Potential maps, Hardy spaces, and tent spaces on special Lipschitz domains

TL;DR

This paper develops a convolution operator on Lipschitz domains that provides optimal regularity potentials for exact forms, and introduces an atomic decomposition of Hardy and tent spaces supported in these domains.

Contribution

It constructs a smoothing potential map preserving support, enabling regularity analysis of the de Rham complex and Hardy spaces on Lipschitz domains.

Findings

Constructed a support-preserving smoothing potential operator.

Established atomic characterizations of Hardy spaces of exact forms.

Developed a new atomic decomposition for tent spaces with support away from the boundary.

Abstract

Suppose that $\Omega$ is the open region in $\mathbb{R}^n$ above a Lipschitz graph and let $d$ denote the exterior derivative on $\mathbb{R}^n$. We construct a convolution operator $T $ which preserves support in $\bar{\Omega$}, is smoothing of order 1 on the homogeneous function spaces, and is a potential map in the sense that $dT$ is the identity on spaces of exact forms with support in $\bar\Omega$. Thus if $f$ is exact and supported in $\bar\Omega$, then there is a potential $u$, given by $u=Tf$, of optimal regularity and supported in $\bar\Omega$, such that $du=f$. This has implications for the regularity in homogeneous function spaces of the de Rham complex on $\Omega$ with or without boundary conditions. The operator $T$ is used to obtain an atomic characterisation of Hardy spaces $H^p$ of exact forms with support in $\bar\Omega$ when $n/(n+1)<p\leq1$. This is done via an atomic decomposition of functions in the tent spaces $\mathcal T^p(\mathbb{R}^n\times\mathbb{R}^+)$ with support in a tent $T(\Omega)$ as a sum of atoms with support away from the boundary of $\Omega$. This new decomposition of tent spaces is useful, even for scalar valued functions.