A T(P) theorem for Sobolev spaces on domains

Abstract

Recently, V. Cruz, J. Mateu and J. Orobitg have proved a T(1) theorem for the Beurling transform in the complex plane. It asserts that given $0<s\le 1$, $1<p<\infty$ with $sp>2$ and a Lipschitz domain $\Omega \subset \mathbb{C}$, the Beurling transform $Bf=-p.v.\frac{1}{\pi z^2}\ast f$ is bounded in the Sobolev space $W^{s,p}(\Omega )$ if and only if $B{\chi }_{\Omega }\in W^{s,p}(\Omega )$. In this paper we obtain a generalized version of the former result valid for any $s\in \mathbb{N}$ and for a larger family of Calderón-Zygmund operators in any ambient space ${\mathbb{R}}^d$ as long as $p>d$. In that case we need to check the boundedness not only over the characteristic function of the domain, but over a finite collection of polynomials restricted to the domain. Finally we find a sufficient condition in terms of Carleson measures for $p\le d$. In the particular case $s=1$, this condition is in fact necessary, which yields a complete characterization.

arXiv comment35 pages, 6 figures

References in corpus (2)

• Beltrami equation with coefficient in Sobolev and Besov spaces
• Regularity of C^1 and Lipschitz domains in terms of the Beurling transform

Cited by in corpus (9)

• A T(1) theorem for fractional Sobolev spaces on domains
• On the Hardy space theory of compensated compactness quantities
• Sobolev regularity of the Beurling transform on planar domains
• Measuring Triebel-Lizorkin fractional smoothness on domains in terms of first-order differences
• Dyadic representation theorem using smooth wavelets with compact support
• Triebel-Lizorkin regularity and bi-Lipschitz maps: composition operator and inverse function regularity
• Global smoothness of quasiconformal mappings in the Triebel-Lizorkin scale
• Weighted Sobolev estimates of the truncated Beurling operator
• T1 theorem for Campanato spaces on domains