.42 (b) in the unweighted case can be found in Bojarski and Iwaniec [52]. See also Hajlasz and Koskela [171, Section 6]. John–Nirenberg lemma in more general contexts has been discussed in Buckley [61], Hyt¨onen [199], Kronz [247] and Mateu, Mattila, Nicolau and Orobitg [313]. See also Bj¨ orn and Bj¨ orn [31, Section 3.3], Heinonen, Kilpel¨ainen and Martio [187] and Str¨omberg and Torchinsky [376, Chapter 3]. Boundedness of the Hardy–Littlewood maximal function in BMO has been studied by Bennett, DeVore and Sharpley [27]. See also Bennett and Sharpley [28, Theorem 7.18, p. 389]. The proof of Theorem 9.46 is by Chiarenza and Frasca [84]. See also Aalto and Kinnunen [1].

10.1090/surv/257/10

CHAPTER 10

Distance Weights and Hardy–Sobolev Inequalities This chapter discusses topics related to the distance function. We begin with analytic and geometric conditions which guarantee that a power of the distance function is a Muckenhoupt weight. By an application of general weighted Poincar´e inequalities from Chapter 9, we obtain Hardy–Sobolev and Hardy–Sobolev–Poincar´e inequalities for distance weights. We give sufficient and necessary conditions for this kind of inequalities through dimensional conditions and we also characterize Hardy–Sobolev inequalities in terms of the properties of the variational capacity. 10.1. Aikawa condition The following integrability condition for the distance function is closely related to Muckenhoupt properties. Definition 10.1. A nonempty set E ⊂ Rn satisfies the Aikawa condition for α ∈ R if there exists a constant C such that (10.1)



d (y, E)−α dy ≤ Cr −α B(x,r)

for every x ∈ E and r > 0. We use here the conventions 00 = 1 and 0−α = ∞ for α > 0. We denote by A(E) the set of α ∈ R for which E satisfies the Aikawa condition. Since d (y, E) = d (y, E) for every y ∈ Rn , it is immediate that α ∈ A(E) if and only if α ∈ A(E). Note that if 0 < α ∈ A(E), then |E| = 0. Example 10.2. Let E = {0} = {(0, 0, . . . , 0)} ⊂ Rn , α ∈ R and r > 0. By integrating over spheres, see (3.1), we obtain   r  d (y, E)−α dy = |y|−α dy = C(n) ρ−α ρn−1 dρ B(0,r)

B(0,r)

 =

0

C(n, α)r ∞,

n−α

,

if α < n, if α ≥ n.

This shows that α ∈ A(E) if and only if α < n. Remark 10.3. Let E ⊂ Rn be a nonempty set. It is easy to see that E satisfies the Aikawa condition for every α ≤ 0. On the other hand, for α ≥ n, we have 

B(x,r)

d (y, E)−α dy ≥ 

|y − x|−α dy = ∞

B(x,r)

for every x ∈ E and r > 0, as in Example 10.2. This shows that E does not satisfy the Aikawa condition for any α ≥ n. Thus we may restrict our attention to the range 0 < α < n in the Aikawa condition. 225

226

10. DISTANCE WEIGHTS AND HARDY–SOBOLEV INEQUALITIES

For a bounded set it is sufficient to require the Aikawa condition for small radii only. Lemma 10.4. Let 0 < α < n and let E ⊂ Rn be a nonempty bounded set. Assume that (10.1) holds with a constant C1 whenever x ∈ E and 0 < r < 14 diam(E). Then (10.1) holds for every x ∈ E and r > 0 with a constant C = C(n, α, C1 ). Proof. By Example 10.2, we may assume that diam(E) > 0. Let x ∈ E and write R = 14 diam(E). We show that (10.1) holds for every r ≥ R with a constant C = C(n, α, C1 ). First assume that R ≤ r ≤ 8R = 2 diam(E) and let R }. By the boundedness of E and Lemma 1.13, there A = {y ∈ Rn : d (y, E) < 10

N exist N = N (n) ∈ N and points x1 , . . . , xN ∈ E such that A ⊂ j=1 B xj , R2 . Since (10.1) holds for all radii up to R, we obtain  N 

d (y, E)−α dy ≤ d (y, E)−α dy B(x,r)∩A

j=1

B(xj , R 2 )

≤ C(n)N C1 2α−n Rn−α ≤ C(n, α, C1 )r n−α . R , and thus On the other hand, for y ∈ B(x, r) \ A we have d (y, E) ≥ 10  d (y, E)−α dy ≤ C(n, α)r nR−α ≤ C(n, α)r n−α . B(x,r)\A

A combination of the two estimates above implies d (y, E)−α dy ≤ C(n, α, C1 )r −α ,



B(x,r)

for every R ≤ r ≤ 8R. Then assume that 8R < r < ∞. By the beginning of the proof we have  d (y, E)−α dy ≤ C(n, α, C1 )(8R)n−α ≤ C(n, α, C1 )r n−α . B(x,8R)

If y ∈ B(x, r) \ B(x, 8R), then d (y, E) ≥ 4R = diam(E), and thus |x − y| ≤ diam(E) + d (y, E) ≤ 2d (y, E). As in Example 10.2, we obtain   d (y, E)−α dy ≤ C(α) B(x,r)\B(x,8R)

|x − y|−α dy ≤ C(n, α)r n−α .

B(x,r)

We conclude that (10.1) holds for every r ≥ R, with a constant C = C(n, α, C1 ), and the proof is complete.  In certain cases it is more convenient to use cubes instead of balls in the Aikawa condition. Lemma 10.5. Let α ∈ R and assume that E ⊂ Rn is a nonempty set. Then α ∈ A(E) if and only if there exists a constant C such that (10.2)



d (y, E)−α dy ≤ Cr −α

Q(x,r)

for all cubes Q(x, r) ⊂ Rn that intersect E. Moreover, the constants in (10.1) and (10.2) only depend on each other, n and α.

10.1. AIKAWA CONDITION

227

Proof. Assume that (10.1) holds with a constant C1 . Let Q(x, r) √ be a cube that intersects E and √let z ∈ Q(x, r) ∩ E. Then Q(x, r) ⊂ B(z, 2 nr) and |Q(x, r)| = C(n)|B(z, 2 nr)|. By (10.1) we have 

Q(x,r)

d (y, E)−α dy ≤ C(n)  ≤

d (y, E)−α dy

√ B(z,2 nr) √ −α C(n)C1 (2 nr)

= C(n, α, C1 )r −α .

This shows that (10.2) holds. Then assume that (10.2) holds with a constant C1 , and let B(x, r) be a ball with x ∈ E and r > 0. Since B(x, r) ⊂ Q(x, r) and |Q(x, r)| = C(n)|B(x, r)|, we have 

B(x,r)

d (y, E)−α dy ≤ C(n) 

d (y, E)−α dy ≤ C(n, C1 )r −α ,

Q(x,r)



and thus (10.1) holds.

Remark 10.6. If α ∈ A(E), then α ∈ A(E) for every α < α. To see this, assume that (10.2) holds with a constant C1 and let Q(x, r) ⊂ Rn be a cube that intersects E. Then 



Q(x,r)

d (y, E)−α dy = 



d (y, E)(α−α )−α dy Q(x,r) 

≤ C(n, α, α )r α−α 



d (y, E)−α dy ≤ C(n, α, α , C1 )r −α . Q(x,r)

It is significantly more difficult to show that if α ∈ A(E) for some 0 < α < n, then there exists α , with α < α < n, such that α ∈ A(E). The proof of this self-improvement result for the Aikawa condition is based on the self-improvement properties of reverse H¨older inequalities. Lemma 10.7. Let E ⊂ Rn and assume that α ∈ A(E) for some 0 < α < n with a constant C1 . If 0 < β < α, then there exists a constant C = C(n, α, β, C1 ) such that  αβ  −α d (y, E) dy ≤C d (y, E)−β dy (10.3)  Q(x,r)

Q(x,r)

for every cube Q(x, r) ⊂ R . n

Proof. Let 0 < β < α and assume first that Q(x, 2r) ∩ E = ∅. Then d (y, E)β ≤ C(n, β)r β for every y ∈ Q(x, r). By (10.2), we obtain

α d (y, E)−α dy ≤ C(n)  d (y, E)−α dy ≤ C(n, α, C1 ) r −β β  Q(x,r)

Q(x,2r)



≤ C(n, α, β, C1 ) 

d (y, E)

−β

 αβ dy

,

Q(x,r)

and (10.3) follows. On the other hand, if Q(x, 2r) ∩ E = ∅ and y ∈ Q(x, r), then C(n)d (y, E) ≤ d (Q(x, r), E) ≤ d (y, E). This implies (10.3) in the case Q(x, 2r) ∩ E = ∅. The following result shows that the Aikawa condition is self-improving.



228

10. DISTANCE WEIGHTS AND HARDY–SOBOLEV INEQUALITIES

Theorem 10.8. Let E ⊂ Rn and 0 < α < n. Assume that α ∈ A(E) with a constant C1 . There exists α = α (n, α, C1 ), with α < α < n, such that α ∈ A(E) with a constant C = C(n, α, C1 ). −β Proof. Let 0 < β < α, p = α for every y ∈ Rn . β > 1 and w(y) = d (y, E) By Lemma 10.7, we have  p1  p w(y) dy ≤ C(n, α, β, C1 )  w(y) dy  Q(x,r)

Q(x,r)

for every cube Q(x, r) ⊂ Rn . Since reverse H¨ older inequalities are self-improving, see Theorem 8.38, there exists q = q(n, α, β, C1 ) > p such that  q1   p1  q p w(y) dy ≤C w(y) dy ≤ C  w(y) dy (10.4)  Q(x,r)

Q(x,r)

Q(x,r)



for every cube Q(x, r) ⊂ R with C = C(n, α, β, C1 ). Let α = qβ > pβ = α. Then  w(y)q = d (y, E)−α and w(y)p = d (y, E)−α . Let Q(x, r) ⊂ Rn be a cube with Q(x, r) ∩ E = ∅. By (10.4) and α ∈ A(E), we obtain   pq q  −α −α d (y, E) dy ≤ C d (y, E) dy ≤ C(r −α ) p = Cr −α ,   n

Q(x,r)

Q(x,r)

where C = C(n, α, β, C1 ). Thus α ∈ A(E), and by Remark 10.3 we have α < n. Finally, by choosing β = α2 , we conclude that the corresponding α and the constant  in the Aikawa condition only depend on n, α and C1 . Since always 0 ∈ A(E) and n ∈ / A(E), the observations in Remark 10.6 and Theorem 10.8 show that the set A(E) is either an interval of the form (−∞, α0 ), with 0 < α0 ≤ n, or A(E) = (−∞, 0]. 10.2. Ap properties of distance functions The distance weight satisfies a reverse H¨older inequality under an appropriate Aikawa condition, see Definition 10.1. Theorem 10.9. Let E ⊂ Rn and 0 ≤ α < n, and let w(y) = d (y, E)−α for every y ∈ Rn . If α ∈ A(E), then w ∈ A∞ . Proof. If α = 0, then w(y) = 1 for every y ∈ Rn , and thus w ∈ A∞ . We may hence assume that 0 < α < n and that (10.1) holds with a constant C1 for every x ∈ E and r > 0. This implies that w is locally integrable. Since α ∈ A(E) and α > 0, we have |E| = 0. Therefore w(x) > 0 for almost every x ∈ Rn , and thus w is a weight, see Definition 8.2. By Theorem 10.8 there exists α = α (n, α, C1 ), with α < α < n, such that  α ∈ A(E) with a constant C = C(n, α, C1 ). Let p = αα > 1. Lemma 10.7 implies  p1   α  α p −α w(y) dy = d (y, E) dy   Q(x,r)

Q(x,r)

≤C

Q(x,r)

d (y, E)−α dy = C 

w(y) dy

Q(x,r)

for every cube Q(x, r) ⊂ Rn , where C = C(n, α, C1 ). The claim w ∈ A∞ follows from Theorem 8.25. 

10.2. Ap PROPERTIES OF DISTANCE FUNCTIONS

229

Theorem 8.25 implies that, under the assumptions of Theorem 10.9, we have w ∈ Ap for some 1 < p < ∞. Next we strengthen this result by showing that w ∈ A1 . Recall that A1 ⊂ Ap ⊂ A∞ whenever 1 < p < ∞, see (8.19). We also extend the range of the Aikawa condition. Theorem 10.10. Let E ⊂ Rn and α ∈ R, and let w(y) = d (y, E)−α for every y ∈ Rn . Then the following assertions hold. (a) If 0 ≤ α ∈ A(E), then w ∈ Ap for every 1 ≤ p ≤ ∞. −α ∈ A(E), then w ∈ Ap . (b) If α < 0 and 1 < p < ∞ are such that p−1 Moreover, the Ap constant of w, in (a) and (b), only depends on n, p, α and the constant in the Aikawa condition. Proof. As in the proof of Theorem 10.9 we see that w is a weight in (a) and (b). Since A1 ⊂ Ap for every 1 ≤ p ≤ ∞ by (8.19), in assertion (a) it suffices to show that w ∈ A1 . This is clear if α = 0, and thus we may assume that 0 < α ∈ A(E). By Lemma 10.5 there is a constant C1 such that (10.2) holds for all cubes Q(x, r) intersecting E. Let Q(x, r) ⊂ Rn be a cube and assume first that Q(x, 2r) ∩ E = ∅. Then there is z ∈ E such that Q(x, r) ⊂ Q(z, 3r). By Lemma 10.5 we obtain 

Q(x,r)

d (y, E)−α dy ≤ C(n, α, C1 )r −α .

w(y) dy ≤ C(n) 

Q(z,3r)

On the other hand, since α > 0, we have 1 = d (y, E)α ≤ |y − z|α ≤ C(n, α)r α w(y) for every y ∈ Q(x, r) \ E. Since |E| = 0, by combining the estimates above we obtain 1 w(y) dy ess sup ≤ C(n, α, C1 ).  w(y) y∈Q(x,r) Q(x,r) This shows that the A1 condition (8.18) holds with a constant C(n, α, C1 ) for the cube Q(x, r) when Q(x, 2r) ∩ E = ∅. Assume then that Q(x, 2r) ∩ E = ∅. In this case C(n)d (y, E) ≤ d (Q(x, r), E) ≤ d (y, E) for every y ∈ Q(x, r), and thus

−α

α 1 ≤ C(n, α)d Q(x, r), E w(y) dy ess sup d Q(x, r), E ≤ C(n, α).  y∈Q(x,r) w(y) Q(x,r) Hence (8.18) holds and the proof of (a) is complete. In part (b), let 

σ(y) = w(y)1−p = w(y)− p−1 = d (y, E) p−1 1

α

p , and the claim w ∈ A for every y ∈ Rn . By part (a) we have σ ∈ A1 ⊂ A p−1 p follows from the duality property of Ap weights, see Theorem 8.13 (b). 

Our next goal is to obtain partial converses to assertions (a) and (b) in Theorem 10.10 for porous sets. Definition 10.11. A set E ⊂ Rn is porous if there exists a constant C such that for every x ∈ Rn and r > 0 there exists y ∈ Rn satisfying B(y, Cr) ⊂ B(x, r) \ E.

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10. DISTANCE WEIGHTS AND HARDY–SOBOLEV INEQUALITIES

Remark 10.12. Assume that E ⊂ Rn is porous. Then also E is porous. Let x ∈ E and r > 0, and let y ∈ Rn be such that B(y, Cr) ⊂ B(x, r) \ E. Then |E ∩ B(x, r)| |B(x, r) \ B(y, Cr)| r n − (Cr)n ≤ = = 1 − C n < 1. |B(x, r)| |B(x, r)| rn The Lebesgue density theorem, see Theorem 1.24, implies that |E| = 0, and hence also |E| = 0. It is easy to characterize porosity in terms of cubes instead of balls. Lemma 10.13. A set E ⊂ Rn is porous if and only if there exists a constant C such that for all cubes Q0 ⊂ Rn there is a cube Q ⊂ Q0 \ E with l(Q) = Cl(Q0 ) and d (Q, E) ≥ l(Q) 2 . Moreover, constant C and the constant C1 in the definition of porosity for E only depend on each other and n. Proof. Assume first that E ⊂ Rn is porous with constant C1 and let Q0 = R n such that B(y, C1 r) ⊂ Q(x, r) ⊂ Rn be a cube. There exists C ar point y ∈ C r B(x, r) \ E ⊂ Q0 \ E. Hence Q = Q y, 4√1 n ⊂ B y, 21 ⊂ Q0 \ E. It follows that l(Q) =

C √1 l(Q0 ) 4 n

and d (Q, E) ≥

C1 r C1 r l(Q) ≥ √ = , 2 2 4 n

as desired. n Conversely, assume that

the condition concerning cubes holds. Let x ∈ R , r > 0 and Q0 = Q x, 2√r n , and let Q = Q(y, r  ) ⊂ Q0 \ E be as in the assumption. Then B(y, r  ) ⊂ Q ⊂ Q0 \ E ⊂ B(x, r) \ E. Since r =

Cl(Q0 ) Cr l(Q) = = √ , 2 2 2 n 

we conclude that E is porous.

The following lemma gives one direction of the connection between porosity and the Aikawa condition; see also Theorem 10.25 below. Lemma 10.14. Assume that E ⊂ Rn satisfies the Aikawa condition for some α > 0 with a constant C1 . Then E is porous with a constant C = C(n, α, C1 ). Proof. Let Q0 ⊂ Rn be a cube. Let j be the smallest integer satisfying j >1+

α 1 log2 (n 2 C2 ), α

where C2 = C2 (n, α, C1 ) ≥ 1 is a constant for which (10.2) holds. Let Dj (Q0 ) = {Q1 , . . . , Q2jn } be the collection of pairwise disjoint half-open dyadic subcubes of Q0 , with side-length l(Qi ) = 2−j l(Q0 ) for each i = 1, . . . , 2jn . We claim that there is a cube Qi ∈ Dj (Q0 ) that does not intersect E. For a contradiction, assume that E intersects all cubes Qi ∈ Dj (Q0 ). For every i = 1, . . . , 2jn and every y ∈ Qi , √ (10.5) d (y, E) ≤ diam(Qi ) = nl(Qi ).

10.2. Ap PROPERTIES OF DISTANCE FUNCTIONS

231

By (10.5) and (10.2), jn

2

√ ( n)−α 2jα l(Q0 )n−α =

i=1 ≤



√

−α nl(Qi ) dy

Qi

d (y, E)−α dy ≤ C2 2α l(Q0 )n−α .

Q0

From this we obtain

α 1 log2 (n 2 C2 ), α which contradicts the original choice of j. Hence, there is Qi = Q(xi , 2−j−1 l(Q0 )) ∈ Dj (Q0 ) such that Qi ∩ E = ∅. Let Q = Q(xi , 2−j−2 l(Q0 )). Then Q ⊂ Qi ⊂ Q0 \ E and d (Q, E) ≥ l(Q) 2 . Thus Lemma 10.13 implies that E is porous, with a constant  C = C(n, α, C1 ).

j ≤1+

We have the following partial converse of Theorem 10.10 (a). This result also contains a strong self-improvement property for the distance weight under the assumption that the set is porous. Theorem 10.15. Assume that E ⊂ Rn is a nonempty porous set. Let α > 0 and let w(y) = d (y, E)−α for every y ∈ Rn . The following conditions are equivalent: (a) α ∈ A(E), (b) w ∈ A1 , (c) w ∈ A∞ . Moreover, the constants in each of the conditions only depend on each other, the porosity constant, n, w and α. Proof. By Theorem 10.10, condition (a) implies both conditions (b) and (c), and the implication from (b) to (c) follows from the inclusion A1 ⊂ A∞ , see (8.19). Thus it suffices to show that (c) implies (a). To this end, assume that w ∈ A∞ and let Q0 ⊂ Rn be a cube. Theorem 8.25 implies that w ∈ Ap for some 1 < p = p(n, w) < ∞, and so  p−1 1 1−p dy w(y) dy w(y) ≤ C(n, w).   Q0

Q0

Since E is porous, by Lemma 10.13 there is a cube Q ⊂ Q0 \ E satisfying l(Q) = C1 l(Q0 ) and d (Q, E) ≥ l(Q) 2 . Thus  p−1  p−1 1 1 1−p 1−p dy ≥ C(n, p, C1 )  w(y) dy  w(y) Q0

Q



≥ C(n, p, α, C1 )  l(Q)

α p−1

p−1 dy

Q α

≥ C(n, p, α, C1 )l(Q) ≥ C(n, α, p, C1)l(Q0 )α . By combining the two estimates above we obtain  1−p 1 −α 1−p dy ≤ Cl(Q0 )−α ,  d (y, E) dy =  w(y) dy ≤ C  w(y) Q0

Q0

Q0

where C = C(n, p, α, w, C1 ), and the claim α ∈ A(E) follows from Lemma 10.5. There is also a partial converse of assertion (b) in Theorem 10.10.



232

10. DISTANCE WEIGHTS AND HARDY–SOBOLEV INEQUALITIES

Theorem 10.16. Assume that E ⊂ Rn is a nonempty porous set. Let α < 0 and let w(y) = d (y, E)−α for every y ∈ Rn . If 1 < p < ∞ and w ∈ Ap , then −α p−1 ∈ A(E). Proof. Let 1 < p < ∞ and w ∈ Ap . The duality property of Ap weights, see Theorem 8.13 (b), implies −α



p ⊂ A∞ . d (·, E)−( p−1 ) = d (·, E) p−1 = w− p−1 = w1−p ∈ A p−1 α

1

Hence the claim follows from the implication from (c) to (a) in Theorem 10.15.  10.3. Assouad dimension Next we connect the Aikawa condition with geometric properties of the sets. The Assouad dimension below is a natural dual of the lower dimension that was discussed in Section 7.3. Recall from Definition 7.11 that for a bounded set A ⊂ Rn and r > 0, we let N (A, r) denote the minimal number of open balls of radius r that are needed to cover the set A. Definition 10.17. Let E ⊂ Rn . The Assouad dimension dimA (E) is the infimum of the numbers λ ≥ 0 for which there exists a constant C such that the  λ

R (10.6) N E ∩ B(x, R), r ≤ C r for every x ∈ E and 0 < r < R. In particular, the estimate in (10.6) holds whenever λ > dimA (E), and possibly also when λ = dimA (E). It is clear that dimA (E) ≤ dimA (E  ) for E ⊂ E  ⊂ Rn . Remark 10.18. Let E ⊂ Rn , x ∈ E and 0 < r < R. The collection {B(z, r5 ) : z ∈ E ∩ B(x, R)} is a cover of E ∩ B(x, R). By Lemma 1.13, there exist points xi ∈ E ∩ B(x, R), i = 1, . . . , N , such that the balls B(xi , r5 ) ⊂ B(x, 2R) are pairwise disjoint and  E ∩ B(x, R) ⊂ N i=1 B(xi , r). This implies N N 





B xi , r = C(n) B xi , r ≤ C(n)|B(x, 2R)| = C(n)Rn , N r n = C(n) 5 5 i=1

i=1

and thus

 n

R N E ∩ B(x, R), r ≤ N ≤ C(n) . r From this we conclude that 0 ≤ dimA (E) ≤ n for every E ⊂ Rn . The Hausdorff dimension always gives a lower bound for the Assouad dimension. Recall from Corollary 7.15 that for the lower dimension we have dimL (E) ≤ dimH (E) for all closed sets E ⊂ Rn . Lemma 10.19. Let E ⊂ Rn . Then dimH (E) ≤ dimA (E). Proof. The Hausdorff dimension is countably stable, see Falconer [125, p. 29], and thus dimH (E) = supk∈N dimH (E ∩ B(x, k)) for every x ∈ Rn . Hence it suffices to show that dimH (E ∩ B(x, R)) ≤ dimA (E) for every x ∈ E and R > 0. Let

10.3. ASSOUAD DIMENSION

233

s > dimA (E) and let dimA (E) < λ < s. Let C1 be the constant in (10.6) for λ, and let R > 0 and x ∈ E. Then E ∩ B(x, R) can be covered by  λ R N ≤ C1 r balls of radius r, for every 0 < r < R. By Definition 1.43, we have

Hrs E ∩ B(x, R) ≤ N r s ≤ C1 Rλ r s−λ .

Letting r → 0 shows that Hs E ∩ B(x, R) = 0. Since this holds for every s >  dimA (E), we conclude that dimH (E ∩ B(x, R)) ≤ dimA (E). In general, and even for closed sets, the inequality in Lemma 10.19 can be strict, as the following example illustrates. Example 10.20. Let E = { k1 : k ∈ N} ∪ {0} ⊂ R. The set E is countable, 1 and thus dimH (E) = 0. We claim that dimA (E) = 1. To see this, let R = K for 1 K ∈ N \ {1} and 0 < r = K 2 < R. The distance between two consecutive points k1 1 and k+1 in E ∩ B(0, R) is 1 1 1 1 − = < 2 = r. k k+1 k(k + 1) K If intervals Bi = B(xi , r) = (xi −r, xi +r), i = 1, 2, . . . , N , cover the set E ∩B(0, R), then all subintervals of [0, R] that are not covered by the intervals Bi have length less than r. On the other hand, we may clearly assume that xi ∈ [0, R] for every i = 1, 2, . . . , N , and thus each of the intervals Bi covers a subinterval of [0, R] of length at least r. On both sides of a subinterval of [0, R] that is not covered by the intervals Bi , there are intervals Bi1 and Bi2 for some i1 , i2 ∈ {1, 2, . . . N }. It follows that the intervals Bi cover at least half of the interval [0, R]. Hence their total length is at least R2 , and thus N 2r ≥ R2 . From this we conclude that there exists a constant C > 0, independent of r and R, such that  

R N E ∩ B(0, R), r ≥ C . r Since Rr = K, letting K → ∞ shows that that dimA (E) ≥ 1. On the other hand, dimA (E) ≤ 1 by Remark 10.18, and thus dimA (E) = 1. By considering the natural embedding of R to Rn , this example can be easily lifted to any Rn . If E ⊂ Rn is Ahlfors–David λ-regular, see Section 7.2, then dimH (E) = dimL (E) = λ, by Theorem 7.12. Next we show that for a λ-regular set E ⊂ Rn also the Assouad dimension is equal to λ, and thus dimL (E) = dimA (E) = dimH (E) = λ. Theorem 10.21. Let 0 < λ ≤ n and assume that a closed set E ⊂ Rn is Ahlfors–David λ-regular. Then dimA (E) = λ. Proof. By Lemma 10.19 we have λ = dimH (E) ≤ dimA (E). Hence it suffices to show that dimA (E) ≤ λ. This can be shown essentially in the same way as the bound dimL (E) ≤ λ in Theorem 7.12.

234

10. DISTANCE WEIGHTS AND HARDY–SOBOLEV INEQUALITIES

Let x ∈ E and 0 < r < R. Assume first that 2R ≤ diam(E). By Lemma 1.13 and boundedness, there exist N ∈ N and points xi ∈ E ∩ B(x, R), i = 1, 2, . . . , N ,  r such that E ∩ B(x, R) ⊂ N i=1 B(xi , r) and the balls B(xi , 5 ) are pairwise disjoint. Hence, by the Ahlfors–David regularity and the additivity and monotonicity of the Hausdorff measure, N rλ =

N

i=1

rλ ≤ C

N





Hλ E ∩ B(xi , 6r ) ≤ CHλ E ∩ B(x, 2R) ≤ CRλ .

i=1

Thus (10.7)







R N E ∩ B(x, R), r ≤ N ≤ C r

λ .

On the other hand, if 2R > diam(E) ≥ r, then a similar argument as above, for E = E ∩ B(x, diam(E)), shows that   λ λ



diam(E) R N E ∩ B(x, R), r ≤ N E ∩ B(x, diam(E)), r ≤ C ≤C . r r

Finally, if r > diam(E), then N E ∩ B(x, R), r = 1. Hence (10.7) holds in all  cases, and the claim dimA (E) ≤ λ follows. Our next results are concerned with the close connection between the Assouad dimension and the Aikawa condition, see Definition 10.1. Theorem 10.22. Let E ⊂ Rn and assume that α ∈ R is such that α ∈ A(E). Then dimA (E) ≤ n − α. Proof. For α ≤ 0, the claim is clear by Remark 10.18. Hence we may assume that α > 0, and by Remark 10.3 we have α < n. Let x ∈ E and 0 < r < R, and write F = E ∩ B(x, R). By Lemma 1.13 and the boundedness of F , there exist N ∈ N and a collection {B(xi , r5 ) : i = 1, . . . , N } of pairwise disjoint open balls,  with xi ∈ F , such that F ⊂ N i=1 B(xi , r). Let Fr = {y ∈ Rn : d (y, F ) < r} ⊂ B(x, 2R) and let C1 be the constant in the Aikawa condition (10.1) for α. Using pairwise disjointness of the balls B(xi , r5 ) ⊂ Fr and the fact that d (y, E) ≤ d (y, F ) < r for every y ∈ Fr , we obtain  N

r B xi , ≤ |Fr | ≤ r α N C(n)r n ≤ d(y, E)−α dy 5 Fr

i=1



d(y, E)−α dy ≤ r α |B(x, 2R)|C1 R−α

≤ rα B(x,2R)

= C(n, C1 )r

n



R r

n−α .

Thus

 N (E ∩ B(x, R), r) = N (F, r) ≤ N ≤ C(n, C1 )

and the claim dimA (E) ≤ n − α follows since n − α > 0.

R r

n−α , 

Notice that if α > 0, then by the self-improvement of the Aikawa condition, see Theorem 10.8, we obtain a strict inequality in Theorem 10.22. See also the proof of

10.3. ASSOUAD DIMENSION

235

Theorem 10.24 below. For the converse direction we assume a strict upper bound for the dimension. Theorem 10.23. Let E ⊂ Rn be a nonempty set and assume that α ∈ R with dimA (E) < n − α. Then α ∈ A(E). Proof. The claim is clear if α ≤ 0, and so we may assume that α > 0. Let λ > 0 with dimA (E) < λ < n − α, and let x ∈ E and r > 0. Define Fj = {y ∈ B(x, r) : d(y, E) < 2−j+1 r}

and

Aj = Fj \ Fj+1 ,

for j ∈ N. Since λ > dimA (E), there is a constant C1 such that the set E ∩ B(x, 2r) can be covered by Nj ≤ C1 2jλ balls of radius 21−j r, for every j ∈ N. It follows that each Fj can be covered by at most Nj balls of radius 22−j r. If Bij , i = 1, . . . , Nj , are such balls, then (10.8)

|Fj | ≤

Nj

|Bij | ≤ Nj C(n)(22−j r)n ≤ C(n, C1 )(2−j )n−λ r n .

i=1

Since E ∩ B(x, r) ⊂ Fj for every j ∈ N and λ < n − α < n, by letting j → ∞ we see in particular that |E ∩ B(x, r)| = 0. Here r > 0 is arbitrary, and thus |E| = 0. If y ∈ Aj , then 2−j r ≤ d(y, E) < 2−j+1 r. In addition, Aj ⊂ Fj for every j ∈ N and the sets Aj cover B(x, r) up to the set E ∩ B(x, r), which has measure zero. By (10.8) we obtain ∞ 

−α −n d(y, E) dy ≤ C(n)r d(y, E)−α dy  B(x,r)

≤ C(n)r −n

j=1 ∞

Aj

|Fj |(2−j r)−α

j=1

≤ C(n, C1 )r −α

∞

(2−j )n−λ−α

j=1

≤ C(n, α, λ, C1 )r −α , where the geometric series converges since λ < n − α. This shows that α ∈ A(E).  Combining the two results above and the self-improvement of the Aikawa condition, see Theorem 10.8, we obtain a characterization connecting the Assouad dimension and the Aikawa condition. Theorem 10.24. Let E ⊂ Rn be a nonempty set and let α > 0. Then α ∈ A(E) if and only if dimA (E) < n − α. Proof. If dimA (E) < n − α, then α ∈ A(E) by Theorem 10.23. Conversely, assume that 0 < α ∈ A(E). By Remark 10.3 we have α < n, and by Theorem 10.8 there is α > α with α ∈ A(E). Thus Theorem 10.22 implies dimA (E) ≤ n − α < n − α, as desired.  Notice that the assumption α > 0 in Theorem 10.24 is essential: if E ⊂ Rn and dimA (E) = n, then 0 ∈ A(E), but dimA (E) ≮ n − 0. For instance, this holds if E = B(0, 1), see Lemma 10.19 and Remark 10.18.

