Weighted Multipolar Hardy Inequalities And Evolution Problems With Kolmogorov Operators Perturbed By Singular Potentials

The main results in the paper are the weighted multipolar Hardy inequalitiesC integral(RN) Sigma(n)(i=1) phi(2)/vertical bar x-a(i)vertical bar 2 mu(x)dx <= integral(RN) vertical bar del(phi)vertical bar(2) mu(x)dx + K integral(RN) phi(2)mu(x)dx,in R-N for any yo in a suitable weighted Sobolev space, with 0 < c <= c(o,) (mu,), a1, ... , a(n) is an element of R-N, K constant. The weight functions p, are of a quite general type.The paper fits in the framework of Kolmogorov operators defined on smooth functionsLu = Delta u + del mu/mu . del u,perturbed by multipolar inverse square potentials, and related evolution problems. Necessary and sufficient conditions for the existence of exponentially bounded in time positive solutions to the associated initial value problem are based on weighted Hardy inequalities. For constants c beyond the optimal Hardy constant c(o/mu) we are able to show nonexistence of positive solutions.