Analytic hypoellipticity of Keldysh operators

We consider Keldysh-type operators, $ P = x_1 D_{x_1}^2 + a (x) D_{x_1} + Q (x, D_{x'} ) $, $ x = ( x_1, x') $ with analytic coefficients, and with $ Q ( x, D_{x'} ) $ second order, principally real and elliptic in $ D_{x'} $ for $ x $ near zero. We show that if $ P u =f $, $ u \in C^\infty $, and $ f $ is analytic in a neighbourhood of $ 0 $ then $ u $ is analytic in a neighbourhood of $ 0 $. This is a consequence of a microlocal result valid for operators of any order with Lagrangian radial sets. Our result proves a generalized version of a conjecture made by the second author and Lebeau and has applications to scattering theory.


Introduction
We consider analytic regularity for generalizations of the Keldysh operator [Ke51], (1.1) The operator P has the feature of changing from an elliptic to a hyperbolic operator at x 1 = 0. It appears in various places including the study of transsonic flows, see for instanceČanić-Keyfitz [CaKe96] or population biology -see Epstein-Mazzeo [EpMa13]. Our interest in such operators comes from the work of Vasy [Va13] where the transition at x 1 = 0 corresponds to the boundary at infinity for asymptotically hyperbolic manifolds (see [Zw16]), crossing the event horizons of Schwartzschild black holes (see [DyZw19a,§5.7]) or the cosmological horizon for de Sitter spaces. The Vasy operator in the asymptotically hyperbolic setting is given by where h(x 1 ) is a smooth family of Riemannian metrics in x ′ , x = (x 1 , x ′ ) ∈ R n and γ ∈ C ∞ (R n ). The resonant states at resonant frequencies λ (see [DyZw19a,Chapter 5]) are the smooth solutions of P (λ)u = 0.
For various reasons reviewed in §1.3 it is interesting to ask if in the case of analytic coefficients the resonant states are real analytic across x 1 = 0. That lead to [Zw17, Conjecture 2] which asked if P (λ)u = f with u smooth and f analytic near x 1 = 0 implies that u is analytic near x 1 = 0. For γ(x) ≡ 0 and h independent of x 1 , this was shown by Lebeau-Zworski [LeZw19] under the assumption that λ / ∈ −iN * . The characteristic varieties, x 1 cos 2 θ + sin 2 θ = 0 and cos 2 θ + x 1 sin 2 θ = 0, respectively, are shown with the direction of the Hamiltonian flow indicated. In the the Keldysh case, the two radial Lagrangians, Λ ± , correspond to θ = π and θ = 0 respectively.
In this paper we prove this result for generalized Keldysh operators with analytic coefficients (1.3). In particular, we do not make any assumptions on lower order terms: Theorem 1. Suppose that U ⊂ R n is a neighbourhood of 0, has analytic coefficients, Q(x, D x ′ ) is a second order elliptic operator in D x ′ with a real valued principal symbol. Then there exists a neighbourhood of 0, U ′ ⊂ U, such that (1.4) We will show in §1.1 that this result follows from a more general microlocal result valid for operators of all orders satisfying a natural geometric condition.
Remarks: 1. In the statement of the theorem 0 can be replaced by any point at which x 1 ≥ 0 and U ′ can be replaced by U provided we include a bicharacteristic convexity condition. That follows from propagation of analytic singularities -see [Ma02,Theorem 4.3.7] or [HiSj18, Theorem 2.9.1]: since there are no singularities near x 1 = 0 there will be no singularities on trajectories hitting x 1 = 0 -see Figure 1.

The result is false for the Tricomi operator
This can be seen using results about propagation of analytic singularities (unlike (1.3) this operator can be microlocally conjugated to D y 1 -see Figure 1) but is also easily demonstrated by the following example: Here, Ai is the Airy function which satisfies We then have and u is not analytic at 0.
