Pluri-potential theory on Grauert tubes of real analytic Riemannian manifolds, I
aa r X i v : . [ m a t h . SP ] J u l PLURI-POTENTIAL THEORY ON GRAUERT TUBES OF REALANALYTIC RIEMANNIAN MANIFOLDS, I
Abstract.
Analogues of the some basic notions of pluri-potential theory on strictly pseudo-convex domains in C m are developed for Grauert tubes M τ in complexifications of realanalytic Riemannian manifolds ( M, g ). In particular, the normalized logarithm of the com-plexified spectral projector Π C I λ ( ζ, ¯ ζ ) is the analogue of the Siciak-Zaharjuta extremal pluri-subharmonic function. It is shown that λ log Π C I λ ( ζ, ¯ ζ ) → √ ρ ( ζ ), where √ ρ is the Grauerttube function. We give several applications to analytic continuations of eigenfunctions: tonorm estimates, triple product integrals and to complex nodal sets. In the study of eigenfunctions of the Laplacian ∆ g on a real analytic Riemannian manifold( M, g ) of dimension m , it is often useful to analytically continue an orthonormal basis { ϕ λ j } of eigenfunctions,∆ g ϕ λ j = λ j ϕ λ j , h ϕ λ j , ϕ λ k i = δ jk , ( λ = 0 < λ ≤ λ ≤ · · · ) , into the complexification M C of M . As recalled in §
1, eigenfunctions admit analytic contin-uations ϕ C λ j to a maximal uniform ’Grauert tube’ M τ = { ζ ∈ M C , √ ρ ( ζ ) < τ } (1)independent of λ j , where the radius is measured by the Grauert tube function √ ρ ( ζ ) corre-sponding to g (see §
1: [LS1, GS1]). Analytic continuation of eigenfunctions and spectral pro-jections (2)-(3) to Grauert tubes have applications to nodal geometry [DF, Lin, Z3, TZ, RZ],analytic wave front sets [Leb, GLS], tunnelling estimates [HS, Mar], Paley-Wiener theo-rems [G], invariant triple products [Sar, BR], random waves [Z2] and Agmon estimates foreigenfunctions in the classically forbidden region (see e.g.[To]).Grauert tubes are strictly pseudo-convex Stein manifolds, and in some ways are analogousto strictly pseudo-convex domains in C m and to Hermitian unit bundles in negative linebundles. The purpose of this article and its sequel [Z1] is to extend to Grauert tubessome of the basic notions and results of PSH (pluri-subharmonic) function theory on strictypseudo-convex domains in C n (cf. [K, BL]), and their recent generalization of this theory toK¨ahler manifolds in [GZ]. The basic theme is to use analytic continuations of eigenfunctions { ϕ C λ j } in place of holomorphic polynomials of degree ∼ λ j on C m or holomorphic sections ofline bundles of degree ∼ λ j over a K¨ahler manifold. The primary objects are the analyticcontinuations of the spectral projections kernels of ∆ g ,Π C I λ ( ζ , ¯ ζ ) = X j : λ j ∈ I λ | ϕ C λ j ( ζ ) | , (2)which are of exponential growth, and their ‘tempered’ analogues, P τI λ ( ζ , ¯ ζ ) = X j : λ j ∈ I λ e − τλ j | ϕ C λ j ( ζ ) | , ( √ ρ ( ζ ) ≤ τ ) , (3) Date : October 15, 2018.Research partially supported by NSF grant where I λ could be a short interval [ λ, λ + 1] of frequencies or a long window [0 , λ ]. In thisarticle, we only consider long windows I λ = [0 , λ ] while in [Z1] we refine the results to shortwindows using the long time behavior of the complexified geodesic flow. The temperedkernels P τI λ ( ζ , ¯ ζ ) are in some ways analogous to the ‘density of states function’ or Bergmankernel on the diagonal in the setting of positive line bundles over K¨ahler manifolds [Z4]. Wegave some initial results on these kernels in [Z2], by somewhat different methods.A basic notion in PSH theory is that of maximal PSH functions satisfying bounds and the(non-obviously) equivalent Siciak-Zaharjuta extremal PSH functions. We define a Grauerttube analogue of the Siciak-Zaharjuta extremal function and show in Theorem 1 that it isthe same as the Grauert tube function. The proof is to relate both to the complexifiedspectral projections (2)which are defined in terms of eigenfunctions. The proof only requiresa one term local Weyl law (see Theorem 2), which also gives improvements on the pointwisebounds on complexified eigenfunctions in [GLS]. The result can be improved to a ratherinteresting two term Weyl law of Safarov-Vasilliev type [SV]; this is carried out in the sequel[Z1].This article also contains a general type of result on integrals of triple products of eigen-functions (Proposition 1). The precise results depend on the radius of the maximal Grauerttube. We point out that there are two possible definitions (see Definition 1.1), an analyticmaximal radius and a geometric maximal radius; in § A Siciak-Zaharjuta extremal function for Grauert tubes.
Before defining theanalogues, let us first recall the definitions of relative maximal or extremal PSH functionssatisfying bounds on a pair E ⊂ Ω ⊂ C m where Ω is a bounded open set. There are twodefinitions: • The pluri-complex Green’s function relative to a subset E ⊂ Ω, defined [Br, Sic] asthe upper semi-continuous regularization V ∗ E, Ω of V E, Ω ( z ) = sup { u ( z ) : u ∈ P SH (Ω) , u | E ≤ , u | ∂ Ω ≤ } . • The Siciak-Zaharjuta extremal function relative to E ⊂ Ω, defined bylog Φ NE ( ζ ) = sup { N log | p N ( ζ ) | : p ∈ P NE } , log Φ E = lim sup N →∞ log Φ NE , where P NE = { p ∈ P N : || p || E ≤ , || p || Ω ≤ e N } . Here, || f || E = sup z ∈ E | f ( z ) | and P N denotes the space of all complex analytic polynomialsof degree N . Siciak proved that log Φ E = V E (see [Sic2] Theorem 1, and [K], Theorem5.1.7). Intuitively, there are enough polynomials that one can obtain the sup by restrictingto polynomials. LURI-POTENTIAL THEORY ON GRAUERT TUBES 3
There are analogous definitions in the case of unit co-disc bundles in the dual of a positiveholomorphic Hermitian line bundle L → M over a K¨ahler manifold. In the case of CP n , onedefines V K ( z ) = sup { u ( z ) : u ∈ L , u ≤ K } where L denotes the Lelong class of all global plurisubharmonic (PSH) functions u on C n with u ( z ) ≤ c u + log (1 + | z | ). We refer to [GZ] for further information in the K¨ahler setting.We now define an analogue of the Siciak-Zaharjuta extremal function for Grauert tubesin the special case where E = M , the underlying real manifold. A generalization to othersets E ⊂ M τ is discussed in § P N is the space H λ = { p = X j : λ j ∈ I λ a j ϕ C λ j , a , . . . , a N ( λ ) ∈ R } spanned by the eigenfunctions with ‘degree’ λ j ≤ λ . Here, N ( λ ) = { j : λ j ∈ I λ } . Asabove, we could let I λ = [0 , λ ] or I λ = [ λ, λ + c ] for some c >
0. It is simpler to work with L based norms than sup norms, and so we define S H λM = { ψ = X j : λ j ≤ λ a j ϕ C λ j , N ( λ ) X j =1 | a j | = 1 } . Definition The Riemannian Siciak-Zaharjuta extremal function (with respect to the reallocus M ) is defined by: log Φ λM ( ζ ) = sup { λ log | ψ ( ζ ) | : ψ ∈ S H λM } , log Φ M = lim sup λ →∞ log Φ λM . (4) Remark:
One could define the analogous notion for any set E ⊂ M τ , with S H λE = { p ∈ H λ , || p || L ( E ) ≤ } . But we only discuss the results for E = M (see § M τ as follows: Definition Let ( M, g ) be a real analytic Riemannian manifold, let M τ be an open Grauerttube, and let E ⊂ M τ . The Riemannian pluri-complex Green’s function with respect to ( E, M τ , g ) is defined by V g,E,τ ( ζ ) = sup { u ( z ) : u ∈ P SH ( M τ ) , u | E ≤ , u | ∂M τ ≤ τ } . It is obvious that V g,M,τ ( ζ ) ≥ √ ρ ( ζ ) and it is almost standard that V g,M,τ ( ζ ) = √ ρ ( ζ ).See Proposition 4.1 of [GZ] or Corollary 9 of [BT2]. The set M = ( √ ρ ) − (0) is often calledthe center. As proved in [LS1], there are no smooth exhaustion functions solving the exactHCMA (Theorem 1.1). Hence u must be singular on its minimum set. In [HW] it is provedthat the minimum set of strictly PSH function is totally real. PLURI-POTENTIAL THEORY ON GRAUERT TUBES
Statement of results.
Our first results concern the logarithmic asymptotics of thecomplexified spectral projections.
Theorem (see also [Z4] ) Let I λ = [0 , λ ] . Then (1) log Φ λM ( ζ ) = λ log Π C I λ ( ζ , ¯ ζ ) . (2) log Φ M = lim λ →∞ log Φ λM = √ ρ. To prove the Theorem, it is convenient to study the tempered spectral projection measures(3), or in differentiated form, d λ P τ [0 ,λ ] ( ζ , ¯ ζ ) = X j δ ( λ − λ j ) e − τλ j | ϕ C j ( ζ ) | , (5)which is a temperate distribution on R for each ζ satisfying √ ρ ( ζ ) ≤ τ. When we set τ = √ ρ ( ζ ) we omit the τ and write d λ P [0 ,λ ] ( ζ , ¯ ζ ) = X j δ ( λ − λ j ) e − √ ρ ( ζ ) λ j | ϕ C j ( ζ ) | . (6)The advantage of the tempered projections is that they have polynomial asymptotics andone can use standard Tauberian theorems to analyse their growth.We prove the following one-term local Weyl law for complexified spectral projections: Theorem On any compact real analytic Riemannian manifold ( M, g ) of dimension n ,we have, with remainders uniform in ζ , (1) For √ ρ ( ζ ) ≥ Cλ ,P [0 ,λ ] ( ζ , ¯ ζ ) = (2 π ) − n (cid:18) λ √ ρ (cid:19) n − (cid:18) λ ( n − / O (1) (cid:19) ;(2) For √ ρ ( ζ ) ≤ Cλ , P [0 ,λ ] ( ζ , ¯ ζ ) = (2 π ) − n λ n (cid:0) O ( λ − ) (cid:3) . This implies new bounds on pointwise norms on complexified eigenfunctions, improvingthose of [GLS]. inequality gives
Corollary Suppose ( M, g ) is real analytic of dimension n , and that I λ = [0 , λ ] . Then, (1) For τ ≥ Cλ and √ ρ ( ζ ) = τ , there exists C > so that Cλ − n − j e τλ ≤ sup ζ ∈ M τ | ϕ C λ ( ζ ) | ≤ Cλ n − + 12 e τλ . (2) For τ ≤ Cλ , and √ ρ ( ζ ) = τ , there exists C > so that | ϕ C λ ( ζ ) | ≤ λ n − ;The lower bound of Corollary 3 (1) combines Theorem 2 with G¨arding’s inequality. Theupper bound sharpens the estimates claimed in [Bou, GLS],sup ζ ∈ M τ | ϕ C λ ( ζ ) | ≤ C τ λ n +1 e τλ . (7)The improvement is due to using spectral asymptotics rather than a crude Sobolev inequality. LURI-POTENTIAL THEORY ON GRAUERT TUBES 5
The next Proposition ties together the work on triple inner products of eigenfunctionsin [Sar, BR] and elsewhere with analytic continuations of eigenfunctions to Grauert tubes.The basic question is the decay rate of the inner products R M ϕ λ j ϕ λ k dV g where dV g is thevolume form of ( M, g ). More generally, one considers integrals where ϕ λ k is replaced by apolynomial in eigenfunctions of fixed eigenvalues. In [Sar], it is proved that |h P, ϕ λ k i| ≤ A ( λ k + 1) B exp( − π √ λ k / π is the radius of the maximal Grauert tube for hyperbolic space andits quotients (see [Sz, KM] and § τ an is the maximal analytic tube radius definedin Definition 1.1. Essentially, it is the largest tube to which all eigenfunctions analyticallycontinue. Its relation to the geometric radius is discussed in § § Proposition Let ( M, g ) be any compact real analytic manifold and let τ an ( g ) be themaximal analytic Grauert tube radius. Then, for all τ < τ an , there exists a constant C τ suchthat | Z M ϕ λ j ϕ λ k dV g | ≤ C τ ( λ k ) e − τλ j . If ∂M τ an ( g ) is a smooth manifold and ϕ C λ k is a distribution of order r on ∂M τ an ( g ) , then thereexists a constant C so that | Z M ϕ λ j ϕ λ k dV g | ≤ C ( λ k ) λ rj e − τ an ( g ) λ j . As will be seen in the proof, C ( λ k ) is a Sobolev W s norm of e τ √ ∆ ϕ λ k . The statementlacks the precision of the hyperbolic case, since we do not determine whether ∂M τ is evena smooth manifold. In § τ an is the usual geometric radius ofthe Grauert tube, and then the estimate of Proposition 1 has almost the same exponentialasymptotics as in the hyperbolic case.Finally, we give an application to complex zeros of the joint eigenfunctions of the algebra D of invariant differential operators on the locally symmetric space SO ( n, R ) \ SL n ( R ) / Γ,where Γ is a co-compact discrete subgroup of SL n ( R ) In [AS], Anantharaman-Silbermanproved a number of results on the entropies of the quantum limit measures of the jointeigenfunctions as the joint eigenvalue tends to infinity. Roughly speaking, the results saythat the quantum limit measures must have a non-trivial Haar component. This result issufficient to determine the limit distribution of complex zeros of the complexifications of thesame eigenfunctions. We denote by [ Z ϕ C λ ] the current of integration over the complex zeroset of ϕ C λ . Theorem Let ( M, g ) be a compact locally symmetric manifold, and let { ϕ λ } be anyorthonormal basis of the joint D -eigenfunctions. Then for τ < τ an . λ [ Z ϕ C λ ] → iπ ∂ ¯ ∂ √ ρ, weakly in D ′ (1 , ( M τ ) , for the entire sequence of eigenfunctions { ψ j } . This proof requires just a small observation improving on the weak convergence result of[Z3], placed on top of the very strong Haar component theorem of Anantharaman-Silberman.
