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

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L=\Delta ^{\alpha /2}+ b\cdot \nabla $$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (1,2)$$\end{document}. We prove the Martin representation and the Relative Fatou Theorem for non-negative singular L-harmonic functions on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{C }^{1,1}$$\end{document} bounded open sets.

From the probabilistic point of view, stable stochastic processes with gradient perturbations on R d , d ≥ 2, i.e. with the infinitesimal 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 financial applications and form a part of Lévy-driven Ornstein-Uhlenbeck processes, cf. [13,43]. The motivations of this paper were to: (i) establish the theory of the Martin representation for singular L-harmonic nonnegative 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 [6,11,21,41,45], see also [37] for a general setting of Markov processes. The Relative Fatou Theorem was proved for α-harmonic functions on C 1,1 sets in [8], on Lipschitz sets in [39], and on the so-called κ-fat sets in [32], see also the survey [7,Chapter 3]. Furthermore, some important variants of stable processes such as relativistic, censored and truncated stable processes were studied from the point of view of topics (i) and (ii), see [17,18,33,35] and [7,Section 3.4]. 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.
On the other hand, Martin representations and boundary properties of harmonic functions were widely studied in the case of diffusion operators. The boundary behavior of harmonic functions for the classical Laplacian = d i=1 ∂ 2 /∂ x 2 i and Lipschitz domains was treated by Hunt and Wheeden in a fundamental and already classical paper [26]. Let us cite e.g. [1,[3][4][5]24,31,46,47,49] for other results on , and [15,22,23,38,42,44,48] for its various generalizations.
Let us mention that the methods of this article give also interesting new results for operators different 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 (Sect. 3.4).
The potential theory of stable stochastic processes with gradient perturbations was started in the Ornstein-Uhlenbeck case by Jakubowski [29,30]. 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 Bogdan and Jakubowski [9,10] and by Chen et al. [19]. Our work is a natural continuation of the research presented in [10].

Preliminaries
In this subsection we recall for the convenience of the reader the basic definitions and results used in the paper. They can be found e.g. in the monography [7] and in the paper [10].
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 If D is C 1,1 at some unspecified scale (hence also at all smaller scales),then we simply say that D is C 1,1 .
The potential theory objects related to the operator L defined in (1) will be denoted with a tilde˜, while those related to the operator α/2 will be denoted without it.
Let p D (t, x, y) andp D (t, x, y) be the heat kernels on D of α/2 and L respectively. ThenG is the Green function of L for D and The Lévy measure of the semigroup generated by α/2 is given by The Poisson kernel of L for D may be introduced, like in [10, (38)], by the Ikeda-Watanabe formulã It is equal to the density of the L-harmonic measure for D, i.e. ifX t is a stochastic process with generator L, then [10, (39)]. The Poisson kernel of α/2 for D is denoted in the paper by P D (x, y) and it may be defined by the formula (2) with G D instead ofG D . If B = B(a, r ) then P B (x, y) is given by where where the last integral is absolutely convergent. If, in addition, h ≡ 0 on D c then it is called singular L-harmonic on D. On the other hand h is called regular L-harmonic The harmonic functions of α/2 on D (called in the paper α-harmonic functions) are defined analogously. Their basic properties may be found in the monography [7].
Throughout this paper, like in [10], we suppose 1 < α < 2, unless stated otherwise. We consider an open bounded set D of class C 1,1 and a vector field b ∈ K α−1 We fix throughout this paper a point x 0 ∈ D and define the Martin kernel of α/2 for D by The L-Martin kernel is defined bỹ and we show in Sect. 3 its existence. The starting point of the research contained in this paper are the following mutual estimates of Green functions and Poisson kernels of L and α/2 (see [10, 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 (4) is the following perturbation formula, that will be also very useful in our present work (see [10,Lemma 12]).
Perturbation formula for Green functions. Let x, y ∈ R d , x = y. We havẽ 1.3 Outline of the paper We start our paper by proving in Sect. 2.1 a generalization of the Comparability Theorem: according to Lemma 1, the constant C in the estimates (4) may be chosen the same for sets D r sufficiently 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 (6) allows us to prove very useful perturbation formulas for Martin kernels (15), Poisson kernels (18) and singular α-harmonic functions (29). In Section 3.4, (6) and (15) are proved in the diffusion case α = 2. Also a perturbation formula (34) for the L-Poisson kernel is derived. Section 4 is devoted to an important fine boundary property of singular L-harmonic functions: the Relative Fatou Theorem (Theorem 23). We provide a proof of this theorem based on the perturbation formula for singular L-harmonic functions (29).

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. In what follows D is a bounded C 1,1 open set.

