Consider the kdv equation (from here) $$\left\{\begin{array}{l} \partial_{t} v+\partial_{x}^{3} v=0 \\ v(x, 0)=v_{0}(x) \end{array}\right.$$ Its solution can be written as $v(t,x)=S_t*v_0(x),$ where $x,t\in \mathbb{R}$ and $$S_{t}(x)=\frac{1}{\sqrt[3]{3 t}} A i\left(\frac{x}{\sqrt[3]{3 t}}\right).$$
I would know already know that the following estimates hold:
\begin{aligned} &|A i(x)| \leq \frac{1}{\left(1+x_{-}\right)^{1 / 4}} e^{-c x_{+}^{3 / 2}}, \\ &\left|A i^{\prime}(x)\right| \leq\left(1+x_{-}\right)^{1 / 4} e^{-c x_{+}^{3 / 2}} \end{aligned}
however, I was wondering if in general one knows how to bound the $k$-th derivative of the Airy function for $k\in \mathbb{N}$ and $k\geq 2?$
