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A compound Poisson process whose jump measure and intensity are unknown isobserved at finitely many equispaced times. We construct a purely data-drivenestimator of the L\'evy density $\nu$ through the spectral approach usinggeneral Calderon--Zygmund integral operators, which include convolution andprojection 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 useLepski\u{i}'s method as it is particularly well-suited for our generality.Thus, novel exponential-concentration inequalities are proved including one forthe uniform fluctuations of the empirical characteristic function. These are ofindependent interest, as are the proof-strategies employed to deal with generalCalderon--Zygmund operators, to depart from the ubiquitous quadratic structureand to show robustness without polynomial-tail conditions. Part of themotivation for such generality is a new insight we include here too that,furthermore, allows us to unify the main two approaches to construct estimatorsused in related literature.

Adaptive nonparametric estimation for compound Poisson processes robust to the discrete-observation scheme