Boundedness of composition operators on Morrey spaces and weak Morrey spaces

Naoya Hatano,Yoshihiro Sawano,Isao Ishikawa,Masahiro Ikeda

doi:10.1186/s13660-021-02599-7

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 this 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.

Highlights

1 Introduction In this study, we investigate the boundedness of composition operators on Morrey spaces and weak Morrey spaces
The composition operator Cφ induced by a mapping φ is a linear operator defined by Cφf ≡ f ◦ φ, where f ◦ φ represents the function composition
The first aim of this study is to investigate a necessary and sufficient condition on the boundedness of the composition operator Cφ on Morrey spaces

Summary

Introduction

We investigate the boundedness of composition operators on Morrey spaces and weak Morrey spaces. The composition operator Cφ induced by φ : X → X is bounded on the Lebesgue space Lp(X, μ) if and only if there exists a constant K = K(φ) such that for all μ-measurable sets E in Rn, μ φ–1(E) ≤ K μ(E). The following theorem provides a sufficient condition on the boundedness of the composition operator Cφ on the Morrey space Mpq(Rn). The composition operator Cφ induced by φ : Rn → Rn is bounded on the Morrey space Mpq(Rn), if φ is a Lipschitz map that satisfies the volume estimate φ–1(E) ≤ K |E|,. Mpq Rn ⊂ WMpq Rn , WMpp Rn = WLp Rn. The following theorem provides a necessary and sufficient condition on the boundedness of the composition operator on weak Morrey spaces.

Mpq Mpq

DW k

According to