Interpolation theorem in generalized Orlicz spaces

Author
Alireza Bagheri Salec
University of Qom
Abstract
Suppose that χ is a Banach function space in the measure space (Ω,A,μ), Φ is an N-function and w is a weight function on Ω. In this article, the Theorem of interpolation in Lebesgue spaces to the weighted generalized Orlicz spaces χΦw corresponding to the Banach function space χ is explained. This Theorem is a generalization of corresponding Theorem in weighted Labesgue and Orlicz spaces. Also, results are presented about the relationship between generalized Orlicz spaces and intermediate spaces, as well as Caldern's product space in weighted generalized Orlicz spaces χΦw.











Keywords
Riesz-Thorian interpolation theorem
Weighted Orlicz space
Banach function space
Intermediate space
Caldero'n product
[1] A. R. Bagheri Salec and S. M. Tabatabaie, An extention of the interpolation theorem, Mathematical Analysis and Convex Optimization, 2 (2021) No. 1, 63-69.

[2] A. R. Bagheri Salec, V. Kumar and S. M. Tabatabaie, Convolution properties of Orlicz spaces on hypergroups, Proc. Amer. Math. Soc., 150(4) (2022), 1685–1696.

[3] J. Bergh and J. Löfstr"om, Interpolation Spaces; An Introduction, Springer-Verlag, Berlin, 1976.

[4] A. P. Caldero'n, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190.

[5] R. del Campo, A. Fernández, F. Mayoral and F. Naranjo, Complex interpolation of Orlicz spaces with respect to a vector measure, Math. Nachr. 287 No.1 (2014). 23-31.

[6] 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. (2020). https://doi.org/10.1007/s13348-020-00295-1.

[7] R. del Campo, A. Fernández, F. Mayoral, F. Naranjo and E. A. Sánchez-Pérez, When and where the Orlicz and Luxemburg (quasi-) norms are equivalent?, J. Math. Anal. Appl. 491 No.1 (2020), 124302, 18, DOI 10.1016/j.jmaa.2020.124302. MR4110269

[8] M. Faragallah, Interpolation of weighted Orlicz spaces, Applied Mathematics and Computation 145 (2003), 613–622

[9] J.E. Gilbert, Interpolation between weighted L^p-spaces, Ark. Math. 10 (1972), 235–249.

[10] C. Goulaouic, Prolongements de foncteurs d’interpolation et applications, Ann. Inst. Fourier, 18 (1968), 1-98.

[11] J. Gustavsson, J. Peetre, Interpolation of Orlicz spaces, Stud. Math. 60 (1977) 33–59.

[12] H.P. Heining, D. Vaughan, Interpolation in Orlicz spaces involving weights, Journal of Mathematical Analysis and Applications 64 (1978), 79-95.

[13] L.V. Kantorovich and G.P. Akilov, Functional Analysis, Pergamon press. 1982.

[14] N. Kalton and M. Mitrea, Stability results on interpolation scales of quasi-Banach spaces and applications, Trans. Amer. Math. Soc. 350 10, (1998), 3903–3922.

[15] Alexei Yu. Karlovich and Lech Maligranda, On the Interpolation Constant for Orlicz Spaces, Proc. Amer. Math. Soc. 129(9), ( 2001), pp. 2727-2739.

[16] M.A. Krasnoselskii ̆,Ya.B. Rutitskii ̆, Convex functions and Orlicz spaces, 1958 (Russian).

[17] A. Lunardi, Interpolation Theory, Scuola Normale Superiore Pisa, 2018.

[18] S. Okada, W. J. Ricker E. A. Sánchez Pérez, Optimal Domain and Integral Extension of Operators, Operator Theory: Advances and Applications, 180, 2007.

[19] A. Osançlıol and S. Öztop, Weighted Orlicz algebras on locally compact groups, J. Aust. Math. Soc. 99 (2015), 399-414.

[20] J. Peetre, On interpolation of L^p-spaces with weight function, Acta Sci. Math. 28 (1967), 61–69.

[21] P. Jain, L.E. Persson, and P. Upreti, Inequalities and properties of some generalized Orlicz classes and spaces, Acta Math. Hungar. 117(no. 1-2) (2007), 161–174.

[22] M.M. Rao, Interpolation, ergodicity, and martingales, J. Math. Mech., 16(6), (1966), 543-567.

[23] M.M. Rao and Z.D. Ren, Theory of Orlicz Spaces, Marcel Dekker, New York, 1991.

[24] M. M. Rao and Z. D. Ren, Applications of Orlicz Spaces, Marcel Dekker, New York, 2002.

[25] Ya. B. Rutitskii, Scales of Orlicz spaces and interpolation theorems, Dokl. Akad. Nauk SSSR, 149(1), (1963), 32–35.

[26] S. M. Tabatabaie and A. R. Bagheri Salec, Convolution of two weighted Orlicz spaces on hypergroups, Revista Colombiana de Matemáticas Revista, 54 (2020) 117-128.

[27] S. M. Tabatabaie, A. R. Bagheri Salec and M. Zare Sanjari, A note on Orlicz algebras, Oper. Matrices, 14(1) (2020) 139–144.

[28] S. M. Tabatabaie, A. R. Bagheri Salec and M. Zare Sanjari, Remarks on weighted Orlicz spaces on locally compact groups, Math. Ineq. Appl. 23(3) (2020) 1015–1025.