We prove in this paper that a sequence M : Z n → L(E) of bounded variation is a Fourier multiplier on the Besov spaceand 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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