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Convolution operators on Banach–Orlicz algebras

Abstract

For a locally compact group G, let \(\mathcal{L}^{\Phi}\)(G) and \(\mathcal{L}_\omega^{\Phi}\)(G) be Orlicz and weighted Orlicz spaces, respectively, where Φ is a Young function and ω is a weight on G. We study the harmonic and convolution operators on Orlicz and weighted Orlicz spaces. We prove that under some conditions the harmonic operators on \(\mathcal{L}^{\Phi}\)(G) and \(\mathcal{L}_\omega^{\Phi}\)(G) are compact. We characterize convolution operators on Orlicz and weighted Orlicz spaces \(\mathcal{L}^{\Phi}\)(G) and \(\mathcal{L}_\omega^{\Phi}\)(G).

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Acknowledgement

The authors would like to thank the referee for careful reading of the paper and for his/her useful suggestions, which greatly improved the presentation of the paper. We revised the whole of the subsection 4.2, in light of his/her comments. In particular, we corrected the proof of Theorem 5.3 and added Theorem 5.4 by his/her suggestions.

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Correspondence to A. Jabbari.

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Ebadian, A., Jabbari, A. Convolution operators on Banach–Orlicz algebras. Anal Math 46, 243–264 (2020). https://doi.org/10.1007/s10476-020-0023-0

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Key words and phrases

  • convolution operator
  • harmonic operator
  • locally compact group
  • Orlicz space
  • weighted Orlicz space

Mathematics Subject Classification

  • 47B48
  • 46E25
  • 46E30