THE POISSON TRANSFORM ON A COMPACT REAL ANALYTIC RIEMANNIAN MANIFOLD

Transcription

1 THE POISSON TRANSFORM ON A COMPACT REAL ANALYTIC RIEMANNIAN MANIFOLD MATTHEW B. STENZEL Abstract. We study the transform mapping an L 2 function f on a compact, real analytic Riemannian manifold X to the analytic continuation of exp( t )f to the interior of a Grauert tube tube M t about X. We show that after precomposing with an elliptic pseudodifferential operator this becomes a unitary map from L 2 (X) onto the holomorphic L 2 functions on M t. If a compact Lie group of isometries acts transitively on X then the inverse of this unitarized map can be constructed by the restriction principle. 1. Introduction The goal of this paper is to study a modified version of the heat kernel transform on a compact, real analytic Riemannian manifold, X. The heat kernel, or Segal- Bargmann, transform on a compact Lie group K maps an L 2 function f on K to the analytic continuation of exp( t )f to K C. In a series of papers [10 12] Brian Hall showed that the heat kernel transform is an isometry onto a weighted L 2 space H of holomorphic functions on K C, found an expression for the inverse transform and proved estimates on the reproducing kernel which give bounds on the phase space probability measure associated with a normalized element of H. Some limited progress has been made in extending the heat kernel transform beyond the compact group setting [13 15,27]. It is known that if X is a symmetric space of noncompact type then there is in general no weighted Hilbert space structure on the image of the heat kernel map which makes the map an isometry [18] (see also [19] for the Heisenberg group). The unitarity of the heat kernel transform seems to depend in Date: July 8, Mathematics Subject Classification. 46F12, 35J05, 43A85. Key words and phrases. Poisson transform, Segal-Bargmann transform, restriction principle. 1

2 2 M. STENZEL an essential way on the existence of an entire Grauert tube, K C, about K and in general such an entire tube doesn t exist. When K is a (non-compact) lie group of complex type it is possible to use a cancellation of singularities argument to let the tube radius go to infinity [13 15], but it is not known whether such cancellations are possible in other contexts. In this paper we will not assume that X has any homogeneity or group structure (although will always assume the manifold X is a compact, real analytic Riemannian manifold). We will show that by replacing the (nonnegative) Laplacian by in the definition of the heat kernel transform one obtains a transform which, modulo an elliptic pseudodifferential operator on X, is a unitary map from L 2 (X) onto the space of L 2 holomorphic functions on a Grauert tube of radius t about X. The motivation for replacing by is that exp( t ) is a Fourier integral operator of complex type and exp( t ) is not. As a consequence exp( t ) propagates analytic singularities in the complex domain with finite speed. This makes it possible to unitarily identify L 2 (X) with the familiar space of L 2 holomorphic functions on a Grauert tube of finite radius t, M t. On the other hand exp( t ) propagates analytic singularities with infinite speed and its image in the complex domain can be quite complicated. Indeed it is not even obvious what is the maximal domain to which all the functions e t f, f L 2 (X), can be analytically continued to. For symmetric spaces of non-compact type the maximal domain is the complex crown [1]. See [18] for a description of the image of the heat kernel transform in this setting. The second advantage is that as a Fourier integral operator in the complex domain exp( t ) has well understood mapping properties between Sobolev spaces. This makes it possible to produce a unitary map from L 2 (X) onto the space of holomorphic L 2 functions M t using exp( t ) and some basic facts about the calculus of Fourier integral operators. Our first result is an analog of Hall s isometry theorem for the Segal-Bargmann transform [10, Theorem 2]. We show that the Poisson transform, the map which sends f C (X) to the analytic continuation of e t f to the interior of the tube

