Quasiregular families bounded in $L^p$ and elliptic estimates

We prove that a family of quasiregular mappings of a domain $\mathrm{\Omega }$ which are uniformly bounded in $L^p$ for some $p>0$ form a normal family. From this we show how an elliptic estimate on a functional differences implies all directional derivatives, and thus the complex gradient to be quasiregular. Consequently the function enjoys much higher regularity than apriori assumptions suggest.