Potential Maps, Hardy Spaces, and Tent Spaces on Domains

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 $\overline\Omega$, 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 $\overline\Omega$. Thus if $f$ is exact and supported in $\overline\Omega$, then there is a potential~$u$, given by $u=Tf$, of optimal regularity and supported in $\overline\Omega$, 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 $\overline\Omega$ when $n/(n+1)<\leq 1$. 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.