Solutions of the divergence equation in Hardy and lipschitz spaces

Abstract

Given a bounded domain $Ø$ and $f$ of zero integral, the existence of a vector fields $\u$ vanishing on $\partial Ø$ and satisfying $\d\u=f$ has been widely studied because of its connection with many important problems. It is known that for $f\in L^p(Ø)$, $1<p<\infty$, there exists a solution $\breve{\in }{W}_0^{1,p}(Ø)$, and also that an analogous result is not true for $p=1$ or $p=\infty$. The goal of this paper is to prove results for Hardy spaces when $\frac{n}{n+1}<p\le 1$, and in the other limiting case, for bounded mean oscillation and Lipschitz spaces. As a byproduct of our analysis we obtain a Korn inequality for vector fields in Hardy-Sobolev spaces.