Variation of Potentials in the Unit Ball of $ {shadC}n$

ISSN： 0278-1077

Source： Complex Variables, Vol.47, Iss.4, 2002-04, pp. : 333-347

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Abstract

In this paper, we study the variation of invariant Green potentials G in the unit ball B of $ {shadC}n$, which for suitable measures are defined by $$ G_{mu}(z) = int_{B}G(z,w), dmu(w), $$ where G is the invariant Green function for the Laplace-Beltrami operator ~ Delta on B. The main result of the paper is as follows.Let be a non-negative regular Borel measure on B satisfying $$ int_{B}(1-|w|2)nlog {1 over (1-|w|2)}, dmu(w)< infty,$$ and let 0 < rho < 1. Then for almost all points zeta on the boundary S of B, G has finite variation on the line segment joining the points rho zeta and zeta .A similar result is also obtained for pluri-Green potentials V on the ball B, which for suitable measures are given by $$ V_{mu}(z) = int_{B}log {1 over |phi_z(w)|}, dmu(w), $$ where for a fixed z isin B, phis _z denotes the holomorphic automorphism of B satisfying phis z(0) = _z, phis _z(z) = 0 and ( phis _z phis _z)(w) = w for every w isin B.