Boundedness of composition operators from Lorentz spaces to Orlicz spaces

TL;DR

This paper establishes necessary and sufficient conditions for the boundedness of composition operators from Lorentz to Orlicz spaces, providing new insights into operator behavior between these complex function spaces.

Contribution

It offers the first comprehensive criteria for boundedness of composition operators between Lorentz and Orlicz spaces, including counterexamples and detailed analysis.

Findings

Conditions for boundedness are fully characterized.

Counterexamples show unboundedness in certain Lebesgue space mappings.

Additional examples illustrate the applicability of the main results.

Abstract

The boundedness (continuity) of composition operators from some function space to another one is significant, though there are few results about this problem. Thus, in this study, we provide necessary and sufficient conditions on the boundedness of composition operators from Lorentz spaces to Orlicz spaces. We also give a counter example of a mapping which implies unboundedness of the composition operators from a Lebesgue space $L^p$ to another Lebesgue space $L^q$ with $p>q$. We emphasize that the measure spaces associated with the Lorentz space may be different from those associated with the Orlicz spaces. We give more examples and counterexamples of the composed mappings in the conditions satisfying our main results.