A matrix differential Harnack estimate for a class of ultraparabolic equations

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Let u be a positive solution of the ultraparabolic equation equation* ∂ₜ u=∑ᵢ₌₁ⁿ ∂ₓᵢ² u+∑ᵢ₌₁ᵏ xᵢ∂ₓₙ₊ᵢu 8mm on 4mm Rⁿ⁺ᵏ× (0,T), equation* where 1≤ k≤ n and 0<T ≤ +∞. 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(u)- H(f) of the Hessian matrices of u and of 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 Δl+n+3k/2t+6k/t³≥ 0, where l= u, and Δ=∑ᵢ₌₁ⁿ⁺ᵏ ∂ₓᵢ².