Kato Classes for Lévy Processes

We prove that the definitions of the Kato class through the semigroup and through the resolvent of the Lévy process in ℝd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {R}^{d}$\end{document} 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.

The definition of the Kato class may differ according to the underlying probabilistic or analytical problem. In the first case the primary definition of the Kato condition is Here q is a Borel function on the state space of the process X = (X t ) t 0 . As shown in [13, section 3.2] through the Khas'minskii Lemma the condition yields sufficient local regularity of the corresponding Schrödinger (Feynman-Kac) semigroup In particular, the existence of a density, strong continuity or strong Feller property are inherited under (1) from properties of the original semigroup P t f (x) = E x f (X t ) (for details and further results see [13,). Moreover, if we denote by L the generator of (P t ) t 0 , we expect the semigroup ( P t ) t 0 to correspond to L − q and to allow for the analysis of the Schrödinger operator H = −L + q [14]. A fact that the Schrödinger operator is essentially self-adjoint and has bounded and continuous eigenfunctions is another consequence of Eq. 1, see [11,32] and [18]. Applications of Eq. 1 to quadratic forms of Schrödinger operators are also known and we describe them shortly after Proposition 3.4.
The condition (1) can be understood as a smallness condition with respect to time. The alternative definition of the Kato condition is given by the following space smallness, for some λ > 0 (equivalently for every λ > 0; see Lemma 3.2). In this paper we obtain a precise description of the equivalence of Eqs. 1 and 2 for Lévy processes in R d , d ∈ N. In order to formulate the result we recall that a point x ∈ R d is said to be regular for a Borel set B ⊆ R d if where T B = inf{t > 0 : X t ∈ B} is the first hitting time of B. Theorem 1.1 Let X be a Lévy process in R d . The conditions (1) and (2) are NOT equivalent if and only if 0 is regular for {0}.
Complete and direct descriptions of Eqs. 1 and 2 in the case of the compound Poisson process are given in Proposition 3.8. When X is not a compound Poisson process and 0 is regular for {0} we fully describe (1) and (2) in Theorems 4.6 and 4.12. To move right away to Section 4 we recommend to read Definition 2 and Section 2.2 first. In Section 2.2 the reader will also find analytic characterization of the situation when 0 is regular for {0}.
In [11, Theorem III.1] Carmona, Masters and Simon declare that Eq. 1 can be expressed by Eq. 2 under additional assumptions on the transition density of the Lévy process. However, the general equivalence of (i) and (iii) from [11,Theorem III.1] that is claimed therein does not hold. As we show in Theorem 4.6 it fails for the Brownian motion in R and for those one-dimensional unimodal Lévy processes for which {0} is not polar. Recall that a Borel set B ⊆ R d is called polar if For example the function q(x) = ∞ k=1 2 k 1 (k,k+2 −k ) (x) satisfies (i), but fails to satisfy (iii) in [11,Theorem III.1] for such processes. The paper [11] was very influential and the mistake reappears in the literature. For instance (1) and (3) of [17,Proposition 4.5] are not equivalent in general.
The special character of the one-dimensional case can also be seen in [25,Remark 3.1]. In [25, Definition 3.1 and 3.2] the authors discuss the Kato class of measures for symmetric Markov processes admitting upper and lower estimates of transition density with additional integrability assumptions, see [25,Theorem 3.2]. Theorem 1.1 allows also for results on the time-dependent Kato class for Lévy processes in R d . Such a class is used for instance in [5,7,9,36,37]. See [31] for a wider discussion of the Brownian motion case, c.f. [31,Theorem 2].

Corollary 1.2 Let X be a Lévy process in
if and only if See Section 4 for the proof. If one uses Corollary 1.2 for time-independent q, i.e., let q : R d → R and put q(u, z) = q(z), then the quantity in Eq. 3 coincides with Eq. 1 and we obtain the following reinforcement of Eq. 1 to a time-space smallness condition.

