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 √ 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 ∈ M, and y ∈ 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 √ operator, exp(−τ ∆), is the solution operator at time t to the pseudodifferential initial value problem √ (∂τ + ∆)u = 0
u|τ=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 √ 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 = {φ < τ}, 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 √ 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
∞ − 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 ∞ ∞ − 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 ψ(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 : √ d eit ∆ = U (t) if t > 0. idt F √ 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 √ 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 √ 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. √ Let P ∈ Cω((0, ∞) × X × X) be the Schwartz kernel of exp(−t ∆). For s 2 s s ∈ 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 ∈ (0, 0):
1. For each fixed y ∈ 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 ∈ 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) ∈ (0, ∞) × X × X
∞ Z 2 dt P (, x, y) = √ 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 ∞ − 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) ∈ {|| < 0}×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 ∂M\0) × (T X\0) generated by φ is the (twisted) graph of an isomorphism of symplectic cones as in [5, Appendix A.6.4]. To show that S is a continuous map between the appropriate Sobolev spaces we verify H¨ormander’scriteria for the L2 continuity of Fourier integral operators with complex phase [14, Theorem 3.5]. To do so requires considering the complex
Lagrangian submanifold Λφ˜. We verify it has the appropriate structure in the proof of Theorem 1, part 3. To verify that S is a bijection from Hs(X) to Os+(n−1)/4(∂M ) we show, √ following the outline in [4], that exp(− ∆) maps Hs(X) onto the space of restrictions to X of functions in Os+(n−1)/4(∂M ) by showing that the inverse √ operator, exp(+ ∆), is well-defined on such restrictions (Lemma 6). The main idea is to use (4) to write the Poisson kernel as the Laplace transform of the
4 symbol a in the Laplace transform variable s = (d2(z, y) + 2)/4 (here s plays the role of Γ in the wave equation approach). This allows us to use the results of [1] to obtain a detailed description of the branched analytic continuation of P to complex values of s in a deleted neighborhood of zero in C (Proposition 1). We fill in the details of the outline in [4], using Stoke’s theorem to move the √ integral which defines exp(−t ∆)(F |X ) into the complex domain and show that the resulting integral is well defined for complex values of t in a neighborhood of zero in C (slit along the nonpositive imaginary axis) containing −.
2 Grauert Tubes
Let M be a connected n-dimensional complex manifold containing X as a totally real embedded submanifold. Such an embedding is always possible, and any two are locally biholomorphically equivalent (see [7]). Let T X = {v ∈ TX : |v| < }. For > 0 sufficiently small, the Riemannian metric on X determines a real analytic diffeomorphism of T X with an open set M ⊂ M by analytic continuation of the exponential map:
v ∈ TxX ∩ T X → Expxiv ∈ M. (5)
This identification is the “adapted complex structure” on T X (or T ∗X; see ∗ [10, 11, 19]). We will refer to the complex manifolds M, T X, and T X as “Grauert tubes.” We will always choose small enough that is less than the maximal value for which this identification is possible, and that ∂M is the image under (5) of {v ∈ TX : |v| = }. The distance squared function, d2(x, y), is real analytic in a neighborhood of the diagonal ∆X ⊂ X × X and can be analytically continued to a holomorphic 2 function, still denoted by d , on a neighborhood of ∆X in M × M. Since X is compact, there is an 0 > 0 such that if v ∈ TyX and |v| < 0, then (Expyiv, y) is in this neighborhood.
2 2 Lemma 1. Let < 0. For all z = Expyiv ∈ ∂M we have d (z, y) = − .
Proof. This follows by analytic continuation in t from the identity
2 2 d (Expy(tv/|v|), y) = t (v 6= 0).
5 3 Analytic Continuation of the Wave, Poisson and Heat Kernels
Let 0 = λ0 < λ1 ≤ λ2 ≤ ... be the spectrum of ∆ (with multiplicity) and let 2 φk be a real valued orthonormal basis of L (X) satisfying ∆φk = λkφk. The
φk are real analytic (see [15, Theorem 8.6.1]). We denote the wave kernel by W , the Poisson kernel by P , and the heat kernel by E. The wave kernel is the √ distribution kernel of the operator exp(it ∆), √ √ X it λk ∞ exp(it ∆)f(x) = φk(x)fke , f ∈ C (X), (6) where fk are the Fourier coefficients, fk =< f, φk > and t ∈ R. In the sense of distributions, √ X it λk W (t, x, y) = φk(x)φk(y)e . (7) For any f ∈ C∞(X), (6) has an analytic continuation in t to the upper half plane {t + is: s > 0}, and so (7) does as well in the sense of distributions. It is well known that the analytic singular support of W is the set of (t, x, y) ∈ R×X ×X such that x and y can be joined by a geodesic of length |t| (see [8, Theorem 1] and the remarks following it). The Poisson kernel is the distribution kernel of √ the operator exp(−s ∆), √ √ X −s λk ∞ exp(−s ∆)f(x) = φk(x)fke , f ∈ C (X), s ≥ 0.
For s > 0, P is given by the convergent sum √ X −s λk P (s, x, y) = φk(x)φk(y)e , and lims→0+ P (s, x, y) = δx(y). In fact the sum converges for Re(s) > 0, and (in the sense of distributions)
lim P (u + iv, x, y) = W (v, x, y). u→0+ √ √ The heat kernel, E, has the same description as P if we replace ∆, resp. λk, by ∆, resp. λk.
3.1 The Analytic Continuation of the Poisson Kernel
Let W˜ denote the formal series √ def ˜ X iζ λk W (ζ, z, w) = φk(z)φk(w)e , (ζ, z, w) ∈ C × M × M. (8) k The following is contained in the work of Boutet de Monvel [3; 4; 5, Appendix 6.4] and gives an analytic continuation of the Poisson kernel.
6 Lemma 2.
1. For all s0 > 0 there is a neighborhood U (depending on s0) of X in M such that the formal series W˜ converges absolutely and uniformly on compacta
to a holomorphic function on {Im ζ > s0} × U × U.
2. There is a neighborhood V of i(0, ∞) × X × X in C × M × M such that W˜ converges absolutely and uniformly on compacta to a holomorphic function on V.
3. The restriction of W˜ to i(0, ∞) × X × X is P .
We note there is no neighborhood U of X in M such that W˜ converges on i(0, ∞) × U × U.
Proof. P is real analytic on (0, ∞) × X × X because it satisfies the elliptic equation with real analytic coefficients,
2 (−2∂t + ∆x + ∆y)P = 0.
