Lecture Notes in Mathematics 3361 Editors: .A Dold, Heidelberg .F Takens, Groningen regnirpS Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore oykoT Hiroaki Aikawa Matts Ess6n laitnetoP yroehT Selected scipoT regnirpS Authors Hiroaki Aikawwa Department of Mathematics Shimane University Matsue 690, Japan E-mail: haikawa@ riko.shimane-u.ac.jp Matts Ess6n Department of Mathematics Uppsala University Box 480 75106 Uppsala, Sweden E-mail: matts.essen @ math.uu.se Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Aikawa, Hiroaki: Potential theory : selected topics / Hiroaki Aikawa ; Matts Essen. - Berlin ; Heidelberg ; New York ; Barcelona ; Budapest ; Hong Kong ; London ; Milan ; Paris ; Santa Clara ; Singapore ; Tokyo : Springer, 1996 (Lecture notes in mathematics ; 1633) ISBN 3-540-61583-0 NE: Essen, Matts:; GT Mathematics Subject Classification (1991 ): 31B05, 31A05, 31B 15, 31B 25 ISSN 0075-8434 ISBN 3-540-61583-0 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. 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Typesetting: Camera-ready TEX output by the authors SPIN: 10479829 46/3142-543210 - Printed on acid-free paper During the academic years 1992-1994, there was a lot of activity on potential theory at the Department of Mathematics at Uppsala University. The main series of lectures were as follows: (cid:12)9 A An introduction to potential theory and a survey of minimal thinness and rarefiedness. (M. EssSn) (cid:12)9 B Potential theory. (H. Aikawa) (cid:12)9 C Analytic capacity. (V. Eiderman) (cid:12)9 D Lectures on a paper of L.-I. Hedberg 24. (M. Ess~n) (cid:12)9 E Harmonic measures on fractals (A. Volberg) These lecture notes contain the lecture series A,B and references for C. The E lec- tures will appear as department report UUDM 1994:32: Zoltan Balogh, Irina Popovici and Alexander Volberg, Conformally maximal polynomial-like dynamics and invariant harmonic measure (to appear, Ergodic Theory and Dynamical Systems). H. Aikawa spent the Spring semester 1993 in Uppsala. V. Eiderman spent the Spring semesters 1993 and 1994 here. A. Volberg was in Uppsala during May 1994. In addition to giving excellent series of lectures, our visitors were also very active participants in the mathematical efil of the department. Uppsala September 21, 1994 Matts Ess~n Contents Part I by M. Ess@n 1 1. Preface 3 2. Introduction 4 2.1. Analytic sets 4 2.2. Capacity 4 2.3. Hausdorff measures 5 2.4. Is mh a capacity? 9 3. The Physical background of Potential theory 10 3.1. Electrostatics in space 01 4. Potential theory 11 4.1. The maximum principle 31 4.2. m-potentials 61 5. Capacity 16 5.1. Equilibrium distributions 17 5.2. Three extremal problems 12 5.3. Every analytic set si capacitable 24 6. Hausdorff measures and capacities 29 6.1. Coverings 30 6.2. Cantor sets 23 6.3. A Cantor type construction 34 7. Two Extremal Problems 36 7.1. The Classical Case 38 8. M. Riesz kernels 39 8.1. Potentials 40 8.2. Properties of ,~U where ~ = .FA 54 8.3. The equilibrium measure 46 8.4. Properties of Ca(.) 47 8.5. Potentials of measures in the whole space 49 8.6. The Green potential 53 8.7. Strong Subadditivity 45 8.8. Metric properties of capacity 57 8.9. The support of the equilibrium measures 57 8.10. Logarithmic capacity 95 viii POTENTIAL THEORY: SELECTED TOPICS 8.11. Polar sets 60 .21.8 A classical connection 61 8.13. Another definition of capacity 61 9. Reduced functions 61 10. Green energy in a Half-space 65 I0.i. Properties of the Green energy 7 66 10.2. Ordinary thinness 67 II. Minimal thinness 68 Ii.i. Minimal thinness, Green potentials and Poisson integrals 68 11.2. A criterion of Wiener type rof minimal thinness 71 12. Rarefiedness 73 13. A criterion of Wiener type for rarefiedness 74 14. Singular integrals and potential theory 75 15. Minimal thinness, rarefiedness and ordinary capacity 81 16. Quasiadditivity of capacity 88 17. On an estimate of Carleson 90 .1 Books on potential theory: a short tsil 93 I.I. Classical potential theory 93 1.2. Potential theory and function theory ni the plane 93 1.3. Abstract potential theory 93 1.4. Nonlinear potential theory 93 1.5. Potential theory and probability 93 1.6. Pluripotential theory 94 Bibliography 95 Index 97 Analytic capacity (references) by V. Eiderman 99 Part II by H. Aikawa 101 1. Introduction 103 2. Semicontinuous functions 105 2.1. Definition and elementary properties 105 2.2. Regularizations 106 2.3. Approximation 106 2.4. Vague convergence 107 3. L p capacity theory 108 3.1. Preliminaries 108 3.2. Definition and elementary properties 109 3.3. Convergence properties 110 3.4. Capacitary distributions 112 3.5. Dual capacity 114 3.6. Duality 115 3.7. Relationship between capacitary distributions 118 3.8. Capacitary measures and capacitary potentials 120 STNETNOC xi 4. Capacity of balls 122 4.1. Introduction 221 4.2. Preliminaries 421 4.3. Kerman-Sawyer inequality 126 4.4. Capacity of balls 129 4.5. Metric Property of Capacity 130 5. Capacity under a Lipschitz mapping 132 5.1. Introduction 231 5.2. Proof of Theorem 5.1.1 133 5.3. Proof of Theorem 5.1.2 136 6. Capacity strong type inequality 137 6.1. Weak maximum principle 731 6.2. Capacity strong type inequality 041 6.3. Lemmas 041 6.4. Proof of Theorem 6.2.1 241 7. Quasiadditivity of capacity 144 7.1. Introduction 144 7.2. How do we get a comparable measure? 