3 POISSON TRANSFORM 3 M t, can be unitarized (with respect to the standard measures) by precomposing with an elliptic pseudodifferential operator on X (Theorem 1). Although this is perhaps well-known to the experts, to our knowledge it has not appeared in the literature (but see the Unitarization Lemma in [31] for a similar result in a different context). Our strategy to prove this is to factor this map by first restricting (the analytic continuation of) exp( t ) to the boundary of M t and then analytically continuing to the interior of M t. This allows us to use a theorem of Boutet de Monvel together with known properties of the operator mapping functions on M t to harmonic functions on M t. There is an integral formula for the inverse of the heat kernel transform on Lie groups and symmetric spaces [11, 27, 28]. Since the Poisson transform and its inverse are Fourier integral operators of complex type there is in principle an integral expression for its inverse. An explicit global expression for this inverse could be quite complicated. Another approach to the inverse of the heat kernel transform is the restriction principle [16, 24]. For compact Lie groups it is known that the unitary part in the polar decomposition of the restriction map, sending F H to F K L 2 (K), is the inverse of the heat kernel transform. Although the straightforward generalization of this seems unlikely to be true in all cases for the Poisson transform, we show that the relevant restriction map does have a polar decomposition, and if X has a transitive action by a compact Lie group of isometries then the unitary part of the restriction map is in fact the inverse of the Poisson transform. We do not require that the metric is the normal metric determined by an Ad-invariant inner product on the Lie algebra so this includes examples with negative sectional curvature and without entire Grauert tubes. This paper is organized as follows. In Section 2 we construct the modified Poisson transform from L 2 (X) onto HL 2 (M t ) and verify that it is unitary. We show that every F HL 2 (M t ) can be expanded in a holomorphic Fourier series (a result already observed in [20]), and identify the reproducing kernel of HL 2 (M t ). In Section 3 we give a number of equivalent conditions for the restriction principle

4 4 M. STENZEL to hold for the modified Poisson transform (Proposition 3). We show that these conditions hold if X is a compact homogeneous space and work out explicitly the torus case. 2. The Poisson Transform Let M be a complex n-dimensional manifold containing the compact, real analytic Riemannian manifold (X, g) as a totally real submanifold. Then there is a neighborhood U of X in M and a real analytic function φ on U such that φ = dφ = 0 on X and i φ is a Kähler metric which pulls back to g on X. This function is unique if u = φ satisfies the homogeneous Monge-Ampère equation ( u) n = 0 away from X and φ is invariant under the complex conjugation fixing X (see [9,21]). Let M ɛ be the corresponding Grauert tube, M ɛ = {φ < ɛ 2 } about X where ɛ is small enough that the closure of M ɛ is contained in U. The Kähler measure, dω n, is a finite measure on M ɛ. Let us now review the construction of [4, Section 1d]. Let H Λ ( M ɛ ) be the completion of C ( M ɛ ) with respect to the norm (2.1) f 2 H Λ ( M ɛ) = M ɛ (Λf)f dν where Λ is a positive (in particular self-adjoint) elliptic pseudodifferential operator and dν is the measure on M ɛ induced by dω n. If the degree of Λ is 2r then H Λ ( M ɛ ) is the Sobolev space of functions with r derivatives in L 2 ( M ɛ ) equipped with the Hilbert space inner product (2.1). Let O Λ ( M ɛ ) = { u H Λ ( M ɛ ): b u = 0 }. This is a closed subspace of H Λ ( M ɛ ). Let (2.2) S ɛ : H (deg Λ)/2 (n 1)/4 (X) O Λ ( M ɛ ) be the operator defined by analytically continuing e ɛ f to M ɛ and restricting to M ɛ. Then there is an ɛ 0 > 0 such that for all ɛ (0, ɛ 0 ], S ɛ is a Fourier integral

