Wave extension problem for the fractional Laplacian

We show that the fractional Laplacian can be viewed as a Dirichlet-to-Neumann map for a degenerate hyperbolic problem, namely, the wave equation with an additional diffusion term that blows up at time zero. A solution to this wave extension problem is obtained from the Schr\"odinger group by means of an oscillatory subordination formula, which also allows us to find kernel representations for such solutions. Asymptotics of related oscillatory integrals are analysed in order to determine the correct domains for initial data in the general extension problem involving non-negative self-adjoint operators. An alternative approach using Bessel functions is also described.


Introduction
In the last decade a lot of attention has been devoted to fractional powers of the Laplacian −∆. The seminal paper by L. Caffarelli and L. Silvestre [CS07] showed that the operator (−∆) σ , with 0 < σ < 1, mediates between Dirichlet and Neumann boundary values on R d for a certain degenerate elliptic problem in the upper half-space R d+1 + . Motivated by their work we show that up to a multiplicative constant the fractional Laplacian can also be viewed as a Dirichlet-to-Neumann map f → lim t→0 t 1−2σ ∂ t u for the hyperbolic problem (1) ∂ 2 t u + 1−2σ t ∂ t u = ∆u, u(·, 0) = f.
Here 0 < σ < 1, and for σ = 1 2 we have the classical wave equation. We use the language of semigroups to study this problem in a parallel way to that used by Stinga and Torrea [ST10] for discussing the elliptic problem of Caffarelli and Silvestre in the generality of non-negative self-adjoint operators. We also wish to point out that an alternative approach to problem (1) by means of Bessel functions is available (see Section 4).
Our main result says that a solution to (1) is given by means of the 'oscillatory' subordination formula (2) u(·, t) = i σ t 2σ 4 σ Γ(σ)ˆ∞ 0 e −i t 2 4s e is∆ f ds s 1+σ , also when ∆ is replaced by a more general non-negative self-adjoint operator. It allows us to make use of the Schrödinger kernel in order to find integral representations for solutions to the associated Neumann problem in different dimensions.
It is perhaps interesting to note that in the classical case when σ = 1 2 , the formula (2) leads to the identity for the wave group (e −it √ −∆ ) t∈R . Recall in particular that the solution u to the wave equation with Neumann initial data g (given by the imaginary part of the wave group) (4) u(·, t) = sin(t √ −∆) √ −∆ g can be expressed in dimensions 3 and 2, respectively, by (5) u(x, t) = t ∂B(x,t) g(y) dS(y) = c t S| ∂B(0,t) * g(x) and (6) u(x, t) = t 2 2 B(x,t) g(y) .
(see [Eva98,Section 2.4,Equations (22) and (27)]). 1 These formulas highlight Huygens' principle concerning the finite propagation speed of solutions to the wave equation. It would be peculiar if the smooth and fully supported Schrödinger kernel would transform by (3) into a singular and spherically supported one such as 1 t S| ∂B(0,t) in (5), and indeed there are restrictions in the relation of σ and the dimension in our kernel representations.
