Potential Theory on Infinite-Dimensional Abelian Groups

✍ Scribed by Alexander Bendikov; Carol Regher

• Publisher: De Gruyter
• Year: 1995
• Tongue: English
• Leaves: 192
• Series: De Gruyter Studies in Mathematics; 21
• Category: Library

✦ Table of Contents

Chapter 1. Introduction
Chapter 2. Elements of potential theory
2.1 Notation
2.2 Harmonic and hyperharmonic sheaves
2.3 The generalized Dirichlet problem
2.4 Harmonic spaces
2.5 Brelot and Bauer spaces
2.6 Smooth Bauer spaces
2.7 Markov processes
2.8 Markov processes on harmonic spaces
2.9 Probability interpretations
2.10 Duality
Chapter 3. Markov processes and harmonic structures
3.1 Markov processes and Brelot spaces
3.2 Markov processes and Bauer spaces
3.3 Projective sequences of harmonic spaces: examples, definitions, statements of theorems
3.4 Projective sequences of harmonic spaces: proofs of theorems
3.5 Projective sequences of harmonic spaces: some remarks on harmonic functions on a Wiener space
Chapter 4. Markov processes and harmonic structures on a group
4.1 Harmonic groups
4.2 Space-homogeneous processes and harmonic functions
4.3 Space homogeneous processes and harmonic functions: quasidiagonal case
4.4 Bony’s theorem on the group ℝp ×T∞
Chapter 5. Elliptic equations on a group
5.1 Admissible distributions and multipliers
5.2 Weak solutions of elliptic equations (Lp-theory)
5.3 Weyl’s lemma and the hypoelliptic property
5.4 Bessel potentials on group T∞
Chapter 6. Special classes of harmonic functions and potentials
6.1 Spaces Mp⃗ of martingales with mixed norm
6.2 Classes hp⃗ of harmonic functions in the semispace T∞+
6.3 Mp⃗ -estimates of potentials. Sobolev inequality on group T∞
Chapter 7. Some thoughts on probability and analysis on locally compact groups
7.1 Dichotomy problem
7.2 Harmonic functions on a group
7.3 The problem of hypoellipticity
7.4 “Can one hear the shape of a drum?”
7.5 Geometry on a group
Bibliography
Index