PLURI-POTENTIAL THEORY ON GRAUERT TUBES
Results of [Z1] . The asymptotics of the complexified spectral projection kernels (2)are complex analogues of those of the diagonal spectral projections in the real domain andreflect the structure of complex geodesics from ζ to ¯ ζ . As in the real domain, one can obtainmore refined asymptotics of P [ λ,λ +1] ( ζ , ¯ ζ ) by using the structure of geodesic segments from ζ to ¯ ζ . This is the subject of the sequel [Z1]. For the sake of completeness, we state the resultshere: There exists an explicit complex oscillatory factor Q ζ ( λ ) depending on the geodesicarc from ζ to ¯ ζ such that(1) For √ ρ ( ζ ) ≥ Cλ ,P τ [0 ,λ ] ( ζ , ¯ ζ ) = (2 π ) − n λ (cid:18) λ √ ρ (cid:19) n − (cid:0) Q ζ ( λ ) λ − + o ( λ − ) (cid:1) ;(2) For √ ρ ( ζ ) ≤ Cλ ,P τ [0 ,λ ] ( ζ , ¯ ζ ) = (2 π ) − n λ n + Q ζ ( λ ) λ n − + o ( λ n − ) , The functions Q ζ ( λ ) depend on whether ( M, g ) is a manifold without conjugate points,or with conjugate points. We refer to [Z1] for the formulae. A special case is that ofZoll manifolds where there exists a complete asymptotic expansion similar to that for linebundles. The two term asymptotics lead to improvement by one order of magnitude onthe bounds in Corollary 3, and are sharp in that they are achieved by complexified zonalspherical harmonics on a standard sphere.1.
Grauert tubes and complex geodesic flow
By a theorem of Bruhat-Whitney, a real analytic Riemannian manifold M admits a com-plexification M C , i.e. a complex manifold into which M embeds as a totally real subman-ifold. Corresponding to a real analytic metric g is a unique plurisubharmonic exhaustionfunction √ ρ on M C satisfying two conditions (i) It satisfies the Monge-Amp`ere equation( i∂ ¯ ∂ √ ρ ) n = δ M,g where δ M,g is the delta function on M with density dV g equal to the volumedensity of g ; (ii) the K¨ahler metric ω g = i∂ ¯ ∂ρ on M C agrees with g along M . In fact, √ ρ ( ζ ) = 12 i q r C ( ζ , ¯ ζ ) , (8)where r ( x, y ) is the square of the distance function and r C is its holomorphic extensionto a small neighborhood of the anti-diagonal ( ζ , ¯ ζ ) in M C × M C . In the case of flat R n , √ ρ ( x + iξ ) = 2 | ξ | and in general √ ρ ( ζ ) measures how far ζ reaches into the complexificationof M . The open Grauert tube of radius τ is defined by M τ = { ζ ∈ M C , √ ρ ( ζ ) < τ } . We usethe imprecise notation M C to denote the open complexificaiton when it is not important tospecify the radius.1.1. Analytic continuation of the exponential map.
The geodesic flow is a Hamiltonianflow on T ∗ M . In fact, there are two standard choices of the Hamiltonian. In PDE it is mostcommon to define the (real) homogeneous geodesic flow g t of ( M, g ) as the Hamiltonian flowon T ∗ M generated by the Hamiltonian | ξ | g with respect to the standard Hamiltonian form ω . This Hamiltonian is real analytic on T ∗ M \
0. In Riemannian geometry it is standard tolet the time of travel equal | ξ | g ; this corresponds to the Hamiltonian flow of | ξ | g , which isreal analytic on all of T ∗ M . We denote its Hamiltonian flow by G t . In general, we denote LURI-POTENTIAL THEORY ON GRAUERT TUBES 7 by Ξ H the Hamiltonian vector field of a Hamiltonian H and its flow by exp t Ξ H . Both ofthe Hamiltonian flows • g t = exp t Ξ | ξ | g ; • G t = exp t Ξ | ξ | g are important in analytic continuation of the wave kernel. The exponential map is the mapexp x : T ∗ M → M defined by exp x ξ = πG t ( x, ξ ) where π is the standard projection.We denote by inj( x ) the injectivity radius of ( M, g ) at x , i.e. the radius r of the largest ballon which exp x : B r M → M is a diffeomorphism to its image. Since ( M, g ) is real analytic,exp x tξ admits an analytic continuation in t and the imaginary time exponential map E : B ∗ ε M → M C , E ( x, ξ ) = exp x iξ (9)is, for small enough ε , a diffeomorphism from the ball bundle B ∗ ε M of radius ε in T ∗ M tothe Grauert tube M ε in M C . We have E ∗ ω = ω T ∗ M where ω = i∂ ¯ ∂ρ and where ω T ∗ M is thecanonical symplectic form; and also E ∗ √ ρ = | ξ | [GS1, LS1]. It follows that E ∗ conjugatesthe geodesic flow on B ∗ M to the Hamiltonian flow exp t Ξ √ ρ of √ ρ with respect to ω , i.e. E ( g t ( x, ξ )) = exp t Ξ √ ρ (exp x iξ ) . Maximal Grauert tubes.
A natural definition of maximal Grauert tube is the max-imum value of ε so that (9) is a diffeomorphism. We refer to this radius as the maximalgeometric tube radius . But for purposes of this paper, another definition of maximality isrelevant: the maximal tube on which all eigenfunctions extend holomorphically. A closelyrelated definition is the maximal tube to which the Poisson kernel (32) extends holomorphi-cally. We refer to the radius as the maximal analytic tube radius .A natural question is to relate these notions of maximal Grauert tube has not been ex-plored. We therefore define the radii more precisely: Definition (1)
The maximal geometric tube radius τ g is the largest radius ε forwhich E (9) is a diffeomorphism. (2) The maximal analytic tube radius τ an M τ an ⊂ M C is the maximal tube to which alleigenfunctions extend holomorphically and to which the anti-diagonal U (2 iτ, ζ , ¯ ζ ) ofthe Poisson kernel admits an analytic continuation. We make:
Conjecture τ g = τ an . In § Model examples. .We consider some standard examples to clarify these analytic continuations. (i)
Complex tori:The complexification of the torus M = R m / Z m is M C = C m / Z m . The adapted complexstructure to the flat metric on M is the standard (unique) complex structure on C m . The PLURI-POTENTIAL THEORY ON GRAUERT TUBES complexified exponential map is exp C x ( iξ ) = z := x + iξ , while the distance function r ( x, y ) = | x − y | extends to r C ( z, w ) = p ( z − w ) . Then √ ρ ( z, ¯ z ) = p ( z − ¯ z ) = ± i | Im z | = ± i | ξ | . The complexified cotangent bundle is T ∗ M C = C m / Z m × C m , and the holomorphic geodesicflow is the entire holomorphic map G t ( ζ , p ζ ) = ( ζ + tp ζ , p ζ ) . (ii) S n [GS1] The unit sphere x + · · · + x n +1 = 1 in R n +1 is complexified as the complexquadric S n C = { ( z , . . . , z n ) ∈ C n +1 : z + · · · + z n +1 = 1 } . If we write z j = x j + iξ j , the equations become | x | − | ξ | = 1 , h x, ξ i = 0. The geodesic flow G t ( x, ξ ) = (cos t | ξ | ) x + (sin t | ξ | ) ξ | ξ | , −| ξ | (sin t | ξ | ) x + (cos t | ξ | ) ξ ) on T ∗ S n complexifies to G t ( Z, W ) = (cos t √ W · W ) Z + (sin t √ W · W )) W √ W · W ) , −√ W · W )(sin t √ W · W )) Z + (cos t √ W · W )) W ) , (( Z, W ) ∈ T ∗ S m C ) . Here, the real cotangent bundle is the subset of T ∗ R n +1 of ( x, ξ ) such that x ∈ S n , x · ξ = 0and the complexified cotangent bundle T ∗ S n C ⊂ T ∗ C n +1 is the set of vectors ( Z, W ) : Z · W = 0. We note that although √ W · W is singular at W = 0, both cos √ W · W ) and √ W · W ) sin t √ W · W ) are holomorphic. The Grauert tube function equals √ ρ ( z ) = i cosh − | z | , ( z ∈ S n C ) . It is globally well defined on S n C . The characteristic conoid is defined by cosh i √ ρ = cosh τ . (iii) (See e.g. [KM] ). H n The hyperboloid model of hyperbolic space is the hypersurfacein R n +1 defined by H n = { x + · · · x n − x n +1 = − , x n > } . Then, H n C = { ( z , . . . , z n +1 ) ∈ C n +1 : z + · · · z n − z n +1 = − } . In real coordinates z j = x j + iξ j , this is: h x, x i L − h ξ, ξ i L = − , h x, ξ i L = 0where h , i L is the Lorentz inner product of signature ( n, C n +1 given by the same equations.We obtain H n C from S n C by the map ( z ′ , z n +1 ) → ( iz ′ , z n +1 ). The complexified geodesic flowis given by G t ( Z, W ) = (cosh t p h W, W i L Z + (sinh t p h W, W i L )) W √ h W,W i L ) , − p h W, W i L )(sinh t p h W, W i L )) Z + (cosh t p h W, W i L )) W ) , (( Z, W ) ∈ T ∗ H m ) . The Grauert tube function is: √ ρ ( z ) = cos − ( || x || L + || ξ || L − π ) / √ . The radius of maximal Grauert tube is ε = 1 or r = π/ √ . LURI-POTENTIAL THEORY ON GRAUERT TUBES 9
2. ∆ g , ✷ g and characteristics The Laplacian of (
M, g ) is given in local coordinates by∆ g = − X i,j ∂∂x i (Θ g ij ) ∂∂x j , is the Laplacian of ( M, g ). Here, g ij = g ( ∂∂x i , ∂∂x j ), [ g ij ] is the inverse matrix to [ g ij ] andΘ = p det[ g ij ] . Since g is fixed we henceforth write the Laplacian as ∆. Note that we haveput a minus sign in front of the sum of squares to make ∆ a non-negative operator. Thisis for later notational convenience. On a compact manifold, ∆ is negative operator withdiscrete spectrum ∆ ϕ j = λ j ϕ j , h ϕ j , ϕ k i = δ jk (10)of eigenvalues and eigenfunctions. Note that the eigenvalues are denoted λ j ; we refer to λ j as the ‘frequency’.In the real domain, ∆ is an elliptic operator with principal symbol σ ∆ ( x, ξ ) =: P ni,j =1 g ij ( x ) ξ i ξ j .Hence its characteristic set (the zero set of its symbol) consists only of the zero section ξ = 0in T ∗ M . But when we continue it to the complex domain it develops a complex characteristicset Ch(∆ C ) = { ( ζ , ξ ) ∈ T ∗ M C : n X i,j =1 g ij ( ζ ) ξ i ξ j = 0 } . (11)The wave operator on the product spacetime ( R × M, dt − g x ) is given by ✷ g = ∂ ∂t + ∆ g . The unusual sign in front of ∆ g is due to the sign normalization above making the Laplaciannon-negative. Again we omit the subscript when the metric is fixed. The characteristicvariety of ✷ is the zero set of its symbol σ ✷ ( t, τ, x, ξ ) = τ − | ξ | x , that is, Ch( ✷ ) = { ( t, τ, x, ξ ) ∈ T ∗ ( R × M ) : τ − | ξ | x = 0 } . (12)The null-bicharacteristic flow of ✷ is the Hamiltonian flow of τ − | ξ | x on Ch( ✷ ). Its graphis thus Λ = { ( t, τ, x, ξ, y, η ) : τ − | ξ | x = 0 , G t ( x, ξ ) = ( y, η ) } ⊂ T ∗ ( R × M × M ) . Characteristic variety and characteristic conoid.
