Let $f : \mathbb C \to \mathbb C$ be an entire function belonging to the Fock space $F\_\alpha^2$, that is,
$$
\int\_\mathbb{C} |f(z)|^2 e^{-\alpha|z|^2} \, dA(z)
$$
with $A$ the Euclidean are measure. To which space does the complex derivative of $f$ belong,
$$
f' \in F\_\beta^2
$$
$\beta > 0$. Can we choose $\beta = \alpha$?



To which space does the derivative of a function in Fock space belong?
It is true for all $\beta>\alpha$. Use Cauchy estimate
$$|f'(z)|\leq \frac{1}{2\pi}\int\_{-\pi}^\pi |f(z+e^{it})|dt,$$
then Cauchy Schwarz,
$$|f'(z)|^2\leq \frac{1}{2\pi}\int\_{-\pi}^\pi|f(z+e^{it})|^2dt,$$
and substitute to your integral. You obtain
$$\int\_{\mathbf{C}}\int\_{-\pi}^\pi e^{-\beta|z|^2}|f(z+e^{it})|^2dt\, dA\_z.$$
Exchange the order of integration and make the change of the variable $w=z+e^{it}$ in the inner integral. Then use the fact that
$$\exp(-\beta|w+e^{it}|^2)\leq \exp(-\alpha|w|^2)$$
when $|w|$ is sufficiently large.


