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We consider Vasy’s operator [10] with analytic coefficients. That operator arises in the study of scattering on asymptotically hyperbolic manifolds and in general relativity. 1. General assumptions and conjectures In his investigation of scattering resolvent on asymptotically hyperbolic manifolds, Vasy [10] introduced a degenerate differential operator switching behaviour from elliptic to hyperbolic along a hypersurface. More general operators with similar properties appear also in the study the of Kerr–de Sitter wave equation. We refer to [4, Chapter 5],[10],[13] and [14, §3.1] for motivation and general results. In this note we present some remarks about Vasy’s operator with analytic coefficients: we ask some general questions about propagation of analytic singularities, prove a conditional analytic hypoellipticity in a model case and give a direct prove of a Cauchy–Kovalevskaya theorem for this operator. After this note was written, Zuily [12] showed that by adapting methods of Bolley–Camus [3], conditional analytic hypoellipticity holds for all such operators settling [14, Conjecture 2]. The method of proof in the model case presented in §2 is however very different and might be open to generalizations. To introduce Vasy’s operator, suppose that M is a compact analytic manifold and (−1, 1) 3 x 7→ Q(x, y,Dy) = ∑ |α|≤2 aα(x, y)D α y , D := 1 i ∂ is an analytic family of self-adjoint (with respect to a density on M) elliptic second order differential operators with analytic coefficients. We are interested in the following family of operators depending on a parameterλ ∈ C: P (λ) := xD x − (λ+ i)Dx + γ(x, y)xDx +Q(x, y,Dy). (1) This is the operator which plays a role in Vasy’s treatment of scattering on asymptotically hyperbolic manifolds and is close to the operator e ge where g is a Kerr–de Sitter-like metric. 1 2 GILLES LEBEAU AND MACIEJ ZWORSKI Here we assume that the coefficients of Q and the function γ are real analytic in a neighbourhood of (−1, 1)×M , aα, γ ∈ C([−1, 1]×M). We denote by WF(u) and WFa(u), C ∞ and analytic wave front sets of u ∈ D′((−1, 1)× M), respectively – see [6, Chapter 8],[8]. We have the following possible analytic regularity statements of increasing strength: for u ∈ C∞((−1, 1)×M) and λ / ∈ −iN∗, P (λ)u = f ∈ C((−1, 1)×M) =⇒ u ∈ C((−1, 1)×M); (2) P (λ)u = f, WFa(f) ∩N∗(M × {0}) = ∅ =⇒WFa(u) ∩N∗(M × {0}) = ∅; (3) P (λ)u = f ∈ C(U), (0, y0) ∈ U ⊂ (−1, 1)×M =⇒ u ∈ C(V ), (4) for some neighbourhood V of (0, y0). Finally, P (λ)u = f, (0, y0, 1, 0) / ∈WFa(f), y0 ∈M =⇒ (0, y0, 1, 0) / ∈WFa(u). (5) In the local versions of the conjecture we can take M = R. The implication (2) is a conditional form of analytic hypoellipticity, that is analytic hypoellipticity under a regularity assumption. If we assume that u ∈ H for s + 1 2 > − Imλ (rather than u ∈ C∞) then the statements (2)–(5) hold in the C∞ category, that is for C replaced by C∞ and WFa replaced by WF – see [13, §4] for (2),(3) and [5] for (4),(5). Consequently we expect (2)–(5) to be valid under that weaker assumption u ∈ H, s > − Imλ − 1 2 (which can be further microlocalized). 