Unique continuation properties for relativistic Schr\"odinger operators with a singular potential

Asymptotics of solutions to relativistic fractional elliptic equations with Hardy type potentials is established in this paper. As a consequence, unique continuation properties are obtained.


Introduction
Let N > 2s with s ∈ (0, 1) and Ω be an open subset of R N . The purpose of the present paper is to establish unique continuation properties for the operator where m ≥ 0, a ∈ C 1 (S N −1 ), and (1.2) h ∈ C 1 (Ω \ {0}), |h(x)| + |x · ∇h(x)| ≤ C h |x| −2s+χ as |x| → 0, for some C h > 0 and χ ∈ (0, 1). Answers to the problem of unique continuation will be derived from a precise description of the asymptotic behavior of solutions to Hu = 0 near 0. From the mathematical point of view, a reason of interest in potentials of the type a(x/|x|)|x| −2s relies in their criticality with respect to the differential operator (−∆ + m 2 ) s ; indeed, they have the same homogeneity as the s-laplacian (−∆) s , hence they cannot be regarded as a lower order perturbation term. The physical interest in the study of properties of the Hamiltonian in (1.1) is manifest in the case s = 1/2; indeed, if s = 1/2 and a ≡ Ze 2 is constant, then the Hamiltonian (1.1) describes a spin zero relativistic particle of charge e and mass m in the Coulomb field of an infinitely heavy nucleus of charge Z, see e.g. [16,18].
Before going further, let us fix our notion of solutions to Hu = 0 in an open set Ω. For every ϕ ∈ C ∞ c (R N ) and s ∈ (0, 1), the relativistic Schrödinger operator with mass m ≥ 0 is defined as where u denotes the unitary Fourier transform of u. We define H s m (R N ) as the completion of C ∞ c (R N ) with respect to the norm induced by the scalar product (1.4). If m > 0, H s m (R N ) is nothing but the standard H s (R N ); then, we will write H s (R N ) without the subscript "m".
By a weak solution to Hu = 0 in Ω, we mean a function u ∈ H s m (R N ) such that We notice that the right hand side of (1.5) is well defined in view of the following Hardy type inequality due to Herbst in [16] (see also [28]): where Λ N,s := 2 2s Γ 2 N +2s A first aim of this paper is to give a precise description of the behavior near 0 of solutions to the equation Hu = 0, from which several unique continuation properties can be derived. The rate and the shape of u can be described in terms of the eigenvalues and the eigenfunctions of the following eigenvalue problem then problem (1.7) admits a diverging sequence of real eigenvalues with finite multiplicity µ 1 (a) ≤ µ 2 (a) ≤ · · · ≤ µ k (a) ≤ · · · , the first one of which coincides with the infimum in (1.9), which is actually attained. Throughout the present paper, we will always assume that Our first result is the following asymptotics of solutions at the singularity, which generalizes to the case m > 0 an analogous result obtained by the authors in [9] for m = 0.
Theorem 1.1. Let u ∈ H s m (R N ) be a nontrivial weak solution to in an open set Ω ⊂ R N containing the origin, with s ∈ (0, 1), N > 2s, m ≥ 0, h satisfying assumption (1.2), and a ∈ C 1 (S N −1 ). Then there exists an eigenvalue µ k0 (a) of (1.7) and an eigenfunction ψ associated to µ k0 (a) such that where S N −1 = ∂S N + . The proof of Theorem 1.1 is based on an Almgren type monotonicity formula (see [1,14]) for a Caffarelli-Silvestre type extended problem. Indeed, for every u ∈ H s (R N ) there exists a unique w = H(u) ∈ H 1 (R N +1 + ; t 1−2s ) weakly solving − div(t 1−2s ∇w) + m 2 t 1−2s w = 0, in R N +1 where R N +1 − lim t→0 + t 1−2s ∂w ∂t (t, x) = κ s a(x/|x|) |x| 2s w + hw , in Ω, in a weak sense. The asymptotics provided in Theorem 1.1 follows from combining an Almgren type monotonicity formula for problem (1.11) with a blow-up analysis; see [10][11][12] for the combination of such methods to prove not only unique continuation but also the precise asymptotics of solutions. We also refer to [4,9] for monotonicity formulas in fractional problems.