236

10. DISTANCE WEIGHTS AND HARDY–SOBOLEV INEQUALITIES

Recall from Theorem 7.22 that if E ⊂ Rn is an unbounded closed set and 1 < p ≤ n, then E satisfies the p-capacity density condition in Definition 6.17 if and only if dimL (E) > n−p. Similarly, for α > 0, Theorem 10.24 shows that a nonempty set E ⊂ Rn satisfies the Aikawa condition for α if and only if dimA (E) < n − α. In the limiting case p = n, the capacity density condition and the positivity of the lower dimension have a simple geometric characterization through uniform perfectness, see Theorem 7.20. In a corresponding manner, porosity of E ⊂ Rn gives a geometric characterization for the properties that the Aikawa condition holds for some α > 0 and that the Assouad dimension is strictly less than the dimension of the space. Theorem 10.25. Let E ⊂ Rn be a nonempty set. The following conditions are equivalent: (a) dimA (E) < n, (b) there is α > 0 such that α ∈ A(E), (c) E is porous. Moreover, the constants and parameters in each of the conditions only depend on each other and n, and if any of these conditions holds, then |E| = 0. Proof. The implication from (a) to (b) follows from Theorem 10.23 by choosing 0 < α < n − dimA (E), and the implication from (b) is (c) is established in Lemma 10.14. Hence it suffices to show that (c) implies (a). To this end, let x ∈ E and R > 0, and write Q0 = Q(x, R). As before, let Dj (Q0 ) be the collection of pairwise disjoint half-open dyadic subcubes of Q0 , with side length 2−j l(Q0 ) = 2−j 2R. Since E is porous, we obtain from Lemma 10.13 that there is j0 ∈ N, independent of Q0 , and a cube Q ∈ Dj0 (Q0 ) such that E ∩ Q = ∅. Hence at most 2nj0 − 1 closed cubes of side length 2−j0 2R are needed to cover E ∩ Q0 . By a simple iteration, we conclude that if k ∈ N, then at most (2nj0 − 1)k closed cubes of side length 2−kj0 2R are needed to cover E ∩ Q0 . Let 0 < r < R and choose k ∈ N satisfying √ √ R < ( n)−1 2kj0 . ( n)−1 2(k−1)j0 ≤ r √ −kj0 Since B(x, R) ⊂ Q0 and r > n2 R, the set E ∩ B(x, R) can be covered by at most (2nj0 − 1)k open balls of radius r; indeed, if Q = Q(z, 2−kj0 R) ∈ Dkj0 (Q0 ), then Q ⊂ B(z, r). Since log2 (2nj0 − 1) < log2 (2nj0 ) = nj0 , we may choose λ > 0 satisfying 1 log2 (2nj0 − 1) < λ < n. j0 Then 2nj0 − 1 < 2λj0 , and we obtain from the considerations above and the choice of k that  λ

k

R . N E ∩ B(x, R), r ≤ (2nj0 − 1)k < 2λj0 ≤ C(n, j0 , λ) r Hence dimA (E) ≤ λ < n, as claimed. Finally, all conditions in the theorem imply that E is porous, and hence |E| = 0 by Remark 10.12. 

10.3. ASSOUAD DIMENSION

237

We restate results concerning the Muckenhoupt properties of the distance function in terms of the Assouad dimension. This leads to the following characterizations for a porous set. Theorem 10.26. Let 1 < p < ∞ and assume that E ⊂ Rn is a nonempty porous set. Let α ∈ R and define w(y) = d (y, E)−α for every y ∈ Rn . Then the following assertions hold. (a) w ∈ A1 if and only if 0 ≤ α < n − dimA (E). (b) w ∈ Ap if and only if

(10.9) (1 − p) n − dimA (E) < α < n − dimA (E). Proof. Since E is porous, dimA (E) < n by Theorem 10.25. We consider first part (b). If 0 ≤ α < n − dimA (E), we have α ∈ A(E) by Theorem 10.23 and thus Theorem 10.10 (a) implies w ∈ Ap . On the other hand, if

(1 − p) n − dimA (E) < α < 0, then 0
0, Theorem 10.15 implies α ∈ A(E), and so α < n − dimA (E) by Theorem 10.24. Note that porosity is needed in Theorem 10.15. If α = 0, then (10.9) holds since dimA (E) < n by −α porosity. Finally, if α < 0, then p−1 ∈ A(E) by Theorem 10.16. Theorem 10.24 gives −α < n − dimA (E), 0< p−1

showing that (10.9) also holds in this case. The proof of (b) is complete. Consider then part (a). If 0 ≤ α < n − dimA (E), the claim w ∈ A1 follows from Theorem 10.23 and Theorem 10.10 (a) just as in case (b). Conversely, if w ∈ A1 and α > 0, then α < n − dimA (E) by Theorem 10.15 and Theorem 10.24. If α = 0 then 0 ≤ α < n − dimA (E) holds since dimA (E) < n by porosity. It remains to show that the condition α ≥ 0 is necessary in (a). To this end, assume that w ∈ A1 and let Q = Q(x, r) be a cube with x ∈ E. Since w(y) > 0 for almost every y ∈ Q, we have Q w(y) dy > 0. If α < 0, then ess sup y∈Q

1 = ess sup d (y, E)α = ∞, w(y) y∈Q

and this clearly violates the A1 condition (8.18). We conclude that w ∈ A1 is possible only when α ≥ 0.  Example 10.27. Consider the porous set E = {0} ⊂ Rn , for which dimA (E) = 0, and let w(x) = d (x, E)−α = |x|−α for every x ∈ Rn \ {0}. Theorem 10.26 implies that w ∈ A1 if and only if 0 ≤ α < n, and w ∈ Ap , for 1 < p < ∞, if and only if (1 − p)n < α < n. Compare to Remark 8.35.

238

10. DISTANCE WEIGHTS AND HARDY–SOBOLEV INEQUALITIES

10.4. Distance weighted Poincar´ e inequalities This section combines the abstract weighted Poincar´e inequalities from Chapter 9 with the Muckenhoupt properties of the distance functions. The following result follows by specializing Theorem 9.11 to a power of the distance function. Theorem 10.28. Let E ⊂ Rn be a nonempty set and assume that 1 < p < n − dimA (E). Let Q ⊂ Rn be a cube and assume that u ∈ Lip(Q). There exists a constant C = C(n, p, E) such that  p1   p1  p −p p −p |u(x) − uQ | d (x, E) dx ≤ Cl(Q) |∇u(x)| d (x, E) dx . (10.10) Q

Q

Proof. Let w(x) = d (x, E)−p for every x ∈ Rn . We have p ∈ A(E) by Theorem 10.23 and thus Theorem 10.10 (a) implies w ∈ Ap . The claim follows from Theorem 9.11.  By applying Theorem 9.21 with suitable weights instead of Theorem 9.11, we obtain an improvement of Theorem 10.28. Note that for q = p the left-hand side of (10.11) is the same as in (10.10), while the right-hand side of (10.11) is significantly smaller on cubes Q which intersect E or are close to E. np and assume that E ⊂ Rn is a Theorem 10.29. Let 1 < p < n and p ≤ q ≤ n−p q n nonempty set with dimA (E) < p (n − p). Let Q ⊂ R be a cube and let u ∈ Lip(Q). There exists a constant C = C(n, p, q, E) such that  1q   p1  q |u(x) − uQ |q d (x, E) p (n−p)−n dx ≤C |∇u(x)|p dx . (10.11) Q

Q

Proof. Let q

w(x) = d (x, E) p (n−p)−n

and

v(x) = 1 = σ(x)

for every x ∈ R . For α = n − − p) ≥ 0 we have dimA (E) < n − α. By Theorem 10.23, Theorem 10.10 (a) and (8.19), we conclude w ∈ A1 ⊂ A∞ . Let Q0 ⊂ Rn be a cube. It suffices to show that there is a constant C1 = C1 (n, p, q, E) > 0 such that  p p 1 (10.12) w(Q) q σ(Q)p−1 ≤ C1 1 1− n |Q| n

q p (n

holds for all dyadic cubes Q ∈ D(Q0 ) since then inequality (10.11) follows from Theorem 9.21. To this end, let Q = Q(x, r) ∈ D(Q0 ) be a dyadic cube. Assume first that Q(x, 2r) ∩ E = ∅. Then there exists z ∈ E such that Q ⊂ Q(z, 3r). Since dimA (E) < pq (n − p), Theorem 10.23 implies that E satisfies the Aikawa condition (10.2) for α = n − pq (n − p), with a constant C = C(n, p, q, E). Thus   pq

p q p q p (n−p)−n q p d (y, E) dy ≤ C r p (n−p) q = Cr n−p = C|Q|1− n , w(Q) ≤ Q(z,3r)

where C = C(n, p, q, E). Since σ(Q)p−1 = |Q|p−1 , we obtain p  p p p 1 q σ(Q)p−1 ≤ C|Q|−p+ n +1− n +p−1 = C, w(Q) 1 |Q|1− n with C = C(n, p, q, E). This proves (10.12) in the case Q(x, 2r) ∩ E = ∅.

´ INEQUALITIES 10.4. DISTANCE WEIGHTED POINCARE

239

Assume then that Q(x, 2r) ∩ E = ∅. In this case C(n)d (y, E) ≤ d (Q, E) ≤ d (y, E) for every y ∈ Q = Q(x, r). Hence p

p

w(Q) q σ(Q)p−1 ≤ C(n, p, q)|Q| q d (Q, E)n−p−

np q

|Q|p−1

p

≤ C(n, p, q)|Q| q +p−1 d (Q, E)n−p− By assumption n − p − 

p

1 1 |Q|1− n

np q

np q

.

≤ 0, and thus p

p

p

p

p

w(Q) q σ(Q)p−1 ≤ C|Q|−p+ n + q +p−1 d (Q, E)n−p− p

np q

p

≤ C|Q|−p+ n + q +p−1+1− n − q = C, where C = C(n, p, q). This shows that (10.12) also holds when Q(x, 2r) ∩ E = ∅, and the proof is complete.  The inequality in (10.11) is a Hardy–Sobolev–Poincar´e inequality, compare to Definition 10.32 below. By applying the weighted local-to-global inequality, see Theorem 9.28, we obtain a variant of the Hardy–Sobolev–Poincar´e inequality on open sets satisfying the Boman chain condition. np Theorem 10.30. Let 1 < p < n and p ≤ q ≤ n−p . Assume that Ω ⊂ Rn is an open set satisfying the Boman chain condition, with τ > 1, and that dimA (∂Ω) < q p (n − p). There exists a constant C = C(n, p, q, τ ) such that



q

|u(x) − c|q d(x, ∂Ω) p (n−p)−n dx

inf

c∈R

 q1

 ≤C

|∇u(x)|p dx

Ω

 p1

Ω

for every u ∈ Liploc (Ω). Proof. Let α = n − pq (n − p) ≥ 0 and let w(x) = d(x, ∂Ω)−α for every x ∈ Rn . Since dimA (∂Ω) < n − α, we have w ∈ A1 ⊂ Ap by Theorems 10.23 and 10.10 (a). Theorem 8.15 implies that w is a doubling weight. Using Theorem 9.28 and (9.20) we obtain 

 q |u(x) − c| w(x) dx ≤ C |u(x) − uQ∗ ;w |q w(x) dx inf c∈R

Ω

Q∈W(Ω)

≤C

(10.13)



|u(x) − uQ∗ |q w(x) dx Q∗

Q∈W(Ω)

≤C

Q∗

l(Q)−α

 |u(x) − uQ∗ |q dx, Q∗

Q∈W(Ω)

where C = C(n, p, q, τ ). By Corollary 9.24 we have 

∗ α

|u(x) − uQ∗ | dx ≤ C(n, p, q)l(Q ) q

Q∗

 |∇u(x)| dx p

Q∗

 pq

240

10. DISTANCE WEIGHTS AND HARDY–SOBOLEV INEQUALITIES

for every Q ∈ W(Ω); observe here that q + n − nq p = α. Since q ≥ p, substitution of these estimates for Whitney cubes to (10.13) gives  pq 

 q p |u(x) − c| w(x) dx ≤ C |∇u(x)| dx inf c∈R

Ω

Q∈W(Ω)

 ≤C

Q∈W(Ω)

Q∗

 pq

 Q∗

|∇u(x)|p dx

,

with C = C(n, p, q, τ ). The desired estimate follows since the dilated Whitney  cubes Q∗ ⊂ Ω have a bounded overlap. Remark 10.31. Let Ω ⊂ Rn be a nonempty open set satisfying the assumptions of Theorem 10.30. Then Ω is necessarily bounded, and thus q n − 1 ≤ dimH (∂Ω) ≤ dimA (∂Ω) < (n − p) p by Lemma 10.19. This poses additional restrictions for the use of Theorem 10.30. However, for instance the domain inside the von Koch snowflake curve of dimension n − 1 < λ < n satisfies the Boman chain condition, and hence Theorem 10.30 can be applied for such open sets. 10.5. Hardy–Sobolev inequalities Hardy–Sobolev inequalities generalize both Hardy and Sobolev inequalities. Definition 10.32. Let 1 ≤ p < ∞ and 1 ≤ q < ∞, and let Ω  Rn be an open set. We say that the (q, p)-Hardy–Sobolev inequality holds in Ω, with a constant C, if   1q   p1 q (10.14) |u(x)|q d (x, ∂Ω) p (n−p)−n dx ≤C |∇u(x)|p dx Ω

Ω

for every u ∈ Lip0 (Ω). Remark 10.33. Let 1 ≤ p < ∞ and 1 ≤ q < ∞, and let Ω  Rn be an open set. Approximation and a straightforward application of Fatou’s lemma, as in the proof of Theorem 3.6, show that if (10.14) holds for every u ∈ Lip0 (Ω), then (10.14) holds for every u ∈ W01,p (Ω). np For q = p∗ = n−p inequality (10.14) reduces to the Sobolev inequality (3.6), and for q = p it reduces to the p-Hardy inequality (6.1). If |Ωc | = 0, then integrals in (10.14) can be taken over the whole Rn . We first examine the validity of such inequalities for functions u ∈ Lipc (Rn ), that is, we do not require u to be zero on Ωc . In this case (10.14) is called a global Hardy–Sobolev inequality. We have the following characterization. np Theorem 10.34. Let 1 < p < n and p ≤ q < n−p and assume that E ⊂ Rn is a nonempty closed set. The following conditions are equivalent.

(a) dimA (E) < pq (n − p).

10.5. HARDY–SOBOLEV INEQUALITIES

241

(b) There exists a constant C = C(n, p, q, E) such that  q1   p1  q (n−p)−n q p |u(x)| d (x, E) p dx ≤C |∇u(x)| dx , (10.15) Rn

Rn

for every u ∈ Lipc (R ). n

Proof. Assume first that (a) holds. Let w, v and σ be as in the proof of Theorem 10.29. Then the same proof shows that the assumptions of Corollary 9.25 are valid, and it follows that (10.15) holds for every u ∈ Lipc (Rn ). Assume then that (10.15) holds for every u ∈ Lipc (Rn ) with a constant C1 . Let x ∈ E, r > 0, and   u(y) = max 0, 1 − 1r d (y, B(x, r)) for every y ∈ Rn . Then u ∈ Lipc (Rn ), u(y) = 1 for every y ∈ B(x, r), u is supported in B(x, 2r) and |∇u(y)| ≤ 1r for almost every y ∈ B(x, 2r). Using (10.15) for u, we obtain   q q (n−p)−n p d (y, E) dy ≤ |u(y)|q d (y, E) p (n−p)−n dy Rn

B(x,r)



≤

C1q

|∇u(y)| dy p

Rn

 pq

q q ≤ Cr −q B(x, 2r) p = Cr p (n−p) , where C = C(n, p, q, C1 ). This shows that the Aikawa condition (10.1) holds for α = n − pq (n − p) > 0. By Theorem 10.24 we have dimA (E) < n − α = pq (n − p).  np Notice that the strict inequality q < n−p is essential in the necessity part of np Theorem 10.34 since the case q = n−p reduces to the Sobolev–Gagliardo–Nirenberg inequality, see Theorem 3.6, and the role of the set E becomes redundant. In particular, the dimensional restriction dimA (E) < n does not follow from the Sobolev inequality. np and assume that Ω  Rn is Remark 10.35. Let 1 < p < n and p ≤ q < n−p q c an open set with dimA (Ω ) < p (n − p). Then the (q, p)-Hardy–Sobolev inequality holds in Ω by Theorem 10.34.

The following special case q = p can be regarded as a dual pair of Theorem 7.17. Corollary 10.36. Let 1 < p < n and assume that Ω  Rn is an open set with dimA (Ωc ) < n − p. Then the p-Hardy inequality (6.1) holds in Ω. Moreover, the p-Hardy inequality holds for every u ∈ Lip0 (Rn ), not only for u ∈ Lip0 (Ω). Proof. The claim follows from Theorem 10.34 for q = p and E = Ωc . Theorem 10.25 implies that |Ωc | = 0 and thus the integrals in (10.15) can be taken over  Ω for u ∈ Lip0 (Rn ). np Hardy–Sobolev inequalities, for p ≤ q ≤ p∗ = n−p , can be obtained by interpolating between the Hardy and Sobolev inequalities. See also Remark 10.53.

Theorem 10.37. Let 1 ≤ p < n and assume that the p-Hardy inequality holds in an open set Ω ⊂ Rn . Then the (q, p)-Hardy–Sobolev inequality holds in Ω for np every p ≤ q ≤ n−p .

242

10. DISTANCE WEIGHTS AND HARDY–SOBOLEV INEQUALITIES

np Proof. For q = p we have the p-Hardy inequality and for q = n−p we have the Sobolev inequality (3.6), and both of these hold in Ω. Hence we may assume np = p∗ . Let that p < q < n−p

α=

p2 >1 np − nq + qp

and

α =

α , α−1

and let u ∈ Lip0 (Ω). By H¨ older’s inequality, we have 

q

|u(x)|q d (x, ∂Ω) p (n−p)−n dx Ω

 ≤

|u(x)| d (x, ∂Ω) p

−p

 q1



p

p∗

|u(x)| α + α d (x, ∂Ω)

= 1   qα

Ω

|u(x)|

dx

Ω

p∗

 dx

−p α

 1q dx

1 qα

.

Ω

The first integral on the right-hand side can be estimated with the p-Hardy in1 n 1 1 equality and the second with the Sobolev inequality. Since qα + n−p qα = p , we conclude that  |u(x)| d (x, ∂Ω) q

Ω

q p (n−p)−n

 ≤C

|∇u(x)|p dx Ω



|∇u(x)|p dx

=C

 1q dx

1   qα

|∇u(x)|p dx

n  n−p

1 qα

Ω

 p1 .



Ω np Remark 10.38. Let 1 < p ≤ q < n−p < ∞. If Ω  Rn is an open set q c and dimA (Ω ) < p (n − p), then the (q, p)-Hardy–Sobolev inequality holds in Ω by Remark 10.35. In this case Theorem 10.37 does not provide new information since for the p-Hardy inequality we need the assumption dimA (Ωc ) < n − p, which is stronger than the assumption for the (q, p)-Hardy–Sobolev inequality. On the other hand, we know from Theorem 7.17 that if dimL (Ωc ) > n − p and Ωc is unbounded, then the p-Hardy inequality holds in Ω. By Theorem 10.37 also the (q, p)-Hardy–Sobolev inequality holds in Ω.

The punctured ball Ω = B(0, 1) \ {0}, considered in Example 6.13, gives a simple model case where a suitable combination of thick and thin parts in the complement Ωc implies that a p-Hardy inequality holds in Ω. Theorem 10.40 below is a generalization of this idea. In the proof we apply the following lemma concerning Hardy–Sobolev inequalities in intersections of open sets. np , and let Ω1 , Ω2 ⊂ Rn be open Lemma 10.39. Let 1 < p < n and p ≤ q ≤ n−p sets with Ω = Ω1 ∩Ω2 = ∅. Assume that the (q, p)-Hardy–Sobolev inequality holds in Ω1 and Ω2 , with constants C1 and C2 , respectively. Then the (q, p)-Hardy–Sobolev inequality holds in the open set Ω with a constant C = C(C1 , C2 ).

Proof. Let u ∈ Lip0 (Ω). Since ∂Ω ⊂ ∂Ω1 ∪ ∂Ω2 and Ω ⊂ Ω1 and Ω ⊂ Ω2 , we have d (x, ∂Ω) = min{d (x, ∂Ω1 ), d (x, ∂Ω2 )} for every x ∈ Ω. This and the assumed

10.6. NECESSARY CONDITIONS FOR HARDY–SOBOLEV INEQUALITIES

243

Hardy–Sobolev inequalities for Ω1 and Ω2 imply  q |u(x)|q d (x, ∂Ω) p (n−p)−n dx Ω   q q ≤ |u(x)|q d (x, ∂Ω1 ) p (n−p)−n dx + |u(x)|q d (x, ∂Ω2 ) p (n−p)−n dx Ω Ω q q (n−p)−n q ≤ |u(x)| d (x, ∂Ω1 ) p dx + |u(x)|q d (x, ∂Ω2 ) p (n−p)−n dx Ω1



≤

C1q

|∇u(x)| dx p

Ω1

≤ C1q + C2q

 pq

Ω2

 +

C2q

 |∇u(x)| dx p

|∇u(x)| dx p

 pq

Ω2

 pq ,

Ω

where the final inequality holds since |∇u| = 0 almost everywhere in Ωc . The claim

1  follows, with C = C1q + C2q q ≤ C1 + C2 . np Theorem 10.40. Let 1 < p < n and p ≤ q ≤ n−p . Assume that an open set n n c Ω1 ⊂ R and a closed set F ⊂ R are such that Ω1 is unbounded, q dimA (F ) < (n − p). dimL (Ωc1 ) > n − p and p

Then the (q, p)-Hardy–Sobolev inequality holds in the open set Ω = Ω1 \ F . Proof. Notice that Ω = ∅ since dimA (F ) < n by the assumptions. Let Ω2 = F c . Then dimA (Ωc2 ) = dimA (F ) < pq (n − p), and Theorem 10.34 implies that the (q, p)-Hardy–Sobolev inequality holds in the open set Ω2 . On the other hand, since dimL (Ωc1 ) > n − p and Ωc1 is unbounded, the p-Hardy inequality holds in Ω1 by Theorem 7.17. Theorem 10.37 implies that the (q, p)-Hardy–Sobolev inequality holds in Ω1 , and the (q, p)-Hardy–Sobolev inequality for Ω = Ω1 ∩ Ω2 follows from Lemma 10.39.  See Example 10.49 for an application of Theorem 10.40. 10.6. Necessary conditions for Hardy–Sobolev inequalities Let 1 < p < n and p ≤ q < (10.16)

np n−p .

dimL (Ωc ) > n − p

By Remark 10.38, the requirement or

dimA (Ωc )
0. By the definition of the Hausdorff content, see Definition 1.43, and the compactness of Ωc ∩ U , there are balls B(zi , ri ), i = 1, . . . , N ,  such that Ωc ∩ U ⊂ N i=1 B(zi , ri ) and N

−j rin−p ≤ u−p ∞ 2 .

i=1

Let

  ψj (x) = min min 1, ri−1 d(x, B(zi , 2ri )) , 1≤i≤N

for every x ∈ Ω. Then |∇ψj (x)|p ≤

N

ri−p χB(zi ,3ri ) (x)

i=1

for almost every x ∈ Ω. Let uj (x) = ψj (x)u(x) for x ∈ Ω. Then uj ∈ Lip0 (Ω) and, by the Leibniz rule (2.4), we have |∇uj (x)| ≤ |∇ψj (x)||u(x)| + |∇u(x)| for almost every x ∈ Ω. Using the (q, p)-Hardy–Sobolev inequality for uj , we obtain  pq  q (n−p)−n q p |uj (x)| d (x, ∂Ω) dx Ω     p p p p p ≤ 2 C1 u∞ |∇ψj (x)| dx + |∇u(x)| dx Ω

Ω

    N

p p −j n−p ≤ C u∞ ri + |∇u(x)| dx ≤ C2 + C |∇u(x)|p dx, i=1

Ω

Ω

where C = C(n, p, C1 ). The claim follows by Fatou’s Lemma since uj (x) → u(x) for almost every x ∈ Ω as j → ∞.  From Lemma 10.41 and Theorem 10.8, we obtain the following result. np and assume that the (q, p)Lemma 10.42. Let 1 < p < n and p ≤ q < n−p C1 . Assume Hardy–Sobolev inequality holds in an open set Ω  Rn with a constant

n−p Ωc ∩ B(x, 2R) = 0. There exists that B(x, R) ⊂ Rn is an open ball such that H∞ δ = δ(n, p, q, C1 ) > 0 such that

q dimA Ωc ∩ B(x, R) ≤ (n − p) − δ. p

10.6. NECESSARY CONDITIONS FOR HARDY–SOBOLEV INEQUALITIES

245

Proof. Let z ∈ Ωc ∩ B(x, R) and 0 < r < R2 . Then B(z, 2r) ⊂ B(x, 2R). Let   u(y) = max 0, 1 − 1r d (y, B(z, r))

for every y ∈ Rn . Then u ∈ Lip0 Ω ∪ B(x, 2R) , u(y) = 1 for every y ∈ B(z, r), and |∇u(y)| ≤ 1r χB(z,2r) (y) for almost every y ∈ Rn . By Lemma 10.41, the (q, p)Hardy–Sobolev inequality holds

in Ω for u with a constant C = C(n, p, C1 ). Since d (y, Ωc ) ≤ d y, Ωc ∩ B(x, R) for every y ∈ B(z, 2r) and |Ωc ∩ B(x, 2R)| = 0, we obtain  

pq (n−p)−n q c d y, Ω ∩ B(x, R) dy ≤ |u(y)|q d (y, Ωc ) p (n−p)−n dy B(z,r)

Ω∩B(z,2r)



≤C

|∇u(y)| dy p

 pq

Ω

q q ≤ Cr −q B(z, 2r) p = Cr p (n−p) , where C = C(n, p, q, C1 ). This shows that the set Ωc ∩ B(x, R) satisfies the Aikawa condition (10.1) for α = n − pq (n − p) > 0 whenever 0 < r < R2 . Lemma 10.4 implies that α ∈ A(Ωc ∩ B(x, R)), with a constant C = C(n, p, q, C1 ). By Theorem 10.8, there is δ = δ(n, p, q, C1 ) > 0 such that α + δ ∈ A(Ωc ∩ B(x, R)). The claim dimA (Ωc ∩ B(x, R)) ≤ pq (n − p) − δ follows from Theorem 10.22.  Remark 10.43. In the case q = p we have the following uniformity concerning δ > 0 in Lemma 10.42. Let 1 < p1 < p2 < ∞ and assume that the p-Hardy inequality holds cin Ω for every

p1 ≤ p ≤ p2 with the same constant C1 . If p1 ≤ p ≤ n−p Ω ∩ B(x, 2R) = 0, then we can choose δ > 0 in Lemma 10.42 to be p2 and H∞ independent of p, that is, δ = δ(n, p1 , p2 , C1 ) > 0. This can be seen by tracing the dependence of δ on p explicitly via Theorem 10.8 and Theorem 8.38 to Lemma 8.37. We have now established enough tools to prove the necessity of a local dimensional dichotomy. np and assume that the (q, p)Theorem 10.44. Let 1 < p < n and p ≤ q < n−p Hardy–Sobolev inequality holds in an open set Ω  Rn . If x ∈ Rn and R > 0, then

q dimA Ωc ∩ B(x, R) < (n − p). dimH Ωc ∩ B(x, 2R) ≥ n − p or p

Proof. Let x ∈ Rn and R > 0. The claim holds if dimH Ωc ∩ B(x, 2R) ≥ n − p. Hence we may assume that dimH (Ωc ∩ B(x, 2R)) < n − p, and consequently n−p Ωc ∩ B(x, 2R) = 0. The assumptions of Lemma 10.42 are satisfied, and we H∞ conclude that

q  dimA Ωc ∩ B(x, R) < (n − p). p

Along the same lines we obtain a global dimension dichotomy result. np Theorem 10.45. Let 1 < p < n and p ≤ q < n−p and assume that the (q, p)Hardy–Sobolev inequality holds in an open set Ω  Rn . Then q dimA (Ωc ) < (n − p). dimH (Ωc ) ≥ n − p or p

246

10. DISTANCE WEIGHTS AND HARDY–SOBOLEV INEQUALITIES

Proof. Let C1 be the constant in the (q, p)-Hardy–Sobolev inequality for Ω. We may assume that dimH (Ωc ) < n − p. Let z ∈ Ωc and r > 0. Arguing as in the proof of Lemma 10.42, we obtain  q q d (y, Ωc ) p (n−p)−n dy ≤ Cr p (n−p) , B(z,r)

where C = C(n, p, q, C1 ). Thus 0 < n − pq (n − p) ∈ A(Ωc ), and the claim follows from Theorem 10.24.  In the case q = p we obtain even better dichotomy results, due to the selfimprovement property of Hardy inequalities, see Theorem 6.16. Theorem 10.46. Let 1 < p < ∞ and assume that the p-Hardy inequality holds in an open set Ω  Rn with a constant C1 . Let x ∈ Rn and R > 0. There exists ε = ε(n, p, C1 ) > 0 such that either



dimH Ωc ∩ B(x, 2R) > n − p + ε or dimA Ωc ∩ B(x, R) < n − p − ε. Proof. By Theorem 6.16, there are ε1 > 0 and C2 > 0, both only depending on C1 , n and p, such that the q-Hardy inequality holds in Ω with the constant C2 for every q with p − ε1 < q ≤ p. Let x ∈ Rn and R > 0, and let 0 < ε < 12 ε1 to be specified later. If

dimH Ωc ∩ B(x, 2R) > n − p + ε,

the claim holds. Hence we may assume that dimH Ωc ∩ B(x, 2R) ≤ n − p + ε,

n−q Ωc ∩ B(x, 2R) = 0 for q = p − 2ε. As the (p − 2ε)-Hardy and consequently H∞ inequality holds in Ω and p−2ε > p−ε1 , we may use Lemma 10.42 and Remark 10.43 to conclude that there exists δ > 0, independent of ε with 0 < ε < 12 ε1 , such that

dimA Ωc ∩ B(x, R) ≤ n − p + 2ε − δ.   The claim follows by choosing 0 < ε < min 12 ε1 , 13 δ .  Along the same lines we obtain a global dimension dichotomy result for the p-Hardy inequality. Theorem 10.47. Let 1 < p < ∞ and assume that the p-Hardy inequality holds in an open set Ω  Rn with a constant C1 . There exists ε = ε(n, p, C1 ) > 0 such that either dimH (Ωc ) > n − p + ε

or

dimA (Ωc ) < n − p − ε.

