Potential Theory on Harmonic Spaces

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- Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen Band 158

Corneliu Constantinescu - Aurel Cornea

Potential Theory on Harmonic Spaces

Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Beriicksichtigung der Anwendungsgebiete Band 158

Herausgegeben von J. L. Doob · A. Grothendieck · E. Heinz· F. Hirzebruch .E. Hopf· W. Maak · S. Mac Lane· W. Magnus· J. K. Moser M. M. Postnikov · F. K. Schmidt· D.S. Scott· K. Stein Geschiiftsfuhrende H erausgeber B. Eckmann und B. L. van der Waerden

Corneliu Constantinescu • Aurel Cornea

Potential Theory on Harmonic Spaces

Springer-Verlag Berlin Heidelberg NewYork 1972

Comeliu Constantinescu · Aurel Cornea Mathematical Institute of the Roumanian Academy, Bucharest

Gescbiftsfilhrende Herausgeber:

B.Eckmann Eidgenossische Technische Hochschule Ziirich

B. L. van der Waerden Mathematisches Institut der Universitat Ziirich

AMS Subject Classifications (1970): 31C05, 31D05

ISBN 3-540-05916-4 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-05916-4 Springer-Verlag New York Heidelberg Berlin

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This work is subject to copyright. All rights reserved, whether the whole or part of the material is concerned, specifically those of translation, reprintin,. re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made fOT other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1972. Library of Congress Catalog Card Number 72-86117. Printed in Germany. Typesetting, printing and binding: Universititsdruckerei H. Stftrtz AG, Wiirzburg.

Preface There has been a considerable revival of interest in potential theory during the last 20 years. This is made evident by the appearance of new mathematical disciplines in that period which now-a-days are considered as parts of potential theory. Examples of such disciplines are: the theory of Choquet capacities, of Dirichlet spaces, of martingales and Markov processes, of integral representation in convex compact sets as well as the theory of harmonic spaces. All these theories have roots in classical potential theory. The theory of harmonic spaces, sometimes also called axiomatic theory of harmonic functions, plays a particular role among the above mentioned theories. On the one hand, this theory has particularly close connections with classical potential theory. Its main notion is that of a harmonic function and its main aim is the generalization and unification of classical results and methods for application to an extended class of elliptic and parabolic second order partial differential equations. On the other hand, the theory of harmonic spaces is closely related to the theory of Markov processes. In fact, all important notions and results of the theory have a probabilistic interpretation. Based on ideas of Brelot, Doob and Tautz, the theory of harmonic spaces has developed so rapidly in the last 15 years that up to now original papers, seminar and lecture notes were the only source for further studies in this field. In view of this situation, C. Constantinescu and A. Cornea who both had considerably influenced the development of the theory, undertook the difficult task to present our present-day knowledge of harmonic spaces as completely as possible. The result of their effort is this book. It introduces the notion of a harmonic space in a somewhat refined form in order to proceed to the main examples of harmonic spaces without too many detours. It offers a wealth of material not only within the main text but also in the form of exercises. With this monograph the authors do not only close a serious gap in the existing mathematical literature, the monograph certainly will also have a strong impact on future research in the field of potential theory. Erlangen, July 1971

Heinz Bauer

Contents Introduction . . . . . . . Terminology and Notation

I 4

PART ONE

Chapter 1. Harmonic Sheaves and Hyperharmonic Sheaves § 1. 1. Convergence Properties § 1.2. Resolutive Sets . . . • § 1.3. Minimum Principle . . Chapter 2. Harmonic Spaces . . § 2.1. Definition of Harmonic Spaces . § 2.2. Superharmonic Functions and Potentials § 2.3. 6-Harmonic Spaces and ~-Harmonic Spaces § 2.4. Resolutive Sets on Harmonic Spaces . Chapter 3. Bauer Spaces and Brelot Spaces . . § 3.1. Definitions and Fundamental Results . § 3.2. The Laplace Equation § 3.3. The Heat Equation . . . . . .

9 9 17 24

30 30 37 42

49 61 61

73 82

PART TWO Chapter 4. Convex Cones of Continuous Functions on Baire Topological Spaces . . . . . . . . § 4.1. Natural Order and Specific Order. . . . . . . . § 4.2. Balayage . . . . . . . . . . . . . . . . . . Chapter 5. The Convex Cone of Hyperharmonic Functions § 5.1. The Fine Topology . . . . . . . . . . . . . . § 5.2. Capacity . . . . . . . . . . . . . . . . . . § 5.3. Supplementary Results on the Balayage of Positive Superharmonic Functions . . . . . . . . . . . Chapter 6. Absorbent Sets, Polar Sets, Semi-Polar Sets § 6.1. Absorbent Sets. . . . . . . § 6.2. Polar Sets . . . . . . . . . § 6.3. Thinness and Semi-Polar Sets

99 99 108 116 116 123

127 137 137 142

149

VIII

Contents

Chapter 7. Balayage of Measures . . . • • . . . . . . § 7.1. General Properties of the Balayage of Measures § 7.2. Fine Properties of the Balayage of Measures . . Chapter 8. Positive Superharmonic Functions. Specific Order . § 8.1. Abstract Carriers . . . . § 8.2. Sets of Nonharmonicity . § 8.3. The Band .,H § 8.4. Quasi-Continuity . . . .

