A criterion for nonzero multiplier for Orlicz spaces of an affine group $\mathbb{R}_{+}\times \mathbb{R}$

Abstract

Let $\mathbb{A}=\mathbb{R}_{+}\times \mathbb{R}$ 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 $(L^{\Phi_1}(\mathbb{A}),L^{\Phi_2}(\mathbb{A}))$ and $(L^{\Psi_2}(\mathbb{A}),L^{\Psi_1}(\mathbb{A}))$ 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 $(L^{\Phi_1}(\mathbb{A}),L^{\Phi_2}(\mathbb{A}))$, i.e., for an affine group $\mathbb{A}$ only the spaces $M(L^{\Phi_1}(\mathbb{A}),L^{\Phi_2}(\mathbb{A}))$, with a concrete condition, are of any interest.