Beltrami equations with coefficient in the fractional Sobolev space $W^{\theta, \frac2{\theta}}$

TL;DR

This paper investigates the regularity of quasiconformal solutions to Beltrami equations with coefficients in fractional Sobolev spaces, establishing that the logarithm of the derivative belongs to the same fractional space under certain conditions.

Contribution

It proves that for Beltrami equations with coefficients in fractional Sobolev spaces, the logarithm of the derivative also resides in the same space when the smoothness parameter exceeds one-half.

Findings

equations with fractional Sobolev coefficients have solutions with logarithmic derivatives in the same space.

The method uses compactness of commutators involving fractional Laplacians and Sobolev symbols.

Results extend regularity theory for quasiconformal maps with fractional Sobolev coefficients.

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 $\mathbb{D}$, $\|\mu\|_\infty<1$ and belongs to the fractional Sobolev space $W^{\alpha, \frac2\alpha}(\mathbb{C})$. Our main result states that $$\log\partial_z\phi \in W^{\alpha, \frac2\alpha}(\mathbb{C})$$ whenever $\alpha>\frac12$. 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 $(-\Delta)^\frac\beta2$ and any symbol $b\in W^{\beta,\frac{n}\beta}(\mathbb{R}^n)$, provided that $1<p<\frac{n}{\beta}$.