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We prove that a family of quasiregular mappings of a domain $\Omega$ whichare uniformly bounded in $L^p$ for some $p>0$ form a normal family. From thiswe show how an elliptic estimate on a functional differences implies alldirectional derivatives, and thus the complex gradient to be quasiregular.Consequently the function enjoys much higher regularity than aprioriassumptions suggest.

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