Bergman-Szegő kernel asymptotics in weakly pseudoconvex finite type cases
aa r X i v : . [ m a t h . C V ] S e p BERGMAN-SZEGŐ KERNEL ASYMPTOTICS IN WEAKLYPSEUDOCONVEX FINITE TYPE CASES
CHIN-YU HSIAO AND NIKHIL SAVALE
Abstract.
We construct a pointwise Boutet de Monvel-Sjöstrand parametrix for the Szegőkernel of a weakly pseudoconvex three dimensional CR manifold of finite type assuming therange of its tangential CR operator to be closed; thereby extending the earlier analysis of Christ[10, 11]. This particularly extends Fefferman’s boundary asymptotics of the Bergman kernel[16] to weakly pseudoconvex domains in C , in agreement with D’Angelo’s example [13]. Finallyour results generalize a three dimensional CR embedding theorem of Lempert [28]. Introduction
Cauchy Riemann (CR) manifolds are natural analogues of complex manifolds in odd dimen-sions. Their structure being modeled on that of a real-hypersurface inside a complex manifold,the natural question of when an abstract CR manifold can be embedded as such into complexspace C N has been long studied. In dimensions at least five a classical embedding theoremfor strongly pseudo-convex CR manifolds was proved by Boutet de Monvel [5]; thereby leavingunresolved the cases of three dimensional manifolds and weakly pseudoconvex manifolds. Indimension three the problem is well known to be more subtle as there are examples of non-embeddable strongly pseudo-convex manifolds [38, 1]. However stronger conditions implyingthree dimensional embeddability are known; Kohn [26, 27] showed that embeddability of astrongly pseudoconvex CR manifold is equivalent its tangential Cauchy-Riemann operator ¯ ∂ b having closed range. Thereafter Lempert [28] (see also Epstein [15]) showed embeddability ofa strongly pseudoconvex CR manifold assuming the existence of transversal CR circle action.In the weakly pseudoconvex case fewer results are known; Christ [10, 11] (see also Kohn [26])showed embeddability of a weakly pseudoconvex CR three manifold of finite type assuming therange of its tangential Cauchy-Riemann operator ¯ ∂ b to be closed.A closely related problem is to study the behavior of the Szegő kernel, the Schwartz kernel ofthe projector from smooth functions onto CR functions. When the manifold is the boundary ofa strictly pseudoconvex domain the singularity of the Szegő kernel was described by Boutet deMonvel and Sjöstrand in [6], the Szegő projector is in this case is a Fourier integral operator withcomplex phase. Combined with the results of [5, 17, 26, 27] this description extends to stronglypseudo-convex manifolds whose tangential CR operator ¯ ∂ b has closed range; and in particularthose of dimension at least five. In particular this description can be used to derive Fefferman’sboundary asymptotics of the Bergman kernel [16] of a strongly pseudoconvex domain. Theweakly pseudoconvex case analog of the problem has been studied by several authors before,with prior results including pointwise bounds on the kernels [10, 30, 32, 35] besides special casesof the asymptotics [13, 3].In the present article we obtain a similar description for the pointwise Szegő kernel of weaklypseudoconvex CR three manifolds of finite type whose tangential CR operator ¯ ∂ b has closed C.-Y. H. is partially supported by Taiwan Ministry of Science and Technology projects 108-2115-M-001-012-MY5 and 109-2923-M-001-010-MY4 .N.S. is partially supported by the DFG funded project CRC/TRR 191. range. Further we prove the boundary asymptotic expansion for the Bergman kernel for weaklypseudoconvex finite type domains in C . Our results thereby extend the aforementioned analysisof Christ, the boundary Bergman kernel asymptotics of Fefferman and the embedding result ofLempert.Let us now state our results more precisely. Let ( X, T , X ) be a compact CR manifoldof dimension three. Thus T , X ⊂ T C X is a complex subbundle of dimension one satisfy-ing T , X ∩ T , X = ∅ , T , X := T , X . Denote by HX := Re ( T , X ⊕ T , X ) the Levi-distribution and J its induced integrable almost complex structure. The Levi form is definedas L ∈ ( HX ∗ ) ⊗ ⊗ ( T x X/H x X ) L ( u, v ) := [[ u, v ]] ∈ T x X/H x X (1.1)for u, v ∈ C ∞ ( HX ) . Given a locally defined vector field T ∈ C ∞ ( T X ) transversal to HX theLevi form can be thought of as a skew-symmetric bi-linear form on HX . We say that the point x is weakly/strongly pseudoconvex iff the corresponding bi-linear form L ( ., J. ) is positivesemi-definite/definite for some choice of orientation for T . The manifold is weakly/stronglypseudoconvex if each point x ∈ X is weakly/strongly pseudoconvex. The CR manifold issaid to be of finite type if the Levi-distribution HX is bracket generating: C ∞ ( HX ) generates C ∞ ( T X ) under the Lie bracket. More precisely, the type of a point x ∈ X is the smallest integer r ( x ) such that HX r ( x ) = T X , where HX j , j = 1 , . . . are inductively defined by HX := HX and HX j +1 := HX + [ HX j , HX ] , ∀ j ≥ . The function x r ( x ) is in general only an upper semi-continuous function. The finite type hypothesis is then equivalent to r := max x ∈ X r ( x ) < ∞ . Note that the type of a strongly pseudoconvex point x is r ( x ) = 2 . For points of higher typeit shall be useful to analogously define the r ( x ) − jet of the Levi-form at xj r x − L ∈ ( HX ∗ ) ⊗ r x ⊗ ( T x X/H x X ) by (cid:0) j r x − L (cid:1) ( u , . . . , u r x ) := (cid:2) ad u ad u . . . ad u r − u r (cid:3) ∈ T x X/H x X (1.2)for u j ∈ C ∞ ( HX ) , j = 1 , . . . , r .Next let ¯ ∂ b : Ω , ∗ ( X ) → Ω , ∗ +1 ( X ) denote the tangential CR operator and choose a smoothvolume form µ on X . The Szegő kernel Π ( x, x ′ ) is by definition the Schwartz kernel of the L projection Π : L ( X ) → ker (cid:0) ¯ ∂ b (cid:1) . To describe our parametrix for Π , first recall the well knownsymbol class S mρ,δ ( U × R t ) , m ∈ R , ρ, δ ∈ (0 , , U ⊂ R , of Hörmander [20]: these are smoothfunctions a ( x, t ) satisfying the estimates ∂ kt ∂ αx a = O (cid:0) t m − ρk + δ | α | (cid:1) , ∀ ( k, α ) ∈ N × N , as t → ∞ ,uniformly on compact subsets of U . Further denote by the notation S mδ ( U × R t ) the special casewhen ρ = 1 . We now introduce the subspace of classical symbols S mδ, cl ( U × R t ) ⊂ S mδ ( U × R t ) as those a ( x, t ) for which there exist functions a j ∈ S ( R ) , j = 0 , , , . . . , satisfying(1.3) a ( x, t ) − t m " N X j =0 t − δj a j (cid:0) t δ x (cid:1) ∈ S m − δNδ ( U × R t ) , ∀ N ∈ N . Our first theorem is now the following.
Theorem 1.
Let X be a compact weakly pseudoconvex three dimensional CR manifold of finitetype for which the range of the tangential CR operator ¯ ∂ b is closed. At any point x ′ ∈ X oftype r = r ( x ′ ) , there exists a set of coordinates ( x , x , x ) centered at x ′ and a classical symbol a ∈ S r r , cl (cid:0) R x ,x × R t (cid:1) , with a > , such that the pointwise Szegő kernel at x ′ satisfies (1.4) Π ( x, x ′ ) = Z ∞ dt e itx a ( x ; t ) + C ∞ ( X ) . ZEGŐ KERNEL 3
We note again that the point x ′ ∈ X above is fixed and thus our ’pointwise parametrix’ is adistribution on the manifold X rather than the product. The direction ∂ x is locally transverseto the Levi distribution HX . More can be said about the amplitude in (1.4): each coefficient a j ∈ S ( R ) in its symbolic expansion (1.3) is a linear combination of functions of the form x α x α a j,α , α + α ≤ jr , with the functions a j,α ∈ S ( R ) further depending only on the firstjet of the Levi-form j r x ′ − L at x ′ and the indices j, α . Furthermore at a strongly pseudoconvexpoint x ′ we may take each a j,α = e − ( x + x ) to be a Gaussian. Following this Theorem 1 isseen to recover the pointwise version of the Boutet de Monvel-Sjöstrand parametrix at stronglypseudoconvex points (see Remark 15 below). At points of higher type however the functions a j,α are no longer Gaussians.An important classical case arises when the CR manifold X = ∂D is the boundary of a do-main, i.e. a relatively compact open subset D ⊂ C . The analogous Bergman kernel Π D ( z, z ′ ) isthe Schwartz kernel of the projector Π D : L ( D ) → ker (cid:0) ¯ ∂ (cid:1) onto the L -holomorphic functionsin the interior. One is then interested in the on-diagonal behavior of the Bergman kernel asone approaches the boundary in terms of a boundary defining function ρ ∈ C ∞ ( C ) , satisfying D = { ρ < } , dρ | ∂D = 0 . This is given as below. Theorem 2.
Let D ⊂ C be a domain with boundary X = ∂D being smooth, weakly pseu-doconvex of finite type. For any point x ′ ∈ X = ∂D on the boundary, of type r = r ( x ′ ) , theBergman kernel satisfies the asymptotics Π D ( z, z ) = N X j =0 − ρ ) r − r j a j + N X j =0 b j ( − ρ ) j log ( − ρ ) + O (cid:16) ( − ρ ) N − − rr (cid:17) , ∀ N ∈ N , as z → x ′ for some set of reals a j , b j with a > . Our description of Szegő kernel Theorem 1 becomes more concrete in the case when the CRmanifold X is circle invariant. In this case one obtains an on diagonal expansion Π m ( x, x ) , m → ∞ , for the m th Fourier mode of the Szegő kernel, we refer to Theorem 21 in Section 5below for the precise statement. The asymptotics of these higher Fourier modes of the Szegőkernel allows one to construct a sufficient number of CR peak functions. These can be used toprove the following embedding theorem. Theorem 3.
