  Using Malliavin operators together with an interpolation technique inspired
by Arratia, Goldstein and Gordon (1989), we prove a new inequality on the
Poisson space, allowing one to measure the distance between the laws of a
general random vector, and of a target random element composed of Gaussian and
Poisson random variables. Several consequences are deduced from this result, in
particular: (1) new abstract criteria for multidimensional stable convergence
on the Poisson space, (2) a class of mixed limit theorems, involving both
Poisson and Gaussian limits, (3) criteria for the asymptotic independence of
$U$-statistics obeying to Gaussian and Poisson asymptotic regimes. Our results
generalize and unify several previous findings in the field. We provide an
application to joint sub-graph counting in random geometric graphs.
