Green function for gradient perturbation of unimodal L\'evy processes in the real line

We prove that the Green function of a generator of symmetric unimodal L\'evy processes with the weak lower scaling order bigger than one and the Green function of its gradient perturbations are comparable for bounded $C^{1,1}$ subsets of the real line if the drift function is from an appropriate Kato class.


Introduction
Perturbations of Markovian generators are widely studied from many years. This theory may be considered from various points of view. Such perturbations appear, e.g., in local and non-local partial differential equations [10,11,32,33], semigroup theory [8,29,25,4,7], stochastic processes [24,30,30], potential theory [9,12,16]. One of the natural question is: how this perturbation affects the solutions of the equations related to the unperturbed operator (e.g., the transition density of the semigroup, the Green function).
In this paper we are interested in the gradient perturbations and the potential theory of the perturbed operator. We briefly recall some results closely related to our research. Cranston and Zhao in [15] considered the operator ∆+b(x)∇ in R d for d 2. They proved that the Green function and the harmonic measure of Lipschitz domains are comparable with those of ∆ for the drift b from the appropriate Kato class. In [23] and [24] Jakubowski studied the α-stable Ornstein-Uhlenbeck process. He proved estimates for the first exit time from the ball and Harnack inequality for this process. In [9] Bogdan and Jakubowski proved similar results as Cranson and Zhao for ∆ α/2 +b(x)∇ in C 1,1 domains in R d , d 2.
In the recent paper [18] these results were generalized to the case of pure-jump symmetric unimodal Lévy processes possessing certain weak scaling properties. We note that in the papers [9,18] the case d = 1 was omitted. The aim of this paper is to fill this gap and prove analogous results in one dimensional case.
We will denote by {X t } a pure-jump symmetric unimodal Lévy process on R. That is, a process with the symmetric density function p t (x) on R \ {0} which is non-increasing on R + . The characteristic exponent of {X t } equals where ν is a Lévy measure, i.e., R (1 ∧ |z| 2 ) ν(dz) < ∞. For general information on unimodal processes, we refer the reader to [5,17,34]. A primary example of the mentioned class of processes is the symmetric α-stable Lévy process having the fractional Laplacian ∆ α/2 as a generator. Let be a generator of the process X t . We will consider a non-empty bounded open C 1,1 set D and the Green function G D (x, y) for L. Now, letG D (x, y) be a Green function for where b is a function from the Kato class K 1 (see Section 2 for details). Our main result is Theorem 1.1. Let b ∈ K 1 , and let D ⊂ R be an union of finitely many open intervals with positive distance between every two intervals. We assume that the characteristic exponent ψ ∈ WLSC(α, 0, c) ∩ WLSC(α 1 , 1, c 1 ) ∩ WUSC(α, 0, C), where α 1 > 1, Then, there exists a constant C such that for x, y ∈ D, Here WLSC and WUSC are the classes of functions satisfying a weak lower and weak upper scaling condition, respectively (see Section 2 for definitions). Set D should be considered as an one-dimensional case of a bounded C 1,1 set, see Definition 4. Generally, we follow the approach of [9] and [18], however there are some important differences. Although the geometry of the set D is much simpler than in higher dimensions, it seems that the one dimensional case sometimes demands more delicate arguments. One of the main difficulties are the proper estimates for derivative of the Green function. As it was mentioned in [9], for d = 1, the available estimates are not integrable near y. We emphasize here that we make no additional assumption on the monotonicity of ν ′ (r)/r as mentioned above. Like in the mentioned papers, our mail tool is the perturbation formula. First, we use it to obtain estimates for sets D with a small radius. Since the Green function G D (x, y) is bounded, we do not use the perturbation series as in [9] and [18]. Instead, we propose a simpler iteration argument. We note also that one of our standing assumptions is α 1 > 1. It may be understood that the rank of the operator L is larger than 1. Without this assumption the drift term may have the stronger effect than L on the behavior of the Green function of theL. Any results concerning the cases α 1 would be interesting, however for α < 1, Theorem 1.1 cannot hold in the form above (see the Introduction of [9] for more details) The paper is organized as follows. In Section 2 we define the process X and present its basic properties. In Section 3, we introduce the Green function of X, prove the estimates for its derivative and some 3G-like inequalities. In Section 4, we define the operatorL and the Green function of the underlying Markov process. Lastly, in Section 5, we prove Theorem 1.1.
When we write f (x) C ≈ g(x), we mean that there is a number 0 < C < ∞ independent of x, i.e. a constant, such that for every x, C −1 f (x) g(x) Cf (x). If the value of C is not important we simply write f (x) ≈ g(x). The notation C = C(a, b, . . . , c) means that C is a constant which depends only on a, b, . . . , c.
We use a convention that numberded constants denoted by capital letters do not change throughout the paper. For a symmetric function f : R → [0, ∞) we shall often write f (r) = f (x) for any x ∈ R with |x| = r.

