The space of multipliers and convolutors of Orlicz spaces on a locally compact group
Volume 219 / 2013
Abstract
Let $G$ be a locally compact group, let $(\varphi , \psi )$ be a complementary pair of Young functions, and let $L^\varphi (G)$ and $L^\psi (G)$ be the corresponding Orlicz spaces. Under some conditions on $\varphi $, we will show that for a Banach $L^\varphi (G)$-submodule $X$ of $L^\psi (G)$, the multiplier space $\mathop {\rm Hom}_{L^\varphi (G)}(L^\varphi (G),X^*)$ is a dual Banach space with predual $L^\varphi (G)\bullet X :=\overline {{\rm span}}\{ ux: u\in L^\varphi (G), x\in X \}$, where the closure is taken in the dual space of $\mathop {\rm Hom}_{L^\varphi (G)}(L^\varphi (G),X^*)$. We also prove that if $\varphi $ is a $\Delta _2$-regular $N$-function, then $\mathop {\rm Cv}_{{\varphi }}(G)$, the space of convolutors of $M^\varphi (G)$, is identified with the dual of a Banach algebra of functions on $G$ under pointwise multiplication.
