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.


Introduction
The Kato class plays an important role in the theory of stochastic processes and in the theory of pseudo-differential operators that emerge as generators of stochastic processes. The definition of the Kato class may therefore 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 Khas'minskii Lemma the condition yields sufficient local regularity of the corresponding Schrödinger (Feynman-Kac) semigroup T t f (x) = E x [exp( t 0 q(X s )ds)f (X t )]. 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,). The fact that the Schrödinger operator is essentially self-adjoint and has bounded and continuous eigenfunctions is another consequence of (1), see [11], [31] and [17]. Applications of (1) to quadratic forms of Schrödinger operators are also known and we describe them shortly in Remark 12.
The condition (1) can be understood as a particular smallness 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 (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 P x (T B = 0) = 1, where Theorem 1.1. Let X be a Lévy process in R d . The conditions (1) and (2) are equivalent if and only if 0 is not regular for {0}.
To complete the picture we note that if 0 is regular for {0} and X is not a compound Poisson process, then we fully describe (1) and (2) in Theorem 4.6 and Theorem 4.12. The descriptions of (1) and (2) in the case of the compound Poisson process X are given in Proposition 3.8. To easily read the results of Section 4 we refer the reader to Definition 2 and Section 2.2.
A similar result is proposed by Carmona, Masters and Simon in [11, Theorem III.1] under additional assumption on the transition density of the Lévy process. However, the equivalence of (i) and (iii) from [11,Theorem III.1] which is claimed therein does not even hold for the Brownian motion in R and, more generally, for a one-dimensional unimodal Lévy process for which {0} is not polar (see Theorem 4.6). 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 resulting confusion reappears in the literature. For instance (1) and (3) of [16,Proposition 4.5] are not equivalent in general.
The special character of the one-dimensional case can also be seen in [24,Remark 3.1]. In [24, 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 [24,Theorem 3.2].
Corollary 1.2. Let X be a Lévy process in R d . For q : R × R d → R we have 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 (3) coincides with (1) and we obtain the following reinforcement of (1) to a time-space smallness condition.
Corollary 1. 3. Let X be a Lévy process in R d . Then (1) holds if and only if In view of the equivalence of (1) and (5) for every Lévy process (see Proposition 3.4 for other description of (1) true for Hunt processes) these conditions should be compared with (2) by its alternative form provided by Proposition 3.6 in a generality of a Hunt process, i.e, (2) holds if and only if for some (every) fixed t > 0. The closeness or possible differences between (1) and (2) are now more evident for Lévy processes through (5) and (6).
The variety of conditions we point out is due to possible applications where one can choose a version suitable to the knowledge on the process and derive a clear analytic description of the Kato condition (1). See also Theorem 4.14 and Theorem 4.15 for other conditions. For instance, in Example 1 we apply Theorem 1.1 and we make use of (6). On the other hand, by Theorem 1.1 and (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 usual that from (2) and (6) one learns about acceptable singularities of q.
A discussion of analytic counterparts of (1) should begin with 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 [ (7) was used by Kato [19] to prove by analytic methods that the operator −∆+q is essentially self-adjoint (see [20] for extensions to second order elliptic operators). The equivalence of (1) with (7) and (8) follow also by Theorem 1.1 (see [36]). The one-dimensional case is covered by Theorem 4.6 of this paper. A major contribution to the understanding of this subject in a general probabilistic manner is made by Zhao [36]. Zhao considers a Hunt process X = (Ω, 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 = inf{t > 0 : X 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 [36] on Figure 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 [36,Lemma 1]. Then (C2) coincides with (1) and as such becomes the principal object of our considerations. We explain the origin and the choice of (2) using the concept of λ-subprocess X λ , λ > 0, of the process X (see [4] for definition). We should first notice that (C2) (and (1)) holds for X if and only if it holds for X λ , λ > 0 (see Remark 10 and Definition 2). A similar statement is not true in general for (C1). In fact for the standard Brownian motion in R d , d 3, (C1) and (7) coincide, but for d = 2 or d = 1 the expectation in (C1) is infinite for constant non-zero q, while this never happens for (C2) (and (1)) and bounded q. This indicates that (C1) is too strong for a general equivalence result and therefore to characterize (1) we need to rely on the relations in Figure 1 for X λ , λ > 0. This results in (2) which is exactly (C1) for X λ , λ > 0. We also observe that (2) holds for X if and only if it holds for X λ ′ , λ ′ > 0 (see Remark 10 and Definition 2). To ultimately clarify the choice of X λ , λ > 0, 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 Lemma 2.8 and 2.9). 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 note that (H2) holds for every Lévy process, and (H1) holds if and only if X is not a compound Poisson process (see Remark 9). As already mentioned, the case of the compound Poisson process can be treated directly and we fully describe conditions (1) and (2) in Proposition 3.8. Thus, in the remaining cases, (H3) for X λ becomes decisive for a complete applicability of Figure 1 to X λ . By Proposition 2.14 the study of the expression 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 the function (see also Lemma 4.2) In Corollary 2.15 we obtain that if X is not a compound Poisson process, then (H3) holds for X λ if and only if {0} is polar, i.e., h λ (x) = 0, x ∈ R d (see Remark 1). We finally summarize the discussion of (H1)-(H3) in Proposition 2.16, which says that all hypotheses (H1), (H2) and (H3) are satisfied for X λ if and only if X is not a compound Poisson process and {0} is polar. Some of the above observations are made by Zhao [36] as motivation for general considerations on (C1), (C2) and (C3). Eventually, 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 (instead of X λ ) in the case of Lévy processes admitting symmetric density with additional assumption on the behaviour of the density time integrals [36,Lemma 5]. Afterwards he applies this to describe (1) for symmetric α-stable processes, d > α, and the relativistic process. We generalize [36,Lemma 5] in Theorem 4.15.
Note that Theorem 1.1 goes much beyond the results of [36] indicated by Figure 1 (see Remark 1 and 2). Nevertheless, we crucially combine the outcomes and methods of Zhao [36] with those of Bretagnolle [10], where the function h λ is examined. The study effects in a list of cases which we additionally classify according to non-degeneracy hypothesis (H0). 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 want to stress that the non-symmetric cases or those close to the compound Poisson process (without (H0)) are more delicate and require more precision.
The paper is organized as follows. In Section 2 we introduce the non-degeneracy hypothesis (H0) for a Lévy process and we use it to classify Lévy processes. This classification provides a detailed plan of our research. In the last part of Section 2 we prove preparatory results on 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 prove other general descriptions of both of those classes and we establish their initial relations for Lévy processes. Next, 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 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 [28]). The characteristic exponent ψ of X defined by E 0 e i x,Xt = e −tψ(x) equals where γ ∈ R d , A is a symmetric non-negative definite matrix and ν is a Lévy measure, i.e., . Further, if γ 0 = 0, A = 0 and ν(R d ) < ∞ then X is called a compound Poisson process (see [28,Remark 27.3]). We say that X is non-Poisson if X is not a compound Poisson process.
, and thus the following hold. If necessary we specify which a Lévy process we have in mind by adding a superscript, for instance h Z,λ is the function given by (10) 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. By M d×d we denote the set of all real d × d dimensional matrices. Consider Remark 3. Let X be a Lévy process in a linear subspace V of R d (see [28,Proposition 24.17]) and denote d 0 = dim(V ). Then there exists a rotation given by 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.
Remark 4. Let X be a Lévy process in R d and let S be a projection on a linear subspace V of R d . If {0} is polar for the process Y = SX then it is polar for X. Indeed, if X t + x = 0 then SX t + Sx = 0, thus inf{t > 0 : where T SX {0} is the first hitting time of {0} by the process SX. The opposite is not true in general.