Preliminaries
In what follows, R denotes the Euclidean space of real numbers, dy stands for the Lebesgue measure on R. Without further mention we will only consider Borelian sets, measures and functions in R. As usual, we write a ∧ b = min(a, b) and a ∨ b = max(a, b). We let B(x, r) = {y ∈ R : |x − y| < r}. For the arbitrary set D ⊂ R, the distance to the complement of D, will be denoted by Definition 1. 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 > 0 and c ∈ (0, 1] such that φ(λθ) cλ α φ(θ) for λ 1, θ > θ. (2.1) In short, we say that φ satisfies WLSC(α, θ, c) and write φ ∈ WLSC(α, θ, c). If φ ∈ WLSC(α, 0, c), then we say that φ satisfies the global weak lower scaling condition.
Throughout the paper, X t will be the pure-jump symmetric unimodal Lévy process on R. The Lévy measure ν of X t is symmetric and non-increasing, so it admits the density ν, i.e., ν(dx) = ν(|x|)dx. Hence the characteristic exponent ψ of X t is radial as well.
We consider a compensated potential kernel By symmetry and [1,Theorem II.19], the monotone convergence theorem implies By [20, Proposition 2.2], K is subadditive.
Definition 2. We say that a function b : R → R belongs to the Kato class K 1 if for small r > 0, in this paper we will use the condition (2.15) in the form We consider the time-homogeneous transition probabilities By Kolmogorov's and Dinkin-Kinney's theorems the transition probability P t define in the usual way Markov probability measure {P x , x ∈ R} on the space Ω of the right-continuous and left-limited functions ω : [0, ∞) → R. We let E x be the corresponding expectations. We will denote by X = {X t } t 0 the canonical process on Ω, X t (ω) = ω(t). Hence, For any open set D, we define the first exit time of the process X t from D, Now, by the usual Hunt's formula, we define the transition density of the process killed when leaving D ( [2], [14], [6]): We briefly recall some well known properties of p D (see [6]). The function p D satisfies the Chapman-Kolmogorov equations Furthermore, p D is jointly continuous when t = 0, and we have In particular,

Green function of L
We define the Green function of X t for D, and the Green operator From now on, every time we will mention the Green function, it should be understand as a Green function of D, and then G = G D .
We say that a function f is a regular L−harmonic (or simply regular harmonic) function on an open bounded set Definition 4 corresponds with the definition of multidimensional C 1,1 set with localization radius r 0 . In what follows, we assume that D is a C 1,1 set with diam D < ∞ and localization radius r 0 = r 0 (D).
Some constants will depend on the ratio diam D/r 0 called the distortion of the set D.
Proof. Note that (see [6,Proposition 4.4 and Theorem 4.5]), where the comparability constant c 1 depends on the scaling characteristics in (2.6) and (2.7) and a distortion of D. Now, by the same calculation as in the proof of [13, Theorem 7.3 (iii) and Corollary 7.4], we obtain where the comparability constant c 2 depends on the scaling characteristics in (2.6) and (2.7), a distortion of D and 1 ∨ diam(D).
Let us consider x, y ∈ D such that Without a loss of generality we may and do assume δ x δ y . Then, . By monotonicity and subadditivity of V we obtain that As a consequence of (2.6), we obtain Again, by (2.6), we get where c 3 = c 3 (α, c). Now, let We only need to show that 2|x − y| 2 δ x δ y . Without the loss of generality we can and do assume δ x < δ y . By monotonicity of V , |x − y| δ x . The case δ y |x − y| is obvious.

