Paper detail

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

Suppose that $Ω$ 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 $\bar{Ω$}, 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 $\barΩ$. Thus if $f$ is exact and supported in $\barΩ$, then there is a potential $u$, given by $u=Tf$, of optimal regularity and supported in $\barΩ$, such that $du=f$. This has implications for the regularity in homogeneous function spaces of the de Rham complex on $Ω$ 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 $\barΩ$ 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 $T(Ω)$ as a sum of atoms with support away from the boundary of $Ω$. This new decomposition of tent spaces is useful, even for scalar valued functions.