Uniform comparability of Green functions and Poisson kernels
For r ≥ 0 define When r is sufficiently small, then D r is also a C 1,1 open set, see [40,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 ( [20,36], see also [28]) of a C 1,1 open set and of the Martin kernel ( [21]) Moreover, in the stable case, the estimates (7) are uniform when we consider the sets D r sufficiently 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 see [28,Theorem 21] and [40,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 [10]. We analyse below its crucial points.
1. Comparison of Green functionsG S (x, y) and G S (x, y) for "small" sets S, [10, Lemma 13], based on estimates from [10, Lemma 11]. Thanks to property (9), we see that the comparison of Green functionsG S r (x, y) and G S r (x, y) for small sets S holds with a common constant c, when r ∈ [0, 0 ]. [10,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 [10].

Harnack inequalities for L and the Boundary Harnack Principle,
The part (ii) is implied by (i), applying the Ikeda-Watanabe formula for the Poisson kernelP D , see [10,Lemma 6 and (39)]. Recall that the Lévy system for the process X t is given by the Lévy measure of the α-stable process X t .
An immediate consequence of Lemma 1 and [28, Theorem 22] is the following uniform estimate of the Poisson kernels 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

Derivatives of the Poisson kernel of α/2
In this section we prove useful gradient estimates for the Poisson kernel of α/2 for We will now show analogous estimate for C 1,1 bounded open sets. Proof ). In view of [11, (29)] we have By (10) and bounded convergence we have

Lemma 4 Let U be an arbitrary open set in R d and let
Since G U (·, y) is α-harmonic in U \{y}, for every y ∈ U we obtain

Lemma 5 G D (y, w)/[δ(w)∧|y −w|] is uniformly in y integrable against |b(w)|dw.
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 C 1,1 property of D and of the estimates (8) and (9) of M D (w, Q) and G D 2 −n (x, w), we can choose N = N (D, x) ∈ N sufficiently large, such that for all n > N , we have where c andc depends only on D, α and x. The first term in the parentheses is integrable against |b(w)|dw independently of Q, n, so we only need to consider the second one.
is uniformly in Q, n integrable against |b(w)|dw, we can restrict our attention to the function and we obtain

Martin kernel and Martin representation
In this section we will discuss first 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 .

Lemma 7
For all x ∈ D and Q ∈ ∂ D we have Proof Let z ∈ D, Q ∈ ∂ D and choose r > 0 such that B(z, r ) ⊂ D and B(z, r ) ∩ B(Q, r ) = ∅. Since G D (·, y) is α-harmonic in B(z, r ) for y ∈ B(Q, r ) ∩ D, by (12), we have Furthermore, by (10) and (7), We now use the estimate [10, (25)] and by considering the cases δ D (w) > |w − y| and δ D (w) ≤ |w − y| we get G D (w,y) δ D (y) α/2 ≤ C|w − y| α/2−d . Hence the last term is uniformly in y ∈ B(Q, r/2) ∩ D integrable against dw, and thus The last equality follows from (12) and the α-harmonicity of the Martin kernel.
Thanks to Lemma 7, we obtain the main result of this subsection.
The function l D (x, Q) is well defined 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 (6) 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 justified by Lemma 11 of [10], 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 sufficiently small ball around Q. We also use the estimates (7), (14) and (4).
The exchange of lim y→Q and ∇ z is justified by Lemma 7. Finally, The strict positivity of the function l D (x, Q) follows from (4), which implies that there exists a > 0 such that Now, we consider the quotient Directly from the definition ofM D (x, Q) and (4) 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. (i) (Perturbation formula for the Poisson kernel) For all x ∈ U, z ∈ (U ) c (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 kernels P U and P U . By (6) 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 (14), (4), Lemma 5 and bounded convergence theorem. In order to prove (ii), we use (i) and insert the formula (18) In the last equality the use of Fubini theorem and the exchange of and ∇ are justified by (11), (8), Lemma 5 and bounded convergence. Let z ∈ D. By (12) and the α-harmonicity of M D (·, Q), for r > 0 sufficiently small, we have From (10) and (8) it follows, that ∇ z P B(z,r ) (z, w)M D (w, Q) is uniformly in Q integrable against dw. This implies that ∇ z M D (z, ·) is continuous on ∂ D for every z ∈ D.
We will now use Lemma 10 to show L-harmonicity ofM D (·, Q).

Theorem 12 For every Q ∈ ∂ D the Martin kernelM D (x, Q) is a singular L-harmonic function of x on D.
Proof First consider a C 1,1 open set U = D r . We note that by the strong Markov property, see e.g. [10, p. 466], By (19), (18), (21) and Fubini's theorem Thus the function l D (x, Q) is regular L-harmonic on each set U = D r for r sufficiently small. By the strong Markov property, it has the mean value property on each open set U ⊂Ū ⊂ D. Lemma 7]. The proof goes along the ideas of [6].