5 POISSON TRANSFORM 5 operator with complex phase of degree n 1 4 and is a continuous linear bijection from H (deg Λ)/2 (n 1)/4 (X) onto O Λ ( M ɛ ) [3, 20, 29, 32]. Let be the Laplace operator associated with the Kähler metric and let K ɛ be the operator which solves the homogeneous Poisson equation for, (K ɛ f) = 0 on M ɛ (K ɛ f) Mɛ = f. K ɛ K ɛ is an elliptic positive pseudo-differential operator of degree 1 on M ɛ ([4, Section 1d]); the adjoint is taken with respect to the L 2 norms on M ɛ and M ɛ determined by dν and dω n respectively). Let Λ = K ɛ K ɛ. Then (2.3) K ɛ : H Λ ( M ɛ ) L 2 (M ɛ ) ker is an isometry of Hilbert spaces (see [4, loc. cit.]). Let HL 2 (M ɛ ) denote the space of square integrable holomorphic functions on M ɛ. In the usual way (using the estimate (2.6) below) we see HL 2 (M ɛ ) is a closed subspace of L 2 (M ɛ ). Since holomorphic functions are harmonic for the Kähler Laplacian, HL 2 (M ɛ ) is contained in L 2 (M ɛ ) ker. As an immediate consequence we obtain Proposition 1. The restriction of K ɛ to O Λ ( M ɛ ) is an isometry of Hilbert spaces K ɛ : O Λ ( M ɛ ) HL 2 (M ɛ ) and the Poisson transform, (2.4) K ɛ S ɛ : H (n+1)/4 (X) HL 2 (M ɛ ), is a bounded isomorphism of vector spaces. Note that K ɛ S ɛ is the complexified wave operator at imaginary time, U C (iɛ) in the notation of [32]. We also note that the technique of [4, Section 1d]) establishes that for s > 1/2, K ɛ S ɛ is a bounded isomorphism from H s (n+1)/4 (X) onto

6 6 M. STENZEL HL s (M ɛ ), the completion of C (M) with respect to the norm F 2 s = M ɛ F 2 φ ɛ 2 2s ω n. Now we unitarize K ɛ S ɛ by composing with a pseudodifferential operator on X. We think of the unitarized map as a generalized Segal-Bargmann transform. Theorem 1. There is an elliptic, positive classical pseudodifferential operator Q ɛ on X of degree (n + 1)/4 such that K ɛ S ɛ Q ɛ : L 2 (X) HL 2 (M ɛ ), mapping f to the analytic continuation of e ɛ Q ɛ f to M ɛ, is an isometry of Hilbert spaces. Proof. Since Kɛ K ɛ is a classical elliptic pseudodifferential operator of degree 1, the same idea as in proof of [30, Lemma 3.1] shows that Sɛ Kɛ K ɛ S ɛ is a classical elliptic pseudodifferential operator of degree (n + 1)/2. (K ɛ S ɛ ) K ɛ S ɛ is a positive (in particular invertible) operator so we can define a positive operator (2.5) Q ɛ def = ((K ɛ S ɛ ) K ɛ S ɛ ) 1/2 by the Spectral Theorem. Then Q ɛ is a classical elliptic pseudodifferential operator of order (n + 1)/4 [25]. K ɛ S ɛ Q ɛ is an isometry because it is the unitary part of K ɛ S ɛ in its UP polar decomposition (the positive part is Q 1 ɛ ). Remark 1. Let X, M ɛ be as above and let φ k be an orthonormal basis of eigenfunctions of the Laplacian on X. It is well known that there is an ɛ 0 > 0 such that for all k and all ɛ (0, ɛ 0 ], φ k can be analytically continued to a holomorphic function φ k on M ɛ (in fact this follows from Proposition 1). Although the φ k will not in general be an orthonormal basis for HL 2 (M) ɛ, we will show that every F HL 2 (M ɛ ) can be expanded in a holomorphic Fourier series (c.f. [10, Lemma 9]). This result has also been obtained in [20]. For F HL 2 (M ɛ ) let F k denote the Fourier coefficients of the restriction of F to X, F k =< F X, φ k > L 2 (X).

7 POISSON TRANSFORM 7 Proposition 2. The series k=0 F k φ k converges to F in the HL 2 (M ɛ ) norm, pointwise uniformly on compact subsets. Proof. By Equation (2.4) there is a (unique) f H (n+1)/4 (X) such that F = K ɛ S ɛ f. The restriction of F to X is e ɛ f. The Fourier coefficients of F X are F k = e ɛ λ k f k where f k is defined by the distributional pairing f(φ k ) = f k. We recall that the Fourier series k=0 f kφ k converges to f in the H (n+1)/4 (X) norm. Then, since K ɛ S ɛ is bounded from H n+1 4 (X) into HL 2 (M ɛ ), F n N F k φk HL2 (M ɛ) = K ɛ S ɛ (f f k φ k ) HL2 (M ɛ) k=0 C f k=0 N f k φ k H (n+1)/4 (X) 0 as N. The pointwise uniform convergence on compact subsets follows from the well-known estimate: for every compact set K M ɛ there are constants C K, k=0 C K such that for all holomorphic functions F on M ɛ (2.6) sup F (z) C K F L 1 (M ɛ) C K F L 2 (M ɛ) z K (for the first inequality see [17, Theorem 2.2.3], the second is Schwarz s inequality and the finite volume of M ɛ ). Remark 2. We describe how the Schwartz kernel of K ɛ S ɛ Q ɛ is related to the reproducing kernel of HL 2 (M ɛ ). A reproducing kernel for a Hilbert space H of functions on a set Y is a function Rep on Y Y such that for each fixed y Y, Rep(, y) is in H, and for each function f H, f(y) =< f, Rep(, y) > H where <, > H is the inner product on H. If {f k } k=1 is an orthonormal basis for H, then Rep(x, y) = f k (x)f k (y). It is well-known that HL 2 (M ɛ ) is a reproducing kernel Hilbert space (the proof is no different from the C n case using the estimate

