  This paper is concerned with adaptive kernel estimation of the L\'evy density
N(x) for bounded-variation pure-jump L\'evy processes. The sample path is
observed at n discrete instants in the "high frequency" context (\Delta =
\Delta(n) tends to zero while n\Delta tends to infinity). We construct a
collection of kernel estimators of the function g(x)=xN(x) and propose a method
of local adaptive selection of the bandwidth. We provide an oracle inequality
and a rate of convergence for the quadratic pointwise risk. This rate is proved
to be the optimal minimax rate. We give examples and simulation results for
processes fitting in our framework. We also consider the case of irregular
sampling.
