Characterizing regularity of domains via the riesz transforms on their boundaries

Dorina Mitrea, Marius Mitrea, Joan Verdera

Research output: Contribution to journalArticleResearchpeer-review

3 Citations (Scopus)

Abstract

© 2016 Mathematical Sciences Publishers. Under mild geometric measure-theoretic assumptions on an open subset Ω of ℝn, we show that the Riesz transforms on its boundary are continuous mappings on the Hölder space C{script}α(∂Ω) if and only if Ω is a Lyapunov domain of order α (i.e., a domain of class C{script} 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 ℝn. 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 C{script}α(∂Ω), and discuss an extension to the scale of Besov spaces.
Original languageEnglish
Pages (from-to)955-1018
JournalAnalysis and PDE
Volume9
Issue number4
DOIs
Publication statusPublished - 1 Jan 2016

Keywords

  • Besov space
  • BMO
  • Cauchy-Clifford operator
  • Clifford algebra
  • Hölder space
  • Lyapunov domain
  • Reifenberg flat
  • Riesz transform
  • Singular integral
  • SKT domain
  • Uniform rectifiability
  • VMO

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