Estimates of the Poisson kernel
If D is C 1,1 , it is known that the harmonic measure of D has a density and we call it the Poisson kernel. By the Ikeda-Watanabe formula [22] it is equal to Lemma 3.2. Let R > 0 and B = B(0, R). Then Proof. By (3.7), Lemma 3.1 and (2.10), there is c 1 = c 1 (σ, 1 ∨ R) such that By Remark 1, we obtain inequality (2.7) for r < 3R with constant c 2 = c 2 (α 1 , c 1 , 1∨R). Hence, for |z| < 2R, we have Here, c 3 = c 3 (c 1 , c 2 , α 1 ) and Sα 1 S refers to the symmetric α-stable process with index of stability α 1 . Similarly, we obtain where c 4 = c 4 (c 1 , α, C). By formula for P Theorem A], we get the assertion of the lemma for |z| < 2R.
If |z| 2R, by (2.5) and [19, Proposition 3.5], we get which implies the claim of the lemma.
where c 2 = c 2 (C 2 , C 3 ). If dist(z, B) = δ z , the lower bound follows by (3.9). Suppose dist(z, B) < δ z and letB be a connected component ofD such that dist(z,B) = δ z . Therefore, by Lemma 3.1, Now, (2.10) and (2.5) imply Hence, we obtain the lower bound in this case. Next, we will prove the upper bound for the second integral. Let λ = δ z ∧ diam(D) and D 1 =D ∩ {y : δ y λ} and D 2 =D ∩ {y : δ y > λ}. By weak scaling conditions, we obtain where c 5 = c 5 (C 2 , C) and c 6 depends only on the scaling characteristics. This completes the proof.

Estimates of ∂ x G(x, y)
Below, we will prove various estimates of ∂ x G(x, y) according to the range of variables x and y. We summarize these results in Theorem 3.10. First, we will need the following auxiliary lemma.
By Lemma 3.1 and Proposition 3.5, we get a weaker but also useful estimate.
Proof. Let 0 < h < δ y /2. Then, Since f ∈ K 1 and the integrand is uniformly in h integrable on and (0, 1) × D, which ends the proof.
Let us fix y ∈ A. For z ∈ B, we define P 1 (y, z) = P A (y, z)1 B(x,ε/2) (z) and P 2 (y, z) = P A (y, z) − P 1 (y, z). Since P 1 is bounded, we have P 1 ∈ K 1 and by Lemma 3.7, Since ∂ x G(x, z) is finite on the support of P 2 (y, ·), by the mean value theorem and the dominated convergence theorem, we get These imply which completes the proof.
Lemma 3.9. Let x, y ∈ D and δ x < 2|x − y|. Then, there exists a constant C 9 = C 9 (σ, diam(D)/r 0 , 1 ∨ diam(D)) such that Proof. Let B ⊂ D be any interval such that B ⊂ D and put A = B c ∩ D. For any x ∈ B and y ∈ D such that x = y, by Propositions 3.8 and 3.5 and harmonicity of G, Therefore, it remains to estimate the integral B M(|x − z|)P A (y, z)dz. (3.14) Let B = B(x, δ x /4). By the assumption y ∈ B, dist(y, B) δ x /4 and |y − z| ≈ |x − y| for z ∈ B. Denote δ A x = dist(x, ∂A). Note that δ A x ≈ δ x and δ A y ≈ δ y . By Proposition 3.3 and Lemmas 3.4, 3.1, we get Since constants c 1 − c 4 depend on D only via constants C 2 , C 4 and C 6 , the proof is completed.
Proof. Due to Corollary 3.6, Lemma 3.9, (2.13) and (2.14) it remains to prove existing of a constant c such that when |x − y| δ x /2. But in this case δ x ≈ δ y and therefore, by Lemma 3.1, Since α 1 > 1, by (2.7), we obtain that s → V 2 (s)/s is almost increasing (bounded from below by an increasing function). Hence, we get the claim.
We end this section we the proof of the uniform intergability of ∂ z G(z, y).
Lemma 3.11. The function ∂ z G(z, y) is uniformly in y integrable against |b(z)|dz.
Proof. It is enough to show that We may assume that the set D is an union of k distinctive intervals. By Proposition 3.5 and monotonicity of M(·), we have By (2.15), lim N →∞ K r N = 0, which completes the proof.

