fix $\gamma>0$. Assume that  is satisfied. Then there exist positive constants $\bar{\varepsilon}$, $L_1'$, $L_2'$, $A'$ and $E$, which depend only on $n$, $\sigma$, $\gamma$, $\tau$, $\alpha$, $L$, $||h||_{\alpha,L,\bar{D}}$, $||f||_{\alpha,L,\bar{D}}$ and on the condition , such that for $\varepsilon \leq \bar{\varepsilon}$, there exists a symplectic transformation $$\Psi : \T^n \times D \rightarrow \T^n \times D$$ whose components belong to $G^{\alpha,\beta}_{L_1',L_2'}(\T^n \times D)$, with $\beta=\alpha(1+\tau)+1$, such that $$H \circ \Psi(\theta,I)=h_*(I)+f_*(\theta,I), \quad h_* \in G^{\beta}_{L_2'}(D), \quad f_* \in G^{\alpha,\beta}_{L_1',L_2'}(\T^n \times D)$$ and a diffeomorphism $\omega : D \rightarrow \Omega$ whose components belong to $G^{\beta}_{L_2'}(D)$, with the following properties:

-   $||\Psi-\mathrm{Id}||_{C^1(\T^n \times D)}\leq E\sqrt{\varepsilon}$ and $||\Psi^{-1}-\mathrm{Id}||_{C^1(\T^n \times D)}\leq E\sqrt{\varepsilon}$, where $||\,\cdot\,||_{C^1(\T^n \times D)}$ denotes the $C^1$-norm on $\T^n \times D$;

-   $||\partial_I^l\omega(I)-\partial_I^l\nabla h(I)||\leq EL_2'^{|l|}l!^\beta \sqrt{\varepsilon}$ for all $I \in D$ and $l \in {\mathbb{N}}^n$;

-   setting $J_{\gamma,\tau}':=\omega^{-1}(\Omega_{\gamma,\tau}')$, we have $\partial_I^l\omega(I)=\partial_I^l\nabla h_*(I)$ for all $I \in J_{\gamma,\tau}'$ and $l \in {\mathbb{N}}^n$;

-   we have $\partial_I^l f_*(\theta,I)=0$ for all $(\theta,I)\in \T^n \times J_{\gamma,\tau}'$ and all $l\in {\mathbb{N}}^n$: hence for any $r>0$ sufficiently small so that $V_{2r}J_{\gamma,\tau}' \subseteq D$, $$||f_*||_{\alpha,\beta,L_1',L_2',V_{2r}J_{\gamma,\tau}'} \leq A'\exp\left(-(2L_2'r)^{-\frac{1}{\alpha(1+\tau)}}\right).$$

This is exactly the content of Theorem 1.1 and Corollary 1.2 of [@Pop04], to which we refer for a proof.

Stably steep polynomials {#s3}
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