As a particular case of Theorem 1.1, if a ≡ 0 we obtain the following result.
with s ∈ (0, 1) and h ∈ C 1 (Ω). Then, for every x 0 ∈ Ω, there exists an eigenvalue µ k0 = µ k0 (0) of problem (1.7) with a ≡ 0 and an eigenfunction ψ associated to µ k0 such that . A relevant application of the asymptotic analysis contained in Theorem 1.1 and Corollary 1.2 is the validity of some unique continuation principles. A direct consequence of Theorem 1.1 is the following strong unique continuation property, which extends to the case m > 0 an analogous result obtained for m = 0 in [9].
in an open set Ω ⊂ R N containing the origin. If u(x) = o(|x| n ) = o(1)|x| n as |x| → 0 for all n ∈ N, then u ≡ 0 in Ω.
We mention that recently some strong unique continuation properties for fractional laplacian have been proved by several authors, see [9,13,21,24,29]. Corollary 1.2 allows also to prove the following unique continuation principle from sets of positive measures, which implies, as an interesting application, that the nodal sets of eigenfunctions for (−∆ + m 2 ) s have zero Lebesgue measure.
Theorem 1.4. Suppose that u is as in Corollary 1.2. If u ≡ 0 on a set E ⊂ Ω of positive measure, then u ≡ 0 in Ω.
A direct application of Theorem 1.4 can be found in [13], where the authors proved the case N = 1 and m = 0. Remark 1.5. We point out that the results presented above still hold for the more general nonlinear problem which was considered in [9] for m = 0. Assuming that x ∈ Ω and all t ∈ R, (1.14) dr, the asymptotics of Theorem 1.1 and the unique continuation principles of Theorems 1.3 and 1.4 still hold. Since the presence of the nonlinear term introduces essentially the same difficulties already treated in [9], we present here the details of proofs only for the linear problem focusing on the differences from [9] due to the introduction of the relativistic correction.
Beside the above unique continuation properties (UCPs), several results of independent interest will be proved in this paper. Indeed, to prove the UCPs, we transform, in the spirit of [9], problems of the type in Ω. Such extension is a generalization of the Caffarelli-Silvestre extension [4] and it is a particular case of more general extension theorems proved in Section 6. We actually derive asymptotics of solutions and unique continuation for problems of type (1.15) as a consequence of asymptotics and unique continuation for the corresponding extended problem (1.16).
In sections 2, 3 and 4 we present some preliminary results including some Hardy type inequalities, Schauder estimates for boundary value problems related to (1.16) and a Pohozaev type identity. These latter preparatory results will be used in the study of the monotonicity properties of the Almgren type frequency function associated to the extended problem (1.11); in section 5 a blowup analysis of the extended problem will be also performed thus leading to the proof of Theorem 1.1 and, as consequences of Theorem 1.1, of Corollary 1.2 and Theorems 1.3 and 1.4. Finally, in Section 7 we describe some properties of the relativistic Schrödinger operator (−∆ + m 2 ) s .