Theorem 1.3 Let X be a Lévy process in R d . Then (1) holds if and only if
In view of the equivalence of Eqs. 1 and 5 for every Lévy process (see Proposition 3.4 for other description of Eq. 1 true for Hunt processes) these conditions should be compared with Eq. 2 by its alternative form provided by Proposition 3.6 in a generality of a Hunt process, i.e., for some (every) fixed t > 0. The closeness or possible differences between Eqs. 1 and 2 are now more evident for Lévy processes through Eqs. 5 and 6. The variety of conditions we point out is due to possible applications where one can choose a suitable version according to the knowledge about the process and derive a clear analytic description of the Kato condition (1). See also Theorems 4.14 and 4.15 for other conditions. For instance, in Example 1 we apply Theorem 1.1 and we make use of Eq. 6. On the other hand, by Theorem 1.1 and Eq. 2 we obtain that for a large class of subordinators (1) is equivalent to where φ is the Laplace exponent of the subordinator. See Section 5.2 for details. This is also usual that from Eqs. 2 and 6 one learns, like through Eq. 7, about acceptable singularities of q. Schrödinger perturbations of subordinators are interesting since they exhibit peculiar properties that indicate complexity of the matter. For instance, we easily see that if q is bounded, then P t f (x) c N P t f (x) for every t ∈ (0, N], x ∈ R, f 0. On the other hand, if −q 0 is time-independent and the above inequality holds for some N > 0 on the level of densities, then necessarily q ∈ L ∞ (R) (see [5,Corollary 3.4]). Nevertheless, perturbation techniques yield an upper bound by means of an auxiliary density for (unbounded) q from the Kato class if an appropriate 4G inequality for the transition density of the subordinator holds (see [5,Proposition 2.4]). Generators of subordinators generalize fractional derivative operators that are used in statistical physics to model anomalous subdiffusive dynamics (see [16]).
A discussion of analytic counterparts of Eq. 1 should contain the fundamental example of the standard Brownian motion in R d , d ∈ N. The famous result of Aizenman and Simon [1,Theorem 4.5] says that in this case (1) is equivalent to Here we also refer to Simon [ [20] to prove by analytic methods that the operator − + q is essentially self-adjoint (see [21] for extensions to second order elliptic operators). The equivalence of Eq. 1 with Eqs. 8 and 9 follows also from Theorem 1.1 (see [38]). The one-dimensional case is also covered by Theorem 4.6 of this paper.
In what follows we present and explain our main ideas in view of the literature. A major contribution to the understanding of the subject in a general probabilistic manner is made by Zhao [38]. Zhao considers a Hunt process X = ( , F t , X t , ϑ t , P x ) with state space (S, ρ) and life-time ζ , where S is a locally compact metric space with a metric ρ (see [4]). For a strong sub-additive functional A t of X, t 0, he discusses relations between the following three conditions in presence of three hypotheses on the process X, Here for any Borel set B in S, T B is the first hitting time of B, τ B = T S\B is the first exit time of B (we let inf ∅ = ∞) and B(x, r) = {y ∈ S : ρ(x, y) < r}, x ∈ S, r > 0. We present the main theorem of Zhao [38] on Fig. 1 below; for instance, under (H3), (C3) implies (C1).
In this paper we assume that A t , t 0, is the additive functional of the form and we note that any additive functional is a strong sub-additive functional; see [38,Lemma 1]. Then (C2) coincides with Eq. 1 and as such becomes the principal object of our considerations. We explain the origin and the choice of Eq. 2 using the concept of λ-subprocess X λ , λ > 0, of the process X (see [4] for the definition). We first notice that (C2) holds for X if and only if it holds for X λ (see Remark 9 and Definition 2). A similar statement is not true in general for (C1). For the standard Brownian motion in R d , d 3, (C2) in fact coincides with (C1), which gives rise to Eq. 8, yet for d = 2 or d = 1 the expectation in (C1) is infinite for constant non-zero q, whereas that never happens for (C2). This shows that (C1) for X is too strong for a general equivalence result. Therefore we rely on the relations of Fig. 1 for X λ , and then (C1) results in Eq. 2. We also observe that Eq. 2 holds for X if and only if it holds for X λ , λ > 0 (see Remark 9). To ultimately clarify the choice of X λ we note that h 1 (X λ ) = h 1 (X), h 2 (X λ ) = h 2 (X) and h 3 (X λ ) h 3 (X) (see Lemmas 2.10 and 2.11). We now restrict ourselves to the case of the Lévy process in R d . Besides being a Hunt process in R d , X is also translation invariant. We point out that (H2) holds for every Lévy process and (H1) holds if and only if X is not a compound Poisson process (see Remark 8). The case of the compound Poisson process is entirely described in Proposition 3.8. Thus, in the remaining cases, (H3) for X λ becomes decisive for understanding the confines of the applicability of Fig. 1 to X λ . By Proposition 2.15 the study of h 3 (X λ ) reduces to the analysis of the first hitting time of a single point set by the original Lévy process X. Namely, we consider (see also Lemma 4.2) Eventually, by Corollary 2.16 and Remark 8 we obtain the following characterization. Therefore Theorem 1.1 goes much beyond the range of [38]. The reason is that in our work we also investigate all the cases that are not covered by Fig. 1. Our initial study effects in a list that classifies Lévy processes according to a non-degeneracy hypothesis (H0) and specific properties of h λ , which is thoroughly examined by Bretagnolle [10] for one-dimensional non-Poisson Lévy processes. A full layout of our development is presented in Section 2.2. Theorem 1.1 results as a summary of Proposition 3.8 and 6 theorems of Section 4. We stress that the non-symmetric cases or those close to the compound Poisson process (without (H0)) are more delicate and require more precision.
In [38,Lemma 4] Zhao proposes sufficient conditions on X under which (H1)-(H3) are satisfied for X λ . He uses them to re-prove the result of Aizenman and Simon [1] for d 2.
He also verifies hypotheses (H1)-(H3) directly for X in the case of Lévy processes admitting rotationally symmetric transition density with additional assumption on the behaviour of the density integrated in time [38,Lemma 5]. Finally he applies that to describe (1) for symmetric α-stable processes, d > α, and the relativistic process. We generalize [38,Lemma 5] in Theorem 4.15.
The paper is organized as follows. In Section 2 we introduce the non-degeneracy hypothesis (H0) for a Lévy process. Next, we give a classification of Lévy processes that provides a detailed plan of our research. In the last part of Section 2 we prove results concerning hypotheses (H1)-(H3). In Section 3, for a Hunt process X, we define Kato classes K(X) and K(X) of functions q satisfying (1) and (2), respectively. We give other general descriptions of both of those classes and we establish their initial relations for Lévy processes. In Section 4 we prove the main description theorems for Lévy processes, separately under and without (H0). Section 4 ends with additional equivalence results involving the class K 0 (X) (see (26)). In Section 5 we present a supplementary discussion on isotropic unimodal Lévy processes and subordinators. The paper finishes with examples.

