REGULARITY FOR THE SUPERCRITICAL FRACTIONAL LAPLACIAN WITH DRIFT

Transcription

1 REGULARITY FOR THE SUPERCRITICAL FRACTIONAL LAPLACIAN WITH DRIFT CHARLES L. EPSTEIN AND CAMELIA A. POP ABSTRACT. We consider the linear stationary equation defined by the fractional Laplacian with drift. In the supercritical case, wherein the dominant term is given by the drift instead of the diffusion component, we prove local regularity of solutions in Sobolev spaces employing tools from the theory of pseudo-differential operators. The regularity of solutions in the supercritical case is as expected from the subcritical case. In the subcritical case the diffusion is at least as strong as the drift, and the operator is an elliptic pseudodifferential operator, which is not the case in the supercritical regime. We also compute the leading part of the singularity of the Green s kernel for the supercritical case, which displays some rather unusual behavior. CONTENTS. Introduction.. Comparison with previous research 4.. Outline of the article 5.3. Notations and conventions 5. A first construction of a two-sided parametrix 5 3. A second construction of a two-sided parametrix Change of coordinates Construction of a two-sided parametrix and regularity of solutions 3... Construction of a diffeomorphism 3... Localization 3 4. The Green s Kernel Asymptotics of E s,b Asymptotics of the Integral Terms Asymptotics of the Residue Term 3 Appendix A. The analysis of ps (x ; x n ) 6 References 8. INTRODUCTION We consider the linear operator defined by the fractional Laplacian with drift, Au(x) := ( ) s u(x) + ν(x) u(x), for u ( n ), the Schwartz space [9, Definition (3.3.3)] consisting of smooth functions whose derivatives of all order decrease faster than any polynomial at infinity. Here ν : n n is a smooth, 99 Mathematics Subject Classification. Primary 35H99; secondary 6G. Key words and phrases. Fractional Laplacian, pseudo-differential operators, Sobolev spaces, jump diffusion processes, symmetric stable processes, Markov processes.

2 C. EPSTEIN AND C. POP tempered vector field. The action of the fractional Laplace operator ( ) s on the space of Schwartz functions is defined through its Fourier representation by ( ) s u(ξ) = ξ s û(ξ). The range of the parameter s of particular interest is the interval (, ). In our article, we study the case when s (, /). Recall that ( n ) is the dual space of ( n ), the space of tempered distributions. The fractional Laplacian plays the same role in the theory of non-local operators that the Laplacian plays in the theory of local elliptic operators. For this reason, the regularity of solutions to equations defined by the fractional Laplacian and its gradient perturbation is intensely studied in the literature. It has applications to the study of a number of nonlinear equations, as for example the quasi-geostrophic equation [4, 7, 8], Burgers equation [5, 4], and to the study of the regularity of solutions and of the free boundary in the obstacle problem defined by the fractional Laplacian with drift [6]. As the infinitesimal generator of a Markov process with jumps described by a α-stable Lévy process, the properties of its fundamental solution and Green s function are studied in articles such as [, 6,, 3], among others. The operator A can be viewed as a pseudo-differential operator using the representation Au(x) = (π) n n e ixξ ( ξ s + iν(x) ξ ) û(ξ) dξ, u ( n ), (.) with associated symbol, a(x,ξ) := ξ s + iν(x) ξ, x,ξ n. (.) Strictly speaking a(x, ξ) is not a symbol, because it is not smooth at ξ =. The Schwartz kernel of A is not properly supported and for this reason A does not map ( n ) to itself. To facilitate our analysis we choose a smooth cut-off function, ϕ : n [, ], such that ϕ(ξ) = for ξ <, and ϕ(ξ) = for ξ >, (.3) and we consider the classical symbol defined by ã(x,ξ) := ξ s ϕ(ξ) + iν(x) ξ, x,ξ n. (.4) Let δ,ρ [, ] and m. We recall the definition of Hörmander class of symbols, S m ρ,δ ( n n ), [, Definition (7..4)] which consists of smooth functions p(x, ξ) such that, for all multi-indices α,β n, there is a positive constant, C α,β, such that the following hold D α x Dβ ξ p(x,ξ) C α,β( + ξ ) m ρ β +δ α, x,ξ n. To any symbol p S m ρ,δ ( n n ), we can associate the operator T p u(x) := (π) n n e ixξ p(x,ξ)û(ξ) dξ, u ( n ), and the class of such operators is denoted by O P S m ρ,δ ( n ). In the language of pseudo-differential operators, we have that the symbol ã defined by (.4) belongs to the Hörmander class S m, ( n n ), where the order of the symbol is m = s, and where we define the operator E by an inverse Fourier transform, Au = Tãu + Eu, u ( n ), (.5) Eu(x) := [ ξ s ( ϕ(ξ))û(ξ) ], u ( n ). (.6) Because the function ξ ξ s is not smooth at ξ =, we see that ξ s û is not necessarily a tempered distribution, when u ( n ). We introduce the space (,s) ( n ) := { u ( n ) : ξ s û ( n ) }.

3 FRACTIONAL LAPLACIAN WITH DRIFT 3 This is largely a condition on the behavior of the distribution at infinity. This space is the largest subspace of ( n ) on which ( ) s makes sense. We also let L loc ( n ) = {u ( n ) : û L loc ( n )}. (.7) If u (,s) ( n ), then ξ s ( ϕ(ξ))û is a compactly supported distribution, and we have that Eu(x) t ( n ), the space of smooth functions all of whose derivatives have tempered growth, that is t ( n ) := ( n ) ( n ). From identity (.5), and the fact that Tã : ( n ) ( n ), we also have that A : (,s) ( n ) ( n ). As ã is a classical symbol, the operator Tã maps ( n ) to itself. We recall that a tempered distribution, u ( n ), belongs to the Sobolev space, H m ( n ), where m is a real number, if ( + ξ ) m/ û(ξ) L ( n ). We say that u Hloc m ( n ), if for any function χ c ( n ), we have that χu belongs to H m ( n ). Using the facts that ϕ(ξ) is a compactly supported, smooth function, and ( + ξ ) m/ Êu(ξ) = ( + ξ ) m/ ξ s ( ϕ(ξ))û(ξ), m, we obtain that if u L loc ( n ), then (+ ξ ) m/ Êu belongs to L ( n ). Thus the preceding argument shows that E : (,s) ( n ) t ( n ) (.8) E : L loc ( n ) H m ( n ), m. (.9) In this article, we establish local regularity in Sobolev spaces of solutions to the equation defined by the fractional Laplacian with drift, Au(x) = f (x), x n, (.) where the source function, f, is assumed to belong locally to the Sobolev space H l ( n ), for some real constant l. Our strategy is to prove existence of a two-sided parametrix for the pseudo-differential operator Tã, and use it to establish the local regularity in Sobolev spaces of solutions to equation (.). The construction of a two-sided parametrix of the pseudo-differential operator Tã is nontrivial in the case when s (, /), the so-called supercritical regime, because the symbol ã(x,ξ) is not elliptic. When s = / or s (/, ), the so-called critical and subcritical regime [7, 4, ], respectively, ã(x,ξ) is an elliptic symbol in the class S, s ( n ), and the existence of a two-sided parametrix follows from well-known results [, 7.4]. In this case, we assume that u (,s) ( n ) is a solution to equation (.) with source function f Hloc l (U), for U an open set. The existence of a two-sided parametrix of the operator Tã, together with identity (.5) and (.8), show that u also belongs to H l+s (V) for V U. In the former case, when s (, /), the drift dominates the diffusion component in the definition (.4) of the symbol ã(x,ξ). The operator ã(x, D) is no longer elliptic and the existence of a parametrix for Tã is not obvious. Even so, by applying a change of coordinates dictated by the vector field ν(x), we are able to build a two-sided parametrix for the operator Tã, in the new system of coordinates. We use this property to prove our main result. We let ψ : n [, ] be a smooth cut-off function such that and we let We can now state ψ(x) = for x <, and ψ(x) = for x >, (.) ψ r (x) = ψ(x/r), r >. (.)

