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R. Bosi, J. Dolbeaut, M.J. Esteban:
"Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators";
in: "ASC Report 25/2007", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2007, ISBN: 978-902627-3-00-1.

By expanding squares, we prove several Hardy inequalities with two critical singularities
and constants which explicitly depend upon the distance between the two singularities.
These inequalities involve the L2 norm. Such results are generalized to an arbitrary number of
singularities and compared with standard results given by the IMS method. The generalized
version of Hardy inequalities with several singularities is equivalent to some spectral information
on a Schr¨odinger operator involving a potential with several inverse square singularities. We also
give a generalized Hardy inequality for Dirac operators in the case of a potential having several
singularities of Coulomb type, which are critical for Dirac operators.

http://www.asc.tuwien.ac.at/preprint/2007/asc25x2007.pdf