Semi-almost periodic Fourier multipliers on rearrangement-invariant spaces with suitable Muckenhoupt weights

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Abstract

Let X(R) be a separable rearrangement-invariant space and w be a suitable Muckenhoupt weight. We show that for any semi-almost periodic Fourier multiplier a on X(R, w) = { f: fw∈ X(R) } there exist uniquely determined almost periodic Fourier multipliers al, ar on X(R, w) , such that a=(1-u)al+uar+a0,for some monotonically increasing function u with u(- ∞) = 0 , u(+ ∞) = 1 and some continuous and vanishing at infinity Fourier multiplier a on X(R, w). This result extends previous results by Sarason (Duke Math J 44:357–364, 1977) for L2(R) and by Karlovich and Loreto Hernández (Integral Equ Oper Theor 62:85–128, 2008) for weighted Lebesgue spaces Lp(R, w) with weights in a suitable subclass of the Muckenhoupt class Ap(R).

Original languageEnglish
Pages (from-to)1135-1162
JournalBoletin de la Sociedad Matematica Mexicana
Volume26
Issue number3
DOIs
Publication statusPublished - 1 Nov 2020

Keywords

  • Almost periodic function
  • Boyd indices
  • Fourier multiplier
  • Muckenhoupt weight
  • Rearrangement-invariant Banach function space
  • Semi-almost periodic function

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