A T(P) theorem for Sobolev spaces on domains

TL;DR

This paper generalizes a T(1) theorem for Sobolev spaces on domains, extending it to higher derivatives and broader operators, providing new criteria for boundedness involving polynomials and Carleson measures.

Contribution

It extends the T(1) theorem to all integer orders of smoothness and a wider class of Calderón-Zygmund operators in higher dimensions.

Findings

Boundedness characterized by polynomial conditions on the domain.

A Carleson measure criterion for p ≤ d.

Complete characterization for the case s=1.

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 $0<s\leq1$, $1<p<\infty$ with $sp>2$ and a Lipschitz domain $\Omega\subset \mathbb{C}$, the Beurling transform $Bf=- {\rm p.v.}\frac1{\pi z^2}*f$ 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 Calder\'on-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 $p\leq d$. In the particular case $s=1$, this condition is in fact necessary, which yields a complete characterization.