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Let X be a UMD space with type t and cotype q, and let T be a Fouriermultiplier operator with a scalar-valued symbol m. If the Mihlin multiplierestimate 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-valuedmultipliers, 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 tooperator-valued multipliers, which are also covered by the Girardi-Weistheorem.

New thoughts on the vector-valued Mihlin-H\"ormander multiplier theorem