Sobolev regularity of quasiconformal mappings on domains

Global smoothness of quasiconformal mappings in the Triebel-Lizorkin scale

“…Finally, the last result in this setting can be found in [Pra15b], which studies the case of integer values for s. The author showed that the parameterizations of the boundary of the domain being…”

“…To verify that this embedding holds we make use of several techniques, which include the approximation of the boundary of the domain by straight lines as Cruz and Tolsa introduced in [CT12] which allow us to swap the transform of the characteristic function of the domain at a given point by a sum of beta coefficients introduced by Dorronsoro in [Dor85]. We also use a recent expression of the kernel of the reflection obtained in [Pra15b] (see Section 6.5) and the techniques on chains of Whitney cubes introduced in [PT15,PS17].…”

“…In this section we prove Theorem 1.5. In Section 6.1 we outline its proof, which follows the steps of [Pra15b] by means of a classical Fredholm argument, reducing the proof to checking that I Ω ´µm pB m q Ω is invertible, and that the commutator rµ, B Ω s and the Beurling reflection χ Ω Bpχ Ω c Bpχ Ω ¨qq is compact (together with a family of related operators).…”

“…That is, we will reduce the proof to the compactness of the commutator. In our context, however, as it happens in [CMO13] and [Pra15b], we will have to deal with the compactness of operators like χ Ω B pχ Ω c B pχ Ω ¨qq as well.…”

“…The idea is to use Stoilow factorization: every µ-quasiconformal mapping g : Ω Ñ Ω can be written as h ˝f where f is the µ-quasiconformal principal mapping and h : f pΩq Ñ Ω is conformal (see Figure 1.1). Now, the first step is to show a particular case of the theorem for f instead of g, which was done in [Pra15b]. Then it remains to show the same for h. To do so first we need to understand the nature of f pΩq.…”