Thus for each fixed s0 > 0 there is a neighborhood Us0 of X in M such that
P (s0, ·, ·) can be analytically continued to a holomorphic function on Us0 × Us0 . The representation
√ Z −s0 λk e φk(x) = P (s0, x, y)φk(y) dy (9) X
shows that all φk can be analytically continued to Us0 . Shrinking Us0 we may assume that P (s0, z, w) is bounded on Us0 × Us0 . We obtain for some Cs0 (independent of k) the rough estimate
√ s0 λk sup |φk(z)| ≤ e Cs0 . (10) z∈Us0
Then for all ζ = t + is with s ≥ 3s0, √ √ sup |φ (z)φ (w)eiζ λk | ≤ e−s0 λk C2 . k k s0 Us0 ×Us0 √ 2/n P iζ λk The estimate λk ∼ (k/C) shows that φk(z)φk(w)e converges ab- solutely and uniformly on compacta to a holomorphic function on {Im ζ >
3s0} × Us0 × Us0 . We may take V = ∪s0>0{Im ζ > 3s0} × Us0 × Us0 . Clearly the restriction of W˜ to i(0, ∞) × X × X is P .
7 3.2 The Analytic Continuation of the Wave Kernel
The following lemma shows that, for small t and (x, y) sufficiently far from the diagonal in X × X, W can be analytically continued in (t, x, y) variables to a holomorphic function equal to W˜ on an open set where both are defined. For any positive number α let
def (X × X)≥α = {(x, y) ∈ X × X : d(x, y) ≥ α}.
Lemma 3. Given any α > 0, there is a β > 0 and a neighborhood Y of (X ×
X)≥α in M × M such that W can be analytically continued from (−α/2, α/2) ×
(X × X)≥α to a holomorphic function, W˜ ≥α, on (−α/2, α/2) × i(−β, β) × Y.
W˜ ≥α is equal to W˜ on a neighborhood of i(0, β) × (X × X)≥α.
Proof. Since no points x, y, with d(x, y) ≥ α can be joined by a geodesic of length |t| < α, the distribution W given by (7) is a real analytic function on
(−α, α) × (X × X)≥α. So W can be analytically continued to a holomorphic function, which we denote by W˜ ≥α, on a neighborhood of the form (−α/2, α/2)× i(−β, β) × Y for some β > 0, where Y is a neighborhood of (X × X)≥α in
M × M. We will show that W˜ ≥α is equal to the series W˜ given by (8) on a neighborhood of i(0, β) × (X × X)≥α (on which W˜ converges by Lemma 2, item
2). Clearly W˜ −W˜ ≥α is holomorphic on a neighborhood of i(0, β)×(X ×X)≥α in ˜ ˜ C×M ×M. Note also that if (x, y) ∈ (X×X)≥α is fixed, then (W −W≥α)(·, x, y) is a holomorphic function on (−α/2, α/2) × i(0, β) (the series in equation (8) is absolutely convergent for ζ ∈ i(0, β), so it certainly converges for ζ ∈ R×i(0, β)). From Equations (7) and (8) we have
lim (W˜ − W˜ ≥α)(t + is, x, y) = 0 s→0+ for (t, x, y) ∈ (−α/2, α/2) × (X × X)≥α, in the sense of distributions. It follows, from the one-dimensional version of the “Edge of the Wedge Theorem” or the distributional version of Painlev´e’sTheorem, that (W˜ − W˜ ≥α)(ζ, x, y) = 0 for all (ζ, x, y) ∈ (−α/2, α/2) × i(0, β) × (X × X)≥α. Since i(0, β) × (X × X)≥α is a totally real submanifold of the domain of definition of W˜ − W˜ ≥α, it follows ˜ ˜ that W = W≥α on a neighborhood of i(0, β) × (X × X)≥α in C × M × M. Corollary 2. Given any α > 0, there is a β > 0 and a neighborhood Y in M×M of (X ×X)≥α such that P can be analytically continued from (0, β)×(X ×X)≥α to (−β, β) × i(−α/2, α/2) × Y.
Proof. W˜ is the analytic continuation of P , and W˜ ≥α provides the desired an- alytic continuation of W˜ .
8 3.3 Analytic Continuation of the Heat Kernel
The essential difference between the analytic continuation of the heat and Pois- son kernels is that there is an 0 such that for all s, the heat kernel at time s can be analytically continued to a Grauert tube M0 (the Poisson kernel can only be continued to a tube whose radius depends on s; see Theorem 1).
Lemma 4. There is a positive 0 such that the heat kernel can be analytically continued from (0, ∞) × X × X to {Re ζ > 0} × M0 × M0 as E(t, z, w) = P −λkt k e φk(z)φk(w), with uniform convergence on compact subsets of {Re ζ >
0} × M0 × M0 .
2/n Proof. This follows from the estimates (10) and λk ∼ (k/C) (the compact- ness of X allows us to find a M0 ⊂ Us0 where Us0 is as in the proof of Lemma 2).
4 Proof of Theorem 1
We will need some results from the proof of [18], Theorem 0.1, on the analytic continuation of the Minakshisundaram-Pleijel parametrix. Although Boutet de Monvel’s theorem was cited in [18], the following result is independent of it. Let
def (X × X)≤α = {(x, y) ∈ X × X : d(x, y) ≤ α}.
Theorem 3 ([18]). For all sufficiently small α > 0, we can find a neighborhood
X in M × X, containing (X × X)≤α with the property that:
1. For (x, y) ∈ (X × X)≤α, x is in the domain of a geodesic coordinate chart centered at y.
2. The distance function squared and the coefficients in the Minakshisundaram-
Pleijel parametrix for the heat equation, uk, can be analytically continued to a neighborhood of X in M × M.
3. There is an L > 0 such that for all k and all (z, y) ∈ X , the estimate k+1 |uk(z, y)| ≤ L k! holds ([18, Proposition 3.1]).
4. Let C > 1 satisfy2 L/(Ce) < 1/4. Then there is an η > 0 such that for all (t, z, y) ∈ (0, 1) × X ,
∞ −n/2 −d2(z,y)/4t X k −1 −η/t E(t, z, y) = (4πt) e t uk(z, y)χ(t − kC) + O(e ) k=0
2Our choice of C is larger than in [18, Definition 4.2], where L/(Ce) < 1/2.
9 where the O(·) is uniform as t → 0+ on X . Here χ ∈ C∞(R, [0, 1]), χ ≡ 0 on (−∞, 0], χ ≡ 1 on [1/2, ∞) so that χ(t−1 − kC) truncates the series after a finite, t-dependent number of terms.
−n/2 P∞ k −1 5. (4πt) k=0 t uk(z, y)χ(t − kC) is an analytic symbol of order n/2 in the parameter 1/t.
6. If (z, y) ∈ X and z = Expxiv, v ∈ TxX, then
2 2 Re d (Expxiv, y) ≥ −|v| . (11)
Proof. For items 1–5 see the proof of Theorem 0.1 in [18, Section 4], in which the result of [3, 4] is used only to show that items 4 and 5 hold for both z and y complex (see the Lemma following [18, Proposition 4.11]); we do not need this part of the result. For item 6, note from [18], Proposition 4.12 and
Remark 4.13, that for z = Expxiv, with v ∈ TxX sufficiently small, the map 2 X 3 y 7→ Re d (Expxiv, y) has a non-degenerate local minimum at y = x with minimum value −|v|2. After possibly shrinking X and α, we may assume that
(11) holds for all (Expxiv, y) ∈ X .