Following [H], we putΓ( t, x, y ) = t − r ( x, y ) . (13)Here, r ( x, y ) is the distance between x, y . It is singular at r = 0 and also when y is in the“cut locus” of x . In this article we only consider ( x, y ) so that r ( x, y ) < inj( x ), where inj(x)is the injectivity radius at x , i.e. is the largest ε so thatexp x : B ∗ x,ε M → M is a diffeomorphism to its image. The injectivity radius inj ( M, g ) is the maximum of inj( x )for x ∈ M . Thus, we work in a sufficiently small neighborhood of the diagonal so that cutpoints do not occur.The squared distance r ( x, y ) is smooth in a neighborhoof of the diagonal. On a simplyconnected manifold ( ˜ M , g ) without conjugate points, it is globally smooth on ˜ M × ˜ M . Werecall that ‘without conjugate points’ means that exp x : T x M → M is non-singular for all x .The characteristic conoid is the set C = { ( t, x, y ) : r ( x, y ) < inj( x ) , r ( x, y ) = t } ⊂ R × M × M. (14)It separates R × M × M into the forward/backward semi-cones C ± = { ( t, x, y ) : t − r ( x, y ) > , ± t > } . The complexificationof C = C R is the complex characteristic conoid C C = { ( t, x, y ) : r C ( x, y ) = t } ⊂ C × M C × M C . (15)We note that C R ⊂ C C is a totally real submanifold. Another totally real submanifold ofcentral importance in this article is the ‘diagonal’ (or anti-diagonal) conoid, C ∆ = { (2 iτ, ζ , ¯ ζ ) : τ ∈ R + , ζ , ¯ ζ ∈ ∂M τ } . (16)By definition, r C ( ζ , ¯ ζ ) = − τ if ζ ∈ ∂M τ .3. Propagators and fundamental solutions
The main ‘wave kernels’ in this article are the half-wave kernel e it √ ∆ and the Poissonkernel e − τ √ ∆ for τ >
0. To put these kernels into context, we now give a brief review ofpropagators and fundamental solutions for the wave equation. We use the term ‘propagator’for a solution operator to a Cauchy problem. It will be a homogeneous solution of ✷ E = 0with special initial conditions. We use the term ‘fundamental solution’ for a solution ofthe inhomogeneous equation ✷ E = δ . We freely use standard notation for homogeneousdistributions on R and refer to [Ho] for notation and background.3.1. Cauchy problem for the wave equation.
The Cauchy problem for the wave equa-tion on R × M is the initial value problem (with Cauchy data f, g ) ✷ u ( t, x ) = 0 ,u (0 , x ) = f, ∂∂t u (0 , x ) = g ( x ) , . The solution operator of the Cauchy problem (the “propagator”) is the wave group, U ( t ) = cos t √ ∆ sin t √ ∆ √ ∆ √ ∆ sin t √ ∆ cos t √ ∆ . The solution of the Cauchy problem with data ( f, g ) is U ( t ) (cid:18) fg (cid:19) . Two of the components of U ( t ) are particularly important: LURI-POTENTIAL THEORY ON GRAUERT TUBES 11 • The even part cos t √ ∆ is the solution operator of the initial value problem, (cid:26) ✷ u = 0 u | t =0 = f ∂∂t u | t =0 = 0 (17) • The odd part sin t √ ∆ √ ∆ is the solution operator of the initial value problem, (cid:26) ✷ u = 0 u | t =0 = 0 ∂∂t u | t =0 = g (18)The kernels of cos t √ ∆ , sin t √ ∆ √ ∆ exhibit finite propagation speed of solutions of the waveequation, i.e. are supported inside the characteristic conoid C where r ≤ | t | . Cauchy problem for the half-wave equation.
The forward half-wave group is thesolution operator of the Cauchy problem( 1 i ∂∂t − √ ∆) u = 0 , u (0 , x ) = u . The solution is given by u ( t, x ) = U ( t ) u ( x ) , with U ( t ) = e it √ ∆ . The Schwartz kernel U ( t, x, y ) of the wave group U ( t ) = e it √ ∆ solves the pseudo-differentialCauchy problem(the half-wave equation), (cid:18) i ∂∂t − √ ∆ x (cid:19) U ( t, x, y ) = 0 , U (0 , x, y ) = δ y ( x ) . (19)Equivalently, it solves the wave equation with pseudo-differential initial condition, ✷ U = 0 ,U (0 , x, y ) = δ y ( x ) , ∂∂t U ( t, x, y ) | t =0 = i √ ∆ x δ x ( y ) . (20)The solution is given by U ( t, x, y ) = cos t √ ∆( x, y ) + i √ ∆ x sin t √ ∆ √ ∆ ( x, y ) . (21)Unlike the even/odd kernels, e it √ ∆ has infinite propagation speed, i.e. is non-zero outsidethe characteristic conoid C ; this is due to the second of its initial condition.The half wave group has the eigenfunction expansion, U ( t, x, y ) = X j e itλ j ϕ λ j ( x ) ϕ λ j ( y ) (22)on R × M × M , which converges in the sense of distributions. Fundamental solutions.
A fundamental solution of the wave equation is a solutionof ✷ E ( t, x, y ) = δ ( t ) δ x ( y ) . The right side is the Schwartz kernel of the identity operator on R × M .There exists a unique fundamental solution which is supported in the forward conoid C + = { ( t, x, y ) : t > , t − r ( x, y ) > } . called the advanced (or forward) propagator. It is given by E + ( t ) = H ( t ) sin t √ ∆ √ ∆ , where H ( t ) = t ≥ is the Heaviside step function. It is well-defined for any curved globallyhyperbolic spacetime, while Cauchy problems and propagators require a choice of “Cauchyhypersurface” like { t = 0 } .In [R] , the forward fundamental solution is constructed in terms of the holomorphic familyof Riesz kernels ( t − r ) α + Γ( α +1) , which are supported in the forward characteristic conoid C + . A morecontemporary treatment using the language of homogeneous distributions on R is given in[Be]. In [J] it is pointed out that the Riesz kernels are Schwartz kernels of complex powers ✷ α of the wave operator on R × M . Unlike complex powers of ∆, ✷ α is only uniquely definedif the Scwhartz kernels are assumed to be supported in C + .3.4. Hadamard-Feynman fundamental solution.
Hadamard and Feynman constructedanother fundamental solution which is a (branched) meromorphic function of ( t, x, y ) nearthe characteristic conoid with the singularity ( t − r ) − m − analogous to the Newtonianpotential r − n in the elliptic case (here m = n + 1 = dim R × M n .) and is not supported in C + . It corresponds to the inverse ( ✷ + i − rather than to the Riesz kernel ✷ − . It is theFor background in the case of R n × R we refer to [IZ] and for the general case we referto [DH]. Hadamard [H] defined this fundamental solution to be the branched meromorphicfundamental solution of ✷ , and referred to it as the ‘elementary solution. We review hisparametrix extensively in § Definition
The Hadamard-Feynman fundamental solution is the operator ( ✷ + i − = Z R e itτ (∆ − τ + i − dt on R × M . Proposition
As a family U F ( t ) of operators on L ( M ) it is given by U F ( t ) = e i | t |√ ∆ √ ∆ . Proof.
The proof is essentially the same as in the case of M = R n (see for instance [IZ]).Using the eigenfunction expansion(∆ − τ + i − = X j ( λ − τ + i − ϕ j ( x ) ϕ j ( y ) LURI-POTENTIAL THEORY ON GRAUERT TUBES 13 it suffices to show that Z R e itτ ( λ j − τ + i − dτ = e i | t | λ j λ j . The evaluation follows by a residue calculation. (cid:3)
Remark:
One can verify that U F ( t ) is a fundamental solution directly by applying ✷ .The analogous expressions for the advanced (resp. retarded) Green’s function are givenby G ret ( t, x, y ) = − X j (cid:18)Z R e − itτ (( τ + iε ) − λ j ) − dτ (cid:19) ϕ j ( x ) ϕ j ( y ) . Since (( τ + iε ) − λ j ) − = 12 λ j (cid:18) τ − λ j + iε − τ + λ j + iε (cid:19) we have G ret ( t, x, y ) = C n H ( t ) X j sin tλ j λ j ϕ j ( x ) ϕ j ( y ) . Fundamental solutions and half-wave propagator on R n . We illustrate the def-initions in the case of R n following [Ho]. We use the notation χ α + ( x ) = x α + Γ( α +1) . The ad-vanced/retarded fundamental solutions of ✷ on R n +1 = R t × R nx is given by E ± ( t, x, y ) = χ − ( n +1)2 ± (Γ) , where we use the Hadamard notation (13) (which unfortunately clashes with the Gammafunction).The Hadamard-Feynamn fundamental solution on R n +1 is a ramified (branched) holomor-phic fundamental solution U F ( t, x, y ) = (Γ + i − ( n +1)2 . (23)There is an associated fundamental solution corresponding to (Γ − i − ( n +1)2 . The half-wave propagator is constructed on R n by the Fourier inversion formula, U ( t, x, y ) = Z R n e i h x − y,ξ i e it | ξ | dξ. (24)The Poisson kernel (extending functions on R n to harmonic functions on R + × R n ) is thehalf-wave propagagor at positive imaginary times t = iτ ( τ > U ( iτ, x, y ) = R R n e i h x − y,ξ i e − τ | ξ | dξ = τ − n (cid:0) x − yτ ) (cid:1) − n +12 = τ ( τ + ( x − y ) )) − n +12 . (25)In the case of R n , the Poisson kernel analytically continues to t + iτ, ζ = x + ip ∈ C + × C n as the integral U ( t + iτ, x + ip, y ) = Z R n e i ( t + iτ ) | ξ | e i h ξ,x + ip − y i dξ, (26) which converges absolutely for | p | < τ. If we substitute τ → τ − it and let τ → U ( t, x, y ) = C n lim τ → it (( t + iτ )) − r ( x, y ) ) − n +12 , (27)for a constant C n depending only on the dimension. Remark:
We observe that the half-wave kernel differs from the Hadamard-Feynman fundamentalsolution not only in the power but also because the former uses powers of the quadratic form(( t + i − r ) while the latter uses ( t − r + i r = | x − y | , • C n ( t − r + i − n − is the Feynman fundamental solution. • C ′ n t (( t + i − r ) − n +12 is the solution operator kernel of the half-wave equation.This difference in the kernels holds for general ( M, g ). In the half-wave kernel, ( t + iε ) = t − ε + 2 itε and the imaginary part only has a fixed sign if we assume that t >
0. This inpart explains the | t | -dependence in the formula of Proposition 3.2.We observe that neither kernel has ‘finite propagation speed’, i.e. neither is supported inthe characteristic conoid.3.6. Subordination of the Poisson kernel to the heat kernel.
There is anotherstandard approach to the Poisson kernel based on the ‘subordination identity’ e − γ = √ π R ∞ e − u √ u e − γ u du. More generally, for any positive operator A , e − tA = t √ π Z ∞ e − t u e − uA u − / du. This recuces the construction of the Poisson kernel to the heat kernel, which is useful sincethere exists a well-known parametrix for the heat kernel (Levi, Minakshisundaram-Pleijel).We follow the exposition in [St] § III.2.The subordination identity follows from two further identities:( i ) Z R n e − πδ | t | e − πi h t,x i dt = δ − n/ e − π | x | δ , and ( ii ) e − γ = 1 √ π Z ∞ e − u √ u e − γ u du. Proof of (ii): We have e − γ = 1 √ π Z ∞−∞ e iγ · x x dx = 1 π Z ∞ e − u Z R e iγx e − ux dx. The formula (25) for the Poisson kernel can be obtained from the subordination identity, e − τ | ξ | = 1 √ π Z ∞ e − u √ u e − ( τ | ξ | )24 u du, LURI-POTENTIAL THEORY ON GRAUERT TUBES 15 giving U ( iτ, x, y ) = 1 √ π Z ∞ Z R n e − u √ u e − ( τ | ξ | )24 u e i h x − y,ξ i dξdu. Interchanging the order of integration then gives, U ( iτ, x, y ) = 1 √ π Z ∞ e − u √ u (cid:18)Z T x M e − ( τ | ξ | )24 u e i h x − y,ξ i dξ (cid:19) du. Substituting the formula for the heat kernel on R n , e − t ∆ = (4 πt ) − n/ e − | x − y | t , we get e − t √ ∆ = t √ π R ∞ e − t u e − u ∆ u − / du = t √ π R ∞ e − t u (4 πu ) − n/ e − | x − y | u u − / du. = C n τ R ∞ e − θ ( τ + | x − y | ) θ ( n − / dθ = C n τ ( τ + | x − y | ) − n +12 . In the last step we put θ = u . Wave kernels and Poisson kernels on spaces of constant curvature.
As furtherillustrations, we consider the Poisson-wave kernels on spaces of non-zero constant curvaturefollowing [T].3.7.1.