Definition 1. We say that (H0) holds for
The hypothesis (H0) excludes compound Poisson process and some other processes in d > 1. We give a precise description in the following remark. iii) For d > 1, (H0) holds if and only if X is not a compound Poisson process and is not of the form (12), below.
Proof. The first part of i) plainly holds. For the proof of the second part we observe that if X is a compound Poisson process, then γ − R d z1 B(0,1) (z)ν(dz) = 0 and for any linear subspace and there exist a linear subspace ii) Y is either zero or a compound Poisson process with the Lévy measure vanishing on V , Proof. Since we assume that X is non-Poisson if (11) holds 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 restricted to a set B. By definition ψ = ψ Y + ψ Z , hence X = Y + Z and i), ii) and iii) are satisfied. The property iv) follows from [28,Proposition 24.17]. Conversely, if X is of the form (12) then its Lévy triplet is given by Proof. Let V be the smallest in dimension linear subspace in R d satisfying Now, let T be the projection on V and define Y = T X the projection of the process X on V .
Observe that by (H0) we have dim(V ) 2. We claim that there is no one-dimensional subspace W ⊂ V such that the projection of Y on W is a compound Poisson process. For the proof assume that there is such W and let S be the projection on W . Then Z = SY = ST X = SX is a compound Poisson process. By [28,Proposition 11.10] we have the following consequences. First, SAS = 0 and by Lemma 2.1 we obtain .
and by (13), which is a contradiction, because then (13)

