Estimates of the derivatives of the fundamental solution of heat equations in $\mathbb{R}^n$

Question

Let $$n_t(z)=\frac{1}{(4\pi t)^{n/2}}e^{-\frac{1}{4t}\|z\|^2}.$$ And the differentiation w.r.t. $x$, denotes as $D_x^{r}$ is defined as $$\frac{\partial^{|r|}}{\partial x_1^{r_1}\cdots\partial x_n^{r_n}}.$$

The claim is $$|D_x^{r}(y-x)|\le C_{r,t}\,n_{t/2}(y-x).$$

Could anyone show me explicitly how? Special attention to $t/2.$

Thanks

Answer 1

Hint:

1. Show (e.g. by induction) that for any fixed $t$, $r$, the derivative $$\frac{\partial^{|r|}}{\partial_{z_1}^{r_1} \cdots \partial_{z_n}^{r_n}} n_t(z)$$ is of the form $$p_{t,r}(z) n_t(z)$$ for some polynomial $p_{t,r}$.
2. Recall that $$z \mapsto p_{t,r}(z) n_{R}(z)$$ is bounded for any $R>0$ since the exponential function $e^{-|z|^2/2R}$ decays faster than any polyomial that
3. Write $$p_{t,r}(z) n_{t}(z) = \left( p_{t,r}(z) \frac{n_t(z)}{n_{t/2}(z)} \right) n_{t/2}(z)$$ and conclude from step 2 that term in the brackets is bounded (in $z$).