Nonlinear Potential Theory on Metric Spaces

EMS Tracts in Mathematics 17

EMS Tracts in Mathematics Editorial Board: Carlos E. Kenig (The University of Chicago, USA) Andrew Ranicki (The University of Edinburgh, Great Britain) Michael Röckner (Universität Bielefeld, Germany, and Purdue University, USA) Vladimir Turaev (Indiana University, Bloomington, USA) Alexander Varchenko (The University of North Carolina at Chapel Hill, USA) This series includes advanced texts and monographs covering all fields in pure and applied mathematics. Tracts will give a reliable introduction and reference to special fields of current research. The books in the series will in most cases be authored monographs, although edited volumes may be published if appropriate. They are addressed to graduate students seeking access to research topics as well as to the experts in the field working at the frontier of research. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

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2-Matrix Compression, Algorithms and Analysis Ronald Brown, Philip J. Higgins and Rafael Sivera, Nonabelian Algebraic Topology. Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids Marek Janicki and Peter Pflug, Separately Analytical Functions

Anders Björn Jana Björn

Nonlinear Potential Theory on Metric Spaces

Authors: Anders Björn Department of Mathematics Linköpings universitet SE-581 83 Linköping Sweden

Jana Björn Department of Mathematics Linköpings universitet SE-581 83 Linköping Sweden

E-mail: [email protected]

E-mail: [email protected]

2010 Mathematical Subject Classification: 31-02, 31E05; 28A12, 30L99, 31C05, 31C15, 31C40, 31C45, 35B45, 35B65, 35D30, 35J20, 35J25, 35J60, 35J67, 35J70, 35J92, 46E35, 47J20, 49J10, 49J27, 49J40, 49J52, 49N60, 49Q20, 58C99, 58J05, 58J32 Key words: Boundary regularity, capacity, Dirichlet problem, doubling measure, interior regularity, metric space, minimizer, Newtonian space, nonlinear, obstacle problem, Perron solution, p-harmonic function, Poincaré inequality, potential theory, Sobolev space, upper gradient

ISBN 978-3-03719-099-9 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2011

European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: info @ems-ph.org Homepage: www.ems-ph.org

Typeset using the author’s TEX files: I. Zimmermann, Freiburg Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321

Preface

In the first half of the 20th century, analysis went from studying smooth functions to nonsmooth ones, introducing such notions as weak solutions, Sobolev spaces and distributions. These concepts were first studied on Rn and later on manifolds and other smooth objects. Around 1970 came the first step towards analysis on nonsmooth objects, when notions such as maximal functions and Lebesgue points were studied on spaces of homogeneous type, i.e. on quasimetric spaces equipped with doubling measures. This theory can be called zero-order analysis, as no derivatives are used. Taking partial derivatives is not possible in metric spaces, but in the 1990s there was a need for studying first-order analysis on nonsmooth spaces. Heinonen and Koskela realized that upper gradients could be used as a substitute for the usual gradient. This gave rise to Newtonian spaces, one of several attempts to define Sobolev spaces on metric spaces, and perhaps the most fruitful one. It turned out that the potential theory of p-harmonic functions can be extended to metric spaces through the use of upper gradients. The nonlinear potential theory of p-harmonic functions on Rn has been developing since the 1960s and has later been generalized to weighted Rn , Riemannian manifolds, graphs, Heisenberg groups and more general Carnot groups and Carnot–Carathéodory spaces, and other situations. Studying potential theory of p-harmonic functions on metric spaces generalizes and gives a unified treatment of all these cases. There are primarily five books devoted to nonlinear potential theory, viz. Adams– Hedberg [5], Heinonen–Kilpeläinen–Martio [171], Malý–Ziemer [258], Mizuta [287] and Turesson [342]. They all study potential theory on unweighted ([5], [258] and [287]) or weighted ([171] and [342]) Rn . In [5] and [342] the focus is on higher order potential theory, whereas the main topic in [171] and [258] is the potential theory of p-harmonic functions (on weighted and unweighted Rn , respectively). The main focus in [287] is on Riesz potentials. The topics covered in our book are closest to [171], but there are also some parts in common with [258]. There is also some overlap, especially in Chapter 8, with the book by Giusti [146] (for unweighted Rn ). Let us also mention the survey papers by Martio [266], which has much in common with this book, and by Björn–Björn [47], which is based on an early version of this book. Moreover, the forthcoming monograph Heinonen–Koskela–Shanmugalingam– Tyson [177] has a certain overlap with the first part of this book. This book consists of two related parts. In the first part we develop the theory of Newtonian (Sobolev) spaces on metric spaces, and in the second part we develop the potential theory associated with p-harmonic functions on metric spaces. Both the Newtonian and the p-harmonic theories on metric spaces have now reached such a maturity that we think they deserve to be written in book form. So far, both theories are scattered over a large number of different research papers published during

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the last two decades, with obvious difficulties for those interested in them. In fact, very few of the results in this book are available in book form. When writing a book, one is faced with many decisions on what to include. Naturally, the choice is influenced by the taste and interests of the authors. Throughout the book we consider solely real-valued Newtonian spaces. We also restrict ourselves to the theory of p-harmonic functions (defined through upper gradients) on complete doubling metric spaces supporting a Poincaré inequality. We do not cover quasiminimizers, nor the noncomplete theory. Neither do we include results that only hold for Cheeger p-harmonic functions, or only in Rn . Related topics that could have been covered but were left out include: Differentiability in metric spaces (in particular, we do not prove Cheeger’s theorem (Theorem B.6)) and the Poincaré inequalities in the various examples discussed in Section 1.7 and Appendix A. Including these topics would have made the book substantially different, and many of these topics would rather deserve books on their own. There are also many recent results which we have not been able to cover, nor have we included a proof of Keith–Zhong’s theorem, Theorem 4.30. (See however the appendices, notes and remarks for some references and comments on the topics mentioned above.) This book is reasonably self-contained and we develop both the Newtonian theory and the p-harmonic theory from scratch. Naturally, we cannot develop all the mathematics needed for this book, and we have most often chosen to omit results which are available elsewhere in book form, but sometimes providing a reference. Thus, the reader is assumed to know measure theory and functional analysis. Apart from comparison results between Newtonian spaces and ordinary Sobolev spaces, there is no background needed in Sobolev space theory. In Chapters 1 and 2 we start developing the theory of upper gradients and Newtonian functions. Here we have collected the theory which works well in general metric spaces, i.e. without any assumptions such as doubling or the validity of a Poincaré inequality. In Chapter 3 we introduce the doubling condition and study some of its consequences. The reader interested only in the Newtonian theory can safely skip Sections 3.3–3.5. The John–Nirenberg lemma and its consequences in Sections 3.3 and 3.4, will be used only in the proof of the weak Harnack inequality for superminimizers (Theorem 8.10). (More specifically, it is Corollary 3.21 which is used there.) The Gehring lemma in Section 3.5, although important, is not used in this book. In Chapter 4 we introduce Poincaré inequalities of various types and look at their relations and some consequences. We discuss, in particular, connections with quasiconvexity. In Chapters 5 and 6 we study various properties of Newtonian spaces, which follow from assuming doubling and a p-Poincaré inequality. Throughout Chapters 1–6 we have taken extra care to see when the proofs are valid for p D 1 and with minimal assumptions, in particular when completeness is not needed. However, as our main interest is in the case p > 1, we have not dwelled further on the case p D 1 when it requires special proofs.

Preface

vii

The second part of the book consists of Chapters 7–14. In it we develop the potential theory associated with p-harmonic functions. Already from the beginning we need to use doubling, a p-Poincaré inequality and that p > 1. We also assume completeness, and these assumptions are general throughout the second part. We have taken special care to cover the case when X is a bounded metric space. In Chapters 7 and 8 we study the basic properties of superminimizers and show interior regularity, whereas Chapter 9 is devoted to superharmonic functions. In Chapter 10 we look closely at the Dirichlet problem, and in Chapter 11 on boundary regularity. In Chapter 12 we consider removable singularities, which are in turn needed to obtain the trichotomy (Theorem 13.2) motivating the study and classification of irregular boundary points in Chapter 13. Finally, in Chapter 14 we show that open sets can be approximated by regular sets and give some consequences thereof, including another formulation of the Dirichlet problem. In Appendix A we give various examples of metric spaces satisfying our basic assumptions (completeness, doubling and the validity of a p-Poincaré inequality). In particular, we show that on weighted Rn our theory coincides with the one developed in Heinonen–Kilpeläinen–Martio [171] and other sources. When specialized to weighted Rn , many of the results in this book appear in [171]. We have refrained from pointing out exactly which ones in the notes at the end of each chapter. Instead, we make [171] as a general reference for comparison throughout the book. The reader is also referred to the comments and references in [171]. In any case, we would like to point out that many of the results in Chapters 10–14 do not appear in [171] (when specialized to weighted Rn ). In Appendix B we take a quick look at Hajłasz–Sobolev and Cheeger–Sobolev spaces, and in Appendix C we give a short overview of the more general potential theory of quasiminimizers. The reader interested in open problems should observe that the item open problem in the index gives references to all open problems stated in the book. We started writing this book when giving a graduate course on this topic during the autumn of 2005 in Linköping. In fact, we gave a similar course in Prague in the autumn of 2003, and the handwritten notes we then obtained are in a sense the very first draft of this book. We thank the participants for their active role in the courses: Jan Kališ, Jan Malý, Petr Pˇrecechtˇel and Jiˇrí Spurný in Prague, and Gunnar Aronsson, Thomas Bäckdahl, Daniel Carlsson, David Färm, Thomas Karlsson, Mats Neymark, Björn Textorius, Johan Thim and Bengt-Ove Turesson in Linköping. Later on we have also given lecture series on these topics: Anders at the Paseky spring school Function Spaces, Inequalities and Interpolation, in June 2009, on Newtonian spaces, and Jana within the Taft Research Seminar at the University of Cincinnati, during the winter term 2010. Again, we thank all the participants. We are also grateful to Zohra Farnana, Heikki Hakkarainen, Stanislav Hencl, David Herron, Tero Kilpeläinen, Juha Kinnunen, Riikka Korte, Visa Latvala, Tero

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Mäkäläinen, Lukáš Malý, Niko Marola, Olli Martio, David Minda, Mikko Parviainen, Nageswari Shanmugalingam, Tomas Sjödin, Heli Tuominen and two anonymous referees for many useful comments given on draft versions of this book, and to Irene Zimmermann for her patience with our typesetting requests. Particular thanks go to Outi Elina Kansanen and Juha Kinnunen who wrote draft versions for Sections 3.3–3.5. Last but not least, we are grateful to Juha Heinonen, who introduced us to analysis on metric spaces during his course in Ann Arbor in 1999. We also encourage all future comments, in particular pointing out any mistakes, for which we apologize. Acknowledgement. The authors have been supported by various sources during their work on this book: the Swedish Science Research Council, Magnuson’s fund of the Royal Swedish Academy of Sciences, the Charles Phelps Taft Research Center at the University of Cincinnati, the Swedish Fulbright Commission, and Linköpings universitet. They also belong to the European Science Foundation Networking Programme Harmonic and Complex Analysis and Applications and to the Scandinavian Research Network Analysis and Application. Linköping, September 2011

Anders and Jana Björn

Contents Preface 1

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Newtonian spaces 1.1 The metric space X and some notation . . . . . . . . 1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . 1.3 Upper gradients and the Newtonian space N 1;p . . . . 1.4 The Sobolev capacity Cp . . . . . . . . . . . . . . . . 1.5 p-weak upper gradients and modulus of curve families 1.6 Banach space and ACCp . . . . . . . . . . . . . . . . 1.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 1 3 8 12 15 24 31 34

Minimal p-weak upper gradients 2.1 Fuglede’s lemma . . . . . . . . . . . 2.2 Minimal p-weak upper gradients . . 2.3 Calculus for p-weak upper gradients 2.4 The glueing lemma . . . . . . . . . . 2.5 N 1;p ./ . . . . . . . . . . . . . . . 1;p p 2.6 Nloc and Dloc . . . . . . . . . . . . 2.7 N01;p . . . . . . . . . . . . . . . . . p 2.8 Gloc . . . . . . . . . . . . . . . . . . 2.9 Dependence on p in gu . . . . . . . 2.10 Representation formulas for gu . . . 2.11 Notes . . . . . . . . . . . . . . . . .

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37 37 40 45 48 50 52 55 58 59 61 64

Doubling measures 3.1 Doubling . . . . . . . . . . . . . . . . . . 3.2 The maximal function . . . . . . . . . . . 3.3 BMO and John–Nirenberg’s lemma . . . . 3.4 Consequences of John–Nirenberg’s lemma 3.5 Gehring’s lemma . . . . . . . . . . . . . . 3.6 Notes . . . . . . . . . . . . . . . . . . . .

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65 65 68 70 75 77 82

Poincaré inequalities 4.1 Poincaré inequalities . . . . . . . . . . . 4.2 Characterizations of Poincaré inequalities 4.3 BiLipschitz invariance . . . . . . . . . . 4.4 .q; p/-Poincaré inequalities . . . . . . . 4.5 Quasiconvexity and connectivity . . . . .

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4.6 4.7 4.8 4.9 4.10 5

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Poincaré inequalities in quasiconvex spaces Inner metric . . . . . . . . . . . . . . . . The relation between L and  . . . . . . . Measurability . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . .

Properties of Newtonian functions 5.1 Density of Lipschitz functions . . . . . 5.2 Quasicontinuity of Newtonian functions 5.3 Continuity of Newtonian functions . . 5.4 Density of Lipschitz functions in N01;p 5.5 Sobolev embeddings and inequalities . 5.6 Lebesgue points for N 1;p -functions . . 5.7 Notes . . . . . . . . . . . . . . . . . .

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116 116 123 134 138 141 147 150

Capacities 6.1 Mazur’s lemma and its consequences . . . . . . . . . . . 6.2 Properties of Cp in complete doubling p-Poincaré spaces 6.3 The variational capacity capp . . . . . . . . . . . . . . . 6.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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154 154 157 161 168

Superminimizers 7.1 Introduction to potential theory . . . . . 7.2 The obstacle problem . . . . . . . . . . . 7.3 Definition of (super)minimizers . . . . . 7.4 Convergence results for superminimizers 7.5 Notes . . . . . . . . . . . . . . . . . . .

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170 170 172 178 183 189

Interior regularity 8.1 Weak Harnack inequalities for subminimizers . . . . . 8.2 Weak Harnack inequalities for superminimizers . . . . 8.3 Hölder continuity for p-harmonic functions . . . . . . 8.4 The need for  in Harnack’s inequality . . . . . . . . 8.5 Lsc-regularized superminimizers . . . . . . . . . . . 8.6 Lsc-regularized solutions of obstacle problems . . . . 8.7 p-harmonic extensions . . . . . . . . . . . . . . . . . 8.8 A sharp weak Harnack inequality for superminimizers 8.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . .

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191 191 197 201 205 206 209 212 213 215

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Superharmonic functions 218 9.1 Definition of superharmonic functions . . . . . . . . . . . . . . . . . 218 9.2 Weak Harnack inequalities for superharmonic functions . . . . . . . 220 9.3 Lsc-regularity and the minimum principle . . . . . . . . . . . . . . . 222

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9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11

Characterizations . . . . . . . . . . . . . . . . . . . . . . Convergence results for superharmonic functions . . . . . Harnack’s convergence theorem for p-harmonic functions Comparison of sub- and superharmonic functions . . . . . New superharmonic functions from old . . . . . . . . . . Integrability of superharmonic functions . . . . . . . . . Lebesgue points for superharmonic functions . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 The Dirichlet problem for p-harmonic functions 10.1 Continuous boundary values . . . . . . . . . 10.2 The Kellogg property . . . . . . . . . . . . . 10.3 Perron solutions . . . . . . . . . . . . . . . 10.4 Resolutivity of Newtonian functions . . . . . 10.5 Resolutivity of continuous functions . . . . . 10.6 Some consequences of resolutivity . . . . . . 10.7 Resolutivity of semicontinuous functions . . 10.8 The p-harmonic measure . . . . . . . . . . . 10.9 Poisson modification . . . . . . . . . . . . . 10.10 The resolutivity problem . . . . . . . . . . . 10.11 Notes . . . . . . . . . . . . . . . . . . . . .

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249 250 251 253 255 261 263 265 268 271 273 275

11 Boundary regularity 11.1 Barrier characterization of regular points . . 11.2 Boundary regularity for the obstacle problem 11.3 Characterizations of regularity . . . . . . . . 11.4 The Wiener criterion . . . . . . . . . . . . . 11.5 Regularity componentwise . . . . . . . . . . 11.6 Fine continuity . . . . . . . . . . . . . . . . 11.7 Notes . . . . . . . . . . . . . . . . . . . . .

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277 277 280 285 288 295 298 301

12 Removable singularities 12.1 Removability . . . . . . . . . . . . . 12.2 Nonremovability . . . . . . . . . . . 12.3 Removable sets with positive capacity 12.4 Nonunique removability . . . . . . . 12.5 Notes . . . . . . . . . . . . . . . . .

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303 303 309 312 314 316

13 Irregular boundary points 13.1 Semiregular and strongly irregular points . . . . . . . 13.2 Characterizations of semiregular points . . . . . . . . 13.3 Characterizations of strongly irregular points . . . . . 13.4 The sets of semiregular and of strongly irregular points

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318 318 320 325 327

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13.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 14 Regular sets and applications thereof 14.1 Regular sets . . . . . . . . . . . . 14.2 Wiener solutions . . . . . . . . . . 14.3 Classically superharmonic functions 14.4 Notes . . . . . . . . . . . . . . . .

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329 329 331 333 334

A Examples A.1 N 1;p in Euclidean spaces . . . . . . . . . . . . . . A.2 Weighted Sobolev spaces on Rn . . . . . . . . . . . A.3 Uniform domains and power weights . . . . . . . . A.4 Glueing spaces together . . . . . . . . . . . . . . . A.5 Graphs . . . . . . . . . . . . . . . . . . . . . . . . A.6 Carnot–Carathéodory spaces and Heisenberg groups A.7 Further examples . . . . . . . . . . . . . . . . . . . A.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . .

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337 337 340 348 349 351 354 357 358

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Appendices

B Hajłasz–Sobolev and Cheeger–Sobolev spaces 360 B.1 Hajłasz–Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . 360 B.2 Cheeger–Sobolev spaces and differentiable structures . . . . . . . . . 363 C Quasiminimizers

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Bibliography

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Index

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Chapter 1

Newtonian spaces

Throughout the book we will study potential theory on the metric space X . We make some general assumptions on X in the beginning of Section 1.1 below. These assumptions will hold throughout the book. In many of the chapters we will make some additional assumptions holding throughout that chapter. These assumptions will always be listed at the very beginning of the chapter. Occasionally we will state some additional assumptions at the very beginning of a section. In the first half of this book, Chapters 1–6, we introduce and study Sobolev type spaces on metric measure spaces. These spaces, called Newtonian spaces, will be the natural setting for p-harmonic functions and potential theory in the second half of the book. In Appendix B we mention some other possible definitions of Sobolev type spaces on metric spaces and relate them to our definition. Let us first take a very quick look at first-order Sobolev spaces on Rn . For 1  p < 1 and f 2 Lp .Rn / we define Z p kf kW 1;p .Rn / D .jf jp C jrf jp / dx; Rn

where rf is the weak or distributional gradient of f . The Sobolev space W 1;p .Rn / is then given by W 1;p .Rn / D ff W kf kW 1;p .Rn / < 1g: To define W 1;p .Rn /, one uses the gradient, i.e. the directional derivatives. In metric spaces we cannot talk about directions, so that is problematic. However, we do not really use the vector rf , only the scalar jrf j is necessary to define W 1;p .Rn /, and for jrf j there is a possible counterpart in metric spaces, called upper gradient and introduced by Heinonen–Koskela [173], [174]. Before defining upper gradients, in Section 1.3, we introduce our metric spaces and give some background results. Let us mention here that in Section 1.7 we give a short overview of some examples of metric spaces on which the theory of Newtonian spaces is useful. A more detailed discussion is given in Appendix A.

1.1 The metric space X and some notation We assume throughout the book that 1  p < 1 and that X D .X; d; / is a metric space endowed with a metric d and a positive complete Borel measure  such that .B/ < 1

for all balls B  X

(we make the convention that balls are nonempty and open).

2

1 Newtonian spaces

We emphasize that to start with we do not assume that X is complete nor locally compact. We will however assume that X contains at least two points (to avoid special formulations needed to cover the pathological cases when X is empty and when X is a singleton). We also emphasize that the -algebra on which  is defined is obtained by the completion of the Borel -algebra. We further extend  as an outer measure on X , so that for an arbitrary set A  X we have .A/ D inff.E/ W E  A is a Borel setg: Proposition 1.1. The measure  is Borel regular, i.e. for every A  X there is a Borel set E  A such that .A/ D .E/. Note that this is the definition of Borel regularity used by Federer [123], Section 2.2.3, Mattila [275], Definition 1.5, and Heinonen [169], p. 3, whereas Rudin [311] has a more restrictive definition of Borel regularity which is not always fulfilled for our spaces. Proof. If .A/ D 1, then we may choose E D X . Otherwise weTcan find Borel sets Ej  A with .Ej / < .A/ C 1=j , j D 1; 2; ::: . Letting E D j1D1 Ej completes the proof. Let us here record some notation used throughout the book. We use the following notation for balls B.x0 ; r/ WD fx 2 X W d.x; x0 / < rg; and for B D B.x0 ; r/ and  > 0, we let B D B.x0 ;  r/. Further, we always assume that , 0 , 00 and  t , t 2 R, are nonempty open subsets of X . In the metric space X , a.e. always means almost everywhere with respect to the given measure , unless otherwise stated. For a measurable set A  X with 0 < .A/ < 1, we define the mean-value integral (or integral average) of f over A as « Z 1 f d WD f d; .A/ A A whenever the right-hand side exists, which it does, in particular, if f 2 L1 .A/ or if f is nonnegative and measurable on A. We shall also use the abbreviation « fA WD f d: A

In general we allow semicontinuous functions to be extended real-valued, i.e. to x D Œ1; 1. On the other hand, continuous functions are always take values in R assumed to be real-valued, unless mentioned otherwise.

1.2 Preliminaries

3

We denote the characteristic function of a set E by E , and let sup ¿ D 1 and inf ¿ D 1. Further, we let fC D maxff; 0g and f D maxff; 0g so that f D fC  f and jf j D fC C f . We define 0  1 WD 0  .1/ WD 0. This will, in particular, be useful when multiplying a function by a characteristic function so that we always let f E D 0 outside of E, even when f may take the values ˙1. Moreover, we make the convention that 1  1 D 1 and .1/  .1/ D 1, see the comments after Definition 1.13 for the usefulness of this. For sequences of numbers or sets we will use the words increasing and decreasing in their nonstrict sense, so that e.g. the sequence faj gj1D1 is increasing if aj C1  aj , j D 1; 2; ::: . We also say that E b A if Ex is a compact subset of A, and let Lipc .A/ D ff 2 Lip.A/ W supp f b Ag be the set of Lipschitz functions on A with compact support. We shall later see that this class of functions will play the important role of test functions for us. Throughout the book we present a number of examples with X being Rn , n  1, or some subset thereof. In these cases, unless otherwise stated, we always equip X with the Euclidean metric and the natural Lebesgue measure (or restrictions thereof). Finally, we make the convention that, unless otherwise stated, the letter C denotes various positive constants whose exact values are not important and may vary with each usage.

1.2 Preliminaries We will need the following consequence of the Borel regularity. x is measurable, then there exist Borel functions Proposition 1.2. If f W X ! R x f1 ; f2 W X ! R such that f1  f  f2 and f1 D f2 a.e. A Borel function f W X ! Y is by definition a function such that f 1 .G/  X is a Borel set for every open set G  Y . It follows that f 1 .E/  X is a Borel set for every Borel set E  Y . The following simple lemma will be useful in the proof of Proposition 1.2. x Then f is a Borel function if and only if Lemma 1.3. Let f W X ! R. Eq D fx W f .x/ > qg is a Borel set for every q 2 Q.

4

1 Newtonian spaces

Proof. The necessity is clear, so let us look at the sufficiency. Let I D .a; b/  R, a < b, be an open interval. Then  [   \  f 1 .I / D Eq n Eq Q3q>a

Q3q
is a Borel set. Moreover, f 1 .1/ D

\

Eq

and

f 1 .1/ D X n

q2Q

[

Eq

q2Q

x are Borel sets. We conclude that f 1 .I / is a Borel set for any open interval in R (which is now allowed to contain the points ˙1). x be open. Then E can be written as a countable union of pairwise Let now E  R S disjoint open intervals, E D j1D1 Ij (where Ij is allowed to contain ˙1), see Lemma 1.4 below. S Since f 1 .E/ D j1D1 f 1 .Ij / is a Borel set, we conclude that f is a Borel function. Proof of Proposition 1.2. For q 2 Q let Eq D fx W f .x/ > qg. A0q be a T Let further 0 0 0 Borel set such that Aq  Eq and .Aq n Eq / D 0, and let Aq D Q3q 0 q Aq 0 which is a Borel set. As A0q 0  Eq 0  Eq for every q 0  q, q 0 2 Q, we see that Eq  Aq  A0q and thus .Aq n Eq /  .A0q n Eq / D 0. Note that Aq  Aq 0 if q 0  q, q 0 2 Q. Let now f2 .x/ D supfq 2 Q W x 2 Aq g  supfq 2 Q W x 2 Eq g D f .x/: Then f2 is a Borel function by Lemma 1.3. Moreover, [ fx W f2 .x/ ¤ f .x/g D fx W f2 .x/ > f .x/g D fx W f2 .x/ > q  f .x/g D

[

q2Q

fx W x 2 Aq and x … Eq g D

q2Q

[

.Aq n Eq /;

q2Q

which is a countable union of sets of measure zero and hence itself of measure zero. Thus f2 has the required properties. The construction of f1 is similar, but its existence may also be obtained by applying the already proved half of the proposition to the function f . x be open. Then G can be written as a countable (or finite) Lemma 1.4. Let G  R union of pairwise disjoint open intervals (which are allowed to contain ˙1). This lemma is well known, but as the proof is short and we have not found a good reference we give the proof here. Let us point out, for later use, that a countable set may be finite.

1.2 Preliminaries

5

Proof. Write G as a union of its components. Each component is necessarily an open interval. Associate with each component one of its rational points. Then we see that the set of components cannot have larger cardinality than Q. Proposition 1.5. Let Y be a metric space. Then the following are equivalent: (a) Y is separable; (b) Y is second countable, i.e. it has a countable open base; (c) Y is Lindelöf, i.e. every open cover of Y has a countable subcover. It is easily seen that these properties are also inherited by any subset of Y (in the induced topology). This is well known, see e.g. Exercise 5, p. 194, in Munkres [296], or Figure 10 in Steen–Seebach [327]. However, we have not found one good reference with full proofs of all implications. Proof. (a) , (b) See Theorem 2, p. 177, in Kuratowski [235]. (b) ) (c) See Theorem 2, p. 176, in [235]. (c) ) (b) See the second paragraph on p. 35 in Steen–Seebach [327]. In our considerations on the metric measure space X , the support of  will be of main interest. The following result is therefore useful. See also Proposition 1.53 and the comment before it. Note, however, that we do not want to assume that X D supp  in general, since we want to be able to consider arbitrary subsets of X as metric spaces on their own and it then may not be true that the support of the restriction of the underlying measure is the whole subset under consideration. Recall that x 2 supp , the support of , if and only if .B.x; r// > 0 for every r > 0. Or in other terms X n supp  is the union of all balls with zero measure, and is the largest open set with zero measure, in the sense that it contains any open set with zero measure. (The support of a function is similarly defined as the complement of the largest open set on which f is 0.) Proposition 1.6. The metric space supp  is separable, second countable and Lindelöf. Before proving Proposition 1.6, let us make some remarks. The following simple but very useful covering lemma is well known. Lemma 1.7 (5-covering lemma). Let B be a family of balls in X with uniformly bounded radii. Then there exists a subfamily B 0  B of pairwise disjoint balls such that [ [ B 5B: B2B

B2B 0

If X is separable, then the subfamily B 0 can be chosen to be countable.

6

1 Newtonian spaces

Proof. For a proof of the first part see e.g. Ambrosio–Tilli [17], Theorem 2.2.3, Heinonen [169], Theorem 1.2, or Federer [123], Theorem 2.8.4 (choose  D 2 and let ı be the radius of the balls in Federer’s formulation). For the second part assume that X is separable. Then X is Lindelöf S by Proposi00 0 tion 1.5, and hence we can find a countable subfamily B  B such that B2B 00 5B D S B2B 0 5B. Remark 1.8. The constant 5 in the 5-covering lemma is not optimal, but has become standard as one wants to have as simple an argument as possible. With just a slight extra thought most proofs actually show that one may replace 5 by 3 C " for any " > 0. In Federer [123], Theorem 2.8.4, this is even a consequence of the statement, just choose  D 1 C 12 ". The following example shows that one cannot replace 5 by 3. Example 1.9. Let X D R and B2j 1 D B.2  2=j; 2  1=j / for j D 1; 2; ::: . In other terms, B2j 1 is the interval .1=j; 4  3=j /. Let also B2j D B2j 1 , j D 1; 2; ::: . Then 0 2 Bj for all j D 1; 2; ::: , and thus any disjoint subfamily must consist of just one ball, say Bk . If k D 2j  S11is odd, then 3Bk is the interval .4 C 1=j; 8  5=j / and therefore does not cover j D1 Bj D .4; 4/. The case when k is even is similar, and hence the constant 5 in the 5-covering lemma cannot be replaced by 3. Proof of Proposition 1.6. Fix x0 2 Y WD supp  and let " > 0 and R > 0. Let B D fB.x; "/ W x 2 Y \ B.x0 ; R/g: By the 5-covering lemma (Lemma 1.7), we can find a subfamily B 0  B of pairwise disjoint balls so that [ [ B 5B: Y \ B.x0 ; R/  B2B

As

X B2B 0

.B/ D 

[

B2B 0

 B  .B.x0 ; R C "// < 1;

B2B 0

and .B/ > 0 for every B 2 B 0 , we conclude that B 0 is a countable set. The centres of the balls B 2 B 0 make a countable "-net in Y \ B.x0 ; R/. S Letting " D 1=n, n D 1; 2; ::: , shows that Y \ B.x0 ; R/ is separable. Thus Y D j1D1 .Y \ B.x0 ; j // is also separable. That Y is second countable and Lindelöf follows from Proposition 1.5. Lemma 1.10 (Cavalieri’s principle). Let be a measure on X . If f W X ! Œ0; 1 is a -measurable function and 0 < q < 1, then Z 1 Z f q d D q q1 .fx W f .x/ > g/ d : X

0

1.2 Preliminaries

7

The Cavalieri principle is a standard tool in integration theory, we state it here for the reader’s convenience. Sometimes it is the case q D 1 that is called the Cavalieri principle, but the general case follows after the change of variable  7! q . The Cavalieri principle can be proved in several different ways, e.g. using the Fubini theorem. The most convenient way actually depends on how the Lebesgue integral is defined and the proof truly belongs to integration theory. We refer the reader in need to Folland [124], Proposition 6.24. Another result which eventually will be needed is that continuous functions can be approximated by Lipschitz functions on compact sets. Recall that f W Y ! R is Lipschitz on the metric space .Y; d / if there exists a constant C such that jf .x/  f .y/j  Cd.x; y/

for all x; y 2 Y:

Proposition 1.11. Let .Y; d / be a compact metric space and let u 2 C.Y / and " > 0. Then there is v 2 Lip.Y / such that supy2Y ju.y/  v.y/j < ". This follows from the Stone–Weierstrass theorem (see e.g. Rudin [312], p. 122), but for the reader’s convenience, we give an elementary proof of this fact. Proof. We can assume that minY u D 0 and let Gk D fx 2 Y W u.x/ > k"g for positive integers k. Clearly, only finitely many Gk are nonempty. As Y is compact, xk , and hence there exists ı > 0 so that dist.Gk ; Y n Gk1 /  ı whenever so is each G Gk ¤ ¿. Let, for x 2 Y , u" .x/ D "

X

k1

1

dist.Gk ; x/ ı



; C

where we only sum over nonempty Gk . Then u" 2 Lip.Y / (with constant "=ı) and ju.x/  u" .x/j  " for all x 2 Y . We will also need to approximate lower semicontinuous functions by Lipschitz functions from below. Proposition 1.12. Let .Y; d / be a compact metric space and let u W Y ! .1; 1 be lower semicontinuous. Then there exists an increasing sequence fuj gj1D1 of Lipschitz functions such that uj .x/ ! u.x/, as j ! 1, for all x 2 Y . If u  0, then we may choose uj  0, j D 1; 2; ::: . In fact it is fairly easy to see that the uj constructed below is the largest j -Lipschitz function  u (apart from the case when u  1 when no such largest function exists). Proof. As u is lower semicontinuous on a compact set, it attains its minimum, which by assumption is not 1. Hence, without loss of generality, we may assume that u  0. If u  1, then we can take uj  j , j D 1; 2; ::: , and we therefore exclude this case below. Fix a positive integer j , and let uj .x/ D inf .u.y/ C jd.x; y//: y2Y

8

1 Newtonian spaces

It is easy to see that uj is real-valued and that 0  uj  uj C1  u. Let us first show that uj is j -Lipschitz. Let x; y 2 Y and " > 0. Then we can find z 2 Y such that uj .x/ > u.z/ C jd.z; x/  ". It follows that uj .y/  u.z/ C jd.z; y/  u.z/ C jd.z; x/ C jd.x; y/ < uj .x/ C jd.x; y/ C ": Letting " ! 0 shows that uj .y/  uj .x/ C jd.x; y/. As we also have uj .x/  uj .y/ C jd.x; y/, we have shown that uj is j -Lipschitz. Let x 2 Y be arbitrary. It remains to show that uj .x/ ! u.x/, as j ! 1. Let  < u.x/. As u is lower semicontinuous we can find a ball B D B.x; r/ in Y such that u >  in B. For j  =r we have   uj .x/ D min inf .u.y/ C jd.x; y//; inf .u.y/ C jd.x; y// y2B y2Y nB    min inf ; inf jr D : y2B

y2Y nB

Hence uj .x/ ! u.x/, as j ! 1.

1.3 Upper gradients and the Newtonian space N 1;p In this section we introduce upper gradients as a substitute for the modulus of the usual gradient. We will later see that under some assumptions it is possible to control a function by its upper gradients. On the other hand, in situations as in Example 1.22 the upper gradients do not give any information about the function. To begin with, we will leave this question aside and just concentrate on developing the theory of upper gradients, and p-weak upper gradients, whatever use they may have. In Chapters 3 and 4 we will introduce some conditions that will be sufficient for the upper gradients to control their functions. By a curve in X we will mean a rectifiable nonconstant continuous mapping from a compact interval. (For us only such curves will be interesting, in general a curve is a continuous mapping from an interval. A rectifiable curve is a curve with finite length.) A curve can thus be parameterized by arc length ds, and we will always assume that all curves are parameterized by arc length, see e.g. Ambrosio–Tilli [17], Section 4.2, or Heinonen [169], Section 7.1. Note that every curve is Lipschitz continuous with respect to its arc length parameterization. We let .X / denote the family of all curves on X . By abuse of notation we also denote the image of a curve  by  . Definition 1.13. A nonnegative Borel function g on X is an upper gradient of an extended real-valued function f on X if for all curves  W Œ0; l  ! X , Z jf ..0//  f ..l //j  g ds: (1.1) 

1.3 Upper gradients and the Newtonian space N 1;p

9

R Recall our convention that 1  1 D 1 and .1/  .1/ D 1, so that  g ds D 1 if at least one Rof f ..0// and f ..l // is infinite. Note that for a Borel function g  0, the integral  g ds is defined (with a value in Œ0; 1) for all curves , x is a nonnegative Borel function. since g ı  W Œ0; l  ! R Note also that upper gradients are not unique. In particular, by adding a nonnegative Borel function to an upper gradient of f we obtain a new upper gradient of f . The function g  1 is an upper gradient of every function. The following simple proposition provides us with plenty nontrivial examples of upper gradients. Proposition 1.14. If f W X ! R is locally Lipschitz, then the lower pointwise dilation lip f .x/ D lim inf

sup

r!0 y2B.x;r/

jf .y/  f .x/j r

(1.2)

is an upper gradient of f . Proof. Let  W Œ0; l  ! X be a curve. Since  is parameterized by arc length, it is 1-Lipschitz, and thus f ı  is Lipschitz and hence absolutely continuous on Œ0; l  (see Definition 1.57 for the definition of absolute continuity). We thus have Z l Z l Z jf ..0//  f ..l //j  j.f ı  /0 .t /j dt  .lip f / ı .t / dt D lip f ds: 0

0



1 Corollary 1.15. If X D Rn and f 2 Cloc .Rn /, then jrf j is an upper gradient of f .

Proof. For all x 2 Rn and all y 2 B.x; r/ we have jf .y/  f .x/j D jru.x/  .y  x/ C o.jy  xj/j  r jru.x/j C o.r/: Inserting this into the definition (1.2) of lip f .x/ finishes the proof. This result will be extended to locally Lipschitz functions in Corollary 1.47, and further refined in Propositions A.3 and A.11. Note that if g and g 0 are upper gradients of u and v, respectively, then g  g 0 is in general not an upper gradient of u  v. (An easy example is to let u.x/ D x, v.x/ D x and g D g 0  1 on R, in which case g  g 0  0 is not an upper gradient of .u  v/.x/ D 2x.) However, we have the following subadditivity result. Lemma 1.16. Let g and g 0 be upper gradients of u and v, respectively, and a 2 R. Then jajg and g C g 0 are upper gradients of au and u C v, respectively. Proof. This follows immediately from Definition 1.13. Now that we have upper gradients, it is possible to define analogues of Sobolev spaces on metric measure spaces.

10

1 Newtonian spaces

Definition 1.17. Whenever u 2 Lp .X /, let Z kukN 1;p .X/ D

juj d C inf X

1=p

Z p

g

p

g d

;

X

where the infimum is taken over all upper gradients g of u. The Newtonian space on X is the space N 1;p .X / D fu W kukN 1;p .X/ < 1g: Let us here point out that we assume that functions are defined everywhere, and not just up to an equivalence class in the corresponding function space. When we say that u 2 N 1;p .X / we thus assume that u is a function defined everywhere. Remark 1.18. A minor point here which is easily overlooked is that one of course can use an equivalent norm, such as kukLp C inf g kgkLp , when studying N 1;p .X /. However, when we define the capacity in Definition 1.24 it is important to use our norm to get a subadditive capacity. In some situations we will also have use for some related spaces. Definition 1.19. Let us define the following spaces Nz 1;p .X / D N 1;p .X /= ; where u v if and only if ku  vkN 1;p .X/ D 0, and Ny 1;p .X / D fu W u D v a.e. for some v 2 N 1;p .X /g: Note that in ku  vkN 1;p .X/ , the infimum is taken over upper gradients of u  v, not over upper gradients of u and v. We equip Nz 1;p .X / and Ny 1;p .X / with the norms induced by N 1;p .X / (so that kukNy 1;p .X/ D kvkN 1;p .X/ if Ny 1;p .X / 3 u D v 2 N 1;p .X / a.e.). It is not obvious that the norm on Ny 1;p .X / is well defined, but this follows from Lemma 1.62 below. Let us already here point out that the equivalence classes in Nz 1;p .X / are not up to measure zero, but are finer than that, in fact, as we will see, they are up to capacity zero. In Proposition 1.65 we will find that Ny 1;p .X /= ae Š Nz 1;p .X /, where u ae v if u D v a.e. The space Ny 1;p .X /= ae is closer to traditional Sobolev spaces since it has a.e.-equivalence classes. It will however turn out to be both useful and important for us to have the more refined q.e.-equivalence classes in Nz 1;p .X / (i.e. to study the functions in N 1;p .X /), see e.g. the discussion following Proposition 1.65. That N 1;p .X /, Nz 1;p .X / and Ny 1;p .X / are vector spaces follows directly from Lemma 1.16. The only difficulty with showing that k  kN 1;p .X/ is a norm on Nz 1;p .X / (and a seminorm on N 1;p .X /) is to prove the triangle inequality. To see this, let u; v 2

1.3 Upper gradients and the Newtonian space N 1;p

11

N 1;p .X / and " > 0 be arbitrary. Find upper gradients g; g 0 2 Lp .X / of u and v, respectively, so that p p 1=p .kukL  kukN 1;p .X/ C "; p .X/ C kgkLp .X/ / p 0 p 1=p .kvkL  kvkN 1;p .X/ C ": p .X/ C kg kLp .X/ /

(1.3)

By Lemma 1.16, g C g 0 is an upper gradient of u C v. Now, note that the left-hand sides in (1.3) are l p -norms (on R2 ) of .kukLp .X/ ; kgkLp .X/ /

.kvkLp .X/ ; kg 0 kLp .X/ /;

and

respectively. Similarly, p 0 p 1=p ku C vkN 1;p .X/  .ku C vkL p .X/ C kg C g kLp .X/ /

 ..kukLp .X/ C kvkLp .X/ /p C .kgkLp .X/ C kg 0 kLp .X/ /p /1=p ; which is the l p -norm of .kukLp .X/ C kvkLp .X/ ; kgkLp .X/ C kg 0 kLp .X/ /: The triangle inequality for the l p -norm now implies that p p p 1=p 0 p 1=p ku C vkN 1;p .X/  .kukL C .kvkL p .X/ C kgkLp .X/ / p .X/ C kg kLp .X/ /

 kukN 1;p .X/ C " C kvkN 1;p .X/ C " and letting " ! 0 proves the triangle inequality for k  kN 1;p .X/ . The completeness of Nz 1;p .X /, i.e. that it is a Banach space, is harder to prove and will be obtained in Theorem 1.71. Theorem 1.20. The space N 1;p .X / is a lattice, i.e. if u; v 2 N 1;p .X /, then maxfu; vg; minfu; vg 2 N 1;p .X /: Proof. Let w D maxfu; vg 2 Lp .X /. Let further g; g 0 2 Lp .X / be upper gradients of u and v, respectively, and g 00 WD g C g 0 2 Lp .X /. Then for a curve  W Œ0; l  ! X we have jw..0//  w..l //j  ju..0//  u..l //j C jv..0//  v..l //j Z Z Z 0  g ds C g ds D g 00 ds; 





and thus g 00 is an upper gradient of w, and w 2 N 1;p .X /. The proof for minfu; vg is similar.

12

1 Newtonian spaces

With a little more care one easily sees that maxfg; g 0 g is an upper gradient of minfu; vg and of maxfu; vg. In Corollary 2.20 we give more precise information about the best upper gradients of minfu; vg and maxfu; vg. The following is an immediate consequence of Theorem 1.20. Corollary 1.21. Assume that u 2 N 1;p .X /. Then uC ; u ; juj 2 N 1;p .X /. The following examples show that in some cases the theory of upper gradients becomes quite pathological. Example 1.22. If the space X contains no nonconstant rectifiable curves, e.g. if X is discrete (or more generally totally disconnected), or the von Koch snowflake curve (see Example 1.23 below), then g  0 is an upper gradient of any function. Hence it follows that N 1;p .X / D Lp .X /. Example 1.23. In this example we consider the von Koch snowflake curve, which is a famous example of a curve of infinite length containing no rectifiable curves. Let K0  R2 , the 0th generation, be an equilateral triangle with side length 1. For each of the three sides, split it into three intervals of equal length and replace the middle one I by two sides I 0 and I 00 of an equilateral triangle (with sides I , I 0 and I 00 ) outside K0 . We have thus produced the 1st generation K1 of the von Koch snowflake curve consisting of 12 pieces of length 13 each. Continuing in this way we obtain the nth generation Kn consisting of 3  4n pieces, each of length 3n . In the limit we obtain the von Koch snowflake curve K, which can formally be defined as K D fx 2 R2 W for every " > 0 there is n > 1=" and y 2 Kn so that jx  yj < "g; or, in other terms, K is the Hausdorff limit of Kn , as n ! 1. We equip K with the distance from R2 . In this case, any curve (in the traditional sense) between two distinct points on K has infinite length and thus is not rectifiable. Hence there are no nonconstant rectifiable curves, and as in Example 1.22, g  0 is an upper gradient of any function, and N 1;p .K/ D Lp .K/. This in fact happens regardless of which measure we equip K with. For instance, one can equip K with the d -dimensional Hausdorff measure, with d D log 4=log 3, making it into an Ahlfors d -regular space, see Definition 3.4.

1.4 The Sobolev capacity Cp Definition 1.24. The capacity of a set E  X is the number Cp .E/ D inf kukpN 1;p .X/ ; where the infimum is taken over all u 2 N 1;p .X / such that u  1 on E.

1.4 The Sobolev capacity Cp

13

This capacity is sometimes referred to as the Sobolev capacity. We say that a property regarding points in X holds quasieverywhere (q.e.) if the set of points for which it fails has capacity zero. Just as sets of zero measure are important in integration theory, sets of zero capacity will be important to us. Most of the time it is the distinction between sets of zero and positive capacity that will matter, though there are exceptions when the actual value of the capacity is important, e.g. in the Wiener criterion and the fine topology, see Sections 11.4 and 11.6. For most parts of the book we have p fixed and do not discuss dependence on p. However, the dependence of the capacity on p is discussed a little in Section 2.9. Remark 1.25. Maybe some words on our choice of terminology can be useful. The capacity depends on p, but we have refrained from making this dependence explicit in the notation and thus do not write “p-capacity” nor “p-q.e.”. This is in contrast to “p-modulus”, “p-a.e. curve” and “p-weak upper gradient” for which we have decided to make the p explicit. This is partly because they are less used and the dependence on p might otherwise be overlooked by some readers; with “capacity” and “q.e.” we see no such risk. In the later part of the book, Chapters 7–14, we have again refrained from making the dependence on p explicit when writing “minimizer”, “subminimizer”, “superminimizer”, “subharmonic” and “superharmonic”. We however do write “p-harmonic” as “harmonic” sounds too linear to us. When calculating capacities it is often convenient to use the following result. Sometimes we will use the intermediate consequence that the capacity is obtained by taking the infimum over functions u  E . Under additional assumptions on X , it is possible to further restrict the collection of functions which are used to test the capacity, see Theorem 6.7. Proposition 1.26. Let E  X . Then Cp .E/ D inf kukpN 1;p .X/ ; where the infimum is taken over all u 2 N 1;p .X / such that E  u  1 on X . Proof. This follows easily by truncation. Let u 2 N 1;p .X / be such that u  1 on E. Let further v D minfu; 1gC ,  W Œ0; l  ! X be a curve, and g be an upper gradient of u. Then Z jv..0//  v..l //j  ju..0//  u..l //j  g ds; 

and thus g is an upper gradient also of v. Since jvj  juj we have that kvkN 1;p .X/ 

14

1 Newtonian spaces

kukN 1;p .X/ . It thus follows that Cp .E/  

inf

kvkpN 1;p .X/ 

inf

kukpN 1;p .X/ D Cp .E/:

E v1

u1 on E

inf

vDminfu;1gC u1 on E

kvkpN 1;p .X/

The capacity satisfies a number of properties. To begin with, we will need the following properties. Theorem 1.27. Let E; E1 ; E2 ; ::: be arbitrary subsets of X . Then (i) Cp .¿/ D 0; (ii) .E/  Cp .E/; (iii) if E1  E2 , then Cp .E1 /  Cp .E2 /; (iv) Cp is countably subadditive and is also an outer measure, i.e. Cp

1 [ iD1

1  X Ei  Cp .Ei /: iD1

In order to prove (iv) we need the following lemma. Lemma 1.28. Let ui  1, i D 1; 2; ::: , be functions with upper gradients gi , and let u D supi ui and g D supi gi . Then g is an upper gradient of u. Remark 1.29. Observe that if we remove the assumption that the functions ui be uniformly bounded from above, then the lemma becomes false. This is due to the special treatment of the function values ˙1 in Definition 1.13, and is most easily seen by letting ui  i, u  1, g  gi  0, and observing that g is not an upper gradient of u (if there are nonconstant curves in X ). See also Lemma 1.52. Proof. Let  W Œ0; l  ! X be a curve. Let us first observe that u..0//  u..l // D sup.ui ..0//  u..l ///  sup.ui ..0//  ui ..l ///: i

i

Together with the corresponding inequality for u..l //  u..0// we get that Z Z ju..0//  u..l //j  sup jui ..0//  ui ..l //j  sup gi ds  g ds: i

i





Proof of Theorem 1.27. The proofs of (i)–(iii) are trivial. (iv) We may assume that the right-hand side is finite. Let " > 0. Choose ui with Ei  ui  1 and upper gradients gi such that p p kui kL p .X/ C kgi kLp .X/  Cp .Ei / C

" : 2i

1.5 p-weak upper gradients and modulus of curve families

15

Let u D sup S i ui and g D supi gi . By Lemma 1.28, g is an upper gradient of u. Clearly u  1 on 1 iD1 Ei . Hence Cp

1 [

 Ei  kukpN 1;p .X/

iD1

Z 



X

 D 

sup ui

p

Z d C

i

Z X 1 X iD1 1 Z X

upi d C

X iD1 1  X

upi

iD1

D"C

1 X

X 1 X

Z

X iD1

Z

d C

Cp .Ei / C



X

sup gi

p

d

i

gip d

gip

 d

" 2i

Cp .Ei /:

iD1

Letting " ! 0 completes the proof of (iv). An easy, but interesting, consequence of the definition of the capacity is the following result. Proposition 1.30. If u 2 N 1;p .X /, then Cp .fx W ju.x/j D 1g/ D 0. Proof. Let E˙ D fx W u.x/ D ˙1g. Then u=k  1 on EC for all k > 0. Thus  u p kukpN 1;p .X/   Cp .EC /    1;p D ! 0; k N .X/ kp

as k ! 1:

Similarly Cp .E / D 0 (consider the function v WD u). Thus, by Theorem 1.27 (iv), Cp .fx W ju.x/j D 1g/  Cp .EC / C Cp .E / D 0.

1.5 p-weak upper gradients and modulus of curve families The following example illustrates a drawback of upper gradients, viz. that they are not preserved by Lp -convergence. This will make it necessary for us to study p-weak upper gradients. Example 1.31. Let E D f0g  X D R2 and 1 < p < 2. Let also f D E and ´ 1=j jxj; jxj  1; gj .x/ D j D 1; 2; ::: : 0; jxj > 1;

16

1 Newtonian spaces

Then gj is an upper gradient of f and since kgj kLp .X/ ! 0, as j ! 1, we see that kf kN 1;p .X/ D 0 (and also that Cp .E/ D 0). However, the zero function is not an upper gradient of f , nor is, in fact, any function g such that g D 0 a.e. To see this, observe that if g D 0 a.e., then for a.e. ˛, g D 0 a.e. on ˛ , where ˛ .t / D .t cos ˛; t sin ˛/, 0  t  1, by Fubini’s theorem (in polar coordinates). For such ˛ we have Z jf ..0//  f ..1//j D 1 > 0 D g ds; ˛

showing that g is not an upper gradient of f . Thus the set of upper gradients of f is not a closed subset of Lp .X / (or more correctly of the cone LpC .X / of nonnegative functions in Lp .X /). To overcome this complication we introduce p-weak upper gradients below. Later, in Section 2.2, we will show that any Newtonian function has a minimal p-weak upper gradient. (In the example above one can alternatively use the upper gradients ´ 1=jxj; jxj  1=j; gQj .x/ D j D 1; 2; ::: :/ 0; jxj > 1=j; Definition 1.32. A nonnegative measurable function g on X is a p-weak upper gradient of an extended real-valued function f on X if for p-a.e. curve  W Œ0; l  ! X (i.e. with the exception of a curve family of zero p-modulus, see below), Z jf ..0//  f ..l //j  g ds: (1.4) 

R

It is implicitly assumed that g is such that  g ds is defined (with a value in Œ0; 1) for p-a.e. curve. However, this is actually a consequence of the fact that g is assumed to be nonnegative and measurable, as we will see in Lemma 1.43. Recall the convention that 1  1 D 1, see also the comments after Definition 1.13. Definition 1.33. Let be a family of curves on X . Then we define the p-modulus of

by Z Modp . / D inf

p d; X

where the infimum is taken over all nonnegative Borel functions such that for all  2 .

R 

ds  1

By Proposition 1.2, weRmay as well take the infimum over all nonnegative measurable functions such that  ds  1 for all  2 . (In fact, using Proposition 1.37 it is easy to show that the infimum can as well be taken over all nonnegative measurable

1.5 p-weak upper gradients and modulus of curve families

17

R functions such that  ds  1 for p-a.e.  2 , but that cannot be used as the definition as the reasoning would be circular.) Just as with the capacity, it is whether the p-modulus is zero or not that will be important to us. The actual value plays a role only in (b) below, but this is a result we will not need; for us the weaker Corollary 1.38 will be sufficient. The reader who so wishes may therefore skip (b) below and its proof. Lemma 1.34. The following are true: (a) if 1  2 , then Modp . 1 /  Modp . 2 /; S  P (b) Modp j1D1 j  j1D1 Modp . j /; (c) if for every  2 there exists a subcurve  0 2 0 of , then Modp . /  Modp . 0 /. For the last part we need to make the following definition. Definition 1.35. A curve  0 is a subcurve of a curve  W Œ0; l  ! X if, after reparameterization and possibly reversion,  0 is equal to  jŒa;b for some 0  a < b  l . We also say that g is an upper gradient of f along  if Z 0 0 jf . .0//  f . .l 0 //j  g ds 0

for every subcurve  0 W Œ0; l 0  ! X of  . Note that, at a first glance, Lemma 1.34 (c) may look counterintuitive. However, as the proof shows, it is actually very natural. A simple illustration of the situation is provided by the following example. Example 1.36. Let X D R, D f g and 0 D f 0 g, where  W Œ0; 2 ! R,  0 W Œ0; 1 ! R and .t / D  0 .t / D t . Then  0 is a subcurve of  and and 0 satisfy the assumptions of Lemma 1.34 (c). For every admissible in the definition of Modp . /, we have Z

Z

2

1

dt  2

11=p

0

1=p

2 p

dt

;

0

with equalities for  12 . It follows that Modp . / D 21p  1 D Modp . 0 /; where Modp . 0 / is calculated similarly. This also shows that, roughly speaking, the longer the curves in , the smaller Modp . /.

18

1 Newtonian spaces

Proof of Lemma 1.34. (a) This is trivial, since the infimum in the definition of Modp . 1 / is taken over a larger set than the infimum defining Modp . 2 /. (b) Let " > 0 and let j be a nonnegative Borel function such that Z j ds  1 for all  2 j and such that Modp



R

p X j

1 [

d  Modp . j / C "=2j . Let D supj j . Then

Z  Z  p

j  p d D sup j d X

j D1



j

X

Z X 1 X j D1

jp

d D

1 Z X j D1 X

jp

d 

1 X

Modp . j / C ":

j D1

Letting " ! 0 completes the proof of this part. (c) Let " > 0 and let be a nonnegative Borel function such that Z ds  1 for all  2 0 

R

and such that X p d  Modp . 0 / C ". Let next  2 . Then there is a subcurve  0 2 0 of  . Hence Z Z ds  ds  1: Thus Modp . / 

R X

0

 0

d  Modp . / C " ! Modp . 0 /, as " ! 0. p

Proposition 1.37. Let x 2 X . The following are equivalent: (a) Modp . / D 0; (b) there is a nonnegative Borel function 2 Lp .X / such that  2 ;

R 

ds D 1 for all

(c) there is a Rnonnegative such that 2 Lp .B.x; j // for all j D 1; 2; ::: , and such that  ds D 1 for all  2 . See also Proposition 2.33 for another equivalent condition. Proof. (a) ) (b) For n D 1; 2; ::: there is a nonnegative Borel function n such that Z n and n ds  1 for all  2 : k n kLp .X/  2 Let D

P1 nD1



n 2 Lp .X /. Then Z ds D 1 

for all  2 :

1.5 p-weak upper gradients and modulus of curve families

19

(b) ) (c) This is trivial. (c) ) (a) By Proposition 1.2 there is a nonnegative Borel function O  such that O D a.e. Let 1 X j O B.x;j / Q D 2 Lp .X /: j 2 k k O Lp .B.x;j // C 1 j D1

R

If  2 , then  Q ds D 1, since  is compact and thus contained in B.x; j / for some j . Thus also Z Q ds D 1  1 for all  2 : n  p Hence Modp . /  k =nk Q ! 0, as n ! 1. Lp .X/

Corollary 1.38. If Modp . j / D 0 for all j , then Modp

S1

j D1



j D 0.

This is a special case of Lemma 1.34 (b), but actually we will only need this special case in this book. We therefore provide a simpler proof, so that the reader who so wishes can skip Lemma 1.34 (b). Proof. By Proposition 1.37, for j D 1; 2; ::: , there exists a nonnegative function j 2 Lp .X / such that Z j ds D 1 for all  2 j : 

Let further D

1 X j D1

Then

j j 2 k j kLp .X/

2 Lp .X /:

Z ds D 1

for all  2



Hence Modp

S1

j D1



1 [

j :

j D1

j D 0, by Proposition 1.37.

The following corollary is a direct consequence of Corollary 1.38. Note that, in the same way as upper gradients, p-weak upper gradients are not unique. Corollary 1.39. Let g and g 0 be p-weak upper gradients of u and v, respectively, and a 2 R. Then jajg and g Cg 0 are p-weak upper gradients of au and uCv, respectively. Note that it is not true in general that g  g 0 is a p-weak upper gradient of u  v. The following lemma strengthens the definition of p-weak upper gradients and will be often used in our proofs, even without notice.

20

1 Newtonian spaces

Lemma 1.40. If g is a p-weak upper gradient of f on X and

D f 2 .X / W g is not an upper gradient of f along  g; then Modp . / D 0. Proof. Let 0 consist of those curves  0 W Œ0; l 0  ! X such that Z 0 0 g ds jf . .0//  f . .l 0 //j 6 0

(which in particular is true if the integral is not defined). Then Modp . 0 / D 0 by assumption. Moreover, for every  2 there is a subcurve  0 2 0 . It follows that Modp . / D 0, by Lemma 1.34 (c). Definition 1.41. Let

E D f 2 .X / W  1 .E/ ¤ ¿g

C and E D f 2 .X / W ƒ1 . 1 .E// ¤ 0g:

Here ƒ1 is the Lebesgue measure on R, extended as an outer measure to all subsets of R. (Note that X D .X /.) C Lemma 1.42. If .E/ D 0, then Modp . E / D 0. C Proof. Let F  E be a Borel set with .F / D 0, and let D 1F . For  2 E , we 1 1 have ƒ1 . .F // ¤ 0. Moreover,  .F / is a Borel set, and thus Z C ds D 1 for all  2 E :



Z

Hence C / Modp . E

p d D 0: X

Lemma 1.43. Let g and gQ be nonnegative measurable functions on X such that g D gQ a.e. Then Z Z g ds D gQ ds for p-a.e. curve  . In particular,

R





 g ds is defined for p-a.e. curve  (with a value in Œ0; 1).

Proof. By Proposition 1.2 there is a nonnegative Borel function Rg 0 which equals g a.e. Let E D fx 2 X W g.x/ ¤ g 0 .x/g. As g 0 is a Borel function,  g 0 ds is defined for C every curve . For curves  2 .X / n E , Z Z g ds D g 0 ds: (1.5) 

 C

Since .E/ D 0, we have Modp . E / D 0, by Lemma 1.42, and thus (1.5) holds for p-a.e. curve R. R Similarly  gQ ds D  g 0 ds for p-a.e. curve  .

1.5 p-weak upper gradients and modulus of curve families

21

Corollary 1.44. Let g be a p-weak upper gradient of f , and let gQ D g a.e., gQ  0. Then gQ is also a p-weak upper gradient of f . In particular, there is a Borel p-weak upper gradient g 0 of f such that g 0 D g a.e. The following example shows that the corresponding result for upper gradients is false. Example 1.45. Let f W R2 ! R be given by f ..x1 ; x2 // D x1 . Then g D 1 is an upper gradient of f . Let further g 0 D gR2 nR . Then g 0 D g a.e. and g 0 is a nonnegative Borel function which is not an upper gradient of f . Nevertheless, g 0 is a p-weak upper gradient of f , by Corollary 1.44. The following result shows that the N 1;p -norm is not changed if the infimum in Definition 1.17 is taken over all p-weak upper gradients of u. Note that even though this result is mainly of interest when g 2 Lp .X /, this is not required. Lemma 1.46. Let g be a p-weak upper gradient of f . Then there exist upper gradients gj so that (1.6) lim kgj  gkLp .X/ D 0: j !1

Proof. First we can find a nonnegative Borel function g 0 such that g 0 D g a.e. By Lemma 1.43, g 0 is also a p-weak upper gradient of f . Let consist of those curves  W Œ0; l  ! X such that Z jf ..0//  f ..l //j 6 g 0 ds: 

By assumption, Modp . / D 0, and hence R by Proposition 1.37, there is a nonnegative Borel function 2 Lp .X / such that  ds D 1 for all  2 . Let finally gj D g 0 C =j . Then gj is an upper gradient of f and (1.6) holds. The following result shows that p-weak upper gradients are a natural generalization of the (modulus of the) usual gradients. It is an improvement of Corollary 1.15 and will be further refined in Propositions A.3 and A.11. Corollary 1.47. If X D Rn and f W Rn ! R is locally Lipschitz, then jrf j (or more precisely any of its nonnegative everywhere defined representatives) is a p-weak upper gradient of f . Proof. Proposition 1.14 shows that lip f is an upper gradient of f . By the Rademacher theorem (see e.g. Theorem 2.2.1 in Ziemer [361]), f is differentiable at a.e. x 2 Rn . For such x and all y 2 B.x; r/ we have jf .y/  f .x/j D jrf .x/  .y  x/ C o.jy  xj/j  r jrf .x/j C o.r/ and hence lip f .x/ D jrf .x/j. The result now follows from Corollary 1.44.

22

1 Newtonian spaces

We shall next see that there is an intimate relation between small sets and small curve families. Proposition 1.48. Let E  X . Then Cp .E/ D 0 if and only if .E/ D Modp . E / D 0. Proof. Assume first that .E/ D Modp . E / D 0. Let u D E . Then on p-a.e. curve u  0, and therefore g  0 is a p-weak upper gradient of u. It follows that p p Cp .E/  kukpN 1;p .X/  kukL p .X/ C kgkLp .X/ D 0:

Assume conversely that Cp .E/ D 0. It follows directly that .E/  Cp .E/ D 0. For each j D 1; 2; ::: , let uj 2 N 1;p .X / be a nonnegative function with upper gradient gj , such that kuj kN 1;p .X/ < 2j ;

kgj kLp .X / < 2j and uj  E : P P Let u D j1D1 uj 2 Lp .X / and g D j1D1 gj 2 Lp .X /. Let further F D fx 2 X W u.x/ D 1g  E; ˚ R

D  W  g ds D 1 ;

.F / D f W   F g: Since g 2 Lp .X / we have Modp . / D 0. Moreover, for  2 .F / we have Z Z u ds D 1 ds D 1: 



As u 2 Lp .X / we have Modp . .F // D 0, by Proposition 1.37. Let now  2 .X /n. [ .F //. Then, in particular, there is y 2  with u.y/ < 1. So for all x 2  we have u.x/ D lim

k!1

k X

uj .x/  lim

j D1

 u.y/ C lim

k!1

k X

k!1

k Z X j D1 

uj .y/ C

j D1

k X



juj .x/  uj .y/j

j D1

Z gj ds D u.y/ C

g ds < 1: 

Thus x … F , and hence   X n F . Therefore E  F  [ .F / has zero p-modulus. Corollary 1.49. If u D v q.e. and g is a p-weak upper gradient of u, then g is also a p-weak upper gradient of v (and thus, u and v have the same set of p-weak upper gradients).

1.5 p-weak upper gradients and modulus of curve families

23

Proof. Let E D fx W u.x/ ¤ v.x/g. By assumption, Cp .E/ D 0, and by Proposition 1.48, Modp . E / D 0. Hence u  v on p-a.e. curve  . It then easily follows that if g is a p-weak upper gradient of u, then it is also of v. Proposition 1.50. Assume that .E/ D 0 and that g  0 is such that for p-a.e. curve  W Œ0; l  ! X it is true that either Z .0/; .l / 2 E or ju..0//  u..l //j  g ds: (1.7) 

Then g is a p-weak upper gradient of u. The following consequence is immediate upon letting E D fx 2 X W ju.x/j D 1g. x be a function which is finite a.e. and assume that Corollary 1.51. Let u W X ! R g  0 is such that for p-a.e. curve  W Œ0; l  ! X it is true that either Z ju..0//j D ju..l //j D 1 or ju..0//  u..l //j  g ds: 

Then g is a p-weak upper gradient of u. The essence of this corollary is that as long as u is real-valued a.e. it makes no difference how we interpret the inequality (1.4) in Definition 1.32 in the special case when the left-hand side is either j1  1j or j.1/  .1/j. Our main interest is in 1;p N 1;p (and Nloc ) functions, and such functions are necessarily real-valued a.e. Proof of Proposition 1.50. Let be the set of the exceptional curves  for which (1.7) does not hold for some subcurve of  . Then Modp . / D 0 by Lemma 1.34. Let also C

.E/ D f W   Eg. Since .E/ D 0 we have Modp . .E//  Modp . E / D 0, by Lemma 1.42. Let  2 .X / n . [ .E//. Then there is t 2 Œ0; l  such that .t/ … E. If t D 0 or t D l , then Z ju..0//  u..l //j  g ds; 

by assumption. Otherwise ju..0//  u..l //j  ju..0//  u..t //j C ju..t //  u..l //j Z Z Z  g ds C g ds D g ds; jŒ0;t 

jŒt;l 



since the second alternative in (1.7) holds for  jŒ0;t and  jŒt;l  . We have thus shown that g is a p-weak upper gradient of u.

24

1 Newtonian spaces

Recall our convention that 1  1 D 1. In particular, the second alternative in (1.7) does not hold for  jŒ0;t if t D 0 and u..0// D ˙1, and therefore the cases t D 0 and t D l needed separate treatment above. Note that in the context of Corollary 1.51 this is not necessary. Lemma 1.28 has a direct counterpart for p-weak upper gradients. (Assuming that gi are merely p-weak upper gradients, we obtain a p-weak upper gradient g with the same proof.) However, using Proposition 1.50 (with E D fx 2 X W u.x/ D 1g), we can show the following stronger result. Lemma 1.52. Let uj be functions with p-weak upper gradients gj , and let u D supj uj , g D supj gj and E D fx W u.x/ D 1g. If .E/ D 0, then g is a p-weak upper gradient of u. Note that unlike in Lemma 1.28, here we do not assume that uj are uniformly bounded from above. Recall also the discussion in Remark 1.29 showing that without any condition on E this lemma would be false. is not an upper Proof. Let j be the set of the exceptional curves along which gj S gradient of uj , j D 1; 2; ::: . Let  W Œ0; l  ! X be a curve not in j1D1 j and let x D .0/ and y D .l /. Assume that either u.x/ < 1 or u.y/ < 1. Then, as in the proof of Lemma 1.28, Z Z ju.x/  u.y/j  sup juj .x/  uj .y/j  sup gj ds  g ds: j

j





It follows from Corollary 1.51 that g is a p-weak upper gradient of u. Finally, we show that for almost all purposes we can forget about X n supp  and replace X by supp . Note, however, that we do not want to assume X D supp  in general, see the comment before Proposition 1.6. Proposition 1.53. Let Z D X n supp . Then Cp .Z/ D 0 and Modp . Z / D 0. Proof. As Z is open, the function g D 1Z is an upper gradient of u D Z as R g ds D 1 for any curve  starting in Z and ending in supp . Hence Cp .Z/   p kukN 1;p .X/ D 0. That also Modp . Z / D 0 follows from Proposition 1.48.

1.6 Banach space and ACCp On Rn it is well known that every Sobolev function has a representative which is ACL, i.e. absolutely continuous on almost every line parallel to the axes, see e.g. Theorem 2.1.4 in Ziemer [361]. On metric spaces we have no preferred lines, but we can obtain a stronger result, Theorem 1.56 below. In fact, for this we only need the existence of a p-integrable upper gradient, which justifies the following definition.

1.6 Banach space and ACCp

25

Definition 1.54. We say that a measurable function belongs to the Dirichlet space D p .X / if it has an upper gradient in Lp .X /. In view of Lemma 1.46 it is equivalent to assume the existence of a p-weak upper gradient in Lp .X /. Clearly, N 1;p .X /  D p .X /, and Lemma 1.16 immediately implies that D p .X / is a vector space. Theorem 1.55. The space D p .X / is a lattice. Proof. The proof of Theorem 1.20 in fact also yields this result. Theorem 1.56. If u 2 D p .X /, then u 2 ACCp .X /, i.e. u is absolutely continuous on p-a.e. curve in the sense that u ı  W Œ0; l  ! R is absolutely continuous for p-a.e. curve  in X . Let us recall the definition of absolute continuity. Definition 1.57. A function f W Œa; b ! R is absolutely continuous on Œa; b if for every " > 0 there is ı > 0 such that n X

jf .bi /  f .ai /j < "

iD1

for any n and any a  a1 < b1  a2 < b2    an < bn  b such that n X

.bi  ai / < ı:

iD1

This is not the right place to study the basic facts about absolutely continuous functions. Let us however recall that a large motivation for their study is the fact that f is absolutely continuous on Œa; b if and only if f 0 2 L1 .Œa; b/ and Z f .x/ D f .a/ C

x

f 0 .t / dt

for all x 2 Œa; b:

a

See e.g. Folland [124], Theorem 3.35, or Rudin [311], Theorem 7.20 together with the comments just before Definition 7.17. We will need the following simple facts. Lemma 1.58. If u and v are absolutely continuous on Œa; b, then u C v, maxfu; vg and uv are also absolutely continuous on Œa; b. If u W Œa; b ! Œc; d  is absolutely continuous and v W Œc; d  ! R is Lipschitz, then v ı u is also absolutely continuous on Œa; b.

26

1 Newtonian spaces

Proof. Let us start with uv: As u and v obviously are continuous they P are bounded on Œa; b, say by M > 0. Let " > 0 and choose ı > 0 so small that if niD1 .bi  ai / < ı for any a  a1 < b1  a2 < b2    an < bn  b, then n X iD1

ju.bi /  u.ai /j <

" 2M

and

n X

jv.bi /  v.ai /j <

iD1

" : 2M

Thus n X

ju.bi /v.bi /  u.ai /v.ai /j 

iD1

n X

.ju.bi /v.bi /  u.bi /v.ai /j

iD1

D

n X iD1

M

C ju.bi /v.ai /  u.ai /v.ai /j/ .ju.bi /j jv.bi /  v.ai /j C jv.ai /j ju.bi /  u.ai /j/

n X

jv.bi /  v.ai /j C M

iD1

n X

ju.bi /  u.ai /j

iD1

" " CM <M 2M 2M D ": To show that u C v and maxfu; vg are absolutely continuous is easier and we leave it to the reader. As for the composition v ı u, let " > 0 and choose ı according to the definition of absolute continuity of u. Let also M > 0 be a Lipschitz constant for v. Then n X

j.v ı u/.bi /  .v ı u/.ai /j  M

iD1

n X

ju.bi /  u.ai /j < M "

iD1

for any n and any a  a1 < b1  a2 < b2    an < bn  b such that n X

.bi  ai / < ı:

iD1

Proof of Theorem 1.56. By Definition 1.54, there is an upper Rgradient g 2 Lp .X / of u. Let be the collection of all curves  in X such that  g ds D 1. Then Modp . / D 0 by Proposition 1.37. Let now  2 .X / n . Then for all a; b 2 Œ0; l , Z g ds < 1; (1.8) j.u ı  /.a/  .u ı  /.b/j  jŒa;b

and, in particular, .u ı  /.a/; .u ı  /.b/ 2 R.

1.6 Banach space and ACCp

27

Assume that u is not absolutely continuous on  , i.e. that f WD uı is not absolutely continuous on Œ0; l . Then there is an " > 0 such that for every j D 1; 2; ::: , there are 0  aj;1 < bj;1    aj;nj < bj;nj  l such that nj X

.bj;i

iD1

Let Ej D

Snj

1  aj;i / < j 2

iD1 Œaj;i ; bj;i .

"

nj X iD1

nj X

and

jf .bj;i /  f .aj;i /j  ":

iD1

Then by (1.8) and dominated convergence, Z

jf .bj;i /  f .aj;i /j 

g ds ! 0;

as j ! 1:

jEj

But this is a contradiction. Hence f is absolutely continuous on  , and u is absolutely continuous on p-a.e. curve. We shall next see that the equivalence classes for Newtonian functions are finer than for Sobolev functions. Remember that functions in N 1;p .X / are defined everywhere and that Nz 1;p .X / is the corresponding quotient space, see Definition 1.19. Proposition 1.59. If u; v 2 ACCp .X / (in particular if u; v 2 N 1;p .X /), and u D v a.e., then u D v q.e. Proof. Without loss of generality we may assume that v  0 (consider otherwise C / D 0. We also u  v). Let E D fx W u.x/ ¤ 0g. Since .E/ D 0, we have Modp . E know that u is absolutely continuous on p-a.e. curve. So on p-a.e. curve  , u is both absolutely continuous and ƒ1 . 1 .E// D 0. On such curves u D 0 a.e. with respect to arc length, and by continuity everywhere. Thus  … E . Hence Modp . E / D 0, and by Proposition 1.48, Cp .E/ D 0. Corollary 1.60. If u; v 2 ACCp .X / (in particular if u; v 2 N 1;p .X /), and u  v a.e., then u  v q.e. Proof. Let f D minfu; vg 2 ACCp .X /. Since u  v a.e. we have f D v a.e. Thus u  f D v q.e., by Proposition 1.59. The following result shows that the equivalence classes in Nz 1;p .X / are up to sets of capacity zero, and not up to measure zero as for usual Sobolev spaces on Rn . This also shows that two functions in the same equivalence class in Nz 1;p .X / have the same set of p-weak upper gradients (by Corollary 1.49). x be a function. Then kukN 1;p .X/ D 0 if and only if Proposition 1.61. Let u W X ! R Cp .fx W u.x/ ¤ 0g/ D 0. x then v D u q.e. if and only if Moreover, if u 2 N 1;p .X / and v W X ! R, 1;p v 2 N .X / and v u.

28

1 Newtonian spaces

Proof. Let E D fx W u.x/ ¤ 0g. Assume first that kukN 1;p .X/ D 0. Then u 2 N 1;p .X / and u D 0 a.e., so by Proposition 1.59, u D 0 q.e., and thus Cp .E/ D 0. Conversely, assume that Cp .E/ D 0. Corollary 1.49 shows that 0 is a p-weak upper gradient of u. Moreover, .E/ D 0, by Theorem 1.27. Hence kukN 1;p .X/ D 0. The last part is obtained by applying the first part to u  v. We are now ready to show that the norm in Ny 1;p .X /, which was introduced after Definition 1.19, is well defined. Lemma 1.62. Assume that u 2 Ny 1;p .X / and that v; w 2 N 1;p .X / are such that u D v D w a.e. Then kvkN 1;p .X/ D kwkN 1;p .X/ , i.e. kukNy 1;p .X/ is well defined and equal to kvkN 1;p .X/ D kwkN 1;p .X/ . Proof. It follows from Proposition 1.59 that v D w q.e. Hence, by Proposition 1.61, kv  wkN 1;p .X/ D 0, which implies that kvkN 1;p .X/ D kwkN 1;p .X/ . As there are sets with zero measure and positive capacity we have shown that N 1;p .X / and Ny 1;p .X / are (in general) different spaces. The following results give some more insight in this direction. Proposition 1.63. Let u 2 Ny 1;p .X /. Then u 2 N 1;p .X / if and only if u 2 ACCp .X /. Proof. The necessity follows from Theorem 1.56. For the sufficiency assume that u 2 ACCp .X /. By assumption, there is v 2 N 1;p .X /  ACCp .X / such that u D v a.e., but then u D v q.e., by Proposition 1.59. Hence, by Proposition 1.61, kuvkN 1;p .X/ D 0, and thus also u 2 N 1;p .X /. Example 1.64. The function R equals zero a.e. in R2 and is therefore a representative of a Sobolev function. It is, however, a bad representative of the zero function, as it is not absolutely continuous on any line parallel to the y-axis. In our notation, it belongs to Ny 1;p .R2 /, but not to N 1;p .R2 /. In view of Proposition 1.61, any Newtonian representative of the zero function must be zero q.e. Proposition 1.65. Ny 1;p .X /= ae Š Nz 1;p .X / WD N 1;p .X /= , where u ae v if and only if u D v a.e. In Rn and many other situations, Sobolev spaces are usually defined as sets of a.e.equivalence classes, which corresponds to the left-hand side above. This result shows that we obtain essentially the same Sobolev spaces with our definition, even though we insist on having the smaller q.e.-equivalence classes. We have just weeded out the bad representatives of Sobolev functions, see Example 1.64. Our q.e.-representatives have the additional property of belonging to ACCp .X /, something not shared by general a.e.-representatives. We will later find additional advantages of q.e.-representatives, e.g. in connection with quasicontinuity (see e.g. Proposition 5.33) and resolutivity (see e.g. Theorem 10.12).

1.6 Banach space and ACCp

29

Proof. Let f1 and f2 be two representatives of one equivalence class in Nz 1;p .X /. Then f1 ; f2 2 N 1;p .X /  Ny 1;p .X /. Moreover, f1 D f2 q.e., by Proposition 1.61, and hence a.e., by Theorem 1.27. Thus they belong to the same equivalence class in Ny 1;p .X /= ae . Conversely, let f1 and f2 be two representatives of one equivalence class in 1;p y N .X /= ae . Then, by the definition of Ny 1;p .X /, there are f10 ; f20 2 N 1;p .X / such that fj0 D fj a.e., j D 1; 2. Hence also f10 D f20 a.e. and consequently, q.e., by Proposition 1.59. Therefore f10 and f20 both define the same equivalence class in Nz 1;p .X /, and there is thus a one-to-one correspondence between the a.e.-equivalence classes in Ny 1;p .X /= ae and the q.e.-equivalence classes in Nz 1;p .X /. Finally, kf1 kNy 1;p .X/ D kf10 kN 1;p .X/ , showing that the norms are the same for the corresponding equivalence classes. Proposition 1.63 gave us a characterization of “good” representatives, but it is not obvious from this characterization that a continuous representative is always “good”. Let us therefore show this, and a bit more. Proposition 1.66. Let u 2 Ny 1;p .X / and E  X be such that Cp .E/ D 0 and ujXnE is continuous. Then u 2 N 1;p .X /. Note that we do not require u to be continuous at all points in X n E, we only require that the restriction of u to X n E is continuous. A function u satisfying our condition is weakly quasicontinuous, see Definition 5.17 (but the converse is not true). In fact, (weak) quasicontinuity characterizes the “good” representatives, under some additional assumptions on X , see Proposition 5.33. We do not know if that characterization holds in general, and therefore the result here is of interest. Proof. By assumption, there is a function v 2 N 1;p .X / such that v D u a.e. Let A D fx W u.x/ ¤ v.x/g so that .A/ D 0. By Lemma 1.42 and Proposition 1.48, Modp . E / D Modp . AC / D 0. As v 2 ACCp .X /, by Theorem 1.56, we thus see that p-a.e. curve  is such that v is absolutely continuous on  and  … E [ AC . Since  … E , u is continuous on  . Moreover, as  … AC , u D v a.e. on  . As u and v are continuous on  and equal a.e. on  , we must have u D v everywhere on  . In particular, u is absolutely continuous on  , and hence u 2 ACCp .X /. By Proposition 1.59, u D v q.e., and thus u 2 N 1;p .X /, by Proposition 1.61. A corresponding result is true also for D p in the following form. x be such that there is v 2 D p .X / with u D v a.e. Let Proposition 1.67. Let u W X ! R further E  X be such that Cp .E/ D 0 and ujXnE is continuous. Then u 2 D p .X /. The proof is the same as above, apart from that rather than applying Proposition 1.61 at the very end, we need to apply its proof. Another consequence of our definition of Newtonian functions is the equality between the essential supremum and the q.e.-essential supremum defined as follows.

30

1 Newtonian spaces

x define Definition 1.68. For A  X and f W A ! R, Cp - ess sup f D inffk 2 R W Cp .fx 2 A W f .x/ > kg/ D 0g; A

Cp - ess inf f D supfk 2 R W Cp .fx 2 A W f .x/ < kg/ D 0g: A

As a corollary of Proposition 1.59 we make the following useful observation. Corollary 1.69. If u 2 ACCp .X / (in particular if u 2 N 1;p .X /), then ess sup u D Cp - ess sup u and X

X

ess inf u D Cp - ess inf u: X

X

Proof. Theorem 1.27 (ii), shows that ess supX u  Cp - ess supX u. In particular, the equality holds if ess supX u D 1. Otherwise, let  > ess supX u,  2 R, be arbitrary. Then u   a.e., and thus u   q.e., by Corollary 1.60, showing that Cp - ess supX u  . Letting  ! ess supX u shows that ess supX u  Cp - ess supX u. The second equality follows by applying the first equality to u. The following generalization of Proposition 1.30 is also worth pointing out. Corollary 1.70. If u 2 ACCp .X / and u is finite a.e., which in particular holds if u 2 N 1;p .X /, then u is finite q.e. Proof. Let E D fx W u.x/ 2 Rg and let v D uE . Let  be a curve on which u is absolutely continuous. Then v D u on  . Thus v 2 ACCp .X /. By Proposition 1.59, v D u q.e., and as v is real-valued the conclusion follows. Theorem 1.71. The space Nz 1;p .X / D N 1;p .X /= is a Banach space. Proof. That Nz 1;p .X / is a normed linear space was observed after Definition 1.19. We need to show the completeness. Let fuj gj1D1 be a Cauchy sequence in N 1;p .X /. By passing to a subsequence, if necessary, we can assume that kuj C1  uj kN 1;p .X/ < 2j.pC1/=p < 2j : Let

(1.9)

Ej D fx 2 X W juj C1 .x/  uj .x/j > 2j g:

Then Cp .Ej /  2jp kuj C1  uj kpN 1;p .X/ < 2j . (Note that if juj C1 .x/j D juj .x/j D 1, then we consider juj C1 .x/  uj .x/j D 1, and thus x 2 Ej . In view of Proposition 1.30 this does not cause any trouble.) Let 1 1 [ \ Fk D Ej and F D Fk : j Dk

kD1

1.7 Examples

31

Then Cp .Fk / < 21k and Cp .F / D 0. Let x 2 X n F . Then x 2 X n Fl for some l and juj C1 .x/  uj .x/j  2j for all j  l. Hence uj .x/ is a Cauchy sequence in R and we can define u.x/ D lim uj .x/ D uk .x/ C j !1

1 X

.uj C1 .x/  uj .x//:

(1.10)

j Dk

The function u is defined q.e. (and we may define it arbitrarily elsewhere) and ku  uk kLp .X/ < 21k , by (1.9). As Cp .F / D 0, p-a.e. curve in X has empty intersection with F , by Proposition 1.48. Let  be one such curve, connecting x and y. Then by (1.10), j.u  uk /.x/  .u  uk /.y/j 

1 X

j.uj C1  uj /.x/  .uj C1  uj /.y/j 

j Dk

Z X 1 

gj ds;

j Dk

where gj is an upper gradient of uj C1  uj such that kgj kLp .X/ < 2j . Hence, P gQ k D j1Dk gj is a p-weak upper gradient of u  uk and kgQ k kLp .X/ < 21k . It follows that ku  uk kN 1;p .X/ ! 0, as k ! 1. As a corollary of the proof we obtain the following result. Corollary 1.72. Assume that uj ! u in N 1;p .X /, as j ! 1. Then there is a subsequence which converges to u pointwise q.e. Moreover, for every " > 0 there is a set E with Cp .E/ < ", such that the subsequence converges uniformly to u outside of E. Furthermore, if all uj are continuous, then the subsequence converges uniformly (though not necessarily to u) outside open sets of arbitrarily small capacity. Proof. In the notation of the proof of Theorem 1.71, we obtain a subsequence (again denoted by fuj gj1D1 ) which converges to uQ 2 N 1;p .X / uniformly outside of Fk for every k, and thus pointwise outside of F . Note that if all uj are continuous, then all Fk are open. Clearly, u uQ and Proposition 1.61 shows that u D uQ q.e. Letting E D Fk [ fx W u.x/ ¤ u.x/g, Q for suitably large k, finishes the proof.

1.7 Examples Let us take the opportunity to mention some examples of metric spaces on which the Newtonian theory can be interesting. (See also Appendix A, where examples of spaces with doubling measures supporting Poincaré inequalities are discussed in greater detail.) The first example is of course Rn . In this case the Newtonian space N 1;p .Rn / is essentially the standard Sobolev space W 1;p .Rn /. More precisely, if   Rn , then

32

1 Newtonian spaces

W 1;p ./ D Ny 1;p ./ and N 1;p ./ is the same space but with less representatives in the equivalence classes, in fact it singles out exactly the nice representatives which are quasicontinuous, as we shall see in Section 5.2. The same is true on weighted Rn with p-admissible weights, see Appendix A.2. The next type of examples that comes to mind are probably Riemannian manifolds. Again, the Newtonian space will be the same as the usual Sobolev space, but with only the good, quasicontinuous, representatives. A particular application here is that one can take limits of sequences of Riemannian manifolds, under Gromov–Hausdorff convergence (which is sometimes called Vietoris convergence). Such limits are typically not manifolds, but are indeed metric spaces and Newtonian spaces are well suited to be defined on them. Another example is the Heisenberg group H1 as well as higher order Heisenberg groups, more general Carnot groups and Carnot–Carathéodory spaces, see Appendix A.6. Yet other examples are simplicial complexes, with simplices glued to each other. Such glueing can be made along different types of faces and it is even possible for the complex to have (highest-order) simplices of different dimension in different parts. The simplest example is probably glueing the segment Œ1; 0 f0g (with the onedimensional Lebesgue measure) to the triangle f.x; y/ W 0  y  x  1g (with a weighted two-dimensional measure) at .0; 0/. See Appendix A.4 for more complicated examples. Even discrete objects, such as graphs, can be included in our theory. Here one has to consider the so-called metric graphs, see Appendix A.5. All of the examples above can be equipped with doubling measures and support Poincaré inequalities, under suitable conditions. So in particular, most of the results we show in Chapters 3–6 are valid for these spaces, and they are also under consideration for the potential theory developed in Chapters 7–14. In many of these examples one has a natural vector-valued gradient, and one can also apply Cheeger’s theorem (Theorem B.6) to show that a vector-valued gradient structure exists. A disadvantage of using Cheeger’s theorem is that it does produce a vector-valued gradient structure, but this structure is not unique. Moreover, one always uses an inner product to take the length of the Cheeger gradient. It is thus not easy to understand what the Cheeger gradient, or its length, really determines, in particular its connection with the geometry of the space. The upper gradient on the other hand has a very geometric definition. Nevertheless, the results we obtain in Chapters 7–14 are valid also for Cheeger p-harmonic functions, see Section 7.1 and Appendix B.2. Further examples include open subsets of the spaces above, as well as closed and other nonopen subsets. Such examples usually do not support Poincaré inequalities, nor are the restricted measures usually doubling, see however Appendix A.3, where uniform domains are discussed. For open subsets one still gets a natural gradient from the gradient in the ambient space, a fact that we exploit to deduce many of our results, such as quasicontinuity, for open subsets of (often complete) doubling p-Poincaré spaces.

1.7 Examples

33

For nonopen sets one cannot just restrict the gradient from the ambient space. Here comes another advantage of the Newtonian approach, the Newtonian spaces are directly definable on nonopen subsets of our ambient space. For other types of Sobolev spaces on X one can of course say that a function f is in the Sobolev space on a subset E if there is some open set G  E such that f is in the Sobolev space on G (with G being allowed to depend on f ). For Newtonian spaces the advantage is that we do not have to go outside of E to define N 1;p .E/. So, in particular, the Newtonian approach is ideal for defining Sobolev spaces on various closed subsets E of Rn . Of course, one has to remember that if E contains too few curves then N 1;p .E/ may be of little use, as e.g. with the von Koch snowflake curve, see Example 1.23. However, in the case when E is the closure of an open set, then the Newtonian space N 1;p .E/ has applications in the boundary value theory, and does make results more general than what can be achieved using classical Sobolev spaces. In such cases we do not know if all functions in N 1;p .E/ are quasicontinuous, but nevertheless they are better representatives than the usual a.e.-Sobolev functions, see the discussion on the zero p-weak upper gradient property in the notes to Chapter 5. There is yet another reason for studying potential theory on subsets of Rn (or other metric spaces). Consider X D R2 n ..1; 1/ .0; 1// which is a complete doubling p-Poincaré space. The potential theory we develop in the later part of the book can thus be applied. Let  D B..0; 0/; 1/ be the unit ball. In X , it has the boundary @X  D f.x; y/ 2 R2 W x 2 C y 2 D 1 and y  0g. Let also E D .1; 1/ f0g and f 2 C.@X /. By the methods in Chapter 10 we can solve the Dirichlet problem and find the p-harmonic function Pf in  having boundary values f on @X . One can actually show that this solution is the p-harmonic function u D Pf in  n E having boundary values f on @X  and satisfying the zero Neumann boundary condition @u D0 @n

on E;

see also Example 8.18. This is true in much more general situations. When the missing boundary, denoted by E above, is less regular, we often cannot talk about normal derivatives in the usual sense. Our way of solving the Dirichlet problem in such situations still gives rise to a very general weak-type zero Neumann boundary condition. Theorem A.21 shows that the closure of any uniform domain in Rn serves as a complete doubling p-Poincaré space and thus the above mixed boundary value problem can be considered on subsets of it. For instance, we can let X be the closed bounded set in R2 , whose boundary is the von Koch snowflake curve K, see Example 1.23. Solving the Dirichlet problem on an open subset  of X corresponds to solving the mixed boundary value problem on  n K with Dirichlet boundary values on @X  and zero Neumann boundary values on K \ . Thus we are actually able to handle rather general mixed boundary value problems within our treatment of the Dirichlet problem.

34

1 Newtonian spaces

1.8 Notes Upper gradients were introduced under the name very weak gradients by Heinonen and Koskela in [173], [174]. They were renamed upper gradients, for obvious reasons, when Koskela and MacManus [229] studied p-weak upper gradients for the first time. In Semmes [314], Definition 1.8, upper gradients are called generalized gradients. Cheeger [91] defines generalized upper gradients, which for p > 1 are roughly the same as p-weak upper gradients. Cheeger also defines Sobolev spaces on general metric spaces that coincide with Ny 1;p .X / if p > 1, see Appendix B.2. In some of the early papers on upper and p-weak upper gradients, it is not stated explicitly if the (p-weak) upper gradients are required to be Borel functions or not. However, since Borelness is sometimes used it may have been implicitly assumed. Requiring that upper gradients be Borel functions has some definite advantages. First of all, the concept of upper gradients becomes totally independent of the measure, and second, the curve integrals involved are always defined. As for p-weak upper gradients the situation is different. At first (when care was taken) it was assumed that p-weak upper gradients be Borel, the reasons for this being that it would prevent any problems with the curve integrals not being defined, and in view of Proposition 1.2 it was not believed to do any harm. However, when working on the book Heinonen–Koskela–Shanmugalingam–Tyson [177], it was observed by Heinonen (as communicated to us by Shanmugalingam [322]), that care has to be taken in glueing formulas such as in Section 2.4, as the glued p-weak upper gradient may fail to be Borel even if the initial p-weak upper gradients are Borel. The immediate response was that one has to take extra care when formulating similar results. In Björn–Björn [44], Section 3 (which incidentally does not appear in Björn– Björn [45]), a different remedy was suggested which we follow here: to omit the assumption of Borelness for p-weak upper gradients. Let us name some of the reasons for this: (a) In Definition 1.32 it is enough if the curve integrals are defined for p-a.e. curve, and in fact this is automatic for nonnegative measurable functions, by Lemma 1.43. (b) Just requiring measurability of course means that the concept of p-weak upper gradients, in contrast to that of upper gradients, is dependent on the measure. However, even the concept of Borel p-weak upper gradients is dependent on the measure as the inequality is required to just hold for p-a.e. curve. (c) The glueing formulas in Section 2.4 become more appealing. Also, how to interpret j1  1j is often not mentioned explicitly. However, taking into account Corollary 1.51, it makes no difference for p-weak upper gradients how j1  1j is interpreted, at least not when only Newtonian functions are under consideration. Lemma 1.34 for metric spaces is in Heinonen–Koskela [174], p. 9, where they observe that the proofs of the corresponding Euclidean results in Fuglede [128] carry over verbatim. Fuglede was the first one to study p-modulus. The 2-modulus and its inverse extremal length were introduced by Beurling in the early 1930s, but first published in the joint papers Ahlfors–Beurling [6], [7]. Proposition 1.37 also goes back

1.8 Notes

35

to Fuglede [128]. A thorough study of extremal length and p-modulus for p  1 in weighted Rn is in Ohtsuka [299]. Newtonian spaces were first defined in Shanmugalingam [318] and [319]. Therein one can find Theorems 1.20 and 1.71, Corollaries 1.21 and 1.72, and Lemma 1.42. Also Theorem 1.56 and Proposition 1.59 (stated only for u 2 N 1;p .X /) can be found in [319], as well as the “only if” part of Proposition 1.48. The “if” part seems to be new here. An equivalence between the modulus and a variational capacity (see Section 6.3) has been obtained by Heinonen–Koskela [174] and Kallunki [Rogovin]– Shanmugalingam [194]. Some estimates for the modulus of curve families in ring domains on metric spaces can be found in Adamowicz–Shanmugalingam [3] and Garofalo–Marola [138]. Lemma 1.46 was first proved by Koskela–MacManus [229], Lemma 2.4. Lemma 1.43 is from Björn–Björn [44] (but is not included in [45]), while Corollary 1.51 is from Björn–Björn–Parviainen [54]. The direction in Proposition 1.63 which is not covered by Theorem 1.56, as well as Propositions 1.66 and 1.67, seems to be new here. If X is a complete doubling p-Poincaré space, then any function with an upper gradient in Lp .X / is measurable, by Theorem 4.52. Thus the measurability assumption can be dropped from the definition of the Dirichlet space D p .X /, Definition 1.54, under these assumptions. In general, however, it is easy to see that measurability is not a redundant assumption in Definition 1.54, consider e.g. the von Koch snowflake curve in Example 1.23. For p D 1, one often studies BV-functions, functions of bounded variation, which in some problems appear as the natural limit of W 1;p , as p ! 1C. We have not pursued that, since our main interest is to study the theory for p > 1 and we include the case p D 1 only when it comes for free. BV-functions have been studied on metric spaces by Ambrosio [14], [15], Camfield [83], Miranda [286], Ambrosio– Miranda–Pallara [16], Kinnunen–Korte–Shanmugalingam–Tuominen [210], [211], Hakkarainen–Kinnunen [156] and Hakkarainen–Shanmugalingam [157]. A further generalization of Newtonian spaces are Orlicz–Sobolev spaces based on upper gradients. They have been studied by e.g. Aïssaoui [12], [13], Heikkinen– Tuominen [167], Mocanu [291] and Tuominen [339], [340], [341]. Estimates and inequalities for Orlicz–Sobolev capacities on metric spaces can be found in Aïssaoui [11], J. Björn–Onninen [71], Costea [102] and Tuominen [339]. Mocanu [288], [289], [290] studies Newtonian spaces and capacities in even more general settings. Variable exponent Newtonian spaces N 1;p.x/ were studied by Harjulehto–Hästö–Pere [160] and Futamura–Harjulehto–Hästö–Mizuta–Shimomura [129]. Throughout the book we have restricted ourselves to real-valued Newtonian functions. Newtonian spaces for Banach-space valued functions were studied by Heinonen–Koskela–Shanmugalingam–Tyson [176], [177]. The (first part of the) 5-covering lemma (Lemma 1.7) has many names in the literature, including the basic covering lemma and the simple Vitali lemma and is often attributed to Vitali [348] or Wiener [353]. However, Vitali [348] obtains a much stronger result under stronger assumptions: he requires every point in the set E under

36

1 Newtonian spaces

consideration to be covered by arbitrarily small balls (a so-called Vitali covering) and obtains a disjoint covering of all of E but for a set of measure zero. (Vitali’s formulation is a bit different, but it is relatively easy to see that it is equivalent to the modern formulation, especially for open E.) It is therefore more correct to reserve the name Vitali covering theorem for this result. Wiener [353], on the other hand, may have been the first to consider coverings which need not be arbitrarily small around every point. He assumes that every point in E is the centre of some ball in the covering and finds a disjoint subfamily whose measure is at least a fixed positive portion of the measure of E. However, he does not consider blow-ups of the disjoint balls (not even in the proof). Morse [292], Theorem 3.5, seems to be the first one proving the 5-covering lemma as formulated here, and he does so in metric spaces. In fact, Morse studies coverings by more general sets and the 5-covering lemma is a special case of his result. This may have been the first time when the conclusion is that certain blow-ups of the disjoint sets (balls in our case) cover E, rather than a measure-theoretic conclusion as in Vitali’s and Wiener’s theorems. However, already Banach [27], when giving a new proof of (the modern formulation of) Vitali’s covering theorem, shows that certain blow-ups of the disjoint balls cover E, but he does not obtain the 5-covering lemma since he has stronger assumptions. He too studies coverings by more general sets than balls. Federer [123], Theorem 2.8.4, proves an even more general result than Morse, from which the 5-covering lemma follows by choosing  D 2 and letting ı be the radius of the balls. Choosing  D 1 C 12 " gives a .3 C "/-covering lemma, cf. Remark 1.8. Let us finally mention that the name Newtonian comes from the fact that the definition of upper gradients is related to the Newton–Leibniz formula, and the notation L1;p was already reserved for other spaces.

Chapter 2

Minimal p-weak upper gradients

By definition, upper gradients are not unique. Nor are the p-weak upper gradients. Indeed, adding a nonnegative Borel (measurable) function to a (p-weak) upper gradient results in a new (p-weak) upper gradient. Later in this book we will study p-harmonic functions as local minimizers of the p-energy integral Z inf g p d (2.1) g

from the definition of k  kN 1;p .X/ . It is not obvious that this infimum can be attained. Example 1.31 shows that in general this is impossible for upper gradients, but fortunately, the situation is different for p-weak upper gradients. In this chapter, we shall prove the existence and study some properties of minimal p-weak upper gradients, which minimize (2.1) and are true substitutes for jruj in metric spaces. Note however that unless there are some additional geometrical assumptions on the metric space, the minimal p-weak upper gradient need not provide any control of the function itself, cf. Example 1.22.

2.1 Fuglede’s lemma We start this section by proving a fundamental lemma which guarantees that a minimizing sequence in (2.1) gives rise to a p-weak upper gradient. We also provide several useful consequences of this lemma, which will be needed later. Lemma 2.1 (Fuglede’s lemma). Assume that gj ! g in Lp .X /, as j ! 1. Then there is a subsequence (again denoted by fgj gj1D1 ) such that for p-a.e. curve  , Z Z gj ds ! g ds; as j ! 1; 



where all the integrals are well defined and real-valued. Moreover, for p-a.e. curve , Z jgj  gj ds ! 0; as j ! 1: 

In fact, we will only use this with g and gj nonnegative. However, it may be worth observing that the positivity of g and gj are not essential for this result. The second part is an easy consequence of the first part, but will actually not be needed here.

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2 Minimal p-weak upper gradients

Proof. By passing toRa subsequence we may assume that kgj  gkLp .X/ < 2j : By Lemma 1.43,  gC ds is well defined with a value in Œ0; 1 for p-a.e. curve. By R R Proposition 1.37,  gC ds < 1 for p-a.e. curve. Similarly  g ds is well defined R and real-valued for p-a.e. curve. Thus  g ds is well defined and real-valued for p-a.e. curve. Arguing similarly for each j and using Corollary 1.38,R we see that there is a curve family with Modp . / D 0 and such that the integrals  gj ds, j D 1; 2; ::: , R and  g ds are well defined and real-valued for  2 .X / n . Let next ² ³ Z Z z

D  2 .X / n W gj ds 6! g ds; as j ! 1 ;   ² ³ Z 1

k D  2 .X / n W lim sup jgj  gj ds > ; k D 1; 2; ::: : k j !1  S z  1 k , and that it is enough to show that Modp . k / D 0 It is easy to see that

kD1 for every k. P1 Let m D k j DmC1 jgj  gj. Then Z m ds > 1 for all  2 k and m D 1; 2; ::: :  p m p So Modp . k /  k m kL / ! 0, as m ! 1. p .X/ < .k2 The last part follows directly as Modp . k / D 0 for every k, but can also be deduced by applying the first part to the functions gj0 D jgj  gj and g 0 D 0.

For us, the most important application of Fuglede’s lemma will be the following convergence results for p-weak upper gradients. For future references, we formulate them as three propositions, even though they are closely related. In particular, Proposition 2.2 is almost a special case of Proposition 2.4, but the assumptions on f are weaker and the proof is simpler. In fact, as we mainly work with Newtonian functions, Proposition 2.4 is not needed later in the book, but we have chosen to include it for completeness and future references. It combines some features of Propositions 2.2 and 2.3. Note also that in view of Example 1.31 there is no hope for similar convergence results for upper gradients. Proposition 2.2. Assume that f 2 D p .X / and that gj 2 Lp .X / are p-weak upper gradients of f , j D 1; 2; ::: . Assume further that gj ! g in Lp .X /, as j ! 1, and that g is nonnegative. Then g is a p-weak upper gradient of f . Proof. By Lemma 1.40 and Fuglede’s lemma (Lemma 2.1), p-a.e. curve R is such that R gj is an upper gradient of f along  for all j D 1; 2; ::: , and  gj ds !  g ds 2 R, as j ! 1. Consider such a curve  W Œ0; l  ! X . Then Z Z jf ..l //  f ..0//j  lim gj ds D g ds: j !1 



39

2.1 Fuglede’s lemma

Hence g is a p-weak upper gradient of f . Sometimes it will be useful to also have the following more general consequence of Fuglede’s lemma (Lemma 2.1). Proposition 2.3. Assume that fj 2 N 1;p .X / and that gj 2 Lp .X / is a p-weak upper gradient of fj , j D 1; 2; ::: . Assume further that fj ! f and gj ! g in Lp .X /, as j ! 1, and that g is nonnegative. Then there is a function fQ D f a.e. such that g is a p-weak upper gradient of fQ, and thus fQ 2 N 1;p .X /. There is also a subsequence such that fjk ! fQ q.e., as k ! 1. ffjk g1 kD1 Moreover, if either f 2 N 1;p .X / or there is a subsequence ffjk g1 such that kD1 fjk ! f q.e., as k ! 1, then we may choose fQ D f . Observe that we do not show that fj ! fQ in N 1;p .X /, as j ! 1. The reason for this is that it is not true in general. To see this we need the concept of minimal p-weak upper gradients, and the counterexample is therefore postponed until the next section, see Example 2.12. Proof. By passing to a subsequence if necessary we may R R assume that fj ! f a.e., and (by Fuglede’s lemma, Lemma 2.1) that  gj ds !  g ds 2 R, as j ! 1, for all  2 .X / n , where Modp . / D 0. Let fQ D lim supj !1 fj , and observe that fQ is defined at every point of X and fQ D f a.e. in X . Let E D fx 2 X W jfQ.x/j D 1g. By Lemma 1.40, p-a.e. curve  is such that gj is an upper gradient of fj along  for all j D 1; 2; ::: , and neither  nor any of its subcurves belong to . Consider such a curve  W Œ0; l  ! X . We see that either .0/; .l / 2 E or Z Z jfQ..l // fQ..0//j  lim sup jfj ..l //fj ..0//j  lim sup gj ds D g ds: j !1

j !1





As .E/ D 0, Corollary 1.51 shows that g is indeed a p-weak upper gradient of fQ. Let now fO D lim infj !1 fj . Arguing as above we see that g is also a p-weak upper gradient of fO 2 N 1;p .X / and that fO D f D fQ a.e. By Proposition 1.59 we see that fO D fQ q.e., and thus fj ! fQ q.e., as j ! 1. On the other hand, if f 2 N 1;p .X /, then by Proposition 1.59, f D fQ q.e. and hence g is a p-weak upper gradient also of f , by Corollary 1.49. Moreover, fj ! f q.e., as j ! 1. Finally, if fjk ! f q.e., as k ! 1, then again f D fQ q.e., and g is a p-weak upper gradient also of f , by Corollary 1.49. In the following proposition, we relax the assumption fj 2 N 1;p .X / and only concentrate on the convergence of p-weak upper gradients.

40

2 Minimal p-weak upper gradients

Proposition 2.4. Assume that fj 2 D p .X / and that gj 2 Lp .X / is a p-weak upper gradient of fj , j D 1; 2; ::: . Assume further that fj ! f q.e. and gj ! g in Lp .X /, as j ! 1, that f is real-valued a.e. and that g is nonnegative. Then g is a p-weak upper gradient of f . Recall the discussion in Remark 1.29 showing that the condition that f is realvalued a.e. is essential. Proof. By passingR to a subsequence if necessary we may assume (by Fuglede’s lemma, R Lemma 2.1) that  gj ds !  g ds 2 R, as j ! 1, for all  2 .X / n , where Modp . / D 0. Let fQ D lim supj !1 fj and E D fx 2 X W jfQ.x/j D 1g. Then fQ D f q.e. By Lemma 1.40, p-a.e. curve  is such that gj is an upper gradient of fj along  for all j D 1; 2; ::: , and neither  nor any of its subcurves belong to . Consider such a curve  W Œ0; l  ! X . We see that either .0/; .l / 2 E or Z Z jfQ..l // fQ..0//j  lim sup jfj ..l //fj ..0//j  lim sup gj ds D g ds: j !1

j !1





As .E/ D 0, Corollary 1.51 shows that g is indeed a p-weak upper gradient of fQ. Corollary 1.49 shows that g is also a p-weak upper gradient of f .

2.2 Minimal p-weak upper gradients In this section, we will single out a p-weak upper gradient which minimizes (2.1) both as an integral and pointwise a.e. It will play the role of jruj in the rest of the book. Our main result here is the following theorem. Theorem 2.5. Let u 2 D p .X /. There exists a minimal p-weak upper gradient gu 2 Lp .X / of u, i.e. gu  g a.e. for all p-weak upper gradients g 2 Lp .X / of u. Moreover, gu is unique up to sets of measure zero. More precisely, the set of minimal p-weak upper gradients of u consists exactly of one equivalence class in the cone LpC .X /. We will exclusively use the notation gu to denote a minimal p-weak upper gradient of u. As before we assume that gu is defined everywhere, even if there is some freedom in its choice. The proof of Theorem 2.5 depends heavily on the following lemma, which is of independent interest and allows us to take the minimum of two p-weak upper gradients. Note that the corresponding result for upper gradients is not true, see Example 2.8.