4 4 C. EPSTEIN AND C. POP Theorem. (Local regularity of solutions). Let s (, /), and let x n. Assume that ν : n n is a smooth vector field in a neighborhood of x, and that ν(x ) =. Let l be a real constant, and assume that u (,s) ( n ) is such that ψ r (x x )Au(x) H l ( n ), for some positive constant r, where ψ r is defined by (.). Then there is a positive constant, r < r, such that for any smooth function, χ : n [, ], with compact support in B r (x ), the function χu H l+s ( n ) and χν u H l ( n ). In the last section we compute the leading order singularity of the Green s Function, i.e. the kernel K(x, z), for A, i.e. A f (x) = K(x, x y) f (y)dy. (.3) n In the supercritical case, this kernel displays a interesting new feature: at z it is more singular in the half plane {y : y ν(x) > } than in the complementary half plane. This is an echo of the fact that the Green s function, for ( ) s + xn, which is essentially a heat kernel, is supported in the half space {(x, x n ) : x n > }. We provide a short appendix where we analyze this kernel as well... Comparison with previous research. Caffarelli and Vasseur [4, Theorem 3] study the regularity of solutions to the evolution equation associated to the fractional Laplacian with drift in the critical case s = /. They assume that the drift coefficient belongs to the BMO class of functions and that it is a divergence free vector field to obtain regularity of solutions in Hölder spaces. Silvestre extends the possible values of the parameter s, which now can be chosen so that s (, ), and relaxes the assumptions on the drift coefficient to obtain Hölder continuity of solutions in space and time in [7, Theorem.]. In particular, the drift coefficient is no longer assumed to be divergence free. Instead, it is required to be bounded, when s [/, ), and to be Hölder continuous s, when s (, /). This result is improved in [8, Theorem.], where the,α Hölder continuity of solutions in space is proved, in the case when s (, /]. In this case, the drift coefficient is required to belong to s+α, for some α (, s). The main differences between our work and that of Caffarelli-Vasseur [4] and Silvestre [7, 8] are that we consider the stationary, instead of the evolution equation defined by the fractional Laplacian with drift, and that we prove regularity of solutions in Sobolev spaces instead of Hölder spaces. We assume that the vector field, ν, is smooth in order to be able to employ techniques from the theory of pseudo-differential operators, while the work of Caffarelli-Vasseur [4] and Silvestre [7, 8] relaxes the regularity assumptions on the drift coefficient and employs De Giorgi s approach to parabolic equations and comparison arguments in order to obtain regularity of solutions. The fundamental solution of the fractional Laplacian with drift is studied in [3], in the case when s (/, ). Assuming that the vector field ν satisfies a certain integral condition [3, Inequality (4)], the authors prove existence [3, Theorem ], and upper and lower bounds of a fundamental solution comparable with the fundamental solution of the fractional Laplacian without drift [3, Theorems and 3]. Similar results are proved in [] and [] in the case when s (/, ), under different integral conditions on the vector field ν. In [], the vector field in assumed to belong to a Kato class of functions [, Definition ], while in [] the vector field is assumed to be divergence free, to satisfy [, Condition ()], and a smallness condition [, Theorem ]. The Green function for the fractional Laplacian with drift on smooth domains, in the case when s (/, ), is studied in [3] and [6], under the assumption that the vector field belongs to the Kato class of functions. Regularity of solutions to the equation defined by the fractional Laplacian with drift may be obtained using upper and lower bounds satisfied by the fundamental solution. The preceding results establish existence and properties of the fundamental solution when s (/, ), while our article establishes regularity of solutions when s (, /), using a method that circumvents the use of pointwise estimates of the

5 FRACTIONAL LAPLACIAN WITH DRIFT 5 fundamental solution. As noted above, in the last section we determine the behavior of the leading singularity of the Green s function in the supercritical case... Outline of the article. In, we build a two-sided parametrix for Tã in Lemma., in the case when s (/4, /). We use Lemma. to prove global regularity of solutions to equation (.), Lemma.. Our method is applicable only in the case when s (/4, /), because we use as a first approximation of the parametrix a pseudo-differential operator with symbol in the Hörmander class S m ρ,δ ( n n ) with δ < ρ, and this property breaks down when s (, /4]. In 3, we take a different approach based on a suitable change of coordinates to prove the existence of a two-sided parametrix for Tã, for all s (, /). In 3., we describe the effect of the coordinate change on the pseudo-differential operator Tã in Lemma 3.3. The second method for constructing the parametrix and a localization procedure is then employed in 3.. to prove the main result of our article, Theorem.. In Section 4 we compute the leading order singularity of the Green s Function, i.e. the kernel K(x, z), for A..3. Notations and conventions. We adopt the following definitions of the Fourier transform and the inverse Fourier transform of a function u ( n ), u(ξ) = û(ξ) = e ix ξu(x) dx, n u(x) = (π) n These operators extend by duality to ( n ). We let n e ix ξ u(ξ) dξ. Denote by O P S ( n ) the set of smoothing operators, that is x = ( + x ). (.4) O P S ( N ) := m= O P S m, ( n ). For n, m positive integers, we let ( n ; m ) be the space of m -valued smooth functions, and we let b ( n ; m ) be the subspace of ( n ; m ) of smooth m -valued functions with bounded derivatives of all orders. The space c ( n ; m ) consists of smooth m -valued functions with compact support. For brevity, when m =, we write ( n ), b ( n ) and c ( n ) instead of ( n ; ), b ( n ; ) and c ( n ; ), respectively. We denote by the extended set of natural numbers, that is := {,,,...}. Given real numbers, a and b, we let a b := min{a, b} and a b := max{a, b}. For a function, φ : n n, we let Jφ(x) denote the Jacobian matrix, and by Jφ(x) the Jacobian determinant. For a matrix, A d d, we let A T denote the transpose matrix.. A FIRST CONSTRUCTION OF A TWO-SIDED PARAMETRIX We build a two-sided parametrix for Tã in the case when s (/4, /). Our method cannot be extended to the case when s /4, because the candidate for a first approximation of the parametrix is a pseudo-differential operator with symbol in the Hörmander class S m ρ,δ ( n n ) with δ ρ, and so the calculus of such symbols does not allow us to build an asymptotic expansion for the composition in terms of symbols of decreasing orders. In 3, we take a different approach to prove the existence of a two-sided parametrix for Tã, for all s (, /).

