Pointwise Multipliers on Weak Morrey Spaces

Ryota Kawasumi,Eiichi Nakai

doi:10.1515/agms-2020-0119

Abstract

Abstract We consider generalized weak Morrey spaces with variable growth condition on spaces of homogeneous type and characterize the pointwise multipliers from a generalized weak Morrey space to another one. The set of all pointwise multipliers from a weak Lebesgue space to another one is also a weak Lebesgue space. However, we point out that the weak Morrey spaces do not always have this property just as the Morrey spaces not always.

Highlights

Let Ω = (Ω, μ) be a complete σ- nite measure space
We say that a function g ∈ L (Ω) is a pointwise multiplier from E to E, if the pointwise multiplication fg is in E for any f ∈ E
We denote by PWM(E, E ) the set of all pointwise multipliers from E to E

Summary

Introduction

To establish the characterization of pointwise multipliers on them, we rst prove a generalized Hölder’s inequality for the generalized weak Morrey spaces. To characterize the pointwise multipliers, we use the fact that all pointwise multipliers on the generalized weak Morrey spaces are bounded operators. This fact follows from Theorem 1.1 and Corollary 1.2 below. If E and E are complete quasi-normed spaces with the lattice property (1.4), all g ∈ PWM(E , E ) are bounded operators. We will show weak Morrey spaces are complete quasi-normed spaces with the lattice property (1.4). We need to investigate the properties of the quasi-norm on the weak Morrey space in the proofs. If f ≤ Cg, we write f g or g f ; and if f g f , we write f ∼ g

Morrey and weak Morrey spaces on spaces of homogeneous type

Main results