Noncompactness of Fourier convolution operators on Banach function spaces

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Let X(R) be a separable Banach function space such that the Hardy-Littlewood maximal operator M is bounded on X(R) and on its associate space X' (R). Suppose that a is a Fourier multiplier on the space X(R) We show that the Fourier convolution operator W-0(a) with symbol a is compact on the space X(R) if and only if a = 0. This result implies that nontrivial Fourier convolution operators on Lebesgue spaces with Muckenhoupt weights are never compact.
Original languageEnglish
Pages (from-to)553-561
JournalAnnals of functional analysis
Issue number4
Publication statusPublished - Nov 2019


  • Fourier convolution operator
  • Compactness
  • Banach function space
  • Hardy-Littlewood maximal operator
  • Lebesgue space with Muckenhoupt weight


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