Operator-valued Fourier multipliers on periodic Besov spaces

Abstract

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 $s\in \mathbb{R}$, $1<p<\infty$, $1\le q\le \infty$ 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.

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