Invariant potential theory in the unit ball of Cn̳ / Manfred Stoll.

On t.p. "n̳" is superscript.

Includes bibliographical references (pages 164-169) and index.

Print version record.

This monograph provides an introduction and a survey of recent results in potential theory with respect to the Laplace-Beltrami operator D in several complex variables, with special emphasis on the unit ball in Cn. Topics covered include Poisson-Szegö integrals on the ball, the Green's function for D and the Riesz decomposition theorem for invariant subharmonic functions. The extension to the ball of the classical Fatou theorem on non-tangible limits of Poisson integrals, and Littlewood's theorem on the existence of radial limits of subharmonic functions are covered in detail. The monograph also contains recent results on admissible and tangential boundary limits of Green potentials, and Lp inequalities for the invariant gradient of Green potentials. Applications of some of the results to Hp spaces, and weighted Bergman and Dirichlet spaces of invariant harmonic functions are included. The notes are self-contained, and should be accessible to anyone with some basic knowledge of several complex variables.

Cover; Half-title; Title; Copyright; Dedication; Contents; Introduction; 1. Notation and Preliminary Results; 1.1 Notation; 1.2 Integral Formulas on B; 1.3 Automorphisms of B; 2. The Bergman Kernel; 2.1 The Bergman Kernel; 2.2 Examples; 2.3 Properties of the Bergman Kernel; 2.4 The Bergman Metric; 3. The Laplace-Beltrami Operator; 3.1 The Invariant Laplacian; 3.2 The Invariant Laplacian for Un; 3.3 The Invariant Laplacian for B; 3.4 The Invariant Gradient; 4. Invariant Harmonic and Subharmonic Functions; 4.1 M.-Subharmonic Functions; 4.2 The Invariant Convolution on B; 4.3 The Riesz Measure

10.3 M-Harmonic Bergman and Dirichlet Spaces10.4 Remarks; References; Index

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