Invariant Potential Theory in the Unit Ball of Cn [Paperback]

The results in potential theory with respect to the LaplaceBeltrami 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*D and the Riesz decomposition theorem for invariant subharmonic functions in this introduction and survey of recent results in potential theory with respect to the Laplace-Beltrami operator ^D*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*D and the Riesz decomposition theorem for invariant subharmonic functions in this introduction and survey of recent results in potential theory with respect to the Laplace-Beltrami operator ^D*D in several complex variables, with special emphasis on the unit ball in Cn.This monograph covers Poisson-Szegö integrals on the ball, the Green's function for ^D*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. It also contains recent results on admissible and tangential boundary limits of Green potentials, and Lp inequalities for the invariant gradient of Greens potentials. Applications of some of the results to Hp spaces, and weighted Bergman and Dirichlet spaces of invariant harmonic functions are included.1. Notation and preliminary results; 2. The Bergman kernel; 3. The LaplaceBeltrami operator; 4. Invariant harmonic and subharmonic functions; 5. PoissonSzeg? 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; Referenl“+