Porosity in the context of hypergroups

S. M. Tabatabaie ; A. R. Bagheri Salec ; H. R. J. Allami

Eurasian mathematical journal, Tome 15 (2024) no. 1, pp. 75-90

Résumé

In this paper we show that the set of all elements $g\in L^p(\mathcal{H})$ for which $(|g|*|g|)(x)\infty$ for a center element $x\in B$, is $\sigma$-$c$-lower porous, where $p > 2$, $\mathcal{H}$ is a non-compact unimodular hypergroup and $B$ is some special symmetric compact neighborhood of the identity element. As an application, we give some new equivalent condition for the finiteness of a discrete Hermitian hypergroup. Moreover, we give some sufficient conditions for the set of all pairs $(f, g)$ in $L^p(\mathcal{H})\times L^q(\mathcal{H})$ for which for a center element $x\in B$, $(|f|*|g|)(x)\infty$, is a $\sigma$-$c$-lower porous, where $p, q > 1$ with $\frac1p+\frac1q1$. Also, we show that the complement of this set is spaceable in $L^p(\mathcal{H})\times L^q(\mathcal{H})$.