AUT Journal of Mathematics and Computing

AUT Journal of Mathematics and Computing

Algebraic property of weighted Lebesgue spaces on a class of hypergroups

Document Type : Original Article

Authors
Seyyed Mohammad Tabatabaie 1
AliReza Bagheri Salec 1
Naeem Mankhi Oudah AlBehadili 1, 2
1 Department of Mathematics, University of Qom, Qom, Iran
2 Accounting Technologies Department, Technical Institute of AMARA, Southern Technical University, Basrah, Iraq
10.22060/ajmc.2025.23979.1345
Abstract
In this paper, in the context of an important class of locally compact hypergroups, namely $\mathcal{K}_{\rho}$, which were introduced by Dunkl and Ramirez, we find some sufficient conditions on a weight $(w_n)_{n=0}^\infty\subseteq (0,\infty)$ such that the set
\begin{equation*}
\left\{\big((f_k)_k,(g_k)_k\big)\in L^p(\mathcal{K}_{\rho})\times L^q(\mathcal{K}_{\rho}):\sum_{k=1}^\infty |f_kg_k|w_k^2\,\rho ^{k-1}<\infty\right\}
\end{equation*}
is a $\sigma$-$c$-lower porous set.
Keywords
Locally compact hypergroups
$\sigma$-lower porous sets
Convolution
Weighted Lebesgue space
Convolution algebra
Subjects
[1]     I. Akbarbaglu and S. Maghsoudi, On certain porous sets in the Orlicz space of a locally compact group, Colloq. Math., 129 (2012), pp. 99–111.
[2]     I. Akbarbaglu, S. Maghsoudi, and J. B. Seoane-Sepulveda´ , Porosity and the ℓp-conjecture for semigroups, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM, 110 (2016), pp. 7–16.
[3]     A. R. Bagheri Salec, V. Kumar, and S. M. Tabatabaie, Convolution properties of Orlicz spaces on hypergroups, Proc. Amer. Math. Soc., 150 (2022), pp. 1685–1696.
[4]     W. R. Bloom and H. Heyer, Harmonic analysis of probability measures on hypergroups, vol. 20 of De Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 1995.
[5]     C. F. Dunkl and D. E. Ramirez, A family of countably compact P∗-hypergroups, Trans. Amer. Math. Soc., 202 (1975), pp. 339–356.
[6]     S. Gl a¸b and F. Strobin, Porosity and the Lp-conjecture, Arch. Math. (Basel), 95 (2010), pp. 583–592.
[7]     R. I. Jewett, Spaces with an abstract convolution of measures, Advances in Math., 18 (1975), pp. 1–101.
[8]     S. M. Tabatabaie and A. Bagheri Salec, Convolution of two weighted Orlicz spaces on hypergroups, Rev. Colombiana Mat., 54 (2020), pp. 117–128.
[9]     S. M. Tabatabaie, A. R. Bagheri Salec, and H. R. J. Allami, Porosity in the context of hypergroups, Eurasian Math. J., 15 (2024), pp. 75–90.
[10]  L. Zaj´ıcekˇ , Porosity and σ-porosity, Real Anal. Exchange, 13 (1987/88), pp. 314–350.
[11]  , On σ-porous sets in abstract spaces, Abstr. Appl. Anal., (2005), pp. 509–534.
AUT Journal of Mathematics and Computing
Volume 7, Issue 4
Autumn 2026
Pages 385-390