8 8 M. STENZEL (2.6) 1 ). Let C ɛ denote the operator K ɛ S ɛ Q ɛ and its Schwartz kernel, K ɛ S ɛ Q ɛ f(w) = C ɛ f(w) = C ɛ (x, w)f(x) dx. Then the reproducing kernel of HL 2 (M ɛ ) is (the complex conjugate of) the integral X kernel of C ɛ C ɛ thought of as the identity operator on HL 2 (M ɛ ), i.e., Rep ɛ (z, w) = C ɛ (x, z)c ɛ (x, w) dx. X The proof (following [16, Remark 3.8]) amounts to translating the fact that C ɛ C ɛ is the identity operator on HL 2 (M ɛ ) into an identity of integral kernels. For F HL 2 (M ɛ ), F (w) = C ɛ (C 1 ɛ F )(w) =< C 1 F, C ɛ,w > L 2 (X) ɛ where C ɛ,w (x) = C ɛ (x, w). Since C ɛ is an isometry, F (w) =< F, C ɛ (C ɛ,w ) > HL2 (M ɛ) =< F, C ɛ (x, )C ɛ (x, w) dx > HL2 (M ɛ) X =< F, Rep(, w) > HL 2 (M ɛ) (note the integral kernel C ɛ C ɛ (z, w) is X C ɛ(x, z)c ɛ (x, w) dx). 3. The restriction principle and the inverse of K ɛ S ɛ Q ɛ A bounded operator L: H 1 H 2 between Hilbert spaces L has a (left) polar decomposition L = P U where P is a positive operator 2 on H 2 and U : H 1 H 2 is a partial isometry. P and U are uniquely determined if we impose the condition ker P = ker U, and this condition can always be satisfied. If L is injective and the range of L is dense then U is a unitary map and P = (LL ) 1/2. Let M C be a complexification of a compact, real analytic Riemannian manifold X, H(M C ) a Hilbert space of holomorphic functions on M C and R ɛ : H(M C ) L 2 (X) the 1 Note that O Id ( M ɛ) is not a reproducing kernel Hilbert space because for fixed z 0 Ω, the Szëgo kernel S(, z 0 ) is not in L 2 ( Ω). Reproducing kernel Hilbert spaces are useful for many applications including phase space bounds, coherent states, and Berezin quantization. 2 A bounded positive operator is automatically self-adjoint. We will require unbounded positive operators to be self-adjoint as part of the definition of positivity.

