Kato classes for L\'evy processes

We prove that the definitions of the Kato class by the semigroup and by the resolvent of the L\'{e}vy process on $\mathbb{R}^d$ coincide if and only if 0 is not regular for {0}. If 0 is regular for {0} then we describe both classes in detail. We also give an analytic reformulation of these results by means of the characteristic (L\'{e}vy-Khintchine) exponent of the process. The result applies to the time-dependent (non-autonomous) Kato class. As one of the consequences we obtain a simultaneous time-space smallness condition equivalent to the Kato class condition given by the semigroup.Comment: 30 pages. We have shortened some argument

Estimates of gradient perturbation series

We give upper and lower bounds of perturbation series for transition densities, corresponding to additive gradient perturbations satisfying certain space-time integrability conditions

Regularity of fundamental solutions for L\'evy-type operators

For a class of non-symmetric non-local L\'evy-type operators $\mathcal{L}^{\kappa}$, which include those of the form \mathcal{L}^{\kappa}f(x):= \int_{\mathbb{R}^d}( f(x+z)-f(x)- 1_{|z|<1} \left)\kappa(x,z)J(z)\, dz\,, we prove regularity of the fundamental solution $p^{\kappa}$ to the equation $\partial_t =\mathcal{L}^{\kappa}$