Lattice Homomorphisms between Sobolev Spaces

TL;DR

This paper characterizes vector lattice homomorphisms between Sobolev spaces, showing they can be represented as composition and multiplication operators, thus providing a structural understanding of such mappings.

Contribution

It establishes a representation theorem for lattice homomorphisms between Sobolev spaces as composition-multiplication operators, a novel structural insight.

Findings

Every lattice homomorphism is of the form $Tu(x)=u(h(x))g(x)$

Representation holds for quasi every or almost every $x$

Provides a structural classification of Sobolev space homomorphisms

Abstract

We show that every vector lattice homomorphism $T$ between Sobolev spaces can be represented by a composition and a multiplication, that is, $T$ is of the form $Tu(x)=u(h(x))g(x)$ for quasi every/almost every $x$ and all $u$.