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.

35 pages, 6 figures

Cited by in corpus (2)

• On the Hardy space theory of compensated compactness quantities
• T1 theorem for Campanato spaces on domains