Proof of Theorem 1, part 1. Choose α > 0 small enough that we can find a neighborhood X as in Theorem 3. From Corollary 2 there is a β > 0 and an open subset Y in M × X containing (X × X)≥α such that for all y ∈ X, the map (, x) → P (, x, y) can be analytically continued from (0, β) × {x ∈ X : d(x, y) ≥ α} to (−β, β) × i(−α/2, α/2) × {z ∈ M :(z, y) ∈ Y}. Let us further assume that β < α/2, so that the disk of radius β, D(β), is contained in (−β, β) × i(−α/2, α/2) (this will be used in the proof of Lemma 6). We will show that there is an 1 ∈ (0, β) such that for all positive less than 1 and all y ∈ X, the map x → P (, x, y) can be analytically continued from
{x ∈ X : d(x, y) ≤ α} to {z ∈ M :(z, y) ∈ X }. Then, since X ∪ Y is an open subset of M × X containing the compact set X × X, we can find an 2 so that
M2 ×X is contained in X ∪Y. If 0 is the smaller of 1, 2 (and smaller than β), then for all y ∈ X and all ∈ (0, 0) the map x → P (, ·, y) can be analytically continued from X to M. This will complete the proof of Theorem 1, part 1.
To show the existence of 1 we use the subordination formula to relate the Poisson and heat kernels. The subordination formula says that for (, x, y) ∈ (0, ∞) × X × X,
∞ Z 2 dt P (, x, y) = √ E(t, x, y)e− /4tt−1/2 (12) 4π 0 t
10 (see [17, Theorem 2.1]). We first consider the integral over [1/2, ∞). On this interval the factor χ(t−1 − kC) is zero for all k > 2/C, so the sum in item 4 of Theorem 3 has no more than b2/Cc terms. Then the integrand is uniformly bounded for (, z, w) in compact subsets of C × M1 × M1 and so the integral ∞ Z 2 dt √ E(t, z, w)e− /4tt−1/2 (13) 4π 1/2 t
is a holomorphic function of (, z, w) ∈ C × M1 × M1 . Thus it suffices to consider the (formal) integral
1/2 Z 2 dt √ E(t, z, y)e− /4tt−1/2 (14) 4π 0 t √ Let us now shrink 1 so that 1 < 2 η. Setting θ = 1/t we can write (14) as, for (, z, y) ∈ (0, 1) × X ∩ (X × X)(z “real”),
1/2 Z 2 dt √ E(t, z, y)e− /4tt−1/2 4π 0 t Z ∞ ∞ − n+1 −(d2(z,y)+2)θ/4 X n−1 −k = (4π) 2 e θ 2 uk(z, y)χ(θ − kC) dθ + R(, z, y) 2 k=0 (15)
R ∞ −(η+2/4)θ −1/2 where R(, z, y) = 2 O(e )θ dθ extends to a smooth function on (, z, y) ∈ {|| < 1} × X , holomorphic in both z and . The integrand of
(15) extends to a holomorphic function of z for (, z, y) ∈ (0, 1) × X , and the sum in (15) is a symbol in θ of order (n − 1)/2. In particular it can be n−1 estimated by a constant times θ 2 , locally uniformly in (z, y). To show that the integral on the right hand side of (15) converges and is holomorphic in z for fixed y,(z, y) ∈ X ∩ (M × X), it suffices to show that for all ∈ (0, 1), 2 2 Re d (z, y)+ > 0 on X ∩(M ×X). Since M = {Expxiv : |v| < } (see Section 2), this follows from Theorem 3, item 6. This completes the proof of Theorem 1, part 1.
Remark 1. We can extend the interval of integration in (15) to (0, ∞) modulo a function which is smooth on (, z, y) ∈ {|| < 1} × X and holomorphic in both z and , because χ vanishes to infinite order at θ = 0. For (θ, z, y) ∈ (0, ∞) × X , let a(θ, z, y) be the analytic symbol of order (n − 1)/2, ∞ n−1 X −k a(θ, z, y) = θ 2 θ uk(z, y)χ(θ − kC). k=0
11 By (15) and Remark 1, the Poisson kernel restricted to the set of all (, z, y) such that (z, y) ∈ X and Re d2(z, y) + 2 > 0 is
− n+1 2 2 P (, z, y) ≈ (4π) 2 L[a(·, z, y)] (d (z, y) + )/4 (16) where L is the Laplace transform and ≈ means modulo a function which is smooth in (, z, y) ∈ {|| < 0} × X , and holomorphic in and z. We will use a result of of Beyer and Heller [1] (see also [13]) to give an explicit expression for the singular part of the Poisson kernel in terms of the Laplace transform variable s = (d2(z, y) + 2)/4 (here s corresponds to Γ in the Hadamard parametrix approach in [23]). Although we could not find a convenient reference to the behavior near the origin of the Laplace transform of an analytic symbol in the analytic microlocal analysis literature, we note the Laplace transform of a symbol has been used in the C∞ setting to obtain the asymptotic expansion of the Bergman kernel at the boundary of a strictly pseudoconvex domain [2; 6, Corollaire 1.7]. Presumably the singularity of the Poisson kernel could also be analyzed by showing that it satisfies a holonomic system of microdifferential equations as in [2, 16]. Instead we will give a classical proof using the result of [1] which we now recall. Let D(η) be the disk of radius η centered at the origin in C. Theorem 4 (W. A. Beyer and L. Heller [1, Theorem 1]). Let F ∈ C0((0, ∞))). Suppose there exists K, σ > 0 such that F is integrable on [0,K] and for all θ > K, " N # −β X −k F (θ) = θ ukθ + RN (θ) k=0 N+1 where RN (θ) = O(N!(σ/θ) ), uniformly in N and θ > K. Then L[F ](s) is analytic for |arg s| < π/2, s 6= 0, and there exists η > 0 (proportional to σ−1) such that L[F ] has a branched analytic continuation to D(η)\{0} of the form:
1. If β 6= Z, then L[F ](s) = sβ−1g(s) + h(s) where g and h are analytic at P∞ i s = 0 and g(s) = i=0 uiΓ(1 − i − β)s .
β−1 P−β i 2. If β ∈ Z, then L[F ](s) = s i=0 uiΓ(1 − i − β)s + (log s)g(s) + h(s) i P∞ (−1) i−1 where g and h are analytic at s = 0 and g(s) = i=1 (i−1)! ui−βs . P∞ k The idea of the proof is the following. Let u(v) = k=0 v uk/k! be the Borel P∞ −k transform of the formal sum k=0 θ uk. This sum converges for |v| < 1/σ. ˜ 1−β R τ −tv For τ = 1/(4σ) let F (t) = t 0 e u(v) dv. Then it is shown using classical analysis that L[F˜] satisfies the conclusion of the Theorem and differs from L[F ] by a function which is analytic at s = 0.
12 Using this result we show that P can be analytically continued in s to a deleted neighborhood of zero in C. This will be used in the proof of Theorem 1, part 3.