2.2 Minimal p-weak upper gradients

41

Lemma 2.6. Let g1 ; g2 2 Lp .X / be p-weak upper gradients of u 2 D p .X /. Then g D minfg1 ; g2 g is also a p-weak upper gradient of u. Proof. By Corollary 1.44, we can assume that g1 and g2 are Borel functions. By R Proposition 1.37 and Lemma 1.40, p-a.e. curve  in X is such that  .g1 Cg2 / ds < 1 and both g1 and g2 are upper gradients along  . Let  W Œ0; l  ! X be one such curve and let E D ft 2 .0; l / W g1 ..t //  g2 ..t //g. Then E  R is a Borel set and there exist open sets .0; l /  U1  U2    E such that ƒ1 .Un n E/ ! 0, as n ! 1, where ƒ1 is the one-dimensional S Lebesgue measure. For a fixed n, write Un as a pairwise disjoint union Un D 1 iD1 Ii of open intervals Ii D .ai ; bi /, see Lemma 1.4. (Here we allow some of the intervals Ii to be empty.) Then ju..0//  u..l //j  ju..0//  u..a1 //j C ju..a1 //  u..b1 //j C ju..b1 //  u..l //j Z Z  g1 ds C g2 ds: jI1

jI1

Continuing in this way, we obtain for all j D 1; 2; ::: , Z Z ju..0//  u..l //j  g1 ds C jSj

g2 ds:

jSj

I i D1 i

I i D1 i

Letting j ! 1, monotone and dominated convergence show that Z Z ju..0//  u..l //j  g1 ds C g2 ds: jUn

jUn

Applying monotone and dominated convergence once more yields Z Z Z ju..0//  u..l //j  g1 ds C g2 ds D g ds: jE

jE



The following example shows that the assumption g1 ; g2 2 Lp .X / is needed in Lemma 2.6. Example 2.7. Let A be a Borel subset of Œ0; 1  R such that 0 < ƒ1 .A \ I / < ƒ1 .I / for every interval I  Œ0; 1 of positive measure, where ƒ1 is the Lebesgue measure on R, see e.g. Exercise 33 in Chapter 1 in Folland [124]. Then both g1 D 1A and g2 D 1.1  A / are upper gradients of any measurable function on Œ0; 1, but their minimum, being zero, is not even a p-weak upper gradient (of any nonconstant function).

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2 Minimal p-weak upper gradients

The upper gradients in the previous example were not integrable (but on the other hand they were upper gradients of all functions). Next, we will see that the minimum of two Lp -integrable upper gradients need not be an upper gradient. So Lemma 2.6 is the best we can hope for. Example 2.8. Let f D f0g 2 N 1;p .R2 /, 1 < p < 2, and ´ 1=jxj; jxj  1; g.x/ D 0; jxj > 1; which is an upper gradient of f . Let also 1 ° ± [ Œ22j ; 212j  ; E1 D x 2 R2 W jxj 2 j D1

E2 D R2 n E1 and gj D gEj , j D 1; 2. We shall show that both g1 and g2 are upper gradients of f . Let  W Œ0; l  ! R2 be a curve. To show that g1 satisfies (1.1), we can assume that .0/ ¤ 0 D .l /. Let J  1 be an integer such that 212J  j.0/j. For j D J; J C 1; ::: , consider the concentric annuli Aj D fx 2 R2 W 22j  jxj  212j g; and let aj D infft 2 Œbj 1 ; l  W .t / 2 Aj g

and

bj D supft 2 Œbj 1 ; l  W .t / 2 Aj g;

where bJ 1 D 0. Note that bj  aj  22j , since  on its way to the origin has to pass through the annulus Aj , j D J; J C 1; ::: . As g1 D g  22j 1 in Aj , we obtain that Z Z 1 g1 ds  g ds  .bj  aj / min g  ; j D J; J C 1; ::: : Aj 2  jŒa ;b  jŒa ;b  \Aj j

j

j

j

Summing up over all j gives Z g1 ds  

1 Z X j DJ

g1 ds D 1;

jŒaj ;bj 

which shows that g1 is an upper gradient of f . A similar argument shows that g2 is also an upper gradient of f . Clearly, g1 ; g2 2 Lp .R2 /. The minimum minfg1 ; g2 g  0 is clearly not an upper gradient of f . In fact, as we observed in Example 1.31, no function vanishing a.e. is an upper gradient of f . Nevertheless, 0 is a p-weak upper gradient of f , by Lemma 2.6.

2.2 Minimal p-weak upper gradients

43

Proof of Theorem 2.5. Let I D inf g kgkLp .X/ , where the infimum is taken over all p-weak upper gradients of u, and choose a minimizing sequence fgj gj1D1 of p-weak upper gradients of u, so that kgj kLp .X/ ! I , as j ! 1. Let gNj D minfg1 ; ::: ; gj g, j D 1; 2; ::: . Then by Lemma 2.6, gNj is a p-weak upper gradient of u and gNj C1  gNj . Let gu D limj !1 gNj . The dominated convergence theorem shows that gNj ! gu in Lp .X /, as j ! 1, and by Proposition 2.2, gu is a p-weak upper gradient of u. Thus, I  kgu kLp .X/ D lim kgNj kLp .X/  lim kgj kLp .X/ D I: j !1

j !1

Let g be another p-weak upper gradient of u. Lemma 2.6 shows that minfgu ; gg is also a p-weak upper gradient and hence kgu kLp .X/ D I  kminfgu ; ggkLp .X/  kgu kLp .X/ : This shows that gu  g a.e. As for the uniqueness, if g 0 is another minimal p-weak upper gradient then clearly gu D g 0 a.e. Moreover, if g 0 is nonnegative and g 0 D gu a.e., then g 0 is also a p-weak upper gradient, by Corollary 1.44, and is obviously minimal as well. The following corollary gives us a more elegant expression for the N 1;p -norm. p p Corollary 2.9. If u 2 N 1;p .X /, then kukpN 1;p .X/ D kukL p .X/ C kgu kLp .X/ .

Proof. By Lemma 1.46, we can find upper gradients gj so that kgj  gu kLp .X/ ! 0, as j ! 1. Hence p p p p kukpN 1;p .X/  kukL p .X/ C lim kgj kLp .X/ D kukLp .X/ C kgu kLp .X/ : j !1

Conversely, gu  g a.e. for every upper gradient g 2 Lp .X / of u. Hence p p p p kukpN 1;p .X/ D kukL p .X/ C inf kgkLp .X/  kukLp .X/ C kgu kLp .X/ ; g

where the infimum is taken over all upper gradients g of u. The following proposition clarifies the problem pointed out in Example 1.31. Proposition 2.10. For f 2 D p .X / let M be the set of all upper gradients of f which S of M in LpC .X / consists precisely of those belong to Lp .X /. Then the closure M p-weak upper gradients of f which are in Lp .X /. Moreover, S D fg 2 LpC .X / W g  gf a.e.g: M By LpC .X / we mean the cone of nonnegative functions (not equivalence classes) in Lp .X /, equipped with the seminorm coming from k  kLp .X/ .

44

2 Minimal p-weak upper gradients

S by Lemma 1.46. Proof. Let g 2 Lp .X / be a p-weak upper gradient of f . Then g 2 M S . Then there exist upper gradients gj 2 Lp .X / of f such Conversely, let g 2 M p that gj ! g in L .X /, as j ! 1. Proposition 2.2 shows that g is a p-weak upper gradient of f . That the set of p-weak upper gradients of f in Lp .X / is contained in M 0 WD fg 2 LpC .X / W g  gf a.e.g follows from the minimality of gf . Conversely, for any function g 2 M 0 , we have g 0 WD minfg; gf g D gf a.e., and thus g 0 is also a p-weak upper gradient of f , by Corollary 1.44. Hence also g  g 0 is a p-weak upper gradient of f . Remark 2.11. The requirement g 2 Lp .X / in Theorem 2.5 is essential, as we shall now see. Write all the rational numbers in X D Œ0; 1  R as a sequence, Q \ Œ0; 1 D fq1 ; q2 ; ::: g. For n D 2; 3; ::: , let En D

1 [

.qj  nj ; qj C nj /

and gn D 1En :

j D1

Note that .En /  2=.n  1/ ! 0, as n ! 1. R y For all 0  x < y  1 and all n we have x gn ds D 1 and hence gn is an upper gradient of every measurable function on X . In particular, if u.x/ D x, then by Proposition A.3, gu D 1 and thus, gn D 0 < gu in X n En . Also, inf n gn D Q\Œ0;1 D 0 a.e. We are now able to show that (in general) fj 6! fQ in N 1;p .X /, as j ! 1, in Proposition 2.3. Example 2.12. Let X D Œ1; 1  R, F .x/ D jxj on X and extend F periodically to R so that F .x C 2/ D F .x/. Let further fj .x/ D F .jx/=j , gj  1, f  0 and g  1, j D 1; 2; ::: . Then the assumptions in Proposition 2.3 are fulfilled and we conclude that g is a p-weak upper gradient of f . However, fj 6! f in N 1;p .X /, as j ! 1, since the minimal p-weak upper gradient gfj f D jfj0 j D 1 a.e. for all j , by Proposition A.3. We can also observe that even though all gj are minimal p-weak upper gradients of fj (again by Proposition A.3), it does not follow that g is a minimal p-weak upper gradient of f . It is not clear whether we can deduce that fj ! f in N 1;p .X /, as j ! 1, if we assume a bit more as in the following question. Open problem 2.13. Assume that f; fj 2 N 1;p .X /, j D 1; 2; ::: , and that fj ! f and gfj ! gf in Lp .X /, as j ! 1. Is it then true that fj ! f in N 1;p .X /, as j ! 1?

2.3 Calculus for p-weak upper gradients

45

2.3 Calculus for p-weak upper gradients Let us point out that after Remark 2.28 we will refer to the results in this section as holding for the corresponding local spaces, whenever necessary. Even though the gradients dealt with in this book are only p-weak upper gradients, they still share many good properties with usual gradients, which makes it possible to use them in the theory of p-harmonic functions. In particular, we shall see in this and the next section that certain Leibniz and chain rules hold for p-weak upper gradients and that the minimal p-weak upper gradient is local. Lemma 2.14. Assume that u 2 ACCp .X / and that g 2 Lp .X / is a p-weak upper gradient of u. Then for p-a.e. curve  W Œ0; l  ! X , we have j.u ı  /0 .t /j  g..t // for a.e. t 2 Œ0; l :

(2.2)

Conversely, if g  0 is measurable, u 2 ACCp .X / and (2.2) holds for p-a.e. curve  W Œ0; l  ! X , then g is a p-weak upper gradient of u. Observe that the converse holds even if g is not in Lp .X /. However, the first part is false if we allow g to be an arbitrary p-weak upper gradient, see Remark 2.11. Proof. Assume first that u 2 ACCp .X / and that g 2 Lp .X / is a p-weak upper gradient of u. Let  W Œ0; l  ! X be Ra curve such that u is absolutely continuous on , g is an upper gradient along  , and  g ds < 1. This holds for p-a.e. curve. For a.e. t 2 .0; l /, u ı  is differentiable at t and t is a Lebesgue point for g ı  (in the sense that the last equality in (2.3) below holds). For such t we have ˇ ˇ ˇ u..t C h//  u..t // ˇˇ ˇ j.u ı  / .t /j D lim ˇ ˇ h h!0 0

1 h!0 h

Z

tCh

 lim

g.. // d  D g..t //:

(2.3)

t

Conversely, assume that g  0 is measurable, u 2 ACCp .X / and (2.2) holds for p-a.e. curve  W Œ0; l  ! X . RLet  W Œ0; l  ! X be a curve on which u is absolutely continuous, (2.2) holds, and  g ds is well defined. This holds for p-a.e. curve, by Lemma 1.43. Then Z l Z l Z 0 j.u ı  / .t /j dt  g..t // dt D g ds: ju..0//  u..l //j  0

0



Next we prove a Leibniz or product rule for p-weak upper gradients. Theorem 2.15 (Leibniz rule). If u; v 2 D p .X /, then jujgv C jvjgu is a p-weak upper gradient of uv.

46

2 Minimal p-weak upper gradients

Proof. As u, v, gu and gv are measurable, so is g D jujgv C jvjgu , and clearly g  0. Moreover, uv 2 ACCp .X /, by Lemma 1.58. Let  W Œ0; l  ! X be a curve on which (2.2) holds for both pairs .u; gu / and .v; gv / (in the place of .u; g/) and such that both u and v are absolutely continuous on  . This is true for p-a.e. curve  . In particular, j.uı /0 .t /j  gu ..t // and j.vı /0 .t /j  gv ..t // for a.e. t 2 Œ0; l . For such t we have letting w D uv, j.w ı  /0 .t /j D ju..t //.v ı  /0 .t / C v..t //.u ı  /0 .t /j  ju..t //jgv ..t // C jv..t //jgu ..t // D g..t //: By Lemma 2.14, g is a p-weak upper gradient of w D uv. Note that the p-weak upper gradient in the Leibniz rule (Theorem 2.15) need not be minimal. This is due to the absolute values, both those explicit and those implicit in gu (and gv ). Recall that gu corresponds to jruj rather than ru. For example, if X D Rn and v D 1=u, then in general, guv D 0 < 2jruj=juj D jujgv C jvjgu . Another example is to let u.x/ D x and v.x/ D x  2 on Œ0; 1 in which case gu .x/ D u0 .x/ D gv .x/ D v 0 .x/ D 1 (so these minimal p-weak upper gradients are equal to the corresponding derivatives, not just their absolute values), while guv .x/ D 2  2x and jujgv C jvjgu D 2. Theorem 2.16 (Chain rule). Let u 2 D p .X / and let ' W I ! R be locally Lipschitz, where I  R is an interval. Assume that Cp .fx 2 X W u.x/ … I g/ D 0. Then j' 0 ı ujgu is a minimal p-weak upper gradient of ' ı u, where we consider ' 0 ı u to be 0 if it is not defined. Note that if I D R and u 2 N 1;p .X /, then Cp .fx 2 X W u.x/ … I g/ D 0 is immediate, by Proposition 1.30. Note also that if ' is continuously differentiable on I , then the use below of the chain rule from Theorem 1.74 in Malý–Ziemer [258], can be replaced by the elementary classical chain rule for differentiable functions. In fact, we will only use the chain rule (Theorem 2.16) for continuously differentiable functions '. Proof. We can redefine u on a set of capacity zero without affecting the p-weak upper gradients of u and ' ı u. We may thus assume that u.X /  I . Moreover, by Proposition 1.48, such a change only affects u on p-almost no curve and thus we still have u 2 ACCp .X /. Let  W Œ0; l  ! X be a curve R on which u is absolutely continuous, (2.2) holds (with gu in the place of g), and  gu ds < 1. Since both u ı  and ' ı u ı  are absolutely continuous, by Lemma 1.58, it is true that for a.e. t 2 Œ0; l  both .u ı  /0 .t / and .' ı u ı /0 .t / exist, and j.u ı  /0 .t /j  gu ..t //

and

j.' ı u ı  /0 .t /j  g'ıu ..t //;

(2.4)

47

2.3 Calculus for p-weak upper gradients

by Lemma 2.14. For a.e. such t , we have j.' ı u ı  /0 .t /j D j' 0 .u..t ///.u ı  /0 .t /j  j' 0 .u..t ///jgu ..t //;

(2.5)

by the chain rule in Theorem 1.74 in Malý–Ziemer [258]. (The chain rule in [258] requires ' to be Lipschitz on R. Here J D u..Œ0; l // is a compact interval since u is continuous on . Thus ', by assumption, is Lipschitz on J . We can therefore use Theorem 1.74 in [258] with any Lipschitz extension of 'jJ to R, e.g. one of the McShane extensions given by Lemma 5.2 (applied to X D R).) Lemma 2.14 and (2.5) imply that j' 0 ı ujgu is a p-weak upper gradient of ' ı u and hence j' 0 ı ujgu  g'ıu a.e. To show minimality, observe that (2.5) and (2.4) yield j.u ı /0 .t/j D j' 0 .u..t ///j1 j.' ı u ı  /0 .t /j  j' 0 .u..t ///j1 g'ıu ..t //; where we interpret j' 0 .u..t ///j1 as 1, whenever j' 0 .u..t ///j D 0. Lemma 2.14 again implies that j' 0 ı uj1 g'ıu is a p-weak upper gradient of u and hence gu  j' 0 ı uj1 g'ıu a.e. It follows that j' 0 ı ujgu  g'ıu a.e., finishing the proof. We finish this section with two concrete applications of Lemma 2.14 and the chain rule (Theorem 2.16), which will be needed later. Proposition 2.17. Assume that u 2 N 1;p .X / is a nonnegative function such that Cp .fx W u.x/ D 0g/ D 0. Then gu =u is a minimal p-weak upper gradient of the function v D log u. If, moreover, .X / < 1 and u  " > 0, then v 2 N 1;p .X /. Proof. The first part follows directly from the chain rule (Theorem 2.16). To prove the second part, we have gv D gu =u  gu =" a.e., and thus gv 2 Lp .X /. As jvj < u C jlog "j and .X / < 1, it is clear that v 2 Lp .X /. Lemma 2.18. Let u; v 2 N 1;p .X / and 2 Lip.X / be such that 0   1. Set w D u C .v  u/ D .1  /u C v. Then g WD .1  /gu C gv C jv  ujg is a p-weak upper gradient of w. Proof. Let  W Œ0; l  ! X be a curve on which u and v are absolutely continuous and such that j.u ı /0 .t/j  gu ..t //;

j.v ı  /0 .t /j  gv ..t //

for a.e. t 2 Œ0; l . This holds for p-a.e. curve.

and

j. ı  /0 .t /j  g ..t //

48

2 Minimal p-weak upper gradients

Then w ı  W Œ0; l  ! R is absolutely continuous, by Lemma 1.58, and for a.e. t 2 Œ0; l  we have j.w ı  /0 .t /j D j.1  . ı  /.t //.u ı  /0 .t / C . ı  /.t /.v ı  /0 .t / C ..v ı  /.t /  .u ı  /.t //. ı  /0 .t /j  .1  . ı  /.t //gu ..t // C . ı  /.t /gv ..t // C j.v ı  /.t /  .u ı  /.t /jg ..t // D g..t //: Lemma 2.14 shows that g is a p-weak upper gradient of w.

2.4 The glueing lemma Let us point out that after Remark 2.28 we will refer to the results in this section as holding for the corresponding local spaces, whenever necessary. One of the important properties of the minimal p-weak upper gradient is that it is local, i.e. if two functions coincide in an open set, then so do their minimal p-weak upper gradients (a.e.). In fact, we shall see in Corollary 2.21 that this is true even if the coincidence set is not open. This will be essential in our study of p-harmonic functions later in the book. The locality is proved using the following lemma which allows us to glue together functions and to obtain the corresponding minimal p-weak upper gradient by glueing gradients in a similar way. Lemma 2.19 (The glueing lemma). Let E  X be a measurable set, f 2 ACCp .X / and u; v 2 D p .X /. Assume that f jE D u and f jXnE D v. Let g 0 2 Lp .X / and g 00 2 Lp .X / be p-weak upper gradients of u and v, respectively. Then g D g 0 E Cg 00 XnE is a p-weak upper gradient of f . Moreover, if g 0 D gu and g 00 D gv are minimal, then g is a minimal p-weak upper gradient of f . Proof. Let g1 D g 0 C g 00 XnE and g2 D g 0 E C g 00 . We will first show that both g1 and g2 are p-weak upper gradients of f . From this and Lemma 2.6 it follows that g D minfg1 ; g2 g is also a p-weak upper gradient of f . By symmetry it is enough to show that g1 is a p-weak upper gradient of f . Take a curve  W Œ0; l  ! X such that  1 .E/ is measurable, u, v and f are all absolutely continuous on  , and g 0 and g 00 are upper gradients of u and v, respectively, along . This is true for p-a.e. curve by Lemmas 1.40 and 1.43 (the latter motivating that  1 .E/ is measurable). If  \ E D ¿, then Z Z jf ..0//  f ..l //j D jv..0//  v..l //j  g 00 ds  g1 ds: 



2.4 The glueing lemma

49

On the other hand, if  \ E ¤ ¿, let ˛ D infft 2 Œ0; l  W .t / 2 Eg

and

ˇ D supft 2 Œ0; l  W .t / 2 Eg:

Then 0  ˛  ˇ  l and Z

g 00 ds 

jf ..0//  f ..˛//j D jv..0//  v..˛//j  j.0;˛/

Z g1 ds; j.0;˛/

where the equality follows by continuity if ˛ > 0 and is trivial otherwise. Similarly, Z jf ..ˇ//  f ..l //j  g1 ds: j.ˇ;l /

We also have, using the continuity of u and f along , that Z Z jf ..˛//  f ..ˇ//j D ju..˛//  u..ˇ//j  g 0 ds  j.˛;ˇ/

g1 ds: j.˛;ˇ/

By the triangle inequality this shows that g1 is a p-weak upper gradient, and hence g is a p-weak upper gradient and g  gf a.e. Now, if g 0 D gu and g 00 D gv , then by applying this lemma (without the minimality part) with the roles of u and f interchanged, we see that gf E C gu XnE is a p-weak upper gradient of u. Since gu and gf are minimal we have that gu  gf  g D gu

a.e. on E:

Hence gf D g a.e. on E and similarly on X n E. Corollary 2.20. Let u; v 2 D p .X /. Then gu fu>vg C gv fvug is a minimal p-weak upper gradient of maxfu; vg, and gv fu>vg C gu fvug is a minimal p-weak upper gradient of minfu; vg. Lemma 2.14 can be used to give an alternative proof for Corollary 2.20 (with the minimality shown as in the proof of Lemma 2.19). As a direct consequence of Corollary 2.20 we get new proofs of Theorems 1.20 and 1.55. Another consequence is the following locality result, advertised at the beginning of this section. Corollary 2.21. If u; v 2 D p .X /, then gu D gv a.e. on fx 2 X W u.x/ D v.x/g: Moreover, if c 2 R is a constant, then gu D 0 a.e. on fx 2 X W u.x/ D cg.

50

2 Minimal p-weak upper gradients

Proof. Let E D fx W u.x/ D v.x/g and ´ u f D v

in X n E; in E:

Then f D u. On the other hand, by Lemma 2.19, gu XnE C gv E is a minimal p-weak upper gradient of f D u. Since gu is also a minimal p-weak upper gradient of u D f , we must have gu XnE C gv E D gu

a.e.

Therefore gu D gv a.e. on E. As for the second part, let v  c, with gv  0, and apply the first part. The following result partially strengthens Theorem 1.27 (iv). Proposition 2.22 (Strong subadditivity). Let E1 ; E2  X . Then Cp .E1 [ E2 / C Cp .E1 \ E2 /  Cp .E1 / C Cp .E2 /: Proof. We may assume that the right-hand side is finite. Let " > 0. We can thus find uj  Ej such that kuj kpN 1;p .X/  Cp .Ej / C ", j D 1; 2. Let v D maxfu1 ; u2 g and w D minfu1 ; u2 g. By Corollary 2.20, we have Cp .E1 [ E2 / C Cp .E1 \ E2 /  kvkpN 1;p .X/ C kwkpN 1;p .X/ Z p D .v p C gvp C w p C gw / d X Z D .up1 C gup1 C up2 C gup2 / d X

 Cp .E1 / C Cp .E2 / C 2": Letting " ! 0 completes the proof.

2.5 N 1;p ./ Let A  X be measurable. Then we naturally can consider A as a metric space in its own right (with the restrictions of d and ). The Newtonian space N 1;p .A/ and the Dirichlet space D p .A/ are then given by Definitions 1.17 and 1.54, respectively. This is also why we did not want to assume that X D supp  from the very beginning. As A is only measurable, it can easily happen that supp jA is a proper subset of A, but the theory presented here is still meaningful on A.

2.5 N 1;p ./

51

A function f 2 N 1;p .X / clearly has a restriction f jA which belongs to N 1;p .A/, and any (p-weak) upper gradient of it remains a (p-weak) upper gradient when restricted. However, the restriction of a minimal p-weak upper gradient is not always minimal. Consider e.g. X D Rn and let A be a totally disconnected subset of Rn of positive Lebesgue measure. Since there are no curves in A, the zero function is a minimal p-weak upper gradient with respect to A of every measurable function u, but it is not the restriction of a corresponding minimal p-weak upper gradient from Rn (except when u D C q.e. on A). Further, for a subset E  A it is always true that Cp0 .E/  Cp .E/, where Cp is taken with respect to X and Cp0 with respect to A, but it is possible to have strict inequality and even Cp0 .E/ D 0 < Cp .E/. Thus the concept “q.e.” depends on whether we consider it with respect to A or X . Unless otherwise said, we will always consider Cp and “q.e.” with respect to X . Let us next look at the case when A D  is open. It then turns out that we have little of these problems. It can still happen that Cp0 .E/ < Cp .E/, but the capacities have the same zero sets. Furthermore, the restriction of a minimal p-weak upper gradient remains minimal. We start by proving the latter fact. Lemma 2.23. Assume that f 2 D p .X / with a minimal p-weak upper gradient gf (with respect to X ). Then gf j is a minimal p-weak upper gradient of f with respect to . Proof. Let g be a minimal p-weak upper gradient of f with respect to . Since clearly gf j is a p-weak upper gradient of f in , we have g  gf a.e. in . Assume that g < gf on a set of positive measure. Then we can find a ball B  2B   such that g < gf on a set of positive measure in B. Now let ´ gf in X n B; g0 D g in B: Let be the set of curves along which g 0 is not an upper gradient of f . Let further

1 D f 2 W   g and 2 D f 2 W   X n Bg. Clearly, g 0 D gf is a p-weak upper gradient of f in X n B. Hence Modp . 2 / D 0, by Lemma 1.40. (Note that as   X n B for all  2 F2 , the modulus Modp . 2 / is the same when taken with respect to X n B and X .) On the other hand, by Lemma 2.19, g 0 is a p-weak upper gradient of f in , and thus Modp . 1 / D 0, by Lemma 1.40. Let now  2 . Since 2B   we can split  into a finite number of pieces 1 ; ::: ; n , each of which lies entirely either in  or in X n B. As g 0 is not an upper gradient of f along  , there must be at least one j for which g 0 is not an upper gradient of f along j . Hence j 2 1 [ 2 . By Lemma 1.34, we have Modp . /  Modp . 1 [ 2 / D 0. Thus g 0 is a p-weak upper gradient of f in X , but this is a contradiction as g 0 < gf on a set of positive measure.

52

2 Minimal p-weak upper gradients

Next, we will see that q.e. is the same when taken with respect to  and X . Lemma 2.24. Let E  . Then Cp .E/ D 0 if and only if Cp0 .E/ D 0, where Cp0 is the capacity with respect to . Proof. If  D X the result is trivial so we may assume that  ¤ X . The necessity has already been observed. Assume therefore that Cp0 .E/ D 0. Let Ej D fx 2 E W dist.x; X n / > 1=j g and fj D Ej . Since Cp0 .Ej / D 0, Proposition 1.48 shows that 0 is a p-weak upper gradient of fj with respect to  and that .Ej / D 0. Arguing as in the proof of Lemma 2.23, we see that 0 is a p-weak upper gradient of fj with respect to all of X . Hence kfj kN 1;p .X/ D 0, and thus Cp .Ej / D 0. Therefore Cp .E/ D Cp

1 [

1  X Ej  Cp .Ej / D 0:

j D1

1;p

j D1

p

2.6 Nloc and Dloc Let us first recall how to define Lploc .X /. Remember that in our theory, X can be replaced by any measurable subset, see the beginning of Section 2.5. A function f on X belongs to Lploc .X / if for every x 2 X there is rx > 0 such that f 2 Lp .B.x; rx //. We similarly say that 1;p f 2 Nloc .X /;

p f 2 Dloc .X /

and

f 2 ACCp;loc .X /

if for every x 2 X there is rx > 0 such that f 2 N 1;p .B.x; rx //;

f 2 D p .B.x; rx //

and

f 2 ACCp .B.x; rx //;

respectively. 1;p p We have trivially that Nloc .X /  Dloc .X /  ACCp;loc .X /, in view of Theo1;p p rem 1.56, and Nloc .X /  Lloc .X /. We will actually show that ACCp;loc .X / D ACCp .X /, see Proposition 2.27 below. As supp  is Lindelöf and Cp .X n supp / D 0 (see Propositions 1.6 and 1.53), the 1;p equivalence classes in Lploc and Nloc remain the same as in Lp and N 1;p , respectively, by Lemma 2.24. p We can now extend Theorem 2.5 to functions in Dloc .X /. p Theorem 2.25. Let u 2 Dloc .X /. There exists a minimal p-weak upper gradient p gu 2 Lloc .X / of u, i.e. gu  g a.e.

for all p-weak upper gradients g 2 Lploc .X / of u. Moreover, gu is unique up to sets of measure zero.

1;p p 2.6 Nloc and Dloc

53

Proof. By assumption, we have frx gx2X such that u 2 D p .B.x; rx // for x 2 X . As supp  is Lindelöf we can find a countable sequence fxj gj1D1 such that supp   S1 j D1 Bj , where Bj D B.xj ; rxj /. Let gj be a minimal p-weak upper gradient of ujBj with respect to Bj . Let now gu .x/ D inf gj .x/; x2Bj

where we say that the infimum is 0 (or arbitrary) if there is no Bj 3 x. By Lemma 2.23, each gj is a restriction of a global minimal p-weak upper gradient of u and hence gj D gk

a.e. in Bj \ Bk ;

which implies that gu D gj a.e. in Bj . Moreover, gu 2 Lploc .X /. If g 2 Lploc .X / is another p-weak upper gradient of u and if B is a ball so that g 2 Lp .B/, then gu is a minimal p-weak upper gradient of ujB by Lemma 2.23 and hence gu  g a.e. in B. Thus gu  g a.e. in X (using again that supp  is Lindelöf). The uniqueness is obtained just as in the proof of Theorem 2.5. 1;p p Proposition 2.26. Let A b X . Then f 2 Lploc .X /, f 2 Nloc .X / and f 2 Dloc .X / p 1;p p implies that f 2 L .A/, f 2 N .A/ and f 2 D .A/, respectively. 1;p Proof. We give the proof for Nloc .X /. The other cases are treated similarly. By assumption, for every x 2 X we have rx > 0 so that f 2 N 1;p .B.x; rx //. As AN is compact we can find a finite subcover fB.xj ; rxj /gjND1 of the open cover fB.x; rx /gx2AN N It follows that of A. p kf kL p .A/ 

n X j D1

and similarly p kgf kL p .A/ 

n X j D1

p kf kL p .B.x ;r j x

j

p kgf kL p .B.x ;r j x

j

//

<1

//

< 1;

where gf is given by Theorem 2.25. Proposition 2.27. If f 2 ACCp;loc .X /, then f 2 ACCp .X /. Proof. By assumption, for every x 2 X we have rx > 0 so that f 2 ACCp .B.x; rx //. As supp  is Lindelöf, there is a countable subcover fBj gj1D1 of balls so that f 2 S ACCp .Bj /, j  1, and supp   j1D1 Bj . Let also B0 D X n supp  even though this is in general not a ball. (It is nevertheless open.) Let be the set of curves in X on which f is not absolutely continuous, and j be the set of curves in Bj on which f is not absolutely continuous, j D 0; 1; ::: . By the definition of ACCp , in Theorem 1.56, we have Modp . j / D 0, j  1. Moreover, as jB0 is the zero measure, Modp . 0 / D 0 as well. Let now  2 . As 

54

2 Minimal p-weak upper gradients

is compact we can split it into pieces 1 ; ::: ; n , so that  D 1 C  C n and k is a curve in Bjk , k D 1; ::: ; n. As f is not absolutely continuous on  , there is at least one k on which f is not absolutely continuous. Thus Lemma 1.34 shows that Modp . / D 0. Remark 2.28. As we have seen, the minimal p-weak upper gradient of a function in p Dloc .X / only depends on the function locally. It is therefore straightforward to extend the results in Sections 2.3 and 2.4 to hold for the corresponding local spaces. Even when we require these extended results we will refer to their original numbers. p 1;p Note, in particular, that the spaces Dloc .X / and Nloc .X / are lattices. The following characterizations of the local spaces are often more useful than the definitions themselves. See also Corollary 2.31. Proposition 2.29. Assume that X is locally compact. Then f 2 Lploc .X /;

1;p p f 2 Nloc .X / and f 2 Dloc .X /

if and only if f 2 Lp ./, f 2 N 1;p ./ and f 2 D p ./, respectively, for all open  b X. 1;p Proof. We give the proof for Nloc .X /. The other cases are treated similarly. The necessity follows directly from Proposition 2.26. As for the sufficiency, assume that f 2 N 1;p ./ for every  b X . Let x 2 X . By local compactness we can find rx > 0 such that B.x; rx / b X , and hence f 2 N 1;p .B.x; rx //. But this shows that 1;p indeed f 2 Nloc .X /.

Definition 2.30. A metric space is proper if all closed and bounded subsets are compact. If X is proper and   X is open, then it follows immediately that  is locally compact. Using this we directly obtain the following corollary of Proposition 2.29. 1;p Corollary 2.31. Assume that X is proper. Then f 2 Lploc ./, f 2 Nloc ./ and f 2 p p 0 1;p 0 p Dloc ./ if and only if f 2 L . /, f 2 N . / and f 2 D .0 /, respectively, for all open 0 b . 1;p Remark 2.32. In the nonproper case an alternative definition of Nloc ./, which mimics the conclusion of Corollary 2.31, was given in A. Björn–Marola [59]. The two definitions coincide when X is proper.

We can now also refine Proposition 1.37. Proposition 2.33. RModp . / D 0 if and only if there is a nonnegative function 2 Lploc .X / such that  ds D 1 for all  2 .

2.7 N01;p

55

Proof. The necessity follows from Proposition 1.37. As for the sufficiency let Z D X n supp . By assumption, for every x 2 X there is a ball Bx 3 x such that 2 Lp .Bx /. By Proposition 1.6, supp  is Lindelöf, and S thus we can find a countable subcover Bj D Bxj , j D 1; 2; ::: , such that supp   j1D1 Bj . Let Aj D

1 2j k kLp .Bj / C 1

; j D 1; 2; ::: ;

and

Q D

1 X

Aj jBj 2 Lp .X /:

j D1

Let further  2 n Z . As  is compact, there is N < 1 such that   Therefore Z Z   Q ds  min Aj ds D 1: 

1j N

SN

j D1

Bj .



Hence, by Proposition 1.37, Modp . n Z / D 0. Moreover, Modp . Z / D 0, by Proposition 1.53, and thus Modp . / D 0.

1;p

2.7 N0

To be able to compare boundary values of Newtonian functions and to solve the Dirichlet problem we shall need the following Newtonian spaces with zero boundary values. These spaces will also be useful as spaces of test functions in the definitions of obstacle problems and (super)minimizers. Definition 2.34. For A; E  X , where A is measurable, we introduce the space of Newtonian functions with zero values in A n E as follows, N01;p .EI A/ D ff jE \A W f 2 N 1;p .A/ and f D 0 in A n Eg: We also let N01;p .E/ D N01;p .EI X /. As N01;p .EI A/ is nothing but N01;p .E \ A/ taken with respect to the surrounding space A we concentrate on studying N01;p .E/ below. Note however the subtle point that the capacity of a set and (p-weak) upper gradients of a function can be different when taken with respect to A and X , see the discussion in the beginning of Section 2.5. Note also that obviously N01;p .E/ does not only depend on E but also on X , but in order not to make the notation too cumbersome we have refrained from making this explicit in the notation. It is easy to see that N01;p .E/ D ff jE W f 2 N 1;p .X / and f D 0 q.e. in X n Eg:

(2.6)

Clearly, N01;p .E/ is a subspace of N 1;p .E/, if E is measurable, and we may also consider it as a subspace of N 1;p .X /, in a natural way. Let us also point out that if

56

2 Minimal p-weak upper gradients

we have use for considering values of a function f 2 N01;p .E/ outside of E, then we often implicitly assume f to be identically 0 outside of E. The following result is worth observing. Proposition 2.35. N01;p .E/ D N 1;p .X / holds if and only if Cp .X n E/ D 0. Note that even though it follows that N01;p .E/ D N 1;p .E/ if Cp .X n E/ D 0, it can happen that N01;p .E/ D N 1;p .E/ even when Cp .X n E/ > 0: Let X consist of two connected components and let E be one of them. Then N01;p .E/ D N 1;p .E/. See also Corollary 5.57. Proof. The sufficiency is clear from (2.6). As for the necessity assume on the contrary that Cp .X n E/ > 0. Then we can find a bounded set F  X n E with Cp .F / > 0. We can then find a function f 2 Lip.X / with bounded support such that f  1 on F . Now f 2 N 1;p .X / n N01;p .E/, which concludes the proof. Theorem 2.36. The space N01;p .E/ is a closed subspace of N 1;p .X / and thus N01;p .E/= is a Banach space. Proof. This follows from Corollary 1.72. The following lemma is useful for showing that a function belongs to N01;p .E/. Lemma 2.37. Assume that E  X is measurable. Let u 2 N 1;p .E/ and v; w 2 N01;p .E/ be such that v  u  w q.e. in E. Then u 2 N01;p .E/. Proof. By subtracting v from all terms and observing that u 2 N01;p .E/ if and only if u  v 2 N01;p .E/, we may assume that v  0. After redefinitions on sets of capacity zero, we may, without loss of generality, assume that 0  u  w everywhere in E, and that w  0 in X n E. 0 2 Lp .X / be an upper Let gu0 2 Lp .E/ be an upper gradient of u in E, and let gw gradient of w in X . Define ´ ´ 0 u in E; gu0 C gw in E; uQ D and g D 0 0 in X n E; gw in X n E: We want to show that g is an upper gradient of uQ in X , from which it follows that uQ 2 N 1;p .X / and thus u 2 N01;p .E/. Let  W Œ0; l  ! X be a curve. If   E, then Z Z 0 ju..0// Q  u..l Q gu ds  g ds:  //j D ju..0//  u..l //j  



On the other hand, if .0/; .l / 2 X n E, then ju..0// Q  u..l Q  //j D 0 

R 

g ds.

2.7 N01;p

57

By splitting  into two parts, if necessary, and possibly reversing the direction, we may thus assume that .0/ 2 E and .l / 2 X n E. Hence, Z Z 0 //j D ju..0//j  jw..0//  w..l //j  ju..0// Q  u..l Q g ds  g ds:   w 



Recall that functions in N01;p .E/ are defined using the global space N 1;p .X /. If E D  is open, then it is possible to restrict both the definition and the norm to . This will be done in the following two propositions. Note that none of them is true for general sets E. Proposition 2.38. For f 2 N01;p ./ we have kf kN 1;p .X/ D kf kN 1;p ./ :

(2.7)

Proof. We may assume that f  0 outside of . Let gf be a minimal p-weak upper gradient of f with respect to X . By Lemma 2.23, gf j is a minimal p-weak upper gradient of f with respect to . On the other hand, by Corollary 2.21, gf D 0 a.e. outside of . Hence p p kf kpN 1;p .X/ D kf kL p .X/ C kgf kLp .X/ p p p D kf kL p ./ C kgf j kLp ./ D kf kN 1;p ./ :

x Proposition 2.39. It is true that N01;p ./ D N01;p .I /. x Conversely, let u 2 N 1;p .I / x and Proof. It is clear that N01;p ./  N01;p .I /. 0 define ´ u in ; uQ D 0 in X n : x we can find an upper gradient g 2 Lp ./ x of u in . x Define As u 2 N01;p .I / ´ x g in ; gQ D x 0 in X n : Let  W Œ0; l  ! X be a curve. If   , then Z

Z

ju..0// Q  u..l Q  //j D ju..0//  u..l //j 

g ds D 

On the other hand, if .0/; .l / 2 X n , then Z ju..0// Q  u..l Q  //j D 0 

gQ ds: 

gQ ds: 

58

2 Minimal p-weak upper gradients

By splitting  into two parts if necessary and possibly reversing the direction, we may thus assume that .0/ 2  and .l / 2 X n . Let c D infft 2 Œ0; l  W .t / 2 X n g: x n , and thus u..l By the continuity of , we have .c/ 2  Q Q D 0.  // D u..c// Hence, Z Z //j D j u..0// Q  u..c//j Q  ju..0// Q  u..l Q g ds  gQ ds:  jŒ0;c



p

2.8 Gloc Sometimes a function is too large to have an upper gradient in Lploc but still there may be use of some sort of minimal p-weak upper gradient. We will apply this to superharmonic functions which can have a minimal p-weak upper gradient not belonging to L1loc , see Remark 9.56. p .X / if .fx 2 X W ju.x/j D 1g/ D 0 and the Definition 2.40. We say that u 2 Gloc 1;p truncations uk WD maxfminfu; kg; kg 2 Nloc .X / for every positive k 2 Z. p If u 2 Gloc .X /, then we let Gu D limk!1 guk and say that Gu is a minimal p-weak upper gradient of u.

is an increasing sequence for a.e. x 2 X (after the repreNote that fguk .x/g1 kD1 sentatives guk have been chosen), in view of Corollary 2.21. The minimality should be understood in the sense described below. p .X /, then Gu is a p-weak upper gradient of u which Proposition 2.41. If u 2 Gloc is minimal in the sense that if g is another p-weak upper gradient of u such that gfxWju.x/jkg is a p-weak upper gradient in Lploc .X / of uk WD maxfminfu; kg; kg for every positive k 2 Z, then Gu  g a.e.

Proof. We may redefine each guk on a set of measure zero without affecting Gu on more than a set of measure zero. To simplify things we may let gu0  0 and then recursively choose gukC1 so that gukC1 D guk on fx W ju.x/j < kg and gukC1 D 0 on fx W ju.x/j  k C 1g. With this choice we will have guk % Gu everywhere in X , as k ! 1. For each positive k 2 Z there is an exceptional set k of curves along which guk S is not an upper gradient of uk . Let D 1

. Then Modp . / D 0. k kD1 For a curve  … ,  W Œ0; l  ! X , we have either ju..0//j D ju..l //j D 1 or Z Z guk ds D Gu ds: ju..0//  u..l //j D lim juk ..0//  uk ..l //j  lim k!1

k!1 



2.9 Dependence on p in gu

59

It thus follows from Corollary 1.51 that Gu is a p-weak upper gradient of u. For the minimality, we have that gfxWju.x/jkg  guk a.e., by the minimality of guk . As .fx W ju.x/j D 1g/ D 0 we see that Gu D lim guk  lim gfxWju.x/jkg D g k!1

k!1

a.e.

p Proposition 2.42. If u; v 2 Gloc .X /, then

Gu D Gv a.e. in fx 2 X W u.x/ D v.x/g: In particular, if c 2 R is a constant, then Gu D 0 a.e. in fx 2 X W u.x/ D cg. Proof. This follows directly from Corollary 2.21 applied to uk and vk . p Lemma 2.43. Assume that .fx 2 X W ju.x/j D 1g/ D 0. Then u 2 Dloc .X / if and p p only if u 2 Gloc .X / and Gu 2 Lloc .X /. In this case gu D Gu a.e.

Proof. Let uk D maxfminfu; kg; kg be the truncations at levels ˙k, k D 1; 2; ::: . p 1;p p Assume first that u 2 Dloc .X /. Then uk 2 Nloc .X / and hence u 2 Gloc .X /. p Moreover, gu D limk!1 guk a.e. in X , and hence Gu D gu 2 Lloc .X /. p .X / and Gu 2 Lploc .X /, then u has a p-weak upper gradient Conversely, if u 2 Gloc p p in Lloc .X /, and hence belongs to Dloc .X /. Note that if X satisfies a p-Poincaré inequality, then one implication follows unconditionally in Lemma 2.43, by Corollary 4.11. This is not true in general by Examp ple 1.22, as in that case u  1 2 D p .X / n Gloc .X / (except in the trivial case when  is the zero measure). p zloc One can build a (sometimes) more general theory by saying that u 2 G .X / if fx 2 X W ju.x/j D 1g can be written as a union E1 [ E2 with .E1 / D 0 and 1;p .X / for every positive Modp . E2 / D 0, and if uk WD maxfminfu; kg; kg 2 Nloc p p z k 2 Z. In this case one obtains the inclusion Dloc .X /  Gloc .X / unconditionally. We p have refrained from this, and also from developing the theory of Gloc further.

2.9 Dependence on p in gu 1;q Assume that 1  p < q. If u 2 Nloc .X /, then Hölder’s inequality shows that 1;p q u 2 Nloc .X /. Indeed if g 2 Lloc .X / is an upper gradient of u, then g 2 Lploc .X /. Even more is true. q p .X /. Then u 2 Dloc .X / and Proposition 2.44. Let 1  p < q and let u 2 Dloc

gu;p  gu;q a.e.; where gu;p and gu;q are the minimal p-weak and q-weak upper gradients of u, respectively.

60

2 Minimal p-weak upper gradients

To prove this we will use the following result. Proposition 2.45. Let 1  p < q and assume that Modq . / D 0. Then Modp . / D 0. Proof. By Proposition 2.33, there is a nonnegative function 2 Lqloc .X / such that R p  ds D 1 for all  2 . By Hölder’s inequality, 2 Lloc .X /, and hence, by Proposition 2.33 again, Modp . / D 0. Proof of Proposition 2.44. Let

D f 2 .X / W gu;q is not an upper gradient of u along  g: Then Modq . / D 0, by Lemma 1.40. By Proposition 2.45, Modp . / D 0, and hence gu;q is a p-weak upper gradient of u. Moreover, gu;q 2 Lqloc .X /  Lploc .X /. Thus the minimality of gu;p shows that gu;p  gu;q a.e. Another consequence of Proposition 2.45 is the following result. Proposition 2.46. Let 1  p < q. If Cq .E/ D 0, then Cp .E/ D 0. Proof. By Proposition 1.48, .E/ D Modq . E / D 0. Hence, by Proposition 2.45, Modp . E / D 0. Thus, Proposition 1.48 shows that Cp .E/ D 0. It is easy to find an example of a function u 2 N 1;p .X / such that u; gu;p 2 Lq .X /, but still u … N 1;q .X /. Example 2.47. Let 1  p < q and let E be a set such that 0 D Cp .E/ < Cq .E/, e.g. let 1 < p  n < q and let E be a singleton in Rn . Other examples of sets such that 0 D Cp .E/ < Cq .E/ can easily be found among Cantor type sets in Rn , see Theorem 5.3.2 in Adams–Hedberg [5] for a criterion when a certain capacity of such sets vanishes. Let u D E . Then kukLp .X/ D kukLq .X/ D 0 and 0 is a p-weak upper gradient of u, so kukN 1;p .X/ D 0. On the other hand, if it were true that u 2 N 1;q .X /, then as u D 0 a.e. we would have u D 0 q-q.e., by Proposition 1.59, which is clearly false. q .X /. Note however that Hence u … N 1;q .X /, and in fact u … ACCq .X / and u … Gloc u 2 Ny 1;q .X /. The problem in this example is that we have chosen a representative of an equivalence class (the 0 equivalence class) which is bad for N 1;q .X /. Let us give another example, of quite different nature. Example 2.48 (Bow-tie). Let n  2 and write x 2 Rn as x D .x1 ; ::: ; xn /. Let XC D fx 2 Rn W xj  0; j D 1; ::: ; ng; X D fx 2 Rn W xj  0; j D 1; ::: ; ng: Let further X D XC [ X , u D XC and 1  p  n < q.

2.10 Representation formulas for gu

61

In Examples 5.6 and A.23 we show that u 2 N 1;p .X /, but u … N 1;q .X /. Moreover gu;p D 0. So despite the fact that u; gu;p 2 Lq .X / (and even u; gu;p 2 L1 .X /), it does not follow that u 2 N 1;q .X / nor that u 2 Ny 1;q .X /. (For the latter, assume that u 2 Ny 1;q .X /. Then there is v 2 N 1;q .X / such that v D u a.e. It follows that gv D 0, which violates the q-Poincaré inequality obtained in Example A.23.) After these examples it is natural to ask the following question. Open problem 2.49. Let 1  p < q and assume that u 2 N 1;q .X /. Does it follow that gu;p D gu;q ? A positive answer to Open problem 2.53 would solve this. In Corollary A.9 we show that the answer is yes, if X is a complete doubling p-Poincaré space (see Section 4.1).

2.10 Representation formulas for gu We finish this chapter with some representation formulas for minimal p-weak upper gradients. For these to hold, or rather for our proofs, we need to assume that Lebesgue’s differentiation theorem holds, i.e. that « lim f d D f .x0 / for a.e. x0 2 X r!0 B.x ;r/ 0

whenever f 2 L1loc .X /. Observe that the left-hand side does not make sense if x0 … supp . On the other hand, the set of all such points has measure zero, so the equality may still be true a.e. Lebesgue’s differentiation theorem is known to hold when  is doubling, see Chapter 3, and more generally when the Vitali covering theorem holds. In particular, this is true if X D Rn , equipped with any measure  satisfying our standard assumption that .B/ < 1 for all balls B  X . See Heinonen [169], Chapter 1, for these facts and some further discussion, and the notes to Chapter 1 for the formulation of the Vitali covering theorem. In fact, Lebesgue’s differentiation theorem is equivalent to the validity of a weaker covering theorem, see Hayes–Pauc [161]. Preiss [306] gave a counterexample of a Gaussian measure on a separable Hilbert space for which the Lebesgue’s differentiation theorem does not hold. See also the discussion and references in Tišer [334]. Theorem 2.50. Assume that Lebesgue’s differentiation theorem holds on X . Let u 2 D p .X /. Then gu can be represented by the formulas « 8
62

2 Minimal p-weak upper gradients

and

8 ˆ <

g2 .x/ WD

inf lim sup g r!0 ˆ :0;

«

1=p p

g d

; x 2 supp ;

B.x;r/

x … supp ;

where the infima are taken over all p-weak upper gradients g 2 Lp .X / of u. Let further g10 and g20 be defined similarly but taking the infima only over all upper gradients g 2 Lp .X / of u. Then gu D g1 D g2 D g10 D g20 a.e., and all of them are minimal p-weak upper gradients of u. Proof. Without loss of generality we may assume that supp  D X . Let «  gu .x/ WD lim sup gu d: r!0

B.x;r/

As gu has Lebesgue points a.e. we have gu D gu a.e. Moreover, for all p-weak upper gradients g 2 Lp .X / of u, «  g d: gu .x/  lim sup r!0

B.x;r/

Hence gu  g1  g10 . Conversely, the proof of Lemma 1.46 provides us with a function 2 Lp .X / such that gj WD gu C =j 2 Lp .X / are upper gradients of u. Thus for j D 1; 2; ::: , « « 1 0  g1 .x/  lim sup gj d  gu .x/ C lim sup d: j r!0 B.x;r/ r!0 B.x;r/ Letting j ! 1 shows that g10  gu a.e. The proof that gu D g2 D g20 a.e. is similar. The representation formulas from Theorem 2.50 can also be extended to local spaces. Theorem 2.51. Assume that Lebesgue’s differentiation theorem holds on X . Let u 2 p Dloc .X /. Then gu can be represented by the following formulas: « 8
2.10 Representation formulas for gu

63

where the infimum is taken over all p-weak upper gradients g 2 Lploc .X / of u; and 8 « 1=p ˆ < p inf lim sup g d ; x 2 supp ; g3 .x/ WD g r!0 (2.11) B.x;r/ ˆ :0; x … supp ; where the infimum is taken over all p-weak upper gradients g of u. Let further gj0 , j D 1; 2; 3, be defined similarly but taking the infima only over upper gradients g (belonging to Lploc .X / for g10 and g20 ) of u. Then gu D g1 D g2 D g3 D g10 D g20 D g30 a.e., and all of them are minimal p-weak upper gradients of u. Proof. Without loss of generality we may assume that supp  D X . That gu D g1 D g10 D g2 D g20 a.e. is obtained as in the proof of Theorem 2.50. As g3  g30  g20 it is enough to show that gu  g3 a.e. Let g be a p-weak upper gradient of u, and define « 1=p  p g d : g .x/ D lim sup r!0

B.x;r/

Then G D fx 2 X W g 2 Lp .B.x; r// for some r > 0g is open, and g   1 in X n G. Since gu jG is a minimal p-weak upper gradient of u with respect to G, by Lemma 2.23, and g 2 Lploc .G/, we have gu  g a.e. in G. Hence gu  g 

in G:

As g   1 in X n G, we see that gu  g  in X . It follows that gu  g3 in X . Since gu D gu a.e., this completes the proof. It is natural to ask the following question. Open problem 2.52. Are the representation formulas in Theorems 2.50 and 2.51 true even when Lebesgue’s differentiation theorem does not hold? Note that in (2.11) we do not require g to be locally Lp -integrable. This leads us to the following open problem. p .X / and p > 1. Is it true, with or without the Open problem 2.53. Let u 2 Dloc assumption that Lebesgue’s differentiation theorem holds, that if g is a (p-weak) upper gradient of u, then «

gu  lim sup r!0

g d

a.e.?

B.x;r/

If this is true it would allow the infimum in (2.9) to be taken over all (p-weak) upper gradients of u, not only those belonging to Lp .X / or Lploc .X /. Observe that for p D 1 this is possible by formula (2.11), at least if Lebesgue’s differentiation theorem holds. This would also give a positive answer to Open problem 2.49, at least under the assumption that Lebesgue’s differentiation theorem holds. Unfortunately, the formulas above are not enough to answer Open problem 2.49.

64

2 Minimal p-weak upper gradients

2.11 Notes Fuglede’s lemma (Lemma 2.1) goes back to Fuglede [128]. It was observed in Shanmugalingam [319] that the proof by Fuglede remains valid in metric spaces. Proposition 2.3 is from Björn–Björn–Parviainen [54]. Theorem 2.5 was obtained by Shanmugalingam [320] for p > 1 and by Hajłasz [151] for general p. Open problem 2.13 is due to Marola [263]. Lemma 2.18 (in a slightly different version) is from Kinnunen–Martio [217], this may also be the first use of the calculus for p-weak upper gradients, as described in Section 2.3, in the literature. Lemma 2.19 and Corollaries 2.20 and 2.21 are from Björn–Björn [44] (but are not included in Björn–Björn [45]) and were motivated by results in Heinonen–Koskela–Shanmugalingam–Tyson [177]. For earlier versions see Shanmugalingam [320]. Proposition 2.22 is probably new in the generality of Newtonian spaces. Lemma 2.23 is folklore. It has been generalized to certain nonopen sets in Björn–Björn [49]. Theorem 2.36 is from Shanmugalingam [320]. Lemma 2.37 is from [49], for open E it appeared in [45]. Farnana [120] has also a version of x and Lemma 2.37 when E is open and v and w are merely quasicontinuous on  v D w D 0 q.e. on @E. The minimal p-weak upper gradients Gu were introduced for superharmonic u by Kinnunen–Martio [219]. Their definition is well adapted to superharmonic functions, but does not apply e.g. to subharmonic functions. Here we have given a more general definition which directly applies to both sub- and superharmonic functions. Neverthep less we will only apply it, in Section 9.9, to superharmonic functions. The spaces Gloc may be called gradient spaces, and this is the reason for our choice of G in the name. Proposition 2.46 is well known to people in the field, but was probably first explicitly written down for Newtonian spaces by Färm [118], the proof here being different. Example 2.48 is from Kinnunen–Korte–Shanmugalingam–Tuominen [211]. They also showed that if X is a complete doubling p-Poincaré space, u 2 N 1;p .X / and u; gu;p 2 Lq .X /, q > p, then u 2 Ny 1;q .X /. Formula (2.8) in Theorem 2.50 is from J. Björn [63], Lemma 2.3. Formulas (2.10) and (2.11) in Theorem 2.51 seem to be new here.

Chapter 3

Doubling measures

Throughout this chapter we assume that  is doubling with doubling constant C , see below. So far we have developed the theory of Newtonian spaces with no special assumptions on the underlying metric space. To get a richer theory it will be essential to make some further assumptions on the metric space. There are three main assumptions that we will eventually add: completeness of X , that  is doubling, and that X supports a p-Poincaré inequality. In the nonlinear potential theory developed in the second part of the book (in Chapters 7–14) these three assumptions will all be assumed. In this chapter we introduce the assumption that  is doubling. Particular consequences are Lebesgue’s differentiation theorem, and the Lp boundedness of the maximal function (for the former see Heinonen [169], pp. 4, 12–13, and for the latter see Section 3.2). A deeper consequence of doubling is the John–Nirenberg lemma and its connection with functions of bounded mean oscillation (BMO functions), see Sections 3.3 and 3.4. The John–Nirenberg lemma will be essential for us in our proof of the Harnack inequality for p-harmonic functions. We end this chapter by giving a proof of the Gehring lemma, an important result which however will not be used in the rest of the book.

3.1 Doubling The measure  is said to be doubling if there exists a constant C  1, called the doubling constant of , such that for all balls B, 0 < .2B/  C .B/ < 1: A metric space is doubling if there exists a constant C < 1 such that every ball B.z; r/ can be covered by at most C balls with radii 12 r. Alternatively and equivalently, for every " > 0 there is a constant C."/ such that every ball B.z; r/ can be covered by at most C."/ balls with radii "r. It is now easy to see that every bounded set in a doubling metric space is totally bounded. A metric space equipped with a doubling measure is doubling, and conversely any complete doubling metric space can be equipped with a doubling measure. See Heinonen [169], Section 10.13, for more on doubling metric spaces. Sections 3.3–3.5 were written with much help from Outi Elina Kansanen and Juha Kinnunen who provided draft versions of them.

66

3 Doubling measures

The following proposition is well known. However, since it does not seem to appear explicitly in the literature, we give a short proof here. Proposition 3.1. Let Y be a doubling metric space. Then Y is proper if and only if Y is complete. Recall that a metric space is proper if all closed and bounded subsets are compact, see Definition 2.30. Let us first show that one direction holds without doubling. Proposition 3.2. Let Y be a proper metric space. Then Y is complete. Proof. Take a Cauchy sequence fxi g1 iD1 in Y . Then for a sufficiently large radius r > 0, xi 2 B.x1 ; r/  Y (where B.x1 ; r/ is a ball in Y ). By the properness of Y , B.x1 ; r/ is compact and the sequence has a limit in Y . Proof of Proposition 3.1. The necessity follows from Proposition 3.2. Conversely, let Y be complete and M  Y be closed and bounded. Then M is totally bounded, and hence compact, see e.g. Rudin [312], Theorem A4, or Folland [124], Theorem 0.25. The following lemma provides us with a counterpart of dimension related to the measure. Note that if (3.1) holds with some s, then it holds also with all s 0  s. Lemma 3.3. Let B.x; R/ be a ball in X , y 2 B.x; R/ and 0 < r  R < 1. Then  r s .B.y; r// ; (3.1) C .B.x; R// R where s D log2 C and C D C2 . The choice s D log2 C may not be optimal. Note also that if (3.1) is satisfied, then  is doubling, so that  is doubling if and only if there is an exponent s such that (3.1) holds (for all balls B.y; r/ and B.x; R/ as above). Proof. By the assumptions, there exists a nonnegative integer k such that 2k1 r < R  2k r. Then B.x; R/  B.y; 2kC1 r/ and hence .B.x; R//  CkC1 .B.y; r//: This and the choice of k imply that  r s .B.y; r// :  Ck1 D C2 .21k /log2 C > C2 .B.x; R// R The (unweighted) Euclidean space Rn is doubling with the doubling constant 2n and the best exponent in (3.1) is s D n. This suggests that the exponent s in (3.1) is a generalization of the notion of dimension to metric spaces. For some results, an opposite inequality is needed, which is indeed true in connected spaces, see Corollary 3.8. In general, one cannot expect the exponents in the lower and upper estimates of the measure to coincide. This leads us to the following definition, which is clearly stronger than requiring that  be doubling.

3.1 Doubling

67

Definition 3.4. The measure  is said to be Ahlfors Q-regular if there exist constants C1 > 0 and C2 such that C1 r Q  .B.x; r//  C2 r Q for all x 2 X and all r > 0. Ahlfors Q-regular sets are sometimes called Ahlfors–David Q-regular, or simply Q-sets in the literature. Example 3.5. Let X be weighted Rn , n  2, equipped with the measure d.x/ D jxj˛ dx, ˛ > n. It is not difficult to verify that  is doubling, see Appendix A.2 for more details. For x ¤ 0, .B.x; r// is comparable to r n with comparison constants depending on jxj, and .B.0; r// is comparable to r nC˛ . It follows that (3.1) holds with s  maxfn; n C ˛g, while (3.2) below holds with   minfn; n C ˛g. Thus, X is not Ahlfors regular (unless ˛ D 0). Note that for ˛ close to n we have  close to 0, and for large ˛, s becomes large. The following two simple lemmas will be useful for further references. Lemma 3.6. Let B D B.x; r/ and B 0 D B.x 0 ; r 0 / be two balls such that d.x; x 0 /  ar and r=a  r 0  ar. Then .B/=C  .B 0 /  C.B/ with the comparison constant depending only on a > 1 and the doubling constant C . Proof. Clearly, B 0  2aB and hence .B 0 /  CN .B/, where N is the smallest integer such that 2a  2N . Similarly, as d.x; x 0 /  a2 r 0 , we have B  .a2 C a/B 0 and hence .B/  C2N 1 .B 0 /, since a2 C a  22.N 1/ C 2N 1 < 22N 1 . Lemma 3.7. Assume that X is connected. For every 0 <  < 1, there exists a constant 0 <  < 1 depending only on  and C such that if B D B.x; r/ is a ball in X with radius r < 12 diam X , then .B/  .B/: Proof. As X is connected, there is y 2 X with d.y; x/ D 12 .1 C  /r. Hence B n B   contains the ball B 0 D B y; 12 .1  /r , and B  B 0 , where  D 4=.1   /. In particular, .B/  CN .B 0 /; where N is the smallest integer such that   2N . Thus .B/  .B/  .B 0 /  .1  CN /.B/. Corollary 3.8. Assume that X is connected. Then there exist constants C > 0 and  > 0 such that for all balls B.x; R/ in X , all y 2 B.x; R/ and all 0 < r  R,  r  .B.y; r// C : .B.x; R// R

(3.2)

68

3 Doubling measures

Proof. As B.y; R/  B.x; 2R/, the doubling property and an iteration of Lemma 3.7 with  D 1=e immediately yield .B.y; r// .B.y; r// C  r  log  C  C  k  ; .B.x; R// .B.y; R//  R where k is the smallest integer such that r  e k R. Corollary 3.9. Assume that X is connected. Then .fxg/ D 0 for every x 2 X . The space X D Z, with .fxg/ D 1 for all x 2 X , shows that we cannot omit the connectedness assumption.

3.2 The maximal function Rather than considering the usual maximal function, we introduce the noncentred maximal function which is often easier to use. We also make the definition more flexible by considering the restricted maximal function, which will be useful later. The nonrestricted maximal function is obtained by letting  D X . Definition 3.10. For f 2 L1 ./, the noncentred maximal function restricted to  is «  M f .x/ WD sup jf j d; B

B

where the supremum is taken over all balls B   containing x. If  D X , we write M  f instead of MX f . Note that the balls considered are balls with respect to X . Remark 3.11. The noncentred maximal function M  f differs somewhat from the usual centred Hardy–Littlewood maximal function « Mf .x/ WD sup jf j d: r>0 B.x;r/

Note, however, that if  is doubling, then Mf  M  f  CMf . This implies that both Lemma 3.12 and Theorem 3.13 below hold for Mf as well (the proof of the lower semicontinuity is a bit more involved for the centred maximal function). Lemma 3.12. For f 2 L1 ./, the noncentred maximal function M f .x/ is lower semicontinuous in  and satisfies Z C3 .E /  jf j d and lim  .E / D 0;

!1  E where E D fx 2  W M f .x/ >  g.

3.2 The maximal function

69

 Proof. Consider first the further restrained noncentred maximal function M;R f in which the supremum is taken only over balls in  with radius less than R. Let  E R D fx 2  W M;R f .x/ >  g:

By definition, for every x 2 E R , there exists a ball Bx 3 x, Bx  , with radius less than R such that Z jf j d: .Bx / < Bx

Note that Bx 

E R

since for every z 2 Bx , «  jf j d > : M;R f .z/  Bx

Hence E R D

[

Bx ;

x2ER  which shows that E R is open and M;R f is lower semicontinuous. By the 5-covering lemma (Lemma 1.7) there exist pairwise disjoint balls Bxj  E R , j D 1; 2; ::: , such S that E R  j1D1 5Bxj . The doubling property and the choice of Bxj imply Z 1 1 X X R 3 3  .E /   .5Bxj /  C  .Bxj /  C jf j d  C: j D1

j D1

ER

 f is lower semicontinuous and that the constant Note that M f D supR>0 M;R C above does not depend on R. Letting R ! 1 proves the first inequality in the 1 statement of the lemma and shows that .E / ! R 0, as  ! 1. As f 2 L ./, the dominated convergence theorem implies that E jf j d ! 0, as  ! 1, which finishes the proof.

The first inequality in Lemma 3.12 says that the operator M  is bounded from L .X / to weak L1 .X / (which is sometimes denoted by L1;1 .X / in the range of Lorentz spaces). The following Hardy–Littlewood theorem shows that for t > 1, the operator M  is bounded on Lt .X /. 1

Theorem 3.13. Let t > 1. If f 2 Lt .X /, then M  f 2 Lt .X / and Z Z 2t t C3  t .M f / d  jf jt d: t 1 X X Proof. For  > 0, let f D f fx2XWjf .x/j> =2g . Then M  f  M  f C =2 and Lemma 3.12 implies that .fx 2 X W M  f .x/ >  g/  .fx 2 X W M  f .x/ > =2g/ Z 2C3  jf j d:  fx2XWjf .x/j> =2g

70

3 Doubling measures

Hence, by Cavalieri’s principle (Lemma 1.10), Z 1 Z .M  f /t d D t  t1 .fx 2 X W M  f .x/ >  g/ d  X 0 Z 1 Z  2t C3  t2 jf j d d  0

Z D 2t C3 D

2t t C3 t 1

fx2XWjf .x/j> =2g

Z

2jf j

jf j X

 t2 d  d 0

Z

jf jt d: X

3.3 BMO and John–Nirenberg’s lemma John–Nirenberg’s lemma and the space of BMO functions, both dating back to the 1961 paper of John and Nirenberg [192], are cornerstones in modern analysis. For us they will play a crucial role when obtaining the interior regularity for p-harmonic functions, more precisely when proving the weak Harnack inequality for superminimizers in Theorem 8.10. Definition 3.14. Let

« kf kBMO.IX/ D sup

jf  fB j d;

B B

ª where fB WD B f d is the mean-value of f on B and the supremum is taken over all balls B   (and we implicitly require that fB is finite for all balls B  ). We define the class of functions of bounded mean oscillation as BMO.I X / D ff W kf kBMO.IX/ < 1g: Note that the balls considered in Definition 3.14 are balls with respect to X . To emphasize this dependence on X , and to avoid misunderstanding, we write X explicitly in the notation. Clearly BMO.I X /  L1loc ./. Theorem 3.15 (John–Nirenberg’s lemma). Let B0 D B.x0 ; R/  X be a ball and f 2 BMO.5B0 I X /. Then for every  > 0, .fx 2 B0 W jf .x/  fB0 j > g/  2.B0 /e A=kf kBMO.5B0 IX/ ; where A D log.2/=4C15 only depends on the doubling constant.

71

3.3 BMO and John–Nirenberg’s lemma

Remark 3.16. A ball in a metric space may not have a unique centre and radius. For instance if X D Œ10; 10  R, then B.5; 6/ D B.6; 7/. Moreover, even if we fix the centre, the radius may not be unique as e.g. X D B.0; r/ for every r > 10 (if X D Œ10; 10), and in this case there is not even an optimal (minimal) radius that can be used. This nonuniqueness does play a role when blowing up (or down) a ball. So that e.g. even if B.x; r/ D B.y; s/ it can happen that 2B.x; r/ D B.x; 2r/ ¤ 2B.y; s/ D B.y; 2s/. For this reason we always require a ball to have a fixed centre and radius, even when none are given explicitly. In certain proofs, such as the one below, when e.g. finding a ball B.x; r/ within a larger ball B.y; s/  B.x; r/ we require, without loss of generality, that r  2s. Note that in general it is not possible to require that r  s, unless x D y, e.g. if X D Œ9; 10 Œ1; 1, then B..8; 0/; 17/  B..0; 0/; 10/ and both balls have unique centres and radii. Proof. Note that as f5B0 is finite and f 2 BMO.5B0 I X /, we have f 2 L1 .5B0 /. Define the (noncentred) maximal function restricted to 5B0 as «  jf  fB0 j d M0 f .x/ D M5B0 .f  fB0 /.x/ D sup B3x B5B0

B

and let G0 D fx 2 5B0 W M0 f .x/ > 0 g; 4C12 kf

kBMO.5B0 IX/ . Applying Lemma 3.12 to jf  fB0 j and 5B0 in where 0 D place of f and  yields Z C3 jf  fB0 j d .G0 /  0 5B0 Z  Z C3  jf  f5B0 j d C jf5B0  fB0 j d : 0 5B0 5B0 For the first term in the sum on the right-hand side we have Z jf  f5B0 j d  .5B0 /kf kBMO.5B0 IX/  C3 .B0 /kf kBMO.5B0 IX/ : 5B0

In the second term the integrand is constant so that ˇ« ˇ Z Z ˇ ˇ .5B0 / ˇ ˇ jf5B0  fB0 j d D .5B0 /ˇ .f  f5B0 / dˇ  jf  f5B0 j d .B0 / 5B0 5B0 B0  C3 .5B0 /kf kBMO.5B0 IX/  C6 .B0 /kf kBMO.5B0 IX/ : By combining the obtained estimates we arrive at .B0 \ G0 /  .G0 / 

2C9 kf kBMO.5B0 IX/ 0

.B0 / 

1 .B0 / < .B0 /; 2

(3.3)

72

3 Doubling measures

from which we conclude that B0 n G0 ¤ ¿. Note that G0 is open by Lemma 3.12. For x 2 B0 define r.x/ D

1 20

dist.x; X n G0 /:

1 Since B0 n G0 ¤ ¿, we have r.x/  10 R and hence B.x; 5r.x//  5B0 \ G0 for all x 2 B0 \ G0 . As [ B.x; r.x//; B0 \ G0  x2B0 \G0

the 5-covering lemma (Lemma 1.7) provides us with countably many pairwise disjoint balls Bzi D B.xi ; r.xi //, i D 1; 2; ::: , such that B 0 \ G0 

1 [

5Bzi  5B0 \ G0 :

(3.4)

iD1

As 25r.xi / > dist.xi ; X n G0 /, we have that 25Bzi n G0 ¤ ¿ for every i D 1; 2; ::: . Note that even though the balls 25Bzi intersect the complement of G0 , we still have 25Bzi  5B0 . This implies that M0 f .x/  0 and, in particular,

for x 2 25Bzi n G0

« zi 25B

jf  fB0 j d  0

for every i:

We denote the balls obtained in the first step of the iteration by Bi D 5Bzi , i D 1; 2; ::: . Then ˇ« ˇ « ˇ ˇ .25Bzi / ˇ jfB0  fBi j D ˇ .f  fB0 / dˇˇ  jf  fB0 j d  C3 0 : (3.5) zi .5Bzi / 25Bzi 5B As Bzi  G0 are pairwise disjoint, we have by (3.3) that 1 X iD1

.Bi / D

1 X

.5Bzi /  C3

iD1

 C3 .G0 / 

1 X

.Bzi / D C3 

iD1 12 2C kf kBMO.5B0 IX/

0

S  (We of course also get the estimate  1 iD1 Bi  stronger estimate above.) S As B0 \ G0  1 iD1 Bi , we have that M0 f .x/  0

for x 2 B0 n

1 [

Bzi



iD1

.B0 / D 1 .B0 /, 2

1 [ iD1

Bi :

1 .B0 /: 2

(3.6)

but we will need the

3.3 BMO and John–Nirenberg’s lemma

In particular,

73

« B

jf  fB0 j d  0

S for all balls B  5B0 which contain some x 2 B0 n 1 iD1 Bi . Hence, by Lebesgue’s differentiation theorem, « 1 [ jf .x/  fB0 j D lim jf  fB0 j d  0 for a.e. x 2 B0 n Bi : (3.7) diam B!0 B

iD1

We would now like to repeat the same argument in every ball Bi . Let «  Mi f .x/ D M5B .f  f /.x/ D sup jf  fBi j d Bi i B

B3x B5Bi

be the noncentred maximal function restricted to 5Bi , Gi D fx 2 5Bi W Mi f .x/ > 0 g

and ri .x/ D

1 20

dist.x; X n Gi / for x 2 Bi :

Recall that 5Bi  5B0 , so that kf kBMO.5Bi IX/  kf kBMO.5B0 IX/ . Repeating the above argument in each ball Bi1 with Mi1 f , Gi1 and ri .x/ in place of M0 f , G0 and r.x/, we obtain the balls Bi1 ;i2 in the second step of the iteration for which (as in (3.4)–(3.7)), 5Bi1 ;i2  5Bi1 ; 1 [

B i1 \ G i1 

Bi1 ;i2  5Bi1 \ Gi1 ;

i2 D1

jfBi1  fBi1 ;i2 j  C3 0 ; 1 X

.Bi1 ;i2 / 

i2 D1

1 .Bi1 / 2

and jf .x/  fBi1 j  0

for a.e. x 2 Bi1 n

1 [

(3.8)

Bi1 ;i2 :

i2 D1

Moreover, from (3.6) and (3.8) we obtain that 1 X i1 ;i2 D1

.Bi1 ;i2 / 

1 .B0 / 4

and by (3.5) and (3.9) we have jf .x/  fB0 j  jf .x/  fBi1 j C jfBi1  fB0 j  0 C C3 0  2C3 0

(3.9)

74

3 Doubling measures

for a.e. x 2 Bi1 n

S1

i2 D1

Bi1 ;i2 . Consequently,

jf .x/  fB0 j  2C3 0

for a.e. x 2 B0 n

1 [

Bi1 ;i2 :

i1 ;i2 D1

In the kth step of the iteration we obtain balls Bi1 ;:::;ik for which jf .x/  fB0 j  kC3 0

1 [

for a.e. x 2 B0 n

Bi1 ;:::;ik

i1 ;:::;ik D1

and

1 X

.Bi1 ;:::;ik /  2k .B0 /:

i1 ;:::;ik D1

Let now  > 0 be arbitrary. If  > C3 0 , we choose a positive integer k such that kC3 0   < .k C 1/C3 0 : In particular, k>

 C3 0

1

3

and 2k < 2e  log.2/=C 0 :

This implies that jf .x/  fB0 j  kC3 0  

1 [

for a.e. x 2 B0 n

Bi1 ;:::;ik :

i1 ;:::;ik D1

Consequently, we have .fx 2 B0 W jf .x/  fB0 j > g/  



1 [

 Bi1 ;:::;ik 

i1 ;:::;ik D1 k

2

.B0 / < 2e

1 X

.Bi1 ;:::;ik /

i1 ;:::;ik D1 3  log.2/=C 0

.B0 /:

If 0 <   C3 0 , then we use the trivial estimate 3

.fx 2 B0 W jf .x/  fB0 j > g/  .B0 /  2e  log.2/=C 0 .B0 /: Hence for any  > 0 we have 3

.fx 2 B0 W jf .x/  fB0 j > g/  2e  log.2/=C 0 .B0 /  2.B0 /e A=kf kBMO.5B0 IX/ ; where A D log.2/kf kBMO.5B0 IX/ =C3 0 D log.2/=4C15 depends only on the doubling constant.

3.4 Consequences of John–Nirenberg’s lemma

75

3.4 Consequences of John–Nirenberg’s lemma In this section, we collect a few consequences of the John–Nirenberg lemma (and the Cavalieri principle), which will be useful for future references. We start by comparing BMO with its Lp -generalization BMOp , defined below. Definition 3.17. Let

«

kf kBMOp .IX/ D sup

1=p jf  fB jp d

;

B

B

where the supremum is taken over all balls B   (and we implicitly require that fB is finite for all balls B  ). We also let BMOp .I X / D ff W kf kBMOp .IX/ < 1g: Lemma 3.18. BMOp .I X /  BMO.I X / and if f 2 BMO.I X /, then kf kBMO.IX/  kf kBMOp .IX/ : Proof. This follows directly from Hölder’s inequality as « 1=p « p jf  fB j d  jf  fB j d : B

B

It is an interesting consequence of the John–Nirenberg lemma that BMOp is essentially independent of p. Proposition 3.19. Let f 2 BMO.11BI X / for some ball B D B.x0 ; r0 /. Then there is a constant c > 0, depending only on C and p, such that kf kBMO.BIX/  kf kBMOp .BIX/  ckf kBMO.11BIX/ : Proof. The first inequality follows from Lemma 3.18. We use the John–Nirenberg lemma (Theorem 3.15) to prove the second inequality. Let B 0 D B.x 0 ; r 0 /  B, r 0  2r0 , be an arbitrary ball in B. Note that 5B 0  11B. By the Cavalieri principle (Lemma 1.10), we have using the change of variable  D skf kBMO.11BIX/ =A, where A is the constant in the John–Nirenberg lemma, « Z 1 p p 0 jf  fB j d D p1 .fx 2 B 0 W jf .x/  fB 0 j > g/ d  0/ 0 .B B Z 10  2p p1 e A=kf kBMO.5B 0 IX/ d  0 Z 1  2p p1 e A=kf kBMO.11BIX/ d  0   Z kf kBMO.11BIX/ p 1 p1 s s e ds  2p A 0 2p .p/ D kf kpBMO.11BIX/ : p A

76

3 Doubling measures

This implies that kf kBMOp .BIX/  ckf kBMO.11BIX/ ; where the constant c D .2p .p//1=p =A depends only on p and C . An even stronger consequence of the John–Nirenberg lemma is the following estimate for exponential integrability of BMO functions, which is often used in applications. In some literature, it is this estimate that is called the John–Nirenberg lemma. We shall use it (or rather its corollary) to “jump over zero” when proving the interior regularity of minimizers by means of the Moser iteration method in Chapter 8. Theorem 3.20. Let B  X be a ball and f 2 BMO.5BI X /. Let further A be the constant in the John–Nirenberg lemma (Theorem 3.15). Then for every 0 < " < A there is a constant c, only depending on " and C , such that «

e "jf fB j=kf kBMO.5BIX/ d  c:

B

Proof. By the Cavalieri principle (Lemma 1.10), we have for h W X ! Œ0; 1, using the change of variable t D e  , that « e h d D

1 .B/

Z

1

.fx 2 B W e h.x/ > t g/ dt 0 Z 1 1 C .fx 2 B W e h.x/ > t g/ dt .B/ 1 Z 1 1 1C e  .fx 2 B W h.x/ > g/ d : .B/ 0

B

This implies, using also John–Nirenberg’s lemma (Theorem 3.15), that « e "jf fB j=kf kBMO.5BIX/ d B Z 1 .fx 2 B W jf .x/  fB j > kf kBMO.5BIX/ ="g/ 1C e d .B/ 0 Z 1 1C2 e .1A="/ d  0

2 A="  1 AC" D ; A" D1C

which only depends on " and C .

3.5 Gehring’s lemma

77

Corollary 3.21. Let B  X be a ball and f 2 BMO.5BI X /. Then « e qjf fB j d  3; B

where q D log.2/=8C15 kf kBMO.5BIX/ . Proof. This follows directly from Theorem 3.20 after letting " D 12 A.

3.5 Gehring’s lemma The Gehring lemma is also of considerable importance, and we have chosen to include it here even though we do not actually need it in this book. Theorem 3.22. Let 1 < p < 1 and let f be a nonnegative function such that f 2 L1 .B/ for every ball B  X . If there exists a constant CH  1 such that f satisfies the reverse Hölder inequality « 1=p « p f d  CH f d for all balls B in X ; (3.10) B

2B

then there exists q > p such that 1=q « « f q d  Cq B

f d for all balls B in X :

(3.11)

2B

The constants Cq and q depend only on p, CH and the doubling constant C . In fact, we may take q D p C .p  1/=22pC1 CHp C3pC17 and Cq D 8CH C31 . It follows directly, from Hölder’s inequality, that if p < t < q, then (3.11) holds with q replaced by t (and C t D Cq ). Note that if X is complete, then the condition that f 2 L1 .B/ for every ball B  X is equivalent to saying that f 2 L1loc .X /, by Corollary 2.31 and Proposition 3.1. In fact, if we had defined L1loc as in A. Björn–Marola [59] this would have been true also in nonproper spaces, see also Remark 2.32. Proof. Let B0 D B.x0 ; r/   X be fixed but arbitrary. We start by constructing a Whitney type covering of B0 . For every x 2 B0 let r.x/ D

dist.x; X n B0 /:  Since the balls B x; 15 r.x/ , x 2 B0 , cover B0 , the 5-covering lemma (Lemma 1.7) provides us with countably many balls Bi D B.xi ; ri /, xi 2 B0 , ri D r.xi /, i D 1; 2; ::: , such that 1 [ Bi B0 D 

1 10

iD1

78

3 Doubling measures

and the balls 51 Bi are pairwise disjoint. Note that if x 2 5Bi , then dist.x; X n B0 /  dist.xi ; X n B0 / C d.x; xi / < 10ri C 5ri D 15ri ; and similarly dist.x; X n B0 /  dist.xi ; X n B0 /  d.x; xi / > 10ri  5ri D 5ri : If 5Bi \ 5Bj ¤ ¿, then, letting y 2 5Bi \ 5Bj , we have 5rj < dist.y; X n B0 / < 15ri and thus 13 ri < rj < 3ri . Moreover, if z 2 5Bj , then d.z; xi /  d.z; xj / C d.xj ; y/ C d.y; xi /  5rj C 5rj C 5ri < 35ri showing that 5Bj  35Bi . Fix x 2 S1Let us next show that the balls 5Bi , i D 1; 2; ::: , have bounded overlap. 1 5B and let I D fi W x 2 5B g. Let further j 2 I be such that r  sup i i j i2I ri . iD1 2 For i 2 I we find that   .5Bj /  .35Bi /  C8  15 Bi : Thus using the pairwise disjointness of the balls 15 Bi and that 15 Bi  5Bi  35Bj we find that   X X  15 Bi .35Bj / 8 1 C  C8  C11 : .5Bj / .5Bj / i2I

i2I

We introduce the auxiliary function h.x/ WD min

5Bi 3x

.5Bi / f .x/: .B0 /

Note that h  f in B0 , as 5Bi  B0 . Moreover, if x 2 12 B0 and x 2 5Bi , then r  dist.x0 ; X n B0 /  d.x0 ; x/ C d.x; xi / C dist.xi ; X n B0 / < 12 r C 5ri C 10ri ; and thus r < 30ri . If further y 2 B0 , then d.y; xi /  d.y; x0 / C d.x0 ; x/ C d.x; xi / < r C 12 r C 5ri < 50ri ; i.e. B0  50Bi . Hence h.x/ D min

5Bi 3x

.5Bi / .5Bi / f .x/ f .x/  min f .x/  5Bi 3x .50Bi / .B0 / C4

for x 2 12 B0 : (3.12)

3.5 Gehring’s lemma

79

Further, if B is a ball such that 2B  5Bi for some i , then by the reverse Hölder inequality (3.10) for f , «

1=p p

h d B

« 1=p .5Bi / p  f d .B0 / B « CH .5Bi /  f d .B0 / 2B « CH .5Bi /  h d min .5Bj / 2B 5Bj \5Bi ¤¿ « CH .35Bj /  max h d .5Bj / 5Bj \5Bi ¤¿ 2B « 3  CH C h d;

(3.13)

2B

i.e. h satisfies the reverse Hölder inequality on B with the constant CH C3 . Fix Bi D B.xi ; ri / and define the noncentred maximal function restricted to the ball 5Bi as «  h d: Mi h.x/ D M5Bi h.x/ D sup B3x B5Bi

B

Let i D inf Mi h Bi

and E D fx 2 5Bi W Mi h.x/ > g,  > 0, which is open by Lemma 3.12. For  > i and every x 2 E let .x/ D dist.x; 5Bi n E /: Note that .x/  2ri for x 2 Bi \ E as Bi n E ¤ ¿.   Since the balls B x; 15 .x/ , x 2 E \ Bi , cover E \ Bi , the 5-covering lemma (Lemma 1.7) provides us with countably many balls Bj0 D B.xj0 ; .xj0 //, xj0 2 E \Bi , S such that E \ Bi  j1D1 Bj0 and the balls 15 Bj0 are pairwise disjoint. It follows that 2Bj0 n E ¤ ¿, 2Bj0  5Bi and Bj0  E . Hence by (3.13) and the choice of .xj0 /, we have for every j D 1; 2; ::: that «

1=p p

Bj0

h d

« 

CH C3

2Bj0

h d  CH C3 :

80

3 Doubling measures

Thus .Bj0 /  CHp C3p p .Bj0 /, where d D hp d. From this we conclude that .E \ Bi / 

1 X

.Bj0 /  CHp C3p p

j D1

1 X

.Bj0 /

j D1

 CHp C3pC3 p

1 X



1 5

 Bj0  CHp C3pC3 p .E /:

j D1

  ˚ Next, writing h D h C h , where h D h  12  C and h D min h; 12  , we have ˚ Mi h  Mi h C 12  and hence E  x 2 5Bi W Mi h .x/ > 12  . Lemma 3.12 then implies that Z Z 2C3 2C3  .E /  h d  h d:  5Bi  fx25Bi Wh.x/>=2g 



For a fixed nonnegative integer k, let hk D minfh; kg and F D fx 2 5Bi W hk .x/ > g: For  < k, we have hk .x/ >  if and only if h.x/ >  and hence F D fx 2 5Bi W h.x/ > g: Since hk  h  Mi h a.e., the last two estimates imply that for  < k, .F \ Bi /  .E \ Bi /  CHp C3pC3 p .E / Z  C 0 p1 h d D C 0 p1 .F Q =2 /;

(3.14)

fx25Bi Wh.x/>=2g

where C 0 D 2CHp C3pC6 and d Q D h d. Since F is empty for   k, this estimate holds for all  > i . The Cavalieri principle (Lemma 1.10), with d D hp d, and (3.14) then yield for q > p, Z Z hq1 d Q  hqp d (3.15) k k F2i \Bi

F2i \Bi

Z

2i

D .q  p/

qp1 .F2i \ Bi / d  Z 1 C .q  p/ qp1 .F \ Bi / d  2i Z 1 qp1 .F \ Bi / d  D .2i /qp .F2i \ Bi / C .q  p/ 2i Z 1  .2i /qp C 0 .2i /p1 .F Q i / C C 0 .q  p/ q2 .F Q =2 / d  0 Z Z 2q1 C 0 .q  p/ q1 0  .2i / C h d C hq1 d : Q k q1 5Bi 5Bi 0

3.5 Gehring’s lemma

At the same time, Z Bi nF2i

hq1 d Q  .2i /q1 .B Q i / D .2i /q1 k

81

Z h d; Bi

which together with (3.15) implies that Z Z Z 2q1 C 0 .q  p/ q1 q 0 q1 hk h d  2 C i h d C hq1 h d: k q1 Bi 5Bi 5Bi

(3.16)

Now, since E \ Bi D Bi for every  < i , Lemma 3.12 shows that Z C3 .Bi /  h d for  < i :  5Bi Letting  ! i then yields that whenever i > 0 we have « « 3 .5Bi / 6 i  C h d  C f d; .Bi / 5Bi B0 which is clearly true also if i D 0. Inserting this into (3.16) and summing over i D 1; 2; ::: together with the bounded overlap C11 of the balls 5Bi gives « B0

hq1 h d  k

2q C 0 C6.q1/ .B0 /

« f d

q1 X 1 Z

B0

iD1

h d 5Bi

1 Z 2q1 C 0 .q  p/ X hq1 h d k .q  1/.B0 / 5B i iD1 q « « 2q C 000 .q  p/ 00 f d C hq1 h d C k q  1 B0 B0

C

(3.17)

< 1; as hk  k and h 2 L1 .B0 /. Here C 00 D 2q C 0 C6.q1/ C11 D 2qC1 CHp C3pC6qC11 and C 000 D 12 C 0 C11 D CHp C3pC17 . Choose now qDpC

p1 22pC1 C 000

:

Note that p < q < 2p and that 2q C 000 .q  p/ 22p .p  1/ 1 < 2pC1 < : q1 2 .q  1/ 2 Subtracting the second term on the right-hand side in (3.17) thus gives us q « « 00 hq1 h d  2C f d : k B0

B0

82

3 Doubling measures

Letting k ! 1 then yields «

hq d  2C 00

q

« f d

B0

:

B0

Using (3.12) and letting B D 12 B0 shows that « B

f q d  C4q

.2B/ .B/

«

hq d  Czq

«

2B

q f d

(3.18)

2B

for all balls B such that 2B ¤ X , where Czq D 2C4qC1 C 00 D 2qC2 CHp C3pC10qC12 .  finally a ball B with 2B D X and let R D diam X . As the balls  Consider 1 R , x 2 X , cover X , the 5-covering lemma (Lemma 1.7) gives us N pairwise B x; 15   S 1 R such that X D N disjoint balls Bi D B xi ; 15 iD1 5Bi (for the moment we may allow for N D 1, but we next give a bound for N ). As .X / D .16Bi /  C4 .Bi /; the disjointness shows that N 

N X C4 .Bi / iD1

.X /

 C4 :

Applying (3.18) to each ball 5Bi , i D 1; 2; ::: ; N , we obtain that « f d  q

B

« N X .5Bi /

N X .2B/

Czq

«

q

f d  f d .B/ 5Bi .B/ 10Bi iD1  q « q « N X .20Bi /  C Czq f d  NC1Cq Czq f d : .10Bi / 2B 2B q

iD1

iD1

Let finally Cq D 8CH C31 . Since 1 < p < q, we have that Cqq D 23q CHq C31q  NC1Cq Czq  Czq , which completes the proof.

3.6 Notes Doubling measures on R appear already in the famous paper by Beurling–Ahlfors [30], where they characterize boundary values of quasiconformal mappings in the upper half plane. The doubling condition makes it possible to prove various classical results such as inequalities for maximal functions and integral operators, Lebesgue’s differentiation theorem and John–Nirenberg’s and Gehring’s lemmas. A lot of analysis has been made

3.6 Notes

83

on so-called spaces of homogeneous type, i.e. on (quasi)metric spaces equipped with a doubling measure, see e.g. Coifman–Weiss [100], [101], Heinonen [169], Kronz [233], Pérez–Wheeden [303] and the references therein. The class BMO was introduced by John and Nirenberg in [192], where they also obtained the first version of the John–Nirenberg lemma. Metric space versions of the John–Nirenberg lemma have been obtained by Buckley [77], Mateu–Mattila–Nicolau– Orobitg [274] and Kronz [233]. Kronz actually obtained this result in the more general spaces of homogeneous type and with metric space targets. Recently, Hytönen [187] proved the John–Nirenberg lemma for certain nondoubling measures on metric spaces. Fix   1 and let ² ³ « BMO -loc .I X / D f W kf kBMO -loc .IX/ ´ sup jf  fB j d < 1 ;

B B

where the supremum is taken over all balls B such that B  . Assuming that X is a length space and that  is doubling, Maasalo [Kansanen] [252] showed that BMO.I X / D BMO -loc .I X /; this is not true for general metric spaces with doubling measures. BMO spaces for nondoubling measures on Rn and metric spaces have been studied by Tolsa [337] and Hytönen [187]. Gehring [139] gave the first version of the Gehring lemma. The Gehring lemma in metric spaces can be found (somewhat hidden) in Strömberg–Torchinsky [330], where they actually work in spaces of homogeneous type. There are also metric space versions of the Gehring lemma in Gianazza [140] (in spaces of homogeneous type) and Zatorska-Goldstein [357]. Martín–Milman [264] gave a proof of the Gehring lemma for nondoubling measures on Rn . The version here is quite close to the metric space version given by Maasalo [Kansanen] [251]. The reverse Hölder inequality in Rn characterizes the class of A1 -weights, which consists of all Ap -weights, 1  p < 1, see e.g. Coifman–Fefferman [99] and Stein [328]. It also implies the open-endedness of the Ap -condition (A.5). In one dimension, the reverse Hölder inequality is closely related to the theory of quasiminimizers, see Martio–Sbordone [273] and Appendix C. The discrete maximal function has been studied by Kinnunen–Latvala [212] and Aalto–Kinnunen [1] on metric spaces equipped with a doubling measure.

Chapter 4

Poincaré inequalities

Upper gradients can be introduced and studied on general metric spaces, as we have done in Chapters 1 and 2. In general, e.g. when there are no curves, the upper gradients of a function give no control of it. Requiring a Poincaré inequality is one possibility of gaining such a control. At the same time as being quite general, this also makes it possible to build a very rich theory. This will be apparent in the second part of the book, where a Poincaré inequality is a standing assumption. In this chapter we introduce different types of Poincaré inequalities and study their connections and consequences, most of the time assuming doubling as well. We shall also see that quasiconvexity and the inner metric are interconnected with the theory of Poincaré inequalities.

4.1 Poincaré inequalities Definition 4.1. Let q  1. We say that X supports a .q; p/-Poincaré inequality if there exist constants CPI > 0 and   1 such that for all balls B  X , all integrable functions u on X and all upper gradients g of u, 1=q

« ju  uB j d q

B

«

1=p

 CPI diam.B/

p

g d

;

(4.1)

B

ª where uB WD B u d. If  D 1, then X supports a strong .q; p/-Poincaré inequality. For this to make sense we also make the implicit assumption that .B/ > 0 for all balls B  X . In the literature, Poincaré inequalities with  > 1 are often called weak Poincaré inequalities, and Poincaré inequality is reserved for the case when  D 1, consequently the word “strong” is often omitted. For most of our results,  > 1 will be sufficient and we therefore simplify the terminology by omitting the word “weak” instead. A point worth mentioning is that the property of supporting a Poincaré inequality is preserved under biLipschitz transformations of X . On the contrary, strong Poincaré inequalities need not be preserved. See Section 4.3. We shall often say p-Poincaré inequality, instead of .1; p/-Poincaré inequality. Further, we say that a space X is a p-Poincaré space if it supports a p-Poincaré inequality, and, somewhat incorrectly, that X is a doubling p-Poincaré space if also  is doubling.

4.1 Poincaré inequalities

85

By the Hölder inequality, it is easy to see that if X supports a .q; p/-Poincaré inequality, then it supports a .q; N p/-Poincaré N inequality for every qN  q and pN  p. There are also other implications between various types of Poincaré inequalities that we will look at throughout the chapter. We first give several geometrical consequences of the p-Poincaré inequality. Proposition 4.2. Let X be a p-Poincaré space. Then X is connected. Recall that a topological space Y is connected if there do not exist two disjoint nonempty open sets U and V with U [ V D Y . Proof. Assume that X is not connected. Then there exist disjoint nonempty open sets U and V such that U [ V D X . Then u D U has g D 0 as an upper gradient (as there can be no curves going between U and V ). Let B be a ball with centre in U and so large that B \ V ¤ ¿. Then «  « 1=p

ju  uB j d  C diam.B/

0< B

g p d

D 0;

(4.2)

B

a contradiction. Corollary 4.3. Assume that X is a doubling p-Poincaré space. Then .fxg/ D 0 for every x 2 X . Proof. By Proposition 4.2, X is connected. Hence the conclusion follows from Corollary 3.9. We will also use the following direct consequence of Proposition 4.2. Corollary 4.4. Let X be a p-Poincaré space. Then @B.x; r/ is nonempty whenever r < 12 diam X . Note that @B.x; r/  fy W d.x; y/ D rg sometimes without equality. Another consequence of the p-Poincaré inequality is that sets with capacity zero cannot separate X . We deduce this fact as a consequence of the following result. Lemma 4.5. Let X be a p-Poincaré space and assume that   X is connected. In particular, we may have  D X . If F    is relatively closed, then Cp .F / D 0 if and only if Cp .@F \ / D 0. Proof. If  D X , then  is nonempty, open and connected, by Proposition 4.2, and thus the case  D X is just a special case. The necessity is clear. As for the sufficiency, let G D int F . If G D ¿, then F D @F \  and we are done. We can therefore assume that G ¤ ¿. Then Cp .F /  Cp .G/ > 0. If @G \  were empty, then both G and  n G would be nonempty and open, contradicting the fact that  is connected. Thus there is x 2 @G \ . Let B D B.x; r/  2B  ,

86

4 Poincaré inequalities

where  is the dilation constant in the p-Poincaré inequality. As B contains points x we have .B \ G/ > 0 and .B n G/ x > 0. both in G and in  n G, Assume that Cp .@F \ / D 0. Then Cp .@G \ /  Cp .@F \ / D 0, and hence Modp . @G\ / D 0, by Proposition 1.48. Thus 0 is a p-weak upper gradient of u D G in B. The p-Poincaré inequality then shows that «

«

1=p

ju  uB j d  C diam.B/

0< B

B

gup d

D 0;

a contradiction. Hence Cp .@F \ / > 0. Lemma 4.6. Let X be a p-Poincaré space and assume that   X is connected. In particular, we may have  D X . If F   is relatively closed with Cp .F / D 0, then  n F is connected. Proof. Assume, on the contrary, that  n F is disconnected, and let G be a component x \  ¤ , as there are other components. Then @E \   F so of  n F . Let E D G Cp .@E \ /  Cp .F / D 0. By Lemma 4.5, Cp .E/ D 0. But 0 < .G/  .E/  Cp .E/, a contradiction. The following examples show that the connectedness and the p-Poincaré inequality in Lemmas 4.5 and 4.6 are essential. Example 4.7. Let  D .0; 1/ [ .2; 3/  X D R and F D .0; 1/. Then @F \  D ¿ and as Cp .F / > 0, the conclusion in Lemma 4.5 fails. With F D ¿ we instead see that the conclusion in Lemma 4.6 fails. Example 4.8. Let  D X be the von Koch snowflake curve, see Example 1.23, which is connected, and let x ¤ y. Then Cp .fx; yg/ D 0, while  n fx; yg is disconnected, showing that Lemma 4.6 fails. Letting A be a component of nfx; yg and F D fx; yg[A gives a counterexample to Lemma 4.5. The following result strengthens Proposition 1.48. Proposition 4.9. Let X be a p-Poincaré space and E  X be measurable. Then Cp .E/ D 0 if and only if Modp . E / D 0. Proof. Assume that Modp . E / D 0. Let us first show that .X n E/ > 0. Assume on C the contrary that .X n E/ D 0. As .X / D E [ XnE , it follows from Lemma 1.42 that Modp . .X // D 0. Hence g  0 is a p-weak upper gradient of every function. Pick x ¤ y 2 X (here we use the assumption from the beginning of Section 1.1 that X is not a singleton). Let r D d.x; y/, u.z/ D .1  d.z; x/=r/C and B D B.x; r/. The p-Poincaré inequality shows as in (4.2) that u D uB a.e. in B. But u > 34 in a neighbourhood of x and u < 14 in a neighbourhood of y, which is a contradiction.

4.1 Poincaré inequalities

87

Thus, we must have .X n E/ > 0. Hence there is a ball B 0 with .B 0 n E/ > 0. Let v D E and E 0 D E \ B 0 . As Modp . E / D 0, g  0 is a p-weak upper gradient of v. So « 1=p « 0 p g d  jv  vB 0 j d 0 D C diam.B / B 0 B0   .E 0 / .E 0 / 0 0 .E / 1  n E/ C .B 2.E 0 /.B 0 n E 0 / .B 0 / .B 0 / D : D 0 .B / .B 0 /2 This yields that .E 0 / D 0, and, by varying B 0 , .E/ D 0. The result now follows from Proposition 1.48. Another consequence of the p-Poincaré inequality is that functions with Lp -integrable upper gradients are finite q.e. This is not true in general, see Remark 4.12. p Proposition 4.10. Let X be a p-Poincaré space and f 2 Dloc .X /. Then

Cp .fx W jf .x/j D 1g/ D 0: p p Corollary 4.11. Let X be a p-Poincaré space. Then Dloc .X /  Gloc .X /.

Proof. This follows directly from Lemma 2.43 and Proposition 4.10. Proof of Proposition 4.10. Let g 2 Lploc .X / be an upper gradient of S f . As X is Lindelöf, we can find a countable family of balls fBj gj1D1 so that X D j1D1 Bj and f; g 2 Lp .Bj / for every j D 1; 2; ::: . Let Ej D fx 2 Bj W jf .x/j D 1g, j D 1; 2; ::: , and E D fx 2 X W jf .x/j D 1g. Let  2 Ej ,  W Œ0; l  ! X . By compactness we can find 0  a < b < c  l so that .Œa; c/  Bj and .b/ 2 Ej . As g is an upper gradient of f and jf ..b//j D 1 we see that Z Z gBj ds  g ds D 1: 

jŒa;b

S Hence, by Proposition 1.37, Modp . Ej / D 0. As E D j1D1 Ej , we also get that Modp . E / D 0 (by Corollary 1.38). Therefore Cp .E/ D 0, by Proposition 4.9. Remark 4.12. If X is a space with no curves (see Example 1.22), then 0 is trivially an upper gradient of every function, including u  1. Assuming that .X / > 0 we have Cp .X / > 0 and we see that the conclusions of Proposition 4.10 and Corollary 4.11 do not hold. Similarly Modp . X / D 0 in this case and the conclusion in Proposition 4.9 does not hold either. This shows that the assumption that X is a p-Poincaré space cannot be omitted in Propositions 4.9 and 4.10, nor in Corollary 4.11.

88

4 Poincaré inequalities

4.2 Characterizations of Poincaré inequalities Before we continue let us give a few simple characterizations of Poincaré inequalities. Proposition 4.13. Let q;   1 and CPI > 0. Then the following are equivalent: (a) X supports a .q; p/-Poincaré inequality with constants CPI and . (b) Inequality (4.1) holds for all balls B  X , all bounded measurable functions u on X and all upper gradients g of u. (c) Inequality (4.1) holds for all balls B  X , all measurable functions u on X and all upper gradients g of u, where the left-hand side is interpreted as 1 if uB is not defined. (d) Inequality (4.1) holds for all balls B  X , all measurable functions u on X and all p-weak upper gradients g of u, where the left-hand side is interpreted as 1 if uB is not defined. Proof. (d) ) (c) ) (a) ) (b) These implications are trivial. (c) ) (d) This implication follows from Lemma 1.46. (Note that we can use Lemma 1.46 even if g … Lp .X /.) (b) ) (a) Let uk D maxfminfu; kg; kg, k D 1; 2; ::: , be the truncations of u at levels ˙k. Then g is an upper gradient also of uk (see the proof of Proposition 1.26). By dominated convergence we get that « « 1=q 1=q D lim ju  uB jq d juk  .uk /B jq d k!1

B

B

«

1=p

 CPI diam.B/

p

g d

;

B

if the last integral is finite, whereas (4.1) is trivial otherwise. (a) ) (c) If uB is finite, then (4.1) follows as in the proof of (b) ) (a). Assume therefore that uB is either infinite or not defined. Then .uC /B D 1 or .u /B D 1 (or both). We may assume that .uC /B D 1. As g is an upper gradient also of uC , we may replace u by uC andªhence assume that u  0. ª We want to show that B g p d D 1. Assume, on the contrary, that B g p d < 1. By Proposition 4.10 (applied to B), we have .fx 2 B W u.x/ D 1g/  Cp .fx 2 B W u.x/ D 1g/ D 0: Hence, by dominated convergence, 1=q 1=q « « ju  uB jq d D lim juk  .uk /B jq d 1D k!1

B

B

«

1=p

 CPI diam.B/ a contradiction. Thus

ª B

g p d B

g p d D 1, and (4.1) holds.

;

4.3 BiLipschitz invariance

89

We can now derive the following consequence of the .p; p/-Poincaré inequality. Proposition 4.14. Assume that X supports a .p; p/-Poincaré inequality. Then p 1;p Dloc ./ D Nloc ./:

Proof. One inclusion is trivial. p For the other, let u 2 Dloc ./ and let g 2 Lploc ./ be an upper gradient of u. Fix x 2 . Then we can find r > 0 such that B D B.x; r/  B   is such that g 2 Lp .B/, where  is the dilation constant in the .p; p/-Poincaré inequality. Proposition 4.13 implies that « 1=p « 1=p p p ju  uB j d  C diam.B/ g d < 1: B

B

In particular, uB 2 R and thus u 2 Lp .B/. Hence u 2 Lploc ./, and therefore 1;p u 2 Nloc ./. Let us give one more characterization due to Keith. Keith gives a few other characterizations as well. The proofs of these, as well as of Theorem 4.15, are however beyond this book. We will not use Theorem 4.15 except when proving Proposition A.17 in Appendix A. Theorem 4.15 (Keith [196], Theorem 2). Assume that X is complete and that  is doubling. Then X is a p-Poincaré space if and only if there exist constants C > 0 and   1 such that for all balls B  X , all functions u 2 Lipc .X / and all upper gradients g 2 Lipc .X / of u, « 1=p « ju  uB j d  C diam.B/ g p d : B

B

4.3 BiLipschitz invariance In this section, we shall see that Poincaré inequalities are preserved under biLipschitz mappings. Note that this is not true for strong Poincaré inequalities, see Example 4.18. Proposition 4.16. Let X D .X; d; / and Y D .Y; d 0 ; 0 / be two metric spaces, and ˆ W X ! Y be a biLipschitz mapping with biLipschitz constant L such that .E/  0 .ˆ.E//  M.E/ M for all -measurable sets E. Assume that X satisfies a .q; p/-Poincaré inequality with constants C 0 and , and that  is doubling. Then Y satisfies a .q; p/-Poincaré inequality with constants C and L2 , where C depends only on p, q, C 0 , M , L and the doubling constant C , but not on .

90

4 Poincaré inequalities

To prove this we will need the following standard lemma, which will often be used in other proofs as well. Lemma 4.17. If u is integrable, 1  q < 1, a 2 R and E is a measurable set with positive measure, then « 1=q « 1=q q q ju  uE j d 2 ju  aj d : E

Recall that uE D

E

ª E

u d.

Proof. Note first that « 1=q « 1=q « q q juE  aj d D juE  aj  ju  aj d  ju  aj d : E

E

Hence, «

1=q ju  uE j d q

E

«

1=q



ju  aj d

E

E

« 2

« C

q

ju  aj d

juE  aj d E

1=q q

1=q q

:

E

Proof of Proposition 4.16. Let B 0 D B.y; r/  Y be a ball (in Y ), v be integrable on Y and g 0 be an upper gradient of v. Let further x D ˆ1 .y/, u D v ı ˆ, g D L.g 0 ı ˆ/ and B D B.x; Lr/. Note that ˆ1 .B 0 /  B  ˆ1 .L2 B 0 /. As distances are distorted by at most a factor L it follows that g is an upper gradient of u. Hence, using Lemma 4.17 we get that Z « 2q q 0 jv  vB 0 j d  0 0 ju  uB jq d.0 ı ˆ/  .B / ˆ1 .B 0 / B0 Z 2q M 2 ju  uB jq d  .ˆ1 .B 0 // ˆ1 .B 0 / Z C  ju  uB jq d .B/ B « q=p q p  C diam.B/ g d B

 C diam.B 0 /q

«

.g 0 /p d0

ˆ.B/

 C diam.B 0 /q

«

L2 B 0

.g 0 /p d0

q=p q=p :

4.4 .q; p/-Poincaré inequalities

91

Example 4.18. Let 3 < T < , X D ŒT; T   R and Y D fz 2 C W jzj D 1 and arg z 2 X g (with the one-dimensional Lebesgue measure along Y , and arg denoting the principal branch). Let further ˆ W X ! Y be given by ˆ.x/ D e ix , where i is the imaginary unit. Then ˆ is biLipschitz, X supports a strong 1-Poincaré inequality, while Y only supports a (weak) 1-Poincaré inequality. Moreover, the dilation constant  for the 1-Poincaré inequality in Y tends to 1 as T ! . Example 4.19. Let X D R2 n ..0; 1/ .0; 1//, Y D R2 n ..0; M / .0; 1=M // and ˆ W X ! Y be given by ˆ.x; y/ D .M x; y=M /. The dilation constants in Poincaré inequalities for Y are the same as for Z D R2 n ..0; M 2 / .0; 1//, which by Example 8.20 is at least 21 M 2 . As ˆ has biLipschitz constant M , we see that the blow-up by L2 in Proposition 4.16 is of the right order.

