Inequality: multivariate normal distribition

Question

Let $p(u,x)=\frac{1}{(4\pi u)^{q/2}}e^{-|x|^2/(4u)},u>0,x \in \mathbb{R}^q.$

1. Prove that for $r\geq 0,c>1$ there exists $C>0$ (depending on $r,c$) such that $$\forall x \in \mathbb{R}^q,u>0,\frac{|x|^{2r}}{u^r}p(u,x) \leq Cp(cu,x).$$
2. Deduce that for $n \in \mathbb{N}^q,k \in \mathbb{N},$ there exists $C'>0$ such that $|\partial_x^n\partial_u^k p(u,x)| \leq C' u^{-|n|/2-k}p(cu,x)$ where $\partial^n_x=\partial x_1^{n_1}...\partial x_q^{n_q}$

1 follows from $\frac{|x|^{2}}{4ur}(1-\frac{1}{c}) \leq e^{\frac{|x|^2}{4ur}(1-1/c)}$ and we choose $C=c^{q/2} 4^r(r+1)^r/(1-1/c)^r$

How to deduce 2 (for simplicity we can take $q=1)$? Is it possible to prove the inequality using induction? If so, how?

Answer 1

$\newcommand\p\partial$By induction, $$\p_u^k p(u,x)=\frac1{u^k}\,P_k\Big(\frac{|x|^2}u\Big)p(u,x) =\frac K{u^{k+q/2}}\,P_k\Big(\frac{|x|^2}u\Big)e^{-|x|^2/(4u)}, \tag{1}\label{1}$$ where $K$ is a real number not depending on $u$ or $x$, and $P_k(z)$ is a polynomial (with coefficients not depending on $u$ or $x$).

Next, for any polynomial $P$ (in $q$ variables) and any $j=1,\dots,q$, there is another polynomial $Q_j$ (in $q$ variables) such that $$\p_{x_j} \Big(P\Big(\frac{x}{u^{1/2}}\Big)e^{-|x|^2/(4u)}\Big) =\frac1{u^{1/2}}\,Q_j\Big(\frac{x}{u^{1/2}}\Big)e^{-|x|^2/(4u)}.$$ So, in view of \eqref{1},for some polynomial $R$ (in $q$ variables) and all real $u>0$, $$|\p_x^n\p_u^k p(u,x)| =\frac1{u^{|n|/2+k+q/2}}\,\Big|R\Big(\frac{x}{u^{1/2}}\Big)\Big|e^{-|x|^2/(4u)} \\ \le \frac C{u^{|n|/2+k+q/2}}\,e^{-|x|^2/(4cu)} =\frac{C_1}{u^{k+|n|/2}}\,p(cu,x),$$ where $C$ and $C_1$ are positive real numbers not depending on $u$ or $x$. $\quad\Box$