Transition densities of one-dimensional Levy processes

In this paper, we study the existence of the transition densities of one-dimensional L\'evy processes. Compared with past results, our results contain the L\'evy processes whose L\'evy symbols have logarithm behavior at infinity. Our results contain the L\'evy symbol induced by the following Laplace exponent $\psi(\xi) := (\ln(1 + \ln(1 + \ln(\cdots \ln(1 + |\xi|)))))^\epsilon$ ($n$ times), $0 < \epsilon < 1$, $2 \le n$. We also show that $\psi(\xi)$ is a L\'evy symbol with transition density.