Affine mappings and multipliers for weighted Orlicz spaces over the affine group $\mathbb{R}_{+}\times \mathbb{R}$

Abstract

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

Keywords

• Affine group
• affine mapping
• multiplier
• weighted Orlicz space

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