Characterizing regularity of domains via the Riesz transforms on their boundaries

Mitrea, Dorina; Mitrea, Marius; Verdera, Joan

doi:10.2140/apde.2016.9.955

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Characterizing regularity of domains via the Riesz transforms on their boundaries

Dorina Mitrea, Marius Mitrea and Joan Verdera

Vol. 9 (2016), No. 4, 955–1018

DOI: 10.2140/apde.2016.9.955

Abstract

Under mild geometric measure-theoretic assumptions on an open subset $Ω$ of $ℝ^{n}$ , we show that the Riesz transforms on its boundary are continuous mappings on the Hölder space $C^{α} (\partial Ω)$ if and only if $Ω$ is a Lyapunov domain of order $α$ (i.e., a domain of class $C^{1 + α}$ ). In the category of Lyapunov domains we also establish the boundedness on Hölder spaces of singular integral operators with kernels of the form $P (x - y) ∕ | x - y |^{n - 1 + l}$ , where $P$ is any odd homogeneous polynomial of degree $l$ in $ℝ^{n}$ . This family of singular integral operators, which may be thought of as generalized Riesz transforms, includes the boundary layer potentials associated with basic PDEs of mathematical physics, such as the Laplacian, the Lamé system, and the Stokes system. We also consider the limiting case $α = 0$ (with $VMO (\partial Ω)$ as the natural replacement of $C^{α} (\partial Ω)$ ), and discuss an extension to the scale of Besov spaces.

Keywords

singular integral, Riesz transform, uniform rectifiability, Hölder space, Lyapunov domain, Clifford algebra, Cauchy–Clifford operator, BMO, VMO, Reifenberg flat, SKT domain, Besov space

Mathematical Subject Classification 2010

Primary: 42B20, 42B37

Secondary: 35J15, 15A66

Milestones

Received: 24 January 2016

Revised: 10 February 2016

Accepted: 11 March 2016

Published: 3 July 2016

Authors

Dorina Mitrea
Department of Mathematics University of Missouri at Columbia Columbia, MO 65211 United States

Marius Mitrea
Department of Mathematics University of Missouri at Columbia Columbia, MO 65211 United States

Joan Verdera
Department de Matemàtiques Universitat Autònoma de Barcelona 08193 Bellaterra, Barcelona Spain

Vol. 9, No. 4, 2016