Operator-valued Fourier multipliers of bounded s-variation

In this paper, we establish an operator-valued Fourier multiplier theorem in weighted Lebesgue spaces, Besov and Triebel–Lizorkin spaces, assuming the multiplier has $\mathcal{R}$-bounded range and satisfies an ${\mathrm{\ell }}^r$-summability condition on its bounded $s$-variation seminorms over dyadic intervals. The exponents $r$ and $s$ reflect the relationship between the geometric properties of the underlying Banach spaces (type and cotype) and the boundedness of Fourier multiplier operators. As our main tool we prove a weighted vector-valued variational Carleson inequality and deduce an estimate of Littlewood–Paley–Rubio de Francia type.