Let X be a compact weakly pseudoconvex three dimensional CR manifold of finitetype admitting a transversal, CR circle action. Then it has an equivariant CR embedding intosome C N , N ∈ N . The Szegő kernel parametrix of Boutet de Monvel-Sjöstrand [6] has had a broad impactin complex analysis and geometry, we refer to [22] for a detailed account of this techniqueand its applications. Particularly this recovered the prior results of Fefferman [16] on the fullboundary asymptotics for the Bergman kernel of a strongly pseudoconvex domain, which inturn refined its leading asymptotics by Hörmander [18]. The weakly pseudoconvex analog ofthe problem has also been considered by several authors before. Prior results have includedpointwise upper [10, 30, 32, 35] and lower [7] bounds on the Bergman and Szegő kernels inlow dimensions, besides particular special cases of the asymptotics for complex ovals [13],h-extendible/semiregular domains [3] and certain toric domains [24]. In higher dimensionalweakly pseudoconvex cases the analogous bounds as well as the asymptotics of Theorem 1 andTheorem 2 are wide open, some known results in higher dimensions include weak estimates onthe Bergman kernel [37] along with estimates on the Bergman metric [33] and distance [14].In the presence of a transversal circle action a weakly pseudoconvex CR manifold is the unitcircle bundle of a semi-positive holomorphic orbifold line bundle over a complex orbifold. When
ZEGŐ KERNEL 4 the action is free and the manifold strongly pseudoconvex, the Szegő kernel expansion Theorem21corresponds to the Bergman kernel expansion of positive line bundles and was first obtainedin [8, 40]. This was recently generalized to the Bergman kernel expansion of semipositive linebundles over a Riemann surface in [31]. The first author had earlier in [23] given a proof ofthe Szegő kernel expansion of a circle invariant weakly pseudoconvex CR manifold, althoughonly on its strictly pseudoconvex part. For general non-free actions on strongly-pseudoconvexmanifolds, the Szegő kernel expansion corresponds to the Bergman kernel expansion of positiveorbifold line bundles and was first proved in [12], [29, Sec. 5.4].As mentioned, the embeddability of strongly pseudoconvex CR manifolds equipped with atransversal CR circle action was shown in [28] and thus generalized by our last Theorem 3. Inthe weakly pseudoconvex case, [11] showed embeddability of a CR three manifold of finite typeassuming the range of its tangential Cauchy-Riemann operator ¯ ∂ b to be closed.The paper is organized as follows. In Section 2 we begin with some preliminaries in CRgeometry including a construction of almost analytic coordinates adapted to the CR structurein 2.1. In Section 3 we construct an appropriate symbol calculus in 3.1 and construct thepointwise Szegő parametrix to prove Theorem 1. In Section 4 we consider the Bergman kernelof a weakly pseudoconvex domain in C and prove Theorem 2. In Section 5 we turn to the circleinvariant case and prove Theorem 21. In the final section Section 6 we prove our embeddingtheorem Theorem 3. 2. CR Preliminaries
Let ( X, T , X ) be a compact CR manifold of dimension three. Thus T , X ⊂ T C X is acomplex subbundle of dimension one satisfying T , X ∩ T , X = ∅ , T , X := T , X . Let HX := Re ( T , X ⊕ T , X ) be the Levi-distribution. This carries an almost complex structure J : HX → HX J ( v + ¯ v ) := i ( v − ¯ v ) , ∀ v ∈ T , X, satisfying J = − and the integrability condition [ J v, u ] + [ v, J u ] ∈ C ∞ ( HX )[ J v, J u ] − [ v, u ] = J ([ J v, u ] + [ v, J u ]) (2.1) ∀ u, v ∈ C ∞ ( HX ) . The antisymmetric Levi-form defined via (1.1) consequently satisfies L ( J u, v ) = L ( u, J v ) . The last contraction L ( ., J. ) is equivalently thought of as a Hermitian form on T , X and denoted by the same notation via(2.2) L ( u, v ) := (cid:20) i [ u, ¯ v ] (cid:21) ∈ ( T X/HX ) ⊗ C ,u, v ∈ T , X. The point x ∈ X is strongly/weakly pseudoconvex if the Levi form above ispositive definite/semi-definite at x for some choice of local orientation for T X/HX . Next onedefines the flag of subspaces HM ,x ⊂ HM ,x ⊂ . . . , at x ∈ X inductively via HM ,x := HM x HM j +1 ,x := HM x + [ HM j,x , HM x ] , j ≥ . The point x is said to be of finite type if HM r ( x ) ,x = T X for some r ( x ) ∈ N , the minimumsuch integer being the type of the point x . The weight vector at the point x is defined to be w ( x ) := (1 , , r ( x )) . The CR structure is of finite type if each point is of finite type. Note thatthe type of a strongly pseudoconvex point x ∈ X is r ( x ) = 2 by definition. ZEGŐ KERNEL 5
The set of horizontal paths of Sobolev regularity one connecting the two points x, x ′ ∈ X isdenoted by(2.3) Ω H ( x, x ′ ) := (cid:8) γ ∈ H ([0 ,
1] ; X ) | γ (0) = x, γ (1) = x ′ , ˙ γ ( t ) ∈ HX γ ( t ) a.e. (cid:9) . Fixing a metric g HX on the Levi distribution HX , one define the obvious the obvious lengthfunctional l ( γ ) := R | ˙ γ | dt on the above path space Ω H . By a classical theorem of Chow-Rashevsky the horizontal path space (2.3) is non-empty when the CR structure is of finitetype; allowing the definition of a distance function(2.4) d H ( x, x ′ ) := inf γ ∈ Ω H ( x,x ′ ) l ( γ ) . The weight w ( f ) of a function f at the point x is defined to be the maximum integer s ∈ N for which a + b = s implies that (cid:0) u a v b f (cid:1) ( x ) = 0 ; where u, v form a local frame for HX near x . Similarly the weight w ( P ) of a differential operator P at the point x ∈ X isthe maximum integer for which w ( P f ) ≥ w ( P ) + w ( f ) holds for each function f ∈ C ∞ ( X ) .It is known that there exists a set of coordinates ( x , x , x ) centered near a point x ∈ X for which ∂∂x , ∂∂x forms a basis for HM x of the canonical flag and moreover each coordinatefunction ( x , x , x ) has weight w ( x ) := (1 , , r ( x )) respectively; such a coordinate system iscalled privileged. In privileged coordinates near x the weight of a monomial x α , α ∈ N , is thus α + α + rα . The weight w ( f ) of an arbitrary function f ∈ C ∞ ( X ) is then the minimumweight of the monomials appearing in its Taylor series in these coordinates. The weight w ( V ) of a vector field V = P j =1 f j ∂ x j is seen to be w ( V ) := min { w ( f ) − , w ( f ) − , w ( f ) − r } .The distance function and volume (with respect to an arbitrary volume form µ ) of a radius ε ball centered at the origin in such coordinates are known to satisfy C (cid:16) | x | + | x | + | x | /r ( x ) (cid:17) ≤ d H ( x, ≤ C ′ (cid:16) | x | + | x | + | x | /r ( x ) (cid:17) Cǫ r ( x ) ≤ µ ( B (0; ε )) ≤ C ′ ǫ r ( x ) ; ε ∈ (0 , . Given a coordinate chart U as above, denote by S mH ( U ) the space of smooth functions p ( x ) on U \ { } satisfying the estimates(2.5) | ∂ αx p | ≤ C α (cid:2) d H ( x, (cid:3) − m + r − α.w x µ (cid:0) B (cid:0) d H ( x, (cid:1)(cid:1) − ,x = 0 , ∀ α ∈ N . It was shown in [10, Sec.12], cf. [35] that the restriction to U of the Szegőkernel lies in the class(2.6) Π ( x, ∈ S r H ( U ) defined above.2.1. Construction of coordinates.
In [11, Prop. 3.2] it was shown that the privileged coor-dinate system near x maybe further chosen so that T , X = C [ Z ] Z = 12 [ ∂ x + ( ∂ x p ) ∂ x + i ( ∂ x − ( ∂ x p ) ∂ x + R )] ; (2.7)where p ( x , x ) is a homogeneous real polynomial of degree/weight r ( x ) , and R = P j =1 r j ( x ) ∂ x j a real vector field of weight w ( R ) ≥ . Furthermore, the pseudoconvexity of X gives ∆ p = (cid:0) ∂ x + ∂ x (cid:1) p ≥ . In this subsection we shall further show how to remove the remainder term R via almost analytic extension. We first have the following. ZEGŐ KERNEL 6
Lemma 4.
There exists a locally defined complex vector field T such that T x = ∂ x and [ T, Z ] vanishes to infinite order at x .Proof. The desired equation for the components of T = P j =1 t j ( x ) ∂ x j is seen to be one of theform ( ∂ x + i∂ x ) t j = − ( ∂ x p − i∂ x p ) ∂ x t j + δ T ( ∂ x − i∂ x ) p + T r j − Rt j + O ( | x | ∞ ) ,j = 1 , , . As the component functions p , r j have degree at least two and one respectively;the degree k homogeneous part on the left hand side above involves the Taylor coefficients of t j for x α , | α | = k − , while those on the right hand side involve those for x α , | α | < k − .We may hence solve the above recursively for the Taylor coefficients of t j , beginning with ( t , t , t ) = (0 , ,
1) + O ( | x | ) , and apply Borel’s construction. (cid:3) Next we complexify the open neighborhood of x ∈ U ⊂ R on which the above coordinatesare defined to an open set U C ⊂ C such that U C ∩ R = U . Denote by z j = x j + iy j , j = 1 , , , the corresponding complex coordinates. For a function f ∈ C ∞ c ( C ) , we write f ∼ if it vanishes to infinite order along the real plane: | f ( z ) | = O ( | Im z | ∞ ) . A function f ∈ C ∞ c ( C ) , is said to be almost analytic iff ∂ ¯ z j f ∼ , j = 1 , , . A complex vector field L = P j =1 (cid:2) a j ∂ z j + b j ∂ ¯ z j (cid:3) is said to be almost analytic iff Lf is almost analytic and L ¯ f ∼ forall almost analytic f ∈ C ∞ c ( C ) . This is seen to be equivalent to a j being almost analytic and b j ∼ for j = 1 , , . For two complex vector fields we write L ∼ L iff L − L is almostanalytic. We choose almost analytic extensions ˜ T , ˜ Z of T, Z respectively. We now have thenext lemma.
Lemma 5.
There exist almost analytic complex coordinates w j = z j + z O ( | z | ) , j = 1 , , , on U C such that ˜ T − ∂ w vanishes to infinite order at x .Proof. Firstly we have ˜ T ∼ P j =1 t j ( z ) ∂ z j for some almost analytic functions t j , j = 1 , , ,satisfying ( t , t , t ) = (0 , ,
1) + O ( | z | ) . Next we find an almost analytic function w ( z ) satisfying P j =1 t j ∂ z j w = O ( | z | ∞ ) or equivalently(2.8) ∂ z w = − [ t ∂ z w + t ∂ z w + ( t − ∂ z w ] + O ( | z | ∞ ) . The degree k homogeneous part on the left hand side above involves the Taylor coefficients of w for z α , | α | = k , while those on the right hand side involve those for z α , | α | < k . We may henceagain solve the above recursively for the Taylor coefficients of w , beginning with w ( z ) = z + O (cid:0) | z | (cid:1) , and apply Borel’s construction. Since solving the equation (2.8) involves integration in z , the higher order terms in the Taylor expansion w ( z ) = z + z O ( | z | ) can be further taken tobe multiples of z . In similar vein, we find almost an analytic functions w ( z ) = z + z O ( | z | ) , w ( z ) = z (1 + O ( | z | )) satisfying P j =1 t j ∂ z j w = O ( | z | ∞ ) and P j =1 t j ∂ z j w = 1 + O ( | z | ∞ ) respectively. Thus ( w , w , w ) is the required coordinate system. (cid:3) We may now prove our main result of this subsection.
Theorem 6.
There exist almost analytic complex coordinates ˜ z j = p j + iq j , j = 1 , , and analmost analytic function ϕ (˜ z , ˜ z ) (of the first two new coordinates) on U C such that (1) ˜ z j = z j + O (cid:0) | z | (cid:1) , j = 1 , , ˜ z = z + z O ( | z | ) + O ( | z | ∞ ) , (2) ˜ Z = ( ∂ ˜ z + i∂ ˜ z ) − i ( ∂ ˜ z ϕ + i∂ ˜ z ϕ ) ∂ ˜ z and ˜ T = ∂ ˜ z + O ( | z | ∞ ) for (3) ϕ (˜ z , ˜ z ) = ϕ (˜ z , ˜ z ) + O (cid:0) | z | r +1 (cid:1) with ϕ a homogeneous polynomial with real coeffi-cients satisfying (cid:0) ∂ p + ∂ p (cid:1) (cid:16) ϕ | q =0 (cid:17) ≥ . ZEGŐ KERNEL 7
Proof.
Firstly we have by definition ˜ Z ∼ P j =1 a j ∂ w j , for some almost analytic functions a j , j = 1 , , satisfying a (0) = 12 , a (0) = i ,a = 12 ( ∂ w ˜ p − i∂ w ˜ p ) + O ( | w | r ) and ˜ p being an almost analytic extension of p . Furthermore, from the preceding Lemma 4,Lemma 5 we have h ˜ T , ˜ Z i = h ∂ w , ˜ Z i + O ( | w | ∞ ) and may assume that a j , j = 1 , , , areindependent of w . Next as in (2.8) we find almost analytic functions ˜ w j ( w , w ) = w j + O ( | w | ∞ ) , j = 1 , , such that a ∂ w ˜ w + a ∂ w ˜ w −
12 = O ( | w | ∞ ) a ∂ w ˜ w + a ∂ w ˜ w − i O ( | w | ∞ ) . Setting ˜ z = w , we have then thus far achieved ˜ Z = ( ∂ ˜ w + i∂ ˜ w ) + a ( ˜ w , ˜ w ) ∂ ˜ w + O ( | ˜ w | ∞ ) .It is then easy to find ϕ ( ˜ w , ˜ w ) satisfying a = − i ( ∂ ˜ w ϕ + i∂ ˜ w ϕ ) + O ( | ˜ w | ∞ ) by a furtherapplication of the Borel construction giving ˜ Z = ( ∂ ˜ w + i∂ ˜ w ) − i ( ∂ ˜ w ϕ + i∂ ˜ w ϕ ) ∂ ˜ w + ˜ Z ∞ forsome almost analytic vector field ˜ Z ∞ = O ( | ˜ w | ∞ ) .Finally to remove this infinite order error term one applies the scattering trick of Nelson [36,Ch. 3]. Choose an almost analytic function χ ∈ C ∞ c (cid:0) U C (cid:1) , equal to one near zero, and set(2.9) ˜ Z = 12 ( ∂ ˜ w + i∂ ˜ w ) − i ∂ ˜ w ϕ + i∂ ˜ w ϕ ) ∂ ˜ w + (1 − χ ) ˜ Z ∞ . It is clear that the almost analytic flows of ˜ Z , ˜ Z starting at U C exit U C in uniformly finitetime, outside which they are equal. Thus the limiting almost analytic map(2.10) W := lim t →∞ e t ˜ Z ◦ e − t ˜ Z exists with the limit achieved in finite time. One then calculates ddt (cid:16) e − t ˜ Z ◦ e t ˜ Z (cid:17) ∗ ˜ w j = (cid:16) e t ˜ Z (cid:17) ∗ (cid:16) ˜ Z − ˜ Z (cid:17) (cid:16) e − t ˜ Z (cid:17) ∗ ˜ w j = O ( | ˜ w | ∞ ) and thus ˜ z j := W ∗ ˜ w j = ˜ w j + O ( | ˜ w | ∞ ) . (2.11)This finally gives ˜ Z = W ∗ ˜ Z = 12 ( ∂ ˜ z + i∂ ˜ z ) − i ∂ ˜ z ϕ + i∂ ˜ z ϕ ) ∂ ˜ z near zero from (2.9), (2.10), (2.11) proving the second part of the theorem. The last part followsby a Taylor expansion and the corresponding subharmonicity of p ( x , x ) (2.7). (cid:3) We remark that although Nelson’s method may also be used to linearize the almost analyticvector field ˜ Z in some almost analytic coordinates, the resulting coordinates thereby will notsatisfy the first property of the above Theorem 6 which shall be used later.Before putting the above coordinates to use in the next section, we shall also need theconstruction of almost analytic continuations of functions in the class S mH defined in (2.5).These shall be defined on the region(2.12) R δ,U := (cid:8) ( x, y ) ∈ U × R | | y | ≤ δ | x | , | y | ≤ δ | x | | x | (cid:9) ZEGŐ KERNEL 8 for any δ > . Lemma 7. (Almost analytic continuations in S mH ) For each δ > there exists almost analyticextension map E : S mH ( U ) → C ∞ ( R δ,U \ { } ) satisfying E f | R = f,∂ ¯ z E f = O ( | Im z | ∞ ) uniformly on R δ,U and Lf ∈ C ∞ ( U ) = ⇒ ˜ L E f ∈ C ∞ ( R δ,U ) (2.13) for each f ∈ S mH and vector field L with almost analytic extension ˜ L .Proof. The map E is defined by the usual Borel-Hörmander construction. Namely with χ ∈ C ∞ c ( R ) and equal to one near zero, set(2.14) ( E f ) ( x, y ) := X α ( iy ) α α ! f ( α ) ( x ) χ (cid:0) λ | α | | y | (cid:1) . Note that on the given region R δ,U (2.12) each successive term above satisfies the estimate ( iy ) α α ! f ( α ) ( x ) = O (cid:16) | y | ( | α |− rm ) / (cid:17) . For a suitable sequence constants λ k → ∞ sufficiently fast,the series above is then seen to be C ∞ convergent, and hence defining a smooth function, oncompact subsets of R δ,U \ { } . The first property in (2.13) then follows immediately from theabove definition. The second property, follows easily on differentiating the definition (2.14)and applying the estimates (2.5) on the region R δ,U . Finally for the last property, note that g := ˜ L E f − E Lf is an almost analytic continuation of zero in the sense g | R = 0 and ∂ ¯ z g = O ( | Im z | ∞ ) uniformly on R δ,U . It furthermore satisfies estimates similar to (2.5) by definition.The Taylor expansion of g in the y -variable is seen to be g ( x, y ) := X | α |≤ N ( iy ) α α ! g ( α ) ( x ) | {z } =0 + O (cid:16) | y | ( N +1 − rm ) / (cid:17) , giving g = O ( | Im z | ∞ ) and g ∈ C ∞ ( R δ,U ) . Since Lf ∈ C ∞ ( U ) is smooth and E Lf ∈ C ∞ ( R δ,U ) by construction, the result follows. (cid:3) Szegő parametrix
In this section we shall prove our main Theorem 1. It shall first be useful to define a requisitesymbol calculus below.3.1.
