Gevrey estimate of derivatives

Question

Let $\phi$ be the $C^\infty$ function defined on $\mathbb R_+^*$ by $e^{-t^{-2}}$ and by $0$ on $\mathbb R_-$.

Question: I think that there exists $\rho>0$ such that $$ \forall t\in \mathbb R,\forall n\in \mathbb N,\quad\vert{\phi^{(n)}(t)}\vert\le (n!)^{3/2} \rho^{1+n}, \tag{$\ast$} $$ but I do not have a simple proof. I would like to know if an elementary argument could provide the above estimates.

As an interesting byproduct of this global Gevrey estimate of order $3/2$, there is this nice counter-example by A.N. Tychonov violating Cauchy uniqueness for the heat equation with $$ u(x,t)=\sum_{n\ge 0}\phi^{(n)}(t)\frac{x^{2n}}{(2n)!},\quad \partial_t u-\partial_x^2 u=0, \quad u_{\vert t\le 0}=0. $$ Of course to prove convergence in $C^\infty$ of the series defining $u$, some estimates are needed and $(\ast)$ is sufficient.

Answer 1

Faa di Bruno's formula for derivatives of compositions of functions says $$ (f\circ g)^{(n)}(t)=n!\sum_{k\ge 0}\ \sum_{n_1,\ldots,n_k\ge 1} \mathbf{1}\left\{\sum_{i=1}^{k} n_i=n\right\} \ \frac{f^{(k)}(g(t))\ g^{(n_1)}(t)\cdots g^{(n_k)}(t)}{k!\ n_1!\cdots n_k!} $$ where $\mathbf{1}\{\cdots\}$ stands for the indicator function of the condition within braces. For $f(x)=e^{-x}$ and $g(t)=\frac{1}{t^2}$, we of course have $f^{(n)}(x)=(-1)^n e^{-x}$ and $g^{(n)}(t)=(-1)^n (n+1)!\ t^{-(n+2)}$ for the derivatives. When $n\ge 1$, we thus get $$ \phi^{(n)}(t)=e^{-t^{-2}} n! \sum_{k\ge 1} \sum_{n_1,\ldots,n_k\ge 1} \mathbf{1}\left\{\sum_{i=1}^{k} n_i=n\right\} \frac{(-1)^{n+k}t^{-(n+2k)}}{k!} (n_1+1)\cdots (n_k+1) $$ Now use the trivial bound $\frac{x^m}{m!}\le e^x$ for $x=t^{-2}$ (followed by taking the square root) and for $x=\frac{1}{2}t^{-2}$ in order to get $$ t^{-n}\le \sqrt{n!}\ e^{\frac{1}{2}t^{-2}} $$ and $$ t^{-2k}\le 2^k\ k!\ e^{\frac{1}{2}t^{-2}}\ . $$ This results in $$ |\phi^{(n)}(t)|\le n!^{\frac{3}{2}}\times \sum_{k\ge 1}\ \sum_{n_1,\ldots,n_k\ge 1} \mathbf{1}\left\{\sum_{i=1}^{k} n_i=n\right\} \ 2^k\ (n_1+1)\cdots (n_k+1)\ . $$ By the arithmetico-geometric inequality and the elementary bound $1+x\le e^x$, we have $$ (n_1+1)\cdots (n_k+1)\le \left(\frac{n+k}{k}\right)^k\le e^n\ . $$ Therefore, $$ |\phi^{(n)}(t)|\le n!^{\frac{3}{2}}\ e^n\times \sum_{k\ge 1}\ 2^k\ \sum_{n_1,\ldots,n_k\ge 1}\mathbf{1}\left\{\sum_{i=1}^{k} n_i=n\right\} $$ $$ =n!^{\frac{3}{2}}\ e^n\times \sum_{k=1}^{n}\ 2^k\ \left( \begin{array}{c} n-1 \\ k-1 \end{array} \right) =2 n!^{\frac{3}{2}} e^n 3^{n-1}\ . $$ So your bound holds with $\rho=3e$.