Geometric theory of composition operators on Sobolev spaces

TL;DR

This paper develops a geometric framework for understanding composition operators on Sobolev spaces, focusing on topological mappings that induce bounded embeddings, with applications to various mathematical and physical theories.

Contribution

It introduces a geometric approach to composition operators on Sobolev spaces, extending quasiconformal mapping theory and linking it to embedding theorems and spectral analysis.

Findings

Defines topological mappings generating bounded Sobolev embeddings

Generalizes quasiconformal mapping theory

Connects composition operators to applications in mechanics and spectral theory

Abstract

In this paper, we present the basic concepts of the geometric theory of composition operators on Sobolev spaces. The main objects of the theory are topological mappings which generate bounded embedding operators on Sobolev spaces by the composition rule. This theory is in some sense a "generalization" of the theory of quasiconformal mappings, but the theory of composition operators is oriented to its applications to the Sobolev embedding theorems, the spectral theory of elliptic operators and continuum mechanics problems.