Pointwise multipliers on weak Morrey spaces
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.
scientific article; zbMATH DE number 7360407
scientific article