A version of Schwarz's lemma for mappings with weighted bounded distortion

Abstract

We consider the class of mappings generalizing qusiregular mappings. Every mapping from this class is de ned in a domain of Euclidean n-space and possesses the following properties: it is open, continuous, and discrete, it belongs locally to the Sobolev classW 1 q , it has nite distortion and nonnegative Jacobian, and its function of weighted (p, q)-distortion is integrable to a certian power depending on p and q, where n − 1 < q 6 p < ∞. We obtain an analog of Schwarz's lemma for such mappings provided that p > n. The technique used is based on the spherical symmetrization procedure and the notion of Gr otzsch condenser.