On the analytic continuation of the Poisson kernel Matthew B. Stenzel Abstract We give a \heat equation" proof of a theorem which says that for p all sufficiently small, the map S : f 7! exp(− ∆)f extends to an s s+(n−1)=4 isomorphism from H (X) to O (@M). This result was announced by L. Boutet de Monvel in 1978 but only recently has a proof, due to S. Zelditch [23], appeared in the literature. The main tools in our proof are the subordination formula relating the Poisson kernel to the heat kernel, and an expression for the singularity of the Poisson kernel in the complex domain in terms of the Laplace transform variable s = d2(z; y) + 2 where d2 is the analytic continuation of the distance function squared on X, z 2 M, and y 2 X. 1 Introduction Let (X; g) be a compact, connected, n-dimensional real analytic Riemannian manifold without boundary and with Laplace operator ∆ ≥ 0. The Poisson p operator, exp(−τ ∆), is the solution operator at time t to the pseudodifferential initial value problem p (@τ + ∆)u = 0 ujτ=0 = f: L. Boutet de Monvel observed in [3] that this operator has a remarkable analytic continuation property. Let X,! M be an embedding of X as a totally real submanifold of a complex manifold M. Then for each sufficiently small τ > 0 there is a complex neighborhood M of X such that the Schwartz kernel p τ of exp(−τ ∆) can be extended to a smooth function on Mτ × X which is holomorphic on Mτ . The restriction to @Mτ × X of this kernel is a Fourier integral distribution of complex type and the corresponding Fourier integral s operator, which we denote by Sτ , is a continuous bijection between H (X) and s+ n−1 the Sobolev space O 4 (@Mτ ) of boundary values of holomorphic functions 1 on Mτ : s ∼= s+ n−1 Sτ : H (X) −! O 4 (@Mτ ): The Mτ are the sublevel sets of a certain non-negative strictly plurisubharmonic −1 function, Mτ = fφ < τg, and X is identified with φ (0). One consequence of this is a Paley-Wiener type theorem on manifolds [9]: a function on X can be analytically continued to a holomorphic function on M with Sobolev regularity τ p n−1 −τ ∆ of order s + 4 at the boundary if and only if it is in the image of e acting on Hs(X). Other applications include the distribution of complex zeros of Laplace eigenfunctions [22], Weyl-type estimates on the growth of eigenfunctions in the complex domain [23], and microlocal analysis of analytic singularities [18]. The announcement in [3] gave a sketch of the proof of this result, but a proof could not be found in the literature until the work of S. Zelditch [23]. Since the goal of this paper is to give another proof we first review Zelditch's proof. Let UF be the Hadamard-Feynman fundamental solution to the wave equation, 2 (@t + ∆)u = 0. Hadamard [12] showed that if (X; g) is real analytic then the 2 2 Schwartz kernel of UF has the following form. Let Γ(t; x; y) = t − d (x; y). If n = dim(X) is even, then 1 − n−1 X j UF (t; x; y) = Γ 2 Uj(t; x; y)Γ (1) j=0 and if n is odd, then n 1 1 − n−1 X j X j X j UF (t; x; y) = Γ 2 Uj(t; x; y)Γ +log(Γ) Vj(t; x; y)Γ + Wj(t; x; y)Γ : j=0 j=0 j=1 (2) If we choose the complexification M small enough, then all of the infinite series above converge to functions which are holomorphic in a neighborhood of the complex zero set of (the analytic continuation of) Γ in C × M × M. Zelditch writes the expressions (1), (2) as a Fourier integrals on X with complex valued phase function (t + iτ; x; y; θ) = θ((t + iτ)2 − d2(x; y)), τ > 0, and shows that the amplitude is a formal analytic symbol of degree (n − 3)=2. He observes that the Poisson-wave operator for t > 0 can be obtained by differentiating UF : p d eit ∆ = U (t) if t > 0: idt F p This gives an expresion for the Schwartz kernel of ei(t+iτ) ∆ of the form (1), (2) (with the exponent −(n − 1)=2 replaced by −(n + 1)=2) and also as Fourier integrals on X with the same phase and formal analytic symbol of degree (n − 2 1)=2 (for t > 0 and τ > 0). The parametrix can be modified (by factoring the phase function t2 − d2) so that it is valid when t ≥ 0, τ > 0 and d 6= 0. Zelditch then