ON A PARABOLIC EQUATION WITH A SINGULAR LOWER ORDER TERM - PART II - THE GAUSSIAN BOUNDS

We establish a lower and an upper Gaussian bound for the fundamental s olutions of some general parabolic equations with potentials under a m inimum integrability assumption. These parabolic equations include the second order uniformly parabolic equations in Euclidean spaces and th e heat equations corresponding to some subelliptic operators. Even in the uniformly parabolic case, the existence of a Gaussian upper bound in the presence of these singular potentials was not known and has bee n questioned by a number of people. We also use these bounds to prove the Harnack inequality, the local boundedness and continuity of weak s olutions. Part II is logically independent of the previous paper [Z].