Hardy type inequalities
Let us denote, for every R > 0, For every R > 0, we define the space H 1 (B + R ; t 1−2s ) as the completion of C ∞ (B + R ) with respect to the norm We also define H 1 (S N + ; θ 1−2s 1 ) as the completion of C ∞ (S N + ) with respect to the norm We recall the Sobolev trace inequality: there exists S N,s > 0 such that, for all w ∈ D 1,2 (R N +1 x)| 2 dt dx 1/2 (see e.g. [9] for details). Using a change of variables and writing , we can easily prove that there exists a well defined continuous trace operator ). In order to construct an orthonormal basis of L 2 (S N + ; θ 1−2s 1 ) for expanding solutions to Hu = 0 in Fourier series, we are naturally lead to consider the eigenvalue problem (1.7), which admits the following variational formulation: we say that µ ∈ R is an eigenvalue of problem (1.7) if there If a ∈ L N/(2s) (S N −1 ) and (1.9) holds, then we can prove that the bilinear form Q is continuous and weakly coercive on H 1 (S N + ; θ 1−2s 1 ). Moreover, since the weight θ 1−2s 1 belongs to the second Muckenhoupt class, the embedding is compact. From classical spectral theory, problem (1.7) admits a diverging sequence of real eigenvalues with finite multiplicity µ 1 (a) ≤ µ 2 (a) ≤ · · · ≤ µ k (a) ≤ · · · the first of which coincides with the infimum in (1.9) and then admits the variational characterization We assume that (1.10) holds. To each k ≥ 1, we associate an L 2 (S N + ; θ 1−2s 1 )-normalized eigenfunction ψ k ∈ H 1 (S N + ; θ 1−2s 1 ), ψ k ≡ 0 corresponding to the k-th eigenvalue µ k (a), i.e. satisfying ).
In the enumeration µ 1 (a) ≤ µ 2 (a) ≤ · · · ≤ µ k (a) ≤ · · · , we repeat each eigenvalue as many times as its multiplicity; thus exactly one eigenfunction ψ k corresponds to each index k ∈ N, k ≥ 1. We can choose the functions ψ k in such a way that they form an orthonormal basis of L 2 (S N + ; θ 1−2s 1 ).
The following results will be useful to prove Hard-type inequalities for the potential a(x/|x|)|x| −2s with a belonging to some L p space; indeed, the Hardy inequality for this potential involves only µ 1 (a) whose corresponding eigenfunction is simple. Lemma 2.1. If a ∈ L N/2s (S N −1 ) and a satisfies (1.9), then µ 1 (a) is attained by a positive minimizer. Moreover, the mapping a → µ 1 (a) is continuous in L q (S N −1 ) for every q > N/(2s).
Proof. The first assertion is classical thanks to the Sobolev-trace inequality on S N + (2.1), so we skip the details. Now let q > N/(2s) and a n ∈ L q (S N −1 ) such that a n → a in L q (S N −1 ) (and a n , a satisfy (1.9)). For every ψ ∈ C ∞ (S N + ), ψ ≡ 0, using Hölder inequality, we can see that So, choosing ψ to be a minimizer for µ 1 (a), we get µ 1 (a n ) ≤ µ 1 (a) + o(1), as n → ∞.
Define C δ = {σ ∈ S N + : dist(σ, ∂S N + ) < δ} for all δ > 0. Let χ δ ∈ C ∞ (S N ) be such that χ δ = 1 on C δ and χ δ = 0 on S N \ C 2δ . Next, let ψ n be a positive minimizer for µ 1 (a n ) normalized so that ∇ S N ψ n · e 1 = κ s a n (θ ′ )ψ n , on ∂S N + . Multiply the above equation by ψ n χ 2 δ and integrate by parts to get Hence by Hölder's inequality and thus for some positive constant C(a, N, s, δ) depending only on a, N, s, δ. Therefore, provided δ is small, by the Sobolev inequality we infer Similar arguments can be performed on geodesic balls of S N + with radius δ. By covering S N + \ C δ/2 with such finite small balls and with a classical argument of partition of unity, we conclude that It turns out that, up to subsequences, ψ n converges weakly in H 1 (S N + ; θ 1−2s 1 ) and strongly in L 2 (S N + ; θ 1−2s 1 ) to some nontrivial function ψ, which can be easily proved to be the positive (or negative) normalized eigenfunction associated to µ 1 (a); it then follows easily that the convergence holds for all the sequence (not only up to subsequences) and that µ 1 (a n ) → µ 1 (a) as n → ∞.
