LAUSR

Abstract Denote by C α ( D ) the space of the functions f on the unit disk D which are Holder continuous with the exponent α, and denote by C 1 , α ( D ) the space which consists of differentiable functions f such that their derivatives are in the space C α ( D ) . Let C be the Cauchy transform of Dirichlet problem. In this paper, we obtain the norm estimates of ‖ C ‖ L p → L q , where 3 / 2 p 2 and q = p / ( p − 1 ) . Suppose g ∈ L p ( 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 ∈ C μ ( D ) , where μ = 2 − 2 / p . We also show that if 2 p ∞ , then f ∈ C 1 , ν ( D ) , where ν = 1 − 2 / p . Finally, for the case p = ∞ , we show that f is not necessarily in C 1 , 1 ( D ) , but its gradient, i.e., | ∇ f | is Lipschitz continuous with respect to the pseudo-hyperbolic metric. This paper is inspired by [3, Chapter 4] and [11] .