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

2010/06/03 by Martin Costabel, Costabel, Martin, Alan McIntosh +3
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Nonlinear Partial Differential Equations

paper · doi:10.48550/arxiv.1006.0562

Abstract

Suppose that Ω is the open region in ℝⁿ above a Lipschitz graph and let d denote the exterior derivative on ℝⁿ. We construct a convolution operator T which preserves support in Ω, 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 Ω. Thus if f is exact and supported in Ω, then there is a potential u, given by u=Tf, of optimal regularity and supported in Ω, such that du=f. This has implications for the regularity in homogeneous function spaces of the de Rham complex on Ω 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 Ω when n/(n+1)

Citations

• Some new function spaces and their applications to Harmonic analysis

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