Potential Theory on Infinite-Dimensional Abelian Groups ( De Gruyter Studies in Mathematics )

Publication series ：De Gruyter Studies in Mathematics

Author： Alexander Bendikov   Carol Regher  

Publisher： De Gruyter‎

Publication year： 1995

E-ISBN: 9783110876840

P-ISBN(Paperback): 9783110142839

Subject： O174.3 harmonic functions and potential theory

Language： ENG

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Description

The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist.

The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level.

The series de Gruyter Studies in Mathematics was founded ca. 30 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics.
While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies

Chapter

Chapter 1. Introduction

pp.：  1 – 7

Chapter 2. Elements of potential theory

pp.：  7 – 13

2.1 Notation

pp.：  13 – 13

2.3 The generalized Dirichlet problem

pp.：  13 – 14

2.2 Harmonic and hyperharmonic sheaves

pp.：  13 – 13

2.4 Harmonic spaces

pp.：  14 – 15

2.5 Brelot and Bauer spaces

pp.：  15 – 16

2.6 Smooth Bauer spaces

pp.：  16 – 17

2.7 Markov processes

pp.：  17 – 18

2.8 Markov processes on harmonic spaces

pp.：  18 – 20

2.9 Probability interpretations

pp.：  20 – 21

2.10 Duality

pp.：  21 – 23

Chapter 3. Markov processes and harmonic structures

pp.：  23 – 25

3.1 Markov processes and Brelot spaces

pp.：  25 – 25

3.2 Markov processes and Bauer spaces

pp.：  25 – 30

3.3 Projective sequences of harmonic spaces: examples, definitions, statements of theorems

pp.：  30 – 39

3.4 Projective sequences of harmonic spaces: proofs of theorems

pp.：  39 – 49

3.5 Projective sequences of harmonic spaces: some remarks on harmonic functions on a Wiener space

pp.：  49 – 73

Chapter 4. Markov processes and harmonic structures on a group

pp.：  73 – 78

4.1 Harmonic groups

pp.：  78 – 79

4.2 Space-homogeneous processes and harmonic functions

pp.：  79 – 87

4.3 Space homogeneous processes and harmonic functions: quasidiagonal case

pp.：  87 – 98

4.4 Bony’s theorem on the group ℝp ×T∞

pp.：  98 – 107

Chapter 5. Elliptic equations on a group

pp.：  107 – 112

5.2 Weak solutions of elliptic equations (Lp-theory)

pp.：  112 – 124

5.1 Admissible distributions and multipliers

pp.：  112 – 112

5.3 Weyl’s lemma and the hypoelliptic property

pp.：  124 – 128

5.4 Bessel potentials on group T∞

pp.：  128 – 136

Chapter 6. Special classes of harmonic functions and potentials

pp.：  136 – 143

6.1 Spaces Mp⃗ of martingales with mixed norm

pp.：  143 – 144

6.2 Classes hp⃗ of harmonic functions in the semispace T∞+

pp.：  144 – 157

6.3 Mp⃗ -estimates of potentials. Sobolev inequality on group T∞

pp.：  157 – 163

Chapter 7. Some thoughts on probability and analysis on locally compact groups

pp.：  163 – 172

7.1 Dichotomy problem

pp.：  172 – 172

7.2 Harmonic functions on a group

pp.：  172 – 175

7.3 The problem of hypoellipticity

pp.：  175 – 176

7.4 “Can one hear the shape of a drum?”

pp.：  176 – 177

7.5 Geometry on a group

pp.：  177 – 178

Bibliography

pp.：  178 – 181

Index

pp.：  181 – 189

LastPages

pp.：  189 – 193

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