We will now describe the content of this paper. Let L be a non-negative self-adjoint operator on a Lebesgue space L 2 and consider the Schrödinger group (e −isL ) s∈R . In Theorem 1 we show that for 0 < σ < 1 the oscillatory integral when interpreted as an improper integral, converges weakly in L 2 for suitable f and solves the equation with the initial data u(·, 0) = f and Theorem 1 is established via the Spectral theorem (see Proposition 1); the requirement that f ∈ Dom(L σ 2 + 3 4 ) is needed to deal with the asymptotics of oscillatory integrals in Lemmas 2 and 3. Our method for analysing these integrals and overcoming the delicate problems of convergence involves 'sidestepping' the imaginary half-axis and move to more suitable complex paths that provide sufficient decay. In Corollary 1, a solution u to (7) with real initial data u(·, 0) = f and ∂ σ t u(·, 0) = g is presented in terms of real and imaginary parts of U σ t as u(·, t) = c σ Re(i 1−2σ U σ t (f )) − c σ Im(U σ t (L −σ g)). For σ = 1 2 this reduces to In Section 3, we return to the case of L = −∆ and find kernel representations for solutions u to the Neumann problem ∂ 2 t u + 1−2σ t ∂ t u = ∆u, u(·, t) = 0, ∂ σ t u(·, t) = g on R d for d = 1, 2, . . . , 5. From the subordination formula (2) we obtain we arrive at the expression the imaginary part of which gives the solution u. Theorem 2 concerns dimensions d = 1, 2, 3 and σ for which 0 < d 2 − σ < 1 and derives from (9) the formula The dimensions d = 3, 4, 5 are considered in Theorem 3, which states that for σ such that 1 < d 2 −σ < 2 we have The limiting cases when d 2 − σ tends to either zero or one are studied in Theorem 4; in dimensions d = 2, 3, 4 we then have In Theorem 5 we show that the solution to Schwartz initial data f and g is unique and can be given in terms of Bessel functions J ±σ , namely, 2 this coincides with the classical formula (8) (with L = −∆); indeed √ rJ −1/2 (r) = 2 π cos r and √ rJ 1/2 (r) = 2 π sin r (and c 1/2 = c 1/2 = π/2). Growth estimates for Bessel functions allow us to deduce, by means of the Fourier transform, fixed-time estimates for solutions of the equation (see Theorem 6). Finally, a classical integral representation for (modified) Bessel functions K σ is converted into an oscillatory integral formula coinciding with (2), thus closing the circle. In order to avoid ambiguity, let us agree that i α = e iπα/2 . Moreover, by saying that an integraĺ ∞ 0 is convergent, we mean that the limit of´R ε exists as ε → 0 and R → ∞. By α β we mean that there exists a constant C such that α ≤ Cβ. Two quantities α and β are comparable, α ∼ β, if α β and β α.
Acknowledgements. The first author gratefully acknowledges the financial support from the Väisälä Foundation and is thankful for the hospitality of the Department of Mathematics at the Autonomous University of Madrid during his stay. The third author is supported by the grant MTM2011-28149-C02-01 from Spanish Government.

An oscillatory subordination formula
Let L be a non-negative self-adjoint operator on a Lebesgue space L 2 and let 0 < σ < 1. In this section, we study when and how a solution to the equation can be obtained from the oscillatory integral The main result of the article is: ) then u is a weak solution to equation (10) in the sense that for all h ∈ L 2 . Moreover, u convergences to the initial data A solution for any combination of Dirichlet and Neumann initial data can be obtained in terms of real and imaginary parts: solves equation (10) with real initial data u(·, 0) = f, ∂ σ t u(·, 0) = 0, whenever f ∈ Dom(L σ 2 + 3 4 ), and • the function solves equation (10) with real initial data u(·, 0) = 0, ∂ σ t u(·, 0) = g, whenever g ∈ Dom(L − σ 2 + 3 4 ).
Proof of Corollary 1. To prove the first claim, it suffices to calculate π 2 )f = sin(σπ)f and to note that For the second claim we begin by noting that The rest of the section is devoted to the proof of Theorem 1, and we start with several auxiliary results. Notice that the following simple lemma is false for σ = 1.
Part I: We consider the truncated integral its convergence as ε → 0 and R → ∞, and to what extent it gives a solution to equation (11). Note first that the integral is absolutely convergent as ε → 0. On the other hand, using integration by parts, we see that Here so that the last integral converges absolutely as R → ∞. Since the integrated term converges as R → ∞, we see that also the integral (12) converges. We then study the convergence of the integral corresponding to the time derivatives, and calculate: λ ds s 2−σ , which converges absolutely as R → ∞, but appears problematic for ε → 0 (see Remark 2). In order to see to what extent the truncated integrals (12) solve equation (11), we note from (14) and (15) that In the light of equations (13) and (16), we infer that At the upper limit R → ∞ the integrated term vanishes, but at the lower limit ε → 0 it diverges and a careful argument will be needed. For now we record that Note here that the integral (12) together with its time derivatives converges locally uniformly in t, which justifies differentiating under the integral sign. Part II: In order to show that (17) from which (11) follows, we view the question in terms of complex path integrals: where the paths γ approximate the segment [0, i] in a suitable way (which we elaborate on below). Let 0 < δ < π 2 be fixed and consider, for 0 < ε < 1, the paths arc ε , arc 1 and ray ε given by θ → εe iθ and θ → e iθ , π 2 ≤ θ ≤ π 2 + δ, and s → se i( π 2 +δ) , ε ≤ s ≤ 1.