3G inequalities
Now, we apply the estimates of the Green function and its derivative to obtain the following 3G-type inequalities.

Green function ofL
Following [8] and [26] we recursively define, for t > 0 and x, y ∈ R, p 0 (t, x, y) = p(t, x, y) , where c T → 1 if T → 0, see [8,Theorem 2]. By Chapman-Kolmogorov equation, there is C 14 > 0 such that We letP,Ẽ be the Markov distributions and expectations defined by transition densitỹ p on the canonical path space. By Hunt formula, (4.4) Except symmetry,p D has analogous properties as p D , i.e. the Chapman-Kolmogorov equation holds R dp D (s, x, z)p D (t, z, y)dz =p D (s + t, x, y) , s, t > 0, x, y ∈ R, 0 p D (t, x, y) p(t, x, y) andp D is jointly continuous on (0, ∞) × D × D.
We denote byG D (x, y) the Green function ofL = L + b∂ on D, As for G, from now on, every time we will mention the Green functionG, it should be understand as a Green function ofL on D, and thenG =G D . By Blumenthal's 0-1 law and (4.
Thus the intensity of jumps of the canonical process X t underP x is the same as under P x . Accordingly, we obtain the following description. We define the Poisson kernel of D forL, By (4.5), (4.7) and (4.6) we havẽ For the case of A ⊂ ∂D, we refer the reader to Lemma 5.8.

Lemma 4.2.G(x, y) is continuous and
Proof. In the same way as in [9,Lemma 7] we get that there are constants c and C such thatp is well defined, integrable and bounded on R. Hence, following [18, Theorem 3.1], we obtain the following perturbation formula (for the proof see [18]). First, we will prove the comparability of G andG for small sets D from the C 1,1 class. For this purpose we could consider the perturbed series forG as it was presented in [18]. We could define by induction the functions G n and show the convergence and estimates of the seriesG However, sinceG is bounded, we present a simpler proof of the following lemma (compare [18,Lemma 3.11]).
We note that the comparison constants in the proof above will improve to 1 if diam(D) → 0 and the distortion of D is bounded. By (4.8), where C 16 = C 16 (σ, b, λ, diam(D)) and diam(D) < ε from Lemma 5.1. Following [9, Proof of Lemma 14], we obtain that the boundary of our general C 1,1 open set D is not hit at the first exit, i.e.
Hence, in the context of Lemma 5.1, theP x distribution of X τ D is absolutely continuous with respect to the Lebesgue measure, and has density functioñ We say that a function f is regularL−harmonic on an open bounded set D ⊂ R, if for every Following [9] and [18], we get the following Harnack inequality.
Now, we have all the tools necessary to prove the main result of our paper. Since in the proof we follow the idea from [9], we only give its basic steps (for details see [9, Proof of Theorem 1]).
To prove (1.2) we will consider x and y in a partitions of D × D. First, we consider y far from the boundary of D, say δ y ρ/4.
Next, suppose that δ D (y) ρ/4. Here, the difficulty lies in the factG is non-symmetric.
In the proof of lower bounds we consider two cases: x close to y and x far away from y.
• For |x − y| ρ, consider the same set F as above. We havẽ G D (x, y) =G F (x, y) + D\FP F (x, z)G D (z, y) dz .
The proof of Theorem 1.1 is complete.