3. Results similar to (1.4) have been obtained in the setting of other operators. In addition to the works [BoCa73], [BCH74] cited above, we mention the work of Baouendi-Sjöstrand [BaSj76] who considered a class of Fuchsian operators generalizing In the case of (1.7), (1.4) holds for any λ, µ ∈ C and [BaSj76] established (1.4) for more general operators satisfying appropriate conditions. 4. The operators (1.3), (1.5) and (1.7) are not C ∞ hypoelliptic, that is, P u ∈ C ∞ ⇒ u ∈ C ∞ . The study of operators which are C ∞ hypoelliptic but not analytic hypoelliptic has a long tradition with a simple example [HöI, §8.6, Example 2] given by x 3 . For more complicated cases, references, and connections to several complex variables, see Christ [Ch96] and for some recent progress and additional references, Bove-Mughetti [BoMu17].
1.1. A microlocal result. We make the following general assumptions. Let P be a differential operator of order m with analytic coefficients: where U is an open neighbourhood of x 0 ∈ R n . We make the following assumptions valid in a conic neighbourhood of (x 0 , ξ 0 ) ∈ T * R n \ 0: p is real valued and there exists a conic Lagrangian submanifold Λ, such that (1.9) Here means that the two vector fields are positively proportional, that is the Lagrangian is radial (the positivity assumptions can be achieved by multiplying P by ±1). Except for the analyticity assumption in (1.8) these are the assumptions made in Haber [Ha14] and Haber-Vasy [HaVa15].
Theorem 1 follows from the following microlocal result. We denote by WF the C ∞wave front set and by WF a the analytic wave front set -see [HöI,§8.1] and [HöI, §8.5,9.3], respectively.
Theorem 2. Suppose that P and (x 0 , ξ 0 ) ∈ T * R n \ 0 satisfy the assumptions (1.8) and (1.9). Then for u ∈ D ′ (R n ), (1.10) The proof is based on the theory of microlocal symbolic weights developed by Galkowski-Zworski [GaZw19b] and based on the work of Sjöstrand -see [Sj96,§2] (and also [HeSj86] and [Ma02, §3.5]). With this theory in place we can use escape functions, G, H p G ≥ 0, which are logarithmically bounded in ξ (hence the C ∞ wave front set assumption on u allows the use of such weights) and which tend to ξ in a neighbourhood of (x 0 , ξ 0 ). The normal form for p constructed in [Ha14] (following much earlier work of Guillemin-Schaeffer [GuSc77] which was based in turn on Sternberg's linearization theorem [St57]) was helpful in the construction of the specific weights needed here. We indicate the method of the proof in §1.2.
In the actual proof, the Fourier transform is replaced by the FBI transform (2.1) and its deformation (2.5) defined using a suitably chosen G ǫ satisfying (1.12) (see Lemma 3.1 which is the heart of the argument). One difficulty not present in the simple one dimensional case is the localization in other variables. It is here that the C ∞ normal forms of [St57], [GuSc77] and [Ha14] are particularly useful. It is essential that no analyticity is needed in the construction of G ǫ .
1.3. Applications to scattering theory. As already indicated in [Zu17] analyticity of smooth solution to the Vasy operator (1.2) implies analyticity of resonant states and of their radiation patterns. We review this here and, in Theorem 3, present a slightly stronger result.
For a detailed presentation of scattering on asymptotically hyperbolic manifolds we refer to [DyZw19a,Chapter 5]. To state Theorem 3, let M be a compact n + 1 dimensional manifold with boundary ∂M = ∅ and let M := M \ ∂M. We assume that M is a real analytic manifold near ∂M. A metric g on M is called asymptotically hyperbolic and analytic near infinity if there exist functions y ′ ∈ C ∞ (M ; ∂M) and is a real analytic diffeomorphism, and near ∂M the metric has the form, is an analytic family of real analytic Riemannian metrics on ∂M. Let Mazzeo-Melrose [MM87] and Guillarmou [Gu05] proved that continues to a meromorphic family of operators for λ ∈ C \ i(− 1 2 − N). In addition, Guillarmou [Gu05] showed that if the metric is even, that is, (1.16) (see [DyZw19a,Theorem 5.6] for an invariant formulation), then R g (λ) is meromorphic in C. In particular, for λ = 0 we have the following Laurent expansion The operator Π(λ) has finite rank and its range consists of generalized resonant states. We then have Theorem 3. Suppose that (M, g) is an even asymptotically hyperbolic manifold (in the sense of (1.16)) analytic near conformal infinity ∂M. Then for λ ∈ C \ 0, Moreover, in coordinates of (1.16), Proof. The metric (1.14) (in the coordinates valid near the boundary) gives the following Laplace operator: (1.18) Following Vasy [Va13] we change the variables to where, near ∂M, P (λ) is given by (1.2). This operator is considered on The key fact is that P (λ) is a Fredholm family operators on suitable spaces, P (λ) −1 is meromorphic and its poles can be studied using microlocal methods -see [Va13], [DyZw19a, Chapter 5] and also [Zw16, §2] for a short self-contained presentation.
