Composition operators on Sobolev spaces and weighted moduli inequalities

TL;DR

This paper explores the relationship between composition operators on Sobolev spaces and weighted moduli inequalities, establishing equivalences and implications between these mathematical concepts.

Contribution

It demonstrates that weighted moduli inequalities induce composition operators on Sobolev spaces and vice versa, revealing a fundamental connection.

Findings

Weighted moduli inequalities lead to composition operators on Sobolev spaces.

Inverse composition operators imply weighted moduli inequalities.

The study establishes a bidirectional relationship between these concepts.

Abstract

In this paper we study connections between composition operators on Sobolev spaces and mappings defined by $p$-moduli inequalities ($p$-capacity inequalities). We prove that weighted moduli inequalities lead to composition operators on corresponding Sobolev spaces and inverse, composition operators on Sobolev spaces imply weighted moduli inequalities.