2. Proof of analytic hypoellipticity in the model case We will prove (2) in the model case of γ = 0 and Q(x, y,Dy) = −∆M where −∆M is the Laplacian for a real analytic metric on M . The proof is based on separation of variables and we will quote some results about Fuchsian differential operators – see §3 – as well as some standard results about Bessel function asymptotics. It is enough to show that u is analytic in (−ε, ε)×M for some ε > 0 as propagation of analytic singularities [8] (which is simply ellipticity for x > 0) implies that u is analytic in (−1, 1)×M . We first recall the following consequence of [2, Theorem 1]. In §3 we present a self-contained argument in our special case. Lemma 2.1. Suppose that λ / ∈ −iN∗. There exists ε > 0 such that for any f ∈ C((−1, 1)×M)) we can solve P (λ)u = f with u ∈ C((−ε, ε)×M). Hence we can assume that f ≡ 0 and it is enough to show that for P (λ)u = 0 near x = 0, u(x, y) is analytic in (−ε, ε)×M for some ε > 0. For that it is in fact sufficient VASY’S OPERATOR WITH ANALYTIC COEFFICIENTS 3 to show that analyticity holds true in [0, ε)×M for some ε > 0. This follows from the following observation: a solution analytic in [0, ε) extends to a solution in ũ in (−ε1, ε) for some ε1 > 0. Hence the difference v := ũ− u is a smooth solution in (−ε1, ε)×M supported in (−ε1, 0]×M . But then [13, Lemma 1] shows that v ≡ 0 and hence u is analytic in (−ε1, ε)×M . Let {uk(y)}k=0 be an orthonormal set of eigenfunctions of −∆M : (−∆M − μk)uk(y) = 0, 〈uk, u`〉L2(M) = δk`. We then write u(x, y) = ∞ ∑ k=1 ak(x)uk(y) where the coefficients ak(x) are smooth and solve (∂xx∂x − iλ∂x − μk)ak(x) = 0, |x| < δ. (6) Let H be the unique solution to (∂xx∂x − iλ∂x − 1)Hλ(x) = 0, Hλ(0) = 1, λ / ∈ −iN∗, where the uniqueness follows from the indicial equation as in the proof of the following Lemma 2.2. For x ∈ C with | arg x| < π, Hλ(x) = 1 √ 2π Γ(1− iλ)2x iλ 2 − 1 2 e √ (1 +Oλ(1/|x|)), |x| → ∞. (7) In addition, for x ∈ C with |x| > 1, |Hλ(x)| ≤ eλ √ |x|. (8) Proof. If x = 1 4 z then (see [13, (2.2)] where the motivation for this change of variables is also presented) x(∂xx∂x − iλ∂x − 1) = 14z (z∂ z + z∂z + λ 2 − z2)z−iλ which is the modified Bessel equation with the parameter α = −iλ. It is solved by I−iλ(z) = 1 Γ(1− iλ) (z 2 )−iλ (1 + zF (z)), z → 0+ Hence Hλ(x) = Γ(1− iλ)x iλ 2 I−iλ(2 √ x), and the expansion follows from the standard asymptotic expansion of I−iλ(z) – see [1, (9.7.1)]. The upper bound (8) is immediate from the recursion relations in the expansion of Hλ(x): Hλ(x) = ∞ ∑ k=0 Hλ,kx , Hλ,0 = 1, Hλ,k+1 = 1 (k + 1)(k + 1− iλ) Hλ,k. 4 GILLES LEBEAU AND MACIEJ ZWORSKI Rescaling (6) by x̃ = μkx we see that ak(x) = ak(0)Hλ(μ 2 kx). (9) Since u(x, y) is smooth in x and y it follows that, uniformly for |x| ≤ 1 2 , ak(x) = 〈u(x, •), uk〉L2(M) = 〈(I −∆M)u, (I −∆M)uk〉L2(M) = ON(〈μk〉), though we only need the case of N = 0. These two facts and (7) give, for large values of k, |ak(0)| = |ak(12)||Hλ( 1 2 μk)| ≤ Cλek. Using this, (8) and the analyticity of Hλ in (9) give |∂ xak(x)| ≤ C`+1``e−μk/3, |x| ≤ δ, (10) where δ depends only on λ. On the other hand, |∂ y uk(y)| ≤ C1+|α|μ n/2 k (μk + |α|) |α|, (11) see for instance [7] for a self-contained presentation. Putting N(r) = max{k : μk ≤ r} ≤ C0r, n = dimM , we obtain |∂ x∂ y u| ≤ C |α|+`+1`` ∞ ∑ k=0 (μk + |α|)|α|e−μk/4 = C |α|+`+1`` ∫ ∞ 0 (r + |α|)|α|e−r/4dN(r) ≤ C |α|+`+1 1 ``|α||α|, ,

Remarks on Vasy’s operator with analytic coefficients