On the composition operator with variable integrability

TL;DR

This paper investigates the boundedness and compactness of composition operators on Lebesgue spaces with variable exponents, providing necessary and sufficient conditions and exploring weak compactness using variable change techniques.

Contribution

It introduces criteria for boundedness and compactness of composition operators on variable exponent Lebesgue spaces, advancing understanding in non-standard function space analysis.

Findings

Characterized boundedness conditions for composition operators.

Established criteria for compactness of these operators.

Demonstrated the application of variable change in studying weak compactness.

Abstract

In this note, we consider a class of composition operators on Lebesgue spaces with variable exponents over metric measure spaces. Taking advantage of the compatibility between the metric-measurable structure and the regularity properties of the variable exponent, we provide necessary and sufficient conditions for this class of operators to be bounded and compact, respectively. In addition, we show the usefulness of the variable change to study weak compactness properties in the framework of non-standard spaces.