Composition operators on Sobolev spaces, $Q$-mappings and weighted Sobolev inequalities

TL;DR

This paper explores the relationship between composition operators, $Q$-mappings, and weighted Sobolev inequalities, establishing measure distortion properties and deriving inequalities with Jacobian-based weights.

Contribution

It introduces new connections between bounded composition operators on Sobolev spaces and $Q$-mappings, leading to novel weighted Sobolev inequalities involving Jacobians.

Findings

Establishes measure distortion properties of $Q$-homeomorphisms.

Derives weighted Sobolev inequalities with Jacobian weights.

Links composition operators on Sobolev spaces to $Q$-mapping theory.

Abstract

In this paper we give connections between mappings which generate bounded composition operators on Sobolev spaces and $Q$-mappings. On this base we obtain measure distortion properties $Q$-homeomorphisms. Using the composition operators on Sobolev spaces we obtain weighted Sobolev inequalities with special weights which are Jacobians of $Q$-mappings.