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 finite measure ν on ∂ D the function u given by is singular L-harmonic on D. Conversely, if u is non-negative singular L-harmonic on D, then there exists a unique non-negative finite measure ν on ∂ D verifying (22).
Proof The L-harmonicity of the Martin integral (22) and the uniqueness of the representation follow from Theorem 12, Lemma 11,(8), (17) and Fubini theorem, in the same way as in the case of the Martin representation for α-harmonic functions in [6, proof of Theorem 1]. We will now focus on the existence part. By L-harmonicity of u and by (18), we have for each n By (11), [10, (72)] and Lemma 5, we have 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 differentiable. In order to justify the exchange of and ∇ in the last integral we fix w ∈ D 1/n . Then, by (11) and (5), for ε > 0 sufficiently 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 difference that in our case the function u under the integral defining u * n is not α-harmonic. By Ikeda-Watanabe formula (2), we have |ξ −y| d+α dydξ . Lemma 1 implies that 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 finite measures with support contained inD is tight. We choose a subsequence μ n k converging to a finite (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 satisfies 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 (12), for x ∈ D 1/n and r > 0 sufficiently small, we have By Lemma 1 and (10), 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 (23) with n k instead of n we observe that the functions G D 1/n k (x, w)b(w) · ∇u * n k (w) are uniformly integrable on D. Clearly, by Lemma 1, 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 (24), Lemma 1 and Lemma 6. Therefore, which, using (24), becomes By the gradient estimates and dominated convergence, we also get Define a measure ν on ∂ D by ν(d Q) = l D (x 0 , Q)dμ(Q). As the function Q → l D (x 0 , Q) is continuous positive, the measure ν is finite positive on ∂ D. Using Fubini theorem in (26) and the perturbation formula forM D from Theorem 8, we obtain Remark 2 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.
Define a singular α-harmonic function v * on D by Then the following formula holds Proof Observe that by (16) there exists δ > 0 such that Thus the measure dμ(Q) = dν(Q) l D (x 0 ,Q) is finite and the function v * is well defined. By the unicity of the Martin representation and the formula (24), the function v * defined by (28) is the same as the function v * defined by a limit procedure and associated to v in the proof of the Theorem 13. Hence, the formula (25) holds for v and v * . It is equivalent to (29).

Perturbation formulas in the diffusion 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 briefly discuss the case α = 2 and d ≥ 3, corresponding to the diffusion operator on R d . The potential theory for such diffusion generators was studied by Cranston and Zhao [23], and more recently by Ifra and Riahi [27], Kim and Song [34] and Luks [38]. 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 [23] worked under this condition and a complementary second condition |b| 2 ∈ K 1 d−1 ; Kim and Song [34] 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 Proof Note that by [34,Theorem 6.2], we have the estimatẽ The proof of the Proposition is the same as the proof of [10,Lemma 12] in the case 1 < α < 2, with (32) replacing [10,Lemma 7].
Let us mention that a perturbation formula for the L-Green function was proposed in [27], but under a restrictive assumption of boundedness of the Kato norm b of b. A simpler direct proof of the estimate (32) without using the precise estimates [34, Theorem 6.2] should be available.
Next we obtain a perturbation formula for the Martin kernel of Laplacians with a gradient perturbation. (33) where l D (x 0 , Q) is a continuous function on ∂ D, equal

. Then the following perturbation formula for the L-Martin kernelM D holds if x ∈ D and Q
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 (31) 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 [27, page 173]) and possible to prove by the Green formula thatP The formula (34) then follows by differentiating of the formula (31) in the direction of the inner normal unit vector n. We omit the technical details.
Let us finish this section by some remarks. The formulaP D (x, Q) = ∂G D ∂n (x, Q) implies, like in the Laplacian case, that the L-Martin and the L-Poisson kernels are related by the formulaM On the other hand, if we insert the formulaM D (x, Q) = P D (x,Q) P D (x 0 ,Q) into (34), we obtain using (33)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 (33) and another proof of the formula (35).

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 [39] 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 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 (29) 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 . 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 [49].
Let f be a -harmonic function on D, corresponding via the Martin representation to a finite measure μ = μ (h) on ∂ D. If Q ∈ suppμ, then lim inf x→Q f (x) > 0, provided x → Q non-tangentially. We have, on D of class C 1,1 , Consequently, and the second part of the Lemma follows.

Lemma 21 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 (29), the quotient I v * (x) v * (x) 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 (4), if we denote J h (x) = D G D (x, w)b(w) · ∇h(w)dw then, 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. 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 suffices 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 [10, Lemma 11]). Hence, there exists δ > 0 such that for λ(F) < δ, where μ and ν are two Borel finite measures concentrated on ∂ D.
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, when x → Q non-tangentially. The formula (40) then follows by the formula (36) and the Lemma 22.