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 Hernandez (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 $$A_p({\mathbb {R}})$$ .