The following corollary follows from Proposition 6.2 and Corollary 2.3.
, satisfies (1.10) and C a,N,s > 0 is as in Corollary 2.3, then From the above characterization of C a,N,s it is then easy to prove that, if a n → a in L q (S N −1 ), then C an,N,s → C a,N,s as n → +∞.

Schauder estimates for degenerate elliptic equations
As stated in Section 1, for u ∈ H s (R N ), the nonlocal equation can be reformulated as a local problem by considering its extension in R N +1 in a weak sense. This will be proved in the appendix A. This naturally leads to the study of regularity properties of solutions to which is the content of this section. Before going on, let us state the following weighted Sobolev inequality whose proof is essentially contained in the book of Opic and Kufner, [20].
We will also need the following result.
Then there exits a constant C > 0 depending only on N, s, The proof is not difficult taking into account the weighted Sobolev inequality (3.1) together with the Sobolev-trace inequality: We skip the details.
Then sup where w + = max{0, w}, and C > 0 depends only on N, s, a + . Following [17,25], testing (3.2) with ϕ, integration by parts, we have for some positive constants δ, θ depending only on N, s and C depending only on N, s, a + L p (B1) . By using Hölder inequality, we get .
, by interpolation and Young's inequalities, we have By the weighted Sobolev inequality (3.1), we have Using the two inequalities above in (3.4), we get Putting this in (3.3), we obtain At this point, the argument in [25, Proposition 3.1] yields the result.
The next result is a weak Harnack inequality.
Then for some p 0 > 0 and any 0 < r < r ′ < 1 we have that where C > 0 depends only on N, s, r, r ′ , a − Proof. Set w = w + k > 0, for some positive k to be determined and v = w −1 . Let Φ be any nonnegative function in Multiplying both sides of the first inequality in (3.5) by w −2 Φ and integrating by parts, we obtain . Otherwise, choose an arbitrary k > 0 which will be sent to zero. Therefore Proposition 3.3 (see also [17]) implies that for any r ′ ∈ (r, 1) and any p > 0 Following exactly the same arguments as in [25], we get the result.
We now prove local Schauder estimates.
Then w ∈ C 0,α (B + 1/2 ) and in addition The proof is a consequence of Propositions 3.3 and 3.4 with a standard scaling argument for which we refer to [15].
with a, b, c, d as in Proposition 3.5 and the matrix A satisfying Then the same conclusion as in Proposition 3.5 holds taking into account the constants C 1 , C 2 .
Then for i = 1, . . . , k we have that w ∈ C i,α (B + r ), for some r ∈ (0, 1) depending only on k, and in addition h , for t ≥ 0. Applying Lemma 3.2, Proposition 3.3 and Proposition 3.5 we get . Hence by Proposition 3.5 and (3.9), we have Iterating this procedure we get the the desired estimate for W i = ∇ i x w.

A Pohozaev type identity
In order to differentiate the Almgren frequency function associated to the extended problem (see section 5), we need to derive a Pohozaev type identity, which first requires the following regularity result.
. Then for every t 0 > 0 sufficiently small there exist positive constants C and α ≥ 0 (with α > 0 if γ > 0), depending only on N, s, t 0 , m, γ such that Proof. If m = 0, this was proved in [3]. We will assume in the following that m > 0. Next pick η ∈ C ∞ c (B ′ 1 ) with η = 1 on B ′ 1/2 and η = 0 on R N \ B ′ 3/2 . Then we have that ηg ∈ L 2 (R N ). By minimization arguments, there exists W ∈ H 1 (R N +1 We define w = −t 1−2s W t and we observe that w ∈ L 2 (R N +1 From Remark 7.3 and Proposition 7.4, it follows that w =P (t, ·) * (ηg), whereP is the Bessel kernel for the conjugate problem given bȳ 2); we refer to Section 6.1 for asymptotics of the Bessel function K ν .
