Potential Theory on Infinite-Dimensional Abelian Groups

Description

ContentsChapter 1. Introduction Chapter 2. Elements of potential theory . . 2.1 Notation . . . . . . . . . . . . . . . . 2.2 Ilarmonic 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 interprctations 2.10 Duality . . . . . . . . . . . . . . . . . Chapter 3. Markov processes and harmonic structures 19 3.1 Markov processes and Brelot spaces ....... 19 3.2 Markov processes and Bauer spaces. . . . . . . . 24 3.3 Projective scqucnccs of hannonic spaces: examplcs. definitions, state- ments of theorems . . . . . . . . . . . . . . . . . . . . . . . . . . .. 33 3.4 Projective sequences of hannonic spaces: proofs of theorems ..... 43 3.5 Projective sequences of hannonic spaces: some remarks on hannonic functions on a Wiener space. . . . . . . . . . . . . . . . . . . . . .. 67 Chapter 4. Markov processes and harmonic structures on a group 72 4.1 Harmonic groups. . . . . . . . . . . . . . . . . . . . . . . . 73 4.2 Space-homogeneous processes and harmonic functions . . . . 81 4.3 Space homogeneous processes and hannonic functions: quasidiagonal case .................. 92 4.4 Bony's theorem on the group IRP x Toe:: . . . . . . . . . . . . . . . . . 101 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 '])'00 ......... Chapter 6. Special classes of harmonic functions and potentials 6.1 Spaces At,:; of martingales with mixed norm ........ 6.2 Classes hp of harmonic functions in the semispace . . . 6.3 ..U p -estimates of potentials. Sobolev inequality on group -roo Chapter 7. Some thouRhts on probability and analysis on locally compact groups . . . . . . . . . 166 7. ] Dichotomy problem . . . . . . 166 7.2 Hannonic functions on a group 169 7.3 The problem of hypoellipticity 170 7.4 "Can one hear the shape of a drum?" 171 7.5 Geometry on a group ........ ]72 Bibliography. 175 Index. . . . . 183