Disjointly strictly singular inclusions between variable Lebesgue spaces

TL;DR

This paper characterizes disjointly strictly singular inclusions between variable Lebesgue spaces on finite measures, providing criteria based on exponents and establishing equivalences with L-weak compactness, while also analyzing infinite measure cases.

Contribution

It offers a complete characterization of disjointly strictly singular inclusions in variable Lebesgue spaces and links them to L-weak compactness, with no restrictions on exponents.

Findings

Characterization of disjointly strictly singular inclusions on finite measure.

Equivalence between L-weak compactness and disjoint strict singularity.

Inclusions on infinite measure are not disjointly strictly singular.

Abstract

Disjointly strictly singular inclusions between variable Lebesgue spaces $L^{p(\cdot)}(\mu)$ on finite measure are characterized. Suitable criteria in terms of the (bounded or unbounded) exponents are given. It is proved the equivalence of $L$-weak compactness (also called almost compactness) and disjoint strict singularity for variable Lebesgue space inclusions. For infinite measure any inclusion $L^{p(\cdot)}(\mu) \hookrightarrow L^{q(\cdot)}(\mu)$ is not disjointly strictly singular. No restrictions on the exponent are imposed.