We have that U : . We deduce that, for some positive constants C, β depending only on N, s, m, by Proposition 3.7. We also observe that Then, by integration, we obtain that, for every x ∈ B ′ 1/4 and 0 < t 0 , t ≤ 1/4, in a weak sense, i.e., for all ϕ ∈ C ∞ c (B + R ∪ B ′ R ), (4.9) The following Pohozaev-type identity holds.
Let ρ < r < R. Integrating by parts (4.12) over the set We now claim that there exists a sequence ε n → 0 such that If no such sequence exists, we would have and thus there exists ε 0 > 0 such that It follows that, for all ε ∈ (0, ε 0 ), and so integrating the above inequality on (0, ε 0 ) we contradict the fact that w ∈ H 1 (B + R ; t 1−2s ). Next, from the Dominated Convergence Theorem, Lemma 4.1, and Proposition 3.7, we have that We conclude (replacing O ε with O εn , for a sequence ε n → 0) that
We prove now that, since w ≡ 0, H(r) does not vanish for r sufficiently small. Lemma 5.4. There exists R 0 ∈ (0, R) such that H(r) > 0 for any r ∈ (0, R 0 ), where H is defined by (5.9).
Proof. From Lemma 2.2 and Corollary 2.6, it follows that for every r ∈ (0, R 0 ). The conclusion follows from the above estimate, choosing r sufficiently small and using Lemma 2.2 and Corollaries 2.3, 2.6.
Proof. From (1.2) and (5.18) we deduce that and, therefore, for any r ∈ (0,R), we have that On the other hand, from (5.18) it also follows that Combining (5.20) with (5.21) we obtain the stated estimate.
Proof. In view of Lemma 5.8, part (iii), it is sufficient to prove that the limit exists. From (5.9), (5.11), and Lemma 5.8 it follows that d dr which, by integration over (r,R), yields ρ 0 ν i (t)dt , i = 1, 2, and ν 1 and ν 2 are as in (5.16) and (5.17). Since, by Schwarz's inequality, ν 1 ≥ 0, we have that lim r→0 + R r f 1 (ρ)dρ exists. On the other hand, by Lemmas 5.7 and 5.8, we have that for all ρ ∈ (0, R), which proves that f 2 ∈ L 1 (0, R). Hence both terms at the right hand side of (5.33) admit a limit as r → 0 + thus completing the proof.
From Lemma 5.9, the following point-wise estimate for solutions to (5.1) follow.
Proof. We first claim that In order to prove (5.34), we argue by contradiction and assume that there exists a sequence τ n → 0 + such that i.e., defining w τ as in (5.23), From Lemma 5.9, along a subsequence τ n k we have that w τn k → |z| γ ψ z |z| in C 0,α loc (S + 1/2 ), for some ψ eigenfunction of problem (1.7), hence passing to the limit in (5.35) gives rise to a contradiction and claim (5.34) is proved. The conclusion follows from combination of (5.34) and part (iii) of Lemma 5.8.
We will now prove that lim r→0 + r −2γ H(r) is strictly positive.
Defining ϕ i and ζ i as in (5.37) and (5.38), from (5.46) it follows that, for any i = j 0 , . . . , j 0 +M −1, As deduced in the proof of Lemma 5.12, for any i = j 0 , . . . , j 0 + M − 1 and τ ∈ (0, R] there holds for some c i 1 ∈ R. Choosing τ = R in the first line of (5.49), we obtain Hence (5.49) yields as τ → 0 + , and therefore from (5.48) we deduce that (5.5) holds; in particular the β i 's depend neither on the sequence {τ n } n∈N nor on its subsequence {τ n k } k∈N , thus implying that the convergences in (5.46) and (5.47) actually hold as τ → 0 + and proving the theorem.
