Let $\varepsilon > 0$, $\Omega = \lbrace \zeta \in \mathbb{C}^n ; |Im\zeta| < \varepsilon |Re\zeta| \rbrace$ and $a(\zeta)$ a holomorphic function on $\Omega$ satisfying an estimate $|a(\zeta)| \leq C(1+|\zeta|^2)^{m/2}$ for large $\zeta $in $\Omega$. By using Cauchy’s integral formula, show that $(1-\varphi(\xi))a(\xi) \in S^m$ for some function $\varphi \in C^\infty_0$ with $\varphi = 1 $ near $\xi =0.$ Here $S^m$ means the set of all functions $a \in C^\infty (\mathbb{R}^{2n})$ satisfying for any $\alpha$ and $\beta$ (multi index) there exists C > 0 such that for any$(x,\xi)\in \mathbb{R}^n \times \mathbb{R}^n$
$ | \partial^\alpha_\xi \partial^\beta_x a(x,\xi)| \leq C|1+|\xi|^2|^{(m-|\alpha|)/2}$
I try to solve this problem using Cauchy’s integral formula for derivatives, and I get $ | \partial^\alpha_\xi \partial^\beta_x a(x,\xi)| \leq |\alpha !|\oint C(1+|z|^2)^{m/2}dz /(2\pi)^n | \prod_{k=1}^n|(z_k- \xi_k)^{\alpha_k +1}|$.
However, I cannot solve after this. Please help me.