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Let X be a Banach space. We show that each m : R \ {0} → L(X) satisfying the Mikhlin condition sup x =0 ( m(x) + xm (x) ) < ∞ defines a Fourier multiplier on B In recent years, operator-valued Fourier multipliers on vector-valued function spaces have been extensively studied [1,3,10,16]. They are needed to establish existence and uniqueness as well as regularity for differential equations in Banach spaces. In the L p spaces case, it was Weis who found the formulation of the operator-valued Fourier multiplier theorem: let X be a Banach space and let m : R \ {0} → L(X) be a C 1 -function, assume that X is a UMD space (see [5,7]), 1 < p < ∞ and that both sets {m(x) : x = 0} and {xm (x) :where we denote by L(X) the set of all bounded linear operators on X. It is known that in Weis' result the R-boundedness may be replaced by the norm-boundedness only when X is isomorphic to a Hilbert space [4].In the vector-valued Besov spaces case, things are different. It was Amann [1] (see also Weis [17]) who discovered that the assumption of R-boundedness is not needed and no hypothesis on the geometry of the underlying Banach space is required in the Besov spaces case: let m : R → L(X) be a C 2 -function, assume thatthen for 1 ≤ p, q ≤ ∞, s ∈ R, m defines a Fourier multiplier on B s p,q (R; X) (see also [10] for further development, and [2] for the corresponding result in the periodic Besov spaces case).In this paper, we are interested in the following questions: what happens when we consider a larger class of multiplier functions m in the Besov spaces case? In view of known Fourier multiplier results in the scalar L p spaces case, one may consider the following two natural classes: the first one is the set of all

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