We are now in position to prove Theorem 1.1 and its corollaries.
Proof of Theorem 1.1. Let u ∈ H s (R N ) be a nontrivial weak solution to Hu = 0 in Ω. By Theorems 6.1 and 7.1 in the appendices there exists a unique w = H(u) ∈ H 1 (R N +1 which also satisfies − lim in a weak sense. Therefore w solves (5.1) in the sense of (4.9). Then Theorem 1.1 follows from Theorem 5.1.
where the subscript t means derivatives with respect to t. In addition in the sense that: for any Ψ ∈Ḣ 1 D (R N +1 and Γ is the usual Gamma function. Extension theorems found useful applications in the study of fractional partial differential equations. For P (D) = −∆, see [4]. We also quote [23] with P (D) a second order differential operator with possibly non constant coefficients, see also [5]. A main point in our result is that the function space is explicitly given.
6.1. Proof of Theorem 6.1. We start with some preliminaries. For any v ∈ C ∞ c (R N ), we define H(v) via its Fourier transform with respect to the variable x as H(v)(t, ξ) = v(ξ)ϑ( P (ξ)t), where ϑ ∈ H 1 (R + ; t 1−2s ) solves the ordinary differential equation: We note that ϑ is a given by a Bessel function: where, K ν denotes the modified Bessel function of the second kind with order ν. It solves the equation We have, see [8], for ν > 0, as r → 0 and K −ν = K ν for ν < 0, while as r → +∞. By using the identity we get Since v ∈ C ∞ c (R N ), v decays faster than any polynomial. Then H(v) ∈Ḣ 1 D (R N +1 + ; t 1−2s ) and in addition it satisfies the equation We start by showing that P (D) satisfies the trace property that any w ∈Ḣ 1 D (R N +1 + ; t 1−2s ) has a trace which belongs toḢ s D (R N ). Proposition 6.2. There exists a (unique) linear trace operator ) and moreover where κ s is given by (6.9). Equality holds in (6.11) for some function w ∈Ḣ 1 D (R N +1 . By (6.10), we have that any w ∈ C ∞ c (R N +1 . Thanks to Parseval identity, we have We conclude that, for any v ∈ C ∞ c (R N ), (6.13) . This with (6.12) implies that The operator T is now defined as the unique extension of the operator w → w(0, ·).
For sake of simplicity, in this paper, we have denoted the trace of a function w ∈Ḣ 1 D (R N +1 + ; t 1−2s ) with the same letter w.
6.1.1. Proof of Theorem 6.1. We first consider u ∈ C ∞ c (R N ). In this case w = H(u) and it is, of course, unique inḢ 1 D (R N +1 By (6.10) and Proposition 6.2, we deduce, after integration by parts, that (6.14) H(u), Ψ H 1 for any Ψ ∈Ḣ 1 D (R N +1 + ; t 1−2s ), and this proves the theorem in this case. For the general case u ∈Ḣ s D (R N ), there exists a sequence u n ∈ C ∞ c (R N ) such that u n → u inḢ s D (R N ). It turns out that H(u n ) ⇀ w inḢ 1 D (R N +1 + ; t 1−2s ) and T r(H(u n )) ⇀ T r( w) = u iṅ H s D (R N ) . In particular for every ψ ∈ C ∞ c (R N +1 This implies that w = w and it is unique inḢ 1 D (R N +1 + ; t 1−2s ). By (6.14) for any Ψ ∈Ḣ 1 D (R N +1 + ; t 1−2s ). Taking the limit as n → ∞, we get the desired result.
Remark 6.3. We note that the trace operator T defined in Proposition 6.2 is surjective. To see that, we argue by density.
