Small-time sharp bounds for kernels of convolution semigroups

We study small-time bounds for transition densities of convolution semigroups corresponding to pure jump L,vy processes in R (d) , d ae<yen> 1, including the processes with jump measures which are exponentially and subexponentially localized at a. For a large class of L,vy measures, not necessarily symmetric or absolutely continuous with respect to Lebesgue measure, we find the optimal upper bound in both time and space for the corresponding heat kernels at a. In case of L,vy measures that are symmetric and absolutely continuous with densities g such that g(x) aei f(|x|) for non-increasing profile functions f, we also prove the full characterization of the sharp two-sided transition densities bounds of the form This is done for small and large x separately. Mainly, our argument is based on new precise upper bounds for convolutions of L,vy measures. Our investigations lead to a surprising dichotomy correspondence of the decay properties at a for transition densities of pure jump L,vy processes. All results are obtained solely by analytic methods, without use of probabilistic arguments.