Characterizing regularity of domains via the Riesz transforms on their boundaries

Abstract

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 (∂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 PDE's 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 α (∂Ω)), and discuss an extension to the scale of Besov spaces.

The spectra of harmonic layer potential operators on domains with rotationally symmetric conical points

“…The proof proceeds by writing the difference of kernels as a sum of products of Riesz kernels with smooth, small kernels. The Riesz transforms are however not bounded on E; since ν r H 1/2 (Γ) this would contradict [39,Eq. (6.50)].…”

“…Recalling that C = P n iξ−1/2 (cos(α)) and comparing the two expressions yields the explicit formula. From (39) we have that…”

Quantitative Hölder Estimates for Even Singular Integral Operators on Patches

“…Going beyond in regularity, the functions are discontinuous on the boundary of the domains, but C σ in each side. This C σ regularity has been bounded polynomially by the C 1`σ norm of the domain [13].Here we provide a sharper bound of type L log L in terms of the C 1`σ regularity of the domain. This new estimate shows explicitly the dependence of the lower order norm and the non-selfintersecting property of the boundary of the domain.…”

“…Going beyond in regularity, the functions are discontinuous on the boundary of the domains, but C σ in each side. This C σ regularity has been bounded polynomially by the C 1`σ norm of the domain [13].…”

“…In this work, we go further in order to control higher regularity for functions given by (1.1) with even kernel. Despite the fact that these functions are discontinuous on BD, it is possible to obtain C σ regularity in D and in R n D. In [13], this regularity has been shown together with qualitative bounds of the form (1.2) }T p1 D q} C σ pDq ď CPp}D} C 1`σ q, with P a polynomial and C ą 0 a constant depending on n, σ, and the geometry of the domain D. Above, the function T p1 D q is extended continuously on D. The paper also characterizes the regularity of the domains in terms of odd singular integrals operators on BD. It uses harmonic analysis techniques and Clifford algebras as a generalization of the field of complex numbers to higher dimensions.…”