Vol. 9, No. 4, 2016

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ISSN: 1948-206X (e-only)
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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
Abstract

Under mild geometric measure-theoretic assumptions on an open subset Ω of n, we show that the Riesz transforms on its boundary are continuous mappings on the Hölder space Cα(Ω) if and only if Ω is a Lyapunov domain of order α (i.e., a domain of class C1+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|n1+l, where P is any odd homogeneous polynomial of degree l in 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(Ω) as the natural replacement of Cα(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