Proposition 1. There is an 0 such that for all fixed (z, y) ∈ X , the map s 7→ L[a(·, z, y)](s) has a (branched) analytic continuation from Re(s) > 0 to 2 D(0)\{0} ∪ Re(s) > 0 of the following form:
1. If n is even, then
− n+1 L[a(·, z, y)](s) = s 2 g(s, z, y) + h(s, z, y)
2 where g and h are smooth on D(0) × X and analytic in s and z.
2. If n is odd, then
n−1 2 − n+1 X k L[a(·, z, y)](s) = s 2 uk(z, y)Γ((n + 1)/2 − k)s k=0 + (log s)g(s, z, y) + h(s, z, y)
2 where g and h are smooth on D(0) × X and analytic in s and z.
Moreover P∞ n+1 k k=0 uk(z, y)Γ( 2 − k)s if n is even g(s, z, y) = k P∞ (−1) k−1 u n−1 (z, y) s if n is odd. k=1 k+ 2 (k−1)! Proof. We verify the hypotheses of Theorem 4 with β = −(n + 1)/2. We must show that there are positive numbers K and σ such that for all (z, y) ∈ X , a(·, z, y) is defined and continuous on (0, ∞), integrable on [0,K], and such that for θ > K, " N # n−1 X −k a(θ, z, y) = θ 2 uk(z, y)θ + RN (θ, z, y) k=0 N+1 where RN (θ, z, y) = O(N!(σ/θ) ), uniformly in N and θ > K, and (z, y) ∈ X . The continuity of a(·, z, y) on (0, ∞) and integrability over [0,K] for any positive K are clear (since χ(t) = 0 for t ≤ 0 and vanishes to infinite order at t = 0). For simplicity we will take K = 1. Write
N ∞ X −k X −k RN (x, y, θ) = uk(x, y)θ (χ(θ − kC) − 1) + uk(x, y)θ χ(θ − kC). k=0 k=N+1 (17)
13 We first estimate the first term on the right hand side of (17). Using item k+1 3 in Theorem 3 we have |uk(x, y)| ≤ L k! for all (x, y) ∈ X . Note that χ(θ − kC) − 1 is zero if θ > kC + 1/2, so that we may assume θ ≤ kC + 1/2. Furthermore we have 1/(2C) < 1. This gives
N N X −k N+1 X N+1−k N+1−k uk(x, y)θ (χ(θ − kC) − 1) ≤ L(L/θ) (C/L) (k + 1) k!. k=0 k=0
N+1−k Note (k + 1) k! ≤ (N + 1)! for k = 0, 1, . . . , N. Then if α1 = max(1, C/L) we have
N X −k N+1 2 |uk(x, y)(χ(θ − kC) − 1)θ ≤ L(α1L/θ) N!(N + 1) . k=0
2 N+1 We can choose α2 so that (N + 1) ≤ α2 for all N ≥ 0. Setting σ1 = Lα1α2 gives the desired estimate for the first term on the right hand side of (17). To estimate the second term on the right hand side of (17), we note that χ(θ − kC) is zero if k ≥ θ/C and write
∞ bθ/Cc X X k! k−N−1 u (x, y)θ−kχ(θ − kC) ≤ L(L/θ)N+1N! (L/θ) . k N! k=N+1 k=N+1
Since k ≤ θ/C, we can estimate k!/N! ≤ (N + 1)(θ/C)k−N−1. Then
bθ/Cc ∞ X −k N+1 X k−N−1 uk(x, y)θ χ(θ − kC) ≤ L(L/θ) N!(N + 1) (L/C) . k=N+1 k=N+1
Since L/C < e/4, the series converges and is bounded independent of θ and N. N+1 We can choose α3 so that N + 1 ≤ α3 for all N ≥ 0. Setting σ2 = Lα3 gives the desired bound for the second term on the right hand side of (17). We can take σ to be the larger of σ1 and σ2. The conclusion of [1, Theorem 1] then gives the existence of 0. Since the estimate on RN holds uniformly, it follows from the proof of [1, Theorem 1] that g and h are smooth on D×X and analytic in z.
Proof of Theorem 1, part 2. Since P (, ·, ·) is smooth on Y, we need only con- sider X ∩(M ×X). As in (15) we can write the restriction of P (, ·, ·) to ∂M ×X formally as
Z ∞ ∞ − n+1 −(d2(z,y)+2)θ/4 X n−1 −k P (, z, y) = (4π) 2 e θ 2 uk(z, y)χ(θ − kC) dθ, 0 k=0 (18)
14 plus a smooth function on ∂M ×X. We will show that (18) is a Fourier integral distribution with complex phase. The proof of [18, Theorem 0.1] shows that the amplitude appearing in (18) is a symbol of order (n−1)/2 on X ∩(∂M×X)×R+. We need to show that the phase function3 iθ φ(z, y, θ) = (d2(z, y) + 2) (19) 4 defined on the cone def V = X ∩ (∂M × X) × R+ is a regular phase function of positive type on V (see [20, Definition 3.5]), and that real locus of the complex Lagrangian submanifold generated by φ is ∗ ∗ 2 contained in (T ∂M\0) × (T X\0). Here d is the analytic continuation of the distance squared function from a neighborhood of the diagonal in X × X to
X ∩ (M0 × X). Let Cφ denote the set of real θ-critical points of φ,
2 2 Cφ = {(z, y, θ) ∈ V : d (z, y) + = 0}.
From (11) and the remarks preceding it,
Cφ = {(Expyiv, y, θ) ∈ V : v ∈ TyX, |v| = , θ > 0.}.
The following lemma shows that dφ 6= 0 on V and d(φθ) 6= 0 on Cφ. Let ρ be the function on M0 defined by ρ(Expyiv) = |v|g, so that M = {ρ(z) < }.
Lemma 5. Let ıX : X ∩ (∂M × X) → X be the inclusion map. Then there is an 1 > 0 such that for all 0 < < 1 and all v ∈ TyX with |v| = , ı∗ d(id2) = 2 −ı∗ Im ∂ρ¯ 2 , v[ X (Expy iv,y) ∂M (Expy iv,y)
[ where v is the covector obtained from the metric identification of TyX and ∗ Ty X.
P i Proof. Let v = v ei where e1, . . . , en is an orthonormal basis for TyX, and let 1 n P k p(s , . . . , s ) = Expy( s ek) be a normal real analytic geodesic chart centered at y. Then
n 2 1 n 1 n X k k ds d (p(iv , . . . , iv ), p(s , . . . , s )) s=0 = −2i v ds s=0 (20) k=1
3We will suppress the dependence on in the notation.
15 P k [ (see [18], proof of Proposition 4.12 (c)). This can be interpreted as −2i( v ek) , and so d (id2) = 2v[. y (Expy iv,y) 2 We now fix y ∈ X and compute d(d (·, y)). Let D,y = {z ∈ ∂M :(z, y) ∈ X }. By (11), 2 2 Re d (z, y) ≥ − for all z ∈ D,y
2 and equality holds if z = Expyiv. Thus d(Re d (·, y)) is zero on vectors tangent 2 to D,y at points of the form z = Expyiv, and so d(Re d (·, y)) must be a multiple of dρ2 at these points. To determine the multiple we evaluate both d t 2 on the vector dt t=0Expyie v and find the multiple is −1. Since d (z, y) is a holomorphic function of z, we have d(Im d2(·, y)) = Jdρ2 at these points and the Lemma follows.