4.4 .q; p/-Poincaré inequalities In this section we shall see that the validity of a p-Poincaré inequality is a self-improving property, i.e. that a doubling p-Poincaré space actually supports a better .q; p/-Poincaré inequality for some q > p. The exponent q is closely related to the “dimension” s of X given by the doubling condition for , see Section 3.1. We start by the following proposition which gives us one direction in the relation between q and s. Note that in Rn , we have s D n and q  p  D np=.n  p/, i.e. the exponents in this section are sharp. See also Remark 9.56. Proposition 4.20. Assume that  is doubling and X satisfies a .q; p/-Poincaré inequality for some q > p. Then for all balls B D B.x; r/  X and B 0 D B.x 0 ; r 0 /, with x 0 2 B and r 0  r, the estimate  r 0 s .B 0 / C .B/ r

(4.3)

holds with s D qp=.q  p/. Proof. Let B D B.x; r/ and B 0 D B.x 0 ; r 0 / be such that x 0 2 B and r 0  r. Note that 2B 0  3B. We shall test the .q; p/-Poincaré inequality on 3B with the .1=r 0 /Lipschitz function u.y/ D minf2  d.y; x 0 /=r 0 ; 1gC . Note that 0  u  1 in X , u D 1 in B 0 and u D 0 outside 2B 0 . We have that u3B 

.2B 0 / .3B/

92

4 Poincaré inequalities

and hence the left-hand side in the .q; p/-Poincaré inequality can be estimated from below as « 1=q     .B 0 / 1=q .2B 0 / ju  u3B jq d  1 : .3B/ .3B/ 3B On the other hand, as gu D 2B 0 nB 0 =r 0 , the right-hand side is « Cr 3B

1=p gup

d



C r .2B 0 /  0 r .3B/

1=p

:

If .2B 0 /  12 .3B/, then these estimates together with the doubling property of  give       C r .2B 0 / 1=p C r .B 0 / 1=p .B 0 / 1=q  0  0 ; .3B/ r .3B/ r .3B/ which yields (4.3) with 1=s D 1=p  1=q. If .2B 0 / > 12 .3B/, then by the doubling property of , .B 0 / 1  r 0 s .2B 0 / 1  :   .B/ C .3B/ 2C 2C r The following theorem shows that a p-Poincaré inequality together with the doubling condition implies the seemingly stronger .q; p/-Poincaré inequality. Together with Proposition 4.20 this also shows that condition (4.3) is both necessary and sufficient for the validity of a .q; p/-Poincaré inequality. Theorem 4.21. Assume that X supports a p-Poincaré inequality with dilation constant  and that the measure  satisfies (4.3) for some s > p. Then X supports a .p  ; p/Poincaré inequality with p  D sp=.s  p/ and dilation constant 2. Before proving Theorem 4.21 we draw some consequences of it. Corollary 4.22. Assume that X supports a p-Poincaré inequality with dilation constant   and that (4.3) holds for some s > p. If u 2 N 1;p .2B/, then u 2 Lp .B/ with p  D sp=.s  p/. Note that the exponent s in (4.3) is not uniquely determined, in particular, as r 0  r, we can always make s larger. Thus, the assumption s > p in Theorem 4.21 and Corollary 4.22 can always be fulfilled. On the other hand, a smaller s gives a better (larger) exponent p  , which is often desirable, e.g. for the embedding results in Section 5.5. We can directly obtain the following Sobolev embedding result. Corollary 4.23 (Local Sobolev embedding). Assume that X is a p-Poincaré space  1;p and that (4.3) holds for some s. Let u 2 Nloc ./. If s > p, then u 2 Lploc ./ with p  D sp=.s  p/, while if s  p, then u 2 Lqloc ./ for all 1  q < 1.

93

4.4 .q; p/-Poincaré inequalities

Proof. If s > p, then this is a direct consequence of Corollary 4.22. On the other hand, if s  p and q > p, then (4.3) also holds with s 0 D qp=.q  p/ > p  s in place of s, and we can apply Corollary 4.22 with s 0 in place of s, obtaining that u 2 Lqloc ./. That u 2 Lqloc ./ also for 1  q  p follows from Hölder’s inequality. The limiting case q D 1 is not possible s D p, as shown in Rn , n > 1, by the   for1  1;n function u.x/ D log log.1=jxj/ 2 W B 0; 2 . However, in this case, a Trudinger 1;p inequality holds and every function in Nloc ./ is locally exponentially integrable, see the notes to this chapter. When s < p it is possible to let q D 1, as we show in the L1 -local Sobolev embedding for large p (Corollary 4.28) below. In fact, if X is complete, even more is true for p > s, viz. that Newtonian functions are locally Hölder continuous, see Corollary 5.49. The following result is another direct consequence of Theorem 4.21. Corollary 4.24. Assume that  is doubling and X supports a p-Poincaré inequality with dilation constant . Then X supports a .q; p/-Poincaré inequality for some q > p and with dilation constant 2. Moreover, X supports a .p; p/-Poincaré inequality with dilation constant 2. Proof. As  is doubling it satisfies (4.3) with s D log2 C , where C is the doubling constant, by Lemma 3.3. It is then trivial that (4.3) is fulfilled for every s  log2 C . With s D maxflog2 C ; 2pg we obtain the desired .q; p/-Poincaré inequality from Theorem 4.21. A .p; p/-Poincaré inequality follows directly from the .q; p/-Poincaré inequality and Hölder’s inequality. To prove the .q; p/-Poincaré inequality, we shall need the following lemma which in various versions goes under the name Maz0 ya’s truncation method. Lemma 4.25. Let G   be an open set, A and E  A  G be measurable with positive measures and q  p be such that q > 1. Assume that for all t  0 and all u 2 N 1;p .G/, the following weak type estimate holds, « q=p q p gu d : (4.4) t .fx 2 A W ju.x/  uE j  t g/  C0 .A/ G

Then for all u 2 N 1;p .G/, 1=q  « 1=q q ju  uA j d  32C0 1C A



.A/ q q  1 .E/

1=p

1=q « G

gup

d

:

Replacing u by u  uE , we may assume that uE D 0. We estimate the integral ªProof. q u d. Let k0 be the largest integer such that A C « uC d; (4.5) 2k0 1  E

94

4 Poincaré inequalities

and for k > k0 , let

uk D minf2k ; .u  2k /C g

be the truncation of u at levels 2k and 2kC1 . Then « « .uk /E D uk d  uC d < 2k0  2k1 E

E

and (4.4) applied to uk with t D 2k1 yields 2.k1/q .fx 2 A W u.x/  2kC1 g/  2.k1/q .fx 2 A W juk .x/  .uk /E j  2k1 g/ « q=p  C0 .A/ gupk d : G

As guk D gu f2k k0 , that Z fx2AWu.x/>2k0 C2 g

uqC d 

1 X

2.kC2/q .fx 2 A W 2kC1  u.x/  2kC2 g/

kDk0 C1

 2 C0 .A/ 3q

« 1 X kDk0 C1

q=p

G

gupk

d

« q=p p  2 C0 .A/ gu d ; 3q

(4.6)

G

where the last inequality follows from the elementary inequality 1 X

aj˛ D

j D1

1 X j D1

aj˛1 aj 

1 X 1 X j D1 iD1

˛1 ai

aj D

1 X

˛ aj

;

j D1

for aj  0, j D 1; 2; ::: , and ˛  1. Using (4.5) we have « q Z uqC d  2.k0 C2/q .A/  23q .A/ uC d : fx2AWu.x/2k0 C2 g

(4.7)

E

Now, for every  > 0, (4.4) implies, by the Cavalieri principle (Lemma 1.10), that « Z 1 1 uC d   C .fx 2 A W ju.x/  uE j  t g/ dt .E/ E q=p Z 1 « C0 .A/ dt p C gu d .E/ tq G   q=p « C0 .A/ p q D 1C : g d  q  1 .E/ G u

95

4.4 .q; p/-Poincaré inequalities

Choosing

 D

we obtain

«

C0 .A/ .E/

1=q «

1=p

G

gup

d

  « 1=p C01=q q .A/ 1=q p uC d  gu d ; q  1 .E/ E G

(4.8)

if  ¤ 0. If  D 0, then u D uE D 0 q.e., and thus (4.8) holds also in this case. Inserting (4.8) into (4.7) now yields 1 .A/

Z fx2AWu.x/2k0 C2 g

uqC d  23q C0



q q1

q

q=p

«

.A/ .E/

G

gup d

:

Adding (4.6) to the last estimate and using the assumption uE D 0, we get 

«

q

A

.u  uE /C d  2 C0

The integral proof.

ª

A .u

3q



q 1C q1

q

.A/ .E/

« G

q=p gup

d

:

 uE /q d is estimated similarly and Lemma 4.17 finishes the

Proof of TheoremR 4.21. Let B D B.x0 ; r/ and u 2 N 1;p .2B/. We can assume that r  diam X and 2B gup d > 0. Let further x 2 B be a Lebesgue point of u, B0 D 2B, r0 D 2r, rj D 2j r and Bj D B.x; rj /, j D 1; 2; ::: . The doubling property of  and the p-Poincaré inequality then show that ju.x/  uB0 j D lim juBj  uB0 j  j !1



C3

1 « X j D0 Bj

1 « X j D0 Bj C1

ju  uBj j d  C

and

ju  uBj j d

1 X

« rj

« juB  uB0 j  C r

Bj

j D0

1=p gup

d

(4.9)

1=p B0

gup d

:

(Observe that the power 3 in C3 above is needed when j D 0.) Condition (4.3) applied to the balls Bj and B0 then yields rj  C r..Bj /=.B0 //1=s and hence Z 1=p 1 X Cr 1=s1=p p ju.x/  uB j  .Bj / gu d : .B0 /1=s j D0 Bj

96

4 Poincaré inequalities

Next, we write the above sum as †0 C †00 , where the summations in †0 and †00 are over j < j0 and j  j0 , respectively (j0  0 will be chosen later). It follows from Lemma 3.7 that there exists  < 1 independent of j such that .Bj /  C  j j0 .Bj0 / for j < j0 and .Bj /  C  j j0 .Bj0 / for j  j0 . Hence, as 1=s  1=p < 0, we get †0 D

jX 0 1

Z

1=p

.Bj /1=s1=p Bj

j D0

gup d

Z  C.Bj0 /

1=s1=p B0

gup

d

1=p jX 0 1

 .j j0 /.1=s1=p/ ;

j D0

where the last sum can be estimated from above by  1=p1=s =.1 1=p1=s /. Similarly, « 1=p 1 1 X X 00 1=s p 1=s 1=p † D .Bj / gu d  C.Bj0 / M.x/  .j j0 /=s ; Bj

j Dj0

j Dj0

where  g p .x/ D sup M.x/ D MB 0 u B0

« B0

gup d

and the supremum is taken over all balls B 0  B0 containing x. Note that M.x/ is finite for a.e. x 2 B, by Lemma 3.12, and the sum on the right-hand side in †00 equals 1=.1   1=s /. ª Next, as 0 < B0 gup d  M.x/ and B0  6B1 , we can for a.e. x 2 B find j0  1 (depending on x) such that Z 1 C2 .Bj0 /  g p d  C3 .Bj0 /: M.x/ B0 u Indeed, j0 is the largest integer such that the right inequality holds, and the left inequality then follows from the doubling property. Inserting this into the above estimates of †0 and †00 yields « 1=s Cr  0 00 p ju.x/  uB j  .† C † /  C r g d M.x/1=p ; u .B0 /1=s B0 where p  D sp=.s  p/. If ju.x/  uB j  t , then this implies that p =s  « C tp p M.x/  p gu d : r B0 Lemma 3.12 shows that .fx 2 B0 W M.x/   g/ 

C 

Z B0

gup d;

(4.10)

4.4 .q; p/-Poincaré inequalities

97

which together with (4.10) implies the following weak type estimate for all t  0 and u 2 N 1;p .2B/, 

«



p =p

t p .fx 2 B W ju.x/  uB j  t g/  C r p .B/ 2B

gup d

;

where C is independent of u and B. An application of Lemma 4.25 with q D p  , A D E D B, G D 2B and  C0 D C r p finishes the proof. Corollary 4.26. If X is a p-Poincaré space and  satisfies (4.3) for some s  p, then X supports a .q; p/-Poincaré inequality for all 1  q < 1. Proof. Let s 0 D p C ", " > 0. Then (4.3) holds with s replaced by s 0 and Theorem 4.21 shows that X supports a .q; p/-Poincaré inequality with qD

s0p .p C "/p D : s0  p "

Choosing " sufficiently small finishes the proof. Corollary 4.26 can be further improved for s < p, so that the case q D 1 is included.   This  is not possible for s D p, as the function u.x/ D log log.1=jxj/ 2 W 1;n B 0; 12 in Rn , n > 1, shows. Instead, an exponential Trudinger inequality holds, see the notes to this chapter. Proposition 4.27. Assume that X is a p-Poincaré space and that (4.3) holds for some s < p. If B  X is a ball with radius r and u 2 N 1;p .2B/, then «

1=p

ess sup ju.x/  uB j  C r x2B

2B

gup d

;

where  is the dilation constant in the p-Poincaré inequality. Note that the left-hand side equals Cp - ess supx2B ju.x/  uB j, by Corollary 1.69. A direct consequence of Proposition 4.27 is the following local Sobolev embedding for the case s < p. Corollary 4.28 (L1 -local Sobolev embedding for large p). Let X be a p-Poincaré 1;p space such that (4.3) holds for some s < p. If u 2 Nloc ./, then u 2 L1 loc ./. We improve upon this result in the local Sobolev embedding for large p (Corollary 5.49), showing that Newtonian functions are locally Hölder continuous under the additional assumption that X is complete.

98

4 Poincaré inequalities

Proof of Proposition 4.27. Let x 2 B be a Lebesgue point of u, rj D 2j r and Bj D B.x; rj /, j D 0; 1; ::: . Note that B0  2B. The doubling property of  and the p-Poincaré inequality show as in (4.9) that 1 X

ju.x/  uB0 j  C

 rj

j D0

1 .Bj /

1=p

Z Bj

gup

d

:

(4.11)

(Note that B0 is defined slightly differently here.) By assumption (4.3) we have that .Bj /=.2B/  C 2js and inserting this into (4.11) gives 1 X

ju.x/  uB0 j  C r

2j.1s=p/

j D0

«



1 .2B/

Bj

1=p

 Cr 2B

gup d

1=p

Z gup d

;

since 1  s=p > 0. Finally, the doubling property of  and the p-Poincaré inequality imply that «

« juB0  u2B j  C

1=p

ju  u2B j d  C r 2B

gup d

2B

:

A similar estimate holds for juB  u2B j, which concludes the proof. The following estimate for the capacity of small sets is the first step in proving that for p > s, singletons have positive capacity, cf. Corollary 5.39. Note that in Corollary 5.39, the space X is assumed to be complete. Here we only obtain a uniform lower bound for all balls B.x0 ; / with > 0. Corollary 4.29. Assume that X is a p-Poincaré space and that (4.3) holds for some s < p. Then for every ball B D B.x0 ; r/ and for every set E  B with positive capacity we have Cp .E/  C.B/=.r p C 1/. In particular, Cp .B.x0 ; //  C.B/=.r p C 1/ for all > 0. Proof. Let u 2 N 1;p .X / be such that 0  u  1 in X and u D 1 on E. Then by Proposition 4.27 and the comment after it, «

1=p

1  j1  uB j C uB  C r  

C.r C 1/ .B/ p

Z X

2B

.gup C up / d

« C

gup d

1=p up d

B

1=p :

Taking infimum over all u admissible in the definition of Cp .E/ finishes the proof of the corollary.

4.5 Quasiconvexity and connectivity

99

We have seen in Theorem 4.21 that the p-Poincaré inequality together with the doubling condition implies the seemingly stronger .q; p/-Poincaré inequality. The next result shows that Poincaré inequalities are self-improving (or open-ended) in another sense as well. Another self-improving (or open-ended) result is the Gehring lemma, Theorem 3.22. Theorem 4.30 (Keith–Zhong [199]). Let p > 1 and assume that X is a complete doubling p-Poincaré space. Then there is q < p such that X is a q-Poincaré space. The proof of this result goes beyond this book, and we refer the reader to [199] for the proof. In fact the proof therein depends on Theorem 4.15 which we have not proved either. We will use this result in Chapter 8, when showing that superminimizers can be regularized by making changes on sets of capacity zero. (More specifically, we use it when applying Theorem 5.62 in the proof of Theorem 8.22.) This result is subsequently used quite extensively. The validity of a better q-Poincaré inequality also implies the equivalence between several different definitions of Sobolev type spaces on metric spaces, see Appendix B. The reader who so prefers may, instead of being dependent on Theorem 4.30, choose to require that X is a complete q-Poincaré doubling space for some q < p instead of our assumption that X is a complete p-Poincaré doubling space, made in the beginning of Chapter 7. In earlier literature, the better q-Poincaré inequality was a standard assumption.

4.5 Quasiconvexity and connectivity We already know, by Proposition 4.2, that any p-Poincaré space is connected. In this section we prove a stronger geometrical consequence of the p-Poincaré inequality (assuming also completeness and doubling), viz. quasiconvexity. This also implies an equivalence between various types of connectivity. Another consequence of quasiconvexity is its influence on the dilation constant in the Poincaré inequality. This will be studied in Section 4.6. Let us start this section with some definitions. Recall that our curves are always parameterized by arc length. Definition 4.31. A metric space X is L-quasiconvex, L  1, if for all x; y 2 X there is a (possibly constant) curve  W Œ0; l  ! X such that .0/ D x, .l / D y and l  Ld.x; y/. A metric space X is quasiconvex if it is L-quasiconvex for some L. A metric space X is geodesic if it is 1-quasiconvex, if it is merely .1C"/-quasiconvex for all " > 0, then it is a length space. The following theorem is the main result of this section.

100

4 Poincaré inequalities

Theorem 4.32. Assume that X is a complete doubling p-Poincaré space. Then X is L-quasiconvex, with L D 192C3 CPI , where CPI is the constant in the p-Poincaré inequality. Note that L is independent of the dilation constant  in the p-Poincaré inequality. See, however, Section 4.8, where relations between L and  are discussed. Theorem 4.39 below shows that one can always obtain a Poincaré inequality with dilation constant  D L. Before proving Theorem 4.32, we need the following definition and lemma. Definition 4.33. We say that .x0 ; ::: ; xm / is an "-chain from x0 to xm if d.xi1 ; xi / < " for i D 1; 2; ::: ; m: Lemma 4.34. Let X be a complete doubling p-Poincaré space. Let .x0 ; x1 ; ::: ; xm / be an "-chain from x0 to x D xm and assume that d.x0 ; x/  ". Then there exists a 3"-subchain .x00 ; x10 ; ::: ; xk0 / from x0 to x such that 0 ; xi0 / < 3" for i D 1; 2; ::: ; k: "  d.xi1

Proof. Let x00 D x0 . If d.x00 ; x/ < 3", then .x00 ; x/ is the required chain. Otherwise, find the largest i such that d.x00 ; xi / < 2" and put x10 D xi . Clearly, d.x10 ; x/  d.x; x00 /  d.x00 ; x10 /  3"  2" D ": We now repeat the argument: Either d.x10 ; x/ < 3", in which case .x00 ; x10 ; x/ is the required chain, or we find the largest j such that d.x10 ; xj / < 2" and put x20 D xj . Again d.x20 ; x/  ". Since j > i , we obtain the required chain in at most m steps. Remark 4.35. For a fixed x0 2 X and " > 0, let U consist of all x 2 X such that there exists an "-chain from x0 to x. Note that U is nonempty and open, since for every x 2 U , we have B.x; "/  U . On the other hand, if x … U , then B.x; "/ \ U is empty and thus X n U is also open. It follows that if X is connected, then X n U must be empty, i.e. there is an "-chain from x0 to x for every x 2 X . Proof of Theorem 4.32. For a fixed x0 2 X and arbitrary x 2 X , let x0 ;" .x/ D inf

m X

d.xi1 ; xi /;

iD1

where the infimum is taken over all "-chains from x0 to x. Note that if d.x; y/ < ", then j x0 ;" .x/  x0 ;" .y/j  d.x; y/ and hence x0 ;" is locally Lipschitz and lip x0 ;" .x/  1 for all x 2 X . It follows that g  1 is an upper gradient of x0 ;" on X , by Proposition 1.14. Hence by the p-Poincaré

4.5 Quasiconvexity and connectivity

101

inequality ªwith B0 D B.x0 ; 2d.x0 ; x//, Bj D B.x; 2j d.x0 ; x//, for j D 1; 2; ::: , and j D Bj x0 ;" d, j x0 ;" .x/  0 j 

1 X

j j C1  j j

j D0



1 « X

j D0 Bj C1

 C3 

j x0 ;"  j j d

1 « X

j D0 Bj

4C3 CPI

j x0 ;"  j j d

1 X

2

j

« d.x0 ; x/

j D0

1=p p

g d Bj

D C 0 d.x0 ; x/; where C 0 D 8C3 CPI . (Observe that the power 3 in C3 above is needed when j D 0 and that diam.B0 /  4d.x0 ; x/.) Similarly, j 0 j D j x0 ;" .x0 /  0 j  C 0 d.x0 ; x/ and the triangle inequality implies that j x0 ;" .x/j  2C 0 d.x0 ; x/. Hence, for fixed x0 and x, x0 ;" .x/ % x0 .x/  2C 0 d.x0 ; x/; as " ! 0: (4.12) Now, let x0 ¤ x 2 X be fixed. For each n D 1; 2; ::: , there exists 0 < "n  2n x0 .x/ such that x .x/ : (4.13) x0 ;"n .x/  0 2 Note that "n  2n x0 .x/  2C 0 2n d.x0 ; x/, i.e. for sufficiently large n (independent of x) we have "n =3  d.x0 ; x/. Pick an "n =3-chain .x0 ; x1 ; ::: ; xm / from x0 to x such that m X d.xi1 ; xi / < 2 x0 .x/ iD1 n and use Lemma 4.34 to thin it out into an "n -chain .x0 D x0n ; x1n ; ::: ; xm D x/ n satisfying for all i D 1; 2; ::: ; mn ;

"n n ; xin / < "n :  d.xi1 3 Let kn be the smallest integer such that 2kn  mn and define ´ if i D 0; 1; ::: ; mn  1; xn; kn n .2 i / D in xmn D x; if i D mn ; mn C 1; ::: ; 2kn :

102

4 Poincaré inequalities

Pmn n n Note that mn "n =3  iD1 d.xi1 ; xi / < 2 x0 .x/ and hence "n  6 x0 .x/=mn . kn This implies that for all i D 1; 2; ::: ; 2 ; n ; xin / < "n  d.xi1

6 x0 .x/ 12 x0 .x/  ; mn 2kn

i.e. n is 12 x0 .x/-Lipschitz on the set Skn , where Sj WD f2j i W i D 0; 1; ::: ; 2j g, j D 1; 2; ::: . Since by (4.13), mn X x .x/ n m n "n > d.xi1 ; xin /  0 ; 2 iD1

we have also 2kn  mn >

x0 .x/  2n1 ; 2"n

i.e. kn  n ! 1, as n ! 1. This means that for every k and every n  k, we have kn  k and since Sk  SkC1 for all k, it follows that all n , n  k, are defined on Sk . Thus, by the Ascoli theorem (see e.g. p. 169 in Royden [310], note that here we need a version of Ascoli’s theorem valid for metric space valued equicontinuous functions), we can find a set N1 of positive integers such that the sequence fn gn2N1 converges uniformly on S1 to a 12 x0 .x/-Lipschitz function  . Continuing in this way, we obtain N1  N2   such that for each j D 1; 2; ::: , the sequence fn gn2Nj converges uniformly on Sj to . A diagonal argument provides us with a sequence n1 ; n2 ; ::: S converging uniformly to  on each Sj . This defines  on the dense subset S D 1 iD1 Si of Œ0; 1. Note also that by the uniform convergence,  is 12 x0 .x/-Lipschitz on each Sj and thus on S. In particular,  can be extended into a 12 x0 .x/-Lipschitz function on the whole Œ0; 1. After reparameterization, it thus provides us with a rectifiable curve from x0 to x with arc length at most 12 x0 .x/. Together with (4.12), this finishes the proof. Definition 4.36. A metric space X is pathconnected if for all x; y 2 X there is a continuous function  W Œ0; 1 ! X with .0/ D x and .1/ D y. A metric space X is rectifiably connected if for all x; y 2 X there is a (possibly constant) curve  W Œ0; l  ! X with .0/ D x and .l / D y. The difference between pathconnected and rectifiably connected is that in the former case, the image of  need not be rectifiable. For us the rectifiability of the connecting curve is important. Recall that every curve, in this book, is implicitly assumed to be rectifiable and parameterized by arc length. Lemma 4.37. Let X be an arbitrary metric space. If X is rectifiably connected, then it is pathconnected. If X is pathconnected, then it is connected.

4.6 Poincaré inequalities in quasiconvex spaces

103

Proof. The first part is trivial. We therefore turn to the second part. Assume that X is pathconnected, but not connected, and let U and V be disjoint nonempty open subsets of X with U [ V D X . Let u D U which is continuous on X . Find points x 2 U and y 2 V . Let further  W Œ0; 1 ! X be a continuous function with .0/ D x and .1/ D y. Then u ı  is continuous, but u ı .Œ0; 1/ D f0; 1g, a contradiction. In quasiconvex spaces the above three notions of connectivity coincide. Lemma 4.38. Assume that X is quasiconvex. Then  is rectifiably connected if and only if it is connected. Note that  itself need not be quasiconvex. Proof. The necessity follows directly from Lemma 4.37. As for the sufficiency, assume that  is not rectifiably connected. Let x 2  and let U be the set of all points in  which are rectifiably connected to x in , i.e. y 2 U if there is a (possibly constant) curve  W Œ0; l  !  with .0/ D x and .l / D y. (Note that the relation “rectifiably connected” between points is an equivalence relation.) Let us show that U is open. Fix y 2 U and find r > 0 so that B.y; Lr/  , where L is such that X is L-quasiconvex. Let z 2 B.y; r/. By the quasiconvexity of X , there is a (possibly constant) curve  W Œ0; l  ! X with .0/ D y, .l / D z and l  Ld.y; z/ < Lr, and hence   B.y; Lr/  . Thus z 2 U , and hence U is open. Similarly V WD  n U is open. Moreover, V is nonempty since  is not rectifiably connected. Hence U and V satisfy the conditions in the definition of connectedness, and  is not connected.

4.6 Poincaré inequalities in quasiconvex spaces If X is quasiconvex, in particular if it is a complete doubling p-Poincaré space, then there is an upper bound for the dilation constant in Poincaré inequalities. Recall from Theorem 4.32 that the quasiconvexity constant L  192C3 CPI and that it does not depend on the dilation constant . In particular, it yields a lower bound on the constant CPI in the p-Poincaré inequality, no matter how large  is. The following result goes in the opposite direction and provides us with an upper bound for the best possible dilation constant in the Poincaré inequality. In view of Example 4.50 it is of the right magnitude. See Section 4.8 for further discussion. Theorem 4.39. Assume that X is L-quasiconvex and supports a p-Poincaré inequality, and that (4.3) holds for some s > p. Then X supports a .p  ; p/-Poincaré inequality with p  D sp=.s  p/ and the dilation constant L.

104

4 Poincaré inequalities

Proof. We can assume that L < 2, as otherwise the result follows directly from Theorem 4.21. Let B.x0 ; r/ be a ball, u 2 N 1;p .B.x0 ; Lr//, and let x 2 B.x R 0 ; r/ be a Lebesgue point of u. We can assume that r  diam.X /=L and that B.x0 ;Lr/ gup d > 0. By the L-quasiconvexity there exists a curve  parameterized by arc length such that .0/ D x0 , .l / D x and its length is l  Ld.x; x0 / < Lr. Let 0 D Lr=2 and i D 2i 0 , i D 1; 2; ::: , where  is the dilation constant in the p-Poincaré inequality. We shall construct a chain of balls along the curve  which connect x0 to x and have substantial overlaps. Find the smallest nonnegative integer ix so that 2 ix  L.r  d.x; x0 //. For each i D 0; 1; ::: ; ix  1 consider all integers j  0 such that ti;j WD .1  2i /l C j i < .1  2.iC1/ /l : We have j < 2.iC1/ l = i < 2.iC1/ Lr= i D  for all such j . Similarly, consider all j  0 such that tix ;j WD .1  2ix /l C j ix < l : This time we have j < 2ix l = ix < 2ix Lr= ix D 2. Let xi;j D .ti;j /. Then the balls Bi;j D B.xi;j ; i / cover  in the direction from x0 to x. Moreover, if i < ix , then d.x0 ; xi;j / C  i  ti;j C  i < .1  2iC1 /l C 2.iC1/ Lr < Lr; while d.x0 ; xix ;j / C  ix  tix ;j C  ix < l C 21 L.r  d.x; x0 //  Ld.x; x0 / C 12 L.r  d.x; x0 // D 12 L.r C d.x; x0 // < Lr; i.e. Bi;j  B.x0 ; Lr/ for all i ’s and j ’s under consideration. Note that d.xi;j ; x/  l  ti;j  2i l  2 i and hence by Lemma 3.6 we have 1 .Bi;j /  .B.x;  i //  C.Bi;j /: C

(4.14)

We complete the chain by adding the balls Bi;0 D B.x; i / for i > ix . Note that again d.x; x0 / C  i  d.x; x0 / C 12 L.r  d.x; x0 //  12 L.r C d.x; x0 // < Lr; so that Bi;0  B.x0 ; Lr/ also for these balls. Order the balls Bi;j lexicographically, i.e. Bi;j comes before Bi 0 ;j 0 if and only if i < i 0 , or i D i 0 and j < j 0 . For a ball B in the chain, let B  be its immediate successor with respect to the lexicographic ordering. Note that the centre of B  lies in Bx and hence we always have B   2B. Moreover, B \ B  contains a ball B 0 with radius  =2, where  is the radius of B  . (Let e.g. B 0 D B.x 0 ;  =2/, where x 0 is

4.6 Poincaré inequalities in quasiconvex spaces

105

obtained by back tracking the distance  =2 along  from the centre of B  .) We can now estimate juB  uB  j as follows, « « ju  uB j d C ju  uB  j d juB  uB  j  juB  uB 0 j C juB 0  uB  j  B0 B0 « « C ju  uB j d C C ju  uB  j d: B

B

The doubling property of  and the p-Poincaré inequality then show that 1=p X X « p  juB uB j  C B gu d ; ju.x/uB0;0 j D lim juBi;0 uB0;0 j  i!1

B

B

B

where B is the radius of B and the sum is over all balls in the chain connecting x0 to x. Condition (4.3) applied to the balls B and B.x0 ; Lr/, together with (4.14) and the fact that L < 2, then yields   .B.x;  B // 1=s B  C r .B.x0 ; Lr// and hence ju.x/  uB0;0 j 

1=p Z X Cr 1=s1=p p .B.x;  // g d : B u .B.x0 ; Lr//1=s B B

Next, we write the above sum as †0 C †00 , where the summations in †0 and †00 are over B with B > i0 and B  i0 , respectively (i0 will be chosen later). It follows from Lemma 3.7 that there exists  < 1 independent of i such that .B.x;  i //  C  ii0 .B.x;  i0 //

for i < i0

.B.x;  i //  C  ii0 .B.x;  i0 //

for i  i0 .

and Hence, as 1=s  1=p < 0, 0

Z

†  C.B.x;  i0 // and

1=s1=p B.x0 ;Lr/

1=p gup

d

†00  C.B.x;  i0 //1=s M.x/1=p ;

 where M.x/ D MB.x gup .x/ is the noncentred maximal function restricted to the 0 ;Lr/ ª ball B.x0 ; Lr/. Next, as 0 < B.x0 ;Lr/ gup d  M.x/ and B.x0 ; Lr/  B.x; 2Lr/ D B.x; 4 0 /, we can find i0  0 such that Z 1 g p d  C2 .B.x;  i0 //: C .B.x;  i0 //  M.x/ B.x0 ;Lr/ u

106

4 Poincaré inequalities

Indeed, i0 is the largest integer for which the right inequality holds, and the left inequality then follows from the doubling property. Inserting this into the above estimates of †0 and †00 yields « 1=s Cr  0 00 p ju.x/  uB0;0 j  .† C † /  C r gu d M.x/1=p ; 1=s .B.x0 ; Lr// B.x0 ;Lr/ (4.15) where p  D sp=.s  p/. Lemma 3.12 shows that Z C .fx 2 B.x0 ; Lr/ W M.x/  g/  g p d:  B.x0 ;Lr/ u In combination with (4.15) this shows that for all u 2 N 1;p .B.x0 ; Lr// and t  0, 

t p .fx 2 B.x0 ; r/ W ju.x/  uB0;0 j  t g/ « p =p p p  C r .B.x0 ; r// gu d ; B.x0 ;Lr/

where C is independent of u, x0 and r. Finally, an application of Lemma 4.25 with q D p  , A D B.x0 ; r/, E D B0;0 ,  G D B.x0 ; Lr/ and C0 D C r p finishes the proof. Note that E D B.x0 ; 0 /  A, as L < 2. Corollary 4.40. Assume that  is doubling and that X is L-quasiconvex and satisfies a .q; p/-Poincaré inequality with some dilation constant. Then X satisfies a .q; p/Poincaré inequality with dilation constant L. In particular, if X is geodesic, then X satisfies a strong .q; p/-Poincaré inequality. Proof. By Lemma 3.3, (4.3) holds with some s > p. If q  p, then Corollary 4.24 implies that X supports a .q; N p/-Poincaré inequality with some qN > p, while if q > p, this is trivially true with qN D q. Hence, by Proposition 4.20, (4.3) holds with s D qp=. N qN  p/ > p. Theorem 4.39 then shows that X supports a .p  ; p/N Finally, Poincaré inequality with dilation constant L and p  D sp=.s  p/ D q. Hölder’s inequality implies that X also supports a .q; p/-Poincaré inequality with dilation constant L.

4.7 Inner metric Assume in this section that X is rectifiably connected. Definition 4.41. The inner metric din is defined by din .x; y/ D inf l ; 

4.7 Inner metric

107

where the infimum is taken over all (arc length parameterized) curves (including constant curves)  W Œ0; l  ! X such that .0/ D x and .l / D y. It is immediate that .X; din / is also a metric space. However, if X were not rectifiably connected, then the inner metric would not be a metric as there would be points x; y 2 X such that din .x; y/ D 1. Note that we always have d.x; y/  din .x; y/. One advantage of the inner metric is that balls are connected. We let Bin .x; r/ D fy 2 X W din .x; y/ < rg and Bin .x; r/ D Bin .x;  r/. Lemma 4.42. The inner metric ball Bin D Bin .x; r/ is rectifiably connected. Proof. Let y 2 Bin . Then din .x; y/ < r and so there is a curve  W Œ0; l  ! X with .0/ D x, .l / D y and l < r. Hence   Bin and Bin is rectifiably connected. Note that this result is true without any assumptions on X . The same is true for the following lemma. Lemma 4.43. Arc lengths with respect to d and din are the same. Proof. Take a curve  W Œ0; l  ! X , parameterized by arc length with respect to the given metric d . Let 0 D t0 < t1 <  < tm D l . As  jŒtj 1 ;tj  , j D 1; ::: ; m, is a curve which is arc length parameterized with respect to d we get that din ..tj 1 /; .tj //  tj  tj 1 : Hence,

m X j D1

din ..tj 1 /; .tj // 

m X

.tj  tj 1 / D tm  t0 D l :

j D1

As this holds for all subdivisions of Œ0; l  it follows that the length of  with respect to din is at most l . The converse inequality is trivial, since d  din , and thus the length of  is the same with respect to d and with respect to din . x and let also Consider now an arbitrary (extended real-valued) function f W X ! R g W X ! Œ0; 1 be arbitrary. As arc length is the same with respect to d and din , g will be an upper gradient of f with respect to .X; d / if and only if it is an upper gradient with respect to .X; din /. As we equip both metric spaces with the same measure  1;p 1;p we see that N 1;p .X / D N 1;p .X; din / and Nloc .X / D Nloc .X; din / (with the same (semi)norms). It also follows that g is a p-weak upper gradient of f if and only if it is a p-weak upper gradient with respect to .X; din /. It is obvious that X is .1 C "/-quasiconvex with respect to din for every " > 0. If X is complete we can say more. Proposition 4.44. Assume that X is complete and rectifiably connected. Then X is 1-quasiconvex, i.e. geodesic, with respect to the inner metric din .

108

4 Poincaré inequalities

Proof. Let x; y 2 X , x ¤ y, and ı D din .x; y/. We can find curves j W Œ0; l0 j  ! X , parameterized by arc length, such that j .0/ D x, j .l0 j / D y and l0 j < ı C 1=j . Let Qj W Œ0; ı ! X be given by Qj .t / D j .t l0 j =ı/, which makes the mappings Qj equicontinuous and with images contained in a bounded set. Thus, by the Ascoli theorem (see e.g. p. 169 in Royden [310], note that here we need a version of Ascoli’s theorem valid for metric space valued equicontinuous functions), a subsequence of fQj gj1D1 converges uniformly to a continuous function Q W Œ0; ı ! X with Q .0/ D x and .ı/ Q D y. The uniform convergence and the fact that each Qj is .1 C 1=j ı/-Lipschitz imply that Q is 1-Lipschitz and hence lQ  ı D din .x; y/. It follows that Q is parameterized by arc length with respect to din . If X is L-quasiconvex, then it is immediate that d.x; y/  din .x; y/  Ld.x; y/:

(4.16)

Hence the topologies of .X; d / and .X; din / coincide. From (4.16) it directly follows that Bin .x; r/  B.x; r/  Bin .x; Lr/: (4.17) Proposition 4.45. Assume that X is L-quasiconvex. Then the measure  is doubling with respect to the given metric d if and only if it is doubling with respect to the inner metric din . The doubling constants depend only on each other and on L (see the proof for explicit estimates). Proof. If  is doubling with doubling constant C , then .Bin .x; 2r//  .B.x; 2r//  C .B.x; r//    CN .B.x; r=L//  CN .Bin .x; r//; where N 2 Z is so large that 2N  2L. Thus  is also doubling with respect to the inner metric. The converse direction is proved similarly. Proposition 4.46. Assume that X is L-quasiconvex and that  is doubling. Then the space X satisfies a .q; p/-Poincaré inequality with respect to the given metric d if and only if it satisfies a .q; p/-Poincaré inequality with respect to the inner metric din . The constants in the Poincaré inequalities depend only on each other, the doubling constant and on L (see the proof for explicit estimates). Proof. Assume that X satisfies a .q; p/-Poincaré inequality with respect to the given metric d with constants C 0 and . Let B D B.x; r/ and Bin D Bin .x; r/. Without

4.7 Inner metric

109

loss of generality, we may assume that r  diamin X  L diam X , where diamin is the diameter with respect to the inner metric. Let u be integrable and g be an upper gradient of u. Then, using Lemma 4.17 and (4.17), we have « 1=q « 1=q q q ju  uBin j d 2 ju  uB j d Bin

Bin

 

2CN=q

«

1=q ju  uB j d q

B

4CN=q C 0 r

«

 4C2N=q C 0 r

1=p p

g d B

«

1=p g p d

;

LBin

where C is the doubling constant and N is so large that 2N  L. The converse direction is proved similarly. Remark 4.47. With Propositions 4.44–4.46 in mind one could replace d by din and therefore assume that X is geodesic if X is complete. Note that all curves would still have the same length, and therefore also the same arc length parameterizations. Moreover, g is a (p-weak) upper gradient of u simultaneously with respect to d and din . Indeed, in many results there would be no loss at all with such an approach. However, in other results the estimates for constants and exponents would be worse. In particular, s in (4.3) could become worse which has consequences in many results, such as Poincaré inequalities, Sobolev embeddings and integrability of superharmonic functions. Another disadvantage would be that the natural (given) balls would be replaced by new distorted and maybe not very natural balls. Indeed, rather than choosing the inner metric it could sometimes be more advantageous to choose a metric minimizing the doubling constant or the “dimension” s in (4.3). Note, however, that the inner metric proves to be useful in connection with boundary regularity, see Lemma 14.2. We also have the following equivalence concerning Poincaré inequalities with different dilation constants in quasiconvex spaces. Proposition 4.48. Assume that X is L-quasiconvex and that  is doubling. Let  D 1 if X is complete, and  > 1 otherwise. Then the following are equivalent: (a) X satisfies a .q; p/-Poincaré inequality with respect to the given metric d ; (b) X satisfies a .q; p/-Poincaré inequality with dilation constant L with respect to the given metric d ; (c) X satisfies a .q; p/-Poincaré inequality with respect to the inner metric din ;

110

4 Poincaré inequalities

(d) X satisfies a .q; p/-Poincaré inequality with dilation constant  with respect to the inner metric din . Proof. Note first that X is -quasiconvex with respect to din , by Proposition 4.44 if X is complete, and the comment before it, otherwise. (a) , (c) This follows from Proposition 4.46. (a) ) (b) and (c) ) (d) These implications follow from Corollary 4.40. (b) ) (a) and (d) ) (c) These implications are trivial. Eventually we will need the following lemma. Lemma 4.49. Assume that X is quasiconvex and proper, that  is connected, and that E b . Then there is a rectifiably connected open set G such that E b G b . Proof. Let d D 12 distin .E; X n / > 0, where dist in is the distance taken with respect to the inner metric din . By compactness we can find x1 ; ::: ; xN 2 E so that Ex  S G1 WD jND1 Bin .xj ; d /. By Lemma 4.38 we can find a curve j connecting xj and S 1 j , which is a compact set, and xj C1 in , j D 1; ::: ; N  1. Let K D jND1 1 ı D 2 dist in .K; X n / > 0. By compactness we can find y1 ; ::: ; yM 2 K so that S S K  jMD1 Bin .yj ; ı/. Let now G WD G1 [ jMD1 Bin .yj ; ı/ which is rectifiably connected as both G1 [ K and Bin .yj ; ı/, j D 1; ::: ; M , are rectifiably connected, by Lemma 4.42, and Bin .yj ; ı/ \ K ¤ ¿, j D 1; ::: ; M .

4.8 The relation between L and  We saw in Theorem 4.39 that if X is an L-quasiconvex doubling p-Poincaré space, then the dilation constant  in the p-Poincaré inequality can always be assumed to satisfy   L. As we shall see later, e.g. in Section 8.4,  plays an important role in certain (weak) Harnack inequalities, and the role can alternatively be taken by L, see the notes to Chapters 8 and 9. It therefore seems natural to discuss the relation between  and L. Observe that the (best) dilation constant  in Poincaré inequalities is not so easily understood, whereas L has a definite geometric meaning. Example 4.50. Let 0 < ˛ < 12  and let X consist of two rays with opening ˛, X D f.t; 0/ W t  0g [ f.t cos ˛; t sin ˛/ W t  0g  R2 ; equipped with the induced distance from R2 and the one-dimensional Lebesgue measure , which is doubling on X . We want to show that X supports a 1-Poincaré inequality. Let B D B..x1 ; x2 /; r/, .x1 ; x2 / 2 X , be arbitrary. We may assume without loss of generality that x2 D 0. We will use that R supports a strong 1-Poincaré inequality, i.e. with dilation constant

4.8 The relation between L and 

111

 D 1. For the space X , let  D 1=sin ˛. Let further f be integrable and g be an upper gradient of f on X . If r  x1 sin ˛ or r > x1 , then B is connected and (when equipped with the inner metric din ) it is isomorphic to an interval in R. Hence we get « « « jf  fB j d  C diamin .B/ g d  C diam.B/ g d; B

B

B

by the strong 1-Poincaré inequality on R and the doubling property of . If x1 sin ˛ < r  x1 , then B is not connected, showing that we cannot have a strong p-Poincaré inequality on X , see below. However, B is connected, and using that B, equipped with the inner metric din , defined in Section 4.7, is isomorphic to an interval on R we find that « « « jf  fB j d  2 jf  fB j d  C jf  fB j d B B B « «  C diamin .B/ g d  C diam.B/ g d; B

B

by Lemma 4.17, the strong 1-Poincaré inequality on R and the doubling property of . Here diamin denotes the diameter with respect to the inner metric. The constant  D 1=sin ˛ is the smallest possible always making B connected, showing that we cannot have a p-Poincaré inequality with any dilation constant 0 < : If 1  0 < , then we can find B such that both B and 0 B are disconnected. Let then f be a Lipschitz function such that f j0 B D R j0 B , yielding «

« jf  fB j d > 0 D B

1=p p

0 B

g d

:

Thus we cannot have a p-Poincaré inequality with dilation constant 0 . A straightforward calculation, or a symmetry argument, shows that LD

2 cos.˛=2/ 1 2 D < D 2: sin.˛=2/ sin ˛ sin ˛

This shows that L can be strictly larger than , but also that it is possible to have arbitrarily large , while  < L  2. Example 4.51. In this example we consider the von Koch snowflake curve, which is a famous example of a curve (in the traditional sense) of infinite length containing no rectifiable curves, and thus not supporting a Poincaré inequality. For our discussion here, it is not the von Koch snowflake curve itself that is useful, but the sets generating it. Let K0  R2 , the 0th generation, be an equilateral triangle with side length 1. For each of the three sides, split it into three intervals of equal lengths and replace the

112

4 Poincaré inequalities

middle one I by the other two sides I 0 and I 00 of an equilateral triangle (with sides I , I 0 and I 00 ) outside K0 . We have thus produced the 1st generation K1 of the von Koch snowflake curve consisting of 12 pieces of length 13 each. Continuing in this way we obtain the nth generation Kn consisting of 3  4n pieces, each of length 3n . Let also En be the set of the end points of the pieces forming Kn . Now let X D Kn for some fixed integer n, equipped with the induced distance from R2 and the one-dimensional Lebesgue measure , which is doubling on X . As in the previous example we will use that R supports a strong 1-Poincaré inequality. Let f be integrable and g be an upper gradient of f on X . Let further B D B.x; r/ and find j such that 3j 1 < r  3j . Assume first that 1  j  1  n. Then we can find y 2 Kj 1 \ .Ej n Ej 1 / such that d.x; y/  3j . Let I be the piece containing y in the .j  1/th generation Kj 1 , and let I 0 and I 00 be its two neighbours in the .j  1/th generation. Let further E be the union of all the pieces in Kn stemming from any of these three pieces. (Hence E is the union of 3  4nj C1 pieces.) Then it is relatively easy to see that B D B.x; r/  B.y; 2  3j /  E  B.y; 5  3j /  B.y; 15r/  B.x; 18r/: Let thus  D 18. As E is connected, and isomorphic to an interval on R, Lemma 4.17 and the strong 1-Poincaré inequality on R imply that « « « jf  fB j d  2 jf  fE j d  C jf  fE j d B B E « « (4.18)  C diamin .E/ g d  C diam.B/ g d; E

B

where diamin .E/ is the diameter of E taken with respect to the inner metric, see Section 4.7. In the cases when j  1 < 1 and when j  1 > n this is easier to obtain, and thus we have shown that X supports a 1-Poincaré inequality with  D 18. Observe that  is independent of n. It is also easy to see that L ! 1 as n ! 1, thus showing that  can be much much smaller than L. An argument similar to the one showing that  D 18 above, shows that there is a common bound, independent of n, for the doubling constant C . As L ! 1, it follows from Theorem 4.32 that the constant C in the 1-Poincaré inequality for Kn must tend to 1, as n ! 1. (In (4.18) above, the blow-up of C occurs in the last inequality, since the comparison between diamin and diam gets worse and worse as n ! 1.) Note that the von Koch snowflake curve is the Hausdorff limit of Kn , see Example 1.23. Moreover, the doubling property and p-Poincaré inequality survive under Gromov–Hausdorff limits, see Appendix A.7. Thus, if there were a common bound for C , as n ! 1, then this would imply that the von Koch snowflake curve satisfied a 1-Poincaré inequality, which it does not by Remark 4.12, as there are no nonconstant rectifiable curves   K, see Example 1.23.

4.9 Measurability

113

4.9 Measurability Even though we neither need nor prove the following result we still want to mention it and some of its consequences. Theorem 4.52 (Järvenpää–Järvenpää–Rogovin–Rogovin–Shanmugalingam [190]). x has a Let X be a complete doubling p-Poincaré space. Assume that u W X ! R p p-weak upper gradient g 2 L .X /. Then u is measurable. In [190], it is assumed that u has an upper gradient in Lp .X /. Nevertheless, Lemma 1.46 allows us to formulate Theorem 4.52 in its present form. A direct consequence is that, under these assumptions on X , the measurability assumption in the definition of D p .X / is redundant. Another consequence is the following improvement upon Propositions 1.48 and 4.9. Proposition 4.53. Let X be a complete doubling p-Poincaré space and E  X be arbitrary. Then Cp .E/ D 0 if and only if Modp . E / D 0. Proof. Assume that Modp . E / D 0. Let v D E . As Modp . E / D 0, we see that g  0 is a p-weak upper gradient of v. Thus, by Theorem 4.52, v is measurable, and hence so is E. Thus Cp .E/ D 0, by Proposition 4.9. The converse implication follows from Proposition 1.48.

4.10 Notes The doubling condition and a p-Poincaré inequality are essential in Moser’s iteration method and other techniques for regularity of solutions to partial differential equations. This observation goes back to Fabes–Jerison–Kenig [116] and Fabes–Kenig– Serapioni [117], where six conditions sufficient for the regularity theory were singled out for weights in Rn . Even before [116], boundary regularity for equations with power weights was treated by Widman [350]. In Heinonen–Kilpeläinen–Martio [171], weights suitable for Moser’s iteration were called p-admissible and the number of sufficient conditions was reduced to four, viz. the doubling condition, a .p; p/-Poincaré inequality, a .q; p/-Sobolev inequality with some q > p, and a uniqueness condition for the gradient. The uniqueness condition was shown redundant by Semmes, see Theorem 5.2 in Heinonen–Koskela [172]. In unweighted Rn , the doubling condition is trivial and the Sobolev and Poincaré inequalities are well known. See e.g. Gilbarg–Trudinger [145], Malý–Ziemer [258] or Ziemer [361] for proofs. For Ap -weights on Rn , Sobolev and Poincaré inequalities were proved in Fabes–Kenig–Serapioni [117] and in Heinonen–Kilpeläinen–Martio [171], where also Jacobians of quasiconformal mappings were shown to be p-admissible weights. A simple proof for Ap -weights, including p D 1, was given in J. Björn [62] and generalized to Orlicz–Poincaré inequalities in J. Björn [68].

114

4 Poincaré inequalities

The self-improving property of the p-Poincaré inequality in metric spaces, implying both a .p; p/-Poincaré inequality and a .q; p/-Sobolev inequality, was proved in Hajłasz–Koskela [153] and strengthened in Hajłasz–Koskela [154], Theorem 5.1. Thus, the number of sufficient conditions for the regularity theory was further reduced to only two: the doubling condition and a p-Poincaré inequality. Earlier, Saloff-Coste [313] showed that a .2; 2/-Poincaré inequality on manifolds implies a .q; 2/-Sobolev inequality for some q > 2. Sharp Orlicz space analogues of these self-improving results have been obtained by Cianchi [96], [97] in Euclidean spaces and by Heikkinen [164], [165] on metric spaces equipped with doubling measures. See also Franchi–Pérez–Wheeden [127] and MacManus–Pérez [255], [256], where generalized Poincaré type inequalities were considered. Our proof of Theorem 4.21 is an adaptation from J. Björn [62] and follows [153] and [154]. Maz0 ya’s truncation method for obtaining strong type estimates from weak type inequalities as in Lemma 4.25 goes back to Maz0 ya [278]. Proposition 4.27 is essentially from Hajłasz–Koskela [154]. An exponential Trudinger inequality with the exponent s=.s  1/ for functions satisfying a p-Poincaré inequality with p D s has been proved in connected metric spaces equipped with doubling measures in [154], Theorem 6.1. Note that under our assumptions, X is connected. For the classical Trudinger inequality on Euclidean spaces see Trudinger [338]. It has been generalized to weighted spaces in Buckley– O’Shea [80] and to generalized Poincaré inequalities in MacManus–Pérez [256] and Heikkinen [164], [165] (also in disconnected spaces). Proposition 4.2 and Corollary 4.4 were observed by Shanmugalingam [318]. The proof of Lemma 4.5 is from A. Björn [43], where it is given under a milder assumption on X . The nontrivial implication was earlier deduced, in Björn–Björn–Shanmugalingam [56] under stronger assumptions, however only the p-Poincaré inequality was really used. Lemma 4.6 belongs to folklore. Estimates for the diameter and Hausdorff content of spheres in Poincaré spaces have recently been obtained by Korte [222]. The necessity of (4.3) for the .q; p/-Poincaré inequality (Proposition 4.20) appeared already in Franchi–Gutiérrez–Wheeden [125] in the setting of -gradients on Euclidean spaces. The proof of quasiconvexity is from Cheeger [91] based on an argument due to Semmes, cf. [315]. The relation between the quasiconvexity constant and the dilation constant in Theorem 4.39, as well as the observation that there is a lower bound on the constant CPI in the p-Poincaré inequality, seems to appear here for the first time, though they were probably known to the experts before. The Examples 4.50 and 4.51 are from Björn– Björn–Marola [53]. The first proof in which a strong Poincaré type inequality with dilation constant 1 is derived from a weak one was given by Jerison [191] for vector fields satisfying Hörmander’s condition. In Hajłasz–Koskela [153] a surprisingly elementary proof of this fact was given in the setting of general metric spaces satisfying a chain condition. On (weighted) Rn , this condition is trivially satisfied. Our proof of Theorem 4.39

4.10 Notes

115

combines J. Björn–Shanmugalingam [72] with J. Björn [62] and has been inspired by Hajłasz–Koskela [154] and Heinonen–Koskela [174]. Heinonen–Koskela [175] had a somewhat weaker version of Theorem 4.15. Another characterization of p-Poincaré inequalities by means of the approximate Lipschitz constant can be found in Keith–Rajala [198]. Counterexamples showing that completeness is needed in Theorems 4.15 and 4.30, and cannot even be replaced by local compactness, can be found in Koskela [228], where removable sets for W 1;p were characterized using Poincaré inequalities. In fact, the counterexamples in [228] are Ahlfors n-regular subsets of Rn . Removable singularities for Poincaré inequalities were studied by Koskela–Shanmugalingam–Tuominen [231] on metric spaces.

Chapter 5

Properties of Newtonian functions

We are now ready to have a look at some consequences of the doubling property and the Poincaré inequality. In this chapter we discuss density of Lipschitz functions, quasicontinuity, Sobolev inequalities, Sobolev embeddings, and existence of Lebesgue points q.e. Consequences for the capacity are postponed to Chapter 6.

5.1 Density of Lipschitz functions In Euclidean spaces, it is well known and often used that C 1 -smooth functions are dense in Sobolev spaces. On metric spaces, the notion of C 1 -smooth (or even C 1 smooth) functions is not available, but it is possible to consider Lipschitz and locally Lipschitz functions instead. Fortunately, this is enough in many applications. We start by the following global theorem, it will be extended to open subsets of X in Theorem 5.47. Theorem 5.1. If X is a doubling p-Poincaré space, then Lipschitz functions are dense in N 1;p .X /. To prove this result we will use McShane extensions and the following lemma. Lemma 5.2. Let E  X and let f W E ! R be L-Lipschitz. Then the upper and lower McShane extensions of f defined by fN.x/ WD inf .f .y/ C Ld.x; y// and f .x/ WD sup .f .y/  Ld.x; y//; y2E

y2E

respectively, are L-Lipschitz and satisfy f  fN in X , and f D f D fN on E. In fact, fN and f are the largest and the smallest, respectively, L-Lipschitz extensions of f to X . This lemma is not very difficult to prove; we refer the reader to the proof of Theorem 3.1.2 in Ambrosio–Tilli [17] or Theorem 6.2 in Heinonen [169]. Proof of Theorem 5.1. Let u 2 N 1;p .X /. Assume first that u is bounded and let g 2 Lp .X / be an upper gradient of u. For  > 0, we let E D fx 2 X W M  g p .x/ >  p g; where M  g p is the noncentred maximal function of g p . Let 0 < 12 r   r and x 2 X n E be arbitrary. The doubling property of  and the p-Poincaré inequality

117

5.1 Density of Lipschitz functions

then imply that «

«

juB.x; /  uB.x;r/ j 

ju  uB.x;r/ j d  C B.x; /

«

1=p

 Cr

p

g d

ju  uB.x;r/ j d B.x;r/

 C r.M  g p .x//1=p  C  r:

B.x;r/

Hence, for arbitrary 0 < < r and a suitable integer k, we have juB.x;r/  uB.x; / j  juB.x;r/  uB.x;r=2/ j C juB.x;r=2/  uB.x;r=4/ j C  C juB.x;r=2k /  uB.x; / j  C .r C r=2 C r=4 C  /  C 0  r:

(5.1)

It follows that every sequence fuB.x;rj / gj1D1 with rj ! 0, as j ! 1, is a Cauchy sequence and the limit u.x/ N D lim uB.x;r/ r!0

exists. We shall show that the function uN is C 0 -Lipschitz on X nE . Let x; y 2 X nE

be arbitrary and rj D 21j d.x; y/, j D 0; 1; ::: . Let also B0 D B.x; r0 / and Bj D B.x; rj /, Bj D B.y; rj /, j D 1; 2; ::: . From (5.1) we get ju.x/ N  uB0 j  C 0  d.x; y/ 0 and similarly ju.y/u N B0 j  C  d.x; y/. (The fact that B0 is not centred at y does not matter here, since we are using the noncentred maximal function M  g p .) It follows that uN is C 0  -Lipschitz on X n E and can be extended to a C 0  -Lipschitz function uN

on the whole of X , e.g. by the upper McShane extension

uN .x/ D

inf .u.y/ N C C 0  d.x; y//:

y2XnE

Let u WD maxfminfuN ; g;  g be the function obtained by truncating uN at the levels ˙ . As u is bounded, we have for large  that juj   in X and hence u D uN D u at all Lebesgue points for u in X n E . It follows that for large  , ku  u kLp .X/  kukLp .E / C ku kLp .E /  2 .E /1=p ! 0;

as  ! 1;

by Lemma 3.12. (Note that E here corresponds to E p in Lemma 3.12). Also, u  u D 0 on X n .E [ E/, where E is the set of non-Lebesgue points for u. Corollary 1.39 and Lemma 2.19 then show that .g C C 0  /E [E is a p-weak upper gradient of u  u . Since .E/ D 0, Lemma 3.12 yields kguu kLp .X/  kgkLp .E / C C 0 .E /1=p ! 0;

as  ! 1;

118

5 Properties of Newtonian functions

and the theorem is proved for bounded functions. If u is unbounded, approximate u by its truncations uk WD maxfminfu; kg; kg at the levels ˙k. Clearly, uk ! u in Lp .X / and .fx W ju.x/j > kg/ ! 0, as k ! 1. Lemma 2.19 shows that gfjuj>kg is a p-weak upper gradient of u  uk and hence kguuk kLp .X/ ! 0, as k ! 1. Remark 5.3. The proof shows that if u is nonnegative, then we can approximate it by nonnegative Lipschitz functions, as in this case u automatically becomes nonnegative. The same is true in Lemma 5.4, Corollary 5.15 and Theorem 5.47 below. The following consequence of Theorem 5.1 will be useful in the proofs of Theorems 6.7 (xi) and 6.19 (x). Lemma 5.4. Assume that X is a doubling p-Poincaré space. Let G  X be open and F be a closed subset of G such that at least one of F and X n G is bounded. If G ¤ X , assume also that dist.F; X n G/ > 0. Let u 2 N 1;p .X / be bounded in X and L-Lipschitz in G. Then there exist bounded Lipschitz functions u such that u ! u in N 1;p .X /, as  ! 1, and u D u on F . Observe that the condition dist.F; X n G/ > 0 is redundant if X is complete, as it is automatically fulfilled. Proof. As u isªL-Lipschitz in G, Proposition 1.14 implies that gu  L < 1 a.e. in G, and hence B gup d  Lp for all balls B  G. Note that if a ball intersects both F and X n G, then its radius must be at least 12 dist.F; X n G/ > 0. Moreover, by Lemma 3.6 and the boundedness of F or X n G, we have .B/  C > 0 for all such balls B. It follows that for x 2 F , the noncentred maximal function ² ³ « Z  p p p 1 p M gu .x/ D sup gu d  max L ; g d ; C X u B3x B i.e. F  fx 2 X W M  gup .x/   p g D X n E for sufficiently large . The proof of Theorem 5.1 now provides us with bounded Lipschitz functions u such that u ! u in N 1;p .X /, as  ! 1, and for large  , u D u at all Lebesgue points of u in X n E , in particular everywhere on F . The following two examples show that the validity of a p-Poincaré inequality is not necessary for the density of Lipschitz functions in N 1;p .X /. Example 5.5. If the space X contains no nonconstant rectifiable curves (e.g. if X is the von Koch snowflake curve), then N 1;p .X / D Lp .X /, see Example 1.22. It is well known that Lipschitz functions are dense in Lp .X /. (The basic ideas for a proof are given in Proposition 7.9 of Folland [124].) In this case the p-Poincaré inequality is obviously not supported, since g D 0 will be an upper gradient of any measurable function.

5.1 Density of Lipschitz functions

119

The next example is perhaps more illuminating than the previous one, since it contains an abundance of nonconstant rectifiable curves. The problem there for the p-Poincaré inequality is that there are not enough curves between the two halves of the space, i.e. that the space is too narrow at one place. In general, narrow passages are bad for Poincaré inequalities, they either destroy them altogether or force the constant CPI to become large. This can be exploited when constructing counterexamples: creating cusps or “tunnels” which are more and more narrow will prevent Poincaré inequalities. Examples of this kind were used in Maz0 ya [283] to disprove Sobolev embeddings on bad domains. Example 5.7 below shows that it is not just the geometric narrowness and its relation to p that is relevant for the p-Poincaré inequality, but the narrowness has to be considered relative to the measure as well. It should also be mentioned that it is not very well understood exactly when a space supports a p-Poincaré inequality. Another way of preventing the p-Poincaré inequality is by violating quasiconvexity, cf. Theorem 4.32. Example 5.6 (Bow-tie). Let n  2 and write x 2 Rn as x D .x1 ; ::: ; xn /. Let XC D fx 2 Rn W xj  0; j D 1; ::: ; ng; X D fx 2 Rn W xj  0; j D 1; ::: ; ng; equipped with the Euclidean metric and the Lebesgue measure, which is doubling (and even Ahlfors n-regular) both on XC and X . Let 1  p  n and X D XC [ X . We shall show that X does not support the p-Poincaré inequality but that Lipschitz functions are dense in N 1;p .X /. The functions ²   ³  log jxj uj D min 1; j C are Lipschitz in X and uj .0/ D 1. Since uj .x/ ! 0, as j ! 1, if x ¤ 0, dominated convergence shows that kuj kLp .X/ ! 0, as j ! 1. Moreover, if p < n, then Z

Z X

gupj D 

1



e j

1 jr

p

r n1 dr 

1 jp

Z

1

r n1p dr ! 0;

as j ! 1:

 log e j D j 1n ! 0; jn

as j ! 1:

0

Similarly, if p D n, then Z

Z X

gupj

D

1 e j



1 jr

n

r n1 dr D

Hence kuj kN 1;p .X/ ! 0, as j ! 1. Thus Cp .f0g/ D 0 and since f0g separates X , the p-Poincaré inequality is violated, by Lemma 4.6. (This can also be seen directly, viz. XC has 0 as a p-weak upper gradient, since Modp . f0g / D 0 by Proposition 1.48, which violates the p-Poincaré inequality.)

120

5 Properties of Newtonian functions

Note, however, that both XC and X support 1-Poincaré inequalities. Indeed, if u 2 N 1;p .XC /, then u and all of its upper gradients g can be extended to the whole of Rn by reflection as follows, u.x N 1 ; ::: ; xn / D u.jx1 j; ::: ; jxn j/

and

g.x N 1 ; ::: ; xn / D g.jx1 j; ::: ; jxn j/;

so that gN is an upper gradient of uN 2 N 1;p .Rn / and kuk N pN 1;p .Rn /  2n kukpN 1;p .X / . C The 1-Poincaré inequality for XC then follows from the 1-Poincaré inequality for Rn . Indeed, for every ball B D B.x0 ; r/  Rn with x0 2 XC we have « « « « ju  uN B j dx  C juN  uN B j dx  C r gN dx  C r g dx B\XC

B

B

B\XC

and Lemma 4.17 makes it possible to replace uN B by uB\XC . To prove the density of Lipschitz functions in N 1;p .X /, let u 2 N 1;p .X / and note that both uXC and uX belong to N 1;p .X /, due to the fact that Cp .f0g/ D 0. Thus, it suffices to approximate uXC by Lipschitz functions in N 1;p .X /. Let " > 0. Since XC supports a p-Poincaré inequality, there exists a Lipschitz function v on XC such that ku  vkN 1;p .XC / < ". Extend v by zero to X n XC . We shall show that v can be approximated by Lipschitz functions in N 1;p .X /. Let w D XC , where 2 Lipc .X / is such that D 1 on B.0; 1/. Then v  v.0/w is Lipschitz on X and as v D .v  v.0/w/ C v.0/w, it suffices to approximate w by Lipschitz functions in N 1;p .X /. It is easily verified that the functions vj .x/ D maxfw; uj g will do. Observe that in Example A.23 we show that for p > n, X is a p-Poincaré space. In particular, it follows from the arguments above that in this case Cp .f0g/ > 0 and XC … N 1;p .X /. Example 5.7 (Weighted bow-tie). Let X be as in Example 5.6, but equip it this time with the measure d D jxj˛ dx, ˛ > n. If p > ˛ C n or p D 1  ˛ C n, then X is a doubling p-Poincaré space, by Example A.24. On the other hand if 1 < p  ˛ C n or p D 1 < ˛ C n, then a similar calculation as in Example 5.6 shows that Cp .f0g/ D 0, and the p-Poincaré inequality is thus violated. Nevertheless, the same argument as in Example 5.6 shows that Lipschitz functions are dense in N 1;p .X /. In view of the preceding examples it is worth pointing out that Lipschitz functions are not always dense in N 1;p .X /. Let us first present three concrete examples and then a more abstract one which contains the first three as special cases. We give a full proof in the last example, which also proves our claims in the first three examples. Example 5.8. Let X D fz 2 C D R2 W jzj D 1g n f1g (with the one-dimensional Lebesgue measure). Then f .z/ D arg z (the principal branch) belongs to N 1;p .X / n Lip.X / (for all p  1).

5.1 Density of Lipschitz functions

121

Example 5.9. Let X be the slit disc B.0; 1/ n .1; 0  C D R2 . Then f .z/ D .2jzj  1/C arg z belongs to N 1;p .X / n Lip.X /. Example 5.10. Let p > n  2, and X D .XC [ X [ fx 2 Rn W jxj  1g/ n f0g; where XC and X are given by Example 5.6. Let further u D minf1; 2  2jxjgC XC . Then u 2 N 1;p .X / n Lip.X /. Example 5.11. Assume that X supports a p-Poincaré inequality. Let F  X be a closed set with zero measure which locally separates X at some point x0 2 F , i.e. there is r > 0 and two nonempty disjoint open sets U and V such that B.x0 ; r/ n F D U [ V and x0 2 @U \ @V . Let u.x/ D minf1; 2  2d.x; x0 /=rgC U as a function on G D X n F . Observe that u D U on B.x0 ; r=2/ \ G and that u D 0 outside B.x0 ; r/. Clearly, u 2 Lp .G/ and gu  .2=r/U nB.x0 ;r=2/ 2 Lp .G/. Hence u 2 N 1;p .G/. We shall show that u … Lip.G/ (with respect to the N 1;p .G/ norm). Assume on the contrary that there are uj 2 Lip.G/ such that kuj ukN 1;p .G/ ! 0, as j ! 1. Since uj is Lipschitz on G (and .F / D 0) it has a unique continuous extension to X (which we also call uj ). It coincides with the McShane extensions given by Lemma 5.2. By Lemma 2.23, guj ;G D guj ;X jG , and as .F / D 0, we have kuj kN 1;p .G/ D kuj kN 1;p .X/ . Since fuj gj1D1 is a Cauchy sequence in N 1;p .G/, and as kuj  uk kN 1;p .G/ D kuj  uk kN 1;p .X/ ;

j; k D 1; 2; ::: ;

it follows that fuj gj1D1 is a Cauchy sequence in N 1;p .X / as well. Hence, by the completeness of N 1;p .X /, given by Theorem 1.71, there is a function v 2 N 1;p .X / such that uj ! v in N 1;p .X /. Since obviously uj ! v also in N 1;p .G/, it follows that u D v q.e. in G. In particular, gv D gu D 0 a.e. in B.x0 ; r=2/ \ G (using again Lemma 2.23), and as .F / D 0 we have gv D 0 a.e. in B.x0 ; r=2/. Hence, by the p-Poincaré inequality on X , we have Z

Z 0< B.x0 ;r=2/

jv  vB.x0 ;r=2/ j d  C r

B.x0 ;r=2/

1=p gvp d

D 0;

where  is the dilation constant in the p-Poincaré inequality, a contradiction. We have thus shown that u cannot be approximated in N 1;p .G/ by Lipschitz functions on G. Observe, however, that u is locally Lipschitz in G. In the last four examples, locally Lipschitz functions are dense in N 1;p .X / (N 1;p .G/ in the last example) and moreover X (G in the last example) is noncomplete. This suggests the following problems.

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5 Properties of Newtonian functions

Open problem 5.12. Is it always true that locally Lipschitz functions are dense in N 1;p .X /? Open problem 5.13. Is it always true that Lipschitz functions are dense in N 1;p .X / if X is proper? In fact, the following result shows that an affirmative answer to Open problem 5.12 would imply an affirmative answer to Open problem 5.13. See also the discussion on Open problems 5.34–5.36. Proposition 5.14. Let X be proper and assume that locally Lipschitz functions are dense in N 1;p .X /. Then Lipc .X / D N 1;p .X /. Proof. If X is bounded, then it is compact and every locally Lipschitz function on X is Lipschitz and the claim follows. If S X is unbounded, then we may write X as an increasing union of compact sets, 1;p .X / and " > 0 be arbitrary, and find a locally Lipschitz XD 1 kD1 Xk . Let u 2 N function v with an upper gradient g 2 Lp .X /, such that ku  vkN 1;p .X/ < ". Then v 2 Lp .X / and Z .jvj C g/p d ! 0;

as k ! 1:

XnXk

Let k .x/ D .1dist.x; Xk //C . Then v k 2 Lipc .X / and .jvjCg/XnXk is a p-weak upper gradient of v  v k . Hence Z kv  v k kN 1;p .X/  21=p

1=p .jvj C g/p d

! 0;

as k ! 1:

XnXk

It follows that for sufficiently large k, ku  v k kN 1;p .X/ < 2" and as " was arbitrary, the result follows. Corollary 5.15. If X is a complete doubling p-Poincaré space, then Lipc .X / is dense in N 1;p .X /. Proof. This follows directly from Theorem 5.1 and Proposition 5.14. The following example shows that Proposition 5.14 is not true without the properness assumption. Example 5.16. Let X D .0; 1  R. In this case X is a noncomplete doubling 1-Poincaré space, and Lip.X / D N 1;p .X /, by Theorem 5.1. Let u  1 2 N 1;p .X /. We shall show that u … Lipc .X /. Let v 2 Lipc .X /. If supX v  12 , then ku  vkN 1;p .X/  ku  vkLp .X/  12 :

5.2 Quasicontinuity of Newtonian functions

123

Otherwise there is x 2 X such that v.x/  12 . As supp v b X , there is 0 < y < x such that v.y/ D 0. Hence 1  jv.y/  v.x/j  2

Z

Z

x

gv ds  y

y

1=p

x

gvp

ds

 ku  vkN 1;p .X/ :

We thus see that u … Lipc .X /.

5.2 Quasicontinuity of Newtonian functions We have seen in Proposition 1.61 that, unlike the usual Sobolev functions which are defined a.e., the equivalence classes in Newtonian spaces are up to sets of capacity zero. In this section we study continuity properties of Newtonian functions in doubling p-Poincaré spaces. In particular, we prove a Luzin type theorem in the capacitary sense, i.e. we show that Newtonian functions are continuous on complements of sets with arbitrarily small capacity. In Section 11.6, we obtain further continuity results for Newtonian functions. x is weakly quasicontinuous if for every " > 0 Definition 5.17. A function u W X ! R there is a set E with Cp .E/ < " such that ujXnE is continuous. x is quasicontinuous if for every " > 0 there is an open set G A function u W X ! R with Cp .G/ < " such that ujXnG is continuous. Recall that a continuous function is assumed to be real-valued. We also point out that it is the restriction ujXnE which is continuous, not that u is continuous at points in X n E. Quasicontinuity is a local property, as the following lemma shows. Lemma 5.18. If u is (weakly) quasicontinuous in j for every j D 1; 2; ::: , then u is S1 (weakly) quasicontinuous in  D j D1 j . Proof. Let " > 0. For each j D 1; 2; ::: , there is an open S (arbitrary) set Gj with Cp .Gj / < 2j " such that ujj nGj is continuous. Let G D j1D1 Gj . Then G is open (arbitrary) and Cp .G/ < ", by Theorem 1.27. Moreover, ujnG is continuous. Theorems 5.20 and 5.21 below form the first step in showing that Newtonian functions correspond to the “good” representatives of Sobolev functions. In doubling p-Poincaré spaces, this is a consequence of Theorem 5.1 and Corollary 1.72, but we formulate the results in more generality, assuming that continuous functions are dense in N 1;p .X /. Note that the doubling condition and the p-Poincaré inequality are not necessary for the density of continuous functions in N 1;p .X /, see Examples 5.5–5.11.

124

5 Properties of Newtonian functions

In the second half of the book (Chapters 7–14) we shall mainly consider Newtonian functions on open subsets  of X . However, as  may not support a p-Poincaré inequality, Theorem 5.1 cannot be applied directly to N 1;p ./. Fortunately, as quasicontinuity is a local property, by Lemma 5.18, this problem can easily be overcome. Since we now work under the assumption that continuous functions are dense in 1;p N .X / (rather than X being a doubling p-Poincaré space), it is possible to have X ¤ supp . This obstacle is however easily overcome using the following lemma. The reader may recall that this possibility was taken into account in Chapters 1 and 2. On the other hand, in Chapters 3 and 4 we either assumed that  was doubling or that X supported some Poincaré inequality (or both), and in both cases a requirement is that X D supp . Lemma 5.19. Let X 0 D supp  and 0 D  \ X 0 , which is open as a subset of X 0 . x be arbitrary, E 0 WD E \ X 0 and u0 WD uj0 . Then Let further E  X and u W  ! R the following are true: (a) u 2 N 1;p ./ if and only if u0 2 N 1;p .0 /, and moreover kukN 1;p ./ D ku0 kN 1;p .0 / I

(5.2)

1;p p 1;p (b) u 2 Nloc ./, u 2 D p ./ and u 2 Dloc ./ if and only if u0 2 Nloc .0 /, p 0 p 0 0 0 u 2 D . / and u 2 Dloc . /, respectively;

(c) .E 0 / D .E/ and Cp0 .E 0 / D Cp .E 0 / D Cp .E/, where Cp0 denotes the capacity with respect to X 0 ; (d) u is (weakly) quasicontinuous in  if and only if u0 is (weakly) quasicontinuous in 0 ; (e) continuous (Lipschitz) functions are dense in N 1;p .X / if and only if continuous (Lipschitz) functions are dense in N 1;p .X 0 /. Proof. Note that one implication in each equivalence is trivial. Let Z D X n X 0 . Then Z is open and .Z/ D Cp .Z/ D 0, by Proposition 1.53. (a) Let g 0 be an upper gradient of u0 (in 0 / and let ´ g 0 in 0 ; gD 1 in  n 0 D  \ Z: Consider a curve  W Œ0; l  ! . Either   0 , in which case g clearly is an upper gradient of u along  , or  hits Z. In the latter case  must hit Z for a positive length, as Z is open, and thus Z g ds D 1  ju..0//  u..l //j: 

Hence g is an upper gradient of u.

5.2 Quasicontinuity of Newtonian functions

125

Since .Z/ D 0 we see that kukLp ./ D ku0 kLp .0 / and kgkLp ./ D kg 0 kLp .0 / from which (5.2) follows. (b) This follows directly from (a) and its proof. (c) The equality of the measures follows directly from the fact that .Z/ D 0. As for the capacity, let " > 0 and v 0 W X 0 ! R be such that v 0  1 on E 0 and kv 0 kpN 1;p .X 0 /  Cp0 .E 0 / C ": ´

Define vD

v0 1

in X 0 ; in Z:

Then, by (a), Cp .E/  kvkpN 1;p .X/ D kv 0 kpN 1;p .X 0 /  Cp0 .E 0 / C ": Letting " ! 0 shows that Cp .E/  Cp0 .E 0 /. Since trivially, Cp0 .E 0 /  Cp .E 0 /  Cp .E/, we are done. (d) Assume that u0 is (weakly) quasicontinuous in 0 and let " > 0. Then there is a relatively open (arbitrary) set G 0  0 such that Cp0 .G 0 / < " and u0 j0 nG 0 is continuous. Let G D G 0 [ .Z \ / which is an open (arbitrary) subset of , since Z \  is open. Moreover, ujnG D u0 j0 nG 0 and is thus continuous. By (c), Cp .G/ D Cp0 .G 0 / < ". Hence u is (weakly) quasicontinuous in . (e) Assume that continuous (Lipschitz) functions are dense in N 1;p .X 0 /. Let u 2 N 1;p .X /, u0 WD ujX 0 and " > 0. Then there is a continuous (Lipschitz) function v 0 W X 0 ! R such that kv 0  u0 kN 1;p .X 0 / < ". By Tietze’s extension theorem (see e.g. Theorem 4.16 in Folland [124]) or McShane’s extension lemma (Lemma 5.2), there is a continuous (Lipschitz) extension v of v 0 to X . By (a), kv  ukN 1;p .X/ D kv 0  u0 kN 1;p .X 0 / < ": Hence continuous (Lipschitz) functions are dense in N 1;p .X /. Theorem 5.20. Assume that continuous functions are dense in N 1;p .X /. (In particu1;p ./, then u is weakly lar, this holds if X is a doubling p-Poincaré space.) If u 2 Nloc quasicontinuous. Note that Theorem 5.20 can also be obtained from Theorem 5.21, whose proof is, however, much more complicated. See Theorem 5.29 for a further improvement in complete doubling p-Poincaré spaces, which will be the standing assumption in the second half of the book (Chapters 7–14). Proof. In view of Lemma 5.19 we may assume that X D supp . For every x 2 , there exists rx > 0 such that u 2 N 1;p .B.x; 2rx //. As  is Lindelöf, S by Proposition 1.6, there exist countably many xj 2 , j D 1; 2; ::: , such that  D j1D1 Bj , where Bj D B.xj ; rxj /.

126

5 Properties of Newtonian functions

Choose j 2 Lip.X / such that j D 1 on Bj and supp j  2Bj , j D 1; 2; ::: . Then u j 2 N 1;p .X / and there exists a sequence of continuous functions converging to u j in N 1;p .X /. Corollary 1.72 provides us with a subsequence converging to u j uniformly outside sets of arbitrarily small capacity. Thus, u j is continuous on the complements of these sets, showing that u D u j is weakly quasicontinuous in each Bj . Lemma 5.18 completes the proof. Theorem 5.21. Assume that continuous functions are dense in N 1;p .X /. (In particu1;p lar, this holds if X is a doubling p-Poincaré space.) Then every u 2 Nloc ./ has a representative v D u q.e. which is quasicontinuous in . Proof. In view of Lemma 5.19 we may assume that X D supp . 1;p ./. By definition, for every x 2  there exists rx > 0 such that Let u 2 Nloc 1;p u 2 N .B.x; 2rx //. As  is Lindelöf, S by Proposition 1.6, there exist countably many xj 2 , j D 1; 2; ::: , such that  D j1D1 Bj , where Bj D B.xj ; rxj /. Choose j 2 Lipc .2Bj / such that j D 1 on Bj , j D 1; 2; ::: . Then u j 2 N 1;p .X / and there exists a sequence of continuous functions converging to u j in N 1;p .X /. Corollary 1.72 provides us, for each fixed j , with a subsequence f'j;i g1 iD1 and open sets Gj;k with Cp .Gj;k / < 2j k , such that 'j;i converge uniformly in S X n Gj;k , k D 1; 2; ::: , and pointwise q.e. in X . Let Gk D j1D1 Gj;k , so that Cp .Gk / < 2k , k D 1; 2; ::: . Let vj D lim supi!1 'j;i . Then vj jXnGk is continuous for allSk D 1; 2; ::: and vj D u j in X n Ej , where Cp .Ej / D 0. Let further E D j1D1 Ej , so that Cp .E/ D 0. Let now i; j; k D 1; 2; ::: be fixed but arbitrary, and Vk D X n .X n .Gk [ E//, which is an open set. Note that Vk n Gk  E and hence Cp .Gk [ Vk /  Cp .Gk / C Cp .Vk n Gk / D Cp .Gk / < 2k : As .E/  Cp .E/ D 0, we see that int E D ¿. Hence, for every x 2 Bi \ Bj n .Gk [ Vk / D Bi \ Bj \ .X n .Gk [ E// n Gk ; there exist yl 2 Bi \ Bj n .Gk [ E/ such that yl ! x, as l ! 1. In Bi \Bj n.Gk [E/ we have vi D u i D u D u j D vj . Since both vi jBi \Bj nGk and vj jBi \Bj nGk are continuous, it follows that vi .x/ D lim vi .yl / D lim vj .yl / D vj .x/; l!1

l!1

in Bi \ Bj n .Gk [ Vk /, for all i; j; k D 1; 2; ::: . i.e. vi D vj everywhere T1 Let F D kD1 .Gk [ Vk / and ´ vj .x/; x 2 Bj n F; v.x/ D 0; x 2 F:

5.2 Quasicontinuity of Newtonian functions

127

As Gk [ Vk is open, v is quasicontinuous in Bj for each j D 1; 2 ::: , and thus in , by Lemma 5.18. Moreover, as Cp .F / D 0 and v D vj D u in Bj n F , j D 1; 2; ::: , we have v D u q.e. in X . Let us next prove a useful rigidity result for the capacity and for quasicontinuous functions. Proposition 5.22. If G  X is open and .E/ D 0, then Cp .G/ D Cp .G n E/: Proof. That Cp .G/  Cp .G n E/ is clear. Let " > 0 and u be such that GnE  u  1 and kukpN 1;p .X/ < Cp .G n E/ C ". Let further g be an upper gradient of u such that p p kukL p .X/ C kgkLp .X/ < Cp .G n E/ C ". Let ´ vD

u 1

in X n G; in G:

Then u D v outside G \ E, so, in particular, kukLp .X/ D kvkLp .X/ . We shall show that g is a p-weak upper gradient of v from which it follows that Cp .G/  kvkpN 1;p .X/ < Cp .G n E/ C "; and the result follows after letting " ! 0. To show that g is a p-weak upper gradient we consider a curve  W Œ0; l  ! X , C . If .0/; .l / … G \ E, then  … G\E Z jv..0//  v..l //j D ju..0//  u..l //j  g ds: 

On the other hand if .0/; .l / 2 G \ E, then Z jv..0//  v..l //j D 0 

g ds: 

Consider next the case when .0/ 2 G \ E and .l / … G \ E. Since G is open there C is t > 0 so that .Œ0; t /  G. As  … G\E , there is  2 Œ0; t  such that . / 2 G nE. Thus v..0// D 1 D u.. // and hence Z Z jv..0//  v..l //j D ju.. //  u..l //j  g ds  g ds: jŒ;l 



Finally, the case when .l / 2 G \ E and .0/ … G \ E is handled by considering the reversed curve. C By Lemma 1.42, Modp . G\E / D 0. Thus g is a p-weak upper gradient of v.

128

5 Properties of Newtonian functions

Proposition 5.23. If u and v are quasicontinuous and u D v a.e., then u D v q.e. Proof. Let " > 0. As u and v are quasicontinuous, we can find an open set G with Cp .G/ < " such that ujXnG and vjXnG are continuous. Thus fx … G W u.x/ ¤ v.x/g is open in the relative topology on X n G, i.e. there is an open set U  X such that U n G D fx … G W u.x/ ¤ v.x/g: Since U [ G is open and .U n G/ D 0, Proposition 5.22 with G and E replaced by U [ G and U n G shows that Cp .fx W u.x/ ¤ v.x/g/  Cp .U [ G/ D Cp .G/ < ": Letting " ! 0 completes the proof. Next, we construct functions with prescribed upper gradients and show that under certain assumptions they are lower semicontinuous. This will be a crucial ingredient when showing that in complete doubling p-Poincaré spaces, all Newtonian functions are quasicontinuous. Definition 5.24. We say that a curve  W Œ0; l  ! X connects two points x and y if .0/ D x and .l / D y. Similarly,  connects x to a set A if .0/ D x and .l / 2 A. Lemma 5.25. Let W X ! Œ0; 1 be a Borel function and A  X . Let ³ ² Z u.x/ D min 1; inf ds ; 



where the infimum is taken over all curves (including constant curves) connecting x to the set A. If there are no such curves, we let the infimum be infinity. Then u D 0 on A and is an upper gradient of u. Moreover, if A is closed, then the function XnA is also an upper gradient of u. Proof. That u D 0 on A is clear (use the constant curves). Let x; y 2 X be arbitrary and let xy be a curve connecting x and y. We may assume that 0  u.y/ < u.x/  1. If  is a curve connecting y to A, then the curve xy C  connects x to A and hence Z Z Z ds D ds C ds: u.x/  xy C

xy



Taking infimum over all curves  connecting y to A shows that Z ds C u.y/; u.x/  xy

i.e. is an upper gradient of u.

5.2 Quasicontinuity of Newtonian functions

129

Now assume that A is closed and let  W Œ0; l  ! X be a curve connecting x and y. The cases   A and   X n A are trivial. If  intersects both A and X n A, then by splitting  into two parts if necessary, and possibly reversing the direction, we can assume that x 2 X n A and y 2 A. Let c D infft 2 Œ0; l  W .t / 2 Ag. As A is closed, we have .Œ0; c//  X n A with z D .c/ 2 A, and hence Z Z ju.x/  u.y/j D ju.x/  u.z/j  ds  XnA ds: jŒa;c



Lemma 5.26. Let F be a nonempty closed subset of a proper space X such that at least one of F and X n F is bounded. Let W X ! Œ0; 1 be lower semicontinuous and such that  c > 0 in X n F . Let Z u.x/ D inf ds; 



where the infimum is taken over all curves (including constant curves) connecting x to F . Then u is lower semicontinuous. Proof. Clearly, u D 0 on F . Let a > 0 be arbitrary and let Fa D fx 2 X W u.x/  ag: We shall show that Fa is closed. Assume that xj 2 Fa and that xj ! x 2 X , as j ! 1. If for sufficiently large j we have xj 2 F , then as F is closed we have x 2 F and hence x 2 Fa . Thus, without loss of generality, we assume that xj … F for j D 1; 2; ::: . Then there exist curves j W Œ0; lj  ! X (parameterized by arc length) such that j .0/ D xj , j .lj / 2 F and Z 1 ds  a C ; j D 1; 2; ::: : j j Let tj D infft 2 .0; lj  W j .t / 2 F g. Since F is closed, j .tj / 2 F and we may assume that tj D lj . Reparameterize j by Qj .t / D j .t lj /, so that Qj W Œ0; 1 ! X , Qj .0/ D xj and Qj .1/ 2 F . Note that each Qj is lj -Lipschitz and Z aC1 lj  ds  : c j c This together with the assumptions on F also shows that all Qj are equicontinuous and have images contained in a bounded set. Thus, by the Ascoli theorem (see e.g. p. 169 in Royden [310], note that here we need a version of Ascoli’s theorem valid for metric space valued equicontinuous functions), a subsequence of fQj gj1D1 converges uniformly to a continuous function Q W Œ0; 1 ! X with Q .0/ D x and Q .1/ 2 F (Q is a curve in the traditional sense, but need not be parameterized by arc length).

130

5 Properties of Newtonian functions

As is lower semicontinuous, we have for all t 2 Œ0; 1, . ı Q /.t /  lim inf . ı Qj /.t / j !1

and the Fatou lemma implies that Z Z a  lim inf ds D lim inf j !1

Z

 L lim inf j !1

j !1

j

lj

. ı j /.t / dt

0

1

Z

1

. ı Qj /.t / dt  L 0

. ı Q /.t / dt; 0

where L D lim infj !1 lj . Note that as each Qj is lj -Lipschitz, the uniform convergence shows that Q is L-Lipschitz and hence the length function sQ .t / WD lj Q Œ0;t  is L-Lipschitz and thus differentiable a.e. in Œ0; 1 with 0  s0Q .t /  L. Reparameterize Q by arc length as Qs .s/ D Q .s1 .s//, where Q s1 Q .s/ D supft W sQ .t / D sg is the generalized right inverse of sQ . It follows that Z lQ Z 1 Z 1 Z 0 ds D . ı Qs /. / d  D . ı Qs ı sQ /.t /sQ .t / dt  L . ı Q /.t / dt  a; Q

0

0

0

i.e. u.x/  a and x 2 Fa . Thus Fa is closed and X n Fa is open. Hence, by definition, u is lower semicontinuous. The following result shows that Cp is an outer capacity at the level of zero sets. This fact will be essential for proving that in complete doubling p-Poincaré spaces all Newtonian functions are quasicontinuous, see Theorem 5.29. We do not know if this is true without the assumption that X is proper. Proposition 5.27. Let X be proper and E  X with Cp .E/ D 0. Then for every " > 0, there exists an open set U  E with Cp .U / < ". Note that as Cp is not additive, it does not immediately follow that Cp is an outer capacity for all sets, cf. Theorem 5.31. Proof. Assume first that E is bounded. As Cp .E/ D 0, we have E 2 N 1;p .X / and there exists an upper gradient g 2 Lp .X / of E . The Vitali–Carathéodory theorem (Proposition 7.14 in Folland [124]) provides us with a lower semicontinuous function 2 Lp .X / such that  g. Let " > 0 be arbitrary. By the definition of capacity, .E/ D 0 and as C 1 2 Lploc .X /, there exists a bounded open set V  E such that Z .V / C . C 1/p d < 2p ": V

5.2 Quasicontinuity of Newtonian functions

131

³ ² Z u.x/ D min 1; inf . C 1/ ds ;

Let





where the infimum is taken over all curves (including constant curves) connecting x to the˚closed set X n V . By Lemma 5.26, u is lower semicontinuous and thus the set U D x 2 X W u > 12 is open. Note that for x 2 E and every curve  connecting x to some y 2 X n V , Z Z . C 1/ ds  g ds  E .x/  E .y/ D 1; 



i.e. u D 1 on E and E  U . Lemma 5.25 shows that the function . C 1/V is an upper gradient of u and u D 0 in X n V . Thus, as u  1 is lower semicontinuous and hence measurable, we have u 2 N 1;p .X / and hence   Z p p p p Cp .U /  2 kukN 1;p .X/  2 .V / C . C 1/ d < "; V

which proves the claim for bounded E. If E is unbounded, write E as a countable union of bounded sets Ej , j D 1; 2; ::: , and using the above Ej such that Cp .Uj / < 2j ". The S1 argument, find open sets Uj P open set U D j D1 Uj contains E and Cp .U /  j1D1 Cp .Uj / < ". A consequence of Proposition 5.27 is that quasicontinuity is preserved under perturbations on sets of capacity zero. Note that a similar result for weak quasicontinuity follows directly from the definition. Proposition 5.28. Assume that X is proper, that u is quasicontinuous in  and that v D u q.e. in . Then v is also quasicontinuous in . Proof. Let " > 0. As u is quasicontinuous in , we can find an open set G with Cp .G/ < " such that ujnG is continuous. Let E D fx 2  W u.x/ ¤ v.x/g. By assumption Cp .E/ D 0, and thus, by Proposition 5.27, there is an open set U  E with Cp .U / < ". Hence vjn.G[U / D ujn.G[U / is continuous, and G [ U is an open set with Cp .G [ U / < 2". As " > 0 was arbitrary, this shows that v is quasicontinuous in . Theorem 5.21 and Proposition 5.28 now imply that in complete doubling p-Poincaré spaces, all Newtonian functions are quasicontinuous. This has important consequences for the capacity (see Theorems 5.31 and 6.7) and for the Dirichlet problem (see Section 10.4). As before, we formulate the result in the more general case, assuming that continuous functions are dense in N 1;p .X /. Note that we prove quasicontinuity for functions on  even though  is neither complete nor supports a p-Poincaré inequality in general.

132

5 Properties of Newtonian functions

Theorem 5.29. Let X be proper and assume that continuous functions are dense in N 1;p .X /. (In particular, this holds if X is a complete doubling p-Poincaré space.) 1;p Then every u 2 Nloc ./ is quasicontinuous in . 1;p Proof. Let u 2 Nloc ./. By Theorem 5.21 there is a quasicontinuous representative v D u q.e. in . By Proposition 5.28, u is also quasicontinuous in .

Example 5.30. If the space X contains no nonconstant rectifiable curves (e.g. if X is the von Koch snowflake curve), then N 1;p .X / D Lp .X / and Cp .E/ D .E/, see Example 1.22. As Lipschitz functions are dense in Lp .X /, see Example 5.5, Theorem 5.29 shows that all functions in N 1;p .X / D Lp .X / are quasicontinuous, if X is proper. However, as Cp .E/ D .E/, the quasicontinuity in this case (assuming that X is locally compact) follows directly from Luzin’s theorem, see e.g. Folland [124], Theorem 7.10, or Rudin [311], Theorem 2.24. (Incidentally, Luzin’s theorem (on R) was first obtained by Vitali [347] in 1905, while Luzin [250] obtained it in 1912, see Bourbaki [74], p. 223.) Quasicontinuity of Newtonian functions implies that Cp is an outer capacity in the following theorem. In particular, this holds if X is proper and continuous functions are dense in N 1;p .X /. Theorem 5.31. Assume that all functions in N 1;p .X / are quasicontinuous. (In particular, this holds if X is a complete doubling p-Poincaré space.) Then Cp is an outer capacity, i.e. for all E  X , Cp .E/ D inf Cp .G/: GE G open

Proof. Let 0 < " < 1 and u  E with kukN 1;p .X/  Cp .E/1=p C ": By assumption, u is quasicontinuous. Hence there is an open set V with Cp .V /1=p < " such that ujXnV is continuous. Thus, there is an open set U such that U n V D fx W u.x/ > 1  "g n V  E n V: We can also find v  V with kvkN 1;p .X/ < ". Let u C v: 1" Then w  1 on .U n V / [ V D U [ V , an open set containing E. Hence wD

Cp .E/1=p  inf Cp .G/1=p  Cp .U [ V /1=p  kwkN 1;p .X/ GE G open

1 1 kukN 1;p .X/ C kvkN 1;p .X/  .Cp .E/1=p C "/ C ": 1" 1" Letting " ! 0 completes the proof. 

5.2 Quasicontinuity of Newtonian functions

133

The following is a consequence of the fact that Cp is an outer capacity. Proposition 5.32. Assume that Cp is an outer capacity. (In particular, this holds if X is a complete doubling p-Poincaré space.) Then a function is weakly quasicontinuous in  if and only if it is quasicontinuous in . Proof. Assume that u is weakly quasicontinuous in . Let " > 0. Then there exists E   such that Cp .E/ < " and ujnE is continuous. Since Cp is an outer capacity there is an open set G  E with Cp .G/ < ". As ujnG is continuous, it follows that u is quasicontinuous in . The converse implication is trivial. We are now ready to give the following characterization of Newtonian functions in relation to arbitrary a.e.-representatives. Recall that Ny 1;p ./ D fu W u D v a.e. for some v 2 N 1;p ./g. Proposition 5.33. Assume that X is proper and that continuous functions are dense in N 1;p .X /. (In particular, this holds if X is a complete doubling p-Poincaré space.) If u 2 Ny 1;p ./, then the following are equivalent: (a) u 2 N 1;p ./; (b) u 2 ACCp ./; (c) u is quasicontinuous; (d) u is weakly quasicontinuous. 1;p ./, then the following are equivalent: Similarly, if u 2 Nyloc 1;p (a0 ) u 2 Nloc ./;

(b0 ) u 2 ACCp ./; (c0 ) u is quasicontinuous; (d0 ) u is weakly quasicontinuous. Proof. Assume first that u 2 Ny 1;p ./. (a) , (b) This was shown in Proposition 1.63 without any additional assumptions on X . (a) ) (c) This is Theorem 5.29. (c) ) (a) As u 2 Ny 1;p ./, there is v 2 N 1;p ./ so that u D v a.e. By Theorem 5.29, v is quasicontinuous. Hence, by Proposition 5.23, u D v q.e., and thus also u 2 N 1;p ./. (c) , (d) This follows from Proposition 5.32. 1;p The characterization for u 2 Nyloc ./ follows directly from the first part and Lemma 5.18. (Recall that ACCp ./ D ACCp;loc ./, by Proposition 2.27.)

134

5 Properties of Newtonian functions

This shows that the equivalence classes in Newtonian spaces consist exactly of the quasicontinuous representatives in the a.e.-equivalence classes for Ny 1;p . So e.g. if   Rn and we consider an equivalence class in W 1;p ./, then the corresponding equivalence class in N 1;p ./ is obtained by weeding out all nonquasicontinuous functions and keeping all quasicontinuous functions, cf. Appendices A.1 and A.2. A natural question to ask is if Newtonian functions are always quasicontinuous. By Theorem 5.29 we know that this is true whenever continuous functions are dense in N 1;p .X /, in particular whenever X is a complete doubling p-Poincaré space. The other extreme is when there are no curves in X , such as on the von Koch snowflake curve, in which case we also know that continuous functions are dense in N 1;p .X / and Newtonian functions are quasicontinuous, see Example 5.30. In fact we do not know of any example when there is a nonquasicontinuous Newtonian function. Let us therefore state the following natural open problems. Open problem 5.34. Is it always true that continuous functions are dense in N 1;p .X /? Open problem 5.35. Is it always true that every function in N 1;p .X / is quasicontinuous? Open problem 5.36. Is it always true that Cp is an outer capacity? By Theorems 5.29 and 5.31, a positive answer to Open problem 5.34 implies a positive answer to Open problem 5.35, which in turn implies a positive answer to Open problem 5.36. An affirmative answer to Open problem 5.12 obviously also answers Open problem 5.34.

5.3 Continuity of Newtonian functions In certain situations, viz. under an additional assumption on the capacity of singletons, it is possible to deduce that all Newtonian functions are continuous, not merely quasicontinuous. This is our aim in this section. See also Section 5.5. Proposition 5.37. Assume that continuous functions are dense in N 1;p .X / (which, in particular, holds if X is a doubling p-Poincaré space), and that for every x 2  there 1;p is ı > 0 such that Cp .fyg/ > ı for all y 2 B.x; ı/. Then every u 2 Nloc ./ is continuous in . One may ask if it is not possible to replace the above condition on the capacity by the assumption that the capacity of all singletons is positive. This is not so, as we show in Examples 5.41 and 5.42 below. However, neither of these examples supports a Poincaré inequality. It is therefore natural to state the following problem. Open problem 5.38. Assume that X is a doubling p-Poincaré space, and further that Cp .fxg/ > 0 for every x 2 . Is it then true that every u 2 N 1;p ./ is continuous?

5.3 Continuity of Newtonian functions

135

As far as we know this problem is open also in weighted Rn . Proof of Proposition 5.37. Let x 2 . Let further ı > 0 be such that Cp .fyg/ > ı for all y 2 B D B.x; ı/ and u 2 N 1;p .B/. By Theorem 5.21 there is a quasicontinuous representative uQ in B such that uQ D u q.e. in B. By the condition on the capacity Cp , we must have uQ D u everywhere in B, i.e. u is quasicontinuous in B. Hence there is E  B such that Cp .E/ < ı and ujBnE is continuous. By the condition on the capacity Cp again we must have E D ¿, and thus u is continuous in B. As x 2  was arbitrary, u is continuous in . The following corollary is a refinement of Corollary 4.29 in complete spaces. Corollary 5.39. If X is complete, (4.3) holds with some s < p and X supports a p-Poincaré inequality, then for every ball B D B.x0 ; r/ and for every x 2 B we have Cp .fxg/  C.B/=.r p C 1/. In particular, Cp .fxg/ > 0 for every x 2 X . 1;p Moreover, every u 2 Nloc ./ is continuous in . We improve upon the last part of Corollary 5.39 in the local Sobolev embedding for large p (Corollary 5.49). Proof. The first part follows directly from Corollary 4.29 and Proposition 5.27, together with Proposition 3.1. The second part now follows from Proposition 5.37 and Theorem 5.1. Example 5.40. Let X D R and p D 1. Let us show that C1 .f0g/ D 2. Let uj .x/ D .1  j jxj/C which has guj D j.1=j;1=j / . Hence C1 .f0g/  kuj kN 1;1 .X/ D kuj kL1 .X/ C kguj kL1 .X/ D

1 C 2 ! 2; j

as j ! 1:

Conversely, assume that u 2 N 1;1 .X / is such that u  f0g . Let " > 0. Since kukL1 .X/ < 1, there must be x > 0 and y < 0 such that ju.x/j < " and ju.y/j < ". Let g 2 L1 .X / be an upper gradient of u. Then Z 0 Z x kukN 1;1 .X/  g ds C g ds  ju.y/  u.0/j C ju.0/  u.x/j  2.1  "/: y

0

Letting " ! 0, shows that indeed C1 .f0g/ D 2. Similarly C1 .fxg/ D 2 for all x 2 X . It is also well known that X is a complete doubling 1-Poincaré space with s D 1. In this case we cannot apply Corollary 5.39. Nevertheless, Lipschitz functions are dense in N 1;1 .X /, by Theorem 5.1. Hence it follows from Proposition 5.37 that every 1;1 u 2 Nloc ./ is continuous in , and we have also obtained the other conclusions in Corollary 5.39, which thus makes it true despite the fact that the condition s < p is not fulfilled in this case. Let us finally remark that if we let p > 1, then using the argument above, together with Hölder’s inequality, one easily sees that Cp .f0g/ > 0, which also follows directly from Proposition 2.46. In this case one can thus apply Corollary 5.39 directly.

136

5 Properties of Newtonian functions

Next, we provide examples showing that the condition on the capacity in Proposition 5.37 cannot be replaced by the assumption that all singletons have positive capacity. Example 5.41. Let p  1 and X DRn

1 [

.22j 1 ; 22j / D .1; 0 [

j D0

1 [

 Ij [ Œ1; 1/  R;

j D1

where Ij D Œ22j ; 212j , j D 1; 2; ::: . As in Example 5.40 we see that Cp .fxg/ > 0 for all x 2 X , and moreover for x ¤ 0 there is ı > 0 such that Cp .fyg/ > ı for 1;p all y 2 B.x; ı/. Hence, by Proposition 5.37 (applied locally), every u 2 Nloc .X / is continuous at all points except possibly at 0. S p 4 2j Let uj D Ej , where Ej D 1 , j D 1; 2; ::: . kDj Ij . Then kuj kN 1;p .X/ D 3 2 1;p Thus there are noncontinuous functions in N .X /. Moreover, Cp .f22j g/  kIj kpN 1;p .X/ D 22j ; showing that the condition on Cp in Proposition 5.37 is not fulfilled at 0. Let us also show that the first condition in Proposition 5.37, that continuous functions are dense in N 1;p .X /, is true in this case. Indeed, let v 2 N 1;p .X /. Then ´ v.0/ in .0; 212j ; j D 1; 2; ::: ; vj D v in X n .0; 212j ; is continuous (in X ) and kv 

vj kpN 1;p .X/

Z

Z

212j

D

212j

.v  v.0// d C p

0

0

gvp d ! 0;

as j ! 1;

by dominated convergence. Here  is the one-dimensional Lebesgue measure restricted to X . Observe that X is complete and  is doubling, but X is not connected nor supports any Poincaré inequality, by Proposition 4.2. Introducing a weight we can provide another example which is connected. Example 5.42. Let p  1 and let X D R, this time equipped with the measure d D w dx, where ´ t ˛ ; 0 < t < 1; w.t / D 1; otherwise; and ˛ > p  1. As in Example 5.40 we see that Cp .fxg/ > 0 for all x 2 X , and moreover for x ¤ 0 there is ı > 0 such that Cp .fyg/ > ı for all y 2 B.x; ı/. Hence, by Proposition 5.37

5.3 Continuity of Newtonian functions

137

1;p .X / is continuous at all points except possibly at 0. (applied locally), every u 2 Nloc 1;p By considering the space X 0 D .1; 0, we also see that every u 2 Nloc .X / is left continuous at 0. Let this time, for j D 2; 3; ::: , 8 8 ˆ ˆ 0 < x  1=j; <1; <1=x; 0 < x  1=j; uj .x/ D 2  jx; 1=j  x  2=j; and gj .x/ D j; 1=j  x  2=j; ˆ ˆ : : 0; otherwise; 0; otherwise:

Then gj 2 Lp .X / is an upper gradient of uj . It follows that uj 2 N 1;p .X / and guj D j.1=j;2=j / . Hence Cp .f1=j g/  kuj kpN 1;p .X/ 

Z

 ˛C1

2=j

.1 C j p / d.x/  0

2j p 2 ˛C1 j

! 0; (5.3)

as j ! 1. Thus there are noncontinuous functions in N 1;p .X / and the condition on Cp in Proposition 5.37 is not fulfilled at 0. Let us also show that the first condition in Proposition 5.37, that continuous functions are dense in N 1;p .X /, remains true in this case. Indeed, let v 2 N 1;p .X /. We can approximate v by its truncations vk WD maxfminfv; kg; kg at levels ˙k, see the last paragraph of the proof of Theorem 5.1. Let further vj;k D vk  uj .vk  vk .0//: Since both vk and uj are continuous at every x ¤ 0 and left continuous at 0, so is vj;k . Moreover, for 0 < x < 1=j , we have that vj;k .x/ D vk .x/  .vk .x/  vk .0// D vk .0/ D vj;k .0/; showing that vj;k is right continuous at 0 as well. Now kvk  vj;k kpN 1;p .X/ D kuj .vk  vk .0//kpN 1;p .X/ Z Z  .jvk  vk .0/jp C 2p gvpk / d C uj >0

uj >0

2p .2k/p gupj d:

The first integral tends to zero, as j ! 1, by dominated convergence, and the second integral is majorized by .4k/p kuj kpN 1;p .X/ which tends to zero, as j ! 1, by (5.3). Note that in this example X is connected. However  is not doubling, nor supports any Poincaré inequality (the functions uj being counterexamples).