Symbol spaces and calculus.
Below we denote by x = (ˆ x, x ) local coordinates on R ,with ˆ x = ( x , x ) denoting the local coordinates on R .A smooth function f (ˆ x, ˆ y ) ∈ C ∞ ( R × R ) is said to lie in the class f ∈ ˆ S ( R × R ) if foreach (cid:16) ˆ α, ˆ β (cid:17) ∈ N there exists N (cid:16) ˆ α, ˆ β (cid:17) ∈ N such that(3.1) (cid:12)(cid:12)(cid:12) ∂ ˆ α ˆ x ∂ ˆ β ˆ y f (ˆ x, ˆ y ) (cid:12)(cid:12)(cid:12) ≤ C N, ˆ α ˆ β (1 + | ˆ x | + | ˆ y | ) N ( ˆ α, ˆ β )(1 + | ˆ x − ˆ y | ) − N , ∀ (ˆ x, ˆ y, N ) ∈ R × R × N . We note that for f ∈ ˆ S ( R × R ) the functions(3.2) f ( ., ˆ y ) , f (ˆ x, . ) ∈ S (cid:0) R (cid:1) are Schwartz for fixed ˆ y and ˆ x respectively.We now introduce some symbol spaces. ZEGŐ KERNEL 9
Definition 8.
Let r ∈ N , r ≥ . A function a ( x, y, t ) ∈ C ∞ (cid:0) R x,y × R t (cid:1) is said to lie in thesymbol class ˆ S m r , m ∈ R , if for each ( α, β, γ ) ∈ N there exists N ( α, β, γ ) ∈ N such that (cid:12)(cid:12) ∂ αx ∂ βy ∂ γt a ( x, y, t ) (cid:12)(cid:12) ≤ C N,αβγ h t i m − γ + r ( | ˆ α | + | ˆ β | ) + α + β (cid:16) (cid:12)(cid:12)(cid:12) t r ˆ x (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) t r ˆ y (cid:12)(cid:12)(cid:12)(cid:17) N ( α,β,γ ) (cid:16) (cid:12)(cid:12)(cid:12) t r ˆ x − t r ˆ y (cid:12)(cid:12)(cid:12)(cid:17) − N , (3.3) ∀ ( x, y, t, N ) ∈ R x,y × R t × N . We further set(3.4) ˆ S m,k r := M p + p ′ ≤ k ( tx ) p ( ty ) p ′ ˆ S m r , ∀ ( m, k ) ∈ R × N . The subset ˆ S m r , cl ⊂ ˆ S m r of classical symbols is those a ( x, y, t ) for which there exist a jpp ′ (ˆ x, ˆ y ) ∈ ˆ S ( R × R ) , j, p, p ′ ∈ N , such that(3.5) a ( x, y, t ) − N X j =0 X p + p ′ ≤ j t m − r j ( tx ) p ( ty ) p ′ a jpp ′ (cid:16) t r ˆ x, t r ˆ y (cid:17) ∈ ˆ S m − ( N +1) r ,N +1 r ∀ N ∈ N . We also set ˆ S m,k r , cl := M p + p ′ ≤ k ( tx ) p ( ty ) p ′ ˆ S m r , cl . The following inclusions are clear ∂ t ˆ S m,k r ⊂ ˆ S m − ,k r ∂ ˆ x ˆ S m,k r , ∂ ˆ y ˆ S m,k r ⊂ ˆ S m + r ,k r ∂ x ˆ S m,k r , ∂ y ˆ S m,k r ⊂ ˆ S m +1 ,k r ˆ S m,k r ⊂ ˆ S m +1 ,k − r , k ≥ , ˆ S m,k r ⊂ ˆ S m,k +1 r , ˆ S m,k r ⊂ ˆ S m ′ ,k r , m < m ′ , (3.6)with similar inclusions applying for ˆ S m,k r , cl .Next we set ˆ S m,k, −∞ r := ∩ j ∈ N ˆ S m − jr ,k + j r ˆ S −∞ r := ∪ m,k ˆ S m,k, −∞ r . Following a standard Borel construction, one has asymptotic summation: for any a j ∈ ˆ S m − r j,k + j r , j = 0 , , . . . , there exists a ∈ ˆ S m,k r such that(3.7) a − N X j =1 a j ! ∈ ˆ S m − r ( N +1) ,k + N +1 r , ∀ N ∈ N , with a similar property being true for the classical symbols ˆ S m r , cl . Moreover the symbol a (3.7)above is unique modulo ˆ S m,k, −∞ r . ZEGŐ KERNEL 10
We now define the quantizations of the symbols in 8.
Definition 9.
An operator G : C ∞ c ( R ) → C −∞ ( R ) is said to be in the class G ∈ ˆ L m,k r , cl if itsdistribution kernel satisfies(3.8) G ( x, y ) ≡ g L := Z ∞ dt e it ( x − y ) g ( x, y, t ) for some g ∈ ˆ S m,k r , cl + ˆ S −∞ r .It is an easy exercise that for G ∈ ˆ L m r , m < − − k , the kernel G ( ., y ) ∈ C k for fixed y . Wenext have a reduction lemma showing that the show that the amplitude g in the quantizationabove (3.8) maybe chosen independent of x or y . Lemma 10.
For any g ∈ ˆ S m,k r , cl there exist g , g ∈ ˆ S m,k r , cl independent of x , y respectively suchthat g L = g L = g L .Proof. By a Fourier transform, it is easy to see that G = g L for g (ˆ x, x , ˆ y, t ) = (cid:2) e i∂ t ∂ y g (cid:3) y = x . Here the above notation follows [21, Sec. 7.6] wherein the partial y , t Fourier transform F y ,t of e i∂ t ∂ y g is given e i∂ t ∂ y g = F − y ,t e iτη F y ,t g by multiplication by the exponential of the dual variables η , τ respectively. From [21, Thm.7.6.5] and (3.6) it is then easy to see that g ∈ ˆ S m +6 ,k r ( U ) . In particular we have g ∈ ˆ S −∞ r for g ∈ ˆ S −∞ r .Next for g ∈ ˆ S m,k r , cl we plug in its classical expansion (3.5) into (3.8). By writing ty = tx + t ( y − x ) and repeated integration by parts using ∂ t e it ( x − y ) = i ( x − y ) e it ( x − y ) weobtain g ,N ∈ ˆ S m,k r , cl , N ∈ N , independent of y such that g − g ,N ∈ ˆ S m − ( N +1) r ,k + N +1 r , ∀ N ∈ N .By asymptotic summation we find g ∼ g , + P ∞ N =1 ( g ,N +1 − g ,N ) ∈ ˆ S m,k r , cl , independent of y ,which satisfies g − g ∈ ˆ S m,k, −∞ r ⊂ ˆ S −∞ r . From this and the first part of the proof the Lemmafollows. The construction of g is similar. (cid:3) Following the above we shall define the principal symbol in ˆ L m,k r , cl via σ L ( G ) = g (ˆ x, ˆ y ) ∈ ˆ S ( R × R ) ,G ∈ ˆ L m,k r , cl , as the leading term in the symbolic expansion (3.5). The following symbol exactsequence is then clear → ˆ L m − r ,k +1 r , cl → ˆ L m,k r , cl σ L −→ ˆ S ( R × R ) → . The class ˆ L m,k r , cl ( U ) is clearly closed under adjoints. The symbol of the adjoint is furthermoreeasily computed(3.9) σ L ( G ∗ ) (ˆ x, ˆ y ) = σ L ( G ) (ˆ y, ˆ x ) . We next have the composition of operators in ˆ L m,k r , cl . ZEGŐ KERNEL 11
Proposition 11.
For any G ∈ ˆ L m,k r , cl , H ∈ ˆ L m ′ ,k ′ r , cl one has the composition G ◦ H ∈ ˆ L m + m ′ − r ,k + k ′ r , cl .Furthermore the leading symbol of the composition is given by (3.10) σ L ( G ◦ H ) (ˆ x, ˆ y ) = Z d ˆ u σ L ( G ) (ˆ x, ˆ u ) σ L ( H ) (ˆ u, ˆ y ) . Proof.
Write G = g L , H = h L in terms of their x , y independent quantizations respectively.From Fourier inversion it is easy to check that ( G ◦ H ) ( x, y ) = Z dt e it ( x − y ) ( g ◦ h ) ( x, y, t ) for ( g ◦ h ) ( x, y, t ) := Z d ˆ u g (ˆ x, x , ˆ u, t ) h (ˆ u, ˆ y, y , t )= t − r Z dt e it ( x − y ) d ˆ vg (cid:16) ˆ x, x , t − r ˆ v, t (cid:17) h (cid:16) t − r ˆ v, ˆ y, y , t (cid:17) upon a change of variables t r ˆ u = ˆ v . The ˆ v integral is seen to be convergent on account of (3.2),which also gives the necessary symbolic estimates for g ◦ h . To obtain the symbolic expansion,we plug x , y independent symbolic expansions for g , h respectively into the above to obtain asymbolic expansion for the composed symbol g ◦ h along with the formula (3.10) for the leadingpart. (cid:3) Finally we show that our algebra of operators is a module over the usual algebra of pseudo-differential operators.
Proposition 12.
Let G ∈ ˆ L m,k r , cl and let P ∈ Ψ m ′ cl be a classical pseudodifferential operator on U of order k . Then, P G ∈ ˆ L m + m ′ ,k r , cl with leading symbol (3.11) σ L ( P G ) (ˆ x, ˆ y ) = σ ( P ) (0 ,
0; 0 , σ L ( G ) (ˆ x, ˆ y ) . Proof.
First write the kernels P ( x, u ) = 1(2 π ) Z dξe i ( x − u ) ξ p ( x, ξ ) G ( u, y ) = Z dte i ( u − y ) t g (ˆ u ; ˆ y, y , t ) p ∈ S m ′ cl , g ∈ ˆ S m,k r , cl using an u independent quantization for G . Then Fourier inversion givesthe composition to be ( P ◦ G ) ( x, y ) = Z dt e i ( x − y ) t q ( x, y, t ) q ( x, y, t ) := 1(2 π ) Z d ˆ ud ˆ ξe i (ˆ x − ˆ u )ˆ ξ p (cid:16) x, ˆ ξ, t (cid:17) g (ˆ u ; ˆ y, y , t ) . (3.12)Again the above amplitude satisfies necessary symbolic estimates on account of (3.2).To obtain the symbolic expansion, a change of variables ˆ v = t r ˆ u , ˆ η = t − r ˆ ξ first gives q (cid:16) t − r ˆ x, x , t − r ˆ y, y , t (cid:17) = Z d ˆ ud ˆ ξe i (cid:16) t − r ˆ x − ˆ u (cid:17) ˆ ξ p (cid:16) t − r ˆ x, x , ˆ ξ, t (cid:17) g (cid:16) ˆ u ; t − r ˆ y, y , t (cid:17) = Z d ˆ vd ˆ ηe i (ˆ x − ˆ v )ˆ η p (cid:16) t − r ˆ x, x , t r ˆ η, t (cid:17) g (cid:16) t − r ˆ v ; t − r ˆ y, y , t (cid:17) . (3.13) ZEGŐ KERNEL 12
Next we plug in the symbolic expansion for g as well as p (cid:16) t − r ˆ x, x , t r ˆ η, t (cid:17) ∼ t k " p (0 , x , ,
1) + ∞ X j =1 t − j/r p j (ˆ x, x , ˆ η ) , obtained from the classical symbolic expansion for p , into (3.12), (3.13). A further Taylorexpansion in x for p (0 , x , , ∼ P ∞ j =0 t − j ( tx ) j (cid:0) ∂ jx p (cid:1) (0 , , , and each p j plugged intothe above completes the proof. (cid:3) Finally we need some mapping properties of operators in ˆ L m,k r , cl . To introduce the functionalspaces first define ˆ S m r (cid:0) R (cid:1) ⊂ ˆ S m r (cid:0) R × R (cid:1) ˆ S m r , cl (cid:0) R (cid:1) ⊂ ˆ S m r , cl (cid:0) R × R (cid:1) (3.14)as the subspace of x , y -independent elements in 8. Note that the above are included ˆ S m r (cid:0) R (cid:1) ⊂ S m r (cid:0) R × R t (cid:1) ˆ S m r , cl (cid:0) R (cid:1) ⊂ S m r , cl (cid:0) R × R t (cid:1) (3.15)in the Hörmander symbol classes from the introduction.We next define the space of partial t -Fourier transforms of the classes (3.14) below S mH (cid:0) R (cid:1) := (cid:26) p ∈ S ′ (cid:0) R (cid:1) | p = Z dte itx a ( t, ˆ x ) , a ∈ ˆ S m r (cid:0) R (cid:1)(cid:27) S mH, cl (cid:0) R (cid:1) := (cid:26) p ∈ S ′ (cid:0) R (cid:1) | p = Z dte itx a ( t, ˆ x ) , a ∈ ˆ S m r , cl (cid:0) R (cid:1)(cid:27) . (3.16)It is an easy exercise using Fourier transforms to see that elements of S mH ( R ) (3.16) above aresmooth outside the origin. While the space S mH ( U ) consists of restrictions to U of elements inthe space S mH ( R ) defined above. It is further easy to see the inclusion(3.17) S mH (cid:0) R (cid:1) ⊂ C α (cid:0) R (cid:1) , m < − − α. We now have the following.