6 6 C. EPSTEIN AND C. POP Lemma. (A two-sided parametrix for Tã). Let s (/4, /), and ν b ( n ; n ). Then the operator Tã has a two-sided parametrix, T q, where q(x,ξ) is a symbol in the Hörmander class S s s, s ( n n ), that is T q Tãu = u mod ( n ), u ( n ), (.) TãT q u = u mod ( n ), u ( n ). (.) Proof. We build a left-parametrix, but the construction is the same for the right-parametrix. We let q (x,ξ) := ϕ(ξ) ã(x,ξ), x,ξ n, where ϕ : n [, ] is a smooth cut-off function chosen as in (.3). We recall that the symbol ã(x,ξ) is defined in (.4). We consider a first approximation of the parametrix given by the pseudodifferential operator T q u(x) = (π) n ϕ(ξ) ξ s + iν(x) ξ û(ξ) dξ, u ( n ), n e ixξ and we set T c = T q Tã. By [, Proposition 7.3.3], we expect the following asymptotic expansion to hold for c, c(x,ξ) i α α! Dα ξ q (x,ξ)dx α ã(x,ξ), x,ξ n. (.3) α To obtain the expansion (.3), we first look at the symbolic properties of ã(x,ξ) and q (x,ξ). Because we assume that s < /, direct calculations give us that D ξ j ã(x,ξ) behaves like a symbol in S, ( n n ), and Dξ α ã(x,ξ) behaves like a symbol in Ss α, ( n n ), for all multi-indices α n such that α. In the case of the symbol p (x,ξ), we obtain D α x q (x,ξ) ( + ξ ) s+ α ( s), α n, D α ξ q (x,ξ) ( + ξ ) s α s, α n. The preceding asymptotic properties of the symbol q (x,ξ) show that q (x,ξ) belongs to the Hörmander class of symbols S s s, s ( n n ). Therefore, [, Proposition 7.3.3] applies to our case and we obtain that the asymptotic expansion for c(x,ξ) given by (.3) holds. We denote, for all j, c j (x,ξ) := {α n : α = j} i j α! Dα ξ q (x,ξ)d α x ã(x,ξ), x,ξ n. By construction, we have that c (x,ξ) = + (ϕ(ξ) ), and the function ϕ belongs to the class S ( n n ). Because the symbol D ξ j q (x,ξ) belongs to S 4s s, s ( n n ) and D x j ã(x,ξ) belongs to S, ( n n ), we obtain by [, Proposition 7.3.3] that c (x,ξ) belongs to S 4s s, s ( n n ). Also, c j (x,ξ) belong to S 4s s, s ( n n ) (even better), for all j, and so we have c(x,ξ) = + r(x,ξ), x,ξ n, where r S 4s s, s ( n n ). Notice that by requiring that s > /4, the order of the symbol r is negative and also δ = s < ρ = s. We obtain that T q Tã = T +r. Because the symbol r has negative order, the pseudo-differential operator T +r is elliptic with symbol in the class S s, s ( n n ). By [, Section 7.4], we can find a two-sided parametrix, T q, with q S s, s ( n n ) such that T q T +r = I + S R, where S R

7 FRACTIONAL LAPLACIAN WITH DRIFT 7 belongs to OPS ( n ). By setting T q = T q T q, we see that the symbol q belongs to the class S s s, s ( n n ), and the pseudo-differential operator T q is a left-parametrix for Tã. Similarly, we can show that Tã admits a right-parametrix, T p, with symbol p which belongs to the class S s s, s ( n n ). We have, for all u ( n ), T p u + S R T p u = ( T q Tã) Tp u = T q ( Tã T p ) u = Tq u + T q S L u, and because S R T p and T q S L are pseudo-differential operators in OPS ( n ), we see that T q u = T p u modulo ( n ) functions, for all tempered distributions u ( n ) in a standard way. The construction of the left-parametrix in Lemma. can be used to prove regularity of solutions to the equation defined by the fractional Laplacian with drift (.). Lemma. (Regularity of solutions). Let s (/4, /) and k, l. Assume that the vector field ν b ( n ; n ). Let u H k ( n ) be such that Au H l ( n ). Then u H l+s ( n ) and ν u H l ( n ). Proof. Let T q be a left-parametrix of Tã constructed as in the proof of Lemma.. Because we assume that s > /4, we have that s > s, and using the fact that the symbol q belongs to the class S s s, s ( n n ), we obtain by [, Proposition 7.5.5] that the pseudo-differential operator T q : H l ( n ) H l+s ( n ) (.4) is bounded. Let u H k ( n ) be such that Au H l ( n ). We know that Tãu = Au Eu by (.5). By (.9) and the fact that u H k ( n ) L loc ( n ), we have that Eu H m ( n ), for all m, and so Tãu H l ( n ). We know that T q Tã = I + R, where the pseudo-differential operator R O P S ( n ), and so the map R : H k ( n ) H m ( n ), m, is bounded. This fact together with the boundedness of the map in (.4) and the assumption u H k ( n ), gives us that u H l+s ( n ). Since we have shown that u H l+s ( n ) we immediately conclude from the fact that Au H l ( n ) that ν u H l ( n ), as well. In fact, using symbolic computations, one can show that ν T p : H l ( n ) H l ( n ) is a bounded operator. This concludes the proof. Remark.3 (Hypotheses of Lemma.). The hypothesis that u H k ( n ), for some real constant k, in the statement of Lemma., can be replaced with the assumption that u L loc ( n ). In this case, we obtain that there is a constant q such that u H l+s ( n ) + x q H m ( n ), m n, (.5) where x q H m ( n ) is the weighted Sobolev space consisting of tempered distributions of the form x q u with u H m ( n ) [5, Definition (.5)]. We can argue that (.5) holds in the following way. From [5, Lemma.], we have that there are real constants, k and q, such that u x q H k ( n ), and from [5, Theorem.4] and the fact that R OPS ( n ), it follows that R : x q H k ( n ) x q H k+m ( n ), m. Thus the identity T q Tã = I + R now gives us (.5).

