Martin representation and Relative Fatou Theorem for fractional Laplacian with a gradient perturbation

Let $L=\Delta^{\alpha/2}+ b\cdot\nabla$ with $\alpha\in(1,2)$. We prove the Martin representation and the Relative Fatou Theorem for non-negative singular $L$-harmonic functions on ${\mathcal C}^{1,1}$ bounded open sets.

From the probabilistic point of view, stable stochastic processes with gradient perturbations on R d , d ≥ 2, i.e. with the innitesimal generator where α ∈ (0, 2), constitute an important class of jump processes, intensely studied in recent years. Their most celebrated case are the Ornstein-Uhlenbeck stable processes with b(x) = λx, λ ∈ R. They have important physical and nancial applications and form a part of Lévy-driven Ornstein-Uhlenbeck processes, cf. [12,34]. The motivations of this paper were to: (i) establish the theory of the Martin representation for singular L-harmonic non-negative functions (ii) study boundary limit properties of L-harmonic functions and to obtain a Relative Fatou Theorem for them (iii) develop the theory of Hardy spaces of L-harmonic functions.
The topics (i) and (ii) are addressed in this article and the subject (iii) in a forthcoming paper.
All these topics are fundamental for the knowledge of L-harmonic functions. The topics (i) and (ii) are well developed for fractional Laplacians. The Martin representation was established in this case in [6,10,15,33]. The Relative Fatou Theorem was proved for α-harmonic functions on Lipschitz sets in [31], see also the survey [7,Chapter 3] and [25]. Also some important variants of stable processes such as relativistic and truncated stable processes were studied from the point of view of topics (i) and (ii), see [7,Section 3.4], [26] and [28]. Nevertheless, the methods of these extensions do not apply to the operator L of the form (1). Let us notice that all our results are also true for Ornstein-Uhlenbeck stable processes.
Let us mention that the methods of this article give also interesting new results for operators dierent from L. In the case of Laplacians with gradient perturbations, i.e. α = 2, we get new perturbation formulas for the Green function and the Martin and Poisson kernels (Section 3.4).
The potential theory of stable stochastic processes with gradient perturbations was started in the OrnsteinUhlenbeck case by T. Jakubowski [22,23]. Next, in the general context of gradient perturbations and α > 1, with a function b from the Kato class K α−1 d it was developed by K. Bogdan and T. Jakubowski [8,9]. This work is a natural continuation of the research presented in [9].
In particular, we send the reader to [9] for the denitions of the fractional Laplacian, C 1,1 sets, Green functions and Poisson kernels, with respect to both operators ∆ α/2 and L. The denitions of α-harmonic, regular α-harmonic and singular α-harmonic functions can be found e.g. in the monography [7, page 61] and are analogous for L-harmonic functions.
Throughout this paper, like in [9], we suppose 1 < α < 2, unless stated otherwise. We consider an open set D of class C 1,1 and a vector eld b The potential theory objects related to the operator L dened in (1) will be denoted with a tilde˜, while those related to the operator ∆ α/2 will be denoted without it. In particularG D is the Green function of D for L and G D is the Green function of D for ∆ α/2 . We x throughout this paper a point x 0 ∈ D and dene the Martin kernel of D for ∆ α/2 by The L-Martin kernel is dened bỹ and we show in Section 3 its existence. The starting point of the research contained in this paper are the following mutual estimates of Green functions and Poisson kernels for L and ∆ α/2 (see [9, Theorem 1 and (72)]). Comparability Theorem. There exists a constant C = C(α, b, D) such that for all x, y ∈ D and z ∈ (D) c , One of the main elements of the proof of (2) is the following perturbation formula, that will be also very useful in our present work (see [9,Lemma 12]).
Perturbation formula for Green functions. Let x, y ∈ R d , x = y. We haveG We start our paper by proving in Section 2.1 a generalization of the Comparability Theorem: according to Lemma 1, the constant C in the estimates (2) may be chosen the same for sets D r suciently close to D. The same phenomenon holds also for the Poisson kernelsP D (x, y) and P D (x, y). In Section 2.2 we prove a uniform integrability result, that will be needed in proving the main results of the paper, contained in Sections 3 and 4.
In Section 3 we develop the Martin theory of L-harmonic functions. We prove the existence of the L-Martin kernel which is L-harmonic (Theorems 8 and 12). Next we obtain the Martin representation of singular L-harmonic non-negative functions on D, see Theorem 13.
The formula (4) allows us to prove very useful perturbation formulas for Martin kernels (14), Poisson kernels (17) and singular α-harmonic functions (28). In Section 3.4, (4) and (14) are proved in the diusion case α = 2. Also a perturbation formula (33) for the L-Poisson kernel is derived. Section 4 is devoted to an important ne boundary property of singular Lharmonic functions: the Relative Fatou Theorem (Theorem 23). We provide a proof of this theorem based on the perturbation formula for singular αharmonic functions (28).