Clearly φ is homogeneous of degree one in θ, and Im φ ≥ 0 by (11). This together with Lemma 5 shows that φ is a regular phase function of positive type on V . The isotropic submanifold generated by φ,
def Λφ = {((z, φz), (y, φy)): (z, y, θ) ∈ Cφ} , can be computed using Lemma 5 as ] θ ∗ ¯ 2 θ ∗ Λφ = ((Exp iη , − ı Im ∂ρ ), (y, η)): η ∈ T X, |η| = , θ > 0 , y 2 ∂M 2 y
−1 ∗ or, replacing η by |ξ| ξ ∈ Ty X\0,
Λ = ((Exp i|ξ|−1ξ], −−1|ξ|ı∗ Im ∂ρ¯ 2), (y, ξ)): y ∈ X, ξ ∈ T ∗X\0, , φ y ∂M y (21) ∗ ∗ which is clearly contained in (T ∂M\0) × (T X\0). Thus (15) is a Fourier integral distribution with complex phase, and the operator S associated with it ∞ ∞ by the Schwartz kernel theorem is a continuous map from C (X) to C (∂M). Its degree of is easily calculated to be −(n − 1)/4 from the definition [20, p. 177–178]. This concludes the proof of Theorem 1, part 2.
Remark 2. Λφ can be interpreted as the twisted graph of an isomorphism of symplectic cones in the following way (see [5, Appendix A.6.4]). Let α0 be the pullback of the canonical one-form on T ∗X to S∗X and let α = ı∗ Im ∂ρ¯ 2. ∂M ∗ Then (S X, α0) and (∂M, α) are contact manifolds. The “adapted complex structure” identification
∗ ] Φ: S X 3 η → Expyiη ∈ ∂M,
16 where y is the base point of η, is a contact isomorphism. The half-line bundle ∗ ∗ Σ0 ⊂ T (S X)\0 consisting of positive multiples of α0 is a symplectic cone ∗ 4 ∗ canonically isomorphic with T X\0. Similarly the half-line bundle Σ ⊂ T ∂M consisting of positive multiples of α is a symplectic cone. Composing with Φ∗ gives an isomorphism of symplectic cones,
χ: T ∗X\0 → Σ (22)
−1 ξ → |ξ|Φ∗ (α0) ξ |ξ| −1 = |ξ|α −1 ] (23) Expy (i|ξ| ξ ) where y is the base point of ξ. Comparing (23) with (21) shows that Λφ is the graph of χ, twisted by multiplication by −1 in the fibers of ∂M. Remark 3. If X is n-dimensional Euclidean space, then the sum in (18) is identically one and d2(x, y) = (x − y)2. For real x, y, and > 0, the integral in (18) gives the classical Poisson kernel,
Z ∞ − n+1 −((x−y)2+2)θ/4 n−1 P (x, y, ) = (4π) 2 e θ 2 dθ 0 − n+1 2 2 − n+1 = π 2 Γ ((n + 1)/2) ((x − y) + ) 2 with the correct constant factor. Remark 4. We can express the Poisson kernel near the diagonal as an oscillatory integral (see (27)) in the following way. We can write (c.f. (15) and Remark 1) for (real) (, x, y) ∈ (0, 0) × (X × X)≤α, Z ∞ − n+1 −(d2(x,y)+2)θ/4 n−1 0 P (, x, y) ≈ (4π) 2 e θ 2 a (x, y, θ) dθ. 0
0 P∞ −k where a (x, y, θ) = k=0 θ uk(x, y)χ(θ − kC) is a symbol of order zero. Let Pn j ej(x) be an orthonormal basis of TxX and write y = Expx( j=1 t ej(x)), t = (t1, . . . , tn) ∈ Rn. Then d2(x, y) =< g(x)t, t > with g(x) the matrix of the metric tensor at x and < ·, · > is the Euclidean inner product. Then Z ∞ − n+1 −
4 ∗ ∗ The isomorphism consists of writing ξ ∈ T X\0 in “polar form” as ξ = θη, with η ∈ S X ∗ ∗ ∗ and θ > 0, and identifying ξ ∈ T X\0 with θ(α0)η ∈ T (S X).
17 Re g(x) ≥ 0, Z −
P (, x, y) ≈ (2π)−n(det g(x))−1/2 Z Z ∞ it·ξ −|ξ|2 /θ−2θ/4 − 1 0 e √ e g θ 2 a (x, y, θ) dθ dξ. n ξ∈R 4π 0 Let Z ∞ def |ξ| −|ξ|2 /θ−2θ/4 − 1 0 b(x, y, ξ, ) = e g √ e g θ 2 a (x, y, θ) dθ. 4π 0 Then b can be expressed as a Laplace transform, b(x, y, ξ, ) = e|ξ|g √ L [F (x, y, ξ, ·)] 2/4 (25) 4π where def −|ξ|2 /θ −1/2 0 F (x, y, ξ, θ) = e g θ a (x, y, θ), (26) so that Z P (, x, y) ≈ (2π)−n(det g(x))−1/2 eit·ξ−|ξ|g b(x, y, ξ, ) dξ (27) n ξ∈R Pn j where y = Expx( j=1 t ej(x)). Note t depends on both y and x but the integral is independent of the choice of orthonormal basis ej(x) because b is invariant under the orthogonal group action on ξ ∈ Rn. It can be shown (using the same technique as in the proof of Proposition 1) that b can be extended to a branched analytic function of d2(x, y) + 2 in a deleted neighborhood of zero in C.
s Proof of Theorem 1, part 3. To prove H -continuity of S we must consider the complex Lagrangian submanifold Λ associated with S . We think of V ⊂ φe ∂M × X × R+ as a real manifold and φ as a complex valued function of the (real) variables in V . Since V and φ are real analytic, we may consider their analytic extensions, which we denote by tildes.5 We may assume Ve has the form Ve = (X ∩ (∂M × X))e× C\0. Let C denote the set of θ-critical points for φ, i.e., φe e e
C = {(z, y, θ) ∈ V : d2(z, y) + 2 = 0}. φe e e e e e e e
5I.e., Ve is a complex manifold containing V as a totally real submanifold and φe is the analytic continuation of φ to Ve (which we may assume exists after possibly shrinking Ve ).
