Dilation-commuting operators on power-weighted Orlicz classes

Abstract

Let ${\Phi }_1$ and ${\Phi }_2$ be nondecreasing functions from ${\mathbb{R}}_{+}=(0,\infty )$ onto itself. For $i=1,2$ and $\gamma \in \mathbb{R}$, define the Orlicz class $L_{{\Phi }_i}({\mathbb{R}}_{+})$ to be the set of Lebesgue-measurable functions $f$ on ${\mathbb{R}}_{+}$ such that \begin{equation*} \int_{\mathbb{R_+}} Φ_{i} \left( k|(Tf)(t)| \right) t^γdt < \infty \end{equation*} for some $k>0$. Our goal in this paper is to find conditions on ${\Phi }_1$, ${\Phi }_2$, $\gamma$ and an operator $T$ so that the assertions \begin{equation} T : L_{Φ_2,t^γ}(\mathbb{R_+}) \rightarrow L_{Φ_1,t^γ}(\mathbb{R_+}), \tag{I} \end{equation} and \begin{equation}\label{modularA} \int_{\mathbb{R_+}} Φ_1 \left( |(Tf)(t)| \right)t^γdt \leq K \int_{\mathbb{R_+}} Φ_2 \left( K|f(s)| \right)s^γds, \tag{M} \end{equation} in which $K>0$ is independent of $f$, say, simple on ${\mathbb{R}}_{+}$, are equivalent and to then find necessary and sufficient conditions in order that (\ref{modularA}) holds.