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

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.

Citation

Download Citation

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

Information

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

zbMATH: 1282.35113
MathSciNet: MR3114771

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

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

JOURNAL ARTICLE
37 PAGES


SHARE
Vol.57 • No. 2 • 2013
Back to Top