Denote by the space of the functions f on the unit disk which are Hölder continuous with the exponent α, and denote by the space which consists of differentiable functions f such that their derivatives are in the space . Let be the Cauchy transform of Dirichlet problem. In this paper, we obtain the norm estimates of , where and . Suppose and is the Green potential of g. By using Sobolev embedding theorem, we show that if , then , where . We also show that if , then , where . Finally, for the case , we show that f is not necessarily in , but its gradient, i.e., is Lipschitz continuous with respect to the pseudo-hyperbolic metric. This paper is inspired by [2, Chapter 4] and .