Preparatory results
In this section we prove some results, interesting independently, that will be useful in proving the main results of the paper, coming in the next sections.

Uniform comparability of Green functions and Poisson kernels
In what follows, R d denotes the Euclidean space of dimension d ≥ 2, dy stands for the Lebesgue measure on R d . Without further mention we will only consider Borelian sets, measures and functions in R d . By x · y we denote the Euclidean scalar product of x, y ∈ R d . Writing f ≈ g we mean that there is a constant C > 0 such that C −1 g ≤ f ≤ Cg. As usual, a ∧ b = min(a, b) and In what follows D is a bounded C 1,1 open set.
When r is suciently small, then D r is also a C 1,1 open set, see [32,Lemma 5], and one may show that the localization radius of D r varies continuously with respect to r.
In the sequel we will often use the estimates of the Green function ( [29], [14], see also [21]) of a C 1,1 open set and of the Martin kernel ( [15]) Moreover, in the stable case, the estimates (5) are uniform when we consider the sets D r suciently close to D, i.e. there exist constants c, 0 > 0 depending only on D and α such that for all r ∈ [0, 0 ] and x, y ∈ D r we have ≤ c|y − z| α−d δ Dr (y) α/2 δ Dr (z) α/2 |y − z| α ∧ 1 , see [21,Theorem 21] and [32, Lemma 5]. We will now show analogous uniformity of constants for the fractional Laplacian with a gradient perturbation. Lemma 1. (i) There exist constants c, 0 > 0 depending only on D and α such that for all r ∈ [0, 0 ] and x, y ∈ D r we have (ii)There exist constants C, 0 > 0 depending only on D and α such that for all r ∈ [0, 0 ], x ∈ D r and y ∈ D c r we have Proof. In order to show (i) we follow the proof of the Theorem 1 in [9]. We analyse below its crucial points.
1. Comparison of Green functionsG S (x, y) and G S (x, y) for "small" sets S, [9, Lemma 13], based on estimates from [9, Lemma 11]. Thanks to property (7), we see that the comparison of Green functionsG Sr (x, y) and G Sr (x, y) for small sets S holds with a common constant c, when r ∈ [0, 0 ].

2.
Harnack inequalities for L and the Boundary Harnack Principle, [9,Lemmas 15,16]. Thanks to 1., we get them uniformly with respect to r ∈ [0, 0 ]. 3. Now the proof of (i) for any C 1,1 open set D is the same as in Section 5 of [9].
The part (ii) is implied by (i), applying the Ikeda-Watanabe formula for the Poisson kernelP D , see [9,Lemma 6 and (39)]. Recall that the Lévy system for the processX t is given by the Lévy measure of the α-stable process X t .
An immediate consequence of Lemma 1 and [21,Theorem 22] is the following estimate of the Poisson kernel of L for D r . Corollary 2. There exist positive constants C, 0 depending only on D, α and b such that for all r ∈ [0, 0 ], x ∈ D r and y ∈ D c r we have Dr (x) δ α/2 Dr (y)(1 + δ Dr (y)) α/2 |x − y| d .

