Invariant Potential Theory in the Unit Ball of Cn

The 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.

Manfred Stoll (Author)

9780521468305, Cambridge University Press

Paperback, published 12 May 1994

184 pages
22.8 x 15.2 x 1.1 cm, 0.271 kg

"The topics included in this book are well chosen and well presented." Walter Rudin, Bulletin of the American Mathematical Society

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.

1. Notation and preliminary results
2. The Bergman kernel
3. The Laplace–Beltrami operator
4. Invariant harmonic and subharmonic functions
5. Poisson–Szegö integrals
6. The Riesz decomposition theorem
7. Admissible boundary limits of Poisson integrals
8. Radial and admissible boundary limits of potentials
9. Gradient estimates and Riesz potentials
10. Spaces of invariant harmonic functions
References.

Subject Areas: Calculus & mathematical analysis [PBK]