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.
Categories: math.FA