Proposition 13.
For G ∈ ˆ L m,k r , cl ( R ) and p ∈ S m ′ H ( R ) we have Gp ∈ S m + m ′ − r H ( R ) . A similarproperty holds for S mH, cl ( R ) .Proof. Again using a y independent quantization for G , the Fourier transform expression p = R dte itx a ( t, ˆ x ) and Fourier inversion gives Gp ( x ) = Z dte ix t ( g ◦ a ) ( x, t )( g ◦ a ) ( x, t ) := Z g ( x, ˆ y, t ) a ( t, ˆ y ) d ˆ y. Next we plugin the symbolic expansion for g into the above and use repeated integration byparts using ∂ t e ix t = ix e ix t to obtain x -independence of the amplitude modulo C ∞ . Pluggingin a classical expansion for p ∈ S mH, cl ( R ) gives a similar expansion for g ◦ a . (cid:3) ZEGŐ KERNEL 13
Local Bergman kernels.
In this section we shall define certain local Bergman kernelsusing the coordinates introduced in Sec. 2.1. Furthermore these shall be shown to lie in thesymbol classes introduced in the previous section.First with the notation as in Theorem 6 one sets V = U C ∩ { q = 0 } ⊂ R p . With χ ( p , p ) ∈ C ∞ c ( R ) of sufficiently small support and equal to one near zero, the function(3.18) ϕ ( p , p ) := ϕ | q =0 + χ ( ϕ − ϕ ) | q =0 | {z } =: ϕ is well defined on R . This equals the restriction of ϕ to V near the origin, and hence we usethe same notation. Next set(3.19) ˆ Z := 12 ( ∂ p + i∂ p ) − i ∂ p ϕ + i∂ p ϕ ) ∂ p and define ¯ ∂ t :Ω , (cid:0) R (cid:1) → Ω , (cid:0) R (cid:1) ¯ ∂ t u := (cid:20)
12 ( ∂ p + i∂ p ) u + 12 t ( ∂ p ϕ + i∂ p ϕ ) u (cid:21) d ¯ z (3.20)with d ¯ z = dp − idp . Define the Kodaira Laplacian via (cid:3) t := ¯ ∂ ∗ t ¯ ∂ t (3.21)acting on Ω , . We denote by(3.22) B t : L (cid:0) R p (cid:1) → ker ( (cid:3) t ) the local Bergman projector onto the kernel of (cid:3) t and B t ( p, p ′ ) it Schwartz kernel.Replacing ϕ with its leading polynomial ϕ in (3.19), (3.20), (3.21) one analogously defines ¯ ∂ t u := (cid:20)
12 ( ∂ p + i∂ p ) u + 12 t ( ∂ p ϕ + i∂ p ϕ ) u (cid:21) d ¯ z (3.23) (cid:3) t := (cid:0) ¯ ∂ t (cid:1) ∗ ¯ ∂ t (3.24)as well as a corresponding Bergman projection B t with kernel B t (ˆ p, ˆ p ′ ) . Theorem 14.
One has B t (ˆ p, ˆ p ′ ) , B t (ˆ p, ˆ p ′ ) ∈ ˆ S r , r , cl , with furthermore B t (ˆ p, ˆ p ′ ) = t r b (cid:16) t r ˆ p, t r ˆ p ′ (cid:17) (3.25) B t (ˆ p, ˆ p ′ ) = t r b (cid:16) t r ˆ p, t r ˆ p ′ (cid:17) + ˆ S r , r , cl (3.26) for some b (ˆ p, ˆ p ′ ) ∈ ˆ S ( R × R ) .Proof. Being symmetric and bounded below, the Kodaira Laplacians (3.21), (3.24) are essen-tially self-adjoint. Furthermore under the rescaling/dilation δ t − /r ( ˆ p, ˆ p ′ ) = (cid:16) t − r ˆ p, t − r ˆ p ′ (cid:17) theseare seen to satisfy ⊡ t := t − /r ( δ t − /r ) ∗ (cid:3) t = (cid:3) ⊡ t := t − /r ( δ t − /r ) ∗ (cid:3) t = (cid:3) + t − /r E, (3.27)where E = a (ˆ p, t ) ¯ ∂ + b (ˆ p, t ) ¯ ∂ + c (ˆ p, t ) ZEGŐ KERNEL 14 is a self-adjoint operator with the coefficients a (ˆ p, t ) , b (ˆ p, t ) and c (ˆ p, t ) being uniformly (in t ) C ∞ bounded.Next, as in [31, Sec. 4.1], see also Prop. 23 below, we haveSpec (cid:0) (cid:3) t (cid:1) ⊂ { } ∪ (cid:2) c t /r , ∞ (cid:1) (3.28) Spec ( (cid:3) t ) ⊂ { } ∪ (cid:2) c t /r − c , ∞ (cid:1) . (3.29)We remark the fact that ϕ here is complex valued makes no difference to the above formulas(3.28), (3.29) so far as the leading part ϕ is real and sub-harmonic. This is because the higherorder Taylor coefficients of ϕ appear at lower order O (cid:0) t − /r (cid:1) after rescaling in (3.27). Hencefor any χ ∈ C ∞ c ( − c , c ) with χ = 1 near , the Bergman kernels equal B t (ˆ p, ˆ p ′ ) = χ (cid:0) t − /r (cid:3) t (cid:1) ( ˆ p, ˆ p ′ ) = t /r χ (cid:0) (cid:3) (cid:1) (cid:0) t /r p, t /r p ′ (cid:1) B t (ˆ p, ˆ p ′ ) = χ (cid:0) t − /r (cid:3) t (cid:1) (ˆ p, ˆ p ′ ) = t /r χ ( ⊡ t ) (cid:0) t /r p, t /r p ′ (cid:1) for t ≫ . By standard elliptic arguments, the Schwartz kernels of ∂ αp ∂ α ′ p ′ χ ( (cid:3) ) , ∂ αp ∂ α ′ p ′ χ ( ⊡ t ) , α, α ′ ∈ N , are rapidly decaying off-diagonal. Regarding their on-diagonal behavior, the growthof χ ( (cid:3) ) (ˆ p, ˆ p ) , χ ( ⊡ t ) (ˆ p, ˆ p ) as ˆ p → ∞ is controlled by the growth of the coefficient functionsof the operators (3.27), which in turn have polynomial growth. Hence B t , B t satisfy estimates3.3 with N (cid:16) ˆ α, ˆ β (cid:17) = r (cid:16) | ˆ α | + (cid:12)(cid:12)(cid:12) ˆ β (cid:12)(cid:12)(cid:12)(cid:17) , cf. also (5.47) below. This gives (3.25) with b (ˆ p, ˆ p ′ ) = χ (cid:0) (cid:3) (cid:1) (ˆ x, ˆ y ) ∈ ˆ S (cid:0) R × R (cid:1) B t (ˆ p, ˆ p ′ ) ∈ ˆ S r , r . (3.30)To show the classical expansion for the above one may use a full expansion of the operator ⊡ t (3.27) as in Section 5 below. We shall however give a different proof consistent with the restof this section. To this end, first begin with ϕ = ϕ + ϕ from (3.18), where(3.31) ϕ (ˆ p ) = O (cid:0) | ˆ p | r +1 (cid:1) ,ϕ (ˆ p ) ∈ C ∞ ( R , C ) . Next define the operator with distributional kernel ˜ B t : L ( R ) → L ( R )˜ B t (ˆ p, ˆ p ′ ) = e − tϕ (ˆ p ) B t (ˆ p, ˆ p ′ ) e tϕ (ˆ p ′ ) . (3.32)It is clear that the above ¯ ∂ t ˜ B t = 0 , (cid:3) t ˜ B t = 0 lies in the kernels of (3.20), (3.21). This gives B t ˜ B t = ˜ B t and(3.33) ˜ B ∗ t B t = ˜ B ∗ t , where ˜ B ∗ t is the adjoint of ˜ B t . Let R t := ˜ B t − ˜ B ∗ t whose Schwartz kernel is computed to be(3.34) R t (ˆ p, ˆ p ′ ) = e − tϕ (ˆ p ) B t (ˆ p, ˆ p ′ ) e tϕ (ˆ p ′ ) − e t ¯ ϕ (ˆ p ) B t (ˆ p, ˆ p ′ ) e − t ¯ ϕ (ˆ p ′ ) . Since (cid:0) ¯ ∂ + t (cid:0) ¯ ∂ϕ (cid:1) ∧ (cid:1) ( e tϕ B t ) = 0 , we have(3.35) ˜ B t B t = B t . From (3.33) and (3.35), we get ( I − R t ) B t = ˜ B ∗ t and hence(3.36) ( I − R Nt ) B t = ( I + R t + R t + · · · + R N − t ) ˜ B ∗ t , ∀ N ∈ N . ZEGŐ KERNEL 15
From the first part (3.30), (3.31), (3.34) and a Taylor expansion, it is easy to see that R t ∈ ˆ S r , r , cl and(3.37) R jt ∈ ˆ S (2 − j ) r , r , cl , ∀ j ∈ N , by an argument similar to Prop. 11. From the above, (3.30), (3.36), (3.37) and ˜ B ∗ t ∈ ˆ S r r , cl , thetheorem follows. (cid:3) Following the above we now prove one of our main theorems Theorem 1.
Proof of Theorem 1.
