For any $\beta \in [0, \frac{1}{2}]$, the fractional derivative of the Airy function satisfies
\begin{align*}
\||D|^\beta \mathrm{Ai}\|_{L^\infty(\mathbb{R})} = \frac{1}{\sqrt{2\pi}} \sup_{x \in \mathbb{R}} \left|\int_{\mathbb{R}} |\xi|^\beta e^{i(x\xi + \xi^3)}\, d\mathcal{L}^1(\xi)\right| < \infty,
\end{align*}
where $\mathrm{Ai}(x) = \frac{1}{\sqrt{2\pi}} \int_{\mathbb{R}} e^{i(x\xi + \xi^3)}\, d\mathcal{L}^1(\xi)$ is the Airy function.
As a consequence, the KdV propagator $e^{-t\partial_x^3}$ satisfies the dispersive decay estimate
\begin{align*}
\||D|^\beta e^{-t\partial_x^3} f\|_{L^\infty(\mathbb{R})} \le C\, t^{-(1+\beta)/3}\, \|f\|_{L^1(\mathbb{R})}
\end{align*}
for all $\beta \in [0, \frac{1}{2}]$, $t > 0$, and $f \in L^1(\mathbb{R})$.