  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 $k$.
