Optimal Hardy inequalities associated with multipolar Schr\"odinger operators

We proved some optimal Hardy inequalities in RNwhich is closely related to multipolar Schr"odinger operators with mean-value type potentials, these sharp inequalities imply some multipolar type Heisenberg inequalities. We also obtained someimproved multipolar Hardy inequalities on bounded domains, moreover, we got the range of the best Hardy constant for a specific Hardy inequality.