Preliminaries
Our main focus in this paper is on a (general) Lévy process X in R d (see [29]). The characteristic exponent ψ of X defined by E 0 e i x,X t = e −tψ(x) equals Further, if γ 0 = 0, A = 0 and ν(R d ) < ∞, then X is called a compound Poisson process (see [29,Remark 27.3]). We say that X is non-Poisson if X is not a compound Poisson process. Recall that E x F (X) = E 0 F (X+x) for x ∈ R d and Borel functions F 0 on paths. In particular h λ (x) = E (−x) e −λT {0} , and thus the following holds.
Proof By [29,Theorem 27.4] it suffices to consider compound Poisson process with nonzero drift. Let then ν and γ 0 be its Lévy measure and drift. According to the decomposition ν = ν d + ν c for discrete and continuous part (see [29,Chapter 5 except for countably many t > 0. We say that a Lévy process X is non-sticky if P 0 (τ {0} > 0) = 0, or equivalently that the hypothesis (H) from [10] holds. Lemma 2.1 reinforces remarks following [38, Lemma 3].

Remark 4 X is non-sticky if and only if X is non-Poisson.
If necessary we specify which Lévy process we have in mind by adding a superscript, for instance h Z,λ is the function given by Eq. 11 that corresponds to the process Z.

Non-Degeneracy Hypothesis (H0) for Lévy Processes
Before we introduce the main non-degeneracy hypothesis on a Lévy process X we recall the basic matrix notation. Let M be a matrix. We let M * to be the transpose of M and M(R d ) the range of M. We call M a projection if it is symmetric and M 2 = M. For a subset V by V ⊥ we denote the orthogonal complement of V in R d . We use the following fact.

