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Orlicz Space on Groupoids (2501.05895v2)
Published 10 Jan 2025 in math.FA
Abstract: Let $G$ be a locally compact second countable groupoid with a fixed Haar system $\lambda={\lambda{u}}_{u\in G{0}}$ and $(\Phi,\Psi)$ be a complementary pair of $N$-functions satisfying $\Delta_{2}$-condition. In this article, we introduce the continuous field of Orlicz space $(L{\Phi}{0},\Delta{1})$ and provide a sufficient condition for the space of continuous sections vanishing at infinity, denoted $E{\Phi}_{0}$, to be an Banach algebra under a suitable convolution. Further, the condition for a closed $C_{b}(G{0})$-submodule $I$ of $E{\Phi}_{0}$ to be a left ideal is established. Moreover, we provide a groupoid analogue of the characterization of the space of convolutors of Morse-Transue space for locally compact groups.