Distributional solutions of the Beltrami equation

We study the distributional solutions to the (generalized) Beltrami equation under Sobolev assumptions on the Beltrami coefficients. In this setting, we prove that these distributional solutions are true quasiregular maps and they are smoother than expected, that is, they have second order derivatives in $L^{1+\varepsilon}_{loc}$, for some $\varepsilon >0$.