Lorentz spaces as $L_1$-modules and multipliers

Abstract

Let $w$ be a weight function on a locally compact group $G$, so that the weighted $L_{1}$-space $L_{1}(w)$ forms a Banach algebra under convolution. Suppose that $G$ acts on a locally compact space $\Omega$, and that $B$ is a Banach space consisting of Radon measures on $\Omega$ which is also a left Banach $L_{1}(w)$-module. Under certain conditions on $B$, we shall characterize those bounded linear operators $T:L_{1}(w)\arrow B$ which satisfy $T(f*g)=f*T(g)$. We shall also show that there are numerous examples of Lorentz spaces which form left Banach $L_{1}(w)$-modules with respect to appropriate weight functions.