Orlicz algebras associated to a Banach function space

Year 2024, , 191 - 200, 29.02.2024

Chung-chuan Chen , Alireza Bagheri Salec , Seyed Mohammad Tabatabaie

https://doi.org/10.15672/hujms.1018098

Abstract

In this paper, we study the spaces ${\mathcal X}^\Phi$ as Banach algebras, where $\mathcal X$ is a quasi-Banach function space and $\Phi$ is a Young function, and extend some well-known facts regarding Lebesgue and Orlicz spaces on this new structure. Also, for each $p\geq 1$, we give some necessary condition for the space $\mathcal{X}^p$ to be a Banach algebra under the pointwise product.

Keywords

convolution algebra, Orlicz space, Young function, PCS-space, Banach algebra, locally compact group, spaceability, pointwise product

References

• [1] A.R. Bagheri Salec and S.M. Tabatabaie, Some necessary and sufficient conditions for convolution weighted Orlicz algebras, Bull. Iranian Math. Soc. 48, 2509-2520, 2022.
• [2] A.R. Bagheri Salec, S. Ivkovic and S.M. Tabatabaie, Spaceability on some classes of Banach spaces, Math. Ineq. Appl. 25(3), 659-672, 2022.
• [3] A.R. Bagheri Salec, V. Kumar and S.M. Tabatabaie, Convolution properties of Orlicz spaces on hypergroups, Proc. Amer. Math. Soc. 150(4), 1685-1696, 2022.
• [4] L. Bernal-González and M.O. Cabrera, Spaceability of strict order integrability, J. Math. Anal. Appl. 385, 303-309, 2012.
• [5] R. del Campo, A. Fernández, F. Mayoral and F. Naranjo, Orlicz spaces associated to a quasi-Banach function space. Applications to vector measures and interpolation, Collect. Math. 72, 481-499, 2021.
• [6] S. Glab and F. Strobin, Dichotomies for $L^p$ spaces, J. Math. Anal. Appl. 368, 382-390, 2010.
• [7] S. Glab and F. Strobin, Spaceability of sets in $L^p\times L^q$ and $C_0\times C_0$, J. Math. Anal. Appl. 440, 451-465, 2016.
• [8] H. Hudzik, Orlicz spaces of essentially bounded functions and Banach-Orlicz algebras, Arch. Math. 44, 535-538, 1985.
• [9] H. Hudzik, A. Kamiska and J. Musielak, On some Banach algebras given by a modular, in: Alfred Haar Memorial Conference, Budapest, Colloquia Mathematica Societatis J anos Bolyai (North Holland, Amsterdam), 49, 445-463, 1985.
• [10] P. Jain, L.E. Persson and P. Upreti, Inequalities and properties of some generalized Orlicz classes and spaces, Acta Math. Hungar. 117, 161-174, 2007.
• [11] V. Kumar, R. Sarma and N. Shravan Kumar, Orlicz algebras on homogeneous spaces of compact groups and their abstract linear representations, Mediterr. J. Math. 15(4), 186, 2018.
• [12] V. Kumar, R. Sarma and N. Shravan Kumar, Orlicz spaces on hypergroups, Publ. Math. Debrecen 94(1-2), 31-47, 2019.
• [13] L. Maligranda and L.E. Persson, Generalized duality of some Banach function spaces, Proc. Konin. Nederlands Akad. Wet. 92, 323-338, 1989.
• [14] S. Okada, W. Ricker, and E.A. Sánchez-Pérez, Optimal domain and integral extension of operators acting in functions spaces, Operator Theory: Advances and Applications, vol. 180, Birkhäuser, Verlag, Besel, 2008.
• [15] A. Osançlıol and S. Öztop, Weighted Orlicz algebras on locally compact groups, J. Aust. Math. Soc. 99, 399-414, 2015.
• [16] S. Öztop and S.M. Tabatabaie, Weighted Orlicz algebras on hypergroups, FILOMAT, 34(7), 2131-2139, 2020.
• [17] L. E. Persson, Some elementary inequalities in connection with Xp-spaces, in: Constructive Theory of Functions, 367-376, 1988.
• [18] L. E. Persson, On some generalized Orlicz classes and spaces, Research Report 1988-3, Department of Mathematics, Lulea University of Technology, 1988.
• [19] T.S. Quek and L.Y.H. Yap, Sharpness of Young’s inequality for convolution, Math. Scand. 53, 221-237, 1983.
• [20] M. Rajagopalan, $L^p$-conjecture for locally compact groups-I, Trans. Amer. Math. Soc. 125, 216-222, 1966
• [21] M. Rajagopalan and W Zelazko, $L^p$-conjecture for solvable locally compact groups, J. Indian Math. Soc. 29, 87-93, 1965.
• [22] M.M. Rao and Z.D. Ren, Theory of Orlicz Spaces, Marcel Dekker, New York, 1991.
• [23] S. Saeki, The $L^p$-conjecture and Young’s inequality, Illinois Journal of Mathematics, 34(3), 614-627, 1990.
• [24] S.M. Tabatabaie and A.R. Bagheri Salec, Convolution of two weighted Orlicz spaces on hypergroups, Revista Colombiana de Matemáticas, 54(2), 117-128, 2020.
• [25] S.M. Tabatabaie, A.R. Bagheri Salec and M. Zare Sanjari, A note on Orlicz algebras, Oper. Matrices, 14(1), 139-144, 2020.
• [26] S.M. Tabatabaie and A.R. Bagheri Salec, On The inclusion of X spaces, Mathematica Bohemica, 148(1), 65-72, 2023.