Lemma 2.2 If A is symmetric non-negative definite and M
Remark 5 Let X be a Lévy process in a linear subspace V of R d (see [29,Proposition 24.17]) and denote d 0 = dim(V ). Then there exists a rotation given by a matrix O ∈ M d×d such that Y = OX is a Lévy process in R d 0 ; the correspondence between X and Y is one-to-one.

Lemma 2.3 Let X be a Lévy process in R d and be a projection. If {0} is polar for the
We give a precise probabilistic description of (H0).

Proposition 2.4 Let d > 1 and X be non-Poisson. Then (H0) does not hold if and only if
and there exist a linear subspace

either zero or a compound Poisson process with the Lévy measure vanishing on
Proof Since we assume that X is non-Poisson, if Eq. 12 holds and dim(V ) min{1, d −1}, then dim(V ) = 1. We let Y to be a compound Poisson process with the Lévy measure ν Y = [ν] R d \V and let Z to be a Lévy process with the Lévy triplet (A, γ − and by Eq. 12, which is a contradiction, because then Eq. 12 holds with

Classification of Lévy Processes
We outline our work-flow to analyze every Lévy process X.
Exclusively one of the following situations holds for a Lévy process in R d .
1. (H0) holds: 2. (H0) does not hold: (a) a compound Poissson process (d 1; then h λ (0) = 1), (b) given by (13) The comment in the case case 1(a) is a consequence of Proposition 2. The analytic counterpart by means of the characteristic exponent or the Lévy triplet is (see [10,Théorèmes 3,7 and 8]) We could similarly reformulate 2(b) for Z, but in proofs of Theorems 4.11 and 4.12 we use the following description.
Thus Remark 5 and [10, Théorèmes 7, 3] end this part of the proof. If {0} is not polar for Z, is not polar for X then by Lemma 2.6 it is not polar for Z and lim sup v∈V ,v→0 h Z,λ (v) = 1. This implies lim sup v∈V ,v→0 P 0 (T Z {v} < t) = 1 for every fixed t > 0. Thus we have for t > 0 which gives lim sup x→0 h λ (x) P 0 (Y t = 0)e −λt . Finally, we let t → 0 + .  The last observation facilitates a discussion of (H3) in the next subsection.

Hypotheses (H1)-(H3)
We start with a general case of a Hunt process X on S with life-time ζ . In the proofs of Lemmas 2.10 and 2.11 all objects corresponding to X λ , the λ-subprocess of X, are indicated with a bar, e.g., Proof Recall that inf ∅ = ∞. For any Borel set B in S and t > 0 we have and Since we may change sup t>0 with lim sup t→0 + , h 1 (X) h 1 (X λ ) h 1 (X) + lim t→0 + (1 − e −λt ) and since we may replace inf t>0 with lim inf t→0 + , h 2 (X) h 2 (X λ ) h 2 (X) + lim t→0 + (e −λt − 1). This ends the proof. (x,r) ) .
Proof Fix x ∈ R d . Define the stopping times T r = T B(x,r) and T = lim r→0 + T r , r > 0. Obviously, T r T T {x} . It suffices to consider (14) on the set {T < ∞}, otherwise both sides of Eq. 14 are infinite. Since T r is non-increasing in r > 0 we have by the quasi-left continuity lim r→0 + X T r = X T a.s. on {T < ∞}. On the other hand, by the right continuity we have X T r ∈ B(x, r) and thus lim r→0 + X T r = x a.s. on {T < ∞}. Finally, X T = x and consequently T T {x} a.s. on {T < ∞}.
It suffices to prove the reverse inequality in the case a = 0, otherwise (15) holds by Eq. 16. Thus let a ∈ (0, 1]. Then for ε > 0 there is u > 0 such that for all r > 0 we have sup |x| u f r (x) > a − ε. Hence, there is a sequence {x n } such that f 1/n (x n ) > a − ε and |x n | u. We will show that {x n } is bounded. For r ∈ (0, 1], m ∈ N and |x| m + 2, we have T B(x,r) τ m thus by Lemma 2.13 and the dominated convergence theorem there is m 0 such that This proves that m 0 + 2 |x n | u > 0 for every n. We let y = 0 to be the limit point of {x n }. Observe that for every r > 0 there is n such that B(x n , 1/n) ⊆ B(y, r), which implies T B(y,r) T B(x n ,1/n) and f r (y) f 1/n (x n ) > a − ε. Finally, by Lemma 2.12 and the dominated convergence theorem we obtain This ends the proof since ε > 0 was arbitrary.