8 8 C. EPSTEIN AND C. POP 3. A SECOND CONSTRUCTION OF A TWO-SIDED PARAMETRIX In this section, we extend the results of from the case when s (/4, /) to that when s (, /). We give a method to build a two-sided parametrix for the pseudo-differential operator Tã which can be applied for all parameters s in the range (, /). From the preceding section, we see that the reason we had to restrict to the case s (/4, /) is that the candidate for a first approximation of the parametrix of Tã is a pseudo-differential operator with symbol, q (x,ξ), that belongs to the Hörmander class S s s, s ( n n ), and we need to have s > s, that is s > /4, in order to obtain a useful symbolic composition formula. We notice that if the vector field ν(x) is a constant, then the symbol q (x,ξ) belongs to the class S s s, ( n n ), and so we do not need to assume any restriction on the values of the parameter s, in order to obtain a two-sided parametrix for the operator Tã. Our strategy for a general smooth vector field ν(x), is to apply a change of coordinates to obtain an operator with a constant drift for which we can construct a two-sided parametrix using the methods of. 3.. Change of coordinates. We first describe the effect of the change of coordinates on the symbols of pseudo-differential operators. While this is a classical subject, we were not able to find the statement of Lemma 3. below in the form required in this article. However, analogous statements which hold for symbols with kernels with compact support can be found in Hörmander [, Theorem 8..7] (see also the comment on [, p. 35] for symbols in the class S m ρ,δ ( n n )), and for symbols in the class S m, ( n n ), can be found in []. For functions : n n and u : n, we define ( u)(x) := u( (x)), x n. Lemma 3. (Change of coordinates). Let a be a symbol in the Hörmander class Sρ,δ m ( n n ), where δ and ρ < ρ. Let : n n be such that () is a bijection, and the functions and belong to b ( n ; n ). () The inverse of the matrix-valued function H : n n n, H(x, y) = J (x + t(y x)) dt, x, y n, (3.) is defined at all points (x, y) n n, and the function (x, y) H (x, y) belongs to b ( n n ; n ). Then there is a symbol, b S m ρ,δ ( n n ) with δ = δ ( ρ), such that T a ( ) = T b, (3.) Moreover, the symbol b satisfies, for all non-negative integers N, b(y,η) α! Dα η Dα w c(y,w,η) w=y S m N(ρ ) ρ,δ ( n n ), (3.3) where α <N c(y,w,η) = a( (y), H(y,w) T η) J (w) H(y,w) T, y,w,η n. (3.4) Remark 3. (Invariance of classes of pseudo-differential operators under change of coordinates). Lemma 3. shows that given a symbol, a S m ρ,δ ( n n ), with ρ < ρ, there is a symbol b S m ρ,δ ( n n ), with δ = δ ( ρ), such that the change of coordinates (3.) holds. To guarantee invariance of symbol classes under change of coordinates, we need to require that ρ δ < ρ, (3.5)

9 FRACTIONAL LAPLACIAN WITH DRIFT 9 When the preceding condition is satisfied, we see that δ = δ, and both symbols a(x,ξ) and b(x,ξ) belong to the same class. Condition (3.5) is consistent with the hypothesis of [, Theorem 8..7] for symbols in the Hörmander class S m, ( n n ) with kernels with compact support (see also the comment on [, p. 35]). Proof of Lemma 3.. Let u ( n ) and define v = u and x = (y). Then u = ( ) v, and the left-hand side of identity (3.) becomes T a ( ) v(y) = T a (v )( (y))(= T a u(x)) = (π) n e i (y) ξ a( (y),ξ) v (ξ) dξ = (π) n e i( (y) z)ξ a( (y),ξ)v( (z)) dzdξ = (π) n e i( (y) (w))ξ a( (y),ξ) J (w) v(w) dwdξ, where we applied the change of variables z = (w) in the penultimate equality. Using identity (3.) we have that (y) (w) = H(y,w)(y w), y,w n, and applying the change of variable η = H(y, z) T ξ, we obtain T a ( ) v(y) = (π) n e i(y w)η a( (y), H(y, z) T η) J (w) H(y,w) T v(w) dwdξ. Thus, the pseudo-differential operator T a ( ) is defined as an operator with a compound symbol c(y,w,η) given by (3.4). The compound symbol c(y,w,η) belongs to the class Sρ,δ, ρ m ( n n ) (see definition [, 7.3, Inequality (3.3)]). This follows from the fact that the symbol a Sρ,δ m ( n n ), the functions and belong to b ( n ; n ) by assumption (), and the map H(y,w) satisfies assumption () in the statement of the lemma. Because we also assume that ρ < ρ, it follows from [, Proposition 7.3.] that there is a symbol b Sρ,δ ( ρ) m ( n n ) such that identity (3.) holds, and the symbol b satisfies the asymptotic expansion (3.3). We apply Lemma 3. to the pseudo-differential operator Tã, defined by the symbol ã in (.4). Lemma 3.3 (Change of coordinates for Tã). Let s (, /), and let : n n be a diffeomorphism which satisfies assumptions () and () of Lemma 3.. Let ν b ( n ; n ) be a smooth vector field such that, for all functions u ( n ), we have that ν(x) u(x) = v yn (y), (3.6) where we let v := u and x = (y). Then there is a symbol, b S, ( n n ), such that and Proof. We write where we let b(y,η) ( (J (y)) T η s ϕ((j (y)) T η) + iη n ) S s, ( n n ), (3.7) Tã( ) = T b. (3.8) ã(x,ξ) = ã (ξ) + ã (x,ξ), (3.9) ã (ξ) = ξ s ϕ(ξ) and ã (x,ξ) = iν(x)ξ. (3.)

10 C. EPSTEIN AND C. POP Clearly the symbol ã belongs to the class S s, ( n n ) and ã belongs to S, ( n n ), and we have that Tã( ) = Tã( ) + Tã( ). Identity (3.6) immediately gives us that Tã( ) = T b with b (y,η) = iη n. Applying (3.3) with N = to the symbol ã, we obtain that and the symbol b satisfies Tã( ) = T b, b (y,η) (J (y)) T η s ϕ((j (y)) T η) S s, ( n n ). To obtain the preceding expression, we used the fact that H(y, y) = (J (y)), and that J (y) H(y, y) T =. Letting now b(y,η) = b (y,η) + b (y,η), we see that the symbol b satisfies (3.7) and identity (3.8). 3.. Construction of a two-sided parametrix and regularity of solutions. We now build a twosided parametrix for the pseudo-differential operator T b (Lemma 3.4), which we then use to prove local regularity of solutions in Sobolev spaces to equation (.) (Lemma 3.5). The hypotheses of Lemmas 3.4 and 3.5 are relaxed in 3.., where in Lemma 3. we give sufficient conditions for the existence of a diffeomorphism,, with suitable local properties. Then, in 3.. we use Lemma 3. to prove the main result of our article, Theorem.. Lemma 3.4 (A two-sided parametrix for T b ). Let s (, /), and let : n n be a diffeomorphism which satisfies assumptions () and () of Lemma 3.. Let ν b ( n ; n ) be a smooth vector field such that identity (3.6) holds for all functions u ( n ), where we let v := u. In addition we assume that there is a positive constant, C, such that C η (J (y)) T η C η, y,η n. (3.) Then there is a symbol, p S s s, ( n n ), such that T p T b = T b T p = I mod O P S ( n ). (3.) Proof. From (3.7), it follows that the leading part of b is given by b (y,η) = (J (y)) T η s ϕ((j (y)) T )η) + iη n, y,η n, and the symbol b S, ( n n ). We use the symbol b(y,η) to build a left- and right-parametrix for the pseudo-differential operator T b exactly as in the proof of Lemma.. We first define p (y,η) := ψ(η) b (y,η), where supp ψ {η : ϕ((j (y)) T )η) = }. Reviewing the proof of Lemma., we see that p defines a symbol in the Hörmander class S s s, ( n n ), while previously we had that p belongs to S s s, s ( n n ) due to the (non-zero) derivatives of the vector field ν(x), which now are zero because of the condition (3.6). We notice that condition (3.) guarantees that the symbol p indeed belongs to the class S s s, ( n n ). The remaining part of the proof of Lemma. applies without the restriction s > s, that is s > /4, to the present setting, and we obtain a two-sided parametrix, T p, with a symbol p(y,η) belonging to the class S s s, ( n n ). We can now state the regularity result in the more general case when < s < /.