By Cauchy's integral theorem we may now writê To see that the first integral on the right-hand side vanishes as ε → 0, it suffices to observe that Re(e ±iθ ) = cos θ ≤ 0 when π 2 ≤ θ ≤ π 2 + δ so that The second integral on the right-hand side iŝ and it converges to a limit J δ (λ, t) as ε → 0, again because Re(e −i( π 2 +δ) ) = cos( π 2 + δ) < 0. Lastly,ˆa Part III: For (17) it suffices to show that It is easy to see that Replacing λ by iλe −iθ δ in (16) and (13), we find that where the integrated term poses no problem at s = 0 because Re(e −iθ δ ) = cos θ δ < 0. As δ → 0, the integrated term therefore tends to λi σ+1 e −i e −i t 2 4 λ , and (18) and (17) follow. Part IV: We address the convergence to the initial value. By Lemma 1, The integral above is absolutely convergent as ε → 0 and therefore by dominated convergence, Using integration by parts, we see that The two integrals on the right-hand side converge absolutely and thus by dominated convergence we haveˆ∞ Performing a change of variables t 2 4s λ = r in the truncated integral, we obtain Since the convergence of the last integral here is locally uniform in t, it is easy to verify that Remark. As ε → 0, the integralˆR ε e −is e −i t 2 4s λ ds s k−σ converges for k = 2 but diverges for k = 3. In both cases, this can be seen by means of the change of variable r = t 2 4s λ. For k = 2, we also get bounds for the integral by means of Lemma 2. Lemma 2. Let 0 < σ < 1. For every t > 0 we have Proof. We denote A = t √ λ 2 and show that ˆR ε e −i(s+ A 2 s ) ds s 1−σ 1, Observe that the phase ϕ(s) = s + A 2 s has its critical point where 0 = ϕ (s) = 1 − A 2 s 2 at s = A. Consider first the case A < 1. We write ˆ2 ε +ˆR 2 e −i(s+ A 2 s ) ds s 1−σ =: I 1 + I 2 , and note that |I 1 | 1. Using integration by parts, we see that and upon calculating the derivative Consider then the case A ≥ 1. We follow the principle of stationary phase and decompose the integral into three pieces For I 1 + I 3 we use integration by parts: and similarly Moreover, because as before. Therefore Lemma 3. Let 0 < σ < 1. For every t > 0 we have Proof. In order to obtain estimates for the derivatives we now split the interval of integration in a way which does not depend on t. We shall prove the estimate for t in a small neighbourhood of a fixed point t 0 > 0, by moving the differentiations into the integrals. This will be uniform in t 0 > 0. With A = t √ λ/2 as before, we write A 0 for the value corresponding to t 0 . Let d = min(A 0 , √ A 0 )/2. The integral will be split at the points a = A 0 − d and b = A 0 + d. We consider only values of t so close to t 0 that |A − A 0 | < d/2. Thus |s − A| > d/2 ∼ min(A, √ A) as soon as s / ∈ (a, b). For such t we write e −i(s+ t 2 4s λ) ds s 1−σ = I 1 + I 2 + I 3 , say. In I 2 and I 3 , it is clear that the derivatives can be taken inside the integrals: To estimate I 2 , we observe that s ∼ A for a ≤ s ≤ b and that b − a = 2d √ A ∼ λ 1/4 . Thus For I 3 the estimate is clear when A ≤ 1: When A > 1 we integrate by parts, with k = 2, 3: Here Moreover, since by simple calculus, as before. Therefore as before. We handle I 1 by switching to complex path integrals as in the proof of Proposition 1, namelŷ where ray a : s → se i 2π 3 , 0 < s ≤ a, and arc a : θ → ae i(θ+ π 2 ) , 0 ≤ θ ≤ π 6 .
In the integrals over ray a and arc a , we shall take the derivatives of the integrand and verify convergence of the resulting integrals. Observe first that for 0 ≤ θ ≤ π/6 |e −ae i(θ+π/2) + A 2 a e −i(θ+π/2) | = e − −a+ A 2 a sin θ , and here For θ = π/6 this implies that and for A > 1ˆr for any M .
We also get ˆa for all A.