Choose B as in (3.22) and χ ∈ C ∞ c ( R ) a cutoff equal to one near zero.Define the operator ˆ B : C ∞ c (cid:0) R (cid:1) → C −∞ (cid:0) R (cid:1) ˆ B := 12 π Z ∞ dt e it ( p − p ′ ) B t ( p, p ′ ) χ (cid:16) t r − ǫ ˆ p ′ , t − r p ′ (cid:17) . (3.38)By definition using Theorem 14 and a Taylor expansion of the cutoff we see ˆ B ∈ ˆ L r , r , cl + ˆ L r , , −∞ r , ,for ˆ L r , , −∞ r , := T j ∈ N ˆ L − jr , j r . Furthermore for ǫ sufficiently small, that Schwartz kernel of theabove ˆ B ( p, p ′ ) can be shown to be smooth away from p = 0 and satisfies estimates similarto (2.5) in p . We let ˜ B ( z, p ′ ) denote the almost analytic continuation of the Schwartz kernel ˆ B ( p, p ′ ) in the p -variable given by Lemma 7. Now consider the coordinates ( x , x , x ) on aneighborhood U centered at the point x ′ ∈ X , along with ( p , p , p ) being (the real parts of)the corresponding almost analytic coordinates given by 6. Letting χ , χ ∈ C ∞ c ( U ) be suchthat χ = 1 on spt ( χ ) and χ = 1 near zero we set(3.39) B ( x, x ′ ) := χ ( x ) (cid:18) ˜ B (cid:12)(cid:12)(cid:12) y,y ′ =0 (cid:19) χ ( x ′ ) ∈ C −∞ ( X × X ) . Since ˆ Z ˆ B = 0 we have Z (cid:18) ˜ B (cid:12)(cid:12)(cid:12) y,y ′ =0 (cid:19) ∈ C ∞ by Lemma 7 from which is it easy to check that ¯ ∂ b B is smooth .Let Π : L ( X ) → H b ( X ) := (cid:8) u ∈ L ( X ) | ¯ ∂ b u = 0 (cid:9) denote the Szegő projection. Assuming ¯ ∂ b has closed range it was shown in [11, Prop. 4.1], [10] that there exists a bounded linearoperator G : Range (cid:0) ¯ ∂ b (cid:1) → L ( X ) such that Π = I − G ¯ ∂ b . Furthermore G is microlocal and itmaps G : Range (cid:0) ¯ ∂ b (cid:1) ∩ H s ( X ) → H s + r ( X ) , ∀ s ∈ R . It now follows that Π is microlocal or thatthe Szegő kernel is smooth away from the diagonal. Furthermore Π B = B − G ¯ ∂ b B = B + C ∞ and(3.40) B ∗ Π = B ∗ + C ∞ . Next, replace
Π ( x, by Π ( x ) = χ ( x ) Π ( x, , which has the same singularities near x = 0 , is compactly supported and satisfies similar bounds to (2.6). We almost analyti-cally continue Π ( x, in the x variable to define ˜Π ( z, . We may further suppose that ˜Π is compactly supported by construction. The restriction ˜Π ( p,
0) = ˜Π (cid:12)(cid:12)(cid:12) q =0 is well-definedand we set ˜Π ,t ( p , p ) := R e − itp ˜Π ( p, dp . Since Z Π ∈ C ∞ it follows that for the al-most analytic extension ˜ Z ˜Π is smooth from Lemma 7. From Theorem 6 it follows that ZEGŐ KERNEL 16 (cid:2) ( ∂ ˜ z + i∂ ˜ z ) − i ( ∂ ˜ z ϕ + i∂ ˜ z ϕ ) ∂ ˜ z (cid:3) ˜Π is smooth. Hence ¯ ∂ t ˜Π ,t := (cid:20)
12 ( ∂ p + i∂ p ) − i t ( ∂ p ϕ + i∂ p ϕ ) (cid:21) ˜Π ,t = O (cid:0) t −∞ (cid:1) in the Schwartz norm. From here it is clear that B t ˜Π ,t = ˜Π ,t + O ( t −∞ ) in the Schwartz norm.Thus one has (cid:16) ˆ B ˜Π − ˜Π (cid:17) ( p, ∈ C ∞ and hence by almost analytic continuation(3.41) (cid:16) ˜ B ˜Π − ˜Π (cid:17) ( x, ∈ C ∞ . Next for each N ∈ N define the operator with kernel B N ( x, x ′ ) = X | α | , | β |≤ N α ! β ! (cid:18) − ∂∂x ′ (cid:19) α " ( iy ( x ′ )) α ( iq ( x )) β (cid:18) ∂∂p (cid:19) β B ( p ( x ) , p ′ ( x ′ )) (cid:12)(cid:12)(cid:12)(cid:12) dpdx ( x ′ ) (cid:12)(cid:12)(cid:12)(cid:12) χ ( x ′ ) where p ( x ) = p ( x,
0) = x + O (cid:0) x (cid:1) q ( x ) = q ( x,
0) = O (cid:0) x (cid:1) y ( x ) = y ( p ( x, ,
0) = O (cid:0) x (cid:1) (3.42)denote the coordinates coming from the change of variables Theorem 6, while the multiplicationfactor (cid:12)(cid:12) dpdx (cid:12)(cid:12) is the Jacobian for the change of variables with respect to the first. Following anintegration by parts argument using (2.14), (3.41), Prop. 12, and Prop. 13 which motivatesthe construction of (3.42), it is easy to see that(3.43) ( B N Π − Π ) ( x, ∈ S − Nr H ∀ N ∈ N . Furthermore writing (3.42) in the p, p ′ coordinates gives B N +1 − B N ∈ ˆ L − Nr ,N r , cl + ˆ L r , , −∞ r , hence B ∞ := B + ∞ X N =0 ( B N +1 − B N ) ∈ ˆ L r , r , cl + ˆ L r , , −∞ r , (3.44)is well-defined by asymptotic summation. The last two equations (3.43), (3.44) then give(3.45) ( B ∞ Π − Π) ( x, ∈ C ∞ . Next the Schwartz kernel of the µ = e g ( x ) dx adjoint B ∗ of B is calculated to be(3.46) B ∗ ( x, x ′ ) = e g ( x ′ ) − g ( x ) χ ( x ′ ) ˜ B ( p ( x ′ ) , p ( x ) + iq ( x )) χ ( x ) . Following Theorem 6, a Taylor expansion, and writing in p, p ′ coordinates, it is easy to see B ∗ ∈ ˆ L r , r , cl + ˆ L r , , −∞ r , . Furthermore from the above and (3.42) it has the same principal symbolas B ∞ σ L ( B ∗ ) = σ L ( B ∞ ) hence R := B ∞ − B ∗ ∈ ˆ L r , r , cl + ˆ L r , , −∞ r , . (3.47) ZEGŐ KERNEL 17
Finally combining (3.40) and (3.45) we have
Π ( x,
0) = [ B ∞ Π] ( x,
0) + C ∞ = ( B ∗ + R ) Π ( x, B ∗ + R Π] ( x,
0) + C ∞ hence(3.48) (cid:2)(cid:0) I − R N (cid:1) Π (cid:3) ( x,
0) = (cid:2) R + R + . . . + R N − (cid:3) B ∗ ( x, (3.49) ∀ N ∈ N . Following 11, (3.47) we have R j ∈ ˆ L (2 − j ) r ,j r , cl + ˆ L r , , −∞ r , , j = 1 , , . . . . Hence byasymptotic summation ∃ P ∈ ˆ L r , r , cl such that(3.50) P N := P − (cid:2) R + R + . . . + R N − (cid:3) ∈ ˆ L (2 − N ) r ,N r , cl + ˆ L r , , −∞ r , ∀ N ∈ N . Since B ∗ ( x, ∈ S r H, cl by definition, we have(3.51) Π ( x,
0) = [
P B ∗ ] ( x,
0) + S − Nr H ∀ N ∈ N , from (2.6), (3.49), (3.50) and Prop. 13. Choosing N large gives Π ( x, ∈ S r H, cl using(3.17) and Prop. 13 which completes the proof on account of (3.14) and (3.15). (cid:3) The next remark shows that our parametrix Theorem 1 recovers the Boutet de Monvel-Sjöstrand parametrix at strongly pseudoconvex points.
Remark . (Strongly pseudoconvex points) Here we show that our main Theorem 1 recoversthe Boutet de Monvel-Sjöstrand description of the Szegő kernel at strongly pseudoconvex points x ′ ∈ X . As noted before, the type of a strongly pseudoconvex point is r x ′ = 2 . The two degree homogeneous polynomials in (2.7) and Theorem 6 can be further taken to be p ( x , x ) = x + x , ϕ (˜ z , ˜ z ) = ˜ z + ˜ z respectively. Following these, the model Bergman kernel is computed tobe an exponential B t (ˆ p, ˆ p ′ ) = tb (cid:16) t p, t p ′ (cid:17) B t (ˆ p, ˆ p ′ ) = tb (cid:16) t p, t p ′ (cid:17) = te − t Φ (ˆ p, ˆ p ′ ) b (ˆ p, ˆ p ′ ) = e − Φ (ˆ p, ˆ p ′ ) Φ (ˆ p, ˆ p ′ ) := 14 (cid:16) p + p + ( p ′ ) + ( p ′ ) + 2 p p ′ + 2 p p ′ + 2 ip ′ p − ip p ′ (cid:17) (3.52)[29, Sec. 4.1.6]. And hence ˜ B t ( p, p ′ ) = te − t [Φ (ˆ p, ˆ p ′ )+ ϕ (ˆ p ) − ϕ (ˆ p ′ )] Next the local Bergman kernel B t (3.22) is by 3.36 modulo C N a finite a sum of terms of theform ˜ B ∗ t (cid:16) ˜ B t ˜ B ∗ t (cid:17) k or (cid:16) ˜ B t ˜ B ∗ t (cid:17) k . Applying the complex stationary phase formula of Melin-Sjöstrand [34, Sec. 2], the kernels ofthe above take the form a k (ˆ p ′ , ˆ p, t ) e − t Φ k (ˆ p ′ , ˆ p ) , where a k ∈ S , cl (cid:0) R p,p ′ × R t (cid:1) is a classical symboland Φ k = Φ + O (cid:0) | ( p, p ′ ) | (cid:1) a phase function agreeing with (3.52) at leading order. To get theBoutet de Monvel-Sjöstrand description however one needs to ensure that all phase functions ZEGŐ KERNEL 18 Φ k agree. To this end, one may replace ˜ B t (3.32) with the alternate approximation for the localBergman kernel B t given by ˜ B t ( ˆ p, ˆ p ′ ) := e t Φ (ˆ p, ˆ p ′ ) B t ( ˆ p, ˆ p ′ )Φ ( ˆ p, ˆ p ′ ) := ϕ (ˆ p ) + ϕ (ˆ p ′ ) − X α,β (cid:16) ∂ αζ ∂ β ¯ ζ ϕ (cid:17) (0) ζ α ¯ ζ ′ β α ! β ! ,ζ := p + ip . The proof of Thm. Theorem 14, Theorem 1 all carry through with ˜ B t replacedby ˜ B t . The above further has the advantage of being self-adjoint ˜ B t (ˆ p, ˆ p ′ ) = ˜ B t ( ˆ p ′ , ˆ p ) and equals ˜ B t (ˆ p, ˆ p ′ ) = e t [Φ +Φ ] in the strongly pseudoconvex case again using (3.52). Furthermore, the composition of complexFourier integral operators and the complex stationary phase formula of Melin-Sjöstrand [34,Sec. 2] in this case gives(3.53) (cid:16) ˜ B t (cid:17) = a (ˆ p, ˆ p ′ , t ) e − t Φ(ˆ p, ˆ p ′ ) with the same phase function Φ for a ∈ S , cl (cid:0) R p,p ′ × R t (cid:1) a classical symbol. Following thisand repeating the argument for Theorem 14 with ˜ B t replaced by ˜ B t , the equations (3.32),3.34 and 3.36 are seen to give a similar form as (3.53) for the local Bergman kernel B t = a (ˆ p, ˆ p ′ , t ) e − t Φ(ˆ p, ˆ p ′ ) , a ∈ S , cl (cid:0) R p,p ′ × R t (cid:1) . Plugging this form for the local Bergman kernel intothe equations (3.39), (3.42), (3.46), (3.49) and (3.51) within the proof of Theorem 1, andanother use of the Melin-Sjöstrand formula gives(3.54) Π ( x,
0) = Z ∞ dt a (ˆ p, t ) e itp − t Φ(ˆ p, for some a ∈ S , cl (cid:0) R p × R t (cid:1) which is the pointwise version of the Boutet de Monvel-Sjöstrandform for the parametrix at strongly pseudoconvex points.We finally note that the reduction to the form (3.54) above is possible on account of theexplicit knowledge of the model Bergman kernel B t (3.52), related to Mehler’s formula for theharmonic oscillator ⊡ t , at a strongly pseudoconvex point. At points of higher type the modelkernel to contend with is less explicit, modeled on anharmonic oscillators, and one has to livewith the description (1.4). 4. Pseudoconvex domains
We now consider the special case when the CR manifold is the boundary of a domain D in C . Thus D ⊂ C is a relatively compact open subset with smooth boundary X = ∂D . TheCR structure on the boundary is simply obtained by restriction T , X = T , C ∩ T C X of thecomplex tangent space of C .We fix a Hermitian metric h · | · i on C T C so that T , C ⊥ T , C . The Hermitian metric h · | · i on C T C induces by duality, Hermitian metrics h · | · i on ⊕ ≤ p,q ≤ T ∗ p,q C , where T ∗ p,q C denote the bundles of ( p, q ) forms. With dv being the induced volume form on C let ( · | · ) D ZEGŐ KERNEL 19 and ( · | · ) C be the inner products on Ω ,q ( D ) and Ω ,q ( C ) defined by ( f | h ) D = Z D h f | h i dv, f, h ∈ Ω ,q ( D ) , ( f | h ) C = Z C h f | h i dv, f, h ∈ Ω ,q ( C ) . (4.1)Also denote by k · k D and k · k C be the corresponding norms and by L ( D ) , L ,q ) ( D ) thecorresponding spaces of square integrable functions. Let ρ ∈ C ∞ ( C , R ) be a defining functionof X satisfying ρ = 0 on X , ρ < on D and dρ | X = 0 . This maybe further chosen to satisfy k dρ k = 1 on X .Let ¯ ∂ : Ω ,q ( C ) → Ω ,q +1 ( C ) be the exterior differential operator and consider its formaladjoint ∂ ∗ f :Ω , ( C ) → C ∞ ( C ) satisfying ( ¯ ∂f | h ) C = ( f | ∂ ∗ f h ) C , f ∈ C ∞ c ( C ) , h ∈ Ω , ( C ) . Also denote by ¯ ∂ ∗ : L , ( D ) → L ( D ) the L adjoint of ¯ ∂ , as an unbounded operator, withrespect to ( · | · ) D . The Bergman kernel of the domain is the distributional kernel Π D ( z, z ′ ) ∈ C −∞ ( D × D ) of the orthogonal projection Π D : L ( D ) → Ker ¯ ∂ ⊂ L ( D ) with respect to ( · | · ) D . The goal of this section is to establish an asymptotic expansion for Π D ( z, z ) as z → x ′ approaches a point on the boundary x ′ ∈ X .This shall use the relation of the Bergman kernel with the Szegő kernel of the boundary viathe Poisson operator [6, Sec. 3b]. To state this, let (cid:3) f = ¯ ∂ ¯ ∂ ∗ f + ¯ ∂ ∗ f ¯ ∂ : C ∞ ( C ) → C ∞ ( C ) denote the complex Laplace-Beltrami operator on functions and γ the operator of restrictionto the boundary X . The Poisson operator is the solution operator to the Dirichlet problem on D defined by P : C ∞ ( X ) → C ∞ ( D ) (4.2) (cid:3) f P u = 0 , γP u = u, ∀ u ∈ C ∞ ( X ) . (4.3)Its adjoint is defined to satisfy P ∗ : C ∞ ( D ) → C −∞ ( X ) (4.4) ( P ∗ u | v ) X = ( u | P v ) D , u ∈ C ∞ ( D ) , v ∈ C ∞ ( X ) (4.5)The microlocal structure of P was described by Boutet de Monvel [4]. Firstly, from [4, pg.29] the operators P, P ∗ extend continuously P : H s ( X ) → H s + ( D ) ,P ∗ : H s ( D ) → H s + ( X ) , ∀ s ∈ R , and in particular map smooth functions onto smooth ones. Furthermore, P ∗ P : C ∞ ( X ) → C ∞ ( X ) is an injective continuous operator and its inverse ( P ∗ P ) − is a classical elliptic pseu-dodifferential operator of order one on X . Its principal symbol is given by(4.6) σ ( P ∗ P ) − = σ ( √−△ X ) , ZEGŐ KERNEL 20 [22] where σ ( √−△ X ) denotes the principal symbol of the square root of the Laplace-Beltramioperator. Next, there is a continuous operator G : H s ( D ) → H s +2 ( D ) , ∀ s ∈ R , satisfying(4.7) G (cid:3) f + P γ = I on C ∞ ( D ) . (4.8)It furthermore follows from the methods of [4] that the Schwartz kernel of G satisfies theestimates(4.9) (cid:12)(cid:12) ∂ αz ∂ βw G ( z, w ) (cid:12)(cid:12) ≤ C | z − w | − −| α |−| β | ∀ z, w ∈ D , α, β ∈ N , along the diagonal.With Γ ∧ : T ∗ ,q C → T ∗ ,q +1 C Γ ∧ , ∗ : T ∗ ,q +1 C → T ∗ ,q C , ∀ Γ ∈ T ∗ , C , denoting the wedge and contraction, adjoint with respect to h · | · i , operators, one has I = 2( ¯ ∂ρ ) ∧ ( ¯ ∂ρ ) ∧ , ∗ + 2( ¯ ∂ρ ) ∧ , ∗ ( ¯ ∂ρ ) ∧ , on Ω ,q ( C ) , ¯ ∂ b = 2 γ ( ¯ ∂ρ ) ∧ , ∗ ( ¯ ∂ρ ) ∧ ¯ ∂P on C ∞ ( X ) . (4.10)In using the above to describe the behavior of Π D near a boundary point x ′ ∈ X one usesthe parametrix construction for the Szegő kernel Π from Theorem 1. Let ( x , x , x ) be thelocal coordinates on an open set x ′ ∈ U ⊂ X on the boundary as in the proof of Theorem 1with B the operator (3.39) therein. Since B is smoothing away the diagonal we may againassume that B is properly supported on U . It is well-known that (cid:3) b is elliptic outside itscharacteristic variety Σ := HX ⊥ ⊂ T ∗ X given by the annihilator of the Levi distribution HX .The characteristic variety Σ carries an orientation given by J t dρ , with J t denoting the dualcomplex structure on T ∗ C , and we denote by Σ − its negatively oriented part. By construction(3.38) we have W F ( Bu ) ⊂ Σ − ∩ T ∗ U, ∀ u ∈ C −∞ ( U ) . Let ˜ U be an open set of C with ˜ U ∩ D = U . We then have the next lemma. Lemma 16.
