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

TL;DR

This paper establishes $L^p$ to $L^q$ norm estimates for the Cauchy transform related to the Dirichlet problem on the unit disk, and explores their implications for the regularity of solutions.

Contribution

It provides new $L^p$ to $L^q$ bounds for the Cauchy transform and applies these to determine the Hölder and Lipschitz regularity of solutions.

Findings

For $3/2<p<2$, solutions are in $C^{rac{2}{p}-1}( ext{disk})$.

For $2<p< ext{infinity}$, solutions are in $C^{1, 1-rac{2}{p}}( ext{disk})$.

For $p= ext{infinity}$, the gradient of solutions is Lipschitz continuous in the pseudo-hyperbolic metric.

Abstract

Denote by $C^{\alpha}(\mathbb{D})$ the space of the functions $f$ on t}he unit disk $\mathbb{D}$ which are H\"older continuous with the exponent $\alpha$, 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 $\|\mathcal{C}\|_{L^p\to L^q}$, where $3/2<p<2$ and $q=p/(p-1)$. As an application, we show that if $3/2<p<2$, then $u\in C^{\mu}(\mathbb{D})$, where $\mu=2/p-1$. We also show that if $2<p<\infty$, then $u\in C^{1, \nu}(\mathbb{D})$, where $\nu=1-2/p$. 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]