  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
$1<p<\frac{n}{\beta}$.