Kato Class
Let X be a Hunt process in R d . For t 0 we define the transition kernel P t (x, dz) and the corresponding transition operator P t by Moreover, for λ 0 and t ∈ (0, ∞] we let to be the (truncated) λ-potential kernel and the (truncated) λ-potential operator G λ t , respectively. We simplify the notation by putting G λ (x, dz) = G λ ∞ (x, dz) and G λ = G λ ∞ .
We write q ∈ K(X) if Eq. 2 holds for some (every) λ > 0, i.e., If the process X is understood from the context we will write in short K, K for K(X), K(X). In the next two lemmas we show that the definition of K is consistent. The first one is an apparent reinforcement of Eqs. 2 and 18.
where θ denotes the usual shift operator. By the right continuity X T ∈ B(x, r) and This ends the proof due to Lemma 3.1. Now, we give alternative characterisations of K(X) and K(X). We easily observe that

Lemma 3.3 For λ > 0 and t ∈ [1/λ, ∞] we have
Proof Actually, the upper bound holds pointwise as follows, We prove the lower bound, Here is a conclusion from Eq. 19 and Lemma 3.3.

Proposition 3.4
The following conditions are equivalent to q ∈ K(X).
For resolvent operators R λ , λ > 0, of a strongly continuous contraction semigroup on a Banach space we have lim λ→∞ λR λ φ = φ. Thus lim λ→∞ R λ φ = 0 in the norm for every element φ of the Banach space. For a Markov process the counterparts of the resolvent operators are the λ-potential operators G λ ∞ . Proposition 3.4 extends the equivalence of (i) and (ii) of [11, Theorem III.1] from a subclass of Lévy processes to any Hunt process. Similar result is proved in [24,Lemma 3.1] where authors discuss the Kato class of measures for Markov processes possessing transition densities that satisfy the Nash type estimate (see [25] for the symmetric case). In Lemma 3.7 we also show that the uniform local integrability of V ([11, Theorem III.1]) is necessary for V ∈ K(X) for any Lévy process X in R d .
We briefly explain the role of Proposition 3.4. For the Brownian motion, as mentioned in [26] (see also [34]), by Stein's interpolation theorem the inequality sup [1,Theorem 4.9]). For a counterpart of such implication for other processes see remarks preceding [17,Theorem 4.10]. The latter inequality with γ < 1 allows to define a selfadjoint Schrödinger operator in the sense of quadratic forms, cf. [27,Theorem 3.17], the analogue of Kato-Rellich theorem.
We use Lemma 3.1 to get a better insight into the result of Lemma 3.3. t (x, dz) and

Lemma 3.5 For
Proof For a fixed y ∈ R d by Lemma 3.3 withq(z) = q(z)1 B(y,r) (z) we have Thus, by Lemma 3.1 we obtain The following is the aftermath of Eq. 19 and Lemma 3.5.
The above truncation in time is useful when the distribution P x (X s ∈ dz) is well estimated only for s ∈ (0, t] near every x ∈ R d . See [19], [ Remark 9 Let λ > 0. Then K(X) = K(X λ ) and K(X) = K(X λ ).

Lemma 3.7 Let X be a Lévy process in
Then there is a constant 0 M < ∞ independent of q such that Proof Let ϕ ∈ C 0 (R d ) be such that ϕ 0, ϕ = 1 on B(0, 1) and where 0 < ε h is such that P u ϕ − ϕ ∞ 1/2 for u ε (see [29,Theorem 31.5]).
Here C 0 (R d ) denotes the set of continuous functions f : Eq. 20 holds. By B(R d ) we denote the set of bounded (Borel) functions on R d . We collect basic properties of K(X) and K(X) for a Lévy process X in R d . Proof The inclusion K ⊆ L 1 unif (R d ) follows from Lemma 3.7. To complete 1. we let q ∈ K(X), which reads as (C1) for X λ , λ > 0. By Remark 8 and Lemma 2.10, (H2) holds for X λ and thus the result of Zhao on Fig. 1 implies that (C2) holds for X λ , i.e., q ∈ K(X λ ) = K(X) (see Remark 9). Plainly, 2. holds. Now, let X be non-Poisson. By Lemma 2.1 we get P t ({0}) = 0 for almost all t > 0 and consequently G λ ({0}) = 0. Further, since G λ (dx) is a finite measure, for q ∈ B(R d ) we have

Proposition 3.8 We have
Hence q ∈ K if and only if q ≡ 0. Moreover, which proves 4.

