Operator-valued Fourier multipliers on periodic Besov spaces

We prove in this paper that a sequence $M:\mathbb{Z}^{n}\to\mathcal{L}(E)$ of bounded variation is a Fourier multiplier on the Besov space $B_{p,q}^{s}(\mathbb{T}^{n},E)$ for $s\in\mathbb{R}$, $1<p<\infty$, $1\leq q\leq\infty$ and $E$ a Banach space, if and only if $E$ is a UMD-space. This extends in some sense the Theorem 4.2 in [AB04] to the $n-$dimensional case. The result is used to obtain existence and uniqueness of solution for some Cauchy problems with periodic boundary conditions.