  Using a new strategy, we extend the classical Nekhoroshev's estimates to the
case of H\"older regular steep near-integrable hamiltonian systems, the
stability times being polynomially long in the inverse of the size of the
perturbation. We prove that the stability exponents can be taken to be
$(\ell-1)/(2n\alpha_1...\alpha_{n-2})$ for the time of stability and
$1/(2n\alpha_1...\alpha_{n-1})$ for the radius of stability, $\ell >n+1$ being
the regularity and the $\alpha_i$'s being the indices of steepness. Our
strategy consists in deriving a perturbation theory which exploits a sharp
analytic smoothing theorem to approximate any H\"older function by an analytic
one. In addition, an appropriate choice of the free parameters in the problem
enables us to have a first grasp on the relation connecting the time and radius
of stability to the threshold that the size of the perturbation must satisfy in
order for the theorem to apply. Particular attention is payed to a geometric
presentation of the construction of the so-called "resonant blocks", in order
to shed a definitive light on the nature of the steepness condition. We also
investigate the convex setting, using a similar approach.
