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Journal Articles Publicacions Matemàtiques Year : 2013

Potential maps, Hardy spaces, and tent spaces on special Lipschitz 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 $\overline{\Omega}$ whenever $n/(n+1) < p \leq 1$.
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Dates and versions

hal-00490854 , version 1 (09-06-2010)

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Cite

Martin Costabel, Alan Mcintosh, Robert Taggart. Potential maps, Hardy spaces, and tent spaces on special Lipschitz domains. Publicacions Matemàtiques, 2013, 57 (2), pp.295-331. ⟨10.5565/PUBLMAT_57213_02⟩. ⟨hal-00490854⟩
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