Beltrami equations with coefficient in the fractional Sobolev space $W^{\theta ,\frac{2}{\theta }}$

Abstract

In this paper, we look at quasiconformal solutions $\varphi :\mathbb{C}\to \mathbb{C}$ of Beltrami equations $${\partial }_{\overline{z}}\varphi (z)=\mu (z){\partial }_z\varphi (z).$$ where $\mu \in L^{\infty }(\mathbb{C})$ is compactly supported on $\mathbb{D}$, $\Vert \mu {\Vert }_{\infty }<1$ and belongs to the fractional Sobolev space $W^{\alpha ,\frac{2}{\alpha }}(\mathbb{C})$. Our main result states that $$\log {\partial }_z\varphi \in W^{\alpha ,\frac{2}{\alpha }}(\mathbb{C})$$ whenever $\alpha >\frac{1}{2}$. Our method relies on an $n$-dimensional result, which asserts the compactness of the commutator $$[b,(-\Delta {)}^{\frac{\beta }{2}}]:L^{\frac{np}{n-\beta p}}({\mathbb{R}}^n)\to L^p({\mathbb{R}}^n)$$ between the fractional laplacian $(-\Delta {)}^{\frac{\beta }{2}}$ and any symbol $b\in W^{\beta ,\frac{n}{\beta }}({\mathbb{R}}^n)$, provided that $1<p<\frac{n}{\beta }$.