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

Abstract

Let \(X({\mathbb {R}})\) 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 \(X({\mathbb {R}},w)=\{f:fw\in X({\mathbb {R}})\}\) there exist uniquely determined almost periodic Fourier multipliers \(a_l,a_r\) on \(X({\mathbb {R}},w)\), such that

$$\begin{aligned} a=(1-u)a_l+ua_r+a_0, \end{aligned}$$

for some monotonically increasing function u with \(u(-\infty )=0\), \(u(+\infty )=1\) and some continuous and vanishing at infinity Fourier multiplier \(a_0\) on \(X({\mathbb {R}},w)\). This result extends previous results by Sarason (Duke Math J 44:357–364, 1977) for \(L^2({\mathbb {R}})\) and by Karlovich and Loreto Hernández (Integral Equ Oper Theor 62:85–128, 2008) for weighted Lebesgue spaces \(L^p({\mathbb {R}},w)\) with weights in a suitable subclass of the Muckenhoupt class \(A_p({\mathbb {R}})\).

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Correspondence to A. Yu. Karlovich.

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Dedicated to Professor Yuri I. Karlovich on the occasion of his 70th birthday.

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This work was partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the projects UID/MAT/00297/2019 (Centro de Matemática e Aplicações).

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Fernandes, C.A., Karlovich, A.Y. Semi-almost periodic Fourier multipliers on rearrangement-invariant spaces with suitable Muckenhoupt weights. Bol. Soc. Mat. Mex. 26, 1135–1162 (2020). https://doi.org/10.1007/s40590-020-00276-1

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Keywords

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

Mathematics Subject Classification

  • Primary 42A45
  • Secondary 46E30