From meromorphy of P (λ) −1 we obtain meromorphy of (1.15) using (1.19): (1.20) Here we make y iλ− n+2 2 1 f into an element of C ∞ c (X) by extending it by zero outside of M. Near any λ, P (ζ) −1 = , with Q j (λ) operators of finite rank and ζ → Q 0 (ζ, λ) is analytic near λ. We then have Hence, the claim about the range of Π(λ) follows from analyticity of functions in the range of Q 1 (λ). This follows from Theorem 1. In fact, Since we already know that the ranges of Q k 's are in C ∞ (see [DyZw19a, (5.6.10)]) we inductively conclude that the ranges are in C ω .

Preliminaries on FBI transforms and their deformations
We will use the FBI transform defined in [GaZw19b] in its R n (rather than T n ) version. Since the weights we use will be compactly supported in x the same theory applies. The constructions there are inspired by the works of Boutet de Monvel-Sjöstrand [BoSj76], Boutet de Monvel-Guillemin [BoGu81], Helffer-Sjöstrand [HeSj86] and Sjöstrand [Sj96]. An alternative approach to using the classes of weights we need here was developed independently and in greater generality by Guedes Bonthonneau-Jézéquel [GuJe20].
recalling that the left inverse of T is given by The first fact we need is the characterization of Sobolev spaces and of the C ∞ wave front set using the FBI transform (2.1). Proposition 2.1. There exists a constant C such that for u ∈ S ′ (R n ), Moreover, Proof. This follows from the characterization of the H s based wave front sets in Gérard [Gé90] as stated in [ where ǫ 0 is small and fixed (so that the constructions below remain valid as in [GaZw19b]). For convenience, we change here the convention from [GaZw19b]: it amounts to to replacing G by −G everywhere.
This provides us with the following new objects: the deformed FBI transform (see [GaZw19b,§4]), and the orthogonal projector The weight H appears naturally in this subject and is given by [GaZw19b,(3 We now prove a slightly modified version of [GaZw19b, Proposition 6.2]: for an open set U and K as in (2.4). Then (2.7) We remark that the expansion remains valid when h is fixed. We can use smallness of h to dominate the lower order terms and then keep it fixed.
Proof. The result follows from the analogue of [GaZw19b, Lemma 6.1] where the operator T Λ h m P S Λ is described in the case where the coefficients of P are globally analytic.
Here we point out that the analyticity of the coefficients is only needed in the neighbourhood U of K ⋐ R n such that in (2.4) supp G ⊂ K × R n and ǫ 0 is small enough depending on the size of the complex neighbourhood to which the coefficients extend holomorphically.
In fact, arguing as in the proof of [GaZw19b, Proposition 6.2] all we need is that for a ∈ C ∞ c (R n ) and a ∈ C ω (U), the Schwartz kernel of T Λ M a S Λ , M a f (x) := a(x)f (x), is given by (2.8) The phase in (2.8) is given by and the amplitude satisfies and A j are supported in a small conic neighbourhood of the diagonal in Λ × Λ. We note that if ǫ 0 is small enough, a extends to some neighbourhood of K in C n and hence To see (2.8) we use the definitions of T Λ and S Λ to write (2.11) We start by showing that the contribution to K a away from the diagonal is negligible. For that let χ ∈ C ∞ c (R) with χ ≡ 1 near 0. Then for all δ > 0 small enough, the operator R 1 with kernel . This amounts to showing that the operator with kernel To see this, we first integrate by parts K times in y, using that on suppχ δ . This reduces the analysis to the case of (2.10) with a is replaced by b(·, α, β) Next, we choose ψ ∈ C ∞ c (R n ; [0, 1]) with ψ ≡ 1 on V and supp ψ ⊂ U, and ψ 1 ∈ C ∞ c (R n ; [0, 1]) with ψ 1 ≡ 1 on V 1 and supp ψ 1 ⊂ V . We then deform the contour This contour deformation is justified since a ∈ C ω (U). The phase in the integrand of (2.10) becomes In particular, for y ∈ V , and (α, β) ∈ suppχ δ , the integrand is bounded by For the integral over y / ∈ V , we consider three cases. First, if both Re α x ∈ K and Re β x ∈ K, then it is easy to see that the integrand is bounded by e −c( α ξ + β ξ )( αx−βx +|y|)/h and hence produces a negligible contribution. Next, if Re α x / ∈ K and Re β x / ∈ K, then H(α) = H(β) = 0, α, β are real, and integration by parts in y shows that the contribution is negligible.
