Quasiregular Families Bounded in \(L^p\) and Elliptic Estimates

  • Aimo Hinkkanen
  • Gaven MartinEmail author


We prove that a family \({{\mathcal {F}}}\) of quasiregular mappings of a domain \(\Omega \) 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 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.


Elliptic estimate Quasiregular mappings Normal family Nonlinear Beltrami equations 

Mathematics Subject Classification

Primary 30C62 Secondary 35J60 



The funding was provided by Marsden Fund (Grant No. MU 2016).


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Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.University of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Massey UniversityAucklandNew Zealand

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