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Generalization of Orlicz spaces

Abstract

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 (pq)-complementary and (pq)-complementary normalized Young pairs, by these definitions; duals of Orlicz \(L^p\)-spaces are investigated and like usual \(L^p\)-spaces, for (pq)-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.

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Acknowledgements

The authors would like to express his deep gratitude to Professors A. T.-M Lau for his encouragements and invaluable suggestions for improving of this paper. Also, the authors thanks to Professor Hästö for introducing the reference [10] and his invaluable suggestions. Moreover, we would like thank to the referee for the careful reading of the paper, the detailed criticism and his/her remarks and suggestions which greatly improved the presentation of the paper.

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Correspondence to Ali Jabbari.

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Ebadian, A., Jabbari, A. Generalization of Orlicz spaces. Monatsh Math 196, 699–736 (2021). https://doi.org/10.1007/s00605-021-01627-4

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Keywords

  • Conjugate convolution
  • Convolution
  • Group algebras
  • Lower porous set
  • Orlicz space
  • Young function

Mathematics Subject Classification

  • 46E30
  • 43A15
  • 54E52