The sphere S n . Let A = q ∆ + ( n − ) . Then the Poisson operator e − tA is given by U ( iτ, ω, ω ′ ) = C n sinh t (cosh τ − cos r ( ω,ω ′ )) n +12 = C n ∂∂τ (cosh τ − cos r ( ω, ω ′ )) − n − . Here, r ( ω, ω ′ ) is the distance between points of S n . This formula is proved in [T] using thePoisson integral formula for a ball. Note that the addition of n − simplifies the formula andmakes the operator strictly positive.The wave kernel is the boundary value: e itA ( x, y ) = U ( t, x, y ) = C n sin t (cos( t + i − cos r ( ω, ω ′ ) − n +12 . We see the kernel structure emphasized in Remark 3.5.The analytically continued Poisson kernel is e ( − τ + it ) A ( ζ , ω ) = C n sinh t (cosh( τ + it ) − cos r ( ω, ζ )) − n +12 . It is singular on the complex characteristic conoidcosh( τ + it ) − cosh r ( ζ , ¯ ζ ′ ) = 0 . Wave kernel on Hyperbolic space.
On hyperbolic space we define A = q ∆ − ( n − ) ,which brings the continuous spectrum down to zero. The wave kernel on hyperbolic space isobtained [T] by analytic continuation of the wave kernel of sin tAA on the sphere by changing r → ir : lim ε → + − C n Im (cos( it − ε ) − cosh r ) − n − + It follows that the Poisson kernel is U ( iτ, x, y ) = sinh t (cosh( t + i − cosh r ) − n +12 . We again see the kernel structure emphasized in Remark 3.5. The Poisson kernel and wavekernels for e it √ ∆ rather than e itA are derived in [JL] for hyperbolic quotients using the sub-ordination method ( § √ ∆ rather than A leads to lowerorder terms.There is an alternative approach using Fourier analysis on hyperbolic space, where theexponential functions e ( iλ +1) h z,b i play the role of plane waves. Here, z is in the interior ofhyperbolic space and b lies on its boundary and h z, b i is the distance of the horospherethrough z, b to 0. For background we refer to [Hel]. We have, e itA ( x, y ) = Z ∞ e itλ (cid:26)Z B e ( iλ +1) h z,b i e ( − iλ +1) h w,b i db (cid:27) dp ( λ ) , where dp ( λ ) is the Planchere measure and db is the standard measure on the boundary (asphere). This formula is the analogue of (24). The inner integral over B is a spherical function ϕ λ ( r ( z, w )) and is the hyperbolic analogue of a Bessel function. The analytic continuationof the Poisson kernel e − τA ( x, y ) = Z ∞ e − τλ (cid:26)Z B e ( iλ +1) h z,b i e ( − iλ +1) h w,b i db (cid:27) dp ( λ ) , can be easily read off from this expression.4. The Hadamard-Feynman fundamental solution and Hadamard’sparametrix
In his seminal work [H], Hadamard constructed a solution of ✷ E = 0 for t > − m +22 , m = n + 1 = dim M × R (recall that Γ is defined by (13)).Note the analogy to the elliptic case where the Green’s function (the kernel of ∆ − ) has thesingularity r − n +2 if n > • The elementary solution in odd spacetime dimensions has the form U Γ − m − , where U = U + Γ U + · · · + Γ h U h + · · · is a holomorphic function. (This U is not the half-wave propagator!). LURI-POTENTIAL THEORY ON GRAUERT TUBES 17 • The elementary solution in even spacetime dimensions has the form U Γ − m − + V log Γ + W, where U = m − X j =0 U j Γ j , V = ∞ X j =0 V j Γ j , W = ∞ X j =1 W j Γ j . Hadamard’s formulae for the fundamental solutions pre-date the Schwartz theory of distri-butions. We follow his approach of describing the fundamental solutions as branched mero-morphic functions (possibly logarithmically branched) on complexified spacetime. In modernterms Γ α (resp. log Γ) would be defined as the distributions (Γ + i α (resp. log(Γ + i x + i Theorem (Hadamard, 1920) With the U j , V k , W ℓ defined as above, • In odd spacetime dimensions, there exists a formal series U as above so that E = U Γ − m solves ✷ E = δ ( t ) δ y ( x ) . If ( M, g ) is real analytic, the series U = P ∞ j =0 U j Γ j converges absolutely for | Γ | < ε sufficient small, i.e. near the characteristic conoidand admits a holomorphic continuation to a complex neighborhood of C C . • In even spacetime dimensions, E = U Γ − m + V log Γ + W solves ✷ E = δ ( t ) δ y ( x ) .If ( M, g ) is real analytic, all of the series for U, V, W converge for | Γ | small enoughand admit analytic continuations to a neighborhood of C C In the smooth case, the series do not converge. But if they are truncated at some j , thepartial sum defines a parametrix, i.e. a fundamental solution modulo functioins in C j . Bythe Levi sums method (Duhamel principle) the parametrix differs from a true fundamentalsolution by a C j kernels. We are mainly interested in real analytic ( M, g ) in this article anddo not go into details on the last point. We note that the singularities of the kernel are dueto the factors Γ − m , log Γ, which are branched meromorphic (and logarithmic) kernels. Theterms are explicitly evaluated in the case of hyperbolic quotients in [JL]. We also refer toChapter 5.2 of [Gar] for a somewhat modern presentation of the proof.It may be of interest to note that this construction only occupies a third of Hadamard’sbook [H]. The rest is devoted to the use of such kernels to solve the Cauchy problem, usingGreen’s formula applied to a domain obtained by intersecting the backward characteristicconoid from a point ( t, x ) of spacetime with the Cauchy hypersurface. The integrals overthe lightlike (null) part of the boundary caused serious trouble since the factors Γ − m areinfinite along them and need to be re-normalized. This was the origin of Hadamard’s finiteparts of divergent integrals. Riesz used analytic continuation methods instead to define theforward fundamental solution in [R].4.1. Sketch of proof of Hadamard’s construction.
Let Θ = p det( g jk ) be the volumedensity in normal coordinates based at y , dV = Θ( y, x ) dx . That is,Θ( x, y ) = (cid:12)(cid:12)(cid:12) det D exp − x ( y ) exp x (cid:12)(cid:12)(cid:12) . Fix x ∈ M and endow B ε ( x ) with geodesic polar coordinates r, θ . That is, use the chartexp − x : B r ( x ) → B ∗ x,r M combined with polar coordinates on T ∗ x M . Then g = 1 , g j = 0 for j = 2 , . . . , n . Also, dV = Θ( x, y ) dy = Θ( x, r, θ ) r n − drdθ . So the volume density J relativeto Lebesgue measure drdθ in polar coordinates is given by J = r n − Θ.In these coordinates,∆ = 1 J n X j,k =1 ∂∂x j (cid:18) J g jk ∂∂x k . (cid:19) = ∂ ∂r + J ′ J ∂∂r + L, where L involves no ∂∂r derivatives. Equivalently,∆ = ∂ ∂r + ( Θ ′ Θ + n − r ) ∂∂r + L, The first step in the parametrix construction is to find the phase function. Hadamardchooses to use Γ (13). In the Lortenzian metric, Γ satisfies ∇ Γ · ∇ Γ = 4Γ . (28)This is not the standard Eikonal equation σ ✷ ( dϕ ) = 0 of geometric optics, but rather hasthe form σ ✷ ( d Γ) = 4Γ . But Γ is a good phase, since the Lagrangian submanifold { ( t, d t Γ , x, d x Γ , y, − d y Γ) } is the graph of the bichafacteristic flow. This is because the d x r ( x, y ) is the unit vectorpointing along the geodesic joining x to y and d y r ( x, y ) is the unit vector pointing along thegeodesic pointing from y to x .To proceed, we introduce the simplifying notation M = ✷ Γ = − − r ( n − r − r Θ r Θ = 2 m + 2 r Θ r Θwhere m = n + 1. We then have, ✷ [ f (Γ) U j ] = ✷ [ f (Γ)] U j + 2 ∇ [ f (Γ)] ∇ U j + f (Γ) ✷ U j = ( f ′′ (Γ) ∇ (Γ) · ∇ (Γ) + f ′ (Γ) ✷ (Γ)) U j + 2 f ′ (Γ) ∇ Γ · ∇ U j + f (Γ) ✷ U j . In addition to (28), we further have ✷ Γ = 4 + J r J r ∇ Γ · ∇ = ∇ ( t − r ) · ∇ = 2( t ∂∂t + r ∂∂r ) = 2 s dds , where we recall that that we are using the Lorentz metric of signature + − −− . Here s = Γ,and the notation s dds refers to differentiation along a spacetime geodesic.We then have ✷ [ f (Γ) U j ] = (cid:0) f ′′ (Γ)(4Γ) + f ′ (Γ)(4 + J r J r )) (cid:1) U j + 2 f ′ (Γ)( − s dds U j ) + f (Γ) ✷ U j . We now apply this equation in the cases f = x − m + j . (and later to f = log x ), in whichcase f ′ = ( 2 − m j ) x − m + j − , f ′′ = ( 2 − m j )( 2 − m j − x − m + j − . LURI-POTENTIAL THEORY ON GRAUERT TUBES 19
We then attempt to solve ✷ Γ − m ∞ X j =0 U j Γ j ! = 0 (29)away from the characteristic conoid by seting the coefficient of each power Γ − m + j − of Γequal to zero. The resulting ‘transport equations’ are[ − (cid:0) ( − m + j )( − m + j − (cid:1) + ( − m + j )( − − J r J r ))+2 (cid:0) ( − m + j )( − s dds ] (cid:1) U j + ✷ U j − = 0 . They are impossible to solve for all j when m is even because the common factor ( − m + j )vanishes when j = m − . We thus first assume that m is odd so that it is non-zero for all j We then recursively solve Hadamard’s transport equations in even space dimensions,4 s dU k ds + ( M − m + 2 r J r J ) U k = − ✷ U k − . When k = 0 we get 2 s dU ds + 2 s Θ s Θ = 0 , which is solved by U = Θ − . The solution of the ℓ th transport equation is then, U ℓ = − U ℓs m + ℓ Z s U − s ℓ + m − ✷ U ℓ − ds. Hence we have a formal solution with the singularity of the Green’s function in the ellipticcase, and by comparison with the Euclidean case we see that it solves ✷ E = δ .We now consider the necessary modifications in the case of even dimensional spacetimes.In this case, Γ − m Γ j is always an integer power. If we could solve the transport equation for j = m − , the resulting term would be regular with power Γ . The problem is that Γ shouldactually be a term with a logarithmic singularity log Γ.Thus the parametrix (29) is inadequate in even spacetime dimensions. Hadamard thereforeintroduced a logarithmic term V log(Γ). By a similar calculation to the above, ✷ [(log Γ) V ] = (cid:0) − Γ − (4Γ) + Γ − ( − − J r J r ) (cid:1) V + 2Γ − ( − s dds V ) + log Γ ✷ V. Due to (28), all terms except the logarithmic term have the same singularity Γ − . On theother hand, the only way to eliminate the logarithmic term is to insist that ✷ V = 0. Wefurther assume that V = ∞ X j =0 V j Γ j . We then return to the unsolvable transport equations for U j for j ≥ m − , which nowacquires the new V term to become:[ − (cid:0) ( − m + j )( − m + j − (cid:1) + ( − m + j )( − − J r J r )) + 2 (cid:0) ( − m + j )( − s dds ] (cid:1) U j + ✷ U j − +Γ − [ (cid:0) − − J r J r ) (cid:1) + 2 dds ] V = 0 . When j = m − , everything cancels in the Γ − term except ✷ U m − . Hence, we drop the U j for j ≥ m − and assume the non logarithmic part is just the finite sum P m − j =0 U j Γ j . Butadding in the V term we get the transport equation, − s dV ds − r J r J V = − ✷ U m − . Here, U m − is known and we solve for V to get, V = − U s m Z s U − s m − ✷ U m − ds. The condition ✷ V = 0 imposed above then determines the rest of the coefficients V j , V ℓ = − U ℓs m + ℓ Z s U − s ℓ + m − ✷ V ℓ − ds. We now have two equations: the original ✷ ( U Γ − m U + V log Γ) = 0 and the new ✷ V = 0.By solving the transport equations for U , . . . , U m − , V , V j ( j ≥
1) we obtain a solution of aninhomogeneous equation of the form, ✷ ( U Γ − m + V log Γ) = X j =0 w j Γ j , where the right side is regular. To complete the construction, we add a new term of the form W = P ∞ ℓ =1 W ℓ ( r − t ) ℓ in order to ensure that ✷ m − X j =0 U j ( r − t ) − m + j + V log( r − t ) + W ) = 0away from the characteristic conoid. It then suffices to find W j so that ✷ ∞ X j =1 W j Γ j = X j =0 w j Γ j . This leads to more transport equations which are always solvable (by the Cauchy-Kowalevskayatheorem). This concludes the sketch of the proof of Theorem 4.1.4.2.