9 POISSON TRANSFORM 9 restriction operator (mapping a holomorphic L 2 function on M ɛ to its restriction to X). If R ɛ is injective with dense range then the unitary part of R ɛ is a unitary map from L 2 (X) to H(M C ). When X is a compact Lie group K and H(M C ) is the image of the heat kernel transform (which consists of holomorphic functions on K C which are square integrable with respect to a certain Gaussian weighted measure) then the unitary part of R ɛ is the inverse of the Segal-Bargmann transform [16,24]. For other applications of this restriction principle see [5 7, 22, 26]. It is natural to ask how the unitary part of R ɛ is related to the unitarized Poisson transform, K ɛ S ɛ Q ɛ. We first verify that R ɛ satisfies the hypothesis of the restriction principle. Lemma 1. R ɛ : HL 2 (M ɛ ) L 2 (X) is bounded, injective and has dense range. Proof. To show R ɛ is bounded, let F HL 2 (M ɛ ). Then since R ɛ F is smooth and X is compact, R ɛ F L2 (X) C sup F (x) C F HL2 (M ɛ) x X (the second inequality is the estimate (2.6)). R ɛ is injective because holomorphic functions are determined by their restriction to a totally real submanifold. The range of R ɛ is dense in L 2 (X) because it contains the algebraic span of the eigenfunctions of the Laplacian. 3 We note that from Equation (2.4) the range of R ɛ acting on HL 2 (M ɛ ) is equal to the range of e ɛ acting on H (n+1)/4 (X), or, the range of e ɛ Q ɛ acting on L 2 (X). Let e ɛ Q ɛ = P U be the polar decomposition. Note R ɛ K ɛ S ɛ Q ɛ = e ɛ Q ɛ so the polar decomposition of R ɛ is R ɛ = P U(K ɛ S ɛ Q ɛ ) 1. This shows that e Q ɛ is a positive operator if and only if K ɛ S ɛ Q ɛ is the inverse of the unitary part of R ɛ. Furthermore e ɛ Q ɛ is positive if and only if e ɛ commutes with Q ɛ (recall e ɛ and Q ɛ are positive; for the if part see for example [8, Theorem 10.7]; the only if part is obvious). 3 We recall there is an ɛ0 > 0 such that for all ɛ (0, ɛ 0 ), all the eigenfunctions of the Laplacian can be analytically continued to M ɛ.

10 10 M. STENZEL Proposition 3. The following are equivalent: (1) The inverse of K ɛ S ɛ Q ɛ is the unitary part of R ɛ. (2) e ɛ Q ɛ is a positive operator. (3) e ɛ commutes with Q ɛ. (4) Let H λ denote the linear span of the eigenfunctions of with eigenvalue λ and let H λ denote the linear span of their analytic continuations to M ɛ, i.e., Hλ ɛ = {F HL 2 (M ɛ ): F X H λ }. If λ λ, then H λ is orthogonal to H λ in HL 2 (M ɛ ). Proof. The equivalence of (1), (2), and (3) have already been established. To show that (3) implies (4) we need the following well-known fact (c.f. [8, Theorem 9.7] in the case of compact operators). Lemma 2. Let A, B, be self-adjoint, elliptic pseuodifferential operators of positive order acting as unbounded operators on L 2 (X). Then there is an orthonormal basis for L 2 (X) of joint eigenfunctions for A and B. If A and B are real in the sense that Af = Af then the eigenfunctions can be taken to be real valued. Proof. L 2 (X) decomposes into an orthogonal direct sum of a countable number of finite dimensional eigenspaces, L 2 (X) = λ spec(a) H λ. Since B commutes with A, the restriction of B to H λ preserves H λ. Then the matrix of B H λ (in an orthonormal basis of H λ of eigenfunctions for A) is a finite dimensional Hermitian symmetric matrix. There is an orthonormal basis for H λ consisting of eigenfunctions for B H λ (which of course are still eigenfunctions for A with eigenvalue λ). If A and B satisfy the reality conditions then there is an orthogonal basis for A consisting of real valued eigenfunctions and the matrix of B H λ is a real symmetric matrix. Then B H λ can be diagonalized by a real orthogonal matrix, i.e., by a real linear combination of the real valued eigenfunctions of A. Applying Lemma 2 to A =, B = Q ɛ we see that (3) implies that there is a countable orthonormal basis {ψ k,ɛ } k=0 of L2 (X) consisting of real valued joint eigenfunctions for e ɛ and Q ɛ with eigenvalues e ɛ λ k, resp., η k,ɛ. To finish the

