We construct the fundamental solution (the heat kernel) p κ to the equation ∂ t = L κ , where under certain assumptions the operator L κ takes one of the following forms,

We construct a fundamental solution of the equation (1, 2) and b satisfying a certain integral space-time condition. We also show it has α-stable upper and lower bounds.

We propose a new general method of estimating Schrödinger perturbations of transition densities using an auxiliary transition density as a majorant of the perturbation series. We present applications to Gaussian bounds by proving an optimal inequality involving four Gaussian kernels, which we call 4G Theorem. The applications come with honest control of constants in estimates of Schrödinger perturbations of Gaussian-type heat kernels and also allow for specific non-Kato perturbations.2010 Mathematics Subject Classification. Primary 47D06, 47D08; Secondary 35A08, 35B25.