18 Note C is C ∩ V . The associated (complex) Lagrangian submanifold is φ φe
Λ def= {((z, φ ), (y, φ )) ∈ (T ∗∂M \0) × (T ∗X\0) :(z, y, θ) ∈ C }. φe e eze e eye e e e e e φe Λ is a positive conic immersed complex Lagrangian submanifold. Note Λ is φe φ def ∗ equal to Λ = Λ ∩ T (∂M × X) (see (21) and [20, Theorem 3.6])). φeR φe s s+ n−1 To verify that S is a continuous map from H (X) to H 4 (∂M), it n−1 2 2 suffices to show that S ◦ (1 + ∆) 8 is continuous from L (X) into L (∂M). n−1 Since (1 + ∆) 8 is a pseudodifferential operator of degree (n − 1)/4, S ◦ n−1 (1 + ∆) 8 is a complex Fourier integral operator of degree zero associated with Λ . According to [14, Theorem 3.5] we must show that for for every φe 0 γ0 = ((z0, ζ0), (y0, −ξ0)) ∈ Λ , the projections from φeR
0 def 0 ∗ (Tγ Λ ) = (Tγ Λ ) ∩ Tγ (T (∂M × X)) 0 φe R 0 φe 0
∗ ∗ 6 to T(z0,ζ0)(T ∂M) and T(y0,ξ0)(T X) are injective. It suffices to prove this for 0 Λ instead of Λ . Let c0 = (z0, y0, θ0) ∈ Cφ, let φe φe
F :(z, y, θ) ∈ Cφ → ((z, φz), (y, φy)) ∈ Λφ = Λ , φeR and let γ = F (c ). Then T Λ = T F (T C ), i.e., 0 0 γ0 φe c0 c0 φe
T Λ = (($, φ $ + φ ϑ + θ−1φ d), (ϑ, φ $ + φ ϑ + θ−1φ d)): γ0 φe zz zy 0 z yz yy 0 y ($, ϑ, d) ∈ T C V ,(dφ ) ($, ϑ, d) = 0 c0 θ c0
(see [21, p. 547-8]; we have used that φ is homogeneous of degree one in θ). Suppose ($, ϑ, d) ∈ T C are complex tangent vectors, T F ($, ϑ, d) is in c0 φe c0 (T Λ ) , and the projection of T F ($, ϑ, d) onto T (T ∗∂M ) is zero. We γ0 φe R c0 (z0,ζ0) will show ($, ϑ, d) = 0. By Lemma 5, φy is real and non-zero on Cφ. Thus
−1 Im (φyy) ϑ + θ0 φy Im d = 0. (28)
The tangency condition d(φθ)c0 ($, ϑ, d) = 0 implies, since $ = 0 and ∂φθ/∂θ = 0, that φyϑ = 0. Since Im (φyy) is positive definite by [18, Proposition 4.12 (a)], taking the inner product of (28) with ϑ gives ϑ = 0 and hence d = 0, so ($, ϑ, d) = 0. ∗ Now suppose Tc0 F ($, ϑ, d) is real and its projection onto T(y0,ξ0)(T X) is zero. Then −1 Im (φzz) $ + θ0 φz Im d = 0. (29)
6 Note (Tγ Λ ) is not a priori equal to Tγ (Λ ). It is obvious that the projections onto 0 φe R 0 φeR each factor of Tγ (Λ ) are injective because Λ = Λφ is the graph of (22). 0 φeR φeR
19 1 n P k Note Im (φzz) is not positive definite. Let (x , . . . , x ) → Expy0 ( x ek) be a normal real analytic geodesic chart centered at y0 as in Lemma 5. Then P k k (x + iτ) → Expy0 ( (x + iτ )ek) are local coordinates on M near y0. By compactness, after shrinking we may assume that the analytic continua- tion exists and gives local coordinates near z0 = Expy0 (iv0), |v0| = . Let (s1, . . . , sn−1) → (τ 1(s), . . . , τ n(s)) be local coordinates on the sphere |v| = centered at v0. Then (x, s) are local coordinates on ∂M centered at z0 and ! φxx φxs Im (φzz) = Im φsx φss
Since Im(φxx) is positive definite when τ = 0, by compactness and shrinking we P k may assume that Im(φxx) is positive definite at z0. Since (Expy0 (i τ (s)ek), y0, θ0) is in Cφ for all s in a neighborhood of zero and dzφ is real on Cφ, we conclude
Im(φxs) = Im(φxs) = Im(φss) = 0 at z0. Write $ in these coordinates as (a, b) and take the inner product of (29) with $. The condition (dφθ)c0 ($, ϑ, d) = 0 t means that φz$ = 0. We obtain a Im(φxx)a = 0 and so a = 0. From this it follows that Im d = 0 and so TcF ($, ϑ, d) ∈ Tγ0 Λφ. Since Λφ is the graph of a ∗ diffeomorphism, the projection onto T(y0,ξ0)(T X) is injective and ($, ϑ, d) = 0. n−1 2 2 This shows that S ◦(1+∆) 8 is a continuous map from L (X) to L (∂M) s s+ n−1 and S is continuous from H (X) to H 4 (∂M). It remains to show that S s s+ n−1 is a bijection from H (X) onto O 4 (∂M ). √ − ∆ If Sf = 0, then e f is a holomorphic function on M whose restriction to ∂M (in the sense of distributions) is zero. Since a holomorphic function is harmonic for the K¨ahlerLaplacian and there is a unique solution to the Dirichlet √ √ − ∆ − ∆ problem on M, we must have e f = 0 on M and so on X. Since e is injective on Hs(X) (as can be seen from the eigenfunction expansion), f = 0.
This shows that S is injective.
We now show, following the outline in [4], that for sufficiently small, S s+ n+1 is onto O 4 (∂M). Since S is continuous, it suffices to show S is onto the dense subspace of restrictions to ∂M of functions holomorphic on some 7 s+ n+1 neighborhood of M. So let F be in this dense subspace of O 4 (∂M).
Since ∂M is compact we may suppose F ∈ O(M 1 ) for some 1 with < 1 < 0 (i.e., F is holomorphic in some neighborhood of M ). It suffices to show that √ 1 there is a g ∈ C∞(X) such that F | = e− ∆g. The following lemma says that √ X + ∆ the operator e is well-defined on F |X .
7 ¯ s Real analytic functions are dense in ker ∂b ∩ H (∂M), and each real analytic function in ¯ ker ∂b extends to a two-sided neighborhood of ∂M, and to the interior also by pseudoconvexity.
20 √ Lemma 6. For each x ∈ X, the map t 7→ exp(−t ∆) (F |X )(x) can be an- alytically continued from t > 0 to a single valued analytic function of t on
D(1)\i(−1, 0], smooth in (t, x) ∈ D(1)\i(−1, 0] × X.
0 Proof. Fix x ∈ X. Let Cut(x) be the cut locus of x and let Xx = X\(Cut(x) ∪ {x}). Since the cut locus has measure zero and P is symmetric in x and y, we have for t > 0 that √ Z exp(−t ∆)(F |X )(x) = P (t, y, x)F |X (y) dy. (30) 0 Xx If x and y are both real then it is not clear that P (t, x, y) can be continued to values of t with Re t < 0. To get around this difficulty we take advantage of the fact that F extends to M1 and deform the integration in y into M1 . We may assume (after possibly shrinking 0) that dy is the pullback to X of a holomorphic volume form, ω, on M0 . Let
0 Cx,1 = {Expy(−isvˆ(y, x)): y ∈ Xx, 0 ≤ s ≤ 1} where v(y, x) is the vector in TyX pointing from y to x andv ˆ(y, x) is the corresponding unit vector. For t > 1, P (t, ·, x)F (·) is holomorphic on M1 ⊂
Mt. Thus the integral of dz (P (t, z, x)F (z)ω(z)) over the (n + 1)-chain Cx,1 is zero. By Stoke’s Theorem, for t > 1 we have Z P (t, ·, x)F (·)ι∗ ω = 0. (31) ∂Cx,1 ∂Cx,1
The boundary ∂Cx,1 consists of four pieces:
0 Cx,1 = ∂Cut(x)Cx,1 ∪ Xx ∪ Γx,1 ∪ Expx(iB(1))
where ∂Cut(x)Cx,1 consists of points of the form Expyiv, y ∈ Cut(x) and |v| ≤ 1, B(1) is the closed ball of radius 1 in TxX, and
0 Γx,1 = Expy(−i1vˆ(y, x)): y ∈ Xx .
