Heat kernel bounds and desingularizing weights

We study the parabolic operator ∂t−Δx+V(t,x), in R+1×Rd, d⩾1, with a potential V=V+−V−, V±⩾0 assumed to be from a parabolic Kato class, and obtain two-sided Gaussian bounds on the associated heat kernel. The constraints on the Kato norms of V+ and V− are completely asymmetric, as they should be. Further improvements to our heat kernel bounds are obtained in the case of time-independent potentials. If V has singularities of the type ±c|x|−2 with a suitably small constant c, we obtain new lower and (sharp) upper weighted heat kernel bounds. The rate of growth of the weights depends (explicitly) on the constant c. The standard bounds and methods (estimates in Lp-spaces without desingularizing weights) fail for singular potentials.