. By (6.13) H(v n ) is bounded and thus converges (up to subsequences) weakly to some function w ∈Ḣ 1 D (R N +1 + ; t 1−2s ) and Theorem 6.1 implies that the convergence is strong and thus w 2Ḣ 7.1. Bessel Kernel. We can observe that the Bessel kernel P m (t, x) is given by the Fourier transform of the mapping ξ → ϑ( |ξ| 2 + m 2 t), where ϑ is the Bessel function solving the differential equation (6.4) and yet we can determine it explicitly. Let U satisfy −div(t 1−2s ∇U ) + m 2 t 1−2s U = 0, in R N +1 We have that V = t 1−2s ∂U ∂t solves the conjugate problem: We look for F (the fundamental solution) which satisfies By direct computations we have where C N,s is a normalizing constant and K ν denotes the modified Bessel function of the second kind with order ν solving (6.6). Hence the choice of the Bessel Kernel in R N +1 Using the identity K ′ ν (r) = ν r K ν − K ν+1 , we obtain (7.2) P m (z) = C ′ N,s t 2s m N +2s 2 |z| − N +2s 2 K N +2s 2 (m|z|).
By using (6.7) we deduce that C ′ N,s is given by where p N,s is the constant for the (normalized) Poisson Kernel with m = 0, see [3]. We refer to [2], [6] for some Green function estimates for relativistic killed process. We also refer to [22] for estimates of the Bessel Kernel. We notice that, since P m (t, x) is the Fourier transform of ξ → ϑ( |ξ| 2 + m 2 t), we have Proof. We know that Or equivalently We now compute the left hand side of the above equality using the Bessel Kernel P m . Given t > 0, again by Parseval identity, we have R N ϑ( |ξ| 2 + m 2 t) − 1 t 2s u 2 (ξ) dξ = 1 t 2s R N u(x)P m (t, ·) * u(x) − u 2 (x) dx, where P m (t, ·) * u(x) = R N u(y)P m (t, x−y)dy. We normalize P m by putting P m (t, x) = 1 ϑ(tm) P m (t, x) so that R N P m (t, x)dx = 1. We therefore have for t > 0 We conclude that for every t > 0 (u(x) − u(y)) 2 (t 2 + |x − y| 2 ) N +2s 4 K N +2s 2 (m(t 2 + |x − y| 2 ) 1/2 ) dxdy. (7.7) We now have to check that we can pass to the limit as t → 0 under all the above three integrals. Firstly, we observe that the function r → ϑ(r)−1 r 2s is decreasing because K s is decreasing and thus since u ∈ H s (R N ), we deduce from (7.5) that For the same reason, we have that (7.9) lim t→0 1 t 2s R N u 2 (x)(ϑ(tm) − 1) dx = −m 2s κ s 2s R N u 2 (x) dx.
Secondly, thanks to the asymptotics of K ν , we have that there exist r, R > 0 such that C|z| −2ν , for |x − y| < r, C, for R ≥ |x − y| ≥ r, C|z| −2ν , for |x − y| > R, where C is a positive constant depending only on N, s, r, R and m. Since u ∈ H s (R N ), we can pass the limit as t → 0 under the integral in (7.7). This with (7.8) and (7.9) in (7.6) yields the result. Finally, to prove (7.4) we use the precise estimate (6.7) and comparing with the Dirichlet form in the case m = 0, see [7]. We observe, using similar arguments as in the the proof of Proposition 7.2, that, for w(t, x) = P m (t, ·) * u, with u ∈ C 2 c (R N ), we have that −div(t 1−2s ∇w)(t, x) + m 2 t 1−2s w(t, x) = 0, for all (t, x) ∈ R N +1 + , − lim t→0 t 1−2s ∂w ∂t (t, x) = κ s (−∆ + m 2 ) s u(x), for all x ∈ R N . We now prove the following result.

Harnack and local Schauder estimates for the relativistic Schrödinger equation.
Let f ∈ L 1 loc (B ′ 1 ). We recall that a solution (resp. subsolution, supersolution) u ∈ H s (R N ) to the equation The following regularity result holds.