Using (30) and (31) with the proper orientations, we have for t > 1 √ Z ∗ exp(−t ∆)(F |X )(x) = P (t, z, x)F (z)ι ω(z). ∂Cut(x)Cx,1 +Γx,1 +Expx(iB(1)) (32) We consider each of the three boundary integrals separately.
I. Integral over ∂Cut(x)Cx,1 . We will show that this does not contribute any analytic singularities for t ∈ D(1). Recall that after having fixed α in
21 the proof of Theorem 1, part 1, we choose 0 less than the β of Corollary 2
(and of course 1 < 0) . After possibly shrinking 0 we may assume that
{(z, x): z ∈ ∂Cut(x)Cx,1 , x ∈ X} ⊂ Y (since {(y, x): y ∈ Cut(x), x ∈ X} ⊂ Y). So modulo a function which is smooth on D(0)×X and (for fixed x) analytic in t ∈ D(0), we can write √ Z ∗ exp(−t ∆)(F |X )(x) ≈ P (t, z, x)F (z)ι ω(z). Γx,1 +Expx(iB(1))
II. Integral over Γx,1 . If (z, x) ∈ Y, then P (t, z, x) is analytic (for fixed x) in
(t, z) ∈ D(0) × {z ∈ M :(z, x) ∈ Y}. So we need only consider points z ∈ Γx,1 such that (z, x) ∈ X .
Lemma 7. Fix z, x such that z ∈ Γx,1 and (z, x) ∈ X . Then P (t, z, x) can be analytically continued to a holomorphic function of t in the disk D(1), depending smoothly on (z, x).
Proof. We have from (16)
− n+1 2 2 P (t, z, x) ≈ t(4π) 2 L[a(·, z, x)] (d (z, x) + t )/4 .
On Γx,1 we have z = Expy(−i1vˆ(y, x)). By analytically continuing the identity 2 2 2 2 2 2 2 d (Expy(lvˆ(y, x)), x) = (|v|g −l) we have d (z, x)+t = |v|g −1 +2i|v|g1 +t . Here 1 is fixed and 0 < |v|g < α. Let us choose the branch of the square root and logarithm that is holomorphic on C\(−∞, 0]i and agrees with the principal branch on (0, ∞). By Proposition 1, L[a(·, z, x)](w) is a holomorphic, single 2 valued function of w on the region C+ ∪ D(0) \i(−∞, 0] where C+ is the open right half-plane (see Figure 1). We must show that for all |v| ∈ (0, α) 2 1 2 and all u ∈ D(1), the quantity 4 (|v| − 1 + 2i|v|1 + u) lies in the region shown in Figure 1 (we’ll write simply |v| instead of |v|g). We will check that 2 2 2 2 if |v| − 1 + Re(u) = 0, then 2|v|1 + Im(u) > 0; and if |v| − 1 + Re(u) ≤ 0 1 2 2 2 2 2 then | 4 (|v| − 1 + 2i|v|1 + u)| < 0. First suppose |v| − 1 + Re(u) = 0. Then 2 4 2 2 2 2 2 2 Im(u) < 1 −Re(u) , and so Im(u) < 21|v| < 41|v| , hence 2|v|1 +Im(u) > 2 2 2 2 2 0. Next suppose |v| − 1 + Re(u) ≤ 0. Then |v| − 1 ≤ 1 and q 1 2 2 1 2 2 2 2 2 1 2 (|v| − + 2i|v|1 + u) < (|v| − ) + 4|v| + 4 1 4 1 1 4 1 1 1 ≤ 32 + 2. 4 1 4 1 The smooth dependence on (z, x) is clear.
22 Im s Im w 2 4 Ε0
Re w Re s 2 2 Ε0 Ε0
Figure 1: Region where Figure 2: Region where 2 2 L[a(·, z, x)](w) is analytic and L[a(·, z, x)]((|v|g −0+s+2i|v|g0)/4) single valued. is analytic for all |v|g ≥ 0 (together 2 with the disk D(0)).
2 2 Remark 5. The set of s such that L[a(·, z, x)]((|v|g − 1 + 2i|v|g1 + s)/4) is 2 analytic in s for all |v|g ≥ 0, together with the disk D(0), is illustrated in Figure 2 in the case where 1 = 0.
So modulo a function which is smooth on D(0)×X and (for fixed x) analytic
in t ∈ D(0), we can write √ Z ∗ exp(−t ∆)(F |X )(x) ≈ P (t, z, x)F (z)ι ω(z). (33) Expx(iB(1))
III. Integral over Expx(iB(1)). Let I(t, x) be the integral on the right hand n side of (33). Identifying TxX with R we have Z I(t, x) = P (t, Expx(iu), x)F (Expx(iu))J(u) du |u|≤1
n where J(u) is the Jacobian of the map (u1, . . . , un) ∈ R 7→ Expx(iu) ∈ M (which is a real analytic diffeomorphism for |u| < 0). Writing this in polar coordinates gives Z 1 I(t, x) = Q(t, r, x)rn−1 dr (34) 0 where Z Q(t, r, x) = P (t, zr, x)F (zr)J(rη) dη η∈Sn−1
23 2 2 and zr = Expx(irη). Note d (z, x) = −r . From Equation (16) and Proposition 1 we can write I = I1 + I2 + I3 where
Z 1 2 2 −(n+1)/2 1 2 2 n−1 I1(t, x) = t (−r + t ) Q1( 4 (−r + t ), zr, x)r dr 0 Z 1 2 2 1 2 2 n−1 I2(t, x) = t log(−r + t )Q2( 4 (−r + t ), zr, x)r dr 0 Z 1 1 2 2 n−1 I3(t, x) = t Q3( 4 (−r + t ), zr, x)r dr 0 Z 1 2 2 1 2 2 Qi( 4 (−r + t ), zr, x) = gi( 4 (−r + t ), zr, x)F (zr)J(rη) dη η∈Sn−1
2 and gi(s, z, x) is smooth in D(0) × X and analytic in s and z, and F ∈ O(M1 ). 1 2 2 2 If t ∈ D(1), then 4 (−r + t ) lies in D(0) for all r ∈ D(1). In particular for 1 2 2 each fixed real r with |r| ≤ 1, Qi( 4 (−r + t ), z, x) is a holomorphic function of t ∈ D(1), smooth in (t, x). It follows I3(t, x) is a holomorphic function of t ∈ D(1), smooth in (t, x). n−1 We claim that for each η ∈ S , zr, F (zr) and J(rη) can be analytically continued in r from [0, 1] to D(1). After possibly shrinking 0 it is clear this is true for zr and J(rη). To see that it is true for F (zr), it suffices to show that the analytic continuation of zr to r ∈ D(1) takes values in M1 . We note that the analytic continuation of the map r 7→ γη(r), where γη is the geodesic associated with η ∈ TxX, into the tangent bundle with the adapted complex structure is r+is 7→ sγ˙ η(r). Since |sγ˙ η(r)|g = s, the analytic continuation of zr = Expx(irη) in r to D(1) takes values in M1 . Then the path of integration in I1 and I2 iα can be changed to the quarter circle {−iα: 0 ≤ α ≤ 1} ∪ {1e : 3π/2 ≤ α ≤
2π}. This shows that there is an analytic continuation of I1 and I2 in t to
D(1)\i(−1, 0], and that (32) is analytic in t on D(1)\i(−1, 0].
