Lp → Lq norm estimates of Cauchy transforms on the Dirichlet problem and their applications

Abstract

Denote by Cα(D) the space of the functions f on the unit disk D which are Hölder continuous with the exponent α, and denote by C1,α(D) the space which consists of differentiable functions f such that their derivatives are in the space Cα(D). Let C be the Cauchy transform of Dirichlet problem. In this paper, we obtain the norm estimates of ‖C‖Lp→Lq, where 3/2<p<2 and q=p/(p−1). Suppose g∈Lp(D) and f=G[g] is the Green potential of g. By using Sobolev embedding theorem, we show that if 1<p≤2, then f∈Cμ(D), where μ=2−2/p. We also show that if 2<p<∞, then f∈C1,ν(D), where ν=1−2/p. Finally, for the case p=∞, we show that f is not necessarily in C1,1(D), but its gradient, i.e., |∇f| is Lipschitz continuous with respect to the pseudo-hyperbolic metric. This paper is inspired by [2, Chapter 4] and [9].