Porosity and products of Orlicz spaces

Abstract

Let \((\Omega , \Sigma , \mu )\) be a measure space and let \(\varphi _1, \ldots , \varphi _n\) and \(\varphi \) be Young functions. In this paper, we, among other things, prove that the set \(E=\{(f_1, \ldots ,f_n)\in M^{\varphi _1}\times \cdots \times M^{\varphi _n}:\, N_\varphi (f_1\cdots f_n)<\infty \}\) is a \(\sigma \)-\(c\)-lower porous set in \(M^{\varphi _1}\times \cdots \times M^{\varphi _n}\), under mild restrictions on the Young functions \(\varphi _1, \ldots , \varphi _n\) and \(\varphi \). This generalizes a recent result due to Gl a b and Strobin (J Math Anal Appl 368:382–390, 2010) to more general setting of Orlicz spaces. As an application of our results, we recover a sufficient and necessary condition for Orlicz spaces to be closed under the pointwise multiplication due to Hudzik (Arch Math 44:535–538, 1985) and Arens et al. (J Math Anal Appl 177:386–411, 1993).