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

Let Ω ⊂ Rn be a strongly Lipschitz domain. In this article, the authors study Hardy spaces, Hpr (Ω)and Hpz (Ω), and Hardy-Sobolev spaces, H1,pr (Ω) and H1,pz,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 ∈ Hpz (Ω) is solvable in H1,pz,0 (Ω) with suitable regularity estimates.

A CHARACTERIZATION OF HARDY SPACE ON STRONGLY LIPSCHITZ DOMAINS OF ℝn BY LITTLEWOOD–PALEY–STEIN FUNCTION

Let Ω be a strongly Lipschitz domain of ℝn and define Hardy spaces on Ω by non-tangential maximal function. In this paper, we will give a characterization of the Hardy spaces on Ω by Littlwood–Paley–Stein function associated to L, where L is an elliptic second-order divergence operator. In order to get our result, we also consider the Lusin area integral characterization of the Hardy spaces on Ω.