We consider the problem of estimating the density of the process associated with the
small jumps of a pure jump Lévy process, possibly of infinite variation, from discrete
observations of one trajectory. The interest of such a question relies on the observation
that even when the Lévy measure is known, the density of the increments of the small
jumps of the process cannot be computed. We discuss results both from low and high
frequency observations. In a low frequency setting, assuming the Lévy density associated
with the jumps larger than 1 in absolute value is known, a spectral estimator relying on the
deconvolution structure of the problem achieves minimax parametric rates of convergence
with respect to the integrated L2 loss. In a high frequency setting it is possible to remove
the assumption that the Lévy measure of the large jumps is known. In that case the rate
of convergence depends on the sampling scheme and on the behaviour of the Lévy measure
in a neighborhood of zero. An adaptive penalized procedure is also proposed to select the
cutoff parameter.