In particular, either dimH (Ωc ) > n − p or dimA (Ωc ) < n − p. Proof. If dimH (Ωc ) ≤ n − p + ε, then, as in the proof of Lemma 10.42, there are δ = δ(n, p, C1 ) > 0 and C = C(n, p, C1 ) such that  d (x, Ωc )−p+2ε−δ dx ≤ Cr n−p+2ε−δ , B(z,r)

for every z ∈ Ω and r > 0. Moreover, both C and δ are independent of ε. Choosing ε as in the proof of Theorem 10.46 proves the claim.  c

10.7. TESTING CONDITIONS

247

Example 10.48. Let 0 < λ < n − 1 and let E ⊂ Rn be an unbounded λregular set. By Theorem 7.12 and Theorem 10.21 we have dimL (E) = dimH (E) = dimA (E) = λ. Let Ω = Rn \ E. The p-Hardy inequality holds in Ω for every p > n − λ, by Theorem 7.17, and for every 1 < p < n − λ, by Corollary 10.36. On the other hand, it follows from Theorem 10.47 that for p = n − λ the p-Hardy inequality does not hold in Ω. Notice that the (q, p)-Hardy–Sobolev inequality np , by Remark 10.38, since at least one of the holds in Ω whenever 1 < p < q ≤ n−p q conditions λ > n − p or λ < p (n − p) holds. Example 10.49. Consider the case 1 < p = q < ∞ and let 0 < λ < n − 1. Let Ω1 = B(0, 1) and let F ⊂ Ω1 be a closed λ-regular set. Then dimL (Ωc1 ) = n and dimL (F ) = dimH (F ) = dimA (F ) = λ, by Theorem 7.12 and Theorem 10.21. Define Ω = Ω1 \ F . Then dimL (Ωc ) = λ, and hence the p-Hardy inequality holds in Ω for every p > n − λ, by Theorem 7.17. On the other hand, if 1 < p < n − λ, then the p-Hardy inequality holds in Ω by Theorem 10.40. Finally, from Theorem 10.46 it follows that the (n − λ)-Hardy inequality does not hold in Ω. In conclusion, the p-Hardy inequality holds in Ω if and only if 1 < p < n − λ or n − λ < p < ∞. Notice that here dimA (Ωc ) = n, and hence the mixed case in Theorem 10.40 is really needed in order to obtain the p-Hardy inequality for 1 < p < n − λ. Observe that for 1 < p < n − λ the p-Hardy inequality holds in Ω but dimL (Ωc ) = λ < n − p and dimA (Ωc ) = n > n − p. Hence neither of the conditions in (10.16), with q = p, is valid. This shows that (10.16) is not necessary for the p-Hardy inequality in Ω and consequently it is not possible to replace dimH (Ωc ) by dimL (Ωc ) in Theorem 10.47. On the other hand, for p = n−λ we have dimH (Ωc ) = n > n − p = λ, but the p-Hardy inequality does not hold in Ω. Hence the condition dimH (Ωc ) > n − p is not sufficient for the p-Hardy inequality in Ω, unlike the condition dimL (Ωc ) > n − p when Ωc is unbounded, see Theorem 7.17. 10.7. Testing conditions In this section we discuss characterizations of Hardy–Sobolev inequalities in terms of capacities. These conditions do not have an immediate connection to the geometry of an open set or its complement, but they give convenient tests for the validity of Hardy–Sobolev inequalities. Theorem 10.50. Let 1 < p ≤ q < ∞ and let Ω  Rn be an open set. The (q, p)-Hardy–Sobolev inequality holds in Ω if and only if there exists a constant C1 such that  q q (10.17) d (x, ∂Ω) p (n−p)−n dx ≤ C1 capp (E, Ω) p E

for every compact set E ⊂ Ω. Proof. Assume first that the (q, p)-Hardy–Sobolev inequality holds in Ω, with a constant C. Let E ⊂ Ω be a compact set and let u ∈ Lip0 (Ω) be such that u(x) = 1 for every x ∈ E. Then the (q, p)-Hardy–Sobolev inequality implies   q q d (x, ∂Ω) p (n−p)−n dx ≤ |u(x)|q d (x, ∂Ω) p (n−p)−n dx E

Ω

≤C

 |∇u(x)| dx

q

p

Ω

 pq .

248

10. DISTANCE WEIGHTS AND HARDY–SOBOLEV INEQUALITIES

Inequality (10.17) with the constant C1 = C q follows by taking infimum over all such functions u, see Theorem 5.44. Assume then that (10.17) holds with a constant C1 for all compact sets E ⊂ Ω, and let u ∈ Lip0 (Ω). For j ∈ Z, let Ej = {x ∈ Ω : |u(x)| > 2j }. By (10.17) and the fact that the set E j ⊂ Ω is compact, we have  q |u(x)|q d (x, ∂Ω) p (n−p)−n dx Ω

≤ (10.18) ≤



∞

2

j=−∞ ∞



Ej+1 \Ej+2 q

d (x, ∂Ω) p (n−p)−n dx

2(j+2)q E j+1

j=−∞

≤ 4q C1

q

d (x, ∂Ω) p (n−p)−n dx

(j+2)q

∞

q

2jq capp (E j+1 , Ω) p

j=−∞

for every j ∈ Z. Define uj : Rn → [0, 1] by ⎧ ⎪ ⎨1, uj (x) = 2−j |u(x)| − 1, ⎪ ⎩ 0,

if |u(x)| ≥ 2j+1 , if 2j < |u(x)| < 2j+1 , if |u(x)| ≤ 2j .

Then uj ∈ Lip0 (Ω), uj = 1 in E j+1 and uj = 0 in Rn \ Ej . By Theorem 5.44, we can test the capacity capp (E j+1 , Ω) with uj , and obtain  |∇uj (x)|p dx capp (E j+1 , Ω) ≤ Ej \Ej+1

−jp



≤2 Since

q p

Ej \Ej+1

≥ 1, it follows that ∞

∞ 

q p

2 capp (E j+1 , Ω) ≤ jq

j=−∞

(10.19)

|∇u(x)|p dx.

j=−∞



|∇u(x)| dx p

Ej \Ej+1

 pq

∞ 

≤

j=−∞

|∇u(x)| dx p

Ej \Ej+1



|∇u(x)| dx p

=

 pq

 pq .

Ω

From (10.18) and (10.19) we conclude that the (q, p)-Hardy–Sobolev inequality 1

in (10.14) holds in Ω with the constant 4C1q .



The variational capacity is countably subadditive by Theorem 5.34 (e). However, the corresponding countable additivity on pairwise disjoint sets does not hold. Next we prove a weaker converse of the subadditivity, where we only consider intersections of a compact set with the Whitney cubes, and allow a constant in the inequality. This is called the quasiadditivity property of the variational capacity.

10.7. TESTING CONDITIONS

249

Theorem 10.51. Let 1 < p ≤ q < ∞. Assume that the (q, p)-Hardy–Sobolev inequality (10.14) holds in an open set Ω ⊂ Rn with a constant C1 , and let W(Ω) = {Qi : i ∈ N} be a Whitney decomposition of Ω. There exists a constant C = C(n, p, q, C1 ) such that ∞

(10.20)

q

q

capp (E ∩ Qi , Ω) p ≤ C capp (E, Ω) p

i=1

for every compact set E ⊂ Ω. Proof. As in Lemma 8.9 we write Qi = Q(xi , ri ) ∈ W(Ω) and Q∗i = Q(xi , 98 ri ) ∞ with i=1 χQ∗i (x) ≤ C(n) for every x ∈ Ω. Let E ⊂ Ω be a compact set and u ∈ Lip0 (Ω) be such that u(x) = 1 for every x ∈ E. For every i ∈ N we choose a Lipschitz function ϕi satisfying ϕi (x) = 1 for every x ∈ Qi , and |∇ϕi (x)| ≤ C(n)ri−1 and 0 ≤ ϕi (x) ≤ χQ∗i (x) for every x ∈ Rn . By Theorem 5.44, we can test the capacity capp (E ∩ Qi , Ω) with ui = uϕi for every i ∈ N. By the Leibniz rule (2.4), we have

|∇ui (x)| ≤ C |∇u(x)| + ri−1 |u(x)| χQ∗i (x) for almost every x ∈ Ω. Here ri−1 ≤ C(n)d (x, ∂Ω)−1 for every x ∈ Q∗i . Thus the left-hand side in (10.20) is bounded from above by  pq ∞ 

p |∇ui (x)| dx i=1

Ω

≤C

(10.21)

≤C

∞ 

Q∗ i i=1  ∞

i=1

Q∗ i

 |∇u(x)|p dx + |∇u(x)| dx p

 pq

Q∗ i

|u(x)|p d (x, ∂Ω)−p dx

+C

∞ 

i=1

 pq

|u(x)| d (x, ∂Ω) p

Q∗ i

−p

 pq dx

.

The first term on the right-hand side of (10.21) can be estimated, by the bounded overlap of the cubes Q∗i , as ∞   pq  pq   pq ∞ 

p p p |∇u(x)| dx ≤ |∇u(x)| dx ≤C |∇u(x)| dx . i=1

Q∗ i

i=1

Q∗ i

Ω

For the second term on the right-hand side of (10.21), we H¨older’s inequality, apply ∞ the estimates |Q∗i | ≤ Cd (x, ∂Ω)n for every x ∈ Q∗i and i=1 χQ∗i (x) ≤ C for every x ∈ Ω, and inequality (10.14). This gives  pq  ∞  ∞

q q−p |u(x)|p d (x, ∂Ω)−p dx ≤ |Q∗i | p q |u(x)|q d (x, ∂Ω)−q dx i=1

Q∗ i

i=1 ∞ 

≤C

Q∗ i

Q∗ i

q

|u(x)|q d (x, ∂Ω)−q+n p −n dx

i=1 q ≤ C |u(x)|q d (x, ∂Ω) p (n−p)−n dx Ω

 ≤C

|∇u(x)| dx p

Ω

 pq .

250

10. DISTANCE WEIGHTS AND HARDY–SOBOLEV INEQUALITIES

After collecting the estimates above and taking infimum over all functions u as above, the claim follows from Theorem 5.44.  The quasiadditivity property in Theorem 10.51 gives the following characterinp zation for Hardy–Sobolev inequalities. Notice that for q = n−p the claim in part (a) of Theorem 10.52 holds for every open set Ω  Rn , and thus also (b) and (c) hold for every open set Ω  Rn . An important ingredient in the proof below is the use of the maximal function MΩ u ∈ W01,p (Ω) as a capacity test function. np Theorem 10.52. Let 1 < p < n and p ≤ q ≤ n−p . Assume that Ω  Rn is an open set and let W(Ω) = {Qi : i ∈ N} be a Whitney decomposition of Ω. The following conditions are equivalent.

(a) The (q, p)-Hardy–Sobolev inequality holds in Ω. (b) There exists a constant C such that inequality ∞

q

q

capp (E ∩ Qi , Ω) p ≤ C capp (E, Ω) p

i=1

holds for every compact set E ⊂ Ω. (c) There exists a constant C such that inequality

 q p

capp (Qi , Ω) ≤ C capp

i∈I



 pq Qi , Ω

i∈I

holds whenever I ⊂ N is a finite set. Moreover, the constants in each of the conditions only depend on each other, n, p and q. Proof. The implication from (a) to (b) is proved  in Theorem 10.51. The implication from (b) to (c) follows by considering E = i∈I Qi . Then assume that condition (c) holds with a constant C1 and fix a compact set E ⊂ Ω. To show that (a) holds, by Theorem 10.50 it suffices to prove that there exists a constant C = C(n, p, C1 ) such that  q q (10.22) d (x, ∂Ω) p (n−p)−n dx ≤ C capp (E, Ω) p . E

Let u ∈ Lip0 (Ω) with u(x) = 1 for every x ∈ E. We partition the Whitney decomposition W(Ω) = {Qi : i ∈ N} as W1 = {Qi ∈ W(Ω) : uQi < 12 } and W2 = W(Ω) \ W1 . Then the left-hand side of (10.22) can be written as  

 q + d (x, ∂Ω) p (n−p)−n dx. (10.23) Q∈W1

Q∈W2

E∩Q

To estimate the first sum, we observe that, for every Q ∈ W1 and every x ∈ E ∩ Q, 1 2

= 1−

1 2

< u(x) − uQ = |u(x) − uQ |.

10.7. TESTING CONDITIONS

251

By the (q, p)-Poincar´e inequality on cubes, see Corollary 9.24, we obtain 



q q (n−p)−n (n−p)−n p p d (x, ∂Ω) dx ≤ C l(Q) |u(x) − uQ |q dx Q∈W1

E∩Q

Q

Q∈W1

≤C

l(Q)

q p (n−p)

 l(Q)

q



|∇u(x)|p dx

 pq

Q

Q∈W1

 ≤C

 pq

Q

Q∈W1

≤C

 |∇u(x)| dx p

|∇u(x)|p dx

 pq ,

Ω

where C = C(n, p, q). The second sum in (10.23) can be estimated by using the boundedness of the local maximal operator MΩ . Let Q ∈ W2 and x ∈ Q. Then Q ⊂ B(x, diam(Q)) ⊂ Ω. The inequalities in (8.5) and the definition of W2 imply (10.24) MΩ u(x) ≥ 

B(x,diam(Q))

|u(y)| dy ≥ C(n)  |u(y)| dy ≥ C(n)uQ ≥ Q

C(n) . 2

Since u ∈ Lip0 (Ω) ⊂ W01,p (Ω), we have by Corollary 4.28 that MΩ u ∈ W01,p (Ω). Moreover, MΩ u is continuous in Ω by Lemma 1.31, and by (10.24) there exists  C2 = C2 (n) > 0 such that C2 MΩ u ≥ 1 in Q∈W2 Q. It follows from Theorem 5.46 

that we may test the capacity capp Q∈W2 Q, Ω with the function C2 MΩ u. Note that the set {x ∈ Ω : u(x) ≥ 12 } is compact since u ∈ Lip0 (Ω), and so W2 is finite. Let Q = Q(z, r) ∈ W2 . As 1 < p < n, we have, by the monotonicity properties of the capacity and Remark 5.37, that capp (Q, Ω) ≥ capp (Q, Rn ) ≥ capp (B(z, r), Rn ) ≥ C(n, p)r n−p . Thus



q

q

q

d (x, ∂Ω) p (n−p)−n dx ≤ C(n, p, q)r p (n−p) ≤ C(n, p, q) capp (Q, Ω) p . E∩Q

From the assumed condition (c), Theorem 4.25, and the maximal function theorem, see Theorem 1.15, we hence obtain

 q q d (x, ∂Ω) p (n−p)−n dx ≤ C capp (Q, Ω) p Q∈W2

E∩Q

Q∈W2



≤ C capp  ≤C



 pq Q, Ω

Q∈W2

|∇MΩ u(x)| dx p

Ω

 ≤C

|∇u(x)| dx p

Ω

 pq

 pq ,

252

10. DISTANCE WEIGHTS AND HARDY–SOBOLEV INEQUALITIES

where C = C(n, p, q, C1 ). By combining the estimates for W1 and W2 we conclude that 

 q q d (x, ∂Ω) p (n−p)−n dx = d (x, ∂Ω) p (n−p)−n dx E

Q∈W

E∩Q



≤ C(n, p, q, C1 )

|∇u(x)|p dx

 pq ,

Ω

and the desired estimate (10.22) follows from Theorem 5.44 by taking infimum over all Lipschitz functions u as above.  Remark 10.53. Theorem 10.52 can be used to give an alternative proof for Theorem 10.37, and in fact we obtain the following more general statement. Let np < ∞ and assume that the (q, p)-Hardy–Sobolev inequality 1 < p ≤ q < p∗ = n−p holds in an open set Ω ⊂ Rn . Then the (q  , p)-Hardy–Sobolev inequality holds in Ω for every q ≤ q  ≤ p∗ . This follows from Theorem 10.52 and the simple observation that if q ≤ q  , then ∞ ∞  p  pq q

q q capp (E ∩ Qi , Ω) p ≤ capp (E ∩ Qi , Ω) p i=1

i=1

whenever W(Ω) = {Qi : i ∈ N} is a Whitney decomposition of Ω and E ⊂ Ω is a compact set. 10.8. Notes The Aikawa condition was introduced and applied in connection with quasiadditivity properties of Riesz capacities by Aikawa [7, 8]. Self-improvement of the Aikawa condition is implicit in the proof of Lemma 2.4 in Koskela and Zhong [246]. This condition was applied for weighted Hardy inequalities by Lehrb¨ ack [258]. Section 10.2 follows the ideas in Dyda, Ihnatsyeva, Lehrb¨ ack, Tuominen and V¨ ah¨akangas [111], but the connection between the Aikawa condition and the Ap properties of distance functions is mentioned already by Aikawa in [8, p. 151]. Horiuchi [193] has similar results in terms of the so-called P (s)-condition, which can also be characterized using the Assouad dimension, see Lehrb¨ ack and V¨ ah¨akangas [265, Theorem 3.4]. The Assouad dimension was studied by Assouad [19, 20], but essentially the same concept has appeared earlier for instance in Larman [255]. See Luukkainen [293] and Fraser [136] for more information concerning the history and theory of Assouad dimension. To emphasize the duality with the lower dimension, sometimes the Assouad dimension is called the upper Assouad dimension and denoted by dimA (E). Assouad dimension of porous sets is discussed in Luukkainen [293, Section 5]. Theorem 10.24 was considered in Lehrb¨ack and Tuominen [263]. Theorem 10.26 is from Dyda, Ihnatsyeva, Lehrb¨ack, Tuominen and V¨ ah¨akangas [111, Corollary 3.8]. See also Dur´an and L´opez Garc´ıa [109, Lemma 3.3] and Aimar, Carena, Dur´an and Toschi [9] concerning Ap properties of distance functions. Theorem 10.29 has been studied by Cruz-Uribe, Moen and Rodney [94] and Kurki and V¨ ah¨ akangas [252]. For Theorem 10.30, see Kurki and V¨ ah¨ akangas [252]. Conditions for Hardy inequalities in terms of the lower dimension and the Assouad

10.8. NOTES

253

dimension were obtained in Lehrb¨ ack [261]. Dyda and V¨ah¨akangas [113] have corresponding results for fractional Hardy inequalities. Similar ideas are also present in the unpublished manuscript by Wannebo [394], see for instance [394, Definition 6.6]. Hardy–Sobolev inequalities have been considered for instance in Dyda, Ihnatsyeva, Lehrb¨ ack, Tuominen and V¨ ah¨ akangas [111], Lehrb¨ack and V¨ ah¨akangas [265], Mazya [318, Section 2.1.6], [321, Section 2.1.7] and Opic and Kufner [340, Section 21]. Horiuchi [192,193] has closely related results for embeddings between weighted Sobolev spaces. It is also possible to deduce a weighted version of the global Hardy–Sobolev inequality (10.15) using Theorem 9.21, see Dyda, Ihnatsyeva, Lehrb¨ack, Tuominen and V¨ ah¨ akangas [111, Section 6]. The mixed case in Theorem 10.40, for p = q, has been considered in Lehrb¨ ack [258, 261] for weighted inequalities. The necessary conditions in Section 10.6 are based on Koskela and Zhong [246], see also Lehrb¨ ack [258, 261] and Lehrb¨ack and Tuominen [263]. The proof of Theorem 10.50 is based on a truncation argument that can be found in Mazya [318, p. 110]. For more information about this kind of characterizations, we refer to Mazya [318–320]. The truncation method has been applied, for example, in Adams and Hedberg [4, Theorem 7.2.1], Bakry, Coulhon, Ledoux and Saloff-Coste [21], Biroli and Mosco [30], Bj¨orn and Bj¨orn [31, Lemma 4.20], Capogna, Danielli and Garofalo [73], Coulhon [90], Franchi, Guti´errez and Wheeden [131], Garofalo and Nhieu [143], Hajlasz and Koskela [171], Heinonen and Koskela [188, 189], Kinnunen and Korte [225], Long and Nie [285], Maheux and Saloff-Coste [297], Saloff-Coste [357, Section 3.2], Semmes [358], Tartar [379] and Turesson [388]. Quasiadditivity of capacity has been studied by Aikawa [7, 8], Dyda and V¨ah¨akangas [114], Hurri-Syrj¨ anen and V¨ ah¨ akangas [198] and Lehrb¨ack and Shanmugalingam [262]. For q > p, the quasiadditivity results in Section 10.7 have not appeared previously, but their counterparts for fractional Hardy inequalities have been obtained by Hurri-Syrj¨ anen and V¨ ah¨akangas [198].

10.1090/surv/257/11

CHAPTER 11

The p-Laplace Equation Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. Assume that u ∈ C 2 (Ω) with ∇u(x) = 0 for every x ∈ Ω. We consider the p-Laplace equation div(|∇u(x)|p−2 ∇u(x)) = 0 for every x ∈ Ω. Here the divergence of a differentiable vector field F : Ω → Rn , F (x) = (F1 (x), . . . , Fn (x)), at x ∈ Ω is div F (x) =

n

∂Fi (x) i=1

∂xi

.

A direct computation shows that div(|∇u(x)|p−2 ∇u(x))   n

∂u ∂u ∂2u (x) (x) (x) = |∇u(x)|p−4 |∇u(x)|2 Δu(x) + (p − 2) ∂xi ∂xj ∂xi ∂xj i,j=1 for every x ∈ Ω. This gives a formula for a classical solution to the p-Laplace equation in Ω. We remark that the assumption ∇u(x) = 0 is not needed when p ≥ 2. For p = 2 we have the Laplace equation, which is linear. When p = 2, the p-Laplace equation is nonlinear and a sum of two solutions is not necessarily a solution. However, a constant can be added to a solution and a solution can be multiplied by a constant without destroying the property of being a solution. This chapter discusses existence and regularity of a weak solution to the p-Laplace equation. 11.1. Weak solutions Assume that u ∈ C (Ω), with ∇u(x) = 0 for every x ∈ Ω, is a classical solution to the p-Laplace equation. An integration by parts gives   0=− div(|∇u(x)|p−2 ∇u(x))ϕ(x) dx = |∇u(x)|p−2 ∇u(x) · ∇ϕ(x) dx 2

Ω

Ω

C0∞ (Ω).

for every ϕ ∈ Observe that there is no boundary term since ϕ is compactly supported in Ω. Conversely, if u ∈ C 2 (Ω) is such that ∇u(x) = 0 for every x ∈ Ω and  |∇u(x)|p−2 ∇u(x) · ∇ϕ(x) dx = 0 Ω

for every ϕ ∈ C0∞ (Ω), then by the computation above  div(|∇u(x)|p−2 ∇u(x))ϕ(x) dx = 0 Ω 255

256

11. THE p-LAPLACE EQUATION

for every ϕ ∈ C0∞ (Ω). This implies div(|∇u(x)|p−2 ∇u(x)) = 0 for every x ∈ Ω. We will use from now on the convention |0|p−2 0 = 0, and with this we may omit the assumption that ∇u(x) = 0 for every x ∈ Ω. The considerations above motivate the definition of a weak solution. Second order derivatives are needed in the definition of a classical solution to the p-Laplace equation, but in the following definition it is enough to only assume that first order weak derivatives exist. Definition 11.1. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. A function 1,p (Ω) is a weak solution to the p-Laplace equation in Ω if u ∈ Wloc  (11.1) |∇u(x)|p−2 ∇u(x) · ∇ϕ(x) dx = 0 Ω

for every ϕ ∈

C0∞ (Ω).

We show that being a weak solution is a local property. Lemma 11.2. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. A function 1,p u ∈ Wloc (Ω) is a weak solution to the p-Laplace equation in Ω if and only if for every x ∈ Ω there exists rx > 0 such that B(x, rx )  Ω and u is a weak solution to the p-Laplace equation in B(x, rx ). 1,p (Ω) is a weak solution in Ω and let B(x0 , r) Proof. Assume first that u ∈ Wloc  Ω. Since C0∞ (B(x0 , r)) ⊂ C0∞ (Ω), we have   p−2 |∇u(x)| ∇u(x) · ∇ϕ(x) dx = |∇u(x)|p−2 ∇u(x) · ∇ϕ(x) dx = 0 B(x0 ,r)

Ω

C0∞ (B(x0 , r)).

for every ϕ ∈ The restriction of u belongs to W 1,p (B(x0 , r)), and thus u is a weak solution in B(x0 , r). Assume then that for every x ∈ Ω there exists 1,p a ball B(x, rx )  Ω, in which u is a weak solution. Observe that u ∈ Wloc (Ω) since  1,p rx ∞ r )) for every x ∈ Ω. Let ϕ ∈ C (Ω). The collection B(x, u ∈ Wloc (B(x, x 0 2 ) :  x ∈ supp ϕ is an open cover of supp ϕ. Since supp ϕ is compact, there exists a finite number of points xi ∈ supp ϕ and corresponding balls B(xi , r2i ) with ri = rxi for k i = 1, 2, . . . , k, such that supp ϕ ⊂ i=1 B(xi , r2i ). We consider a partition of unity k ψi ∈ C0∞ (B(xi , ri )) with 0 ≤ ψi ≤ 1 for every i = 1, 2, . . . , k, and i=1 ψi (x) = 1 for every x ∈ supp ϕ. Then ϕψi ∈ C0∞ (B(xi , ri )) for every i = 1, 2, . . . , k, and k i=1 ∇ψi (x) = 0 for almost every x ∈ supp ϕ by Lemma 2.13. By summing up we obtain k 

0= |∇u(x)|p−2 ∇u(x) · ∇(ϕ(x)ψi (x)) dx i=1

B(xi ,ri )



|∇u(x)|p−2 ∇u(x) ·

= Ω

|∇u(x)|p−2 ∇u(x) · 

Ω

k



∇ϕ(x)ψi (x) + ϕ(x)∇ψi (x) dx

i=1

|∇u(x)|

p−2

=

∇(ϕ(x)ψi (x)) dx

i=1

 =

k

∇u(x) · ∇ϕ(x) dx.

Ω

This shows that u is a weak solution to the p-Laplace equation in Ω.



11.1. WEAK SOLUTIONS

257

Example 11.3. The function Φ : Rn \ {0} → R,  p−n p = n, |x| p−1 , Φ(x) = log|x|, p = n, is a classical solution to the p-Laplace equation in Rn \ {0} and it is called the fundamental solution of the p-Laplace equation. However, if 1 < p ≤ n, then Φ fails to be a weak solution to the p-Laplace equation in Rn . Indeed, direct computations show that in this case Φ ∈ Lqloc (Rn ) for every 1 ≤ q < n(p−1) n−p , which in the case q n p = n means that Φ ∈ Lloc (R ) for every 1 ≤ q < ∞. Moreover, ∇Φ ∈ Lqloc (Rn ) for 1,q every 1 ≤ q < n(p−1) (B(0, 1)) for every 1 ≤ q < n(p−1) n−1 , and thus Φ ∈ W n−1 . This upper bound is optimal since 1, n(p−1) n−1

Φ∈ / Wloc for 1 < p ≤ n. Observe that 1,p (Rn ) if p > n. Φ ∈ Wloc

n(p−1) n−1

(B(0, 1))

≤ p for 1 < p ≤ n. On the other hand,

The following example illustrates that a weak solution to the p-Laplace equation does not necessarily belong to W 1,p (Ω). Example 11.4. Consider the fundamental solution Φ in Example 11.3 with 1 < p < n. Then u = 1 − Φ is a weak solution to the p-Laplace equation in Ω = B(0, 1) \ {0}. By Example 11.3 and Theorem 5.30 applied with v = min{0, 1 − Φ}, / W 1,q (Ω) for q ≥ n(p−1) we have u ∈ W01,q (Ω) for 1 < q < n(p−1) n−1 , but u ∈ n−1 . In 1,p particular, u ∈ W (Ω). On the other hand, also the constant function u = 0 is a weak solution to the p-Laplace equation in Ω with the zero boundary values. This shows that the p-Laplace equation with the boundary condition u ∈ W01,q (Ω) for q < p does not have a unique solution in general. Observe that in this case we cannot use u as a test function in the definition of a weak solution and thereby conclude that   |∇u(x)|p dx = |∇u(x)|p−2 ∇u(x) · ∇u(x) dx = 0, Ω

Ω

which would imply that ∇u = 0 almost everywhere in Ω and consequently u = 0 almost everywhere in Ω, by Lemma 3.15. Next we show that − div(|∇Φ|p−2 ∇Φ) = C(n, p)δ0 in the weak sense in Rn , where δ0 is Dirac’s delta at the origin. Theorem 11.5. Let 1 < p < ∞ and let Φ be as in Example 11.3. There exists a constant C = C(n, p) = 0 such that  |∇Φ(x)|p−2 ∇Φ(x) · ∇ϕ(x) dx = Cϕ(0) Rn

for every ϕ ∈ C0∞ (Rn ).

258

11. THE p-LAPLACE EQUATION

Proof. Let ϕ ∈ C0∞ (Rn ), let B(0, R) be a ball with supp ϕ  B(0, R), and let 0 < ε < R. Then   p−2 |∇Φ(x)| ∇Φ(x) · ∇ϕ(x) dx = |∇Φ(x)|p−2 ∇Φ(x) · ∇ϕ(x) dx Rn B(0,R)\B(0,ε)  + |∇Φ(x)|p−2 ∇Φ(x) · ∇ϕ(x) dx. B(0,ε)

For the second integral on the right-hand side we have   p−2 ≤ C(n, p) |∇Φ(x)| ∇Φ(x) · ∇ϕ(x) dx B(0,ε)

|x|1−n |∇ϕ(x)| dx

B(0,ε)

≤ C(n, p)∇ϕL∞ (Rn )



ε

r 1−n r n−1 dr 0 ε→0

= C(n, p)ε∇ϕL∞ (Rn ) −−−→ 0. To estimate the other integral, we observe by a direct computation that |∇Φ(x)|p−2 ∇Φ(x) = C(n, p)

x |x|n

for every x = 0. Since Φ is a classical solution to the p-Laplace equation in Rn \{0}, using the divergence theorem we obtain  |∇Φ(x)|p−2 ∇Φ(x) · ∇ϕ(x) dx B(0,R)\B(0,ε)  = div(|∇Φ(x)|p−2 ∇Φ(x)ϕ(x)) dx B(0,R)\B(0,ε)  x =− |∇Φ(x)|p−2 ∇Φ(x)ϕ(x) · dσ(x) |x| ∂B(0,ε)  |x|1−n ϕ(x) dσ(x) = C(n, p) ∂B(0,ε)  = C(n, p)ε1−n ϕ(x) dσ(x) ∂B(0,ε)

C(n, p) = σ(∂B(0, ε))



ε→0

ϕ(x) dσ(x) −−−→ C(n, p)ϕ(0).