Classification of Lévy processes
We now outline our workflow to analyze every Lévy process X. Though in what follows we assume that X is not a compound Poisson process, since we deal with this process separately. We start with X satisfying (H0). For d > 1 by Proposition 2.3 and Remark 1 we have h λ (x) = 0, x ∈ R d , which is satisfactory. According to Remark 6 and [10, Theoreme 3 and 6] one of the following excluding situations holds. For d = 1 and λ > 0 This translates equivalently into probabilistic properties of X, see [10,Theoreme 6,8] and Remark 13. We have (A) {0} is polar, (B) X has finite variation and non-zero drift, The analytic counterpart by means of characteristic exponent or Lévy triplet is (see [10, Theoreme 3, 7 and 8]) Next, we consider the case when (H0) does not hold. Since we assume that X is not a compound Poisson process, by Remark 5 and Proposition 2.2 we have d > 1 and X is given by (12). According to Remark 3 we can treat Z from the decomposition as a non-Poisson process in R and thus by [10] there are three excluding cases for Z: We could similarly reformulate these cases for Z but in proofs of Theorem 4.11 and Theorem 4.12 we just use the following description.
Thus Remark 3 and [10, Theoreme 7, 3] end this part of the proof. If {0} is not polar for Z, is not polar for X if and only if lim sup x→0 h λ (x) = 1.
is not polar for X then by Lemma 2.4 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 Proof. We just observe that T {0} = 0 if and only if T Z {0} = 0 on the set {Y s = 0 for all s ∈ [0, δ] for some δ > 0}, which is of measure one with respect to P 0 (see Remark 8).
The next observation follows by summarizing the above considerations, especially by Lemma 2.5. It will facilitate the discussion of (H3) in the next section.

Hypotheses (H1)-(H3)
We start with a general case of a Hunt process X on S with life-time ζ. In the proofs of Lemma 2.8 and 2.9 all objects corresponding to X λ , the λ-subprocess of X, are indicated with a bar, e.g., T B = inf{t > 0 : X λ t ∈ B}. and Since we may change sup t>0 with lim sup t→0 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.
Now, let S = R d be the Euclidean space and ζ = ∞. The following lemmas and corollary address the question whether 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 (14) are infinite. Since T r is non-increasing in r > 0 we have by the quasi-left continuity lim r→0 + X Tr = X T a.s. on {T < ∞}. On the other hand, by the right continuity we have X Tr ∈ B(x, r) and thus lim r→0 + X Tr = x a.s. on {T < ∞}. Finally, X T = x and consequently T T {x} a.s. on {T < ∞}.
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 Bm thus by Lemma 2.11 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(xn,1/n) and f r (y) f 1/n (x n ) > a − ε. Finally, by Lemma 2.10 and the dominated convergence theorem we obtain This ends the proof since ε > 0 was arbitrary.
We continue discussing (H1)-(H3) for a Lévy process X in R d .
Proof. By [28,Theorem 27.4] it suffices to consider compound Poisson process with non-zero 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 [28,Chapter 5 except for countably many t > 0. Proposition 2.14. Let X be a Lévy process in R d and λ > 0. For h λ defined by (10) we have Proof. By Lemma 2.9, B(x, r/2) ⊆ B(x, r) ⊆ B(x, r) and Lemma 2.12 As a consequence of Proposition 2.14, Remark 7 and Remark 1 we obtain the following improvement of [

