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

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

2010-06-03

arxiv.org/abs/1006.0562v2

Mathematics

Analysis of PDEs

Scientific paper

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)

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