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

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 is an element of X(R)} there exist uniquely determined almost periodic Fourier multipliers a(l), a(r) on X(R, w), such that a = (1 - u)a(1) + ua(r) +a(0), for some monotonically increasing function u with u(-infinity) = 0, u(+infinity) = 1 and some continuous and vanishing at infinity Fourier multiplier a(0) on X(R, w). This result extends previous results by Sarason (Duke Math J 44:357-364, 1977) for L-2(R) and by Karlovich and Loreto Herna ' ndez (Integral Equ Oper Theor 62:85-128, 2008) for weighted Lebesgue spaces L-p(R, w) with weights in a suitable subclass of the Muckenhoupt class A(p)(R).