Sharp two-sided heat kernel estimates for critical Schrodinger operators on bounded domains

On a smooth bounded domain Omega subset of R-N we consider the Schrodinger operators - Delta - V, with V being either the critical borderline potential V( x) = (N 2)(2) /4 vertical bar x vertical bar(-2) or V(x) = (1/ 4) dist( x,delta 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 Schrodinger 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 Poincare. As a byproduct of our technique we are able to answer positively to a conjecture of E. B. Davies.