Convergence in the real analytic case.
The above parametrix construction wasformal. However, when the metric is real analytic, Hadamard proved that the formal seriesconverges for | t | and | Γ | sufficiently small. The convergence proof based on the method ofmajorants. Theorem [H] (see also [Gar] ) Assume that ( M, g ) is real analytic. Then there exists K > so that the Hadamard parametrix converges for any ( t, y ) such that t = 0 , r ( x, y ) <ε = inj ( x ) and | t − r | ≤ (cid:16) (1 − || y || ε (cid:17) (cid:16) m ε + m ε (cid:17) K , ( m = m −
22 ) . (30) LURI-POTENTIAL THEORY ON GRAUERT TUBES 21
It follows that the Hadamard fundamental solutions holomorphically extend to a neigh-borhood of C C as branched meromorphic functions iwith C C as branch locus. To obtain singlevalued distributions, one then needs to restrict the kernels to regions where a unique branchcan be defined.4.3. Away from C R . A further complication is that the fundamental solution has only beenconstructed in a neighborhood of C R . But it is known to be real analytic on ( R × M × M ) \C R and that it extends to a holomorphic kernel away from a neighborhood of C C . To prove thisit suffices to note that the analytic wave front set of the fundamental solution is C R . A moredetailed proof is given by Mizohata [Miz, Miz2] for ‘elementary solutions’, i.e. solutions E ( t, x, y ) of ✷ E = 0 such as cos t √ ∆ or sin t √ ∆ √ ∆ whose Cauchy data is either zero or a deltafunction. Here, ✷ operates in the x variable with y as a parameter. To analyze the wavekernels away from the characteristic conoid, Mizohata makes the decomposition E ( t, x, y ) = E N ( t, x, y ) + w N ( t, x, y ) + z N ( t, x, y ) , (31)where • ✷ E N = f N with f N ∈ C N − ( R × M × M ). Also, E N ( t, x, y ) | t =0 = δ y + a ( x, y ) with a ( x, y ) ∈ C ω ( M × M ) (in fact it is independent of y ); • ✷ w N ( t, x, y ) = − f N ( t, x, y ) , with w (0 , x, y ) = 0; • ✷ z N = 0 , z N (0 , x, y ) = − a ( x, y ).The same kind of decomposition applies to the Hadamard fundamental solution. The termconstructed by the parametrix method is E N . By solving the above equations, it is shownin [Miz] that the sum is analytic away from C R . One can see that the Hadamard method isonly a branched Laurent type expansion near C R by considering the examples in § Hadamard parametrix for the Poisson-wave kernel
We are most interested in the Hadamard parametrix for the half-wave kernel, which doesnot seem to have been discussed in the literature. We are more generally interested in thePoisson-wave semi-group e i ( t + iτ ) √ ∆ for τ >
0. The Poisson-wave lernel U ( t + iτ, x, y ) = X j e ( i ( t + iτ ) λ j ϕ j ( x ) ϕ j ( y ) (32)is a real analytic kernel which possesses an analytic extension to a Grauert tube. Thus, thereexists a non-zero analytic radius τ an > U ( t + iτ, ζ , y ) to M τ × M for τ ≤ τ an . Since U ( iτ ) ϕ λ = e − τλ ϕ C λ , (33)the eigenfunctions analytically extend to the same maximal tube as does U ( iτ ).We would like to construct a Hadamard type parametrix for (32). We may derive it fromthe Feynman-Hadamard fundamental solution in Proposition 3.2 using that ddt e i | t |√ ∆ √ ∆ = i sgn( t ) e i | t |√ ∆ (34) and e it √ ∆ = 1 i H ( t ) ddt e i | t |√ ∆ √ ∆ − i H ( − t ) ddt e − i | t |√ ∆ √ ∆ . (35)Hence, didt U F ( t ) = e it √ ∆ , ( t > . (36)The restriction to t > e it √ ∆ ( x, y ) has the singularity (( t + i − r ) − m (in odd spacetime dimensions) while U F ( t ) has the singularity ( t − r + i − m . We note (again) that (( t + i − r ) α = ( t − r + i α for t > Corollary
Let ( M, g ) be real analytic. Then with the U j , V k , W ℓ defined as in Theorem4.1, we have: • In odd spacetime dimensions, for t > the Poisson-wave kernel U ( t + iτ, x, y ) ( τ > ) has the form A Γ − m where A = P ∞ j =0 A j Γ j with A j holomorphic. The seriesconverges absolutely to a holomorphic function for | Γ | < ε sufficient small, i.e. nearthe characteristic conoid. • In even spacetime dimensions, for t > , the Poisson-wave kernel has the form B Γ − m + C log Γ + D where the coefficients B, C, D are holomorphic in a neighborhoodof C C , and have the same Γ expansions as A . We use this parametrix to prove Theorem 2 (1).5.1.
Hadamard parametrix as an oscillatory integral with complex phase.
Corol-lary 5.1 gives a precise description of the singularities of the Poisson-wave propagator. Itimplicitly describes the kernel as a Fourier integral kernel. We now make this description ex-plicit in the real domain. In the following sections, we extend the description to the complexdomain.We first express Γ − m + j as an oscillatory integral with one phase variable using the well-known identity Z ∞ e iθσ θ λ + dλ = ie iλπ/ Γ( λ + 1)( σ + i − λ − . (37)At least formally, this leads to the representation Z ∞ e iθ ( t − r ) θ n − − j + dθ = ie i ( n − − j ) π/ Γ( n − − j + 1)( t − r + i j − n − − for the principal term of the Poisson-wave. Here, the notation Γ = t − r unfortunatelyclashes with that for the Gamma function, and we temporarily write out its defintion.In even space dimensions, the Hadamard parametrix for the Hadamard-Feynman funda-mental solution thus has the form ∞ X j =0 U j ( t, x, y )Γ − n + j = Z ∞ e iθ ( t − r ) ∞ X j =0 U j ( t, x, y )( ie i ( n − − j ) π/ ) − θ n − − j + Γ( n − − j + 1) ! dθ. (38) LURI-POTENTIAL THEORY ON GRAUERT TUBES 23
Here we follow Hadamard’s notation, but it is simpler to re-define the coefficients U j so thatthe Γ-factors appear on the left side as in [Be] (7). We thus define U j ( t, x, y ) = (cid:18) ( ie i ( n − − j ) π/ ) − n − − j + 1) (cid:19) U j ( t, x, y ) . By the duplication formula Γ( z )Γ(1 − z ) = π sin πz with z = m − k − α , i.e.Γ( m − j − α − j π sin π ( m − α ) 1Γ( − m + 1 + j + α ) , it follows that U j ( t, x, y ) = (cid:18) ( − j π sin π ( m − α ) 1Γ( − m + 1 + j + α ) (cid:19) U j ( t, x, y ) , so that the formula in odd spacetime dimensions becomes C n π ( m − α ) P ∞ j =0 ( − j U j ( t,x,y )Γ( − m +1+ j + α ) ( t − r ) − m − + j = R ∞ e iθ ( t − r ) (cid:16)P ∞ j =0 U j ( t, x, y ) θ n − − j + (cid:17) dθ. (39)The amplitude in the right side of (39) is then a formal analytic symbol, A ( t, x, y, θ ) = ∞ X j =0 U j ( t, x, y ) θ n − − j + , (40)Due to the Gamma-factors appearing in the identity (37), convergence of the series on the leftside of (39) does not imply convergence of the series (40). However, there exists a realizationof the formal symbol (40) by a holmorphic symbol A ( t, x, y, θ ) = X ≤ j ≤ θeC U j ( t, x, y ) θ n − − j + , and one obtains an analytic parametrix U ( t, x, y ) = Z ∞ e iθ Γ A ( t, x, y, θ ) dθ (41)which approximates the wave kernel for small | t | and ( x, y ) near the diagonal up to a holo-morphic error, whose amplitude is exponentially decaying in θ . Here, we recall (see [Sj], p.3 and section 9) that a classical formal analytic symbol ([Sj], page 3) on a domain Ω ⊂ C n is a formal semi-classical series a ( z, λ ) = ∞ X k =0 a k ( z ) λ − k , where a k ( z, λ ) ∈ O (Ω) for all λ >
0. Then for some
C >
0, the a k ( z ) ∈ O (Ω) satisfy | a k ( z ) | ≤ C k +1 k k , k = 0 , , , . . . . A realization of the formal symbol is a genuine holomorphic symbol of the form, a ( z, λ ) = X ≤ k ≤ λeC a k ( z ) λ − k . It is an analytic symbol since, with the index restriction, | a k ( z ) λ − k | ≤ C Ω ( Ckλ ) k ≤ Ce − k . Hence the series converges uniformly on Ω to a holomorphic function of z for each λ .Returning to (40), the Hadamard-Riesz coefficients U j are determined inductively by thetransport equations Θ ′ U + ∂ U ∂r = 04 ir ( x, y ) { ( k +1 r ( x,y ) + Θ ′ ) U j +1 + ∂ U j +1 ∂r } = ∆ y U j . , (42)whose solutions are given by: U ( x, y ) = Θ − ( x, y ) U j +1 ( x, y ) = Θ − ( x, y ) R s k Θ( x, x s ) ∆ U j ( x, x s ) ds (43)where x s is the geodesic from x to y parametrized proportionately to arc-length and where∆ operates in the second variable.As discussed above, the representation (39) does not suffice when n is odd, since Γ( z ) and θ z + have poles at the negative integers. To rescue the representation when n is odd, we needto use the distributions θ − n + with n = 1 , , . . . , defined as follows (see [Ho] Vol. I): θ − k + ( ϕ ) = Z ∞ (log θ ) ϕ ( k ) ( θ ) dx/ ( k − ϕ ( k − (0)( k X j =1 /j ) / ( k − . This family behaves in an unusal way under derivation, ddθ θ − k + = − kθ − k − + ( − k δ ( k )0 /k !(see [Ho] Vol. I (3.2.2)”) and is therefore sometimes avoided in the Hadamard-Riesz parametrixconstruction (as in [Be]).However, we have already constructed the parametrices and only want to express themin terms of the above oscillatory integrals to make contact with Fourier integral operatortheory. In odd space dimensions, the Hadamard parametrices can be written in the form R ∞ e iθ Γ (cid:0) U ( t, x, y ) θ m + + · · · + U m θ (cid:1) dθ + R ∞ e iθ Γ (cid:0) U m +1 θ − + U m +2 θ − + · · · (cid:1) dθ (44)Again the amplitude is a formal symbol. To produce a genuine amplitude it needs to bereplace by a realization which approximates it modulo a holomorphic symbol which is expo-nentially decaying in θ .We are paying close attention to the regularization of the integral at θ = 0, but only thebehavior of the amplitude as θ → ∞ is relevant to the singularity. The terms with θ − k + for LURI-POTENTIAL THEORY ON GRAUERT TUBES 25 k > θ = 0, we obtaindistributions of the form u µ (Γ) = Z R e iθ Γ χ ( θ ) θ µ dθ where χ ( θ ) = 1 for θ ≥ χ ( θ ) = 0 for θ ≤ . Then u − k (Γ) = i k +1 Γ k − log Γ , modulo C ∞ , u − k ( − Γ) = ( − i ) n +1 Γ n − log Γ . Hence the terms with negative powers of θ + in (44) produce the logarithmic terms and theholomorphic terms.Above, we discussed the Hadamard-Feynman fundamental solution, but for t > t (according to Proposition 5.1 ) to obtain the parametrices forthe Poisson-wave group. Away from the characteristic conoid the Schwartz kernels of thePoisson-wave group and Hadamard-Feynman fundamental solution are holomorphic by thetheorem on propagation of analytic wave front sets [Ho, Sj] (see also [Miz, Miz2]). TheFourier integral structure and mapping properties follow immediately from the order of theamplitude and from the exact formula for the phase.5.2. Modified Hadamard parametrix with phase t − r when t ≥ . We now assumethat t ≥ τ >
0. We note that the phase t − r for U ( t, x, y ) factors as ( t − r )( t + r )with t + r ≥ t ≥
0. Of course, r ( x, y ) ≥ θ → ( t + r ) θ in (41) to obtain the modified Hadamard parametrix, U ( t, x, y ) = ( t + r ) − Z ∞ e iθ ( t − r ) A ( t, x, y, θt + r ) dθ, ( t ≥
0) (45)with phase t − r . We note that it is singular when r = 0 but we only intend to use it for r C = 0. The amplitude = ( t + r ) − A ( t, x, y, θt + r ) is a representative of an analytic symbol aslong as r + t = 0 and r = 0.6. H¨ormander parametrix for the Poisson-wave kernel
A more familiar construction of U ( t, x, y ) and its analytic continuation which is particularlyuseful for small | t | is the one (24) based on the Fourier inversion formula. Its generalizationto Riemannian manifolds is given by U ( t, x, y ) = Z T ∗ y M e it | ξ | gy e i h ξ, exp − y ( x ) i A ( t, x, y, ξ ) dξ, (46)for ( x, y ) sufficiently close to the diagonal. We use this parametrix to prove Theorem 2 (2).The amplitude is a polyhomogeous symbol of the form A ( t, x, y, ξ ) ∼ ∞ X j = A j ( t, x, y, ξ ) , (47)where the asymptotics are in the sense of the symbol topology and where A j ( t, x, y, τ ξ ) = τ − j A j ( t, x, y, ξ ) , for | ξ | ≥ . The principal term A ( t, x, y, ξ ) equals 1 when t = 0 on the diagonal, and the higher A j aredetermined by transport equations discussed in [DG]. It can be verified that in the case of real analytic (
M, g ), the amplitude is a classical formalanalytic symbol (see § A ( t, x, y, ξ ) is a realization of the amplitude A ( t, x, y, ξ ),then one obtains an analytic parametrix U ( t, x, y ) = Z T ∗ y M e it | ξ | gy e i h ξ, exp − y ( x ) i A ( t, x, y, ξ ) dξ, (48)which approximates the wave kernel for small | t | and ( x, y ) near the diagonal up to a holo-morphic error, whose amplitude is exponentially decaying in | ξ | .6.1. Subordination to the heat kernel.
