  For commuting contractions $T_1,\dots ,T_n$ acting on a Hilbert space
$\mathcal H$ with $T=\prod_{i=1}^n T_i$, we find a necessary and sufficient
condition under which $(T_1,\dots ,T_n)$ dilates to commuting isometries
$(V_1,\dots ,V_n)$ on the minimal isometric dilation space $T$, where
$V=\prod_{i=1}^nV_i$ is the minimal isometric dilation of $T$. We construct
both Sch$\ddot{a}$ffer and Sz. Nagy-Foias type isometric dilations for
$(T_1,\dots ,T_n)$ on the minimal dilation spaces of $T$. Also, a different
dilation is constructed when the product $T$ is a $C._0$ contraction, that is
${T^*}^n \rightarrow 0$ as $n \rightarrow \infty$. As a consequence of these
dilation theorems we obtain different functional models for $(T_1,\dots ,T_n)$
in terms of multiplication operators on vectorial Hardy spaces. One notable
fact about our models is that the multipliers are analytic functions in one
variable. The dilation, when $T$ is a $C._0$ contraction, leads to a
conditional factorization of a $T$. Several examples have been constructed.