11 FRACTIONAL LAPLACIAN WITH DRIFT Lemma 3.5 (Regularity of solutions). Let s (, /), and let : n n be a diffeomorphism which satisfies the hypotheses of Lemma 3.4. Let u H k ( n ) be such that Au H l ( n ), for some real constants, k and l. Then u H l+s ( n ) and ν u H l ( n ). Remark 3.6. The hypotheses of Lemma 3.5 are relaxed in Theorem.. To prove Lemma 3.5, we use the following Proposition 3.7. [9, 4.] Let : n n be a diffeomorphism with bounded derivatives of all orders. Assume there is a positive constant, C, such that Then, for all l, is a continuous linear map. J (x) C, x n. : H l ( n ) H l ( n ) Proof of Lemma 3.5. From Lemma 3.4, we obtain that there is a symbol p S s s, ( n n ) such that T p T b = I mod O P S ( n ). Thus, if w H k ( n ) is such that T b w H l ( n ), then the previous identity implies that w belongs to H l+s ( n ) and w yn belongs to H l ( n ). We will use this fact together with Proposition 3.7 to prove that u H l+s ( n ) and ν u H l ( n ). We first show that T b v belongs to the Sobolev space H l ( n ). Recall from identity (3.8) that T b v(y) = Tãu(x), where we recall that v = u and x = (y), which implies that T b v(y) = Au(x) Eu(x), where Eu is defined in (.6). Therefore, we have that T b v(y) = (Au)(y) (Eu)(y). By hypothesis, we have that Au H l ( n ), and so Proposition 3.7 implies that (Au) also belongs to H l ( n ). From our assumption that u H k ( n ) L loc ( n ) and property (.9), we see that because Eu belongs to H l ( n ). From Proposition 3.7, it follows that (Eu) H l ( n ), and so we can conclude that T b v belongs to H l ( n ) also. Thus the function v belongs to H l+s ( n ) and v yn belongs to H l ( n ). Recall that we have u = ( ) v and ν u = ( ) v yn. Applying Proposition 3.7 again implies that the function u is in H l+s ( n ) and ν u is in H l ( n ). Remark 3.8 (Hypothesis of Lemma 3.5). As in Remark.3, the hypothesis that u H k ( n ), for some real constant k, in the statement of Lemma 3.5, can be replaced with the assumption that u L loc ( n ), and we obtain that (.5) holds. Remark 3.9 (Parametrix for Tã). In Lemma 3.4, we prove the existence of a two-sided parametrix for the pseudo-differential operator T b, which is given by the operator T p, with symbol p in the Hörmander class S s s, ( n n ). Because the pseudo-differential operator T b satisfies identity (3.8), we can see that the operator Q = ( ) T p, (3.3) is a two-sided parametrix for Tã, that is QTãu = Tã Qu = u mod ( n ), u ( n ). We now discuss to what extent the two-sided parametrix of the operator Tã, Q, can be represented as a pseudo-differential operator.

12 C. EPSTEIN AND C. POP When s (/4, /), we have that s < s <. Therefore Lemma 3. shows that Q can be represented as a pseudo-differential operator with symbol in the class S s s, s ( n n ), which is consistent with Lemma.. Conversely, we can start from the two-sided parametrix, T q, of the pseudo-differential operator Tã, given by Lemma.. Now q is a symbol in the Hörmander class S s s, s ( n n ). According to the previous argument, P = T q ( ), is a two-sided parametrix for T b. Because the symbol q belongs to S s s, s ( n n ) and we assume that s < s, Lemma 3. gives us that P can be represented as a pseudo-differential operator with symbol p S s s, s ( n n ). In Lemma 3.4, we show that p is actually in a better symbol class, that is it belongs to S s s, ( n n ). When s (, /4], the condition s < s is no longer fulfilled, and so we cannot guarantee that we can represent Q as a pseudo-differential operator. This explains the difficulty in the construction of a two-sided parametrix for the operator Tã using the direct approach of Construction of a diffeomorphism. We now want to build a diffeomorphism, : n n, which satisfies assumptions () and () of Lemma 3., verifies locally identity (3.6) of Lemma 3.3, and verifies condition (3.) of Lemma 3.4. Let ν be a smooth vector field. Because we want identity (3.6) to be satisfied only locally, we fix a point x n, and we assume that ν(x ) =. We may assume choose a system of coordinates such that x = O and ν e n =. We define the smooth vector field µ by µ(x) = ψ r (x)ν(x) + ( ψ r (x))ν(o), x n, (3.4) where the positive constant r will be suitably chosen below. We recall that the cut-off function ψ r is defined in (.). The vector field µ is a globally Lipschitz function and it is constant outside a compact set. By the Picard-Lindelöf Theorem [, Theorem II..], for any y n, there is a unique global solution to { d dt (y, t) = µ( (y, t)), t =, (y, ) = (y (3.5), ). We want to use (y, t) to introduce a new system of coordinates, that is, for all x n, we want to show that there is a unique (y, t) n such that x = (y, t). For this purpose, let G : n n n be defined by G(x, y, t) = x (y, t), x n, (y, t) n. From our assumption that ν n (O) =, we also have that µ n (O) =. We see that J (y,t) (O, O) =, and so, the Implicit Function Theorem gives that there are neighborhoods of O, which we denote by W and V, such that for all x W, there is a unique (y, t) V such that x = (y, t). (3.6) By choosing r small enough, we can make the oscillation of the vector field µ small enough, so that the proof of the Implicit Function Theorem implies that B r (O) W. Because the vector field µ is constant outside the ball B r (O) by identity (3.4), we can extended the - correspondence (3.6), between points x W and (y, t) V, to all points x n and (y, t) n. To see this, let S = W ({x n = }\W).

