Hardy and Hardy-Sobolev Spaces on Strongly Lipschitz Domains and Some Applications

Let Omega subset of R-n be a strongly Lipschitz domain. In this article, the authors study Hardy spaces, H-r(p)(Omega) and H-z(p)(Omega), and Hardy-Sobolev spaces, H-r(1),p (Omega) and Pi(1)(,p)(Z)(0),(Omega) on Omega, for p is an element of(n/n+1,) . The authors establish grand maximal function characterizations of these spaces. As applications, the authors obtain some div-curl lemmas in these settings and, when Omega is a bounded Lipschitz domain, the authors prove that the divergence equation div u = f for f is an element of H-z(p)(Omega) is solvable in H-z,(0)1,(p),(Omega) with suitable regularity estimates.