Harnack inequality and heat kernel estimates for the Schroedinger operator with Hardy potential

In this preliminary note we outline some results of the forthcoming paper [11], concerning positive solutions of the equation \partial_t u=\Delta u+ c/|x|^2 u (0<c<(n-2)^2/4; n\ge 3). A parabolic Harnack inequality is proved, which in particular implies a sharp two-sided estimate for the associated heat-kernel. Our approach relies on the unitary equivalence of the Schroedinger operator Hu=-\Delta u -c/|x|^2 u with the opposite of the weighted Laplacian \Delta_\lambda v=1/|x|^2 div(|x|^\lambda \nabla v) when \lambda=2-n+2\sqrt[c_0-c}.