On the structure of variable exponent spaces

TL;DR

This paper surveys the lattice structure of variable exponent Lebesgue spaces and investigates properties of certain operators between these spaces, providing new insights into their structure and operator behavior.

Contribution

It offers a comprehensive survey of the lattice structure of variable exponent Lebesgue spaces and introduces new results on the disjoint strict singularity of inclusion operators.

Findings

New results on disjoint strict singularity of inclusion maps

Insights into the lattice structure of variable exponent spaces

Characterization of strictly singular operators in these spaces

Abstract

The first part of this paper surveys several results on the lattice structure of variable exponent Lebesgue function spaces (or Nakano spaces) $\lpv$. In the second part strictly singular and disjointly strictly singular operators between spaces $\lpv$ are studied. New results on the disjoint strict singularity of the inclusions $ L^{p(\cdot)}(\Omega) \hookrightarrow L^{q(\cdot)}(\Omega)$ are given.