Derivatives of the Poisson kernel for ∆ α/2
In this section we prove useful gradient estimates for the Poisson kernel of ∆ α/2 for D, 0 < α < 2.
Consider a ball B = B(ξ 0 , r) ⊂ B ⊂ D and let P B be the Poisson kernel of B and equal to 0 elsewhere. By [11,Lemma 3.1], We will now show analogous estimate for C 1,1 bounded open sets. Proof ). In view of [10, (29)] we have By (9) and bounded convergence we have From (9) and the dominated convergence theorem it follows, that if f is α-harmonic in D then The estimate (9) and (11) gives ([11, Lemma 3.2]) Lemma 4. Let U be an arbitrary open set in R d and let α ∈ (0, 2). For every non-negative function u on R d which is α-harmonic in U , we have Since

A uniform integrability result
One of the important results of [9] is In the next lemma we will show a similar property for the family of functions are uniformly in Q ∈ ∂D and n > N integrable against |b(w)|dw.
Proof. In view of the properties of D and of the estimates of G D 2 −n (x, w) and where c andc depends only on D, α and x. The rst term in the parentheses is integrable against |b(w)|dw independently of Q, n, so we only need to consider the second one. For w ∈ D 2 −N and Q ∈ ∂D we have we can restrict our attention to the function and we obtain

Martin kernel and Martin representation
In this section we will discuss rst the existence and the properties of the Martin kernel of L for a C 1,1 bounded open set D. Next we will investigate the Martin representation for non-negative singular L-harmonic functions on D.

Existence and Perturbation formula for the L-Martin kernel
In order to prove the existence of the L-Martin kernel, we will need the following property of the Green function for ∆ α/2 .
Proof. Let z ∈ D, Q ∈ ∂D and choose r > 0 such that B(z, r) ⊂ D and Furthermore, by (9) and (5), We now use the estimate [9, (25)] and by considering the cases δ D (w) > |w −y| Hence the last term is uniformly in y ∈ B(Q, r/2) ∩ D integrable against dw, and thus The last equality follows from (11) The function l D (x, Q) is well dened for x ∈ D and Q ∈ ∂D and l D (x, Q) > 0. Moreover the following limit exists and equals: Thus the Martin kernel of L = ∆ α/2 + b · ∇ for D exists and equals Proof. We divide the perturbation formula (4) for the Green functionG D (x, y) by G D (x 0 , y) and let y → Q.
The exchange of lim y→Q and D is justied by Lemma 11 of [9], see the formula (49) in its proof. Note that by the Boundary Harnack Principle, G D (x 0 , y) ≈ G D (x, y) when y ∈ B(Q, 0 ), a suciently small ball around Q. We also use the estimates (5), (13) and (2).
The exchange of lim y→Q and ∇ z is justied by Lemma 7. Finally The strict positivity of the function l D (x, Q) follows from (2), which implies that there exists a > 0 such that Now we consider the quotient when y → Q.
Directly from the denition ofM D (x, Q) and (2) we obtain the following corollary. Corollary 9. There is a constant c such that for all x ∈ D and Q ∈ ∂D,

