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$L^p\to L^q$ norm estimates of Cauchy transforms on the Dirichlet problem and their applications

Denote by $C^α(\mathbb{D})$ the space of the functions $f$ on t}he unit disk $\mathbb{D}$ which are Hölder continuous with the exponent $α$, and denote by $C^{1, α}(\mathbb{D})$ the space which consists of differentiable functions $f$ such that their derivatives are in the space $C^α(\mathbb{D})$. Let $\mathcal{C}$ be the Cauchy transform of Dirichlet problem. In this paper, we obtain the norm estimates of $\|\mathcal{C}\|_{L^p\to L^q}$, where $3/2<p<2$ and $q=p/(p-1)$. As an application, we show that if $3/2<p<2$, then $u\in C^μ(\mathbb{D})$, where $μ=2/p-1$. We also show that if $2<p<\infty$, then $u\in C^{1, ν}(\mathbb{D})$, where $ν=1-2/p$. Finally, for the case $p=\infty$, we show that $u$ is not necessarily in $C^{1, 1}(\mathbb{D})$, but its gradient, i.e., $|\nabla u|$ is Lipschitz continuous with respect to the pseudo-hyperbolic metric. This paper is inspired by Chapter 4 of [Astala, Iwaniec, Martin: Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton Mathematical Series, Vol. 48, Princeton University Press, Princeton, NJ, 2009, p. xviii+677] and [Kalaj, Cauchy transform and Poisson's equation, Adv. Math. \textbf{231} (2012), 213--242]