Kato class
Let X be a Hunt process in R d . For t 0, λ 0 we define the transition kernel P t (x, dz) and λ-potential kernel G λ (x, dz) by The corresponding transition operator P t and λ-potential operator G λ are given by whenever the integrals exist. Moreover, we use the following (truncated) measures We unify the notation by putting G λ ∞ = G λ .
Definition 2. Let q : R d → R. We say that q ∈ K(X) if (1) holds, i.e., We say that q ∈ K(X) if for some λ > 0 (all λ > 0) (2) holds, i.e., If the process X is understood from the context we will write in short K, K for K(X), K(X), respectively.
In the next two lemmas we show that the definition of K is consistent. The first lemma is an apparent reinforcement of (2) and (18).
Proof. Let T = T B(x,r) . The strong Markov property leads to where θ denotes the usual shift operator. By the right continuity X T ∈ B(x, r) and B(x, r) ⊆ B(X T , 2r) on {T < ∞}. Thus eventually Lemma 3.2. If (2) or (18) holds for some λ 0 > 0, then it holds for every λ > 0.
Proof. Clearly, by the resolvent formula (see [4, Chapter 1, (8.10)]) it suffices to consider the This ends the proof due to Lemma 3.1.
Remark 10. Let λ > 0. Then K(X) = K(X λ ) and K(X) = K(X λ ). The first equality follows by the comparability of E x |q(X λ u )| = e −λu E x |q(X u )| and E x |q(X u )| for 0 u t. The other arises as a straightforward consequence of the definition of K(X). Now, we give alternative characterisations of K(X) and K(X).

Lemma 3.3. We have
Proof. Actually, the upper bound holds pointwise as follows, t 0 P s |q|(x)ds e t 0 e −s/t P s |q|(x)ds e G 1/t |q|(x).
We prove the lower bound, Recall from a general theory that 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. By Lemma 3.3, below we express (17) throughout the behaviour of λ-potential operators at infinity (λ → ∞). Proposition 3.4. q ∈ K(X) if and only if lim λ→∞ sup x∈R d G λ |q|(x) = 0.

Remark 11. 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 [23, Lemma 3.1]
where authors discuss the Kato class of measures for Markov processes possessing transition densities which satisfy the Nash type estimate (see also [24] for symmetric case). Later in Lemma 3.7 we also show that the assumption of uniform local integrability of V ([11, Theorem III.1]) is a necessary condition for V ∈ K(X) for any Lévy process X in R d .
We explain briefly the importance of Proposition 3.4.
We can use Lemma 3.1 to get a better insight into the result of Lemma 3.3.
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 As a consequence in Proposition 3.6 we obtain a description of K(X) by G 0 t (x, dz). This result is about cutting at time 0 < t < ∞ rather than taking λ = 0, since the measures G λ t (x, dz) and G 0 t (x, dz) are comparable if t is finite. This truncation in time is useful when the distribution P x (X s ∈ dz) is well estimated for s ∈ (0, t] near x ∈ R d . See [18], [ for some (equivalently for all) 0 < t < ∞, We use Proposition 3.6 in Example 1 below.
Lemma 3.7. Let X be a Lévy process in R d . Assume that there are t > 0 and 0 M < ∞ such that for all x ∈ R d , t 0 P s |q|(x) ds M .
We write q ∈ (L 1 loc ) uni (R d ) if (19) holds. At the end of this section we collect basic properties of K(X) and K(X) for a Lévy process X in R d . Proof. The inclusion K ⊆ (L 1 loc ) uni (R d ) follows by Lemma 3.7. To complete the proof of 1. we let q ∈ K(X), which reads as (C1) for X λ , λ > 0, and A t = t 0 |q(X λ s )|ds. Since by Remark 9 and Lemma 2.8 h 2 (X λ ) = 0 the hypothesis (H2) holds for X λ and we can apply the result of Zhao [36] presented on Figure 1. Thus, (C2) holds for X λ . This means q ∈ K(X λ ), and q ∈ K(X) follows by Remark 10. Plainly, 2. holds. Now, let X be non-Poisson. By Lemma 2.13 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 lim r→∞ sup x∈R d Br 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 we start we give a short proof of Corollary 1.2 directly from Theorem 1.1.

