Kato classes for L\'evy processes

Abstract

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.