SHARP TWO-SIDED HEAT KERNEL ESTIMATES FOR CRITICAL SCHRODINGER OPERATORS ON BOUNDED DOMAINS

On a smooth bounded domain \Omega \subset R^N we consider the Schrödinger operators ?\Delta? V, with V being either the critical borderline potential V(x) = (N -2)^2/4 |x|^(?2) or V(x) = (1/4) dist(x, ?\Omega)^(?2), under Dirichlet boundary conditions. In this work we obtain sharp two-sided estimates on the corresponding heat kernels. To this end we transform the Schrödinger operators into suitable degenerate operators, for which we prove a new parabolic Harnack inequality up to the boundary. To derive the Harnack inequality we have established a series of new inequalities such as improved Hardy, logarithmic Hardy Sobolev,Hardy-Moser and weighted Poincaré. As a byproduct of our technique we are able to answer positively to a conjecture of E. B. Davies.