Denote by $C^{\alpha }(\mathbb{D})$ the space of the functions f on the unit disk $\mathbb{D}$ which are Hölder continuous with the exponent α, and denote by $C^{1,\alpha }(\mathbb{D})$ the space which consists of differentiable functions f such that their derivatives are in the space $C^{\alpha }(\mathbb{D})$. Let $\mathcal{C}$ be the Cauchy transform of Dirichlet problem. In this paper, we obtain the norm estimates of ${\Vert \mathcal{C}\Vert }_{L^p\to L^q}$, where $3/2<p<2$ and $q=p/(p-1)$. Suppose $g\in L^p(\mathbb{D})$ and $f=G[g]$ is the Green potential of g. By using Sobolev embedding theorem, we show that if $1<p\le 2$, then $f\in C^{\mu }(\mathbb{D})$, where $\mu =2-2/p$. We also show that if $2<p<\infty$, then $f\in C^{1,\nu }(\mathbb{D})$, where $\nu =1-2/p$. Finally, for the case $p=\infty$, we show that f is not necessarily in $C^{1,1}(\mathbb{D})$, but its gradient, i.e., $|\mathrm{\nabla }f|$ is Lipschitz continuous with respect to the pseudo-hyperbolic metric. This paper is inspired by [3, Chapter 4] and [11].

表示为 $C^{\alpha }（\mathbb{d}）$功能f在单元盘上的空间$\mathbb{d}$它们是与指数α连续的Hölder并表示为$C^{\mathrm{1个}，\alpha }（\mathbb{d}）$由微分函数f组成的空间，使得它们的导数在空间中$C^{\alpha }（\mathbb{d}）$。让$\mathcal{C}$是Dirichlet问题的柯西变换。在本文中，我们获得了${\Vert \mathcal{C}\Vert }_{{\mathrm{大号}}^p\to {\mathrm{大号}}^q}$， 在哪里 $3/\mathrm{2个}<p<\mathrm{2个}$ 和 $q=p/（p-\mathrm{1个}）$。认为$G\in {\mathrm{大号}}^p（\mathbb{d}）$ 和 $F=G[G]$是g的绿色势。通过使用Sobolev嵌入定理，我们证明了$\mathrm{1个}<p\le \mathrm{2个}$， 然后 $F\in C^{\mu }（\mathbb{d}）$， 在哪里 $\mu =\mathrm{2个}-\mathrm{2个}/p$。我们还表明，如果$\mathrm{2个}<p<\infty$， 然后 $F\in C^{\mathrm{1个}，\nu }（\mathbb{d}）$， 在哪里 $\nu =\mathrm{1个}-\mathrm{2个}/p$。最后，对于这种情况$p=\infty$，我们证明f不一定在$C^{\mathrm{1个}，\mathrm{1个}}（\mathbb{d}）$，但它的渐变，即 $|\mathrm{\nabla }F|$关于伪双曲度量，Lipschitz是连续的。本文的灵感来自于[3，第4章]和[11]。