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

Abstract

Abstract Let Ω ⊂ Rn be a strongly Lipschitz domain. In this article, the authors study Hardy spaces, Hp r (Ω)and Hp z (Ω), and Hardy-Sobolev spaces, H1,p r (Ω) and H1,p z,0 (Ω) on , for p ∈ ( n/n+1, 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 is a bounded Lipschitz domain, the authors prove that the divergence equation div u = f for f ∈ Hp z (Ω) is solvable in H1,p z,0 (Ω) with suitable regularity estimates.