13 FRACTIONAL LAPLACIAN WITH DRIFT 3 For each x W c, let x S be the closest point to x, with respect to the Euclidean distance, with the property that x x and ν(o) are linearly dependent. Because x S, there is (y, t ) V such that x = (y, t ). Now let t be such that (t t )ν(o) = x x, then x = (y, t). Therefore, the function : n n is a bijection. Because µ b ( n ; n ), [, Corollary V.4.] shows that belongs to b ( n ; n ). We notice that J (y, ) =, y n. Because µ is a constant vector field outside the ball B r (O), we can choose r small enough so that J (y, t) =, (y, t) n, (3.7) and we can also arrange so that condition (3.) and assumption () of Lemma 3. hold. From (3.7), the Inverse Function Theorem now shows that is a C function on n with bounded derivatives of first order. From the identity, J = (J ), we see that, the fact that the function is smooth and the inverse function is C, and both have bounded derivatives, implies that is a C function with bounded derivatives up to order two. Inductively, we obtain that is a C function with bounded derivatives of any order. Moreover, for any function u ( n ), applying the change of variable v = u and x = (y), we have that v yn (y) = n i= u x i ( (y)) i y n (y) = µ(x) u(x) (by (3.5), and the fact that x = (y)). Therefore, identity (3.6) is satisfied locally, for all x B r (O), because µ(x) = ν(x) on B r (O) by construction. The preceding argument proves Lemma 3. (Construction of a diffeomorphism with suitable local properties). Let x n be such that ν(x ) =, and let ν ( n ; n ). Then there is diffeomorphism, : n n, which satisfies assumptions () and () of Lemma 3., and verifies condition (3.) of Lemma 3.4, and there is a positive constant, r, such that identity (3.6) of Lemma 3.3 holds, for all x B r (x ) Localization. Because we are interested in the local regularity of u, we only need to study χu, where χ : n [, ] is a smooth function with compact support. Remark 3. (Comparison between Lemma 3.5 and Theorem.). The difference between Lemma 3.5 and Theorem. is that Lemma 3.5 assumes the existence of a diffeomorphism,, satisfying suitable properties, while in Theorem. this assumption is replaced by the condition that ν(x ) =. The result that allows us to weaken the hypotheses of Lemma 3.5, to only assume that ν(x ) = in the statement of Theorem., is Lemma 3.. Proof of Theorem.. For clarity, we divide the proof into three steps. The first step is an application of Lemma 3., which allows us to locally change the coordinates so that we can construct a two-sided parametrix for the pseudo-differential operator in the new system of coordinates. In the second step, we collect various intermediate results which are ingredients in the iteration procedure employed to obtain the local regularity of solutions. The iteration procedure is described in the third step.

14 4 C. EPSTEIN AND C. POP Step (Change of coordinates). Without loss of generality, we may assume that r =, that is ψ r = ψ, where we recall that the cut-off function ψ r is defined in (.) and ψ is defined by (.). Therefore, we have that ψ Au H l ( n ). Because we assume that ν(x ) =, we may choose a system of coordinates such that x = O and ν(o) e n = ν n (O) =. By Lemma 3. there is a diffeomorphism, : n n, which satisfies assumptions () and () of Lemma 3., and verifies condition (3.) of Lemma 3.4. Moreover, there is a positive constant, r, such that identity (3.6) of Lemma 3.3 holds, for all x B r (x ). Let r := ( r)/4, and let the vector field µ be defined by (3.4), where the constant r is replaced by r. We consider the new symbol, α(x,ξ), obtained from ã by replacing the drift coefficient ν by µ, that is α(x,ξ) = ξ s ϕ(ξ) + iµ(x) ξ, x,ξ n. Then the diffeomorphism satisfies the hypotheses of Lemma 3.3, with the vector field ν replaced by µ. We obtain that there is a symbol β S, ( n n ) such that T α ( ) = T β. (3.8) Step (Intermediate results). Since ψ c ( n ), there is a real k so that ψu H k ( n ). Let J be the smallest integer such that k + s J l. We choose a family of smooth cut-off functions, {χ j : j =,..., J + }, with values in [, ] such that χ j = on B r (O), j =,..., J +, (3.9) χ j = on B c r (O), j =,..., J +, (3.) supp χ j+ {χ j = }, j =,..., J. (3.) From [9, Property (4..9)] and using the fact that supp χ j {ψ = }, it follows that Our goal is to show that χ j u H k ( n ), j =,..., J +. (3.) χ j+ T α χ j u H l ( n ), j =,..., J. (3.3) Because ν = µ on B r (O), and the support of each χ j is contained in B r (O), we have χ j T α u = χ j Tãu, j =,..., J +. We recall that Tãu = Au + Eu, where Eu is defined as in (.6). From our assumption that ψ Au H l ( n ), we have that χ j Au is in H l ( n ), by (3.), definition (.) of ψ, and the choice of the constant r. Because we assume that u (,s) ( n ), we know that Eu t ( n ), from (.8). Therefore, we can conclude that We can write χ j T α u H l ( n ), j =,..., J +. (3.4) χ j+ T α u = χ j+ T α χ j u + χ j+ T α ( χ j )u, j =,..., J. The preceding identity together with (3.4) shows that to obtain (3.3), it is enough to establish χ j+ T α ( χ j )u H l ( n ), j =,..., J +. (3.5) We can write T α = T α + T α as the sum of two pseudo-differential operators with symbols α (ξ) = ξ s ϕ(ξ) and α (x,ξ) = iµ(x) ξ, x,ξ n. We see that T α v(x) = µ(x) v(x), v ( n ).

15 FRACTIONAL LAPLACIAN WITH DRIFT 5 From (3.), we obtain that χ j+ T α ( χ j )u = on n. We also see that T α is a classical pseudo-differential operator, and hence pseudolocal, so [, Proposition 7.4.] implies that the function χ j+ T α ( χ j )u is smooth with compact support, hence contained in H l ( n ). Because both (3.4) and (3.5) hold, we obtain that property (3.3) holds. Step 3 (Iteration procedure). We now employ an iteration procedure to prove the local regularity of solutions. We denote w j = χ j u, where the cut-off functions, {χ j : j =,..., J + }, are chosen in Step. We consider the change of coordinates w j (x) = v j (y) and x = (y). From identity (3.8), we have that T β v j+ (y) = T α w j+ (y), j =,..., J. Any smooth function with compact support, χ : n [, ], can be viewed as a symbol in the class S, ( n n ). Because α is a symbol in S, ( n n ), the commutator rule [, Chapter 7, Identity (3.4)] gives that there is symbol, e S, ( n n ), such that Using the fact that it follows that We now prove inductively that T α (χu) = χt α u + T e u, u ( n ). (3.6) T α w j+ (y) = T α χ j+ χ j u(x) = χ j+ T α χ j u(x) + T e χ j u(x) (by identity (3.6)) = χ j+ T α w j (x) + T e w j (x), T β v j+ (y) = (χ j+ T α w j + T e w j )(y), j =,..., J. (3.7) v j+ H l (k+( j )s)+s ( n ). (3.8) If j =, we have shown in (3.3) that the function χ j+ T α w j belongs to H l ( n ), and because the symbol e S, ( n n ) and the function w j H k ( n ) by (3.), we have that T e w j belongs to H k ( n ). Therefore, the function χ j+ T α w j (y) + T e w j belongs to H l k ( n ). Applying Proposition 3.7, we obtain that the right-hand side in identity (3.7) also belongs to H l k ( n ). By Lemma 3.4, the operator T β admits a left-parametrix, T p, with symbol p S s s, ( n n ). Thus, it follows that property (3.8) holds when j =. Assume now that property (3.8) holds for j = j. We want to prove that (3.8) holds for j = j +. Using the fact that v j + H l (k+( j )s)+s ( n ), and that w j + = ( ) v j +, we obtain from Proposition 3.7 that w j + H l (k+( j )s)+s ( n ). To prove that property (3.8) holds in the case j = j +, we can apply the same argument that we employed to prove (3.8) in the case when j =, with the observation that we replace the role of w with that of w j +, and the role of k is replaced with that of l (k + ( j )s) + s. When j = J +, we see that k + s J l, and so l (k + ( j )s) = l. From (3.8), it follows that v J+ H l+s ( n ). As T β v J+ H l ( n ), we easily conclude, as before, that yn v J+ H l ( n ). (3.9)

