Journal Article

Potential Maps, Hardy Spaces, and Tent Spaces on Domains

Costabel, Martin, McIntosh, Alan, Taggart, Robert J

2013

Overview

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.

• Publisher: Universitat Autònoma de Barcelona, Departament de Matemàtiques
• Subject: 35B65 / 35C15 / 42B30 / 47G10 / 58J10 / differential forms / Exterior derivative / Hardy space / Lipschitz domain / potential map / Sobolev space / tent space