Solutions of the divergence equation in Hardy and lipschitz spaces

Given a bounded domain $\text{Ø}$ and $f$ of zero integral, the existence of a vector fields \u vanishing on $\mathrm{\partial }\text{Ø}$ and satisfying $\text{d}\breve{=}f$ has been widely studied because of its connection with many important problems. It is known that for $f\in L^p(\text{Ø})$, $1<p<\mathrm{\infty }$, there exists a solution $\breve{\in }{W}_0^{1,p}(\text{Ø})$, and also that an analogous result is not true for $p=1$ or $p=\mathrm{\infty }$. 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.