Boundedness of composition operators on Morrey spaces and weak Morrey spaces

TL;DR

This paper characterizes when composition operators are bounded on Morrey and weak Morrey spaces, revealing new phenomena and linking boundedness to bi-Lipschitz continuity and volume estimates.

Contribution

It provides necessary and sufficient conditions for the boundedness of composition operators on Morrey and weak Morrey spaces, including new insights into their behavior.

Findings

Boundedness characterized by bi-Lipschitz continuity.

Introduction of a new phenomenon not seen in Lebesgue spaces.

Boundedness criteria established for weak Morrey spaces.

Abstract

In this study, we investigate the boundedness of composition operators acting on Morrey spaces and weak Morrey spaces. The primary aim of this study is to investigate a necessary and sufficient condition on the boundedness of the composition operator induced by a diffeomorphism on Morrey spaces. In particular, detailed information is derived from the boundedness, i.e., the bi-Lipschitz continuity of the mapping that induces the composition operator follows from the continuity of the composition mapping. The idea of the proof is to determine the Morrey norm of the characteristic functions, and employ a specific function composed of a characteristic function. As the specific function belongs to Morrey spaces but not to Lebesgue spaces, the result reveals a new phenomenon not observed in Lebesgue spaces. Subsequently, we prove the boundedness of the composition operator induced by a mapping that satisfies a suitable volume estimate on general weak-type spaces generated by normed spaces. As a corollary, a necessary and sufficient condition for the boundedness of the composition operator on weak Morrey spaces is provided.