Cauchy transform and Poisson's equation

Abstract

Let $u\in W_0^{2,p}$, $1\le p\le \infty$ be a solution of the Poisson equation $\Delta u=h$, $h\in L^p$, in the unit disk. It is proved that $\Vert \nabla u{\Vert }_{L^p}\le a_p\Vert h{\Vert }_{L^p}$ with sharp constant $a_p$ for $p=1$ and $p=\infty$ and that $\Vert \partial u{\Vert }_{L^p}\le b_p\Vert h{\Vert }_{L^p}$ 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}.