New thoughts on the vector-valued Mihlin-Hörmander multiplier theorem

Let $X$ be a UMD space with type $t$ and cotype $q$, and let $T_m$ be a Fourier multiplier operator with a scalar-valued symbol $m$. If $\vert\partial^{\alpha}m(\xi)\vert\lesssim\vert{\xi}\vert^{-\vert\alpha\vert}$ for all $\vert\alpha\vert\leq\lfloor{n/\max(t,q')\rfloor}+1$, then $T_m$ is bounded on $L^p(\mathbb{R}^n;X)$ for all $p\in(1,\infty)$. 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 $\lfloor{n/r}\rfloor+1$, where $r\leq\min(t,q')$ 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.