Given any uniform domain Ω, the Triebel-Lizorkin space F s p,q pΩq with 0 ă s ă 1 and 1 ă p, q ă 8 can be equipped with a norm in terms of first order differences restricted to pairs of points whose distance is comparable to their distance to the boundary.Using this new characterization, we prove a T(1)-theorem for fractional Sobolev spaces with 0 ă s ă 1 for any uniform domain and for a large family of Calderón-Zygmund operators in any ambient space R d as long as sp ą d. arXiv:1507.03935v5 [math.CA]

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 ă s ď 1, 1 ă p ă 8 with sp ą 2 and a Lipschitz domain Ω Ă C, the Beurling transform Bf "´p.v. 1 πz 2˚f is bounded in the Sobolev space W s,p pΩq if and only if BχΩ P W s,p pΩq. In this paper we obtain a generalized version of the former result valid for any s P N and for a larger family of Calderón-Zygmund operators in any ambient space R d as long as p ą 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 ď d. In the particular case s " 1, this condition is in fact necessary, which yields a complete characterization.

Consider a Lipschitz domain $\Omega$ and the Beurling transform of its characteristic function $\mathcal{B} \chi_\Omega(z)= - {\rm p.v.}\frac1{\pi z^2}*\chi_\Omega (z) $. It is shown that if the outward unit normal vector $N$ of the boundary of the domain is in the trace space of $W^{n,p}(\Omega)$ (i.e., the Besov space $B^{n-1/p}_{p,p}(\partial\Omega)$) then $\mathcal{B} \chi_\Omega \in W^{n,p}(\Omega)$. Moreover, when $p>2$ the boundedness of the Beurling transform on $W^{n,p}(\Omega)$ follows. This fact has far-reaching consequences in the study of the regularity of quasiconformal solutions of the Beltrami equation.Comment: 33 pages, 8 figures. arXiv admin note: text overlap with arXiv:1507.0433

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Consider a Lipschitz domain Ω and a measurable function µ supported in Ω with µ L ∞ < 1. Then the derivatives of a quasiconformal solution of the Beltrami equation ∂f = µ ∂f inherit the Sobolev regularity W n,p (Ω) of the Beltrami coefficient µ as long as Ω is regular enough. The condition obtained is that the outward unit normal vector N of the boundary of the domain is in the trace space, that is, N ∈ B n−1/p p,p (∂Ω).