Gaussian bounds for degenerate parabolic equations

Let A be a real symmetric, degenerate elliptic matrix whose degeneracy is controlled by a weight w in the A2 or QC class. We show that there is a heat kernel Wt (x, y) associated to the parabolic equation wut = divA∇u, and Wt satisfies classic Gaussian bounds: Wt (x, y) C1 tn/2 exp −C2 |x − y|2 t . We then use this bound to derive a number of other properties of the kernel.