The Schwartz kernel ¯ ∂P B ( z, x ) ∈ C ∞ (cid:16)(cid:16) ˜ U × U (cid:17) ∩ (cid:0) D × X (cid:1)(cid:17) Proof.
From (4.10), we have γ ( ¯ ∂ρ ) ∧ , ∗ ( ¯ ∂ρ ) ∧ ¯ ∂P B = ¯ ∂ b B ∈ C ∞ . Combining this with (cid:0) ¯ ∂ ∗ f ¯ ∂ (cid:1) P = 0 , P γ ¯ ∂P = ¯ ∂P , we have ∂ ∗ f ¯ ∂P B = ¯ ∂ ∗ f P γ ¯ ∂P B = ¯ ∂ ∗ f P γ (cid:0) I −
2( ¯ ∂ρ ) ∧ , ∗ ( ¯ ∂ρ ) ∧ (cid:1) ¯ ∂P B + C ∞ = ¯ ∂ ∗ f P γ (cid:0)
2( ¯ ∂ρ ) ∧ ( ¯ ∂ρ ) ∧ , ∗ (cid:1) ¯ ∂P B + C ∞ . (4.11)From (4.11), we deduce that γ ( ¯ ∂ρ ) ∧ ¯ ∂ ∗ f P γ (2( ¯ ∂ρ ) ∧ ( ¯ ∂ρ ) ∧ , ∗ ) ¯ ∂P B ∈ C ∞ . It is known that [22, Propsition 4.2] that γ ( ¯ ∂ρ ) ∧ ¯ ∂ ∗ f P : C ∞ ( X, I , T ∗ C ) → C ∞ ( X, I , T ∗ C ) ZEGŐ KERNEL 21 is elliptic near Σ − , where I , T ∗ C is the vector bundle over C with fiber I , T ∗ C = (cid:8) ( ¯ ∂ρ )( z ) ∧ g ; g ∈ T ∗ , z C (cid:9) . Since
WF ( Bu ) ⊂ Σ − ∩ T ∗ U , u ∈ C −∞ ( U ) , we get γ (2( ¯ ∂ρ ) ∧ ( ¯ ∂ρ ) ∧ , ∗ ) ¯ ∂P Bu is smooth and hence γ (2( ¯ ∂ρ ) ∧ ( ¯ ∂ρ ) ∧ , ∗ ) ¯ ∂P B ∈ C ∞ as required. (cid:3) In view of [22, Lemmas 4.1, 4.2], we see that P and ( P ∗ P ) − P ∗ are smoothing away fromthe diagonal. Hence, they maybe replaced by continuous properly supported operators L : C ∞ c ( ˜ U ∩ D ) → C ∞ c ( U ) , ˆ P : C ∞ c ( U ) → C ∞ c ( ˜ U ∩ D ) such that L − ( P ∗ P ) − P ∗ ≡ C ∞ (( U × ˜ U ) ∩ ( X × D )) , ˆ P − P ≡ C ∞ (( ˜ U × U ) ∩ ( D × X )) . (4.12)We now set(4.13) A := ˆ P BL : C ∞ ( ˜ U ∩ D ) → C ∞ ( ˜ U ∩ D ) . From Lemma 16, we see that(4.14) ¯ ∂A ≡ C ∞ (( ˜ U × ˜ U ) ∩ ( D × D )) . Since the boundary X is of finite type, it was proved by Kohn [25] that there is a pseudolocalcontinuous operator N : L ( M ) → L , ( M ) ∩ Dom ¯ ∂ ∗ ,N : C ∞ ( D ) → Ω , ( D ) , such that(4.15) Π D = I − ¯ ∂ ∗ N ¯ ∂. From (4.14) and (4.15), we deduce that(4.16) Π D A ≡ A mod C ∞ (( ˜ U × ˜ U ) ∩ ( D × D )) . Next we take ˜ U small enough enough so that z = ( x, ρ ) form local coordinates of ˜ U .The Bergman kernel Π D is then known to satisfy the bounds(4.17) (cid:12)(cid:12)(cid:12) ∂ γρ ∂ γ ′ ρ ′ ∂ αx Π D (( x, ρ ) , (0 , ρ ′ )) (cid:12)(cid:12)(cid:12) ≤ C αγγ ′ (cid:0) | ρ | + | ρ ′ | + d H ( x ) (cid:1) − w.α − γ − γ ′ − r ( x ′ ) − , ∀ ( α, γ, γ ′ ) ∈ N , similar to (2.5) (see [32, 35]). This gives corresponding estimates for the kernel γ Π D (( x, ρ ) , (0 , ρ ′ )) which satisfies ¯ ∂ b γ Π D = 0 . Following these, we can repeat the procedurein the proof of Theorem 1 to conclude(4.18) ( B ∞ γ Π D )(( x, , (0 , ρ ′ )) ≡ γ Π D (( x, , (0 , ρ ′ )) mod C ∞ (( ˜ U × ˜ U ) ∩ ( X × D )) , as with eqn. (3.45).Next Π D = P γ Π D gives P ∗ Π D = P ∗ P γ Π D and hence ( P ∗ P ) − P ∗ Π D = γ Π D . This combines with (4.18) to give ( A ∞ Π D )( z, (0 , ρ ′ )) ≡ Π D ( z, (0 , ρ ′ )) mod C ∞ (( ˜ U × ˜ U ) ∩ ( D × D )) (4.19) for A ∞ := ˆ P B ∞ L. We now have the following proposition.
ZEGŐ KERNEL 22
Lemma 17.
One has Π D ( z, (0 , ρ )) = ( P QP ∗ )( z, (0 , ρ )) + C ∞ (cid:16) ˜ U × R ρ (cid:17) for some properly supported Q ∈ ˆ L r r , cl .Proof. Denote by(4.20) A ∗ : C ∞ c ( ˜ U ∩ D ) → C ∞ ( ˜ U ∩ D ) , be the adjoint of A with respect to ( · | · ) . From (4.16), we have(4.21) A ∗ Π D ≡ A ∗ mod C ∞ (( ˜ U × ˜ U ) ∩ ( D × D )) . Thus ( A ∞ Π D )( z, (0 , ρ ′ ))= ( A ∗ Π D )( z, (0 , ρ ′ )) + (( A ∞ − A ∗ )Π D )( z, (0 , ρ ′ )) ≡ A ∗ ( z, (0 , ρ ′ )) + ( R Π D )( z, (0 , ρ ′ )) mod C ∞ (( ˜ U × ˜ U ) ∩ ( D × D )) , (4.22)for R := A ∞ − A ∗ = P B ∞ ( P ∗ P ) − P ∗ − P ( P ∗ P ) − B ∗ P ∗ mod C ∞ (( ˜ U × ˜ U ) ∩ ( D × D ))= P ( P ∗ P ) − (cid:16) ( P ∗ P ) B ∞ ( P ∗ P ) − − B ∗ (cid:17)| {z } E := P ∗ mod C ∞ (( ˜ U × ˜ U ) ∩ ( D × D )) . (4.23)From (4.19) and (4.22), we get ( I − R )Π D ( z, (0 , ρ ′ )) ≡ A ∗ ( z, (0 , ρ )) mod C ∞ (( ˜ U × R ρ ) ∩ ( D × R ρ )) (4.24) (cid:16) ( I − R N )Π D (cid:17) ( z, (0 , ρ ′ )) ≡ (cid:16) ( I + R + R + · · · + R N − ) A ∗ (cid:17) ( z, (0 , ρ )) mod C ∞ (( ˜ U × R ρ ) ∩ ( D × R ρ )) . ∀ N ∈ N .From (3.9) and (3.11) one has E ∈ ˆ L r , r , cl and thus E N ∈ ˆ L (2 − N ) r ,N r , cl , ∀ N ∈ N . By Prop. 12,Prop. 11 and asymptotic summation ∃ Q ∈ ˆ L r r , cl such that Q − ( P ∗ P ) − (cid:16) ( I + E + E + · · · E N ) B ∗ (cid:17) ∈ ˆ L − N ) r ,N r , cl . Thus for each l , ∃ N l ∈ N such that P QP ∗ − (cid:16) ( I + R + R + · · · + R N − ) A ∗ (cid:17) ( z, (0 , ρ ))= P h Q − ( P ∗ P ) − (cid:16) ( I + E + E + · · · + E N − ) B ∗ (cid:17)i P ∗ ∈ C l (( ˜ U × R ρ ) ∩ ( D × R ρ )) (4.25) ∀ N ≥ N l . Finally from the kernel estimates (4.17), for each l ≥ , ∃ N ′ l ∈ N such that(4.26) ( R N Π D )(( x, , (0 , ρ )) ∈ C ℓ (( ˜ U × R ρ ) ∩ ( D × R ρ )) . ∀ N ≥ N ′ l . From 4.24, (4.25) and 4.26 the Lemma follows. (cid:3) Next let (cid:3) f = ¯ ∂ ¯ ∂ ∗ f + ¯ ∂ ∗ f ¯ ∂ : C ∞ ( C ) → C ∞ ( C ) denote the complex Laplace-Beltrami operator on functions and we denote q the principlesymbol of (cid:3) f . Repeating the proof of [22, Prop. 7.6] one has the following. ZEGŐ KERNEL 23
Lemma 18.
There exists a smooth function φ ( z, y ) ∈ C ∞ (( ˜ U × U ) ∩ ( D × X )) such that φ ( x, y ) = x − y ,φ ( z, y ) = x − y − iρ q − σ △ X ( x, (0 , , O ( | ρ | ) ,q ( z, d z φ ) vanishes to infinite order on ρ = 0 , (4.27) where Re p − σ △ X ( x, (0 , , > , z = ( x, ρ ) . For our next result we shall need an extension of the symbol spaces Definition 8. Namelyone may similarly define the classes ˆ S m r (cid:0) C × R × R t (cid:1) , ˆ S m,k r (cid:0) C × R × R t (cid:1) , ˆ S m,k r , cl (cid:0) C × R × R t (cid:1) (4.28) ˆ S m r (cid:0) C × C × R t (cid:1) , ˆ S m,k r (cid:0) C × C × R t (cid:1) , ˆ S m,k r , cl (cid:0) C × C × R t (cid:1) (4.29)as functions depending on additional ρ or ρ, ρ ′ variables. The additional variables appear in afashion similar to the x , y variables in the symbolic estimates and expansions. That is theequations 3.3, (3.4) and (3.5) are replaced by (cid:12)(cid:12) ∂ αx ∂ α ρ ∂ βy ∂ γt a ( x, y, t ) (cid:12)(cid:12) ≤ C N,αβγ h t i m − γ + r ( | ˆ α | + | ˆ β | ) + α + α + β (cid:16) (cid:12)(cid:12)(cid:12) t r ˆ x (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) t r ˆ y (cid:12)(cid:12)(cid:12)(cid:17) N ( α,β,γ ) (cid:16) (cid:12)(cid:12)(cid:12) t r ˆ x − t r ˆ y (cid:12)(cid:12)(cid:12)(cid:17) − N , (4.30) ∀ ( x, y, t, N ) ∈ R x,y × R t × N . (4.31) ˆ S m,k r := M p + q + p ′ ≤ k ( tx ) p ( tρ ) q ( ty ) p ′ ˆ S m r , ∀ ( m, k ) ∈ R × N . (4.32) a ( x, y, t ) − N X j =0 X p + q + p ′ ≤ j t m − r j ( tx ) p ( tρ ) q ( ty ) p ′ a jpp ′ (cid:16) t r ˆ x, t r ˆ y (cid:17) ∈ ˆ S m − ( N +1) r ,N +1 r in defining ˆ S m r ( C × R × R t ) , ˆ S m,k r ( C × R × R t ) , ˆ S m,k r , cl ( C × R × R t ) respectively. Andsimilarly for the classes (4.29).We now have the following Lemma 19.
Let H = h L ∈ ˆ L m,k r , cl be an operator in the class 9 with distribution kernel H ( x, y ) = h L ( x, y ) = Z ∞ e i ( x − y ) t h ( x, y, t ) dt. Then there exists α ( z ; y, t ) ∈ ˆ S m,k r , cl ( C × R × R t ) , with α ( x, y, t ) = h ( x, y, t ) , such that Λ( z, y ) = Z ∞ e iφ ( z,y ) t α ( z, y, t ) dt with ( P H − Λ) ((0 , ρ ) , y ) ∈ C ∞ ( R ρ × U ) . Proof.
