Analytic Smoothing and Nekhoroshev estimates for H\"older steep Hamiltonians

In this paper we prove the first result of Nekhoroshev stability for steep Hamiltonians in H\"older class. Our new approach combines the classical theory of normal forms in analytic category with an improved smoothing procedure to approximate an H\"older Hamiltonian with an analytic one. It is only for the sake of clarity that we consider the (difficult) case of H\"older perturbations of an analytic integrable Hamiltonian, but our method is flexible enough to work in many other functional classes, including the Gevrey one. The stability exponents can be taken to be $(\ell-1)/(2n{\mathbf{\alpha}}_1...{\mathbf{\alpha}}_{n-2})+1/2$ for the time of stability and $1/(2n{\mathbf{\alpha}}_1...{\mathbf{\alpha}}_{n-1})$ for the radius of stability, $n$ being the dimension, $\ell >n+1$ being the regularity and the ${\mathbf{\alpha}}_i$'s being the indices of steepness. Crucial to obtain the exponents above is a new non-standard estimate on the Fourier norm of the smoothed function. As a byproduct we improve the stability exponents in the $C^k$ class, with integer $k$.