Martí Prats and Xavier Tolsa: A T(P) theorem for Sobolev spaces on domains

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 [katex] 0< s \leq 1 [/katex], [katex] 1 < p < \infty [/katex] with [katex]sp>2[/katex] and a Lipschitz domain [katex]\Omega\subset \mathbb{C}[/katex], the Beurling transform [katex]Bf=- {\rm p.v.}\frac1{\pi z^2}*f[/katex] is bounded in the Sobolev space [katex]W^{s,p}(\Omega)[/katex] if and only if [katex]B\chi_\Omega\in W^{s,p}(\Omega)[/katex].

In this paper we obtain a generalized version of the former result valid for any [katex]s\in \mathbb{N}[/katex] and for a larger family of Calderón-Zygmund operators in any ambient space [katex]\mathbb{R}^d[/katex] as long as [katex]p>d[/katex]. 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 [katex]p\leq d[/katex]. In the particular case [katex]s=1[/katex], this condition is in fact necessary, which yields a complete characterization.