Main Theorems
In this section we consider a Lévy process X in R d and we pursue according to the cases of Section 2.2. Before that, we prove Corollary 1.2 directly from Theorem 1.1.

Proof of Corollary 1.2 Consider a Lévy process
Since for Y 0 is not regular for {0} Theorem 1.1 applies to Y . Finally, we use (2) taking into account that 1 B d+1 ((s,x),r) (s + u, X u ), where B d+1 (x, r) denotes a ball in R d+1 , can be replaced with 1 [0,r) (u)1 B(x,r) (X u ) and that e −λu is comparable with one for u ∈ [0, r).

Under (H0)
In this subsection we consider a Lévy process X satisfying (H0). Proof By Proposition 3.8 we concentrate on K(X) ⊆ K(X). Let q ∈ K(X) = K(X λ ), λ > 0. This reads as (C2) for X λ . Since X is non-Poisson, by Remark 8 and Lemma 2.10 the hypothesis (H1) holds for X λ . To obtain (C1) for X λ , that is to prove q ∈ K(X), it remains to verify (H3) for X λ . In view of Corollary 2.16 it suffices to justify that {0} is a polar set. For d > 1 this is assured by Proposition 2.5. For d = 1 it is our assumption.
From now on in this subsection we discuss the case of d = 1. For simplicity we recall from [10, Théorèmes 7, 1, 5, 6 and 8] the following facts. We investigate the properties of G λ t (dz), λ > 0, t ∈ (0, ∞). Proof According to Lemma 4.2 we define F λ (z) := G λ (z) on R \ {0} and F λ (0) := lim sup z→0 F λ (z). Then F λ (z) is a density of G λ (dz). Since G λ t (B) G λ (B) and is absolutely continuous and its density G λ t (x) can be chosen as To prove the lower semi-continuity of G λ t we observe that for x 0 ∈ R \ {0}, Then by continuity of F λ on R \ {0} and the bounded convergence theorem

Lemma 4.5 Let 0 be regular for
Further, G λ t (x) given by Eq. 21 is continuous on R and Proof Let F λ be defined as in the proof of Lemma 4.3. By Lemma 4.2 the functions G λ and F λ are equal and continuous on R. Further, Lemma 2.13 implies that the function Since h λ (x + y) h λ (x)h λ (y), x, y ∈ R (see remarks after [10, Lemma 2]), we get By positivity and continuity of h λ we obtain (22) with Eventually, by Eq. 21, Proof For K(X) we just observe that G λ (z) is bounded and G λ (z) ε > 0 if |z| 1. Now, we describe K(X). The condition q ∈ L 1 unif (R) is necessary by Lemma 3.7. We show that it is sufficient. Let λ > 0 and denote c t = λt Since f ∈ C 0 (R) we get c t → 0 as t → 0 + (see [29,Theorem 31.5]).

Without (H0)
In this subsection we assume that (H0) does not hold. In view of Proposition 3.8 we assume that d > 1 and X is given by Eq. 13. We use results of Section 4.1 and analyze the cases (A'), (B') and (C').
where dv is the one-dimensional Lebesgue measure on V and C = Proof By Eqs. 24 and 25 we have where the last equality follows by the translation invariance of the Lebesgue measure on V . This ends the proof.

