×

Generalization of Orlicz spaces. (English) Zbl 1487.46028

Summary: Let \(\Phi\) be a Young function and \(\mathcal{L}^\Phi(\mu)\) be an Orlicz space. For \(1\le p<\infty\), we consider Orlicz spaces such as \(\mathcal{L}^\Phi(\mu)\) such that \(\Phi\) satisfies (aInc)\(_p\) (almost increasing classes); we denote this space by \(\mathcal{L}_p^\Phi(\mu)\) (or \(\mathbb{L}_p^\Phi(\mu))\) that we call them generalized Orlicz spaces. Some basic results related to these spaces are given and by introducing Orlicz \(p\)-norm on, we show, that they are Banach space. We introduce new versions of Young functions that we call them \((p,q)\)-complementary and \((p,q)\)-complementary normalized Young pairs, by these definitions; duals of Orlicz \(L^p\)-spaces are investigated and like usual \(L^p\)-spaces, for \((p,q)\)-complementary normalized Young pair \((\Phi,\Psi)\) is shown that \(\mathbb{L}_p^\Phi(\mu)^*=\mathbb{L}_q^\Psi(\mu)\), where \(1< p,q<\infty\) and \(1/p+1/q=1\). Finally, for a locally compact group \(G\), we investigate algebraic properties of Orlicz space \(\mathcal{L}^\Phi (G)\) as a Banach algebra and generalized Orlicz spaces, \(\mathcal{L}_p^\Phi (G)\) as a Banach \(L^1(G)\)-bimodule with two products convolution and conjugate convolution.