Finally, we consider the case Re α x ∈ K, Re β x / ∈ K, (the case Re β x ∈ K and Re α x / ∈ K being similar). In this case, we have H(β) = 0 and β real. Since y / ∈ V , we have that the integrand is bounded by e −c α ξ αx−y /h h K β ξ −K and hence this term is also negligible.
Since R is negligible, we may assume from now on that In particular, there are three cases: Re α x ∈ K and Re β x ∈ V 1 , Re β x ∈ K and Re α x ∈ V 1 , or Re α x / ∈ K and Re β x / ∈ K.
The first two cases are similar, so we consider only one of them. Since Re α x ∈ K and Re β x ∈ V 1 , the contribution from y / ∈ V is negligible. Therefore, we may deform the contour to y → y + ψ(y)y c (α, β), The proof in this case then follows from the method of complex stationary phase.
When, both Re α x / ∈ K and Re β x / ∈ K, α = Re α, β = Re β, and H(α) = H(β) = 0. In order to handle this situation, we will Taylor expand a(y) around y = α x . For that we first consider (2.10) with a = O(|y − α x | 2N ). In that case, we consider the integral (1 −χ δ (α, β))dy. (2.12) Changing variables y → y + α x , Therefore, using the Schur test for boundedness, the operator K N with kernel K N (α, β) satisfies where a N (y) is a polynomial of order 2N − 1 in (y − α x ). In particular, Since a N is analytic and the integrand is exponentially decaying in y, we may deform the contour with y → y + y c (α, β) in the integral forming the kernel of K a N and apply complex stationary phase as in the case where Re α x ∈ K or Re β x ∈ K. This finishes the proof of the proposition after taking N large enough.
2.2. Analytic wave front set. We now relate weighted estimates to analyticity. Proposition 2.3. Let T be the FBI transform defined in (2.1) for some fixed h, and let ψ ∈ S 1 (T * R n ) satisfy where U ⊂ R n and Γ ⊂ R n \ 0 is an open cone. Then, for u ∈ H −N (R n ), (2.14) Conversely, suppose u ∈ H −N (R n ), Γ 0 ⊂ R n is a conic open set such that Γ 0 ∩ S n−1 ⋐ Γ ∩ S n−1 , U 0 ⋐ U. Then for any ψ ∈ S 1 (R n × R n ) with supp ψ ⊂ U 0 × V 0 , (2.15) Remark: Here we do not consider uniformity in h in the L 2 bounds. If we demanded that, than we would only need ψ ∈ C ∞ c (T * R n ), ψ > 0 on U × (Γ ∩ S n−1 ). The proof is based on the following Lemma 2.4. Let T and S be given by (2.1) and (2.2), respectively, with h fixed. Suppose that χ,χ ∈ S 0 (R n × R n ) and supp χ, supp χ 1 ⊂ K × R n , K ⋐ R n . Then for any a > 0 there exists b > 0 such that for any N.
If in addition χ 1 ≡ 1 on a a conic neighbourhood of the support of χ, then there exists b > 0 such that (2.17) for any N.
Proof of Proposition 2.3. We start by recalling the characterization of the analytic wave front set using the standard FBI/Bargmann-Segal transform: Then see [HöI,Theorem 9.6.3] for a textbook presentation; note the somewhat different convention: T u(x, ξ; h) = e − 1 2h ξ 2 T 1/h u(x − iξ). We first prove (2.14). Hence suppose that (x 0 , ξ 0 ) ∈ U × Γ. Let χ ∈ S 0 be supported in a small conic neighbourhood, U 0 × Γ 0 , of (x 0 , ξ 0 ) and choose χ 1 ∈ S 0 which is supported in U × Γ and is equal to 1 on a conic neighbourhood of the support of χ and χ 2 ∈ S 0 supported in U × Γ and equal to 1 on a conic neighborhood of the support of χ 1 . Our assumptions then show that e a ξ /h χ 2 T u ∈ L 2 (R 2n ) for some a > 0. We now write χe b ξ T u = χe b ξ T S χ 1 e −a ξ e a ξ χ 2 T u + (1 − χ 1 ) ξ N ξ −N T u .