Denote the Riemannian metric on T C , induced from the Hermitian metric h · | · i , by g = X j,k =1 g j,k ( z ) dx j ⊗ dx k , dx = dρ ZEGŐ KERNEL 24 and let ( g j,k ( z )) − ≤ j,k ≤ = (cid:0) g j,k ( z ) (cid:1) ≤ j,k ≤ be the inverse metric on T ∗ C . In the local coordinates z = ( x, ρ ) chosen one has (cid:3) f = − (cid:16) a , ( z ) ∂ ∂ρ + 2 X j =1 a ,j ( z ) ∂ ∂ρ∂x j + T ( ρ ) (cid:17) + first order , where(4.33) T ( ρ ) = X j,k =1 a j,k ( z ) ∂ ∂x j ∂x k . (4.34)Furthermore, T (0) − △ X is a first order operator on the boundary with a , ( x ) = 1 ,a ,j ( x ) = 0 , j = 1 , . . . , . (4.35)From the above (4.27), (4.33), (4.34) and (4.35) we now compute (cid:3) f (cid:20)Z ∞ e iφ ( z,y ) t β ( x, y, t ) dt (cid:21) ≡ Z ∞ e iφ ( z,y ) t (cid:20) t q − σ △ X ( x, (0 , , ∂ ρ β + Lβ (cid:21) dt mod C ∞ (( ˜ U × ˜ U ) ∩ ( D × X )) , where L = ( tρb ( z, y ) + b ( z, y )) ∂ ρ + L ,x + tL ,x (4.36)for some smooth t − independent functions b , b and second/first order differential operators L ,x , L ,x respectively in the ( x , x , x ) variables. It is easy to check that the above maps ρt L : ˆ S m,k r , cl → ˆ S m,k r , cl (cid:16) ρt L (cid:17) N : ˆ S m,k r , cl → ρ N − k ˆ S m,k r , cl + ˆ S m − Nr ,k + N r , cl , N ≥ k. Next setting h ( z, y, t ) := h ( x, y, t ) − ρt p − σ △ X ( x, (0 , , Lh = h ( x, y, t ) − ρt p − σ △ X ( x, (0 , , L ,x + tL ,x ) h ∈ ˆ S m,k r , cl and following (4.36) one computes (cid:3) f (cid:20)Z ∞ e iφ ( z,y ) t h ( z, y, t ) dt (cid:21) ≡ Z ∞ e iφ ( z,y ) t r ( z, y, t ) dt mod C ∞ (( ˜ U × ˜ U ) ∩ ( D × X )) ,r ∈ ρ − k ˆ S m +2 ,k r , cl + ˆ S m +2 − r ,k +1 r , cl , ≥ k. Continuing in this way, we can find h N ( z, y, t ) ∈ ˆ S m,k r , cl such that h N +1 − h N ∈ ρ N − k ˆ S m,k r , cl + ˆ S m − Nr ,k + N r , cl and (cid:3) f (cid:20)Z ∞ e iφ ( z,y ) t h N ( z, y, t ) dt (cid:21) ≡ Z ∞ e iφ ( z,y ) t r N ( z, y, t ) dt mod C ∞ (( ˜ U × ˜ U ) ∩ ( D × X )) r N ∈ ρ N − k ˆ S m +2 ,k r , cl + ˆ S m +2 − Nr ,k + N r , cl , N ≥ k. ZEGŐ KERNEL 25
By asymptotic summation we can find α := h + P ∞ N =1 ( h N +1 − h N ) ∈ ˆ S m,k r , cl satisfying (cid:3) f Z ∞ e iφ ( z,y ) t α ( z, y, t ) dt | {z } =Λ ≡ Z ∞ e iφ ( z,y ) t r ∞ ( z, y, t ) dt mod C ∞ (( ˜ U × ˜ U ) ∩ ( D × X )) r ∞ ∈ ρ ∞ ˆ S m +2 ,k r , cl + ˆ S m +2 ,k, −∞ r , cl γ Λ = H. Finally we apply the Green’s operator G (4.7) to both sides of the above equation to get G (cid:3) f ((0 , ρ ) , y ) = Z dw Z ∞ dtG ((0 , ρ ) , w ) e iφ ( w,y ) t r ∞ ( w, y, t ) dt . Writing w = ( u , u , u , ρ ′ ) and repeated integration by parts using e iu t = it ∂ u e iu t , (4.9) and(4.27) gives the above G (cid:3) f ((0 , ρ ) , y ) ∈ C ∞ ( R ρ × U ) and hence [ P H − Λ] ((0 , ρ ) , y ) ∈ C ∞ ( R ρ × U ) from (4.8) as required. (cid:3) Similarly, as Lemma 18, we can find Φ( z, w ) ∈ C ∞ (( ˜ U × ˜ U ) ∩ ( D × D )) such that Φ( z, y ) = φ ( z, y ) and Φ( z, w ) = φ ( z, y ) − iρ ′ q − σ △ X ( y, (0 , , O ( | ρ ′ | ) ,q ( w, − d w Φ) vanishes to infinite order on ρ ′ = 0 . (4.37)A similar argument to Lemma 19 then gives the following. Lemma 20.
Let H = h L ∈ ˆ L m,k r , cl be an operator in the class 9 with distribution kernel H ( x, y ) = h L ( x, y ) = Z ∞ e i ( x − y ) t h ( x, y, t ) dt. Then there exists α ( z ; w, t ) ∈ ˆ S m,k r , cl ( C × R × R t ) , with α ( x, y, , t ) = h ( x, y, t ) , such that Λ( z, w ) = Z ∞ e i Φ( z,w ) t α ( z, w, t ) dt with ( P HP ∗ − Λ) ((0 , ρ ) , y ) ∈ C ∞ ( R ρ × R ρ ′ ) . Finally from the above we can prove one of our the main theorems Theorem 2. Setting z = (0 , ρ ) , w = (0 , ρ ) in 17 and the above 20 gives Π D ((0 , ρ ) , (0 , ρ )) = Z ∞ e i Φ((0 ,ρ ) , (0 ,ρ )) t α ((0 , , ρ )) , (0 , , ρ ); t ) dt + C ∞ ( R ρ ) . for α ∈ ˆ S r r , cl ( ˜ U × ˜ U × R + ) . Plugging the classical symbolic expansion for α into the above andusing Φ((0 , ρ ) , (0 , ρ )) = − iρ q − σ △ X (0 , (0 , , O ( | ρ | ) ZEGŐ KERNEL 26 gives Π D ((0 , ρ ) , (0 , ρ )) = N X j =0 − ρ ) r − r j a j + N X j =0 b j ( − ρ ) j log ( − ρ ) + O (cid:16) ( − ρ ) N − − rr (cid:17) , ∀ N ∈ N , proving Theorem 2. 5. S invariant case In this section we investigate the Szegő kernel parametrix in the circle invariant case obtaininga more concrete version of our main theorem Theorem 1.Thus we now assume that X is equipped with a CR S -action which is transversal; that isthe generator T of the S action satisfies (cid:2) T, C ∞ (cid:0) T , X (cid:1)(cid:3) ⊂ C ∞ (cid:0) T , X (cid:1) (5.1) C [ T ] ⊕ T , X ⊕ T , X = T X ⊗ C . (5.2)Denote by s x := | S x | cardinality of the stabilizer S x := (cid:8) e iθ ∈ S | e iθ x = x (cid:9) of the point x ∈ X with respect to the circle action. For a locally free circle action as above the function x s x is also upper semi-continuous with the compactness of X implying s := max x ∈ X s x < ∞ . Wefurther set s := min x ∈ X s x , X i,j := { x ∈ X | s x = j, r x = i } to obtain a decomposition of themanifold X = S s,ri =1 ,j =2 Y i,j where each X ≤ i, ≤ j := S i,ji ′ =1 ,j ′ =2 Y i,j is open and X s , ≤ r ⊂ X is dense.The m -th Fourier mode of the Szegő kernel Π m ( x, x ′ ) , m ∈ Z , taken with respect to an S -invariant volume form µ , is now a smooth function on the product. What corresponds to thesingularity in (1.4) is its asymptotic behavior as m → ∞ . This is described below and is themain theorem of this section. Theorem 21.
Let X be a compact pseudoconvex three dimensional CR manifold of finite typeadmitting a transversal, CR circle action. The m -th Fourier mode of the Szegő kernel has thepointwise expansion on diagonal (5.3) Π m ( x, x ) = φ m ( s x ) " m /r x N X j =0 c j ( x ) m − j/r x + O (cid:0) m − N/r x (cid:1) , ∀ N ∈ N , as m → ∞ . Here each c j is a smooth function on X , with the leading term c = Π g HXx ,j rx − L ,J HX (0 , > given in terms of certain model Bergman kernels on the Levi-distribution HX at x and thephase factor φ m ( s x ) := P s x − l =0 e i πlmsx = ( s x ; s x | m, s x ∤ m. We first begin with some requisite CR geometry in the circle invariant case.5.1. S invariant CR geometry. Let HX := Re ( T , X ⊕ T , X ) ⊂ T X be the real Levidistribution. The volume form µ is further assumed to be S invariant. We let h T , X be an S invariant Hermitian metric on T , X and denote by h T , X the invariant Hermitian metricon T , X . This gives one h T X on T X ⊗ C for which (5.1) is an orthogonal decomposition with | T | = 1 . We also denote by g T X the induced Riemannian metric on the real tangent spaceand by h , i the corresponding C -bilinear form on T X ⊗ C . It is easy to see that h T , X may bechosen so that the volume form µ arises as the Riemannian volume of such an invariant metric.Such a S -invariant CR manifold is locally the unit circle bundle of a Hermitian, holomorphicline bundle (cid:0) L, h L (cid:1) X = S L π −→ Y ; (5.4) ZEGŐ KERNEL 27 on a complex manifold Y , dim C Y = 1 . To describe the equivalence, choose a local hypersurface Y ⊂ X through a given point x ∈ X transversal to generator T ⋔ T Y satisfying T x Y = ( HX ) x .The map T Y ֒ → T X → T X/ R [ T ] ∼ = HX is then an isomorphism inducing a integrable complexstructure on T Y and a corresponding Hermitian metric h T , Y on its complex tangent spacefrom h T , X . We choose S × Y =: X → Y to be the trivial circle bundle with the map ι : X → X ; ι (cid:0) y, e iθ (cid:1) = ye iθ being a local diffeomorphism between collar neighborhoods ι : (cid:18) − πs x , πs x (cid:19) × Y ∼ −→ N Y ⊂ Xs x := | S x | (5.5)of the zero section of X and Y ⊂ X . The Levi-distribution then defines a horizontal distri-bution on X giving corresponding connections ∇ L on X and the associated Hermitian linebundle L := C × Y → Y corresponding to the trivial representation of S . By the integrabilitycondition the curvature of the corresponding connection is a (1 , form on Y and hence the (0 , part of the connection prescribes a holomorphic structure on L . It is now clear that X it the unit circle bundle of L with ι being the required CR isomorphism by definition. We alsonote that pseudoconvexity of the CR structure corresponds to semi-positivity of the curvature R L of ∇ L .We may also obtain a local coordinate expression for the CR structure. To this end, startwith a local orthonormal basis { e , e = J e } of T x Y = ( HX ) x . Using the exponential mapobtain a geodesic coordinate system on a geodesic ball B ̺ ( x ) centered at x ∈ Y . The point x corresponding to point ( y, l y ) ∈ ( S L ) y in the fiber above y , one may parallel transport l y along geodesic rays in Y to obtain a local orthonormal frame l for L . In such a parallel framethe connection on the tensor product is of the following form ∇ Λ , ∗ ⊗ L m = d + a Λ , ∗ + ma L a Λ , ∗ j = Z dρ (cid:16) ρy k R Λ , ∗ jk ( ρx ) (cid:17) a Lj = Z dρ (cid:0) ρy k R Ljk ( ρx ) (cid:1) (5.6)in terms of the respective curvatures of ∇ T , X , ∇ L see [31, Sec. 4]. The connection one formmay further be written a L + ia L = ∂ z ϕ = 1 r x ¯ z X | α | = r x − R Lα y α + O ( y r x ) (5.7)in terms of a potential function and a Taylor expansion with the tensor R Lα denoting the firstnon-vanishing jet of the curvature at x . In some local coordinate system ( θ, y ) ∈ (cid:16) − πs x , πs x (cid:17) × B ̺ ( x ) , s x := | S x | , on an open set U ⊂ X the CR structure on X is then locally given by T , X = C [ ∂ z + i ( ∂ z ϕ ) ∂ θ ] ,T = ∂ θ , (5.8) z = y + iy , by construction. A trivialization/coordinate system in which the CR looks asabove (5.8) is referred to as a BRT trivialization [2, Thm II.1].
