## Analysis & PDE

• Anal. PDE
• Volume 9, Number 4 (2016), 955-1018.

### Characterizing regularity of domains via the Riesz transforms on their boundaries

#### Abstract

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 $Cα(∂Ω)$ 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 $P(x − y)∕|x − y|n−1+l$, 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 $Cα(∂Ω)$), and discuss an extension to the scale of Besov spaces.

#### Article information

Source
Anal. PDE, Volume 9, Number 4 (2016), 955-1018.

Dates
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
https://projecteuclid.org/euclid.apde/1510843304

Digital Object Identifier
doi:10.2140/apde.2016.9.955

Mathematical Reviews number (MathSciNet)
MR3530198

Zentralblatt MATH identifier
06607581

#### Citation

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

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