for $r>0$ sufficiently small with respect to $\bar{r}, ||h||_{C^{m_0+1}(V_{\bar{r}}K)}, \rho', \varpi, m_0, C'$ and $\delta'$, the function $h$ is $(\kappa,C,\delta,m_0-1)$-steep on $V_{2r}K$ with $$\kappa=\varpi/2, \quad C=C'/2, \quad \delta=r.$$

The proof is completely analogous to the proof of Proposition \[propsteep1\], so we do not repeat the argument.

Proof of the main results {#s4}
=========================

In this section, we give the proofs of Theorem \[Th1\] and Theorem \[Th2\]. Theorem \[Th1\] follows from Theorem \[thmMP\], Proposition \[propsteep1\] and a version of Nekhoroshev estimates for perturbations of steep integrable Hamiltonians in Gevrey classes, which is stated as Theorem \[thmNek\] in Appendix \[app\]. Theorem \[Th2\] follows from Theorem \[thmP\], Proposition \[propsteep2\] and Theorem \[thmNek\].

Recall that we are considering $H \in \mathcal{H}^\alpha_L(\omega)$, with $\omega$ satisfying , and that we are assuming that $H_{m_0} \in SS(n,m_0)$, for $m_0:=[n^2/2+2]$. We can apply Theorem \[thmMP\]: setting $\beta=\alpha(1+\tau)+1$, there exist positive constants $\bar{r},L_1,L_2,A$, which depend only on $n$, $\gamma$, $\tau$, $\alpha$, $L$ and $||H||_{\alpha,L,B}$ and a symplectic transformation $$\Phi : \T^n \times B_{\bar{r}} \rightarrow \T^n \times B$$ whose components belong to $G^{\alpha,\beta}_{L_1,L_2}(\T^n \times B_{\bar{r}})$, such that $$H \circ \Phi(\theta,I)=H_*(I)+R_*(\theta,I), \quad H_* \in G^{\beta}_{L_2}(B_{\bar{r}}), \quad R_* \in G^{\alpha,\beta}_{L_1,L_2}(\T^n \times B_{\bar{r}})$$ and with the following properties:

-   for any $(\theta,I) \in 