A matrix differential Harnack estimate for a class of ultraparabolic equations

Abstract

Let $u$ be a positive solution of the ultraparabolic equation \begin{equation*} \partial_t u=\sum_{i=1}^n \partial_{x_i}^2 u+\sum_{i=1}^k x_i\partial_{x_{n+i}}u \hspace{8mm} \mbox{on} \hspace{4mm} \mathbb{R}^{n+k}\times (0,T), \end{equation*} where $1\leq k\leq n$ and $0<T \leq +\infty$. 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 $\Delta l+\frac{n+3k}{2t}+\frac{6k}{t^3}\geq 0$, where $l=\log u$, and $\Delta=\sum_{i=1}^{n+k} \partial_{x_i}^2 $.