Beltrami equations with coefficient in the fractional Sobolev space $W^{θ, \frac2θ}$

In this paper, we look at quasiconformal solutions $ϕ:\mathbb{C}\to\mathbb{C}$ of Beltrami equations $\partial_{\overline{z}} ϕ(z)=μ(z)\,\partial_z ϕ(z).$ where $μ\in L^\infty(\mathbb{C})$ is compactly supported on $\mathbb{D}$, $\|μ\|_\infty<1$ and belongs to the fractional Sobolev space $W^{α, \frac2α}(\mathbb{C})$. Our main result states that $\log\partial_zϕ\in W^{α, \frac2α}(\mathbb{C})$ whenever $α>\frac12$. Our method relies on an $n$-dimensional result, which asserts the compactness of the commutator $[b,(-Δ)^\fracβ{2}]:L^\frac{np}{n-βp}(\mathbb{R}^n)\to L^p(\mathbb{R}^n)$ between the fractional laplacian $(-Δ)^\frac\beta2$ and any symbol $b\in W^{β,\frac{n}β}(\mathbb{R}^n)$, provided that $1<p<\frac{n}β$.

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