  Existing results for the estimation of the L\'evy measure are mostly limited
to the onedimensional setting. We apply the spectral method to multidimensional
L\'evy processes in order to construct a nonparametric estimator for the
multivariate jump distribution. We prove convergence rates for the uniform
estimation error under both a low- and a high-frequency observation regime. The
method is robust to various dependence structures. Along the way, we present a
uniform risk bound for the multivariate empirical characteristic function and
its partial derivatives. The method is illustrated with simulation examples.