considers the analytic continuation of the Schwartz kernel p ei(·) ∆(t + iτ; x; y), to complex values of x (with t ≥ 0, τ > 0). Since the formal series in (1), (2) converge to a holomorphic function in a neighborhood of the complex zero set of Γ in ×M ×M, the amplitude of the Fourier integral p C expression for ei(·) ∆(t + iτ; x; y) is also a formal analytic symbol for complex values of x. Setting t = 0 Zelditch concludes that the operator Sτ described above is a complex Fourier integral operator. In this article we re-prove this result using a slightly different approach and fill in some details which have not appeared in the literature to date. Instead of starting with the Hadamard parametrix we start with the Minakshisundaram- Pleijel parametrix for the heat equation and use the subordination formula to relate the Poisson kernel to the heat kernel. This method, already suggested by Zelditch in [23], has much in common with the wave equation approach and leads to the same results. A possible advantage is it shows how the well- known Minakshisundaram-Pleijel parametrix is related to the parametrix for 1 the Poisson kernel. We also prove the bijectivity of Sτ (following the outline in [4]) which does not seem to have appeared in the literature. p Let P 2 C!((0; 1) × X × X) be the Schwartz kernel of exp(−t ∆). For s 2 s s 2 R, let O (@M) be the closure in the L -Sobolev space H (@M) of the set of restrictions to @M of holomorphic functions on M which are smooth on the s closure of M (or, equivalently, the closure of the kernel of @b in H (@M)). Our goal is to give a \heat equation" proof of the following theorem. Theorem 1 (Boutet de Monvel [3; 4; 5, Appendix 6.4; 11, Theorem 5.1]). There exists an 0 > 0 such that for each fixed 2 (0; 0): 1. For each fixed y 2 X, the map x ! P (, x; y) can be analytically continued to M. 2. The restriction of P (, ·; ·) to @M × X is a Fourier integral distribution with complex phase, of degree −(n − 1)=4. 3. Let S be the operator whose Schwartz kernel is the restriction of P (, ·; ·) s to @M × X. For all s 2 R, S is a continuous bijection from H (X) onto s+(n−1)=4 O (@M). 1 It's easy to see that the kernel and cokernel of Sτ are finite dimensional by elliptic theory (we thank S. Zelditch for this observation). 3 To establish the analytic continuation of P (, ·; y) to M, we use the subor- dination formula to relate the Poisson kernel to the heat kernel: if E(t; x; y) is the heat kernel, then for (, x; y) 2 (0; 1) × X × X 1 Z 2 dt P (, x; y) = p E(t; x; y)e− =4tt−1=2 (3) 4π 0 t (see [17, Theorem 2.1]). For z complex and close to y the properties of the analytic continuation of the heat kernel proved in [18] can be used to construct the analytic continuation of P . If z is not close to y then the relationship between P and the wave kernel together with the finite propagation speed of analytic singularities of the wave kernel gives the analytic continuation of P (Corollary 2). The restriction of P (, ·; y) to @M will become singular and we must show that the restriction is a Fourier integral distribution of complex type. To do so we use the results of [18] (see Theorem 3) to re-write (3) as Z 1 − n+1 −(d2(z;y)+2)θ=4 P (, z; y) ≈ (4π) 2 e a(θ; z; y) dθ (4) 0 where d2(·; y) is the analytic continuation of the distance function in the first variable and a(θ; z; y) is an analytic symbol of order (n+1)=2. Here z, y are in a complex neighborhood X ⊂ M × X of a neighborhood of the diagonal in X × X (see Remark 1, and Remark 4 for a oscillatory integral description), and ≈ means modulo a function which is smooth in (, z; y) 2 fjj < 0g×X and holomorphic in and z. In the proof of Theorem 1, part 2, we show that φ(θ; z; y) = i(d2(z; y) + 2)θ=4 is a regular complex phase function of positive type and that the operator S associated with P by the Schwartz kernel theorem is in fact a Fourier integral operator of complex type. In Remark 2 we verify that the ∗ ∗ isotropic submanifold of (T @Mn0) × (T Xn0) generated by φ is the (twisted) graph of an isomorphism of symplectic cones as in [5, Appendix A.6.4].