138

5 Properties of Newtonian functions 1;p

5.4 Density of Lipschitz functions in N0

In Section 5.1 we saw that in doubling p-Poincaré spaces, Lipschitz functions (and even compactly supported Lipschitz functions in the complete case) are dense in N 1;p .X /, but that this is not true for general X . In particular, Lipschitz functions need not be dense in N 1;p ./ even if they are dense in N 1;p .X /. In this section we show that the situation is different for functions vanishing on @, viz. that under rather mild assumptions on X and no assumptions on , we have Lipc ./ D N01;p ./, see Theorems 5.45 and 5.46. We shall also see that under the same assumptions, locally Lipschitz functions are dense in N 1;p ./ for every open . We start by the following approximation lemma. Lemma 5.43. Assume that all functions in N 1;p .X / are quasicontinuous. (In particular, this holds if X is a complete doubling p-Poincaré space.) Then every u 2 N01;p ./ can be approximated in N 1;p .X / by bounded functions with closed bounded support in . 1;p Proof. To start with, we make several S1 obvious reductions. Note that u 2 N0 ./  1;p N .X /. We can write X D nD1 B.x0 ; n/ for some x0 2 . Then the se1;p quence u n is in N0 ./ with u n ! u in N 1;p .X /, as n ! 1, where n .x/ D .1  dist.x; B.x0 ; n//C . Hence, without loss of generality we may assume that u has bounded support in X . Next, as both uC and u belong to N01;p ./, we can assume that u  0. Lemma 2.19 and dominated convergence show that

guminfu;kg D gu fu>kg ! 0

in Lp .X /;

and hence minfu; kg ! u in N 1;p .X /, as k ! 1. We can thus assume that u is bounded. Similarly, gu.u"/C D gu f0
in Lp .X /;

and u" WD .u  "/C ! u in N 1;p .X /, as " ! 0. By assumption, u is quasicontinuous in X , and so there exist open sets Uj , j D 1; 2; ::: , with limj !1 Cp .Uj / D 0, such that ujXnUj is continuous. It follows that for each j , the set fx 2 X n Uj W u.x/ < "g is open in X n Uj and hence Vj D Uj [ fx 2 X W u.x/ < "g is open in X . Thus, X n Vj is closed with X n Vj  . Moreover, as u has bounded support in X and u  " in X n Vj , this set is also bounded. Now, by the choice of Uj , there exist wj such that 0  wj  1, wj D 1 on Uj , limj !1 kwj kN 1;p .X/ D 0 and wj ! 0 a.e. in X . Let u";j D .1  wj /u" . The Leibniz rule (Theorem 2.15) shows

5.4 Density of Lipschitz functions in N01;p

139

that ku"  u";j kpN 1;p .X/ D kwj u" kpN 1;p .X/ Z  Z Z p p p p 2 .wj u" / d C .u" gwj / d C .wj gu" / d : X

X

X

As u" is bounded, the first two integrals on the right-hand side tend to zero as j ! 1. As for the last integral, we have wj ! 0 a.e. in X and as wj gu"  gu" 2 Lp .X /, dominated convergence shows that the last integral tends to zero, as j ! 1. A diagonal argument now shows that among the functions u";j , " > 0, j D 1; 2; ::: , we can find a subsequence uk WD u"k ;jk such that uk ! u in N 1;p .X /, as k ! 1. To complete the proof, we observe that each u";j has closed bounded support in . Indeed, if u";j .x/ ¤ 0, then u.x/ > " and thus x 2 X n Vj , which is a closed bounded subset of . We remark that the proof above shows that for quasicontinuous u 2 N01;p ./ the conclusion of Lemma 5.43 holds without any assumptions on X . The proof also implies the following useful result. Corollary 5.44. If u in Lemma 5.43 is nonnegative, then 0  u";j  u"  u, i.e. the approximations can be chosen to be nonnegative and bounded above by u. Theorem 5.45. Let X be proper and assume that locally Lipschitz functions are dense in N 1;p .X /. Then Lipc ./ D N01;p ./. Furthermore, if nonnegative locally Lipschitz functions are dense among nonnegative N 1;p .X /-functions, and 0  u 2 N01;p ./, then the Lipschitz approximations of u can be chosen nonnegative. Theorem 5.46. Assume that X is a complete doubling p-Poincaré space. Then Lipc ./ D N01;p ./. Furthermore, if 0  u 2 N01;p ./, then the Lipschitz approximations of u can be chosen nonnegative. Proof. This follows directly from Theorems 5.1 and 5.45, together with Remark 5.3. Proof of Theorem 5.45. One inclusion is clear. To prove the other, let u 2 N01;p ./. Without loss of generality u 6 0. If  D X , then the result follows from Proposition 5.14. Assume therefore that  ¤ X . By Lemma 5.43, together with Theorem 5.29, we can assume that u is bounded and that supp u is a compact subset of . Let " > 0 be arbitrary, ı D dist.supp u; X n / and .x/ D .1  2 dist.x; supp u/=ı/C . Note that 2 Lipc ./, 0   1, D 1 on supp u and g  2=ı. As u 2 N 1;p .X /, by Proposition 5.14 there exists a Lipschitz function v on X such that kuvkpN 1;p .X/ < ".

140

5 Properties of Newtonian functions

Then v 2 Lipc ./ and gv.1/  jvjg C gv . Thus we get, using that u D 0 in X n supp u, Z p kv  v kN 1;p .X/  .jvjp C .jvjg C gv /p / d 

Xnsupp u  p 2 2p p C ı





1 kvkpN 1;p .X nsupp u/ 

2p 2 C 1 ku  vkpN 1;p .X/ p ı  p  p 2 <2 C 1 ": ıp  ku  vkN 1;p .X/ C kv  v kN 1;p .X/ , the result follows. p

As ku  v kN 1;p .X/

The following result is a refinement of Theorem 5.1. Note that  itself is not (in general) proper and that it may not support a p-Poincaré inequality, cf. Examples 5.8 and 5.9 and Open problems 5.12 and 5.13. Theorem 5.47. Let X be proper and assume that locally Lipschitz functions are dense in N 1;p .X /. (In particular, this holds if X is a complete doubling p-Poincaré space.) 1;p If u 2 Nloc ./, and " > 0, then there exists a locally Lipschitz function v W  ! R such that ku  vkN 1;p ./ < ". Moreover, if u 2 N 1;p ./, then also v 2 N 1;p ./, i.e. locally Lipschitz functions are dense in N 1;p ./. Proof.SLet " > 0. As X is proper, there are open sets 1 b 2 b  b  such that  D j1D1 j . Choose j 2 Lipc .j C1 / so that j D 1 on j and 0  j  1  P 1  everywhere. Let u1 D u 1 and uj D u  jiD1 ui j , j D 2; 3; ::: . It is easily shown by induction that u

j X

ui D u.1  1 /.1  2 /  .1  j /;

iD1

P and hence u D j1D1 uj in . x j 1 /, where we let Now, for each j D 1; 2; ::: , we have uj 2 N01;p .j C1 n  x j 1 / such that 0 D ¿. By Theorem 5.45, there exist vj 2 Lipc .j C1 n  kuj  vj kN 1;p ./ < 2j ":

P Let v D j1D1 vj . Note that for each x 2  there is a neighbourhood U 3 x such that at most three terms in the sum are nonzero in U , and hence v is locally Lipschitz in . Finally, 1 X kuj  vj kN 1;p ./ < "; ku  vkN 1;p ./  j D1

which concludes the proof.

5.5 Sobolev embeddings and inequalities

141

5.5 Sobolev embeddings and inequalities It follows from Section 4.1 that Newtonian functions in doubling p-Poincaré spaces have additional integrability and boundedness properties. If X is complete and p is sufficiently large, then we get the following pointwise estimate which improves upon Proposition 5.37. Proposition 5.48. Assume that X is complete and supports a p-Poincaré inequality, with dilation constant , and that (4.3) holds for some s < p. Then there exists C > 0 such that if B D B.x0 ; r/ is a ball in X and u 2 N 1;p .9B/, then for every x; y 2 B, 1=p « ju.x/  u.y/j  Cd.x; y/1s=p r s=p gup d : 9B

Proof. Let x; y 2 B and B0 D B.x; 2d.x; y//. Then 2B0  9B. Condition (4.3) yields   .2B0 / d.x; y/ s C .B/ r and Corollary 5.39 implies that u is continuous in 9B. Proposition 4.27 then yields « 1=p p ju.x/  uB0 j  Cd.x; y/ gu d 2B0

 Cd.x; y/

«

1s=p s=p

r

9B

1=p gup

d

:

The same estimate holds for ju.y/  uB0 j, which finishes the proof. We are now able to improve upon the L1 -local Sobolev embedding for large p (Corollary 4.28), under the additional assumption that X is complete. Recall also that for s  p we obtained a local Sobolev embedding in Corollary 4.23. Corollary 5.49 (Local Sobolev embedding for large p). If X is complete, supports a 1;p p-Poincaré inequality and (4.3) holds for some s < p, then every u 2 Nloc ./ is locally .1  s=p/-Hölder continuous in . Under an additional growth assumption for large balls we obtain the following global embedding theorem. In particular, this holds if X is Ahlfors s-regular, see Definition 3.4. Theorem 5.50 (Global Sobolev embedding). Assume that X supports a p-Poincaré inequality, that (4.3) holds and that there exist x0 2 X and a sequence frj gj1D1 such that limj !1 rj D 1 and .B.x0 ; rj //  C rjs for all j:

(5.4)



(a) If s > p, then N 1;p .X / continuously embeds into Lp .X / with p  D sp=.sp/. (b) If s < p and X is complete, then N 1;p .X / continuously embeds into C 1s=p .X /.

142

5 Properties of Newtonian functions

Proof. (a) For each Bj D B.x0 ; rj / with rj  1 we have by the .p  ; p/-Poincaré inequality, provided by Theorem 4.21, and (5.4) that « 1=p 1=p Z  1=p  p p  .Bj / juj d ju  uBj j d C .Bj /1=p juBj j Bj

Bj

 C rj .Bj /1=p

 1=p

Z

1=p Bj

gup d

1=p 

Z C

jujp d Bj

 C kukN 1;p .X/ : Letting j ! 1 finishes this part of the proof. (b) Proposition 5.48 and (5.4) imply that for all x; y 2 B.x0 ; rj /, Z 1=p Cd.x; y/1s=p rjs=p p ju.x/  u.y/j  g d u .B.x0 ; rj //1=p B.x0 ;9rj /  Cd.x; y/1s=p kukN 1;p .X/ : In the rest of this section we give several Sobolev type inequalities for functions vanishing at the boundary or on sets with positive capacity. See also Theorem 6.21 for Maz0 ya’s inequality for the variational capacity. Theorem 5.51 (Sobolev inequality). Assume that  is doubling and that X supports a .q; p/-Poincaré inequality. Then there exists C > 0 such that if B D B.x0 ; r/ is a ball in X with 0 < r < 14 diam X and u 2 N01;p .B/, then « 1=q « 1=p q p juj d  Cr gu d : (5.5) B

B

See Remark 9.56 for a discussion about the sharpness of this Sobolev inequality. Proof. By Corollary 4.24 we can assume that q > 1. Then « 1=q « 1=q jujq d  ju  u2B jq d C ju2B j: 2B

2B

By the Hölder inequality and the fact that u vanishes on 2B n B, we have « 1=q  «  .B/ 11=q ju2B j  jujB d  jujq d : .2B/ 2B 2B Lemma 3.7 implies that .B/=.2B/   < 1, where  only depends on the doubling constant C . The last two estimates and the .q; p/-Poincaré inequality now give 1=q « 1=q « 1=p «   1   11=q jujq d  ju  u2B jq d  Cr gup d ; 2B

2B

2B

where  is the dilation constant in the .q; p/-Poincaré inequality. As u and thus gu vanish a.e. outside B, the ball 2B in the last integral can be replaced by B.

143

5.5 Sobolev embeddings and inequalities

If X supports a p-Poincaré inequality and (4.3) holds for some s > p, then Theorem 4.21 and the Sobolev inequality (Theorem 5.51) imply (5.5) with q D p  D sp=.s  p/. If instead s  p, then Corollary 4.26 and the Sobolev inequality (Theorem 5.51) show that (5.5) holds for all 1  q < 1. Moreover, in the case when s < p, the following proposition shows that every u 2 N01;p .B/ is (essentially) bounded. Proposition 5.52 (L1 -Sobolev inequality for large p). Assume that X supports a p-Poincaré inequality and that (4.3) holds for some s < p. Then there exists C > 0 such that if B D B.x0 ; r/ is a ball in X with 0 < r < 14 diam X and u 2 N01;p .B/, then «  1=p

ess sup juj  C r B

B

gup d

:

(5.6)

If moreover X is complete, then u is continuous and « 1=p p sup juj  C r gu d : B

B

Note that the left-hand side in (5.6) equals Cp - ess supB juj, by Corollary 1.69. Proof. Since u vanishes on 2B n B, we have as in the proof of Theorem 5.51 that ess sup juj  ess sup ju  u2B j C ju2B j 2B

2B

 ess sup ju  u2B j C 2B

.B/ ess sup juj .2B/ 2B

 ess sup ju  u2B j C  ess sup juj; 2B

2B

where  < 1 is the constant from Lemma 3.7 which only depends on the doubling constant C . An application of Proposition 4.27 to 2B together with the fact that gu D 0 a.e. outside B finishes the proof of (5.6). If X is complete, then u is continuous by the local Sobolev embedding for large p (Corollary 5.49). It is clear that neither (5.5) nor (5.6) can hold for general u 2 N 1;p .B/. This is best seen by considering nonzero constant functions. In the following theorem we replace the assumption u 2 N01;p .B/ by the requirement that u vanishes on a set of positive capacity. Theorem 5.53 (Maz0 ya’s inequality). Assume that X supports a .q; p/-Poincaré in1;p equality for some q  p. For u 2 Nloc .X /, let S D fx 2 X W u.x/ D 0g. Then for all balls B D B.x0 ; r/, « p=q Z C.r p C 1/ q juj d  g p d; (5.7) Cp .B \ S / 2B u 2B where  is the dilation constant in the .q; p/-Poincaré inequality.

144

5 Properties of Newtonian functions

Note that the Hölder inequality makes it possible to replace q  p in the left-hand side of (5.7) by any smaller exponent. The validity of a .p; p/-Poincaré inequality is however needed in the proof even for small exponents in the left-hand side. By Corollary 4.24, the assumption that X supports a .q; p/-Poincaré inequality is satisfied for some q > p in doubling p-Poincaré spaces. Note, however, that here we do not need that  is doubling. The space and the function constructed in Example 8.20 show that the factor  on the right-hand side in Theorem 5.53 cannot be omitted. In Euclidean spaces, Theorem 10.1.2 in Maz0 ya [283] shows that Cp .B \ S / can be replaced by Cp .2B \ S / giving a stronger result. (Of course,  D 1 in this case.) The argument is based on extending Sobolev functions from B to 2B (e.g. by reflection), a method which is difficult to apply in metric spaces. However, the proof below can easily be modified to obtain a similar result with Cp .B \ S/ and the constant C depending on  < 2. Proof. By splitting u into its positive and negative parts and considering them separately, we can assume that u  0. If u D 0 a.e. in 2B, then (5.7) is trivial. Otherwise, let «  1=q

uN D

uq d

:

2B

Let .x/ D .1  dist.x; B/=r/C . Note that is 1=r-Lipschitz, 0   1 in 2B,

D 1 in B and D 0 outside 2B. Then v WD .1  u=u/ N D 1 in B \ S and hence Z Z Cp .B \ S/  v p d C gvp d X X Z Z 1 1 p  p ju  uj N d C p .gu C ju  ujg N  /p d: uN 2B uN 2B As g  1=r, this becomes Cp .B \ S/ 

2p .1 C r p / uN p

Z ju  uj N p d C 2B

2p uN p

Z 2B

gup d:

(5.8)

Next, Z

1=p ju  uj N d p

2B

Z

1=p



ju  u2B j d

C juN  u2B j .2B/1=p : (5.9)

p

2B

The first term is estimated directly by the .p; p/-Poincaré inequality and for the second term we have ˇ ˇ juN  u2B j .2B/1=q D ˇkukLq .2B/  ku2B kLq .2B/ ˇ « 1=q q  ku  u2B kLq .2B/ D ju  u2B j d .2B/1=q : 2B

5.5 Sobolev embeddings and inequalities

145

Estimating the integral on the right-hand side by the .q; p/-Poincaré inequality yields « 1=p p juN  u2B j  C r gu d : 2B

Inserting this into (5.9) shows that Z Z ju  uj N p d  C r p 2B

2B

This together with (5.8) now gives C.r p C 1/ Cp .B \ S/  uN p

gup d:

Z 2B

gup d

(5.10)

and the result follows. The following inequality is sometimes called the Poincaré inequality for N01;p . For doubling measures, a similar conclusion follows already from the Sobolev inequality (Theorem 5.51). Corollary 5.54 (Poincaré inequality for N01;p ). Assume that X supports a .p; p/Poincaré inequality. Let E be bounded with Cp .X n E/ > 0. Then there exists CE > 0 such that for all u 2 N01;p .E/, Z Z jujp d  CE gup d: (5.11) X

kukpN 1;p .X/

X

CzE kgup kLp .X/

 In particular, for u 2 N01;p .E/. If E is measurable, then the integrals and norms can equivalently be taken with respect to E. Proof. As E is bounded, there exists a ball B D B.x; r/  E such that Cp .B nE/ > 0. (If X is bounded we may choose B D X .) Let u 2 N01;p .E/, i.e. u 2 N 1;p .X / and u D 0 q.e. in X n E. Maz0 ya’s inequality (Theorem 5.53) and the fact that gu D 0 a.e. in X n B show that Z Z Z Z p p p juj d D juj d  CE gu d D CE gup d; X

2B

2B

X

where CE D C.r C 1/=Cp .B n E/ < 1. As u D gu D 0 a.e. outside E the integrals can equivalently be taken over E, if E is measurable. p

Example 5.55. We shall show that the assumption that E is bounded in the Poincaré inequality for N01;p (Corollary 5.54) cannot be omitted. Let X D R, E D .0; 1/ and uj .x/ D .1  jx  j j=j /C , j D 1; 2; ::: . Then Z Z 2j p juj j dx D gupj dx D 2j 1p : ; while pC1 E E Letting j ! 1 shows that (5.11) cannot hold for any CE < 1.

146

5 Properties of Newtonian functions

Example 5.56. Let us modify the example above and show that the boundedness assumption in the Poincaré inequality for N01;p (Corollary 5.54) cannot be omitted in any unbounded metric space X equipped with a doubling measure  with doubling constant C . We do not even need to assume that X supports some Poincaré inequality, nor even that X is connected. Let B D B.x0 ; 1/ be a ball in X and E D X n B. Since X is unbounded, there are xk 2 X such that dk WD d.xk ; x0 / > k, k D 2; 3; ::: . Let further uk .x/ D .1  2d.x; xk /=dk /C 2 N01;p .E/ so that Z  p 1 upk d  .B.xk ; dk =4// 2 X while

Z X



gupk

d 

2 dk

p



.B.xk ; dk =2//  C

2 dk

p

.B.xk ; dk =4//

for k D 2; 3; ::: . Letting k ! 1 shows that (5.11) cannot hold for any CE < 1. Corollary 5.57. Assume that X supports a .p; p/-Poincaré inequality and that E  X . Then N01;p .E/ D N 1;p .E/ if and only if Cp .X n E/ D 0 or Cp .E/ D 0. Proof. Note first that it is always true that N01;p .E/  N 1;p .E/ if E is measurable. We split the proof into four cases. Case 1. Cp .X n E/ D 0. In this case .X n E/  Cp .X n E/ D 0. Thus X n E is measurable, and hence also E is measurable. If u 2 N 1;p .E/ has an upper gradient g 2 Lp .E/ with respect to E, then gE is a p-weak upper gradient of u, extended by 0 to X n E, by Proposition 1.48. Hence u 2 N01;p .E/. Case 2. Cp .E/ D 0. In this case .E/  Cp .E/ D 0, and thus again E is measurable. Let u be an arbitrary function on E which we extend by 0 to X n E. Then u D 0 q.e. in X , and hence u 2 N01;p .E/. Case 3. .E/ D 0 < Cp .E/. Again, E is measurable. Let u D E 2 N 1;p .E/. Since u D 0 a.e. but not q.e., u … N 1;p .X / by Proposition 1.59, and hence u … N01;p .E/. Case 4. .E/ > 0 and Cp .X n E/ > 0. If E is not measurable, then N 1;p .E/ is not defined and hence not equal to N01;p .E/. Assume therefore that E is measurable. We can then find a ball B such that Cp .B n E/ > 0 and .B \ E/ > 0. Let

2 Lipc .X / be such that D 1 on 2B. Clearly, u WD E 2 N 1;p .E/. If N 1;p .E/ D N01;p .E/, then by the definition of N01;p .E/, u 2 N 1;p .X /. Moreover, gu D 0 a.e. in 2B, by Corollary 2.21 (as u  1 in E \ 2B and u  0 in X n E). Maz0 ya’s inequality (Theorem 5.53) then implies that « Z .B \ E/ p 0<  u d  C gup d D 0; .2B/ 2B 2B a contradiction. Thus, N 1;p .E/ ¤ N01;p .E/.

5.6 Lebesgue points for N 1;p -functions

147

5.6 Lebesgue points for N 1;p -functions As N 1;p .X /  Lp .X /, the Lebesgue differentiation theorem implies that every Newtonian function has Lebesgue points a.e., if  is doubling. We shall see that if X is sufficiently nice, then more is true, viz. the set of non-Lebesgue points has capacity zero. The proof is based on finding an upper gradient for the following integral operator. Definition 5.58. For u 2 Lqloc .X /, q  1, let « 1=q jujq d ; uj .x/ D

j D 0; 1; ::: ;

B.x;2j /

Tq u.x/ D lim sup uj .x/: j !1

Recall that the noncentred maximal function of f 2 L1 .X /, is «  M f .x/ WD sup jf j d; B

B

where the supremum is taken over all balls B containing x. Lemma 5.59. Assume that  is doubling, and that X supports a .q; p/-Poincaré N inequality for some q  1 and pN  1. Let g be an upper gradient of a function u 2 Lq .X / and let p  1 be arbitrary. Then C.M  g pN /1=pN is a p-weak upper gradient of Tq u. Here, C depends only on the doubling constant C and the constants in the .q; p/-Poincaré N inequality. Note that p and pN in Lemma 5.59 can be arbitrary and unrelated. Proof. Since Tq u  .M  jujq /1=q , Lemma 3.12 implies that Tq u < 1 a.e. Let  W Œ0; l  ! X be a curve. Let j be such that rj D 2j  2l . By splitting  into parts if necessary, we can assume that 12 rj  l  rj . Let x D .0/ and y D .l / be the endpoints of  . Since N inequality and the u 2 Lq .X /, both uj .x/ and uj .y/ are finite. The .q; p/-Poincaré doubling property imply that for all z 2 B WD B.x; 2rj /, ˇ« ˇ ˇ« ˇ 1=q 1=q ˇ ˇ ˇ ˇ q q ˇ ˇ ˇ juj d  uB ˇ C ˇ juj d  uB ˇˇ juj .x/  uj .y/j  ˇ «

B.x;rj /



1=q

ju  uB j d q

B.x;rj /

« C

1=q

ju  uB jq d B

«

g pN d

 C rj 

B

 C rj M  g pN

1=pN

1=pN

.z/;

«

B.y;rj /

C

1=q

ju  uB j d q

B.y;rj /

148

5 Properties of Newtonian functions

where  is the dilation constant in the .q; p/-Poincaré N inequality. As   B and l  12 rj , we get that Z   pN 1=pN M g ds: (5.12) juj .x/  uj .y/j  C 

Glueing together all the parts of , we see that (5.12) holds for all j such that 12 rj  l . Finally, we have either Tq u.x/ D Tq u.y/ D 1 or Z jTq u.x/  Tq u.y/j  sup juj .x/  uj .y/j  C.M  g pN /1=pN ds; j



where the supremum is taken over sufficiently large j . Corollary 1.51 then shows that C.M  g pN /1=pN is a p-weak upper gradient of Tq u. Lemma 5.60. Assume that X is a p 0 -Poincaré space with 1  p 0 < p and that (4.3) holds with some s. Let 1  q < q0 , where 8 sp < ; if s > p, (5.13) q0 D s  p :1; if s  p. If u 2 N01;p .B/ for some ball B  X with diam B < 18 diam X , then Tq u 2 N .X / and kTq ukN 1;p .X/  C kukN 1;p .B/ , where C depends only on B, s, p, p 0 , q, the doubling constant C and the constants in the p 0 -Poincaré inequality, but not on u. 1;p

Proof. If s < p, then (4.3) holds with s replaced by p, so we can assume that s  p. Observe also that we always have q0 > p. Find pN  p 0 so that pN  ps=.p C s/ and qs=.q C s/ < pN < p  s. Further, find " > 0 such that qN WD q.1 C "/ < s p=.s N  p/ N and qN > p. By Theorem 4.21 (and Hölder’s inequality), X supports a .q; N p/-Poincaré N inequality. Note that supp Tq u  Bx and Tq u  .M  jujq /1=q . Theorem 3.13 and the Sobolev inequality (Theorem 5.51) yield Z 1=qN Z 1=qN qN  q 1C"  jTq uj d .M juj / d X

X

Z C

qN

juj d

1=qN  C kgu kLp .B/ :

B

Since supp Tq u  Bx and qN > p, this implies that kTq ukLp .X/  C kukN 1;p .B/ :

(5.14)

Let g be an upper gradient of u. By Lemma 5.59, C.M  g pN /1=pN is a p-weak upper gradient of Tq u. As p=pN > 1, Theorem 3.13 together with (5.14) finishes the proof.

5.6 Lebesgue points for N 1;p -functions

149

The next corollary follows directly from the definition of capacity and Lemma 5.60, since Tq u= is admissible for calculating the capacity of fx 2 X W Tq u.x/ >  g. Corollary 5.61. Assume that X is a p 0 -Poincaré space with 1  p 0 < p and that (4.3) holds with some s. Let 1  q < q0 , where q0 is given by (5.13). If u 2 N01;p .B/ for some ball B  X , then for all  > 0, Cp .fx 2 X W Tq u.x/ >  g/ 

C kukpN 1;p .X/ ; p

where C depends only on B, s, p, p 0 , q, the doubling constant C and the constants in the p 0 -Poincaré inequality, but not on u. We are now ready to prove the main result about Lebesgue points of Newtonian functions. Theorem 5.62. Assume that X is a p 0 -Poincaré space with 1  p 0 < p and that (4.3) holds with some s. Let 1  q < q0 , where q0 is given by (5.13). 1;p .X /, then for q.e. x 2 X , If u 2 Nloc « lim ju  u.x/jq d D 0: r!0 B.x;r/

In particular, q.e. x 2 X is a Lebesgue point of u. Observe that if X is a complete doubling p-Poincaré space and 1 < p < 1, then X supports a p 0 -Poincaré inequality for some 1  p 0 < p, by Theorem 4.30 (which we have not proved). Moreover, there is some s such that (4.3) holds. Proof. As Lebesgue points are a local issue, after multiplying u by a cut-off function, we can assume that u 2 N01;p .B/ for some ball B  X . By Theorem 5.1 and Corollary 1.72, there exist Lipschitz functions uk , k D 1; 2; ::: , such that uk ! u both in N 1;p .X / and pointwise q.e., as k ! 1. Writing ju  u.x/j  ju  uk j C juk  uk .x/j C juk .x/  u.x/j and using the fact that uk has Lebesgue points everywhere, we have for all x 2 X and all k D 1; 2; ::: , that « lim sup j !1

1=q

B.x;2j /

ju  u.x/jq d

 Tq .u  uk /.x/ C juk .x/  u.x/j: (5.15)

The last term on the right-hand side tends to zero as k ! 1 for q.e. x 2 X . To estimate the first term on the right-hand side, we have by Corollary 5.61 for every  > 0, Cp .fx 2 X W Tq .u  uk /.x/ >  g/ 

C ku  uk kpN 1;p .X/ : p

150

5 Properties of Newtonian functions

This estimate and (5.15) imply that ² Cp

« x 2 X W lim sup j !1

1=q ju  u.x/j d q

B.x;2j /

 Cp .fx 2 X W juk .x/  u.x/j >  g/ C

³ > 2

C ku  uk kpN 1;p .X/ ! 0; p

as k ! 1. As  > 0 was arbitrary, the doubling property of  then implies that for q.e. x 2 X , « « q lim sup ju  u.x/j d  C lim sup ju  u.x/jq d D 0: r!0

B.x;r/

j !1

B.x;2j /

5.7 Notes Theorem 5.1 is in Shanmugalingam [319], and Lemma 5.43 and Theorem 5.46 (without the part for u  0) are in Shanmugalingam [320]. In Euclidean domains, Sobolev spaces with zero boundary values are usually defined as the closure of C01 ./ under the Sobolev norm. It is then a question of spectral synthesis to show that they are exactly those global Sobolev functions vanishing outside , which is exactly our definition of N01;p ./. See Theorem 4.5 in Heinonen–Kilpeläinen–Martio [171] or Theorem 9.1.3 in Adams–Hedberg [5]. Here, we have reversed the process, taking advantage of the fact that Newtonian functions are defined pointwise everywhere and in decent spaces are automatically quasicontinuous. Thus, in our setting, spectral synthesis was established in Theorems 5.45 and 5.46. The history of spectral synthesis goes back to Beurling and Deny. Hedberg [162] showed the corresponding result for higher order Sobolev spaces on Rn . See Adams– Hedberg [5], Section 9.13, for a historical account as well as an explanation of the name spectral synthesis. For spectral synthesis in very general function spaces on Rn , including e.g. Besov and Lizorkin–Triebel spaces, see Hedberg–Netrusov [163]. In Remark 4.4 in Shanmugalingam [319] it is claimed that if X is a doubling p-Poincaré space, then all functions in N 1;p .X / are quasicontinuous. Unfortunately, there is a flaw in the argument, see Björn–Björn–Shanmugalingam [58], the remark at the end of Section 4. Nevertheless, the remark in [319] was the main inspiration for the work [58], and as we saw in Theorem 5.29 the claimed result is true under the additional assumption that X is complete, but remains open in the noncomplete case. Recently Kinnunen–Korte–Shanmugalingam–Tuominen [211] showed that Newtonian functions have a stronger type of quasicontinuity, assuming that X is a complete doubling p-Poincaré space which in addition supports a strong relative isoperimetric inequality.

5.7 Notes

151

Many of the results and examples in Sections 5.1 and 5.2 are from Björn–Björn– Shanmugalingam [58], viz. Theorems 5.29 and 5.31, a special case of Theorem 5.21, Propositions 5.14 and 5.27, Lemmas 5.25 and 5.26, Examples 5.5, 5.6, 5.8 and 5.9 and Open problem 5.12. Moreover, (the first parts of) Theorems 5.45 and 5.47 are also from [58]. The implication (c) ) (a) in Proposition 5.33 does not seem to have been proved earlier in the literature, although it has been stated several times at least on Rn . The implication (b) ) (a), which we obtained in Proposition 1.63, seems to be new here. Proposition 5.23 and its proof are from Kilpeläinen [201]. Note that the proof is much shorter than the corresponding proofs in Euclidean spaces given earlier (even in the linear situation p D 2). Proposition 5.22, which is essential in the proof of Proposition 5.23, is usually much easier to obtain than here, e.g. in (weighted) Rn . In fact, it is an easy consequence of Theorem 6.7 (x), which we obtain in the next chapter. However, the proof of Theorem 6.7 (x) uses that the capacity is an outer capacity, whereas our proof of Proposition 5.22, and hence of Proposition 5.23, holds for arbitrary X . See the notes to Chapter 4 in Heinonen–Kilpeläinen–Martio [171] and the references therein for a historical account on quasicontinuity for Sobolev functions in Euclidean spaces. Often capacities are defined in a way which automatically makes them outer, as e.g. the capacities in Kinnunen–Martio [215], [216] and Heinonen–Kilpeläinen– Martio [171]. In our situation this has not been desirable as the Cp capacity is closely connected with the equivalence classes in Nz 1;p .X /, a connection we do not want to lose. Instead, we obtained the outer regularity of the capacity in Theorem 5.31 by using the quasicontinuity of Newtonian functions and an argument close to Kilpeläinen– Kinnunen–Martio [202]. If a capacity is outer, then quasicontinuity and weak quasicontinuity coincide, as we saw in Proposition 5.32. Therefore one usually does not distinguish between quasicontinuity and weak quasicontinuity, but for us the distinction is important. Sobolev embeddings for X D Rn go back to Sobolev [326]. Various forms of Sobolev type inequalities and sufficient conditions for their validity in Euclidean domains can be found in Maz0 ya’s book [283]. It also contains many examples of domains for which the Sobolev embeddings fail. Simply connected planar domains supporting Sobolev embeddings were characterized by Buckley–Koskela [78], [79]. For p D 2, a Sobolev inequality on manifolds was proved by Saloff-Coste [313]. Sobolev inequalities for -gradients associated with vector fields were studied by Franchi–Gutiérrez–Wheeden [125]. In general metric spaces, Sobolev type inequalities on balls and local Sobolev embeddings for functions satisfying a p-Poincaré inequality were thoroughly studied by Hajłasz–Koskela [154] using Riesz potentials. For example, Proposition 5.48 is a part of Theorem 5.1 in [154]. See the references in Chapter 9.2 in [154] for more on Sobolev type inequalities and embeddings in various settings. Embeddings of gener-

152

5 Properties of Newtonian functions

alized Sobolev type spaces defined by means of Poincaré inequalities were obtained in Heikkinen–Koskela–Tuominen [166]. Isocapacitary characterizations of Sobolev inequalities on metric spaces have been given by Kinnunen–Korte [207]. The global Sobolev embedding N 1;p .X / ,! C 1s=p .X / in metric spaces (as in Theorem 5.50) was proved in Shanmugalingam [319] under the global assumption .B/  C r s for all balls B D B.x0 ; r/. See Adachi–Tanaka [2], and the references therein, for global embeddings in the borderline case p D n on X D Rn . As far as we know such embeddings have not been studied in metric spaces for p D s, while for p < s they may be new here. The proof of the Sobolev inequality (Theorem 5.51) comes from J. Björn [63] but the arguments are as in Theorem 13.1 in Hajłasz–Koskela [154], see also Lemma 2.8 in Kinnunen–Shanmugalingam [220]. Maz0 ya’s inequality (Theorem 5.53) was first proved by Maz0 ya in the Euclidean setting, see [276] or Theorem 10.1.2 in [283]. The original proof goes through in metric spaces as well, see [63]. In fact, Maz0 ya showed that positivity of the capacity of the set where u vanishes is necessary and sufficient for inequalities such as (5.7) to hold. Another inequality, in which the Lp -norm of a gradient majorizes an Lp -norm of a function weighted by a distance to the boundary, is the Hardy inequality. Hardy inequalities and the closely related p-fat sets have been studied on metric spaces by Kilpeläinen–Kinnunen–Martio [202], J. Björn–MacManus–Shanmugalingam [70], Koskela–Zhong [232], Kinnunen–Korte [208], Korte–Shanmugalingam [227] and Korte–Lehrbäck–Tuominen [225]. We define and briefly mention p-fat sets in connection with boundary regularity in Corollary 11.25 (b). Lebesgue points for quasicontinuous Hajłasz–Sobolev functions were studied by Kinnunen–Latvala [212] and, in view of Theorem 5.29, they obtained Theorem 5.62 under the extra assumption that X is complete. (See Appendix B.1 for more on Hajłasz– Sobolev spaces and their relation to Newtonian spaces.) Our approach is different and Section 5.6 follows Björn–Björn–Parviainen [54]. In [54] one can also find weaker versions of Corollary 5.39 and the local Sobolev embedding for large p (Corollary 5.49). Lebesgue points for N 1;1 -functions in complete doubling 1-Poincaré spaces were studied by Kinnunen–Korte–Shanmugalingam–Tuominen [209]. They obtained Theorem 5.62 for p D q D 1 (under the additional assumption that X is complete). In A. Björn [43], the following definition was introduced: X has the zero p-weak upper gradient property at x 2 X if every bounded measurable function, which has 0 as a p-weak upper gradient in some ball B.x; r/, is essentially constant in some (possibly smaller) ball B.x; ı/. A space X has the zero p-weak upper gradient property if it has it at every x 2 X . The zero p-weak upper gradient property is strictly weaker than supporting a p-Poincaré inequality. In fact there are nonconnected spaces having the zero p-weak upper gradient property, see [43]. Under the assumption that X has the zero p-weak upper gradient property we do not know if all Newtonian functions are quasicontinuous. Nevertheless it was shown in A. Björn [43] that Newtonian functions are still much better than arbitrary a.e.representatives, and in particular restrictions to @ are natural and have important x are restrictions properties. (Recall that if .@/ D 0, then all functions f W @ ! R

5.7 Notes

153

of a.e.-representatives of Sobolev functions.) We refer to [43] for the cluster set results obtained therein, some of which are true if and only if X has the zero p-weak upper gradient property. They are new also on Rn and also utilize Newtonian spaces on closed subsets in a natural way, thus giving an example of the use of Newtonian spaces on Rn in a situation when standard Sobolev spaces would not do.

Chapter 6

Capacities

In this chapter we deduce a number of different properties for the capacity Cp . We have already seen in Chapters 1 and 2 that Cp is countably and strongly subadditive, which is true under no extra assumptions. The new properties here require some extra conditions to hold, or at least for our proofs to work. In particular, we will most of the time require X to be a complete doubling p-Poincaré space, assumptions which will be standard in the forthcoming Chapters 7–14 as well. In Section 6.3 we introduce the variational capacity capp , which in many ways is equivalent to Cp . In particular, under our standard assumptions, they have the same zero sets. The variational capacity has both advantages and disadvantages to the Cp capacity, but sometimes it is essential that the former is used.

6.1 Mazur’s lemma and its consequences We start this section by some auxiliary results about convergence of Newtonian functions, which will be useful both for proving some properties of the capacity and later in our study of p-harmonic functions. They are based on the following lemma from functional analysis. Lemma 6.1 (Mazur’s lemma). Assume that xj ! x, as j ! 1, weakly P in a normed linear space V and let " > 0. Then there exists a convex combination jnD1 aj xj such that n   X   aj xj  < ": x  j D1

V

For a proof see e.g. Yosida [355], pp. 120–121, or Rudin P [312], Theorem 3.12. Recall that, by definition, a convex combination v D x2A aP x x of a set A is such that ax  0, only finitely many coefficients ax are nonzero, and x2A ax D 1. Lemma 6.2. Assume that 1 < p < 1. Assume further that gj is a p-weak upper gradient of uj , j D 1; 2; ::: , and that both sequences fuj gj1D1 and fgj gj1D1 are bounded PNj in Lp .X /. Then there are u; g 2 Lp .X /, convex combinations vj D iDj aj;i ui PNj with p-weak upper gradients gNj D iDj aj;i gi , and a strictly increasing sequence of , such that indices fjk g1 kD1 (a) both ujk ! u and gjk ! g weakly in Lp .X /, as k ! 1;

6.1 Mazur’s lemma and its consequences

155

(b) both vj ! u and gNj ! g in Lp .X /, as j ! 1; (c) vj ! u q.e., as j ! 1; (d) g is a p-weak upper gradient of u. Note that we do not show that vj ! u in N 1;p .X /, as j ! 1. However, if N 1;p .X / is reflexive, one can show, using Mazur’s lemma, that there are convex combinations vj ! u in N 1;p .X /, as j ! 1. It is a deep result of Cheeger [91] that, indeed, N 1;p .X / is reflexive, if X is a complete doubling p-Poincaré space and p > 1. This result, however, goes beyond this book. Note also that the convex combination vj starts from uj . This is important in the applications of this result later on in this section. Proof. Since Lp .X / is reflexive, its unit ball is weakly compact (by Banach–Alaoglu’s theorem) and thus there is a subsequence of fuj gj1D1 which converges weakly in Lp .X /. Taking a subsequence of this subsequence and again using Banach–Alaoglu’s theorem we obtain a subsequence (again denoted by fuj gj1D1 ) such that both fuj gj1D1 and fgj gj1D1 converge weakly in Lp .X /, say to v and g (where g is not necessarily a p-weak upper gradient of v). As gj , j D 1; 2; ::: , are nonnegative we may choose g nonnegative. Applying Mazur’s lemma repeatedly to the sequences fui g1 iDj , j D 1; 2; ::: , we find PNj0 0 0 0 convex combinations vj D iDj ai;j ui such that kvj  vkLp .X/ < 1=j . Lemma 1.39 PNj0 0 implies that gj0 D iDj ai;j gi is a p-weak upper gradient of vj0 . Since moreover 0 p gj ! g weakly in L .X /, as j ! 1, we can again apply Mazur’s lemma (repeatedly) PNj ai;j ui with p-weak upper gradients gNj D to obtain convex combinations vj D iDj PNj Nj ! g in Lp .X /, as j ! 1. By Proposition 2.3, iDj ai;j gi such that vj ! v and g there is a function u D v a.e. with the required properties. Corollary 6.3. Assume that 1 < p < 1. Assume further that fui g1 iD1 is bounded in N 1;p .X / and that ui ! u q.e., as i ! 1. Then u 2 N 1;p .X / and Z Z p gu d  lim inf gupi d: i!1

X

X

Proof. We can find a strictly increasing sequence fij gj1D1 such that Z Z p lim gui d D lim inf gupi d: j !1 X

j

i!1

X

By Lemma 6.2 there are convex combinations vj of fuij gj1D1 and functions u; Q g 2 Lp .X / such that vj ! uQ q.e., as j ! 1, g is a p-weak upper gradient of u, Q and (after possibly taking another subsequence) both uij ! uQ and guij ! g weakly in Lp .X /, as j ! 1.

156

6 Capacities

Since vj must tend to u q.e., as j ! 1, we have uQ D u q.e. and thus, by Corollary 1.49, g is also a p-weak upper gradient of u. As guij ! g weakly in Lp .X /, as j ! 1, we have Z Z Z Z gup d  g p d  lim gupi d D lim inf gupi d < 1: X

j !1 X

X

j

i!1

X

Finally, as kukLp .X/ D kuk Q Lp .X/ < 1, we get that u 2 N 1;p .X /. The following continuity result for the capacity is a consequence of the above convergence results, and holds without any extra assumptions on X . Note that a corresponding result for the intersection in Theorem 6.7 (viii) is only known to be valid under additional assumptions on X and the sets under consideration, see Examples 6.8–6.10. Theorem 6.4. Assume that 1 < p < 1. If E1  E2    X , then Cp

1 [ iD1

 Ei D lim Cp .Ei /: i!1

S Proof. Let E D 1 iD1 Ei . That lim i!1 Cp .Ei /  Cp .E/ follows from monotonicity, and monotonicity also shows that the limit always exists. Conversely, assume that limi!1 Cp .Ei / < 1. We can find ui  Ei with upper gradients gi such that p p kui kL p .X/ C kgi kLp .X/ < Cp .Ei / C 1= i:

P By Lemma 6.2 there are u; g 2 Lp .X /, convex combinations vj D 1 iDj aj;i ui and such that both u ! u and gik ! g a strictly increasing sequence of indices fik g1 i k kD1 weakly in Lp .X /, as k ! 1, vj ! u q.e., as j ! 1, and g is a p-weak upper gradient of u. It is clear that vj  Ej and thus u  E q.e. Let v WD maxfu; E g D u q.e., and thus uik ! v weakly in Lp .X /, as k ! 1, and g is a p-weak upper gradient also of v. Hence p p p p Cp .E/  kvkL p .X/ C kgkLp .X/  lim inf .kuik kLp .X/ C kgik kLp .X/ / k!1

 lim .Cp .Eik / C 1= ik / D lim Cp .Ei /: k!1

i!1

The following examples show that the assumption p > 1 is essential for the results in this section. It is easy to give similar examples in Rn . Example 6.5. Let X D R. Let further ui .x/ D .1  i jxj/C . Then fui g1 iD1 is a bounded sequence in N 1;1 .R/ and ui ! u WD f0g , as i ! 1, everywhere on R. However, u … N 1;1 .R/. This shows that neither Lemma 6.2 nor Corollary 6.3 holds for p D 1.

6.2 Properties of Cp in complete doubling p-Poincaré spaces

157

Example 6.6. Let X D R and d D w dx, where ´ 1; x > 0; w.x/ D 2; x  0: Let further Ei D Œ1= i; 1, i D 1; 2; ::: , and E D .0; 1. We want to calculate C1 .Ei / and C1 .E/. Let u  Ei , u 2 N 1;1 .X /. In the same way as in Example 5.40 we see that u has to be continuous. Without loss of generality we can assume that 0  u  1. As kukL1 .R/ < 1, there exists a decreasing sequence tj ! 1 such that u.tj / ! 0, as j ! 1. We then have for each j D 1; 2; ::: , Z 1=i Z 1=i ju.1= i /  u.tj /j  gu ds  gu d; and letting j ! 1 yields point 1 we see that

R 1=i 1

tj

tj

gu d  1. Using a similar estimate at the other end 



1 1 C2D3 ; i i and taking infimum over all u admissible in the definition of C1 .Ei / we obtain C1 .Ei /  3  1=i . At the same time, the functions uj D .1  j dist.x; Ei //C , j > i , show that kukN 1;1 .R/  1 

C1 .Ei /  kuj kN 1;1 .R/ D 3 

1 1 1 C !3 ; i j i

as j ! 1:

Doing the same type of calculation for E D .0; 1 gives C1 .E/ D 4, as this time u  E is forced to have u.0/  1, and thus, Z 0 Z 0 Z 1 1 0 1 D lim ju.tj /  u.0/j  lim gu ds D lim gu d D gu d: j !1 j !1 t 2 j !1 tj 2 1 j Hence C1

1 [ iD1

 Ei D 4 > 3 D lim C1 .Ei /: i!1

Thus Theorem 6.4 fails for p D 1. It also follows that C1 is not a Choquet capacity, as defined in Theorem 6.11 below. We remark that .R; / is a complete doubling p-Poincaré space.

6.2 Properties of Cp in complete doubling p-Poincaré spaces Our main goal in the second part of the book (Chapters 7–14) will be to study p-harmonic functions on complete doubling p-Poincaré spaces. Let us therefore, for further reference, here collect all the basic properties of the capacity in such spaces. Note, however, that properties (i)–(vi) hold in general metric measure spaces.

158

6 Capacities

Theorem 6.7. Let X be a complete doubling p-Poincaré space and E; E1 ; E2 ; :::  X . Then the following properties hold: (i) Cp .¿/ D 0; (ii) .E/  Cp .E/; (iii) if E1  E2 , then Cp .E1 /  Cp .E2 /; (iv) Cp is strongly subadditive, i.e. Cp .E1 [ E2 / C Cp .E1 \ E2 /  Cp .E1 / C Cp .E2 /I (v) Cp is countably subadditive (and is also an outer measure), i.e. Cp

1 [

1  X Ei  Cp .Ei /;

iD1

iD1

(vi) if 1 < p < 1 and E1  E2   , then Cp

1 [

 Ei D lim Cp .Ei /; i!1

iD1

(vii) Cp is an outer capacity, i.e. for all E  X , Cp .E/ D inf Cp .G/; GE G open

(viii) if K1  K2   are compact subsets of X , then Cp

1 \

 Ki D lim Cp .Ki /; i!1

iD1

(ix) Cp .E/ D inf kukpN 1;p .X/ ; u

where the infimum is taken over all functions u 2 N 1;p .X / such that u  1 in an open set containing E; (x) Cp .E/ D inf kukpy 1;p u

N

.X/

;

where the infimum is taken over all functions u 2 Ny 1;p .X / such that u  1 in an open set containing E;

6.2 Properties of Cp in complete doubling p-Poincaré spaces

159

(xi) if K  X is compact, then Cp .K/ D

inf

u1 on K u2Lip.X/

kukpN 1;p .X/ ;

(xii) if G  X is open and .E/ D 0, then Cp .G/ D Cp .G n E/; (xiii) if  is connected and F    is relatively closed, then Cp .F / D 0

”

Cp .@F \ / D 0;

(xiv) if  is connected, F   is relatively closed and Cp .F / D 0, then  n F is connected. The following examples illustrate that compactness is essential in (viii). Note that in all three examples, X is a doubling p-Poincaré space. Moreover, in Examples 6.8 and 6.10, X is also complete. k D 1;2; ::: , which are Example 6.8. Let p  1, X D Rn and Fk D Rn n B.0; k/, T1 closed but unbounded. Then Cp .Fk / D 1 for all k, but Cp kD1 Fk D Cp .¿/ D 0. Example 6.9. Let p  1, X D .0; 1/  R and Fk D .0; 1=k, k D 1; 2; ::: , which are bounded and closed as subsets of X , though not compact,  T1as X is not proper. As in Example 6.6, we see that Cp .Fk /  1 for all k, but Cp kD1 Fk D Cp .¿/ D 0. Example 6.10. Let p  1, X D Rn and Gk D B.0; 1/ n B.0; 1  1=k/, k D 1; 2; ::: , which are open sets. Then Cp

1 \

 Gk D Cp .¿/ D 0 < lim Cp .Gk /: k!1

kD1

A set function satisfying (iii), (vi) and (viii) is a Choquet capacity. An important consequence is the following result. Theorem 6.11 (Choquet’s capacitability theorem). If C.  / is a Choquet capacity (which, in particular, holds for Cp if X is a complete doubling p-Poincaré space and p > 1), then all Borel sets (and even all Suslin sets) E  X are capacitable, i.e. C.E/ D

sup KE K compact

C.K/ D inf C.G/: GE G open

Suslin sets are sometimes called analytic sets (although analytic sets in complex analysis is an entirely different concept). The interested reader should look elsewhere for more on Suslin sets and Choquet’s capacitability theorem, e.g. in Aikawa–Essén [8], Part 2, Section 10.

160

6 Capacities

Remark 6.12. The properties (ix) and (x) in Theorem 6.7 show that our definition of the capacity is compatible with the usual definition of Sobolev capacity on Rn , which is for general sets defined as in (x), see Section 2.35 in Heinonen–Kilpeläinen–Martio [171]. Note also that by Theorem 6.7 (vii), we have Cp .E/ D inf Cp .G/ GE G open

for all sets, not only for capacitable sets. Proof of Theorem 6.7. We established properties (i)–(iii) and (v) in Theorem 1.27, (iv) in Proposition 2.22, (xii) in Proposition 5.22, and (vi) in Theorem 6.4 (without any extra assumptions on our underlying space X ). (vii) This is Theorem 5.31. T (viii) Let K D 1 K iD1 i . By monotonicity we S1have Cp .K/  limi!1 Cp .Ki /. Conversely, let G  K be open. Then G [ iD1 .X n Ki / is an open cover of the compact set K1 . Thus, there is a finite subcover, i.e. an N such that K1  G [

N [

.X n Ki / D G [ .X n KN /:

iD1

As KN  K1 , it follows that KN  G. So limi!1 Cp .Ki /  Cp .G/. By (vii) we obtain the equality sought for. (ix) Let " > 0. By (vii) we can find an open set G  E and a function u  G such that kukpN 1;p .X/  Cp .E/ C ": Letting " ! 0 shows that Cp .E/  inf kukpN 1;p .X/ : u

The converse implication is trivial. (x) It is clear that the infimum in (x) is at most the infimum in (ix). Let u 2 Ny 1;p .X / be at least 1 in an open set G  E. Then there is v 2 N 1;p .X / so that u D v a.e. Thus v  1 a.e. in G, and hence, by Proposition 1.59, v  1 q.e. in G. Let ´ v in X n G; wD max¹v; 1º in G: Then w D v q.e. and hence w 2 N 1;p .X /. As u D w a.e., we have kukNy 1;p .X/ D kwkN 1;p .X/ and hence the infimum in (x) is at least the infimum in (ix). The equality under consideration now follows from (ix). (xi) Let " > 0. By (vii), there exists an open set G  K such that Cp .G/ < Cp .K/ C ". Let u 2 N 1;p .X / be such that u D 1 on G and kukpN 1;p .X/ < Cp .K/ C ":

6.3 The variational capacity capp

161

Lemma 5.4 provides us with a bounded Lipschitz function u with kuu kN 1;p .X/ < " and u D 1 on K. Hence, we have Cp .K/1=p  ku kN 1;p .X/  kukN 1;p .X/ C ku  u kN 1;p .X/ < .Cp .K/ C "/1=p C "; and letting " ! 0 finishes the proof. (xiii) This follows from Lemma 4.5. (xiv) This follows from Lemma 4.6.

6.3 The variational capacity capp In this section we assume that   X is bounded. Definition 6.13. Let E  . Then we define the variational capacity Z capp .E; / D inf gup d; u

where the infimum is taken over all u 2



N01;p ./

such that u  1 on E.

The variational capacity is sometimes referred to as the relative capacity. Recall, that we consider the infimum to be 1 if there are no such functions u. The infimum is always finite if E b , but it can happen that it is finite even if dist.E; X n / D 0. Indeed, the function f in Example 11.10 is continuous in  and belongs to N01;p ./ for 1 < p < n, and dist.fx 2  W f .x/ D 1g; X n / D 0. As with the Sobolev capacity, the infimum can be taken only over u 2 N01;p ./ such that E  u  1. To see this, just use truncation as in the proof of Proposition 1.26. Lemma 6.14. Assume that E   and Cp .E/ D 0. Then capp .E; / D 0. This result is true even if  is unbounded. Proof. As Cp .E/ D 0, we have that u WD E 2 N 1;p .X / and gu D 0. Consequently R u 2 N01;p ./, and capp .E; /   gup d D 0. The converse is not always true. For instance, if X is a space without curves, then Cp .E/ D 0 if and only if .E/ D 0, while capp .E; / D 0 for all E (if  is allowed to be unbounded, then capp .E; / D 0 if and only if .E/ < 1). In connections with Newtonian spaces, Cp .E/ D 0 is the right condition for defining quasieverywhereness. On the other hand, for the Wiener condition, p-potentials and fine continuity in Chapter 11 it is more natural to work with the variational capacity capp . The following lemma shows that, under suitable assumptions, sets of zero Cp capacity coincide with sets of zero capp -capacity, and hence that quasieverywhere is the same with respect to both capacities.

162

6 Capacities

Lemma 6.15. Assume that X supports a .p; p/-Poincaré inequality, that  is bounded and that Cp .X n/ > 0. Let E  . Then Cp .E/ D 0 if and only if capp .E; / D 0. Observe that the condition Cp .X n / > 0 is essential for if Cp .X n / D 0, then 1 2 N01;p ./ showing that capp .E; / D 0 for all E  . Proof. The necessity follows directly from Lemma 6.14. As for the sufficiency, assume that capp .E; / D 0 and let " > 0. Then there is u 2 N01;p ./ such that u D 1 on E and kgu kpN 1;p .X/ < ". The Poincaré inequality for N01;p (Corollary 5.54) then yields that Z p z Cp .E/  kukN 1;p .X/  C gup d < Cz ": X

Letting " ! 0 concludes the proof. The following result gives some more precise estimates for the variational capacity, in particular for balls. It also implies that Cp and capp have the same zero sets. Note that the comparison constant on the right-hand side in the second formula explodes as r ! 0. Proposition 6.16. Let X be a doubling p-Poincaré space with doubling constant C . Then there exists C > 0 such that if E  B D B.x0 ; r/ with 0 < r < 18 diam X , then C .B/ .E/  capp .E; 2B/  p Cr rp and





1 Cp .E/  capp .E; 2B/  2p 1 C p Cp .E/: p C.1 C r / r

Proof. If v is admissible in the definition of capp .E; 2B/, then by the Sobolev inequality (Theorem 5.51) with q D p, Z Z .E/  jvjp d  C r p gvp d; 2B

Z

and Cp .E/ 

2B

Z jvj d C p

X

X

Z gvp

d  C.1 C r / p

2B

gvp d:

Taking infimum over all admissible v yields the left inequalities in the lemma. Conversely, let .x/ D minf2  d.x; x0 /=r; 1gC . Note that is a 1=r-Lipschitz function vanishing outside 2B such that 0   1 and D 1 on B. If u 2 N 1;p .X / is admissible in the definition of Cp .E/, then and u are admissible in the definition of capp .E; 2B/ and hence Z .2B/ C .B/ capp .E; 2B/  gp d   ; p r rp 2B

6.3 The variational capacity capp

and

163

Z

Z p gu d  .jujg C gu /p d 2B 2B Z Z 2p p  p juj d C 2p gup d: r X X

capp .E; 2B/ 

Taking infimum over all u 2 N 1;p .X / finishes the proof. The variational capacity satisfies more or less the same properties as the Sobolev capacity. Let us start with properties not requiring extra conditions on the space. Theorem 6.17. Assume that  is bounded and let E1 ; E2 ; :::  . Then the following properties hold: (i) capp .¿; / D 0; (ii) if E1  E2  , then capp .E1 ; /  capp .E2 ; /; (iii) capp is strongly subadditive, i.e. capp .E1 [ E2 ; / C capp .E1 \ E2 ; /  capp .E1 ; / C capp .E2 ; /I (iv) capp is countably subadditive (and is also an outer measure), i.e. capp

1 [ iD1

1  X Ei ;   capp .Ei ; /; iD1

(v) if 1 < p < 1 and E1  E2    , then capp

1 [ iD1

 Ei ;  D lim capp .Ei ; /; i!1

(vi) if F   is closed as a subset of X , then capp .F; / D capp .@F; /. It is natural to ask if (v) holds for p D 1. In Example 6.6 we saw that the corresponding identity fails for C1 . Example 6.18. Let the notation be as in Example 6.6 and let  D .1; 2/. Then cap1 .Ei ; / D 2 for i D 1; 2; ::: , while cap1

1 [

 Ei ;  D cap1 ..0; 1; / D 3;

iD1

showing that (v) fails for p D 1.

164

6 Capacities

Proof of Theorem 6.17. (i), (ii) and (iv). This is similar to the proof of Theorem 1.27, the only difference being that the functions admissible for testing the capacity are chosen from N01;p ./. The same is true for (iii) and (v) below. (iii) This is similar to the proof of Proposition 2.22. (v) This is similar to the proof of Theorem 6.4. (vi) Let u be admissible in the definition of capp .@F; /. Without loss of generality we can assume that 0  u  1 and that u D 0 in X n . Let ´ 1 in F; vD u in X n F: Then kvkLp .X/  kukLp .X/ C.F / < 1. Let  W Œ0; l  ! X be a curve such that gu is an upper gradient of u along . If   F or   X n F , then it is straightforward that gu is an upper gradient of v along . If  intersects both F and X n F , we can, by splitting  into parts if necessary, and possibly reversing the direction, assume that x D .0/ 2 F and y D .l / 2 X n F . Letting t D supf W . / 2 F g we have .t/ 2 @F and hence Z Z gu ds  gu ds; jv.x/  v.y/j D ju..t //  u.y/j  jŒt;l 



i.e. gu is an upper gradient of v along  as well. As this holds for p-a.e. curve  it follows that gu is a p-weak upper gradient of v. Hence v 2 N 1;p .X /. As v D u D 0 in X n , we have v 2 N01;p ./ and also Z Z gvp d  gup d: capp .F; /  



Taking infimum over all u, we see that capp .F; /  capp .@F; /. The converse inequality is trivial. Let us next look at properties for which we require additional conditions on the space. Theorem 6.19. Assume that X is a complete doubling p-Poincaré space and that  is bounded. Then the following properties hold: (vii) capp is an outer capacity, i.e. for all E b , capp .E; / D

inf

G open E G

capp .G; /;

(viii) if K1  K2   are compact subsets of , then capp

1 \ iD1

 Ki ;  D lim capp .Ki ; /; i!1

6.3 The variational capacity capp

(ix) if E b , then

165

Z capp .E; / D inf u



gup d;

where the infimum is taken over all functions u 2 N01;p ./ such that u  1 in an open set containing E; (x) if K   is compact, then capp .K; / D

inf

u1 on K u2Lip0 ./

p kgu kL p .X/ ;

where Lip0 ./ D ff 2 Lip.X / W f D 0 on X n g. Proof. The proofs of (viii) and (ix) use (vii) and are similar to the corresponding proofs of (viii) and (ix) in Theorem 6.7. For (vii) and (x), the proofs are similar to the proofs of Theorems 5.31 and 6.7 (xi), but more elaborate since special care has to be taken near the boundary of . (vii) Let 0 < " < 1 and find u 2 N01;p ./ such that u  E and p kgu kL p .X/  capp .E; / C ":

By Theorem 5.29, u (extended by zero outside ) is quasicontinuous in X and hence there is an open set V with Cp .V /1=p < " such that ujXnV is continuous. Thus, there is an open set U such that U n V D fx W u.x/ > 1  "g n V  E n V: We can also find v  V with kvkN 1;p .X/ < ". Let .x/ D minf1; 2dist.x; E/=d gC , where d D 12 dist.E; X n / > 0. Note that 2 Lip0 ./, 0   1 in , g  1=d and D 1 in the open neighbourhood W WD fx 2  W dist.x; E/ < d g of E. Then 

kgv kLp .X/  kgv kLp .X/ C



1 1 " kvkN 1;p .X/ < " C : kvkLp .X/  1 C d d d

Let w D u=.1  "/ C v, so that w 2 N01;p ./ and w  1 on ..U n V / [ V / \ W D .U [ V / \ W; an open set containing E. Hence capp .E; /1=p 

inf

G open E G

capp .G; /1=p

 capp ..U [ V / \ W; /1=p  kgw kLp .X/ kgu kLp .X/ C kgv kLp .X/  1" .capp .E; / C "/1=p C " C "=d  : 1"

166

6 Capacities

Letting " ! 0 completes this part of the proof. (x) Let " > 0. By (vii), there exists an open set G  K such that capp .G; / < capp .K; / C ". Let u 2 N01;p ./ be such that G  u  1 in  and p kgu kL p .X/ < capp .K; / C ":

By Theorem 5.29, the function u (extended by zero outside ) is quasicontinuous in X and hence there is an open set V with Cp .V /1=p < " such that ujXnV is continuous. Thus, there is an open set U such that U n V D fx W u.x/ < "g n V  .X n / n V: We can also find v  V with kvkN 1;p .X/ < ". Let w D min

²

³

.u  "/C ;1  v : 1"

Then w D 0 on .U n V / [ V D U [ V , an open set containing X n . Moreover, 1 kgu kLp .X/ C kgv kLp .X/ : 1" Let G 0 D G [ U [ V and note that X n G 0  X n U   is bounded. Lemma 5.4 applied to F D K [ .X n / and G 0  F provides us with a bounded Lipschitz function w such that kw  w kN 1;p .X/ < ", w D 1 on K and w D 0 in X n . Hence, we have kgw kLp .X/ 

kgw kLp .X/  kgw kLp .X/ C kw  w kN 1;p .X/ <

.capp .K; / C "/1=p 1"

C 2"

and letting " ! 0 finishes the proof. The following result is an analogue of Corollaries 4.29 and 5.39 for the variational capacity. Corollary 6.20. Assume that X supports a p-Poincaré inequality and that (4.3) holds for some s < p. Then for every ball B D B.x0 ; r/ with 0 < r < 14 diam X and every set E  B with capp .E; B/ > 0 we have capp .E; B/  C r p .B/: If moreover X is complete, then this holds for all nonempty sets E  B. Proof. This is trivial if capp .E; B/ D 1. Let u 2 N01;p .B/ be such that 0  u  1 in B and u D 1 on E. Then by Lemma 6.15, the L1 -Sobolev inequality for large p (Proposition 5.52) and the comment after it, we have Z C rp 1 D ess sup u  g p d .B/ B u B and taking infimum over all u admissible in the definition of capp .E; B/ finishes the first part of the proof. The second statement now follows from Theorem 6.19 (vii).

6.3 The variational capacity capp

167

The following inequality is an analogue of Maz0 ya’s inequality (Theorem 5.53), this time with the variational capacity. Note that, when  is doubling, Theorem 5.53 for small balls follows from Theorem 6.21 but not vice versa, by the estimates in Proposition 6.16. Theorem 6.21 (Maz0 ya’s inequality). Assume that X supports a .q; p/-Poincaré in1;p equality for some q  p. For u 2 Nloc .X /, let S D fx 2 X W u.x/ D 0g. Then for all balls B D B.x0 ; r/, p=q « Z C q juj d  g p d; capp .B \ S; 2B/ 2B u 2B where  is the dilation constant in the .q; p/-Poincaré inequality. As in Theorem 5.53, the Hölder inequality makes it possible to replace q  p in the left-hand side of (5.7) by any smaller exponent, even though the proof requires a .p; p/Poincaré inequality also for small exponents in the left-hand side. By Corollary 4.24, the assumption that X supports a .q; p/-Poincaré inequality is satisfied for some q > p in doubling p-Poincaré spaces. Note, however, that here we do not assume that  is doubling. The space and the function constructed in Example 8.20 show that the factor  on the right-hand side in Theorem 6.21 cannot be omitted. In Euclidean spaces, Theorem 10.1.2 in Maz0 ya [283] shows that the balls 2B and 2B in the integrals can be replaced by B, giving a stronger result. (Of course,  D 1 in this case.) The argument is based on extending Sobolev functions from B to 2B (e.g. by reflection), a method which is difficult to apply in metric spaces. As in Theorem 5.53, the factor 2 in 2B and 2B is not essential and the proof below can easily be modified to obtain a similar result with B and B and the constant C depending on  > 1. Proof. The proof is similar to the proof of Theorem 5.53. The only changes needed are replacing (5.8) and (5.10) by Z Z Z 2p r p 2p p p capp .B \ S; 2B/  gv d  ju  uj N d C p g p d uN p uN 2B u 2B 2B and capp .B \ S; 2B/ 

C uN p

Z 2B

gup d;

respectively. To obtain Adams’ criterion (Theorem 7.3) later on we will need the following lemma, often called Maz0 ya’s capacitary inequality. Lemma 6.22 (Maz0 ya’s capacitary inequality). Assume that  is bounded. Then for all u 2 N01;p ./, Z Z 1 t p1 capp .fx W ju.x/j > t g; / dt  2p log.2/ gup d: (6.1) 0



168

6 Capacities

Remark 6.23. The Cavalieri principle (Lemma 1.10) says that if f W X ! Œ0; 1 is a -measurable function, then Z Z 1 f p d D p t p1 .fx W f .x/ > t g/ dt: X

0

By analogy it is natural to write (6.1) as Z Z jujp d capp .  ; /  2p p log.2/ gup d; 



even though capp .  ; / is not a measure. Such integrals are called Choquet integrals and their study goes back to Choquet [94]. Proof. As gu D gjuj a.e., we may assume that u  0. For t > 0, let E t D fx 2  W u.x/ > t g

and u t D minf.u  t =2/C ; t =2g:

Note that u t is the truncation of u at the levels t =2 and t , shifted by t =2, and that gu t D gu ft=2
Z

1

t 0

p1

capp .E t ; / dt 

1 0

Z

 p

Z

D 2p

2 t

t

p1

Z

gup ft=2
gu .x/p X

u.x/

d dt

dt d.x/ D 2p log.2/ t

Z X

gup d;

where to get the last equality we used the fact that gu D 0 a.e. in fx W u.x/ D 0g, by Corollary 2.21.

6.4 Notes Most of the properties for our capacities have not been deduced explicitly in the Newtonian literature, instead references have been given to similar results for capacities in other situations. In particular, Kinnunen–Martio [215], while studying capacities for Hajłasz–Sobolev spaces on metric spaces, gave proofs which are easily modified to give Theorem 6.7 (v) and (viii). Theorem 6.7 (vi) was established in Kinnunen–Martio [216] for Hajłasz–Sobolev spaces (the corresponding result for open sets appeared already in [215]). The main ingredient in the proof in [216] was the double usage of Mazur’s lemma, which we have here distilled into Lemma 6.2, which is of use also elsewhere. Its proof is from Björn–Björn–Parviainen [54].

6.4 Notes

169

Theorem 6.7 (xi) is a special case of much more general results in Kallunki [Rogovin]–Shanmugalingam [194], but the proof given here is different and shorter. Proposition 6.16 is from J. Björn [63], and Example 6.6 is due to Korte [223]. Lemmas 6.14 and 6.15, Theorems 6.17 and 6.19 (except for (x)), and Maz0 ya’s capacitary inequality (Lemma 6.22) are from Björn–Björn [49], [50], where they are obtained also for nonopen . Capacitary inequalities, such as the one in Lemma 6.22, were originally obtained for the p-capacity in Rn by Maz0 ya in [281] and [283]. For further generalizations and applications, see Maz0 ya [284] and the references therein, and also [49]. Sets of capacity zero have measure zero. In fact, it follows from Theorem 4 in J. Björn–Onninen [71] that if X is a complete p-Poincaré space and (4.3) holds for some s > p, then sets of capacity zero have .s  p/-dimensional Hausdorff measure zero. Sharp estimates for capp .B.x; r/; B.x; R//, 0 < r < R, have been given in metric spaces by Garofalo–Marola [138]. For p D 1, the C1 -capacity as well as the closely related capacity for BV-functions has been studied by Hakkarainen–Kinnunen [156].

Chapter 7

Superminimizers From now on (i.e. in Chapters 7–14) we assume that 1 < p < 1 and that X is a complete doubling p-Poincaré space, with doubling constant C and dilation constant  in the p-Poincaré inequality. By Theorem 4.30 it follows that X is a q-Poincaré space for some q 2 Œ1; p/. The reader who prefers may instead of being dependent on Theorem 4.30, which is not proved in this book, choose to require that X is a complete doubling q-Poincaré space for some q < p. We will use the q-Poincaré inequality when applying Theorem 5.62 in the proof of Theorem 8.22. This result is subsequently used quite extensively. Recall also that throughout the book,  is a nonempty open subset of X . In some sections we will add further assumptions on , such as boundedness and that the complement has positive capacity. This is e.g. crucial for the obstacle and Dirichlet problems, as otherwise there would not be any boundary data to talk of, cf. the notes to Chapter 10. Other notions, such as superminimizers, p-harmonic and superharmonic functions do not require these additional assumptions.

7.1 Introduction to potential theory With this chapter we start the second part of the book. In this part we study the potential theory associated with p-harmonic functions (in metric spaces). The two main themes are interior and boundary regularity. Intimately connected are also the theories for superminimizers, superharmonic functions and the obstacle problem. In (unweighted) Rn , a p-harmonic function u is a continuous weak (or distributional) solution of the p-Laplace equation p u D div.jrujp2 ru/ D 0: Their theory has been extensively studied in Heinonen–Kilpeläinen–Martio [171], also in weighted Rn . (Even though many of our results, when specialized to weighted Rn , are not covered in [171], especially those in Chapters 10–14.) Solutions of the p-Laplace equation coincide with the minimizers of the p-Dirichlet integral as in Definition 7.7, see Theorem 5.13 in [171] (and Appendix A.1). The supersolutions of the p-Laplace equation also coincide with the superminimizers of the p-Dirichlet integral, see again Theorem 5.13 in [171]. In the generality of metric spaces we do not have a partial differential equation, only the p-Dirichlet integral. For

7.1 Introduction to potential theory

171

this reason we speak about minimizers instead of (weak) solutions, and superminimizers instead of supersolutions, etc. A difference to the situation in Rn is that in metric spaces we only have a scalarvalued gu corresponding to jruj, whereas in Rn , ru is vector-valued. The vector structure is needed in the p-Laplace equation, but not in the p-Dirichlet integral. To develop our potential theory we will need much of the Newtonian theory from the first part of the book. On the other hand, p-harmonic functions are minimizers in the definition of the capacity (more precisely of the variational capacity, see Lemmas 11.17 and 11.19), and hence play a role also in the Newtonian theory. This is perhaps most apparent in Theorem 11.40 about fine continuity of Newtonian functions. Various types of potentials, such as Riesz, Bessel, Green and Wolff potentials, play a vital role in potential theories of various types. However, as we have no equation defining p-harmonic functions such tools are not available to us. The only potentials we will use are capacitary potentials. Under our assumptions, Cheeger’s theorem (Theorem B.6) shows that there is a vector-valued gradient structure on X of so-called Cheeger gradients. Moreover, this gradient structure leads to Cheeger–Sobolev spaces which coincide Ny 1;p .X /. One R with ˛ p Cheeger minimizers, etc., by minimizing jd uj d instead of Rcanpnaturally define gu d, where d ˛ u is the Cheeger gradient of u. The results we derive in the rest of this book remain valid with this change. An advantage of using Cheeger gradients is that a (super)minimizer is the same as a weak (super)solution of the Cheeger p-Laplace equation div.jd ˛ ujp2 d ˛ u/ D 0: This makes a lot more of the classical techniques from Rn available. A disadvantage of using Cheeger’s theorem is that it does produce a vector-valued gradient structure, but this structure is not unique. Moreover, one always uses an inner product to take the length of the Cheeger gradient. It is thus only implicitly understood what the Cheeger gradient, or its length, really determines. In particular, its connection to the geometry of the space is not so clear. The upper gradient on the other hand has a very geometric definition. Another reason for choosing upper gradients as the basis for the nonlinear potential theory is to see how much of the theory can be developed only based on the variational integrals, with no equation available. This necessitates new ideas and proofs of various results, adding new insight also to the theory on Rn . The interest in this development should also be larger than if it was primarily a copying of the results from Rn . One can go one step further and study quasiminimizers for which there is also no equation, only variational integrals. This theory is also interesting and many of the techniques are available in metric spaces. However, quasiminimizing causes additional problems. The most obvious is that there is no uniqueness when solving the Dirichlet problem, so many of the results we obtain are simply not true in such a generality. But there are also many results that are true but which need more complicated proofs in the quasiminimizing case.

172

7 Superminimizers

Here we have chosen to consider only the theory of minimizers (or p-harmonic functions) not quasiminimizers, but without an equation. Along the way we make some comments on certain additional results available in the Cheeger gradient case, when the equation can be used. In Appendix C we also briefly discuss the current status of the theory of quasiminimizers. As mentioned, the main aim in the rest of the book is to study p-harmonic functions. To do so we need superminimizers, superharmonic functions and solutions of obstacle problems as tools. In this chapter, we start developing the theories for obstacle problems and superminimizers. In Chapter 8 we concentrate on interior regularity, which makes it possible to fully develop the superharmonic theory in Chapter 9. In Chapter 10 we turn to the Dirichlet problem and boundary regularity, which is the main theme in the remaining chapters as well.

7.2 The obstacle problem In this section we make the additional assumption that  is bounded and such that Cp .X n / > 0. If X is unbounded, then the condition Cp .X n / > 0 is of course immediately fulfilled. Definition 7.1. Let V  X be a nonempty bounded open set with Cp .X n V / > 0. x Then we define Let also f 2 N 1;p .V / and W V ! R. K

;f

.V / D fv 2 N 1;p .V / W v  f 2 N01;p .V / and v 

q.e. in V g:

Furthermore, a function u 2 K ;f .V / is a solution of the K ;f .V /-obstacle problem if Z Z gup d  gvp d for all v 2 K ;f .V /: V

We also let K

;f

V

DK

;f ./.

Observe that we solve the obstacle problem with boundary data f 2 N 1;p ./. Such functions are a priori defined only in , not on @. So, strictly speaking, we have boundary data without boundary values, and the definition should be understood in a weak Sobolev sense. See also Section 8.7, where the Dirichlet problem is solved in a similar way. Note also that we only define the obstacle problem for nonempty bounded open sets V with Cp .X n V / > 0. If we consider the obstacle problem in V with Cp .X n V / D 0, using the obvious definition, then the condition vf 2 N01;p .V / becomes void. Hence if M D Cp - ess supV , then v  c is a solution for any M  c < 1, and these are all solutions. This is easy to see if M < 1. If M RD 1, v 2 K R ;fp.V / and p c > Cp - ess inf V v, then vQ WD maxfv; cg 2 K ;f .V / and V gvQ d < V gv d (by

7.2 The obstacle problem

173

the Poincaré inequality), showing that v is not a solution, and thus that there is no solution. With the assumption that Cp .X n V / > 0 we next see that we have solubility and uniqueness under a very natural assumption: that the set K ;f is nonempty, a condition which we characterize in Adams’ criterion (Theorem 7.3) below, see also Proposition 7.4 for a simpler characterization in the case when 2 N 1;p ./. Theorem 7.2. Assume that  is bounded and such that Cp .X n / > 0. Let f 2 x If K ;f ¤ ¿, then there is a unique solution (up to sets of N 1;p ./ and W  ! R. capacity zero) of the K ;f -obstacle problem. For the proof of the uniqueness part we need to use the strict convexity of Lp ./. A Banach  space Y (with norm k  k) is strictly convex if x D y whenever kxk D kyk D  21 .x C y/ D 1. A stronger condition is that Y is uniformly convex, i.e. that kxj  yjk ! 0, as j ! 1, whenever kxj k D kyj k D 1, j D 1; 2; ::: , and   1 .xj C yj / ! 1, as j ! 1. 2 Clarkson [98] introduced the notion of uniformly convex spaces and showed that all Lp spaces, 1 < p < 1, are uniformly convex, see also Adams–Hedberg [5] and the references therein. Proof. Let

Z I D

inf

v2K

;f



gvp d:

Since K ;f ¤ ¿, we have 0  I < 1. Let fuj gj1D1  K sequence such that Z gupj d & I; as j ! 1:

;f

be a minimizing



is bounded in Lp ./. Using the Poincaré inequality for N01;p It follows that (Corollary 5.54) we find that Z Z Z Z p p p juj  f j d  C guj f d  C guj d C C gfp d: fguj gj1D1









Thus fuj gj1D1 is bounded in N 1;p ./. By Lemma 6.2, we find convex combinations PNj PNj a u with p-weak upper gradients gj D a g and limit vj D kDj j;k k kDj j;k uk functions v and g such that vj ! v and gj ! g in Lp ./, vj ! v q.e., as j ! 1, and g is a p-weak upper gradient of v. It follows that v 2 N 1;p ./. Since clearly vj  q.e. in , also v  q.e. in . Further, wj WD vj  f 2 N01;p ./ and we can thus consider wj to be identically zero outside of . Let also w D v  f , gj0 D gj C gf and g 0 D g C gf , all three considered to be identically zero outside of . Then wj ! w and gj0 ! g 0 in Lp .X /, and wj ! w q.e. in X , as j ! 1. By Proposition 2.3, w 2 N 1;p .X / and thus v  f 2 N01;p ./.

174

7 Superminimizers

We have thus shown that v 2 K ;f . Since Z Z Z p p gv d  g d D lim gjp d D I; I  

j !1 



we conclude that v is the desired minimizer. It remains to prove the uniqueness. Assume that u1 and u2 are solutions. Then also u0 D 12 .u1 C u2 / 2 K ;f and thus   kgu1 kLp ./  kgu0 kLp ./   12 .gu1 C gu2 /Lp ./  12 kgu1 kLp ./ C 12 kgu2 kLp ./ D kgu1 kLp ./ : Hence gu1 D gu2 a.e. in  by the strict convexity of Lp ./. We shall show that gu1 u2 D 0 a.e. in . The Poincaré inequality for N01;p (Corollary 5.54) then yields ku1  u2 kLp ./ D 0 and hence u1 D u2 q.e. in , by Proposition 1.59. To show that gu1 u2 D 0 a.e. in , let c 2 R and u D maxfu1 ; minfu2 ; cgg: Then u 2 N 1;p ./ and u 

q.e. in . Also,

u  f  maxfu1 ; u2 g  f D maxfu1  f; u2  f g 2 N01;p ./ and u  f  u1  f 2 N01;p ./. Lemma 2.37 shows that u  f 2 N01;p ./ and hence u 2 K ;f . Let Vc D fx 2  W u1 .x/ < c < u2 .x/g and note that Vc  fx 2  W u.x/ D cg and hence gu D 0 a.e. in Vc , by Corollary 2.21. The minimizing property of gu1 then implies that Z Z Z Z p p p gu1 d  gu d D gu d D gup1 d; (7.1) 



nVc

nVc

since by Corollary 2.20, gu .x/ D gu1 .x/ D gu2 .x/ for a.e. x 2  n Vc . From (7.1) we conclude that gu2 D gu1 D 0 a.e. in Vc for all c 2 R. Now, [ fx 2  W u1 .x/ < u2 .x/g  Vc c2Q

and hence gu2 D gu1 D 0 a.e. in fx 2  W u1 .x/ < u2 .x/g, and similarly for fx 2  W u1 .x/ > u2 .x/g. It follows that gu1 u2  .gu1 C gu2 /fx2Wu1 .x/¤u2 .x/g D 0

a.e. in ;

which concludes the proof of the uniqueness and of the whole theorem. Let us now give Adams’ criterion for when K the obstacle problem is soluble.

;f

¤ ¿, and hence, by Theorem 7.2,

7.2 The obstacle problem

175

Theorem 7.3 (Adams’ criterion). Assume that  is bounded and that Cp .X n / > 0. x Then K ;f ¤ ¿ if and only if Let f 2 N 1;p ./ and W  ! R. Z 1 t p1 capp .fx W .x/  f .x/ > t g; / dt < 1: (7.2) 0

As in Remark 6.23, the condition (7.2) can be written as the Choquet integral Z .  f /pC d capp .  ; / < 1: 

If the obstacle K ;f ¤ ¿.

belongs to N 1;p ./ there is a much easier criterion for when

Proposition 7.4. Assume that  is bounded and such that Cp .X n / > 0. Let f; 2 N 1;p ./. Then K ;f ¤ ¿ if and only if .  f /C 2 N01;p ./. Proof. Assume first that there is u 2 K 0.

;f

. Then

 f /C  .u  f /C 2 N01;p ./:

Hence, .  f /C 2 N01;p ./, by Lemma 2.37. Conversely, assume that . f /C 2 N01;p ./ and let u D maxf ; f g 2 N 1;p ./. Then u  f D .  f /C 2 N01;p ./. As u  in , it follows that u 2 K ;f . Proof of Theorem 7.3. As K ;f D f C K f;0 we may assume, without loss of generality, that f  0. Assume first that there is some uQ 2 K ;0 . Then u WD maxfu; Q g D uQ q.e. in , and thus also u 2 K ;0 . Hence, by Maz0 ya’s capacitary inequality (Lemma 6.22) we have Z 1 Z 1 t p1 capp .fx W .x/ > t g; / dt  t p1 capp .fx W u.x/ > t g; / dt 0 0 Z p  2 log.2/ gup d < 1: 

Conversely, assume that (7.2) holds. It follows that capp .fx W .x/ > t g; / < 1 for all t > 0, as capp .fx W .x/ > t g; / is nonincreasing with respect to t . Thus we can find uk 2 N01;p ./, for k 2 Z, such that f >2k g  uk  1 and Z gupk d < capp .fx W .x/ > 2k g; / C 2jkj.kC1/p : 

Let vN D sup 2kC1 uk ; gN D sup 2kC1 guk ; kN

v D sup 2kC1 uk ; k2Z

N 2 Z;

(7.3)

kN

g D sup 2kC1 guk : k2Z

(7.4)

176

7 Superminimizers

(Here we take the same representative of guk in all places.) Then v  2kC1 when > 2k , in particular when 2k <  2kC1 , k 2 Z, from which it follows that v  in . By Lemma 1.52, gN is a p-weak upper gradient of vN . Moreover, Z Z g p d D sup.2kC1 guk /p d 

Z 

 k2Z 1 X



D

<

.2kC1 guk /p d

kD1

1 X

Z

2.kC1/p

kD1 1 X



gupk d

kD1

3C

.x/ > 2k g; / C 2jkj.kC1/p /

2.kC1/p .capp .fx W

1 X

« 2.kC1/p

kD1 Z 1 p p1

D3C4

t

0



2k 2k1

p1

t 2k1

capp .fx W

capp .fx W

.x/ > t g; / dt

.x/ > t g; / dt

DW A < 1: Thus gN ! g pointwise and in Lp .X /, by monotone convergence. By monotone convergence again and the Poincaré inequality for N01;p (Corollary 5.54), Z Z Z Z p v p d D lim vN d  C gvpN d  C g p d  C A: 

N !1 





Thus vN ! v both pointwise and in L .X /. Hence, Lemma 6.2 shows that g is a p-weak upper gradient of v, and thus v 2 N 1;p .X /. Consequently v 2 N01;p ./, as v D 0 in X n , and therefore v 2 K ;0 . p

The following is an easy observation. Proposition 7.5. Assume that  is bounded and such that Cp .X n / > 0. Let x be such that K ;f ¤ ¿. Let further u be a solution of f 2 N 1;p ./ and W  ! R the K ;f -obstacle problem. Then ˚ ess sup u  max Cp - ess sup ; ess sup f ;    ˚ ess inf u  max Cp - ess inf ; ess inf f ; 





7.2 The obstacle problem

D inffk 2 R W Cp .fx 2  W

where Cp - ess sup is defined similarly.

177

.x/ > kg/ D 0g and Cp - ess inf

Recall that as both u and f are Newtonian functions, Corollary 1.60 implies that the essential suprema of u and f equal their capacitary suprema Cp - ess sup. Note however that for we may have ess sup < Cp - ess sup . With this in mind and not to overload the notation, we shall continue to use ess sup whenever possible (in particular for Newtonian functions) and only use Cp - ess sup when necessary. Proof. Clearly, ess inf  u  Cp - ess inf  . Let ˚ A D max Cp - ess sup ; ess sup f : 



Then both v D minfu; Ag and w D maxfu; ess inf  f g belong to K ;f . As gv  gu and gw  gu a.e. in , by Corollary 2.20, both v and w are also solutions of the K ;f -obstacle problem. Thus u D v D w q.e. in , by the uniqueness in Theorem 7.2, from which the conclusion follows. The following comparison principle will be of much use to us. We formulate it for boundary values in N 1;p ./, but if f1 ; f2 2 N 1;p .X /, then .f1 f2 /C 2 N01;p ./ just x means that f1  f2 q.e. on @, by Proposition 2.39. Similarly, if f1 ; f2 2 N 1;p ./, 1;p 0 then .f1  f2 /C 2 N0 ./ if and only if Cp .fx 2 @ W f1 .x/ < f2 .x/g/ D 0, where x Cp0 is the capacity taken with respect to N 1;p ./. Lemma 7.6 (Comparison principle). Assume that  is bounded with Cp .X n / > 0. x and fj 2 N 1;p ./ be such that K ;f ¤ ¿, and let uj be a solution Let j W  ! R j j of the K j ;fj -obstacle problem, j D 1; 2. If 1



2

q.e. in  and .f1  f2 /C 2 N01;p ./;

then u1  u2 q.e. in . Proof. Let u D minfu1 ; u2 g and h D u1  f1  .u2  f2 / 2 N01;p ./. It follows that .f1  f2 /C  h D .f2  f1 /  h  minff2  f1 ; hg  h: By Lemma 2.37, minff2  f1 ; hg 2 N01;p ./ and thus u  f1 D minfu2  f1 ; u1  f1 g D u2  f2 C minff2  f1 ; hg 2 N01;p ./: As u  1 q.e. in , we get that u 2 K 1 ;f1 . Similarly v D maxfu1 ; u2 g 2 K 2 ;f2 . Let A D fx 2  W u1 .x/ > u2 .x/g. Since u2 is a solution of the K 2 ;f2 -obstacle problem, we have that Z Z Z Z gup2 d  gvp d D gup1 d C gup2 d: 



A

nA

178

7 Superminimizers

A

It follows that Z Z Z p p gu d D gu2 d C 

Z

Z

Thus

A

gup2 d 

A

gup1 d:

Z nA

gup1

d  A

Z gup1

Z

d C nA

gup1

d D 

gup1 d:

As u1 is a solution of the K 1 ;f1 -obstacle problem, so is u. By the uniqueness in Theorem 7.2, u1 D u D minfu1 ; u2 g q.e. in ; and thus u1  u2 q.e. in .

7.3 Definition of (super)minimizers As mentioned in the introduction to this chapter, the role of solutions to the p-Laplace equation will for us be played by minimizers of the p-Dirichlet integral, given by the following definition. 1;p ./ is a minimizer in  if for all ' 2 Lipc ./ Definition 7.7. A function u 2 Nloc we have Z Z '¤0

gup d 

'¤0

p guC' d:

(7.5)

1;p A function u 2 Nloc ./ is a superminimizer in  if (7.5) holds for all nonnegative ' 2 Lipc ./, and a subminimizer in  if (7.5) holds for all nonpositive ' 2 Lipc ./. We also say that a function is p-harmonic if it is a continuous minimizer.

At this point, the existence of nontrivial p-harmonic functions is not obvious, i.e. it is not clear that there are any nonconstant continuous minimizers. This regularity issue will be clarified in Section 8.3. If u is a superminimizer, ˛; ˇ 2 R and ˛  0, then it is easy to see that ˛u C ˇ is also a superminimizer, whereas ˛u C ˇ is a subminimizer. Similarly if u is a minimizer (or a p-harmonic function) and ˛; ˇ 2 R, then ˛u C ˇ is also a minimizer (or a p-harmonic function). Note, however, that the sum of two (super)minimizers (or p-harmonic functions) need not be a (super)minimizer (or a p-harmonic function), and thus our theory is nonlinear. Instead we will often use that the minimum of two superminimizers is a superminimizer, see Proposition 7.12. Proposition 7.8. A function u is a minimizer in  if and only if it is both a subminimizer and a superminimizer in .

7.3 Definition of (super)minimizers

179

Proof. The necessity is clear. As for the sufficiency let u be both a subminimizer and a superminimizer in . Let also ' 2 Lipc ./. Since u is a superminimizer we have Z Z Z p p p gu d  guC'C d D guC' d: 'C ¤0

'C ¤0

'C ¤0

Similarly (using that u is a subminimizer) we get Z Z Z p p gu d  gu' d D ' ¤0

' ¤0

' ¤0

p guC' d:

From this it follows that Z Z Z gup d D gup d C gup d '¤0 ' ¤0 ' ¤0 Z C Z Z p p  guC' d C guC' d D 'C ¤0

' ¤0

'¤0

p guC' d:

We next give a few characterizations of superminimizers. 1;p Proposition 7.9. Let u 2 Nloc ./. Then the following are equivalent:

(a) The function u is a superminimizer in . (b) For all nonnegative ' 2 Lipc ./ we have Z Z p gu d  supp '

supp '

p guC' d:

(7.6)

(c) For all open 0   and all nonnegative ' 2 N01;p .0 / we have Z Z p gup d  guC' d: 0

0

(d) For all nonnegative ' 2 N01;p ./ we have Z Z gup d  '¤0

'¤0

p guC' d:

Remark 7.10. (1) If we omit “super” from (a) and “nonnegative” from (b)–(d) we have corresponding characterizations for minimizers. The proof of these equivalences is the same as the proof below. (2) Note that in (c) and (d) we can replace “nonnegative” with “nonnegative a.e.” This follows since any function in N 1;p ./ which is nonnegative a.e. is actually nonnegative q.e., and then we can replace such a function by a nonnegative representative in the same equivalence class without changing any of the integrals involved.

180

7 Superminimizers

(3) Note that some of the integrals occurring in (c) and (d) may be infinite, but when this happens both sides are always simultaneously 1. (4) Any of these statements can be used to define superminimizers. The seemingly weakest requirements are made in (b). On the other hand, it is sometimes useful to know that (d) holds, in which the strongest requirements are made. Proof. (d) ) (c) Since gu D guC' a.e. on A WD fx 2 0 W '.x/ D 0g, we get Z Z Z p p gu d D gu d C gup d 0 A '¤0 Z Z Z p p p  guC' d C guC' d D guC' d: A

0

'¤0

(c) ) (a) This is trivial by letting 0 D fx 2  W '.x/ ¤ 0g. (a) ) (d) Let " > 0. By Theorem 5.46 there is a nonnegative f 2 Lipc ./ such that kf  'kN 1;p .X/ < ". Using (a) together with the fact that gu D guCf a.e. on fx W f .x/ D 0g we obtain 1=p Z 1=p Z Z 1=p p p p gu d  guCf d  guC' d C ": '¤0 or f ¤0

'¤0 or f ¤0

'¤0 or f ¤0

A WD fx W '.x/ D 0 ¤ f .x/g and A b  we can subtract As R gpu D guC' R on p g d D g A u A uC' d < 1 from the brackets in the left- and right-hand sides to obtain 1=p Z Z 1=p p p gu d  guC' d C ": '¤0

'¤0

Letting " ! 0 completes the proof of this implication. (a) , (b) Let ' 2 Lipc ./ be nonnegative and let A D fx 2 supp ' W '.x/ D 0g. Then Z Z Z p p gu d D gu d C gup d supp '

and

Z supp '

p guC'

'¤0

Z d D '¤0

A

p guC'

Z d C A

p guC' d:

As gu D guC' a.e.R in A b , Rwe can obtain (7.5) from (7.6), and vice versa, by p subtracting/adding A gup d D A guC' d < 1 from/to the left- and right-hand sides. The following localization lemma is another useful characterization of superminimizers. Note, however, that it is not known whether a similar result holds for general j , without the monotonicity assumption. It is not even known for a union of two sets, see Open problems 9.22 and 9.23.

7.3 Definition of (super)minimizers

181

S Lemma 7.11. Let 1  2     D j1D1 j . Then u is a superminimizer in  if and only if u is a superminimizer in j for j D 1; 2; ::: . Proof. Assume first that u is a superminimizer in . That u is a superminimizer in j follows directly from the fact that Lipc .j /  Lipc ./. Conversely, assume that u is a superminimizer in j for j D 1; 2; ::: , and let 0  ' 2 Lipc ./. By compactness, supp '  j . As u R there is j such R that p is a superminimizer in j we have '¤0 gup d  '¤0 guC' d; and thus u is a superminimizer in . The following two results are very useful tools in the nonlinear potential theory. They make it possible to construct new superminimizers (and later superharmonic functions) and will be used many times throughout this book, in particular when one of the functions is constant or p-harmonic. Proposition 7.12. Let u1 and u2 be superminimizers in . Then u WD minfu1 ; u2 g is a superminimizer in . Proof. This is a special case, upon letting 1 D 2 D , of the following pasting lemma. Lemma 7.13 (Pasting lemma for superminimizers). Assume that 1  2 and that u1 and u2 are superminimizers in 1 and 2 , respectively. Let ´ in 2 n 1 ; u2 uD min¹u1 ; u2 º in 1 : 1;p If u 2 Nloc .2 /, then u is a superminimizer in 2 . 1;p Note that if .u2  u1 /C 2 N01;p .1 I 2 /, then u D u2  .u2  u1 /C 2 Nloc .2 /. 1;p 1;p In fact u 2 Nloc .2 / if and only if .u2  u1 /C 2 N0 .1 I 2 \ G/ for every open G b 2 .

Proof. Let ' 2 Lipc .2 /, '  0, and let G WD fx 2 2 W '.x/ > 0g b 2 . Let further v D u C '. By the definition, it is enough to show that Z Z p gu d  gvp d: G

G

Let A D fx 2 2 W u2 .x/ < v.x/g  G: As .v  u2 /C 2 N01;p .A/ and v  u2 D .v  u2 /C in A, we have v  u2 2 N01;p .A/. Since u2 is a superminimizer we thus get that Z Z A

gup2 d 

A

gvp d:

182 Let next

7 Superminimizers 1;p D minfu2 ; vg 2 Nloc .2 /. Then

E WD fx 2 2 W

.x/ > u.x/g D fx 2 G \ 1 W u1 .x/ < u2 .x/g:

So > u D u1 in E. Now .  u/C 2 N01;p .E/, as E  G b 2 . Moreover,  u1 D .  u/C in E, and hence  u1 2 N01;p .E/. Since u1 is a superminimizer in 1 , we get that Z Z Z Z p p p gu1 d  g d D gu2 d C gvp d: E

E

E \A

E nA

It is easy to see that G D E [ A. Collecting all of these observations we obtain that Z Z Z p p gu d D gu2 d C gup1 d G GnE E Z Z Z p p  gu2 d C gu2 d C gvp d GnE E \A E nA Z Z  gup2 d C gvp d A E nA Z Z  gvp d C gvp d A E nA Z p D gv d: G

The following four results provide useful links between superminimizers and obstacle problems. Proposition 7.14. If  is bounded and Cp .X n / > 0, then a solution u of the K ;f -obstacle problem is a superminimizer in . Proof. Let 0  ' 2 Lipc ./. Then u C ' 2 K ;f . Let A D fx 2  W '.x/ D 0g. Since u is a solution of the K ;f -obstacle problem, we get that Z Z Z Z Z Z p p p gup dC gup d D gup d  guC' d D guC' dC guC' d: A

'¤0



As gu D guC' a.e. on A we right-most and left-most sides



R

A

'¤0

R p can subtract A gup d D A guC' d < R R p to obtain '¤0 gup d  '¤0 guC' d.

1 from the

Proposition 7.15. Let u be a superminimizer in  and let 0   be a nonempty open subset such that Cp .X n 0 / > 0 and u 2 N 1;p .0 /. Then u is a solution of the Ku;u .0 /-obstacle problem. The condition Cp .X n 0 / > 0 is needed since we have not defined the obstacle problem otherwise (as the solution would not be q.e. unique). For 0 b , it is trivially satisfied in all cases except when  D X and X is bounded. Similarly, the condition u 2 N 1;p .0 / holds for all 0 b . If, moreover, u 2 N 1;p ./,  is bounded and Cp .X n / > 0, then we can allow 0 D .

7.4 Convergence results for superminimizers

183

Proof. As u 2 N 1;p .0 /, we directly have that u 2 Ku;u .0 /. Furthermore, let v 2 Ku;u .0 / and w WD maxfu; vg. Then w D v q.e. in 0 and, by Proposition 7.9 (c), Z Z Z p p gu d  gw d D gvp d: 0

0

0

Hence u is a solution of the Ku;u .0 /-obstacle problem. In fact, superminimizers can be characterized by obstacle problems as in the following result. Note that in this case, Cp .X n 0 / > 0 for every 0 b . Proposition 7.16. Assume that either X is unbounded or  ¤ X . Then u is a superminimizer in  if and only if it is a solution of the Ku;u .0 /-obstacle problem for all 0 b . We generalize this result to arbitrary  in Proposition 9.25. Proof. The necessity follows directly from Proposition 7.15. As for the sufficiency, let ' 2 Lipc ./, '  0, and let 0 D fx 2  W '.x/ > 0g b . Since u is a solution of the Ku;u .0 /-obstacle problem and ' 2 N01;p .0 / is nonnegative, we have Z Z p p gu d  guC' d; 0

0

which shows that u is a superminimizer in . Another consequence of Proposition 7.15 is the following relation between superminimizers and obstacle problems, viz. that a solution of an obstacle problem is the smallest superminimizer with the prescribed boundary values, which lies above the obstacle. Corollary 7.17. Assume that  is bounded and Cp .X n / > 0. Let u be a solution of the K ;f -obstacle problem and v 2 K ;f be a superminimizer. Then u  v q.e. in . Proof. As v 2 K ;f , we have v 2 N 1;p ./ and Proposition 7.15 implies that v is a solution of the Kv;v -obstacle problem. The claim now follows immediately from the comparison principle (Lemma 7.6).

7.4 Convergence results for superminimizers I this section we consider sequences of superminimizers and study when they converge to a superminimizer. Our main interest lies in monotone sequences, but more general sequences are also considered. One part of the problem is to show that the limiting 1;p function belongs to Nloc ./, which is necessary for it to be a superminimizer. For this, we will need Lemma 7.19 below, whose proof uses the following algebraic lemma.

184

7 Superminimizers

Lemma 7.18. Let f be a bounded nonnegative function defined on ŒR1 ; R2 . Assume that for all R1  r1 < r2  R2 , f .r1 /  f .r2 / C

A C B; .r2  r1 /˛

(7.7)

where A; B  0, ˛ > 0 and 0   < 1. Then there exists C > 0 depending only on ˛ and  such that for all R1  r1 < r2  R2 , f .r1 /  C





A CB : .r2  r1 /˛

Proof. Let r1 and r2 be fixed. For 0 <  < 1, let tj D r1 C .1   j /.r2  r1 /, j D 0; 1; ::: . An iteration of (7.7) shows that f .r1 / D f .t0 /   j f .tj / C

jX 1  iD0



A  ˛ ˛ .1   / .r2  r1 /  ˛

i



C B i :

Choosing  D  1=.˛C1/ and letting j ! 1 finishes the proof. Lemma 7.19. Let fui g1 iD1 be a sequence of superminimizers in  such that ui ! u 1;p q.e. in , as i ! 1. Assume that there is a function f 2 Nloc ./ such that jui j  f 1;p a.e. in , i D 1; 2; ::: . Then u 2 Nloc ./. Moreover, for every 0 b , we have Z Z p gu d  lim inf gupi d: (7.8) i!1

0

0

In particular, Lemma 7.19 applies if the sequence fui g1 iD1 is decreasing (or increasing) and the limit function is locally essentially bounded from below (or above). p Proof. If we can show that fgui g1 iD1 is uniformly bounded in L .B/ for all balls 1;p B b , then we can deduce that u 2 Nloc ./ using Corollary 6.3, as clearly fui g1 iD1 is bounded in Lp .B/. Let B D B.x0 ; R/ b . We can find R0 > R so that also B 0 D B.x0 ; R0 / b . Let next 0 < r1 < r2  R0 , Bj D B.x0 ; rj /, j D 1; 2, and ² ³ r2  d.x0 ; x/

.x/ D min ;1 2 N01;p .B2 /: r 2  r1 C

Note that D 1 on B1 and g 

1 B nB : r 2  r1 2 1

Set 'i D .f  ui / 2 N01;p .B2 /, which is nonnegative q.e. in , by Corollary 1.60. Since 'i and .'i /C are representatives of the same equivalence class in N01;p .B2 /, we can assume that 'i is nonnegative everywhere in .

7.4 Convergence results for superminimizers

185

By Lemma 2.18, we have that a.e. in B 0 :

gui C'i  .1  /gui C gf C jf  ui jg

As ui is a superminimizer we have that Z Z gupi d  gupi d B1 B Z 2 gupi C'i d  B2 Z  Z Z p p p p p p p .1  / gui d C jf  ui j g d C

gf d 3 B2 B2 B2 Z  Z Z 2p p p gupi d C f d C g d :  3p f .r2  r1 /p B 0 B2 nB1 B0 We continue by “hole-filling”. Adding 3p times the left-hand side to both sides we obtain that Z  Z Z Z 2p p p gupi d  3p .1 C 3p / gupi d C f d C g d : f .r2  r1 /p B 0 B1 B2 B0 After dividing by 1 C 3p we get, with  D 3p =.1 C 3p / < 1, that Z Z Z Z 2p  p p p gui d   gui d C f d C  gfp d: .r2  r1 /p B 0 B1 B2 B0 Lemma 7.18 then implies that  Z gupi d  C B1

1 .r2  r1 /p

Z B0

Z f p d C

B0

 gfp d

for 0  r1 < r2  R0 . By choosing r1 D R and r2 D R0 we see that fgui g1 iD1 is 1;p bounded in Lp .B/. By Corollary 6.3, u 2 N 1;p .B/, and hence u 2 Nloc ./. Let next 0 b . Then, by compactness, its closure can be covered with a finite number of balls compactly contained in , and by the above we see that fgui g1 iD1 is bounded in Lp .0 /. Corollary 6.3 now implies (7.8). We next give a sufficient condition for when an increasing limit of superminimizers 1;p is a superminimizer. Since superminimizers belong to Nloc ./ and ui  u, the condition is clearly also necessary. Theorem 7.20. Let fui g1 iD1 be an increasing sequence of superminimizers in  and 1;p assume that there is a function f 2 Nloc ./ such that u WD limi!1 ui  f in . Then u is a superminimizer in .

186

7 Superminimizers

Note that if all ui are lsc-regularized, then so is u. (For the definition of lscregularization, see Section 8.5.) This is true also in the following immediate corollary. Theorem 7.21. Let fui g1 iD1 be an increasing sequence of superminimizers in  and 1;p assume that u D limi!1 ui is locally bounded from above in  or that u 2 Nloc ./. Then u is a superminimizer in . The following corollary is a direct consequence of Proposition 7.12 and Theorem 7.21 and gives a useful characterization of superminimizers. 1;p ./, then u is a superminimizer in  if and only if Corollary 7.22. If u 2 Nloc uk WD minfu; kg is a superminimizer in  for k D 1; 2; ::: . 1;p Proof of Theorem 7.20. As jui j  ju1 j C jf j 2 Nloc ./, Lemma 7.19 implies that 1;p u 2 Nloc ./, so we only need to show the superminimizing property. Let ' 2 Lipc ./ be nonnegative, v WD u C ' and " > 0. Let also 0 and 00 be open sets such that supp ' b 0 b 00 b  and Z Z p gv d D gup d < ": 00 n0

00 n0

Let 2 Lipc .00 / be such that 0   1, and let By Lemma 2.18, we have gui C

i

i

WD .v  ui / 2 N01;p .00 /.

 .1  /gui C gv C .v  ui /g

a.e. in 00 :

Since ui is a superminimizer we get Z 00

gupi

1=p Z  d

gupi C

00

i

1=p d

Z 

00

..1  /gui

1=p Z C gv / d C p

00

.v 

ui /p gp

1=p d

DW ˛i C ˇi : Using the elementary inequality .˛ C ˇ/p  ˛ p C pˇ.˛ C ˇ/p1 ;

˛; ˇ  0;

together with the convexity of t 7! t p we obtain that Z Z gupi d  ..1  /gui C gv /p d C pˇi .˛i C ˇi /p1 00 00  Z Z  .1  /gupi d C

gvp d C pˇi .˛i C ˇi /p1 : 00

00

7.4 Convergence results for superminimizers

Subtracting the first term on the right-hand side we obtain that Z Z Z gupi d 

gupi d 

gvp d C pˇi .˛i C ˇi /p1 0 00 00 Z  gvp d C " C pˇi .˛i C ˇi /p1 : 0

187

(7.9)

The sequence f˛i g1 iD1 is bounded by the proof of Lemma 7.19. At the same time, as g is bounded and 00 b , we have by dominated convergence that Z 1=p p p ˇi D .u  ui / g d ! 0; as i ! 1: 00 n0

Altogether this shows that the last term in (7.9) tends to 0, as i ! 1. Together with (7.8) we obtain that Z Z 0

gup d 

0

gvp d C ":

As " > 0 was arbitrary this completes the proof. In the following theorem we treat decreasing sequences of superminimizers. Note that even if all ui are lsc-regularized, then the limit u need not be (in contrast to the situation with increasing limits), see Example 9.35. Theorem 7.23. Let fui g1 iD1 be a decreasing sequence of superminimizers in  such 1;p that ui  f a.e. in  for some f 2 Nloc ./. Then u WD limi!1 ui is a superminimizer in . In the proof below we use Proposition 9.25 which has not yet been deduced. However, there is no circular reasoning here. Indeed, Theorem 7.23 is in this book only used when proving Corollary 7.24, which is subsequently only used to obtain Theorem 9.31 after Proposition 9.25. Thus we could just have postponed Theorem 7.23, and Corollary 7.24, until after obtaining Proposition 9.25. Moreover, apart from the exceptional case when X is bounded and  D X , we can replace the use of Proposition 9.25 by an appeal to Proposition 7.16, and thus it is only in that exceptional case that we are not able to deduce Theorem 7.23, and Corollary 7.24, here. 1;p 1;p Proof. Since ui  u1 2 Nloc ./, Lemma 7.19 shows that u 2 Nloc ./. Let G b  be open and such that Cp .X n G/ > 0, and let v be a solution of the Ku;u .G/-obstacle problem. As ui is a solution of the Kui ;ui .G/-obstacle problem, the comparison Lemma 7.6 implies that v  ui q.e. in G. Furthermore, as this holds for all i we have v  u q.e. in G. On the other hand, by the definition of the obstacle problem, v  u q.e. in G, and thus u D v q.e. in G. Hence, u is also a solution of the Ku;u .G/-obstacle problem. Therefore, u is a superminimizer in , by Proposition 7.16 or 9.25 (if  D X is bounded).

188

7 Superminimizers

By combining the increasing and decreasing convergence results, we can prove a convergence result for any pointwise converging sequence of superminimizers, not necessarily monotone, which is bounded from above and from below by functions in 1;p Nloc ./. In particular, this is true if the sequence is locally uniformly essentially bounded from below and the limiting function is locally essentially bounded from above. Corollary 7.24. Let fui g1 iD1 be a sequence of superminimizers in  and let v D lim inf ui : i!1

1;p ./ such that ui  f1 a.e. in , i D 1; 2; ::: , and v  f2 If there exist f1 ; f2 2 Nloc a.e. in , then v is a superminimizer in .