16 6 C. EPSTEIN AND C. POP Recall that w J+ = ( ) v J+ and ν w J+ = ( ) yn v J+. Proposition 3.7 implies that w J+ is in H l+s ( n ) and ν w J+ is in H l ( n ), and since χ J+ can be any smooth function with values in [, ] and compact support in B r (O), the conclusion follows immediately. This concludes the proof. 4. THE GREEN S KERNEL In [3] and [6] estimates are given for the Green s kernel on a bounded domain, assuming that < s <, but allowing a somewhat singular coefficient for the vector field. In our setting for this range of s, with a smooth coefficient for the vector field, the operator A is a classical, elliptic pseudodifferential operator of order s, so the calculation of the leading singularity of the Green s function is trivial. In the previous sections we employed a change of coordinates y = (x), to obtain a parametrix for the fundamental solution for the operators Au(x) = ( ) s u(x) + ν(x) u(x), for < s <, under the assumption that ν(x) is non-vanishing. In this section we examine the leading term in the asymptotic expansion, in the y -coordinates, of this operator s kernel along the diagonal. The kernel of the leading term, after applying the change of variables, y = (x), is given by K(y, z) = ψ(η)e iη z dη, (4.) (π) n (J (y)) t η s + iη n n where z = y ỹ as usual. After changing variables with (J (y)) t χ = η, this becomes K(y, z) = ψ((j (y)) t χ)e iχ J (y)z J (y) dχ. (4.) (π) n χ s + i((j (y)) t χ) n n We finally choose an orthogonal transformation U(y) so that b(y)(u(y)χ) n = ((J (y)) t χ) n, where we normalize so that b(y) >. Setting ξ = U(y)χ, the integral becomes K(y, z) = ψ((j (y)) t U(y) t ξ)e iξ U(y)J (y)z J (y) dξ. (4.3) (π) n ξ s + ib(y)ξ n n Up to a linear transformation, the leading order singularities in K(y, z) as z, are determined by asymptotic evaluations of the Fourier transforms E s,b (x) = e ix ξ dξ, (4.4) (π) n ξ s + ibξ n as x. Here b is a positive constant. In fact, equation (4.3) shows that, up to a smooth error: n K(y, z) = E s,b(y) (U(y)J (y)z) J (y). (4.5) E s,b (x) is the kernel for Green s function of the constant coefficient operator ( n) s + b xn. The Green s function for the operator ( n ) s + b xn is the kernel, ps (x ; x n ), of the heat operator e xn b ( n ) s, which is supported in the half space {(x, x n ) : < x n }. As we shall see, when < s <, there is an echo of this in the kernel for E s,b(x). For simplicity we state this in terms of the leading orders of these singularities.

17 Theorem 4.. If < s < and b >, then ( O E s,b (x, x n ) = ( O FRACTIONAL LAPLACIAN WITH DRIFT 7 [x s n + x ] n [xn + x ] n +s ) where x n >, ) where x n <. Remark 4.. Note that when s <, then n + s < n. Notice also that the homogeneities of these terms are different. A more precise description of the leading singularity follows from formula (4.8), along with the asymptotic evaluations in (4.39), (4.43), and (4.49), from which this theorem follows immediately. 4.. Asymptotics of E s,b. To evaluate the asymptotic behavior of the integral in (4.4), as x, we split it into an integral over the first n variables and an integral over ξ n : E s,b (x, x n ) = (π) n = ω n (π) n = c n ω n where x = (x, x n ), ξ = (ξ,ξ n ), and To simplify the computation we let obtaining n π c n = E s,b (x, x n ) = c nω n x n 3 e ix ξ e ix nξ n dξ n dξ ( ξ + ξn )s + ibξ n e ix nξ n dξ n e ir x cosθ sin n 3 θdθr n dr (r + ξ n )s + ibξ n e ix nξ n dξ n (r + ξ n )s + ibξ n J n 3 π n 3 (π) n ( n Ŵ (r x )r n dr, (r x ) n 3 (4.6) (4.7) ). (4.8) ξ n = rτ, and β = br s, (4.9) e irxnτ dτ J n 3 (r x )r n ( + τ ) s + iβτ (+τ ) s +iβτ r s dr. (4.) To evaluate the τ-integral, we consider the function to be a single valued analytic function in the slit region D = \ ( i, i] [i, i ). (4.) A careful examination of the denominator on the boundary of D reveals that it has a single pole. This pole is of the form τ = i y(β), where y(β) (, ) satisfies the equation: As β, we see that ( y (β)) s = βy(β). (4.) y(β) = β s + a j β j s, (4.3) j=

18 8 C. EPSTEIN AND C. POP where the sum is convergent in some neighborhood of β =. As β, we can show that y(β) = b j +, (4.4) β β j where again the series in convergent for /β in a neighborhood of zero. To compute the τ-integral for x n > we use the contour Ŵ + R shown in Figure ; letting R go to infinity, we easily show that lim R R R j= r s e irxnτ dτ ( + τ ) s + iβτ = πe rxn y(β) ( y (β)) s + 4πr b( ( s)y (β)) e rx nτ sin(πs)(τ ) s dτ (τ ) s e πis βτ. A similar calculation using the reflection of Ŵ + R across the real axis gives that for x n <, lim R R R r s e irx nτ dτ ( + τ ) s + iβτ = 4πr s (4.5) e r x n τ sin(πs)(τ ) s dτ (τ ) s e πis + βτ. (4.6) Recall that β = br s, and y(β) has the expansion in (4.4), so, as r, the residue term in (4.5) behaves like πe rs x n /b, (4.7) b which decays at a slower exponential rate than the contribution from the integral. To simplify the formulae that follow we let e r xn I s ± (r, x n) denote the integral terms on the right hand sides of (4.5) and (4.6), and H(r, x n ) denote the residue term in (4.5) where β = br s. With this notation we see that c n ω n J E s,b (x x, x n ) = n 3 n 3 (r x ) [ e r xn I s + (r, x n) + H(r, x n ) ] r n dr if x n >, (4.8) c n ω n x n 3 J n 3 (r x )e r xn I s (r, x n)r n dr if x n <. This explains why, if s <, E s,b (x, x n ) has a stronger singularity approaching (, ) from x n > than from x n <. Where x n > the principal term in E s,b (x, x n ) is essentially a multiple of p s (x ; x n ). In the next two sections we derive asymptotics for the terms on the right hand side of (4.8) as x + x n tends to. 4.. Asymptotics of the Integral Terms. To understand the behavior of the integral terms when r is small, we let σ = r(τ ), to obtain that I ± s (r, x n) = 4π sin(πs)e xn σ σ s (σ + r) s dσ σ s (σ + r) s e πis b(σ + r). (4.9) From these expressions and the fact that s <, it is immediate that the limits C s ± = lim I (r,x n ) ( +, ± s ± (r, x n) (4.) )