Properties of the L-Martin kernel
We will now study further properties of the Martin kernel of L for D. We start with the following useful formulas.
(ii) Let Q ∈ ∂D. We have the following expression for the L-Poisson integral of the Martin kernel of ∆ α/2 : Proof. In the following we apply the Ikeda-Watanabe formula for the Poisson kernelsP U and P U . By (4) and Fubini's theorem, for any x ∈ U and z ∈ U c , For the necessary exchanges of order of integration and derivation in the last formula, we apply (13), (2), Lemma 5 and bounded convergence theorem. In order to prove (ii), we use (i) and insert the formula (17) In the last equality the use of Fubini theorem and the exchange of and ∇ are justied by (10), (6), Lemma 5 and bounded convergence. Let z ∈ D. By (11) and the α-harmonicity of M D (·, Q), for r > 0 suciently small, we have From (9) and (6) it follows, that ∇P B(z,r) (z, w)M D (w, Q) is uniformly in Q integrable against dw. This implies that ∇M D (z, ·) is continuous on ∂D for every z ∈ D. Let now x ∈ D and choose r > 0 such that B(x, r) ⊂ D. By (2), (5), (12) and (6), for all y ∈ B(x, r), z ∈ D and Q ∈ ∂D, we havẽ Hence,G D (y, z)|∇M D (z, Q)| is uniformly in y ∈ B(x, r) and Q ∈ ∂D integrable against |b(z)|dz, which gives the continuity of f (·, ·).
We will now use Lemma 10 to show L-harmonicity ofM (x, Q). Proof. First consider a C 1,1 open set U = D r . We note that By (18), (17), (20) and Fubini's theorem Thus the function l D (x, Q) is regular L-harmonic on each set U = D r for r suciently small. By the strong Markov property, it has the mean value property on each open set U ⊂Ū ⊂ D.

L-Martin representation
The objective of this section is to prove the following Martin representation theorem for non-negative singular L-harmonic functions on D.
Theorem 13. For every non-negative nite measure ν on ∂D the function u given by is singular L-harmonic on D. Conversely, if u is non-negative singular Lharmonic on D, then there exists a unique non-negative nite measure ν on ∂D verifying (21). Proof. The L-harmonicity of the Martin integral (21) and the uniqueness of the representation follow from Theorem 12, Lemma 11, (6), (16) and Fubini theorem, in the same way as in the case of the Martin representation for αharmonic functions in [6, proof of Theorem 1]. We will now focus on the existence part. By L-harmonicity of u and by (17) we have for each n where C = C(α, b, D 1/n ) > 0. Hence, by Fubini theorem The function u * n is α-harmonic on D 1/n , so it is dierentiable. In order to justify the exchange of and ∇ in the last integral we x w ∈ D 1/n . Then by (10) and [9, (72)], for ε > 0 suciently small and all w ∈ B(w, ε) and y ∈ D c 1/n we have where C = C(α, b, D 1/n , ε) > 0. Since the last term is integrable on D c 1/n , by the dominated convergence we obtain We now study the sequence u * n (x) in the same way as K. Bogdan [6] in the proof of the existence part of the ∆ α/2 −Martin representation, with the dierence that in our case the function u under the integral dening u * n is not α-harmonic. Like in [6, (2.27)] we have The only other property of the function u intervening in the proof of the existence part of the ∆ α/2 −Martin representation in [6] is and it also holds in our case: the L-harmonic function u is integrable on D c 1/n for every n. The sequence (µ n ) of simultaneously bounded nite measures with support contained inD is tight. We choose a subsequence µ n k converging to a nite (perhaps zero) measure µ. This choice is common for all x. Without loss of generality, we may suppose that (n k ) is a subsequence of (2 −n ). The limit measure µ satises supp(µ) ⊂ ∂D.
Exactly as in the proof of the existence part of the ∆ α/2 −Martin representation in [6], we deduce that for all x ∈ D the limit exists and Furthermore, in view of (11), for x ∈ D 1/n and r > 0 suciently small we have By Lemma 1 and (9) we have and by the dominated convergence we get ∇u * n k (x) → ∇u * (x) as k → ∞. We also haveG D 1/n (x, w) G D (x, w). In order to justify the passage with the limit under the integral sign in (22) with n k instead of n we observe that the functionsG D 1/n k (x, w)b(w) · ∇u * n k (w) are uniformly integrable on D. Clearly, by Lemma 1 we have c −1 u * n (w) ≤ u(w) ≤ cu * n (w), where c does not depend on n, thus u * n (w) ≤ cu * (w). By the gradient estimates we get , and the uniform integrability follows from (23), Lemma 1 and Lemma 6. Therefore which, using (23), becomes By the gradient estimates and dominated convergence we also get Dene a measure ν on ∂D by ν(dQ) = l D (x 0 , Q)dµ(Q). As the function Q → l D (x 0 , Q) is continuous positive, the measure ν is nite positive on ∂D. Using Fubini theorem in (25) and the perturbation formula forM D from Theorem 8, we obtain Remark 1. We point out that the proof of Theorem 13 is based on the perturbation formula. In fact, the methods used in [6] in order to prove the Martin representation theorem for singular α-harmonic functions can not be applied in the present case because the Green functionG D (x, y) is not L-harmonic on D \ {x} as a function of y.
Dene a singular α-harmonic Then the following formula holds Proof. Observe that by (15) there exists δ > 0 such that for all Q ∈ ∂D. Thus the measure dµ(Q) = dν(Q) l D (x 0 ,Q) is nite and the function v * is well dened. By the unicity of the Martin representation and the formula (23), the function v * dened by (27) is the same as the function v * dened by a limit procedure and associated to v in the proof of the Theorem 13. Hence the formula (24) holds for v and v * . It is equivalent to (28). Corollary 15. Let v(x) ≥ 0 be a singular L-harmonic function on D. The functions v and v * are comparable: there exists c > 0 such that for all Proof. We use the Martin representations (26), (27), the Corollary 9 and the fact that l D (x 0 , Q) > δ > 0 for all Q ∈ ∂D.