The corresponding result with w ´ lim supi!1 ui is not true. This is most easily seen by letting ui .x/ D .1/i x on  D .1; 1/  R, since w.x/ D jxj is not a superminimizer, see Remark 9.56. Proof. For every k D 1; 2 ::: , the functions vk;i D minfuk ; ::: ; ui g;

i  k;

are superminimizers by Proposition 7.12 and uk  vk;i  f1 a.e. in . Theorem 7.23 implies that vk D limi!1 vk;i is also a superminimizer in . The sequence fvk g1 kD1 is increasing and by Theorem 7.20, v D limk!1 vk is a superminimizer in . The following convergence result is a special case of Corollary 7.24, but let us provide a different and more straightforward proof, which also applies to superharmonic functions in Theorem 9.28 below. Theorem 7.25. Let fui g1 iD1 be a sequence of superminimizers which converges locally uniformly to u in . Then u is a superminimizer in . S Proof. Let 1 b 2 b  b  D j1D1 j . Then ui ! u, as i ! 1, uniformly in j . We can then find a subsequence fuik g1 such that kuik  ukL1 .j / < 2k : kD1 1 2k Let vk D uik  2 : Then fvk gkD1 is an increasing sequence of superminimizers converging to u in j . Moreover, u  v1 C2 2 N 1;p .j /. Thus u is a superminimizer in j by Theorem 7.20, and hence in , by Lemma 7.11. For the reader’s convenience let us point out that Proposition 10.18 and Corollary 10.20 contain further convergence results for solutions of obstacle problems and for p-harmonic extensions (as defined in Definition 8.31).

7.5 Notes

189

7.5 Notes Definition 7.1 and Theorem 7.2 are essentially from Kinnunen–Martio [217]. They defined the obstacle problem using a.e. instead of q.e. Our definition has some advantages, in particular in connection with Adams’ criterion (Theorem 7.3), see below, and capacitary potentials, see Remark 11.18. Note however that by Corollary 1.60, the 1;p two types of obstacle problems coincide if the obstacle belongs to Nloc ./ or more generally to ACCp ./. See also Farnana [119], [120], [121], [122] for the double obstacle problem, and Björn–Björn [49] for obstacle problems on nonopen sets and in noncomplete metric spaces. Adams’ criterion (Theorem 7.3) is from [49], where it is given for more general obstacle problems on nonopen sets. In the linear case on unweighted Rn , it was obtained by Adams [4]. For Adams’ criterion to hold it is important that the obstacle problem is defined using q.e.-inequalities, see the discussion in [49]. Shanmugalingam [320] had earlier solved the obstacle problem without an obstacle (or with obstacle 1), i.e. the Dirichlet problem for p-harmonic functions, assuming that f 2 N 1;p .X /. The idea for proving uniqueness in Theorem 7.2 goes back to Cheeger [91]. The comparison principle (Lemma 7.6) is from Björn–Björn [45]. It had earlier been obtained for the Dirichlet problem in [320], i.e. with 1 D 2 D 1 and f1 ; f2 2 N 1;p .X /. In the literature, various definitions have been used to define superminimizers, including conditions which are intermediate to the conditions in Proposition 7.9, often without mentioning whether the given definition is equivalent to other definitions in the literature. The first definition of superminimizers for metric spaces was given by Kinnunen–Martio [217]. Their definition is obviously between (c) and (a) in Proposition 7.9, and thus equivalent to our definition. Proposition 7.9 is from A. Björn [36]. Some characterizations of (quasi)superminimizers had earlier been obtained by Kinnunen–Martio [218]. Proposition 7.12 is from [217], while most of the other results in Section 7.3 are folklore. Lemma 7.13 was obtained (for quasisuperminimizers) in A. Björn–Martio [60]. Theorems 7.21 and 7.25, as well as Corollary 7.22, are from Kinnunen–Martio [217], whereas Theorem 7.20 is due to Ono [300]. Lemma 7.19, Theorem 7.23 and Corollary 7.24 are from Björn–Björn–Parviainen [54]. The well-known hole-filling technique used in the proof of Theorem 7.19, as well as in the proofs of the De Giorgi type Caccioppoli inequality (Lemma 8.1) and the boundary weak Harnack inequality (Lemma 11.4) below, is due to Widman [349]. The proof of Lemma 7.18 is from Giaquinta [142]. Various types of convergence results for single and double obstacle problems have been obtained by Farnana [120], [121]. Linear potential theory has a probabilistic counterpart, and in particular there is a very close relation between 2-harmonic functions and Brownian motion on unweighted Rn , see e.g. Doob [114]. For the nonlinear potential theory no such connection has been known until very recently, when such connections were found first for the 1-harmonic

190

7 Superminimizers

functions by Peres–Schramm–Sheffield–Wilson [301], and then for p-harmonic functions by Peres–Sheffield [302] and Manfredi–Parviainen–Rossi [259], [260]. These connections are to tug-of-war games (with noise in the latter case). They are however, so far, limited to unweighted Rn .

Chapter 8

Interior regularity

Recall that in Chapters 7–14 we assume that 1 < p < 1 and that X is a complete doubling p-Poincaré space, with doubling constant C and dilation constant  in the p-Poincaré inequality. In the previous chapter we solved the obstacle problem and defined superminimizers and p-harmonic functions. Our next goal is to study regularity of superminimizers and solutions of the obstacle problem. In particular, we shall see that there exist nontrivial p-harmonic functions, i.e. continuous representatives of minimizers. Similarly, superminimizers and solutions of obstacle problems will be shown to have lower semicontinuous representatives. We also obtain the strong maximum principle and various Harnack inequalities.

8.1 Weak Harnack inequalities for subminimizers In this section we use the De Giorgi method to estimate the essential supremum of subminimizers by their averaged Lq -norms. In particular, this shows that subminimizers are essentially locally bounded from above. The argument applies also to solutions of some obstacle problems, so both cases are treated simultaneously. We start by proving the following De Giorgi type Caccioppoli inequality on level sets. 1;p Lemma 8.1 (De Giorgi type Caccioppoli inequality). Let u 2 Nloc ./ and let B.x; r1 /  B.x; r2 / b  be such that Cp .X n B.x; r2 // > 0. Assume that either

(a) u is a subminimizer in  and k 2 R; or (b)  is bounded, Cp .X n / > 0, u is a solution of the K and k  q.e. in B.x; r2 /. Then

Z A1

gup d 

C .r2  r1 /p

;f

-obstacle problem,

Z .u  k/p d; A2

where Aj D fy 2 B.x; rj / W u.y/ > kg, j D 1; 2; and C only depends on p. Proof. Let

²

.y/ D min 1;

r2  d.x; y/ r 2  r1

³

: C

(8.1)

192

8 Interior regularity

Note that 0   1, D 1 on B.x; r1 /, D 0 outside B.x; r2 / and g  1=.r2  r1 /. Let v D u  .u  k/C . Then u  v 2 N01;p .B.x; r2 // and v  u. Hence, v 2 Ku;u .B.x; r2 // in case (a). In case (b), we have, q.e. in , either v D u  or v D .1  /u C k  , and hence v 2 K ;f ./. Also, as v D .1  /.u  k/ C k in A2 and v D u elsewhere, the function ..1 /gu C.uk/g /A2 Cgu nA2 is a p-weak upper gradient of v by Lemma 2.19. It then follows from the definitions of subminimizers and obstacle problems that Z Z Z gup d  gup d  gvp d A1 A2 A2 Z Z p p p p 2 .1  / gu d C 2 .u  k/p gp d A A2 Z 2 Z p 2 p p 2 gu d C .u  k/p d: .r2  r1 /p A2 A2 nA1 As in Lemma 7.19, we continue by “hole-filling”. Adding 2p -times the left-hand side to both sides of the inequality yields Z Z Z  gup d   gup d C .u  k/p d; p .r  r / 2 1 A1 A2 A2 where  D 2p =.2p C 1/ < 1. Lemma 7.18 now finishes the proof. The De Giorgi type Caccioppoli inequality (Lemma 8.1) shows that subminimizers and solutions of certain obstacle problems belong to the so-called De Giorgi class, which consists of functions satisfying (8.1) for all sufficiently large k. Next, we show that functions in the De Giorgi class are essentially locally bounded from above, and obtain an estimate of their essential supremum. It would be possible to pursue these arguments further to prove Hölder continuity, maximum principle and a Harnack inequality for minimizers, as in Kinnunen– Shanmugalingam [220], but we refrain from this approach here. Instead, in the next two sections, we shall use arguments from Moser’s method to derive weak Harnack inequalities for superminimizers and finally, the Harnack inequality, the strong maximum principle and Hölder continuity for minimizers. Proposition 8.2 (Weak Harnack inequality for De Giorgi classes). Assume that u 2 N 1;p .B.x; R// and that (8.1) holds for all k  k  and 0 < r1 < r2  R, where R < 14 diam X . Let 0 <   p=s if s > p in (4.3) and 0 <  < 1 otherwise. Then for all k0  k  and r < R, 1=p  1= « R ess sup u  k0 C C .u  k0 /pC d ; Rr B.x;R/ B.x;r/ where C depends only on p, , C , the constants in the p-Poincaré inequality, and the constant in (8.1).

193

8.1 Weak Harnack inequalities for subminimizers

Note that the left-hand side equals Cp - ess supB.x;r/ u, by Corollary 1.69. Similar observations can be made in connection with several of the results in this chapter. Proof. Let r  < r2  R, r1 D 12 . C r2 / and ²

r1  d.x; y/

.y/ D min 1; r1 

³

: C

Note that 0   1, D 1 on B.x; /, D 0 outside B.x; r1 / and g 

1 2 D : r1  r2 

Let further k > l  k  , A D fy 2 B.x; r1 / W u.y/ > kg and v D .u  k/C 2 N01;p .B.x; r1 //. Then Z B.x;r2 /

.u  l/pC d 

Z .u  l/p d  .k  l/p .A/ A

and the Hölder inequality yields for q > p, Z

Z

p=q

p

B.x; /

.u  k/C d 

q

.A/1p=q

v d B.x;r1 /

Z 

p=q  q

v d B.x;r1 /

1 .k  l/p

(8.2) 1p=q

Z

p

B.x;r2 /

.u  l/C d

:

Corollary 2.21 and the Leibniz rule (Theorem 2.15) show that gv  . gu C .u  k/g /A : The Sobolev inequality (Theorem 5.51) (together with Theorem 4.21) and the assumption (8.1) then imply that Z

p=q q

v d B.x;r1 /

C r1p  .B.x; r1 //1p=q C r1p  .B.x; r1 //1p=q

Z B.x;r1 /

gvp d

Z

A

gup d C

C r1p  .r2  /p .B.x; r1 //1p=q

Z

2p .r2  /p

B.x;r2 /

Z B.x;r1 /

.u  k/pC d;

 .u  k/pC d (8.3)

194

8 Interior regularity

where p < q  p  D sp=.s  p/ if s > p in (4.3) and p < q < 1 otherwise. The estimates (8.2) and (8.3) then yield Z Z C r1p .u  k/pC d  .u  l/pC d p .B.x; r //1p=q .r  / 2 1 B.x; / B.x;r2 /  1p=q Z 1 p

.u  l/ d : (8.4) C .k  l/p B.x;r2 / If moreover r2  2 , then by the doubling property of , the measures .B.x; //, .B.x; r1 // and .B.x; r2 // are comparable. Let « u.k; / D B.x; /

1=p .u  k/pC d

and choose q > p so that  D 1  p=q. Then (8.4) can be written as u.k; / 

C r2 u.l; r2 /1C : .r2  /.k  l/

(8.5)

For n D 0; 1; ::: , let n D r C 2n .R  r/  R and kn D k0 C d.1  2n /  k  , where d > 0 will be chosen later. Then 0 D R, n & r and kn % k0 C d , as n ! 1. Note that n  2 nC1 , so that (8.5) holds for D nC1 and r2 D n . We now show by induction that with a suitable d , u.k0 C d; r/  u.kn ; n /  2n u.k0 ; R/ ! 0;

as n ! 1;

(8.6)

where  D .1 C /=. Indeed, (8.6) is trivially true for n D 0 and assuming (8.6) for n  0, we have by (8.5) with r2 D n , D nC1 , k D knC1 and l D kn , CR u.kn ; n /1C . n  nC1 /.knC1  kn / 2nC1 CR  2n.1C / u.k0 ; R/1C D 2.nC1/ u.k0 ; R/; .R  r/.2n1 d /

u.knC1 ; nC1 / 

provided that d D .21C C CR=.R  r//1= u.k0 ; R/, where C is the same constant as in the last inequality. It follows that u.k0 C d; r/ D 0, i.e. that u  k0 C d a.e. in B.x; r/. We combine the last two results to obtain bounds for subminimizers and solutions of obstacle problems. In fact, we can do even better and estimate the local essential supremum of u by arbitrarily small powers of u. Proposition 8.3 (Weak Harnack inequality for subminimizers and solutions of obstacle 1;p problems). Let u 2 Nloc ./ and B.x; R/   with R < 14 diam X . Assume that either

8.1 Weak Harnack inequalities for subminimizers

195

(a) u is a subminimizer in  and k 2 R; or (b)  is bounded, Cp .X n / > 0, u is a solution of the K and k  q.e. in B.x; R/.

;f

-obstacle problem

Let further 0 <   p=s if s > p in (4.3) and 0 <  < 1 otherwise. Then for all 0 < q  p and 0 < r < R, 

R ess sup u  k C C Rr B.x;r/

p= q «

1=q q

B.x;R/

.u  k/C d

;

where C only depends on p, q, , C and the constants in the p-Poincaré inequality. Proof. Replacing R by R " and letting " ! 0, we can assume that B.x; R/ b . For r  r1 < r2  R, the weak Harnack inequality for De Giorgi classes (Proposition 8.2), together with Lemma 8.1 and the Young inequality 0

ab 

ap bp C 0; p p

where

1 1 C 0 D 1; p p

implies that ess sup.u  k/  C B.x;r1 /



r2 1= r 2  r1

«

1=p

B.x;r2 /

.u  k/qC d



1 R  ess sup.u  k/ C C 2 B.x;r2 / r 2  r1

ess sup.u  k/1q=p

p= q «

B.x;r2 /

1=q q

B.x;r2 /

.u  k/C d

:

Lemma 7.18 with ˛ D p=q then finishes the proof. We are now ready to prove the following weak Harnack inequality for nonnegative subminimizers. It is primarily, but not only, (8.7) below that we will use later on. Theorem 8.4 (Weak Harnack inequality for subminimizers and solutions of obstacle 1;p problems). Let u 2 Nloc ./ and B D B.x; R/  . Assume that either (a) u is a subminimizer in  and k 2 R; or (b)  is bounded, Cp .X n / > 0, u is a solution of the K and k  q.e. in B.

;f

-obstacle problem

Then for all q > 0 and R > 0, « ess sup u  k C C 1 2B

B

1=q .u  k/qC d

;

where C only depends on p, q, C and the constants in the p-Poincaré inequality.

196

8 Interior regularity

In particular, if u is a nonnegative subminimizer, then «

1=q

ess sup u  C

q

u d

1 2B

:

(8.7)

B

Proof. We can clearly assume that q  p, by Hölder’s inequality. Moreover, if R  3 diam X , then 21 B D B D X and we may replace R by 3 diam X . We therefore assume that R  3 diam X . If R < 14 diam X , then the theorem follows by letting r D R=2 in Proposition 8.3. For 14 diam X  R  3 diam X , let " > 0 and find z 2 12 B such that ess sup u  ess sup u  ": (8.8) 1 2B

B.z;R=30/

Proposition 8.3 and the doubling property of  then imply that «

1=q q

ess sup u  k C C B.z;R=30/

B.z;R=15/

.u  k/C d

«

1=q q

kCC B

.u  k/C d

:

Inserting this into (8.8) and letting " ! 0 finishes the proof. Remark 8.5. Inequalities of the type (8.7) are sometimes called weak Harnack inequalities in the literature, as we do here. However, some people prefer to refer to (8.7) as a local maximum principle and reserve the name “weak Harnack inequality” for inequalities of the type obtained in Theorem 8.10. Corollary 8.6. If u is a subminimizer in , then u is essentially locally bounded from above in . Similarly any superminimizer in  is essentially locally bounded from below in . That u is locally bounded in  is defined by saying that for every x 2  there is rx such that u is bounded in B.x; rx /. This is however equivalent to saying that u is bounded in 0 for every 0 b . By saying that u is essentially locally bounded we allow for an exceptional set of measure zero, which in fact, by Corollary 1.60, has to be of capacity zero. (Cf. Section 2.6.) Proof. Let u be a subminimizer in  and let x 2 . We can without loss of generality assume that u is nonnegative, as also uC is a subminimizer, by Proposition 7.12. Let 1;p further R > 0 be such that B D B.x; R/ b . Since u 2 Nloc ./ we have that u 2 N 1;p .B/. By the weak Harnack inequality for subminimizers (Theorem 8.4), « ess sup u  C 1 2B

1=p p

u d B

< 1:

8.2 Weak Harnack inequalities for superminimizers

197

8.2 Weak Harnack inequalities for superminimizers By approximating the function x 7! 1=x by piecewise linear functions, it is possible to use the weak Harnack inequality for subminimizers (Theorem 8.4) to derive a lower bound for nonnegative superminimizers. This is the first step in our proof of the weak Harnack inequality for superminimizers (Theorem 8.10). In fact, the convexity of x 7! 1=x is essential in our argument. In Section 9.8 we will give further convexity results for superminimizers and superharmonic functions. Proposition 8.7 (Very weak Harnack inequality for superminimizers). Assume that u is a nonnegative superminimizer in . Then for every ball B with 2B   and any q > 0 we have « 1=q q ess inf u  C u d ; (8.9) B

2B

where C only depends on p, q, C and the constants in the p-Poincaré inequality. Proof. Let '.x/ D 1=x, " > 0 and u" D u C ". By the convexity of x 7! 1=x, we can find a nonnegative function of the form .x/ D max .aj x C bj /; 1j N

where aj  0, j D 1; ::: ; N , so that '.x/ 

.x/  '.x/ C "

for x  ":

Since aj  0, the functions aj u" C bj are subminimizers, j D 1; ::: ; N , and it follows from Proposition 7.12 that ı u" is a nonnegative subminimizer. Thus using the weak Harnack inequality for subminimizers (Theorem 8.4) we find that « 1=q 1 D ess sup ' ı u"  ess sup ı u"  C . ı u" /q d ess inf u" 2B B B B « 1=q «  1=q q 1 q C .' ı u" C "/ d C : C " d 2B 2B u Since the constant C is independent of " we obtain the result, using monotone convergence, by letting " ! 0. Our next aim is to replace the negative exponent q in the very weak Harnack inequality for superminimizers (Proposition 8.7) by positive ones. This will be done by showing that the logarithm of a superminimizer belongs to BMO and by applying the John–Nirenberg lemma. For this we will need a logarithmic Caccioppoli type inequality which is a special case of the following Caccioppoli inequality for superminimizers.

198

8 Interior regularity

Proposition 8.8 (Caccioppoli inequality for superminimizers). Assume that u > 0 is a superminimizer in  and let ˛ > 0. Then for all nonnegative 2 Lipc ./, Z  p p Z u1˛ gup p d  up1˛ gp d: (8.10) ˛   Proof. Using the homogeneity of (8.10) we can replace by a and can therefore assume that 0   1 in . Let us also first assume that u  ˛ 1=.˛C1/ in . This assumption will be removed at the end of the proof. Let next w D u C p u˛ and g D .1  ˛ p u1˛ /gu C p p1 u˛ g . In order to show that g is a p-weak upper gradient of w, we let  W Œ0; l  ! X be a curve such that u is absolutely continuous on , gu and g are upper gradients along  , and R g ds < 1. This holds for p-a.e. curve. u  Let h D w ı  , which is absolutely continuous, by Lemma 1.58, and thus for a.e. t 2 Œ0; l , with x D .t /, jh0 .t/j D j.u ı  /0 .t / C p .x/p1 . ı  /0 .t /u.x/˛  ˛ .x/p u.x/1˛ .u ı  /0 .t /j D j.1  ˛ .x/p u.x/1˛ /.u ı  /0 .t / C p .x/p1 u.x/˛ . ı  /0 .t /j  .1  ˛ .x/p u.x/1˛ /j.u ı  /0 .t /j C p .x/p1 u.x/˛ j. ı  /0 .t /j: In the last inequality we used the temporary assumption that u  ˛ 1=.˛C1/ , so that 1  ˛ .x/p u.x/1˛  0. Lemma 2.14 shows that j.u ı  /0 .t /j  gu .x/ and j. ı  /0 .t /j  g .x/. Hence jh0 .t/j  .1  ˛ .x/p u.x/1˛ /gu .x/ C p .x/p1 u.x/˛ g .x/ D g.x/ and g is a p-weak upper gradient of w, again by Lemma 2.14. We then have by the convexity of the function t 7! t p that  p1 ˛ p p u g g p  .1  ˛ p u1˛ /gup C ˛ p u1˛ p ˛ u1˛ D .1  ˛ p u1˛ /gup C ˛ 1p p p up1˛ gp : Since u is a superminimizer and 0  w  u 2 N01;p .0 /, where 0 D fx W .x/ > 0g b ; we have Z Z Z p gup d  gw d  g p d 0 0 0   Z Z Z p  gu d  ˛

p u1˛ gup d C ˛ 1p p p 0

0

0

up1˛ gp d:

8.2 Weak Harnack inequalities for superminimizers

199

R

gup d < 1 from both sides we obtain Z  p p Z ˛

p u1˛ gup d  ˛ up1˛ gp d; 0 0 ˛  

After subtracting

0

and after division by ˛, the result follows under the assumption that u  ˛ 1=.˛C1/ in . In the general case, let " > 0 be arbitrary and let v D ˛ 1=.˛C1/ .u C "/=". Then v  ˛ 1=.˛C1/ is a superminimizer in  and hence (8.10) holds with u replaced by v. Using the homogeneity of (8.10), we obtain (8.10) for u C " and letting " ! 0 finishes the proof. The following lemma is the logarithmic Caccioppoli inequality for superminimizers which will play a crucial role in the proof of the weak Harnack inequality for superminimizers (Theorem 8.10) using Moser’s method. Proposition 8.9 (Logarithmic Caccioppoli inequality for superminimizers). Assume that u > 0 is a superminimizer in  which is locally bounded away from 0. Let 1;p v D log u. Then v 2 Nloc ./ and gv D gu =u a.e. in . Furthermore, for every ball B D B.x; r/ with 2B   we have « C gvp d  p ; r B where C D C .2p=.p  1//p . The assumption that u is locally bounded away from 0 can actually be omitted, since this follows (essentially) from the weak Harnack inequality for superminimizers (Theorem 8.10), for the proof of which we however need Proposition 8.9 in its present form. See Corollary 9.47 for an improvement upon Proposition 8.9. It follows from Theorem 9.42 below that v is in fact also a superminimizer. 1;p Proof. Proposition 2.17 shows that v 2 Nloc ./ and that gv D gu =u a.e. Let .x/ D .1  2 dist.x; B/=r/C . Note that 2 Lipc .2B/, 0   1, D 1 on B and g  2=r. The Caccioppoli inequality for superminimizers (Proposition 8.8) with ˛ D p  1 yields Z Z Z Z  p 2 gvp d  gvp p d D up gup p d  C 0 gp d  C 0 .2B/; r B   

where C 0 D .p=.p  1//p . From this and the doubling property we obtain the required inequality. We are now ready to provide the proof of the weak Harnack inequality for superminimizers. A sharp version of the following result is proved in Theorem 8.34.

200

8 Interior regularity

Theorem 8.10 (Weak Harnack inequality for superminimizers). If u is a nonnegative superminimizer in , then there are q > 0 and C > 0, only depending on p, C and the constants in the p-Poincaré inequality, such that 1=q « uq d  C ess inf u (8.11) B

2B

for every ball B D B.z; r/ such that 50B  . Recall that  is the dilation constant in the p-Poincaré inequality. Proof. Assume first that u > 0 is bounded away from 0. By the very weak Harnack inequality for superminimizers (Proposition 8.7) we have « 1=q « 1=q « 1=q « ess inf u  C uq d DC uq d uq d uq d : B

2B

2B

2B

2B

To complete the proof, we have to show that for some q > 0, « « q u d uq d  C: 2B

2B

Let v D log u. We shall show that v 2 BMO.10BI X /. (Recall that BMO was defined in Definition 3.14.) Let B.x; r 0 /  10B. We may assume that r 0  20r, as if r 0 > 20r we must have 10B D X and hence B.x; r 0 / D X D B.x; 20r/, see also Remark 3.16. It is easy to see that 2B.x; r 0 /  B.z; 40r C d.x; z//  50B  : The p-Poincaré inequality and the logarithmic Caccioppoli inequality for superminimizers (Proposition 8.9) imply that « 1=p « p jv  vB.x;r 0 / j d  C r gv d  C 0; B.x;r 0 /

B.x;r 0 /

0

where C only depends on p, C and the constants in the p-Poincaré inequality. Thus kvkBMO.10BIX/  C 0 . Let now q D log.2/=8C15 C 0 . By Corollary 3.21, « « « « e qv d e qv d D e q.v2B v/ d e q.vv2B / d 2B

2B

2B

« 

2B

e qjvv2B j d

2

 9;

2B

from which the claim follows for u bounded away from 0. If u is an arbitrary nonnegative superminimizer, then clearly uˇ WD u C ˇ  ˇ is a superminimizer for all constants ˇ > 0. Hence we may apply (8.11) to uˇ . Letting ˇ ! 0 and using Fatou’s lemma completes the proof.

8.3 Hölder continuity for p-harmonic functions

201

Proposition 8.11. Let u > 0 be a superminimizer in . Then log u 2 BMO -loc .I X /; where  D 5. Recall that the definition of BMO -loc was given in the notes to Chapter 3. Proof. This follows directly from the proof of the weak Harnack inequality for superminimizers (Theorem 8.10).

8.3 Hölder continuity for p-harmonic functions Combining the weak Harnack inequalities for sub- and superminimizers, we easily obtain the Harnack inequality for minimizers. Theorem 8.12 (Harnack’s inequality). Assume that u is a nonnegative minimizer in . Then there exists a constant C  1, only depending on p, C and the constants in the p-Poincaré inequality, such that ess sup u  C ess inf u B

B

for every ball B  50B  . Proof. Combine the weak Harnack inequalities for sub- and superminimizers (Theorems 8.4 and 8.10). From Harnack’s inequality it follows that minimizers are locally Hölder continuous (after modification on a set of capacity zero) and satisfy the strong maximum principle. The strong maximum principle for p-harmonic functions is a special case of the strong minimum principle that we will obtain for superharmonic functions, see Theorem 9.13. It is however possible to obtain it directly from Harnack’s inequality (Theorem 8.12) in the following way. Theorem 8.13 (The strong maximum principle). If  is connected, u is p-harmonic in  and u attains its maximum in , then u is constant in . Proof. We may assume that the maximum is 0. Let A D fx 2  W u.x/ D 0g, a relatively closed subset of , since u is continuous. Let further x 2 A. Then we can find a ball B 3 x such that 50B b . As u is a nonnegative p-harmonic function in , we have by Harnack’s inequality (Theorem 8.12) that  inf u D sup.u/  C u.x/ D 0; B

B

i.e. 0  inf B u  supB u D 0. Hence B  A, i.e. A is open. Since  is connected, A must be the only nonempty relatively closed open subset of , viz.  itself. Thus u  0.

202

8 Interior regularity

Theorem 8.14. Let u be a minimizer in . Then u can be modified on a set of capacity zero so that it becomes locally ˛-Hölder continuous in , where 0 < ˛ < 1. More precisely, if B D B.x0 ; r0 /  2B b  is a ball, then for all x; y 2 B,   d.x; y/˛ ju.x/ Q  u.y/j Q  C sup uQ  inf uQ ; 2B r0˛ 2B where uQ D u q.e. in , and C and ˛ only depend on p, C and the constants in the p-Poincaré inequality. Theorem 8.14 shows, in particular, that the class of p-harmonic functions is nontrivial and that every p-harmonic function is locally Hölder continuous in the domain of its p-harmonicity. In the proof below we could have used Theorem 5.62 to deduce directly that u D uQ q.e., see the notes to this chapter why it may be of interest not to do so. Proof. By Lebesgue’s differentiation theorem, the function « u.x/ Q D lim sup u d r!0

B.x;r/

equals u a.e. Assume that B.x; r/ b  and let m.r/ D inf uQ D ess inf u B.x;r/

and

B.x;r/

M.r/ D sup uQ D ess sup u: B.x;r/

B.x;r/

As u  m.r/  0 a.e. in B.x; r/, Corollary 1.60 shows that in fact u  m.r/  0 q.e. in B.x; r/. Hence .u  m.r//C is a nonnegative minimizer in B.x; r/. The Harnack inequality (Theorem 8.12) then implies that, with  D 50, M. 1 r/  m.r/ D

sup

.uQ  m.r//

B.x; 1 r/

C

inf

.uQ  m.r// D C.m. 1 r/  m.r//:

B.x; 1 r/

It follows that .C  1/m.r/  C.m. 1 r/  M. 1 r// C .C  1/M.r/ and hence M. 1 r/  m. 1 r/ 

C 1 .M.r/  m.r//; C

where C > 1 depends only on p, C and the constants in the p-Poincaré inequality. If y 2 B.x; r/ and  n1 r  d.x; y/ <  n r, an iteration of the last inequality yields 



C 1 n .M.r/  m.r// C Q C.supB.x;r/ uQ  inf B.x;r/ u/ d.x; y/˛ ; (8.12)  ˛ .C  1/r

ju.x/ Q  u.y/j Q  M. n r/  m. n r/ 

8.3 Hölder continuity for p-harmonic functions

203

where ˛ D log.C =.C  1//=log  > 0 is independent of x and r. (Note that  ˛ D C =.C  1/.) Let now B D B.x0 ; r/ be such that 2B b  and x; y 2 B arbitrary. If d.x; y/ < r, then y 2 B.x; r/  2B b  and (8.12) shows that ju.x/ Q  u.y/j Q 

C.sup2B uQ  inf 2B u/ Q d.x; y/˛ : ˛ .C  1/r

On the other hand, if r  d.x; y/ < 2r, then ju.x/ Q  u.y/j Q  sup uQ  inf uQ  B

B

sup2B uQ  inf 2B uQ d.x; y/˛ : r˛

1;p ./, Proposition 1.66 This shows that uQ is locally Hölder continuous in . As uQ 2 Nyloc 1;p shows that uQ 2 Nloc ./. Hence uQ D u q.e.

Since X is proper, the local Hölder continuity can equivalently be formulated in 1;p a more global way, just as for Lploc and Nloc , cf. Proposition 2.29. In particular, for p-harmonic functions we get the following more global formulation of the local Hölder continuity. Theorem 8.15. Let u be a p-harmonic function in  and G b G 0 b . Then for x; y 2 G,   ju.x/  u.y/j  C sup u  inf0 u d.x; y/˛ ; G0

G

where C and 0 < ˛ < 1 only depend on G, G 0 , p, C and the constants in the p-Poincaré inequality. Proof. Let

´ rD

dist.G; X n G 0 /; if G 0 ¤ X; 1; if G 0 D X: 1 3

Let x; y 2 G. If d.x; y/ < r, then y 2 B.x; r/ and Theorem 8.14 implies that ˛  d.x; y/˛    z sup u  inf u d.x; y/ ;  C ju.x/  u.y/j  Cz sup u  inf u 2B G0 r˛ r˛ 2B G0

where Cz and ˛ only depend on p, C and the constants in the p-Poincaré inequality. On the other hand, if d.x; y/  r, then   d.x; y/˛ ju.x/  u.y/j  sup u  inf u  sup u  inf0 u : G G r˛ G G0 This proves the theorem with C D maxf1; Cz gr ˛ . Another corollary of Harnack’s inequality (Theorem 8.12) is Liouville’s theorem.

204

8 Interior regularity

Corollary 8.16 (Liouville’s theorem). If u is p-harmonic and bounded from below on all of X , then u is constant. Proof. Let v D u  infX u  0. For x 2 X we thus get, by Harnack’s inequality (Theorem 8.12), that v.x/  sup v  C inf v ! 0; B.x;r/

B.x;r/

as r ! 1:

Thus v  0, and u is constant. Example 8.17. It may seem that there is something wrong with Corollary 8.16 if X is bounded, but there is no problem here. If X is bounded (and thus compact), then Lipc .X / consists of all Lipschitz functions on X and hence p-harmonic functions compete in Definition 7.7 with all possible functions. This forces them to have zero energy, i.e. to be constant. See also Proposition 8.23, Corollary 9.14 and Example 9.17. x  Rn for Example 8.18. A similar argument as in Example 8.17 shows that if X D G n x \V a sufficiently nice domain G  R and   X (as an open set in X , i.e.  D G for some open set V  Rn ), then p-harmonic functions in  necessarily have zero Neumann boundary data on V \ @G, see also the discussion in Section 1.7. This is particularly obvious in the one-dimensional example X D Œ0; 2  R and  D .1; 2. Here, p-harmonic functions in  are necessarily constant, i.e. affine functions are not p-harmonic in this case, see Example 12.19. The following Harnack inequality also follows from Theorem 8.12. Corollary 8.19 (Harnack’s inequality). If  is connected and E b , then there is a constant C such that for all nonnegative p-harmonic functions u on , sup u  C inf u: E

E

Proof. By Lemma 4.49 we can find a rectifiably connected open set G so that E  x find rz > 0 such that B.z; 50rz /  . By compactness, the G b . For each z 2 G, x has a finite subcover, consisting of N balls for some N . open cover fB.z; rz /gz2Gx of G Take now two points x; y 2 E. Since G is rectifiably connected, x and y can be connected by a curve  W Œ0; l  ! G. Using that  is connected, we can find a x so that x 2 B1 , y 2 Bn and sequence fBj gjnD1 , n  N , from the N balls covering G, Bj \ Bj C1 ¤ ¿, j D 1; ::: ; n  1. (Note that we do not need that fBj gjND1 cover  , nor that the balls Bj are connected.) Find points xj 2 Bj \ Bj C1 , j D 1; ::: ; n  1. Let A be the constant (called C ) in the Harnack inequality (Theorem 8.12), which only depends on p, C and the constants in the p-Poincaré inequality. Then, by the Harnack inequality (Theorem 8.12), u.x/  Au.x1 /  A2 u.x2 /    An1 u.xn1 /  An u.y/  AN u.y/: Letting C D AN concludes the proof.

8.4 The need for  in Harnack’s inequality

205

8.4 The need for  in Harnack’s inequality It may seem that a better proof could eliminate the need for  in the weak Harnack inequality for superminimizers (Theorem 8.10) and consequently also in the Harnack inequality (Theorem 8.12), in particular after noting that no  is needed in the weak Harnack inequality for subminimizers and solutions of obstacle problems (Theorem 8.4) and in the very weak Harnack inequality for superminimizers (Proposition 8.7). However,  is really essential both in the weak Harnack inequality for superminimizers (Theorem 8.10) and in the Harnack inequality (Theorem 8.12). The following example illustrates this. Example 8.20. Let XM D R2 n ..M; M / .0; 1//, M > 1, equipped with the Euclidean metric and the restriction of the Lebesgue measure, which is doubling. By Theorem A.21, XM supports a 1-Poincaré inequality. By considering the balls Br D B..0; 0/; r/, 1 < r < M (which are disconnected as balls in X ), ´ 1; if y  1; u.x; y/ D 0; if y  0; and letting r & 1, we see that the dilation constant 0 in any Poincaré inequality must satisfy 0  M . On the other hand, as XM is clearly .2M C 1/-quasiconvex, Corollary 4.40 implies that one can take 0 D 2M C 1. Let us fix M  2 and let X D XM . Let next  D .M; M /2 \ X (which is disconnected). Since gu  0 in , we see that u is p-harmonic in  (for all p). As M r 1 Br   for all 1 < r < 2, this shows that the constant 50 cannot be replaced by anything smaller than M , neither in the weak Harnack inequality for superminimizers (Theorem 8.10) nor in the Harnack inequality (Theorem 8.12). By varying M we see that the constant 50 in Theorems 8.10 and 8.12 cannot be replaced by any absolute constant. In this example,  was disconnected. We next make a modification of  to obtain a connected counterexample as well. Let   " D  [ B..M; 0/; M / n B..M; 0/; M  "/ ; where 0 < " < 1 and the balls are taken within X . Note that " is a connected subset of X . Let next f" .x; y/ D minfyC ; 1g on @" and let u" be the solution to the Dirichlet problem with boundary values f" on @" for p D 2, i.e. the 2-harmonic function such that u"  f" 2 N01;p ." /. In the notation of Definition 8.31, we have u D H" f" . The harmonic measure of @ n @" with respect to  tends to 0, as " ! 0. Hence u" ! u uniformly on , as " ! 0. Alternatively, one can use Theorem 11.23 to see that all u" are close to 1 in a neighbourhood of the point .M; M / and applying the maximum principle to the set C D f.x; y/ 2  W y > 0g, we get that u" ! 1, as " ! 0, uniformly in C . Similarly, u" ! 0, as " ! 0, uniformly in the set  D f.x; y/ 2  W y  0g. Considering the balls Br , with r & 1, again shows

206

8 Interior regularity

that the constant 50 cannot be replaced by anything smaller than M , in the weak Harnack inequality for superminimizers (Theorem 8.10) and the Harnack inequality (Theorem 8.12), even if  is required to be connected. As we can have  D 2M C 1, this also shows that the constant 50 in Theorems 8.10 and 8.12 has the right growth. The factor 50 is however not optimal. Remark 8.21. As in Remark 4.47 one could as well replace the metric d by the inner metric din , and thus work in a geodesic space. Indeed, as the p-weak upper gradients of a function remain the same, we would still have the same minimizers, sub- and superminimizers, solutions to obstacle problems, and also sub- and superharmonic functions, see Chapter 9. An advantage would be that we could avoid having  in some of our results, see Corollary 4.40. On the other hand, our new balls with respect to din may be less natural than the original balls with respect to d . Also, such a change may result in a larger doubling constant and a larger s in (4.3) and also in larger constants in the Poincaré inequalities. In some results this of course does not play any role, while in others it does.

8.5 Lsc-regularized superminimizers We saw in Theorem 8.14 that every minimizer has a canonical representative, viz. a p-harmonic function, which is locally Hölder continuous. In this section we study a similar question for superminimizers. Since we cannot hope for continuity, the right notion in this context is lower semicontinuity. This will become even more important in Chapter 9 when dealing with superharmonic functions. Let us first define the lsc-regularization u .x/ WD ess lim inf u.y/ WD lim ess inf u: y!x

r!0 B.x;r/

1;p We will mainly apply this to functions u 2 Nloc , in which case the exceptional sets in the ess inf are not only of zero measure, but of zero capacity, by Corollary 1.60.

Theorem 8.22. Let u be a superminimizer in . Then u is lower semicontinuous in 1;p  and u u in Nloc ./. Moreover, u never takes the value 1, and u .x/ D lim inf u .y/ D ess lim inf u .y/; y!x

Since

y!x

u .x/ D ess lim inf u .y/; y!x

x 2 :

x 2 ;

we say that u is an lsc-regularized superminimizer. We remark that the proof below shows that u D u at all Lebesgue points of u. Note also that u is the unique lsc-regularized representative of u, i.e. such that  u D u q.e. However, there are (in general) many different lower semicontinuous representatives of u.

8.5 Lsc-regularized superminimizers

207

Proof. It follows from Corollary 8.6 that u .x/ > 1. Let us next show that u is lower semicontinuous. This holds for all u, not only for superminimizers. Let a 2 R and u .x/ > a. Then there is r > 0 such that B.x; r/   and ess inf B.x;r/ u > a. It follows that u > a in B.x; r/. Thus, the set fx W u .x/ > ag is open and hence u is lower semicontinuous. By Theorem 5.62, the set ³ ² « ju.x0 /  uj d D 0 E D x0 2  W ju.x0 /j < 1 and lim R!0 B.x0 ;R/

differs from  only by a set of capacity zero. Choose x0 2 E and observe that .u.x0 /u/C is a nonnegative subminimizer by Proposition 7.12. By the weak Harnack inequality for subminimizers (Theorem 8.4) we have that for B.x0 ; R/ b , « ess sup .u.x0 /  u/C  C .u.x0 /  u/C d: B.x0 ;R/

B.x0 ;R=2/

Let " > 0. Since x0 2 E, there exists R0 > 0 such that « « .u.x0 /  u/C d  ju.x0 /  uj d < " B.x0 ;R/

for all 0 < R < R0 :

B.x0 ;R/

We deduce that for 0 < R < R0 , C " > ess sup .u.x0 /  u/C  ess sup .u.x0 /  u/ D u.x0 /  ess inf u: B.x0 ;R=2/

B.x0 ;R=2/

B.x0 ;R=2/

Since this holds for every 0 < R < R0 and " > 0 was arbitrary, it follows that u.x0 /  u .x0 /. On the other hand, x0 is a Lebesgue point and thus, «  u d D u.x0 /: u .x0 / D ess lim inf u.y/  lim y!x0

R!0 B.x0 ;R/

Consequently, u D u q.e. in . Therefore, we see that u .x/  lim inf u .y/  ess lim inf u .y/ D ess lim inf u.y/ D u .x/; y!x

y!x

y!x

x 2 :

Hence, we must have equality throughout. The following observation may be worth knowing, cf. Liouville’s theorem (Corollary 8.16) and Example 8.17. Proposition 8.23. Assume that X is bounded and that u is a superminimizer on X . Then u D C q.e. in X for some C 2 R. If moreover u is lsc-regularized, then u  C in X .

208

8 Interior regularity

Proof. By Theorem 8.22, u is an lsc-regularized superminimizer which does not take the value 1. As X is compact in this case, u attains its minimum, which we may assume to be 0, at some point x0 . It follows from the weak Harnack inequality for superminimizers (Theorem 8.10), applied to X D B D B.x0 ; 2 diam X /, that for some q > 0, « 1=q  q 0 .u / d  C ess inf u D C inf u D 0: B

2B

X

Hence u D 0 a.e. in X . As u is lsc-regularized, we get u  0 in X and u D 0 q.e. in X . If u is lsc-regularized, then of course u D u , which completes the proof. The proof above can also be based on the logarithmic Caccioppoli inequality for superminimizers (Proposition 8.9) instead of the weak Harnack inequality for superminimizers (Theorem 8.10) In Section 5.6 we showed that N 1;p -functions have Lebesgue points q.e. For locally bounded lsc-regularized superminimizers we can improve upon this by showing that they have Lebesgue points everywhere, in fact in Lq -sense for every q > 0. Proposition 8.24. Assume that u is a locally bounded lsc-regularized superminimizer in . Then « lim ju  u.x0 /j d D 0 for all x0 2 : r!0 B.x ;r/ 0

Proof. Let x0 2 . We may assume that 0 < u < 1 in a neighbourhood of x0 . Let 0 < " < u.x0 /. As u is lower semicontinuous there is r 0 such that u.x0 /  " < u < 1 in B.x0 ; r 0 /. Let q > 0 be given by the weak Harnack inequality for superminimizers (Theorem 8.10),  D minfq; 1g and r < r 0 =50. Let v D u  .u.x0 /  "/. Then 0 < v < 1 in B.x0 ; r 0 / and by the weak Harnack inequality for superminimizers (Theorem 8.10), « « « ju  u.x0 /j d D jv  v.x0 /j d  v.x0 / C v 1 v  d B.x0 ;r/ B.x0 ;r/ B.x0 ;r/ « "C v  d  " C C v.x0 / D " C C " : B.x0 ;r/

Letting r ! 0 and then " ! 0 completes the proof. Proposition 8.24 can be improved using the following general result. Note that it holds for an arbitrary locally essentially bounded function. Proposition 8.25. Assume that u is essentially locally bounded in  and let x0 2 . If « lim

r!0 B.x ;r/ 0

ju  u.x0 /jq d D 0

for some q D q0 > 0, then (8.13) holds for all q > 0.

(8.13)

8.6 Lsc-regularized solutions of obstacle problems

209

For superminimizers we thus have the following immediate corollary of Propositions 8.24 and 8.25. Corollary 8.26. Assume that u is a locally bounded lsc-regularized superminimizer in  and that q > 0. Then « lim ju  u.x0 /jq d D 0 for all x0 2 : r!0 B.x ;r/ 0

Proof of Proposition 8.25. Assume that (8.13) holds for q D q0 . If q < q0 , then it follows from Hölder’s inequality that (8.13) holds, even for unbounded u. Assume therefore that q > q0 . We may also assume that 0  u  1 in B.x0 ; r 0 / for some r 0 > 0. Then, for 0 < r < r 0 we get that « « q ju  u.x0 /j d D ju  u.x0 /jqq0 ju  u.x0 /jq0 d B.x0 ;r/ B.x0 ;r/ «  ju  u.x0 /jq0 d ! 0; B.x0 ;r/

as r ! 0.

8.6 Lsc-regularized solutions of obstacle problems Since solutions of obstacle problems are superminimizers we can now obtain the following regularity result, which is a direct consequence of Theorem 8.22. Theorem 8.27. Assume that  is bounded and such that Cp .X n / > 0. Let f 2 x If K ;f ¤ ¿, then there is a unique lsc-regularized N 1;p ./ and W  ! R. solution of the K ;f -obstacle problem. Proof. By Theorem 7.2, the K ;f -obstacle problem has a solution u, and all solutions are equal to u q.e. By Proposition 7.14, u is a superminimizer and hence, by Theorem 8.22, u D u q.e. Thus, u is also a solution of the K ;f -obstacle problem, and hence the unique lsc-regularized solution. If the obstacle is continuous, then more can be said about the regularity of the solution. Theorem 8.28. Assume that  is bounded and such that Cp .X n / > 0. Assume x further that W  ! Œ1; 1/ is continuous (as an R-valued function), that f 2 N 1;p ./ and that K ;f ¤ ¿. Then the lsc-regularized solution u of the K ;f obstacle problem is continuous (as a real-valued function). Moreover, u is p-harmonic (and thus locally Hölder continuous) in the open set A D fx 2  W u.x/ > .x/g (with boundary values u).

210

8 Interior regularity

In the notation introduced in Definition 8.31 below, u D HA u in A. For the first part we have the following corresponding pointwise result. Theorem 8.29. Assume that  is bounded and such that Cp .X n / > 0. Let f 2 x be such that K ;f ¤ ¿, and let u be the lsc-regularized N 1;p ./ and W  ! R solution of the K ;f -obstacle problem. Assume further, that is continuous (as an x R-valued function) at x 2 . x Then u is continuous (as an R-valued function) at x 2 . If .x/ D 1, then u.x/ D 1. On the other hand, if .x/ < 1, then u.x/ 2 R (or in other terms u is continuous as a real-valued function at x). Observe, that the case when Proof. Assume first that

.x/ D 1 is allowed, in which case u.x/ 2 R.

.x/ D 1. As u is lsc-regularized, we see that

lim inf u.y/  u.x/ D ess lim inf u.y/  ess lim inf y!x

y!x

.y/ D

y!x

.x/ D 1:

Hence we must have u.x/ D 1 and u must be continuous at x. Assume therefore in the rest of the proof that .x/ < 1. Let " > 0. By Theorem 8.22, we already know that u is a lower semicontinuous function which does not take the value 1. So it is enough to show that lim sup u.y/  u.x/ < 1:

(8.14)

y!x

´

Let kD

.x/ C "; if min¹u.x/; 0º; if

.x/ 2 R; .x/ D 1:

Using the continuity of and the lower semicontinuity of u, we can find R > 0 such that B D B.x; R/ b 2B b ; sup  k 2B

and either u.x/ D 1 or inf u > u.x/  ":

(8.15)

B

Next we show that u.x/ < 1, and hence (8.15) holds. Indeed, the weak Harnack inequality for solutions of obstacle problems (Theorem 8.4) and the regularity of u show that «  1=p

sup u  k C C B

2B

p .u  k/C d

In particular, (8.15) holds. If .x/ 2 R, then we have by the continuity of u.x/ D ess lim inf u.y/  ess lim inf y!x

y!x

< 1:

(8.16)

that

.y/ D

.x/  sup 2B

 ":

8.6 Lsc-regularized solutions of obstacle problems

On the other hand, if

211

.x/ D 1, then u.x/  k  sup : 2B

Thus, the weak Harnack inequality for solutions of obstacle problems (Theorem 8.4), applied with k D u.x/ C " and q D 1, yields for B 0 D B.x; r/, 0 < r < R, « « sup u  .u.x/ C "/  C .u  .u.x/ C "//C d  C .u  .u.x/  "//C d 2B 0 2B 0 B0 «  « DC .u  .u.x/  "// d D C u d  u.x/ C " : 2B 0

2B 0

By Proposition 8.24 together with (8.16) we see that « lim u d D u.x/ r!0 B.x;r/

and thus lim sup u.y/  u.x/  "  C ": y!x

Letting " ! 0, we obtain (8.14). Proof of Theorem 8.28. The first part follows directly from Theorem 8.29. As for the second part, A is open since and u are continuous. Let ' 2 Lipc .A/. Since supp ' b A is compact, there exists " > 0 such that u  C " in supp '. As ' is bounded, there is 0 < t < 1 such that w WD .1  t /u C t .u C '/ D u C t '  and thus w 2 K Z

in ;

. Hence, using the convexity of the function x 7! x p , Z Z p p gu d  gw d  ..1  t /gu C tguC' /p d '¤0 '¤0 '¤0 Z Z p  .1  t / gup d C t guC' d: ;f

'¤0

'¤0

Subtracting the first term in the right-hand side and dividing by t shows that Z Z p gup d  guC' d: '¤0

'¤0

Thus, u is a continuous minimizer, i.e. a p-harmonic function, in A. The following comparison principle improves upon Lemma 7.6.

212

8 Interior regularity

Lemma 8.30 (Comparison principle). Assume that  is bounded and such that x and fj 2 N 1;p ./ be such that K ;f ¤ ¿, Cp .X n / > 0. Let j W  ! R j j and let uj be the lsc-regularized solution of the K j ;fj -obstacle problem, j D 1; 2. If 1  2 q.e. in  and .f1  f2 /C 2 N01;p ./, then u1  u2 in . Proof. By Lemma 7.6, u1  u2 q.e. in . Since both u1 and u2 are lsc-regularized it follows that u1 .x/ D ess lim inf u1 .y/  ess lim inf u2 .y/ D u2 .x/ y!x

y!x

for all x 2 :

8.7 p-harmonic extensions The following definition is the first step in solving the Dirichlet problem for p-harmonic functions, a topic which will be thoroughly studied in Chapter 10. Definition 8.31. Let V be a bounded open set with Cp .X n V / > 0, and let f 2 N 1;p .V /. The p-harmonic extension HV f of f to V is the continuous solution of the K1;f .V /-obstacle problem. When V D  we usually suppress the index and merely write Hf D H f . The unique lsc-regularized solution of the K1;f -obstacle problem, given by Theorem 8.27, is continuous by Theorem 8.28, and hence the definition makes sense. Theorem 8.28 moreover shows that Hf is p-harmonic in . It also follows that Hf is the unique p-harmonic function in  such that f  Hf 2 N01;p ./. Thus we have solved the Dirichlet (boundary value) problem for p-harmonic functions with boundary values in N 1;p ./, and in particular for Lipschitz boundary values. Since f  Hf 2 N01;p ./, these solutions are said to take the boundary values in Sobolev sense. Note that we solve the Dirichlet problem for functions merely in N 1;p ./, which are not even defined on the boundary. If f is defined outside of  it is sometimes useful to define Hf D f wherever f is defined outside of . We leave the further discussion of the Dirichlet problem to Chapter 10, but we need a few more observations before that to be able to study superharmonic functions, whose definition depends on p-harmonic extensions, see Chapter 9. The following result is a special case of Lemma 8.30. Note that by letting f2 D max@0 u, for a suitable 0 b , we obtain a weak maximum principle for p-harmonic functions. Lemma 8.32 (Comparison principle). Assume that  is bounded and such that Cp .X n / > 0. If f1 ; f2 2 N 1;p ./ and .f1  f2 /C 2 N01;p ./, then Hf1  Hf2 in . x and f1  f2 q.e. on @, then Hf1  Hf2 In particular, if f1 ; f2 2 N 1;p ./ in .

8.8 A sharp weak Harnack inequality for superminimizers

213

x Hf only depends on f j@ . A Lipschitz function It follows that for f 2 N 1;p ./, x such that f D fQ on @ (e.g. f on @ can be extended to a function fQ 2 Lip./ using one of the McShane extensions in Lemma 5.2). As HfQ does not depend on the choice of extension, we define Hf WD HfQ. Theorem 8.33 (Poisson modification for superminimizers). Assume that u is a superminimizer in  and let G b  be open and such that Cp .X n G/ > 0. Then ´ u in  n G; vD HG u in G is a superminimizer in . Moreover, v  u q.e. in . 1;p Proof. As u is a superminimizer, it belongs to Nloc ./, by definition. Let ´ u in  n G; v 0 D minfu; vg D min¹u; HG uº in G:

As u is a solution of the Ku;u .G/-obstacle problem, by Proposition 7.15, the comparison Lemma 8.30 shows that HG u  u D u q.e. in G. Hence v D v 0  u q.e. in . 1;p Moreover, HG u  u 2 N01;p .G/, which shows that v 0 D u  .u  HG u/C 2 Nloc ./ 0 0 and thus by Lemma 7.13, v is a superminimizer in . As v D v q.e. in , also v is a superminimizer in .

8.8 A sharp weak Harnack inequality for superminimizers The weak Harnack inequality for superminimizers (Theorem 8.10) shows that the infimum of a nonnegative superminimizer u is locally bounded away from zero by an Lq -norm of u, for some q > 0. In this section we improve this estimate and obtain a sharp bound for these exponents, uniformly for all nonnegative superminimizers. Theorem 8.34 (Sharp weak Harnack inequality for superminimizers). Assume that X supports a .~p; p/-Poincaré inequality for some ~ > 1 and that 0 <  < ~.p  1/. If u is a nonnegative superminimizer in , then there is C > 0, only depending on  , p, ~, C and the constants in the .~p; p/- and p-Poincaré inequalities, such that « 1= u d  C ess inf u (8.17) B

B

for every ball B such that 50B  . A similar result for superharmonic functions is proved in Theorem 9.7. See Remark 9.56 for a discussion about the sharpness of Theorem 8.34. Here  is the dilation constant in the p-Poincaré inequality, we do not need to use the possibly larger dilation constant in the .~p; p/-Poincaré inequality.

214

8 Interior regularity

Proof. Let q be as given by the weak Harnack inequality for superminimizers (Theorem 8.10). If   q, then the result follows by Hölder’s inequality, assume therefore that  > q. Let B D B.x; r/. 1 If r  10 diam X , then  D X is bounded, and by Proposition 8.23, u is constant 1 q.e. in X , from which (8.17) follows directly. We therefore assume that r < 10 diam X in the rest of the proof. Assume first that inf 50B u > 0. Let k be a positive integer such that ~ k   q < ~ 1k :

(8.18)

Let Bj D .1 C j=k/B for j D 0; 1; ::: ; k. Fix j D 0; 1; ::: ; k  1, and let

j .y/ D .1  2k dist.y; Bj /=r/C so that j 2 Lipc .Bj C1 /, Bj  j  1 and gj  2k=r. Let further ˛j D p  1  ~ 1j  > 0;

ˇj D 1 

1 C ˛j ~ 1j  D >0 p p

and vj D j uˇj . The Leibniz rule (Theorem 2.15) and the chain rule (Theorem 2.16) yield gvj  uˇj gj C ˇj u.1C˛j /=p gu j

a.e.,

and thus gvpj  2p upˇj gpj C 2p ˇjp u1˛j gup jp

a.e.

By the doubling property and the Sobolev inequality (Theorem 5.51) we obtain that « Bj

«

1=~p vj~p

1=~p

C

d

Bj C1

vj~p

d

«

1=p

 Cr

gvpj

Bj C1

d

«

1=p

 Cr Bj C1

upˇj gpj d

«

C Cr

u

1˛j

Bj C1

1=p gup jp

d

:

The last integral is estimated using the Caccioppoli inequality for superminimizers (Proposition 8.8), as follows « u Bj C1

1˛j

1=p gup jp

d

p  ˛j

«

1=p u

Bj C1

p1˛j

gpj

d

:

8.9 Notes

215

Note that p  1  ˛j D pˇj . This yields «

«

1=~p

Bj

vj~p

1=p

 Cr

d

u

pˇj

Bj C1

« C

gpj

d 1=p

up~ˇj C1 d Bj C1

«

1=p vj~p C1

DC Bj C1

d

:

We iterate this estimate k times, and then use (8.18) together with Hölder’s inequality and the weak Harnack inequality for superminimizers (Theorem 8.10) to obtain «

«

1= u d

1=

D

B

B0

v0~p d

« C

B1

~= v1~p

d

  «

~ k =

C Bk

vk~p

« DC

u «

d

~ k 

Bk

C

~ k = d

1=q q

u d 2B

 C ess inf u: B

In the case when inf 50B u D 0, we apply the result to u C ı for ı > 0, and let ı ! 0 to obtain the result for u.

8.9 Notes There are primarily two ways of proving Harnack inequalities and Hölder continuity for minimizers, and along the way obtaining the weak Harnack inequalities for sub- and superminimizers: De Giorgi’s method, see De Giorgi [110], and Moser’s iteration, see Moser [293], [294] and Serrin [316]. Simultaneously with De Giorgi [110], Nash [298] obtained interior regularity results for solutions of elliptic equations as a special case of parabolic equations. In metric spaces, interior regularity for minimizers was first

216

8 Interior regularity

obtained by Kinnunen–Shanmugalingam [220] using De Giorgi’s method. A. Björn– Marola [59] later showed that also Moser’s method is available in metric spaces. Here we have chosen to use a combination of these methods. First we deduce a weak Harnack inequality for subminimizers and solutions of obstacle problems (Theorem 8.4) using De Giorgi’s method as in Kinnunen–Shanmugalingam [220]. Then we use a convexity argument to obtain the very weak Harnack inequality for superminimizers (Proposition 8.7), after which we follow Moser’s method more or less as in A. Björn–Marola [59]. A subtle point perhaps worth mentioning is that when running De Giorgi’s method it is necessary to use a q-Poincaré inequality for some q < p. On the contrary, Moser’s method, as well as our combined approach here, only requires the p-Poincaré inequality. This is why we in the proof of Theorem 8.14 used the Lebesgue differentiation theorem and Proposition 1.66 rather than Theorem 5.62 about Lebesgue points of Newtonian functions, which uses the q-Poincaré inequality. However, we do use the q-Poincaré inequality, through the use of Theorem 5.62, when obtaining the lsc-regularization of superminimizers in Theorem 8.22 (which is then used quite extensively). In view of Theorem 4.30 the difference is little, as we assume completeness, which is however not assumed in the papers Kinnunen–Shanmugalingam [220] and A. Björn–Marola [59]. By an iteration argument the blow-up constant 50 in Theorems 8.10 and 8.12 was replaced by 2L in Björn–Björn–Marola [53], if X is L-quasiconvex, under the modification that the integral in the left-hand side of (8.11) is taken over B. There is however a price to pay when  is replaced by L, see Section 4.8 for a discussion on the relationship between L and . The same iteration argument can directly be applied to show that 50 may be replaced by 2L also in Theorem 8.34. Let us mention that in unweighted Rn , p-harmonic functions are not only Hölder 1;˛ continuous, but Cloc . This was shown by Uraltseva [344] for p > 2 and independently by Lewis [243], DiBenedetto [112] and Tolksdorf [335] for general p. Moreover, a p-harmonic function u is real-analytic off the set S D fx W ru.x/ D 0g, see k;˛ regularity for p-harmonic functions on R2 was Lewis [242], p. 208. Optimal Cloc obtained by Aronsson [20], for p > 2, and Iwaniec–Manfredi [189] for general p. 2;˛ In particular, it is shown in [189] that p-harmonic functions on R2 are Cloc when 1 < p < 2. Bojarski–Iwaniec [73] showed that S is discrete for p-harmonic functions on R2 unless ru  0 in some component, see also Aronsson–Lindqvist [21]. That 2-harmonic functions are real-analytic everywhere is a much older result. On the other hand, an example by Koskela–Rajala–Shanmugalingam [230], p. 149, shows that 2-harmonic functions on weighted Rn need not be Lipschitz continuous. Under some additional assumptions, Cheeger p-harmonic functions are known to be Lipschitz continuous, see [230]. In Heisenberg groups Hn (see Appendix A.6), the best local regularity results are due to Zhong [358] when p ¤ 2. See [358] for the earlier history including the case p D 2.

8.9 Notes

217

The Caccioppoli inequalities in Propositions 8.8 and 8.9 are from Kinnunen– Martio [219]. The sharp weak Harnack inequality for superminimizers (Theorem 8.34) was also proved in [219]. The proofs of the De Giorgi type Caccioppoli inequality (Lemma 8.1) and the weak Harnack inequality for De Giorgi classes (Proposition 8.2) follow J. Björn [63], rather than the original proof from Kinnunen–Shanmugalingam [220]. The use of convexity in the proof of Proposition 8.7 may be new here for minimizers not associated with equations. The main idea is related to Theorem 9.41 and a result by Tolksdorf [336]. Example 8.20 is essentially from A. Björn–Marola [59], while Proposition 8.23 is from A. Björn [35], Remark 4.2 (5). Theorem 8.22 is from Kinnunen–Martio [217], but the proof used here is from Björn–Björn–Parviainen [54]. Proposition 8.24 is also from [54], an earlier slightly weaker result can be found in [217]. Also Proposition 8.25 and Corollary 8.26 are from [54]. Theorem 8.28 was proved in [217] and a somewhat stronger version, not given here, is in Björn–Björn–Mäkäläinen–Parviainen [52]. Theorem 8.29 is a special case of a result by Farnana [122]. Therein she also shows that solutions of obstacle problems with Hölder continuous obstacles are locally Hölder continuous. Maasalo [Kansanen] [252] showed that if u is a positive superminimizer in   X and X is geodesic, then log u 2 BMO.I X /. In fact this follows from her result that BMO.I X / D BMO -loc .I X / in geodesic spaces, see the notes to Chapter 3, together with Proposition 8.11. Björn–Björn–Marola [53] showed that one may take  D 4L in Proposition 8.11, if X is L-quasiconvex (when they obtained the corresponding result for quasisuperminimizers, see Appendix C). Latvala–Marola–Pere [239] studied the first eigenvalue and eigenfunction for the p-Laplacian on metric spaces even though there is no equation in this case (they used a characterization using only integrals). In particular, they obtained a Harnack inequality.

Chapter 9

Superharmonic functions

Recall that in Chapters 7–14 we assume that 1 < p < 1 and that X is a complete doubling p-Poincaré space, with doubling constant C and dilation constant  in the p-Poincaré inequality. Superharmonic functions are roughly speaking lsc-regularized superminimizers and their increasing limits, however their definition is quite different. This means that superminimizers (and superharmonic functions) can be defined in two rather different but equivalent ways. Often in mathematics such nontrivial equivalences are very fruitful, and this is so also here. In the first part of this chapter, our aim is to establish this equivalence, which is fully true for bounded functions and for Newtonian functions. This is completed when we obtain Theorem 9.24, which also contains a (necessarily more complicated) characterization of unbounded superharmonic functions in terms of superminimizers. Note that there are superharmonic functions which are too large to be superminimizers and even too large to belong to L1loc , see Remark 9.56. In the rest of this chapter we will study various other aspects of superharmonic functions, such as convergence results, local integrability and Lebesgue points. In Chapter 10, superharmonic functions will play a vital role in our study of the Dirichlet problem for p-harmonic functions.

9.1 Definition of superharmonic functions Definition 9.1. A function u W  ! .1; 1 is superharmonic in  if (i) u is lower semicontinuous; (ii) u is not identically 1 in any component of ; (iii) for every nonempty open set 0 b  with Cp .X n 0 / > 0, and all functions v 2 Lip.@0 /, we have H0 v  u in 0 whenever v  u on @0 . A function u W  ! Œ1; 1/ is subharmonic if u is superharmonic. Here, H0 v is the p-harmonic extension of v from Definition 8.31. One also defines u to be hyperharmonic if it satisfies (i) and (iii), and hypoharmonic if u is hyperharmonic. This is used frequently in some parts of the literature. On the other hand, both superharmonic and hyperharmonic functions are well behaved from below, by the next proposition.

9.1 Definition of superharmonic functions

219

Proposition 9.2. A superharmonic function u is locally bounded from below. Proof. This follows directly from the lower semicontinuity of u as u never takes the value 1 (and X is locally compact). The following easy lemma demonstrates a certain flexibility of superharmonic functions and, together with Proposition 9.2, forms a first step on the way to a characterization of superharmonic functions by superminimizers, see Theorem 9.24. Note also that if u and v are superharmonic, ˛; ˇ 2 R and ˛  0, then ˛u C ˇ is superharmonic, but in general u C v is not superharmonic. For other ways of making new superharmonic functions, see Sections 9.8 and 10.9 and the pasting lemma (Lemma 10.27). Lemma 9.3. (a) If u and v are superharmonic in , then so is minfu; vg. (b) If u is not identically 1 in any component of , then u is superharmonic in  if and only if minfu; kg is superharmonic in  for k D 1; 2; ::: . (c) The function u is superharmonic in  if and only if u is superharmonic in every component of . We can show this directly from the definition. Proof. (a) Let w D minfu; vg. That w fulfills conditions (i) and (ii) in Definition 9.1 is clear. As for (iii), let 0 b  be such that Cp .X n 0 / > 0, and let ' 2 Lip.@0 / satisfy '  w on @0 . Then '  u on @0 and hence H0 '  u in 0 , as u is superharmonic. Similarly, H0 '  v in 0 , and thus H0 '  w in 0 . (b) Assume first that u is not identically 1 in any component of  and that uk WD minfu; kg is superharmonic in  for k D 1; 2; ::: . Then u.x/ D lim uk .x/  lim lim inf uk .y/  lim inf u.y/; k!1

k!1 y!x

y!x

i.e. (i) (and (ii)) in Definition 9.1 is fulfilled. As for (iii), let 0 b  be such that Cp .X n 0 / > 0, and let ' 2 Lip.@0 / satisfy '  u on @0 . Let k D sup0 '. Then '  uk on @0 and hence H0 '  uk  u in 0 . The converse follows directly from (a). (c) This follows directly from the definition. Proposition 9.4. If u is a superminimizer in , then u is superharmonic in . In particular, all lsc-regularized superminimizers are superharmonic. This is the first result connecting (lsc-regularized) superminimizers and superharmonic functions. Theorem 9.12 below implies that the lsc-regularized superminimizers are the only superminimizers which are superharmonic. Proof. By Theorem 8.22, u is also a superminimizer, and we may assume that u D u . Also by Theorem 8.22, u is lower semicontinuous and never takes the value 1. As 1;p u 2 Nloc ./, u is not identically 1 in any component of .

220

9 Superharmonic functions

It remains to show (iii). Let 0 b  satisfy Cp .X n 0 / > 0, and let v 2 Lipc ./ be such that v  u on @0 . Since u and H0 v are lsc-regularized solutions of the Ku;u .0 /- and K1;v .0 /-obstacle problems, by Proposition 7.15 and Definition 8.31, respectively, the comparison principle (Lemma 8.30) shows that H0 v  u

in :

9.2 Weak Harnack inequalities for superharmonic functions In this section we assume that either  ¤ X or X is unbounded. After we have deduced Corollary 9.14 it will directly follow that all the results in this section are valid also without this restriction, see Remark 9.15. Hence, after Remark 9.15 we will use the results in this section for arbitrary . First, we continue our study of the relation between superharmonic functions and superminimizers, and then use the results to extend weak Harnack inequalities for superminimizers to superharmonic functions. Theorem 9.5. Let u be superharmonic in  and let 0 b . Then there is an increasing sequence of continuous superminimizers fuj gj1D1 in 0 such that u D limj !1 uj in 0 . If u is nonnegative, then uj can be chosen nonnegative. x 0 , and therefore bounded from below in Proof. Since u is lower semicontinuous in  x 0 , Proposition 1.12 shows that there is an increasing sequence f'j g1 of Lipschitz  j D1 x 0 such that functions on  x 0: u D lim 'j on  (9.1) j !1

If u  0, then we may choose 'j  0. Let uj be the lsc-regularized solution of the K'j ;'j .0 /-obstacle problem, which, by Theorem 8.28, is continuous and such that 'j < uj D HAj 'j

in the open set Aj WD fx 2 0 W 'j .x/ ¤ uj .x/g:

As u is superharmonic, we have uj  u

in Aj :

It thus follows that 'j  uj  u

in 0 ;

these inequalities being trivial outside of Aj . Together with (9.1), this shows that u D limj !1 uj and the comparison principle (Lemma 8.30) implies that the sequence fuj gj1D1 is increasing.

9.2 Weak Harnack inequalities for superharmonic functions

221

A consequence of Theorem 9.5 is the following sufficient condition for superhar1;p monic functions to be superminimizers. Since superminimizers belong to Nloc ./, it is clearly also necessary, cf. Remarks 9.20 and 9.56. 1;p Corollary 9.6. Let u be superharmonic in . If u  f in  for some f 2 Nloc ./, in particular if u is locally bounded in , then u is a superminimizer in , and hence 1;p u 2 Nloc ./.

Together with Lemma 9.3, this shows that for every k 2 R, minfu; kg is a superminimizer. Proof. Let 0 b . By Theorem 9.5, u is a limit of an increasing sequence of superminimizers in 0 . It follows from Theorem 7.20 that u is a superminimizer in 0 . Since 0 b  was arbitrary it follows from Lemma 7.11, that u is a superminimizer in . The weak Harnack inequalities for superminimizers can now be extended to superharmonic functions. Theorem 9.7 (Sharp weak Harnack inequality for superharmonic functions). Assume that X supports a .~p; p/-Poincaré inequality for some ~ > 1 and that 0 <  < ~.p  1/. Let u  0 be superharmonic in . Then there is C > 0, only depending on , p, ~, C and the constants in the .~p; p/- and p-Poincaré inequalities, such that «

1=  C inf u



u d

B

B

for every ball B  50B b . See Remark 9.56 for a discussion about the sharpness of the sharp weak Harnack inequality for superharmonic functions (Theorem 9.7). Here and below,  is the dilation constant in the p-Poincaré inequality. As in the sharp weak Harnack inequality for superminimizers (Theorem 8.34), we do not need to use the possibly larger dilation constant in the .~p; p/-Poincaré inequality. Proof. By Theorem 9.5 there is an increasing sequence fuj gj1D1 of nonnegative continuous superminimizers tending to u in 50B. By the sharp weak Harnack inequality for superminimizers (Theorem 8.34), and the monotone convergence theorem, « u d B

«

1= 

D lim

j !1

B

1= uj

d

 C lim ess inf uj D C lim inf uj  C inf u: j !1

B

j !1 B

B

222

9 Superharmonic functions

Theorem 9.8 (Weak Harnack inequality for superharmonic functions). Let u  0 be superharmonic in . Then there are q > 0 and C > 0, only depending on p, q, C and the constants in the p-Poincaré inequality, such that « 1=q uq d  C inf u (9.2) B

2B

for every ball B  50B b . Proof. The proof is almost identical to the proof of Theorem 9.7, but uses the weak Harnack inequality for superminimizers (Theorem 8.10) instead of Theorem 8.34. Note that if we assume that B  100B b , then we can obtain Theorem 9.8 as a corollary of the sharp weak Harnack inequality (Theorem 9.7) and Corollary 4.24.

9.3 Lsc-regularity and the minimum principle By definition, superharmonic functions are lower semicontinuous. We also know from Proposition 9.4 that a superminimizer is superharmonic, provided that it is lscregularized. In this section, we shall see that all superharmonic functions are lscregularized. Thus, Proposition 9.4 is optimal in the sense that the lsc-regularized superminimizers are the only superminimizers which are superharmonic. We shall also obtain the strong minimum principle and two Liouville type theorems for superharmonic functions. First, we need three auxiliary results, which seem natural, but their proofs are by far not trivial, see the comment below. By definition, superharmonic functions (unlike hyperharmonic functions) cannot be identically 1 in any component of , but the following result shows that much more is true. For an even better estimate of the size of the set where a superharmonic function is infinite or large see Corollary 9.51 and Proposition 9.58. Lemma 9.9. Let u be superharmonic in . Then u < 1 a.e. in . A complication in the proof below is that we do not yet have the weak Harnack inequality for superharmonic functions (Theorem 9.8) at our disposal with respect to , as it has not yet been obtained for the case when  D X is bounded. Another complication is that we do not, yet, know that a restriction of a superharmonic function is superharmonic. This will however follow directly once this lemma has been obtained. Proof. Let G be a component of  and A D fx 2 G W u.x/ D 1g. As u is superharmonic in , we have A ¤ G. Let us first show that int A D ¿. Assume not. Then we can find x 2 @.int A/ \ G, as G is connected. Let r 0 > 0 be so small that Bin D Bin .x; r 0 / b G

and

Cp .X n Bin / > 0;

9.3 Lsc-regularity and the minimum principle

223

where Bin .x; r 0 / WD fy 2 X W din .y; x/ < r 0 g and din is the inner metric, see Definition 4.41. Let m D inf Bin u < 1, as Bin 6 A. Hence u 6 1 in Bin , and thus u satisfies (ii) in Definition 9.1 with respect to Bin , as Bin is connected, by Lemma 4.42. That u also satisfies (i) and (iii) with respect to Bin follows directly from those conditions with respect to . Therefore u is superharmonic in Bin and Bin ¤ X (which makes the weak Harnack inequality for superharmonic functions (Theorem 9.8) at our disposal with respect to Bin ). As u is lower semicontinuous it attains its minimum on Bxin which must be finite as u never takes the value 1. Hence also m is finite. Let further r > 0 be so small that 50B b Bin , where B D B.x; r/. As u  m  0 in 50B, and B 6 A, the weak Harnack inequality for superharmonic functions (Theorem 9.8) shows that « 1=q .u  m/q d  C inf .u  m/ < 1; B

2B

showing that u < 1 a.e. in B. But this contradicts the fact that x 2 @.int A/. Hence int A D ¿. As X , and hence G, is separable, we can cover G by balls Bj D B.xj ; rj / b 51Bj b  such that Cp .X n 51Bj / > 0. Fix j D 1; 2; ::: . Since int A D ¿, u satisfies (ii) in Definition 9.1 with respect to 51Bj . That u also satisfies (i) and (iii) with respect to 51Bj follows directly from those conditions with respect to . Therefore u is superharmonic in 51Bj . Again, u attains its minimum mj on 51Bj , which must be real. The weak Harnack inequality for superharmonic functions (Theorem 9.8) shows that  « 1=q

.u  mj /q d

 C inf .u  mj / < 1; Bj

2Bj

showing that u < 1 a.e. in Bj , and hence a.e. in G. Applying this to every component of  finishes the proof. Corollary 9.10. Let u be superharmonic in  and let G   be a nonempty open subset. Then u is superharmonic in G. Proof. That u satisfies (i) and (iii) in Definition 9.1 with respect to G is clear. That it also satisfies (ii) follows from Lemma 9.9. To show that superharmonic functions are lsc-regularized, we still need the following lemma. Lemma 9.11. Let u be superharmonic in  and such that u D 0 a.e. in . Then u D 0 everywhere in . Proof. Let x0 2  be arbitrary. The lower semicontinuity of u yields u.x0 /  lim inf u.y/  0: y!x0

224

9 Superharmonic functions

Let " > 0. Again, since u is lower semicontinuous, there is a ball B D B.x0 ; r/  51B   such that Cp .X n 51B/ > 0 and u.y/  u.x0 /  "

for y 2 51B:

Then v D u  u.x0 / C " is a nonnegative superharmonic function in 51B, by Corollary 9.10, and v D u.x0 / C " a.e. in 51B. Note that 51B ¤ X , and thus, by the weak Harnack inequality for superharmonic functions (Theorem 9.8), « 1=q v q d  C inf v  C v.x0 / D C ": u.x0 /  u.x0 / C " D B

2B

Letting " ! 0 gives u.x0 /  0, i.e. u.x0 /  0. Hence u.x0 / D 0. Theorem 9.12. Let u be superharmonic in . Then u is lsc-regularized and u.x/ D lim inf u.y/ D ess lim inf u.y/; y!x

y!x

x 2 :

Proof. Let x 2  be arbitrary. Since u is lower semicontinuous, we have u.x/  lim inf u.y/  ess lim inf u.y/ DW : y!x

y!x

(9.3)

Let a <  be real. Then there is a ball B D B.x; r/   such that u.y/ > a

for a.e. y 2 B:

By Lemma 9.3, v D minfu  a; 0g is superharmonic in  and v D 0 a.e. in B. By Corollary 9.10, v is superharmonic in B, and Lemma 9.11 implies that v D 0 everywhere in B. Thus u.x/  a. Letting a !  shows that u.x/ D  . Hence, we have equality throughout (9.3). Theorem 9.13 (The strong minimum principle). If  is connected, u is superharmonic in  and u attains its minimum in , then u is constant in . Proof. We may assume that the minimum is 0. Let A D fx 2  W u.x/ D 0g, which is a nonempty relatively closed subset of , since u is lower semicontinuous. Let further x 2 A. Then we can find r > 0 such that B D B.x; r/  51B b  and Cp .X n 51B/ > 0. By Corollary 9.10, u is superharmonic in 51B. Note that 51B ¤ X , and thus it follows from the weak Harnack inequality for superharmonic functions (Theorem 9.8) that « 1=q q 0 u d  C inf u D 0: 2B

B

Hence u D 0 a.e. in 2B. As u is lsc-regularized, by Theorem 9.12, u  0 in 2B and thus 2B  A. Therefore A is open, and since  is connected, A must be the only nonempty relatively closed open subset of , viz.  itself. Hence u  0.

9.3 Lsc-regularity and the minimum principle

225

The following result is an analogue of Liouville’s theorem (Corollary 8.16) and a generalization of Proposition 8.23. Note that here we assume X to be bounded. On unbounded spaces, there may be plenty of nonconstant superharmonic functions, see Remark 9.56. Corollary 9.14. Assume that u is superharmonic in X and that X is bounded. Then u is constant. Proof. As u is lower semicontinuous and X is compact, u attains its infimum in X . Since X is connected, the strong minimum principle (Theorem 9.13) shows that u must be constant. Remark 9.15. After Corollary 9.14 it is immediate that the results in Section 9.2 are all valid also in the case when  D X and X is bounded. In fact, the proof of the strong minimum principle (Theorem 9.13) makes it possible to give the following generalization of Corollary 9.14. Note, however, that in general metric spaces it is not known whether p-harmonic and superharmonic functions have the sheaf property, see Open problems 9.22 and 9.23. Proposition 9.16. Assume that X is bounded and that for every x 2 X , u is superharmonic in some ball Bx 3 x. Then u is constant, and in particular superharmonic, in X . Proof. It follows directly that u is lower semicontinuous in X , and as X is compact it attains its infimum in X , which we may assume to be 0. Let A D fx 2 X W u.x/ D 0g, a closed subset of X , since u is lower semicontinuous. Let further x 2 A. Then we can find a ball B 3 x such that u is superharmonic in 51B and Cp .X n 51B/ > 0. It follows from the weak Harnack inequality for superharmonic functions (Theorem 9.8) that «  1=q

0

uq d 2B

 C inf u D 0: B

Hence u D 0 a.e. in 2B. As u is lsc-regularized in 2B, by Theorem 9.12, u  0 in 2B and thus 2B  A. Therefore A is open, and since X is connected, A must be the only nonempty closed open subset of X , viz. X itself. Hence u  0 in X . If X is bounded and    X is such that Cp .X n / D 0, then a superharmonic function in  which is bounded from below (or belongs to N 1;p ./) is constant, by Theorem 12.3 and Corollary 9.14. However, it may be worth pointing out that there still may exist unbounded superharmonic functions on . Example 9.17. Let X D B.0; 1/ in Rn , n  2, 1 < p  n,  D X n f0g and ´ log jxj; if p D n; u.x/ D 1  jxj.pn/=.p1/ ; if 1 < p < n:

226

9 Superharmonic functions

Let us show that u is superharmonic in . (Note that as  is not open in Rn , this does not follow directly from the Euclidean theory.) Let 0 b  and let v 2 Lipc .@0 / be such that v  u on @0 . Then clearly, H0 v D v  u  0 on @0 . The comparison principle (Lemma 8.32) shows that H0 v  0 and hence H0 v  u

on @B.0; 1/:

(9.4)

Let further G D 0 \ B.0; 1/. Since @G  @0 [ @B.0; 1/, we get that H0 v  u on @G. Thus, using the comparison principle again and the p-harmonicity of u in G (see Remark 9.56), we obtain that H0 v D HG .H0 v/  HG u D u

in G:

Together with (9.4) we get that H0 v  u

in 0 ;

showing that u is superharmonic in . Note however that by e.g. the maximum principle, u is not p-harmonic in , cf. Example 8.17.

9.4 Characterizations In this section, we relate superharmonic functions to p-harmonic functions and superminimizers. We also show that our definition of superharmonic functions is equivalent to another definition appearing in the literature. See also Section 14.3 for yet another equivalent definition of superharmonic functions, which however is not used anywhere else in this book. Proposition 9.18. A function u is p-harmonic in  if and only if it is both sub- and superharmonic in . Proof. The necessity follows from the comparison principle, Lemma 8.32. As for the sufficiency, it directly follows that u is continuous and hence locally bounded, and thus by Corollary 9.6 is both a sub- and a superminimizer, i.e. a continuous minimizer, by Proposition 7.8. Proposition 9.19. Let u be a superharmonic function in  and v be a superminimizer in . Then w D minfu; vg is a superminimizer in . Proof. By Proposition 9.4, v  is superharmonic, and thus, by Lemma 9.3, w 0 D 1;p ./, Corollary 9.6 implies minfu; v  g is also superharmonic. As w 0  v  2 Nloc 0 0 that w is a superminimizer. Since w D w q.e., it is also a superminimizer.

9.4 Characterizations

227

Remark 9.20. We may consider the set S of superharmonic functions as a partially ordered set under the partial order . By Proposition 9.4, Corollary 9.6 and Theorem 9.12, 1;p the set S 0 of all lsc-regularized superminimizers satisfies S 0 D S \ Nloc ./. In fact, 1;p 0 by Corollary 9.6, S contains all functions in S which have a majorant in Nloc ./. Thus the nonsuperminimizer superharmonic functions are simply too large to be superminimizers. Moreover, S n S 0 is the top part of S , in the sense that if u 2 S 0 , v 2 S and u  v, then v 2 S 0 . Observe that in general S ¤ S 0 , see Remark 9.56. The following localization lemma will be useful. Note that it is not known whether the monotonicity assumption for the j ’s can be removed, not even for two sets, see Open problems 9.22 and 9.23 below. S Proposition 9.21. Let 1  2     D j1D1 j . Then u is superharmonic in  if and only if it is superharmonic in j , j D 1; 2; ::: . Observe that if  ¤ X orSif  D X is unbounded, then we can exhaust  by 1 b 2 b  b  D j1D1 j , and moreover j ¤ , see the proof of Theorem 9.24. On the other hand, if  D X is bounded, then there is no such nontrivial exhaustion, as for any such exhaustion we will have by compactness that  D j for some j . Proof. The necessity follows from Corollary 9.10. For the sufficiency, it is clear that u is lower semicontinuous in  and that it satisfies condition (ii) in Definition 9.1. As for condition (iii), let 0 b , with Cp .X n 0 / > 0, and let v 2 Lip.@0 / be such that v  u on @0 . By compactness, there is j such that 0 b j . As u is superharmonic in j , it follows that H0 v  u in 0 . Hence u is superharmonic in . In view of Proposition 9.21 it is natural to ask the following question. Open problem 9.22 (The sheaf property). Is it true that if f is p-harmonic (or superharmonic) in the open S sets j , j D 1; 2; ::: , then it follows that f is p-harmonic (or superharmonic) in j1D1 j ? S If j1D1 j D X and X is bounded, then this is true by Proposition 9.16, but otherwise it remains open in general. For Cheeger p-harmonic functions this follows from a partition of unity argument, see Appendix B.2. The situation is similar in other situations when gu D jruj for some type of vector-valued gradient, and we thus have a p-Laplace equation, as e.g. in (weighted) Rn and open subsets thereof, on manifolds, graphs and Carnot–Carathéodory spaces, see Appendix A. In fact the following finite version also remains open in general. Open problem 9.23. Is it true that if f is p-harmonic (or superharmonic) in the open sets 1 and 2 , then it follows that f is p-harmonic (or superharmonic) in 1 [ 2 ? We are now ready to provide some useful characterizations of superharmonic functions.

228

9 Superharmonic functions

Theorem 9.24. Let u W  ! .1; 1 be a function which is not identically 1 in any component of . Then the following are equivalent: (a) u is superharmonic in ; (b) minfu; kg is superharmonic in  for all k D 1; 2; ::: ; (c) u is lsc-regularized, and minfu; kg is a superminimizer in  for all k D 1; 2; ::: ; (d) minfu; kg is an lsc-regularized superminimizer in  for all k D 1; 2; ::: ; (e) u is superharmonic in 0 for every 0 b  with Cp .X n 0 / > 0. Proof. (a) , (b) This is Lemma 9.3 (b). (a) ) (c) That u is lsc-regularized follows from Theorem 9.12. That minfu; kg is a superminimizer follows from Proposition 9.19, alternatively one can use Lemma 9.3 together with Corollary 9.6. (c) ) (d) It follows directly that also minfu; kg is lsc-regularized. (d) ) (b) This follows from Proposition 9.4. (a) ) (e) This follows from Corollary 9.10. (e) ) (a) If  D X is bounded, then this implication follows directly from Proposition 9.16. Assume therefore that either  ¤ X or X is unbounded. Fix x0 2  and let dj D dist.x0 ; X n /=j and ´ ¹x 2 B.x0 ; j / W dist.x; X n / > dj º; if  ¤ X; j D if  D X is unbounded, B.x0 ; j /; for j D 2; 3; ::: . By assumption, u is superharmonic in j , and it follows from S Proposition 9.21 that u is superharmonic in  D j1D2 j . We are also able to finally deduce the following characterization of superminimizers, which extends Proposition 7.16 to the case when  D X is bounded. Proposition 9.25. The following are equivalent: (a) u is a superminimizer in ; (b) u is a superminimizer in 0 for every 0 b  with Cp .X n 0 / > 0; (c) u is a solution of the Ku;u .0 /-obstacle problem for every 0 b  with Cp .X n 0 / > 0. Proof. (a) ) (c) This follows directly from Proposition 7.15. (c) ) (b) This follows directly from Proposition 7.14. (b) ) (a) Let first 0 b  with Cp .X n 0 / > 0 be arbitrary. By Proposition 9.4,  u is superharmonic in 0 . Moreover, u D u q.e. in 0 , by Theorem 8.22, and thus 1;p u 2 Nloc .0 /. As 0 was arbitrary, u is superharmonic in , by Theorem 9.24.

9.5 Convergence results for superharmonic functions

229

1;p ./, as X , and hence , is Lindelöf, by Moreover, u D u q.e. in  and u 2 Nloc  Proposition 1.6. Thus, u is a superminimizer in , by Corollary 9.6. As u D u q.e. in , also u is a superminimizer in .

Proposition 9.26. Let u W  ! .1; 1. Then the following are equivalent: (iii) For every nonempty open set 0 b  with Cp .X n 0 / > 0, and all functions v 2 Lip.@0 /, we have H0 v  u in 0 whenever v  u on @0 . (iii0 ) For every nonempty open set 0 b  with Cp .X n 0 / > 0, and all functions x 0 / \ N 1;p .0 /, we have H0 v  u in 0 whenever v  u on @0 . v 2 C. This shows that we can replace (iii) in Definition 9.1 by (iii0 ). In fact, when Kinnunen–Martio [217] first defined superharmonic functions on metric spaces they used (iii0 ). Proof. (iii0 ) ) (iii) Let 0 b , with Cp .X n 0 / > 0, and let v 2 Lip.@0 /, with x 0 /, e.g. one v  u on @0 , be given. Then v has an extension V as a function in Lip. x 0 / \ N 1;p .0 /, it follows from of the McShane extensions in Lemma 5.2. As v 2 C. (iii0 ) that H0 v D H0 V  u in 0 . x 0 / \ N 1;p .0 /, (iii) ) (iii0 ) Let 0 b , with Cp .X n 0 / > 0, and let v 2 C. with v  u on @0 , be given. Let also " > 0. As continuous functions (on a compact set) can be uniformly approximated by Lipschitz functions (by Proposition 1.11), we x 0 / such that v  "  w  v in  x 0 . By (iii) and the comparison can find w 2 Lip. principle (Lemma 8.32) we have H0 v D H0 .v  "/ C "  H0 w C "  u C "

in 0 :

Letting " ! 0 completes the proof. We provide further characterizations of superharmonic functions in Proposition 10.44 and Theorem 14.10. For characterizations in terms of solutions to the obstacle problem, see A. Björn [35].

9.5 Convergence results for superharmonic functions In this section, we use Theorem 9.24 to extend the convergence results from Section 7.4 to superharmonic functions and prove the fundamental convergence theorem of potential theory. Theorem 9.27. Let fui g1 iD1 be an increasing sequence of superharmonic functions in  such that u D limi!1 ui is not identically 1 in any component of . Then u is superharmonic in .

230

9 Superharmonic functions

Let us give two proofs of this result, one shorter and one more elementary. First proof. Let k 2 Z be fixed. Then minfui ; kg is an lsc-regularized superminimizer, by Theorem 9.24. By Theorem 7.21, minfu; kg D limi!1 minfui ; kg is an lscregularized superminimizer, and is thus superharmonic by Proposition 9.4. Another application of Theorem 9.24 shows that u is superharmonic. Second proof. It is clear that u is lower semicontinuous, so it is enough to verify condition (iii) in Definition 9.1. Let 0 b , with Cp .X n 0 / > 0, and v 2 Lip.@0 / be such that v  u on @0 . Let further " > 0. As ui is lower semicontinuous and v is continuous, the set Ai D fx 2 @0 W v.x/ < ui .x/ C "g S 0 0 is relatively open. Moreover, A1  A2    1 iD1 Ai D @ . Since @ is 0 0 compact, there is some j so that @ D Aj . As v < uj C " on @ and uj C " is superharmonic, we see that H0 v  uj C " < u C " in 0 . Letting " ! 0 shows that H0 v  u in 0 , as required. Theorem 9.28. Let fui g1 iD1 be a sequence of superharmonic functions which converges locally uniformly to u in . Then u is superharmonic in . Proof. This follows from Theorem 9.27 in the same way as in the proof of Theorem 7.25. By combining Theorems 9.27 and 9.28 with Proposition 9.21 we immediately obtain the following two generalizations, which are sometimes useful in applications. S Theorem 9.29. Let 1  2     D j1D1 j and let uj be superharmonic in j , j D 1; 2; ::: . Assume that uj  uj C1 in j , j D 1; 2; ::: , and that u D limj !1 uj is not identically 1 in any component of . Then u is superharmonic in . S Theorem 9.30. Let 1  2     D j1D1 j and let uj be superharmonic in j , j D 1; 2; ::: . Assume that uj ! u locally uniformly in , as j ! 1. Then u is superharmonic in . When dealing with nonincreasing sequences, we have to be more careful, since the limiting function need not be lower semicontinuous, see Example 9.35. However, its lsc-regularization is and does the job. We have the following general theorem and its corollary for decreasing sequences. Note that Theorem 9.27 is also a special case of the following general theorem. Since no regularization is needed in Theorem 9.27, we have chosen to present the above two proofs as well. Theorem 9.31. Let fuj gj1D1 be a sequence of superharmonic functions in  such that 1;p uj  f a.e. in  for some f 2 Nloc ./ and all j D 1; 2; ::: . Let v D lim infj !1 uj .  If v is not identically 1 in any component of , then it is superharmonic in  and v D v  q.e. in .

9.5 Convergence results for superharmonic functions

231

Theorem 9.32. Let fuj gj1D1 be a decreasing sequence of superharmonic functions in 1;p  such that v WD limj !1 uj  f for some f 2 Nloc ./. Then v  is superharmonic  in  and v D v q.e. in .

Proof of Theorem 9.31. Let k 2 Z and vk WD minfv; kg D lim infj !1 minfuj ; kg. Then each minfuj ; kg is a superminimizer in , by Theorem 9.24. Furthermore, 1;p minfuj ; kg  minff; kg a.e. in  and vk  k 2 Nloc ./. Corollary 7.24 then implies that vk is a superminimizer in  and Theorem 8.22 shows that vk D vk q.e. in . Moreover, vk is superharmonic in  by Proposition 9.4. Since vk D minfv  ; kg in , Theorem 9.24 again shows that v  is superharmonic in . Moreover, v  D limk!1 vk D limk!1 vk D v q.e. in . In the rest of this section we prove the fundamental convergence theorem of potential theory. x as First, we need to define the lim inf-regularization of a function f W  ! R fO.x/ D lim

inf

r!0 \B.x;r/

f;

x 2 :

Observe that, unlike for f  .x/, the value f .x/ is taken into account when defining fO.x/. It follows that fO  f , and it is easy to show that fO is lower semicontinuous, see the beginning of the proof of Theorem 8.22. Theorem 9.33 (The fundamental convergence theorem). Let F be a nonempty family 1;p ./ such that u  f of superharmonic functions in . Assume that there is f 2 Nloc a.e. in  for all u 2 F . Let w D inf F . Then the following are true: (a) w y is superharmonic; (b) w y D w  in ; (c) w y D w q.e. in . Usually, in the fundamental convergence theorem it is assumed that the functions in F are locally uniformly bounded from below, rather than the (slightly) more general condition here. In fact, it follows from our result that under our condition, the functions in F are locally uniformly bounded from below. The advantage with our formulation is that it can be applied in more general situations without knowing a priori that the functions in F are locally uniformly bounded from below. We will need Choquet’s topological lemma. We say that a family of functions U is downwards directed if for each u; v 2 U there is w 2 U with w  minfu; vg. Lemma 9.34 (Choquet’s topological lemma). Let U D fu W  2 I g be a nonempty x Let further u D inf U. If U is downwards directed, family of functions u W  ! R. then there is a decreasing sequence of functions vj 2 U with v D limj !1 vj such that vO D u. O

232

9 Superharmonic functions

Proof. As X is separable, there exists a countable set Z   which is dense in . Consider all balls, with centres in Z and rational radii, which are contained in . Let B1 ; B2 ; ::: , be an enumeration of these balls. For each k D 1; 2; ::: , choose xk 2 Bk such that 1 u.xk /  inf u C Bk k if the infimum is not 1, and u.xk /  k  1 otherwise. Next, let k 2 I be an index with 1 uk .xk /  u.xk / C k if u.xk / > 1, and uk .xk /  k otherwise. Now, for w WD inffuj W j D 1; 2; ::: g and all k D 1; 2; ::: , we have inf w  uk .xk /  inf u C Bk

Bk

2 k

if inf Bk u > 1, and inf Bk w  uk .xk /  k otherwise. As each ball B.x; r/ with y  u. O x 2  contains infinitely many Bk , it follows that w Now, let v1 D u1 and for j D 2; 3; ::: , choose vj 2 U such that vj  minfvj 1 ; uj g: Then vj is a decreasing sequence and u  v D limj !1 vj  w. Hence, uO  vO  w y  u, O which finishes the proof. Proof of Theorem 9.33. (b) It is clear that w   w. y For the converse inequality let B D B.x; r/   be a ball and " > 0. If w.x/ y D 1, then w  .x/  w.x/ y trivially, so we may assume that w.x/ y < 1. Let us assume that w.x/ y 2 R (the case w.x/ y D 1 can be treated similarly). We can then find y 2 B such that w.y/ < w.x/ y C " and hence also u 2 F such that u.y/ < w.x/ y C ". As u is superharmonic and hence lsc-regularized, it follows that ess inf w  ess inf u  u.y/ < w.x/ y C ": B

B

Since this is true for all balls B D B.x; r/   we see that w  .x/  w.x/ y C ". Letting " ! 0 shows that w   w y and thus w  D w. y (a) and (c) Let U D fu W u is superharmonic in  and u  w in g  F . Then w  inf u  inf u D w: u2U

u2F

As U is downwards directed, Choquet’s topological lemma (Lemma 9.34) implies that there is a decreasing sequence of superharmonic functions vj 2 U with v D limj !1 vj such that vO D w. y By (b) applied to v we have that w  D w y D vO D v  everywhere 1 in . Theorem 9.32 applied to the sequence fvj gj D1 shows that v  is superharmonic and v  D v q.e. in . Finally, as vj  w, j D 1; 2; ::: , we get that w y D v D v  w  w y

q.e. in :

9.6 Harnack’s convergence theorem for p-harmonic functions

233

Example 9.35. The lower semicontinuous regularization is necessary in the fundamental convergence theorem (Theorem 9.33), as well as in Theorems 9.31 and 9.32. To see this, consider for 1 < p < n the sequence of superharmonic functions uj .x/ D minfjxj.pn/=.p1/ =j; 1g;

j D 1; 2; ::: ;

(with uj .0/ D 1) in Rn , see Remark 9.56. Then clearly u WD infj uj D f0g , which is not lower semicontinuous and hence not superharmonic. The lsc-regularization u is identically zero.

9.6 Harnack’s convergence theorem for p-harmonic functions For p-harmonic functions, we have the following direct consequence of Theorem 9.30. S Theorem 9.36. Let 1  2     D j1D1 j and let uj be p-harmonic in j , j D 1; 2; ::: . If uj ! u locally uniformly in , as j ! 1, then u is p-harmonic in . Proof. By Theorem 9.30, u is both super- and subharmonic in  (the latter after applying Theorem 9.30 to uj ! u, as j ! 1). Hence u is p-harmonic in  by Proposition 9.18. Another important convergence result for p-harmonic functions is Harnack’s convergence theorem. Theorem 9.37 (Harnack’s convergence theorem). Assume that  is connected. Let fuj gj1D1 be a sequence of nonnegative p-harmonic functions in . If there is some x 2  and a constant Cz such that uj .x/  Cz for j D 1; 2; ::: , then a subsequence of fuj gj1D1 converges locally uniformly to a p-harmonic function in . Proof. Let 1 b 2 b  b  be such that x 2 1 . For each i; j D 1; 2; ::: the Harnack inequality (Corollary 8.19) implies that sup uj  Ci uj .x/  Ci Cz : i

Hence, by Theorem 8.15, for all x; y 2 i , juj .x/  uj .y/j  Ci0 d.x; y/˛ sup uj  Ci0 CiC1 Cz d.x; y/˛ ; i C1

i.e. the sequence is equicontinuous on each i . The Ascoli theorem (see e.g. Theorem 4.43 in Folland [124], p. 169 in Royden [310], or Theorem 11.28 in Rudin [311]) and a diagonal argument provide us with a subsequence (also denoted by fuj gj1D1 ) converging locally uniformly to a continuous function u. Theorem 9.36 shows that u is p-harmonic in .

234

9 Superharmonic functions

Corollary 9.38. Assume that  is connected. Let fui g1 iD1 be an increasing sequence of p-harmonic functions in  such that u D limi!1 ui is not identically 1. Then u is p-harmonic in . Proof. Note that since the sequence fui g1 iD1 is not increasing, this does not follow directly from Theorem 9.27 as in Theorem 9.36. However, as u is not identically 1, there exists x 2  such that ui .x/  u.x/ < 1 for all i . Moreover, for every 0 b , u1 is bounded from below on 0 , and hence (“vertical translations” of) the functions ui satisfy the assumptions of Theorem 9.37 (for 0 ). It follows that u is p-harmonic in 0 , and as 0 was arbitrary this finishes the proof.

9.7 Comparison of sub- and superharmonic functions The following theorem will play an important role in the development of the Perron method for solving the Dirichlet problem for p-harmonic functions in Chapter 10. Theorem 9.39 (Comparison principle). Assume that    X is bounded. Assume further that u is superharmonic and v is subharmonic in . If 1 ¤ lim sup v.y/  lim inf u.y/ ¤ 1 for all x 2 @; 3y!x

3y!x

(9.5)

then v  u in .

S Proof. Let 1 b 2 b  b  D 1 kD1 k and " > 0. For every x 2 @, it follows from (9.5) that lim inf .u.y/  v.y//  0 3y!x

and hence there is a ball Bx 3 x such that u  v  " Thus, x D 

1 [ kD1

in Bx \ :

k [

[

Bx ;

x2@

x  k [ Bx1 [  [ BxN for and by compactness, there exists k > 1=" such that  some N . It follows that v  uC" on @k . As v is upper semicontinuous (and does not take the value 1), Proposition 1.12 shows that there is a decreasing sequence f'j gj1D1 , x k /, such that 'j ! v in  x k , as j ! 1. 'j 2 Lip. Since uC" is lower semicontinuous, the compactness of @k shows that there is i so that 'i  u C " on @k . By (iii) in Definition 9.1 of superharmonicity, Hk 'i  u C " in k . Similarly, v  Hk 'i , and thus v  u C " in k . Letting " ! 0 (and hence k ! 1) completes the proof.

9.8 New superharmonic functions from old

235

Open problem 9.40. Assume that  is bounded and Cp .X n / > 0. Assume further that u is superharmonic and bounded from below in , that v is subharmonic and bounded from above in , and that lim sup v.y/  lim inf u.y/ 3y!x

3y!x

for q.e. x 2 @:

(9.6)

Does it follow that v  u? It is equivalent to consider this problem for bounded u and v, as we can replace u by minfu; sup vg and v by maxfv; inf  ug. A positive answer to this question would have important consequences, in particular for Perron solutions, see the discussion in Section 10 in Björn–Björn–Shanmugalingam [56]. Using some of the results on Perron solutions we are able to give a positive answer to a special case of Open problem 9.40 in Proposition 10.28. When the exceptional set in (9.6) consists only of semiregular points (see Chapter 13) a positive answer to Open problem 9.40 was obtained by A. Björn [42]. Observe that unlike in Theorem 9.39 we cannot allow for Cp .X n / D 0, as the answer would then directly be negative, consider e.g. v  1 and u  0.

9.8 New superharmonic functions from old One of the ingredients in Moser’s proof in [293] of regularity for elliptic differential equations was the observation that for every solution u, the function v D f ı u, with f nondecreasing and convex, is a subsolution. A similar result holds for superharmonic functions. We have already seen this phenomenon in the proof of the very weak Harnack inequality for superminimizers (Proposition 8.7). Theorem 9.41. Let 1  a < 1 and let u W  ! .a; 1 be superharmonic in . Let further ' W .a; 1/ ! R be a nondecreasing concave function and let '.1/ D lim t!1 '.t/. Then ' ı u is also superharmonic in . In particular, if u > 0, then log u is superharmonic. See Corollary 9.48 for an improvement of the last part of Theorem 9.41. Let us explain why we only consider intervals of the type .a; 1. First of all, if ' is nondecreasing and concave on the interval .a; b/, for some b < 1, then by letting '.t/ D lims!b  '.s/ < 1 for t 2 Œb; 1, ' becomes nondecreasing and concave on .a; 1. Secondly, if a superharmonic function u W  ! Œa; 1 satisfies u.x/ D a for some x 2 , then, by the strong minimum principle (Theorem 9.13), u is constant in the component of  containing x, and this case can easily be handled in more trivial ways.

236

9 Superharmonic functions

Proof. Note first that 'j.a;1/ is continuous and real-valued. It follows that ' ı u is lsc-regularized. Let 0 b  and m D inf  x 0 u. By the lower semicontinuity of u, the value m is 0 x , i.e. m D u.x/ > a. attained for some x 2  Let next k 2 Z and 'k .t / D minf'.t /; kg. For " > 0 we can find constants aj;"  0 and bj;" , 1  j  N" , so that j'k .t /  f" .t /j  "

for m < t < 1;

where f" .t / D min .aj;" t C bj;" /: 1j N"

By Lemma 9.3 (a) (applied N"  1 times), f" ı u is superharmonic. Thus, the superharmonic functions f1=i ı u converge uniformly to 'k ı u in 0 , as i ! 1. By Theorem 9.30, it follows that minf' ı u; kg D 'k ı u is also superharmonic in 0 . Hence ' ı u is superharmonic in 0 , by Lemma 9.3 (b). Since 0 b  was arbitrary, it follows from Proposition 9.21 that ' ı u is superharmonic in . We have a corresponding result for superminimizers as well. Theorem 9.42. Let 1  a < 1 and let u W  ! .a; 1 be a superminimizer in . Let further ' W .a; 1/ ! R be a nondecreasing concave function and let '.1/ D lim t!1 '.t /. Then ' ı u is also a superminimizer in . In particular, if u > 0, then log u is a superminimizer. Proof. By Proposition 9.4, u is superharmonic. Clearly u ./  Œa; 1. If u .x/ D a for some x 2 , then by the strong minimum principle (Theorem 9.13), u  a in the component containing x, but this contradicts the fact that u > a and that u D u q.e. Hence u ./  .a; 1 and ' ı u is superharmonic by Theorem 9.41. As ' is concave, there exist ˛; ˇ 2 R such that '.x/  ˛x C ˇ for x 2 .a; 1. Thus 1;p ' ı u  ˛u C ˇ 2 Nloc ./, and by Corollary 9.6, ' ı u is a superminimizer. As  u D u q.e., also ' ı u D ' ı u q.e. and ' ı u is a superminimizer in . The following result is a direct consequence of Theorem 9.41. We record it for the reader’s convenience. There is, of course, also a corresponding result for sub- and superminimizers. Corollary 9.43. Let 1  a  1. Then the following hold: (a) If u W  ! .a; 1 is superharmonic in  and ' W .a; 1/ ! R is a nonincreasing convex function, then ' ı u is subharmonic in . In particular, if u > 0, then 1=u is subharmonic. (b) If u W  ! Œ1; a/ is subharmonic in  and ' W .1; a/ ! R is a nondecreasing convex function, then ' ı u is also subharmonic in .

9.8 New superharmonic functions from old

237

(c) If u W  ! Œ1; a/ is subharmonic in  and ' W .1; a/ ! R is a nonincreasing concave function, then ' ı u is superharmonic in . (Here we define '.˙1/ D lim t!˙1 '.t /.) Note that the converse of (a) in Corollary 9.43 is not true, i.e. if u > 0 is subharmonic, then 1=u need not be superharmonic. (This is best seen by taking u p-harmonic. Combining this with (a) in Corollary 9.43 would mean that 1=u would be p-harmonic, which is not true.) We can thus say that there are “fewer” positive superharmonic functions than positive subharmonic (or equivalently, negative superharmonic) functions. We are also able to obtain the following Poisson modification for superharmonic functions. It generalizes Theorem 8.33, where a Poisson modification was proved for superminimizers. Here, extra care has to be taken to make the new function lower semicontinuous. For regular G (see Definition 10.4), this is not needed, see Theorem 10.43. The Poisson modification will be needed to develop the theory of Perron solutions, in the proof of Theorem 10.10. When the theory of Perron solutions has been developed, we will be able to obtain Poisson modifications also for superharmonic functions not 1;p in Nloc ./ in Theorem 10.42, see also Theorem 10.43. Theorem 9.44 (Poisson modification for Newtonian superharmonic functions). As1;p sume that u 2 Nloc ./ is superharmonic in  and let G b  be nonempty, open and such that Cp .X n G/ > 0. Then ´ u in  n G; vD HG u in G is a superminimizer in . Furthermore, 8 x ˆ x 2  n G; ˆ
(9.7)

G3y!x

is superharmonic in , and v   v  u in . Proof. By Corollary 9.6, u is a superminimizer in . It thus follows from Theorem 8.33 that v is a superminimizer, and by Proposition 9.4 that v  is superharmonic in . By Propositions 7.15 and 9.12, u is the lsc-regularized solution of the Ku;u .G/-obstacle problem. The comparison principle (Lemma 8.30) then shows that HG u  u in G, and hence v  u in . As both u and HG u are lsc-regularized, it follows that v  is given by the formula (9.7). That v   v in  is trivial.

