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

Abstract:

In this paper, we look at quasiconformal solutions $\phi :\mathbb {C}\to \mathbb {C}$ of Beltrami equations \[ \partial _{\overline {z}} \phi (z)=\mu (z) \partial _z \phi (z), \] where $\mu \in L^\infty (\mathbb {C})$ is compactly supported on $\mathbb {D}$, and $\|\mu \|_\infty <1$ and belongs to the fractional Sobolev space $W^{\alpha , \frac 2\alpha }(\mathbb {C})$. Our main result states that \[ \log \partial _z\phi \in W^{\alpha , \frac 2\alpha }(\mathbb {C})\] whenever $\alpha \ge \frac 12$. 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 $1<p<\frac {n}{\beta }$.