  We study lower and upper bounds for the density of a diffusion process in
${\mathbb{R}}^n$ in a small (but not asymptotic) time, say $\delta$. We assume
that the diffusion coefficients $\sigma_1,\ldots,\sigma_d$ may degenerate at
the starting time $0$ and point $x_0$ but they satisfy a strong H\"ormander
condition involving the first order Lie brackets. The density estimates are
written in terms of a norm which accounts for the non-isotropic structure of
the problem: in a small time $\delta$, the diffusion process propagates with
speed $\sqrt{\delta}$ in the direction of the diffusion vector fields
$\sigma_{j}$ and with speed $\delta=\sqrt{\delta}\times \sqrt{\delta}$ in the
direction of $[\sigma_{i},\sigma_{j}]$. In the second part of this paper, such
estimates will be used in order to study lower and upper bounds for the
probability that the diffusion process remains in a tube around a skeleton path
up to a fixed time.
