ON SOME MEASURE RELATED WITH POISSON INTEGRAL ON THE UNIT BALL

Abstract

Let µ be a finite positive Borel measure on the unit ball B ⊂ Cn and ν be the Euclidean volume measure such that ν(B) = 1. For the unit sphere S = {z : |z| = 1}, σ is the rotation-invariant measure on S such that σ(S) = 1. Let P[f ] be the invariant Poisson integral of f . We will show that there is a constant M > 0 such that R B |P[f ](z)|pdν(z) for all f ∈ Lp(σ) if µ(E(z,r)) and only if k µ kr = supz∈B ν(E(z,r)) < ∞.

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