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
54E52 Baire category, Baire spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Akbarbaglu, I.; Maghsoudi, S., On certain porous sets in the Orlicz space of a locally compact group, Colloq. Math., 129, 1, 99-111 (2012) · Zbl 1267.46042 · doi:10.4064/cm129-1-7
[2] Akbarbaglu, I.; Maghsoudi, S., Banach-Orlicz algebras on a locally compact group, Medit. J. Math., 10, 4, 1937-1934 (2013) · Zbl 1290.46020 · doi:10.1007/s00009-013-0267-z
[3] Arai, T., Convex risk measures on Orlicz spaces: inf-convolution and shortfall, Math. Financ. Econ., 3, 2, 73-88 (2010) · Zbl 1255.91173 · doi:10.1007/s11579-010-0028-8
[4] Armstrong, S., Kuusi, T., Mourrat, J.Ch.: The additive structure of elliptic homogenization. Invent. Math. (2016). doi:10.1007/s00222-016-0702-4 · Zbl 1377.35014
[5] Baron, K.; Hudzik, H., Orlicz spaces which are \(L^p\)-spaces, Aequ. Math., 48, 254-261 (1994) · Zbl 0805.46025 · doi:10.1007/BF01832988
[6] Byun, SS; Lee, M., Weighted estimates for nondivergence parabolic equations in Orlicz spaces, J. Funct. Anal., 269, 8, 2530-2563 (2015) · Zbl 1323.35056 · doi:10.1016/j.jfa.2015.07.009
[7] Cao, J.; Chang, D-C; Yang, D.; Yang, S., Riesz transform characterizations of Musielak-Orlicz-Hardy spaces, Trans. Am. Math. Soc., 368, 6979-7018 (2016) · Zbl 1338.42024 · doi:10.1090/tran/6556
[8] Dales, HG, Banach Algebras and Automatic Continuity. London Math Society Monographs (2000), Oxford: Clarendon Press, Oxford · Zbl 0981.46043
[9] Ebadian, A.; Jabbari, A., Convolution operators on Banach-Orlicz algebras, Anal. Math., 46, 2, 243-264 (2020) · Zbl 1463.47116 · doi:10.1007/s10476-020-0023-0
[10] Harjulehto, P.; Hästö, P., Orlicz Spaces and Generalized Orlicz Spaces. Lecture Notes in Mathematics (2019), New York: Springer, New York · Zbl 1436.46002 · doi:10.1007/978-3-030-15100-3
[11] Hästö, P., The maximal operator on generalized Orlicz spaces, J. Funct. Anal., 269, 4038-4048 (2015) · Zbl 1338.47032 · doi:10.1016/j.jfa.2015.10.002
[12] Iwaniec, T.; Koskela, P.; Onninen, J., Mappings of finite distortion: monotonicity and continuity, Invent. Math., 144, 507-531 (2001) · Zbl 1006.30016 · doi:10.1007/s002220100130
[13] Johnson, BE, Cohomology in Banach algebras, Mem. Am. Math. Soc., 127, 1-96 (1972) · Zbl 0256.18014
[14] Junge, M.; Xu, Q., Representation of certain homogeneous Hilbertian operator spaces and applications, Invent. Math., 179, 75-118 (2010) · Zbl 1215.46039 · doi:10.1007/s00222-009-0210-x
[15] Krasnosel’skiĭ, M.A., Rutickiĭ, Ya.B.: Convex functions and Orlicz spaces, Translated from the first Russian edition by L. F. Boron, P. Noordhoff, Ltd. Groningen, Nethelands (1961) · Zbl 0095.09103
[16] Labuschagne, LE; Majewski, WA, Maps on non-commutative Orlicz spaces, Ill. J. Math., 55, 1053-1081 (2011) · Zbl 1272.46052
[17] Majewski, WA; Labuschagne, LE, On applications of Orlicz spaces to statistical physics, Ann. Henri Poincaré, 15, 1197-1221 (2014) · Zbl 1295.82019 · doi:10.1007/s00023-013-0267-3
[18] Majewski, WA; Labuschagne, LE, Why are Orlicz spaces useful for statistical physics?, Oper. Theory Adv. Appl., 252, 271-283 (2016) · Zbl 1368.46055
[19] Martínez, S.; Wolanski, N., A minimum problem with free boundary in Orlicz space, Adv. Math., 218, 6, 1914-1971 (2008) · Zbl 1170.35030 · doi:10.1016/j.aim.2008.03.028
[20] Mastyło, M., Rodriguez-Piazza, L.: Convergence almost everywhere of multiple Fourier series over cubes. Trans. Am. Math. Soc. 370, 1629-1659 (2018) · Zbl 1385.43002
[21] Osançlıol, A.; Öztop, S., Weighted Orlicz algebras on locally compact groups, J. Aust. Math. Soc., 99, 3, 399-414 (2015) · Zbl 1338.46039 · doi:10.1017/S1446788715000257
[22] Öztop, S.; Samei, E., Twisted Orlicz algebras, I, Stud. Math., 236, 3, 271-296 (2017) · Zbl 1368.43006 · doi:10.4064/sm8562-9-2016
[23] Öztop, S.; Samei, E., Twisted Orlicz algebras, II, Math. Nach., 292, 5, 1122-1136 (2019) · Zbl 1419.43001 · doi:10.1002/mana.201700362
[24] Rao, MM; Ren, ZD, Theory of Orlicz Spaces (1991), New York: Marcel Dekker Inc, New York · Zbl 0724.46032
[25] Rao, MM; Ren, ZD, Applications of Orlicz Spaces (2002), New York: Marcel Dekker, Inc, New York · Zbl 0997.46027 · doi:10.1201/9780203910863
[26] Rickert, NW, Convolution of Lp functions, Proc. Am. Math. Soc., 18, 762-763 (1967) · Zbl 0166.11702
[27] Runde, V., Lectures on Amenability. Lecture Notes in Mathematics (2002), New York: Springer, New York · Zbl 0999.46022 · doi:10.1007/b82937
[28] Streater, RF, Quantum Orlicz spaces in information geometry, Open Syst. Inf. Dyn., 11, 359-375 (2004) · Zbl 1084.81024 · doi:10.1007/s11080-004-6626-2
[29] Takahasi, S-E, BSE Banach modules and multipliers, J. Funct. Anal., 125, 67-68 (1994) · Zbl 0848.46034 · doi:10.1006/jfan.1994.1117
[30] Takahasi, S-E; Hatori, O., Commutative Banach algebras which satisfy a Bochner-Schoenberg-Eberlein-type theorem, Proc. Am. Math. Soc., 110, 149-158 (1990) · Zbl 0722.46025
[31] Treil, S.; Volberg, A., Entropy conditions in to weight inequalities for singular integral operators, Adv. Math., 301, 499-548 (2016) · Zbl 1377.42024 · doi:10.1016/j.aim.2016.06.021
[32] Wendel, JG, Left centralizers and isomorphisms of group algebras, Pac. J. Math., 2, 251-261 (1952) · Zbl 0049.35702 · doi:10.2140/pjm.1952.2.251
[33] Yuan, CK, Conjugate convolutions and inner invariant means, J. Math. Anal. Appl., 157, 166-178 (1991) · Zbl 0744.43004 · doi:10.1016/0022-247X(91)90142-M
[34] Zelazko, W., On algebras \(L^p\) of locally compact groups, Colloq. Math., 8, 112-120 (1961) · Zbl 0095.10301
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.