Quasiregular Families Bounded in L^p and Elliptic Estimates

We prove that a family F of quasiregular mappings of a domain Ω 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 difference implies all directional derivatives, and thus the complex gradient to be quasiregular. Consequently the function enjoys much higher regularity than apriori assumptions suggest. We present applications in the theory of Beltrami equations and their nonlinear counterparts.