The next proposition relates weighted estimates to deformed FBI transform: In addition, For a simpler version of this result in the case of compactly supported weights see [GaZw19a,§8].
To obtain (2.23) we apply the same analysis to T Λ S and we need to show that two weights coincide. That is done as in [GaZw19a,§8].
We will then put Λ ǫ := Λ Gǫ so that the assumption u ∈ C ∞ will give u ∈ H Λǫ . On the other hand the assumption that Γ ∩ WF a (P u) shows that P u H Λǫ ≤ C with the constant C independent of ǫ. But then [GaZw19b, Proposition 6.2] and the properties of G ǫ show that u H Λǫ is bounded independently of ǫ. Propositions 2.3 and 2.5 then show that WF a (u) ∩ Γ 0 = ∅.
3.1. Construction of the weight. We now construct a family of weights, G ǫ , satisfying (3.1). In fact, we need more precise conditions on G ǫ given in the following Lemma 3.1. Suppose that p satisfies (1.9) at ρ 0 = (x 0 , ξ 0 ) ∈ T * R n \ 0 and Γ is an open conic neighbourhood of ρ 0 . Then, there exists G ǫ ∈ S 1 (T * R n ), supp G ǫ ⊂ Γ, such that We stress that the constants C αβ and c 0 are independent of ǫ and M 1 .
3.2. Microlocal analytic hypoelliticity. We will have bounds which are uniform in ǫ but not in h. We start with the following Lemma 3.2. Suppose that P is of the form (1.8) with real valued principal symbol p and suppose that Γ ⊂ U × R n \ is an open cone, Γ ∩ S n−1 ⋐ U × S n−1 and G ∈ S 1 (Γ; R), |G| ≤ C log ξ , (3.14) Then for T Λ , H Λ , Λ = Λ θG defined in (2.4) and (2.6), h and θ sufficiently small, and where M depends only on P and the semi-norms of G in S 1 .
Therefore, applying the Schur test for L 2 boundedness completes the proof that χ ξ K e −aG/h T Λ S ξ −K = O(1) : L 2 (T * R n ) → L 2 Λ and the lemma follows.
With these two lemmas in place we can prove the main result: Proof of Theorem 2. By multiplying u by a C ∞ c -function which is 1 in a neighbourhood of x 0 , we can assume that u ∈ H −N +m , for some N, is compactly supported in U and ρ 0 := (x 0 , ξ 0 ) / ∈ WF(u). By Proposition 2.1, there existsχ ∈ S 0 withχ ≡ 1 in an open conic neighborhood, Γ, of ρ 0 such that for any K > 0, (3.20) Also, since u ∈ H −N +m , ξ −N +m T u L 2 ≤ C. (3.21) Let Γ 1 ⋐ Γ be an open conic neighborhood of ρ 0 and χ ∈ S 1 with χ ≡ 1 on Γ 1 and supp χ ⊂ Γ.
We choose θ small enough so that (2.4) and (3.16) hold. We then fix 0 < h ≤ 1 small enough so that (3.16) holds. From now we neglect the dependence on h which is considered to be a fixed parameter. We choose for G = G ǫ constructed in Lemma 3.1 and supported in Γ 1 . We recall that the estimates depend only on the S 1 seminorms of G and these are uniform in ǫ. We now claim that u ∈ H −N +m Λǫ , Λ ǫ := Λ θGǫ .
Next, note that P u ∈ H −N is supported in U and ρ 0 / ∈ WF a (P u) .Propositions 2.3 and 2.5 (see (2.15) and (2.23) respectively) then show that for G ǫ satisfying the assumptions of Lemma 3.2 and θ sufficiently small P u H −N Λǫ ≤ C 0 , where C 0 depends only on P u and S 1 -seminorms of θG ǫ .