Pointwise multipliers on weak Morrey spaces
type: Item
instance of: scholarly article
title: Pointwise multipliers on weak Morrey spaces
zbMATH Open document ID: 1475.42037
author: Ryota Kawasumi
author: Eiichi Nakai
published in: Analysis and Geometry in Metric Spaces
publication date: 2021-06-17T00:00:00Z
review text: The aim of this paper is to characterize the pointwise multipliers on generalized weak Morrey spaces with variable growth condition on spaces of homogeneous type. Let \(X=(X,d,\mu)\) be a space of homogeneous type. For an index \(p\in(0,\infty]\), a variable growth function \(\phi: X\times(0,\infty)\rightarrow (0,\infty)\) and a ball \(B\), let \[ \|f\|_{p,\phi,B,weak}=\frac1{\phi(B)}\bigg( \frac{\sup_{t>0}t^p\mu(\{x\in{B}:|f(x)|>t\})}{\mu(B)}\bigg)^{1/p}, \quad 0<p<\infty, \] \[ \|f\|_{\infty,\phi,B,weak} =\frac1{\phi(B)}\mathop{\mathrm{ess\,sup}}_{x\in{B}}|f(x)|, \quad {p}=\infty. \] The generalized weak Morrey spaces with variable growth condition, \(wL_{p,\phi}(X)\), are the sets of all functions \(f\) such that \[ \|f\|_{wL_{p,\phi}}=\sup_{B}\|f\|_{p,\phi,B,weak}<\infty, \] where the supremum is taken over all balls \(B\) in \(X\). Denote by \({\mathrm{PWM}}\big(wL_{p_1,\phi_1}(X),wL_{p_2,\phi_2}(X)\big)\) the set of all pointwise multipliers from \(wL_{p_1,\phi_1}(X)\) to \(wL_{p_2,\phi_2}(X)\). The authors give a sufficient condition for \[ {\mathrm{PWM}}\big(wL_{p_1,\phi_1}(X),wL_{p_2,\phi_2}(X)\big)=wL_{p_3,\phi_3}(X). \] A necessary condition for it when \(X={\mathbb{R}}^n\) is also given.
reviewed by: Pu Zhang
Mathematics Subject Classification ID: 42B35
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 7360407
zbMATH Keywords: Morrey space
zbMATH Keywords: weak Morrey space
zbMATH Keywords: pointwise multiplier
MaRDI profile type: Publication
full work available at URL: agms-2020-0119
OpenAlex ID: W163097270
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DOI: 10.1515/AGMS-2020-0119
description: scientific article; zbMATH DE number 7360407