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+ε loc , for some ε > 0.1 K −1 ) −1 shows that for any p < 2K K+1 there exists a weakly Kquasiregular map in W 1,p loc which is not K-quasiregular. Later, Petermichl and Volberg [13] proved that the weaker assumption p ≥ 2K K+1 is already sufficient for weakly K-quasiregular maps to be K-quasiregular. (See the monograph [3] for a complete and detailed information).