Kato Classes for Lévy Processes

Abstract

We prove that the definitions of the Kato class through the semigroup and through the resolvent of the Lévy process in 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 (Lévy-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.

Heat kernels of non-symmetric Lévy-type operators

“…Thus the Kato class for X and Y is the same. The function Kato classes that consist of absolutely continuous measures are for Lévy processes well studied [30]. Remark 1.7.…”

“…In view of (41) and (30) it suffices to consider the following expression for δ > 0 as t → s + , By (38) for any ε > 0 there is δ > 0 such that |q(s, z, y) − q(s, x, y)| < ε if |z − x| < δ. Together with Proposition 2.1 and Lemma 5.6 we get I 1 cε.…”

Majorization, 4G Theorem and Schrödinger perturbations

“…Last but not least, it may be cumbersome to point out p * suitable for p and q, because this essentially requires guessing the rate of inflation ofp. In this connection we note that (5) trivially yields (12) Thus, for perturbations of p by ηq ≥ 0 with 0 ≤ η < 1 one may take p * =p, hence estimatingp and finding an appropriate majorant p * are closely related problems.…”

Fall-Off of Eigenfunctions for Non-Local Schrödinger Operators with Decaying Potentials

“…It is known that(2.24) is equivalent to the property that lim η→∞ (−L + η Id) −1 |V | ∞ = 0 (see e.g [24,. Prop.…”