  Let X be a UMD space with type t and cotype q, and let T be a Fourier
multiplier operator with a scalar-valued symbol m. If the Mihlin multiplier
estimate holds for all partial derivatives of m up to the order n/max(t,q')+1,
then T is bounded on the X-valued Bochner spaces. For scalar-valued
multipliers, this improves the theorem of Girardi and Weis (J. Funct. Anal.,
2003) who required similar assumptions for derivatives up to the order n/r+1,
where r is a Fourier-type of X. However, the present method does not apply to
operator-valued multipliers, which are also covered by the Girardi-Weis
theorem.
