A note on operator‐valued Fourier multipliers on Besov spaces

Let X be a Banach space. We show that each m : ? \ {0} → L (X ) satisfying the Mikhlin condition sup_{x ≠0}(‖m (x )‖ + ‖xm ′(x )‖) < ∞ defines a Fourier multiplier on B ^s _p,q (?; X ) if and only if 1 < p < ∞ and X is isomorphic to a Hilbert space; each bounded measurable function m : ? → L (X ) having a uniformly bounded variation on dyadic intervals defines a Fourier multiplier on B ^s _p,q (?; X ) if and only if 1 < p < ∞ and X is a UMD space. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)