Cauchy transform and Poisson's equation

摘要

Let u∈W ~(1,p)∩W0 ~(1,p), 1≤p≤∞ be a solution of the Poisson equation δu=h, h∈L _p, in the unit disk. We prove {norm of matrix}?u{norm of matrix}L _p≤a _p{norm of matrix}h{norm of matrix}L _p and {norm of matrix}u{norm of matrix}L _p≤b p{norm of matrix}h{norm of matrix}L _p with sharp constants ap and b _p, for p=1, p=2, and p=∞. In addition, for p>2, with sharp constants c _p and C _p, we show {norm of matrix}?u{norm of matrix}L∞≤c _p{norm of matrix}h{norm of matrix}L _p and {norm of matrix}?u{norm of matrix}L∞≤C _p{norm of matrix}h{norm of matrix}L _p. We also give an extension to smooth Jordan domains. These problems are equivalent to determining a precise value of the L _p norm of the Cauchy transform of Dirichlet's problem.