11 POISSON TRANSFORM 11 proof of (3) implies (4) we recall that K ɛ S ɛ Q ɛ is an isometry, that K ɛ S ɛ Q ɛ ψ k,ɛ = e ɛ λ k η k,ɛ ψk,ɛ where the tilde denotes analytic continuation to M ɛ, and that Q ɛ has no zero eigenvalues. Then if ψ k,ɛ and ψ l,ɛ are orthogonal in L 2 (X) we have e ɛ( λ k + λ l ) η k,ɛ η l,ɛ < ψ k,ɛ, ψ l,ɛ > HL2 (M ɛ)= 0 so that ψ k,ɛ and ψ l,ɛ are orthogonal in HL 2 (M ɛ ). Proof of (4) implies (3): We claim the hypothesis implies that (K ɛ S ɛ ) K ɛ S ɛ maps H λ to itself. If λ λ and f H λ, g H λ, then, since K ɛ S ɛ f is the analytic continuation of e ɛ f to M ɛ, < (K ɛ S ɛ ) K ɛ S ɛ f, g > L2 (X) =< K ɛ S ɛ f, K ɛ S ɛ g > HL2 (M ɛ) = e ɛ( λ+ λ ) < f, g > HL2 (M ɛ) = 0. This shows that the image of H λ under (K ɛ S ɛ ) K ɛ S ɛ is orthogonal to all the other H λ with λ λ, and so (K ɛ S ɛ ) K ɛ S ɛ maps each H λ into itself. Since Q ɛ = ((K ɛ S ɛ ) K ɛ S ɛ ) 1/2 it suffices to show that e ɛ commutes with (K ɛ S ɛ ) K ɛ S ɛ. But this is clear because (K ɛ S ɛ ) K ɛ S ɛ preserves each H λ and e ɛ acts as a multiple of the identity on H λ. We don t have an example where the analytically continued eigenspaces are not orthogonal to each other but it seems unlikely that they are in general. We can show that the orthogonality condition is satisfied by compact homogeneous spaces. Corollary 2. If a compact Lie group acts transitively on X by isometries then the inverse of K ɛ S ɛ Q ɛ is the unitary part of R ɛ. Proof. We will verify condition (4) in Proposition 3. The idea of the proof is the same is in [10, Lemma 10]. We can identify (X, g) with the homogeneous space K/H equipped with a left K-invariant metric. There exists a metric on K such that the coset projection π : K K/H is a Riemannian fibration with totally

12 12 M. STENZEL geodesic fibers [2, Section 2.2], so for all smooth functions φ on K/H we have K (φ π) = ( K/H φ) π. Let H λ K/H, resp. Hλ K, denote the eigenspaces in L2 (K/H), resp. L 2 (K), corresponding to the eigenvalue λ. Then the above shows that if f H λ K/H, then f π Hλ K. Now suppose φ H λ K/H and ψ Hλ K/H for λ λ. The assumptions imply that for all ɛ > 0 the action of K on X lifts to a holomorphic action on M ɛ which preserves the Kähler metric. The Riemannian measure on K/H is a positive multiple of Haar measure, dk. Then using the K-invariance of the inner product on HL 2 (M ɛ ), integrating over K and changing the order of integration (which is valid because we have chosen ɛ small enough that all eigenfunctions are analytic on a slightly larger Grauert tube) < φ, ψ > HL2 (M ɛ)= c φ(k z), ψ(k z) dk dz. M ɛ K The inner integral over K is a real analytic function of z M ɛ. We will show the inner integral is zero for z K/H, which implies that it must be zero for all z M ɛ. If z is the coset lk K/H then by the bi-invariance of Haar measure φ(k z), ψ(k z) dk = (φ π)(kl)(ψ π)(kl) dk K = = 0 K K (φ π)(k)(ψ π)(k) dk because φ π H λ K and ψ π Hλ K which are orthogonal if λ λ. We note that by changing the inner product on HL 2 (M ɛ ) we can simplify the polar decomposition of R ɛ. For example let B ɛ = K ɛ S ɛ (1 + ) (n+1)/8. By (2.4) this is a vector space isomorphism from L 2 (X) onto HL 2 (M ɛ ). Define a new inner product on HL 2 (M ɛ ) by (F, G) =< Bɛ 1 F, Bɛ 1 G > L 2 (X).

