Sobolev regularity of the Beurling transform on planar domains

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

Triebel-Lizorkin regularity and bi-Lipschitz maps: composition operator and inverse function regularity

We study the stability of Triebel-Lizorkin regularity of bounded functions and Lipschitz functions under bi-Lipschitz changes of variables and the regularity of the inverse function of a Triebel-Lizorkin bi-Lipschitz map in Lipschitz domains. To obtain the results we provide an equivalent norm for the Triebel-Lizorkin spaces with fractional smoothness in uniform domains in terms of the first-order difference of the last weak derivative available averaged on balls.Comment: 34 pages, 1 figur

Global smoothness of quasiconformal mappings in the Triebel-Lizorkin scale

We give sufficient conditions for quasiconformal mappings between simply connected Lipschitz domains to have H\"older, Sobolev and Triebel-Lizorkin regularity in terms of the regularity of the boundary of the domains and the regularity of the Beltrami coefficients of the mappings. The results can be understood as a counterpart for the Kellogg-Warchawski Theorem in the context of quasiconformal mappings.Comment: 45 pages, 3 figure

Characterization for stability in planar conductivities

We find a complete characterization for sets of uniformly strongly elliptic and isotropic conductivities with stable recovery in the L2 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 as conjectured by Alessandrini in his 2007 paper, giving explicit quantitative control for every pair of conductivities