238

9 Superharmonic functions

9.9 Integrability of superharmonic functions 1;p Some superharmonic functions are too large to belong to Nloc , or even L1loc , see Remarks 9.20 and 9.56. However, they cannot be arbitrarily large, as we shall see in this section. We start this section by showing that superharmonic functions belong to p p the gradient space Gloc . Recall the basic theory of the space Gloc from Section 2.8. p Lemma 9.45. Let u be superharmonic in . Then u 2 Gloc ./.

Proof. This follows from Lemma 9.9 together with Lemma 9.3 and Corollary 9.6. The following two logarithmic Caccioppoli inequalities improve Proposition 8.9 and extend it to superharmonic functions. Here, Gu is a p-weak upper gradient of u, which is minimal in the sense of Proposition 2.41. Proposition 9.46 (Logarithmic Caccioppoli inequality for superharmonic functions). Assume that u  0 is superharmonic in  and that there is no component G   1;p such that u  0 in G. Let v D log u. Then v 2 Nloc ./ and gv D Gu =u a.e. in . Furthermore, for every ball B D B.x; r/  2B   we have « C gvp d  p ; r B where C D C .2p=.p  1//p . Proof. As u takes positive values in every component, the strong minimum principle (Theorem 9.13) shows that u > 0 in . By lower semicontinuity, u attains its minimum x which is thus positive. Hence, v D log u is bounded from below in B and as on B, p vC  Cq uq , where q is the exponent in the weak Harnack inequality for superharmonic functions (Theorem 9.8), it follows from Theorem 9.8 that v 2 Lp .B/, as long as 50B b . Thus, v 2 Lploc ./. As v is bounded from below on B we have, using the logarithmic Caccioppoli inequality for superminimizers (Proposition 8.9), that Gv D lim glog minfu;j g D lim j !1

j !1

gminfu;j g Gu D minfu; j g u

a.e. in B:

Hence Gv D Gu =u a.e. in . Moreover, by the logarithmic Caccioppoli inequality for superminimizers (Proposition 8.9) again, we see that « « C p p Gv d D lim glog d  p ; minfu;j g j !1 B r B 1;p ./ and gv D Gv . and hence v 2 Nloc

9.9 Integrability of superharmonic functions

239

Corollary 9.47 (Logarithmic Caccioppoli inequality for superminimizers). Assume that u  0 is a superminimizer in  and that there is no component G   such 1;p that u D 0 a.e. in G. Let v D log u. Then v 2 Nloc ./ and gv D gu =u a.e. in . Furthermore, for every ball B D B.x; r/  2B   we have « C gvp d  p ; r B where C D C .2p=.p  1//p . Proof. Let u be the lsc-regularization of u and v  D log u . Then u is superharmonic and takes positive values in every component. Moreover, v  D v q.e. and gv D gv a.e. The conclusion now follows from the logarithmic Caccioppoli inequality for superharmonic functions (Proposition 9.46). Corollary 9.48. Let u > 0 be a superharmonic function in . Then log u is an lsc-regularized superminimizer in . Proof. By Theorem 9.41, log u is superharmonic and hence lsc-regularized. Since 1;p log u 2 Nloc ./, by the logarithmic Caccioppoli inequality for superharmonic functions (Proposition 9.46), Corollary 9.6 shows that log u is a superminimizer. We know by Theorems 1.56 and 5.29 that Newtonian functions are absolutely continuous on p-a.e. curve and quasicontinuous. In particular, this holds for superminimizers. Superharmonic functions are in general too large to belong to Newtonian spaces, see Remarks 9.20 and 9.56. It is therefore not obvious that they have the same continuity properties as Newtonian functions. This is shown in the following proposition. See also Section 11.6, where the direction is reversed and fine continuity of superharmonic functions is used to obtain continuity results for Newtonian functions. Proposition 9.49. Let u be superharmonic in . Then u 2 ACCp ./, u is quasicontinuous in  and Z Gu ds < 1

(9.8)



for p-a.e. curve  in . S Proof. Let 1 b 2 b  b  D j1D1 j . Fix a positive integer j and let 0 m D min x j u > 1, by the lower semicontinuity of u. Let also u D u  m C 1.

1;p .j /, by the logarithmic Caccioppoli Then u0  1 in j and 0  v D log u0 2 Nloc inequality for superharmonic functions (Proposition 9.46). R Proposition 1.37 and Theorem 1.56 then yield that for p-a.e. curve  2 .j /,  gv ds < 1 and v is absolutely continuous (and in particular bounded) on  . Here we have chosen gv D Gu =u0 everywhere.

240

9 Superharmonic functions

It follows that

Z

Z Gu ds D



e v gv ds < 1: 

Moreover, u D e v C m  1 is absolutely continuous on  , by Lemma 1.58. As every curve  in  lies in some j , (9.8) holds for p-a.e. curve  in  and u 2 ACCp ./. Theorem 5.29 shows that v is quasicontinuous in j , and as compositions of continuous functions are continuous, also u is quasicontinuous in j , and thus in , by Lemma 5.18. The sets where superharmonic functions are infinite can be characterized as sets of zero capacity, see Theorem 9.52 below. This shows that there is a strong link between superharmonic functions and Newtonian spaces, see also Section 11.6. Definition 9.50. A set E  X is polar if there is an open set   X and a superharmonic function u on  such that E  fx 2  W u.x/ D 1g. The following corollary of Proposition 9.49 improves upon Lemma 9.9. Corollary 9.51. Let u be superharmonic in . Then Cp .fx 2  W u.x/ D 1g/ D 0. In particular, any polar set has capacity zero. Proof. By Proposition 9.49, u 2 ACCp ./, and by Lemma 9.9, u < 1 a.e. in . The conclusion now follows from Corollary 1.70. We also have the following characterization of polar sets. Theorem 9.52. Assume that  is bounded, Cp .X n / > 0 and E  . Then the following are equivalent: (a) E is polar; (b) Cp .E/ D 0; (c) there is a nonnegative superharmonic function u on  such that u D 1 in E; (d) there is a nonnegative superharmonic u 2 N 1;p ./ such that u D 1 in E; (e) there is a function u 2 N 1;p ./ such that u D 1 in E. Proof. (a) ) (b) This follows directly from Corollary 9.51. (b) ) (d) By Theorem 5.31, Cp is an outer capacity, i.e. there exists, for j D 1; 2; ::: , an open set Gj  E with Cp .Gj / < 2jp and thus aP nonnegative 'j 2 1;p j N .X / such that k'j kN 1;p .X/ < 2 and 'j  Gj . Let ' D j1D1 'j and let w be the lsc-regularized solution of the K';' -obstacle problem. Then w 2 N 1;p ./ is a nonnegative superharmonic function and w D 1 in E. (d) ) (c) ) (a) These implications are trivial. (d) ) (e) This implication is trivial. (e) ) (b) This follows from Proposition 1.30.

9.9 Integrability of superharmonic functions

241

The following integrability result is a consequence of the sharp weak Harnack inequality for superharmonic functions (Theorem 9.7). Note that it is possible to have ~.p  1/ < 1, in which case u need not be a distribution, see Remark 9.56. Theorem 9.53. Assume that X supports a .~p; p/-Poincaré inequality for some ~ > 1 and that 0 <  < ~.p  1/. Let u be superharmonic in . Then u 2 Lloc ./. Proof. Let B  51B b  be a ball. Since u is lower semicontinuous, it attains its minimum m on 51B. Hence u  m is a nonnegative superharmonic function on 51B. By Lemma 9.9, u is finite a.e. Thus, the sharp weak Harnack inequality for superharmonic functions (Theorem 9.7), applied to u  m, shows that u 2 L .B/. As B was arbitrary, u 2 Lloc ./. A similar integrability result holds for the p-weak upper gradient Gu as follows. Theorem 9.54. Assume that X supports a .~p; p/-Poincaré inequality for some ~ > 1 and that 0 < q < ~p.p  1/=.~.p  1/ C 1/. Let u be superharmonic in . Then Gu 2 Lqloc ./. Proof. Let B  2B b  and 0 < " < ~.p1/.pq/=q1. Let m D min2B u > 1 and uk D minfu  m; kg C 1, 0 < k 2 Z, which is a positive superminimizer. Then, using the Caccioppoli inequality for superminimizers (Proposition 8.8) with a suitable cut-off function 2 Lipc .2B/, we obtain Z Guq d B Z D lim guqk d k!1 B Z D lim guqk u.1C"/q=p u.1C"/q=p d k k k!1 B

Z

 lim

k!1

Z

B

gupk u.1C"/ d k

q=p Z B

1q=p u.1C"/q=.pq/ d k

q=p Z

.u  m C 1/p.1C"/ d

C 2B

1q=p .u  m C 1/.1C"/q=.pq/ d

B

< 1; where the last two integrals are finite by Theorem 9.53. Corollary 9.55. There exists q > p  1 such that if u is superharmonic in , then u; Gu 2 Lqloc ./. In particular, we have u; Gu 2 Lp1 loc ./. 1;p1 ./. Moreover, if p  2, then u 2 Nloc In fact the optimal range for q is the range in Theorem 9.54, the range for  in Theorem 9.53 being always larger.

242

9 Superharmonic functions

Proof. The first and second parts follow directly from Theorems 9.53 and 9.54 (together with Corollary 4.24). As for the last part, Lemma 9.45 and Proposition 2.41 show that Gu is a p-weak upper gradient, and thus also a .p  1/-weak upper gradient, by Proposition 2.45. It 1;p1 follows that u 2 Nloc ./ if p  1  1. Remark 9.56. In unweighted Rn , n  2, we have s D n in Lemma 3.3. Thus, for 1 < p < n we have p  D np=.n  p/ in Theorem 4.21 and q D p  in the Sobolev inequality (Theorem 5.51). We can thus have any positive  < p  .p  1/=p D n.p  1/=.n  p/ in the sharp weak Harnack inequalities in Theorems 8.34 and 9.7 as well as in Theorem 9.53, and any positive q<

p  .p  1/ n.p  1/ D  p .p  1/=p C 1 n1

in Theorem 9.54. On the other hand, the function u.x/ D jxj.pn/=.p1/ is superharmonic in Rn , see Example 7.47 in Heinonen–Kilpeläinen–Martio [171]. Observe that u 2 Lloc .Rn / if and only if  < n.p  1/=.n  p/, and gu D jruj 2 Lqloc .Rn / if and only if 1;p q < n.p  1/=.n  1/. In particular, u … Nloc .Rn /. Thus in this sense, the sharp weak Harnack inequalities in Theorems 8.34 and 9.7, as well as the exponents in Theorems 9.53 and 9.54 are all sharp. This also shows the sharpness of Theorem 4.21 and the Sobolev inequality (Theorem 5.51). (Note that in the proof of the sharp weak Harnack inequality for superminimizers (Theorem 8.34), the exponent  was depending solely on the exponent in the Sobolev inequality (Theorem 5.51).) We also see that if p  2n=.n C 1/, then u … L1loc .Rn / and hence u is not a 1;1 distribution (nor belongs to Wloc .Rn /). Moreover, if 2n=.n C 1/ < p  2  1=n, then 1 1 n u 2 Lloc .R /, but ru … Lloc .Rn /. Hence ru as a function is not the distributional gradient of u and u does not have a distributional gradient which is just a function. Similarly, if p  n (in unweighted Rn , n  2), then we can take p  and q arbitrarily large in Theorem 4.21 and the Sobolev inequality (Theorem 5.51), respectively. We can thus have any positive  in the sharp weak Harnack inequalities in Theorems 8.34 and 9.7 as well as in Theorem 9.53, and any positive p  .p  1/ Dp !1 p  .p  1/=p C 1

q < lim p

in Theorem 9.54. For p D n (in unweighted Rn , n  2), the superharmonic function u.x/ D  log jxj shows the sharpness of Theorem 9.54 in this case. Note further that in unweighted Rn with p > n, all superharmonic functions are continuous, by Proposition 9.49 and Corollary 5.39, and thus belong to Lqloc .Rn / for all q  1. This is exactly what is obtained from the sharp weak Harnack inequalities

9.9 Integrability of superharmonic functions

243

in Theorems 8.34 and 9.7, as well as in Theorem 9.53, apart from the case q D 1 1;p which is not studied therein. In this case, u 2 Nloc ./ by Corollary 9.6, and thus p gu 2 Lloc ./, whereas from Theorem 9.54 we can only deduce that gu 2 Lqloc ./ for all q < p. Finally look at X D R (unweighted) and any 1 < p < 1. Then p-harmonic functions are just affine functions x 7! ax C b, from which it follows that superharmonic functions are just concave functions (simultaneously for all p). It thus follows that if u is superharmonic in I D .a; b/, then gu D ju0 j is locally bounded in I , and hence gu 2 Lqloc .I / for all q. However, for a fixed p, Theorem 9.54 is not sharp in this case as it only shows that gu 2 Lqloc .I / for all q < p. That gu 2 Lqloc .I / for all q < 1 can however still be obtained from Theorem 9.54 by varying p. On graphs (see Appendix A.5), we similarly find that gu is locally bounded, but in this case we have a dependence on p and thus cannot vary p to obtain sharp results as we can on the real line. Superharmonic functions may be too large to be superminimizers, but by taking suitable powers, we can obtain superminimizers from superharmonic functions. Proposition 9.57. Let u > 0 be a superharmonic function in  and 0 < ˇ < 1  1=p. Then uˇ is a superminimizer in . Proof. Theorem 9.41 shows that v WD uˇ is superharmonic, and by Theorem 9.53, v 2 Lploc ./. By definition, Gu D limj !1 gminfu;j g and Gv D limj !1 gminfv;j g . The chain rule (Theorem 2.16), shows that gminfv;j g D ˇ minfu; j gˇ 1 gminfu;j g ;

j D 1; 2; ::: :

Let 0  2 Lipc ./ be arbitrary. By the Caccioppoli inequality for superminimizers (Proposition 8.8), with ˛ D p.1  ˇ/  1 > 0, we see that Z Z p Gv d D lim gp d j !1 D1 minfv;j g D1 Z D ˇ p lim gp minfu; j gp.ˇ 1/ p d j !1 D1 minfu;j g Z  C lim minfu; j gpˇ gp d j !1  Z DC v p gp d < 1; 

1;p ./. since v 2 Lploc ./. By varying , we see that Gv 2 Lploc ./, and hence v 2 Nloc Corollary 9.6 implies that v is a superminimizer.

For Rn , the functions u.x/ D jxj.pn/=.p1/ , 1 < p < n, and u.x/ D  log jxj, p D n, show the sharpness of this result, see Remark 9.56.

244

9 Superharmonic functions

On the other hand, the result is not sharp when p > n, as then all superharmonic functions are continuous, by Proposition 9.49, and thus are superminimizers by Corollary 9.6. In this case, any ˇ with 0 < ˇ  1 can be used. The situation is similar in spaces such that Cp .fxg/ > C > 0 for all x 2 X (or for which this holds locally), see Corollary 5.39. As a consequence of Proposition 9.57 we can obtain an estimate for the growth of superharmonic functions. Proposition 9.58. Let u be a superharmonic function in  and let 0 b  and 0 < " < p  1. Then there is a constant C > 0 such that Cp .fx 2 0 W u.x/ > kg/ 

C

for k > 0:

k p1"

By considering the p-harmonic function u.x/ D jxj.pn/=.p1/ in Rn , n > p, one sees that the proposition is false for " < 0. In fact, the result is true also for " D 0, which was shown by Korte [224], Remark 3.5, using quite a different technique. Proof. Let 00 be such that 0 b 00 b . As u is lower semicontinuous, u is bounded x 00 , and after possibly adding a positive constant to u (which will only from below on  make the set fx 2 0 W u.x/ > kg larger) we may assume that u > 0 in 00 . Let ˇ D 1  .1 C "/=p, so that ˇp D p  1  ". By Proposition 9.57, v WD uˇ is 1;p a superminimizer in 00 , and hence uˇ 2 Nloc .00 /. Let 2 Lipc .00 / be such that 0 ˇ 1;p

D 1 in  . Then u 2 N .X /. Therefore Cp .fx 2 0 W u.x/ > kg/ D Cp .fx 2 0 W uˇ .x/ > k ˇ g/  ˇ p  u   ˇ  1;p

 

k

N

 .X/

C C D p1" : ˇp k k

9.10 Lebesgue points for superharmonic functions Every locally bounded superharmonic function (or, which is the same, locally bounded lsc-regularized superminimizer) has Lebesgue points everywhere, as we showed in Proposition 8.24 and Corollary 8.26. For unbounded superharmonic functions this is not true, as they need not belong to L1loc , see Remark 9.56, but we can show that also unbounded superharmonic functions have Lq -Lebesgue points everywhere for certain q. Moreover, our range for q is sharp, at least for p  2. Theorem 9.59. Assume that X supports a .~p; p/-Poincaré inequality for some ~ > 1 and that 0 <  < ~.p  1/. Assume further that u is a superharmonic function in  and x0 2 . Then « ju  u.x0 /j d D 0; if u.x0 / < 1; (9.9) lim r!0 B.x ;r/ 0

9.10 Lebesgue points for superharmonic functions

and, for every q > 0, « lim

r!0 B.x ;r/ 0

uq d D lim

inf u D 1;

r!0 B.x0 ;r/

if u.x0 / D 1:

245

(9.10)

Proof. When u.x0 / < 1, the proof is fairly similar to the proof of Proposition 8.24. Indeed, let " > 0. As u is lower semicontinuous, there is r 0 such that u > u.x0 /  " in B.x0 ; r 0 /. Let v D u  .u.x0 /  "/ and r < r 0 =50. As v > 0 in B.x0 ; r/, we have by the sharp weak Harnack inequality for superharmonic functions (Theorem 9.7) that « « « ju  u.x0 /j d D jv  v.x0 /j d  v.x0 / C v  d B.x0 ;r/



B.x0 ;r/ v.x0 / C

B.x0 ;r/

C v.x0 /  " C C " : 





Letting r ! 0 and then " ! 0 completes the proof of (9.9). If u.x0 / D 1 and q > 0, then the lower semicontinuity of u shows that  1=q « q lim u d  lim inf u D 1: r!0 B.x ;r/ 0

r!0 B.x0 ;r/

Example 9.60. Let us demonstrate the sharpness of Theorem 9.59. Theorem 4.21 or Remark 9.56 show that Rn with 2  p < n supports a .p  ; p/-Poincaré inequality with p  D np=.n  p/. We can thus have any positive  <   WD p  .p  1/=p D n.p  1/=.n  p/ in Theorem 9.59. Let v.x/ D jxj.pn/=.p1/ , xj D .2j ; 0; ::: ; 0/, j D 1; 2; ::: , uN .x/ D

N X j D1

2j

v.x C xj / v.xj /

and

u.x/ D

1 X j D1

2j

v.x C xj / : v.xj /

By Theorem 3.1 in Crandall–Zhang [103], the function uN  0 is superharmonic in Rn n fx1 ; ::: ; xN g (here we use that p  2). As Cp .fx1 ; ::: ; xN g/ D 0, Theorem 12.3 below shows that uN is superharmonic in Rn . It then follows from Theorem 9.27 that u D limN !1 uN is superharmonic in Rn . As u.0/ D 1 and «  u d D 1; B.0;r/

we see that (9.9) fails for  D   . This example also demonstrates the sharpness of the following result (for p  2). Proposition 9.61. Assume that X supports a .~p; p/-Poincaré inequality for some ~ > 1 and that 0 <  < ~.p  1/. Assume further that u is a superharmonic function in . Then «  1=

u.x0 / D lim

r!0

u d B.x0 ;r/

for all x0 2 ;

246

9 Superharmonic functions

where we interpret u as juj sign u, and similarly for other powers. Let further q > 0 and uk WD minfu; kg. Then «

1=q

u.x0 / D lim lim

k!1 r!0

B.x0 ;r/

uqk

for all x0 2 :

d

Proof. The first part follows directly from (9.10) when u.x0 / D 1. When u.x0 / < 1 and   1, it follows from (9.9) by the triangle (Minkowski) inequality. For  < 1, we instead need to use the elementary inequality ja  b  j  21 ja  bj

(9.11)

(the extreme case is when b D a). In the second part, we use the triangle (Minkowski) inequality or (9.11) together with Proposition 8.25 to get that 1=q

« uk .x0 / D lim

r!0

B.x0 ;r/

uqk d

for all x0 2 ;

from which the last part follows directly.

9.11 Notes Superharmonic functions on metric spaces were first studied by Kinnunen–Martio [217]. Their definition used (iii0 ) from Proposition 9.26, which is equivalent to our definition, by Proposition 9.26. They obtained Theorems 9.5, 9.12, 9.27, 9.28 and 9.39, Propositions 9.4 and 9.18, a result similar to our weak Harnack inequality for superharmonic functions (Theorem 9.8) and a weaker version of Corollary 9.6. Corollary 9.6 in its present form, Proposition 9.19, Theorems 9.31–9.33 and Section 9.10 are from Björn– Björn–Parviainen [54]. The strong minimum principle (Theorem 9.13) seems to have been first pointed out explicitly in Björn–Björn [46]. Example 9.17 (for n D 3 and p D 2) is from A. Björn [37], while some of the characterizations in Section 9.4 come from A. Björn [35], which also contains Corollary 9.14. We have taken special care to cover even the case when Cp .X n / D 0, which is often not covered in the literature. As in Chapter 8, if X is L-quasiconvex, then the blow-up constant 50 in Theorems 9.7 and 9.8 can be replaced by 2L, under the modification that the integral in the left-hand side of (9.2) is taken over B, see the notes to Chapter 8. For Theorem 9.8 this was observed in Björn–Björn–Marola [53], while the same argument can also directly be applied to Theorem 9.7. Our proof of the fundamental convergence theorem (Theorem 9.33) is from Björn– Björn–Parviainen [54]. Earlier proofs, in the nonlinear Euclidean theory, e.g. in Heinonen–Kilpeläinen–Martio [171], use more advanced tools from the theory of balayage

9.11 Notes

247

(also called sweeping). Having the fundamental convergence theorem at disposal already from the beginning makes it possible to derive the theory of balayage in a different way than earlier. This was done in Björn–Björn–Mäkäläinen–Parviainen [52], to which we refer the reader for the present knowledge of balayage in the nonlinear potential theory on metric spaces. There is also a global version of Theorem 9.5 (i.e. with 0 D ) in [52]. The proof of Choquet’s topological lemma (Lemma 9.34) is from Heinonen–Kilpeläinen–Martio [171]. Harnack’s convergence theorem (Theorem 9.37) is from Shanmugalingam [321]. Open problem 9.40 is from Björn–Björn–Shanmugalingam [56]. A part of Theorem 9.44 for locally bounded u was obtained in [56], the proof given here being quite different, see also Theorems 10.42 and 10.43. The rest of Section 9.8 seems to be new here. Polar sets were studied extensively in Kinnunen–Shanmugalingam [221], where they in particular obtained Proposition 9.49 (the quasicontinuous part) and Corollary 9.51, and also the implication (b) ) (a) in Theorem 9.52 (for relatively closed E). Theorem 9.52 was obtained by Björn–Björn–Mäkäläinen–Parviainen [52], where also further characterizations of polar sets, in terms of balayage, can be found. Kinnunen and Martio obtained Proposition 9.49 (apart from the quasicontinuity), Theorems 9.53, 9.54, and (part of) the logarithmic Caccioppoli inequality for superharmonic functions (Proposition 9.46) in [219]. Therein they also obtained a version of the sharp weak Harnack inequality for superharmonic functions (Theorem 9.7) and pointed out that Corollary 9.48 holds. Maasalo [Kansanen] [252] showed that if u is a positive superharmonic function in   X and X is geodesic, then log u 2 BMO.I X /. In fact her method also shows that if X is not geodesic, then log u still belongs to BMO -loc .I X / with  D 2. Björn–Björn–Marola [53] showed that one may take  D 4L, if X is L-quasiconvex (when they obtained the corresponding result for quasisuperharmonic functions, see Appendix C). On Rn , Corollary 9.55 is the basis for saying that a superharmonic function is a 1;p very weak supersolution of the p-Laplace equation. Recall that u 2 Wloc ./ is a weak supersolution of the p-Laplace equation in  if Z jrujp2 ru  r' dx  0 (9.12)  1;p1 for all nonnegative ' 2 Lipc ./. If we merely require that u 2 Wloc ./, which is enough for the integral to be defined, u is said to be a very weak supersolution of the p-Laplace equation. In general metric spaces we do not have an equation and hence no such weak formulation as in (9.12) and thus we do not know how to define very weak superminimizers. For Cheeger p-harmonic functions, as well as in weighted Rn , one can of course proceed as above, and in this case Corollary 9.55 says that a superharmonic function is a very weak supersolution. Global integrability results for superharmonic functions in metric spaces have been obtained by Maasalo [Kansanen] [252]. She states her results for geodesic spaces,

248

9 Superharmonic functions

but remarks that they hold in general spaces, which also follows from Remark 8.21. Maasalo [Kansanen]–Zatorska-Goldstein [253] studied global integrability of p-weak upper gradients of superharmonic functions. Green functions are a topic which we have not considered here. They are sometimes called singular functions due to the fact that uniqueness is an open problem in many cases. Green functions on (more or less general) metric spaces have been studied by Holopainen [180], Holopainen–Shanmugalingam [181], Balogh–Holopainen– Tyson [24], Garofalo–Marola [138] and Danielli–Garofalo–Marola [105]. Singular measures, which are associated with Green functions, were also studied in Björn– Björn–Shanmugalingam [55]. Propositions 9.16, 9.25 and 9.57, Remark 9.20 and Corollary 9.55 may be new observations here.

Chapter 10

The Dirichlet problem for p-harmonic functions

Recall that in Chapters 7–14 we assume that 1 < p < 1 and that X is a complete doubling p-Poincaré space, with doubling constant C and dilation constant  in the p-Poincaré inequality. In this chapter, apart from in Section 10.9, we also make the additional assumption that  is bounded and such that Cp .X n / > 0. Given a function f W @ ! R, the Dirichlet (boundary value) problem asks for a p-harmonic function u taking f as boundary values. If f is not continuous, then it is rarely reasonable to require that u attains its boundary values f as limits (i.e. that lim3y!x u.y/ D f .x/ for all x 2 @), but even when f is continuous this is not possible in general. (The first examples showing this are Examples 13.3 and 13.4 of irregular boundary points.) We therefore need to consider boundary values in some weaker sense in the Dirichlet problem. For Newtonian boundary values, we have already solved the Dirichlet problem by Definition 8.31 as a special case of the obstacle problem. (These solutions are said to take the boundary values in Sobolev sense.) In particular, this means that we have solutions for Lipschitz boundary values. Next, we go on to more general functions, in particular we consider continuous boundary values which play a role in the boundary regularity of p-harmonic functions, see Chapter 11. A large part of this chapter is devoted to studying the Perron method for solving the Dirichlet problem, which provides us with an upper and a lower solution for arbitrary boundary data. A major question is when these two solutions coincide, which turns out to be the case at least for Newtonian and continuous boundary values. Moreover, for Newtonian functions we show that the solutions we obtained earlier (by Definition 8.31) coincide with Perron solutions. In fact, using uniform approximations it is possible to define solutions in the Sobolev sense also for continuous boundary values, and again these turn out to coincide with the corresponding Perron solutions. For continuous boundary values, yet another way of defining the Dirichlet problem is given in Theorem 10.24, where instead of requiring that the boundary values are taken as limits at all boundary points we only require this at q.e. boundary point. This provides a unique solution which coincides with the earlier two solutions. In Section 14.2 we give a fourth way (Wiener solutions) of solving the Dirichlet problem, again only for continuous boundary data, and also this time leading to the same solution as before. Thus, for continuous boundary values we have four equivalent, but fundamentally different, formulations of the Dirichlet problem.

250

10 The Dirichlet problem for p-harmonic functions

Another theme in this section is invariance results of the type that the (Perron) solutions Pf1 and Pf2 coincide if f1 D f2 q.e. on @ (under some assumptions on f1 ). Along the way we need and prove our first boundary regularity results, a topic which we further refine in Chapters 11 and 13, see also Section 14.1.

10.1 Continuous boundary values Definition 10.1. Given f 2 C.@/, define Hf W  ! R by Hf .x/ D

H '.x/;

sup

x 2 :

Lip.@/3'f

Here we abuse the notation, since if f 2 N 1;p ./, then Hf has already been defined by Definition 8.31. However, since continuous functions can be uniformly approximated by Lipschitz functions (by Proposition 1.11), the comparison principle (Lemma 8.32), together with the fact that H.f C a/ D Hf C a for a 2 R, shows that the two definitions of Hf coincide in this case. The comparison principle extends immediately to p-harmonic extensions of functions in C.@/ in the following way. Lemma 10.2 (Comparison principle). If f1 ; f2 2 C.@/ and f1  f2 q.e. on @, then Hf1  Hf2 in . Let us next show that Hf is indeed p-harmonic. Lemma 10.3. Let f 2 C.@/. Then Hf is p-harmonic in  and Hf .x/ D

inf Lip.@/3'f

H '.x/ D lim Hfj .x/; j !1

x 2 ;

for every sequence ffj gj1D1 of functions in Lip.@/ converging uniformly to f . Proof. Let fj 2 Lip.@/ be such that sup@ jf  fj j < 1=j , j D 1; 2; ::: . Then sup@ jfj 0  fj 00 j  2=j whenever j 0 ; j 00  j and the comparison principle implies that for all x 2 , Hfj 0 .x/ 

2 2  Hfj 00 .x/  Hfj 0 .x/ C ; j j

i.e. the sequence fHfj .x/gj1D1 is a Cauchy sequence. Hence, the limit h.x/ WD limj !1 Hfj .x/ exists, and by Theorem 9.36 it is a p-harmonic function in . Using the comparison principle again, it follows that h.x/ D lim H.fj  1=j /.x/  j !1



inf Lip.@/3'f

sup

H '.x/

Lip.@/3'f

H '.x/  lim H.fj C 1=j /.x/ D h.x/: j !1

10.2 The Kellogg property

251

10.2 The Kellogg property The p-harmonic extensions Hf from Definitions 8.31 and 10.1 attain their boundary values in the Sobolev sense. By Theorem 8.14, Hf is continuous in , but even if the boundary data f are continuous, it is in general not possible to have continuity up to the boundary, see Examples 13.3 and 13.4. Definition 10.4. A point x 2 @ is regular if lim Hf .y/ D f .x/

3y!x

for all f 2 C.@/. If x 2 @ is not regular, then it is irregular. The set  is regular if all x 2 @ are regular. Note that a regular set G is, by definition, a nonempty bounded open set with Cp .X n G/ > 0, since otherwise Hf is not defined. This is just one of many equivalent ways of defining regularity. The perhaps most correct and common way is to use Perron solutions as in Theorem 10.29 (c). See Theorems 10.29, 11.2 and 11.11 for various characterizations of regularity. It is not obvious that there exist regular boundary points nor that some particular boundary point is regular, but the following theorem shows that in fact, most boundary points are regular. See Chapters 11 and 13 and Section 14.1 for more on regular points and regular sets. Theorem 10.5 (The Kellogg property). If Ip denotes the set of all irregular points in @, then Cp .Ip / D 0. Moreover, Ip is an F set. The proof of Theorem 10.5 is based on the following lemma. Lemma 10.6. Let x 2 @ and B D B.x; r/. Let further f 2 Lip.@/ be such that 0  f  1 and f D 1 on B \ @. Then ´ Hf in ; uD 1 in B n  is a superminimizer in B. This is essentially just a special case of the pasting lemma for superminimizers (Lemma 7.13). x e.g. using one of the McShane extensions Proof. Extend f as a Lipschitz function to , x in Lemma 5.2, and as 1 to B n . As 1  f 2 N01;p .B \ I B/, by Proposition 2.39, we see that f 2 N 1;p .B/. (Alternatively one can use that the lower pointwise dilation lip f is bounded in B, together with Proposition 1.14.) Since Hf  f 2 N01;p ./  N 1;p .X /  N 1;p .B/, it follows that u 2 N 1;p .B/. As Hf  H1  1, by the comparison principle (Lemma 10.2), the conclusion follows from the pasting lemma for superminimizers (Lemma 7.13).

252

10 The Dirichlet problem for p-harmonic functions

Proof of Theorem 10.5. For each j D 1; 2; ::: , we can cover @ by a finite number of balls Bj;k D B.xj;k ; 1=j /, 1  k  Nj . Let 'j;k 2 Lipc .3Bj;k / with 0  'j;k  1 and 'j;k D 1 on 2Bj;k . Consider the sets ˚ Ij;k D x 2 Bxj;k \ @ W lim inf H 'j;k .y/ < 'j;k .x/ D 1 : 3y!x

Note that Ij;k contains only irregular points. Let further ´ H 'j;k in ; uj;k D 1 in 2Bj;k n ; which is a superminimizer in 2Bj;k by Lemma 10.6. As uj;k is continuous in , we   D H 'j;k in . By Theorem 8.22, uj;k D uj;k q.e. in 2Bj;k and hence have uj;k   1 D uj;k .x/ D uj;k .x/  lim inf uj;k .y/ D lim inf H 'j;k .y/ 3y!x

3y!x

for q.e. x 2 Bxj;k \ @. Thus Cp .Ij;k / D 0. Now consider a function ' 2 C.@/ and assume that we do not have lim H '.y/ D '.x/

3y!x

for some x 2 @. By considering ' if necessary, and adding a constant, we can assume that '  1 and that lim inf 3y!x H '.y/ < '.x/. Since ' is continuous we can find a ball Bj;k containing the point x so that M WD

inf

3Bj;k \@

' > lim inf H '.y/  1: 3y!x

Thus, 'j;k  '=M on @, and hence, by the comparison principle (Lemma 8.32), lim inf H 'j;k .y/ 

3y!x

1 lim inf H '.y/ < 1 D 'j;k .x/; M 3y!x

i.e. x 2 Ij;k . Thus Ip D

Nj 1 [ [

Ij;k ;

(10.1)

j D1 kD1

a countable union of sets of zero capacity, and hence itself of zero capacity. Finally, it is easy to see that Ij;k D

1 [

.Bxj;k \ @ \ fy 2  W H 'j;k .y/ < 1  1= lg/;

lD1

a countable union of compact sets. Together with the identity (10.1) this shows that Ip is an F set.

10.3 Perron solutions

253

10.3 Perron solutions By Corollary 7.17, solutions of obstacle problems can be characterized as the smallest superminimizer with the prescribed boundary values, which lies above the given obstacle. In this section, we use a similar idea to define solutions of the Dirichlet problem for arbitrary boundary data by means of superharmonic functions. Definition 10.7. Let V  X be a nonempty bounded open set with Cp .X n V / > 0. x let Uf .V / be the set of all superharmonic functions u Given a function f W @V ! R, on V which are bounded below and such that lim inf u.y/  f .x/

V 3y!x

for all x 2 @V:

(10.2)

Define the upper Perron solution of f by PxV f .x/ D

inf

u2Uf .V /

u.x/;

x 2 V:

Similarly, let Lf .V / be the set of all subharmonic functions u on V which are bounded above and such that lim sup u.y/  f .x/

for all x 2 @V;

V 3y!x

and define the lower Perron solution of f by PV f .x/ D

sup

u.x/;

x 2 V:

u2Lf .V /

If PxV f D PV f , then we let PV f WD PxV f . If moreover PV f is real-valued, then f is said to be resolutive. If V D , we usually drop V from the notation and write e.g. Pf . Let us first list some trivial but important consequences of the definition. Proposition 10.8. Let f and h be arbitrary functions on @ such that f  h on @. Then the following are true: x ; (a) Pf  Pf x D P .f /; (b) Pf x  Px h and Pf  P h; (c) Pf (d) If f and h are resolutive, then Pf  P h; (e) Pf D limk!1 P minff; kg.

254

10 The Dirichlet problem for p-harmonic functions

Proof. (a) This follows directly from the comparison principle in Theorem 9.39. (b) This follows from the fact that u 2 Uf if and only if u 2 Lf . (c) This follows from the trivial inclusions Uf  Uh and Lf  Lh . (d) This follows directly from (c). (e) Let u 2 Lf . As u is bounded from above, there is k such that u 2 Lminff;kg . Hence u  lim P minff; kg; k!1

where the limit exists by (c). Hence, again by (c), Pf D sup u  lim P minff; kg  Pf; u2Lf

k!1

which concludes the proof. In view of the duality given in (b), we will formulate most results just for upper Perron solutions. It is also natural to ask if (e) holds for upper Perron solutions, this is however an open problem. x D limk!1 Px minff; kg for all functions f ? Open problem 10.9. Is it true that Pf Our next result shows that Perron solutions are indeed p-harmonic, and thus reasonable candidates for solutions of the Dirichlet problem. See also Sections 10.4 and 10.5, where Perron solutions are related to Sobolev solutions (p-harmonic extensions) Hf and to the boundary data. x be given and G be a component of . Then Pf x is Theorem 10.10. Let f W @ ! R either p-harmonic in G or identically ˙1 in G. x  1 in some component G  , then Uf D ¿ and hence Remark 10.11. If Pf x x  1 in , or Pf x W  ! Œ1; 1/. Pf  1 in all of . Thus either Pf x When Pf  1 in some component G  , then the situation is different. The example  D B.0; 1/ [ B.3; 1/  C D R2 with f D 1@B.0;1/ shows that we can x  1 in one component B.0; 1/, while Pf x  0 is p-harmonic in the other have Pf component B.3; 1/. One can avoid this difference in behaviour at 1 and 1 by letting Uf consist of all hyperharmonic functions bounded from below and satisfying (10.2), which we however have refrained from here. (Recall that hyper- and hypoharmonic functions were defined after Definition 9.1.) If we would have done so, then we would have x D PxG f in every component G of . In our situation this is true under the had Pf x 6 1 in  (or in G). As we are primarily interested in additional assumption that Pf situations when the Perron solutions are p-harmonic, we prefer our definition. x  1. Assume therefore that Uf ¤ ¿. Proof of Theorem 10.10. If Uf D ¿, then Pf 00 Let  b G. By Lemma 4.49, we can find a connected set 0 such that 00 b 0 b .

10.4 Resolutivity of Newtonian functions

255

Let v 2 Uf be arbitrary and vm D minfv; mg, m D 1; 2; ::: . Then vm is su1;p perharmonic and vm 2 Nloc ./, by Corollary 9.6. The Poisson modification for Newtonian superharmonic functions (Theorem 9.44) applied to vm provides us with 0 0 0 such that vm D H0 vm in 0 and vm a new superharmonic function vm  vm in 0 0 . Let v D limm!1 vm , which is superharmonic in  by Theorem 9.27 (note that 0 1 fvm gmD1 is an increasing sequence of functions, and that v 0 is not identically 1 in any 0 component of  since v 0  v). The functions vm are p-harmonic in 0 , and hence x 0, by Corollary 9.38 so is v 0 . Therefore, v 0 is continuous in 0 . As v 0 D v in  n  x is upper x D inf v2U v 0 . It follows that the upper Perron solution Pf we have Pf f 0 semicontinuous in  . Recall that by Proposition 1.6, X is a separable metric space. Let Z D fz1 ; z2 ; ::: g be a countable dense subset of  and for each j D 1; 2; ::: , find superharmonic x .zj /: As the minimum of two functions uj;k 2 Uf so that limk!1 uj;k .zj / D Pf superharmonic functions is also superharmonic, the sequence uj D min ui;k i;kj

belongs to Uf , is pointwise decreasing and has the limit x .z/ lim uj .z/ D Pf

j !1

for all z 2 Z:

Let uj0 be the superharmonic functions obtained from uj by the construction at the x  u0 for all j , we get that beginning of this proof, and u D limj !1 uj0 . As Pf j x  u in . At the same time, we have for all z 2 Z, Pf x .z/ D lim uj .z/  lim uj0 .z/ D u.z/; Pf j !1

j !1

x on Z. i.e. u D Pf The functions uj0 are all p-harmonic in 0 . Applying Corollary 9.38 to fuj0 gj1D1 shows that u is p-harmonic or identically 1 in 0 . x  1 in 0 . Otherwise, u is continuous in 0 If u  1 in 0 , then also Pf x , we find that and using the upper semicontinuity of Pf x .x/  lim sup Pf x .z/ D lim sup u.z/ D u.x/ u.x/  Pf Z3z!x

for x 2 0 ;

Z3z!x

x D u is p-harmonic in 0 . By Propositions 9.18 and 9.21, Pf x is p-harmonic i.e. Pf in G.

10.4 Resolutivity of Newtonian functions As mentioned in the introduction to this chapter, a main question for Perron solutions is to decide when the upper and the lower solutions coincide, i.e. when the boundary data are resolutive.

256

10 The Dirichlet problem for p-harmonic functions

We are now ready to state our main resolutivity result. Our other resolutivity results are more or less direct consequences. x is quasicontinuous (with respect to ), x Theorem 10.12. Assume that f 2 N 1;p ./ 1;p which in particular holds if f 2 N .X /. Then f is resolutive and Pf D Hf . We postpone the proof of Theorem 10.12 until the end of this section. Instead, we first give some examples and comments. Here and throughout this section, quasicontinuity, Cp and q.e. are taken with respect x instead of X , and so are relatively open sets. Thus e.g. when saying that something to  holds q.e. we allow for a possibly slightly larger exceptional set than usual, see the discussion at the beginning of Section 2.5. In Theorem 10.12, it is important that we use Newtonian spaces with q.e.-equivalence classes. If we merely assume that f 2 Ny 1;p .X /, then it is easy to construct counterexamples. Example 10.13. Let  be such that .@/ D 0 and let f D @ 2 Ny 1;p .X /. Then Pf  1 and Hf  0. Example 10.14. Let  D B.0; 1/  C D R2 and p D 2. If E  @ is a nonmeasurable set with respect to the one-dimensional Lebesgue measure on @, or equivalently with respect to the harmonic measure, then the function E 2 Ny 1;p .X / is not resolutive by Brelot’s theorem (Theorem 10.45). Since Hf is independent of which (quasicontinuous) representative we choose from a given equivalence class, it also follows that Pf is independent of the choice of representative. x is quasicontinuous (with respect to ), x Theorem 10.15. Assume that f 2 N 1;p ./ 1;p which in particular holds if f 2 N .X /. Assume further that h D f q.e. on @. Then Pf D P h. In particular, h is resolutive. When requiring that f D h q.e., it is enough to consider the capacity with respect x (but it is not enough to consider it with respect to @). to  x by letting h D f in . Then h D f q.e. in  x and hence Proof. Extend h to  1;p x x h 2 N ./. We need to show that also h is quasicontinuous (with respect to ). x (Note that since  need not support any Poincaré inequality, the quasicontinuity of h does not follow from the general theory of Newtonian spaces.) x Indeed, let " > 0. As f is quasicontinuous, we can find a relatively open set U   1 such that Cp .U / < 2 " and f jnU is continuous. By Proposition 5.27 we can also find x x a relatively open set V   such that V  fx 2 @ W f .x/ ¤ h.x/g and Cp .V / < 12 ". Letting W D U [V it follows that Cp .W / < " and that hjnW D f jnW is continuous. x x Since h D f in  we have H h D Hf . Theorem 10.12 applied to both f and h shows that h is resolutive and Pf D Hf D H h D P h:

10.4 Resolutivity of Newtonian functions

257

We also have the following uniqueness result. Note however that, unlike for continuous boundary data in Theorem 10.24 below, here we do not know for general Newtonian functions whether lim3y!x Pf .y/ D f .x/ for q.e. x 2 @. x is quasicontinuous (with respect to ), x Corollary 10.16. Assume that f 2 N 1;p ./ 1;p which in particular holds if f 2 N .X /. Assume also that f is bounded, that u is a bounded p-harmonic function in  and that there is a set E  @ with Cp .E/ D 0 such that lim u.y/ D f .x/ for all x 2 @ n E: 3y!x

Then u D Pf . As before, when requiring that Cp .E/ D 0, it is enough to consider Cp with respect x to  (but it is not enough to consider it with respect to @). Note that if the word bounded is omitted, the result becomes false even for p D 2; consider e.g. the Poison kernel 1  jzj2 j1  zj2 in the unit disc B.0; 1/  C D R2 with a pole at 1, which is zero on @B.0; 1/ n f1g. Proof. By adding a sufficiently large constant to both f and u, and then rescaling them simultaneously we may assume without loss of generality that 0  u  1 and 0  f  1. Hence u 2 Uf E and u 2 Lf CE . Therefore, by Theorem 10.15, we see that u  Px .f  E / D Pf D P .f C E /  u. In order to prove Theorem 10.12 we will need the following results. Lemma 10.17. Let  be an arbitrary nonempty open subset of X . Let fUk g1 kD1 x such that Cp .Uk / < 2kp . be a decreasing sequence of relatively open sets in  Then there exists a decreasing sequence of nonnegative functions f j gj1D1 such that k j kN 1;p ./ < 2j and j  k  j in Uk whenever k > j . In particular, j D 1 T1 x on kD1 Uk . 1;p x Proof. Since Cp .Uk / < 2kp there is a nonnegative function fk 2 NP ./ such 1 k that fk D 1 in Uk and kfk kN 1;p ./ , k D 1; 2; ::: . Let j D kDj C1 fk , x < 2 j j D 1; 2; ::: . Then k j kN 1;p ./ and j  k  j in Uk , whenever k > j . x <2

Proposition 10.18. Let ffj gj1D1 and f j gj1D1 be q.e. decreasing sequences such that K j ;fj ¤ ¿ and both fj ! f and j ! in N 1;p ./, as j ! 1. Let uj be a solution of the K j ;fj -obstacle problem, j D 1; 2; ::: . Then there exists u 2 N 1;p ./ such that fuj gj1D1 decreases q.e. in  to u and u is a solution of the K ;f -obstacle problem.

258

10 The Dirichlet problem for p-harmonic functions

Here we assume that the K j ;fj -obstacle problem is soluble for all j D 1; 2; ::: . The solubility of the K ;f -obstacle problem then follows from the proposition. Observe that it does not follow that uj ! u everywhere in , as the following example shows, see also Example 9.35. Example 10.19. Consider the functions uj .x/ D minf1=j jxj; 1g, j D 1; 2; ::: , which are superharmonic in  D B.0; 1/  R3 with p D 2, see Remark 9.56. Then uj ! 0 in L2 ./ and Z

Z 

gu2j dx D 4

1 1=j



1 jr 2

2

Z

1

r 2 dr D 4 1=j



dr 1 1 D 4  2 2 2 j r j j



! 0;

as j ! 1. Hence, uj ! 0 in N 1;p ./. Moreover, uj D uj is the continuous solution of the Kuj ;uj -obstacle problem. Then u WD infj uj D f0g , which is not lower semicontinuous. However the lsc-regularization u  0 is the continuous solution of the K0;0 -obstacle problem. For p-harmonic extensions we have a similar result, where we do obtain convergence everywhere. Corollary 10.20. Let ffj gj1D1 be a q.e. decreasing sequence such that fj ! f in N 1;p ./, as j ! 1. Then Hfj decreases to Hf locally uniformly in . Proof. Observe first that by the comparison principle (Lemma 8.32), Hfj  Hfj C1  Hf in , j D 1; 2 ::: . Let j D D Hf , j D 1; 2; ::: . It then follows that Hfj and Hf are the continuous solutions of the Kfj ; j - and Kf; -obstacle problems, j D 1; 2; ::: . Proposition 10.18 implies that Hfj ! Hf q.e. in , as j ! 1. As Hf is p-harmonic and thus locally bounded from below, it follows from Harnack’s convergence theorem (Theorem 9.37) that Hfj ! Hf locally uniformly in , as j ! 1. In fact, Kinnunen–Marola–Martio [214], Theorem 3, have shown that if ffj gj1D1 is a bounded sequence in N 1;p ./ and fj ! f q.e. in , as j ! 1, then Hfj ! Hf locally uniformly in , as j ! 1. (Note that f 2 N 1;p ./ by Corollary 6.3.) It follows from this that if fj ! f in N 1;p ./, as j ! 1, then Hfj ! Hf locally uniformly in , as j ! 1. See also Shanmugalingam [321] for an earlier result in this direction. This motivates the following stronger question. Open problem 10.21. Assume that f; h 2 N 1;p .X /. Is it then true that there is a constant C.x/, independent of f and h, such that jHf .x/  H h.x/j  C.x/kf  hkN 1;p .X/ ;

x 2 ‹

259

10.4 Resolutivity of Newtonian functions

Proof of Proposition 10.18. Without loss of generality we may assume that all uj are lsc-regularized, j D 1; 2; ::: . As j  j C1 and fj  fj C1 q.e. in , it follows from the comparison principle (Lemma 8.30) that uj  uj C1 in . Hence fuj gj1D1 is a decreasing sequence and converges to a function u on . Let wj D uj  fj and w D u  f , all extended by zero outside of . We have wj 2 N01;p ./  N 1;p .X /, j D 1; 2; ::: . Using the Poincaré inequality for N01;p (Corollary 5.54), and the fact that uj is a solution of the K j ;fj -obstacle problem, it follows that kwj kN 1;p .X/  kwj kLp ./ C kgwj kLp ./  C kgwj kLp ./  C.kguj kLp ./ C kgfj kLp ./ /  C kfj kN 1;p ./  C and hence kuj kN 1;p ./  kwj kN 1;p ./ C kfj kN 1;p ./  C kfj kN 1;p ./  C: By Lemma 6.2, a subsequence (also denoted by uj ) converges weakly in Lp ./ to u 2 Lp ./ and guj converge weakly, as j ! 1, to a p-weak upper gradient g 2 Lp ./ of u. In particular, u 2 N 1;p ./ and kgu kLp ./  kgkLp ./  lim inf kguj kLp ./ :

(10.3)

j !1

Since wj ! w q.e. in X , as j ! 1, Corollary 6.3 shows that w 2 N 1;p .X / and as w D 0 q.e. in X n , we get that u  f 2 N01;p ./. Since uj  j  q.e. in , we have u  q.e. in , and hence u 2 K ;f . Let v be the lsc-regularized solution of the K ;f -obstacle problem and 'j D maxfv C fj  f; j g. Then 'j  fj D maxfv  f;

 fj g  v  f 2 N01;p ./ q.e.

j

and  fj /C g 2 N01;p ./; R by Proposition 7.4. Lemma 2.37 then yields 'j 2 K j ;fj , and hence  gupj d  R p  g'j d. Letting Ej D fx 2  W j .x/ > .v C fj  f /.x/g, we have 'j  fj D maxfv  f;

Z 

j

 fj g  maxfv  f; .

j

1=p g'pj v d Z D Z

nEj

 

gfpj f d C

gfpj f

1=p d

1=p

Z Ej

g pj 

C

v d

Z

C

1=p g



p j

d

Z C

1=p g

Ej

p v

d

:

260

10 The Dirichlet problem for p-harmonic functions

The first two integrals on the right-hand side tend to zero since fj ! f and in N 1;p ./, as j ! 1. As g v D 0 a.e. on the set where D v, we have Z Z g p v d  g p v d; Ej

j

!

Aj

T  where Aj D fx 2  W j .x/ > v.x/ > .x/g. Since  j1D1 Aj D 0, we get that .Aj / ! 0, as j ! 1. Dominated convergence then implies that Z g p v d ! 0; as j ! 1: Aj

Hence g'j ! gv in Lp ./, as j ! 1, and therefore, using (10.3), Z Z Z Z p p p gu d  lim inf guj d  lim inf g'j d D gvp d: 

j !1

j !1



Thus, u is also a solution of the K

;f





-obstacle problem.

Proof of Theorem 10.12. To begin with, assume that f  0. We first show that Hf x Let h D f  Hf , extended by zero outside of . Then is quasicontinuous on . x Since Hf is the h 2 N01;p ./  N 1;p .X / is quasicontinuous on X and thus on . x sum of two quasicontinuous functions, it is also quasicontinuous on . 1 x We can now find a decreasing sequence fUk gkD1 of relatively open subsets of  kp such that Cp .Uk / < 2 and Hf jnU x k is continuous. (Here Cp is taken with respect x to .) Consider the decreasing sequence of nonnegative functions f j gj1D1 given by Lemma 10.17. Let fj D Hf C j and let 'j be the lsc-regularized solution of the Kfj ;fj -obstacle problem. If m is a positive integer, then by Lemma 10.17, fj 

j

m

on UmCj :

(10.4)

Let " > 0 and x 2 @ be arbitrary. If x … UmCj , then by the continuity of Hf jnU x mCj x of x such that there is a relative neighbourhood Vx   fj .y/  Hf .y/  Hf .x/  " D f .x/  ";

if y 2 Vx n UmCj :

(10.5)

Combining (10.4) and (10.5) we see that fj  minff .x/  "; mg

in Vx :

On the other hand, if x 2 UmCj , then setting Vx D UmCj , we see by inequality (10.4) that in the relative neighbourhood Vx of x, fj  m  minff .x/  "; mg:

10.5 Resolutivity of continuous functions

261

Since 'j  fj q.e. and 'j is lsc-regularized, it follows that 'j .y/  minff .x/  "; mg for y 2 Vx \ : Hence lim inf 'j .y/  minff .x/  "; mg:

3y!x

Letting " ! 0 and m ! 1, we see that lim inf 'j .y/  f .x/

3y!x

for all x 2 @:

As 'j is superharmonic and nonnegative, it follows that 'j 2 Uf , and hence that x . 'j  Pf Since Hf clearly is a solution of the KHf;Hf -obstacle problem, we see by Propox  Hf q.e. in , provided sition 10.18 that f'j gj1D1 decreases q.e. to Hf . Hence Pf that f is bounded from below. Finally, let f 2 N 1;p ./ be arbitrary. Then, by Corollary 10.20 and the above argument, x  lim Px maxff; mg  lim H maxff; mg D Hf Pf m!1

m!1

q.e. in :

x and Hf are continuous, we have Pf x  Hf everywhere in . It then Since both Pf follows that x  Hf D H.f /  Px .f / D Pf  Pf; x Pf x . and hence that Hf D Pf D Pf

10.5 Resolutivity of continuous functions The resolutivity results from the previous section can now be extended to continuous functions. We thus obtain another way of solving the Dirichlet problem with continuous boundary data, cf. Definition 10.1, which has been traditionally one of the main interests in the classical theory of harmonic and p-harmonic functions. For another equivalent way of solving the Dirichlet problem with continuous boundary data, see Section 14.2. Yet another equivalent possibility is to use the uniqueness in Theorem 10.24 below to define a solution. Theorem 10.22. Let f 2 C.@/ and h be a function which is zero q.e. on @. Then f C h is resolutive, and P .f C h/ D Hf D Pf: x the When requiring that h D 0 q.e. it is enough to consider Cp with respect to , same is true throughout this section.

262

10 The Dirichlet problem for p-harmonic functions

Proof. For each j D 1; 2; ::: , there is a Lipschitz function fj 2 Lipc .X /  N 1;p .X / such that f  1=j  fj  f C 1=j on @. Using the comparison principle (Lemma 10.2) we see that Hf  1=j  Hfj  Hf C 1=j: Hence Hfj ! Hf uniformly, as j ! 1. It follows directly from Definition 10.7 that x  1=j  Pf x j  Pf x C 1=j; Pf x j ! Pf x uniformly, as j ! 1. The uniform convergences of Pfj , and thus Pf x P .fj C h/ and P .fj C h/ are proved in the same way. As fj 2 N 1;p .X /, we have by Theorem 10.15 that P .fj C h/ D H.fj C h/ D Hfj D Pfj ;

j D 1; 2; ::: :

Letting j ! 1 completes the proof. A direct consequence is the following uniqueness result. Corollary 10.23. Let f 2 C.@/. Assume that u is a bounded p-harmonic function in  and that there is a set E  @ with Cp .E/ D 0 such that lim u.y/ D f .x/ for all x 2 @ n E:

3y!x

Then u D Pf . As in Corollary 10.16 the word bounded is essential. The proof of Corollary 10.23 is exactly the same as the proof of Corollary 10.16, with Theorem 10.22 playing the role of Theorem 10.15. A further consequence of Corollary 10.23 and the Kellogg property (Theorem 10.5) is the following uniqueness result. It shows that Perron solutions are the only reasonable candidates for solving the Dirichlet problem with continuous boundary data. Theorem 10.24. Let f 2 C.@/. Then there exists a unique bounded p-harmonic function U in  such that lim U.y/ D f .x/ for q.e. x 2 @;

3y!x

(10.6)

moreover U D Pf . x or X , giving slightly different Here q.e. may be taken either with respect to  results. Proof. By the Kellogg property (Theorem 10.5) and Theorem 10.22, we see that the function U D Pf D Hf satisfies (10.6). On the other hand, if U satisfies (10.6), then Corollary 10.23 shows that U D Pf .

10.6 Some consequences of resolutivity

263

We can actually obtain the following stability results, the proofs being similar to the proof of Theorem 10.22. Theorem 10.25. Let fj , j D 1; 2; ::: , be resolutive functions and assume that fj ! f uniformly on @. Then f is resolutive and Pfj ! Pf uniformly in . x be quasicontinuous (with respect to ). x Assume Theorem 10.26. Let fj 2 N 1;p ./ that fj ! f uniformly on @, as j ! 1. Let also h be a function which is zero q.e. on @. Then f C h is resolutive and Pf D P .f C h/.

10.6 Some consequences of resolutivity We shall now use Perron solutions and our results on resolutivity to obtain some consequences for superharmonic functions. See also Section 10.9 for yet further implications. The following pasting lemma generalizes the pasting lemma for superminimizers (Lemma 7.13) and will be needed when proving the Carleman principle for the (lower) p-harmonic measure in Theorem 10.41. Lemma 10.27 (Pasting lemma for superharmonic functions). Assume that 1  2 are arbitrary nonempty open sets and that u1 and u2 are superharmonic in 1 and 2 , respectively. Let ´ in 2 n 1 ; u2 uD min¹u1 ; u2 º in 1 : If u is lower semicontinuous, then it is superharmonic in 2 . Proof. Let G b 2 be open and such that Cp .X n G/ > 0. Let further v 2 Lip.@G/ satisfy v  u on @G, and w D HG v (with w D v on @G). It is enough to show that w  u in G. x As u2 is superharmonic and v  u2 on @G we directly obtain that w  u2 in G. x the lower semicontinuity of u yields For x 2 @1 \ G, w.x/  u2 .x/ D u.x/  lim inf u.y/  lim inf u1 .y/: 1 3y!x

1 3y!x

On the other hand, for x 2 1 \ @G, we have that w.x/ D v.x/  u.x/  u1 .x/  lim inf u1 .y/: 1 3y!x

Hence u1 2 Uw .1 \ G/, and thus, by Theorem 10.12, we get that u1  P1 \G w D H1 \G w D w Therefore w  u in G, and we are done.

in 1 \ G:

264

10 The Dirichlet problem for p-harmonic functions

Using Theorem 10.15 we can also give an alternative comparison principle between sub- and superharmonic functions, cf. Theorem 9.39. This result answers Open problem 9.40 in a special case. Proposition 10.28. Assume that u is superharmonic and bounded from below in , that x is quasicontinuous v is subharmonic and bounded from above in , that f 2 N 1;p ./ 1;p x (with respect to ), which in particular holds if f 2 N .X /, and that lim sup v.y/  f .x/  lim inf u.y/ for q.e. x 2 @: 3y!x

3y!x

(10.7)

Then v  u in . x When requiring (10.7) to hold q.e. it is enough to consider Cp with respect to . Proof. Let for x 2 @, ˚ '.x/ D max f .x/; lim sup v.y/

and

˚ .x/ D min f .x/; lim inf u.y/ : 3y!x

3y!x

Then ' D f D yields

q.e. on @. Moreover, v 2 L' and u 2 U . Theorem 10.15 then v  P ' D Pf D P

u

in :

We finish this section by some implications of regularity which will be needed in Section 10.7. To fit with the topic in Chapter 11 we formulate them as a number of characterizations of boundary regularity. In Theorems 11.2 and 11.11 we provide many more characterizations. Theorem 10.29. Let x0 2 @ and d.x/ WD d.x; x0 /. Then the following are equivalent: (a) The point x0 is a regular boundary point. (b) It is true that lim

3y!x0

P d.y/ D 0:

(c) It is true that lim

3y!x0

Pf .y/ D f .x0 /

for all continuous f W @ ! R. (d) It is true that lim

3y!x0

x .y/ D f .x0 / Pf

for all bounded f W @ ! R which are continuous at x0 .

10.7 Resolutivity of semicontinuous functions

(e) It is true that

265

x .y/  f .x0 / lim sup Pf 3y!x0

for all functions f W @ ! R which are bounded from above on @ and upper semicontinuous at x0 . Recall that continuous functions are real-valued, whereas semicontinuous functions may take the values ˙1. Proof. (a) , (c) This follows directly from Theorem 10.22. (c) ) (b) This is trivial. (b) ) (e) Let A > f .x0 / be real and M D sup@ .f  A/C . Let further r > 0 be such that f .x/ < A for x 2 B.x0 ; r/ \ @. Then f  A C M d=r on @. It follows that x .y/  A C M lim P d.y/ D A: lim sup Pf r 3y!x0 3y!x0 x .y/  f .x0 /. Letting A ! f .x0 / gives lim sup3y!x0 Pf (e) ) (d) Applying (e) to f yields x .y/  lim inf Pf .y/ D  lim sup Px .f /.y/  .f .x0 // D f .x0 /: lim inf Pf

3y!x0

3y!x0

3y!x0

Together with (e) this gives the desired conclusion. (d) ) (c) This follows directly, as continuous functions are resolutive by Theorem 10.22.

10.7 Resolutivity of semicontinuous functions So far, we have studied resolutivity of Newtonian and continuous functions. Our next step is to look at semicontinuous functions. In this section we formulate a number of propositions for upper semicontinuous functions. There are immediate analogues for lower semicontinuous functions. In particular, we will show that bounded semicontinuous functions on regular sets are resolutive. Proposition 10.30. Let f be an upper semicontinuous function on @ bounded from above and let h be a nonnegative function which is zero q.e. on @. Then x D Px .f C h/ D Pf

inf Lip.@/3'f

P' D

inf

H ':

(10.8)

Lip.@/3'f

x instead As in Sections 10.4 and 10.5 we can take Cp and q.e. with respect to  of X , throughout this section. Recall that a family F of functions is downwards directed if for every pair of functions u; v 2 F there exists a function w 2 F such that w  minfu; vg.

266

10 The Dirichlet problem for p-harmonic functions

Lemma 10.31. Assume that F is a downwards directed family of upper semicontinuous functions ' W @ ! Œ1; 1/. Let f D inf F . Then x D inf Px ': Pf '2F

x  h. Conversely, Proof. Let h D inf '2F Px '. As f  ' for ' 2 F we get that Pf fix " > 0 and let u 2 Uf . Since the functions ' 2 F are upper semicontinuous and u is lower semicontinuous, the sets ˚ E' D x 2 @ W lim inf u.y/ C " > '.x/ ; ' 2 F ; 3y!x

form an open covering of @. Since @ is compact, there exists a finite subcovering fE'j gjND1 . As f is downwards directed, we can find a function ' 2 F such that '  minf'1 ; ::: ; 'N g and hence lim inf u.y/ C " > '.x/

3y!x

for all x 2 @:

Thus, u C " 2 U' and u C "  Px '  h. Taking infimum over all u 2 Uf implies that x C "  h: Pf Letting " ! 0 concludes the proof. Proof of Proposition 10.30. If F D f' 2 Lip.@/ W '  f g, then F is downwards directed. Proposition 1.12 and the upper semicontinuity of f imply that f D inf F . x D inf '2F P '. Using Theorem 10.22, we find Hence by Lemma 10.31, we have Pf that x  Px .f C h/  inf Px .' C h/ D inf P ' D Pf: x Pf '2F

'2F

The last equality in (10.8) follows directly from Theorem 10.22. Proposition 10.32. Let f be an upper semicontinuous function on @ bounded from above and h be a nonnegative function which is zero q.e. on @ and such that f .x/ C h.x/  sup f

for all irregular x 2 @:

@

Assume further that one of the following conditions is satisfied: (a) f is bounded; x is p-harmonic in ; (b) Pf x  1 in ; (c) Pf (d)  is connected. x . Then f C h is resolutive and P .f C h/ D Pf

10.7 Resolutivity of semicontinuous functions

267

Note that it is necessary to have some condition on f as in (a)–(d). In fact, if both (b) x  1 in some component G (but not in all) of , and in view of and (c) fail, then Pf Proposition 10.30, the same holds for Px .f Ch/. Hence P .f Ch/  Px .f Ch/  1 in G and it follows that P .f C h/  1 in , see Remark 10.11 (applied to f ). Thus, P .f C h/ ¤ Px .f C h/. This anomaly would not have taken place, if we had defined the lower Perron solutions using hypoharmonic functions. Proof. Observe first that (a) ) (b) by Theorem 10.10. Also, if (d) holds, then either (b) or (c) holds, again by Theorem 10.10. Thus we only need to consider the two cases (b) and (c). Assume first that (b) holds. Then x .y/  sup f  f .x/ C h.x/ lim sup Pf 3y!x

for irregular x 2 @:

@

Moreover, by Theorem 10.29, x .y/  f .x/  f .x/ C h.x/ lim sup Pf

for regular x 2 @:

@3y!x

x is p-harmonic and bounded from above, Pf x 2 Lf Ch , and thus As Pf x D Px .f C h/  P .f C h/; P .f C h/  Pf by Proposition 10.30. Assume finally that (c) holds. Then, by Proposition 10.30, we have that x D 1: 1  P .f C h/  Px .f C h/ D Pf Corollary 10.33. Assume that  is regular and let f be an upper semicontinuous function on @ bounded from above. If either  is connected, or f is bounded (or x is real-valued in ), then f is resolutive. Pf Again, had we defined Perron solutions using hyper- and hypoharmonic functions, then all upper semicontinuous functions bounded from above would have been resolutive, assuming that  is regular. We also have the following consequence of Proposition 10.32 for characteristic functions, cf. Corollary 10.39. Corollary 10.34. Let K  @ be compact, and Ej  @ be such that Cp .Ej / D 0, j D 1; 2. Assume that E1 contains all irregular points of . Then K[E1 is resolutive and P K[E1 D Px K D Px K[E2 : Similarly, if G  @ is a relatively open set, then GnE1 is resolutive and P GnE1 D P G D P GnE2 :

268

10 The Dirichlet problem for p-harmonic functions

In particular, if K contains all irregular boundary points and G only contains regular boundary points, then both K and G are resolutive and P K D P K[E

and P G D P GnE

for all E  @ with Cp .E/ D 0. If  is regular, then this holds for all compact K and relatively open G.

10.8 The p-harmonic measure Harmonic measure is an important tool in linear potential theory. In particular, it can be used to solve the Dirichlet problem, cf. Section 10.10. There are several possible generalizations to the nonlinear theory depending on which properties you want to preserve. However, due to the nonlinearity, it is not possible to preserve both the p-harmonicity and the measure additivity. Here we have chosen to preserve the p-harmonicity. Hence, despite the name, the p-harmonic measure is in general not a measure. Having said this, let us just point out that in the literature there exist other generalizations of the harmonic measure under the name p-harmonic measure, see e.g. Bennewitz–Lewis [29]. Definition 10.35. Let V  X be a nonempty bounded open set with Cp .X n V / > 0. We define the upper and lower p-harmonic measures of E  @V as !.EI x V / D PxV E

and

!.EI V / D PV E ;

respectively. If !.EI x V / D !.EI V / we also denote the common value by !.EI V /. x V /.x/, ! x .EI V / D !.EI V /.x/ and For x 2 V we further let ! xx .EI V / D !.EI !x .EI V / D !.EI V /.x/. If V D , we usually drop V from the notation and write e.g. !.E/. For p-harmonic measures Proposition 10.8 takes the following form. Proposition 10.36. If E  E 0  @, then x (a) !.E/  !.E/; (b) !.E/ x  !.E x 0 / and !.E/  !.E 0 /; (c) !.E/ x D 1  !.@ n E/. The following result is a special case of Proposition 10.30. Proposition 10.37. Let K  @ be compact. Then !.K/ x D

inf Lip.@/3'K

P ':

10.8 The p-harmonic measure

269

Similarly, if G  @ is relatively open, then !.G/ D

P ':

sup Lip.@/3'G

The following regularity result is a more or less immediate corollary. Proposition 10.38. Let K  @ be compact. Then !.K/ x D

inf

GK G relatively open

!.G/:

Similarly, if G  @ is relatively open, then !.G/ D

sup

!.K/: x

KG K compact

Proof. Assume first that G  K is relatively open. Then we can find a Lipschitz function f such that K  f  G . As f is resolutive it follows that !.K/ x  Pf  !.G/ and one inequality is established. For the converse inequality, fix x 2  and " > 0. By Proposition 10.37, we can find ' 2 Lip.@/ such that '  K and P '.x/  ! xx .K/ C ". We can also find G  K relatively open so that '  1  " in G. Hence ! x .G/ 

P '.x/ ! xx .K/ C "  : 1" 1"

Letting " ! 0 gives the other inequality. The second part follows by duality. We can now reformulate Corollary 10.34 as follows. Corollary 10.39. Let K  @ be compact, and Ej  @ be such that Cp .Ej / D 0, j D 1; 2. Assume that E1 contains all irregular points of . Then x D !.K x [ E2 /: !.K [ E1 / D !.K/ Similarly, if G  @ is a relatively open set, then !.G n E1 / D !.G/ D !.G n E2 /: In particular, if K contains all irregular boundary points and G only contains regular boundary points, then both !.K/ and !.G/ are defined and !.K/ D !.K [ E/ and !.G/ D !.G n E/ for all E  @ with Cp .E/ D 0. If  is regular, then this holds for all compact K and relatively open G.

270

10 The Dirichlet problem for p-harmonic functions

One of the important properties of harmonic measures is that they are mutually absolutely continuous when taken with respect to different points in the same component. This generalizes also to p-harmonic measures and takes the following form. Proposition 10.40. Assume that x and y belong to the same component of . Then there is a constant C > 0 so that for all E  @, 1 xx .E/  C ! xy .E/: ! xy .E/  ! C In particular, xy .E/ D 0: ! xx .E/ D 0 if and only if ! The corresponding statements for the lower p-harmonic measure are also true. Proof. This follows directly from the Harnack inequality (Corollary 8.19) and the fact that !.E/ x is p-harmonic as a function on . Let us end this section by showing Carleman’s principle. Proposition 10.41 (Carleman’s principle). If 1  2 are bounded and such that Cp .X n 2 / > 0, and E  @1 \ @2 , then !.EI x 1 /  !.EI x 2 / and !.EI 1 /  !.EI 2 / in 1 : Proof. Let u 2 UE .2 /. Then u is superharmonic in 1 , by Corollary 9.10. As u  0, this implies that u 2 UE .1 /. Thus, !.EI x 1 / D

inf

u2UE .1 /

u

inf

u2UE .2 /

u D !.EI x 2 / in 1 :

The Carleman principle for the lower p-harmonic measure is a little harder to deduce. Let u 2 LE .1 /. Then also uC 2 LE .1 /, so we can assume that u  0. Let ´ u in 1 ; vD 0 in 2 n 1 : Since E \ .2 \ @1 / D ¿, we have lim sup v.y/ D lim sup u.y/  E .x/ D 0 D v.x/

1 3y!x

1 3y!x

for x 2 2 \ @1 ;

showing that v is upper semicontinuous. Hence by the pasting lemma for superharmonic functions (Lemma 10.27) (applied to v) v is subharmonic in 2 . As ˚ lim sup v.y/  max 0; lim sup v.y/  E .x/ for x 2 @2 \ @1 ; 2 3y!x

1 3y!x

and lim sup v.y/ D 0 D E .x/

2 3y!x

for x 2 @2 n @1 ;

we see that v 2 LE .2 /. Thus !.EI 1 / D

sup u2LE .1 /

uD

sup u2LE .1 /

v  !.EI 2 / in 1 :

10.9 Poisson modification

271

10.9 Poisson modification In this section we combine several results on Perron solutions to obtain Poisson modifications of arbitrary superharmonic functions, cf. Theorem 9.44, where it was only proved for Newtonian superharmonic functions. We have thus in this chapter used superharmonic functions to define Perron solutions, which we in turn use to construct new superharmonic functions. Theorem 10.42 (Poisson modification for superharmonic functions). Let   X be an arbitrary nonempty open set. Assume that u is superharmonic in  and let G b  be open and such that Cp .X n G/ > 0. Let further ´ u in  n G; vD PG u in G: Then

8 x ˆ x 2  n G; ˆ
(10.9)

G3y!x

Moreover, v  is superharmonic in  and p-harmonic in G, and v   v  u in . Let E D fx 2 @G W x is irregular with respect to Gg. Then v  D v in  n E, in particular v  D v q.e. in . Proof. As u is superharmonic, it is lower semicontinuous and does not take the value x Thus u 2 Uu .G/ and u  PxG u  PG u 1. Hence u is bounded from below in G. in G. Therefore v  u in . As v is lsc-regularized in  n @G, it is easy to see that v  is given by (10.9) and that v   v in . Next we want to show that v  is superharmonic. Let uk D minfu; kg, k 2 Z, and ´ in  n G; uk vk D PG uk in G: It follows from Proposition 10.8 (e) that vk ! v in , as k ! 1. Moreover, vk is given by 8 x ˆuk .x/; x 2  n G; ˆ < x 2 G; vk .x/ D PG u®k .x/; ¯ ˆ ˆ :min uk .x/; lim inf PG uk .y/ ; x 2 @G: G3y!x

1;p ./. Hence Theorem 9.44, together with Corollary 9.6 shows that uk 2 Nloc  Theorem 10.12, shows that vk is superharmonic in . Let vQ D limk!1 vk , which is superharmonic by Theorem 9.27. By Theorem 9.44 again, vk D vk q.e. in , and

272

10 The Dirichlet problem for p-harmonic functions

thus v D vQ q.e. in . Since vQ is lsc-regularized, it follows that v  D vQ and hence v  is superharmonic in . That v  is p-harmonic in G follows from Theorem 10.10. x Theorem 10.29 (e) As u is lower semicontinuous and bounded from below on G, applied to u shows that u.x/  lim inf PG u.y/ G3y!x

for x 2 @G n E:

Hence, v  D u D v in @G n E and by the Kellogg property (Theorem 10.5), v  D v q.e. in . For regular sets G, the special care taken at @G in the Poisson modification is not necessary, as we will now see. Note that u is resolutive on G by Corollary 10.33 (applied to u), but this also follows from the proof below. (To deduce resolutivity of u from Corollary 10.33 in the case when G is not connected, one needs to observe that P u is bounded from below and that P u  Px u  u which makes it real-valued a.e. by Lemma 9.9 (or even q.e. by Corollary 9.51). Thus P u is real-valued by Theorem 10.10, and the resolutivity can be deduced from Corollary 10.33.) Theorem 10.43 (Poisson modification in regular sets). Let   X be an arbitrary nonempty open set. Assume that u is superharmonic in  and let G b  be a regular open set such that Cp .X n G/ > 0. Let further ´ u in  n G; wD PG u in G: Then w is superharmonic in  and p-harmonic in G, and w  u in . Proof. Let v, v  and E be as given by Theorem 10.42. As G is regular, E D ¿ and thus v  D v in . It also follows that lim inf PG u.y/  u.x/

G3y!x

for x 2 @G:

x so is PG u. Hence PG u 2 Uu .G/ and so Since u is bounded from below on G, PxG u  PG u from which we conclude that they are equal and that w D v D v  . That w is superharmonic in  and p-harmonic in G, and that w  u in , now follows directly from Theorem 10.42. For superharmonic functions we can also obtain the following result. Proposition 10.44. Let  be an arbitrary nonempty open subset of X . Assume that u is superharmonic in . Then (iii0 ) for every nonempty open set 0 b  with Cp .X n 0 / > 0, and all functions 1;p v 2 Nloc ./ we have H0 v  u in 0 whenever v  u q.e. on @0 .

10.10 The resolutivity problem

273

This shows that (iii) in Definition 9.1 can equivalently be replaced by (iii0 ). (That (iii ) implies (iii) is obvious.) 0

1;p ./ Proof. Let 0 b  be a nonempty open set with Cp .X n 0 / > 0 and v 2 Nloc 0 be such that v  u q.e. on @ . Let further ´ min¹v; uº on @0 ; vQ D v in  n @0 : 1;p Then vQ D v q.e. in , and thus also vQ 2 Nloc ./. Moreover, vQ  u on @0 . Since u x 0 . Moreover, is lower semicontinuous, it is bounded from below on 

Q lim inf u.y/  u.x/  v.x/

0 3y!x

for x 2 @0 ;

from which it follows that u 2 UvQ .0 /. Using Theorem 10.12, we see that u  P0 vQ D H0 vQ D H0 v

in 0 :

10.10 The resolutivity problem So far, we have seen that Newtonian and continuous functions are resolutive. On regular sets, also semicontinuous functions are resolutive. Let us further discuss the resolutivity problem: which functions are resolutive? In the linear case, the harmonic measure (i.e. the 2-harmonic measure) !x0 is really a measure and we have the following complete characterization of resolutive functions in this case. This is one of the most important results in linear potential theory which has not yet been generalized to the nonlinear potential theory, not even on unweighted Rn . x Theorem 10.45 (Brelot’s theorem). Assume that  is connected. Let f W @ ! R and let x0 2 . Then f is resolutive for Cheeger 2-harmonic functions if and only if f 2 L1 .@I !x0 /. Moreover, the Perron solution for Cheeger 2-harmonic functions is given by the formula Z Pf .x/ D f d!x ; x 2 : @

For the definition of Cheeger p-harmonic functions, see Appendix B.2. Observe that the harmonic measures !x and !x0 are comparable, by e.g. Proposition 10.40, and thus f equivalently belongs to L1 .@I !x / for all x 2 . This latter condition is the right characterization also in the case when  is not necessarily connected, a case which we avoided above just for simplicity. On unweighted Rn , this result was obtained by Brelot [76] in 1939. The reader interested in the proof may prefer to take a look at Theorem 6.4.6 in Armitage–Gardiner [19].

274

10 The Dirichlet problem for p-harmonic functions

Brelot’s theorem (Theorem 10.45) was generalized to the form above by Björn– Björn–Shanmugalingam [56]. As this result is a linear result we omit the proof here. The proof is essentially the same as in the Euclidean case, see the comments in [56]. Note that for p D 2 we do not know that our 2-harmonic functions give rise to a linear theory, this is due to the nonlinearity of taking upper gradients. Open problem 10.46. Is it true that if f and g are two 2-harmonic functions, then f C g is 2-harmonic? Open problem 10.47. Is it true that if p D 2 and f and g are two superharmonic functions, then f C g is superharmonic? For Cheeger 2-harmonic functions, one can use the Cheeger 2-Laplace equation to show linearity. In particular, it answers the two questions above in the affirmative for Cheeger 2-(super)harmonic functions, enabling the proof of Brelot’s theorem (Theorem 10.45) in this case. Thus in most of the examples considered in Appendix A the situation is linear when p D 2. The general theory is nonlinear also for p D 2, but linearity can be used when constructing counterexamples. In the nonlinear case, the resolutivity problem seems much harder. One may hope for a positive answer to the following question. Open problem 10.48. Are all bounded Borel functions f W @ ! R resolutive? In the linear case the step from this to Brelot’s theorem (Theorem 10.45) is simply a completion. In the general case, due to the nonlinearity, it is not likely that one can find a simple description of which functions h produce the same Perron solution as a given resolutive function f . Thus Open problem 10.48 seems to be the natural counterpart of Brelot’s theorem (Theorem 10.45) in the nonlinear case. We have presented various resolutivity results in this chapter, some of which are rather new, but all are nevertheless far from answering Open problem 10.48. What we have shown is that continuous and Newtonian functions, as well as their perturbations on sets of capacity zero and uniform limits of such functions, are resolutive. When  is regular we have also shown that all bounded semicontinuous functions are resolutive. We have also presented a few more resolutivity results for semicontinuous functions and perturbations of such functions on sets of capacity zero. These results are all that is known also on unweighted Rn , apart from some very recent results obtained in A. Björn [42]. Let us discuss these. Theorem 10.49 (Baernstein’s problem). Let  D B.0; 1/ be the unit disc in R2 . Let E be a union of finitely many open arcs on @. Then x !.E/ D !.E/: Note that as  is regular, both E and Ex are resolutive by Corollary 10.33. Baernstein [23], p. 548, asked if Theorem 10.49 holds. (Strictly speaking he states this question for the case of two arcs.) For the linear case p D 2 the positive answer

10.11 Notes

275

is well known and easy to obtain, it follows directly from Brelot’s theorem (Theorem 10.45) as !x0 .Ex n E/ D 0. In Björn–Björn–Shanmugalingam [57], Baernstein’s problem was answered, in the affirmative, for 1 < p  2 and also for one arc when p > 2. The method used in [57] for 1 < p < 2 was to show that E is a restriction of a Newtonian function, and then use Theorem 10.12 together with the fact that E D Ex q.e. on @. Neither of these two facts are true for p > 2, but in A. Björn [42] a proof for general p was given using quite a different technique. This proof, on the other hand, uses the scaling structure of unweighted R2 and thus is quite restricted to this case. A proof of Theorem 10.49 was also obtained independently by Kim [205], [206]. His probabilistic game-theoretic proof uses the connection between p-harmonicity and tug-of-war with noise discovered by Peres–Sheffield [302]. In [42] also the following result was obtained. Theorem 10.50. Assume that  D B.0; 1/ is the unit ball in Rn , n  2, f 2 C.@/ and h D f on @ n E, where E is a finite or countable set. Then P h D Pf . Note that h is allowed to be infinite on E. When 1 < p  n, this is just a special case of Theorem 10.22. On the other hand, for p > n it is quite different, as the perturbation set E has positive capacity (when nonempty). A. Björn [42] obtained various generalizations of these two results, but we omit them here as they are a bit technical to describe fully. Kim [205], [206] has also further improved some of the results in [42]. We have already mentioned that the p-harmonic measure is not a measure, and in fact it is very far from being a measure. Llorente–Manfredi–Wu [247] showed that for 2 any p ¤ 2 there are sets E1 ; ::: ; Ek  R such that !p .Ej I RC / D ƒ1 .R n Ej / D 0, Sk j D 1; ::: ; k, but j D1 Ej D R, where ƒ1 is the one-dimensional Lebesgue measure 2 and RC D f.x; y/ 2 R2 W y > 0g is the upper half plane. Thus, the p-harmonic measure is not finitely subadditive on zero sets. Kaufman–Llorente–Wu [195] had earlier obtained related results for p-harmonic measures on trees.

10.11 Notes Perron solutions were first introduced by Perron [304] and independently by Remak [308], see footnote 1 in [308]. They both considered the usual harmonic functions on R2 . The (linear) theory for Perron solutions in connection with the Laplace operator was further developed by Wiener, and Brelot [76] obtained his complete characterization of the resolutive functions, Brelot’s theorem (Theorem 10.45). For this reason the Perron method is often called the Perron–Wiener–Brelot (PWB) method, especially (but not only) in the linear literature. As we here generalize the Perron method in a

276

10 The Dirichlet problem for p-harmonic functions

direction in which Brelot’s result does not apply (or at least is not available now), we prefer to use the name Perron solution. Sections 10.1, 10.2 and part of 10.8 are based on Björn–Björn–Shanmugalingam [55]. For a weak Kellogg property for quasiminimizers see A. Björn [36]. Sections 10.3–10.5 and 10.7 are based on Björn–Björn–Shanmugalingam [56], but many of the results are a bit more general here. Proposition 10.18 with j D fj is a special case of a result by Farnana [120] for the double obstacle problem. Theorem 10.24 follows directly from the results in [55] and [56], but may have been first pointed out explicitly in Björn–Björn [46], p. 347. A generalization of Corollary 10.33 to certain nonregular sets was obtained in A. Björn [42]. Carleman’s principle (Proposition 10.41) does not seem to have been formulated in metric spaces earlier. Alternative definitions of Perron solutions have been given in [42] and [56]. Theorem 10.29 is a combination of results from [56] and Björn–Björn [45]. Lemma 10.27 is from Björn–Björn–Mäkäläinen–Parviainen [52]; the proof therein uses Theorem 14.10, while our proof, although a bit more involved, is only based on our definition of superharmonicity. Theorem 10.42 is also from [52], while Proposition 10.44 is from A. Björn [35]. In weighted Rn , many of the results in this chapter are given in Heinonen–Kilpeläinen–Martio [171]. Kurki [236] obtained the equality !.K/ x D !.K x [ E2 / in Corollary 10.39 for weighted Rn . Similar results with a condition on the Hausdorff dimension of E2 were obtained by Avilés–Manfredi [22] for unweighted Rn . Finally, let us note that we only study the Dirichlet problem in bounded  with Cp .X n / > 0. Without the condition Cp .X n / > 0 there is not much of a boundary to have boundary values at, and in particular we lose uniqueness in the obstacle problem. Thus, for our purposes, it is not desirable to study the Dirichlet problem, nor boundary regularity, without this assumption. For unbounded , the situation is different. This case has not yet been treated in metric spaces, but deserves to be studied. In weighted Rn , the unbounded case was studied in Heinonen–Kilpeläinen–Martio [171]. They had a boundary condition also at 1. Gardiner [131] characterized both solubility and uniqueness in the Dirichlet problem for harmonic functions (i.e. p D 2) on unbounded sets in (unweighted) Rn without any condition at 1, see also the book Gardiner [132]. The Martin boundary was studied for p-harmonic functions on metric spaces by Holopainen–Shanmugalingam [181], Holopainen–Shanmugalingam–Tyson [182] and Aikawa–Shanmugalingam [9]. See also Lewis–Nyström [244] for recent progress on Rn . Let us finally mention that short introductions to the topic in this chapter are in A. Björn [34] and J. Björn [64].

Chapter 11

Boundary regularity

Recall that in Chapters 7–14 we assume that 1 < p < 1 and that X is a complete doubling p-Poincaré space, with doubling constant C and dilation constant  in the p-Poincaré inequality. In this chapter, apart from in Section 11.6, we also assume that  is bounded and such that Cp .X n / > 0. We saw in Chapter 10 that the Dirichlet problem is uniquely soluble for continuous boundary values. We also know from Chapter 8 that the solution is locally Hölder continuous in its domain of p-harmonicity. A natural question is for which boundary points the solution attains its boundary value in a continuous way, i.e. as a limit. In Section 10.2 we defined regular boundary points, i.e. points where the right boundary value is attained for all continuous boundary data. We also proved the Kellogg property, i.e. that the set of irregular boundary points has capacity zero. In Section 10.6, we gave some characterizations of regular boundary points. In this chapter, we continue our study of regular boundary points and characterize them by means of barriers and obstacle problems. We also obtain the sufficiency part of the Wiener criterion. Finally, we use the boundary regularity results to show that Newtonian functions are finely continuous q.e.

11.1 Barrier characterization of regular points In this section we characterize regular boundary points by means of barriers, from which it follows that regularity is a local property of the boundary. Definition 11.1. A function u is a barrier (with respect to ) at x0 2 @ if (i) u is superharmonic in ; (ii) lim3y!x0 u.y/ D 0; (iii) lim inf 3y!x u.y/ > 0 for every x 2 @ n fx0 g. By the strong minimum principle (Theorem 9.13) a barrier is always nonnegative. Moreover, a barrier is positive if every component G   has a boundary point in @G n fx0 g. The zero function is a barrier if and only if @ D fx0 g, see also Example 13.9. Theorem 11.2. Let x0 2 @ and d.x/ WD d.x; x0 /. Then the following are equivalent: (a) The point x0 is a regular boundary point.

278

11 Boundary regularity

(b) There is a barrier at x0 . (c) There is a positive continuous barrier at x0 . (d) It is true that lim

3y!x0

P d.y/ D 0:

Note that other regularity characterizations are given in Theorems 10.29 and 11.11. Proof. (a) ) (c). Let us first consider the case when Cp .fx0 g/ D 0. As Cp .X n/ > 0, by assumption, we can find B 0 D B.x0 ; r/ so small that Cp .X n . [ 2B 0 // > 0. Let u be the continuous solution of the Kf;f . [ 2B 0 /-obstacle problem, where f D d . By Theorem 8.28, ujA D HA f , where A D fy 2  [ 2B 0 W u.y/ > f .y/g and A is open. It is clear that u  0, that u is bounded from below and that u.x0 / D 0. Let G be a component of A. Then ujG D HG f . Since Cp .X n G/  Cp .X n . [ 2B 0 // > 0, Lemma 4.5 shows that Cp .@G/ > 0. As Cp .fx0 g/ D 0, it follows from the Kellogg property (Theorem 10.5) that there exists x1 2 @G nfx0 g which is regular for G. Hence lim

G3y!x1

u.y/ D f .x1 / < 0:

Thus u 6 0 in G. By the strong maximum principle (Theorem 8.13) it follows that u < 0 in G, and thus that u < 0 in . [ 2B 0 / n fx0 g. Let now m D sup@B 0 , which is negative by compactness. Note that m  r. Since 0 uj.[2B 0 /nBx 0 is the continuous solution of the Kf;u .. [ 2B 0 / n Bx /-obstacle problem 0 and f  r in . [ 2B 0 / n Bx , we see that sup 0 x 0 u D m. It follows that .[2B /nB

lim sup u.y/  m < 0

0

for all x 2 @ n Bx :

3y!x

As u is continuous in 2B 0 , we also have that lim sup u.y/ D u.x/ < 0

0

for all x 2 .@ \ Bx / n fx0 g;

3y!x

and lim3y!x0 u.y/ D u.x0 / D 0. Let now 8 <u.x/; x 2 ; v.x/ D : lim inf u.y/; x 2 @: 3y!x

Then v is subharmonic in  and continuous at x0 . Let w D P v. Since v 2 Lv , we see that w  v, and thus lim inf w.y/  lim inf v.y/ > 0

3y!x

3y!x

11.1 Barrier characterization of regular points

279

for every x 2 @ n fx0 g. On the other hand, since v is continuous at x0 and bounded, and x0 is regular, Theorem 10.29 (d) shows that lim3y!x0 w.y/ D v.x0 / D 0. We have thus shown that w is a positive p-harmonic barrier at x0 (and thus continuous). Let us finally consider the case when Cp .fx0 g/ > 0. Using Theorem 8.28, we let u be the continuous solution of the Kd;d -obstacle problem, and observe that ujA D HA d , where A D fy 2  W u.y/ > d.y/g. If x0 2 @A, then as Cp .fx0 g/ > 0, the Kellogg property (Theorem 10.5) implies that x0 is regular for A, and hence lim

A3y!x0

u.y/ D d.x0 / D 0:

On the other hand, if x0 2 @. n A/, then lim

nA3y!x0

u.y/ D

lim

nA3y!x0

d.y/ D d.x0 / D 0:

It follows that lim3y!x0 u.y/ D 0 regardless of the location of x0 on @. (Note that it is possible that x0 belongs to both @A and @. n A/.) Moreover, lim inf u.y/  lim inf d.y/ D d.x/ > 0

3y!x

3y!x

for x 2 @ n fx0 g:

Since u is superharmonic, it is a positive continuous barrier at x0 . (c) ) (b) This is trivial. (b) ) (d) If Cp .fx0 g/ > 0, then x0 is regular by the Kellogg property (Theorem 10.5) and (d) holds by Theorem 10.29. Assume therefore that Cp .fx0 g/ D 0. Let u be a barrier at x0 and G be a component of . Then Cp .X n G/  Cp .X n / > 0, and by Lemma 4.5, Cp .@G/ > 0. Thus, there exists x 2 @G n fx0 g, and the strong minimum principle (Theorem 9.13) implies that u > 0 in G. Since u > 0 in every component, u > 0 in . Let " > 0 be so small that 0 WD  n B.x0 ; "/ ¤ ¿. Let m D inf 0 u. We want to 0 show that m > 0. Let fyn g1 nD1 , yn 2  , be a sequence such that lim n!1 u.yn / D m. By compactness, there is a convergent subsequence (also denoted by fyn g1 nD1 ) with a x 0 . If y0 2 , then the lower semicontinuity of u yields m  u.y0 / > 0. limit y0 2  On the other hand if y0 2 @, then m  lim inf 3y!y0 u.y/ > 0, since y0 ¤ x0 . Thus m > 0. Let now M D sup d . Then M u=m C " 2 Ud and thus P d  M u=m C ", from which it follows that lim sup P d.y/  3y!x0

M lim sup u.y/ C " D ": m 3y!x0

Letting " ! 0 shows that lim sup3y!x0 P d.y/  0. That lim inf 3y!x0 P d.y/  0 is trivial. (d) ) (a) This is part of Theorem 10.29.

280

11 Boundary regularity

Corollary 11.3. Let x0 2 @ be regular, and let V   be open and such that x0 2 @V . Then x0 is regular also with respect to V . Proof. By Theorem 11.2, (a) ) (c), there is a positive continuous barrier u at x0 (with respect to ). Let v D ujV . Then limV 3y!x0 v.y/ D 0. For x 2 .@ \ @V / n fx0 g we have lim inf v.y/  lim inf u.y/ > 0: V 3y!x

3y!x

And for x 2 @V n @ we have lim v.y/ D u.x/ > 0:

V 3y!x

Hence v is a positive continuous barrier at x0 with respect to V . It thus follows from Theorem 11.2, (c) ) (a), that x0 is regular with respect to V .

11.2 Boundary regularity for the obstacle problem By comparing p-harmonic functions with solutions of obstacle problems, we can extend our regularity results to obstacle problems. For this we need the following boundary weak Harnack inequality. Lemma 11.4 (Boundary weak Harnack inequality). Let u be the lsc-regularized sox and f 2 N 1;p ./ are such lution of the K ;f -obstacle problem, where W  ! R that K ;f ¤ ¿. Let x0 2 @ and assume that there are a ball B D B.x0 ; r/ with r < 14 diam X and k0 2 R such that .f  k0 /C 2 N01;p .I B/ and  k0 q.e. in B \ . Then for all k  k0 , 

1 sup u  k C C .B/ \ 1 B 2

1=p

Z

p

\B

.u  k/C d

:

In particular, u is bounded from above in  \ 12 B. Proof. Let 0 < r1 < r2  r and k  k0 be arbitrary. Let v D u  .u  k/C , where ²

r2  d.x0 ; y/

.y/ D min 1; r 2  r1

³ C

is a 1=.r2  r1 /-Lipschitz function vanishing outside B.x0 ; r2 / such that 0   1 in B.x0 ; r2 / and D 1 in B.x0 ; r1 /. We shall show that v 2 K ;f . Indeed, q.e. in  either v D u  or v D .1  /u C k  . To show that v  f 2 N01;p ./, we observe that u  k D u  f C f  k0  .k  k0 /

281

11.2 Boundary regularity for the obstacle problem

and hence 0  .u  k/C  .u  f /C C .f  k0 /C 2 N01;p . \ B/: Thus, by Lemma 2.37, v  f D u  f  .u  k/C 2 N01;p ./ and v 2 K Let Aj D fx 2  \ B.x0 ; rj / W u.x/ > kg; j D 1; 2:

;f

.

We have v D .1  /.u  k/ C k in A2 , and hence gv  .1  /gu C .u  k/g a.e. in A2 . In  n A2 we have gv D gu a.e. Since u is a solution of the K ;f -obstacle problem, we get as in Lemma 8.1, Z Z Z Z Z p p p p p p gu d  gu d  gv d  2 gu d C 2 .u  k/p gp d: A1

A2

A2 nA1

A2

A2

As g  1=.r2  r1 /, adding 2p times the left-hand side to both sides and dividing by 2p C 1, we obtain that Z Z Z  gup d   gup d C .u  k/p d; p .r  r / 2 1 A1 A2 A2 where  D 2p =.2p C 1/ < 1. Lemma 7.18 then shows that for all k  k0 and 0 < r1 < r2  r, Z Z C p gu d  .u  k/p d: p .r  r / 2 1 A1 A2 ´

Let further wD

max¹u; kº in ; k in B n :

Then Aj D fx 2 B.x0 ; rj / W w.x/ > kg, j D 1; 2, and u D w in A2 . Hence, Z Z C p gw d  .w  k/p d: p .r  r / 2 1 A1 A2 As

0  .u  k/C  .u  f /C C .f  k0 /C 2 N01;p .I B/;

Lemma 2.37 shows that w D k C .u  k/C 2 N 1;p .B/. The weak Harnack inequality for De Giorgi classes (Proposition 8.2) together with the lower semicontinuity of u then yields « 1=p p .w  k/C d sup u  ess sup w  k C C \ 1 2B

\ 1 2B

B



1 DkCC .B/

1=p

Z

p

\B

.u  k/C d

:

282

11 Boundary regularity

To formulate our results on boundary regularity of obstacle problems, we need the following notion of limits in the capacitary sense. x and x 2 , x let Definition 11.5. For f W  ! R Cp - ess lim sup f .y/ D lim Cp - ess sup f: r!0 B.x;r/\

3y!x

Observe that the value f .x/ plays a role in the definition above, if x 2  and 1;p ./ (or more generally if f 2 ACCp ./) and Cp .fxg/ > 0. However, if f 2 Nloc x 2 , then Cp - ess lim sup f .y/ D ess lim sup f .y/; y!x

y!x

by Corollary 1.69 and Lemma 2.24. It follows that in this case, Cp - ess lim sup f .y/ D lim Cp - ess sup f; r!0 B.x;r/nfxg

y!x

i.e. the value of f .x/ may be disregarded. x and f 2 N 1;p ./ be such that K ;f ¤ ¿. Let u Theorem 11.6. Let W  ! R be the lsc-regularized solution of the K ;f -obstacle problem and let x0 2 @ be a regular boundary point. Let further m D supfk 2 R W .f  k/ 2 N01;p .I B/ for some ball B 3 x0 g; M 0 D inffk 2 R W .f  k/C 2 N01;p .I B/ for some ball B 3 x0 g; ˚ M D max M 0 ; Cp - ess lim sup .y/ : 3y!x0

Then m  lim inf u.y/  lim sup u.y/  M: 3y!x0

3y!x0

Remark 11.7. Roughly speaking, m is the lim inf of f at x0 in the Sobolev sense and M 0 is the corresponding lim sup. Observe also that it is not possible to replace M by M 0 , since the lsc-regularized solution must satisfy lim sup u.y/  Cp - ess lim sup u.y/  Cp - ess lim sup .y/; 3y!x0

3y!x0

and it can happen that Cp - ess lim sup3y!x0

3y!x0

.y/ > M 0 , see Example 11.10.

Proof. Let k > M and find a ball B D B.x0 ; r/, r < N01;p .I B/ and k  Cp - ess supB\ . As

1 4

diam X , such that .f  k/C 2

0  .u  k/C  .u  f /C C .f  k/C 2 N01;p .I B/;

11.2 Boundary regularity for the obstacle problem

283

Lemma 2.37 shows that .u  k/C 2 N01;p .I B/. Let ´ max¹u; kº D k C .u  k/C in ; vD k in B n : Then v 2 N 1;p .B/. Let further G D  \ 31 B and v 0 D HG v D PG v, by Theorem 10.12. The maximum principle (Theorem 8.13) yields v 0  k  Cp - ess sup

in G

G

and hence v 0 is a solution of the K ;v .G/-obstacle problem. Now, u is clearly a solution of the K ;u .G/-obstacle problem, and the comparison principle (Lemma 8.30) shows that u  v 0 in G. The boundary weak Harnack inequality (Lemma 11.4) shows that v x By Corollary 11.3, x0 is regular also with respect to G. Hence, as is bounded on G. v  k on B \ @, Theorem 10.29 (d) shows that lim sup u.y/ D lim sup u.y/  3y!x0

G3y!x0

lim

G3y!x0

v 0 .y/ D v.x0 / D k:

Taking infimum over all k > M shows one inequality of the theorem. To prove the other inequality, note that u  Hf D H.f /, by the comparison principle (Lemma 8.30), and that H.f / is the lsc-regularized solution of the K1;f -obstacle problem. The first part of the proof applied to f with M replaced by m then shows that lim inf u.y/   lim sup H.f /.y/  m:

3y!x0

3y!x0

In the following theorem, we give a necessary and sufficient condition on the obstacle for continuity of the solution at a regular boundary point. x and f 2 N 1;p ./ be such that K ;f ¤ ¿. Let u Theorem 11.8. Let W  ! R be the lsc-regularized solution of the K ;f -obstacle problem and let x0 2 @ be a regular boundary point. Assume further that either (a) f .x0 / WD lim3y!x0 f .y/ exists ; or x \ B/ for some ball B 3 x0 , and f j@ is continuous at x0 . (b) f 2 N 1;p . Then lim

3y!x0

u.y/ D f .x0 /

if and only if f .x0 /  Cp - ess lim sup3y!x0

.y/.

Note that it is possible to have f .x0 / < Cp - ess lim sup3y!x0 a soluble obstacle problem, see Example 11.10 below. The following corollary is a special case of Theorem 11.8.

.y/ and still have

284

11 Boundary regularity

x let u be the continuous solution of the Corollary 11.9. Let f 2 N 1;p ./ \ C./, Kf;f -obstacle problem and let x0 2 @ be regular. Then lim3y!x0 u.y/ D f .x0 /. x In particular, if  is regular and we let u D f on @, then u 2 C./. Proof of Theorem 11.8. Assume first that f .x0 /  Cp - ess lim sup3y!x0 .y/, and let m and M be defined as in Theorem 11.6. Let " > 0, and let B 0 D B.x0 ; r/, with B 0  B in case (b), be such that ´ in case (a); x 2 B0 \  jf .x/  f .x0 /j < " for 0 x 2 B \ @ in case (b): Then .f  .f .x0 / C "//C 2 N01;p .I B 0 /. It follows that M  f .x0 / C " and letting " ! 0 shows that M  f .x0 /. Similarly, f .x0 /  m, and by Theorem 11.6, m  M . Hence lim3y!x0 u.y/ D m D M D f .x0 /, by Theorem 11.6. Conversely, assume that f .x0 / < Cp - ess lim sup3y!x0 .y/. As u  q.e., we see that f .x0 / < Cp - ess lim sup .y/  Cp - ess lim sup u.y/  lim sup u.y/: 3y!x0

3y!x0

We shall now see that the K

;f

3y!x0

-obstacle problem can be soluble even if

f .x0 / < Cp - ess lim sup .y/:

(11.1)

3y!x0

Example 11.10. (a) Let  D .1; 1/n1 .0; 1/  Rn , n  2, 8x 8 n 0 ˆ ˆ ˆ jxj  13 ; < jx 0 j ; 0 < xn  jx j  1; <1; h.x 0 ; xn / D 1;

.x/ D 2  3jxj; 13  jxj  23 ; jx 0 j < xn  1; ˆ ˆ ˆ : : 0; jxj  23 ; 0; otherwise; and f D h, where x D .x 0 ; xn / 2 Rn1 R. After observing that gh D jrhj  2=jx 0 j, it is straightforward to check that f 2 N 1;p .Rn / for 1 < p < n. Hence f 2 Kf;f , and thus by Theorem 7.2, the Kf;f -obstacle problem is soluble. In this case (with x0 D 0) we have m D M 0 D 0, M D 1 and lim sup u.y/  Cp - ess lim sup f .y/ D 1 > 0 D M 0 D f .0/ D 0: 3y!0

3y!0

This shows that it is not possible to replace M by M 0 in Theorem 11.6, and also that (11.1) holds (with x0 D 0). Let k be a positive integer and ak D 1=k. As  is regular, we have by Theorem 11.6 (or Theorem 11.8) that lim

3y!.ak ;0;:::;0/

u.y/ D f .ak ; 0; ::: ; 0/ D 0:

11.3 Characterizations of regularity

285

Hence there is 0 < bk < 1=k such that u.ak ; 0; ::: ; 0; bk / < 1=k. It follows that lim inf u.y/  lim u.ak ; 0; ::: ; 0; bk / D 0:

3y!0

m!1

Since 0  f  1 we know that 0  u  1. Thus lim inf u.y/ D 0

3y!0

and

lim sup u.y/ D 1 3y!0

and u is not continuous at the regular point x0 D 0. (b) If we let fk .x/ D f .kx/, k  1, then kfk kpN 1;p .Rn /  k pn kf kpN 1;p .Rn / . It P follows that fQ D j1D0 f2j 2 N 1;p .Rn /, 1 < p < n, and we get m D M 0 D 0 and M D 1 (with respect to fQ and x0 D 0). In this case we get lim inf u.y/ D 0 and

3y!0

lim sup u.y/ D 1: 3y!0

11.3 Characterizations of regularity We are now prepared to give further characterizations of boundary regularity. Let us at the same time collect the earlier characterizations, given in Theorems 10.29 and 11.2, as well. Theorem 11.11. Let x0 2 @, ı > 0, B D B.x0 ; ı/ and d.x/ WD d.x; x0 /. Then the following are equivalent: (a) The point x0 is a regular boundary point. (b) There is a barrier at x0 . (c) There is a positive continuous barrier at x0 . (d) It is true that lim

3y!x0

P d.y/ D 0:

(e) It is true that lim

3y!x0

Pf .y/ D f .x0 /

for all continuous f W @ ! R. (f) It is true that lim

3y!x0

x .y/ D f .x0 / Pf

for all bounded f W @ ! R which are continuous at x0 .

286

11 Boundary regularity

(g) It is true that

x .y/  f .x0 / lim sup Pf 3y!x0

for all functions f W @ ! R which are bounded from above on @ and upper continuous at x0 . (h) The point x0 is regular with respect to G WD B \ . (i) It is true that lim

3y!x0

Hf .y/ D f .x0 /

for all f 2 N 1;p ./ such that f .x0 / WD lim3y!x0 f .y/ exists. (j) It is true that lim

3y!x0

Hf .y/ D f .x0 /

x such that f j@ is continuous at x0 . for all f 2 N 1;p . [ .B \ // x such that K W!R

(k) For all f 2 N 1;p ./ and all f .x0 / WD

lim

3y!x0

;f

¤ ¿ and

f .y/  Cp - ess lim sup .y/ 3y!x0

(where the limit in the middle is assumed to exist), the lsc-regularized solution u of the K ;f -obstacle problem satisfies lim

3y!x0

u.y/ D f .x0 /:

x such that f j@ is continuous at x0 , and all (l) For all f 2 N 1;p . [ .B \ // x such that f .x0 /  Cp - ess lim sup3y!x .y/ and K ;f ¤ ¿, W!R 0 the lsc-regularized solution u of the K ;f -obstacle problem satisfies lim

3y!x0

u.y/ D f .x0 /:

(m) The continuous solution u of the Kd;d -obstacle problem satisfies lim

3y!x0

u.y/ D 0;

i.e. u is a positive continuous barrier at x0 . x which is superharmonic in  and such that f j@ (n) For any function f 2 N 1;p ./ is lower semicontinuous at x0 , lim inf f .y/  f .x0 /:

3y!x0

11.3 Characterizations of regularity

287

Remark 11.12. Condition (h) shows that regularity is a local property of the boundary. Note that it is not possible to replace lim inf by lim in (n) even if we require f j@ to be continuous at x0 , see Example 11.10. See also the notes to this chapter for further comments on (n). 1;p x The function d in (d) and (m) can be replaced by any function d 0 2 C./\N ./ 0 0 0 x with d .x0 / D 0 and d .x/ > 0 for all x 2 nfx0 g. In particular, we can have d D d ˛ with ˛ > 0. Proof. The equivalences of (a)–(g) were deduced in Theorems 10.29 and 11.2. (a) ) (k) and (a) ) (l) These implications follow from Theorem 11.8. (k) ) (i) and (l) ) (j) These implications are trivial, as Hf is the continuous solution of the K1;f -obstacle problem. (i) ) (d) and (j) ) (d) These implications follow from Theorem 10.12, since x d 2 N 1;p ./. (k) ) (m) The first part is trivial, after noting that u is continuous, by Theorem 8.28. As u  d everywhere in , we see that lim inf u.y/  d.x; x0 / > 0

3y!x

for all x 2 @ n fx0 g:

Since u is superharmonic, it is a positive continuous barrier. (m) ) (c) This is trivial. (a) ) (h) This follows from Corollary 11.3. (h) ) (n) Let " > 0. Then there is 0 < r < ı such that inf

B.x0 ;r/\@G

f  f .x0 /  ":

Then h WD minff; f .x0 /  "g is superharmonic in G and h  .f .x0 /  "/ 2 N01;p .GI B.x0 ; r//: Since h 2 N 1;p .G/, it is the lsc-regularized solution of the Kh;h .G/-obstacle problem. We can therefore apply Theorem 11.6 (with h and G in the place of f and ). Observing that m  f .x0 /  ", where m is as in Theorem 11.6, gives lim inf f .y/ D lim inf f .y/  lim inf h.y/  f .x0 /  ":

3y!x0

G3y!x0

G3y!x0

Letting " ! 0 gives the desired conclusion. (n) ) (d) Let ´ Hd in ; f D d in X n : Then both f and f satisfy the conditions in (n), and thus (d) follows, since P d D Hd , by Theorem 10.12.

