Hardy spaces and regularity for the inhomogeneous Dirichlet and Neumann problems

Abstract

This article has three aims. First, we study Hardy spaces, $h_L^p(\Omega )$, associated with an operator $L$ which is either the Dirichlet Laplacian ${\Delta }_D$ or the Neumann Laplacian ${\Delta }_N$ on a bounded Lipschitz domain $\Omega$ in ${\mathbb{R}}^n$, for $0<p\le 1$. We obtain equivalent characterizations of these function spaces in terms of maximal functions and atomic decompositions. Second, we establish regularity results for the Green operators, regarded as the inverses of the Dirichlet and Neumann Laplacians, in the context of Hardy spaces associated with these operators on a bounded semiconvex domain $\Omega$ in ${\mathbb{R}}^n$. Third, we study relations between the Hardy spaces associated with operators and the standard Hardy spaces $h_r^p(\Omega )$ and $h_z^p(\Omega )$, then establish regularity of the Green operators for the Dirichlet problem on a bounded semiconvex domain $\Omega$ in ${\mathbb{R}}^n$, and for the Neumann problem on a bounded convex domain $\Omega$ in ${\mathbb{R}}^n$, in the context of the standard Hardy spaces $h_r^p(\Omega )$ and $h_z^p(\Omega )$. This gives a new solution to the conjecture made by D.-C. Chang, S. Krantz and E. M. Stein regarding the regularity of Green operators for the Dirichlet and Neumann problems on $h_r^p(\Omega )$ and $h_z^p(\Omega )$, respectively, for all $\frac{n}{n+1}<p\le 1$.