COMPACT OPERATOR RELATED WITH POISSON-SZEG¨o INTEGRAL

Abstract

Suppose that µ is a finite positive Borel measure on the unit ball B ⊂ C n. The boundary of B is the unit sphere |z| = 1}. Let σ be the rotation-invariant measure S = {z : on S such that σ(S) = 1. In this paper, we will show that if supζ∈S B P (z, ζ)dµ(z) < ∞ where P (z, ζ) is the Poission-Szeg¨o kernel for B, then µ is a Carleson measure. We will also show that if supζ∈S B P (z, ζ)dµ(z) < ∞, then the operator T such that T (f ) = P [f ] is compact as a mapping from Lp(σ) into Lp(B, dµ).

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