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 defined in a domain of Euclidean $n$-space and possesses the following properties: it is open, continuous, and discrete, it belongs locally to the Sobolev class $W^{1}_{q}$, it has finite distortion and nonnegative Jacobian, and its function of weighted $(p,q)$-distortion is integrable to a certian power depending on $p$ and $q$, where $n-1<q\leqslant p<\infty$. We obtain an analog of Schwarz's lemma for such mappings provided that $p\geqslant n$. The technique used is based on the spherical symmetrization procedure and the notion of Gr\"{o}tzsch condenser.