Perturbation formulas in the diusion case
In the present article we exploit the perturbation formulas in the case of the singular operator L = ∆ α/2 + b · ∇, 1 < α < 2. In this short chapter we make a parenthesis and briey discuss the case α = 2 and d ≥ 3, corresponding to the diusion operator on R d . The potential theory for such diusion generators was studied by Cranston and Zhao [16], and more recently by Ifra and Riahi [20], Kim and Song [27] and Luks [30]. Our methods allow to enrich this theory by some new perturbation formulas. We suppose that b ∈ K 1 d and we assume additionally that D is connected, i.e. it is a domain. Recall that Cranston and Zhao [16] worked under this condition and a complementary second condition |b| 2 ∈ K 1 d−1 ; Kim and Song [27] suppressed the condition on |b| 2 and considered signed measures in the place of b.
Then the following perturbation formula for the L-Green functionG D holds if x, y ∈ R d , x = y.
Proof. Note that by [27,Theorem 6.2], we have the estimatẽ The proof of the Proposition is the same as the proof of [9,Lemma 12] in the case 1 < α < 2, with (31) replacing [9,Lemma 7].
Let us mention that a perturbation formula for the L-Green function was proposed in [20], but under a restrictive assumption of boundedness of the Kato norm b of b. A simpler direct proof of the estimate (31) without using the precise estimates [27, Theorem 6.2] should be available.
Next we obtain a perturbation formula for the Martin kernel of Laplacians with a gradient perturbation.
Then the following perturbation formula for the L-Martin kernelM D holds if x ∈ D and Q ∈ ∂D. (32) where l D (x 0 , Q) is a continuous function on ∂D, equal Proof. We follow the proof of the Theorem 8 in the case α = 2.
The next perturbation formula concerns the L-Poisson kernelP D (x, Q).
Then the following perturbation formula for the L-Poisson kernelP D holds if x ∈ D and Q ∈ ∂D.
Proof. Observe that by the formula (30) the functionG D has the same differentiability properties as the function G D . In particular the inner normal derivative ∂G D ∂n (x, Q) exists for x ∈ D and Q ∈ ∂D. It is known (see [20, page 173]) and possible to prove by the Green formula that The formula (33) then follows by dierentiating of the formula (30) in the direction of the inner normal unit vector n. We omit the technical details.
On the other hand, if we insert the formula M D (x, Q) = P D (x,Q) P D (x 0 ,Q) into (33), we obtain using (32)P Evaluating the last equation at x 0 we obtain a formula for the function l D (x 0 , Q) intervening in the perturbation formula (32) and another proof of the formula (34). 20 Relative Fatou Theorem for L-harmonic functions We prove in this section an important boundary property of L-harmonic functions: the Relative Fatou Theorem. As in the preceding sections, we consider a nonempty bounded C 1,1 open set D. Recall the Relative Fatou Theorem in the α-stable case. It was proved in [31] for Lipschitz sets D.
Theorem 19. Let g and h be two non-negative singular α-harmonic functions on D, with Martin representations Then, for µ (h) -almost all Q ∈ ∂D, where f is the density of the absolute continuous part of µ (g) in the decomposition µ (g) = f dµ (h) + µ (g) sing with respect to the measure µ (h) , and x → Q non-tangentially.
Our objective in this section is to prove an analogous limit property for non-negative singular L-harmonic functions u and v on D.
If we denote the integral part of the perturbation formula (28) by where u * and v * are singular α-harmonic non-negative functions. We write The limit boundary behaviors of the quotients u(x) v(x) and u * (x) v * (x) will be related if we control the limit behavior of the quotients I u * (x) u * (x) and I v * (x) v * (x) . Thus we start with discussing the properties of the quotient I h (x) h(x) for a singular α-harmonic non-negative function h. Proof. The limit in the case Q ∈ supp(µ (h) ) follows easily from the Martin representation of h and the Lebesgue theorem. In the case Q ∈ supp(µ (h) ) we use the following result of Wu [38].
Let f be a ∆-harmonic function on D, corresponding via the Martin representation to a nite measure µ = µ (h) on ∂D. If Q ∈ suppµ, then and the second part of the Lemma follows. Lemma