∂B(0,ε)

If u ∈ W 1,p (Ω) is a weak solution, then in Definition 11.1 the class of test functions can be taken to be the Sobolev space W01,p (Ω) with zero boundary values. Lemma 11.6. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. Assume that u ∈ W 1,p (Ω) is a weak solution to the p-Laplace equation in Ω. Then  |∇u(x)|p−2 ∇u(x) · ∇ϕ(x) dx = 0 Ω

for every ϕ ∈ W01,p (Ω).

11.1. WEAK SOLUTIONS

259

Proof. Let ϕi ∈ C0∞ (Ω), i ∈ N, be such that ϕi → ϕ in W 1,p (Ω) as i → ∞. By the Cauchy–Schwarz inequality and H¨ older’s inequality, we obtain  |∇u(x)|p−2 ∇u(x) · ∇ϕ(x) dx Ω   p−2 p−2 = |∇u(x)| ∇u(x) · ∇ϕ(x) dx − |∇u(x)| ∇u(x) · ∇ϕi (x) dx Ω Ω p−2 = |∇u(x)| ∇u(x) · (∇ϕ(x) − ∇ϕi (x)) dx Ω ≤ |∇u(x)|p−1 |∇ϕ(x) − ∇ϕi (x)| dx Ω

 |∇u(x)|p dx

≤

 p−1  p

Ω

|∇ϕ(x) − ∇ϕi (x)|p dx

 p1

i→∞

−−−→ 0.



Ω

1,p Remark 11.7. Let 1 < p < ∞ and assume that u ∈ Wloc (Ω) is a weak solution n to the p-Laplace equation in an open set Ω ⊂ R . Then  |∇u(x)|p−2 ∇u(x) · ∇ϕ(x) dx = 0 Ω

W01,p (Ω)

for every ϕ ∈ with supp ϕ  Ω. Indeed, by convolution approximation we obtain a compact set K ⊂ Ω and functions ϕi ∈ C0∞ (Ω) such that supp ϕ∪supp ϕi ⊂ K, for every i ∈ N, and ∇ϕi → ∇ϕ in Lp (Ω) as i → ∞. The rest of the proof is similar to the proof of Lemma 11.6 using the fact that K |∇u(x)|p dx < ∞. We remark that if ϕ ≥ 0, then also ϕi ≥ 0 for every i ∈ N. Remark 11.8. Let 1 < p < ∞ and assume that u ∈ W 1,p (Rn ) is a weak solution to the p-Laplace equation in Rn . By Lemma 11.6, we may use ϕ ∈ W01,p (Rn ) as a test function in (11.1). Since u ∈ W 1,p (Rn ) = W01,p (Rn ), we obtain   |∇u(x)|p dx = |∇u(x)|p−2 ∇u(x) · ∇u(x) dx = 0. Rn

Rn

Thus ∇u = 0 almost everywhere in Rn , and by Lemma 3.15 we conclude that u = 0 almost everywhere in Rn . As an application of Lemma 11.6, we obtain a Sobolev space version of the comparison principle. In the proof we apply the following vectorial inequalities: if 2 ≤ p < ∞, then (|ζ|p−2 ζ − |η|p−2 η) · (ζ − η) ≥ 22−p |ζ − η|p

(11.2)

for every ζ, η ∈ Rn ; and if 1 < p ≤ 2, then (11.3)

(|ζ|p−2 ζ − |η|p−2 η) · (ζ − η) ≥ (p − 1)(1 + |ζ| + |η|)p−2 |ζ − η|2

for every ζ, η ∈ Rn , ζ = η. See Lindqvist [280, Chapter 12]. Theorem 11.9. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. Assume that u, v ∈ W 1,p (Ω) are weak solutions to the p-Laplace equation in Ω. If min{v −u, 0} ∈ W01,p (Ω), then u ≤ v almost everywhere in Ω. Proof. Since u, v ∈ W 1,p (Ω) are weak solutions, by Lemma 11.6 we have 

|∇v(x)|p−2 ∇v(x) − |∇u(x)|p−2 ∇u(x) · ∇ϕ(x) dx = 0 Ω

260

11. THE p-LAPLACE EQUATION

for every ϕ ∈ W01,p (Ω). Let ϕ = min{v − u, 0} ∈ W01,p (Ω). Assume first that p ≥ 2. By Lemma 2.25 and (11.2) we obtain   p |∇ min{v(x) − u(x), 0}| dx = |∇v(x) − ∇u(x)|p dx Ω {u>v} 

≤ C(p) |∇v(x)|p−2 ∇v(x) − |∇u(x)|p−2 ∇u(x) · (∇v(x) − ∇u(x)) dx {u>v} 

= C(p) |∇v(x)|p−2 ∇v(x) − |∇u(x)|p−2 ∇u(x) · ∇ϕ(x) dx = 0. Ω

Consequently ∇ min{v − u, 0} = 0 almost everywhere in Ω. Since min{v − u, 0} ∈ W01,p (Ω), Lemma 3.15 implies that min{v − u, 0} = 0 almost everywhere Ω. This shows that u ≤ v almost everywhere in Ω. For 1 < p < 2, we use a similar reasoning as above but apply (11.3) instead of (11.2), and obtain  (1 + |∇v(x)| + |∇u(x)|)p−2 |∇ min{v(x) − u(x), 0}|2 dx Ω  = (1 + |∇v(x)| + |∇u(x)|)p−2 |∇v(x) − ∇u(x)|2 dx = 0. {u>v}

From this we conclude as above that u ≤ v almost everywhere in Ω.



The comparison principle above implies that a weak solution to the p-Laplace equation with given Sobolev boundary values is unique whenever it exists. Corollary 11.10. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. Assume that u, v ∈ W 1,p (Ω) are weak solutions to the p-Laplace equation in Ω with u − v ∈ W01,p (Ω). Then v = u almost everywhere in Ω. Proof. Theorem 2.30 implies min{v − u, 0} ∈ W01,p (Ω) and min{u − v, 0} ∈ Theorem 11.9 shows that u ≤ v almost everywhere in Ω and v ≤ u almost everywhere in Ω. Thus v = u almost everywhere in Ω.  W01,p (Ω).

11.2. A variational approach Next we consider a variational approach to the Dirichlet problem for the pLaplace equation. We show that the p-Laplace equation is the Euler–Lagrange equation of the p-Dirichlet integral  I(u) = |∇u(x)|p dx. Ω

In other words, a function with prescribed boundary values is a minimizer of the p-Dirichlet integral if and only if it is a weak solution to the p-Laplace equation. Theorem 11.11. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. Assume that u, g ∈ W 1,p (Ω) and u − g ∈ W01,p (Ω). Then   I(u) = inf I(v) : v ∈ W 1,p (Ω), v − g ∈ W01,p (Ω) if and only if

 |∇u(x)|p−2 ∇u(x) · ∇ϕ(x) dx = 0

for every ϕ ∈ C0∞ (Ω).

Ω

11.2. A VARIATIONAL APPROACH

261

Proof. First assume that   I(u) = inf I(v) : v ∈ W 1,p (Ω), v − g ∈ W01,p (Ω) . Let ϕ ∈ C0∞ (Ω) and 0 < ε ≤ 1. Then v = u + εϕ ∈ W 1,p (Ω) and v − g ∈ W01,p (Ω). Thus   p |∇u(x)| dx = I(u) ≤ I(v) = |∇u(x) + ε∇ϕ(x)|p dx Ω

and so

Ω

 Ω

1 |∇u(x) + ε∇ϕ(x)|p − |∇u(x)|p dx ≥ 0. ε

On the other hand, since ∇(|ζ|p ) = p|ζ|p−2 ζ for ζ ∈ Rn , ζ = 0, we have

ε→0 1 |∇u(x) + ε∇ϕ(x)|p − |∇u(x)|p −−−→ p|∇u(x)|p−2 ∇u(x) · ∇ϕ(x), ε if ∇u(x) = 0. By the mean value theorem for ε → |∇u(x) + ε∇ϕ(x)|p and Young’s inequality, there is a constant γ ∈ (0, 1) such that 1 |∇u(x) + ε∇ϕ(x)|p − |∇u(x)|p ≤ p|∇u(x) + γε∇ϕ(x)|p−1 |∇ϕ(x)| ε ≤ (p − 1)|∇u(x) + γε∇ϕ(x)|p + |∇ϕ(x)|p ≤ C(p)(|∇u(x)|p + |∇ϕ(x)|p ) for every x ∈ Rn since the estimate above is trivial in the case ∇u(x) = 0. The dominated convergence theorem gives 

 |∇u(x)|

p−2

p

1 lim |∇u(x) + ε∇ϕ(x)|p − |∇u(x)|p dx ε 

1 |∇u(x) + ε∇ϕ(x)|p − |∇u(x)|p dx = lim ε→0 Ω ε ≥ 0.

∇u(x) · ∇ϕ(x) dx =

Ω

Thus

Ω ε→0

 |∇u(x)|p−2 ∇u(x) · ∇ϕ(x) dx ≥ 0 Ω

for every ϕ ∈ C0∞ (Ω). The reverse inequality follows by replacing ϕ with −ϕ. This shows that u is a weak solution to the p-Laplace equation in Ω. Then assume that u ∈ W 1,p (Ω) is a weak solution in Ω with u − g ∈ W01,p (Ω), and let v ∈ W 1,p (Ω) with v − g ∈ W01,p (Ω). We have u − v = (u − g) − (v − g) ∈ W01,p (Ω) and by Lemma 11.6  |∇u(x)|p−2 ∇u(x) · (∇u(x) − ∇v(x)) dx = 0. Ω

262

11. THE p-LAPLACE EQUATION

Young’s inequality implies   p |∇u(x)| dx = |∇u(x)|p−2 ∇u(x) · ∇u(x) dx Ω Ω = |∇u(x)|p−2 ∇u(x) · ∇v(x) dx Ω ≤ |∇u(x)|p−1 |∇v(x)| dx Ω   p−1 1 p ≤ |∇u(x)| dx + |∇v(x)|p dx, p p Ω Ω and by reorganizing the terms we obtain   p |∇u(x)| dx ≤ |∇v(x)|p dx. Ω

This shows that

Ω

  I(u) = inf I(v) : v ∈ W 1,p (Ω), v − g ∈ W01,p (Ω) .



Remark 11.12. The proof of Theorem 11.11 also gives a local version of the 1,p (Ω) is a weak solution to the p-Laplace equation in theorem: a function u ∈ Wloc Ω if and only if   |∇u(x)|p dx ≤ |∇u(x) + ∇ϕ(x)|p dx Ω

for every ϕ ∈

Ω

C0∞ (Ω).

Next we prove existence and uniqueness of a solution to the Dirichlet problem using the direct method in the calculus variations. This means that, instead of the partial differential equation, the argument uses the variational integral. Theorem 11.13. Let 1 < p < ∞, let Ω ⊂ Rn be a bounded open set, and assume that g ∈ W 1,p (Ω). There exists a unique u ∈ W 1,p (Ω) such that u − g ∈ W01,p (Ω) and   |∇u(x)|p dx ≤ |∇v(x)|p dx Ω

for every v ∈ W

1,p

(Ω) with v − g ∈

Ω

W01,p (Ω).

Proof. Let

  I = inf I(v) : v ∈ W 1,p (Ω) and v − g ∈ W01,p (Ω) .

We note that I is finite since I ≤ I(g) < ∞. There exists a minimizing sequence (ui )i∈N , with ui ∈ W 1,p (Ω) and ui − g ∈ W01,p (Ω), such that I(ui ) → I as i → ∞. The existence of the limit implies that the sequence (I(ui ))i∈N is bounded. Thus there exists a constant M < ∞ such that  I(ui ) = |∇ui (x)|p dx ≤ M Ω

11.2. A VARIATIONAL APPROACH

263

for every i ∈ N. Using Corollary 3.7 with q = p, we obtain 

|ui (x)|p + |∇ui (x)|p dx ui pW 1,p (Ω) =  Ω

≤ 2p−1 |ui (x) − g(x)|p + |g(x)|p + |∇ui (x)|p dx Ω  

p p p−1 ≤ C diam(Ω) |g(x)|p + |∇ui (x)|p dx |∇ui (x) − ∇g(x)| dx + 2 Ω Ω 

p p p |∇ui (x)| + |g(x)| + |∇g(x)|p dx ≤ C(diam(Ω) + 1)

Ω p ≤ C(diam(Ω) + 1) M + gpW 1,p (Ω) < ∞ for every i ∈ N, with C = C(n, p). This shows that (ui )i∈N is a bounded sequence in W 1,p (Ω). By Theorem 2.39, there exists a subsequence (uik )k∈N and u ∈ W 1,p (Ω), with u − g ∈ W01,p (Ω), such that (uik , ∇uik ) → (u, ∇u) weakly in Lp (Ω; Rn+1 ) as k → ∞. Lemma 2.34 gives I(u) ≤ lim inf I(uik ) = I k→∞

and consequently u is a minimizer. To show the uniqueness, assume that u1 , u2 ∈ W 1,p (Ω) are minimizers of the variational integral I with u1 − g ∈ W01,p (Ω) and u2 − g ∈ W01,p (Ω). For a contradiction, we assume that |{x ∈ Ω : u1 (x) = u2 (x)}| > 0. By Corollary 3.7 with q = p, we have   0 < |u1 (x) − u2 (x)|p dx ≤ C(n, p) diam(Ω)p |∇u1 (x) − ∇u2 (x)|p dx Ω

Ω

and thus |{x ∈ Ω : ∇u1 (x) = ∇u2 (x)}| > 0.

(11.4)

By Theorem 11.11, both u1 and u2 are weak solutions to the p-Laplace equation in Ω. Lemma 11.6 implies  |∇ui (x)|p−2 ∇ui (x) · ∇ϕ(x) dx = 0 Ω

W01,p (Ω)

for every ϕ ∈ and i = 1, 2. By subtracting these equations for i = 1, 2, choosing ϕ = u1 − u2 , and applying (11.4), we obtain the contradiction 

|∇u1 (x)|p−2 ∇u1 (x) − |∇u2 (x)|p−2 ∇u2 (x) · ∇ϕ(x) dx 0= Ω

|∇u1 (x)|p−2 ∇u1 (x) − |∇u2 (x)|p−2 ∇u2 (x) · (∇u1 (x) − ∇u2 (x)) dx > 0. = Ω

The final inequality follows from the fact that

|∇u1 (x)|p−2 ∇u1 (x) − |∇u2 (x)|p−2 ∇u2 (x) · (∇u1 (x) − ∇u2 (x)) > 0 in the set {x ∈ Ω : ∇u1 (x) = ∇u2 (x)}, by (11.2) and (11.3).



264

11. THE p-LAPLACE EQUATION

11.3. Weak super- and subsolutions We consider not only weak solutions but also weak super- and subsolutions to the p-Laplace equation. As we shall see, weak super- and subsolutions are more flexible than weak solutions. Another advantage is that the properties of superand subsolutions can be considered separately. Definition 11.14. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. A function 1,p (Ω) is a weak supersolution to the p-Laplace equation in Ω if u ∈ Wloc  |∇u(x)|p−2 ∇u(x) · ∇ϕ(x) dx ≥ 0 Ω

C0∞ (Ω)

1,p with ϕ ≥ 0. Analogously, a function u ∈ Wloc (Ω) is a weak for every ϕ ∈ subsolution to the p-Laplace equation in Ω if  |∇u(x)|p−2 ∇u(x) · ∇ϕ(x) dx ≤ 0 Ω

for every ϕ ∈

C0∞ (Ω)

with ϕ ≥ 0.

Remark 11.15. We can use functions ϕ ∈ W01,p (Ω) satisfying supp ϕ  Ω and ϕ ≥ 0 almost everywhere in Ω as test functions for weak super- and subsolutions in Ω. Indeed, by convolution approximation, we obtain a compact set K ⊂ Ω and nonnegative functions ϕi ∈ C0∞ (Ω) such that supp ϕ∪supp ϕi ⊂ K, for every i ∈ N, and ∇ϕi → ∇ϕ in Lp (Ω) as i → ∞. The rest of the proof is similar to the proof of Lemma 11.6. Lemma 11.16. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. A function 1,p (Ω) is a weak solution to the p-Laplace equation in Ω if and only if u is u ∈ Wloc both weak super- and weak subsolution to the p-Laplace equation in Ω. Proof. It is clear that a weak solution is both super- and subsolution. Assume 1,p (Ω) is a weak super- and subsolution in Ω. Let ϕ ∈ C0∞ (Ω). then that u ∈ Wloc Theorem 2.30 implies that ϕ = ϕ+ − ϕ− with ϕ+ = max{ϕ, 0} ∈ W01,p (Ω) and ϕ− = − min{ϕ, 0} ∈ W01,p (Ω). Since ϕ+ and ϕ− are compactly supported in Ω, by Remark 11.15 we obtain  |∇u(x)|p−2 ∇u(x) · ∇ϕ+ (x) dx ≥ 0 Ω



and

|∇u(x)|p−2 ∇u(x) · ∇ϕ− (x) dx ≤ 0. Ω

Thus  |∇u(x)|p−2 ∇u(x) · ∇ϕ(x) dx Ω   = |∇u(x)|p−2 ∇u(x) · ∇ϕ+ (x) dx − |∇u(x)|p−2 ∇u(x) · ∇ϕ− (x) dx ≥ 0. Ω

Ω

The reverse inequality follows by replacing ϕ with −ϕ. This shows that u is a weak solution.  1,p (Ω) is a weak suIt follows immediately from Definition 11.14 that u ∈ Wloc persolution if and only if −u is a weak subsolution in Ω. Next we discuss another symmetry property of super- and subsolutions.

11.3. WEAK SUPER- AND SUBSOLUTIONS

265

Lemma 11.17. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. Let ε > 0 1,p (Ω) is a weak supersolution to the p-Laplace equation in and assume that u ∈ Wloc Ω with u(x) ≥ ε for every x ∈ Ω. Then u1 is a weak subsolution to the p-Laplace equation in Ω. Proof. Let ϕ ∈ C0∞ (Ω) with ϕ ≥ 0 and let v = u1 . Then 0 < ε2 v ≤ u and by the chain rule ∇v = −u−2 ∇u almost everywhere in Ω. Thus ε2 |∇v| ≤ |∇u| almost 1,p everywhere in Ω and so v ∈ Wloc (Ω). This implies |∇v(x)|p−2 ∇v(x) = −u2(1−p) |∇u(x)|p−2 ∇u(x)

(11.5)

for almost every x ∈ Ω. Since u ≥ ε > 0 and ϕu2(1−p) ∈ W01,p (Ω) is compactly supported in Ω, by Remark 11.15 we may apply it as a test function. This, together with (11.5), gives 

0 ≤ |∇u(x)|p−2 ∇u(x) · ∇ϕ(x)u(x)2(1−p) + 2(1 − p)u(x)1−2p ∇u(x)ϕ(x) dx Ω u(x)2(1−p) |∇u(x)|p−2 ∇u(x) · ∇ϕ(x) dx = Ω  + 2(1 − p) ϕ(x)u(x)1−2p |∇u(x)|p dx Ω  ≤ − |∇v(x)|p−2 ∇v(x) · ∇ϕ(x) dx Ω

for every ϕ ∈ C0∞ (Ω) with ϕ ≥ 0. Hence v is a weak subsolution in Ω.



The class of weak subsolutions is closed with respect to truncation from below. Lemma 11.18. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. Assume 1,p (Ω) is a weak subsolution to the p-Laplace equation in Ω. Then that u ∈ Wloc 1,p u+ = max{u, 0} ∈ Wloc (Ω) is a weak subsolution to the p-Laplace equation in Ω. 1,p Proof. By Theorem 2.25, we have u+ ∈ Wloc (Ω). Let ϕ ∈ C0∞ (Ω) with ϕ ≥ 0, and define uk = min{ku+ , 1} for every k ∈ N. Then (uk )k∈N is a pointwise 1,p (Ω), 0 ≤ uk ≤ 1 for every k ∈ N and increasing sequence in Wloc

lim uk (x) = χ{x∈Ω:u(x)>0} (x)

k→∞

for every x ∈ Ω. Notice that uk ϕ ≥ 0 is compactly supported in Ω and  k∇u a.e. in {x ∈ Ω : 0 < ku(x) < 1}, ∇uk = 0 a.e. in {x ∈ Ω : ku(x) ≥ 1} ∪ {x ∈ Ω : u(x) ≤ 0}. By Remark 11.15 we can use uk ϕ as a test function, and the Leibniz rule (2.17) gives  0 ≥ |∇u(x)|p−2 ∇u(x) · ∇(uk ϕ)(x) dx Ω  p−2 = ϕ(x)|∇u(x)| ∇u(x) · ∇uk (x) dx + uk (x)|∇u(x)|p−2 ∇u(x) · ∇ϕ(x) dx Ω Ω   =k ϕ(x)|∇u(x)|p dx + uk (x)|∇u(x)|p−2 ∇u(x) · ∇ϕ(x) dx. 1 {x∈Ω:0 0, 1,p (Ω) is a weak supersolution to the p-Laplace equation in Ω assume that u ∈ Wloc with u(x) ≥ ε for every x ∈ Ω, and define v = log u. There exists a constant C = C(n, p) such that (11.7)



|v(x) − vB(z,r) | dx ≤ C

B(z,r)

for every ball B(z, r) with B(z, 2r)  Ω.

11.4. ENERGY ESTIMATES

269

Proof. Let ϕ ∈ C0∞ (B(z, 2r)) be such that 0 ≤ ϕ ≤ 1, ϕ = 1 in B(z, r) and |∇ϕ| ≤ Cr . By Theorem 11.23, we obtain    |∇v(x)|p dx ≤ ϕ(x)p |∇v(x)|p dx ≤ C(p) |∇ϕ(x)|p dx B(z,r)

Ω

C(p) ≤ p r

Ω



1 dx = C(n, p)r n−p , B(z,2r)

and thus |∇v| ∈ Lploc (Ω). On the other hand, since u ≥ ε and u ∈ Lploc (Ω), we 1,p (Ω). By the (1, p)-Poincar´e inequality, see have v ∈ Lploc (Ω), and so v ∈ Wloc Theorem 3.14, we conclude that   p1 p |v(x) − v | dx ≤ C(n, p)r |∇v(x)| dx B(z,r)   B(z,r)

B(z,r)

≤ C(n, p)r(r

−n n−p

r

1

) p = C(n, p).



The energy estimate in Theorem 11.20 can be generalized by incorporating suitable powers of the function u. Theorem 11.25. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. Assume that 1,p (Ω) is a weak subsolution to the p-Laplace equation in Ω and let α > 0. u ∈ Wloc There exists a constant C = C(p) such that   C u(x)α−1 |∇u(x)|p ϕ(x)p dx ≤ p u(x)p+α−1 |∇ϕ(x)|p dx α {u>0} {u>0} for every ϕ ∈ C0∞ (Ω) with ϕ ≥ 0. Here {u > 0} = {x ∈ Ω : u(x) > 0}. Proof. By Lemma 11.18 and Theorem 2.25, we may consider u = u+ . We would like to apply uα ϕp as a test function, but it is not clear whether this function belongs to W01,p (Ω). Thus we define ψk = ϕp min{uα , ku} for every k ∈ N. Then (11.8)

∇ψk = pϕp−1 ∇ϕ min{uα , ku} + ϕp ∇ min{uα , ku}

almost everywhere in Ω. Let Ωk = {x ∈ Ω : 0 < uα (x) ≤ ku(x)} for every k ∈ N. Since ∇u = 0 almost everywhere in the set {u = 0}, we have  αuα−1 ∇u a.e. in Ωk , (11.9) ∇ min{uα , ku} = k∇u a.e. in Ω \ Ωk . It follows that ψk ∈ W01,p (Ω) for every k ∈ N. Observe that uα−1 ≤ k in Ωk . Fix k ∈ N. Since u is a weak subsolution in Ω and ψk ≥ 0 is compactly supported in Ω, we have by Remark 11.15 and (11.8) that  0 ≥ |∇u(x)|p−2 ∇u(x) · ∇ψk (x) dx Ω = ϕ(x)p |∇u(x)|p−2 ∇u(x) · ∇ min{u(x)α , ku(x)} dx (11.10) Ω  ϕ(x)p−1 min{u(x)α , ku(x)}|∇u(x)|p−2 ∇u(x) · ∇ϕ(x) dx. +p Ω

270

11. THE p-LAPLACE EQUATION

By (11.9), (11.10), and Young’s inequality we obtain  α

 ϕ(x)p u(x)α−1 |∇u(x)|p dx + k ϕ(x)p |∇u(x)|p dx Ωk Ω\Ωk  = ϕ(x)p |∇u(x)|p−2 ∇u(x) · ∇ min{u(x)α , ku(x)} dx Ω ≤ p ϕ(x)p−1 min{u(x)α , ku(x)}|∇u(x)|p−2 ∇u(x) · ∇ϕ(x) dx Ω ≤p ϕ(x)p−1 u(x)α |∇u(x)|p−1 |∇ϕ(x)| dx Ωk  ϕ(x)p−1 u(x)|∇u(x)|p−1 |∇ϕ(x)| dx + pk Ω\Ωk   α k ≤ ϕ(x)p u(x)α−1 |∇u(x)|p dx + ϕ(x)p |∇u(x)|p dx 2 Ωk 2 Ω\Ωk   + C(p)α1−p u(x)p+α−1 |∇ϕ(x)|p dx + C(p)k u(x)p |∇ϕ(x)|p dx. Ωk

Ω\Ωk

1,p Since 0 < uα ≤ ku in Ωk and u ∈ Wloc (Ω), we have



 p

ϕ(x) u(x)

α−1

|∇u(x)| dx ≤ k

Ωk

and

ϕ(x)p |∇u(x)|p dx < ∞

p

Ω



 ϕ(x)p |∇u(x)|p dx ≤ Ω\Ωk

ϕ(x)p |∇u(x)|p dx < ∞. Ω

Hence these terms can be absorbed into the left-hand side. This gives 

 α

p

ϕ(x)p |∇u(x)|p dx  p+α−1 p u(x) |∇ϕ(x)| dx + C(p)k u(x)p |∇ϕ(x)|p dx,

α−1

ϕ(x) u(x)  1−p ≤ C(p)α Ωk

|∇u(x)| dx + k p

Ω\Ωk

Ωk

Ω\Ωk

where



k→∞

ϕ(x)p |∇u(x)|p dx −−−− →0

k Ω\Ωk

and



 u(x)p |∇ϕ(x)|p dx ≤

k Ω\Ωk

k→∞

u(x)p+α−1 |∇ϕ(x)|p dx −−−−→ 0, Ω\Ωk

since ku ≤ uα in Ω \ Ωk . Here we may assume that up+α−1 |∇ϕ|p ∈ L1 (Ω) since otherwise there is nothing to prove. By the monotone convergence theorem we

11.4. ENERGY ESTIMATES

conclude that   u(x)α−1 |∇u(x)|p ϕ(x)p dx = lim {u>0}

k→∞

 k→∞

u(x)α−1 |∇u(x)|p ϕ(x)p dx Ωk

ϕ(x)p u(x)α−1 |∇u(x)|p dx +

= lim

Ωk

271

k α



 ϕ(x)p |∇u(x)|p dx Ω\Ωk

   C(p) C(p)k p+α−1 p p p ≤ lim u(x) |∇ϕ(x)| dx + u(x) |∇ϕ(x)| dx k→∞ α p Ωk α Ω\Ωk  C(p) = u(x)p+α−1 |∇ϕ(x)|p dx.  αp {u>0} 

Next we prove for weak supersolutions a similar estimate as in Theorem 11.25. Theorem 11.26. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. Assume that 1,p u ∈ Wloc (Ω) is a nonnegative weak supersolution to the p-Laplace equation in Ω and let α > 0. There exists C = C(p) such that   C |∇u(x)|p u(x)−1−α ϕ(x)p dx ≤ p u(x)p−1−α |∇ϕ(x)|p dx α Ω Ω for every ϕ ∈ C0∞ (Ω) with ϕ ≥ 0. ∈ W01,p (Ω) as a Proof. Let uk = u + k1 for every k ∈ N and apply ϕp u−α k test function. Observe that this is compactly supported in Ω. Since u is a weak supersolution, by Remark 11.15 we have 

0 ≤ |∇u(x)|p−2 ∇u(x)· pϕ(x)p−1 ∇ϕ(x)uk (x)−α − αϕ(x)p uk (x)−1−α ∇uk (x) dx Ω  = p |∇uk (x)|p−2 ∇uk (x) · ϕ(x)p−1 ∇ϕ(x)uk (x)−α dx Ω  − α |∇uk (x)|p uk (x)−1−α ϕ(x)p dx, Ω

where the last term is finite. By Young’s inequality   p |∇uk (x)|p uk (x)−1−α ϕ(x)p dx ≤ |∇uk (x)|p−1 uk (x)−α ϕ(x)p−1 |∇ϕ(x)| dx α Ω Ω  p−1 ≤ |∇uk (x)|p uk (x)−1−α ϕ(x)p dx p Ω  p−1 −p +p α |∇ϕ(x)|p uk (x)p−1−α dx. Ω

The first term on the right-hand side is finite, and absorption of this term gives   p p  |∇uk (x)|p uk (x)−1−α ϕ(x)p dx ≤ |∇ϕ(x)|p uk (x)p−1−α dx. α Ω Ω By Fatou’s lemma, we obtain   p −1−α p |∇u(x)| u(x) ϕ(x) dx ≤ lim inf |∇uk (x)|p uk (x)−1−α ϕ(x)p dx k→∞ Ω Ω  p p  ≤ lim inf |∇ϕ(x)|p uk (x)p−1−α dx k→∞ α Ω  p p  |∇ϕ(x)|p u(x)p−1−α dx, = α Ω

272

11. THE p-LAPLACE EQUATION

where the final step follows from the monotone convergence theorem if p−1−α ≤ 0, and from the dominated convergence theorem otherwise; in the latter case, we may clearly assume that the last integral is finite.  11.5. Local boundedness of weak solutions 1,p For p > n, Morrey’s inequality implies that every function u ∈ Wloc (Ω) is continuous and thus locally bounded in Ω, see Remark 3.24. In this section we 1,p (Ω) to the p-Laplace equation is locally bounded show that a weak solution u ∈ Wloc for every 1 < p < ∞. Moreover, this result comes with a useful estimate that is sometimes called a weak maximum principle, see Theorem 11.30. First we show that a weak subsolution is locally bounded from above. The proof is based on the Moser iteration technique together with a Caccioppoli inequality and a Sobolev inequality. We will later extend this result for every β > 0, see Lemma 11.32.

Theorem 11.27. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. Assume that 1,p (Ω) is a weak subsolution to the p-Laplace equation in Ω and let β > p − 1. u ∈ Wloc There exist constants C = C(n, p, β) and τ = τ (n, p) > 0 such that   β1 R τ β u (x) dx (11.11) ess sup u+ (x) ≤ C + R − r B(z,R) x∈B(z,r) whenever B(z, R)  Ω and 0 < r < R. Remark 11.28. By choosing r =

R 2

and β = p in Theorem 11.27, we have   p1 p ess sup u(x) ≤ ess sup u+ (x) ≤ C(n, p)  u+ (x) dx

x∈B(z,r)

x∈B(z,r)

B(z,2r)



≤ C(n, p) 

|u(x)|p dx

 p1

< ∞.