The parametrix (53) can also be obtained bysubordinating the Poisson-wave kernel to the heat kernel in the sense of § M τ . This analytic continuation wasstudied by Golse-Leichtnam-Stenzel in [GLS]., who proved the following: For any x ∈ M there exists ε, ρ > W of x in M ε such that for 0 < t < x, y ) ∈ W × W , E ( t, x, y ) = N ( t, x, y ) e − r x,y )4 t + R ( t, x, y ) , where N ( t, x, y ) = X ≤ j ≤ Ct W j ( x, y ) t j as t ↓ + where W j ( x, y ) are the Hadamard-Minakshisundaram-Pleijel heat kernel coeffi-cients. is an analytic symbol of ordern n/ t − in the sense of [Sj]. As above,the remainder is exponentially small, | R ( t, x, y ) | ≤ Ce − ρ t with a uniform C in ( x, y ) as t ↓ + . The heat kernel itself obviously admits a holmoorphicextension in the open subset Re r C ( x, y ) > M C × M C .7. Complexified Poisson kernel as a complex Fourier integral operator
We now consider the Fourier integral operator aspects of the analytic continuation of thePoisson-wave kernel U ( t + iτ, ζ , y ) for τ > ζ , y ) ∈ M τ × M , developing analoguesof the results of § O ( M ε )the space of holomorphic functions on the Grauert tube and by a slight abuse of notation wealso denote by O ( ∂M ε ) the CR holomorphic functions on the boundary ∂M ε of the strictlypseudo-convex domain M ε (the null space of the boundary Cauchy-Riemann operator ¯ ∂ b .)In particular, we denote by O ( ∂M ε ) = H ( ∂M ε ) the Hardy space of boundary values ofholomorphic functions of M ε which lie in L ( ∂M ε ) relative to the natural Liouville measure dµ τ = ( i∂ ¯ ∂ √ ρ ) m − ∧ d c √ ρ. (49)We further denote by O s + n − ( ∂M τ ) the Sobolev spaces of CR holomorphic functions on ∂M τ , i.e. O s + m − ( ∂M τ ) = W s + m − ( ∂M τ ) ∩ O ( ∂M τ ) , (50)where W s is the s th Sobolev space.The spray Σ τ = { ( ζ , rd c √ ρ ( ζ ) : r ∈ R + } ⊂ T ∗ ( ∂M τ ) (51) LURI-POTENTIAL THEORY ON GRAUERT TUBES 27 of the contact form d c √ ρ defines a symplectic cone. There exists a symplectic equivalence(cf. [GS2]) ι τ : T ∗ M − → Σ τ , ι τ ( x, ξ ) = ( E ( x, τ ξ | ξ | ) , | ξ | d c √ ρ E ( x,τ ξ | ξ | ) ) . (52)The following theorem is stated in [Bou] (see also [Z3]): Theorem (see [Bou, GS2, GLS] ) For sufficiently small τ > , U C ( iτ ) : L ( M ) →O ( ∂M τ ) is a Fourier integral operator of order − m − with complex phase associated to thecanonical relation Λ = { ( y, η, ι τ ( y, η ) } ⊂ T ∗ M × Σ τ . Moreover, for any s , U C ( iτ ) : W s ( M ) → O s + m − ( ∂M τ ) is a continuous isomorphism. The proof of Theorem 7.1 is barely sketched in [Bou]. However, the theorem follows almostimmediately from the construction of the branched meromorphic Hadamard parametrix inCorollary 5.1, or alternatively from the analytic continuation of the H¨ormander parametrixof § U C ( iτ, ζ , y ), i.e. differs from it byan analytic kernel (smooth would be sufficient by analytic wave front set considerations).But the Hadamard parametrix construction is an exact formula and actually gives a moreprecise description of the singularities of U C ( iτ, ζ , y ) than is stated in Theorem 7.1. Webriefly explain how either the Hadamard or H¨ormander parametrix can be used to completethe proof.7.1. Fourier integral distributions with complex phase.
First, we review the relevantdefinitions (see [Ho] IV, § X is a distribution that can locally be represented by an oscillatory integral A ( x ) = Z R N e iϕ ( x,θ ) a ( x, θ ) dθ where a ( x, θ ) ∈ S m ( X × V ) is a symbol of order m in a cone V ⊂ R N and where the phase ϕ is a positive regular phase function, i.e. it satisfies • Im ϕ ≥ • d ∂ϕ∂θ , . . . , d ∂ϕ∂θ N are linearly independent complex vectors on C ϕ R = { ( x, θ ) : d θ ( x, θ ) = 0 } . • In the analytic setting (which is assumed in this article), ϕ admits an analytic con-tinuation ϕ C to an open cone in X C × V C .Define C ϕ C = { ( x, θ ) ∈ X C × V C : ∇ θ ϕ C ( x, θ ) = 0 } . Then C ϕ C is a manifold near the real domain. One defines the Lagrangian submanifoldΛ ϕ C ⊂ T ∗ X C as the image ( x, θ ) ∈ C ϕ C → ( x, ∇ x ϕ C ( x, θ )) . Analytic continuation of the Hadamard parametrix.
As in § § U C ( iτ, ζ , y ) as a local Fourier integral distribution with complex phase by rewritingthe Hadamard series in Corollary 5.1 as oscillatory integrals. Here we assume that τ > , t ≥
0. A complication is that we can only use the complexified phase Γ = t − r in regions ofcomplexified R × M × M where its imaginary part is ≥
0. As in § t − r (resp. t + r ) in regions where t + r = 0 (resp. t − r = 0) and where the contour R + can be deformed back to itself after the the change of variables θ → ( t + r ) θ .7.3. Analytic continuation of the H¨ormander parametrix.
As was the case in R n (26), the parametrix (48) admits an analytic continuation in time to a strip { t + iτ : τ <τ an , | t | < } . In the space variables, the parametrix then admits an analytic continuation tocomplex x, y satisfying | r C ( x, y ) | ≤ τ. The analytically continued parametrix (53) approximates the true analytically continuedPoisson kernel up to a holomorphic kernel. More preicsely, for any x ∈ M and τ > ε, ρ > W of x in M τ such that for | t | < x, y ) ∈ W × W , U ( t + iτ, x, y ) = Z T ∗ y M e − τ | ξ | gy e i h ξ, exp − y ( x ) i A ( t + iτ, x, y, ξ ) dξ + R ( t, x, y ) , (53)where R ( t, x, y ) is holomorphic for small | t | and for ( x, y ) near the diagonal.The parametrix is only defined near the diagonal where exp − y is defined. However one canextend it to a global holomorphic kernel away from C C by cutting off the first term of (53)with a smooth cutoff χ ( x, y ) supported near the diagonal in M τ × M τ and then solving a ¯ ∂ problem on the Grauert tube (or a ¯ ∂ b problem on its boundary) to extend the kernel to beglobally holmorphic (resp. CR). We refer to [Z1] for a more detailed discussion. This givesan alternative to the Hadamard parametrix construction of Corollary 5.1.This concludes the sketch of proof of Theorem 7.1.8. Tempered spectral projector and Poisson semi-group as complexFourier integral operators
To study the tempered spectral projection kernels (2), we further need to continue U C ( t, ζ , y )anti-holomorphically in the y variable. The discussion is similar to the holomorphic case ex-cept that we need to double the Grauert tube radius to obtain convergence. We thus have, U C ( t + 2 iτ, ζ , ¯ ζ ) = P j e ( − τ + it ) λ j | ϕ C j ( ζ ) | = R R e itλ d λ P τ [0 ,λ ] ( ζ , ¯ ζ ) . (54)Properties of these kernels may be obtained from kernels which are analytically continuedin one variable only from the formula, U C ( t + 2 iτ, ζ , ¯ ζ ′ ) = R M U ( t + iτ, ζ , y ) U C ( iτ, y, ¯ ζ ′ ) dV g ( x )= P j e ( − τ + it ) λ j ϕ C j ( ζ ) ϕ C j ( ζ ′ ) . (55)We have, LURI-POTENTIAL THEORY ON GRAUERT TUBES 29
Proposition
For small t, τ > and for sufficiently small τ ≥ √ ρ ( ζ ) > , thereexists a realization B ( t, ζ , ¯ ζ, θ ) of a formal analytic symbol B ( t, ζ , ¯ ζ, θ ) so that as tempereddistributions on R × M τ , U C ( t + 2 iτ, ζ , ¯ ζ ) = Z ∞ e iθ (( t +2 iτ ) − i √ ρ ( ζ )) B ( t, ζ , ¯ ζ, θ ) dθ + R ( t + 2 iτ, ζ , ¯ ζ ) , (56) where R ( t +2 iτ, ζ , ¯ ζ ) is the restriction to the anti-diagonal of a holomorphic kernel. Moreover • θ (( t + 2 iτ ) − i √ ρ ( ζ )) is a phase of positive type. • If √ ρ ( ζ ) < τ the entire kernel is locally holomorphic. • If √ ρ ( ζ ) = τ then U C ( t + 2 iτ, ζ , ¯ ζ ) = Z ∞ e iθt B ( t, ζ , ¯ ζ, θ ) dθ + R ( t + 2 iτ, ζ , ¯ ζ ) . (57) Proof.
We use the Hadamard parametrix (Corollary 5.1) for U ( t + 2 iτ, ζ , ¯ ζ ) and use (8) tosimplify the phase, i.e. we writeΓ( t + 2 iτ, ζ , ¯ ζ ) = ( t + 2 iτ − i √ ρ )( t + 2 iτ + 2 i √ ρ )in the Hadamard parametrix in Corollary 5.1. The factors of ( t + 2 iτ + 2 i √ ρ ) are non-zero when τ > B to distinguish it from the amplitude in Corollary 5.1. We then express eachterm as a Fourier integral distribution of complex type with phase t + 2 iτ − i √ ρ . It ismanifestly of positive type. On ∂M τ , t + 2 iτ − i √ ρ simplifies to t . (cid:3) Complexified wave group and Szeg˝o kernels.
As in [Z3] it will also be necessaryfor us to understand the composition U C ( iτ ) ∗ U C ( iτ ). In this regard, it is useful to introducethe Szeg˝o kernels Π τ of M τ , i.e. the orthogonal projectionsΠ τ : L ( ∂M τ , dµ τ ) → H ( ∂M τ , dµ τ ) , (58)where dµ τ is the natural volume form (49). Here as above, H ( ∂M τ , dµ τ ) is the Hardyspace of boundary values of holomorphic functions in M τ which belong to L ( ∂M τ , dµ τ ). Itis simple to prove that the restrictions of { ϕ C λ j } to ∂M τ is a basis of H ( ∂M τ , dµ τ ). TheSzeg˝o projector Π τ is a complex Fourier integral operator with a positive complex canonicalrelation. The real points of its canonical relation form the graph ∆ Σ of the identity map onthe symplectic cone Σ τ ⊂ T ∗ ∂M τ (51). We refer to [Z3] for further background. We onlyneed the first statement in the following: Lemma
Let Ψ s ( X ) denote the class of pseudo-differential operators of order s on X .Then, • U C ( iτ ) ∗ U C ( iτ ) ∈ Ψ − m − ( M ) with principal symbol | ξ | − ( m − ) g . • U C ( iτ ) ◦ U C ( iτ ) ∗ = Π τ A τ Π τ where A τ ∈ Ψ m − ( ∂M τ ) has principal symbol | σ | ( m − ) g asa function on Σ τ . Proof.
