Solutions of the divergence equation in Hardy and Lipschitz spaces

Given a bounded domain $\varOmega $ and $f$ of zero integral, the existence of a vector field ${\bf u}$ vanishing on $\partial \varOmega $ and satisfying $\operatorname{div}{\bf u}=f$ has been widely studied because of its connection with many important problems. It is known that for $f\in L^p(\varOmega )$, $1 \lt p \lt \infty $, there exists a solution ${\bf u}\in W^{1,p}_0(\varOmega )$, 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 $\frac{n}{n+1} \lt p\le 1$, 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.