Cauchy transform and Poisson's equation

TL;DR

This paper establishes sharp bounds for the gradients of solutions to the Poisson equation in the unit disk, linking these bounds to the $L^p$ norms of the data and extending results to smooth Jordan domains.

Contribution

It provides the first sharp constants for gradient estimates of Poisson solutions in the disk for various $p$, and relates these to the $L^p$ norms of the Cauchy transform of Dirichlet's problem.

Findings

Sharp $L^p$ bounds for $ abla u$ and $ abla u$ derivatives for $p=1,2, ext{and}\infty$.

Extension of bounds to smooth Jordan domains.

Connection between gradient estimates and the $L^p$ norm of the Cauchy transform.

Abstract

Let $u\in W^{2,p}_0$, $1\le p\le \infty$ be a solution of the Poisson equation $\Delta u = h$, $h\in L^p$, in the unit disk. It is proved that $\|\nabla u\|_{L^p} \le a_p\|h\|_{L^p}$ with sharp constant $a_p$ for $p=1$ and $p=\infty$ and that $\|\partial u\|_{L^p} \le b_p\|h\|_{L^p}$ with sharp constant $b_p$ for $p=1$, $p=2$ and $p=\infty$. In addition is proved that for $p>2$ $||\partial u||_{L^\infty}\le c_p\Vert h\Vert_{L^p} $, and $||\nabla u||_{L^\infty}\le C_p\Vert h\Vert_{L^p}, $ with sharp constants $c_p$ and $C_p$. An extension to smooth Jordan domains is given. These problems are equivalent to determining the precise value of $L^p$ norm of {\it Cauchy transform of Dirichlet's problem}.