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We prove in this paper that a sequence M : Z n → L(E) of bounded variation is a Fourier multiplier on the Besov space B s p,q(T n , E) for s ∈ R, 1 < p < ∞, 1 ≤ q ≤ ∞ and E a Banach space, if and only if E is a UMDspace. This extends the Theorem 4.2 in [3] to then−dimensional case. As illustration of the applicability of this results we study the solvability of two abstract Cauchy problems with periodic boundary conditions. [ABSTRACT FROM AUTHOR]