  The main results in the paper are the weighted multipolar Hardy inequalities
\begin{equation*} c\int_{\R^N}\sum_{i=1}^n\frac{u^2}{|x-a_i|^2}\,d\mu
\leq\int_{\R^N}|\nabla u |^2d\mu+ K\int_{\R^N} u^2d\mu, \end{equation*} in
$\R^N$ for any $u$ in a suitable weighted Sobolev space, with $0<c\le
c_{o,\mu}$, $a_1,\dots,a_n\in \R^N$, $K$ constant. The weight functions $\mu$
are of a quite general type.
  The paper fits in the framework of the study of Kolmogorov operators
\begin{equation*} Lu=\Delta u+\frac{\nabla \mu}{\mu}\cdot\nabla u,
\end{equation*} perturbed by multipolar inverse square potentials, and of the
related evolution problems.
  The necessary and sufficient conditions for the existence of positive
exponentially bounded in time solutions to the associated initial value problem
are based on weighted Hardy inequalities. The optimality of the constant
constant $c_{o,\mu}$ allow us to state the nonexistence of positive solutions.
  We follow the Cabr\'e-Martel's approach. To this aim we state some properties
of the operator $L$, of its corresponding $C_0$-semigroup and density results.