288

11 Boundary regularity

Let us end the discussion on characterizations of regular points by stating an open problem. We first need the following definition. Definition 11.13. A function u is a weak barrier at x0 2 @ if it is a positive superharmonic function in  such that lim

3y!x0

u.y/ D 0:

If x0 is regular, then Theorem 11.11 (m) shows that there is a weak barrier at x0 . In our generality it is an open problem if the existence of a weak barrier suffices to conclude that a point is regular. Note also that Example 13.9 shows that not all barriers are weak barriers. Open problem 11.14. Is it true that if there is a weak barrier at x0 2 @, then x0 is regular? In weighted Rn this was shown by Kilpeläinen–Lindqvist [203], Theorem 3.1, and for Cheeger p-harmonic functions on metric spaces it was shown by A. Björn [40], Theorem 1.1.

11.4 The Wiener criterion In this section we prove the sufficiency part of the Wiener criterion for regularity of boundary points. Definition 11.15. Let E  . The capacitary potential u for E in  is the lscregularized solution of the KE ;0 -obstacle problem (if it exists). We also let u  0 in X n . Note that if dist.E; X n / > 0, then the KE ;0 -obstacle problem is soluble. In particular, this means that the capacitary potential always exists for E b . In fact, it is easy to verify, e.g. using Adams’ criterion (Theorem 7.3), that the KE ;0 -obstacle problem is soluble if and only if capp .E; / < 1, see also Lemma 11.19. Note further that u is lower semicontinuous on X , but may fail to be lsc-regularized at some points on @. Example 11.16. Let  D B.0; 2/ n f0g  R2 , K D @B.0; 1/, 1 < p  2, and let u be the capacitary potential for K in . Then u  1 in B.0; 1/ n f0g, while u.0/ D 0. Hence u is not lsc-regularized at 0. The following lemma shows that for compact E, the capacitary potential could be equivalently defined as a solution to the Dirichlet problem for p-harmonic functions. However, our definition by means of obstacle problems is more general, as it works for arbitrary sets.

11.4 The Wiener criterion

289

Lemma 11.17. Let u be the capacitary potential for a compact subset K  . Then u D HnK K in  n K: Proof. Clearly, 0  u  1 and hence u D 1 q.e. on K. Let v D HnK K , extended by 1 on K. Then v 2 N01;p ./ and v  K in , i.e. v 2 KK ;0 . As u D v D 1 q.e. on K, we have gu D gv D 0 a.e. on K and hence by the definition of u, Z Z Z Z gup d D gup d  gvp d D gvp d: nK





nK

This shows that u is a solution of the K1;K . n K/-obstacle problem and hence u D v q.e. in  n K. As u is lsc-regularized and v continuous, also u D v everywhere in  n K. Remark 11.18. This lemma also shows one reason why we use the obstacle problem with q.e. rather than a.e. inequality. If K   is a compact set of zero measure but positive capacity, then the solution of the q.e.-obstacle problem is the capacitary potential for K in , while the a.e.-obstacle problem has the trivial solution u D 0. The following lemma justifies the name “capacitary potential” and relates the capacitary potentials to the capacity. It also provides estimates for level sets of capacitary potentials. Lemma 11.19. Let u be the capacitary potential for a set E in . Then Z gup d D capp .E; /

(11.2)



and for all 0 < M < 1, we have Z 1 0 g p d D capp .EM ; / D capp .EM ; /; M p u<M u 0 where EM D fx 2  W u.x/ > M g and EM D fx 2  W u.x/  M g.

Proof. First, let v 2 N01;p ./ be admissible in the definition of capp .E; /. Then v 2 KE ;0 ./ and taking infimum over all such v shows that Z gup d  capp .E; /: 

On the other hand, since u  1 q.e. on E, a representative of u is admissible in the definition of capp .E; / and (11.2) follows. 0 Similarly, minfu=M; 1g is admissible in the definition of capp .EM ; /, which proves that Z 1 0 g p d  capp .EM ; /  capp .EM ; /: M p u<M u

290

11 Boundary regularity

For the converse, let v be admissible in the definition of capp .EM ; /. As Z Z p p gv d  gminfv;1g d; 



we can assume that v  1 and hence v D 1 on EM . It follows that for a.e. x 2 , at least one of g.uM /C and gv vanishes. Since .u  M /C C M v 2 KE ;0 ./, we obtain Z Z Z Z Z p p p p p p gu d  g.uM /C d C M gv d D gu d C M gvp d: 





uM



Subtracting the first term in the right-hand side from both sides of the inequality, and taking infimum over all v admissible in the definition of capp .EM ; / finishes the proof. The following pointwise estimate for the capacitary potential will be the crucial ingredient in the proof of the sufficiency part of the Wiener criterion. Lemma 11.20. Let B be a ball in X , E  Bx and B0 D 100B. Assume that Cp .X n B0 / > 0 and let u be the capacitary potential for E in B0 . Then   capp .E; B0 / 1=.p1/ : inf u  C 2B capp .B; B0 / Recall that  is the dilation constant in the p-Poincaré inequality. Proof. If Cp .E/ D 0, then capp .E; B0 / D 0, and the estimate is trivial. We therefore assume that Cp .E/ > 0 in the rest of the proof. Let M D sup@2B u. As u D 1 q.e. on E, we can find x 2 E such that u.x/ D 1. Let G be the component of B0 containing x. By Theorem 4.32, there exists a curve  such that .0/ D x and .l / 2 X n B0 . This curve must intersect @2B, so there is y 2 G \ @2B. By the strong minimum principle (Theorem 9.13), M  u.y/ > 0. Assume first that M < 1. Let u1 D minfu=M; 1g and u2 D .u  M u1 /=.1  M /. Note that both u1 D 1 and u2 D 1 q.e. on E. It follows that for each 0  a  1, a representative of the function au1 C .1  a/u2 is admissible in the definition of capp .E; B0 /. Since for a.e. x 2 B0 , either gu1 D 0 or gu2 D 0, Lemma 11.19 yields Z Z Z p p p p capp .E; B0 / D gu d  a gu1 d C .1  a/ gup2 d: B0

B0

B0

Denote the above integrals by I , I1 and I2 , respectively, and rewrite the last estimate as I  ap I1 C .1  a/p I2 : The right-hand side attains its minimum for a D I11=.1p/ =.I11=.1p/ C I21=.1p/ / and we obtain that I 1=.1p/  I11=.1p/ C I21=.1p/ : (11.3)

11.4 The Wiener criterion

291

Note that gu2 D gu fu>M g =.1  M / and hence I2  I =.1  M /p . Inserting this into (11.3) shows that I1 .1  .1  M /p=.p1/ /1p   I1 pM p1  D I1 : .1  .1  pM=.p  1///1p p1

capp .E; B0 / D I 

As u D HB0 n2B u in B0 n 2B, the comparison principle (Lemma 8.32) shows that u  M in B0 n 2B. Thus EM WD fx 2 B0 W u.x/ > M g  2B. Lemma 11.19 and Proposition 6.16, together with the doubling property of , then imply that Z I1 D gup1 d D capp .EM ; B0 /  capp .2B; B0 /  C capp .B; B0 /; B0



and hence M

p1

C

capp .E; B0 / capp .B; B0 /

 :

(11.4)

If M D 1, then (11.4) holds trivially. Now, let B 0 be a ball with centre on @2B and the same radius as B such that M  sup 1 B 0 u. Since u is p-harmonic in B 0 and an lsc-regularized superminimizer in 2 B0 , the weak Harnack inequalities (Theorems 8.4 and 8.10) together with the doubling property of  imply that for some q > 0, « 1=q « 1=q q q M  sup u  C u d C u d  C inf u: 1 0 2B

B0

4B

2B

Together with (11.4) this finishes the proof. We shall now iterate Lemma 11.20 to obtain a stronger estimate on arbitrarily small balls. Note that the right-hand side in Theorem 11.21 tends to zero if the sum there diverges. Theorem 11.21. Let B be a ball in X , E  Bx and B0 D 100B. Assume that Cp .X n B0 / > 0 and let u be the capacitary potential for E in B0 . Then for all Bk D .100/k B0 , k D 1; 2; ::: ,    k  X capp .Bxj \ E; Bj 1 / 1=.p1/ : sup .1  u/  exp C capp .Bj ; Bj 1 / Bk j D1

Proof. Let uj denote the capacitary potential for Bxj \ E with respect to Bj 1 . Note that u1 D u. Let   capp .Bxj \ E; Bj 1 / 1=.p1/ aj WD : capp .Bj ; Bj 1 /

292

11 Boundary regularity

By Lemma 11.20, we have for all x 2 Bxj , uj .x/  C 0 aj  1  e C

0a j

:

(11.5)

Let v1 D u1 and for j D 1; 2 ::: , vj C1 D 1  e C

0a j

.1  vj / D 1  e C

0 .a Ca CCa / 1 2 j

.1  v1 /:

(11.6)

We shall show by induction that vj  uj in Bj 1 for all j D 1; 2 ::: . We have v1 D u1 in B0 . Assume that vk1  uk1 in Bk2 for some k D 2; 3; ::: . Then by (11.5) and 0 (11.6), we have vk  1  e C ak1 .1  uk1 /  0 in Bk1 . Note that vk D 1 q.e. on E by (11.6), and hence vk  E \Bxk q.e. in Bk1 . By Proposition 7.14 and (11.6), vk is an lsc-regularized superminimizer in B0 . Proposition 7.15 then yields that vk is the lsc-regularized solution of the Kvk ;vk .Bk1 /obstacle problem. As uk is the lsc-regularized solution of the KE\Bx ;0 .Bk1 /k obstacle problem, the comparison principle (Lemma 8.30) shows that vk  uk  0 in Bk1 . The induction is thus completed. From this, (11.6) and (11.5), we now infer that in Bk , 1  u D 1  v1 D e C e

0 .a Ca CCa 1 2 k1 /

C 0 .a1 Ca2 CCak1 /

.1  vk /

.1  uk /  e C

0 .a Ca CCa / 1 2 k

;

which finishes the proof. Before formulating a pointwise estimate for solutions of the Dirichlet problem, we first prove the following simple lemma which makes it possible to get rid of the factor 100 in the Wiener integral. x Then for every 1 < s < t with Lemma 11.22. Let B D B.x0 ; r/ and E  B. 1 t r < 4 diam X , 

capp .E; tB/  capp .E; sB/  C 1 C



tp capp .E; tB/: .s  1/p

Proof. The first inequality is obvious. As for the second inequality, let u 2 N01;p .tB/ be a function admissible in the definition of capp .E; tB/. Let v D u , where ²

.y/ D min 1;

sr  d.x0 ; y/ .s  1/r

³ C

is a Lipschitz function with Lipschitz constant 1=.s  1/r such that D 1 on B and

D 0 outside sB. Then v 2 N01;p .sB/ is admissible in the definition of capp .E; sB/, and hence Z  Z Z jujp gvp d  C gup d C d : capp .E; sB/  p p sB sB sB .s  1/ r

293

11.4 The Wiener criterion

The last integral is estimated using the Sobolev inequality (Theorem 5.51) on tB and we obtain  Z tp capp .E; sB/  C 1 C gup d: .s  1/p tB Taking infimum over all functions u admissible in the definition of capp .E; tB/ finishes the proof. We can now prove a pointwise estimate for solutions of the Dirichlet problem, which leads to the sufficiency part of the Wiener criterion. Theorem 11.23. Let f W @ ! R be resolutive and bounded from above. Let further x0 2 @ and 0 <  r < diam.X /=400. Then sup

.Pf /C 

\B.x0 ; /

sup

fC

(11.7)

@\B.x0 ;100r/

   Z r capp .B.x0 ; t / n ; B.x0 ; 2t // 1=.p1/ dt C sup fC exp C : capp .B.x0 ; t /; B.x0 ; 2t // t @

Proof. Let u be the capacitary potential of B.x0 ; r/ n  in B0 WD B.x0 ; 100r/, extended by zero outside of B0 . Let also M D sup@ fC and m D sup@\B0 fC . Note that Pf  M in  and that f  m C M.1  u/ on @. Hence, Pf  m C M.1  P u/

in :

Now, as every v 2 Uu ./ also belongs to Uu .G/, where G D  \ B0 , it follows directly from the definition of Perron solutions that P u  PG u D HG u D u in G. Thus, Pf  m C M.1  u/ in G: (11.8) Choose an integer k so that .100/k r <  .100/1k r and let Bj D .100/j B0 , j D 1; 2; ::: . Then B.x; /  Bk . Estimating the numerator by Lemma 11.22 and the denominator by Proposition 6.16 and using the fact that  is doubling, we obtain that   capp .Bj n ; Bj 1 / 1=.p1/ capp .Bj ; Bj 1 /  Z .100/1j r  capp .B.x0 ; t / n ; B.x0 ; 2t // 1=.p1/ dt C : capp .B.x0 ; t /; B.x0 ; 2t // t .100/j r Summing up over j D 1; 2; ::: ; k; and inserting this into Theorem 11.21, together with (11.8), finishes the proof. Theorem 11.24 (Sufficiency of the Wiener criterion). A point x0 2 @ is regular if for some ı > 0,  Z ı capp .B.x0 ; t / n ; B.x0 ; 2t // 1=.p1/ dt D 1: (11.9) t capp .B.x0 ; t /; B.x0 ; 2t // 0

294

11 Boundary regularity

Proof. Let 0 <  r < diam.X /=400 be arbitrary. Theorem 11.23 with f .x/ D d.x/ D d.x; x0 / implies that sup \B.x0 ; /

P d  100r

   Z r capp .B.x0 ; t / n ; B.x0 ; 2t // 1=.p1/ dt : C diam./ exp C capp .B.x0 ; t /; B.x0 ; 2t // t

By (11.9), the second term on the right-hand side tends to zero as ! 0. Thus, for sufficiently small , sup P d  200r \B.x0 ; /

and letting r ! 0, together with Theorem 10.29, (b) ) (a), finishes the proof. Next, we formulate several simple geometrical conditions which are sufficient for boundary regularity. These conditions are easier to verify than the Wiener criterion and are often sufficient in applications. Corollary 11.25. Let x0 2 @ and assume that one of the following conditions holds: (a) (Corkscrew) The complement of  has a corkscrew at x0 , i.e. there exists c > 0 such that for all sufficiently small r, the set B.x0 ; r/ n  contains a ball with radius cr. (b) (Fatness) The complement of  is p-fat at x0 , i.e. there exists c > 0 such that for all sufficiently small r, capp .B.x0 ; r/ n ; B.x0 ; 2r//  c capp .B.x0 ; r/; B.x0 ; 2r//: (c) (Geometric porosity) There is c > 0 and a sequence rj ! 0 such that for every j D 1; 2; ::: , the set B.x0 ; rj / n  contains a ball with radius crj . (d) (Measure porosity) There is c > 0 and a sequence rj ! 0 such that for every j D 1; 2; ::: , .B.x0 ; rj / n /  c.B.x0 ; rj //. (e) (Capacitary porosity) There is c > 0 and a sequence rj ! 0 such that for every j D 1; 2; ::: , capp .B.x0 ; rj / n ; B.x0 ; 2rj //  c capp .B.x0 ; rj /; B.x0 ; 2rj //: Then x0 is regular. These are all various types of porosity conditions, the last one being the weakest. Proof. That (a) ) (c) and (b) ) (e) are obvious. That (c) ) (d) follows from Lemma 3.3, and that (d) ) (e) follows from Proposition 6.16.

11.5 Regularity componentwise

295

1 Assume therefore that (e) holds. We may also assume that r1 < 16 diam X and 1 that rj C1 < 2 rj , j D 1; 2; ::: . It then follows from Lemma 11.22, Proposition 6.16 and the doubling property of  that for all rj  t  2rj , j D 1; 2; ::: ,

capp .B.x0 ; t / n ; B.x0 ; 2t //  capp .B.x0 ; rj / n ; B.x0 ; 2t //  C capp .B.x0 ; rj / n ; B.x0 ; 2rj //  C capp .B.x0 ; rj /; B.x0 ; 2rj // C.B.x0 ; rj //  rjp C.B.x0 ; t // tp 0  C capp .B.x0 ; t /; B.x0 ; 2t //; 

where C 0 > 0 is independent of j . Hence  Z 2r1  capp .B.x0 ; t / n ; B.x0 ; 2t // 1=.p1/ dt 0



capp .B.x0 ; t /; B.x0 ; 2t // 1 X Z 2rj  capp .B.x0 ; t / n j D1 rj



1 X

t ; B.x0 ; 2t //

capp .B.x0 ; t /; B.x0 ; 2t //

1=.p1/

dt t

.C 0 /1=.p1/ log 2

j D1

D 1: Thus x0 is regular, by Theorem 11.24. Open problem 11.26. (Necessity of the Wiener criterion) Is it true that if x0 is regular and ı > 0, then (11.9) holds? This is known to be true in Rn , see the notes to this chapter for the references. For Cheeger p-harmonic functions on metric spaces (see Appendix B.2), a positive answer was obtained by J. Björn [65]. For p-harmonic functions defined using minimal upper gradients, only a weaker result is available so far, viz. that the Wiener condition with the exponent 1=p instead of 1=.p1/ is necessary for boundary regularity, see J. Björn [67]. For examples of irregular boundary points, see Examples 13.3, 13.4, 14.3 and 14.4.

11.5 Regularity componentwise Recall that in our considerations,  need not be connected. Even if  is connected, then it can easily happen that it is not locally connected at a certain boundary point, i.e.

296

11 Boundary regularity

for every sufficiently small neighbourhood V of this point,  \ V falls apart into two or more components. Since boundary regularity is a local property by Theorem 11.11, we may equivalently consider regularity with respect to  \ V . The following result is thus of fundamental interest. Theorem 11.27. Assume that x0 is irregular with respect to . Then there is exactly one component G of  with x0 2 @G such that x0 is irregular with respect to G. Example 11.28. Let B.0; 1/ be the unit disc in R2 D C and let ˚  D B.0; 1/ n z D re i 2 C W 0  r  12 and  2 F ; where F  Œ0; 2 is closed and consists of at least  two  points (0 and 2 are considered to be the same point). The open set  \ B 0; 12 consists of at most countably many conical components, each of which is regular at the origin by e.g. the corkscrew condition in Corollary 11.25. Theorem 11.27 then implies that the origin is a reg ular boundary point of  \ B 0; 12 and hence of , as well. In particular, letting F D f0; g shows that the origin is regular with respect to the “cut” disc. The following interesting consequence follows directly. Corollary 11.29. Let x0 2 @ and assume that there is no component G of  such that x0 2 @G. Then x0 is regular with respect to . Example 11.30. Let B.0; 1/ be the unit ball in Rn , n  1, and let faj gj1D1 be a strictly decreasing sequence such that aj ! 0, as j ! 1. Then the set  D fx 2 B.0; 1/ W 0 < jxj ¤ aj ; j D 1; 2 ::: g is open and 0 2 @, but 0 does not belong to the closure of any of ’s components. Corollary 11.29 then implies that 0 is regular with respect to . In this example,  is not connected, but adding one extra dimension produces the connected set 0 D . .1; 1/ [ .B.0; 1/ .1; 2// in RnC1 , with the origin as a regular boundary point. Another way of making  connected in Rn is by adding small relatively open sets from each sphere in the boundary of . Example 11.31. Let Bj  B.0; 1/  Rn , n  1, be a sequenceS of pairwise disjoint balls such that 0 … @Bj , j D 1; 2; ::: ; but 0 2 @, where  D j1D1 Bj : Then 0 is regular, by Corollary 11.29. If we let 0 D  .1; 1/  RnC1 , then again the origin will be regular with respect to 0 , by Corollary 11.29.    Neither  nor 0 is connected, but 00 D 0 [ B.0; 1/ 12 ; 1 is connected with the origin as a regular boundary point.

11.5 Regularity componentwise

297

In order to prove Theorem 11.27 we will need the following consequence of (the sufficiency part of) the Wiener criterion. Lemma 11.32. Let 1 ; 2  X be two nonempty disjoint bounded open sets and let x0 2 @1 \ @2 . Then x0 is regular with respect to at least one of 1 and 2 . Proof. By the subadditivity of the variational capacity capp we obtain that  Z 1 capp .B.x0 ; r/; B.x0 ; 2r// 1=.p1/ dr 1D capp .B.x0 ; r/; B.x0 ; 2r// r 0 1=.p1/ Z 1 capp .B.x0 ; r/ n 1 ; B.x0 ; 2r// dr C capp .B.x0 ; r/; B.x0 ; 2r// r 0 1=.p1/ Z 1 capp .B.x0 ; r/ n 2 ; B.x0 ; 2r// dr CC : cap .B.x ; r/; B.x r ; 2r// 0 0 0 p Hence at least one of the integrals in the right-hand side must equal 1, and x0 is therefore regular with respect to at least one of 1 and 2 , by Theorem 11.24. (Observe that Cp .X n 1 /  Cp .2 / > 0 and similarly for X n 2 .) Proof of Theorem 11.27. As x0 is irregular, Theorem 10.29 shows that lim sup P d.y/ > 0: 3y!x0

Thus, there is a sequence fyj gj1D1 such that  3 yj ! x0 ; as j ! 1;

and

lim P d.yj / D lim sup P d.y/:

j !1

3y!x0

Assume first that there are infinitely many components of  containing points from the sequence fyj gj1D1 . Then we can find a subsequence fyjk g1 such that each kD1 . Let Gk be component of  contains at most one point from the sequence fyjk g1 kD1 the component of  containing yjk . We see that lim P d.yj2k / D lim P d.yj2kC1 / > 0;

k!1

k!1

S and thus x0 is irregular both with respect to 1 D 1 kD1 G2k and with respect to S1 2 D kD1 G2kC1 . As 1 and 2 are disjoint this contradicts Lemma 11.32. Hence there are only finitely many components of  containing points from the sequence fyj gj1D1 . So there must be a component G containing infinitely many of the points from the sequence fyj gj1D1 . We can thus find a subsequence fyjk g1 such that kD1 yjk 2 G for k D 1; 2; ::: . As lim P d.yjk / > 0;

k!1

we find that x0 2 @G and that it is irregular with respect to G. Finally, if G 0 is any other component of  with x0 2 @G 0 , then, by Lemma 11.32, x0 is regular with respect to G 0 .

298

11 Boundary regularity

11.6 Fine continuity In Section 5.2 we saw that all Newtonian functions are quasicontinuous, i.e. that they are continuous on complements of open sets with arbitrarily small capacity. Another notion of continuity, related to capacity and the Wiener criterion, is fine continuity, based on the following notion of thinness. Definition 11.33. A set E  X is thin at x 2 X if  Z 1 capp .E \ B.x; t /; B.x; 2t // 1=.p1/ dt 0

capp .B.x; t /; B.x; 2t //

t

< 1:

(11.10)

A set U  X is finely open if X n U is thin at every x 2 U . Equivalently, a set is finely closed if it contains all points at which it is not thin. The finely open sets generate the fine topology on X . In the definition of thinness we make the convention that the integrand is 1 whenever capp .B.x; t/; B.x; 2t // D 0. This happens e.g. if X D B.x; 2t / is bounded, but never if t < 18 diam X , by Proposition 6.16. It is clear that the definition of thinness can be equivalently stated using a sum instead of an integral, and that the upper limit on the integral can be replaced by any positive number. This shows that thinness is a local property, i.e. if ı > 0, then E is thin at x if and only if E \ B.x; ı/ is thin at x. Moreover, by Lemma 11.22, the factor 2 in B.x; 2t/ can be replaced by any other number greater than 1. Note also that the concepts above depend on p, even though we have refrained from making this explicit in the notation. Example 11.34. The Lebesgue spine E in Example 13.4 is thin at the origin and hence the set .R3 n E/ [ f0g is finely open, though not open. Example 11.35. If G is an open set, then G is finely open, as the integrand in (11.10) is zero for small t. More generally, if G is an open or finely open set and E is an arbitrary set with Cp .E/ D 0, then G n E is finely open, since sets of zero capacity have no influence on the integral in (11.10). Proposition 11.36. The fine topology is indeed a topology on X . Proof. We need to show that arbitrary unions and finite intersections of finely open sets are finely open. S Let first x 2 U WD ˛2A U˛ , where U˛ , ˛ 2 A, are finely open sets, and A is some index set. Then there is ˇ 2 A such that x 2 Uˇ . It follows that  Z 1 capp .B.x; t / n U; B.x; 2t // 1=.p1/ dt t capp .B.x; t /; B.x; 2t // 0 1=.p1/ Z 1 capp .B.x; t / n Uˇ ; B.x; 2t // dt  < 1; capp .B.x; t /; B.x; 2t // t 0

11.6 Fine continuity

299

as Uˇ is finely open. Hence X n U is p-thin at x and as x 2 U was arbitrary, U is finely open. T Let now instead x 2 U WD jND1 Uj , where U1 ; U2 ; ::: ; UN are finely open. Then, by the subadditivity of the variational capacity, Z 1 0

capp .B.x; t / n U; B.x; 2t //

1=.p1/

dt capp .B.x; t /; B.x; 2t // t   N Z 1 X capp .B.x; t / n Uj ; B.x; 2t // 1=.p1/ dt C < 1; capp .B.x; t /; B.x; 2t // t 0 j D1

where C depends on N and p. Thus, as above, U is finely open. x defined on a finely open set U , is finely Definition 11.37. A function u W U ! R, x with continuous if it is continuous when U is equipped with the fine topology, and R the usual topology. Note that u is finely continuous in U if and only if it is finely continuous at every x 2 U in the sense that for all " > 0 there exists a finely open set V 3 x such that ju.y/  u.x/j < " for all y 2 V , if u.x/ 2 R, and such that ˙u.y/ > 1=" for all y 2 V , if u.x/ D ˙1, or equivalently if and only if the sets fx 2 U W u.x/ > kg and fx 2 U W u.x/ < kg are finely open for all k 2 R. Every open set is finely open and thus the fine topology generated by the finely open sets is finer than the metric topology. In fact, it is so fine that all superharmonic functions become finely continuous. Theorem 11.38. Let   X be an arbitrary nonempty open set, and let u be a superharmonic function in . Then u is finely continuous in . Proof. Let k be an arbitrary real number. By the lower semicontinuity of u, the set fx 2  W u.x/  kg is closed (and hence finely closed). We shall show that the set Ek WD fx 2  W u.x/  kg is also finely closed. It suffices to show that Ek is thin at every x 2  for which u.x/ < k. Consider such an x and find a ball B D B.x; r/ so that B0 D 100B b  and Cp .X n B0 / > 0. As u is locally bounded from below, we may assume that u  0 in B0 and that k > 0. Let uk D minfu; kg. Lemma 9.3 and Theorem 9.24 show that uk is an lsc-regularized superminimizer in  and hence the lsc-regularized solution of the Kuk ;uk .B0 /-obstacle problem, by Proposition 7.15. Let v be the capacitary potential for Ek \ B in B0 , i.e. the lscregularized solution of the KEk \B ;0 .B0 /-obstacle problem. As uk  kEk \B  0 in B0 , the comparison principle (Lemma 8.30) shows that uk  kv in B0 . Assume now that Ek is not thin at x. Theorem 11.21 together with Lemma 11.22 then yields v.x/ D 1. It follows that u.x/  uk .x/  kv.x/ D k, which contradicts the assumption u.x/ < k and finishes the proof.

300

11 Boundary regularity

Corollary 11.39. Assume that  is bounded and Cp .X n / > 0. Let E b  and let p Ex be the fine closure of E, i.e. the smallest finely closed set containing E. Then capp .Ex ; / D capp .E; /: p

(11.11)

Note that the corresponding result for the usual topology is far from true. Let e.g. E be a countable dense subset of B.0; 1/ and  D B.0; 2/  Rn , 1 < p  n. Then x / D capp .B.0; 1/; / > 0 D capp .E; /. capp .E; Proof. One inequality is trivial. To prove the other one, assume first that E is open and let u be the capacitary potential for E in , i.e. the lsc-regularized solution of the KE ;0 ./-problem. Note that as E is open, u D 1 in E. By Theorem 9.24, u is superharmonic and hence finely continuous in , by Theorem 11.38. Thus, the set p F D fx 2 X W u.x/  1g is finely closed and contains E. Hence Ex  F . It follows p that u is admissible in the definition of capp .Ex ; / and Lemma 11.19 yields Z p x gup d D capp .E; /; capp .E ; /  

proving the corollary for open E. For general E, Theorem 6.19 (vii) implies that capp .E; / D

inf

G open E G

capp .G; / D

inf

G open E G

x ; /  capp .Ex ; /; capp .G p

p

x p is the fine closure of G. where G Another consequence of the fine continuity of superharmonic functions is the following theorem showing that Newtonian functions are finely continuous at q.e. point. Theorem 11.40. Let   X be an arbitrary nonempty open set. Then every quasicontinuous function u W  ! Œ1; 1 is finely continuous at q.e. x 2 . In particular, 1;p this is true for all u 2 Nloc ./. In Rn , quasicontinuous functions are exactly those functions which are finely continuous at q.e. point, see Malý–Ziemer [258]. Proof. By the subadditivity of the capacity, it suffices to prove the theorem on small balls. Let B be a ball such that 2B b  and Cp .X n 2B/ > 0. As u is quasicontinuous, there exist open sets Uj  B, j D 1; 2; ::: , such that the restriction ujBnUj is continuous and capp .Uj ; 2B/ < 1=j: p p (Here we have used Proposition 6.16.) Let Uxj be the fine closure of Uj . If x 2 B n Uxj p for some j , then for all ı > 0, the set B.x; ı/ n Uxj is a fine neighbourhood of x and as ujBnUj is continuous, we get that u is finely continuous at x.

11.7 Notes

Thus, in view of Proposition 6.16, it suffices to show that capp Corollary 11.39 implies that capp

1 \

 T1

j D1

301

 p Uxj ; 2B D 0.

 p p Uxj ; 2B  capp .Uxj ; 2B/ D capp .Uj ; 2B/ < 1=j

j D1

and letting j ! 1 finishes the proof.

11.7 Notes In 1924, Wiener [352] characterized regular boundary points for the Laplace equation by the so-called Wiener criterion containing the Wiener integral (11.9) (with p D 2). In fact, Wiener [351] provided a very general sufficient condition (similar to (e) in Corollary 11.25) a few months earlier. The same year Lebesgue [241] characterized regular boundary points in terms of barriers. The Wiener criterion was generalized to linear uniformly elliptic equations by Littman–Stampacchia–Weinberger [246] and to weighted linear equations by Widman [350] (for power weights) and Fabes–Jerison–Kenig [116]. Meanwhile, Maz0 ya [277], [279], obtained pointwise boundary estimates similar to (11.7) for solutions of linear uniformly elliptic equations, which imply the sufficiency part of the Wiener criterion for such equations. They also lead to sufficient conditions for boundary Hölder regularity of the solutions. Similar capacitary estimates (and thus the sufficiency part of the Wiener criterion) for nonlinear equations of p-Laplace type were obtained by Maz0 ya [280] and, using a different method, Gariepy–Ziemer [134]. A partial result on boundary regularity appeared earlier in Gariepy–Ziemer [133]. The weighted situation was treated in Heinonen–Kilpeläinen–Martio [171] and subelliptic equations were considered by Danielli [104]. On metric spaces, the sufficiency part of the Wiener criterion was first proved in J. Björn–MacManus–Shanmugalingam [70] under the additional geometrical assumption that X is linearly locally convex. This assumption was removed in J. Björn [66]. The proof given here is a combination of [66] and [70]. The proof of Lemma 11.20 has been inspired by Lindqvist–Martio [245]. Hölder boundary regularity for p-harmonic functions on metric spaces was studied in J. Björn [63] and Aikawa–Shanmugalingam [10]. The necessity part of the Wiener criterion for nonlinear equations of p-Laplace type was proved by Kilpeläinen–Malý [204] for unweighted Rn , and extended to the weighted case in Mikkonen [285]. In the special case p > n  1, on unweighted Rn , it was obtained already in Lindqvist–Martio [245]. For nonlinear subelliptic problems with measure data, the necessity of the Wiener criterion was proved by Biroli– Tchou [31]. On metric spaces, it was obtained for Cheeger p-harmonic functions

302

11 Boundary regularity

in J. Björn [65], whereas a weaker condition was given for p-harmonic functions in J. Björn [67], see the discussion after Open problem 11.26. The regularity characterizations presented in Sections 11.1–11.3 are based on Björn– Björn [45]. Barriers (of all types) are often defined in a local way, but we prefer global definitions. That also local barriers characterize regularity follows directly using that regularity is a local property. In A. Björn–Martio [60], the following condition was shown to characterize regux which is larity (cf. condition (n) in Theorem 11.11): For any function f 2 N 1;p ./ superharmonic in  and such that f j@ is continuous at x0 , lim inf f .y/ D f .x0 /:

3y!x0

To obtain this characterization, the cluster set results of A. Björn [43] are used. As we do not include these results, we cannot provide the proof here. Lemma 11.19 is essentially from J. Björn [66] and Björn–Björn–Mäkäläinen– Parviainen [52]. Theorem 11.27 is from A. Björn [43], it had earlier been obtained (without the word “exactly”) for Cheeger p-harmonic functions in metric spaces and for A-harmonic functions in weighted Rn , with a completely different proof, in A. Björn [40]. The results in Section 11.6 are from J. Björn [66]. Theorems 11.38 and 11.40 were also obtained by Korte [224] at the same time. Her proof of Theorem 11.38 is quite different and uses Caccioppoli type estimates. Weaker versions of Theorems 11.38 and 11.40 (with q-capacity, q < p, instead of p-capacity) were obtained by Kinnunen– Latvala [213]. For a historical account on fine continuity and topology (in nonlinear potential theory), we refer the reader to the books by Heinonen–Kilpeläinen–Martio [171] and Malý–Ziemer [258], and to the references therein. For p-fatness, as defined in Corollary 11.25 (b) see the notes to Chapter 5.

Chapter 12

Removable singularities

Recall that in Chapters 7–14 we assume that 1 < p < 1 and that X is a complete doubling p-Poincaré space, with doubling constant C and dilation constant  in the p-Poincaré inequality. In this chapter we also let E    be relatively closed and such that no component of  is completely contained in E, and K    be compact. In the previous chapters, we saw that sets of capacity zero on the boundary often do not have any influence on solutions of the Dirichlet problem. For example, if  is the punctured ball B.x0 ; r/ n fx0 g and Cp .fx0 g/ D 0, then the value at the origin does not play any role, the solution is determined by its values at @B and will be p-harmonic in the whole of B, see Theorem 10.22 and Corollary 10.23. This is the simplest example of a removable singularity for p-harmonic functions. In this chapter we study which sets are removable for p-harmonic and superharmonic functions. First, we consider sets of capacity zero and provide conditions under which they are removable. This result will play a crucial role, together with the Kellogg property, in establishing the trichotomy (Theorem 13.2) in the next chapter. We also give some nonremovability results and in some cases obtain a complete characterization of removable sets. Finally, we provide examples of removable sets with positive capacity, and show that such sets can have rather peculiar properties.

12.1 Removability Definition 12.1. The set E is (uniquely) removable for bounded p-harmonic functions in  n E if every bounded p-harmonic function in  n E has a (unique) bounded p-harmonic extension to . Similarly, E is (uniquely) removable for p-harmonic functions in N 1;p . n E/ if every p-harmonic function u 2 N 1;p . n E/ has a (unique) p-harmonic extension in N 1;p ./. Removability for superharmonic functions is defined analogously. Here, when we e.g. talk about a superharmonic function u 2 N 1;p ./, we mean that it is superharmonic in . Also, if u is defined on  n E, then U is an extension of u to  if U jnE D u and U is defined on . In view of Theorems 12.2 and 12.3 below one might get the feeling that it is not essential to stress the set  n E in the definition, and that the notation could therefore

304

12 Removable singularities

be simplified. However, in Example 12.22 (a) we see that for removable singularities of positive capacity it is essential to always stress on which set we discuss removability. In this section we show that sets of capacity zero are removable for p-harmonic and superharmonic functions that are either bounded or in N 1;p . Moreover, the extensions are always unique in these cases. Theorem 12.2. Assume that Cp .E/ D 0. Let u be a p-harmonic function in  n E. Assume further that one of the following conditions holds: (a) u 2 N 1;p . n E/; 1;p (b) u is the restriction of a function in Nloc ./;

(c) u 2 N 1;p .B n E/ for every ball B b ; (d) u is bounded. Then u has a unique superharmonic extension U to . Moreover, U is p-harmonic in , U is bounded if u is bounded, and U 2 N 1;p ./ if u 2 N 1;p . n E/. The proof of Theorem 12.2 is based on the following theorem. Theorem 12.3. Assume that Cp .E/ D 0. Let u be a superharmonic function in  n E. Assume further that one of the conditions (a)–(d) in Theorem 12.2 holds, or that (e) u is bounded from below. Then u has a unique superharmonic extension U to . Furthermore, U is bounded (bounded from below) if u is bounded (bounded from below), and U 2 N 1;p ./ if u 2 N 1;p . n E/. Note that to find the extension U , we only need to find the unique lsc-regularized extension of u to ; U.x/ D ess lim inf nE 3y!x u.y/. In Theorem 12.2, part of the conclusion is that this extension is continuous in . Note also that some condition of the type (a)–(e) is needed: Consider a bounded open set   Rn (unweighted), n  3, p D 2, and let v be the Green function of  with respect to some y 2  (or v.x/ D jx  yj2n ). Then v is a p-harmonic function in  n fyg with no superharmonic extension to . Let us next observe that unique removability for bounded superharmonic functions is the same as for superharmonic functions bounded from below. Note that in Section 12.4 we give examples of sets which are nonuniquely removable for p-harmonic functions, but we do not know if there are any nonuniquely removable sets for superharmonic functions. Proposition 12.4. The set E is uniquely removable for superharmonic functions bounded from below in  n E if and only if it is uniquely removable for bounded superharmonic functions in  n E.

12.1 Removability

305

Proof. Assume first that E is uniquely removable for superharmonic functions bounded from below in  n E, and let u be a bounded superharmonic function in  n E. By assumption, u has a unique superharmonic extension U to  which is bounded from below. We need to show that U is bounded. Let Uz D minfU; supnE ug. Then Uz is also a superharmonic extension of u to . By the assumed uniqueness we must have Uz D U in , and hence U is bounded. As for the converse, assume that E is uniquely removable for bounded superharmonic functions in  n E. Let u be a superharmonic function bounded from below in  n E. Then uj WD minfu; j g has a superharmonic extension Uj to . As minfUj C1 ; Uj g D uj in  n E, both Uj and minfUj C1 ; Uj g are superharmonic extensions of uj to . By uniqueness, Uj D minfUj C1 ; Uj g and thus Uj C1  Uj in . Let U D limj !1 Uj . Then U D u in  n E and hence U is not identically 1 in any component of . By Theorem 9.27, U is a superharmonic extension of u to . As for the uniqueness, let U be a superharmonic extension of u to . Then for every k D 1; 2; ::: , the function minfU; kg is the unique superharmonic extension of minfu; kg to . It follows that U is unique. Proof of Theorem 12.3. The uniqueness follows from the observation above, i.e. we let U.x/ D ess lim inf u.y/; x 2 : nE 3y!x

It is easily verified that U is lsc-regularized and that U D u in  n E. Assume first 1;p that (c) holds. It follows that u 2 Nloc . n E/. Hence u is a superminimizer in  n E, by Corollary 9.6. Moreover, gu is a p-weak upper gradient of U in  n E. By Proposition 1.48, Modp . E / D 0, and hence gu is a p-weak upper gradient of U in  (where we define gu  0 arbitrarily in E). The argument at the beginning of the proof of Lemma 2.23 shows that it is also minimal. Let B b . Since .E/ D 0, we 1;p see that kU kN 1;p .B/ D kukN 1;p .BnE / < 1, and thus U 2 Nloc ./. Moreover, if 1;p 1;p u 2 N . n E/, then U 2 N ./ . We shall now show that U is a superminimizer in . Let ' 2 N01;p ./ be nonnegative. Then also ' 2 N01;p . n E/, as Cp .E/ D 0. Since u is a superminimizer in  n E, we see that (using characterization (d) in Proposition 7.9) Z Z Z Z p p p p gU d D gu d  guC' d D gU C' d; '¤0

A

A

'¤0

where A D fx 2  n E W '.x/ ¤ 0g and we have used that u C ' D U C ' a.e. Thus, U is a superminimizer in , and as it is lsc-regularized, it is superharmonic in , by Proposition 9.4. That (a) ) (c) and (b) ) (c) hold is clear. We shall next show that (d) ) (c). Let B b  and 2 Lipc ./ be such that 0   1 in  and D 1 in B. Since E \ supp is compact and has zero capacity, Theorem 6.7 (xi) shows that there exists a sequence j 2 Lip.X / such that k j kN 1;p .X/ ! 0, as j ! 1, 0  j  1 in X and j D 1 on E \ supp .

306

12 Removable singularities

We may also assume that j ! 0 a.e., as j ! 1. Let j D .1  2 j /C . Then

j 2 Lipc . n E/, 0  j  1 and j ! in N 1;p .X /, as j ! 1. Without loss of generality, 21  u  1. Using the Caccioppoli inequality for superminimizers (Proposition 8.8) with ˛ D p  1 we see that Z Z Z p p p p p gu j d  u gu j d  C gpj d: supp nE

nE

By Fatou’s lemma Z Z gup d  lim inf BnE

j !1

supp nE

gup jp d

Z

 C lim inf j !1

nE

nE

Z gpj

d D C nE

gp d < 1:

Since u is bounded we conclude that u 2 N 1;p .B n E/, and thus (c) holds. Assume now that (e) holds. As we have shown that E is uniquely removable for bounded superharmonic functions in  n E, it follows from Proposition 12.4 that u has a unique superharmonic extension to , which must equal U . In order to obtain Theorem 12.2 from Theorem 12.3 we formulate the following result. For future use we make it more general than what is needed here. Proposition 12.5. Assume that .E/ D 0 and that u is a p-harmonic function in  n E, which has a superharmonic extension U and a subharmonic extension V to . Then both U and V are unique and U D V is p-harmonic in . If u is bounded, then U is also bounded, and if u 2 N 1;p . n E/, then U 2 N 1;p ./. Proof. Since U is lsc-regularized and .E/ D 0, we have U.x/ D ess lim inf U.y/ D ess lim inf u.y/; 3y!x

nE 3y!x

x 2 :

Thus, U is unique. Moreover, if u is bounded, then U is bounded. Similarly, V is unique. Let B b . As V is upper semicontinuous in , it is bounded from above in B. It follows that u is bounded from above in B nE, and hence also U is bounded from above 1;p in B. By Corollary 9.6, U is a superminimizer in  and U 2 Nloc ./. Similarly, 1;p V 2 Nloc ./ is a subminimizer in . Since U D V a.e. in , Proposition 1.59 shows that U D V q.e. in . Thus, U is also a subminimizer in , and hence a minimizer in , by Proposition 7.8, and there exists a p-harmonic function W such that W D U D u a.e. in . Since both W and u are continuous in  n E, they coincide in  n E. By the uniqueness of U and V , we have U D W D V in . As  n E is open, the minimal p-weak upper gradient gU of U in  is also minimal as a p-weak upper gradient in  n E, by Lemma 2.23. Hence kU kN 1;p ./ D kukN 1;p .nE / .

12.1 Removability

307

Proof of Theorem 12.2. This now follows directly by combining Theorem 12.3 and Proposition 12.5. By combining Theorems 12.2, 12.3 and Propositions 12.11 and 12.12 below, we obtain the following characterizations of removability for (almost) compact subsets of . (Note that the condition Cp .X n / is essential, see Example 12.19.) Corollary 12.6. Assume that  is bounded, Cp .X n / > 0 and E has empty interior. Then the following are equivalent: (a) Cp .K [ E/ D 0; (b) K [ E is removable for bounded p-harmonic functions in  n .K [ E/; (c) K [ E is removable for p-harmonic functions in N 1;p . n .K [ E//. If moreover .E/ D 0, then also the following statements are equivalent to those above: (d) K [ E is removable for bounded superharmonic functions in  n .K [ E/; (e) K [ E is removable for superharmonic functions in N 1;p . n .K [ E//. In particular, letting E D ¿ provides equivalent characterizations of removability for compact subsets K of . The following characterization of removable singularities, in spaces where singletons have zero capacity, is an immediate consequence of Theorem 12.2 and Proposition 12.14 below. (Recall that we do not allow E to contain any component of .) Corollary 12.7. Assume that  is bounded and Cp .fxg/ D 0 for every x 2  \ @E. Then the following are equivalent: (a) Cp .E/ D 0; (b) E is removable for bounded p-harmonic functions in  n E; (c) E is removable for p-harmonic functions in N 1;p . n E/. We end this section by observing that removability for p-harmonic (superharmonic) functions is the same as removability for (super)minimizers. By saying that removability is unique for (super)minimizers we mean that the extensions are unique up to capacity zero. Proposition 12.8. The set E is (uniquely) removable for bounded (super)minimizers in  n E if and only if it is (uniquely) removable for bounded p-harmonic (superharmonic) functions in  n E.

308

12 Removable singularities

The proof shows that the word “bounded” in Proposition 12.8 can be replaced by any of the conditions (a)–(d) from Theorems 12.2 and 12.3. (See Proposition 12.9 and Open problem 12.10 for results with condition (e).) Proof. Assume first that E is removable for bounded (super)minimizers in  n E, and that u is a bounded p-harmonic (superharmonic) function in  n E. Then u is a (super)minimizer in  n E, and has a bounded (super)minimizer extension U to . Let U  be the lsc-regularization of U , a p-harmonic (superharmonic) function in . Since both U  and u are lsc-regularized in  n E and U  D u q.e. in  n E, we see that U  D u everywhere in  n E, i.e. U  is a bounded p-harmonic (superharmonic) extension of u to . Conversely, assume that E is removable for bounded p-harmonic (superharmonic) functions in  n E, and that u is a bounded (super)minimizer in  n E. Then u is p-harmonic (superharmonic) in nE and has a bounded p-harmonic (superharmonic) extension U  to . Thus ´ U  in E; U D u in  n E is a (super)minimizer extending u to . If the removability is not unique in one case, then the constructions above can be used to obtain two extensions which differ in a set of positive capacity from which the nonuniqueness of the other case follows. The following removability result for superminimizers is an easy consequence of Propositions 12.4 and 12.8. Proposition 12.9. The following are equivalent: (a) E is uniquely removable for bounded superharmonic functions in  n E; (b) E is uniquely removable for superharmonic functions bounded from below in  n E; (c) E is uniquely removable for bounded superminimizers in  n E; (d) E is uniquely removable for superminimizers bounded from below in  n E. Proof. (a) , (b) This follows from Proposition 12.4. (a) , (c) This follows from Proposition 12.8. (c) , (d) This is shown just as in Proposition 12.4. We do not know if the corresponding result without uniqueness is true. Open problem 12.10. Is Proposition 12.9 true if the word “uniquely” is deleted (four times)?

12.2 Nonremovability

309

12.2 Nonremovability In the previous section we proved that sets of zero capacity are removable. In this section, we shall see that sets of positive capacity are in many cases nonremovable. Note, however, that there are situations when some sets of positive capacity are removable, see Section 12.3. Proposition 12.11. Assume that  is bounded, Cp .X n / > 0 and .E/ D 0 < Cp .K [ E/. Then there is a bounded p-harmonic function in N 1;p . n .K [ E// with no superharmonic extension to . When we just consider p-harmonic extensions we can allow for slightly more general sets E. Proposition 12.12. Assume that  is bounded, Cp .X n / > 0, Cp .K [ E/ > 0 and that E has empty interior. Then there exists a bounded p-harmonic function in N 1;p . n .K [ E// with no p-harmonic extension to . Note that in neither Proposition 12.11 nor 12.12 the requirement Cp .X n / > 0 can be omitted, not even when E D ¿, see Example 12.19. Proof of Proposition 12.11. Let Kj D fx 2 K [ E W dist.x; X n /  1=j g;

j D 1; 2; ::: ;

which are compact sets. By Theorem 6.4, limj !1 Cp .Kj / D Cp .K [ E/ > 0. Hence for large enough j , we have K  Kj and Cp .Kj / > 0. Let K 0 D Kj for such a j . Let v be the capacitary potential of K 0 in , i.e. the lsc-regularized solution of the KK 0 ;0 -obstacle problem. Then u D 1  v is p-harmonic in  n K 0 and 0  u  1. Assume that there is a superharmonic function U in  such that U D u in the open set  n .K [ E/. The continuity of u and the lsc-regularity of U together with the condition .E/ D 0 imply that U D u in  n K 0 . Since u  0, it follows that U  0 in , by the minimum principle (Theorem 9.13). Lemma 4.5 implies that Cp .@K 0 / > 0 and by the Kellogg property (Theorem 10.5), there exists a regular point x0 2 @K 0 , and thus lim

nK 0 3y!x0

U.y/ D

lim

nK 0 3y!x0

u.y/ D u.x0 / D 0:

By the lower semicontinuity of U , U.x0 / D 0. The strong minimum principle (Theorem 9.13) implies that U  0 in the component G   containing x0 . Since Cp .X n G/  Cp .X n / > 0, Lemma 4.5 and the Kellogg property (Theorem 10.5) however imply that there is a regular point x1 2 @G (with respect to G). Hence lim

G3y!x1

U.y/ D 1;

and thus U is nonconstant in G, a contradiction.

310

12 Removable singularities

Proof of Proposition 12.12. This proof is almost identical to the proof of Proposition 12.11, we only need to modify the second paragraph a little as follows: Assume that there is a p-harmonic function U in  such that U D u in the open set  n .K [ E/. The continuity of u and U together with the condition that E has no interior implies that U D u in  n K 0 . The rest of the proof is identical to the proof of Proposition 12.11. If  is bounded and Cp .X n / D 0, then we have the following result. (Observe x is bounded and  is connected, by Lemma 4.6.) that in this case X D  Proposition 12.13. Assume that  is bounded, Cp .X n / D 0 and Cp .E/ > 0. Assume further that either (a) the capacity of  \ @E is not concentrated to one point, i.e. for all x 2  \ @E we have Cp .. \ @E/ n fxg/ > 0; or (b)  n E is disconnected. Then there is a bounded p-harmonic function in N 1;p . n E/ with no superharmonic extension to  which is bounded from below, nor any superharmonic extension in N 1;p ./. When the (positive) capacity of E is concentrated to one point and  n E is connected, it is possible both to have removability and nonremovability for p-harmonic functions, see Examples 12.15 and 12.19. Proof. In case (a), we can find two disjoint compact sets K1 ; K2   \ @E with positive capacity. Let f D 1 on K1 , f D 0 on K2 and let u D HXn.K1 [K2 / f . By the Kellogg property (Theorem 10.5) there are regular points x1 2 @K1 and x2 2 @K2 . It follows that u is nonconstant. In case (b), we let u  1 in one component of  n E and u  0 in all other components of  n E. Thus in both cases we have a nonconstant bounded p-harmonic function u in nE. Assume that u has a superharmonic extension U to , which is either bounded from below or belongs to N 1;p ./. Then, by Theorem 12.3, there is a superharmonic function V on X which is an extension of U and hence of u. By Corollary 9.14, V is constant, which contradicts the fact that u is nonconstant. When singletons have zero capacity we can treat even more general  and E. Proposition 12.14. Assume that  is bounded and that Cp .E/ > 0. If Cp .fxg/ D 0 for all x 2  \ @E, then there exists a bounded p-harmonic function in N 1;p . n E/ which has no p-harmonic extension to .

12.2 Nonremovability

311

Proof. There is a component G of  such that Cp .G \E/ > 0. Since by Lemma 4.5 we x \ @E/ > 0, have Cp .G \ @E/ > 0, Theorem 6.4 shows that there is  > 0 with Cp .G where G WD fx 2 G W dist.x; X n G/ > g (if G D X we set G D G). By the Kellogg property (Theorem 10.5) and by the fact that finite subsets of G \@E x have zero capacity, there exists a sequence fxn g1 nD1 of points in G \ @E which are x regular for the open set G n E. Since G is compact, without loss of generality we x \ @E, has no other may assume that this sequence converges to a point x1 2 G limit points, and moreover consists of distinct points. For each xn in this sequence, let Bn D B.xn ; rn / be a ball so that Bxn  G. We can also choose the balls Bn to be pairwise disjoint. It follows from Proposition 6.16 and Theorem 6.19 (x) that we can find 'n 2 Lipc .Bn / so that 'n .xn / D 1, 0  'n  1 and kg'n kLp .X/ < 2j . Let ˆD

1 X

'2n :

nD1

It is easy to see that ˆ 2 N 1;p .X / is a bounded lower semicontinuous function which is continuous except at x1 . Let u D HnE ˆ, which is a bounded p-harmonic function in N 1;p . n E/. We will show that u has no p-harmonic extension to . Since ˆ is continuous at xn , we see by Theorem 11.11 that u.y/ D ˆ.xn /:

lim

nE 3y!xn

´

Note that ˆ.xn / D

1; if n is even; 0; if n is odd.

1 Hence as x1 is the limit point of the sequence fxn g1 nD1 , we obtain a sequence fyn gnD1 3 1 in  n E converging to x1 and such that u.yn /  4 if n is even and u.yn /  4 if n is odd. Thus ujnE has no continuous extension to the point x1 2  \ @E. Since p-harmonic functions are continuous, u has no p-harmonic extension to .

Example 12.15. Let X D fz 2 C D R2 W jzj D 1g (with the one-dimensional Lebesgue measure), K D f1g and  D X n K. Then u.z/ D arg z (the principal branch) is a nonconstant p-harmonic function in . Since superharmonic functions on X are constant, by Corollary 9.14, u has no superharmonic extension to X , and K is not removable. Note that in this case no (nonempty) set is removable. The results in this section have so far been based on solving the Dirichlet problem for p-harmonic functions and are thus restricted to bounded . For unbounded , these tools are not available, but we are anyway able to give one nonremovability result which holds for general open sets.

312

12 Removable singularities

Proposition 12.16. Assume that for some component G   the set G n K is disconnected. Then there is a bounded p-harmonic function in  n K with no superharmonic extension to . Since sets of capacity zero cannot separate the space, by Lemma 4.6, this result only applies to sets of positive capacity. In view of Theorems 12.2 and 12.3, this is not surprising. Note that by Example 12.27 there are relatively closed removable sets E   for which  is connected and  n E is disconnected. Proof. Let u  1 in one component of G n K and u D 0 elsewhere in  n K. Then u is a bounded p-harmonic function in  n K. Assume that it has a superharmonic extension U to . Since U is lower semicontinuous there is x 2 G \ K such that U.x/ D inf G\K U . Thus, ˚ inf U D min U.x/; inf u D minfU.x/; 0g G

GnK

is attained at some point in G. By the strong minimum principle (Theorem 9.13), U is constant in G. But this contradicts the fact that u is nonconstant in G n K.

12.3 Removable sets with positive capacity In this section we construct some examples of sets with positive capacity which are removable for bounded p-harmonic functions. We also provide counterexamples to several properties of removable singularities which seem plausible but are not true in general. Example 12.20 is a little simpler than our first example, but we prefer to start with the following example. Example 12.17. Let X D R. In this one-dimensional setting it is easy to see that p-harmonic functions are exactly the linear functions, i.e. of the form x 7! ax C b, moreover this is true simultaneously for all p. We actually prove this in the second paragraph of the proof of Lemma A.27. (Observe that on bounded sets all p-harmonic functions are automatically bounded in this case.) Let  D .1; 1/ and E D .1; 0. Note that Cp .E/ > 0 and even .E/ > 0. Thus, if f W  n E ! R is a p-harmonic function, it is linear and thus directly extends to a p-harmonic (linear) function in , i.e. E is removable for p-harmonic functions. Note that the extension is unique. Let us now look at bounded superharmonic functions. In this setting, superharmonic functions are exactly the concave functions (again simultaneously for all p). Thus, f .x/ D x ˛ , with 0 < ˛ < 1, is a bounded superharmonic function in  n E, and since limx!0C f 0 .x/ D 1 we cannot find a superharmonic (i.e. concave) extension of f to all of . Thus, E is not removable for bounded superharmonic functions.

12.3 Removable sets with positive capacity

313

This example shows that the sets removable for bounded p-harmonic functions do not coincide with the sets removable for bounded superharmonic functions. A natural question to ask is the following question. Open problem 12.18. If E is removable for bounded superharmonic functions in  n E, does it then follow that E is removable for bounded p-harmonic functions in  n E? If .E/ D 0, then this follows from Proposition 12.5. Otherwise, if f is a bounded p-harmonic function in  n E, then it, of course, has a subharmonic extension and a superharmonic extension to all of , but it is not clear whether these can always be made to coincide. In the special case when  D X is bounded, we can find a compact set with positive capacity which is removable for p-harmonic functions. Example 12.19. Let X D  D Œ0; 1  R and K D f1g. As in Example 5.40 we see that Cp .K/ > 0. Let u be a p-harmonic function in X n K. Again u.x/ D ax C b. However, as both u and u obey the strong maximum principle (Theorem 8.13) and u.0/ is an extreme value, we see that a D 0 and u is constant. Thus, u is extendable to all of X as a p-harmonic function. Hence, K is removable for p-harmonic functions in X n K. Superharmonic functions in X n K are exactly concave nonincreasing functions, whereas all superharmonic functions in X are constant, by Corollary 9.14. Thus K is not removable for bounded superharmonic functions in X n K. In this case, the sets E removable for bounded p-harmonic functions in X n E are exactly the sets Œ0; a and Œa; 1 for 0  a  1, except for X itself. See also Example 8.18. Example 12.20. Let X D Œ0; 1/  R and  D Œ0; 1/. Then by the arguments in the previous example, a relatively closed set E   is removable for bounded p-harmonic functions in  n E if and only if E D Œa; 1/ for some 0 < a < 1. We have now seen several examples of sets with positive capacity, which are removable for bounded p-harmonic functions, but none of them is removable for bounded superharmonic functions. It is therefore natural to state the following problem. Open problem 12.21. Do there exist E,  and X such that Cp .E/ > 0 and E is removable for bounded superharmonic functions on  n E? Example 12.22. The space X D Œ0; 1  R in Example 12.19 provides us also with the following three counterexamples to seemingly plausible properties: (a) (Removability depends on the surrounding set.) Let K D f0g and  D Œ0; 1/. Then K is removable for bounded p-harmonic functions in X n K, but not for bounded p-harmonic functions in  n K. Observe that u.x/ D x is a bounded p-harmonic function on  n K with no p-harmonic extension to .

314

12 Removable singularities

 (b) (A subset of a removable set need not be removable.) Let K D 0; 12 and ˚ K 0 D 12  K. Then K is removable for bounded p-harmonic functions in X n K, but K 0 is not removable for bounded p-harmonic functions in X n K 0 . (c) (The union of removable sets need not be removable.) Let K1 D f0g and K2 D f1g. Then Kj is removable for bounded p-harmonic functions in X n Kj , j D 1; 2, but K1 [ K2 is not removable for bounded p-harmonic functions in X n .K1 [ K2 /. In all of these cases, the singularities have positive capacity. If we instead consider singularities with zero capacity, then it follows directly from Theorems 12.2 and 12.3, together with the monotonicity and the countable subadditivity of the capacity, that removability for p-harmonic (superharmonic) functions, which are bounded or belong to N 1;p , is independent of the surrounding set and is preserved for subsets and for countable unions of removable sets of zero capacity. In particular, this is true in the situation of Corollary 12.6, i.e. if  is bounded, Cp .X n / > 0 D Cp .K/ and K b  is compact. However, there is one general property that holds also for removable singularities of positive capacity. Proposition 12.23. Let E1 ; E2   be relatively closed and such that no component of  is contained in E1 [ E2 . Assume that E1 and E2 are (uniquely) removable for bounded p-harmonic (superharmonic) functions in  n E1 and  n .E1 [ E2 /, respectively. Then E1 [ E2 is (uniquely) removable for bounded p-harmonic (superharmonic) functions in  n .E1 [ E2 /. It is implicitly assumed that 0 WD . n E1 / [ E2 is open. Proof. Let u be a bounded p-harmonic function in  n .E1 [ E2 / D 0 n E2 . By assumption, it extends to 0 as a bounded p-harmonic extension U 0 . Its restriction U 00 WD U 0 jnE1 has a bounded p-harmonic extension U in . It is immediate that U is a bounded p-harmonic extension of u to . In the case when both E1 and E2 are uniquely removable we see that U j0 is a p-harmonic extension of u to 0 , and hence is unique (and equal to U 0 ). Therefore U jnE1 is also unique (and equal to U 00 ). Using that E1 is uniquely removable, the uniqueness of U follows. The superharmonic case is proved analogously.

12.4 Nonunique removability We will now give an example of a set which is removable for p-harmonic functions, but in which the extensions are not unique. We will consider p-harmonic functions on graphs, see Appendix A.5.

12.4 Nonunique removability

315

Example 12.24. Consider the graph G D .f1; 2; 3; 4g; f.1; 2/; .1; 3/; .1; 4/g/. Let X be the corresponding metric graph,  D X n f2; 3; 4g and E D .1; 3/ [ .1; 4/ [ f1g, i.e.  n E is just the open edge .1; 2/. A p-harmonic function u in  n E is linear and thus can be described by giving its boundary values u.1/ and u.2/. Let now U1 .1/ D U2 .1/ D u.1/; U1 .2/ D U2 .2/ D u.2/; U1 .3/ D U2 .4/ D u.1/; U1 .4/ D U2 .3/ D 2u.1/  u.2/ and let U1 and U2 be linear on the edges. By Theorem A.26, U1 and U2 are p-harmonic in  (simultaneously for all p). Thus, E is removable for p-harmonic functions in  n E (which are automatically bounded). However, the extensions are not unique in this case. Finally, the same argument as in Example 12.17 shows that E is not removable for bounded superharmonic functions. Example 12.25. Example 12.24 also gives a counterexample to unique continuation: There are two (different) p-harmonic functions on a connected open set  such that they coincide on some nonempty open subset, in this case  n E. Unique continuation is open even in Rn . Open problem 12.26. Let X D Rn (unweighted), n  3, and p ¤ 2. Does the unique continuation property hold for p-harmonic functions? In Rn , 2-harmonic functions are real-analytic, see e.g. Hörmander [186], Theorem 4.4.3, from which an affirmative answer to this problem follows. When n D 2 this question has an affirmative answer for every p with 1 < p < 1, see the discussion on p. 130 in Heinonen–Kilpeläinen–Martio [171] where they also discuss some related counterexamples. In the next example we show that it is possible to have removability even if E disconnects . Example 12.27. Consider the graph G D .f1; ::: ; 6g; f.1; 2/; .1; 3/; .1; 4/; .4; 5/; .4; 6/g/: Let X be the corresponding metric graph,  D X n f2; 3; 5; 6g and E D f1; 4g [ .1; 3/ [ .1; 4/ [ .4; 6/: Then  n E is disconnected and consists of the two edges .1; 2/ and .4; 5/. Let u be any p-harmonic function in  n E. It can be described by its values at the vertices 1, 2, 4 and 5. We want to extend it to a p-harmonic function U in . Again,

316

12 Removable singularities

U is described by its values at the vertices. Apart from being linear at each edge we only need to require that U satisfies (A.8) at the internal vertices, and this holds if and only if 0 D '.U.2/  U.1// C '.U.3/  U.1// C '.U.4/  U.1// D '.U.1/  U.4// C '.U.5/  U.4// C '.U.6/  U.4//; where '.t/ D jt jp2 t . Since ' W R ! R is onto and we can freely choose U.3/ and U.6/ this can always be achieved. We can modify this construction to get a set E that disconnects  and is nonuniquely removable. Example 12.28. Consider the graph G D .f1; ::: ; 7g; f.1; 2/; .1; 3/; .1; 4/; .4; 5/; .4; 6/; .4; 7/g/: Let X be the corresponding metric graph,  D X n f2; 3; 5; 6; 7g and E D f1; 4g [ .1; 3/ [ .1; 4/ [ .4; 6/ [ .4; 7/: (Thus we have added the extra edge .4; 7/ to Example 12.27.) Arguing as in Example 12.27 we see that every p-harmonic function in  n E is extendable to . Moreover, since we now have the freedom to choose both U.6/ and U.7/ we can actually prescribe one of them arbitrarily, which shows that the removability is nonunique.

12.5 Notes This chapter is based on A. Björn [37]. For bounded functions and bounded  (with Cp .X n / > 0), Theorems 12.2 and 12.3 had been obtained earlier in Björn–Björn– Shanmugalingam [56]. In weighted Rn , Theorems 12.2 and 12.3 were obtained for bounded functions and functions bounded from below, respectively, by Heinonen–Kilpeläinen–Martio [171], Theorems 7.35 and 7.36. They also showed, see p. 143 in [171], that compact sets are removable if and only if they have capacity zero. (See Serrin [316], [317] and Maz0 ya [282] for earlier proofs of this fact for unweighted Rn .) A. Björn [32], [33], [38] has constructed examples of removable singularities for various classes of analytic functions with a similar character as (a)–(c) in Example 12.22, see also Hejhal [178], Example 1, p. 19. Example 12.25, as well as an early version of Proposition 12.11, was given in Björn–Björn–Shanmugalingam [56]. Proposition 12.14, without the N 1;p part, also appeared in [56], the proof is due to Heinonen.

12.5 Notes

317

Mäkäläinen [257] characterized removable singularities for Hölder continuous Cheeger p-harmonic functions using weighted Hausdorff measures. In unweighted Rn , Pokrovskii [305] characterized removable singularities for C 1;˛ p-harmonic functions.

Chapter 13

Irregular boundary points

Recall that in Chapters 7–14 we assume that 1 < p < 1 and that X is a complete doubling p-Poincaré space, with doubling constant C and dilation constant  in the p-Poincaré inequality. In this chapter we also make the additional assumption that  is bounded and such that Cp .X n / > 0. In Chapter 11, we studied regular boundary points and characterized them in several ways. In this chapter, we instead concentrate on the irregular points and show that they can be divided into two categories with very different behaviour.

13.1 Semiregular and strongly irregular points That a boundary point is regular can be rephrased in the following odd way: A point x0 2 @ is regular if the following two conditions hold: (a) for all f 2 C.@/ the limit lim

3y!x0

Pf .y/

existsI

(b) for all f 2 C.@/ there is a sequence fyj gj1D1 such that  3 yj ! x0

and

Pf .yj / ! f .x0 /;

as j ! 1:

It turns out that for irregular boundary points exactly one of these two properties fails; a priori one would assume that it is possible that both fail but this can never happen. Definition 13.1. We will say that x0 2 @ is semiregular if (a) holds but not (b); and strongly irregular if (b) holds but not (a). Let us first show the following trichotomy. Theorem 13.2 (Trichotomy). Let x0 2 @. Then x0 is either regular, semiregular or strongly irregular. Proof. Case 1. There is r > 0 such that Cp .B \ @/ D 0, where B D B.x0 ; r/. Let Bin D Bin .x0 ; r/ be a ball with respect to the inner metric din (see Definition 4.41), which is connected, by Lemma 4.42, and such that Bin  B. By Lemma 4.5,

13.1 Semiregular and strongly irregular points

319

x Let f 2 C.@/. By Theorem 12.2, the Perron Cp .Bin n / D 0 and thus Bin  . solution Pf has a p-harmonic extension U to  [ Bin . Since U is continuous we have lim

3y!x0

Pf .y/ D U.x0 /;

i.e. (a) holds and x0 is either regular or semiregular. Case 2. The capacity Cp .B.x0 ; r/ \ @/ > 0 for all r > 0. (Note that this is complementary to case 1.) For every j D 1; 2; ::: , we thus have Cp .B.x0 ; 1=j /\@/ > 0, and by the Kellogg property (Theorem 10.5) there is a regular boundary point xj 2 B.x0 ; 1=j / \ @. (We do not require the xj to be distinct.) As xj is regular, we can find yj 2 B.xj ; 1=j / \  so that jPf .yj /  f .xj /j < 1=j . It follows directly that yj ! x0 and Pf .yj / ! f .x0 /, as j ! 1, i.e. (b) holds, and thus x0 is either regular or strongly irregular. With a little extra work one can actually say more. In case 1, we can deduce that x0 is always semiregular: Let f .x/ D .1  dist.x; x0 /=r/C on @. Since f D 0 q.e. on @, Corollary 10.23 implies that Pf  0. Hence lim

3y!x0

Pf .y/ D 0 ¤ 1 D f .x0 /;

which shows that x0 is not regular, and thus must be semiregular. We observe that semiregular points are only obtained in the first case, and thus the relatively open set S D fx 2 @ W there is r > 0 such that Cp .B.x; r/ \ @/ D 0g consists exactly of all semiregular boundary points. On the other hand, the closed set @ n S consists of all points which are either regular or strongly irregular, and is moreover the closure of the set of all regular boundary points. Also in case 2, it is possible to improve upon the above result. Namely, one can show that the sequence fyj gj1D1 can be chosen independently of f , see the proof of :(e) ) :(c) in Theorem 13.10. The following two examples illustrate the semiregular and strongly irregular points. As a matter of fact, these were the first examples of irregular boundary points, see the notes to this chapter. Example 13.3 (Zaremba’s punctured ball). The easiest example of a semiregular point is the punctured ball  D B.0; 1/ n f0g in R2 for p D 2, where it is easy to see that for x0 D 0 we are in case 1 above and thus 0 is a semiregular point. Example 13.4 (The Lebesgue spine). Let X D R3 , E D f.x; t / 2 R2 R W t > 0 and jxj < e 1=t g and

x  D B.0; 1/ n E:

320

13 Irregular boundary points

Then it follows from the Wiener criterion with p D 2 that 0 is an irregular boundary point for harmonic functions, and as we are clearly in case 2, it must be strongly " irregular. In fact, e 1=t may be replaced by e t for any " > 0. n In X D R , n  4, one can even use powers, viz. let E D f.x; t / 2 Rn1 R W t > 0 and jxj < t ˛ g and

x  D B.0; 1/ n E:

Then it follows from the Wiener criterion that 0 is an irregular boundary point if and only if ˛ > 1, and then it is strongly irregular. See e.g. Armitage–Gardiner [19], Remark 6.6.17, for further details.

13.2 Characterizations of semiregular points Similarly to regular points, semiregular points can be characterized by a number of equivalent conditions. This will be done in Theorem 13.10, but before that we obtain the following characterizations of relatively open sets of semiregular points. Theorem 13.5. Let V  @ be relatively open. Then the following are equivalent: (a0 ) The set V consists entirely of semiregular points. (b0 ) The set V does not contain any regular point. (c0 ) It is true that Cp .V / D 0. (d0 ) The upper p-harmonic measure !.V x /  0. (e0 ) The lower p-harmonic measure !.V /  0. (f 0 ) The set  [ V is open in X , and every bounded p-harmonic function on  has a p-harmonic extension to  [ V . (g0 ) The set  [ V is open in X , .V / D 0, and every bounded superharmonic function on  has a superharmonic extension to  [ V . (h0 ) For every f 2 C.@/, the Perron solution Pf depends only on f j@nV (i.e. if f; h 2 C.@/ and f D h on @ n V , then Pf  P h). Together with the implication (a) ) (f) in Theorem 13.10 this theorem shows that the set S of all semiregular boundary points is a relatively open set which can be characterized as the largest relatively open subset of @ having any of the properties above, or e.g. as [ SD fV  @ W Cp .V / D 0 and V is relatively openg: x i.e. S  @ n @. x Note By (f 0 ) we also see that S is contained in the interior of , x however that it can happen that S ¤ @ n @.

13.2 Characterizations of semiregular points

321

Example 13.6. If Cp .fxg/ > 0 and G 3 x is a regular set, then  WD G n fxg is also x regular and hence S D ¿, but x 2 @ n @. Example 13.7. Let X D C D R2 ,  be the slit disc B.0; 1/ n .1; 0 and p D 2. It x D .1; 0. is well known that  is regular, and hence S D ¿, However, @ n @ Proof of Theorem 13.5. (a0 ) ) (b0 ) This is trivial. (b0 ) ) (c0 ) This follows directly from the Kellogg property (Theorem 10.5). (c0 ) ) (g0 ) Let x 2 V and let G be a connected neighbourhood of x, so small that G \ @  V . Sets of capacity zero cannot separate space, by Lemma 4.6, and hence x from which it follows that  [ V is open. G n @ must be connected, i.e. G  , That .V / D 0 follows directly from the fact that Cp .V / D 0. The extension is now provided by Theorem 12.3. (g0 ) ) (f 0 ) The first part is immediate. Let u be a bounded p-harmonic function on . By assumption, u has a superharmonic extension U to  [ V . Also u has a superharmonic extension W to  [ V . Thus W is a subharmonic extension of u to  [ V . By Proposition 12.5, U D W is p-harmonic. (f 0 ) ) (a0 ) Let x0 2 V and f 2 C.@/. Then Pf has a p-harmonic extension U to  [ V . It follows that lim

3y!x0

Pf .y/ D U.x0 /;

and thus the limit in the left-hand side always exists. It remains to show that x0 is irregular. Let h.x/ D dist.x; @ n V /. Then P h has a p-harmonic extension U to  [ V and by the Kellogg property (Theorem 10.5), lim

[V 3y!x

U.y/ D h.x/ D 0

for q.e. x 2 @. [ V / D @ n V:

Hence U  0 in  [ V , by Corollary 10.23. Therefore lim

3y!x0

P h.y/ D 0 ¤ h.x0 /;

and x0 is irregular. (c0 ) ) (d0 ) This follows from Theorem 10.22. (d0 ) ) (e0 ) This is trivial. (e0 ) ) (b0 ) Let f D V . As f is continuous at x 2 V , Theorem 10.29 shows that if x were regular than we would have 0D

lim !y .V / D

3y!x

lim P V .y/ D 1;

3y!x

a contradiction. Hence V consists entirely of irregular boundary points. (c0 ) ) (h0 ) This follows from Theorem 10.22. (h0 ) ) (e0 ) This follows from Proposition 10.37, as if '  V , ' 2 Lip.@/, then (h0 ) implies that P ' D P .' /  P 0  0.

322

13 Irregular boundary points

In Theorem 11.2, it was shown that the existence of a barrier is equivalent to regularity of a boundary point. Here we introduce semibarriers and weak semibarriers. We shall show that the existence of a semibarrier or of a weak semibarrier for a boundary point is equivalent to the fact that the point is not semiregular. Recall also that in our generality we do not know if the existence of a weak barrier is sufficient to conclude regularity, see Open problem 11.14. Definition 13.8. A function u is a semibarrier (with respect to ) at x0 2 @ if (i) u is superharmonic in ; (ii) lim inf 3y!x0 u.y/ D 0; (iii) lim inf 3y!x u.y/ > 0 for every x 2 @ n fx0 g. A function u is a weak semibarrier (with respect to ) at x0 2 @ if it is a positive superharmonic function in  such that (ii) holds. Let us observe that not all semibarriers are weak semibarriers. Example 13.9. Let X D Œ0; 2  R and  D Œ0; 1/. Then u  0 is a semibarrier at 1 which is not a weak semibarrier. In fact, u is also a barrier but not a weak barrier. We can now characterize semiregular points by means of capacity, p-harmonic measures, removable singularities and semibarriers. In particular, we show that semiregularity is a local property. Theorem 13.10. Let x0 2 @, ı > 0 and d.y/ D d.y; x0 /. Then the following are equivalent: (a) The point x0 is semiregular. (b) The point x0 is semiregular with respect to G WD  \ B.x0 ; ı/. (c) There is no sequence fyj gj1D1 such that  3 yj ! x0 , as j ! 1, and lim Pf .yj / D f .x0 / for all f 2 C.@/:

j !1

(d) The point x0 is not regular nor strongly irregular. (e) It is true that x0 … fx 2 @ W x is regularg. (f) There is a neighbourhood V of x0 such that Cp .V \ @/ D 0. (g) There is a neighbourhood V of x0 such that Cp .V n / D 0. (h) There is a neighbourhood V of x0 such that !.V x \ @/  0. (i) There is a neighbourhood V of x0 such that !.V \ @/  0.

13.2 Characterizations of semiregular points

323

(j) There is a neighbourhood V of x0 such that every bounded p-harmonic function x (or equivaon  has a p-harmonic extension to  [ V , and moreover V   lently V n  has empty interior). (k) There is a neighbourhood V of x0 such that every bounded p-harmonic function on  has a p-harmonic extension to  [ V , and moreover x0 is irregular. (l) There is a neighbourhood V of x0 such that every bounded superharmonic function on  has a superharmonic extension to [V , and moreover .V n/ D 0. (m) There is a neighbourhood V of x0 such that for every f 2 C.@/, the Perron solution Pf depends only on f j@nV (i.e. if f; h 2 C.@/ and f D h on @nV , then Pf  P h). (n) It is true that lim

3y!x0

P d.y/ > 0:

(o) It is true that lim inf P d.y/ > 0:

3y!x0

(p) There is no weak semibarrier at x0 . (q) There is no semibarrier at x0 . (r) The continuous solution of the Kd;d -obstacle problem is a not a semibarrier at x0 . The reader may easily get the impression that the superficial looking conditions x and “x0 is irregular” in (j) and (k) are artifacts of our proof. However, V   removing them makes the theorem false, as the following example shows. The authors do not know if the condition .V n / D 0 is essential in (l). Example 13.11. Let X D R,  D .0; 1/, x0 D 0 and V D .1; 1/. Since p-harmonic functions on  are linear (x 7! ax C b), the set V n  is removable for p-harmonic functions in  (but not for superharmonic functions in ), see Example 12.17. Morex .V n / > 0 and x0 is regular. over, V 6 , On the other hand, when singletons have zero capacity, then the conditions (j) and (k) in Theorem 13.10 can be simplified. Theorem 13.12. Assume that Cp .fxg/ D 0 for all x 2 X (which in particular holds for unweighted Rn , p  n). Let x0 2 @. Then the following statement is equivalent to the statements in Theorem 13.10: (j00 ) There is a neighbourhood V of x0 such that every bounded p-harmonic function on  has a p-harmonic extension to  [ V .

324

13 Irregular boundary points

Proof. (j) ) (j00 ) This implication is trivial. (j00 ) ) (j) We may assume that V is connected (replace otherwise V by the component of V containing x0 ). x then V n  contains a ball and therefore Cp .V n / > 0. If V is not a subset of , Proposition 12.14 shows that there is a bounded p-harmonic function on  with no x p-harmonic extension to  [ V , but this contradicts (j00 ). Thus V  . Proof of Theorem 13.10. (e) , (f) , (h) , (i) , (m) ) (a) This follows directly from Theorem 13.5, with V in Theorem 13.5 corresponding to V \ @ here. (a) ) (n) The limit c WD lim P d.y/ 3y!x0

exists. If c were 0, then x0 would be regular, by Theorem 10.29, a contradiction. Thus c > 0. (n) ) (o) ) (d) ) (c) This is trivial. :(e) ) :(c) For each j  1, the set B.x0 ; 1=j / \ @ contains a regular boundary point xj . Let fj D minf.2  jd /C ; 1g. Then we can find yj 2 B.xj ; 1=j / \  so that 1 > jfj .xj /  Pfj .yj /j D j1  Pfj .yj /j: j Then yj ! x0 and Pfj .yj / ! 1, as j ! 1. Let now f 2 C.@/. Without loss of generality we may assume that 0  f  2 and that f .x0 / D 1. Let " > 0. Then we can find k such that f 1"

on B.x0 ; 2=k/ \ @:

It follows that f  fj  " for j  k, and thus lim inf Pf .yj /  lim inf Pfj .yj /  " D 1  ": j !1

j !1

Letting " ! 0 gives lim infj !1 Pf .yj /  1. Applying this to fQ WD 2  f instead gives lim supj !1 Pf .yj /  1, and the implication is proved. (f) , (b) Note first that (f) is equivalent to the existence of a neighbourhood W of x0 with Cp .W \ @G/ D 0. But this is equivalent to (b), by the already proved equivalence (f) , (a) applied to G instead of . (f) ) (g) By Theorem 13.5, (c0 ) ) (g0 ), the set  [ .V \ @/ is open, and we can use this as our set V in (g). (g) ) (f) This is trivial. x Thus their (g) , (j) , (l) In all three statements it follows directly that V  . equivalence follows directly from Theorem 13.5, with V in Theorem 13.5 corresponding to V \ @ here.

13.3 Characterizations of strongly irregular points

325

(j) ) (k) The first part is obvious and we only need to show that x0 is irregular, but this follows from the already proved implication (j) ) (a). (k) ) (a) Let f 2 C.@/. Then Pf has a p-harmonic extension U to  [ V for some neighbourhood V of x0 . It follows that lim

3y!x0

Pf .y/ D U.x0 /;

and thus the limit in the left-hand side always exists. Since x0 is irregular it follows that x0 must be semiregular. (l) ) (p) Let u be a positive superharmonic function on . Then minfu; 1g has a superharmonic extension U to  [ V . Since U is lsc-regularized and .V n / D 0, it follows that U  0 in  [ V . If U.x0 / were 0, then it would follow from the strong minimum principle (Theorem 9.13) that U  0 in the component of  [ V containing x0 , but this would contradict the fact that u is positive in . Thus 0 < U.x0 /  lim inf u.x0 /; 3y!x0

and hence there is no weak semibarrier. :(q) ) :(p) Let u be a semibarrier at x0 . If u > 0 in all of , then it is also a weak semibarrier and the implication holds. On the other hand, if there is x 2  with u.x/ D 0, then u is not a weak semibarrier. In this case, the strong minimum principle (Theorem 9.13) shows that u  0 in the component 0 of  containing x. Since u is a semibarrier it follows that x0 is the only boundary point of 0 . As Cp .X n 0 /  Cp .X n / > 0, we obtain, by Lemma 4.5, that Cp .fx0 g/ D Cp .@0 / > 0. The Kellogg property (Theorem 10.5) implies that x0 is regular and Theorem 11.2 shows that there is a positive barrier v at x0 , and thus v is a weak semibarrier. (q) ) (r) This is trivial. : (e) ) : (r) Let u be the continuous solution of the Kd;d -obstacle problem. It is clear that u satisfies (i) and (iii) in Definition 13.8. Let fxj gj1D1 be a sequence of regular boundary points such that d.xj / < 1=j (we do not require the xj to be distinct). By Corollary 11.9, lim3y!xj u.y/ D d.xj /. We can therefore find yj 2 B.xj ; 1=j / \  such that u.yj / < 2=j . Thus 0  lim inf u.y/  lim inf u.yj / D 0: 3y!x0

j !1

13.3 Characterizations of strongly irregular points In this section, we complete our descriptions of boundary points and provide several characterizations of the strongly irregular points. Similarly to regularity and semiregularity, strong irregularity is a local property.

326

13 Irregular boundary points

Theorem 13.13. Let x0 2 @, ı > 0 and d.y/ D d.y; x0 /. Then the following are equivalent: (a) The point x0 is strongly irregular. (b) The point x0 is strongly irregular with respect to G WD  \ B.x0 ; ı/. (c) The point x0 is irregular and there is a sequence fyj gj1D1 such that  3 yj ! x0 , as j ! 1, and lim Pf .yj / D f .x0 / for all f 2 C.@/:

j !1

x n R, where R D fx 2 @ W x is regularg. (d) It is true that x0 2 R (e) There exists f 2 C.@/ such that lim

3y!x0

Pf .y/

does not exist. (f) It is true that lim inf P d.y/ D 0 < lim sup P d.y/:

3y!x0

3y!x0

(g) The continuous solution u of the Kd;d -obstacle problem satisfies lim inf u.y/ D 0 < lim sup u.y/:

3y!x0

3y!x0

(h) There is a semibarrier (or equivalently there is a weak semibarrier) but no barrier at x0 . Proof. We will use the fact that a boundary point is either regular, semiregular or strongly irregular, by Theorem 13.2 or Theorem 13.10, (a) , (d). (a) , (b) This follows since semiregularity and regularity are local properties, by Theorems 11.11 and 13.10. (a) , (c) , (d) This follows from Theorem 13.10, (a) , (c) , (e). (a) ) (f) Since P d is nonnegative, we have lim inf P d.y/ D 0  lim sup P d.y/:

3y!x0

3y!x0

If lim sup3y!x0 P d.y/ were 0, then Theorem 10.29 would show that x0 were regular, a contradiction. Hence lim sup P d.y/ > 0: 3y!x0

(f) ) (e) This is trivial.

13.4 The sets of semiregular and of strongly irregular points

327

(e) ) (a) By definition, x0 is neither regular nor semiregular, and hence must be strongly irregular. (a) , (g) It follows directly from Theorem 11.11 that x0 is regular if and only if lim sup3y!x0 u.y/ D 0. On the other hand, Theorem 13.10 shows that x0 is semiregular if and only if lim inf 3y!x0 u.y/ > 0. Combining these two facts gives the equivalence. (a) , (h) By Theorem 13.10, x0 is semiregular if and only if there is no (weak) semibarrier at x0 . On the other hand, by Theorem 11.2, there is a barrier at x0 if and only if x0 is regular. Combining these two facts gives the equivalence.

13.4 The sets of semiregular and of strongly irregular points Let us consider the partition of @ into R D fx 2 @ W x is regularg; S D fx 2 @ W x is semiregularg; I D fx 2 @ W x is strongly irregularg: By the Kellogg property (Theorem 10.5), Cp .S / D Cp .I / D 0, and thus R is the significantly largest of these three sets. It is natural to ask if one can compare the sizes of S and I . We first observe that it is possible to have S D I D ¿, and also that any of them can be empty while the other one is a singleton set, see Examples 13.3 and 13.4 and Lemma 14.2, where the existence of regular sets is guaranteed. Using the characterization of S after Theorem 13.5 together with the existence of regular sets (Theorem 14.1), one easily obtains that for any given compact K with Cp .K/ D 0, one can find a set  such that S D K and I D ¿. (Let 0 be a regular set containing K and let  D 0 n K.) On the other hand, since strong irregularity is a local property, it is easy to construct examples where S D ¿ and I is countable. It is less obvious that I can be equal to any prescribed compact set with zero capacity. This is indeed possible, at least as long as Cp .fxg/ D 0 for all x, or Cp .fxg/ > 0 for all x, in which case I D ¿. In particular, this is true in unweighted Rn . For the proof of this fact we refer the reader to A. Björn [41], Theorem 4.1 and Section 5. The construction uses a Wiener type condition necessary for regularity, with the exponent 1=p instead of 1=.p  1/, which was recently obtained by J. Björn [67], but is not included in this book. See the comments after Open problem 11.26. In Rn , one can of course use the usual Wiener criterion.

328

13 Irregular boundary points

13.5 Notes This chapter is primarily based on A. Björn [41] (and its preprint version [39], in particular Theorem 13.13 is from [39]). The distinction between semiregular and strongly irregular points has appeared in some earlier papers in the literature. In the linear case, the trichotomy was developed in detail in Lukeš–Malý [249]. Example 13.3 was given by Zaremba [356] in 1911 and was the very first example of an irregular boundary point. Lebesgue spines, as in Example 13.4, were first given by Lebesgue [240]. The equivalence of (b0 ), (c0 ) and (e0 ) in Theorem 13.5 was obtained in Björn–Björn– Shanmugalingam [55]. As with barriers, (weak) semibarriers are often defined in a local way, but we prefer global definitions. That local (weak) semibarriers also characterize semiregularity follows directly using that semiregularity is a local property. The cluster set C .f; ; x0 / for a function f with respect to  and a boundary point x0 2 @ is defined as \ C .f; x0 ; / D f .B.x0 ; r/ \ /; r>0

x We can thus reformulate (c) in Theorem 13.13 as: x0 where the closure is taken in R. is irregular and f .x0 / 2 C .Pf; ; x0 / for all f 2 C.@/: (13.1) In fact it is possible to say more, viz. in A. Björn [41] it is shown that if f 2 C.@/ and x0 is a strongly irregular point, then C .Pf; ; x0 / is a closed interval with f .x0 / as an end point. For unweighted Rn in the borderline case p D n, this was shown by Martio [265] under some restrictions on f . The fact that it is not possible to have lim inf Pf .x/ < f .x0 / < lim sup Pf .x/

3x!x0

3x!x0

at any x0 2 @, follows already from Theorem 2.5 in Gariepy–Ziemer [134], where it was proved for rather general quasilinear elliptic equations.

Chapter 14

Regular sets and applications thereof

Recall that in Chapters 7–14 we assume that 1 < p < 1 and that X is a complete doubling p-Poincaré space, with doubling constant C and dilation constant  in the p-Poincaré inequality. In this short chapter we first show that open sets can be approximated by regular open sets, after which we look at some consequences of this approximation. See the notes to this chapter for further comments.

14.1 Regular sets In Rn , it follows from Corollary 11.25 that polyhedra and balls are regular. This, in particular, means that every nonempty Euclidean open set can be approximated by regular sets. A similar result holds in metric spaces, i.e. open sets can be approximated from inside by bounded regular open sets. Theorem 14.1. Assume that either X is unbounded or  ¤ X . Then there exist S regular sets 1 b 2 b  such that  D j1D1 j . Observe that if G is a regular set, then G is bounded and Cp .X n G/ > 0, by definition, see the comments S after Definition 10.4. If  D X and X is bounded, and 1 b 2 b  b  D j1D1 j , then we have  D j for some j , by compactness, and hence j is not regular. Thus, this case has to be omitted in Theorem 14.1. Our construction is based on the inner metric din , see Definition 4.41. By Theorem 4.32 we have d.x; y/  din .x; y/  Ld.x; y/; (14.1) i.e. X is L-quasiconvex, where L only depends on the doubling constant C and the constants in the p-Poincaré inequality. Lemma 14.2. Let  be bounded with nonempty complement X n . Let ı > 0 and assume that 0 WD fx 2  W dist in .x; X n / > ıg ¤ ¿; where distin is the distance taken with respect to the inner metric din . Then the complement X n 0 has a corkscrew at every boundary point. In particular, 0 is regular. Recall that X n 0 has a corkscrew at x0 if for all sufficiently small r, the set B.x0 ; r/ n 0 contains a ball with radius C r, and that this implies that x0 is regular with respect to 0 , by Corollary 11.25.

330

14 Regular sets and applications thereof

Proof. Let x0 2 @0 and 0 < < 13 ı be arbitrary. Find a point y 2 X n  and a curve  W Œ0; l  ! X such that .0/ D x0 , .l / D y and l < ı C . Let z D .2 / and let L  1 be as in (14.1). We shall show that B.z; =L/  B.x0 ; 3 / n 0 , i.e. that X n 0 has a corkscrew at x0 . Clearly, d.x0 ; z/  2 , and hence B.z; =L/  B.x0 ; 3 /. Let x 2 0 be arbitrary. Then Ld.x; z/  din .x; z/  din .x; y/  din .y; z/  distin .x; X n /  .l  2 / > ı  .ı C  2 / D : Thus, B.z; =L/\0 D ¿ and hence B.z; =L/  B.x0 ; 3 /n0 . As was arbitrary, X n 0 has a corkscrew at x0 , and thus by Corollary 11.25, x0 is regular with respect to 0 . Since x0 was arbitrary, 0 is regular. Proof of Theorem 14.1. If  is bounded, then the theorem follows from Lemma 14.2 by taking ı D 1=j , j D 1; 2; ::: . If  is unbounded, fix z0 2  and let j0 D fx 2  W din .x; z0 / < j g

and

j D fx 2 j0 W dist in .x; X n j0 / > 1=j g:

Then j b j0 b j C1 , j D 2; 3; ::: . Moreover, for large enough j , we have j ¤ ¿, and Lemma 14.2 implies that each such j is regular, which concludes the proof. We finish this section by giving two examples of metric spaces, satisfying our assumptions, but having nonregular balls. In the second example (Example 14.4) there are no regular balls. Example 14.3. Consider the cone X D f.x1 ; ::: ; xn / 2 Rn W xj  0 for all j D 1; ::: ; ng: Then X is a complete doubling 1-Poincaré space. This follows from Theorem A.21 p or by the direct calculation in Example 5.6. Let x D .1; ::: ; 1/ 2 X , r D n, B D B.x; r/ and 1 < p  n. Then the origin is an isolated boundary point with zero capacity, which is thus semiregular, by Theorem 13.10. Therefore, B is not regular. This example can be iterated in the following way to obtain a sequence of shrinking balls which are not regular: Let Tj , j D 1; 2; ::: , be the closed isosceles triangles in R2 with bases Œ2j ; 21j   R and heights 21j . Let X D .Œ0; 1 Œ1; 0/ [

1 [

Tj  R2 ;

j D1

equipped with the two-dimensional Lebesgue measure. It is not difficult to verify that X is a uniform domain in R2 , see Definition A.20. Theorem A.21 then implies that X is a doubling 1-Poincaré space. Now for rj D 5  2j 1 , the balls B.0; rj / are not regular, since @B.0; rj / contains the isolated boundary point xj D .3  2j 1 ; 21j / of zero capacity.

14.2 Wiener solutions

331

Example 14.4. Another example of nonregular balls can be found in the Heisenberg groups Hn , n  2 (see Appendix A.6). It is shown in Proposition 7 in Capogna– Garofalo [88] that the complement of the Carnot–Carathéodory ball B.0; r/ in Hn , n  1, near each of its poles .0; ˙r 2 / is contained in a Euclidean cone. At the same time, Proposition 8 in [88] shows that the interior corkscrew condition (with respect to the Carnot–Carathéodory metric) fails for the Euclidean cone. Theorem 3.4 in Hansen–Hueber [158] shows that if n  2 and a > 0, then the generalized cone f.z; t / 2 Hn W ajzj˛  t g is thin (in the linear case p D 2) at the origin if and only if 0 < ˛ < 2. Thus, any Euclidean cone (with ˛ D 1) is thin (with respect to Hn ) at its vertex. As both implications in the Wiener criterion hold in this case (see Theorem 11.24, the comments after Open problem 11.26 and Appendix A.6), this implies that the Carnot–Carathéodory ball B.0; r/ in Hn , n  2, p D 2, is not regular. Due to the group structure this means that in Hn , n  2, p D 2, there do not exist any regular balls, and in particular no base of regular balls. Theorem 14.1 of course shows that there is a base of regular sets. On the other hand, Example 2.16 in J. Björn [63] shows that if X is a geodesic space such that all geodesics can be continued as geodesics beyond every point, then every ball in X satisfies the exterior corkscrew condition at every boundary point and is therefore regular. To relate this to the results from [88] and [158], note that the geodesic in the Heisenberg group from the origin to the north pole .0; r 2 / cannot be continued as a geodesic.

14.2 Wiener solutions Assume in this section that  is bounded and Cp .X n / > 0. If  is not regular, then the Dirichlet problem cannot be solved in the classical sense for general continuous boundary data f 2 C.@/. (Classical in the sense that the boundary values are really attained at all boundary points as limits. We still consider weak solutions of the equation, even when our minimization problem corresponds to a partial differential equation.) Definition 14.5. Assume that  is bounded and Cp .X n / > 0. Let f 2 C.@/. A Wiener solution u of the Dirichlet problem in  with boundary values f is obtained by the following construction: Extend f arbitrarily to a continuousS function (also called x let 1 b 2 b  b  be regular sets such that  D 1 j , and let f ) on , j D1 u D lim Pj f: j !1

Observe that since j is regular, the solutions Pj f are classical solutions of the corresponding boundary value problems, i.e. classical in the sense that the boundary

332

14 Regular sets and applications thereof

values are really attained at all boundary points as limits. Theorem 14.7 below shows that the Wiener solution is p-harmonic in . Theorem 14.6. Assume that  is bounded and Cp .X n / > 0. Let f 2 C.@/. Then there exists a unique Wiener solution of the Dirichlet problem in  with boundary values f . Proof. Let us first look at the existence. The first step, the extension of f , is directly obtained by Tietze’s extension theorem (see e.g. Theorem 4.16 in Folland [124] or Theorem 20.4 in Rudin [311]). The next step is to approximate  by regular sets, which is obtained by Theorem 14.1. Finally one needs to show that the limit limj !1 Pj f exists everywhere in . We combine the existence and uniqueness parts of the proof and make them into a theorem of its own below. Theorem S 14.7. Assume that  is bounded and Cp .X n / > 0. Let 1  2    x Then  D j1D1 j and let f 2 C./. lim Pj f D Pf:

j !1

This result shows that one could define Wiener solutions also with respect to nonregular exhaustions of . However, that would defy the purpose of Wiener solutions, that to define Wiener solutions we only need to use classical solutions of boundary value problems. Nevertheless, it is an interesting stability result. x by letting Proof. Let u 2 Uf and " > 0. Extend u to , u.x/ D lim inf u.y/; 3y!x

x 2 @;

x Let further which makes u lower semicontinuous on . x W u.x/ C " > f .x/g; A D fx 2  which is an open set (in the relative topology) by the lower semicontinuity of u  f . The set A contains @ by assumption. By compactness, there is some k such that x and hence @k  A. It follows that A [ k D , .u C "/jj 2 Uf .j /

for j  k;

and thus that lim supj !1 Pj f  u C ". Letting " ! 0 and taking infimum over all u 2 Uf , shows that lim sup Pj f  Pf: j !1

Applying this also to f we obtain Pf D P .f /   lim sup Pj .f / D lim inf Pj f  lim sup Pj f  Pf:  j !1

j !1

j !1

14.3 Classically superharmonic functions

333

14.3 Classically superharmonic functions In this section, we compare our definition of superharmonic functions with another definition which often appears in Euclidean literature. Definition 14.8. A function u W  ! .1; 1 is classically superharmonic in  if (i) u is lower semicontinuous; (ii) u is not identically 1 in any component of ; (iii) for every nonempty open set 0 b  with Cp .X n 0 / > 0, and all functions x 0 / such that v is p-harmonic in 0 and v  u on @0 , we have v  u v 2 C. in 0 . This definition was used in Heinonen–Kilpeläinen–Martio [171] as well as in most other studies before Kinnunen–Martio [217], where superharmonic functions on metric spaces were defined for the first time. At that time, it was not known whether open sets in metric spaces can be approximated by regular sets. This fact is used to prove that a function is classically superharmonic if and only if it is superharmonic, which we will do in this section. Without sufficiently many regular sets, Definition 14.8 becomes rather weak, as superharmonicity is then tested by fewer p-harmonic functions (only those continuous up to the boundary of 0 ). Indeed, the first part of the proof of Theorem 14.10 shows that superharmonic functions are classically superharmonic even without any knowledge of regular sets. The following theorem, on the other hand, requires 0 to be regular, or in view of Proposition 9.21 that 0 be exhaustible by regular sets. Theorem 14.9. Let u be classically superharmonic in  and let 0 b  be regular. Then u is superharmonic in 0 . Recall that, by definition, a regular set 0 is bounded and satisfies Cp .X n 0 / > 0. x 0 , and hence bounded from below in  x 0, Proof. As u is lower semicontinuous in  Proposition 1.12 shows that there is an increasing sequence f'j gj1D1 of Lipschitz funcx 0 such that tions on  x 0: u D lim 'j on  (14.2) j !1

Let uj be the lsc-regularized solution of the K'j ;'j .0 /-obstacle problem, which, by Theorem 8.28, is continuous and such that 'j < uj D HAj 'j

in the open set Aj WD fx 2 0 W 'j .x/ ¤ uj .x/g:

x 0 / and therefore In particular, uj is p-harmonic in Aj . By Corollary 11.9, uj 2 C. uj D 'j  u everywhere on @Aj . Since u is classically superharmonic and uj is p-harmonic in Aj , we have uj  u in Aj :

334

14 Regular sets and applications thereof

It thus follows that 'j  uj  u

in 0 ;

these inequalities being trivial outside of Aj . Together with (14.2), this shows that u D limj !1 uj and the comparison lemma (Lemma 8.30) implies that the sequence fuj gj1D1 is increasing. By Propositions 7.14 and 9.4, each uj is superharmonic in 0 . Theorem 9.27 then implies that so is u. Theorem 14.10. A function u is superharmonic in  if and only if it is classically superharmonic in . Proof. Assume first that u is superharmonic. If  D X and X is bounded, then u is constant by Corollary 9.14 and thus u is classically superharmonic. Otherwise, let x 0 / be p-harmonic 0 b  be a nonempty open set. Then Cp .X n0 / > 0. Let v 2 C. in 0 and such that v  u on @0 . As v is subharmonic in 0 , the comparison principle for sub- and superharmonic functions (Theorem 9.39) yields v  u in 0 . Hence u is classically superharmonic. Conversely, assume that u is classically superharmonic. Let G   be a nonempty open subset with Cp .X S n G/ > 0. Theorem 14.1 shows that there exist regular sets G1 b G2 b  b G D j1D1 Gj . By Theorem 14.9, u is superharmonic in each Gj , j D 1; 2; ::: . Hence, u is superharmonic in G, by Proposition 9.21. Finally we obtain by Theorem 9.24, (e) ) (a), that u is superharmonic in .

14.4 Notes Sections 14.1 and 14.2 are based on Björn–Björn [46]. Theorem 14.10 is from A. Björn [35], but the proof here is considerably shorter. Example 14.4 answers Open problem 3.2 in Björn–Björn [46]. Had we defined superharmonic functions using Definition 14.8 it would have been possible to “copy” more proofs from the Euclidean theory. Indeed, we could have built up the theory in this way. To do so we would have had to obtain Theorem 14.1 earlier. For this, it is enough to have a weaker version of the Wiener condition, as long as it is strong enough to show that corkscrews imply regularity. Such a Wiener type condition was obtained in J. Björn [63] using results which were at our disposal already after Chapter 8. Theorem 9.39 in the proof of Theorem 14.10 follows from the definition of superharmonic functions. The rest of the proofs of Theorems 14.9 and 14.10 (apart from the use of Theorem 14.1) only uses results from Sections 9.1–9.4. Theorem 14.10 could therefore have been proved already in Section 9.4. We have chosen an alternative route and have avoided using Theorem 14.1 in this book, except in this chapter. One reason was to show that the theory of superharmonic functions does not require an abundance of regular sets, and perhaps in the future this

14.4 Notes

335

theory will be generalized to some situation in which one cannot approximate by regular sets, while (modifications of) our proofs will still work. Theorem 14.10 is given only for completeness and future applications. Wiener solutions of the Dirichlet problem for harmonic functions (p D 2) were introduced by Wiener [351] at about the same time as Perron [304] constructed what we now call Perron solutions. At that time, the Dirichlet problem was soluble only for nice domains and Wiener used exhaustions by such domains to solve the Dirichlet problem in general domains. Farnana [121] proved aSresult similar to Theorem 14.7 for f 2 N 1;p ./, viz. that if 1  2     D j1D1 j , then lim Hj f D Hf:

j !1

She also proved convergence results for solutions of single and double obstacle problems in [120] and [121].

Appendix A

Examples

In this appendix we assume that 1  p < 1, unless otherwise specified. In this appendix we primarily look at various examples of spaces which satisfy our three axioms: completeness, doubling and Poincaré inequality. We do not aim to be self-contained, and some of the examples are not even defined completely, as there are other sources in the literature, where they are studied thoroughly.

A.1 N 1;p in Euclidean spaces The aim in this section is to show that Newtonian spaces are the right generalization of the usual Sobolev spaces, i.e. that both spaces coincide for domains in Rn . We start by treating the unweighted Rn by rather elementary methods. In the next section we turn to more general results for weighted Rn and give the corresponding references. In particular, we shall consider the so-called p-admissible weights from Heinonen– Kilpeläinen–Martio [171]. The key for proving the equality between Newtonian and Sobolev spaces in Rn is the following lemma from Ohtsuka [299], Lemma 4.2.1, whose roots go back to Fuglede [128], Theorem 12. Lemma A.1. Let be a collection of curves in Rn . If Modp . / D 0, then a.e. (with respect to the .n  1/-dimensional Lebesgue measure) line parallel to the x1 -axis contains no curve from . Proof. Let Qk D Œk; kn , k D 1; 2; ::: , be an increasing sequence of cubes in Rn and kSbe the collection of curves from which are contained in Qk . Note that

D 1 kD1 k . Let Lk be the collection of all lines parallel to the x1 -axis which contain a curve in k . R As Modp . / D 0, there exists nonnegative 2 Lp .Rn / such that  ds D 1 for all  2 . By assumption, each l 2 Lk contains a curve  2 k and hence Z Z ds  ds D 1 for all l 2 Lk : R

l\Qk



R

At the same time, as Qk dx  Qk .1 C p / dx < 1, the Fubini theorem shows that R l\Qk ds < 1 for a.e. line l passing through Qk and parallel to the x1 -axis. This shows that a.e. line passing through Qk and parallel to the x1 -axis contains no curve from k , and letting k ! 1 finishes the proof.

