Let X X be a UMD space with type t t and cotype q q , and let T m T_m be a Fourier multiplier operator with a scalar-valued symbol m m . If | ∂ α m ( ξ ) | ≲ | ξ | − | α | |\partial ^{\alpha }m(\xi )|\lesssim |{\xi }|^{-|\alpha |} for all | α | ≤ ⌊ n / max ( t , q ′ ) ⌋ + 1 |\alpha |\leq \lfloor {n/\max (t,q’)\rfloor }+1 , then T m T_m is bounded on L p ( R n ; X ) L^p(\mathbb {R}^n;X) for all p ∈ ( 1 , ∞ ) p\in (1,\infty ) . 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 \lfloor {n/r}\rfloor +1 , where r ≤ min ( t , q ′ ) r\leq \min (t,q’) is a Fourier-type of X X . However, the present method does not apply to operator-valued multipliers, which are also covered by the Girardi–Weis theorem.