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