I'm trying to follow the proof of the Paley-Wiener-Schwartz theorem, but I don't know how do I obtain the following inequality:
$$\sum_{|\alpha|\leq N} \sup_{x\in K} |D^\alpha(\chi_\delta e^{-i<x,\zeta>})| \leq C' \exp(H(Im (\zeta))+\delta|Im(\zeta)|)\sum_{\alpha\leq N} \delta^{-|\alpha|}(1+|\zeta|)^{N-|\alpha|}$$
Where
$H$ is the supporting function of the convex compact set K.
$\chi_\delta \in C^{\infty}_0(K_\delta)$, $\chi_\delta=1$ in $K_{\delta/2}$
$K_\delta=\{x+y; x\in K,|y|\leq \delta\}$
I also have a estimative for $|D^\alpha \chi_\delta|\leq C_\alpha \delta^{-|\alpha|}$
I tried expanding the derivative in the left side using the Leibniz formula, but I can't see where come the $\exp(\delta |Im(\zeta)|$ term in the right side of the inequality.
By the Leibniz formula, \begin{align} |D^\alpha (\chi_\delta(x) e^{-i\langle x, \zeta \rangle})| &= \left| \sum_{\beta \leq \alpha} \binom{\alpha}{\beta} \left(\partial^{\alpha-\beta} e^{-i\langle x, \zeta \rangle}\right) \left( \partial^{\beta}\chi_\delta (x) \right) \right|\\ &\leq \sum_{\beta \leq \alpha} \binom{\alpha}{\beta} |\zeta^{\alpha-\beta}| |e^{-i\langle x, \zeta \rangle}| |\partial^{\beta}\chi_\delta (x)|\\ &\leq C_\alpha \, 2^{|\alpha|}\, e^{\langle x, \text{Im}\zeta\rangle}\sum_{\beta \leq \alpha} |\zeta|^{|\alpha|-|\beta|} \delta^{-|\beta|}. \end{align} Taking the supremum over $x \in K_\delta$ gives \begin{align} \sup_{x \in K_\delta}|D^\alpha (\chi_\delta(x) e^{-\langle x, \zeta \rangle})| &\leq C_\alpha' \exp\left({H_{K_\delta}(\text{Im}\zeta)}\right) \sum_{\beta \leq \alpha} \ |\zeta|^{|\alpha|-|\beta|} \delta^{-|\beta|}\\ &\leq C_\alpha' \exp\left({H_{K}(\text{Im}\zeta) + |\delta||\text{Im}\zeta|}\right) \sum_{\beta \leq \alpha} |\zeta|^{|\alpha|-|\beta|} \delta^{-|\beta|}. \end{align} Can you finish?