  We characterize the local smoothness and the asymptotic growth rate of the
L\'evy white noise. We do so by characterizing the weighted Besov spaces in
which it is located. We extend known results in two ways. First, we obtain new
bounds for the local smoothness via the Blumenthal-Getoor indices of the L\'evy
white noise. We also deduce the critical local smoothness when the two indices
coincide, which is true for symmetric-alpha-stable, compound Poisson, and
symmetric-gamma white noises to name a few. Second, we express the critical
asymptotic growth rate in terms of the moment properties of the L\'evy white
noise. Previous analyses only provided lower bounds for both the local
smoothness and the asymptotic growth rate. Showing the sharpness of these
bounds requires us to determine in which Besov spaces a given L\'evy white
noise is (almost surely) not. Our methods are based on the wavelet-domain
characterization of Besov spaces and precise moment estimates for the wavelet
coefficients of the L\'evy white noise.