Altogether, this implies that Proof of Theorem 1. Assume first that and denote by E the spectral measure of L. In order to see that the family converges weakly in L 2 as ε → 0 and R → ∞ note first that for any h ∈ L 2 , Here, by Lemma 2, ˆt2 /4ε 2 , from which the convergence follows.
Assume then that f ∈ Dom(L σ 2 + 3 4 ). By Lemma 3 and Proposition 1, we have Weak convergence to the initial data, holds by Proposition 1 and Lemma 2. Moreover, taking into account that we have, by Proposition 1, that Remark. Note that the proof of Theorem 1 entails the fixed-time norm estimate

Kernel representations in the case of the Laplacian
Let 0 < σ < 1 and consider the Neumann problem for Schwartz initial data g on R d . In this section we prove the following three results: Theorem 2. Suppose that d ∈ {1, 2, 3} and let 0 < σ < 1 be such that 0 < d 2 − σ < 1. A solution to the Neumann problem (21) with Schwartz initial data g is given by Theorem 3. Suppose that d ∈ {3, 4, 5} and let 0 < σ < 1 be such that 1 < d 2 − σ < 2. A solution to the Neumann problem (21) with Schwartz initial data g is given by This can be continued to higher dimensions, but the formulas will be more complicated.
Theorem 4. The solution u σ of the Neumann problem (21) with Schwartz initial data g has the limit Before the proofs, we use the Schrödinger kernel to rewrite the solution formulas. According to Corollary 1, a solution to (21) is given by where e −i t 2 4s e is∆ g ds s 1−σ , by change of variable. Hence u(·, t) is the imaginary part of the integral The Schrödinger group is given, for Schwartz functions g, by the left-hand side converges absolutely and uniformly in x for all d and σ (restricting to σ < 1 2 for d = 1). Indeed, ImV γ g(x, t).
In the inner integral here, we make a change of variable t 2 −|x−y| 2 4s = ±r, separating the cases t > |x−y| and t < |x − y|. Doing so we obtain, respectively, where we also used Lemma 1. Consequently, we have by dominated convergence Proof of Theorem 2. The formula for the solution u follows from (23) and (25) with γ = d 2 − σ. In order to calculate ∂ t ImV β g(x, t) for 0 < β < 1 we argue by change of variables y = y−x t : Proof of Theorem 3. The formula for the solution follows by means of (24) from (23) with γ = d 2 − σ and (26) with β = γ − 1, so that Limiting cases γ 1 and γ 1. For the case γ 1, recall first that By the change of variable z = y − x and writing z = tz |z| , the integral can be written Here and, with polar coordinates, After the transformation ρ = r 2 /t 2 , the inner integral here will be .
Noting that d = ∆h and y = ∇h for h(y) = |y| 2 2 , we make use of Green's formulâ to see that also Proof of Theorem 4. Note first that γ = d 2 − σ → 1 is possible in dimensions d = 2, 3 and 4 when σ 0, σ → 1 2 and σ 1, respectively. Since Remark. The one-dimensional problem is not covered by Theorem 2 when σ > 1 2 . A solution formula can be derived by the method of descent by viewing u and g with an additional spatial variable and using Theorem 2 for d = 2. Doing so we obtain u(x, t) = σ π t 2σ¨y 2 1 +y 2 2 <1 g(x + ty 1 ) (1 − y 2 1 − y 2 2 ) 1−σ dy 1 dy 2 . Here the double integral can be calculated as follows: (1 − s 2 ) σ−1 ds dy 1 , . Therefore u(x, t) = σΓ(σ) which converges to 1 2ˆx +t x−t g(y) dy, as σ 1 2 . This coincides with the limit as σ 1 2 . Remark. In a similar vein one may consider the modified Klein-Gordon equation arising from L = −∆ + m 2 with m > 0. The solution to (21) is then given by where the quantity inside Im coincides with −im 2 s ds s 1+γ g(y) dy, Assuming that 0 < γ < 1, we may use the formula (cf. Lemma 1) > 0 to deduce finite speed of propagation:

A Bessel function approach
In this section we study the wave extension problem for the Laplacian L = −∆ on R d by means of Bessel functions and obtain some elementary estimates for the solutions. In order to simplify the presentation, we make stronger decay and regularity assumptions than necessary.