B(z,2r)

In particular, weak subsolutions are locally essentially bounded from above in Ω. 1,p (Ω) is a Proof of Theorem 11.27. Lemma 11.18 implies that u+ ∈ Wloc weak subsolution in Ω. There are two stages in the proof. First we show that u+ ∈ Lβloc (Ω) for every 0 < β < ∞. As a byproduct, we obtain a reverse H¨ older inequality for u+ . In the second stage, we prove inequality (11.11) by using the so-called Moser iteration technique, where the reverse H¨ older inequality is applied recursively. In the first stage, we begin by assuming that u+ ∈ Lβloc (Ω) for some p − 1 < β < ∞. Observe that for β = p, and hence also for 0 < β ≤ p, this follows 1,p (Ω). Let B(z, R)  Ω and 0 < r < R, and let from the assumption u ∈ Wloc C ϕ ∈ C0∞ (B(z, R)) be such that ϕ = 1 in B(z, r), 0 ≤ ϕ ≤ 1 and |∇ϕ| ≤ R−r . Using Theorem 11.25 with α = β − (p − 1) > 0 and {u > 0} = {x ∈ Ω : u(x) > 0}, we obtain p   β β −1 p β ϕ(x)∇ u+ (x) p p dx = p ϕ(x) u(x) ∇u(x) dx p Ω {u>0}  β p  ϕ(x)p u(x)β−p |∇u(x)|p dx = p {u>0}  p  β u+ (x)β |∇ϕ(x)|p dx. ≤ C(p) β − (p − 1) Ω

11.5. LOCAL BOUNDEDNESS OF WEAK SOLUTIONS

273

By the Leibniz rule, β β β ∇ ϕ(x)u+ (x) p ≤ ϕ(x)∇ u+ (x) p + u+ (x) p ∇ϕ(x) , and thus



β ∇ ϕ(x)u+ (x) p p dx Ω    p β p β p−1 p p ϕ(x)∇ u+ (x) u+ (x) ∇ϕ(x) dx ≤2 dx + Ω Ω   p β β u+ (x) p ∇ϕ(x) p dx. ≤ C(p) +1 β − (p − 1) Ω

Since p > 1, we have  p β β p + (β − (p − 1))p  2β − p + 1 p +1= ≤ , β − (p − 1) (β − (p − 1))p β−p+1 and using the properties of ϕ we obtain p  β β ∇ ϕ(x)u+ (x) p p dx ≤ C(p) 2β − p + 1 u+ (x) p ∇ϕ(x) p dx   β−p+1 B(z,R) B(z,R)  2β − p + 1 1 p u+ (x)β dx. ≤ C(p) β−p+1 R−r  B(z,R)

β

By the estimates above, we find that ϕu+p ∈ W01,p (B(z, R)). The Sobolev inequality in Theorem 3.17 implies 1  κp   p1  β β p ϕ(x)u+ (x) p κp dx p ∇ ϕ(x)u+ (x) ≤ C(n, p)R  dx ,  B(z,R)

B(z,R)

where κ = κ(n, p) > 1. By combining the previous estimates and using the properties of ϕ, we obtain 1 1   κβ    κβ β κp R n κβ p ϕ(x)u+ (x) u+ (x) dx ≤ dx   r B(z,r) B(z,R) n   β1   κβ p β p R p β p (11.12) ≤ C(n, p) dx ∇ ϕ(x)u+ (x) R  r B(z,R)  β1   nβ  p R 2β − p + 1 R p β ≤ C(n, p) β u (x) dx . + r β − p + 1 R − r B(z,R) This is a reverse H¨older inequality, which holds whenever B(z, R)  Ω and 0 < r < R. From (11.12) and the assumption u+ ∈ Lβloc (Ω) we conclude that u+ ∈ Lκβ loc (Ω) with κ > 1. This observation makes it possible to increase the level of local integrability stepwise. In particular, since initially u+ ∈ Lploc (Ω) by the assumptions of the theorem, we may iterate (11.12) starting with β = p and conclude that u+ ∈ Lβloc (Ω) for every 0 < β < ∞. Next we fix β0 > p − 1 and show that (11.11) holds with this β0 . Note that if β ≥ β0 , then 2β − p + 1 2β0 − p + 1 ≤ = C(p, β0 ) β−p+1 β0 − p + 1

274

11. THE p-LAPLACE EQUATION

and u+ ∈ Lβloc (Ω) by the first part of the proof. Using (11.12) we obtain 1  κβ  β1   R  nβ  R  βp  p κβ β β u (x) dx ≤ C(n, p, β ) u (x) dx . + 0 +   r R−r B(z,r) B(z,R) We apply this estimate recursively. Let r0 = R and rk = r + R−r for every k ∈ N. 2k rk rk 2k+1 R Then r < rk+1 < rk ≤ R, rk+1 ≤ 2 and rk −rk+1 ≤ R−r for every k ∈ N0 , and thus 1  κβ  κβ u (x) dx +  B(z,rk+1 )

(11.13)

p β

≤ C(n, p, β0 ) · 2

n β

 2k+1 R  βp  R−r

 β1

β

u+ (x) dx



B(z,rk )

for every β ≥ β0 and k ∈ N0 . In the first step we have, by (11.13), 

 u+ (x)



κβ0

dx

1 κβ0

B(z,r1 )

≤ C(n, p, β0 )

p β0

·2

n β0

 β1  2R  βp  0 0 β0 u+ (x) dx .  R−r B(z,r0 )

In the second step, by applying (11.13) twice,  u+ (x)



κ2 β0

 dx

1 κ2 β0

B(z,r2 )

≤ C(n, p, β0 )

p κβ0

·2

p

n κβ0

p

 κβ1  22 R  κβp  0 0 κβ0 u+ (x) dx  R−r B(z,r1 ) n

n

≤ C(n, p, β0 ) β0 + κβ0 · 2 β0 + κβ0  β1  R  βp + κβp  p 2p 0 0 0 β0 · 2 β0 + κβ0 u (x) dx . +  R−r B(z,r0 ) By applying (11.13) recursively we obtain  u+ (x)

 (11.14)

κk β0

 dx

1 κk β0

p

≤ C(n, p, β0 ) β0

B(z,rk ) p

· 2 β0

k i=1

i κi−1

 R  βp 0 R−r

k i=1

1 κi−1

k i=1

1 κi−1

i=1

1

k→∞

κi−1

−−−− →

∞

i=1



u+ (x)β0 dx B(z,r0 )

1 1 = κi−1 1−

1 κ

=

κ , κ−1

and by recognizing the derivative of a geometric series we obtain k

i=1

i κi−1

k→∞

−−−− →

∞

i=1

i κi−1

=

k



for every k ∈ N. The sum of a geometric series gives k

n

· 2 β0

1 . (1 − κ1 )2

i=1

 β1

0

1 κi−1

11.5. LOCAL BOUNDEDNESS OF WEAK SOLUTIONS

275

Hence we conclude from (11.14) that  k1  k1    κ β0 κ β0 k rk n κk β0 u+ (x)κ β0 dx ≤ lim sup u (x) dx lim sup  +  r k→∞ k→∞ B(z,r) B(z,rk )  β1  κ  R  βp κ−1 0 0 ≤ C(n, p, β0 ) u+ (x)β0 dx .  R−r B(z,r0 ) Thus u is essentially bounded in the ball B(z, r) and   β1 pκ 0 R  κ−1 β0 ess sup u+ (x) ≤ C(n, p, β0 ) u (x) dx . +  R−r x∈B(z,r) B(z,R) The claim follows with τ = κ=



pκ κ−1 .

Remark 11.29. For 1 < p < n, by the proof of Theorem 11.27, we may choose pκ pκ n n−p and thus τ = κ−1 = n. When p ≥ n, the proof gives τ = κ−1 > p ≥ n. Next we conclude that every weak solution is locally essentially bounded.

Theorem 11.30. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. Assume that 1,p (Ω) is a weak solution to the p-Laplace equation in Ω and let β > p − 1. u ∈ Wloc There exist constants C = C(n, p, β) and τ = τ (n, p) > 0 such that   β1 R τ β ess sup |u(x)| ≤ C |u(x)| dx R − r B(z,R) x∈B(z,r) whenever B(z, R)  Ω and 0 < r < R. Proof. As a weak solution, u is a weak subsolution the p-Laplace equation 1,p (Ω) is a weak subsolution in Ω and thus by in Ω. By Lemma 11.18, u+ ∈ Wloc ∞ Theorem 11.27 we have u+ ∈ Lloc (Ω) with   β1 R τ β u+ (x) dx . ess sup u+ (x) ≤ C(n, p, β) R − r B(z,R) x∈B(z,r) On the other hand, since u is a weak solution −u is a weak solution as well. Again by 1,p Lemma 11.18, (−u)+ = u− ∈ Wloc (Ω) is a weak subsolution, and by Theorem 11.27 ∞ we have u− ∈ Lloc (Ω) with   β1 R τ β u (x) dx . ess sup u− (x) ≤ C(n, p, β) − R − r B(z,R) x∈B(z,r) This shows that u = u+ − u− ∈ L∞ loc (Ω). Moreover, ess sup |u(x)| = ess sup (u+ (x) + u− (x)) ≤ ess sup u+ (x) + ess sup u− (x) x∈B(z,r)

x∈B(z,r)

x∈B(z,r)

x∈B(z,r)

  β1 R τ β ≤ C(n, p, β) u (x) dx + R − r B(z,R)   β1 R τ β + C(n, p, β) u (x) dx − R − r B(z,R)   β1 R τ β ≤ C(n, p, β) |u(x)| dx . R − r B(z,R)



276

11. THE p-LAPLACE EQUATION

The following technical lemma will be used when proving that Theorem 11.27 holds for every β > 0. Lemma 11.31. Let Ψ : [0, R0 ] → R be a nonnegative bounded function and assume that there exist C1 > 0, α > 0 and 0 < ε < 1 such that Ψ(r) ≤ εΨ(R) + C1 (R − r)−α for every 0 ≤ r < R ≤ R0 . Then there exists a constant C = C(α, ε) such that Ψ(r) ≤ C(R − r)−α C1 for every 0 ≤ r < R ≤ R0 . Proof. Let 0 < τ < 1 and 0 ≤ r < R ≤ R0 . We define t0 = r and ti+1 = ti + (1 − τ )τ i (R − r), for every i ∈ N0 . Then r ≤ ti < ti+1 ≤ R for every i ∈ N0 and Ψ(t0 ) ≤ εΨ(t1 ) + C1 (t1 − t0 )−α = εΨ(t1 ) + C1 (1 − τ )−α (R − r)−α ≤ ε(εΨ(t2 ) + C1 (t2 − t1 )−α ) + C1 (1 − τ )−α (R − r)−α = ε2 Ψ(t2 ) + εC1 τ −α (1 − τ )−α (R − r)−α + C1 (1 − τ )−α (R − r)−α = ε2 Ψ(t2 ) + C1 (1 − τ )−α (R − r)−α (ετ −α + 1). Recursively, we obtain Ψ(r) = Ψ(t0 ) ≤ ε Ψ(tk ) + C1 (R − r) k

−α

(1 − τ )

−α

k−1

εi τ −iα

i=0

for every k ∈ N. Since Ψ is bounded, here εk Ψ(tk ) → 0 as k → ∞. By choosing τ = τ (ε, α) with τεα < 1, we conclude that   k−1

Ψ(r) ≤ lim sup εk Ψ(tk ) + C1 (R − r)−α (1 − τ )−α εi τ −iα k→∞

= C(α, ε)C1 (R − r)

i=0 −α

.



Lemma 11.32. Theorems 11.27 and 11.30 hold for every β > 0. Proof. We may assume that 0 < β ≤ p − 1 since for β > p − 1 the results are covered by Theorem 11.27 and Theorem 11.30, respectively. By Remark 11.29 we may also assume that τ = τ (n, p) ≥ n in Theorem 11.27. Let B(z, R0 )  Ω, 0 < r < R ≤ R0 , and 0 < ε < 1. Since B(z, R)  Ω, Theorem 11.27 and Young’s

11.5. LOCAL BOUNDEDNESS OF WEAK SOLUTIONS

277

inequality imply   p1 R τ p u+ (x) dx ess sup u+ (x) ≤ C(n, p) R − r B(z,R) x∈B(z,r)   p1 R τ β p−β ≤ C(n, p) u (x) ess sup u (x) dx + + R − r B(z,R) x∈B(z,R)   p1 β R τ β = C(n, p) u+ (x) dx ess sup u+ (x)1− p R − r B(z,R) x∈B(z,R)   β1  τ R β ≤ ε ess sup u+ (x) + C(n, p, ε, β) u+ (x) dx R − r B(z,R) x∈B(z,R) ≤ ε ess sup u+ (x) + C1 (R − r)− β , τ

x∈B(z,R)

where τ −n

 u+ (x)β dx

C1 = C(n, p, ε, β)R0 β

 β1

< ∞.

B(z,R0 )

Here we used the facts that τ ≥ n and that by Theorem 11.27 we have u+ ∈ L∞ loc (Ω). It is important to use R0 instead of R above since C1 is not allowed to depend on R. Without loss of generality, we may assume that C1 > 0. We let Ψ(0) = 0 and Ψ(r) = ess sup u+ (x) ≤ ess sup u+ (x) < ∞ x∈B(z,r)

x∈B(z,R0 )

for every 0 < r ≤ R0 . By Lemma 11.31 we then have  τ −n τ −β β ess sup u+ (x) ≤ C(n, p, ε, β)(R − r) R0 x∈B(z,r)

β

 β1

u+ (x) dx

B(z,R0 )

≤ C(n, p, ε, β)

  β1 R0 τ β u (x) dx + R − r B(z,R0 )

whenever 0 < r < R ≤ R0 . By choosing R = R0 we conclude that Theorem 11.27 holds for every β > 0. Finally, the proof of Theorem 11.30 together with the knowledge that Theorem 11.27 holds for every β > 0 shows that also Theorem 11.30 holds for every β > 0.  Next we consider a version of Theorem 11.27 for nonnegative weak supersolutions. Lemma 11.33. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. Assume that 1,p u ∈ Wloc (Ω) is a nonnegative weak supersolution to the p-Laplace equation in Ω and let β > 0. There exist constants C = C(n, p, β) and τ = τ (n, p) > 0 such that − β1  R τ −β u(x) dx ≤ C ess inf u(x) (11.15) R − r B(z,R) x∈B(z,r) whenever B(z, R)  Ω and 0 < r < R. Moreover, the constants C and τ are the same as in Theorem 11.27 and Lemma 11.32. Proof. We may assume that the left-hand side of (11.15) is not zero. Since the function uk = u+ k1 ≥ k1 is a weak supersolution in Ω for every k ∈ N, Lemma 11.17

278

11. THE p-LAPLACE EQUATION

implies that u1k = ( u1k )+ is a weak subsolution in Ω for every k ∈ N. For a fixed k ∈ N, Theorem 11.27 and Lemma 11.32 give  β1   β 1 R τ 1 dx 0 < ess sup ≤ C(n, p, β) R − r B(z,R) uk (x) x∈B(z,r) uk (x) and equivalently  − β1  −1 R τ 1 −β 0< uk (x) dx ≤ C(n, p, β) ess sup R − r B(z,R) x∈B(z,r) uk (x) = C(n, p, β) ess inf uk (x) x∈B(z,r)   1 = C(n, p, β) ess inf u(x) + . k x∈B(z,r) The claim follows from the monotone convergence theorem by letting k → ∞ since − β1  − β1  u(x)−β dx = lim  uk (x)−β dx 0<  k→∞

B(z,R)

B(z,R)

 = lim

k→∞

uk (x)



−β

− β1 dx

.



B(z,R)

Another way to prove Lemma 11.33 is to apply the Moser iteration technique as in the proof of Theorem 11.27 using Theorem 11.26 for weak supersolutions. This alternative approach avoids the use of Lemma 11.17. 11.6. Harnack’s inequality Harnack’s inequality states that locally the supremum of a nonnegative solution is bounded by a constant times the infimum of the solution. However, since a 1,p function u ∈ Wloc (Ω) is defined only up to a set of measure zero, we consider the essential supremum and infimum. For a continuous function, these can be replaced by the standard supremum and infimum. Harnack’s inequality can be seen as a quantitative version of the maximum principle. We begin with Harnack’s inequality for p > n. In this case, Harnack’s inequality holds for nonnegative weak supersolutions. Theorem 11.34. Let n < p < ∞ and let Ω ⊂ Rn be an open set. Assume that 1,p u ∈ Wloc (Ω) is a nonnegative weak supersolution to the p-Laplace equation in Ω. There exists a constant C = C(n, p) such that ess sup u(x) ≤ C ess inf u(x) x∈B(z,r)

x∈B(z,r)

whenever B(z, r) is a ball with B(z, 2r)  Ω. Proof. The function uk = u + k1 ≥ k1 is weak supersolution in Ω for every 1,p (Ω) for every k ∈ N. k ∈ N. As in the proof of Theorem 11.24, we have log uk ∈ Wloc Let k ∈ N and let B(z, r) be a ball with B(z, 2r)  Ω. Since p > n, by Morrey’s inequality, see Remark 3.24, there is a set Ek ⊂ B(z, r) such that |Ek | = 0 and n

|log uk (x) − log uk (y)| ≤ C(n, p)|x − y|1− p ∇ log uk Lp (B(z,r)) n

≤ C(n, p)r 1− p ∇ log uk Lp (B(z,r))

11.6. HARNACK’S INEQUALITY

279

for every x, y ∈ B(z, r) \ Ek . Theorem 11.23 implies   |∇ log uk (x)|p ϕ(x)p dx ≤ C(p) |∇ϕ(x)|p dx Ω

Ω

C0∞ (Ω)

C0∞ (B(z, 2r))

with ϕ ≥ 0. By taking ϕ ∈ such that 0 ≤ ϕ ≤ 1, for every ϕ ∈ ϕ = 1 in B(z, r) and |∇ϕ| ≤ Cr , we obtain   |∇ log uk (x)|p dx ≤ C(p) |∇ϕ(x)|p dx ≤ C(n, p)r n−p , B(z,r)

B(z,2r)

and thus ∇ log uk Lp (B(z,r)) ≤ C(n, p)r p −1 . n

By the estimates above, log uk (x) = log uk (x) − log uk (y) ≤ C(n, p)r 1− np ∇ log uk Lp (B(z,r)) uk (y) ≤ C(n, p)r 1− p r p −1 = C(n, p) n

n

and consequently uk (x) ≤ eC(n,p) uk (y) for every x, y ∈ B(z, r) \ Ek . Finally, by taking the limit k → ∞, we see that u(x) ≤ eC(n,p) u(y)  for every x, y ∈ B(z, r) \ E, where E = k∈N Ek satisfies |E| = 0. This proves Harnack’s inequality ess sup u(x) ≤ eC(n,p) ess inf u(x). x∈B(z,r)

x∈B(z,r)



Remark 11.35. In the case p = n, there is a direct proof for Harnack’s inequality for nonnegative weak solutions to the n-Laplace equation, see Granlund, Lindqvist and Martio [161, Theorem 4.15]. Next we consider Harnack’s inequality in the case 1 < p < ∞. In the proof we apply BMO and the John–Nirenberg inequality, see Section 9.7. Since it was convenient to use cubes in Section 9.7, we also use cubes in the proof below. However, this is just a technical point and we could work either with cubes or balls throughout. Theorem 11.36. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. Assume that 1,p (Ω) is a nonnegative weak solution to the p-Laplace equation in Ω. There u ∈ Wloc exists a constant C = C(n, p) such that ess sup u(x) ≤ C ess inf u(x) x∈B(z,r)

x∈B(z,r)

√ for every ball B(z, r) with B(z, 6 nr)  Ω.

√ Proof. Let B(z, r) be a ball with B(z, 6 nr)  Ω. Since u = u+ is a weak sub- and supersolution, Lemma 11.32 and Lemma 11.33 respectively imply   β1 u(x)β dx ess sup u(x) ≤ C(n, p, β)  x∈B(z,r)

B(z,2r)

280

11. THE p-LAPLACE EQUATION

and

 u(x)



−β

− β1 dx

≤ C(n, p, β) ess inf u(x), x∈B(z,r)

B(z,2r)

for every β > 0. It suffices to show that there exists β = β(n, p) > 0 satisfying the reverse H¨ older inequality  (11.16)

u(x)β dx



 β1

B(z,2r)

 ≤ C(n, p) 

u(x)−β dx

− β1 ,

B(z,2r)

since this, together with the estimates above, gives  ess sup u(x) ≤ C(n, p) 

x∈B(z,r)

 ≤ C(n, p) 

β

 β1

u(x) dx B(z,2r)

u(x)

−β

− β1 dx

≤ C(n, p) ess inf u(x). x∈B(z,r)

B(z,2r)

For k ∈ N, let

vk = log u + k1 , √ and let Q = Q(z  , r  ) ⊂ Q(z, 2r). Since B(z, 6 nr)  Ω and u + weak supersolution in Ω for every k ∈ N, Theorem 11.24 implies 

Q

1 k

≥

1 k

> 0 is a

|vk (x) − (vk )Q | dx ≤ C(n)  vk (x) − (vk )B  dx ≤ C(n, p), B

√ where B  = B(z  , nr  ). It follows that vk is of bounded mean oscillation over such cubes Q , and thus vk ∈ BMOd (Q(z, 2r)) with vk BMOd (Q(z,2r)) ≤ C(n, p). By Theorem 9.41, there exists β = β(n, p) > 0, independent of k ∈ N, such that 

eβ|vk (x)−(vk )Q(z,2r) | dx ≤ C(n).

Q(z,2r)

Hence, 

B(z,2r)

eβvk (x) dx 

B(z,2r)

= C(n) 

Q(z,2r)

Q(z,2r)

eβ(vk (x)−(vk )Q(z,2r) ) dx 



≤ C(n) 

e−βvk (x) dx ≤ C(n) 

eβ|vk (x)−(vk )Q(z,2r) | dx

eβvk (x) dx 

e−βvk (x) dx

Q(z,2r)

e−β(vk (x)−(vk )Q(z,2r) ) dx

Q(z,2r) 2

≤ C(n).

Q(z,2r)

We emphasize that the constants in this estimate are independent of k ∈ N. Since u + 1 ∈ Lβloc (Ω), we can apply both dominated and monotone convergence theorems

11.6. HARNACK’S INEQUALITY

to conclude that  

β

u(x) dx

B(z,2r)

 β1

 = lim

k→∞



 = lim

k→∞



B(z,2r)

281



β u(x) + k1 dx eβvk (x) dx

 β1

 β1

B(z,2r)



≤ C(n, β) lim

k→∞



 = C(n, β) lim

k→∞



e−βvk (x) dx

− β1

B(z,2r)





u(x) +

B(z,2r)

= C(n, β) 

u(x)

−β

1 −β k

− β1 dx

− β1 dx

.

B(z,2r)

Since β = β(n, p), the desired reverse H¨ older inequality (11.16) follows, and the proof is complete.  Harnack’s inequality implies the following strong maximum principle for weak solutions. An analogous argument gives a strong minimum principle as well. Theorem 11.37. Let 1 < p < ∞ and let Ω ⊂ Rn be a connected open set. 1,p (Ω) is a continuous representative of a weak solution to the Assume that u ∈ Wloc p-Laplace equation in Ω. If u attains its maximum in Ω, then u is constant in Ω. Proof. Assume that there exists x0 ∈ Ω such that u(x0 ) = max u(x). x∈Ω

Then the function x √ → u(x0 )−u(x) is a nonnegative weak solution in Ω. Let r0 > 0 be such that B(x0 , 6nr0 )  Ω. By Harnack’s inequality in Theorem 11.36, we have sup (u(x0 ) − u(x)) ≤ C(n, p) min (u(x0 ) − u(x)) = 0. x∈B(x0 ,r0 )

x∈B(x0 ,r0 )

Thus u(x) = u(x0 ) for every x ∈ B(x0 , r0 ). Let then x ∈ Ω. Since Ω is connected, √ there exists a finite chain of balls B(xi , ri ), i = 0, . . . , k, such that B(xi , 6 nri )  Ω, xk = x and B(xi , ri ) ∩ B(xi+1 , ri+1 ) = ∅, for every i = 0, 1, . . . , k − 1. By repeating the first part of the proof in each of these balls we conclude that u is constant in each of the balls,  and thus u(x) = u(x0 ) for every x ∈ Ω. Remark 11.38. The strong maximum principle implies the standard maximum 1,p principle: if 1 < p < ∞ and u ∈ C(Ω) ∩ Wloc (Ω) is a weak solution to the p-Laplace n equation in a bounded open set Ω ⊂ R , then max u = max u. Ω

∂Ω

Remark 11.39. Let 1 < p < ∞ and let Ω ⊂ Rn be a connected open set. 1,p (Ω) is a nonnegative weak solution to the p-Laplace Assume that u ∈ C(Ω) ∩ Wloc equation in Ω and let Ω  Ω. A chaining argument and Harnack’s inequality in Theorem 11.36 give the pointwise estimate 1 u(y) ≤ u(x) ≤ Cu(y) C

282

11. THE p-LAPLACE EQUATION

for every x, y ∈ Ω , with a constant C = C(n, p, Ω , Ω) ≥ 1. In particular, if u(x0 ) = 0 for some x0 ∈ Ω, then u(x) = 0 for every x ∈ Ω. 11.7. Local H¨ older continuity 1,p For p > n, a weak solution u ∈ Wloc (Ω) to the p-Laplace equation in Ω ⊂ Rn has a locally H¨ older continuous representative solely based on the Sobolev regular1,p (Ω) is not necessarily ity, see Remark 3.24. For 1 < p ≤ n, a function u ∈ Wloc even continuous. However, in this section we show that if u is a weak solution to the p-Laplace equation for any 1 < p < ∞, then u has a locally H¨ older continuous representative. We begin with the case p = n. 1,n (Ω) is a weak Lemma 11.40. Let Ω ⊂ Rn be an open set. Assume that u ∈ Wloc solution to the n-Laplace equation in Ω. Then u has a locally H¨ older continuous representative in Ω.

Proof. First assume that B(x, 2R)  Ω. Let 0 < r < R, let ϕ ∈ C0∞ (B(x, 2r)) be such that 0 ≤ ϕ ≤ 1, ϕ = 1 in B(x, r) and |∇ϕ| ≤ Cr , and define A(x, r) = 1,n B(x, 2r) \ B(x, r). Then u − uA(x,2r) ∈ Wloc (Ω) is a weak solution to the n-Laplace equation in Ω. Theorem 11.20 implies   n |∇u(y)| dy ≤ ϕ(y)n |∇(u − uA(x,r) )(y)|n dy B(x,r) Ω  ≤ C(n) |u(y) − uA(x,r) |n |∇ϕ(y)|n dy Ω C(n) ≤ n |u(y) − uA(x,r) |n dy. r A(x,r) It is straightforward to verify that A(x, r) satisfies the Boman chain condition with τ = τ (n), see Definition 9.2. By Theorem 9.32 and H¨ older’s inequality we obtain   |u(y) − uA(x,r) |n dy ≤ C(n)r n |∇u(y)|n dy. A(x,r)

A(x,r)





Thus

|∇u(y)|n dy ≤ C1 B(x,r)

|∇u(y)|n dy, A(x,r)

where C1 = C(n). This implies    (1 + C1 ) |∇u(y)|n dy ≤ C1 |∇u(y)|n dy + C1 |∇u(y)|n dy B(x,r) B(x,r) A(x,r)  = C1 |∇u(y)|n dy, B(x,2r)

and consequently  |∇u(y)|n dy ≤ B(x,r)

C1 1 + C1

 |∇u(y)|n dy. B(x,2r)



Let

|∇u(y)|n dy

Ψ(r) =

and γ =

B(x,r)

Then 0 < γ < 1 and Ψ(r) ≤ γΨ(2r). By iteration Ψ(r) ≤ γ k Ψ(2k r)

C1 . 1 + C1

¨ 11.7. LOCAL HOLDER CONTINUITY

283

for every k ∈ N with 2k−1 r < R. Let k ∈ N with 2k−1 r < R ≤ 2k r. Then  r δ Ψ(2R) < ∞, Ψ(r) ≤ γ k Ψ(2k r) ≤ R where δ = − log2 γ > 0. Moreover, by considering min{δ, 1} instead of δ, we may assume that 0 < δ ≤ 1. By H¨ older’s inequality,   n1 1 n |∇u(y)| dy ≤  |∇u(y)| dy = C(n)r −1 Ψ(r) n  (11.17) B(x,r) B(x,r)

1 δ ≤ C(n) R−δ Ψ(2R) n r n −1 . Next we use the fractional maximal function with α = 1 − nδ > 0, see Definition 1.38. By taking supremum over 0 < r < R in (11.17), we obtain   n1  −δ n |∇u(y)| dy < ∞, Mα,R |∇u|(x) ≤ C(n) R B(x,2R)

which holds for every x ∈ Ω and R > 0 satisfying B(x, 2R)  Ω. Let B(z, 3R0 )  Ω and let x, x ∈ B(z, R80 ). Then B(x, 2R0 )  Ω and thus   n1  −δ n Mα,4|x−x | |∇u|(x) ≤ Mα,R0 |∇u|(x) ≤ C(n) R0 |∇u(y)| dy B(x,2R0 )

  ≤ C(n) R0−δ

|∇u(y)|n dy

 n1

< ∞.

B(z,3R0 )

A corresponding estimate holds for Mα,R0 |∇u|(x ) as well. By a localized version of Theorem 4.17, there exists E ⊂ Rn such that |E| = 0 and

|u(x) − u(x )| ≤ C(n, α)|x − x |1−α Mα,4|x−x | |∇u|(x) + Mα,4|x−x | |∇u|(x )  n1   |∇u(y)|n dy ≤ C(n, α)|x − x |1−α R0−δ B(z,3R0 ) 

for every x, x ∈ B(z, representative in Ω.

R0 8 )

\ E. This shows that u has a locally H¨ older continuous 

The following theorem shows that a weak solution to the p-Laplace equation has a locally H¨ older continuous representative for every 1 < p < ∞. The proof is based on Harnack’s inequality and an iteration scheme. Theorem 11.41. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. Assume 1,p (Ω) is a weak solution to the p-Laplace equation in Ω. Then u has a that u ∈ Wloc locally H¨ older continuous representative in Ω. Proof. Assume that B(x, 2R)  Ω. For 0 < r ≤ R, we define m(x, r) = ess inf u(y) y∈B(x,r)

and M (x, r) = ess sup u(y). y∈B(x,r)

The oscillation of u in B(x, r) is then osc u = M (x, r) − m(x, r).

B(x,r)

By Theorem 11.30, we have −∞ < m(x, r) ≤ M (x, r) < ∞ for every 0 < r ≤ R.

