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The $M$-harmonic Dirichlet space on the ball

Miroslav Engliš, El-Hassan Youssfi

Abstract

We~describe the Dirichlet space of $M$-harmonic functions, i.e.~functions annihilated by the invariant Laplacian on~the unit ball of the complex $n$-space, as~the limit of the analytic continuation (in~the spirit of Rossi and Vergne) of the corresponding weighted Bergman spaces. Characterizations in terms of tangential derivatives are given, and the associated inner product is shown to be Moebius invariant. The pluriharmonic and harmonic cases are also briefly treated.

The $M$-harmonic Dirichlet space on the ball

Abstract

We~describe the Dirichlet space of -harmonic functions, i.e.~functions annihilated by the invariant Laplacian on~the unit ball of the complex -space, as~the limit of the analytic continuation (in~the spirit of Rossi and Vergne) of the corresponding weighted Bergman spaces. Characterizations in terms of tangential derivatives are given, and the associated inner product is shown to be Moebius invariant. The pluriharmonic and harmonic cases are also briefly treated.
Paper Structure (6 sections, 18 theorems, 167 equations)

This paper contains 6 sections, 18 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. $M$-harmonic Dirichlet space
  4. Moebius invariance
  5. The pluriharmonic Dirichlet space
  6. The harmonic Dirichlet space

Key Result

Corollary 1

(Corollary PX) In terms of the Peter-Weyl decomposition $f=\sum_{p,q}f_{pq}$, $f_{pq}\in\mathbf H^{pq}$,

Theorems & Definitions (36)

  • Corollary
  • Theorem
  • Corollary
  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Proposition 1
  • proof
  • Lemma 2
  • ...and 26 more