159 159 166 184 184 194 201 211

PART THREE Chapter 9. Axiom of Polarity and Axiom of Domination § 9.1. Axiom of Polarity . . . . . . . . . . . § 9.2. Axiom of Domination . . . . . . . . . Chapter 10. Markov Processes on Harmonic Spaces § 10.1. Sub-Markov Semi-Groups . . . . . . . § 10.2. Sub-Markov Semi-Groups on Harmonic Spaces . Chapter 11. Integral Representation of Positive Superharmonic Functions . . . . . . . . . . . . . . . . . . . § 11.1. Locally Convex Vector Spaces of Harmonic Functions § 11.2. Locally Convex Topologies on the Convex Cone of Positive Superharmonic Functions . . § 11.3. Abstract Integral Representation . . . . . . . . . . § 11.4. Riesz-Martin Kernels . ; . . . . . . . . . . . . . . § 11.5. Integral Representation of Positive Superharmonic Functiom. References . . Bibliography . Index . . Notation

219 219 227 238 238 246 272 272 285 299 309 3~ 337 346 351 355

Introduction Potential theory is a very old area of mathematics. Its origins may be placed in the 18th century when J. Lagrange remarked in 1773 that the gravitational forces derive from a function (called a potential function by G. Green in 1828 and simply a potential by C. F. Gauss in 1840) and when P. S. Laplace showed in 1782 that in a mass free region this function satisfies the partial differential equation which today bears his name. The fundamental principles of this theory were elaborated during the last century and this theory constitutes today classical potential theory. In 1823 S. D. Poisson introduced his integral formula on the disc and on the ball in order to solve the first boundary-value problem for the Laplace equation (also called the Dirichlet problem); in 1828 G. Green invented the Green function and, with its aid, solved the Dirichlet problem for domains with sufficiently smooth boundary; in 1839 S. Earnshaw proved the minimum principle for the solution of the Laplace equation and in 1840 C. F. Gauss, in a celebrated paper, resolved the equilibrium problem, developed a capacity theory and gave a new solution for the Dirichlet problem. Needless to say, the rigour of these proofs left much to be desired. This prompted many mathematicians to come back to the problems: W. Thomson studied the Poisson integral formula and gave it the form known today; H. A. Schwarz, in 1870, gave the first rigorous proof of the behaviour of this integral at the boundary. The Dirichlet problem was also studied by L. Dirichlet and B. Riemann in 1853, but it was again H. A. Schwarz in 1870 who succeeded in giving the first rigorous proof for the two dimensional case, using the alternating method; the corresponding problem in three dimensions having to wait until 1887 for its solution via the balayage method ofH. Poincare. The proof in 1886 by A. Harnack of the inequality to which he lent his name must be considered an important contribution to potential theory. From his inequality he deduced the convergence property of monotone sequences. Thus by the end of the last century the three basic principles of potential theory, namely the Dirichlet problem, the minimum principle and the convergence property were established. Gradually it became clear that these properties of the Laplace equation are also shared by other partial differential equations such as

2

Introduction

the heat equation or, more generally, linear elliptic or parabolic equation.s of second order. On the other hand, step by step, it was remarked that ,a large part of the results of potential thepry could be obtained using only the above three principles. It seemed therefore quite natural to develop an axiomatic system which would unify these theories and extend potential theory to these partial differential equations. This axiomatic theory was constructed around the nineteen fifties, by G. Tautz, J. L. Doob, M. Brelot and H. Bauer. Their theory started with a linear sheaf of continuous real functions defined on a locally compact space (this sheaf playing the role of the sheaf of solutions of a partial differential equation) for which a convergence property, a minimum principle and the possibility of solving the Dirichlet problem for sufficiently many open sets was given. Such a construction had the advantage of being more elegant and more general and of giving better insight into the implications of the various results. It was used also as a guide for much research in the field of partial differential equations, and it drew attention to the key properties that had to be proved. Since it can be shown that under rather mild conditions there exist suitable Markov processes associated with the theory, it may be also used as a link between partial differential equations and Markov processes. For the parabolic equations the proof that it is possible to solve the Dirichlet problem for a sufficiently large class of open sets is very difficult and even needs the development of a part of potential theory:· We therefore considered it convenient to weaken the corresponding axiom, which can be done without giving up any interesting results of the theory. This leads to a more general axiomatic system but obliges us to take as the starting point the sheaf of hyperharmonic functions instead of the sheaf of harmonic functions. The book is divided in three parts. In the first one (Chapter 1-3) harmbnic spaces are presented, in the second, (Chapters 4-8) general problems are treated (balayage, natural and specific order, negligible sets) and in the last one (Chapters 9-11) three special problems (the axiom of domination and the axiom of polarity, Markov processes and the integral representation) are discussed. Two other special topics, namely duality and ideal boundaries are not treated here, since they are not yet sufficiently developed. The connections between the chapters are roughly indicated by the following diagram:

Introduction

3

However, with the exception of the last three chapters which are independent from each other, it is assumed in each chapter that the definitions and the theorems of the preceding ones are known. A difficult problem was to decide what attitude to take concerning the countable base of the underlying space. The interesting examples of the theory always have a countable base and the theory becomes simpler with this assumption. Nevertheless many of the theorems still hold in the uncountable case and therefore except for some sections we did not assume that the harmonic space had a countable base. The future evolution of the theory will show whether or not this decision was right. Nearly all of the sections contain exercises. Their principal aim is to present supplementary results closely related to the material treated in the section. A few exercises give counterexamples, which indicate the limitations of the theory as well as the necessity of some of the hypotheses appearing in some of the theorems. There are also some exercises, which present to the reader important results (or sometimes, outlines of whole theories), that have been obtained in neighbouring fields. In the exercises, the references, besides their historical role, have the mission to indicate places were proofs may be found. We draw attention to the fact that the hypotheses assumed in a section are also supposed to hold in the exercises (unless the contrary is explicitly mentioned). The historical notes and the bibliographical indications are certainly not complete. The emergence of a notion or of a result in mathematics is usually made in a long series of small steps. We tried to indicate those works where we felt that the principal contributions were made. Frequently we quoted the classical papers for theorems about harmonic spaces. It was done so when we thought that the essential difficulty was solved in the classical case. We apologise for the mistakes and the omissions. Besides the References (to which the reader is sent via the bibliographical indications) there exists also a list of papers, entitled Bibliography, which were not quoted in the book, but may present interest for the theory of harmonic spaces. The book is intended for those wishing to do research within the theory itself and those wishing to use it in other fields (as for instance semi-elliptic partial differential equations or Markov processes). Elementary notions of general topology and integration theory on locally compact spaces are assumed; in these fields we quote N. Bourbaki. Some exercises may require more knowledge from other fields of analysis. We express our gratitude to Prof. Heinz Bauer for his kindness in writing a preface to the book.

Terminology and Notation The purpose of this section is to make precise the terminology and the notation for some of the notions which we use in the present book, especially for those terms and notations which either are not very usual or are used with different meanings in contemporary mathematical literature. Generally however, we have followed Bourbaki's terminology and notation. If A, B are sets, A ,B denotes the set of elements belonging to A and not belonging to B. If R (x) is a proposition concerning x and A is a set, we denote by {xeAIR(x)}, the set of elements x of A for which R (x) holds. If A, B are sets and for any xe A, T(x) is an element of B, we shall denote by x f--+ T(x) the map of A into B which associates with any xeA the element T(x)eB and by {T(x)lxeA} the image of A through this map. Let A be a set. We say that a function is defined on A if its domain contains A. Iffis a function defined on A (resp. if IF is a set of functions defined on A) and if B is a subset of A we denote by f IB (resp. IFl 8 ) the restriction off to B (resp. the set of restrictions to B of the functions of §). Let A be an ordered set and B a subset of A. We call the least upper bound or supremurn (resp. the greatest lower bound or infimum) of B in A, if it exists, the smallest (resp. greatest) element of A which dominates every element of B (resp. is dominated by any element of B). The ordered set A is called inductive if any chain (i.e. linearity ordered subset) of A possesses a majorant in A. The ordered set A is called upper (resp. lower) directed if any finite non-empty subset possesses a majorant (resp. minorant). If A is upper directed we caU the filter generated by the filter base {{yeAly~x}lxeA}

the section filter of A. The ordered set A is called a· lattice if any finite non-empty subset of A possesses a supremum and an infimum. The ordered set A is called upper complete (resp. conditionally complete) if any non-empty subset of A (resp. any non-empty subset of A possessing a majorant) possesses a least upper bound. A subset B of a lattice A is called a sublattice if the supremum and the infimum in A of any finite non-empty subset of B

Terminology and Notation

5

belongs to B. If x, y are elements of an ordered set A we denote: [x, y]•= {zeAlx::;;z::;;y},

[x, y[:= {zeAlx::;;z lg(x)- f(x)I