A criterion for nonzero multipliers for Orlicz spaces of an affine group $\R_{+}\times \R$

Abstract

Let $\A=\R_{+}\times \R$ be an affine group with right Haar measure $d\mu$ and $\Phi_i$, $i=1,2$, be Young functions. We show that there exists an isometric isomorphism between the multiplier of the pair $(L^{\Phi_1}(\A),L^{\Phi_2}(\A))$ and $(L^{\Psi_2}(\A),L^{\Psi_1}(\A))$ where $\Psi_i$ are complementary pairs of $\Phi_i$, $i=1,2$, respectively. Moreover we show that under some conditions there is no nonzero multiplier for the pair $(L^{\Phi_1}(\A),L^{\Phi_2}(\A))$, i.e., for an affine group $\A$ only the spaces $M(L^{\Phi_1}(\A),L^{\Phi_2}(\A))$, with a concrete condition, are of any interest.