A Matrix Differential Harnack Estimate for a Class of Ultraparabolic Equations

Let u be a positive solution of the ultraparabolic equation partial derivative(t)u Sigma(kn)(i=1) partial derivative(2)(xi)u Sigma(k)(i=1) x(i)partial derivative(xn) (i)u on R-n (k) x 0 T where 1 <= k <= n and 0 < T <= infinity. Assume that u and its derivatives (w.r.t. the space variables) up to the second order are bounded on any compact subinterval of (0, T). Then the difference H(log u) - H (log f) of the Hessian matrices of log u and of log f (both w.r.t. the space variables) is non-negatively definite, where f is the fundamental solution of the above equation with pole at the origin (0, 0). The estimate in the case n = k = 1 is due to Hamilton. As a corollary we get that , where l = log u, and Delta Sigma(n)(i) 1(k) partial derivative(2)(xi).