19 FRACTIONAL LAPLACIAN WITH DRIFT 9 are finite. Hence from (4.7), (4.), and the asymptotic behavior of Bessel functions near zero we see that, for any finite r, the integrals c n ω n x n 3 r J n 3 (r x )r n e r x n I ± s (r, x n)dr, c n ω n x n 3 r J n 3 (r x )r n H(r, x n )dr (4.) remain bounded as ( x, x n ) tends to zero. In determining the asymptotics of E s,b (x, x n ) we are therefore free to restrict attention in (4.) to << r < r. In this regime we rewrite I ± s as I ± s (r, x n) = 4π Expanding these integrands in powers of gives the convergent expansions: (τ r ) s (τ r ) s e πis bτ = (τ r ) s (bτ) r sin(πs)e x n (τ r) (τ r ) s dτ (τ r ) s e πis bτ. (4.) (τ r ) s, (4.3) bτ + j= a ± j ( (τ r ) s ) j, (4.4) bτ valid for sufficiently large τ. The functions I s ± (r, x n), then have expansions of the form I s ± (r, x n) a ± f j (r x n ) j r ( s)( j+), (4.5) where f j (y) = j= e y(σ ) [ (σ ) s σ ] j+ dσ σ. (4.6) Using a standard remainder estimate for a geometric series, it follows easily that for given < b and < s < there is an r so that, for each N there is a C N, for which N I s ± (r, x n) a ± f j (r x n ) j r ( s)( j+) C N r ( s)(n+), if r r. (4.7) j= The functions { f j } are non-negative, and f j () is positive. As y, f j (y) satisfies a standard symbolic estimate. Lemma 4.3. For each j the function f j (y) is in ((, )) and for each k there is a constant C j,k so that, k y f j(y) Proof. Letting τ = σ, we can rewrite f j as f j (y) = C k, j for y [, ). (4.8) yk++s( j+) [ τ e yτ s (τ + ) s ] j+ dτ τ + τ +. (4.9)

20 C. EPSTEIN AND C. POP We can differentiate under the integral sign to obtain k y f j(y) = ( ) k If we let yτ = w, then we see that k y f j(y) = ( )k yk++s( j+) e yτ τ k [ τ s (τ + ) s τ + The Lebesgue dominated convergence theorem easily implies that lim y yk++s( j+) y k f j(y) = ( ) k s j+ from which the lemma follows easily. ] j+ dτ τ +. (4.3) [ ( + w/y) e w w k+s( j+) s ] j+ dw + w/y + w/y. (4.3) e w w k+s( j+) dw <, (4.3) We also have the following result describing the behavior of f j (y) as y +. Lemma 4.4. For j, if ( j + )s /, then there are functions f j (y), f j (y) [, ) so that f j (y) = f j (y) + f j (y)y ( s)( j+). (4.33) Remark 4.5. Throughout the remainder of this section we assume that s /. This is not an essential restriction, but allows us to avoid considering many special cases. Proof of Lemma 4.4. We use formula (4.33) for f j (y). It is clear that the function is smooth for y (, ) and that the possible singularities as y + only result from the large τ behavior of the integrand. Thus it suffices to show that [ τ f + j (y) = e yτ s (τ + ) s ] j+ dτ (4.34) τ + τ + has the behavior given in (4.33). To that end we rewrite this integral in the form [ ( + /τ) f + j (y) = e yτ τ (s )( j+) s ] j+ dτ + /τ τ( + /τ) = k= a k e yτ τ (s )( j+) τ k dτ. As s <, to prove the lemma it therefore suffices to examine functions of the form (4.35) e yτ τ α l dτ, (4.36) where < α <, and l. Repeatedly integrating by parts we can show that there is a constant c l,α so that e yτ τ α l dτ = c l,α y l+α + g l,α (y), (4.37)

21 FRACTIONAL LAPLACIAN WITH DRIFT where g l,α ([, )). Using this relation repeatedly, with l +α = ( s)( j +)+k, and noting that k, we easily obtain the assertion of the lemma. We now derive the leading order asymptotics of E s,b (x, x n ) of x as x. We begin by obtaining the asymptotics for the contributions of the I s ± -terms. As noted above it suffices to consider the integrals x n 3 r We split this analysis into the two cases: x / x n and. x n 3 J n 3 (r x )r n e r x n I s ± (r, x n)dr = J n 3 (r x )r n e r x n I ± s (r, x n)dr for << r. (4.38) Proposition 4.6. For < b and < s <, there are functions L± s,b (y) ([, )), so that, as x n + x tends to zero, with x / x n bounded above, ( ) L ± x ( s,b x n (xn + x ) n +s + O ( x n + x ) n 3+4s ). (4.39) Proof. We choose an r so that we can apply the remainder estimate in (4.7). If we denote the quantity on the left of (4.39) by ± s ( x, x n ), then it follows that, for each N there is a C N so that ± s ( x, x n ) = N j= a ± j x n 3 r J n 3 (r x )r n e r x n f j(r x n )dr r C N x n 3 r ( s)(+ j) + J n 3 (r x )r n e r x n dr + O(). (4.4) r ( s)(n+) We choose N so that n ( s)( + N) > and n ( s)( + N) <. (Here we use our standing assumption that s /. We leave the equality case to the interested reader.) To estimate the remainder we use the very crude estimate J n 3 (r x ) C(r x ) n 3. (4.4) This estimate, along with our choice of N shows that, without regard for the behavior of the ratio x / x n, the remainder term is O() as x + x n tends to zero. To estimate the terms in the finite sum we change variables letting r x n = ρ to obtain that ± s ( x, x n ) = N j= a ± j x ( s)(+ j) n x n 3 x n n+ r x n J n 3 (ρ x / x n )ρ n e ρ f j (ρ)dρ + O(). (4.4) ρ ( s)(+ j) Recalling that as x / x n is bounded or tending to zero, these integrals are easily evaluated by replacing the Bessel function with its power series. Using the estimates for f j that follow from Lemmas (4.3) and (4.4), we obtain the formula in (4.39). Note that only the j = term contributes to the leading order behavior. The analysis of the case with x / x n tending to infinity is rather different. Nonetheless we obtain the same leading order behavior in this limit as well.