UID:
almafu_9959239119802883
Format:
1 online resource (x, 173 pages) :
,
digital, PDF file(s).
ISBN:
1-139-88505-7
,
1-107-36701-8
,
1-107-37164-3
,
1-107-36210-5
,
1-107-36928-2
,
1-299-40471-5
,
1-107-36455-8
,
0-511-52618-0
Series Statement:
London Mathematical Society lecture note series ; 199
Content:
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.
Note:
In the title "n" is superscript.
,
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
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4.4 Remarks5. Poisson-Szego Integrals; 5.1 The Poisson-Szego Kernel; 5.2 The Dirichlet Problem for Δ; 5.3 Poisson-Szego Integrals; 5.4 The Dirichlet Problem for rB; 5.5 Remarks; 6. The Riesz Decomposition Theorem; 6.1 Harmonic Majorants for M-Subharmonic Functions; 6.2 The Green's Function for Δ; 6.3 The Riesz Decomposition Theorem; 6.4 Green Potentials; 6.5 A Characterization of Hp Spaces; 6.6 Remarks; 7. Admissible Boundary Limits of Poisson Integrals; 7.1 Admissible Limits of Poisson Integrals; 7.2 Maximal Functions of Measures; 7.3 Differentiation Theorems
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10.3 M-Harmonic Bergman and Dirichlet Spaces10.4 Remarks; References; Index
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English
Additional Edition:
ISBN 0-521-46830-2
Language:
English