13 Since B ɛ and B 1 ɛ POISSON TRANSFORM 13 are bounded (, ) and <, > HL2 (X) generate equivalent norms and of course B ɛ is unitary with respect to (, ). Let R ɛ be the restriction operator as above. Note R ɛ B ɛ = e ɛ (1 + ) (n+1)/8. Since e ɛ commutes with (1 + ) (n+1)/8, e ɛ (1 + ) (n+1)/8 is a positive operator. Then the polar decomposition is R ɛ = e ɛ (1 + ) (n+1)/8 B 1 ɛ so that B 1 ɛ with respect to the inner products (, ) and <, > L2 (X). is the unitary part of R ɛ Finally we show how this works in the case of the torus T n = R n /(2πZ) n with its Grauert tube M ɛ = {x+iy C n : y < ɛ}/(2πz) n and the standard inner products. An orthonormal basis of eigenfunctions of for L 2 (X) is e k (x) = (2π) n/2 e ik x, k Z n. The analytic continuation of the Poisson operator is K ɛ S ɛ e k = e ɛ k ẽ k where ẽ k is the analytic continuation of e k to M ɛ. The inner products of the ẽ k are (3.1) < ẽ k, ẽ l > HL 2 (M ɛ)= δ k,l y <ɛ An orthonormal basis for HL 2 (M ɛ ) is f k e 2k y dy def = δ k,l J ɛ (k). def = J 1/2 ɛ (k)ẽ k. Note (K ɛ S ɛ ) ẽ k = e ɛ k J ɛ (k)e k so Q ɛ e k = ((K ɛ S ɛ ) K ɛ S ɛ ) 1/2 e k = e ɛ k J 1/2 ɛ (k)e k. Then K ɛ S ɛ Q ɛ e k = f k. On the other hand the restriction map is R ɛ f k = K ɛ S ɛ Q ɛ e k X = J 1/2 (k)e k = J 1/2 (k)(k ɛ S ɛ Q ɛ ) 1 f k. Since K ɛ S ɛ Q ɛ is unitary and the map on L 2 (X) determined by e k J 1/2 (k)e k is positive, 4 the unitary part of the restriction map is (K ɛ S ɛ Q ɛ ) 1. References [1] D. N. Akhiezer and S. G. Gindikin, On Stein extensions of real symmetric spaces, Math. Ann. 286 (1990), no. 1-3, 1 12, DOI /BF MR (91a:32047) [2] L. Bérard-Bergery and J.-P. Bourguignon, Laplacians and Riemannian submersions with totally geodesic fibres, Illinois J. Math. 26 (1982), no. 2, MR (84m:58153) 4 The map is bounded because Jɛ(k) 1/2 c n,ɛe ɛ k ( k (n+1)/4 + lower order terms) (using stationary phase for complex valued phase functions; see [23, Theorem 2.3]).

14 14 M. STENZEL [3] L. Boutet de Monvel, Convergence dans le domaine complexe des séries de fonctions propres, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 13, A855 A856 (French, with English summary). MR (80i:58040) [4], On the index of Toeplitz operators of several complex variables, Invent. Math. 50 (1978/79), no. 3, MR (80j:58063) [5] M. Davidson and G. Ólafsson, The generalized Segal-Bargmann transform and special functions, Acta Appl. Math. 81 (2004), no. 1-3, 29 50, DOI /B:ACAP d7. MR (2005k:33004) [6] M. Davidson, G. Ólafsson, and G. Zhang, Laplace and Segal-Bargmann transforms on Hermitian symmetric spaces and orthogonal polynomials, J. Funct. Anal. 204 (2003), no. 1, , DOI /S (03) MR (2004j:43012) [7], Laguerre polynomials, restriction principle, and holomorphic representations of SL(2, R), Acta Appl. Math. 71 (2002), no. 3, , DOI /A: MR (2003f:22015) [8] D. H. Griffel, Applied functional analysis, Dover Publications Inc., Mineola, NY, Revised reprint of the 1985 edition. MR (2003c:00002) [9] V. Guillemin and M. Stenzel, Grauert tubes and the homogeneous Monge-Ampère equation, J. Differential Geom. 34 (1991), no. 2, MR (93e:32018) [10] B. C. Hall, The Segal-Bargmann coherent state transform for compact Lie groups, J. Funct. Anal. 122 (1994), no. 1, MR (95e:22020) [11], The inverse Segal-Bargmann transform for compact Lie groups, J. Funct. Anal. 143 (1997), no. 1, , DOI /jfan MR (98e:22004) [12], Phase space bounds for quantum mechanics on a compact Lie group, Comm. Math. Phys. 184 (1997), no. 1, , DOI /s MR (98g:81108) [13] B. C. Hall and J. J. Mitchell, The Segal-Bargmann transform for noncompact symmetric spaces of the complex type, J. Funct. Anal. 227 (2005), no. 2, MR (2006g:58051) [14], Isometry theorem for the Segal-Bargmann transform on a noncompact symmetric space of the complex type, J. Funct. Anal. 254 (2008), no. 6, MR [15], The Segal-Bargmann transform for compact quotients of symmetric spaces of the complex type, Taiwanese J. Math. 16 (2012), no. 1, MR [16] J. Hilgert and G. Zhang, Segal-Bargmann and Weyl transforms on compact Lie groups, Monatsh. Math. 158 (2009), no. 3, , DOI /s MR (2011a:22011)