Proof of Corollary 1.2. Consider a Lévy process Y in
Since for Y 0 is not regular for {0} Theorem 1.1 applies to Y . Finally, we use (2) taking into account that and that e −λt is comparable with one for t ∈ [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 9 and Lemma 2.8 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 2.15 it suffices to justify that {0} is a polar set. For d > 1 this is assured by Proposition 2.3. For d = 1 it is our assumption.
From now on in this subsection we discuss the case of d = 1 and we analyze K(X λ ) rather than K(X) (see Remark 10). To this end we use G λ t (dz) , λ > 0, 0 < t < ∞, and observe that For convenience we recall from [10, Theoreme 7, 1, 5, 6 and 8] the following facts.
is absolutely continuous and its density G λ t (x) can be chosen to satisfy To prove the semi-continuity we observe that for x 0 ∈ R \ {0}, and by the bounded convergence theorem

Without (H0)
In this subsection we assume that (H0) does not hold. In view of Proposition 3.8 we assume that X is non-Poisson. Remark 5 and Proposition 2.2 imply then that d > 1 and X is given by (12). Thus the transition kernel of X equals The characteristic exponent ψ of X can be written as ψ = ψ Y +ψ Z . We note that We use results of Section 4.1 and analyze the cases (A'), (B') and (C').
Proof. By Remark 3 we assume that V = R and we observe that the Fourier transform of G Z,λ,n equals Since Re(1/z) = Re(z)/|z| 2 and Re[ψ] 0 we obtain .
This implies that the Fourier transform is integrable and (25) follows by the inversion formula.
where dv is the one-dimensional Lebesgue measure on V and Proof. By (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 3 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 G Z,λ (v) ε > 0 if v ∈ B(0, 1) ∩ V (ε may depend on λ). Thus, which proves the necessity. Further, the necessity of the condition proposed to describe K(X) follows by Remark 3, Lemma 3.7 and Since G λ (dz) G 0 (dz), (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, that is the implication from q ∈ K(X) to (26), and this can only to be the subcase of K(X) = K(X). Our basic assumptions will be that X is transient and {0} is polar (in Theorem 4.15 polarity will follow implicitly by other assumptions). The former one is necessary, otherwise G 0 (dz) is a locally unbounded measure (see [28,Theorem 35.4]) and non-zero constant functions do not belong to K 0 (X), but are always included in K(X). The polarity of {0} assures K(X) = K(X). Moreover, if {0} is not polar, then the class K(X) is described explicitly by our theorems. As shown in section 2.2 the polarity of {0} is to some extent encoded in the characteristic function ψ. It is even more so for the transience property of X (see [28,Remark 37.7]). Finally, we note that q ∈ K 0 (X) is equivalent to (C1) and q ∈ K(X) to (C2) (and with A t given by (9)). Thus according to Figure 1, we focus on showing (H3) for X (see also Remark 9).
Such statement is not true in general, but here it follows by P 0 (T B(x,r) < ∞) , Lemma 2.10 and lim t→∞ |X t | = ∞ P 0 a.s.
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 any f ∈ C c (R d ).
Remark 15. Observe that under a certain assumption on the group of the Lévy process [28,Definition 24.21] G 0 (dz) tends to zero for every transient X if d 2, see [28,Exercise 39.14]. As indicated in [28,Exercise 39.14] the case d = 1 is more complicated, see also Remark 18.
Lemma 4.13. Let X be transient. If G 0 (dz) tends to zero at infinity then Proof. The statement follows by the same proof as for Proposition 2.14 but with λ = 0 and a version of Lemma 2.12 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 (27) in place of Lemma 2.10 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 [28, Theorem 42.8 and Definition 41.6] for g ∈ C c (R d ) such that 1 B(0,1) g we get since m B(0,2) (dw) is finite and supported on B(0, 2) and G(dv) tends to zero at infinity. This contradicts (28) and ends the proof.
In the next result we improve [36, 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).

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 finally we present a number of 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 In short we say that φ satisfies WLSC(α, θ, c) and write φ ∈ WLSC(α, θ, c). Similarly, we consider θ ∈ [0, ∞). The weak upper scaling condition holds if there are numbers α ∈ R and C∈ [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 function (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 Lévy processes are characterized in [33] by isotropic unimodal Lévy measures ν(dx) = ν(x)dx = ν(|x|)dx. The distribution (transition probability) of X t has a radial non-increasing density p(t, x) on R d \ {0}, and atom at the origin, with mass exp[−tν(R d )] (no atom if ψ is unbounded).
Proof. We use [9,Lemma 4.2] with k(x) = t 0 p(s, x)ds and K(x) = G 0 t 0 (x). In what follows we assume that d 3 and since X is (isotropic) unimodal with an unbounded Lévy-Khintchine exponent thus the assumptions of Theorem 4.15 are satisfied by [28,Theorem 37.8] and radial monotonicity of G 0 . Hence K 0 (X) = K(X) = K(X). Under additional assumptions we strengthen this relation.