This follows from Proposition 7.1. The first statement is a special case of the followingLemma from [Z3] (Lemma 3.1): Let a ∈ S ( T ∗ M − < τ < τ max ( g ), wehave: U ( iτ ) ∗ Π τ a Π τ U ( iτ ) ∈ Ψ − m − ( M ) , with principal symbol equal to a ( x, ξ ) | ξ | − ( m − ) g . The second statement follows from Theorem 7.1 and the composition theorem for complexFourier integral operators. We do not use it in this article and refer to [Z1] for the proof.We note that U C ( iτ ) ◦ U C ( iτ ) ∗ ( ζ , ζ ′ ) = X j e − τλ j ϕ C λ j ( ζ ) ϕ C λ j ( ζ ′ ) . (59) (cid:3) One term local Weyl law
In this section, we prove Theorem 2 (1). To prove the local Weyl law we employ paramet-rices for the Poisson-wave kernel adapted to e i ( t + iτ ) √ ∆ for τ > Proof of the local Weyl law.
Proof.
As in the real domain, we obtain asymptotics of P τ [0 ,λ ] ( ζ , ¯ ζ ) by the Fourier-Tauberianmethod of relating their asymptotics to the singularities in the real time t of the Fouriertransform (54). We refer to [SV] (see also the Appendix of [Z1]) for background on Tauberiantheorems. We follow the classical argument of [DG], Proposition 2.1, to obtain the local Weyllaw with remainder one degree below the main term.The proof is based on the oscillatory integral representation of Proposition 8.1. We areworking in the case where √ ρ ( ζ ) = τ and hence can simplify it to (57).We then introduce a cutoff function ψ ∈ S ( R ) with ˆ ψ ∈ C ∞ supported in sufficiently smallneighborhood of 0 so that no other singularities of U C ( t + 2 iτ, ζ , ¯ ζ ) lie in its support. Wealso assume ˆ ψ ≡ θ → λθ andapply the complex stationary phase to the integral, R R ˆ ψ ( t ) e − iλt U C ( t + 2 iτ, ζ , ¯ ζ ) dt = R R R ∞ ˆ ψ ( t ) e − iλt e iθt (cid:0) B ( t, ζ , ¯ ζ, θ ) dθ + R ( t + 2 iτ, ζ , ¯ ζ )) (cid:1) dt. (60)The second R term can be dropped since it is of order λ − M for all M >
0. In the firstwe change variables θ → λθ to obtain a semi-classical Fourier integral distribution of realtype with phase e iλt ( θ − . The critical set consists of θ = 1 , t = 0. The phase is clearlynon-degenerate with Hessian determinant one and inverse Hessian operator D θ,t . Takinginto account the factor of λ − from the change of variables, the stationary phase expansiongievs X j ψ ( λ − λ j ) e − τλ j | ϕ C j ( ζ ) | ∼ ∞ X k =0 λ n − − k ω k ( τ ; ζ ) (61)where the coefficients ω k ( τ, ] ζ ) are smooth for ζ ∈ ∂M τ . However the coefficients are notuniform as τ → + due to the factors of ( t +2 iτ +2 i √ ρ ( ζ )) which were left in the denominatorsof the modified Hadamard parametrix. Since t = 0 at the stationary phase point, the LURI-POTENTIAL THEORY ON GRAUERT TUBES 31 resulting expansion is equivalent to one with the large parameter τ λ (or √ ρ ( ζ ) λ ). Theuniform expansion is then X j ψ ( λ − λ j ) e − τλ j | ϕ C j ( ζ ) | ∼ ∞ X k =0 (cid:18) λτ (cid:19) n − − k ω k ( ζ , ¯ ζ ) , (62)where ω j are smooth in ζ , and ω = 1. The remainder has the same form.To complete the proof, we apply the Fourier Tauberian theorem (see the Appendix ([SV]):Let N ∈ F + and let ψ ∈ S ( R ) satisfy the conditions: ψ is even, ψ ( λ ) > λ ∈ R ,ˆ ψ ∈ C ∞ , and ˆ ψ (0) = 1. Then, ψ ∗ dN ( λ ) ≤ Aλ ν = ⇒ | N ( λ ) − N ∗ ψ ( λ ) | ≤ CAλ ν , where C is independent of A, λ . We apply it twice, first in the region √ ρ ( ζ ) ≥ Cλ − andsecond in the complementary region.In the first region, we let N τ,ζ ( λ ) = P τ,λ ( ζ , ¯ ζ ). It is clear that for √ ρ = τ , N τ,ζ ( λ ) is amonotone non-decreasing function of λ of polynomial growth which vanishes for λ ≤
0. For ψ ∈ S positive, even and with ˆ ψ ∈ C ∞ ( R ) and ˆ ψ (0) = 1, we have by (62) that ψ ∗ dN τ,ζ ( λ ) ≤ C (cid:18) λτ (cid:19) n − , (63)where C is independent of ζ , λ . It follows by the Fourier Tauberian theorem that N τ,ζ ( λ ) = N τ,ζ ( λ ) ∗ ψ ( λ ) + O (cid:18) λτ (cid:19) n − . Further, by integrating (62) from 0 to λ we have N τ,ζ ( λ ) ∗ ψ ( λ ) = (cid:18) λτ (cid:19) n − (cid:18) λ n − + 1 + O (1) (cid:19) , proving (1).To obtain uniform asymptotics in τ down to τ = 0, we use instead the analytic continuationof the H¨ormander parametrix (53). We choose local coordinates near x and write exp − x ( y ) =Ψ( x, y ) in these local coordinates for y near x , and write the integral T ∗ y M as an integralover R m in these coordinates. The holomorphic extension of the parametrix to the Grauerttube | ζ | < τ at time t + 2 iτ has the form U C ( t + 2 iτ, ζ , ¯ ζ ) = Z R n e ( it − τ ) | ξ | gy e i h ξ, Ψ( ζ, ¯ ζ ) i A ( t, ζ , ¯ ζ, ξ ) dξ. (64)Again, we use a cutoff function ψ ∈ S ( R ) with ˆ ψ ∈ C ∞ supported in sufficiently smallneighborhood of 0 so that no other singularities of E ( t + 2 iτ, ζ , ¯ ζ ) lie in its support and sothat ˆ ψ ≡ R R ˆ ψ ( t ) e − iλt U C ( t + 2 iτ, ζ , ¯ ζ ) dt = λ m R ∞ R R ˆ ψ ( t ) e − iλt R S n − e ( it − τ ) λr e irλ h ω, Ψ( ζ, ¯ ζ ) i A ( t, ζ , ¯ ζ, λrω ) r n − drdω. (65) We then apply complex stationary phase to the drdt integral, regarding Z S n − e irλ h ω, Ψ( ζ, ¯ ζ ) i A ( t, ζ , ¯ ζ, λrω ) r m − dω as the amplitude. When √ ρ ( ζ ) ≤ Cλ the exponent is bounded in λ and the integral definesa symbol. Applying stationary phase again to the dtdθ integral now gives X j ψ ( λ − λ j ) e − τλ j | ϕ C j ( ζ ) | ∼ ∞ X k =0 λ n − − k ω k ( ζ , ¯ ζ ) , (66)where ω k ( ζ , ¯ ζ ) is smooth down to the zero section.We apply the Fourier Tauberian theorem again, but this time with the estimates ψ ∗ dN τ,ζ ( λ ) ≤ Cλ n − , where C is independent of ζ . We conclude that N τ,ζ ( λ ) = Cλ n + O ( λ n − ) , proving (2). (cid:3) Corollary
For all ζ ∈ M C , and with τ = √ ρ ( ζ ) , cλ n +12 ≤ P τ [0 ,λ ] ( ζ , ¯ ζ ) ≤ Cλ n . Proof of Corollary 3.
Proof.
For the upper bound, we use thatsup ζ ∈ ∂M τ | ϕ C λ ( ζ ) | ≤ sup ζ ∈ ∂M τ Π I λ ( ζ , ζ ) | ≤ sup ζ ∈ ∂M τ e λ √ ρ ( ζ ) | P I λ ( ζ ) | . The upper bound stated in Corollary 3 then follows from Corollary 9.1 to Theorem 2.For the lower bound in (2) of Corollary 3, we use that || ϕ C j || L ( ∂M τ ) = e τ j h U ( iτ ) ∗ U ( iτ ) ϕ j , ϕ j i L ( M ) . By Lemma 8.2, the operator U ( iτ ) ∗ U ( iτ ) is an elliptic pseudodifferential operator of order µ = − n − (or so). Let C > h ξ i µ . Then by Garding’sinequality, h U ( iτ ) ∗ U ( iτ ) ϕ j , ϕ j i L ( M ) ≥ Cλ − µj , and so || ϕ C j || L ( ∂M τ ) ≥ Cλ − µj e τλ j . (67) (cid:3) Siciak extremal functions: Proof of Theorem 1 (1)
In this section we prove Theorem 1. First we prove a pointwise local Weyl law in thecomplex domain.
LURI-POTENTIAL THEORY ON GRAUERT TUBES 33
Proof of Theorem 1(2).
This follows from Theorem 2 together with the following
Lemma [Z4]
For any τ = √ ρ ( ζ ) > , and for any δ > , √ ρ ( ζ ) − log | δ | λ + O ( log λλ ) ≤ λ log Π [0 ,λ ] ( ζ , ¯ ζ ) ≤ √ ρ ( ζ ) + O ( log λλ ) hence lim λ →∞ λ log Π [0 ,λ ] ( ζ , ¯ ζ ) = 2 √ ρ ( ζ ) . Proof.
For the upper bound, we use thatΠ [0 ,λ ] ( ζ , ¯ ζ ) ≤ e λ √ ρ ( ζ ) P j : λ j ∈ [0 ,λ ] e − √ ρ ( ζ ) λ j | ϕ C λ j ( ζ ) | = e λ √ ρ ( ζ ) P ]0 ,λ ] ( ζ , ¯ ζ ) . . (68)We then take λ log of both sides and apply Theorem 2 to conclude the proof.The lower bound is subtler for reasons having to do with the distribution of eigenvalues(see the Remark below). It is most natural to prove two-term Weyl asymptotics for P [0 ,λ ] ( ζ , ¯ ζ )and to deduce Weyl asymptotics for short spectral intervals [ λ, λ + 1]. But that requires ananalysis of the singularity of the trace of the complexified wave gropup for longer times thana short interval around t = 0 and we postpone the more refined analysis until [Z1].Instead we use the longer intervals [(1 − δ ) λ, λ ] for some δ >
0. We clearly have e − δ ) λ √ ρ ( ζ ) X j : λ j ∈ [(1 − δ ) λ,λ ] e − √ ρ ( ζ ) λ j | ϕ C λ j ( ζ ) | ≤ Π [0 ,λ ] ( ζ , ¯ ζ ) (69)By Theorem 2, P j : λ j ∈ [(1 − δ ) λ,λ ] e − √ ρ ( ζ ) λ j | ϕ C λ j ( ζ ) | = P [0 ,λ ] ( ζ , ¯ ζ ) − P [0 , (1 − δ ) λ ] ( ζ , ¯ ζ )= C n ( τ )[1 − (1 − δ ) n ] λ n +12 + O ( λ n − )Taking λ log then gives λ log Π [0 ,λ ] ( ζ , ¯ ζ ) ≥ − δ ) √ ρ ( ζ ) − | log δ | λ + O ( log λλ ) . It follows that for all δ > λ →∞ λ log Π [0 ,λ ] ( ζ , ¯ ζ ) ≥ − δ ) √ ρ ( ζ ) . The conclusion of the Lemma follows from the fact that the left side is independent of δ. (cid:3) Remark:
The problematic issue in the lower bound is the width of I λ . If ( M, g ) is aZoll manifold, the eigenvalues of √ ∆ form clusters of width O ( λ − ) around an arithmeticprogression { k + β } for a certain Morse index β . Unless the intervals I λ are carefully centeredaround this progression, P I λ could be zero. Hence we must use long spectral intervals if wedo not analyze the long time behavior of the geodesic flow; for short ones no general lowerbound exists. Proof of Theorem 1 (1).
Proof.
We need to show thatΠ C I λ ( ζ , ¯ ζ ) = sup {| ϕ ( ζ ) | : ϕ = X j : λ j ∈ I a j ϕ C λ j , || a || = 1 } . We define the ‘coherent state’, Φ zλ ( w ) = Π C I λ ( w, ¯ z ) q Π C I λ ( z, ¯ z ) , satisfying, Φ zλ ( w ) = X j : I λ a j ϕ C j ( w ) , a j = ϕ C j ( ζ ) q Π C I λ ( z, ¯ z ) , X j | a j | = 1 . Hence, Φ ζI λ is a competitor for the sup and since | Φ ζI λ ( ζ ) | = Π I λ ( ζ , ¯ ζ ) one hasΠ C I λ ( ζ , ¯ ζ ) ≤ sup {| ψ ( ζ ) | : ψ = X j : λ j ∈ I a j ϕ C j , || a || = 1 } . On the other hand, by the Schwartz inequality for ℓ , for any ψ = P j : λ j ∈ I a j ϕ C j one has | X j : λ j ∈ I a j ϕ C j | = |h a, ψ i| ≤ || a || X | ϕ C j | = Π I λ ( ζ , ¯ ζ )and one has Π C I ( ζ , ¯ ζ ) ≥ sup {| ψ ( ζ ) | : ψ = X j : λ j ∈ I a j ϕ C j , || a || = 1 } . (cid:3) Remark:
Since N ( I λ ) ∼ λ m − ,1 λ log Π I λ ( ζ , ¯ ζ ) = 1 λ log X j : λ j ∈ I λ | ϕ C λ j ( ζ ) | = max j : λ j ∈ I λ { λ log | ϕ C λ j ( ζ ) | } + O ( log λλ ) . We recall (see [Z3]) that a sequence of eigenfunctions is called ergodic if h Aϕ j , ϕ j i → µ ( S ∗ g M ) R S ∗ g M σ A dµ . The complexified eigenfunctions then satisfy λ j log | ϕ j ( ζ ) | → √ ρ ( ζ ). Itfollows that ergodic eigenfunctions are asymptotically maximal, i.e. have the same logarith-mic asymptotics as Φ λM .10.3. Remarks on more general extremal PSH functions.
