  The distributional support of the sample paths of L\'evy processes is an
important issue for the construction of sparse statistical models, theories of
integration in infinite dimensions and the existence of generalized solutions
of stochastic partial differential equations driven by L\'evy white noise. Here
one considers a family K_\alpha (0<\alpha<2) of L\'evy processes which have no
support in S'. For 1<\alpha<2 they are supported in K', the space of
distributions of exponential type and for 0<\alpha=<1 on similar spaces of
power exponential type.
