  A compound Poisson process whose jump measure and intensity are unknown is
observed at finitely many equispaced times. We construct a purely data-driven
estimator of the L\'evy density $\nu$ through the spectral approach using
general Calderon--Zygmund integral operators, which include convolution and
projection kernels. Assuming minimal tail assumptions, it is shown to estimate
$\nu$ at the minimax rate of estimation over Besov balls under the losses
$L^p(\mathbb{R})$, $p\in[1,\infty]$, and robustly to the observation regime
(high- and low-frequency). To achieve adaptation in a minimax sense, we use
Lepski\u{i}'s method as it is particularly well-suited for our generality.
Thus, novel exponential-concentration inequalities are proved including one for
the uniform fluctuations of the empirical characteristic function. These are of
independent interest, as are the proof-strategies employed to deal with general
Calderon--Zygmund operators, to depart from the ubiquitous quadratic structure
and to show robustness without polynomial-tail conditions. Part of the
motivation for such generality is a new insight we include here too that,
furthermore, allows us to unify the main two approaches to construct estimators
used in related literature.
