Dilation-commuting operators on power-weighted Orlicz classes

TL;DR

This paper investigates conditions under which dilation-commuting operators act boundedly on power-weighted Orlicz classes, establishing equivalences between operator boundedness and modular inequalities.

Contribution

It provides necessary and sufficient conditions for operators to be bounded on weighted Orlicz classes, linking modular inequalities with operator boundedness.

Findings

Derived conditions for operator boundedness on weighted Orlicz classes.

Established equivalence between modular inequalities and boundedness.

Identified specific criteria involving $\

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_+}} \Phi_{i} \left( k|(Tf)(t)| \right) t^{\gamma}dt < \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_{\Phi_2,t^{\gamma}}(\mathbb{R_+}) \rightarrow L_{\Phi_1,t^{\gamma}}(\mathbb{R_+}), \tag{I} \end{equation} and \begin{equation}\label{modularA} \int_{\mathbb{R_+}} \Phi_1 \left( |(Tf)(t)| \right)t^{\gamma}dt \leq K \int_{\mathbb{R_+}} \Phi_2 \left( K|f(s)| \right)s^{\gamma}ds, \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.