In this note we give equivalent characterizations for a fractional Triebel-Lizorkin space [katex]F_{p,q}^s(\Omega)[/katex] in terms of first-order differences in a uniform domain [katex]\Omega[/katex]. The characterization is valid for any positive, non-integer real smoothness [katex]s\in \mathbb{R}_+\setminus \mathbb{N}[/katex] and indices [katex] 1 \leq p < \infty [/katex], [katex]1\leq q \leq \infty[/katex] as long as the fractional part [katex]s[/katex] is greater than [katex]d/p-d/q[/katex].

Consider a Lipschitz domain [katex] \Omega [/katex] and a measurable function [katex] \mu [/katex] supported in [katex] \overline\Omega [/katex] with [katex] \left\|{\mu}\right\|_{L^\infty}< 1 [/katex]. Then the derivatives of a quasiconformal solution of the Beltrami equation [katex] \overline{\partial} f =\mu\, \partial f [/katex] inherit the Sobolev regularity [katex] W^{n,p}(\Omega) [/katex] of the Beltrami coefficient [katex] \mu [/katex] as long as [katex] \Omega [/katex] is regular enough. The condition obtained is that the outward unit normal vector [katex] N [/katex] of the boundary of the domain is in the trace space, that is, [katex] N\in B^{n-1/p}_{p,p}(\partial\Omega) [/katex].

We find a complete characterization for sets of isotropic conductivities with stable recovery in the [katex] L^2 [/katex] norm when the data of the Calderón Inverse Conductivity Problem is obtained in the boundary of a disk and the conductivities are constant in a neighborhood of its boundary. To obtain this result, we present minimal a priori assumptions which turn out to be sufficient for sets of conductivities to have stable recovery in a bounded and rough domain. The condition is presented in terms of the integral moduli of continuity of the coefficients involved and their ellipticity bound.

Some new results regarding the Sobolev regularity of the principal solution of the linear Beltrami equation [katex]\bar\partial f = \mu \partial f + \nu\, \overline{\partial f}[/katex] for discontinuous Beltrami coefficients [katex]\mu[/katex] and [katex]\nu[/katex] are obtained, using Kato-Ponce commutators. A conjecture on the cases where the limitations of the method do not work is raised.

Let [katex] \phi [/katex] be a quasiconformal mapping, and let [katex] T_\phi [/katex] be the composition operator which maps [katex] f [/katex] to [katex] f\circ\phi [/katex]. Since [katex] \phi [/katex] may not be bi-Lipschitz, the composition operator need not map Sobolev spaces to themselves. The study begins with the behavior of [katex] T_\phi[/katex] on [katex]L^p [/katex] and [katex]W^{1,p} [/katex] for [katex]1 < p < \infty [/katex]. This cases are well understood but alternative proofs of some known results are provided. Using interpolation techniques it is seen that compactly supported Bessel potential functions in [katex]H^{s,p} [/katex] are sent to [katex]H^{s,q} [/katex] whenever [katex]0 < s < 1 [/katex] for appropriate values of [katex]q [/katex]. The techniques used lead to sharp results and they can be applied to Besov spaces as well.

Given any uniform domain [katex] \Omega [/katex], the Triebel-Lizorkin space [katex] F^s_{p,q}(\Omega) [/katex] with [katex] 0 < s < 1 [/katex] and [katex] 1 < p,q < \infty [/katex] 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 characterization, originally due to Seeger and reproven here, we prove a T(1)-theorem for fractional Sobolev spaces with [katex] 0 < s < 1 [/katex] for any uniform domain and for a large family of Calderón-Zygmund operators in any ambient space [katex] \mathbb{R}^d [/katex] as long as [katex] sp>d [/katex].

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