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2013 Potential Maps, Hardy Spaces, and Tent Spaces on Domains
Martin Costabel, Alan McIntosh, Robert J. Taggart
Publ. Mat. 57(2): 295-331 (2013).


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.


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Martin Costabel. Alan McIntosh. Robert J. Taggart. "Potential Maps, Hardy Spaces, and Tent Spaces on Domains." Publ. Mat. 57 (2) 295 - 331, 2013.


Published: 2013
First available in Project Euclid: 12 December 2013

zbMATH: 1282.35113
MathSciNet: MR3114771

Primary: 35B65
Secondary: 35C15 , 42B30 , 47G10 , 58J10

Keywords: differential forms , Exterior derivative , Hardy space , Lipschitz domain , potential map , Sobolev space , ‎tent space

Rights: Copyright © 2013 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.57 • No. 2 • 2013
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