284

11. THE p-LAPLACE EQUATION

√ Let b = 7 n and fix 0 < r < Rb . Then functions y → u(y) − m(x, br) and y → M (x, br) − u(y) are weak solutions to the p-Laplace equation in B(x, br) and u − m(x, br) ≥ 0 and M (x, br) − u ≥ 0 almost everywhere in B(x, br). By Harnack’s inequality in Theorem 11.36, applied in the open set B(x, br), we have M (x, r) − m(x, br) = ess sup (u(y) − m(x, br)) y∈B(x,r)

≤ C(n, p) ess inf (u(y) − m(x, br)) y∈B(x,r)

= C(n, p)(m(x, r) − m(x, br)). A similar argument gives M (x, br) − m(x, r) = ess sup (M (x, br) − u(y)) y∈B(x,r)

≤ C(n, p) ess inf (M (x, br) − u(y)) y∈B(x,r)

= C(n, p)(M (x, br) − M (x, r)). By combining these estimates we obtain M (x, r) − m(x, br) + M (x, br) − m(x, r)

≤ C(n, p) m(x, r) − m(x, br) + M (x, br) − M (x, r) . Reorganization of the terms gives M (x, r) − m(x, r) ≤

C(n, p) − 1 (M (x, br) − m(x, br)), C(n, p) + 1

that is, (11.18)

osc u ≤ γ osc u,

B(x,r)

B(x,br)

with γ =

C(n, p) − 1 ∈ (0, 1). C(n, p) + 1

Let k ∈ N with bk r < R ≤ bk+1 r. By iterating (11.18), we obtain 1  r α (11.19) osc u ≤ γ k osc u ≤ osc u, γ R B(x,R) B(x,r) B(x,bk r) where α = − logb γ. This estimate holds whenever B(x, 2R)  Ω and 0 < r < Rb √ with b = 7 n. Let B(z, 3R0 )  Ω and let x, x ∈ B(z, R4b0 ), x = x , be Lebesgue points of u. Then B(x, 2R0 )  Ω and 2|x − x | < Rb0 . Since x and x are Lebesgue points of u, Theorem 1.21 and (11.19) imply  α 2|x − x | |u(x) − u(x )| ≤ osc  u ≤ C(n, p) osc u R0 B(x,2|x−x |) B(x,R0 ) ≤ C(n, p)R0−α |x − x |α ess sup |u(y)| y∈B(x,R0 )

≤

C(n, p)R0−α |x

 α

−x|

ess sup |u(y)|. y∈B(z,2R0 )

Here ess sup |u(y)| < ∞ y∈B(z,2R0 ) 1,p by Theorem 11.30, and therefore the function u ∈ Wloc (Ω) has a locally H¨ older continuous representative in Ω. 

¨ 11.7. LOCAL HOLDER CONTINUITY

285

1,p Remark 11.42. Let 1 < p < ∞ and let u ∈ Wloc (Ω) be a weak solution to the n p-Laplace equation in an open set Ω ⊂ R . Theorem 11.41 shows that u is locally 0,α (Ω). By regularity theory even the gradient H¨older continuous in Ω, that is, u ∈ Cloc 1,α (Ω). See DiBenedetto [101], ∇u is locally H¨ older continuous in Ω, that is, u ∈ Cloc Evans [120], Lewis [270], Tolksdorf [382], Uhlenbeck [389] and Uraltseva [390].

Remark 11.43. Let 1 < p < ∞, let Ω ⊂ Rn be a bounded open set, and assume that g ∈ W 1,p (Ω). By Theorem 11.13 and Theorem 11.11 there exists a unique weak solution u ∈ W 1,p (Ω) to the p-Laplace equation with u − g ∈ W01,p (Ω). Recall from Remark 6.21 that a point y ∈ ∂Ω satisfies the Wiener criterion if

 1  1 capp Ωc ∩ B(y, r), B(y, 2r) p−1 dr

= ∞. r capp B(y, r), B(y, 2r) 0 This is a sufficient condition the for regularity of a boundary point for the Dirichlet problem for the p-Laplace equation. That is, if y ∈ ∂Ω satisfies the Wiener criterion, then for every g ∈ C(∂Ω) the solution u to the Dirichlet problem with boundary values g satisfies lim u(x) = g(y). x→y

This result was obtained by Mazya [317]. See also Gariepy and Ziemer [140, 141]. Note that if the complement Ωc satisfies the p-capacity density condition in Definition 6.17, then the Wiener criterion is satisfied at every point of the boundary. The Wiener criterion in fact characterizes regular boundary points. The necessity was proved by Lindqvist and Martio [281] for p > n − 1 and by Kilpel¨ainen and Mal´ y [222] for p > 1. See also Adams and Hedberg [4], Heinonen, Kilpel¨ainen and Martio [187], Mal´ y and Ziemer [303] and Mikkonen [327]. For the Laplace equation such a characterization was obtained by Wiener [396], for uniformly elliptic linear equations by Littman, Stampacchia and Weinberger [282] and for linear equations in a weighted case by Fabes, Kenig and Serapioni [123]. Remark 11.44. Let 1 < p < ∞ and let Ω ⊂ Rn be a bounded open set. Assume that g ∈ W 1,p (Ω) ∩ C(Ω) and that u is the weak solution to p-Laplace equation in Ω with u − g ∈ W01,p (Ω). There exists a constant C = C(n, p) such that osc g osc u ≤ Ω∩B(y,r1 )

∂Ω∩B(y,2r2 )



+ osc g exp −C ∂Ω



 c

 1 Ω ∩ B(y, r), B(y, 2r) p−1 dr

r capp B(y, r), B(y, 2r)

r2  cap p

r1

for every y ∈ ∂Ω and 0 < r1 ≤ r2 < ∞. Sufficiency of the Wiener criterion for the boundary regularity follows form this estimate, see Bj¨orn [34], Bj¨orn and Bj¨ orn [31, Chapter 11], Bj¨ orn, MacManus and Shanmugalingam [36], DiBenedetto and Gianazza [102], Heinonen, Kilpel¨ainen and Martio [187, Theorem 6.18, Theorem 6.27], Mazya [316, 317], Stredulinsky [374] and Ziemer [406]. In particular, if Ωc satisfies the p-capacity density condition in Definition 6.17 with a constant C1 , then  r δ 1 osc u ≤ osc g + osc g ∂Ω r2 Ω∩B(y,r1 ) ∂Ω∩B(y,r2 ) for every y ∈ ∂Ω and 0 < r1 ≤ r2 < ∞ with δ = δ(n, p, C1 ) > 0. Here the oscillation of a function is as in the proof of Theorem 11.41. This estimate implies that if Ωc

286

11. THE p-LAPLACE EQUATION

satisfies the p-capacity density condition, then a solution to the Dirichlet problem for H¨ older continuous boundary values is H¨ older continuous in Ω, but possibly with a different H¨ older continuity exponent. For further details, we refer to Heinonen, Kilpel¨ainen and Martio [187, Theorem 6.44]. 11.8. Notes Our presentation of the theory of the p-Laplace equation is based on Heinonen, Kilpel¨ainen and Martio [187], Lindqvist [280] and Mal´ y and Ziemer [303]. For the corresponding theory on metric measure spaces, we refer to Bj¨orn and Bj¨orn [31]. The Moser iteration scheme is based on the works of Moser [331, 332], Serrin [359] and Trudinger [386]. It can be found in Bj¨orn and Bj¨orn [31, Chapter 8], Bj¨orn and Marola [33], Chen and Wu [82, Chapter 4], Fabes, Kenig and Serapioni [124], Giaquinta [148, Section 5.2], Giaquinta and Martinazzi [149, Section 8.3], Gilbarg and Trudinger [152, Chapter 8], Han and Lin [178], Heinonen, Kilpel¨ainen and Martio [187, Chapter 3], Lindqvist [280, Chapter 3], Mal´ y and Ziemer [303, Section 2.3.5] and Stredulinsky [374, Chapter 3]. It is possible to avoid the use of the John–Nirenberg inequality in the proof of the Harnack inequality, see Bombieri and Giusti [54], Han and Lin [178, Section 4.4], Saloff-Coste [357, Chapter 2] and Trudinger [387]. For the parabolic case we refer to Ivert, Marola and Masson [205], Kinnunen and Kuusi [228] and Marola and Masson [307]. The hole filling technique in the proof of Theorem 11.40 is from Widman [395]. See also Granlund, Lindqvist and Martio [161] and Lindqvist [280]. De Giorgi’s method [97] gives an alternative approach to H¨older continuity and Harnack inequalities. See also DiBenedetto and Trudinger [103], Giaquinta [148, Chapter 5], Giaquinta and Martinazzi [149, Section 8.3], Giusti [153, Chapter 7], Kinnunen and Shanmugalingam [238], Ladyzhenskaya and Uraltseva [253] and Mal´ y and Ziemer [303, Section 2.3.4].

10.1090/surv/257/12

CHAPTER 12

Stability Results for the p-Laplace Equation In this chapter we apply results and techniques from the previous chapters of the book for three themes related to Sobolev space properties of weak solutions to the p-Laplace equation. These themes are higher integrability of the gradient of a weak solution, stability of weak solutions with given boundary values with respect to a varying p, and very weak solutions, which are assumed to belong to a Sobolev space below the natural exponent p. In particular, these results show that for a solution to the p-Laplace equation, we can reach a Sobolev space higher than the natural one from a Sobolev space below than the natural one. The capacity density condition for the complement, see Definition 6.17, turns out to be essentially optimal for the global higher integrability and stability results, but these are nontrivial already for sets with smooth boundaries. 12.1. Higher integrability of the gradient We begin with a local higher integrability result for the gradient of a weak 1,p (Ω) to the p-Laplace equation solution, showing that a weak solution u ∈ Wloc 1,p+δ belongs to a slightly higher Sobolev space, that is, u ∈ Wloc (Ω) for some δ > 0. The proof is based on the energy estimate in Lemma 11.22 and a Sobolev–Poincar´e inequality in Theorem 3.14, which give a reverse H¨older inequality for the weak gradient. Local higher integrability then follows from the self-improvement property of reverse H¨ older inequalities, see Theorem 8.43. Theorem 12.1. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. Assume 1,p (Ω) is a weak solution to the p-Laplace equation in Ω. There exists that u ∈ Wloc δ = δ(n, p) > 0 such that |∇u| ∈ Lp+δ loc (Ω). Moreover, there exists a constant C = C(n, p) such that  

|∇u(x)|p+δ dx

1  p+δ

B(z,r)

 ≤C 

|∇u(x)|p dx

B(z,2r)

whenever B(z, 2r)  Ω. Proof. Let

  np  ,1 , max n+p q= p 2,

and

 ∗

q =

1 < p < n, p ≥ n, 1 ≤ q < n, q ≥ n.

nq n−q ,

2q, 287

 p1

288

12. STABILITY RESULTS FOR THE p-LAPLACE EQUATION

1,p 1,q Then 1 ≤ q < p ≤ q ∗ < ∞ and u ∈ Wloc (Ω) ⊂ Wloc (Ω). By Theorem 3.14, we have  q1∗   q1  q∗ q |u(x) − u | dx ≤ C(n, q)r |∇u(x)| dx (12.1) B(z,2r)   B(z,2r)

B(z,2r)

whenever B(z, 2r)  Ω. Since q = q(n, p), we conclude that the constant C in the Sobolev–Poincar´e inequality above only depends on n and p. By Lemma 11.22, H¨ older’s inequality and (12.1), we obtain  p1  p1   C(n, p) p p |∇u(x)| dx ≤ |u(x) − uB(z,2r) | dx   r B(z,r) B(z,2r)  q1∗  C(n, p) q∗ ≤ |u(x) − uB(z,2r) | dx  r B(z,2r)   q1 q ≤ C(n, p)  |∇u(x)| dx . B(z,2r)

Let f = |∇u| . The estimate above can be rewritten as  pq  p q dx f (x) ≤ C(n, p)  f (x) dx  q

B(z,r)

B(z,2r)

for every B(z, 2r)  Ω. The claim follows from Remark 8.44.



Remark 12.2. It is known that the gradient of a weak solution to the p-Laplace equation is locally H¨older continuous and thus ∇u ∈ L∞ loc (Ω), see Remark 11.42. However, Theorem 12.1 comes with a reverse H¨ older inequality for the weak gradient. Moreover, a weak solution to a more general partial differential equation of the p-Laplacian type is not more regular than locally H¨ older continuous and the argument also applies in this case. An example in Meyers [324] and Serrin [360] shows that the higher integrability exponent in Theorem 12.1 is not very large and depends on the structural constants for general elliptic elliptic equations. See also Chen and Wu [82, p. 189] and Giaquinta [147, p. 157]. Corollary 12.3. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. Assume 1,p (Ω) is a weak solution to the p-Laplace equation in Ω. There exists that u ∈ Wloc 1,p+δ δ = δ(n, p) > 0 such that u ∈ Wloc (Ω). Proof. By Theorem 3.14, we obtain q = q(n, p) > p such that   q1   p1 q p |u(x) − uB(z,r) | dx ≤ C(n, q)r  |∇u(x)| dx  B(z,r)

B(z,r)

for every B(z, r)  Ω, which implies  q1   1q  q q |u(x)| dx ≤  |u(x) − uB(z,r) | dx + |uB(z,r) |  B(z,r)

B(z,r)



≤ C(n, q)r 

|∇u(x)| dx p

 p1

+ |u|B(z,r) < ∞.

B(z,r)

This shows that u ∈ Lqloc (Ω) for some q = q(n, p) > p. Together with Theorem 12.1 1,p+δ this implies that u ∈ Wloc (Ω) for some δ = δ(n, p) > 0. 

12.1. HIGHER INTEGRABILITY OF THE GRADIENT

289

Remark 12.4. Theorem 11.41 asserts that a weak solution to the p-Laplace equation is continuous, and thus u ∈ L∞ loc (Ω). This fact together with Theorem 12.1 can be used to give an alternative proof of the previous corollary. Next we consider a global higher integrability result over the entire open set Ω. In the argument we need the following variant of the energy estimate given in Theorem 11.20. Observe that the support of the test function ϕ in the following lemma need not be a compact subset of Ω. Lemma 12.5. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. Assume that g ∈ W 1,p (Ω) and that u ∈ W 1,p (Ω) is a weak solution to the p-Laplace equation in Ω with u − g ∈ W01,p (Ω). There exists a constant C = C(p) such that     p p p p p p ϕ(x) |∇u(x)| dx ≤ C |u(x) − g(x)| |∇ϕ(x)| dx + ϕ(x) |∇g(x)| dx Ω

Ω

for every ϕ ∈

C0∞ (Rn )

Ω

with ϕ ≥ 0.

Proof. Let ϕ ∈ C0∞ (Rn ) with ϕ ≥ 0 and let v = ϕp (u − g). Since u − g ∈ 1,p W0 (Ω), Corollary 5.29 implies v ∈ W01,p (Ω). By Leibniz’s rule ∇v(x) = ϕ(x)p (∇u(x) − ∇g(x)) + pϕ(x)p−1 ∇ϕ(x)(u(x) − g(x)) for almost every x ∈ Ω. Lemma 11.6 gives  0 = |∇u(x)|p−2 ∇u(x) · ∇v(x) dx Ω = ϕ(x)p |∇u(x)|p−2 ∇u(x) · (∇u(x) − ∇g(x)) dx Ω  +p ϕ(x)p−1 (u(x) − g(x))|∇u(x)|p−2 ∇u(x) · ∇ϕ(x) dx Ω   = ϕ(x)p |∇u(x)|p dx − ϕ(x)p |∇u(x)|p−2 ∇u(x) · ∇g(x) dx Ω Ω  ϕ(x)p−1 (u(x) − g(x))|∇u(x)|p−2 ∇u(x) · ∇ϕ(x) dx +p Ω

and thus   p p ϕ(x) |∇u(x)| dx ≤ ϕ(x)p |∇u(x)|p−1 |∇g(x)| dx Ω Ω  ϕ(x)p−1 |u(x) − g(x)||∇u(x)|p−1 |∇ϕ(x)| dx, +p Ω

where we also used the fact that the two terms on the right-hand side are finite. By Young’s inequality   p p−1 ϕ(x) |∇u(x)| |∇g(x)| dx = ϕ(x)p−1 |∇u(x)|p−1 ϕ(x)|∇g(x)| dx Ω Ω   1 ϕ(x)p |∇u(x)|p dx + C(p) ϕ(x)p |∇g(x)|p dx ≤ 4 Ω Ω and  p ϕ(x)p−1 |u(x) − g(x)||∇u(x)|p−1 |∇ϕ(x)| dx Ω   1 ≤ ϕ(x)p |∇u(x)|p dx + C(p) |u(x) − g(x)|p |∇ϕ(x)|p dx. 4 Ω Ω

290

12. STABILITY RESULTS FOR THE p-LAPLACE EQUATION

This implies    1 ϕ(x)p |∇u(x)|p dx ≤ ϕ(x)p |∇u(x)|p dx + C(p) ϕ(x)p |∇g(x)|p dx 2 Ω Ω Ω  p + C(p) |u(x) − g(x)| |∇ϕ(x)|p dx. Ω

The first term on the right-hand side is finite by the assumption u ∈ W 1,p (Ω), and the claim follows by absorbing this term.  We state a global higher integrability result on open sets whose complement satisfies the capacity density condition in Definition 6.17. The self-improvement of the capacity density condition, see Theorem 7.21, is applied in the proof. Theorem 12.6. Let 1 < p < ∞ and let Ω ⊂ Rn be a bounded open set such that the complement Ωc satisfies the p-capacity density condition with a constant C1 . Assume that g ∈ W 1,s (Ω) for some s > p and that u ∈ W 1,p (Ω) is a weak solution to the p-Laplace equation in Ω with u − g ∈ W01,p (Ω). There exists δ = δ(n, p, s, C1 ) > 0 with p + δ ≤ s such that |∇u| ∈ Lp+δ (Ω). Moreover, there exists a constant C = C(n, p, s, C1, Ω) such that 1  p+δ

 |∇u(x)|

p+δ

dx

! ≤C

Ω

|∇u(x)| dx p

 p1

1 "  p+δ

 |∇g(x)|

p+δ

+

Ω

dx

.

Ω

Proof. Let B(z, r) be a ball with B(z, 2r) ∩ Ωc = ∅ and let ϕ ∈ C0∞ (B(z, 2r)) be such that ϕ = 1 in B(z, r), 0 ≤ ϕ ≤ 1 and |∇ϕ| ≤ Cr . Since Ω is bounded, we have u − g ∈ W01,p (Ω) and g ∈ W 1,s (Ω) ⊂ W 1,p (Ω). Lemma 12.5 implies   p |∇u(x)| dx ≤ ϕ(x)p |∇u(x)|p dx (12.2)

B(z,r)∩Ω

C(p) ≤ p r



Ω



|u(x) − g(x)|p dx + C(p) B(z,2r)∩Ω

|∇g(x)|p dx. B(z,2r)∩Ω

Theorem 7.21 gives an exponent 1 < q0 = q0 (n, p, C1 ) < p such that Ωc satisfies the q0 -capacity density condition with a constant C = C(n, p, C1 ). By increasing the exponent q in the proof of Theorem 12.1, if necessary, we obtain exponents nq for 1 < q < n q0 ≤ q = q(n, p, C1 ) < p and p ≤ q ∗ = q ∗ (n, p, C1 ) with 1 ≤ q ∗ ≤ n−q and 1 ≤ q ∗ < ∞ for n ≤ q < ∞. Since q0 ≤ q, Ωc satisfies the q-capacity density condition with a constant C = C(n, p, C1 ), see Remark 6.20. By H¨ older’s inequality u − g ∈ W01,p (Ω) ⊂ W01,q (Ω). By considering a qquasicontinuous representative v ∈ W 1,q (Rn ) of u − g ∈ W01,q (Ω), given by Corollary 5.29, we have v = 0 q-quasieverywhere in Ωc . Since B(z, 2r) ∩ Ωc = ∅, there exists a point y ∈ B(z, 2r) ∩ Ωc . Let A = x ∈ B(y, 4r) : v(x) = 0 . By Theorem 5.34, Theorem 5.38 and Lemma 5.35

capq (A, B(y, 8r)) ≥ capq B(y, 4r) ∩ Ωc , B(y, 8r)

≥ C(n, p, C1 ) capq B(y, 4r), B(y, 8r) ≥ C(n, p, C1 )r n−q .

12.1. HIGHER INTEGRABILITY OF THE GRADIENT

291

Theorem 5.47 gives  |v(x)|



q∗

B(y,8r)

 q1∗ dx 

 q1  1 q ≤ C(n, p, C1 ) |∇v(x)| dx capq (A, B(y, 8r)) B(y,8r)   1q q ≤ C(n, p, C1 )r  |∇v(x)| dx . B(y,8r)

Since v = u − g almost everywhere in Ω and v = 0 almost everywhere in Ωc , it follows that  p1   1 1 p |u(x) − g(x)| dx r |B(z, r)| B(z,2r)∩Ω  q1∗   q1  C q∗ q ≤ |v(x)| dx ≤C  |∇v(x)| dx r B(y,8r) B(y,8r)   1q  1 q =C |∇u(x) − ∇g(x)| dx |B(y, 8r)| B(y,8r)∩Ω   q1  1 ≤C |∇u(x)|q dx |B(z, 10r)| B(z,10r)∩Ω   p1  1 +C |∇g(x)|p dx , |B(z, 10r)| B(z,10r)∩Ω with C = C(n, p, C1 ). By (12.2) and the estimates above we have 

(12.3)

 p1  1 p |∇u(x)| dx |B(z, r)| B(z,r)∩Ω !  q1  1 ≤C |∇u(x)|q dx |B(z, 10r)| B(z,10r)∩Ω   p1 "  1 p + |∇g(x)| dx , |B(z, 10r)| B(z,10r)∩Ω

with C = C(n, p, C1 ), whenever B(z, 2r) ∩ Ωc = ∅. By the proof of Theorem 12.1, we conclude that (12.3) is also valid when B(z, 2r) ∩ Ωc = ∅. Let f = |∇u|q χΩ and h = |∇g|q χΩ . Then (12.3) implies  

p q

 pq

f (x) dx

B(z,r)

!



≤ C(n, p, C1 ) 

B(z,10r)

f (x) dx + 

B(z,10r)

p q

h(x) dx

 pq "

292

12. STABILITY RESULTS FOR THE p-LAPLACE EQUATION

for every B(z, r) ⊂ Rn . By Remark 8.44, there exists δ = δ(n, p, s, C1 ) > 0 with p + δ ≤ s and C = C(n, p, C1 ) such that 1  p+δ   1 p+δ |∇u(x)| dx |B(z, r)| B(z,r)∩Ω !  p1  1 p ≤C |∇u(x)| dx |B(z, 2r)| B(z,2r)∩Ω 1 "   p+δ  1 p+δ + |∇g(x)| dx |B(z, 2r)| B(z,2r)∩Ω for every B(z, r) ⊂ Rn . The claim follows by considering a ball B(z, r) with z ∈ ∂Ω and r = diam(Ω).  Remark 12.7. Let 1 < p0 < ∞. Assume that Ω ⊂ Rn is a bounded open set such that the complement Ωc satisfies the p0 -capacity density condition with a constant C1 and that g ∈ W 1,s (Ω) for some s > p0 . There exists δ = δ(n, p0 , s, C1 ) > 0 such that the conclusion of Theorem 12.6 holds for every p0 − δ < p < p0 + δ and for every weak solution u ∈ W 1,p (Ω) to the p-Laplace equation in Ω with u − g ∈ W01,p (Ω), with uniform constants 2δ and C = C(n, p0 , s, C1 , Ω). This follows by carefully tracking the constants in the proof of Theorem 12.6. Corollary 12.8. Let 1 < p < ∞ and let Ω ⊂ Rn be a bounded open set such that the complement Ωc satisfies the p-capacity density condition with a constant C1 . Assume that g ∈ W 1,s (Ω) for some s > p and that u ∈ W 1,p (Ω) is a weak solution to the p-Laplace equation in Ω with u−g ∈ W01,p (Ω). There exists δ = δ(n, p, s, C1 ) > 0 with p + δ ≤ s such that u ∈ W 1,p+δ (Ω). Proof. Since u − g ∈ W01,p (Ω), Corollary 3.7 implies that there exists q = q(n, p, s) > p with q ≤ s such that   q1   p1 q p |u(x) − g(x)| dx ≤ C(n, p, s, Ω) |∇u(x) − ∇g(x)| dx < ∞. Ω

Ω

It follows that   1q   1q   q1 q q q |u(x)| dx ≤ |u(x) − g(x)| dx + |g(x)| dx 0. Together with Theorem 12.6  this implies that u ∈ W 1,p+δ (Ω) for some δ = δ(n, p, s, C1 ) > 0. p+δ

Next example shows that the capacity density condition in Definition 6.17 is essentially the weakest possible condition for the global higher integrability in Theorem 12.6. Example 12.9. Let 1 < p < n. By Example 6.28 and Theorem 5.38 there exists a compact set K ⊂ B(0, 1) ⊂ Rn such that capp (K, B(0, 2)) > 0 and

dimH (K) = n − p.

Without loss of generality, we may assume that 0 ∈ K. Theorem 5.42 and Theorem 5.38 show that (12.4)

capq (K, B(0, 2)) = 0

12.1. HIGHER INTEGRABILITY OF THE GRADIENT

293

for every 1 < q < p. Let Ω = B(0, 2) \ K. Then Ωc cannot satisfy the p-capacity density condition since otherwise Theorem 7.21 would imply the existence of 1 < q < p and C > 0 such that

capq (K, B(0, 2)) = capq Ωc ∩ B(0, 1), B(0, 2) ≥ C > 0, contradicting (12.4). Let g ∈ C0∞ (B(0, 2)) with g = 1 on K. By Theorem 11.13 and Theorem 11.11, there exists a unique weak solution u ∈ W 1,p (Ω) to the p-Laplace equation in Ω with u − g ∈ W01,p (Ω). That is, u is the weak solution to the Dirichlet problem for the p-Laplace equation in Ω with the Sobolev boundary values zero on ∂B(0, 2) and one on K. By Theorem 11.41, we may assume that u is continuous in Ω. Corollary 5.29 implies the existence of a p-quasicontinuous function v ∈ W 1,p (Rn ) such that v = u − g almost everywhere in Ω and v = 0 p-quasieverywhere in Rn \ Ω. Thus v = 0 p-quasieverywhere in K and consequently v + g = 1 pquasieverywhere in K. By Theorem 5.46 we may use the p-quasicontinuous function v+g ∈ W01,p (B(0, 2)) as a test function for the capacity capp (K, B(0, 2)). This gives 



0 < capp (K, B(0, 2)) ≤

|∇v(x) + ∇g(x)| dx =

|∇u(x)|p dx.

p

B(0,2)

B(0,2)

The final equality holds since v + g = u almost everywhere in Ω and |K| = 0. We conclude that u is not constant. By Theorem 11.9 we have 0 ≤ u ≤ 1 everywhere in Ω. Since dimH (K) = n − p < n − 1, the set Ω is connected and Theorem 11.37 implies that 0 < u(x) < 1 for every y ∈ Ω. We claim that ∇u ∈ / Lp+δ (Ω) for any δ > 0. For a contradiction, assume that  |∇u(x)|p+δ dx < ∞

(12.5) Ω

1,p for some δ > 0 and let q = p+δ 1+δ < p. We prove that w = v + g ∈ W0 (B(0, 2)) is a weak solution to the p-Laplace equation in B(0, 2). Since w = u almost everywhere in Ω, w is a p-quasicontinuous extension of u to B(0, 2). Recall from (12.4) that capq (K, B(0, 2)) = 0. Hence, by Theorem 5.44 there exist functions ψj ∈ Lip0 (B(0, 2)), j ∈ N, such that 0 ≤ ψj ≤ 1, ψj = 1 in K for every j ∈ N, and



j→∞

|∇ψj (x)|q dx −−−→ 0.

(12.6) B(0,2)

Let ϕ ∈ C0∞ (B(0, 2)) and fix j ∈ N. Then ϕ(1 − ψj ) ∈ Lip0 (Ω) ⊂ W01,p (Ω). Since u ∈ W 1,p (Ω) is a weak solution in Ω and w = u almost everywhere in Ω, Lemma 11.6 implies  |∇w(x)|p−2 ∇w(x) · ∇((1 − ψj (x))ϕ(x)) dx = 0. Ω

294

12. STABILITY RESULTS FOR THE p-LAPLACE EQUATION

Consequently, since |K| = 0,  |∇w(x)|p−2 ∇w(x) · ∇ϕ(x) dx B(0,2)  = |∇w(x)|p−2 ∇w(x) · ∇((1 − ψj (x))ϕ(x)) dx B(0,2)  + |∇w(x)|p−2 ∇w(x) · ∇(ψj (x)ϕ(x)) dx B(0,2)  = |∇w(x)|p−2 ∇w(x) · ∇((1 − ψj (x))ϕ(x)) dx Ω  |∇w(x)|p−2 ∇w(x) · ∇(ψj (x)ϕ(x)) dx + B(0,2)  = |∇w(x)|p−2 ∇w(x) · ∇(ψj (x)ϕ(x)) dx. B(0,2)

By H¨ older’s inequality, we have  p−2 |∇w(x)| ∇w(x) · ∇ϕ(x) dx B(0,2)  p−2 |∇w(x)| ∇w(x) · ∇(ψj (x)ϕ(x)) dx = B(0,2)  (12.7) ≤ |∇w(x)|p−1 |∇(ψj (x)ϕ(x))| dx B(0,2)

 ≤

|∇w(x)|

p+δ

B(0,2)

 p−1  p+δ dx

|∇(ψj (x)ϕ(x))|

p+δ 1+δ

1+δ  p+δ

dx

.

B(0,2)

The first integral on the right-hand side of (12.7) is finite by (12.5) since w = u almost everywhere in Ω and |K| = 0. To estimate the second integral on the right-hand side of (12.7), we recall that q = p+δ 1+δ . Theorem 3.17 implies ψj Lq (B(0,2)) ≤ C(n, p, δ)∇ψj Lq (B(0,2)) , and thus, by (12.6), 1+δ  p+δ  p+δ 1+δ |∇(ψj (x)ϕ(x))| dx ≤ ϕ∇ψj Lq (B(0,2)) + ψj ∇ϕLq (B(0,2)) B(0,2)

≤ ϕL∞ (B(0,2)) ∇ψj Lq (B(0,2)) + ∇ϕL∞ (B(0,2)) ψj Lq (B(0,2))

j→∞ ≤ C(n, p, δ) ϕL∞ (B(0,2)) + ∇ϕL∞ (B(0,2)) ∇ψj Lq (B(0,2)) −−−→ 0. We conclude that w ∈ W01,p (B(0, 2)) is a weak solution to the p-Laplace equation in B(0, 2) with the property that 0 < w(x) < 1 for almost every y ∈ Ω. This is however a contradiction, since by Theorem 11.13 the zero function is the unique solution to the p-Laplace equation in B(0, 2) that belongs to W01,p (B(0, 2)). Thus (12.5) cannot be true. In conclusion, u ∈ W 1,p (Ω) is a weak solution to the p-Laplace equation in Ω with a smooth boundary function g and ∇u ∈ / Lp+δ (Ω) for any δ > 0. This shows that we can not omit in Theorem 12.6 the assumption that Ωc satisfies the p-capacity density condition. Observe from Lemma 6.19 that this construction is not possible for p > n.

