Derivative of the fundamental solution of the heat equation

Question

Let $\Gamma$ be the fundamental solution of the heat equation in $(0,\infty)\times\mathbb{R}^n$, that is \begin{equation} \Gamma(t,x)=\frac{1}{(4\pi t)^{\frac{n}{2}}}e^{\frac{-|x|^2}{4t}} \mbox{ per }x\in\mathbb{R}^n,t>0. \end{equation}

In the book by Ladyzhenskaya,Solonnikov and Ural'tseva "Linear and Quasi-linear Equations of Parabolic Type" at page 274 is stated this inequality \begin{equation} |D_t^rD_x^s\Gamma(t,x)|\leq C_{r,s}t^{-\frac{n}{2}-r-\frac{s}{2}}e^{C\frac{-|x|^2}{t}}. \end{equation} It's correct? The exponent of $t$ in the inequality won't be $-\frac{n}{2}-2r-s$ or I have made an error in the computations?

Answer 1

Yes, it is correct. Say, each differentiation w.r.t. $t$ adds $-1$ to the exponent of $t$.