941 7.3. Green energy 251 7.4. Application 155 8. Fine limit approach to the Nagel-Stein boundary limit theorem 158 8.1. Introduction 851 8.2. Boundary behavior of singular harmonic functions 261 8.3. Proof of Theorem 8.1.1 561 8.4. Sharpness of Theorem 8.1.1 and Theorem 8.2,1 167 8.5. Necessity of an approach region 961 8.6. Further results 071 9. Integrability of superharmonic functions 171 9.1. Integrability for smooth domains 171 9.2. Integrability for Lipschitz domains 172 9.3. Integrability for nasty domains 174 9.4. Sharp integrability for plane domains 175 9.5. Sharp integrability for Lipschitz domains 571 9.6. Lower estimate of the gradient of tile Green function 081 10. Appendix: Choquet's capacitability theorem 182 10.1. Analytic sets are capacitable 281 10.2. Borel sets are analytic 183 11. Appendix: Minimal fine limit theorem 184 11.1. Introduction 481 11.2. Balayage (Reduced function) 581 11.3. Minimal thinness 187 11.4. PWB h solution 091 11.5. Minimal fine boundary limit theorem 193 Bibliography 195 Index 198 POTENTIAL THEORY PART I Matts Ess~n 1. Preface The first part of these notes were written to prepare the audience for lectures by H. Aikawa on recent developments in potential theory and by A. Volberg on har- monic measure. It was assumed that the participants were familiar with the theory of integration, distributions and with basic functional analysis. Section 2 to 8 give an introduction to potential theory based on the books 7 and 29. We begin by discussing two definitions of capacity. In the first case, the capacity of a set is the supremum of the mass which can be supported by the set if the potential is at most one on the set (cf. Section 5; also the remarks at the end of Section 8.3). This definition is taken from L. Carleson's book 7: the book deals with general kernels, takes us quickly to interesting problems but is short on details. In the second case, the inverse of the capacity of a set is the infimum of the energy integral if the total mass is one (cf. Section 8.1). This definition is taken from the book of Landkof 29. Landkof considers c~-potentials and the corresponding a-capacity and gives many details. When discussing ~c potentials in Section 8, we can often use results from the general theory in the first six sections: it applies without change in the case 0 < a < 2. When 2 < ~( < N, where N is the dimension of the space, it is no longer possible to use the definition in Section 5, since there is no strong maximum principle in this case (cf. Theorem 4.3 and the remarks preceding Theorem 5.11). We can always use results which do not depend on the strong maximum principle: we have therefore tried to make clear what can be proved without applying this result. In Sections 9 - 16, there is a survey of minimal thinness and rarefiedness. Minimal thinness of a set E at infinity in a half-space is defined in terms of properties of the regularized reduced function RE of a minimal harmonic function with pole at infinity (cf. Section 11). A set E is defined to be rarefied at infinity in a half- space D if certain positive superharmonic functions in D dominate 1* on E (cf. Definition 12.4). Characterizations of these exceptional sets in terms of conditions of Wiener type involving Greeen energy and Green mass appear here as Theorems 11.3 and 13.1. In 2, Aikawa uses singular integral techniques to study problems in potential theory. In Section 14, we consider Green potentials and Poisson integrals in a half- space and carry through the program of Aikawa for these kinds of potentials. In Section 15, we show that Green capacity and Green mass can in the Wiener-type conditions be replaced by ordinary capacity. This is one of the starting points of the work of Aitmwa on "quasi-additivity" of capacity (cf. Section 16). There are also other interesting consequences (see the remark at the end of Section 15). It was a pleasure to give these lectures. The participants were very active and our discussions led to many improvements in the final version of these notes. I am particularly grateful to TorbjSrn Lundh for typing these notes using the AA/I$-b~I~X - -system. There has been a lot of interaction between us, many preliminary versions have been circulated in the class and TorbjSrn has with enthusiasm done a tremendous amount of work. These lecture notes are dedicated to the memory of Howard Jackson of McMas- ter University who died in 1986 at the age of .25 Together, we tried to understand exceptional sets of the type discussed in Sections 11-15.