12.2. STABILITY WITH RESPECT TO THE EXPONENT

295

12.2. Stability with respect to the exponent This section discusses a stability result, with respect to a varying exponent p, for weak solutions to the p-Laplace equation with given boundary values. This turns out to be a rather delicate problem. The main challenge is that the underlying Sobolev space changes as p varies and the associated energies are not necessarily finite. We show that under suitable assumptions solutions with varying exponent converge to the solution of the limit problem in the Sobolev space. The argument is based on the variational approach, and the equivalence in Theorem 11.11 between weak solutions to the p-Laplace equation and minimizers of the p-Dirichlet integral, see Theorem 11.13, is used throughout. The global higher integrability in Theorem 12.6 serves as a starting point for the proof of the stability. Theorem 12.10. Let 1 < p < ∞ and assume that Ω ⊂ Rn is a nonempty bounded open set such that Ωc satisfies the p-capacity density condition with a constant C1 . Let (pi )i∈N be a sequence with 1 < pi < ∞ for every i ∈ N and pi → p as i → ∞. Assume that g ∈ W 1,s (Ω) for some s > p and let ui ∈ W 1,pi (Ω) be a weak solution to the pi -Laplace equation in Ω with ui − g ∈ W01,pi (Ω) for every i ∈ N. Then there exist a unique weak solution u ∈ W 1,p (Ω) to the p-Laplace equation in Ω with u − g ∈ W01,p (Ω) and a subsequence of (ui )i∈N that converges to u locally uniformly in Ω as i → ∞. The lengthy proof of Theorem 12.10 is divided into several lemmas, which are of independent interest. We shall pass to a subsequence of (ui )i∈N several times and for simplicity the obtained subsequences will be denoted again by (ui )i∈N . Lemma 12.11. Under the assumptions in Theorem 12.10, there exist a subsequence of (ui )i∈N denoted by (ui )i∈N , 0 < δ ≤ s − p, and a function u ∈ W 1,p+δ (Ω) such that (a) (ui )i∈N is a bounded sequence in W 1,p+δ (Ω), (b) ∇ui → ∇u weakly in Lp+δ (Ω; Rn ), (c) ui → u locally uniformly in Ω as i → ∞. Proof. First we show that (ui − g)i∈N is a bounded sequence in W01,p+δ (Ω) for some δ > 0. By Theorem 12.6 and Remark 12.7, there exist δ = δ(n, p, s, C1 ) > 0 and C = C(n, p, s, C1 , Ω) such that, for all sufficiently large i ∈ N, the exponents q = p + δ and p − δ satisfy the assumptions of Corollary 3.7, 1 < p − δ < pi < p + δ < pi + 2δ ≤ s

(12.8) and  (12.9)

|∇ui (x)|pi +2δ dx Ω ! ≤C

p

1 i +2δ

|∇ui (x)| dx pi

Ω

 p1



i

|∇g(x)|

pi +2δ

+

p

1 i +2δ

dx

" .

Ω

By passing to a subsequence, we may assume that both (12.8) and (12.9) hold for every i ∈ N. Since ui − g ∈ W01,pi (Ω) ⊂ W01,p−δ (Ω) for every i ∈ N, Corollary 3.7

296

12. STABILITY RESULTS FOR THE p-LAPLACE EQUATION

and H¨ older’s inequality imply that 1 1   p+δ   p−δ p+δ p−δ |ui (x) − g(x)| dx ≤C |∇ui (x) − ∇g(x)| dx Ω Ω (12.10)  1 ≤C

|∇ui (x) − ∇g(x)|pi dx

pi

Ω

with C = C(n, p, s, C1, Ω). By (12.8), we have g ∈ W 1,s (Ω) ⊂ W 1,pi (Ω) for every i ∈ N. Theorem 11.11 asserts that ui is a minimizer of the pi -Dirichlet integral with ui − g ∈ W01,pi (Ω) and thus  p1   p1  i i |∇ui (x)|pi dx ≤ |∇g(x)|pi dx , (12.11) Ω

Ω

for every i ∈ N. By H¨ older’s inequality and (12.9) we obtain 1 1   p+δ   p +2δ i p+δ pi +2δ |∇ui (x)| dx ≤C |∇ui (x)| dx Ω Ω ! 1   ≤C

(12.12)

|∇ui (x)| dx pi

pi

+

Ω

1 pi +2δ

"

dx

Ω

 |∇g(x)| dx

≤C

|∇g(x)|

pi +2δ

s

 1s

Ω

for every i ∈ N with C = C(n, p, s, C1 , Ω). This shows that (∇ui )i∈N is a bounded sequence in Lp+δ (Ω; Rn ). It follows that 1   p+δ p+δ |∇ui (x) − ∇g(x)| dx Ω

 ≤

(12.13)

|∇ui (x)|

p+δ

1 1  p+δ   p+δ p+δ dx + |∇g(x)| dx

Ω

 ≤C

|∇g(x)|s dx

Ω

 1s

Ω

for every i ∈ N with C = C(n, p, s, C1 , Ω). Using (12.10) and (12.13) we obtain 1   p+δ   p1 i p+δ pi |ui (x) − g(x)| dx ≤C |∇ui (x) − ∇g(x)| dx Ω

Ω

 ≤C

|∇ui (x) − ∇g(x)|

p+δ

Ω

 ≤C

|∇g(x)| dx s

1  p+δ

dx

 1s

Ω

for every i ∈ N with C = C(n, p, s, C1 , Ω). Thus (ui − g)i∈N is a bounded sequence in Lp+δ (Ω). Together with (12.13) and the fact that g ∈ W 1,s (Ω) ⊂ W 1,p+δ (Ω) we conclude that (ui )i∈N is a bounded sequence in W 1,p+δ (Ω). By Theorem 2.39 there exists u ∈ W 1,p+δ (Ω) and a subsequence (ui )i∈N such that ui → u weakly in Lp+δ (Ω) and ∇ui → ∇u weakly in Lp+δ (Ω; Rn ) as i → ∞.

12.2. STABILITY WITH RESPECT TO THE EXPONENT

297

By Theorem 11.41 we may assume that ui is continuous in Ω, for every i ∈ N. Let Ω  Ω be an open set. By Theorem 11.30 there exists a constant C = C(n, p, s, C1) such that 1   p+δ p+δ |ui (x)| dx sup |ui (x)| ≤ C  x∈B(z,r)

B(z,2r)

for every i ∈ N whenever B(z, 2r)  Ω. By covering Ω with finitely many such balls and using the fact that (ui )i∈N is a bounded sequence in Lp+δ (Ω), we conclude that 1  p+δ  p+δ (12.14) sup |ui (x)| ≤ C sup |ui (x)| dx < ∞, x∈Ω

i∈N

Ω

for every i ∈ N, with C = C(n, p, s, C1 , Ω ). Assuming B(z, 3r)  Ω, the proof of Theorem 11.41 and Theorem 11.30 show that there exist an exponent α = α(n, p, s, C1 ) > 0 and a constant C = C(n, p, s, C1 ) such that |ui (x) − ui (x )| ≤ Cr −α |x − x |α

sup

|ui (y)|

y∈B(z,2r)

 ≤ Cr −α |x − x |α 

(12.15)

|ui (x)|p+δ dx

1  p+δ

B(z,3r)

for every i ∈ N and every x, x ∈ B z, 28r√n . A covering argument using (12.15) and (12.14) gives 1  p+δ    α p+δ (12.16) sup |ui (x) − ui (x )| ≤ C|x − x | sup |ui (x)| dx 1j }  Ω  for every j ∈ N. Then Ω = ∞ j=1 Ωj . We take a subsequence of (ui )i∈N that converges uniformly to u in Ω1 and then proceed recursively: in the jth step we choose from the previously chosen subsequence a new subsequence that converges uniformly in Ωj to u. At the end, we take the diagonal subsequence. This diagonal subsequence converges uniformly to u in all compact subsets K ⊂ Ω since each such K is contained in Ωj for some j ∈ N. Clearly the diagonal subsequence also satisfies the conditions (a) and (b).  Remark 12.12. Let Ω and (ui )i∈N be as in Theorem 12.10. By the regular1,α (Ω) for every i ∈ N, see Remark 11.42. A careful analysis ity theory ui ∈ Cloc of constants in the regularity theory shows that (ui )i∈N and (∇ui )i∈N are locally equicontinuous and, in addition, locally uniformly bounded. Thus the Arzela– Ascoli theorem implies the existence of a subsequence (ui )i∈N such that ui → u and ∇ui → ∇u locally uniformly in Ω as i → ∞. If the boundary function g is H¨ older continuous on ∂Ω, then ui ∈ C 1,α (Ω) for every i ∈ N since Ωc satisfies the

298

12. STABILITY RESULTS FOR THE p-LAPLACE EQUATION

p-capacity density condition, see Remark 11.44. Moreover, we have ui → u and ∇ui → ∇u uniformly in Ω as i → ∞. The limit function u in Lemma 12.11 has the correct boundary values. The p-capacity density condition is applied in the proof below. Lemma 12.13. Under the assumptions in Theorem 12.10 and notation as in Lemma 12.11, we have u − g ∈ W01,p (Ω). Proof. Let 0 < ε < p − 1 and let (ui )i∈N be the subsequence given by Lemma 12.11. For sufficiently large i ∈ N we have pi > p − ε and ui − g ∈ W01,pi (Ω). It follows that ui − g ∈ W01,p−ε (Ω) for every such i ∈ N, and by taking a further subsequence we may assume that this holds for every i ∈ N. By H¨ older’s inequality, Lemma 12.11 (a), and the fact that g ∈ W 1,s (Ω) with p − ε < p + δ ≤ s, we find that (ui − g)i∈N is a bounded sequence in W01,p−ε (Ω). Lemma 12.11 (c) asserts that ui − g → u − g locally uniformly and thus almost everywhere in Ω as i → ∞. Hence Theorem 2.40 implies that u − g ∈ W01,p−ε (Ω), and Lemma 12.11 gives u − g ∈ W 1,p (Ω). According to Remark 6.33 and the assumption that Ωc satisfies the p-capacity density condition, we obtain W01,p−ε (Ω) = W01,p (Ω),

W 1,p (Ω) ∩ 0 0. If |ζ| ≥ 1, then (12.18) implies log|ζ| (|ζ|a + |ζ|b )|a − b| ≤ 2 · 1 · |ζ|max{a,b}+ε |a − b| ≤ 1 · |ζ|max{a,b}+ε |a − b|. e ε ε 1 On the other hand, if 0 < |ζ| ≤ 1, then (12.18) with t = |ζ| gives   log|ζ| (|ζ|a + |ζ|b )|a − b| ≤ 1 1 + 1 |a − b|. e a b The claim follows by combining the estimates above.  (12.18)

0≤

Next we prove a lower semicontinuity result in the varying exponent case.

12.2. STABILITY WITH RESPECT TO THE EXPONENT

299

Lemma 12.15. Under the assumptions in Theorem 12.10 and notation as in Lemma 12.11, we have   |∇u(x)|p dx ≤ lim inf |∇ui (x)|pi dx i→∞

Ω

Ω



whenever Ω ⊂ Ω is an open set. Proof. We consider the subsequence of (ui )i∈N given by Lemma 12.11. By the proof of Lemma 12.11, we may assume that the inequalities in (12.8) hold for every i ∈ N. Let 0 < ε < p−1 2 . For sufficiently large i ∈ N, we have pi > p − ε. By taking a further subsequence, we may assume that this holds for every i ∈ N. Observe that pi    p+δ  pi |∇ui (x)|pi dx ≤ |∇ui (x)|p+δ dx |Ω |1− p+δ . Ω

Ω

Since (ui )i∈N is a bounded sequence in Lp+δ (Ω) and Ω ⊂ Ω has a finite measure, the left-hand side is uniformly bounded over i ∈ N. Inequality (12.17) implies   p−ε   p−ε pi p pi pi |∇ui (x)| dx − |∇ui (x)| dx Ω Ω qi #  $ p − ε p − ε (12.19) pi p 1 pi + |∇ui (x)| dx + ≤ pi − p e p−ε p−ε Ω i→∞

−−−→ 0, p−ε  where qi = max{ p−ε pi , p } + 1. Since 1 < p − ε < p + δ and Ω ⊂ Ω, Lemma 12.11 and the fact that Ω is bounded implies that ∇ui → ∇u weakly in Lp−ε (Ω ; Rn ) as i → ∞. Using Lemma 2.34, H¨older’s inequality and (12.19), we obtain   p−ε |∇u(x)| dx ≤ lim inf |∇ui (x)|p−ε dx i→∞ Ω Ω ! "  p−ε

≤ lim inf i→∞

(12.20)

Ω

|∇ui (x)|pi dx

 = lim inf i→∞

|∇ui (x)| dx pi

Ω

pi

 p−ε p

|Ω |

1− p−ε p i

|Ω | p ε

  p−ε  p ε pi = lim inf |∇ui (x)| dx |Ω | p . i→∞

Ω

We apply (12.17) with δ > 0, ζ = ∇u(x), a = p − ε and b = p, and obtain  |∇u(x)|p−ε − |∇u(x)|p dx Ω    1 1 1 1 ε→0 ≤ |ε| |∇u(x)|p+δ dx + |Ω | + −−−→ 0. δ Ω e p−ε p Here we also used the fact that by Lemma 12.11, we have  |∇u(x)|p+δ dx < ∞. Ω

300

12. STABILITY RESULTS FOR THE p-LAPLACE EQUATION

From this, together with (12.20), we conclude that    |∇u(x)|p dx = lim |∇u(x)|p−ε dx ≤ lim inf |∇ui (x)|pi dx. ε→0

Ω

i→∞

Ω

Ω



We obtain a local uniform integrability estimate for the gradients. Lemma 12.16. Let K be a compact subset of Ω and let Kε = {x ∈ Ω : d (x, K) < ε} for 0 < ε < d (K, ∂Ω). Under the assumptions in Theorem 12.10 and notation as in Lemma 12.11, we have   pi p |∇ui (x)| dx ≤ 3 |∇u(x)|p dx lim sup i→∞

Kε

Kε

for a dense subset of ε with 0 < ε < d (K, ∂Ω). Proof. We consider δ > 0, u ∈ W 1,p+δ (Ω) and the subsequence of (ui )i∈N given by Lemma 12.11. Taking a further subsequence, if necessary, we may assume that δ (12.21) 1 < p − δ < pi ≤ p + < p + δ ≤ s 2 for every i ∈ N. Let 0 < ε < ε < d (K, ∂Ω) and let ψ ∈ Lip0 (Kε ) be a cutoff function with 0 ≤ ψ ≤ 1 and ψ = 1 in Kε . For every i ∈ N, we define vi = ui + ψ(u − ui ) = ψu + (1 − ψ)ui . Let i ∈ N. By (12.21), vi ∈ W 1,pi (Ω) with vi − g ∈ W01,pi (Ω). By the minimizing property of ui , see Theorem 11.11, and properties of vi , we have     |∇ui (x)|pi dx + |∇ui (x)|pi dx = |∇ui (x)|pi dx ≤ |∇vi (x)|pi dx Ω\Kε Kε Ω Ω   = |∇ui (x)|pi dx + |∇vi (x)|pi dx Ω\Kε

and therefore

Kε



 |∇ui (x)|pi dx ≤ Kε

|∇vi (x)|pi dx. Kε

This implies    pi pi |∇ui (x)| dx ≤ |∇ui (x)| dx ≤ |∇vi (x)|pi dx K ε Kε Kε  (1 − ψ(x))∇ui (x) + (u(x) − ui (x))∇ψ(x) + ψ(x)∇u(x) pi dx = Kε #  pi pi pi (1 − ψ(x)) |∇ui (x)| dx + ψ(x)pi |∇u(x)|pi dx ≤3 Kε Kε $  pi pi |ui (x) − u(x)| |∇ψ(x)| dx . + Kε

Next we apply a hole filling technique. Using the facts that ψ = 1 in Kε ⊂ Kε and 0 ≤ ψ ≤ 1, and by adding the term  pi |∇ui (x)|pi dx 3 K ε

12.2. STABILITY WITH RESPECT TO THE EXPONENT

to the both sides, we obtain #  |∇ui (x)|pi dx ≤ 3pi (1 + 3pi ) K ε

301

 |∇ui (x)|pi dx +

Kε

ψ(x)pi |∇u(x)|pi dx Kε



$

|ui (x) − u(x)| |∇ψ(x)| dx . pi

+

pi

Kε

This, together with 0 ≤ ψ ≤ 1, implies #   3pi pi pi |∇ui (x)| dx ≤ |∇u (x)| dx + |∇u(x)|pi dx i 1 + 3pi Kε K ε Kε (12.22) $  + |ui (x) − u(x)|pi |∇ψ(x)|pi dx . Kε

Since pi ≤ p + δ and ui → u locally uniformly in Ω, see Lemma 12.11, we have   |ui (x) − u(x)|pi |∇ψ(x)|pi dx ≤ ∇ψpLi∞ (Rn ) |ui (x) − u(x)|pi dx Kε

Kε

 ≤ ∇ψpLi∞ (Rn )

|ui (x) − u(x)|p+δ dx

pi  p+δ

pi

i→∞

|Kε |1− p+δ −−−→ 0.

Kε

By taking limes superior on both sides of (12.22), we obtain  lim sup |∇ui (x)|pi dx i→∞

≤

K ε p

$ #   3 pi pi |∇u (x)| dx + lim sup |∇u(x)| dx . lim sup i 1 + 3p i→∞ Kε i→∞ Kε

Let

 |∇ui (x)|pi dx,

Ψ(ε) = lim sup i→∞

Kε

for 0 < ε < d (K, ∂Ω). Estimate (12.11) and inequality (12.21) give   p1   p1   1s i i |∇ui (x)|pi dx ≤ |∇ui (x)|pi dx ≤C |∇g(x)|s dx 0 such that max{pi , p} + δ  < p + δ for every i ∈ N. Lemma 12.11 implies   |∇u(x)|max{pi ,p}+δ dx lim sup i→∞

Kε

!

≤ lim sup i→∞

|∇u(x)|

p+δ

dx

|Kε |

1−

max{pi ,p}+δ  p+δ

"

Kε   p+δ p+δ

 |∇u(x)|

p+δ

=

 i ,p}+δ  max{pp+δ

dx

p+δ 

|Kε |1− p+δ < ∞.

Kε

By (12.17) with ζ = ∇u(x), a = pi and b = p, we conclude that  |∇u(x)|pi − |∇u(x)|p dx Kε    (12.25) 1 1 1 1 i→∞ max{pi ,p}+δ  ≤ |pi − p|  |∇u(x)| dx + |Kε | + −−−→ 0. δ Kε e pi p Thus (12.24) holds and this proves the claim for a dense set of ε with 0 < ε < d (K, ∂Ω).  We show that u is a minimizer of the p-Dirichlet integral among compactly supported smooth variations. Lemma 12.17. Under the assumptions in Theorem 12.10 and notation as in Lemma 12.11, we have   p |∇u(x)| dx ≤ |∇u(x) + ∇ϕ(x)|p dx Ω

for every ϕ ∈

Ω

C0∞ (Ω).

Proof. We consider δ > 0, u ∈ W 1,p+δ (Ω) and the subsequence of (ui )i∈N given by Lemma 12.11. Without loss of generality, we may assume that (12.21) holds for every i ∈ N. Fix ε > 0. Let ϕ ∈ C0∞ (Ω) and let Ω  Ω  Ω be open sets such that supp ϕ ⊂ Ω and  ε |∇u(x)|p dx < . p · 3p 2 · 3  Ω\Ω Let ψ ∈ C0∞ (Ω ) be a cutoff function with 0 ≤ ψ ≤ 1 and ψ = 1 in Ω , and let vi = ϕ + ψ(u − ui ) for every i ∈ N. By Lemma 12.11 we have ui ∈ W 1,p+δ (Ω) for every i ∈ N and u ∈ W 1,p+δ (Ω). From this and the fact that pi ≤ p + δ for every i ∈ N we conclude that vi ∈ W01,pi (Ω ) for every i ∈ N. Since ϕ = 0 in Ω \ Ω and ψ = 1 in Ω , we have  u+ϕ in Ω , ui + vi = in Ω \ Ω . ui + ψ(u − ui )

12.2. STABILITY WITH RESPECT TO THE EXPONENT

303

The minimizing property of ui in Ω , see Theorem 11.11, implies   pi |∇ui (x)| dx ≤ |∇ui (x) + ∇vi (x)|pi dx Ω Ω   |∇ui (x) + ∇vi (x)|pi dx + |∇ui (x) + ∇vi (x)|pi dx = Ω Ω \Ω  (12.26) = |∇u(x) + ∇ϕ(x)|pi dx  Ω 

∇ui (x) + ∇ ψ(x)(u(x) − ui (x)) pi dx. + Ω \Ω

For the second integral on the right-hand side of (12.26), we have 

∇ui (x) + ∇ ψ(x)(u(x) − ui (x)) pi dx Ω \Ω

#

≤3



pi

(12.27)

(1 − ψ(x)) |∇ui (x)| dx + pi

Ω \Ω

pi

|∇ψ(x)| |u(x) − ui (x)| dx . pi

Ω \Ω

ψ(x)pi |∇u(x)|pi dx

$

 +

Ω \Ω

pi

We take limit superior on both sides of (12.27) and estimate the terms separately. Let K = Ω \ Ω and Kε = {x ∈ Ω : d (x, K) < ε } with 0 < ε < d (K, ∂Ω) as in Lemma 12.16. By Lemma 12.16 there exists a dense set of ε satisfying   lim sup (1 − ψ(x))pi |∇ui (x)|pi dx ≤ lim sup |∇ui (x)|pi dx i→∞ i→∞ Ω \Ω K ε  |∇u(x)|p dx. ≤ 3p K ε

By letting ε → 0 along such a dense set, and using dominated convergence theorem, we obtain   ε pi pi p (1 − ψ(x)) |∇ui (x)| dx ≤ 3 |∇u(x)|p dx < . lim sup 2 · 3p i→∞ Ω \Ω K For the second integral on the right-hand side of (12.27) we apply (12.17) as in the proof of Lemma 12.16 and obtain   lim sup ψ(x)pi |∇u(x)|pi dx ≤ lim sup |∇u(x)|pi dx i→∞ i→∞ Ω \Ω Ω\Ω  ε ≤ |∇u(x)|p dx < . 2 · 3p · 3p Ω\Ω For the third integral on the right-hand side of (12.27), we have   |∇ψ(x)|pi |u(x) − ui (x)|pi dx ≤ ∇ψpLi∞ (Rn ) |u(x) − ui (x)|pi dx Ω \Ω

≤ ∇ψpLi∞ (Rn )

Ω \Ω

 Ω \Ω

|u(x) − ui (x)|p+δ dx

pi  p+δ

pi

|Ω \ Ω |1− p+δ −−−→ 0, i→∞

304

12. STABILITY RESULTS FOR THE p-LAPLACE EQUATION

since ui → u locally uniformly in Ω by Lemma 12.11. Thus we conclude from (12.27) that 

|∇ui (x) + ∇ ψ(x)(u(x) − ui (x)) |pi dx < ε. lim sup i→∞

Ω \Ω

Together with (12.26) and again (12.17) as in the proof of Lemma 12.16, this implies   pi lim sup |∇ui (x)| dx ≤ lim sup |∇u(x) + ∇ϕ(x)|pi dx + ε i→∞ i→∞ Ω Ω  (12.28) = |∇u(x) + ∇ϕ(x)|p dx + ε. Ω

Finally, by (12.28) and Lemma 12.15 applied in Ω ,    |∇u(x)|p dx = |∇u(x)|p dx + |∇u(x)|p dx Ω Ω\Ω Ω   ≤ |∇u(x)|p dx + lim inf |∇ui (x)|pi dx i→∞ Ω\Ω Ω   ≤ |∇u(x)|p dx + lim sup |∇ui (x)|pi dx   i→∞ Ω\Ω Ω   p ≤ |∇u(x)| dx + |∇u(x) + ∇ϕ(x)|p dx + ε   Ω\Ω Ω  p = |∇u(x) + ∇ϕ(x)| dx + ε, Ω

where the final equality holds since supp ϕ ⊂ Ω . The claim follows by letting ε → 0.  The function u is a minimizer of the p-Dirichlet integral also in sense of Theorem 11.13. Lemma 12.18. Under the assumptions in Theorem 12.10 and notation as in Lemma 12.11, we have   p |∇u(x)| dx ≤ |∇v(x)|p dx Ω

for every v ∈ W

1,p

(Ω) with v − g ∈

Ω

W01,p (Ω).

Proof. Let v ∈ W 1,p (Ω) with v − g ∈ W01,p (Ω). By Lemma 12.13 we have v − u = (v − g) + (g − u) ∈ W01,p (Ω). There exist functions ϕj ∈ C0∞ (Ω), j ∈ N, such that ϕj → v − u in W 1,p (Ω) as j → ∞. Lemma 12.17 implies   |∇u(x)|p dx ≤ |∇u(x) + ∇ϕj (x)|p dx Ω

Ω

for every j ∈ N, and thus   |∇u(x)|p dx ≤ lim |∇u(x) + ∇ϕj (x)|p dx j→∞ Ω Ω   = |∇u(x) + (∇v(x) − ∇u(x))|p dx = |∇v(x)|p dx. Ω

We are ready to prove the main result of this section.

Ω



12.2. STABILITY WITH RESPECT TO THE EXPONENT

305

Proof of Theorem 12.10. By Lemma 12.11 there exists u ∈ W 1,p (Ω) such that a subsequence of (ui )i∈N converges to u locally uniformly in Ω as i → ∞. Lemma 12.13 asserts that u − g ∈ W01,p (Ω). From Lemma 12.18 we conclude that u is a minimizer of the p-Dirichlet integral with the boundary values g in Ω and Theorem 11.11 implies that u is a weak solution to the p-Laplace equation in Ω.  We have also Sobolev convergence in the setting of Theorem 12.10. Lemma 12.19. Under the assumptions in Theorem 12.10 and notation as in 1,p (Ω) as i → ∞. Lemma 12.11, we have ui → u in Wloc Proof. We use same notation as in the proof of Lemma 12.17. Let ε > 0. By Lemma 12.15 with Ω and (12.28) with ϕ = 0 we have   p |∇u(x)| dx ≤ lim inf |∇ui (x)|pi dx i→∞ Ω Ω   pi |∇ui (x)| dx ≤ |∇u(x)|p dx + ε ≤ lim sup i→∞

Ω

Ω

and thus, by letting ε → 0, we conclude that   |∇ui (x)|pi dx = lim i→∞

Ω

Ω

|∇u(x)|p dx.

Since (ui )i∈N is a bounded sequence in W 1,p+δ (Ω), an application of (12.17) as in connection with (12.25) gives    p pi lim |∇ui (x)| dx = lim |∇ui (x)| dx = |∇u(x)|p dx. i→∞

Ω

i→∞

Ω

Ω

Hence, i→∞

∇ui Lp (Ω ;Rn ) −−−→ ∇uLp (Ω ;Rn ) . Moreover, it follows from Lemma 12.11 that ∇ui → ∇u weakly in Lp (Ω ; Rn ) as i → ∞. By Lemma 2.37, we obtain i→∞

∇ui − ∇uLp (Ω ;Rn ) −−−→ 0, and consequently ∇ui → ∇u in Lploc (Ω; Rn ) as i → ∞. Since ui → u locally 1,p (Ω) as i → uniformly in Ω and thus in Lploc (Ω), we conclude that ui → u in Wloc ∞.  Next we show that the capacity density condition in Definition 6.17 is essentially the weakest possible condition for the stability result in Theorem 12.10. If the complement does not satisfy the p-capacity density condition, it may happen that a sequence of solutions to the pi -Laplace equation, even with smooth boundary values, converges to a wrong solution as pi → p. Example 12.20. We modify Example 12.9 to the varying exponent case. Let K and Ω = B(0, 2) \ K be as in Example 12.9 and let (pi )i∈N be an increasing sequence with 1 < pi < pi+1 < p for every i ∈ N and pi → p as i → ∞. For every i ∈ N, let ui ∈ W 1,pi (Ω) be the unique continuous weak solution to the pi -Laplace equation in Ω with ui − g ∈ W01,pi (Ω), where g ∈ C0∞ (B(0, 2)) with g = 1 in K. By Corollary 5.29 there exists a pi -quasicontinuous function vi ∈ W 1,pi (Rn ) such that vi = ui − g almost everywhere in Ω and vi = 0 pi -quasieverywhere in Rn \ Ω. This implies that vi = 0 pi -quasieverywhere in B(0, 2)c . By Corollary 5.29 we conclude

306

12. STABILITY RESULTS FOR THE p-LAPLACE EQUATION

that vi ∈ W01,pi (B(0, 2)) and consequently vi + g ∈ W01,pi (B(0, 2)) for every i ∈ N. We claim that wi = vi + g is a weak solution to the pi -Laplace equation in B(0, 2) for every i ∈ N. Observe that wi = ui almost everywhere in Ω and thus wi is an extension of ui to B(0, 2) for every i ∈ N. Fix i ∈ N. Let (ψj )j∈N be as in (12.6) for q = pi < p and let ϕ ∈ C0∞ (B(0, 2)). Then ϕ(1 − ψj ) ∈ Lip0 (Ω) ⊂ W01,pi (Ω) for every j ∈ N. Since ui ∈ W 1,pi (Ω) is a weak solution to the pi -Laplace equation in Ω and wi = ui almost everywhere in Ω, Lemma 11.6 implies  |∇wi (x)|pi −2 ∇wi (x) · ∇((1 − ψj (x))ϕ(x)) dx = 0. Ω

As in (12.7), also using the fact that |K| = 0, we have  pi −2 |∇wi (x)| ∇wi (x) · ∇ϕ(x) dx B(0,2)  = |∇wi (x)|pi −2 ∇wi (x) · ∇(ψj (x)ϕ(x)) dx B(0,2)  (12.29) ≤ |∇wi (x)|pi −1 |∇(ψj (x)ϕ(x))| dx B(0,2)

 ≤

|∇wi (x)| dx pi

B(0,2)

 pip−1  i

|∇(ψj (x)ϕ(x))| dx pi

 p1

i

.

