Fig\(\grave{\hbox {a}}\)–Talamanca–Herz–Orlicz Algebras and Convoluters of Orlicz Spaces

Abstract

Let G be a locally compact group and \((\Phi , \Psi )\) be a complementary pair of N-functions. In this paper, the Fig\(\grave{\hbox {a}}\)–Talamanca–Herz algebras and the space of p-pseudomeasures on G are extended to Orlicz spaces. Indeed, the Banach algebra of \(\Phi \)-pseudomeasures \({\mathop {\mathrm{PM}}}_{\Phi }(G)\) and the Fig\(\grave{\hbox {a}}\)–Talamanca–Herz–Orlicz algebras \({\mathop {\mathrm{A}}}_{\Phi }(G)\) are defined. Then, it is shown that \({\mathop {\mathrm{A}}}_{\Phi }(G)^*={\mathop {\mathrm{PM}}}_{\Psi }(G)\). Furthermore, we characterize \(\mathop {\mathrm{Cv}}_{\Phi }(G)\), the space of \(\Phi \)-convoluters, in terms of right translation invariant operators on \(M^{\Phi }(G)\). Then when G is amenable, we show that \(\mathop {\mathrm{Cv}}_{\Phi }(G)\), is equal to \(\mathop {\mathrm{PM}}_{\Phi }(G)\), a generalization of the classical p-version. Finally, we study \(B_{\Phi }(G)\), the space of pointwise multipliers of \(\mathop {\mathrm{A}}_{\Phi }(G)\) when G is amenable.

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Acknowledgements

The authors are very grateful to the anonymous referee(s) for a very careful reading of the paper, several invaluable remarks and pointing out some misprints in an earlier version of the paper.

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Correspondence to Ibrahim Akbarbaglu.

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Aghababa, H.P., Akbarbaglu, I. Fig\(\grave{\hbox {a}}\)–Talamanca–Herz–Orlicz Algebras and Convoluters of Orlicz Spaces. Mediterr. J. Math. 17, 106 (2020). https://doi.org/10.1007/s00009-020-01547-x

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Keywords

  • Orlicz spaces
  • Young functions
  • Pseudomeasures
  • Locally compact groups
  • Fig\(\grave{\hbox {a}}\)–Talamanca–Herz algebras
  • Representations

Mathematics Subject Classification

  • Primary 43A15
  • 43A22
  • Secondary: 46E30
  • 47L10