338

A Examples

Theorem A.2. If   Rn , then N 1;p ./ D W 1;p ./ as Banach spaces, with equivalent norms. More precisely, if u 2 N 1;p ./, then u 2 W 1;p ./, and conversely, for every u 2 W 1;p ./ there exists uN 2 N 1;p ./ such that uN D u a.e. in , i.e. W 1;p ./ D Ny 1;p ./. Moreover, if u 2 W 1;p ./ is quasicontinuous, then u 2 N 1;p ./. See Proposition A.12 for a similar characterization in weighted Euclidean spaces. Note that in Corollary A.4 we improve upon this result by showing that the norms are equal, not only equivalent, and that in Proposition A.13 we characterize the functions in W 1;p ./ which belong to N 1;p ./. p Proof. Let u 2 N 1;p ./ with a p-weak upper R gradient g 2 L ./. By Theorem 1.56 and Proposition 1.37, u 2 ACCp ./ and  g ds < 1 for p-a.e. curve  in . Lemmas 2.14 and A.1 now imply that u is absolutely continuous on a.e. (with respect to the .n  1/-dimensional Lebesgue measure) line segment l in  parallel to the x1 -axis and for a.e. x 2 l, ˇ ˇ ˇ @u.x/ ˇ ˇ ˇ  g.x/: ˇ @x1 ˇ

The Fubini theorem then shows that j@u=@x1 j  g a.e. in  and hence @u=@x1 2 Lp ./  L1loc ./. The absolute continuity of u at l implies that @u=@x1 is the one-dimensional distributional derivative of u on l. Another application of the Fubini theorem to the integrals Z Z @'.x/ @u.x/ u.x/ dx D  '.x/ dx @x @x1 1   with ' 2 C01 ./, shows that @u=@x1 is the distributional derivative of u in . The 1;p other partial derivatives are handled similarly. Hence u 2 Wloc ./ and kukW 1;p ./  nkukN 1;p ./ . Conversely, let u 2 W 1;p ./. By e.g. Theorem 2.3.2 in Ziemer [361] there exist uj 2 C 1 ./ such that uj ! u in W 1;p ./, as j ! 1. Corollary 1.15 shows that jruj j are upper gradients of uj . Hence, uj 2 N 1;p ./ and kuj kN 1;p ./  kuj kW 1;p ./ , j D 1; 2; ::: . Proposition 2.3 provides us with a function uN 2 N 1;p ./ such that uN D u a.e. and jruj is a p-weak upper gradient of u. N Moreover, kuk N N 1;p ./  1;p y kukW 1;p ./ . Thus, by definition, u 2 N ./. If u 2 W 1;p ./ is quasicontinuous, then uN D u q.e., by Proposition 5.23, and hence u 2 N 1;p ./. We saw in Corollary 1.47 that for locally Lipschitz functions u on Rn , jruj is a p-weak upper gradient of u. Next, we show that it is in fact the minimal p-weak upper gradient of u. This further justifies the definition of upper gradients and gives equality of the norms in N 1;p and W 1;p .

A.1 N 1;p in Euclidean spaces

339

Proposition A.3. Let   Rn and let u be locally Lipschitz in . Then gu D jruj a.e. in . Proof. It remains to show the minimality. By Theorem 2.51, « gu .x/ D inf lim sup g dx a.e., g

r!0

B.x;r/

where the infimum is taken over all upper gradients g 2 Lploc ./ of u. At the same time, a.e. x 2  is a Lebesgue point of jruj, i.e. « jruj dx a.e. jru.x/j D lim r!0 B.x;r/

Hence it suffices to show that for all upper gradients g 2 Lploc ./ of u, jruj  g a.e. in . Fix an upper gradient g 2 Lploc ./ of u. By the Rademacher theorem (see e.g. Theorem 2.2.1 in Ziemer [361]), u is differentiable at a.e. x 2 , i.e. for  close to 0, u.x C /  u.x/ D ru.x/   C o.jj/: For q 2 Qn and 0 < " < 12 , let Aq;" D fx 2  W jru.x/  qj  "jru.x/jg: S Note that jqj  .1 C "/jruj on Aq;" . Clearly,   q2Qn Aq;" for all " > 0. Hence, it suffices to show that for a.e. x 2 Aq;" , jru.x/j 

1C" g.x/: 1"

(A.1)

Letting " ! 0 then yields jruj  g a.e. in . Note that A0;" D fx 2  W jru.x/j D 0g and (A.1) is trivially satisfied for all x 2 A0;" . Let q ¤ 0 and 0 < " < 12 be fixed but arbitrary. Let further L be the collection of all line segments in  with direction q. Since g 2 Lploc ./, the Fubini theorem shows that g 2 L1loc .l/ for a.e. l 2 L (with respect to the .n  1/-dimensional Lebesgue measure). For such l, a.e. x 2 l is a Lebesgue point of gjl . Another application of the Fubini theorem shows that a.e. x 2 Aq;" is a Lebesgue point of gjl for some l 2 L. By the Rademacher theorem, we can also assume that u is differentiable at x. For such x 2 Aq;" and t close to 0, we have Z g ds C o.t jqj/; (A.2) jru.x/  tqj D ju.x C t q/  u.x/j C o.t jqj/  x;xCtq

where x; x C tq is the line segment between x and x C t q. Now, the definition of Aq;" yields that for a.e. x 2 Aq;" , jru.x/j2  jru.x/  .ru.x/  q/j C jru.x/  qj  "jru.x/j2 C jru.x/  qj;

340

A Examples

and hence using (A.2),   Z jru.x/  qj 1 jqj .1  "/jru.x/j   g ds C o jru.x/j jtru.x/j x;xCtq jru.x/j Z 1C"  g ds C o.1/: jt qj x;xCtq Letting t ! 0, and using that x is a Lebesgue point for gjl , proves (A.1). We can now improve upon Theorem A.2. Corollary A.4. For every u 2 N 1;p ./ we have kukN 1;p ./ D kukW 1;p ./ . Proof. By Proposition A.3, the result is true for locally Lipschitz functions. Theorem A.2 shows that the norms in N 1;p ./ and W 1;p ./ are equivalent. Since they coincide on a dense subset (here we use Theorem 5.1), they must be equal. We have now seen that for Euclidean domains, the Newtonian spaces coincide with the usual Sobolev spaces. Moreover, as all Newtonian functions are ACCp and quasicontinuous, Newtonian spaces in fact pick out only the best representatives of Sobolev functions, i.e. they correspond (exactly) to the p-precise functions of Beppo Levi type considered by Deny–Lions [111] (p D 2), Fuglede [128], Ziemer [359], Evans– Gariepy [115] and Ohtsuka [299] in Rn . In weighted Sobolev spaces, quasicontinuous functions were studied in Chapter 4 in Heinonen–Kilpeläinen–Martio [171] and their “refined Sobolev spaces” coincide with our N 1;p spaces. See also Proposition A.13. Remark A.5. Both Lemma A.1 and Proposition A.3 can be proved in the same way for weighted Euclidean spaces with d D w dx provided that the weight w satisfies the n integrability condition w 1=.1p/ 2 L1loc .Rn / if p > 1, and w 1 2 L1 loc .R /, if p D 1. In particular, this holds for Ap -weights, for which also Theorem A.2 and Corollary A.4 are true if p > 1. Their proofs use the equivalence between two different definitions of weighted Sobolev spaces, obtained by Kilpeläinen [200], see also below. However, in the next section, we treat more general weights for p > 1 by a different method based on Cheeger’s theorem (Theorem A.7).

A.2 Weighted Sobolev spaces on Rn There are essentially two approaches to weighted Sobolev spaces on Rn , one using distributional derivatives as in Kufner [234] and Ohtsuka [299], and the other using converging sequences of smooth functions as in Heinonen–Kilpeläinen–Martio [171]. In general, these two definitions do not generate the same space, but Kilpeläinen [200] showed that for weights belonging to the Muckenhoupt class Ap (see (A.5) below), p > 1, they do.

A.2 Weighted Sobolev spaces on Rn

341

Next, we follow the exposition from [171], where p-admissible weights for Sobolev spaces on Rn were introduced. Even earlier, such weights were used to study regularity of degenerate elliptic equations with p D 2 by Fabes–Jerison–Kenig [116] and Fabes– Kenig–Serapioni [117]. See the notes to Chapter 4 for some more history. Definition A.6. A nonnegative locally integrable function w on Rn is a p-admissible weight if the measure  defined by d WD w dx is doubling and such that the following p-Poincaré inequality «

« ju  uB j d  C diam.B/

B

1=p jruj d

for all u 2 Lipc .Rn /

p

B

(A.3)

is satisfied. We show, in Proposition A.17 below, that the p-Poincaré inequality (A.3) is equivalent to our usual p-Poincaré inequality (and moreover that it is equivalent to require that  D 1), under the assumption that  is doubling. That our definition of p-admissible weights is equivalent to the definition given by Heinonen–Kilpeläinen–Martio [171], pp. 7–8, was shown by Corollary 20.9 in [171] (which only appears in the second edition of [171]). In [171] only p > 1 was considered but 1-admissible weights have been studied elsewhere. Sobolev spaces on Euclidean domains with p-admissible weights were defined in [171] (for p > 1) as the closure of C 1 -functions on  under the norm Z kukW 1;p .;/ D

1=p

Z juj d C

jruj d

p



p

:



For p D 2, this construction of weighted Sobolev spaces appeared already in Fabes– Kenig–Serapioni [117]. We denote the Sobolev spaces obtained in this way by W 1;p .; /, also for p D 1. In this definition, a vector-valued function v 2 Lp .; / is the (Sobolev) gradient of u 2 W 1;p .; / if there exist 'j 2 C 1 ./ with k'j kW 1;p .;/ < 1, such that both 'j ! u and r'j ! v in Lp .; /, as j ! 1. Note that this definition guarantees that W 1;p .; / is always a Banach space. n If d D w dx and w 1=.1p/ 2 L1loc .Rn /, if p > 1, or w 1 2 L1 loc .R /, if p D 1, 1;p then every u 2 W .; / is locally integrable and v is its distributional gradient, but in general this is not true, see Section 1.9 in Heinonen–Kilpeläinen–Martio [171]. However, for locally Lipschitz functions, Lemma 1.11 in [171] shows that ru is the distributional gradient, with any , at least for p > 1. The spaces W 1;p .; / are the natural spaces for studying partial differential equations of degenerate p-Laplace type, p > 1. In [171], a rich potential theory was developed for them. A nice feature of Newtonian spaces is that they (as Banach spaces) coincide with W 1;p .; /. We have already seen this for the classical Sobolev spaces W 1;p ./, where  is the Lebesgue measure.

342

A Examples

The following deep result is due to Cheeger [91]. For a proof, see Theorem 6.1 in [91]. The proof there is for  D X , but the general case then follows since gu only depends on u locally, by Lemma 2.23. (Note that completeness is needed as the proof in [91] uses Chapter 17 from [91].) Theorem A.7 (Cheeger [91], Theorem 6.1). Let X be a complete doubling p-Poincaré space, p > 1. Let further u be a locally Lipschitz function in , and define the upper pointwise dilation by Lip u.x/ WD lim sup r!0

ju.y/  u.x/j : r y2B.x;r/ sup

(A.4)

Then the minimal p-weak upper gradient gu D Lip u a.e. in . Note that the p-weak upper gradient is taken with respect to the measure  and can a priori depend both on  and p, cf. Section 2.9. Cheeger’s theorem shows that there is no such dependence. Corollary A.8. If X is a complete doubling p-Poincaré space, then the minimal q-weak upper gradient of a locally Lipschitz function in  is independent of q  p > 1. Corollary A.9. Assume that X is a complete doubling p-Poincaré space, and let 1;q u 2 Nloc ./, q  p > 1. Then the minimal p-weak upper gradient and the minimal q-weak upper gradient coincide a.e. 1;p Proof. First observe that u 2 Nloc ./, as if g 2 Lqloc ./ is an upper gradient of u, p then u; g 2 Lloc ./. Thus, u has a minimal p-weak upper gradient. Let G b . By Theorem 5.47 there exists a sequence of locally Lipschitz functions fj on G such that fj ! u in N 1;q .G/. It then follows that fj ! u also in N 1;p .G/. By passing to a subsequence, if necessary, we can assume that gfj ;q ! gu;q a.e. Hence

gu;q D lim gfj ;q D lim gfj ;p D gu;p j !1

j !1

a.e.;

by Corollary A.8. Let us mention that Cheeger’s result (Theorem A.7) can be reformulated in the following way. Theorem A.10. Let X be a complete doubling p-Poincaré space, p > 1. Let further u be a locally Lipschitz function in , and define the lower pointwise dilation by lip u.x/ WD lim inf r!0

ju.y/  u.x/j : r y2B.x;r/ sup

Then the minimal p-weak upper gradient gu D lip u a.e. in . Moreover, lip u D Lip u a.e., and both lip u and Lip u are upper gradients of u.

A.2 Weighted Sobolev spaces on Rn

343

Proof. By Proposition 1.14, lip u is an upper gradient of u. Since Lip u  lip u, it is also an upper gradient of u. Cheeger’s theorem (Theorem A.7) implies that gu  lip u  Lip u D gu

a.e.;

and hence we must have a.e. equality throughout. Another consequence of Cheeger’s result (Theorem A.7) is the following equality between the minimal p-weak upper gradient and the distributional gradient on weighted Euclidean spaces, which generalizes Proposition A.3. Proposition A.11. Assume that X D Rn , equipped with a measure , is a doubling p-Poincaré space, p > 1. If u is a locally Lipschitz function in , then gu D jruj a.e. in . Proof. Since both gu and jruj are local notions, we can assume that  D Rn . Corollary 1.47 shows that jruj is a p-weak upper gradient of u and hence gu  jruj a.e. By the Rademacher theorem (see e.g. Theorem 2.2.1 in Ziemer [361]), u is differentiable at a.e. x 2 , i.e. for  close to 0, u.x C /  u.x/ D ru.x/   C o.jj/: Choosing  D t ru.x/, t > 0, gives with y D x C , ju.y/  u.x/j D t jru.x/j2 C o.t jru.x/j/: Inserting this into (A.4) shows that Lip u.x/  jru.x/j. Hence, by Cheeger’s theorem (Theorem A.7), gu  jruj a.e. We can now prove the equality between the Newtonian spaces on Euclidean domains and the weighted Sobolev spaces. Note that for unweighted Rn this was proved without the use of Cheeger’s theorem (Theorem A.7) by more elementary methods in Theorem A.2 and Corollary A.4, see also Remark A.5. Proposition A.12. Assume that X D Rn , equipped with a measure , is a doubling p-Poincaré space, p > 1. Then W 1;p .; / D Ny 1;p ./ with the same norms. 1;p 1;p .; / D Nyloc ./. Similarly, Wloc Our definition of capacity, and hence of quasicontinuity, is slightly different from the definition of (Sobolev) capacity in Heinonen–Kilpeläinen–Martio [171], Section 2.35. However, by Theorem 6.7 (ix) and Proposition A.12, they coincide. Proof. Locally Lipschitz functions are dense both in Ny 1;p ./, by Theorem 5.1, and W 1;p .; /, by definition, and have the same norm in both spaces by Proposition A.11. It follows that Ny 1;p ./ D W 1;p .; /. The corresponding result for the local spaces follows directly.

344

A Examples

The following characterization shows exactly which representatives of W 1;p .; / functions belong to N 1;p ./. Proposition A.13. Assume that X D Rn , equipped with a measure , is a doubling p-Poincaré space, p > 1, or that X is unweighted Rn and p  1. If u 2 W 1;p .; / D Ny 1;p ./, then the following are equivalent: (a) u 2 N 1;p ./; (b) u 2 ACCp ./; (c) u is quasicontinuous; (d) jruj is a p-weak upper gradient of u; (e) jruj is a minimal p-weak upper gradient of u. 1;p 1;p Similarly, if u 2 Wloc .; / D Nyloc ./, then the following are equivalent: 1;p ./; (a0 ) u 2 Nloc

(b0 ) u 2 ACCp ./; (c0 ) u is quasicontinuous; (d0 ) jruj is a p-weak upper gradient of u; (e0 ) jruj is a minimal p-weak upper gradient of u. Proof. Assume first that u 2 W 1;p .; / D Ny 1;p ./. (a) , (b) , (c) This follows directly from Proposition 5.33. (a) ) (e) By Theorem 5.1, we can find a sequence uj of locally Lipschitz functions converging to u in N 1;p ./. By Proposition A.11 (Proposition A.3 for the case p D 1), guj D jruj j a.e. Moreover, guj ! gu in Lp .; /, and also ruj ! ru in Lp .; /, as j ! 1, by the definition of ru. It follows that gu D jruj a.e., and hence jruj is a minimal p-weak upper gradient of u. (e) ) (d) This is trivial. (d) ) (a) This follows directly from the definition of N 1;p ./. 1;p 1;p .; / D Nyloc ./ follows directly from The characterization for u 2 Wloc the first part and Lemma 5.18. (Recall that ACCp ./ D ACCp;loc ./, by Proposition 2.27.) In connection with W 1;p ./ (unweighted) one often considers ACL representatives, i.e. representatives which are absolutely continuous on almost every line parallel to the axes, see e.g. Theorem 2.1.4 in Ziemer [361]). Lemma A.14. Let X D Rn (unweighted) and u 2 ACCp ./. Then u 2 ACL./. Note that this is true also for p D 1.

A.2 Weighted Sobolev spaces on Rn

345

Proof. This follows directly from Lemma A.1. The converse is however not true. Example A.15. Let X D Rn (unweighted) and 1  q < p < 1. Let further E be a set with 0 D Cq .E/ < Cp .E/, cf. Example 2.47. Such examples can be found among Cantor type sets, see Theorem 5.3.2 in Adams–Hedberg [5] for a criterion when a certain capacity of such sets vanishes. When 1 < p  n < q one can choose E to be a singleton. Let u D E . Then u D 0 a.e., but not p-q.e., and hence u … N 1;p .Rn /. Thus, by Proposition A.13, u … ACCp .Rn /. On the other hand, u 2 N 1;q .Rn / and thus u 2 ACCq .Rn /. Hence, by LemmaA.14, u 2 ACL.Rn /. Example A.16. Let C .0/ D Œ0; 1 and let C .1/ be the set remaining after having removed the open middle part of length 12 from the interval C .0/ . Continue in this way by letting C .mC1/ be the set remaining after having removed the open middle parts of lengths 212m from each of the intervals in C .m/ . The setTC .m/ consists of 2m .m/ disjoint closed intervals, each of length 22m . Finally let C D 1

C .m/ /: mD0 .C Then C is a (planar self-similar) Cantor set. Let u D C . Then u 2 ACL.Rn / and its partial derivatives are zero a.e. Thus u 2 W 1;p .Rn /, p  1, by Theorem 2.1.4 in Ziemer [361]. On the other hand, the projection of C onto the line L D f.x; y/ W 2x C y D 0g is a line segment. Hence, u is not absolutely continuous on any line perpendicular to L and intersecting Œ0; 12 , i.e. u … ACL.R2 / when taken with respect to a coordinate system with L as one axis. Thus, by Lemma A.14, u does not belong to ACCp .R2 / for any p  1. Note that the ACCp condition is obviously invariant under rotations of the coordinate system (and even under biLipschitz transformations), while this example shows that the same is not true for the ACL condition. (We also see that C has positive one-dimensional Hausdorff measure.) Let us finally show that X D Rn supports our usual p-Poincaré inequality if and only if it supports a p-Poincaré inequality of the type (A.3). Note that this result is true also for p D 1. Proposition A.17. Let X D Rn be equipped with a doubling measure  given by d D w dx for some weight function w. Then the following are equivalent: (a) X supports a p-Poincaré inequality; (b) X supports a strong p-Poincaré inequality; (c) X supports a p-Poincaré inequality of the type (A.3); (d) X supports a strong p-Poincaré inequality of the type (A.3), i.e. with  D 1.

346

A Examples

Observe that when proving (c) ) (a) below we do not have Proposition A.11 at our disposal. Proof. (a) ) (b) This follows from Corollary 4.40. (b) ) (d) Let u 2 Lipc .Rn /. By Corollary 1.47, gu  jruj a.e., and Proposition 4.13 (d) yields « 1=p « 1=p « p p ju  uB j d  C diam.B/ gu d  C diam.B/ jruj d : B

B

B

(d) ) (c) This is trivial. (c) ) (a) Let u; g 2 Lipc .Rn /, where g is an upper gradient of u. By Theorem 4.15, it is enough to prove that « 1=p « p ju  uB j d  C diam.B/ g d B

B

(with C and  independent of u and g). Let l be a line parallel to the x1 -axis. Then gjl is a Lipschitz upper gradient of ujl . Thus by Lemma 2.14, ˇ ˇ ˇ @u ˇ ˇ ˇg ˇ ˇ

a.e. on l; @x1 where a.e. is taken with respect to the one-dimensional Lebesgue measure. Fubini’s theorem yields that ˇ ˇ ˇ @u ˇ ˇ ˇ  g a.e. on Rn : ˇ @x1 ˇ Similarly, ˇ ˇ ˇ @u ˇ ˇ ˇ  g a.e. on Rn ; j D 2; ::: ; n: ˇ @xj ˇ Hence jruj  ng a.e. on Rn . We thus have « 1=p « 1=p « p p ju  uB j d  C diam.B/ jruj d  C n diam.B/ g d : B

B

B

We finish this section by giving some examples of p-admissible weights. It was observed already in Fabes–Kenig–Serapioni [117] that weights from the Muckenhoupt class Ap , p > 1, and certain powers of Jacobians of quasiconformal mappings satisfy the conditions needed for Moser’s iteration and other regularity methods, i.e. that they are p-admissible in the terminology of Heinonen–Kilpeläinen–Martio [171] and according to our definition. (See also Chapter 15 in [171] for proofs that these weights are p-admissible.) Recall that a weight w is an Ap -weight if for all balls B  Rn , 8 « 1p ˆ « 1=.1p/ 1, w dx < (A.5) B ˆ B :C ess inf w; if p D 1. B

A.2 Weighted Sobolev spaces on Rn

347

The Ap -weights were introduced by Muckenhoupt [295] who showed that they are exactly those weights for which the unweighted Hardy–Littlewood maximal operator is bounded on weighted Lp .Rn ; w dx/, p > 1. Since then, Ap -weights have turned out to be useful in many areas of analysis. See e.g. the monographs García-Cuerva–Rubio de Francia [130] and Stein [328]. For n D 1, the Ap -weights are the only p-admissible weights, as was shown by J. Björn–Buckley–Keith [69]. More precisely, any doubling measure on R which supports a p-Poincaré inequality is necessarily absolutely continuous with respect to the Lebesgue measure, and its Radon–Nikodym derivative is an Ap -weight. Similar results for measures supporting Orlicz–Poincaré inequalities can be found in J. Björn [68]. In higher dimensions, non-Ap examples of p-admissible weights were given by e.g. Chanillo–Wheeden [90], Franchi–Gutiérrez–Wheeden [125], J. Björn [62], Keith [196] and J. Björn–Shanmugalingam [72]. Other examples of p-admissible weights come from the class of strong A1 -weights introduced by David–Semmes [109]. In particular, if w is a strong A1 -weight in Rn and 1  p < 1, then the weight w 1p=n is p-admissible, see Heinonen–Koskela [172], Franchi–Gutiérrez–Wheeden [125], J. Björn [62] and Heinonen–Kilpeläinen–Martio [171], Section 20.2 (which is only in the second edition). As Jacobians of quasiconformal mappings are strong A1 -weights according to Gehring [139], this also applies to them. The simplest examples of such weights are the powers w.x/ D jxj˛ , which are Jacobians of radial stretchings. Thus, for every ˛ > n, the weight jxj˛ is 1-admissible, see Corollary 15.35 in Heinonen–Kilpeläinen–Martio [171] and Theorem 1 in J. Björn [62]. Note that jxj˛ is an Ap -weight on Rn if and only if p > 1 C n˛ or ˛ D 0. Products of the type jx  a1 j˛1  jx  am j˛m were shown to be 1-admissible in Chanillo–Wheeden [90] and J. Björn [62]. Even more general examples of 1-admissible weights not coming from strong A1 weights were obtained by Keith [196] and J. Björn–Shanmugalingam [72]. See Section A.3 for details. We finish this discussion by noting that if Rn1 and Rn2 , equipped with measures 1 and 2 , respectively, are doubling p-Poincaré spaces, then Fubini’s theorem implies that Rn1 Rn2 with the product measure 1 2 is also a doubling p-Poincaré space, cf. Lemma 11 in [62] and Lemma 2 in Lu–Wheeden [248]. We end this section by stating two open problems. Open problem A.18. Does there exist a doubling measure on Rn , n  2, which supports a p-Poincaré inequality but is not absolutely continuous with respect to the Lebesgue measure? The next problem concerns Theorem A.7 for p D 1. A positive answer would directly imply that Corollaries A.8 and A.9, Theorem A.10 and Propositions A.11, A.12 and A.13 also hold for p D 1.

348

A Examples

Open problem A.19. Let X be a complete doubling 1-Poincaré space and u be a locally Lipschitz function in . Is it then true that the minimal 1-weak upper gradient gu D Lip u a.e. in ?

A.3 Uniform domains and power weights In Section 1.7 and Example 8.18, we pointed out that Neumann boundary data can be included into our study of Dirichlet problems, by letting X be the closure of a suitable Euclidean domain. In this short section we provide some examples of such domains. Definition A.20. A domain    X is A-uniform, A  1, if for every pair x; y 2  there is a curve  W Œ0; l  !  with .0/ D x and .l / D y such that 1 minft; l  t g for 0  t  l : A By definition, a domain is a nonempty connected open set. Here it actually follows from the definition that  is rectifiably connected. Uniform domains in Euclidean spaces were first studied by Martio–Sarvas [272] in the context of quasiconformal mappings. The following theorem is proved in Keith [196], for X D Rn and ˛ > 0, and in J. Björn–Shanmugalingam [72], Theorem 4.4, in the general case. It provides us with many examples of doubling p-Poincaré spaces. dist..t /; X n / 

Theorem A.21. Let X be a complete doubling p-Poincaré space and let   X be an A-uniform domain. Then there exists ˛0 > 0 such that for all ˛  ˛0 , the measure d .y/ WD dist.y; X n/˛ d.y/ is doubling on  and supports a p-Poincaré inequality on  with the dilation constant 3A. To obtain complete spaces, apply Proposition 7.1 from Aikawa–Shanmugalingam [9], which asserts that a closure of a doubling p-Poincaré space is also a doubling p-Poincaré space. The examples, which we can obtain by Theorem A.21, include e.g. balls and half spaces in Rn , with or without weights. The interior of the von Koch snowflake curve (see Example 1.23) is also a uniform domain. Theorem A.21 also yields examples of p-admissible weights on Rn . The following example is from Keith [196], Theorem 4. Example A.22. Let E  Rn be of Lebesgue measure zero and Rn n E be a uniform domain. Then Rn equipped with the measure d.x/ D dist.x; E/˛ dx, ˛  0, is a complete doubling 1-Poincaré space. Hence, by Proposition A.17, w.x/ D dist.x; E/˛ is a 1-admissible weight. Note that these weights may vanish on rectifiable curves (such as a circle in R3 ) and are therefore not strong A1 -weights, see David–Semmes [109]. For other examples of p-admissible weights which vanish on large sets, see J. Björn [62].

A.4 Glueing spaces together

349

A.4 Glueing spaces together In this section, we discuss another way of making new doubling Poincaré spaces from old ones. We start by the following familiar example. Example A.23 (Bow-tie). Let n  2 and write x 2 Rn as x D .x1 ; ::: ; xn /. Let XC D fx 2 Rn W xj  0; j D 1; ::: ; ng; X D fx 2 Rn W xj  0; j D 1; ::: ; ng; equipped with the Euclidean metric and the Lebesgue measure (which is doubling both on XC and X ). In Example 5.6 we saw that X D XC [ X does not support the p-Poincaré inequality for 1  p  n. As we shall now see, the situation is quite different for p > n. In Example 5.6 we showed, using a reflection argument, that both XC and X support 1-Poincaré inequalities with dilation constant  D 1. Let p > n. Corollary 6.20 shows that for all x0 2 X and r > 0, capp .fx0 g; B.x0 ; r//  C r np ;

(A.6)

where capp stands for the variational capacity on XC or X , depending on whether x0 2 XC or x0 2 X , respectively. Let u 2 N 1;p .X / be arbitrary. We can assume that u.0/ D 0. If B.x0 ; r/  XC or B.x0 ; r/  X , then the p-Poincaré inequality for B.x0 ; r/ follows directly from the p-Poincaré inequality on XC and X , respectively. If 0 2 B.x0 ; r/, then B.x0 ; r/  B.0; 2r/ DW 2B and 2B  B.x0 ; 3r/. Maz0 ya’s inequality (Theorem 6.21) applied to 2B \ XC then implies that « « Z C p p p juj dx  jruj dx  C r jrujp dx; capp .f0g; 2B/ 2B\XC 2B\XC B.x0 ;3r/ ª where the last inequality follows from (A.6). The integral 2B\X jujp dx is estimated similarly and hence «  « « p p p juj dx  C juj dx C juj dx B.x0 ;r/ 2B\XC 2B\X «  C rp jrujp dx: B.x0 ;3r/

Lemma 4.17 implies that X is a p-Poincaré space. By putting weights on XC and X in Example A.23, it is possible to glue sets with different topological dimensions, such as a half-line and a two-dimensional sector. Example A.24. For k D 1; 2; ::: ; m let nk  1 and Xk D fx 2 Rnk W xj  0; j D 1; ::: ; nk g;

350

A Examples

equipped with the measure dk D jxj˛k dx, where 0 < ˛1 C n1 D ˛2 C n2 D  D ˛m C nm : Let also p  1 be such that p > ˛1 C n1 or p D 1  ˛1 C n1 . From Corollary 15.35 in Heinonen–Kilpeläinen–Martio [171] and Theorem 4 in J. Björn [62] one can deduce that k is doubling and supports a p-Poincaré inequality on Rnk . A reflection argument as in Example 5.6 shows that k is doubling and supports a p-Poincaré inequality also on Xk , k D 1; 2; ::: ; m: A simple calculation shows that 1 ˛k Cnk  k .Xk \ B.0; r//  C r ˛k Cnk r C

for all r > 0:

Let u 2 C01 .B.0; r// be admissible in the definition of capp;k .f0g; Xk \ B.0; r//. Lemma 7.4 in Gilbarg–Trudinger [145] yields Z ru.x/  .y  x/ dx u.y/ D C n jy  xjnk R k for all y 2 Rnk . Hölder’s inequality then implies that 1 D u.0/ Z C

jru.x/j jxj1nk dx

B.0;r/

Z C

11=p Z jxj

.p.1nk /˛k /=.p1/

B.0;r/

 Cr

1.˛k Cnk /=p

jru.x/j jxj

dx

Z

1=p p

1=p

˛k

dx

B.0;r/

jru.x/j dk .x/ p

Rnk

and taking infimum over all such u shows that capp;k .f0g; Xk \ B.0; r//  C r ˛k Cnk p : It is then shown as in Example A.23 that X D X1 [ X2 [  [ Xm (e.g. embedded in some higher-dimensional Euclidean space and inheriting its Euclidean metric) is doubling and supports a p-Poincaré inequality. The above examples are special cases of a general technique, where new spaces are obtained by glueing spaces which support a p-Poincaré inequality: Let X and Y be two locally compact Ahlfors Q-regular metric measure spaces (see Definition 3.4). Let also A be a closed subset of X which has an isometric copy inside Y and form the space X [A Y which is the disjoint union of X and Y with points in the two copies of A identified. The distance between x 2 X and y 2 Y is inf .dX .x; z/ C dY .y; z//

z2A

A.5 Graphs

351

(where dX and dY are the distance functions in X and Y , respectively), and the measures on X and Y add to a Q-regular measure on X [A Y . The following result is from Heinonen–Koskela [174], Theorem 6.15. Here, ƒs stands for the s-dimensional Hausdorff measure. Theorem A.25. Assume that Q  p < s  Q and that ƒs .A \ Br /  C r s for all balls Br in X or in Y with centre in A and radius 0 < r < minfdiam X; diam Y g. If both X and Y are p-Poincaré spaces, then so is X [A Y . More intricate examples of this phenomenon can be found in Remark 6.19 in Heinonen–Koskela [174].

A.5 Graphs In this section we assume that p > 1. Holopainen and Soardi studied p-harmonic functions on graphs in [183] and [184]. It is interesting that these discrete objects can be included in the continuous theory considered here. Let G D .V ; E/ be a (finite or infinite) graph, where V is the set of vertices and E  V V is the set of edges. If x and y are endpoints of an edge we say that they are neighbours and write x y. Let us first look at the definitions used by Holopainen–Soardi. The metric on G is the length of the shortest path between two vertices and the measure on G is the counting measure # on the vertices. It is assumed that the counting measure is doubling and that the following p-Poincaré inequality holds for all balls B and functions u W V ! R,  1=p 1 X 1 X ju.x/  uB j  C diam.B/ ju.y/  u.x/jp ; #B #B x2B

x2B y x

where uB D

1 X u.x/: #B x2B

A function u W V ! R is p-harmonic in U  V with boundary values uj@U if X ju.y/  u.x/jp2 .u.y/  u.x// D 0 for all x 2 U: (A.7) y x

Here the boundary @U is defined by @U D fx 2 V n U W x y for some y 2 U g; and we define ja  ajp2 .a  a/ to be 0 for a 2 R and 1 < p < 2.

352

A Examples

Note that if p D 2 then (A.7) is nothing but the mean-value property for u, i.e. for every u 2 U , u.x/ is the average of the values at its neighbours. The definitions above have several similarities with our theory, but do not fall directly into the theory. For us the following definitions are more natural. Let again G D .V ; E/ be a finite or infinite graph. Consider S an edge as a geodesic closed ray of length 1 between its endpoints, and let X D e2E e be the metric graph, where we glue the edges at the vertices as given by G . We equip X with the inner metric and the one-dimensional Lebesgue measure , and assume that G equipped with  is a doubling p-Poincaré space, which in particular requires G to be connected. Let  b X be a nonempty open set and assume that @  V . x ! R is p-harmonic in  (in the sense of DefiniTheorem A.26. The function u W  tion 7.7), with the boundary values uj@ taken continuously, if and only if it is linear on each edge in  and satisfies X ju.y/  u.x/jp2 .u.y/  u.x// D 0 for all x 2 V \ : (A.8) y x

Thus we see that we obtain exactly the same p-harmonic functions as Holopainen– Soardi, but as a special case of our theory. To prove Theorem A.26 we first obtain the following characterization. x ! R is p-harmonic in  (in the sense of DefiniLemma A.27. The function u W  tion 7.7), with the boundary values uj@ taken continuously, if and only if it is linear on each edge in  and satisfies X ju.x/  u.y/jp2 .u.x/  u.y//.w.x/  w.y// D 0 (A.9) x y

for all w 2 N01;p ./. Here the summation is over all edges .x; y/  , x; y 2 V , and the same is true for all similar summations below when both variables are free. Observe that it is only the open edge .x; y/ that is required to belong to , the endpoints may belong to @. Proof. Let

Z I.u/ D 

gup d:

Assume first that u is p-harmonic in . For each edge x y, parameterized as  7! .1   /x C y, 0    1, we get using Lemma 2.14 and Hölder’s inequality that ˇZ 1 ˇ Z y Z y 1=p ˇ du..1  /x C y/ ˇˇ ju.x/  u.y/j D ˇˇ gu ds  gup ds ; dˇ  d 0 x x

A.5 Graphs

353

with equality throughout if and only if u is linear on the edge. As u is p-harmonic and thus minimizes the energy over all functions with the boundary values uj@ , it must minimize the energy on each edge among all functions with the same values at the end points, i.e. u must be linear on the edge. Let u t D u C t w, where t 2 R, w 2 N01;p ./, and assume, without loss of generality, that w is linear on each edge. As u minimizes the p-energy we get, using Proposition 7.9 (d) and the fact that u t is linear on each edge, dI.u t / ˇˇ ˇ tD0 dt ˇ d X ˇ D ju.x/  u.y/ C t .w.x/  w.y//jp ˇ tD0 dt x y X Dp ju.x/  u.y/jp2 .u.x/  u.y//.w.x/  w.y//;

0D

x y

and thus (A.9) is satisfied. Conversely, assume that u is linear on each edge and satisfies (A.9) for all w 2 N01;p ./. Letting v D u C w yields 0D

X

ju.x/  u.y/jp2 .u.x/  u.y//..v.x/  v.y//  .u.x/  u.y///:

x y

Rewriting this and using the linearity of u on each edge, we obtain that I.u/ D

X

ju.x/  u.y/jp

x y

D

X

ju.x/  u.y/jp2 .u.x/  u.y//.v.x/  v.y//

x y



X

ju.x/  u.y/jp1 jv.x/  v.y/j

x y



X

ju.x/  u.y/jp

x y

D I.u/

11=p

11=p  X

jv.x/  v.y/jp

1=p

x y

I.v/

1=p

:

Hence, I.u/  I.v/ and as w D vu 2 N01;p ./ was arbitrary, the result follows. Proof of Theorem A.26. Assume first that u is p-harmonic. Lemma A.27 implies that it is linear on each edge. Moreover, if we fix z 2 V \, then we can find w 2 N01;p ./ such that w.z/ D 1 and w.y/ D 0 for all z ¤ y 2 V \ . Again using Lemma A.27,

354

A Examples

we see that X 0D ju.x/  u.y/jp2 .u.x/  u.y//.w.x/  w.y// x y

D

X

ju.z/  u.y/jp2 .u.z/  u.y// 

y z

D2

X

X

ju.x/  u.z/jp2 .u.x/  u.z//

x z

ju.z/  u.y/j

p2

.u.z/  u.y//:

y z

Hence (A.8) is satisfied. Conversely, assume that u is linear P on each edge and that (A.8) is satisfied. Let w 2 N01;p ./. We can write w D z2\V wz , where wz 2 N01;p ./, wz .y/ D 0 for all y 2  \ V n fzg and wz .z/ D w.z/. We then get that X ju.x/  u.y/jp2 .u.x/  u.y//.w.x/  w.y// x y

D

X x y



ju.x/  u.y/jp2 .u.x/  u.y//w.x/ X

ju.x/  u.y/jp2 .u.x/  u.y//w.y/

x y

X X

D

ju.z/  u.y/jp2 .u.z/  u.y//wz .z/

z2\V y z



X X

z2\V x z

X

D2

z2\V

wz .z/

ju.x/  u.z/jp2 .u.x/  u.z//wz .z/ X

ju.z/  u.y/jp2 .u.z/  u.y//

y z

D 0:

A.6 Carnot–Carathéodory spaces and Heisenberg groups Some of the best studied examples of non-Euclidean spaces falling within the scope of our theory are Heisenberg groups, which are special cases of Carnot–Carathéodory spaces. Let X1 ; X2 ; ::: ; Xm be C 1 -smooth vector fields on Rn . The sub-Riemannian metric associated with these vector fields is 2 g.x; v/ D inffc12 C  C cm W .c1 ; ::: ; cm / 2 Rm and c1 X1 .x/ C  C cm Xm .x/ D vg:

The length of an absolutely continuous curve (in the traditional sense)  W Œa; b ! Rn with respect to the sub-Riemannian metric g is then defined by Z b g..t /;  0 .t //1=2 dt: length. / D a

A.6 Carnot–Carathéodory spaces and Heisenberg groups

355

Note that if v … spanfX1 .x/; ::: ; Xm .x/g, then g.x; v/ D 1 and hence length. / is finite only if for a.e. t 2 Œa; b, the tangent  0 .t / is a linear combination of the vector fields X1 ; X2 ; ::: ; Xm . Such curves are called admissible or horizontal. The Carnot–Carathéodory distance between two points x and y in Rn is then d.x; y/ D inf length. /; where the infimum is taken over all admissible curves. It is clearly symmetric and satisfies the triangle inequality. However, it may be infinite, so it is a metric only if every pair of points can be joined by an admissible curve. The following condition, called Chow’s condition, guarantees this. Theorem A.28. Assume that at every x 2 Rn , the vector fields X1 ; X2 ; ::: ; Xm and their iterated brackets ŒXi ; Xj ; ŒŒXi ; Xj ; Xk ; ::: , up to some order, span the whole of Rn . Then any two points x; y 2 Rn can be joined by an admissible curve. For a proof of Theorem A.28, see Chow [95], Rashevski˘ı [307] or Bellaïche [28]. An early version of Chow’s condition can be traced back to Carathéodory [89]. If Chow’s condition holds, then length. / coincides with the length l of an admissible curve. By Proposition 11.6 in Hajłasz–Koskela [154], jXuj D j.X1 u; ::: ; Xm u/j is an upper gradient (with respect to the Carnot–Carathéodory metric) of u 2 C 1 . Theorem 11.7 in [154] shows that it is also minimal. 2 Chow’s condition also guarantees that the differential operator X12 C  C Xm is hypoelliptic, see Hörmander [185], and it is therefore often also called Hörmander’s condition. It was shown by Nagel–Stein–Wainger [297] that under this condition, the n-dimensional Lebesgue measure is locally doubling on balls in Rn taken with respect to the Carnot–Carathéodory metric, see Theorem 1 in [297]. A fundamental theorem due to Jerison [191] shows that Rn , equipped with the n-dimensional Lebesgue measure and the Carnot–Carathéodory metric generated by a system of vector fields satisfying Hörmander’s condition, supports a 1-Poincaré inequality. Together with the local doubling condition from [297] this means that Carnot– Carathéodory spaces are suitable for the potential theory considered in this book. The local doubling condition is enough since we only consider potential theory on bounded sets. In this setting, p-harmonic functions are solutions of the equation m X

Xj .jXujp2 Xj u/ D 0;

(A.10)

j D1

where Xj is the formal adjoint of Xj , j D 1; ::: ; m; defined by Z Z  vXj u dx D uXj v dx for all v 2 C01 .Rn /: Hypoelliptic equations, such as (A.10), associated with vector fields have been studied by many authors. The nonlinear theory has been considered by e.g. Balogh–

356

A Examples

Manfredi–Tyson [25], Capogna–Danielli–Garofalo [85], [86], Chernikov–Vodop0 yanov [92], [93], Danielli–Garofalo–Nhieu [106], [107], [108], Garofalo [136], Gianazza–Marchi [141] and Hajłasz–Strzelecki [155]. Potential theory for hypoelliptic equations is studied in Markina–Vodop0 yanov [261], [262]. We refer the reader to the survey Balogh–Tyson [26] and the forthcoming monograph Garofalo [137] for further references and results on Carnot–Carathéodory spaces. Note that due to the vector structure, p-harmonic and Cheeger p-harmonic functions (for the gradient u 7! .X1 u; ::: ; Xm u/) coincide in Carnot–Carathéodory spaces, making the whole theory closer to the Euclidean situation. In particular, the Wiener criterion and the sheaf property hold in this case, see Appendix B.2. Some additional results that are only known for Cheeger p-harmonic functions are also mentioned in Appendix B.2. Example A.29 (The Grushin plane). Consider R2 with XD

@ @x

and Y D x

Note that ŒX; Y  D

@ : @y

@ @y

and hence, Hörmander’s condition holds. Thus, R2 , equipped with the two-dimensional Lebesgue measure and the Carnot–Carathéodory metric generated by X and Y , supports a 1-Poincaré inequality. A ball centred at the point .a; b/ 2 R2 is of size r in the x-direction and roughly of size maxfar; r 2 g in the y-direction, and thus has volume comparable to maxfar 2 ; r 3 g. This implies that the doubling condition holds globally, making the Grushin plane into a doubling 1-Poincaré space. In this case, harmonic functions are solutions of the hypoelliptic equation 2 @2 u 2@ u C x D 0: @x 2 @y 2

Example A.30. Prime examples of Carnot–Carathéodory spaces satisfying Hörmander’s condition are the Heisenberg and Carnot groups. Note that due to the group structure, the local doubling condition from Nagel–Stein–Wainger [297] becomes global. We refer the reader to Balogh–Tyson [26], Capogna–Danielli–Pauls–Tyson [87], Garofalo [137], Hajłasz–Koskela [154], Heinonen [168] and Stein [328] for more details. Here we restrict ourselves to presenting the simplest Heisenberg group H1 , which is C R with the group law .z; t /.w; s/ D .z C w; t C s C 2 Im z w/ N and the identity element .0; 0/. By Proposition 11.15 in [154], the Carnot–Carathéodory metric generated by the vector fields XD

@ @ C 2y @x @t

and Y D

@ @  2x @y @t

A.7 Further examples

357

is equivalent to the metric induced by the group law from the homogeneous norm k.z; t/k D .jzj4 C t 2 /1=4 . Note that ŒX; Y  D 4

@ ; @t

so that Hörmander’s condition holds. Balls in H1 are roughly of size r in the X and Y directions and of size r 2 in the ŒX; Y  direction. The Lebesgue measure makes H1 into an Ahlfors 4-regular space which supports a 1-Poincaré inequality by Jerison’s theorem, see also Proposition 11.17 in Hajłasz– Koskela [154] and the references therein. This doubling 1-Poincaré space is topologically 3-dimensional, measure theoretically Ahlfors 4-regular, and the gradient of smooth functions is given by the two vector fields X and Y . Harmonic functions are solutions of the hypoelliptic equation 

2 @2 u @2 u @2 u @2 u 2 2 @ u C C 4.x C y / C 4 y  x @x 2 @y 2 @t 2 @x@t @y@t



D 0:

A.7 Further examples If u is a Lipschitz function on a Riemannian manifold, with the natural measure, then by Proposition 10.1 in Hajłasz–Koskela [154], jruj is the minimal upper gradient of u. It then follows from Buser’s inequality [81] that complete Riemannian manifolds with nonnegative Ricci curvature are doubling 1-Poincaré spaces. If the Ricci curvature is merely bounded from below (e.g. constant negative as in the hyperbolic plane), then for every R > 0, the doubling condition and the 1-Poincaré inequality hold for all balls with radii at most R, which is sufficient for most applications concerning p-harmonic functions and potential theory. See [154] for the details and further references. As in Carnot–Carathéodory spaces, the vector structure of the gradient implies that p-harmonic and Cheeger p-harmonic functions on Riemannian manifolds coincide, making the whole theory closer to the Euclidean situation. See Appendix B.2 for some additional results that are only known for Cheeger p-harmonic functions. The doubling property and the p-Poincaré inequality reasonably well survive under Gromov–Hausdorff limits (sometimes called Vietoris limits), which provides further examples, see Theorem 9.6 in Cheeger [91] and Theorem 3 in Keith [196]. Semmes [314] showed that every Ahlfors n-regular space with sufficiently many rectifiable curves supports a 1-Poincaré inequality, n D 1; 2; ::: . Hinde–Petersen [179] have defined a generalized doubling condition which on its own implies the 1-Poincaré inequality. Bourdon–Pajot [75], Hanson–Heinonen [159] and Laakso [237] created some exotic examples of doubling p-Poincaré spaces. Laakso’s examples are particularly interesting, as he shows that for every Q  1 there is a complete 1-Poincaré space which

358

A Examples

is Ahlfors Q-regular (see Definition 3.4). The spaces are obtained from Cartesian products of Cantor sets with an interval, by identifying points from different copies of the interval. Other constructions of the Laakso spaces have been given by Steinhurst [329] and Romeo–Steinhurst [309]. Recently, Mackay–Tyson–Wildrick [254] showed that certain non-self-similar Sierpi´nski carpets are doubling p-Poincaré spaces. For a more in-depth discussion of when metric spaces support Poincaré inequalities see Heinonen [170].

A.8 Notes Theorem A.2 was obtained by Shanmugalingam [319]. Hajłasz [151] gave a more detailed proof and also proved Corollary A.4. Theorems A.7 and A.10 are from Cheeger [91], where a different, but equivalent, definition of minimal p-weak upper gradient is used, cf. Appendix B.2. For weighted Rn , a full proof of Proposition A.12 does not seem to be available in the literature. Some of the implications in Proposition A.13 are either new here or proved here for the first time, see also the discussion on Proposition 5.33 in the notes to Chapter 5. The equivalence (a) , (c) in Proposition A.17, as well as Remark A.5, also seems to be new here. It was Shanmugalingam, Section 3 in [321], who first looked at the metric graphs and showed that they fit directly into our theory, and that the p-harmonic functions agree with those of Holopainen–Soardi [183], [184]. The proofs of Theorem A.26 and Lemma A.27 are relatively close to the proofs by Andersson [18] and Holopainen– Soardi [183] given for the discrete setting. Metric graphs have later been used for easy construction of examples with certain properties, such as the examples of nonunique removability and nonunique continuation in Section 12.4. To facilitate calculations for p-harmonic functions on graphs Andersson [18] wrote a computer program solving the Dirichlet problem (approximately). The program is available freely on the Internet at http://www.mai.liu.se/~anbjo/pharmgraph Example A.24 in the case m D 2, n1 D 1, n2 D 2, ˛1 D 0 and ˛2 D 1 appeared in A. Björn [35], Section 8, where the doubling condition and the strong 1-Poincaré inequality were given direct, although somewhat tedious, proofs. The Cantor set C in Example A.16 is sometimes called the Garnett–Ivanov set as they showed, independently, that it is removable for bounded analytic functions, see Garnett [135] and Ivanov [188], footnote on p. 346. For the historically interested reader it may be worth noting that Veltmann [346] considered planar Cantor sets similar to C in 1882 (see also Veltmann [345]), before Cantor [84], p. 590 (p. 407 in Acta Math.),

A.8 Notes

359

published his ternary set in 1883. Cantor type sets were constructed already in 1875 by Smith [325], Sections 15 and 16.

Appendix B

Hajłasz–Sobolev and Cheeger–Sobolev spaces

In this chapter we shall briefly present two other approaches to Sobolev type spaces on metric spaces: the Hajłasz–Sobolev space M 1;p .X / and the Cheeger–Sobolev space. Yet another way of defining Sobolev type spaces on metric spaces is by considering Lp -integrable functions which in a pair with some other Lp -integrable function satisfy an abstract Poincaré inequality, as in Koskela–MacManus [229]. In Franchi–Hajłasz– Koskela [126], Sobolev spaces on metric spaces were obtained as a closure of the set of locally Lipschitz functions together with a linear operator playing the role of a gradient. If X is a complete doubling q-Poincaré space for some q < p and p > 1, all four definitions result in the same space which coincides with N 1;p .X / (as Banach spaces), see Theorems 4.9 and 4.10 in Shanmugalingam [319], Theorem 4.5 in [229] and Theorem 10 in [126]. In view of Theorem 4.30, this holds also for complete doubling p-Poincaré spaces. A thorough discussion of all these different definitions and relations between them can be found in Hajłasz [151]. Some topics related to Sobolev functions on general metric spaces were also studied in Semmes [315]. Even more generally, the Poincaré inequality can be replaced by a more general inequality involving the mean oscillation on the left-hand side and a functional satisfying certain summability conditions on the right-hand side, see Heikkinen–Koskela– Tuominen [166]. Yet another generalization is to replace the mean value in the left-hand side by an integral operator. Such Sobolev spaces were studied by Yan–Yang [354]. Axiomatic Sobolev spaces and axiomatic potential theory on metric spaces have been studied by Gol0 dshtein–Troyanov [147], [148] and Timoshin [331], [332], [333]. Shanmugalingam [323] has shown that under some mild geometrical assumptions (such as the doubling condition, a Poincaré inequality and strong locality of the axiomatic D-structure), the axiomatic Sobolev spaces are the same as Newtonian spaces. For p D 2, they also coincide with the Sobolev spaces defined by means of Dirichlet forms, see [323].

B.1 Hajłasz–Sobolev spaces Hajłasz–Sobolev spaces are often called Hajłasz spaces in the literature. Here we have chosen to call them Hajłasz–Sobolev spaces to emphasize that they are a type of Sobolev spaces. The idea of Sobolev type spaces on general metric measure spaces appeared for the first time in Hajłasz [149]: A function f 2 Lp .X / belongs to the Hajłasz–Sobolev

B.1 Hajłasz–Sobolev spaces

361

space M 1;p .X / if there exist g 2 Lp .X / and E  X such that .E/ D 0 and for all x; y 2 X n E, jf .x/  f .y/j  d.x; y/.g.x/ C g.y//: (B.1) The Hajłasz–Sobolev space equipped with the norm kf kLp .X/ C inf kgkLp .X/ ; g

where the infimum is taken over all g 2 Lp .X / satisfying (B.1), is a Banach space. A function g satisfying (B.1) is a Hajłasz gradient of f . If p > 1, then there exists a unique g (up to measure zero) which minimizes the Lp -norm among all functions satisfying (B.1). In contrast to minimal p-weak upper gradients, the minimal Hajłasz gradient is not pointwise minimal a.e., which is easy to see in the case when X D f0; 1g has exactly two points. Example B.1. Let X D f0; 1g and f D f0g . Then ´ t; x D 0; 0  t  1; g t .x/ D 1  t; x D 1; are Hajłasz gradients of f . If p > 1, then g1=2 has minimal Lp -norm among all Hajłasz gradients of f , but it is not pointwise minimal. If p D 1, all g t , 0  t  1 have minimal L1 -norm. This definition of Sobolev type spaces stems from a similar characterization of the classical Sobolev space W 1;p .Rn /, p > 1, proved in Hajłasz [149]. Similar characterizations involving maximal functions were for p D 1 and Orlicz–Sobolev spaces given by Hajłasz [150] and Tuominen [340], respectively. It was also shown in [149] that M 1;p ./ D W 1;p ./ for bounded extension domains. This is not true in general, as we shall now see. Example B.2. Let  be the slit disc B.0; 1/ n .1; 0  C D R2 , cf. Example 5.9. Then f .z/ D .2jzj  1/C arg z belongs to W 1;p ./. Let g be a Hajłasz gradient of f , ˚ E˙ D x C yi 2  W x <  34 and ˙ y > 0 and E D EC [ E ; where i here is the imaginary unit. If z 2 EC , then f .z/ > 12 .=2/ > 12 , and similarly f .z/ N <  12 . It follows that for a.e. z D x C yi 2 EC , g.z/ C g.z/ N  Hence for p  1,

Z

Z



g p d  E

Thus f … M 1;p ./.

EC

1 jf .z/  f .z/j N > : d.z; zN / 2y 1 4y

p

d.x C yi / D 1:

362

B Hajłasz–Sobolev and Cheeger–Sobolev spaces

Example B.3 (Bow-tie). Let p D n  2, and let X , X˙ and w be as in Example 5.6. It was shown in Example 5.6 that w 2 N 1;n .X /. Let g be a Hajłasz gradient of w. As in Example B.2, we see that for a.e. x 2 XC with jxj < 1 we have g.x/ C g.x/  Hence

Z

1 jw.x/  w.x/j  : d.x; x/ 2jxj

Z

1

g p d  C X

Thus w … M

1;n

0



1 4r

n

r n1 dr D 1:

.X /.

Example B.4. Yet another example is the von Koch snowflake curve K, see Example 1.23 where it is shown that N 1;p .K/ D Lp .K/. On the other hand, it is easily verified from the definition that M 1;p .K/ ¤ Lp .K/. Thus in general, we see that M 1;p .X / ¤ Ny 1;p .X /. Nevertheless, Shanmugalingam [319] showed that M 1;p .X / always continuously embeds into Ny 1;p .X /, and also that if X is a doubling q-Poincaré space for some q < p (cf. Theorem 4.30), then M 1;p .X / D Ny 1;p .X /, with equivalent norms. It is easily seen by integrating (B.1) twice over a ball B, that functions in Hajłasz– Sobolev spaces a priori satisfy a Poincaré inequality, while for Newtonian functions this is true only under suitable geometrical conditions on the underlying metric space, cf. Appendix A. From the point of view of p-harmonic functions and potential theory, a drawback of the Hajłasz–Sobolev space is that the minimal Hajłasz gradient is not local, i.e. if f 2 M 1;p .X / and g 2 Lp .X / satisfy (B.1) and f is constant on some set A  X , then f and gQ D gXnA need not satisfy (B.1). In particular, the minimal Hajłasz gradient does not vanish a.e. on a set where f is constant. (Note that by the glueing lemma (Lemma 2.19), a similar property holds for N 1;p .X / and p-weak upper gradients.) The reason for this is that g in (B.1) corresponds to the maximal function of the usual gradient rather than to its modulus. Another drawback is that the minimal Hajłasz gradient does depend on p, cf. Corollary A.8. Example B.5. Let X D f0; 1; 2g  R (with the induced distance) and let u D f1g . If p > 1, then a straightforward calculation shows that ´ 1 a; x D 0; 2; where a D g.x/ D ; 1 C 21=.p1/ 1  a; x D 1; is the minimal p-Hajłasz gradient of u. For p D 1 it is easy to see that g D u is the minimal 1-Hajłasz gradient of u, which is thus unique in this case.

B.2 Cheeger–Sobolev spaces and differentiable structures

363

Motivated by a characterization of W 1;p .Rn /, due to Calderón [82], Shvartsman [324] used the fractional sharp maximal function to define Calderón–Sobolev spaces on metric spaces. By Theorem 3.4 in Hajłasz–Kinnunen [152] these spaces coincide with the Hajłasz–Sobolev spaces if p > 1 and  is doubling.

B.2 Cheeger–Sobolev spaces and differentiable structures Cheeger [91] defines Sobolev spaces on general metric spaces as those u 2 Lp .X / for which there exists a sequence ui with upper gradients gi , i D 1; 2; ::: , such that ui ! u in Lp .X /, as i ! 1, and lim inf i!1 kgi kLp .X/ < 1. It was shown by Shanmugalingam [319], Theorem 4.10, that the Cheeger–Sobolev space coincides with Ny 1;p .X / when p > 1, with the same norms. (Note however that the equivalence classes in Cheeger’s definition are a.e.-equivalence classes, not q.e. as for N 1;p .X /, so that the Cheeger–Sobolev space is equal to Ny 1;p .X /.) For p D 1 the Cheeger–Sobolev space coincides with BV.X /, the space of functions of bounded variation, rather than Ny 1;1 .X /, see the notes to Chapter 1 for some comments on BV. Cheeger’s definition yields the notion of partial derivatives in the following deep result, see Theorem 4.38 in [91]. Theorem B.6 (Cheeger). Let X be a complete doubling p-Poincaré space, p > 1. Then there exists N and a countable collection .U˛ ; X ˛ / of measurable sets U˛ and “coordinate” functions X ˛ W X ! Rk.˛/ , 1  k.˛/  N , such that  Lipschitz  S  X n ˛ U˛ D 0 and for every Lipschitz function f W X ! R there exist unique bounded vector-valued functions d ˛ f W U˛ ! Rk.˛/ such that for a.e. x 2 U˛ , jf .y/  f .x/  hd ˛ f .x/; X ˛ .y/  X ˛ .x/ij D 0; r!0 y2B.x;r/ r lim

sup

where h  ;  i denotes the usual inner product in Rk.˛/ . Cheeger further shows that for a.e. x 2 U˛ , there is an inner product norm j  jx on Rk.˛/ such that for all Lipschitz f , 1 gf .x/  jd ˛ f .x/jx  Cgf .x/; C

(B.2)

where C is independent of f and x, see p. 460 in Cheeger [91]. As Lipschitz functions are dense in Ny 1;p .X /, the “gradients” d ˛ f extend uniquely to the whole Ny 1;p .X /, by Theorem 10 in Franchi–Hajłasz–Koskela [126] or Keith [197]. Moreover, (B.2) holds even for functions in N 1;p .X / (but not in Ny 1;p .X /, as functions in Ny 1;p .X /nN 1;p .X / do not have p-weak upper gradients in Lp .X /). For the vector-valued Cheeger gradient  d ˛ f , it was shown in J. Björn [61] that Newtonian functions are Lp -differentiable

364

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at a.e. point, where p  D sp=.s  p/ is the exponent from the Sobolev–Poincaré inequality, Theorem 4.21. Using Cheeger gradients, one can define Cheeger p-harmonic functions in the same way as here, just by replacing the minimal p-weak upper gradient by the modulus of the Cheeger gradient. All properties that can be proved for p-harmonic functions hold also for Cheeger p-harmonic functions (with the same proofs). However, because of the vector structure of Cheeger gradients, the Cheeger p-harmonic functions actually satisfy a partial differential equation (in a weak sense), viz. Z jd ˛ f jp2 d ˛ f  d ˛ ' d D 0 for all ' 2 Lipc ./; x 

where  denotes the inner product giving rise to the norm j  jx from (B.2) (note that it depends on x). This makes it possible to prove some stronger results for Cheeger p-harmonic functions, much like those in Euclidean spaces. In particular, a partition of unity argument shows that Cheeger p-harmonicity is a local property, i.e. that a function S which is Cheeger p-harmonic in j , j D 1; 2; ::: , is Cheeger p-harmonic in j1D1 j , see Björn–Björn [46]. This sheaf property is not known for p-harmonic functions defined by means of upper gradients, see Open problems 9.22 and 9.23. Under some additional assumptions, Cheeger p-harmonic functions are known to be Lipschitz continuous, see Koskela–Rajala–Shanmugalingam [230]. The Wiener criterion in its full strength (sufficiency and necessity) was proved for Cheeger p-harmonic functions in J. Björn [65]. In J. Björn–MacManus–Shanmugalingam [70], Cheeger superminimizers were related to Radon measures and estimated by means of Wolff potentials. Recall also that Brelot’s theorem (Theorem 10.45) is valid for Cheeger 2-harmonic functions. See also Keith [197] for a generalization of Theorem B.6.

Appendix C

Quasiminimizers

In this appendix we assume that 1 < p < 1 and that X is a complete doubling p-Poincaré space. Nonlinear potential theory on metric spaces has not only been studied in connection with p-harmonic functions, but also in the more general context of quasiminimizers. Some of the results in this book hold also for quasiminimizers with hardly any changes in the proofs given. In other cases, the results remain true but have considerably more complicated proofs for quasiminimizers. Yet other results remain open or are false for quasiminimizers. In this book we have chosen not to deal with the potential theory associated with quasiminimizers. One reason for this is that the extra complications needed would have made the book unnecessarily difficult for all those not really interested in quasiminimizers. Also, the potential theory of quasiminimizers is considerably less understood than the p-harmonic theory we cover. Nevertheless, we want to give a very brief overview of the potential theory of quasiminimizers. 1;p Definition C.1. Let Q  1. A function u 2 Nloc ./ is a Q-quasiminimizer in  if for all ' 2 Lipc ./ we have Z Z p p gu d  Q guC' d: '¤0

'¤0

1;p A function u 2 Nloc ./ is a Q-quasisuper(sub)minimizer in  if (7.5) holds for all nonnegative (nonpositive) ' 2 Lipc ./. We also say that a Q-quasiharmonic function is a continuous Q-quasiminimizer.

Quasiminimizers have a rigidity that minimizers lack, viz. if c p  F .; x/  C  p ; then the quasiminimizers from Definition C.1 coincide with the quasiminimizers of Z F .gu ; x/ d: (The quasiminimizing constant Q may of course change.) The following result was recently obtained by Martio [268], Theorem 4.1. 1;p Theorem C.2. Let u be a Q-quasiminimizer in  and f 2 Wloc ./,   Rn (unweighted), be such that jrf j  cjruj a.e. in , where 0 < c < Q1=p . Then u C f is a Q0 -quasiminimizer in , where Q0 D .1 C c/p =.Q1=p  c/p .

366

C Quasiminimizers

This result shows that quasiminimizers are much more flexible under perturbations than solutions of differential equations. This flexibility can be useful in applications and in particular shows that results obtained for quasiminimizers are very robust. On the other hand, one-dimensional examples show that being a quasiminimizer is not a local property, see Judin [193] and Björn–Björn [48]. (Recall that p-harmonicity on Rn , as well as for Cheeger p-harmonic functions on metric spaces (see Section B.2), is a local property. This, so-called sheaf property, remains open for p-harmonic functions defined by upper gradients on general metric spaces, see Open problems 9.22 and 9.23.) Also, there is obviously no uniqueness for solutions of the Dirichlet problem for quasiminimizers, and hence also no comparison principle (as in Lemma 8.32) nor even a possibility to define Perron solutions in any reasonable way. Quasiminimizers were introduced by Giaquinta and Giusti [143], [144], as a tool for a unified treatment of variational integrals, elliptic equations and quasiregular mappings. They realized that De Giorgi’s method could be extended to quasiminimizers. In [144] they showed that quasiminimizers (after redefinition on a set of measure zero) are locally Hölder continuous (cf. Theorem 8.14) and obey the weak maximum principle. They also obtained Liouville’s theorem (cf. Corollary 8.16). DiBenedetto– Trudinger [113] obtained the Harnack inequality for quasiminimizers, as well as weak Harnack inequalities for quasisub- and quasisuperminimizers (cf. Theorems 8.4, 8.10 and 8.12). Observe that the strong maximum principle for quasiharmonic functions is a direct consequence of the Harnack inequality (cf. Theorem 8.13). Ziemer [360] obtained a Wiener type condition sufficient for the regularity of a boundary point (though more restrictive than the one in Theorem 11.24). Tolksdorf [336] obtained a Caccioppoli inequality and a convexity result for quasisubminimizers (cf. Theorem 9.42) and also showed that sets of capacity zero are removable for bounded quasisuperminimizers (cf. Theorem 12.3). All of the results above were obtained for unweighted Rn . The results of Giaquinta–Giusti [144] and DiBenedetto–Trudinger [113] were generalized to metric spaces by Kinnunen–Shanmugalingam [220]. A. Björn–Marola [59] made an attempt to use Moser’s method to obtain such results, but at one point the crucial logarithmic Caccioppoli inequality (cf. Proposition 8.9) is missing and remains open. Björn–Björn–Marola [53] gave various counterexamples showing that the exponents in Caccioppoli inequalities and weak Harnack inequalities necessarily are different for quasisuperminimizers and superminimizers. These counterexamples are based on the power-type quasi(super)minimizer results obtained by Björn–Björn [48], see below. The results of Ziemer [360] were generalized by J. Björn [63] to metric spaces. In particular, it is shown that if x0 satisfies any of the conditions in Corollary 11.25, with p replaced by some smaller number in (b) and (e), then x0 is regular for quasiminimizers. Twosided pointwise boundary estimates for quasisub- and quasiharmonic functions were obtained in J. Björn [63], [67]. In particular, they imply a sufficient condition for boundary regularity for quasiminimizers. Hölder boundary continuity for quasiharmonic functions was studied in [63]. Capacitary estimates for quasimini-

C Quasiminimizers

367

mizers, similar to those in Lemma 11.20, were obtained in Martio [268], [271]. They also lead to a (better) regularity condition. Kinnunen–Martio [218] showed that quasiminimizers have an interesting potential theory, in particular they introduced quasisuperharmonic functions and showed that the corresponding quasipolar sets have zero capacity. They pointed out that this theory fits very well into the metric space generality. They also gave a number of characterizations of quasisuperminimizers, and A. Björn [36] provided a few more (cf. Proposition 7.9). In [218] it was also shown that quasisuperminimizers have unique lsc-regularized representatives (cf. Theorem 8.22). Kinnunen–Martio [218] obtained the following result. Proposition C.3. Assume that uj is a Qj -quasisuperminimizer in , j D 1; 2. Then minfu1 ; u2 g is a Q-quasisuperminimizer in , where Q D minfQ1 C Q2 ; Q1 Q2 g:

(C.1)

The best constant (instead of Q) in this theorem is not known. Clearly, one cannot do any better than maxfQ1 ; Q2 g, and thus the constant above is optimal when Q1 D 1 or Q2 D 1. Open problem C.4. Find the best constant, instead of Q, or show that Q is best possible, in Proposition C.3. In Björn–Björn–Korte [51] progress has been made on this problem. The removability result of Tolksdorf was extended and generalized to metric spaces by A. Björn [37], where most of the results in Chapter 12 were obtained for quasi(super)-harmonic functions. This was partly motivated by the need of having certain removability results to obtain the trichotomy (Theorem 13.2) for quasiharmonic functions, which was obtained in A. Björn [41], wherein it was also shown that semiregular points remain the same with respect to p-harmonic functions and quasiharmonic functions. Generalizations of Theorems 13.5 and 13.10 to quasiharmonic functions can also be found in [41] although some parts are missing and remain open. Our proof of the trichotomy for p-harmonic functions (Theorem 13.2) uses the Kellogg property (Theorem 10.5) which remains open for quasiharmonic functions. However, a weak Kellogg property for quasiharmonic functions, sufficient for the proof of Theorem 13.2, was obtained in A. Björn [36]. Various characterizations of regular boundary points for quasiharmonic functions were obtained by A. Björn–Martio [60] (cf. Theorem 11.11), in particular it was shown that regularity for quasiharmonic functions is a local property. They also obtained pasting lemmas (cf. Lemmas 7.13 and 10.27). As in Proposition C.3, there is a certain blowup of the quasisuperminimizer constant. Martio [265] gave cluster set results similar to (13.1) in Section 13.5 for quasiminimizers in the borderline case when X D Rn (unweighted) and p D n, which have later been extended to arbitrary metric spaces by A. Björn [43]. Theorem 11.27 and Corollary 11.29 are also in [43] for quasiminimizers.

368

C Quasiminimizers

Björn–Björn [48] showed that jxj˛ , ˛ < 0, is quasisuperharmonic in the unit ball in unweighted Rn , when 1 < p < n. They also determined the optimal quasisuperharmonicity constant, and obtained similar results when p  n. These are the only examples known, except for the one-dimensional case, when the optimal quasiminimizer constant has been determined and is not 1. These results were used in Björn–Björn– Marola [53] to show that local integrability is necessarily different for quasisuperharmonic and superharmonic functions. There is very little of the theory of quasiminimizers which is restricted to the Euclidean case. However, in the one-dimensional case X D R (unweighted) substantially more can be said. Nevertheless, even in this case, the theory is by no means fully understood. The one-dimensional case was already studied by Giaquinta–Giusti [144], and the theory has been further developed by Judin [193], Martio [267], [271], Martio–Sbordone [273] and Uppman [343]. A quasiharmonic version of Harnack’s convergence theorem (Theorem 9.37) was obtained by Kinnunen–Marola–Martio [214]. Martio [267] studied the reflection principle for quasiminimizers on Rn , again obtaining a stronger result in the onedimensional case. Uppman [343] improved upon the quasiminimizing constant for the reflected function in the one-dimensional case and also showed that his result is sharp. Latvala [238] studied BMO-invariance of quasiminimizers, and Korte–Marola– Shanmugalingam [226] showed that homeomorphisms f W X ! Y preserve quasiminimizers if and only if they are quasiconformal, under some assumptions on the metric spaces X and Y . The quasiminimizing theory was used to treat Riccati type equations in Rn by Martio [269], [270].

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Index ª

,2 b, 3 , 10 ae , 10 1-admissible weight, 347, 348 1-Poincaré inequality, 91, 110–112, 120, 205, 349, 355–357 1-Poincaré space, 122, 135, 152, 330, 348, 356, 357 5-covering lemma, 5, 6, 35, 36, 69, 72, 77, 79, 82 A1 weight, 83 strong, 347, 348 Ap weight, 83, 113, 340, 346, 347 Aalto, D., 83, 369 absolutely continuous, 25 on a.e. line parallel to the axes, see ACL on p-a.e. curve, see ACCp ACCp , 25, 27, 28, 30, 45, 48, 53, 133, 189, 340, 344, 345 for superharmonic functions, 239 ACCp;loc , 52, 53 ACL, 24, 344, 345 Adachi, S., 152, 369 Adamowicz, T., 35, 369 Adams, D. R., v, 60, 150, 173, 189, 345, 369 Adams’ criterion, 167, 173–175, 189 admissible curve, 355 admissible weight, see p-admissible weight a.e., 2 Ahlfors, L. V., 34, 82, 369, 370 Ahlfors regular, 12, 67, 141, 350, 357, 358 Ahlfors–David regular, 67 Aikawa, H., 159, 276, 301, 348, 369

Aïssaoui, N., 35, 369 almost everywhere, 2 Ambrosio, L., 6, 8, 35, 116, 369, 370 Andersson, T., 358, 370 arc length, 8 Armitage, D. H., 273, 320, 370 Aronsson, G., vii, 216, 370 Ascoli’s theorem, 102, 108, 129, 233 Avilés, P., 276, 370 Bin , 107–110, 222, 223, 318, 319 Bäckdahl, T., vii Baernstein II, A., 274, 370 Baernstein’s problem, 274, 275 balayage, 246, 247 Balogh, Z. M., 248, 355, 356, 370 Banach, S., 36, 370 Banach space, 30, 56 Banach–Alaoglu’s theorem, 155 Banach-space valued Newtonian space, 35 barrier, 277, 285, 322 semibarrier, 322, 323 weak, 288 basic covering lemma, 35, see also 5-covering lemma Bellaïche, A., 355, 370 Bennewitz, B., 268, 370 Beppo Levi type function, 340 Besov space, 150 Bessel potential, 171 Beurling, A., 34, 82, 150, 369, 370 biLipschitz invariance, 89, 345 Biroli, M., 301, 370 Björn, A., v, vii, 34, 35, 54, 64, 77, 114, 150–152, 168, 169, 189, 216, 217, 229, 235, 246–248, 274–276, 288, 302, 316, 327, 328, 334, 358, 364, 366–368, 370–372

390

Index

Cp - ess lim sup, 282–284, 286 Cp - ess sup, 30, 97, 143, 172, 176, 177, 193, 282, 283 CPI , 84 Caccioppoli inequality, 217 De Giorgi type, 189, 191 for quasiminimizers, 366 logarithmic (missing), 366 for superharmonic functions logarithmic, 238 for superminimizers, 198 logarithmic, 199, 239 Caccioppoli type estimates, 302 calculus, 45–47 Calderón, A. P., 363, 373 Calderón–Sobolev space, 363 Camfield, C., 35, 373 Cantor, G., 358, 373 Cantor set, 60, 345, 358, 359 capp , 154, 161–169, 175, 176, 288–295, 297–301, 349 fine closure, 300 capacitable, 159 capacitary inequality, 167, 169 potential, 171, 189, 288–291 capacity, 12, 13 C1 , 135, 157, 169 Cp , 12, 14, 15, 22, 24, 50, 52, 60, 86, 87, 113, 127, 132, 156, 158, 168, 169, 240, 244 capp , 154, 161–169, 175, 176, 288–295, 297–301, 349 fine closure, 300 Choquet, 157, 159 C1 , 135, 157, 169 countably subadditive, 14, 158, 163 C 1;˛ -regularity for p-harmonic for BV-functions, 169 functions, 216 outer, 132–134, 151, 158, 164 Cp , 12, 14, 15, 22, 24, 50, 52, 60, 86, 87, outer measure, 14, 158, 163 113, 127, 132, 156, 158, 168, 169, 240, properties of, 14, 158, 163, 164 244 relation to Modp , 22, 86, 113 relation to Modp , 22, 86, 113 relative, 161 Sobolev, 12, 13 Cp - ess inf, 30, 172, 176, 177

Björn, J., v, vii, 34, 35, 64, 113–115, 150–152, 168, 169, 189, 216, 217, 235, 246–248, 274–276, 295, 301, 302, 316, 327, 328, 331, 334, 347, 348, 350, 363, 364, 366–368, 371–373 BMO, 70, 75–77, 83, 217, 247, 368 BMOp , 75 BMO -loc , 83, 201, 217, 247 Bojarski, B., 216, 373 Borel function, 3 regular measure, 2 boundary Hölder regularity, 301 for quasiharmonic functions, 366 boundary regularity, see regular point and regular set boundary value problem mixed, 33, 204 see also Dirichlet problem boundary weak Harnack inequality, 280 bounded mean oscillation, see BMO bounded variation, 35, 169, 363 Bourbaki, N., 132, 373 Bourdon, M., 357, 373 bow-tie, 60, 119, 349, 362 weighted, 120 Brelot, M., 273, 275, 276, 373 Brelot’s theorem, 256, 273–275, 364 Brownian motion, 189 Buckley, S. M., 83, 114, 151, 347, 372, 373 Buser, P., 357, 373 Buser’s inequality, 357 BV-function, 35, 169, 363

Index

strongly subadditive, 50, 158, 163 variational, 161 Capogna, L., 331, 356, 373, 374 Carathéodory, C., 355, 374 Carleman’s principle, 270, 276 Carlsson, D., vii Carnot group, v, 32, 356 Carnot–Carathéodory ball, 331 distance, 355 metric, 331, 355, 356 space, v, 32, 227, 354–357 Cavalieri’s principle, 6, 7, 70, 75, 76, 80, 94, 168 chain rule, 46 Chanillo, S., 347, 374 characteristic function, 3 Cheeger, J., vi, 32, 34, 114, 155, 171, 189, 340, 342, 343, 357, 363, 374 Cheeger 2-harmonic, 273, 274, 364 2-Laplace equation, 274 gradient, 32, 171, 172, 363, 364 minimizer, 171 p-harmonic, vi, 32, 216, 227, 247, 288, 295, 301, 302, 356, 357, 364, 366 p-Laplace equation, 171 removability for Hölder continuous Cheeger p-harmonic functions, 317 superminimizer, 364 Cheeger’s theorem, 342, 363 Cheeger–Sobolev space, vii, 171, 360, 363 Chernikov, V. M., 356, 374 Choquet, G., 168, 374 Choquet capacity, 157, 159 integral, 168, 175 Choquet’s capacitability theorem, 159 Choquet’s topological lemma, 231 Chow, W.-L., 355, 374 Chow’s condition, 355, see also Hörmander’s condition Cianchi, A., 114, 374

391

Clarkson, J. A., 173, 374 classical sense, 331 classically superharmonic function, 333, 334 cluster set, 153, 328, 367 Coifman, R. R., 83, 374 comparison principle for obstacle problems, 177, 212 for Perron solutions, 253 for p-harmonic extensions, 212, 250 for sub- and superharmonic functions, 234, 235, 264 not for quasiminimizers, 366 concave/convex composition for quasisubminimizers, 366 for sub/superharmonic functions, 235, 236, 239, 243 for sub/superminimizers, 236 connect, 128 connected, 85, 86, 102, see also pathconnected and rectifiably connected convex, see concave/convex corkscrew, 294, 329–331, 334 Costea, S., ¸ 35, 374 countable set, 4 countably subadditive, 14, 158, 163 covering 5-covering lemma, 5, 6, 35, 36, 69, 72, 77, 79, 82 Vitali covering theorem, 36, 61 Whitney type, 77 Crandall, M. G., 245, 374 curve, 8 rectifiable, 8 din , 106–111, 206, 223, 318, 329, 330 D p , 25, 29, 35, 38, 40, 41, 43, 45, 46, 48–54, 61, 113, 124 p Dloc , 52–54, 59, 62, 63, 87, 89, 124 Danielli, D., 248, 301, 356, 373–375 David, G., 67, 347, 348, 375 De Giorgi, E., 191, 215, 375

392

Index

De Giorgi class, 192 Caccioppoli inequality, 189, 191 weak Harnack inequality, 192 De Giorgi’s method, 191, 215, 216, 366 decreasing, 3 density of Lipschitz functions in N 1;p , 116, 118–122 in N01;p , 138, 139 of locally Lipschitz functions in N 1;p , 122, 140 Deny, J., 150, 340, 375 diamin , 111, 112 DiBenedetto, E., 216, 366, 375 dilation lower pointwise, 9, 342, see also lip upper pointwise, 342, see also Lip Dirichlet problem in classical sense, 331 in Sobolev sense, see p-harmonic extension Perron solution, see Perron solution p-harmonic extension, see p-harmonic extension q.e.-unique solution, 262 Wiener solution, see Wiener solution Dirichlet space, 25 discrete maximal function, 83 dist in , 110, 329, 330 domain, 348 Doob, J. L., 189, 375 doubling constant, 65 generalized doubling condition, 357 locally, 355, 356 measure, 65 metric space, 65 p-Poincaré space, 84 downwards directed, 231, 265 ds, 8 "-chain, 100 eigenfunction, 217

eigenvalue, 217 equivalence class, 28, see also N 1;p -equivalence class ess lim inf, 206 Essén, M., 159, 369 essentially locally bounded, 196 Evans, L. C., 340, 375 extension, 303, see also McShane extension and p-harmonic extension extension domain, 361 extremal length, 34, 35 Fabes, E. B., 113, 301, 341, 346, 375 Färm, D., vii, 64, 375 Farnana, Z., vii, 64, 189, 217, 276, 335, 375 fat set, 152, 294 Federer, H., 2, 6, 36, 375 Fefferman, C., 83, 374 fine closure, 300 topology, 13, 298 finely closed, 298 continuous, 161, 277, 299 N 1;p -function, 300 quasicontinuous function, 300 superharmonic function, 299 open, 298, 299 Folland, G. B., 7, 25, 41, 66, 118, 125, 130, 132, 233, 332, 375 Franchi, B., 114, 151, 347, 360, 363, 375, 376 Fuglede, B., 34, 35, 64, 337, 340, 376 Fuglede’s lemma, 37, 64 fundamental convergence theorem, 231, 233, 246 Futamura, T., 35, 376

E , 20 C

E , 20

.X /, 8

Index

gu , 40 p Gloc , 58, 59, 64, 87, 238 Gu , 58, 59, 64, 238, 239, 241 game-theoretic, 275 García-Cuerva, J., 347, 376 Gardiner, S. J., 273, 276, 320, 370, 376 Gariepy, R. F., 301, 328, 340, 375, 376 Garnett, J. B., 358, 376 Garnett–Ivanov set, 358 Garofalo, N., 35, 169, 248, 331, 356, 373–376 Gaussian measure, 61 Gehring, F. W., 83, 347, 376 Gehring’s lemma, vi, 77, 82, 83, 99 generalized doubling condition, 357 geodesic, 99 with respect to inner metric, 107 Gianazza, U., 83, 356, 376 Giaquinta, M., 189, 366, 368, 376 Gilbarg, D., 113, 350, 377 Giusti, E., v, 366, 368, 376, 377 global integrability for superharmonic functions, 247 global Sobolev embedding, 141 glueing lemma, 48, see also pasting lemma glueing spaces, 349, 351 Gol0 dshtein, V., 360, 377 gradient space, 64 graphs, v, 227, 243, 315, 316, 351 Green function, 248, 304 potential, 171 Gromov–Hausdorff limit, 32, 112, 357 Grushin plane, 356 Gutiérrez, C. E., 114, 151, 347, 375 H1 , 356, 357 Hn , v, 32, 216, 331, 354 Hf , 212, 250 Hajłasz, P., 64, 114, 115, 151, 152, 355–358, 360, 361, 363, 376, 377 Hajłasz

393

gradient, 361, 362 minimal, 361, 362 space, 360 Hajłasz–Sobolev space, vii, 152, 168, 360–363 Hakkarainen, H., vii, 35, 169, 377 Hansen, W., 331, 377 Hanson, B., 357, 377 Hardy inequality, 152 Hardy–Littlewood maximal function, see maximal function Harjulehto, P., 35, 376, 377 harmonic measure, 273 Harnack inequality, 201, 204 for eigenfunctions, 217 for quasiminimizers, 366 weak, see weak Harnack inequality Harnack’s convergence theorem for p-harmonic functions, 233 for quasiharmonic functions, 368 Hästö, P., 35, 376, 377 Hausdorff content, 114 dimension, 276 limit, 12, 112 measure, 12, 169, 345, 351 weighted, 317 Hayes, C. A., Jr., 61, 377 Hedberg, L. I., v, 60, 150, 173, 345, 369, 377 Heikkinen, T., 35, 114, 152, 360, 377, 378 Heinonen, J., v, vii, viii, 1, 2, 6, 8, 34, 35, 61, 64, 65, 83, 113, 115, 116, 150, 151, 160, 170, 242, 246, 247, 276, 301, 302, 315, 316, 333, 337, 340, 341, 343, 346, 347, 350, 351, 356–358, 377, 378 Heisenberg group, v, 32, 216, 331, 354, 356 Hejhal, D. A., 316, 378 Hencl, S., vii Herron, D., vii Hinde, C., 357, 378

394

Index

Hölder continuity, see also boundary Hölder regularity for minimizers, 202 for Newtonian functions, 141 for obstacle problems, 209, 217 for p-harmonic functions, 202, 203, 215 for quasiminimizers, 366 removability for Hölder continuous Cheeger p-harmonic functions, 317 hole-filling, 185, 189, 192 Holopainen, I., 248, 276, 351, 352, 358, 370, 378, 379 homeomorphism, 368 homogeneous type, v, 83 horizontal curve, 355, see also admissible curve Hörmander, L., 315, 355, 379 Hörmander’s condition, 114, 355–357, see also Chow’s condition Hueber, H., 331, 377 hyperbolic plane, 357 hyperharmonic function, 218, 254, 267 hypoelliptic equation, 355–357 hypoharmonic function, 218, 267 Hytönen, T., 83, 379 increasing, 3 inner metric, 106–111, 206, 223, 318, 329, 330, 352 integrability, see also Sobolev embedding global for superharmonic functions, 247 local for quasisuperharmonic functions, 368 local for superharmonic functions, 238, 239, 241 sharpness, 242 integral average, 2 interior regularity, see Hölder continuity and lsc-regularized irregular point, 251, see also regular point, semiregular point and strongly irregular point isocapacitary, 152

isoperimetric inequality, 150 Ivanov, L. D., 358, 379 Iwaniec, T., 216, 373, 379 Jacobians of quasiconformal mappings, 113, 346, 347 Järvenpää, E., 113, 379 Järvenpää, M., 113, 379 Jerison, D., 113, 114, 301, 341, 355, 375, 379 John, F., 70, 83, 379 John–Nirenberg’s lemma, vi, 70, 82, 83 consequence of, 75 Judin, P. T., 366, 368, 379 K ;f , 172 Kališ, J., vii Kallunki, S., 35, 169, 379, see also Rogovin, S. Kansanen, O. E., viii, 65, 83, 217, 247, 248, 382 Karlsson, T., vii Kaufman, R., 275, 379 Keith, S., vi, 89, 99, 115, 347, 348, 357, 363, 364, 372, 379 Kellogg property, 251 weak for quasiharmonic functions, 276, 367 Kenig, C. E., 113, 301, 341, 346, 375 Kilpeläinen, T., v, vii, 113, 150–152, 160, 170, 242, 246, 247, 276, 288, 301, 302, 315, 316, 333, 337, 340, 341, 343, 346, 347, 350, 378–380 Kim, S., 275, 380 Kinnunen, J., vii, viii, 35, 64, 65, 83, 150–152, 168, 169, 189, 192, 216, 217, 229, 246, 247, 258, 302, 333, 363, 366–369, 377, 379–381 von Koch snowflake curve, 12, 33, 35, 86, 111, 112, 118, 132, 134, 348, 362 Korte, R., vii, 35, 64, 114, 150, 152, 169, 244, 302, 367, 368, 371, 380, 381

Index

Koskela, P., v, 1, 34, 35, 64, 113–115, 151, 152, 216, 347, 351, 355–357, 360, 363, 364, 373, 376–378, 381 Kronz, M., 83, 381 Kufner, A., 340, 381 Kuratowski, K., 5, 381 Kurki, J., 276, 381

395

Lindelöf, 5, 6 Lindqvist, P., 216, 288, 301, 370, 380, 382 Lions, J. L., 340, 375 Liouville’s theorem, 204 for quasiminimizers, 366 lip, 9, 21, 100, 251, 342 Lip, 3, 342, 343, 348 Lip0 , 165 ƒ1 , 20, 41 Lipc , 3 ƒs , 351 Lipschitz L1;1 , 69 continuity, 216, 364 L1 -local Sobolev embedding for large p, function, 7 97 Littman, W., 301, 382 L1 -Sobolev inequality for large p, 143 Lizorkin–Triebel space, 150 LpC , 43 Llorente, J. G., 275, 379, 382 Lploc , 52–54 local Hölder continuity, Lf , 253 see Hölder continuity local integrability Laakso, T., 357, 381 for quasisuperharmonic functions, 368 Laakso space, 358 for superharmonic functions, 238, 239, lattice, 11, 25 241 Latvala, V., vii, 83, 152, 217, 302, 368, sharpness, 242 380, 381 local maximum principle, 196, Lebesgue, H., 301, 328, 382 see also weak Harnack inequality Lebesgue points, v local Sobolev embedding, 92 for Hajłasz–Sobolev functions, 152 for large p, 141, 152 for N 1;1 -functions, 152 locally bounded, 196 for N 1;p -functions, 147, 149 for superharmonic functions, 244, 245 locally doubling, 355, 356 locally Lipschitz function, 9, 21, 116, 121, for superminimizers, 208, 209 122, 140, 339, 342, 343, 360 Lq -Lebesgue points, 208, 209, 244, 245 locally uniform convergence, 188, 230, Lebesgue spine, 298, 319, 328 233, 258 Lebesgue’s differentiation theorem, Lorentz space L1;1 , 69 61–63, 65, 73, 82, 202 Lehrbäck, J., 152, 381 lower Perron solution, 253 Leibniz rule, 45 lower pointwise dilation, 9, 342, length see also lip arc length, 8 lsc-regularized, 206 space, 83, 99 quasisuperminimizer, 367 with respect to the sub-Riemannian solution of obstacle problem, 209 metric, 354 superminimizer, 206 Levi type function, 340 Lu, G., 347, 382 Lewis, J. L., 216, 268, 276, 370, 382 Lukeš, J., 328, 382 lim inf-regularization, 231 Luzin, N. N., 132, 382

396

Index

Luzin type theorem, 123 Luzin’s theorem, 132 M 1;p , 361, 362 Mf , 68 M  f , 68, 69 M f , 68 Maasalo, O. E., 83, 217, 247, 248, 382, see also Kansanen, O. E. Mackay, J., 358, 382 MacManus, P., 34, 35, 114, 152, 301, 360, 364, 373, 381, 382 McShane extension, 116, 121 Mäkäläinen, T., viii, 217, 247, 276, 302, 317, 371, 382 Malý, J., v, vii, 46, 47, 113, 300–302, 328, 380, 382, 383 Malý, L., viii Manfredi, J. J., 190, 216, 275, 276, 356, 370, 379, 382, 383 manifolds, v, 32, 114, 227, 357 Marchi, S., 356, 376 Markina, I. G., 356, 383 Marola, N., viii, 35, 54, 64, 77, 114, 169, 216, 217, 246–248, 258, 366, 368, 372, 374, 376, 380, 381, 383 Martín, J., 83, 383 Martin boundary, 276 Martio, O., v, vii, viii, 64, 83, 113, 150–152, 160, 168, 170, 189, 217, 229, 242, 246, 247, 258, 276, 301, 302, 315, 316, 328, 333, 337, 340, 341, 343, 346–348, 350, 365, 367, 368, 372, 378–380, 382, 383 Mateu, J., 83, 384 Mattila, P., 2, 83, 384 maximal function, v, 65, 68, 82, 347, 361, 362 centred, 68 discrete, 83 fractional sharp, 363 noncentred, 68, 69, 147 maximum principle

strong, 201 for quasiminimizers, 366 weak, 212 for quasiminimizers, 366 Mazur’s lemma, 154, 168 Maz0 ya, V. G., 114, 119, 144, 151, 152, 167, 169, 301, 316, 384 Maz0 ya’s capacitary inequality, 167, 169 Maz0 ya’s inequality, 143, 152, 167 Maz0 ya’s truncation method, 93, 114 McShane extension, 116, 121 mean-value integral, 2 measure doubling, 65 harmonic, 273 p-harmonic, 268–270, 274, 275 Mikkonen, P., 301, 384 Milman, M., 83, 383 Minda, D., viii minimal p-weak upper gradient, 40, 48, 49, 51, 52, 58 representation formula, 61–64 minimizer, 13, 178 Harnack inequality, 201, 215 Hölder continuity, 202, 215 minimum principle strong, 224, 246 Miranda, M., Jr., 35, 370, 384 mixed boundary value problem, 33, 204 Mizuta, Y., v, 35, 376, 384 Mocanu, M., 35, 384 Modp , 16–20, 24, 54, 337 Modp . E /, 22, 86, 113 C Modp . E /, 20 relation to Cp , 22, 86, 113 modulus, see p-modulus and Modp Morse, A. P., 36, 385 Moser, J., 215, 235, 385 Moser’s iteration method, 76, 113, 192, 199, 215, 216, 235, 346, 366 Muckenhoupt, B., 347, 385

Index

Muckenhoupt class, 340, 346, see also weight Munkres, J. R., 5, 385

397

Hölder continuity, 209, 217 solubility, 173 unique lsc-regularized solution, 209 uniqueness, 173 N 1;p , 10 weak Harnack inequality, 194, 195 N 1;p ./, 50 Ohtsuka, M., 35, 337, 340, 385 N 1;p.x/ , 35 Onninen, J., 35, 169, 373 N 1;p -equivalence class, 10, 27, 123, 134, Ono, T., 189, 385 151, 256, 363 open problem, 44, 61, 63, 122, 134, 227, N 1;p -function 235, 254, 258, 274, 288, 295, 308, 313, finely continuous, 300 315, 347, 348, 367 N 1;p -norm, 10, 43 Orlicz–Poincaré inequality, 113 N01;p , 55–57, 139, 146 Orlicz–Sobolev space, 35, 361 1;p , 52–54, 89, 124 Nloc Orobitg, J., 83, 384 Ny 1;p , 10, 28, 29, 32, 34, 133, 134, 158, O’Shea, J., 114, 373 171, 203, 256, 338, 343, 344, 362, 363 outer capacity, 132–134, 151, 158, 164 Ny 1;p = ae , 10, 28 outer measure, 2 Nz 1;p , 10, 11, 27, 28, 30, 151 Nagel, A., 355, 356, 385 Pf , 253 Nash, J., 215, 385 x , 253 Pf Netrusov, Yu., 150, 377 Pf , 253 Neumann boundary data, 33, 204 p-admissible weight, 113, 337, 341, Newtonian, 36 346–348 space, 10 p-a.e. curve, 13, 16 Banach-space valued, 35 Pajot, H., 357, 373 variable exponent, 35 Pallara, D., 35, 370 Neymark, M., vii parabolic equation, 215 Nhieu, D. M., 356, 374, 375 Parviainen, M., viii, 35, 64, 152, 168, 189, Nicolau, A., 83, 384 190, 217, 246, 247, 276, 302, 371, 372, Nirenberg, L., 70, 83, 379 383 nonregular ball, 330, 331 pasting lemma, see also glueing lemma nonremovable set, 309–312 for quasiharmonic functions, 367 Nyström, K., 276, 382 for quasiminimizers, 367 for superharmonic functions, 263 obstacle problem, 172, 182, 183, 228, 280, for superminimizers, 181, 251 286, 288, 289 pathconnected, 102 boundary regularity, 282–285 Pauc, C. Y., 61, 377 boundedness, 176 Pauls, S. D., 356, 374 comparison principle, 177, 212 Pere, M., 35, 217, 377, 381 continuous obstacle, 209, 210 Peres, Y., 190, 275, 385 continuous solution, 209, 210 Pérez, C., 83, 114, 376, 382, 385 convergence result, 257 double, 189, 276, 335 Perron, O., 275, 335, 385

398

Index

Perron solution, 253, 263 Baernstein’s problem, 274, 275 Brelot’s theorem, 256, 273–275, 364 comparison principle, 253 not for quasiminimizers, 366 of Borel function, 274 of characteristic function, 267, 274, 275 of continuous function, 261 of N 1;p -function, 256 of semicontinuous function, 265–267 perturbation, 263 of continuous function, 261, 275 of N 1;p -function, 256 p-harmonic, 254 q.e.-invariance, 256, 261, 263 uniqueness, 257, 262 Petersen, P., 357, 378 p-fat set, 152, 294 p-harmonic, 13 p-harmonic extension, 212, 250, 289 comparison principle, 212, 250 convergence result, 258, 335 p-harmonic function, 178, 226 C 1;˛ -regularity, 216 convergence result, 233, 234 Harnack inequality, 201, 204 Harnack’s convergence theorem, 233 Hölder continuity, 202, 203, 215 Liouville’s theorem, 204 maximum principle strong, 201 on Œ0; 1, 313 on R, 243, 312 sheaf property, 227, 356 strong maximum principle, 201 p-harmonic measure, 268–270, 274, 275 p-Laplace equation, 170, 171, 227, 247 p-modulus, 13, 16, 17, 34, 35, see also Modp Poincaré inequality, 84 1-, 91, 110–112, 120, 205, 349, 355–357 biLipschitz invariance, 89

characterization, 88, 89 for N01;p , 145 in Rn , 113, 341, 345 on graphs, 351 open-ended, 99 p-, 84 .p  ; p/-, 92, 103 sharpness, 91, 242 .q; p/-, q > p, 93, 97 self-improving, 99 strong, 84, 106, 109 weak, 84 with respect to inner metric, 108, 109 Poincaré space, 1-, 122, 135, 152, 330, 348, 356, 357 p-, 84 Poisson modification for Newtonian superharmonic functions, 237 for superharmonic functions, 271, 272 for superminimizers, 213 Pokrovskii, A. V., 317, 385 polar set, 240, 247 porosity, 294 potential Bessel, 171 capacitary, 171, 189, 288–291 Green, 171 Riesz, 171 Wolff, 171 p-Poincaré space, 84 p-precise function, 340 Pˇrecechtˇel, P., vii Preiss, D., 61, 385 probabilistic, 189, 275 product rule, 45 proper, 54 punctured ball, 319 PWB method, 275, see also Perron solution p-weak upper gradient, 13, 16, 19–24, 34, 37–43, 47 calculus, 45–47

Index

chain rule, 46 dependence on p, 59 glueing lemma, 48 Leibniz rule, 45 minimal, 40, 48, 49, 51, 52, 58 representation formula, 61–64

399

rectifiably connected, 102, 103, 106, 107, 110, 204 reflection principle, 368 regular point, 251, 280, 282–284 characterization, 264, 277, 285 for quasiharmonic functions, 367 componentwise, 296 for quasiharmonic functions, 367 q.e., 13, 161 irregular, 251, see also semiregular point q.e.-equivalence class, 1;p and strongly irregular point see N -equivalence class Hölder regularity, 301 quasiconformal mapping, 82, 113, for quasiharmonic functions, 366 346–348, 368 Kellogg property, see Kellogg property quasicontinuous, 32, 33, 123, 124, 126, Wiener criterion, see Wiener criterion 128, 131–134, 138, 150–152, 256, 257, and Wiener type criterion 263, 264, 298 regular set, 251, 284, 329 characterization of, 133, 344 regularity finely continuous, 300 boundary, see regular point and N 1;p -function, 126, 132 regular set superharmonic function, 239 interior, see Hölder continuity and weakly, 29, 123, 125, 133, 151 lsc-regularized quasiconvex, vi, 84, 99, 100, 103, relative capacity, 161 106–110, 114, 119, 205 Remak, R., 275, 385 with respect to inner metric, 107, 110 removable set quasieverywhere, 13, 161 characterization, 307, 313 quasiharmonic function, 365 disconnecting, 315, 316 quasiminimizer, vi, vii, 83, 171, 172, 276, for C 1;˛ p-harmonic functions, 317 365–368 for Hölder continuous Cheeger quasipolar set, 367 p-harmonic functions, 317 quasiregular mapping, 366 for minimizers, 307 quasisubminimizer, 365, 366 for p-harmonic functions, 303, 304, 306, quasisuperharmonic function, 367, 368 307 local integrability, 368 for quasisuperminimizers, 366, 367 quasisuperminimizer, 365–367 for superharmonic functions, 303, 304, characterization, 367 307, 308 lsc-regularized, 367 for superminimizers, 307, 308 for W 1;p , 115 x R, 2 nonremovability, 309–312 Rademacher’s theorem, 21, 339, 343 nonuniquely, 315, 316 Radon–Nikodym derivative, 347 uniquely, 303, 304, 306–308, 314 Rajala, K., 115, 216, 364, 379, 381 with positive capacity, 312 Rashevski˘ı, P. K., 355, 385 removable singularities for Poincaré inequalities, 115 rectifiable curve, 8

400

Index

resolutive, 253, 263 Baernstein’s problem, 274, 275 Borel function, 274 Brelot’s theorem, 256, 273–275, 364 characteristic function, 267, 274, 275 continuous function, 261 N 1;p -function, 256 perturbation, 263 of continuous function, 261, 275 of N 1;p -function, 256 semicontinuous function, 266, 267 reverse Hölder inequality, 77, 79, 83 Riccati type equation, 368 Ricci curvature, 357 Riemannian manifold, v, see also manifold Riesz potential, v, 151, 171 rigidity for quasiminimizers, 365 Rogovin, K., 113, 379 Rogovin, S., 35, 113, 169, 379 Romeo, K., 358, 385 Rossi, J. D., 190, 383 Royden, H. L., 102, 108, 129, 233, 385 Rubio de Francia, J. L., 347, 376 Rudin, W., 2, 7, 25, 66, 132, 154, 233, 332, 385

Shanmugalingam, N., v, viii, 34, 35, 64, 113–115, 150–152, 169, 189, 192, 216, 217, 235, 247, 248, 258, 274–276, 301, 316, 328, 347, 348, 358, 360, 362–364, 366, 368, 369, 372, 373, 377–381, 386 sheaf property, 227, 356, 364, 366 Sheffield, S., 190, 275, 385 Shimomura, T., 35, 376 Shvartsman, P., 363, 386 Sierpi´nski carpet, 358 simple Vitali lemma, 35, see also 5-covering lemma simplicial complex, 32 singular function, 248 measure, 248 Sjödin, T., viii slit disc, 121, 321, 361 Smith, H. J. S., 359, 386 snowflake, see von Koch snowflake curve Soardi, P., 351, 352, 358, 379 Sobolev, S. L., 151, 386 Sobolev capacity, 12, 13, see also capacity embedding, 151 global, 141, 152 Saloff-Coste, L., 114, 151, 385 L1 -local for large p, 97 Sarvas, J., 348, 383 local, 92 Sbordone, C., 83, 368, 383 local for large p, 141, 152 Schramm, O., 190, 385 inequality, 141, 142 second countable, 5 L1 for large p, 143 Seebach, J. A., Jr., 5, 386 sense, 172, 212, 249, 251 semibarrier, 322, 323 spaces of homogeneous type, v, 83 semicontinuous, 2, 7 semiregular point, 318, 319, 327, 330, 367 spectral synthesis, 150 Spurný, J., vii characterization, 320, 322, 323 Stampacchia, G., 301, 382 for quasiharmonic functions, 367 Steen, L. A., 5, 386 Semmes, S., 34, 113, 114, 347, 348, 357, Stein, E. M., 83, 347, 355, 356, 385, 386 360, 375, 386 Steinhurst, B., 358, 385, 386 separable, 5 Stone–Weierstrass’s theorem, 7 Serapioni, R. P., 113, 341, 346, 375 Serrin, J., 215, 316, 386 strictly convex, 173

Index

Strömberg, J.-O., 83, 386 strong A1 -weight, 347, 348 strong maximum principle, 201 for quasiminimizers, 366 strong minimum principle, 224, 246 strongly irregular point, 318, 319, 327 characterization, 326 strongly subadditive, 50, 158, 163 Strzelecki, P., 356, 377 subadditive countably, 14, 158, 163 strongly, 50, 158, 163 subcurve, 17 subharmonic function, 13, 218, 226 subminimizer, 13, 178 local boundedness, 196 weak Harnack inequality, 194, 195 sub-Riemannian metric, 354 superharmonic function, 13, 218, 226 ACCp , 239 Caccioppoli inequality logarithmic, 238 characterization, 219, 227–229, 272, 334 classically, 333, 334 concave composition, 235, 236, 239, 243 convergence result, 229–231, 233, 246 finely continuous, 299 fundamental convergence theorem, 231, 233, 246 p Gloc , 238 global integrability, 247 growth, 244 integrability global, 247 local, 238, 239, 241 sharpness, 242 Lebesgue points, 244, 245 level sets, 244 local boundedness from below, 219 local integrability, 238, 239, 241 sharpness, 242 log of, 235, 238, 239

401

lsc-regularized, 224 minimum principle strong, 224, 246 on bounded X , 225 on R, 243, 312 on Rn , 242 pasting lemma, 263 Poisson modification, 237, 271, 272 power of, 243 quasicontinuous, 239 relation to superminimizers, 219, 221 sheaf property, 227, 356 strong minimum principle, 224, 246 weak Harnack inequality, 222 sharp, 221 sharpness, 242 superminimizer, 13, 178, 181, 182, 226, 228, 239 Caccioppoli inequality, 198 logarithmic, 199, 239 characterization, 179, 181, 183, 186, 228 concave composition, 236 convergence result, 184–188 Lebesgue points, 208, 209 local boundedness, 196 lsc-regularized, 206 pasting lemma, 181, 251 Poisson modification, 213 relation to superharmonic functions, 219, 221 weak Harnack inequality, 197, 200 sharp, 213 sharpness, 242 supp , 5 support, 5 Suslin set, 159 sweeping, 247 Tanaka, K., 152, 369 Tchou, N. A., 301, 370 Textorius, B., vii Thim, J., vii

402

Index

thin, 298 Tietze’s extension theorem, 125, 332 Tilli, P., 6, 8, 116, 370 Timoshin, S. A., 360, 386, 387 Tišer, J., 61, 387 Tolksdorf, P., 216, 217, 366, 367, 387 Tolsa, X., 83, 387 Torchinsky, A., 83, 386 trichotomy, 318 for quasiharmonic functions, 367 Triebel–Lizorkin space, 150 Troyanov, M., 360, 377 Trudinger, N. S., 113, 114, 350, 366, 375, 377, 387 Trudinger inequality, 93, 97, 114 tug-of-war, 190, 275 Tuominen, H., viii, 35, 64, 115, 150, 152, 360, 361, 378, 380, 381, 387 Turesson, B.-O., v, vii, 387 Tyson, J. T., v, 34, 35, 64, 248, 276, 356, 358, 370, 374, 378, 382

Vitali covering theorem, 36, 61 Vodop0 yanov, S. K., 356, 374, 383 von Koch snowflake curve, 12, 33, 35, 86, 111, 112, 118, 132, 134, 348, 362

W 1;p , 1, 31, 32, 35, 93, 97, 115, 134, 242, 247, 338, 340, 341, 343–345, 361, 363, 365 Wainger, S., 355, 356, 385 weak barrier, 288 weak Harnack inequality, 196 for De Giorgi classes, 192 for quasisubminimizers, 366 for quasisuperminimizers, 366 for solutions of obstacle problems, 194, 195 for subminimizers, 194, 195 for superharmonic functions, 222 sharp, 221 sharpness, 242 for superminimizers, 197, 200 sharp, 213 Uf , 253 sharpness, 242 uniform domain, 348 on boundary, 280 uniformly convex, 173 weak Kellogg property for unique continuation, 315 quasiharmonic functions, 276, 367 upper gradient, 8, 9, 14, 16, 34, 42, 43 weak L1 , 69 along a curve, 17 weak maximum principle, 212 p-weak, see p-weak upper gradient for quasiminimizers, 366 upper Perron solution, 253 weak semibarrier, 322, 323 upper pointwise dilation, 342, see also Lip weak upper gradient, see p-weak upper Uppman, H., 368, 387 gradient Uraltseva, N., 216, 387 weakly quasicontinuous, 29, 123–125, 133, 151 variable exponent Newtonian space, 35 weight variational capacity, 161 1-admissible, 347, 348 Veltmann, W., 358, 387 A1 , 83 very weak gradient, 34 Ap , 83, 113, 340, 346, 347 very weak superminimizer, 247 p-admissible, 113, 337, 341, 346–348 very weak supersolution, 247 strong A1 , 347, 348 Vietoris limit, 32, 357 Weinberger, H. F., 301, 382 Vitali, G., 35, 36, 132, 387 Weiss, G., 83, 374 Vitali covering, 36

Index

Wheeden, R. L., 83, 114, 151, 347, 374–376, 382, 385 Whitney type covering, 77 Widman, K.-O., 113, 189, 301, 387 Wiener, N., 35, 36, 275, 301, 335, 388 Wiener criterion, 13, 161, 301, 356, 364 necessity, 295 sufficiency, 277, 288, 293 solution, 331, 332 type criterion necessity with exponent 1=p, 295, 327 sufficiency for quasiminimizers, 366 Wildrick, K., 358, 382 Wilson, D. B., 190, 385 Wolff potential, 171 Wu, J.-M., 275, 379, 382

403

Yan, L., 360, 388 Yang, D., 360, 388 Yosida, K., 154, 388 Young inequality, 195 Zaremba, S., 328, 388 Zaremba’s punctured ball, 319 Zatorska-Goldstein, A., 83, 248, 382, 388 zero Neumann boundary condition, 33, 204 zero p-weak upper gradient property, 33, 152, 153 Zhang, J., 245, 374 Zhong, X., vi, 99, 152, 216, 379, 381, 388 Ziemer, W. P., v, 21, 24, 46, 47, 113, 300–302, 328, 338–340, 343–345, 366, 376, 383, 388