Theorem 4.11 Under (B') we have
where dv is the one-dimensional Lebesgue measure on V .
Proof The condition postulated for the description of K(X) is sufficient by Lemma 4.10. Next, by Remark 5 and Lemma 4.2 the λ-potential kernel of Z, that is G Z,λ (dv) = G Z,λ,0 (dv), has a density G Z,λ (v) with respect to the Lebesgue measure on V , such that may depend on λ). Thus, which proves the necessity. Further, the necessity of the condition proposed to describe K(X) follows from Remark 5, Lemma 3.7 and which G 0 = G 0 ∞ can be used to describe K(X). The condition we want to analyze now is q ∈ K 0 (X) defined by Since G λ (dz) G 0 (dz), Eq. 26 implies q ∈ K(X) and thus K 0 (X) ⊆ K(X) ⊆ K(X) by Proposition 3.8. Our aim is to obtain the equivalence, i.e., the implication from q ∈ K(X) to Eq. 26, and this is the subcase of K(X) = K(X). We will assume that X is transient and {0} is polar (in Theorem 4.15 polarity follows implicitly from other assumptions). The transience is necessary, otherwise G 0 (dz) is locally unbounded (see [29,Theorem 35.4]) and non-zero constant functions do not belong to K 0 (X), which shows K 0 (X) K(X). The polarity of {0} assures K(X) = K(X). Moreover, if {0} is not polar, the class K(X) is explicitly described by our previous theorems. Both, transience and polarity of {0} are to some extent encoded in the characteristic exponent ψ (see [29,Remark 37.7] and Section 2.2). Finally, we note that q ∈ K 0 (X) is equivalent to (C1) and q ∈ K(X) to (C2). Thus according to Fig. 1 and Remark 8, we focus on showing (H3) for X.
We say that a measure G 0 (dz) tends to zero at infinity if lim |x|→∞ R d f (z+x)G 0 (dz) = 0 for all f ∈ C c (R d ) (i.e., f is continuous with compact support). Under certain assumptions on the group of the Lévy process [ Proof The statement follows by the same proof as for Proposition 2.15 but with λ = 0 and a version of Lemma 2.14 for λ = 0. To prove the latter one we also repeat its proof with functions f r extended to λ = 0, i.e., f r (x) = P 0 (T B(x,r) < ∞) up to a moment when a > 0 and a sequence {x n } such that f 1/n (x n ) > a − ε are chosen. The rest of the proof easily applies with Eq. 27 in place of Lemma 2.12 as soon as we can show that {x n } is bounded. To this end assume that the sequence is unbounded. Since f r (x) = P y (T B(x+y,r) < ∞), r > 0, y ∈ R d , for r ∈ (0, 1] and |x − x n | < 1 we have Next, by [29,Theorem 42.8] there is a finite measure ρ supported on B(0, 2) (see also [29,Definition 42.1]) such that for any g ∈ C c (R d ) satisfying 1 B(0,1) g we get since G 0 (dv) tends to zero at infinity. This contradicts (28) and ends the proof.
In the next result we improve [38, Lemma 5] and we cover some cases when G 0 (dz) may not tend to zero at infinity.

Theorem 4.15
Let X be transient and let G 0 (dz) have a density G 0 (z) with respect to the Lebesgue measure which is unbounded and bounded on |z| r for every r > 0. Then K 0 (X) = K(X) = K(X).
Proof We note that the polarity of {0} follows by our assumptions (see [ Next, for u > 0, |x| u and 0 < r < u/2 we obtain,

Further Discussion and Applications
In this section we give additional results for isotropic unimodal Lévy processes concerning (the implication) K(X) ⊆ K(X), we apply general results to a subclass of subordinators and we present examples. We recall from [6] the definition of weak scaling. Let θ ∈ [0, ∞) and φ be a non-negative non-zero function on (0, ∞). We say that φ satisfies the weak lower scaling condition (at infinity) if there are numbers α ∈ R and c ∈ (0, 1], such that φ(ηθ) ≥ cη α φ(θ) for η ≥ 1, θ > θ.

Isotropic Unimodal Lévy Processes
A measure on R d is called isotropic unimodal, in short, unimodal, if it is absolutely continuous on R d \ {0} with a radial non-increasing density (such measures may have an atom at the origin). A Lévy process X is called (isotropic) unimodal if all of its one-dimensional distributions P t (dx) are unimodal. Unimodal pure-jump Lévy processes are characterized in [35] by isotropic unimodal Lévy measures ν(dx) = ν(x)dx = ν(|x|)dx. The distribution of X t has a radial non-increasing density p(t, x) on with the convention that inf ∅ = ∞. The function is increasing and càglàd where finite. Notice that φ(φ − (u)) = u for u ∈ [0, φ(∞)] and φ − (φ(s)) s for s ∈ [0, ∞). Moreover, by the continuity of φ we have φ − (φ(s) + ε) > s for ε > 0 and s ∈ [0, ∞). We also define f * (u) = sup |x| u |f (x)| for f : R d → R.