Lattice Lipschitz superposition operators on Banach function spaces

TL;DR

This paper characterizes lattice Lipschitz operators between Banach function spaces, linking them to measurable functions and multiplication operators, thus enhancing understanding of their structure and relationships to tensor products.

Contribution

It provides a characterization of lattice Lipschitz operators as pointwise compositions with measurable functions within Banach function spaces, extending classical multiplication operator theory.

Findings

Lattice Lipschitz operators can be represented by measurable functions in certain conditions.

These operators are linked to multiplication operators and belong to specific Bochner spaces.

The results facilitate understanding of the structure and relationships of these operators.

Abstract

We analyse and characterise the notion of lattice Lipschitz operator (a class of superposition operators, diagonal Lipschitz maps) when defined between Banach function spaces. After showing some general results, we restrict our attention to the case of those Lipschitz operators which are representable by pointwise composition with a strongly measurable function. Mimicking the classical definition and characterizations of (linear) multiplication operators between Banach function spaces, we show that under certain conditions the requirement for a diagonal Lipschitz operator to be well-defined between two such spaces $X(\mu)$ and $Y(\mu)$ is that it can be represented by a strongly measurable function which belongs to the Bochner space $\mathcal M(X,Y) \big(\mu, Lip_0(\mathbb R) \big). $ Here, $\mathcal M(X,Y) $ is the space of multiplication operators between $X(\mu)$ and $Y(\mu),$ and $Lip_0(\mathbb R)$ is the space of real-valued Lipschitz maps with real variable that are equal to $0$ in $0. $ This opens the door to a better understanding of these maps, as well as finding the relation of these operators to some normed tensor products and other classes of maps.