The quotient I h (x)
h(x) is bounded. More exactly, there exists c > 0 such that Proof. Observe that by Corollary 15 and the formula (28), the quotient is bounded. More exactly, there exists c > 0 such that As the function l D (x 0 , Q) is bounded, any singular α-harmonic non-negative function h is of the form v * for a singular L-harmonic non-negative function v.
By (2), if we denote In particular, by Lemma 21, the quotient J h (x)/h(x) is bounded. We prove a much stronger property of this quotient in the following lemma. Lemma 22. Let h be a non-negative singular α-harmonic function on D, with the Martin representation h(x) = ∂D M D (x, Q)dµ (h) (Q) for a nite measure µ (h) on ∂D. Then, when Q ∈ supp(µ (h) ) and x → Q non-tangentially, we have Proof. We will show that G D (x,w)h(w) h(x)δ D (w) is uniformly integrable in x ∈ D against the measure |b(w)|dw. Let ε > 0. Since J h (x)/h(x) is bounded it suces to show that there is δ > 0 such that provided λ(F ) < δ. Here, λ denotes the Lebesgue measure on R d . First, we note that The function G D (x,w)G D (w,y) is uniformly integrable in x, y ∈ D against |b(w)|dw (see the proof of [9, Lemma 11]). Hence, there exists δ > 0 such that for λ(F ) < δ, We decompose the measure µ into its absolutely continuous and singular parts with respect to the measure ν dµ = f dν + dµ sing with a non-negative function f ∈ L 1 (ν) and ν(supp(µ sing )) = 0. when x → Q non-tangentially. Proof. We will use the Relative Fatou Theorem for the singular α-harmonic functions u * and v * dened according to (27). Let Q ∈ supp(ν) \ supp(µ). Then, if x → Q, v * (x) → ∞ and u * (x) → 0, so lim x→Q u * (x) v * (x) = 0. The formulas (36) and (35) imply that in this case lim x→Q u(x) v(x) = 0.
Let us consider the case Q ∈ supp(ν) ∩ supp(µ). As the Relative Fatou Theorem for the singular α-harmonic functions u * and v * says that for ν-almost every point Q ∈ ∂D