We now conclude the proof of Theorem 1, part 3, by showing S is onto. As √ noted above it suffices to show that F | is in the range of e− ∆. Fix > 0 and √ X √ √ − ∆ −s ∆ −(+s) ∆ x ∈ X. For Re(s) > 0 we have e (e (F |X ))(x) = e (F |X )(x). Both sides are analytic functions of s ∈ D( )\i(− , 0], so equality continues to 1 √ √1 hold for such s. Letting s → −+ gives e− ∆(e ∆(F | ))(x) = F | (x), i.e., √ X X − ∆ F |X is in the range of e .
References
[1] W. A. Beyer and L. Heller, Analytic continuation of Laplace transforms by means of asymptotic series, J. Mathematical Physics 8 (May, 1967), no. 5, 1004–1018.
24 [2] L. Boutet de Monvel, Compl´ementsur le noyau de Bergman, S´eminairesur les ´equations aux d´eriv´eespartielles, 1985–1986, Ecole´ Polytech., Palaiseau, 1986, pp. Exp. No. XX, 13 (French). MR874579 (88c:32036) [3] Louis Boutet de Monvel, Convergence dans le domaine complexe des s´eriesde fonctions propres, C. R. Acad. Sci. Paris S´er.A-B 287 (1978), no. 13, A855–A856 (French, with English summary). MR551763 (80i:58040) [4] L. Boutet de Monvel, Convergence dans le domaine complexe des s´eriesde fonctions propres, Journ´ees: Equations´ aux D´eriv´eesPartielles (Saint-Cast, 1979), Ecole´ Polytech., Palaiseau, 1979, pp. Exp. No. 3, 2 (French). MR544506 (80i:35185) [5] L. Boutet de Monvel and V. Guillemin, The spectral theory of Toeplitz operators, An- nals of Mathematics Studies, vol. 99, Princeton University Press, Princeton, NJ, 1981. MR620794 (85j:58141) [6] L. Boutet de Monvel and J. Sj¨ostrand, Sur la singularit´edes noyaux de Bergman et de Szeg¨o, Journ´ees: Equations´ aux D´eriv´eesPartielles de Rennes (1975), Soc. Math. France, Paris, 1976, pp. 123–164. Ast´erisque,No. 34-35 (French). MR0590106 (58 #28684) [7] F. Bruhat and H. Whitney, Quelques propri´et´es fondamentales des ensembles analytiques-r´eels, Comment. Math. Helv. 33 (1959), 132–160 (French). MR0102094 (21 #889) [8] J. J. Duistermaat and V. W. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975), no. 1, 39–79. MR0405514 (53 #9307) [9] Victor Guillemin, Paley-Wiener on manifolds, The Legacy of Norbert Wiener: A Cen- tennial Symposium (Cambridge, MA, 1994), Proc. Sympos. Pure Math., vol. 60, Amer. Math. Soc., Providence, RI, 1997, pp. 85–91. MR1460277 (98j:58107) [10] Victor Guillemin and Matthew Stenzel, Grauert tubes and the homogeneous Monge- Amp`ere equation, J. Differential Geom. 34 (1991), no. 2, 561–570. MR1131444 (93e:32018) [11] , Grauert tubes and the homogeneous Monge-Amp`ere equation. II, J. Differential Geom. 35 (1992), no. 3, 627–641. MR1163451 (94e:32032) [12] Jacques Hadamard, Lectures on Cauchy’s problem in linear partial differential equations, Dover Publications, New York, 1953. MR0051411 (14,474f) [13] Richard A. Handelsman and John S. Lew, Asymptotic expansion of a class of inte- gral transforms via Mellin transforms, Arch. Rational Mech. Anal. 35 (1969), 382–396. MR0252930 (40 #6145) [14] Lars H¨ormander, L2 estimates for Fourier integral operators with complex phase, Ark. Mat. 21 (1983), no. 2, 283–307, DOI 10.1007/BF02384316. MR727350 (85h:47058) [15] , The analysis of linear partial differential operators. I, Grundlehren der Mathe- matischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR717035 (85g:35002a) [16] M. Kashiwara, Analyse micro-locale du noyau de Bergman, S´eminaire Goulaouic- Schwartz (1976/1977), Equations´ aux d´eriv´eespartielles et analyse fonctionnelle, Exp. No. 8, Centre Math., Ecole´ Polytech., Palaiseau, 1977, pp. 10 (French). MR0590108 (58 #28685)
25 [17] Jay Jorgenson and Serge Lang, Analytic continuation and identities involving heat, Pois- son, wave and Bessel kernels, Math. Nachr. 258 (2003), 44–70. MR2000044 (2005f:58037) [18] Eric Leichtnam, Fran¸coisGolse, and Matthew Stenzel, Intrinsic microlocal analysis and inversion formulae for the heat equation on compact real-analytic Riemannian manifolds, Ann. Sci. Ecole´ Norm. Sup. (4) 29 (1996), no. 6, 669–736. MR1422988 (97h:58153) [19] L´aszl´oLempert and R´obert Sz¨oke, Global solutions of the homogeneous complex Monge- Amp`ere equation and complex structures on the tangent bundle of Riemannian manifolds, Math. Ann. 290 (1991), no. 4, 689–712. MR1119947 (92m:32022) [20] Anders Melin and Johannes Sj¨ostrand, 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. 120–223. Lecture Notes in Math., Vol. 459. MR0431289 (55 #4290) [21] Fran¸coisTr`eves, Introduction to pseudodifferential and Fourier integral operators. Vol. 2, Plenum Press, New York, 1980. Fourier integral operators; The University Series in Mathematics. MR597145 (82i:58068) [22] Steve Zelditch, Complex zeros of real ergodic eigenfunctions, Invent. Math. 167 (2007), no. 2, 419–443. MR2270460 (2008b:58034) [23] , 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. 299–339, DOI 10.1090/pspum/084/1363. MR2985323
26