Analysis & PDE
- Anal. PDE
- Volume 9, Number 4 (2016), 955-1018.
Characterizing regularity of domains via the Riesz transforms on their boundaries
Under mild geometric measure-theoretic assumptions on an open subset of , we show that the Riesz transforms on its boundary are continuous mappings on the Hölder space if and only if is a Lyapunov domain of order (i.e., a domain of class ). In the category of Lyapunov domains we also establish the boundedness on Hölder spaces of singular integral operators with kernels of the form , where is any odd homogeneous polynomial of degree in . 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 (with as the natural replacement of ), and discuss an extension to the scale of Besov spaces.
Anal. PDE, Volume 9, Number 4 (2016), 955-1018.
Received: 24 January 2016
Revised: 10 February 2016
Accepted: 11 March 2016
First available in Project Euclid: 16 November 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B37: Harmonic analysis and PDE [See also 35-XX]
Secondary: 35J15: Second-order elliptic equations 15A66: Clifford algebras, spinors
Mitrea, Dorina; Mitrea, Marius; Verdera, Joan. Characterizing regularity of domains via the Riesz transforms on their boundaries. Anal. PDE 9 (2016), no. 4, 955--1018. doi:10.2140/apde.2016.9.955. https://projecteuclid.org/euclid.apde/1510843304