We can define a moregeneral Siciak extremal function of a subset E ⊂ M τ by,Φ λE ( z ) = sup {| ψ ( z ) | /λ : ψ ∈ H λ ; k ψ k E ≤ } , and Φ E ( z ) = sup λ Φ λE ( z ) . It would be interesting to determine this function and the associated equilibrium measureof E , i.e. Monge-Amp`ere mass of V ∗ E . LURI-POTENTIAL THEORY ON GRAUERT TUBES 35
This is of interest even when E ⊂ M (i.e. is totally real). Suppose that instead oforthonormalizing the eigenfunctions ϕ j on M , we orthonormalize them on a ball B ⊂ M .Let { ϕ Bλ j ( x ) } be the resulting orthonormal basis. We have simply changed the inner productto R B f f dV g . We then obtain a spectral projections kernelΠ B [0 ,λ ( x, y ) := X j : λ j ≤ λ ϕ Bλ j ( x ) ϕ Bλ j ( y ) . (70)The growth of Π B [0 ,λ ] ( ζ , ¯ ζ ) determines doubling estimates for eigenfunctions. Its exponentialgrowth rate should be that of the associated pluri-complex Green’s function log Φ B ( z ) =lim λ →∞ λ log Π B [0 ,λ ( ζ , ¯ ζ ). It would be interesting to determine this analogue of √ ρ . Its Monge-Amp`ere mass should concentrate on B , so should be the metric delta-function on B .11. Analytic continuation of eigenfunctions
In this section, we briefly review some results about analytic continuations of eigenfunc-tions to Grauert tubes and then prove Proposition 1. A more detailed analysis will appearin [Z1, Z5].A function f on a real analytic manifold M is real analytic, f ∈ C ω ( M ), if and only if itsatisfies the Cauchy estimates | D α f ( x ) | ≤ K L | α | α ! (71)for some K, L >
0. In place of all derivatives it is sufficient to use powers of ∆. In thelanguage of Baouendi-Goulaouic [BG, BG2, BG3], the Laplacian of a compact real analyticRiemannian manifold has the property of iterates, i.e. the real analytic functions are preciselythe functions satisfying Cauchy estimates relative to ∆, C ω ( M ) = { u ∈ C ∞ ( M ) : ∃ L > , ∀ k ∈ N , || ∆ k u || L ( M ) ≤ L k +1 (2 k )! } . (72)It is classical that all of the eigenfunctions extend holomorphic to a fixed Grauert tube. Theorem (Morrey-Nirenberg Theorem) Let P ( x, D ) be an elliptic differential operatorin Ω with coefficients which are analytic in Ω . If u ∈ D ′ (Ω) and P ( x, D ) u = f with f ∈ C ω (Ω) , then u ∈ C ω (Ω) . The proof shows that the radius of convergence of the solution is determined by the radiusof convergence of the coefficients.In Theorem 2 of [BG2] and Theorem 1.2 of [BGH] it is proved that the operator ∆ hasthe iterate property if and only if, for all b >
1, each eigenfunction extends holomorphicallyto some Grauert tube M τ and satisfiessup z ∈ M τ | ϕ C λ j ( z ) | ≤ b λ j sup x ∈ M | ϕ λ j ( x ) | . (73)The concept of Grauert was not actually used in these articles, so the relation between thegrowth rate and the Grauert tube function was not stated. But it again shows that alleigenfunctions extend to some fixed Grauert tube. Maximal holomorphic extension.
The question then arises if all eigenfunctionsextend to the maximal Grauert tube allowed by the geometry as in Definition 1.1. Weconjectured in the introduction that this does hold, and now explain how it should followfrom known theorems on extensions of holomorphic solutions of holomorphic PDE acrossnon-characteristic hypersurfaces.
Theorem [Zer, Ho3, BSh]
Let f be analytic in the open set Z ⊂ C n and suppose that P ( x, D ) u = f in the open set Z ⊂ Z . If z ∈ Z ∩ ∂Z and if Z has a C non-characteristicboundary at z , then u can be analytically continued as a solution of P ( x, D ) u = f in aneighborhood of z . The idea of the proof is to rewrite the equation as a Cauchy problem with respect tothe non-characteristic hypersurface and to apply the Cauchy Kowaleskaya theorem. Toemploy the theorem we need to verify that the hypsurfaces ∂M τ are non-characteristic forthe complexified Laplacian ∆ C , i.e. that P i,j g ij ( ζ ) ∂ √ ρ∂ζ i ∂ √ ρ∂ζ j = 0 . To prove this, we observethat in the real domain g ( ∇ r , ∇ r ) = 4 r , an identity that was used in (13). In thisformula r = r ( x, y ) and we differentiate in x . We now analytically continue the identity in x → ζ , y, → ¯ ζ and differentiate only with the holomorphic derivatives ∂∂ζ j . From (8), we get g C ( ∂r C ( ζ , ¯ ζ ) , ∂r C ( ζ , ¯ ζ )) = − r C ( ζ , ¯ ζ ) = ρ ( ζ , ¯ ζ ) > . Hence the Theorem applies and we can analytically continue eigenfunctions across anypoint of any ∂M τ for τ < τ g , the maximal radius of a Grauert tube in which the coefficientsof ∆ C are defined and holomorphic. We can take the union of the open sets where ϕ C j hasa holmomorphic extension to obtain a maximal domain of holomorphy. If it fails to be M τ g there exists a point ζ with √ ρ ( ζ ) < τ g so that ϕ C j cannot be holomorphically extended across ∂M τ at ζ . This contradicts the Theorem above and shows that the maximal domain mustbe M τ g . Triple inner products of eigenfunctions: Proof of Proposition 1.
We startwith the identity, Z M ϕ λ j ϕ λ k dV g = e − τλ j h e τ √ ∆ ϕ λ j , ϕ λ k i , (74)and then choose the largest value of τ for which e τ √ ∆ ϕ λ j , e τ √ ∆ ϕ λ k ∈ W s ( M ) for some s ∈ R .Since h e τ √ ∆ ϕ λ j , ϕ λ k i = h ϕ λ j , e τ √ ∆ ϕ λ k i , the assumption that e τ √ ∆ ϕ λ k ∈ W s ( M ) implies that Z M ϕ λ j ϕ λ k dV g ≤ Ce − τλ j || ϕ λ j || W − s ≤ || e τ √ ∆ ϕ λ k || W s λ sj e − τλ j . To complete the proof it suffices to show that e τ √ ∆ ϕ λ j ∈ W s ( M ) and e τ √ ∆ ϕ λ k ∈ W s ( M )for some s ∈ R as long as τ < τ an ( g ). This is obvious for all τ for ϕ λ j since e τ √ ∆ ϕ λ j = e τλ j ϕ λ j . To see that it also holds for ϕ λ k , we note that the analytic continuation operator A ( τ ) isgiven by e τ √ ∆ f = ( U C ( iτ )) − A ( τ ) f. (75) LURI-POTENTIAL THEORY ON GRAUERT TUBES 37
Since U C ( iτ ) is an elliptic Fourier integral operator of finite order by Theorem 7.1, its inverseis an elliptic Fourier integral of the opposite order. In particular, it is clear that e τ √ ∆ f ∈ W s ( M ) for some s if and only if A ( τ ) f ∈ O t ( ∂M τ ) for some t . In fact, A ( τ ) ϕ λ k is realanalytic on M τ for any τ < τ an ( g ) . To go beyond this result, one would need to know the structure of ∂M τ an ( g ) and about therestriction of analytic continuations of eigenfunctions to it.12. Complex zeros of eigenfunctions: Proof of Theorem 4
The real distribution of zeros is by definition the measure supported on the real nodalhypersurfaces Z ϕ j = { x ∈ M : ϕ j ( x ) = 0 } defined by h [ Z ϕ j ] , f i = Z Z ϕj f ( x ) d H n − , (76)where d H n − is the ( n − M, g ). The complex nodal hypersurface of an eigenfunction is defined by Z ϕ C λ = { ζ ∈ M τ : ϕ C λ ( ζ ) = 0 } . (77)There exists a natural current of integration over the nodal hypersurface, given by h [ Z ϕ C λ ] , ϕ i = i π Z M τ ∂ ¯ ∂ log | ϕ C λ | ∧ ϕ = Z Z ϕ C λ ϕ, ϕ ∈ D ( m − ,m − ( M τ ) . (78)In the second equality we used the Poincar´e-Lelong formula. The notation D ( m − ,m − ( M τ )stands for smooth test ( m − , m − M τ . The nodal hypersurface Z ϕ C λ also carries a natural volume form | Z ϕ C λ | as a complex hypersurface in a K¨ahler manifold. ByWirtinger’s formula, it equals the restriction of ω m − g ( m − to Z ϕ C λ . Hence, one can regard Z ϕ C λ asdefining the measure h| Z ϕ C λ | , ϕ i = Z Z ϕ C λ ϕ ω m − g ( m − , ϕ ∈ C ( M τ ) . (79)For background we refer to [Z3]. In that article, we proved: Theorem
Let ( M, g ) be any real analytic compact Riemannian manifold with ergodicgeodesic flow. Then λ j k [ Z ϕ C jk ] → iπ ∂ ¯ ∂ | ξ | g , weakly in D ′ (1 , ( B ∗ ε M ) , for a full density subsequence { ϕ j k } . In this section, we show that the same limit formula is valid for the entire sequence ofeigenfunctions on higher rank locally symmetric manifolds studied in [AS].
Plurisubharmonic functions.
We put ϕ ελ = ϕ C λ | ∂M ε ∈ H ( ∂M ε ) u ελ := ϕ ελ ( z ) || ϕ ελ || L ∂Mε ) ∈ H ( ∂M ε ) U λ ( z ) := ϕ C λ ( z ) || ϕ ελ || L ∂Mε ) , z ∈ ∂M ε . (80)Of these, U λ will play the central role. We note that U λ is CR holomorphic on ∂M τ .However, the normalizing factor || ϕ ελ || − L ( ∂M ε ) depends on ε , so U λ / ∈ O ( M ε ) . Lemma
Let { ϕ j } be an orthonormal basis of eigenfunctions on any compact analyticRiemannian manifold ( M, g ) . Then for τ < τ an , { λ j log | U j | } is pre-compact in L ( M τ ) :every sequence has a convergent subsequence in L ( M τ ) .Proof. As in [Z3], we use the following fact about subharmonic functions (see [Ho, Theo-rem 4.1.9]): • Let { v j } be a sequence of subharmonic functions in an open set X ⊂ R m which have auniform upper bound on any compact set. Then either v j → −∞ uniformly on everycompact set, or else there exists a subsequence v j k which is convergent in L loc ( X ) . • If v is subharmonic and v j → v weakly in D ′ ( M C ) then v j → v in L . We note that λ j log | ϕ C j | is plurisubharmonic and uniformly bounded above on the Grauerttube. Therefore, it either tends to −∞ uniformly on compact sets of the Gruaert tube or ispre-compact in L . The first possibility is ruled out by the fact that it has the form U ( iτ ) C ϕ j on ∂M τ . Hence, || ϕ C j || L ( ∂M τ ) = e τλ j h U ( iτ ) ∗ U ( iτ ) ϕ j , ϕ j i L ( M ) ≥ e τλ j λ − m − j , by Garding’s inequality (67). This contradicts the hypothesis that λ j log | ϕ C j | tends to zerouniformly on all compact sets, i.e. that | ϕ C j ( ζ ) | ≤ e − ε τ λ j . (cid:3) We thus have two different and independent types of weak limit problems: • Weak limits of the L -normalized shell functions U j ; • Weak limits of λ log | u j | . Lemma
Suppose that { ϕ j } is a sequence of eigenfunctions with a unique limit measure dµ and suppose that dµ = ρdµ L + ν where ρ ≥ C > and ν ⊥ µ L . Then λ j Z λ j → i∂ ¯ ∂ | ξ | .Proof. We claim that in this case λ j log | U j | → . Indeed, it is clear that the limsup of theleft side is ≤
0. On the other hand, suppose that the limsup is negative on an open set U .Then R U | U j | →
0. This contradicts the assumption that limit measure has an everywherepositive Liouville component. The rest of the proof is exactly the same as in [Z3]. (cid:3)
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