Operator-valued Fourier multipliers on periodic Besov spaces and applications

Let 1 less than or equal to p, q less than or equal to infinity, s epsilon R and let X be a Banach space. We show that the analogue of Marcinkiewicz's Fourier multiplier theorem on L-P (T) holds for the Besov space B-p(s),(q) (T; X) if and only P if 1 < p < infinity and X is a UMD-space. Introducing stronger conditions we obtain a periodic Fourier multiplier theorem which is valid without restriction on the indices or the space (which is analogous to Amann's result (Math. Nachr. 186 (1997), 5-56) on the real line). It is used to characterize maximal regularity of periodic Cauchy problems.