Solutions of the divergence equation in Hardy and lipschitz spaces

TL;DR

This paper investigates solutions to the divergence equation within Hardy and Lipschitz spaces, extending classical results from Lebesgue spaces and establishing new existence theorems and inequalities in these function spaces.

Contribution

It extends the existence theory of divergence solutions to Hardy and Lipschitz spaces, including a Korn inequality for Hardy-Sobolev spaces, filling a gap in the mathematical understanding.

Findings

Existence results for divergence in Hardy spaces with n/(n+1)<p≤1

Solutions in BMO and Lipschitz spaces for limiting cases

A Korn inequality for vector fields in Hardy-Sobolev spaces

Abstract

Given a bounded domain $\O$ and $f$ of zero integral, the existence of a vector fields $\u$ vanishing on $\partial\O$ and satisfying $\d\u=f$ has been widely studied because of its connection with many important problems. It is known that for $f\in L^p(\O)$, $1<p<\infty$, there exists a solution $\u\in W^{1,p}_0(\O)$, 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.