  In this paper we develop the theory of Fourier multiplier operators
$T_{m}:L^{p}(\mathbb{R}^{d};X)\to L^{q}(\mathbb{R}^{d};Y)$, for Banach spaces
$X$ and $Y$, $1\leq p\leq q\leq \infty$ and $m:\mathbb{R}^d\to
\mathcal{L}(X,Y)$ an operator-valued symbol. The case $p=q$ has been studied
extensively since the 1980's, but far less is known for $p<q$. In the scalar
setting one can deduce results for $p<q$ from the case $p=q$. However, in the
vector-valued setting this leads to restrictions both on the smoothness of the
multiplier and on the class of Banach spaces. For example, one often needs that
$X$ and $Y$ are UMD spaces and that $m$ satisfies a smoothness condition. We
show that for $p<q$ other geometric conditions on $X$ and $Y$, such as the
notions of type and cotype, can be used to study Fourier multipliers. Moreover,
we obtain boundedness results for $T_m$ without any smoothness properties of
$m$. Under smoothness conditions the boundedness results can be extrapolated to
other values of $p$ and $q$ as long as $\tfrac{1}{p}-\tfrac{1}{q}$ remains
constant.
