Affine mappings and multipliers for weighted Orlicz spaces over an affine group $\R_{+}\times \R$

Abstract

Let $\A=\R_{+}\times \R$ be the affine group with right Haar measure $d\mu$, $\omega$ be a weight function on $\A$ and $\Phi$ be a Young function. We characterize the affine continuous mappings on the subsets of $L^\Phi(\A,\omega)$. Moreover we show that there exists an isometric isomorphism between the multiplier for the pair $(L^{1}(\A,\omega),L^{\Phi}(\A,\omega))$ and the space of bounded measures $M(\A,\omega)$.