15 POISSON TRANSFORM 15 [17] L. Hörmander, An introduction to complex analysis in several variables, 3rd ed., North- Holland Mathematical Library, vol. 7, North-Holland Publishing Co., Amsterdam, MR (91a:32001) [18] B. Krötz, G. Ólafsson, and R. J. Stanton, The image of the heat kernel transform on Riemannian symmetric spaces of the noncompact type, Int. Math. Res. Not. 22 (2005), MR (2006e:58036) [19] B. Krötz, S. Thangavelu, and Y. Xu, Heat kernel transform for nilmanifolds associated to the Heisenberg group, Rev. Mat. Iberoam. 24 (2008), no. 1, MR (2009e:35032) [20] G. Lebeau, FBI transform and the complex Poisson kernel on a compact analytic Riemannian manifold, Minicourse at the Workshop on Microlocal Analysis, Northwestern University (2013). [21] L. Lempert and R. Szöke, Global solutions of the homogeneous complex Monge-Ampère equation and complex structures on the tangent bundle of Riemannian manifolds, Math. Ann. 290 (1991), no. 4, MR (92m:32022) [22] S.-L. Luo, On the Bargmann transform and the Wigner transform, Bull. London Math. Soc. 30 (1998), no. 4, , DOI /S MR (99e:81121) [23] A. Melin and J. Sjöstrand, Fourier integral operators with complex-valued phase functions, Fourier integral operators and partial differential equations (Colloq. Internat., Univ. Nice, Nice, 1974), Springer, Berlin, 1975, pp Lecture Notes in Math., Vol MR (55 #4290) [24] G. Ólafsson and B. Ørsted, Generalizations of the Bargmann transform, Lie theory and its applications in physics (Clausthal, 1995), World Sci. Publ., River Edge, NJ, 1996, pp MR (99e:22032) [25] R. T. Seeley, Complex powers of an elliptic operator, Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), Amer. Math. Soc., Providence, R.I., 1967, pp MR (38 #6220) [26] S. B. Sontz, The C-version Segal-Bargmann transform for finite Coxeter groups defined by the restriction principle, Adv. Math. Phys. (2011), Art. ID , 23. MR (2012j:47022) [27] M. B. Stenzel, The Segal-Bargmann transform on a symmetric space of compact type, J. Funct. Anal. 165 (1999), no. 1, MR (2000j:58053) [28], An inversion formula for the Segal-Bargmann transform on a symmetric space of non-compact type, J. Funct. Anal. 240 (2006), no. 2, , DOI /j.jfa MR (2007i:58034) [29], On the analytic continuation of the Poisson kernel, Manuscripta Math. 144 (2014), no. 1-2, , DOI /s MR

16 16 M. STENZEL [30] S. Zelditch, Complex zeros of real ergodic eigenfunctions, Invent. Math. 167 (2007), no. 2, MR (2008b:58034) [31] S. Zelditch, Index and dynamics of quantized contact transformations, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 1, (English, with English and French summaries). MR (99a:58082) [32] S. Zelditch, Pluri-potential theory on Grauert tubes of real analytic Riemannian manifolds, I, Spectral geometry, Proc. Sympos. Pure Math., vol. 84, Amer. Math. Soc., Providence, RI, 2012, pp , DOI /pspum/084/1363. MR Ohio State University, Newark Campus address: