  For $n$ equidistant observations of a L\'evy process at time distance
$\Delta_n$ we consider the problem of testing hypotheses on the volatility, the
jump measure and its Blumenthal-Getoor index in a non- or semiparametric
manner. Asymptotically as $n\to\infty$ we allow for both, the high-frequency
regime $\Delta_n=\frac1n$ and the low-frequency regime $\Delta_n=1$ as well as
intermediate cases. The approach via empirical characteristic function unifies
existing theory and sheds new light on diverse results. Particular emphasis is
given to asymptotic separation rates which reveal the complexity of these
basic, but surprisingly non-standard inference questions.