B(0,2)

The first term on the right-hand side of (12.29) is finite. By (12.6) with q = pi < p we obtain  p1  i |∇(ψj (x)ϕ(x))|pi dx ≤ ϕ∇ψj Lq (B(0,2)) + ψj ∇ϕLq (B(0,2)) B(0,2)

≤ ϕL∞ (B(0,2)) ∇ψj Lq (B(0,2)) + ∇ϕL∞ (B(0,2)) ψj Lq (B(0,2))

j→∞ ≤ C(n, pi ) ϕL∞ (B(0,2)) + ∇ϕL∞ (B(0,2)) ∇ψj Lq (B(0,2)) −−−→ 0. Thus it follows from (12.29) that wi ∈ W01,pi (B(0, 2)) is a weak solution to the pi -Laplace equation in B(0, 2). On the other hand, by Theorem 11.13 the zero function is the unique solution to the pi -Laplace equation in B(0, 2) that belongs to W01,pi (B(0, 2)). Hence wi = 0 in B(0, 2) and consequently ui = 0 in Ω for every i ∈ N. Finally, let u ∈ W 1,p (Ω) be the unique weak solution to the p-Laplace equation in Ω, with u − g ∈ W01,p (Ω). Arguing as in Example 12.9, we find that 0 < u(x) < 1 for every x ∈ Ω, and thus there is no subsequence of (ui )i∈N that would converge to u pointwise almost everywhere in Ω. In conclusion, the claim in Theorem 12.10 does not hold in this case since the solutions ui converge to the solution of a wrong limit problem. This is possible since Ωc does not satisfy the p-capacity density condition. 12.3. Very weak solutions In this section we take another look at the Sobolev space assumption in the definition of a weak solution to the p-Laplace equation. Let 1 < p < ∞ and let 1,p (Ω). However, Ω ⊂ Rn be an open set. In Definition 11.1 we assumed that u ∈ Wloc

12.3. VERY WEAK SOLUTIONS

307

1,q the integral in (11.1) makes sense under the weaker assumption that u ∈ Wloc (Ω) for some q with max{1, p − 1} ≤ q < p since   |∇u(x)|p−2 ∇u(x) · ∇ϕ(x) dx ≤ |∇u(x)|p−1 |∇ϕ(x)| dx Ω

 |∇u(x)| dx

≤

q

Ω

 p−1  q

supp ϕ

|∇ϕ(x)|

q q−p+1

 q−p+1 q dx

0. The proof of Theorem 12.23 is based on localization, Lipschitz truncation and weighted norm inequalities. Also the following version of Lemma 11.6 is applied in the proof. Lemma 12.22. Let 1 < p < ∞ and max{1, p − 1} < q < p, and let Ω ⊂ Rn 1,q be an open set. Assume that u ∈ Wloc (Ω) satisfies (12.30) for every ϕ ∈ C0∞ (Ω). Then (12.30) holds for every ϕ ∈ Lipc (Ω). Proof. Let ϕ ∈ Lipc (Ω). By convolution approximation, we obtain a compact set K ⊂ Ω and functions ϕi ∈ C0∞ (Ω) such that supp ϕ ∪ supp ϕi ⊂ K for every q i ∈ N and ∇ϕi → ∇ϕ in L q−p+1 (Ω) as i → ∞. As in the proof of Lemma 11.6, see also Remark 11.7, H¨ older’s inequality implies   |∇u(x)|p−2 ∇u(x) · ∇ϕ(x) dx = |∇u(x)|p−2 ∇u(x) · (∇ϕ(x) − ∇ϕi (x)) dx Ω Ω  ≤ |∇u(x)|p−1 |∇ϕ(x) − ∇ϕi (x)| dx K

 ≤

|∇u(x)| dx q

K

 p−1  q

|∇ϕ(x) − ∇ϕi (x)|

q q−p+1

 q−p+1 q dx

i→∞

−−−→ 0.



Ω

Theorem 12.23. Let 1 < p < ∞ and let Ω ⊂ Rn be an open set. There exists 1,p−ε (Ω) satisfies (12.30) for every 0 < ε = ε(n, p) < p − 1 such that if u ∈ Wloc 1,p ∞ ϕ ∈ C0 (Ω), then u ∈ Wloc (Ω). Proof. Let ε with (12.31)



1 p−1 1 , 0 < ε < min ,  2 2 2(p − 1)



1,p−ε p . Assume that u ∈ Wloc (Ω) satisfies (12.30) to be chosen later. Here p = p−1 ∞ for every ϕ ∈ C0 (Ω). Let B(z, r) be a ball with B(z, 4r) ⊂ Ω, and let ψ ∈

308

12. STABILITY RESULTS FOR THE p-LAPLACE EQUATION

Lip0 (B(z, 2r)) be such that 0 ≤ ψ ≤ 1, ψ = 1 in B(z, r) and |∇ψ| ≤ everywhere in Rn . Let

1 r

almost

v = (u − uB(z,2r) )ψ and Et = {x ∈ Rn : M |∇v|(x) > t}, for t > 0. Then v ∈ W 1,1 (Rn ), and by Theorem 4.2 and Theorem 2.7 there exists a C(n)t-Lipschitz function vt : Rn → R such that vt = v almost everywhere in Rn \ Et = {x ∈ Rn : M |∇v|(x) ≤ t}. Since v = 0 in Rn \ B(z, 2r), we have M |∇v|(x) = sup  ρ>0

B(x,ρ)

|∇v(y)| dy ≤ C(n) 

|∇v(y)| dy = t0

B(z,2r)

for every x ∈ Rn \B(z, 3r). It follows that if t ≥ t0 , then vt = v = 0 in Rn \B(z, 3r), and thus vt is compactly supported in Ω for t ≥ t0 . By Lemma 12.22 we may use vt ∈ Lipc (Ω) as a test function in (12.30) for every t ≥ t0 . Thus  |∇u(x)|p−2 ∇u(x) · ∇v(x) dx B(z,2r)\Et  = |∇u(x)|p−2 ∇u(x) · ∇v(x) dx Ω\Et  = |∇u(x)|p−2 ∇u(x) · ∇vt (x) dx (12.32) Ω\Et  =− |∇u(x)|p−2 ∇u(x) · ∇vt (x) dx Ω∩Et  ≤ C(n)t |∇u(x)|p−1 dx B(z,4r)∩Et

for every t ≥ t0 , where the final inequality holds since |∇vt | ≤ C(n)t almost everywhere in Rn and ∇vt = 0 in Ω \ B(z, 4r). We multiply (12.32) by t−1−ε , integrate over (t0 , ∞), and change the order of integration on the right-hand side. This and the assumption 0 < ε < 12 from (12.31) give   ∞ −1−ε t |∇u(x)|p−2 ∇u(x) · ∇v(x) dx dt B(z,2r)\Et  ∞  −ε

t0

≤ C(n) (12.33)

≤ C(n)

t

t0  ∞

t−ε

C(n) 1−ε



|∇u(x)|p−1 dx dt B(z,4r)∩Et

0

=

|∇u(x)|p−1 dx dt

B(z,4r)∩Et



(M |∇v|(x))1−ε |∇u(x)|p−1 dx 

B(z,4r)

≤ C(n)

(M |∇v|(x))1−ε |∇u(x)|p−1 dx. B(z,4r)

12.3. VERY WEAK SOLUTIONS

309

On the other hand, 

∞

ε t0

t−1−ε  =

 |∇u(x)|p−2 ∇u(x) · ∇v(x) dx dt B(z,2r)\Et

(M |∇v|(x))−ε |∇u(x)|p−2 ∇u(x) · ∇v(x) dx

B(z,2r)∩Et0

+ t−ε 0



|∇u(x)|p−2 ∇u(x) · ∇v(x) dx. B(z,2r)\Et0

By substituting this equality to (12.33), we obtain I1 = t−ε 0

 |∇u(x)|p−2 ∇u(x) · ∇v(x) dx B(z,2r)\Et0



(M |∇v|(x))−ε |∇u(x)|p−2 ∇u(x) · ∇v(x) dx

+

(12.34)

B(z,2r)∩Et0

 ≤ C(n)ε

(M |∇v|(x))1−ε |∇u(x)|p−1 dx = εI2 . B(z,4r)

By the properties of ψ and Theorem 3.5, we have

|∇v(x)| ≤ |∇u(x)|ψ(x) + |u(x) − uB(z,2r) ||∇ψ(x)| ≤ C(n)M |∇u|χB(z,4r) (x) for almost every x ∈ Rn . Recall from (12.31) that 0 < ε < min{ 12 , p−1 2 }. Hence, (12.35)

5n 2p−ε

p−ε p p ≤ 5n 2p 1 = C1 (n, p) > 1. = 5n 2p+1 p−ε−1 p−1 2 (p − 1)

Thus we can apply Theorem 1.15 (c) with p − ε > 1 and this constant C1 (n, p) which is independent of ε; see the end of the proof of Theorem 1.15. Using also p−ε Young’s inequality, with exponents p−ε 1−ε > 1 and p−1 > 1, we obtain 



I2 ≤ C(n) B(z,4r)



1−ε M M |∇u|χB(z,4r) (x) |∇u(x)|p−1 dx

 ≤ C(n) (12.36)

|∇u(x)|p−ε dx B(z,4r)



+ C(n)  ≤ C(n, p)

B(z,4r)



p−ε M M |∇u|χB(z,4r) (x) dx

|∇u(x)|p−ε dx.

B(z,4r)

By combining (12.34) and (12.36), we have  (12.37)

I1 ≤ εI2 ≤ C(n, p)ε

|∇u(x)|p−ε dx. B(z,4r)

310

12. STABILITY RESULTS FOR THE p-LAPLACE EQUATION

Since M |∇v|(x) ≤ t0 for every x ∈ B(z, 2r) \ Et0 , we estimate I1 from below as follows  |∇u(x)|p−2 ∇u(x) · ∇v(x) dx I1 = t−ε 0 B(z,2r)\Et0



(M |∇v|(x))−ε |∇u(x)|p−2 ∇u(x) · ∇v(x) dx

+ B(z,2r)∩Et0

≥

−t−ε 0



|∇u(x)|p−1 |∇v(x)| dx



B(z,2r)\Et0

(M |∇v|(x))−ε |∇u(x)|p−2 ∇u(x) · ∇v(x) dx

+ B(z,2r)



(M |∇v|(x))−ε |∇u(x)|p−2 ∇u(x) · ∇v(x) dx

− B(z,2r)\Et0



(M |∇v|(x))−ε |∇u(x)|p−2 ∇u(x) · ∇v(x) dx

≥ B(z,2r)



−2

(M |∇v|(x))−ε |∇u(x)|p−1 |∇v(x)| dx

B(z,2r)\Et0

= J1 − J2 . To estimate J1 , we consider

  D1 = x ∈ B(z, 2r) \ B(z, r) : M |∇v|(x) ≤ εM |∇u|χB(z,2r) (x) and D2 = (B(z, 2r) \ B(z, r)) \ D1 . Then  J1 = (M |∇v|(x))−ε |∇u(x)|p−2 ∇u(x) · ∇v(x) dx B(z,r)  + (M |∇v|(x))−ε |∇u(x)|p−2 ∇u(x) · ∇v(x) dx B(z,2r)\B(z,r)  = (M |∇v|(x))−ε |∇u(x)|p dx B(z,r)  + (M |∇v|(x))−ε |∇u(x)|p−2 ∇u(x) · ∇v(x) dx D1 + (M |∇v|(x))−ε |∇u(x)|p−2 ∇u(x) · ∇u(x)ψ(x) dx D2 

(M |∇v|(x))−ε |∇u(x)|p−2 ∇u(x) · ∇ψ(x) u(x) − uB(z,2r) dx + D2  ≥ (M |∇v|(x))−ε |∇u(x)|p dx B(z,r)  − (M |∇v|(x))−ε |∇u(x)|p−1 |∇v(x)| dx D1  1 (M |∇v|(x))−ε |∇u(x)|p−1 |u(x) − uB(z,2r) | dx − r D2 = I3 − I 4 − I 5 .

12.3. VERY WEAK SOLUTIONS

To estimate I3 , we recall from (12.31) that 0 < ε < 

311

where p =

1 2(p −1) ,

p p−1 .

Theorem 8.33 implies σ = (M |∇v|)ε(p −1) ∈ A1 with [σ]A1 ≤ C(n, p); observe that σ(x) > 0 for every x ∈ Rn unless |∇v| = 0 almost everywhere in Rn , in which case the claim of the theorem is trivial. Thus σ ∈ Ap and [σ]Ap ≤ [σ]A1 , and by the Ap duality in Lemma 8.13 (b),

−ε

− 1  w = M |∇v| = (M |∇v|)ε(p −1) p −1 = σ 1−p ∈ Ap with [w]Ap ≤ C(n, p). The weighted norm inequality for the Hardy–Littlewood maximal operator with a Muckenhoupt Ap weight in Theorem 8.28 implies  

p M (|∇u|χB(z,r) )(x) w(x) dx |∇u(x)|p w(x) dx ≥ C(n, p) I3 = B(z,r) B(z,r) 

−ε

p = C(n, p) M |∇v|(x) M (|∇u|χB(z,r) )(x) dx. B(z,r)

Since ∇u = ∇v almost everywhere in B(z, r) and |∇v| = 0 almost everywhere in Rn \ B(z, 2r), we have M |∇v|(x) ≤

sup 0 0 and for all balls B(z, r) satisfying B(z, 4r) ⊂ Ω. To conclude the proof, assume that θ = θ(n, pq ) > 0 is given and choose 0 < ε1 , δ, γ < 1, only depending on n and p, such that 1 1 q C(n, p) ε + ε 2 + δ + γ 2 p−ε ≤ θ for every 0 < ε < ε1 . Observe that the constant C1 = C1 (n, p, q, δ, γ) = C1 (n, p) in (12.45) is independent of ε. Remark 8.45, with exponent 1 < pq < ∞ and functions f = |∇u|q and g = 0, gives ε = ε(n, p) > 0 and C > 0 such that q  pq   p−ε  p p−ε |∇u(x)| dx ≤C  |∇u(x)| dx  B(z,r)

B(z,2r)

1,p−ε whenever B(z, 2r) ⊂ Ω. Since u ∈ Wloc (Ω), it follows that |∇u| ∈ Lploc (Ω). 1,p (Ω).  Finally, inequality (12.41) implies that also u ∈ Lploc (Ω), and thus u ∈ Wloc

Remark 12.24. The proof above gives some exponent q with p−1 ≤ q < p such 1,q that a very weak solution u ∈ Wloc (Ω) to a general partial differential equation of 1,p ainen and Martio [187] p-Laplacian type belongs to Wloc (Ω), see Heinonen, Kilpel¨ and Mal´ y and Ziemer [303]. For this kind of equations the exponent q depends on structural conditions, see Meyers [324] and Serrin [360]. 12.4. Notes The local higher integrability result in Theorem 12.1 has been studied by Bojarski [47], Meyers [324] and Meyers and Elcrat [325]. See also Chen and Wu [82, Section 12.4], Giaquinta [147, Chapter 5], [148, Section 6.2], Giaquinta and Martinazzi [149, Section 9.2.2], Giaquinta and Modica [150], Giusti [153, Section 6.4], Stredulinsky [374, Section 3.3.0] and Zhikov [403]. For the global higher integrability result in Theorem 12.6, see Granlund [160] in the case of the measure density condition and Kilpel¨ ainen and Koskela [220] in the case of the capacity density condition. See also Giaquinta [147, p. 151–154], Giusti [153, Section 6.5] and Maasalo and Zatorska-Goldstein [295]. Example 12.9 is from Kilpel¨ainen and Koskela [220, Remark 3.3]. For local higher integrability results for parabolic problems, see Antontsev and Zhikov [17], Arkhipova [18], B¨ ogelein [37], B¨ ogelein and Duzaar [39], B¨ ogelein, Duzaar, Kinnunen and Scheven [40], B¨ ogelein, Duzaar, Korte and Scheven [41], B¨ ogelein, Duzaar and Scheven [43], Gianazza and Schwarzacher [146,146], Giaquinta and Struwe [151], Habermann [163], Kinnunen and Lewis [230], Masson, Miranda, Paronetto and Parviainen [311], Zhikov and Pastukhova [404] and Wieser [398]. For global higher integrability in the parabolic case, see B¨ ogelein and Parviainen [45], Fujishima and Habermann [137], Habermann [162], Masson and Parviainen [312], Moring, Scheven, Schwarzacher and Singer [330] and Parviainen [342, 343].

12.4. NOTES

315

For stability results, we refer to Lindqvist [278,279] and Maasalo and ZatorskaGoldstein [295]. Example 12.20 is from Lindqvist [279]. Related stability results are studied by Kinnunen, Marola and Martio [233] and Li and Martio [273–276]. Stability results for parabolic problems can be found in Fujishima, Habermann, Kinnunen and Masson [138] and Kinnunen and Parviainen [236]. Very weak solutions have been studied by Bojarski [47], Meyers [324], Iwaniec and Sbordone [209] with the Hodge decomposition and Lewis [272] by the Lipschitz truncation. Our presentation is based on Lewis [272] and Mikkonen [327]. See also B¨ ogelein and Zatorska-Goldstein [46] and Zatorska-Goldstein [402]. For very weak solutions in the parabolic case, see Adimurthi and Byun [6], B¨ ogelein [38], B¨ ogelein and Li [44], Kinnunen and Lewis [231] and Li [277]. The Lipschitz truncation method has been applied not only in connection with very weak solutions but also in other questions in partial differential equation and the calculus of variations. For example, see Acerbi and Fusco [2], Breit, Diening and Schwarzacher [55], Bul´ıˇcek, Burczak and Schwarzacher [66], Bul´ıˇcek, Diening and Schwarzacher [67], Bul´ıˇcek and Schwarzacher [68], B¨ ogelein, Duzaar and Mingione [42], Diening, R˚ uˇziˇcka and Wolf [105], Diening, Schwarzacher, Stroffolini and Verde [106] and Duzaar and Mingione [110].

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Index

chain decomposition, 195 chain rule for Lipschitz functions, 27 for Sobolev functions, 34 Coifman–Rochberg lemma, 179, 180 for measures, 183 comparison principle, 259 convolution, 31 convolution approximation, 31, 32 covering lemma, 7 cube, 161 dyadic decomposition, 162 dyadic maximal function on a cube, 163 half-open, 161 maximal dyadic cubes, 163

5r-covering lemma, 7 ∂A, xi Ac , xi A, xi Ap (E), 85 Ap (E), 85 A∗p (E), 93 |A|, xi Ap , 168 A1 , 179 A∞ , 173 Ahlfors–David regular set, 138 Aikawa condition, 225 self-improvement, 227 Assouad dimension, 232 and Hardy’s inequality, 241

d(A, B), xi d(x, A), xi D(u), 65 dσ, 47 De Giorgi’s method, 286 derivative, 27 weak, 29 diam(A), xi difference quotient, 43 integrated, 43 differentiability, 27 for Lipschitz functions, 27 for Sobolev functions, p > n, 62 dimA , 232 dimH , 22 dimL , 140 Dirac’s delta, 257 Dirichlet integral, 260 minimizer, 260 Dirichlet problem, 260, 262 distance function, 25, 68, 225 distribution set, 2 divergence, 255 doubling weight, 161 dyadic child, 163 cube, 162 parent, 163

Boman chain condition, 195 boundary Poincar´ e inequality, 126 boundary values in the Sobolev sense, 35 bounded mean oscillation (BMO), 71, 216, 279 dyadic, 218 exponential integrability, 220 self-improvement, 220 C(Ω), xi C0 (Ω), xii C k (Ω), xii C0k (Ω), xii 0,α (Ω), 285 Cloc Caccioppoli estimate, 266, 269 Calder´ on–Zygmund lemma, 163, 171 capacitary weak type estimate, 94 capacity and Hausdorff dimension, 108 Sobolev, 85, 93 variational, 101 capacity density condition, 124, 137, 147 case p = n, 145 self-improvement, 147 Cavalieri’s principle, 7 χA , xi 335

336

INDEX

energy estimate, 266, 269, 289 logarithmic, 267 Euler–Lagrange equation, 260

for weak solutions, 283 Morrey condition, 72 hole filling technique, 300

f ∗ g, 31 f+ , xi f− , xi fA , xii fQ;σ , 174 f |B , xi f ∗ , 11 Fefferman–Stein inequality, 203 fractional maximal function, 20, 72, 79 and Riesz potential, 21 weak type estimate, 22

Iα f , 17 integration over spheres, 47

Gehring lemma, 186 Green’s identity, 76 H λ , 47 Hλ , 22 Hλ δ , 22 Hajlasz–Sobolev space, 65 Hardy’s inequality, 115, 144, 241 and Assouad dimension, 241 and lower dimension, 144 case p = n, 145 one-dimensional weighted, 118 pointwise, 116, 147 self-improvement, 121, 124 weighted, 129 Hardy–Littlewood maximal function, 1 Hardy–Littlewood–Sobolev theorem for dyadic fractional maximal function, 208 for fractional maximal function, 21 for Riesz potential, 19 Hardy–Littlewood–Wiener maximal function theorem, 8 Hardy–Sobolev inequality, 240 global, 240 Hardy–Sobolev–Poincar´e inequality, 239 on domains, 239 Harnack’s inequality, 283 for 1 < p < ∞, 279 for p > n, 278 Hausdorff content, 22 and variational capacity, 105 density condition, 137, 147 Hausdorff dimension, 22 and capacity, 108 Hausdorff measure, 22 normalization, 47 Havin–Bagby characterization of W01,p (Ω), 100 Heaviside function, 12 H¨ older continuity Campanato characterization, 70 for Sobolev functions, 60

John domain, 196 John–Nirenberg inequality, 219 Lp (A), 6 Lp (E; w dx), 161 Lploc (A), 6 Lp (A; Rm ), 37 L log+ L(Rn ), 14 Leb(f ), 11 Lebesgue density theorem, 13 Lebesgue differentiation theorem, 12 for Sobolev functions, 95 Lebesgue point, 11 Leibniz rule for Lipschitz functions, 27 for Sobolev functions, 34 Lip, 25 Lip0 , 26 Lipc , 26 Liploc , 25 Lipschitz constant, 25 Lipschitz function, 25 compactly supported, 26 differentiability, 27 extension, 26, 27 locally, 25 weak differentiability, 30 Lipschitz truncation, 66, 122, 307 local-to-global inequality, 196 weighted, 213 lower dimension, 140, 147 and Hardy’s inequality, 144 lower semicontinuity, 2 M 1,p (Rn ), 65 M c f , 162 M c,w f , 162 d,w f , 198 MQ 0 d f , 163 MQ 0 Mf, 1 Mα f , 20 Mα,R f , 20 M ∗f , 1 MR f , 15 MΩ f , 17, 75  f , 69 Mβ,R M w f , 162 maximal function, 1 basic properties, 3 centered, 1 continuity, 4 dyadic fractional on a cube, 205

INDEX

dyadic on a cube, 163 dyadic sharp, 203 finiteness, 8 fractional, 20, 79 fractional sharp, 69 L∞ -boundedness, 6 Lp -boundedness for 1 < p ≤ ∞, 8 lower semicontinuity, 2 noncentered, 1 on Sobolev spaces, 73, 77, 79 on Sobolev spaces with zero boundary values, 79 restricted, 15 weak type estimate, 8 weak type estimate, reverse, 13 weighted, 162 weighted dyadic on a cube, 198 weighted with respect to cubes, 162 with respect to an open set, 17, 75, 250 with respect to cubes, 162, 180 maximal function theorem, 8 maximum principle, 281 strong, 281 weak, 272 Mazur’s lemma, 40 Mazya inequality, 112 McShane extension, 26, 27 measure density condition, 118 minimum principle strong, 281 Morrey’s inequality, 58, 72 for Lipschitz functions, 58 for Sobolev functions, 60 Moser iteration, 272, 278 Muckenhoupt class, 168 A1 weight, 179 A∞ weight, 173 Ap weight, 168 self-improvement, 173, 177 N (A, r), 140, 232 noncontinuous maximal function, 3 normalized Hausdorff measure, 47 ωn , xi oscA u, 283 oscillation, 283, 285 p-Laplace equation, 255 Dirichlet problem, 260, 262 fundamental solution, 257 very weak solution, 307 weak solution, 256 weak subsolution, 264 weak supersolution, 264 Poincar´ e inequality, 52, 55, 71 (q, p)-version, 55 Ap -weighted, 200 for 1 < p < ∞, 52

337

for cubes, 197, 210 for Sobolev functions, 55 for two weights, 209 for two weights on an open set, 215 for zero boundary values, 57 on an open set, 197 pointwise, 49 reverse, 267 pointwise capacity density condition, 134 pointwise Hardy inequality, 116 porous set, 229 quasiadditivity, 248 quasicontinuity, 91, 110 quasieverywhere, 90 Rademacher’s theorem, 27 regular boundary point, 285 restricted maximal function, 15, 75 continuity, 16 reverse H¨ older inequality, 273, 280 self-improvement, 183 reverse Poincar´ e inequality, 267 Riesz potential, 17, 47 and maximal function, 18, 21 Lp -boundedness for 1 ≤ p < ∞, 18 weak type estimate, 20 self-improvement of Aikawa condition, 227 BMO, 220 capacity density condition, 147 Hardy’s inequality, 121 Muckenhoupt class, 173, 177 reverse H¨ older inequality, 183, 189 weighted norm inequality, 166 shadow of a cube, 195 slit domain, 25 Sobolev capacity, 85, 93 admissible function, 85 is outer measure, 86 of balls, 88 outer regularity, 87 Sobolev conjugate exponent, 19 Sobolev function Lebesque points, 95 quasicontinuous representative, 91 Sobolev inequality, 52 for Riesz potential, 19 for two weights, 211 Sobolev norm, 30 Sobolev space, 30, 71 and difference quotients, 43, 65 Campanato approach, 69 Characterization by a Poincar´ e inequality, 71 density of smooth functions, 34 differentiability for p > n, 62 Hajlasz type characterization, 65, 71

338

local, 30 Morrey approach, 72 with zero boundary values, 35, 68, 79, 97, 100, 132 Sobolev–Gagliardo–Nirenberg inequality, 50, 57 Sobolev–Poincar´ e inequality for 1 < p < n, 53 for p = 1, 54 for cubes, 210 on an open set, 216 sparse domination, 201, 206 supp f , xi strong maximum principle, 281 strong minimum principle, 281 support, xi surface measure, 47 Trudinger’s inequality, 220 truncation, 26, 50, 66 in Sobolev spaces, 34, 35, 37 uniformly perfect set, 144 variational approach, 260, 295 variational capacity, 101 and Hausdorff content, 105 Lipschitz test functions, 108 of balls, 103 quasicontinuous test functions, 110 very weak solution, 307 von Koch snowflake, 240 W 1,∞ , 30, 33 W 1,p , 30 W01,p , 35, 68, 97 1,p , 30 Wloc weak convergence, 38 in Sobolev spaces, 40, 41 weak derivative, 29 weak gradient, 29 weak maximum principle, 272 weak solution, 256 existence, 262 uniqueness, 260, 262 weak subsolution, 264 weak supersolution, 264 weak type estimate, 8, 20, 22 capacitary, 94 reverse, 13 weight, 161 doubling, 161 weighted average, 174, 212 weighted Poincar´ e inequality, 200, 209 Whitney decomposition, 165, 190, 195 Wiener criterion, 126, 285 Young’s convolution inequality, 31 zero boundary values, 26

INDEX

in the Sobolev sense, 35 zero extension, 37 Zygmund’s class, 14

Selected Published Titles in This Series 257 Juha Kinnunen, Juha Lehrb¨ ack, and Antti V¨ ah¨ akangas, Maximal Function Methods for Sobolev Spaces, 2021 256 Michio Jimbo, Tetsuji Miwa, and Fedor Smirnov, Local Operators in Integrable Models I, 2021 255 Alexandre Boritchev and Sergei Kuksin, One-Dimensional Turbulence and the Stochastic Burgers Equation, 2021 254 252 251 250

Karim Belabas and Henri Cohen, Numerical Algorithms for Number Theory, 2021 Julie D´ eserti, The Cremona Group and Its Subgroups, 2021 David Hoff, Linear and Quasilinear Parabolic Systems, 2020 Bachir Bekka and Pierre de la Harpe, Unitary Representations of Groups, Duals, and Characters, 2020

249 Nikolai M. Adrianov, Fedor Pakovich, and Alexander K. Zvonkin, Davenport–Zannier Polynomials and Dessins d’Enfants, 2020 248 Paul B. Larson and Jindrich Zapletal, Geometric Set Theory, 2020 247 Istv´ an Heckenberger and Hans-J¨ urgen Schneider, Hopf Algebras and Root Systems, 2020 246 Matheus C. Bortolan, Alexandre N. Carvalho, and Jos´ e A. Langa, Attractors Under Autonomous and Non-autonomous Perturbations, 2020 245 244 243 242

Aiping Wang and Anton Zettl, Ordinary Differential Operators, 2019 Nabile Boussa¨ıd and Andrew Comech, Nonlinear Dirac Equation, 2019 Jos´ e M. Isidro, Jordan Triple Systems in Complex and Functional Analysis, 2019 Bhargav Bhatt, Ana Caraiani, Kiran S. Kedlaya, Peter Scholze, and Jared Weinstein, Perfectoid Spaces, 2019 241 Dana P. Williams, A Tool Kit for Groupoid C ∗ -Algebras, 2019 240 Antonio Fern´ andez L´ opez, Jordan Structures in Lie Algebras, 2019 239 Nicola Arcozzi, Richard Rochberg, Eric T. Sawyer, and Brett D. Wick, The Dirichlet Space and Related Function Spaces, 2019 238 Michael Tsfasman, Serge Vlˇ adut ¸, and Dmitry Nogin, Algebraic Geometry Codes: Advanced Chapters, 2019 237 Dusa McDuff, Mohammad Tehrani, Kenji Fukaya, and Dominic Joyce, Virtual Fundamental Cycles in Symplectic Topology, 2019 236 Bernard Host and Bryna Kra, Nilpotent Structures in Ergodic Theory, 2018 235 Habib Ammari, Brian Fitzpatrick, Hyeonbae Kang, Matias Ruiz, Sanghyeon Yu, and Hai Zhang, Mathematical and Computational Methods in Photonics and Phononics, 2018 234 Vladimir I. Bogachev, Weak Convergence of Measures, 2018 233 N. V. Krylov, Sobolev and Viscosity Solutions for Fully Nonlinear Elliptic and Parabolic Equations, 2018 232 Dmitry Khavinson and Erik Lundberg, Linear Holomorphic Partial Differential Equations and Classical Potential Theory, 2018 231 Eberhard Kaniuth and Anthony To-Ming Lau, Fourier and Fourier-Stieltjes Algebras on Locally Compact Groups, 2018 230 Stephen D. Smith, Applying the Classification of Finite Simple Groups, 2018 229 Alexander Molev, Sugawara Operators for Classical Lie Algebras, 2018 228 Zhenbo Qin, Hilbert Schemes of Points and Infinite Dimensional Lie Algebras, 2018 227 Roberto Frigerio, Bounded Cohomology of Discrete Groups, 2017 226 Marcelo Aguiar and Swapneel Mahajan, Topics in Hyperplane Arrangements, 2017

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/survseries/.

This book discusses advances in maximal function methods related to Poincaré and Sobolev inequalities, pointwise estimates and approximation for Sobolev functions, Hardy’s inequalities, and partial differential equations. Capacities are needed for fine properties of Sobolev functions and characterization of Sobolev spaces with zero boundary values. The authors consider several uniform quantitative conditions that are self-improving, such as Hardy’s inequalities, capacity density conditions, and reverse Hölder inequalities. They also study Muckenhoupt weight properties of distance functions and combine these with weighted norm inequalities; notions of dimension are then used to characterize density conditions and to give sufficient and necessary conditions for Hardy’s inequalities. At the end of the book, the theory of weak solutions to the p-Laplace equation and the use of maximal function techniques is this context are discussed. The book is directed to researchers and graduate students interested in applications of geometric and harmonic analysis in Sobolev spaces and partial differential equations.

For additional information and updates on this book, visit www.ams.org/bookpages/surv-257

SURV/257