ZEGŐ KERNEL 28
Following its local description as a unit circle bundle, several notions/formulas from complexgeometry carry over to the S -invariant CR geometry of X . Firstly, the tangential CR operator ¯ ∂ b locally corresponds to Dolbeault differential ¯ ∂ on Y (5.4) under pullback ¯ ∂ b ( π ∗ ω ) = π ∗ ¯ ∂ω , ω ∈ Ω , ∗ ( Y ) and similarly for their adjoints. An analog of the Chern connection then existson T , X . This is the unique ( S -invariant) connection ∇ T , X compatible with h , i satisfying ∇ T , XT = L T and whose (0 , -component agrees with the tangential CR operator ∇ T , XU = i U ¯ ∂ b , U ∈ T , X. Locally, ∇ T , XT is just the pullback of the Chern connection ∇ T , Y from Y . Complex conju-gation then defines a connection ∇ T , X on ( T , X ) ∗ . We denote by the same notation dualconnections on T , X/T , X and set ˜ ∇ T X := d ⊕ ∇ T , X ⊕ ∇ T , X to be a connection on T X ⊗ C with d denoting the trivial connection on C [ T ] under the decomposition (5.1). The connection ˜ ∇ T X preserves the real tangent space
T X ⊂ T X ⊗ C and we denote by T its torsion. Thetorsion T maps T , X ⊗ T , X into T , X (respectively for T , X ), T , X ⊗ T , X into C [ T ] and vanishes on HX ⊗ C [ T ] . Indeed its components involving the generator T are T ( ., T ) = T ( T, . ) = 0 (cid:10) T (cid:0) U, ¯ V (cid:1) , T (cid:11) = i L (cid:0) U, ¯ V (cid:1) , U, V ∈ T , X. Next with ∇ T X being the Levi-Civita connection and P HX being the horizontal projectiononto HX , define a new connection on T X via ∇ HX := d ⊕ P HX ∇ T X with respect to thedecomposition
T X = R ⊕ HX . Locally; ∇ HX is the pullback of the Levi-Civita connection ∇ T Y from Y (5.4). The torsion T HX of ∇ HX thus has T HX ( U, V ) = i L (cid:0) U, ¯ V (cid:1) T, U, V ∈ HX, as its only non-vanishing component. The difference S := ˜ ∇ T X − ∇ HX maybe computed h S ( U ) V, W i = 0 , (cid:10) S ( U ) ¯ V , W (cid:11) = − (cid:10) T (cid:0) U, ¯ V (cid:1) , W (cid:11) ; (5.9) (cid:10) S ( T ) U, ¯ V (cid:11) = i L (cid:0) U, ¯ V (cid:1) , (cid:10) S ( U ) ¯ V , T (cid:11) = − (cid:10) S ( U ) T, ¯ V (cid:11) = 0 , (5.10) U, V, W ∈ T , X, in terms of the torsion and Levi forms.One next defines the Bismut connection ∇ B on T X via ∇ B := ∇ HX + S B ; (cid:10) S B ( T ) U, ¯ V (cid:11) = i L (cid:0) U, ¯ V (cid:1)(cid:10) S B ( U ) V, W (cid:11) :=0 ,U, V, W ∈ T , X. Its horizontal projection P HX ∇ B locally agrees with the pullback of theBismut connection of Y [29, Def. 1.2.9]. The connection ∇ B preserves the decomposition (5.1)and hence induces a connection on T , X , ( T , X ) ∗ and their exterior powers which we againdenote by ∇ B . Finally set(5.11) ∇ B, Λ , ∗ := ∇ B + h S ( . ) w, ¯ w i ,w ∈ T , X , | w | = 1 . We now define the Clifford multiplication endomorphism c : ( T X ) ∗ → End (cid:0) Λ ∗ T , X (cid:1) c ( v ) := √ (cid:0) v , ∧ − i v , (cid:1) , ∀ v ∈ ( HX ) ∗ ,c ( θ ) ω := ± ω, ∀ ω ∈ Λ even/odd T , X. ZEGŐ KERNEL 29
Next, the Kohn-Dirac operator D b := √ (cid:0) ¯ ∂ b + ¯ ∂ ∗ b (cid:1) (5.12) = c ◦ ∇ B, Λ , ∗ ,H maybe written as the composition of Clifford multiplication with the horizontal component ofthe Bismut connection ∇ B, Λ , ∗ ,H := π H ◦ ∇ B, Λ , ∗ : C ∞ (cid:0) X ; Λ , ∗ (cid:1) → C ∞ (cid:0) X ; H ∗ X ⊗ Λ , ∗ (cid:1) (cf. [29, Thm 1.4.5]). We then have the following Lichnerowicz formula. Theorem 22.
The Kohn Laplacian (5.12) satisfies (5.13) (cid:3) b = D b = (cid:16) ∇ B, Λ , ∗ ,H (cid:17) ∗ ∇ B, Λ , ∗ ,H | {z } ∆ B, Λ0 , ∗ := + 12 r X ¯ wi ¯ w + L ( w, ¯ w ) [2 ¯ wi ¯ w − i L T where r X := R T , X ( w, ¯ w ) for w ∈ T , X , | w | = 1 .Proof. On account of S invariance, both sides of the formula commute with the generator L T . It then suffices to check their equality on sections that are locally of the form s (cid:0) y, e iθ (cid:1) = s ( y ) e imθ , s ∈ Ω , ∗ ( Y ) , eigenspaces of L T , on the unit circle bundle (5.4). Further one locallyhas the correspondence(5.14) C ∞ m ( X ) ∼ = C ∞ ( Y ; L m ) between sections on X that are m -eigenspaces of L T and sections of L m on Y for example.Under this correspondence D b , ∇ B, Λ , ∗ ,H act by the Dolbeault-Dirac operators and Bismutconnection on tensor powers L m . The Lie derivative L T acts by multiplication by m while thecurvature of L is identified with the Levi form by definition. The curvatures r X , R T , X arepulled back from the scalar and Chern curvatures on Y while the horizontal components of Θ agree with the components of the corresponding tensor on Y . With these identifications, theLichnerowicz formula (5.13) is locally the same as [29, Thm 1.4.7]. (cid:3) The (horizontal) Bochner Laplacian appearing in (5.13) can be written ∆ B, Λ , ∗ := m X j =1 (cid:20) − (cid:16) ∇ B, Λ , ∗ U j (cid:17) s + ( div U j ) ∇ B, Λ , ∗ U j s (cid:21) , in terms of a real orthonormal basis { U j } mj =1 for HX . It is a sub-Riemannian Laplacian associ-ated to the metric (bracket generating) distribution HX and the natural Riemannian volumesee [31, Sec. 2]. From the above expression it is clearly a hypoelliptic operator of Hörmandertype [19]. It satisfies a hypoelliptic estimate: ∃ C > such that(5.15) D ∆ B, Λ , ∗ u, u E + k u k ≥ C k u k H /r ∀ u ∈ Ω , ∗ ( X ) (see [39]). Here r is the maximal type of a point on X . This is also referred to asthe step or degree of non-holonomy of the Levi distribution HX in sub-Riemannian geometry. ZEGŐ KERNEL 30
Spectral gap and closed range.
In this subsection we show that the spectral gapproperty for the Kohn Laplacian as well as the closedness of the range for ¯ ∂ b in three dimensionsautomatically follow in the circle invariant case.First, each Ω ,q ( X ) has an orthogonal decomposition (with respect to the chosen invariantmetric) into the Fourier modes for the S action Ω ,q ( X ) = ⊕ m ∈ Z Ω ,qm ( X ) where(5.16) Ω ,qm ( X ) := (cid:8) ω ∈ Ω ,qm ( X ) |L T ω = imω (cid:9) . Indeed the orthogonal projection of any ω ∈ Ω ,q ( X ) onto its m th Fourier mode is given by P m :Ω ,q ( X ) → Ω ,qm ( X )( P m ω ) ( x ) := Z S dθ ω (cid:0) x.e iθ (cid:1) e − imθ (5.17)Since the S action is assumed to be CR, we have (cid:2) ¯ ∂ b , T (cid:3) = 0 . Hence the tangential CRoperator preserves the Fourier modes (5.16) (cid:2) ¯ ∂ b , P m (cid:3) = 0 , ¯ ∂ b : Ω ,qm ( X ) → Ω ,q +1 m ( X ) . Onemay then define the m -th equivariant Kohn-Rossi cohomology H ∗ b,m ( X ) := ker (cid:2) ¯ ∂ b : Ω ,qm ( X ) → Ω ,q +1 m ( X ) (cid:3) Im (cid:2) ¯ ∂ b : Ω ,qm ( X ) → Ω ,q +1 m ( X ) (cid:3) . Its adjoint ¯ ∂ ∗ b with respect to the invariant metric further commutes (cid:2) T, ¯ ∂ ∗ b (cid:3) = 0 with thegenerator. A similar commutation then applies to the Kohn Laplacian [ T, (cid:3) b ] = 0 , thus givinga decomposition (cid:3) b = ⊕ q =0 , ⊕ m ∈ Z (cid:3) qb,m (cid:3) qb,m := (cid:3) b | Ω ,qm : Ω ,qm ( X ) → Ω ,qm ( X ) . The equivariant version of the Hodge theorem holds(5.18) ker (cid:0) (cid:3) qb,m (cid:1) = H ∗ b,m ( X ) [9, Thm 3.7].The S -invariant operator (cid:3) X := − T + (cid:3) b is elliptic and self-adjoint with respect to h , i , dx .There is thus a complete orthonormal basis ϕ qj,m , j = 0 , , . . . , m ∈ Z , of L ( X ; Λ ,q ) consistingof joint eigenvectors (cid:3) X ϕ qj,m = λ qj,m ϕ qj,m ; T ϕ qj,m = imϕ qj,m with ≤ m ≤ λ q ,m ≤ λ q ,m ≤ . . . ր ∞ , ∀ m ∈ Z . Thus, for each fixed m ∈ Z the set (cid:8) ϕ qj,m (cid:9) ∞ j =0 is then an orthonormal basis of L m ( X ; Λ ,q ) := { ω ∈ L m ( X ; Λ ,q ) | T ω = imω } of eigenvectors for (cid:3) qb,m . Note that (cid:3) qb,m is anunbounded operator on L m ( X ; Λ ,q ) with domain Dom (cid:0) (cid:3) qb,m (cid:1) = (cid:8) ω ∈ L m ( X ; Λ ,q ) | (cid:3) qb,m ω ∈ L m ( X ; Λ ,q ) (cid:9) .Similarly one also hasDom (cid:0) ¯ ∂ b (cid:1) = (cid:8) ω ∈ L (cid:0) X ; Λ , ∗ (cid:1) | ¯ ∂ b ω ∈ L (cid:0) X ; Λ , ∗ (cid:1)(cid:9) , Dom (cid:0) ¯ ∂ ∗ b (cid:1) = (cid:8) ω ∈ L (cid:0) X ; Λ , ∗ (cid:1) | ¯ ∂ ∗ b ω ∈ L (cid:0) X ; Λ , ∗ (cid:1)(cid:9) , as unbounded operators on L .We now have the following spectral gap property for (cid:3) qb,m . ZEGŐ KERNEL 31
Proposition 23.
There exists a constants c , c > such thatSpec (cid:0) (cid:3) b,m (cid:1) ⊂ { } ∪ h c | m | /r − c , ∞ (cid:17) (5.19) Spec (cid:0) (cid:3) b,m (cid:1) ⊂ h c | m | /r − c , ∞ (cid:17) (5.20) for each m ∈ Z .Proof. The Lichnerowicz formula (5.13) when restricted for the restriction to the m -th Fouriermode Ω , m ( X ) gives (cid:3) b,m = ∆ B, Λ , m | {z } =: ∆ B, Λ0 , ∗ | Ω0 , m ( X ) + 12 r X + L ( w, ¯ w ) m in terms of an invariant orthonormal frame w ∈ T , X . Following the subelliptic estimate(5.15) and pseudoconvexity gives λ qj.m = (cid:10) (cid:3) b,m ϕ j,m , ϕ j,m (cid:11) = (cid:28)(cid:20) ∆ B, Λ , m + 12 r X + L ( w, ¯ w ) m (cid:21) ϕ j,m , ϕ j,m (cid:29) ≥ c k ϕ j,m k H /r − c k ϕ j,m k = c (cid:13)(cid:13) T /r ϕ j,m (cid:13)(cid:13) − k ϕ j,m k = c m /r − c for m ≥ . Thus Spec (cid:0) (cid:3) b,m (cid:1) ⊂ (cid:2) c m /r − c , ∞ (cid:1) for m ≥ proving the second part (5.20).Similarly one proves Spec (cid:0) (cid:3) b,m (cid:1) ⊂ h c | m | /r − c , ∞ (cid:17) for m ≤ . The first part (5.19) followson noting that the Dirac operator D b,m := D b | Ω , ∗ m ( X ) gives an isomorphism between the non-zeroeigenspaces of (cid:3) b,m and (cid:3) b,m for each m ∈ Z . (cid:3) It follows immediately from (5.18), (5.20) that H b,m ( X ) = 0 for m ≫ . This gives d m := dim H b,m ( X ) = φ m ( s ) Z X Td b (cid:0) T , X (cid:1) e − mdθ ∧ θ = φ m ( s ) m (cid:20)Z X dθ ∧ θ (cid:21) + O (1) following the index theorem of [9, Cor. 1.13], where Td b ( T , X ) denotes the tangential/invariantTodd class of T , X [9, Sec. 2.3] and θ ( T ) = 1 , θ ( HX ) = 0 .As another application of Proposition 23 one has the estimate Spec + (cid:0) D b,m (cid:1) ⊂ h c | m | /r − c , ∞ (cid:17) for each m ∈ Z on the positive spectrum of the Dirac operator. One then sees (cid:13)(cid:13) D b ω (cid:13)(cid:13) ≥ c k D b ω k , ∀ ω ∈ Ω , ( X ) , on its decomposition into Fourier modes. The last inequality is rewritten (cid:13)(cid:13) ¯ ∂ b ¯ ∂ ∗ b ω (cid:13)(cid:13) ≥ c (cid:13)(cid:13) ¯ ∂ ∗ b ω (cid:13)(cid:13) , ∀ ω ∈ Ω , ( X ) and hence(5.21) (cid:13)(cid:13) ¯ ∂ b ω (cid:13)(cid:13) ≥ c k ω k , ∀ ω ∈ Dom (cid:0) ¯ ∂ b (cid:1) ∩ Range (cid:0) ¯ ∂ ∗ b (cid:1) , which is equivalent to the closed range property for ¯ ∂ b (see [18] Sec. 1). ZEGŐ KERNEL 32
Szegő kernel expansion.
We now investigate the asymptotic expansion of the Szegőkernel on diagonal. In the presence of a locally free circle action, its m -th Fourier componentof the Szegő kernel Π b,m ( x, x ′ ) := π R Π b (cid:0) x, x ′ e iθ (cid:1) e imθ dθ is given as the Schwartz kernel of theorthogonal projector(5.22) Π b,m := Π b ◦ P m : L ( X ) → ker (cid:0) (cid:3) b,m (cid:1) ⊂ L ( X ) . It is also written in terms of orthonormal zero eigenfunctions (cid:8) ψ m , . . . , ψ md m (cid:9) = (cid:8) ϕ j,m ∈ L ( X ) | λ j,m = 0 (cid:9) of (cid:3) b,m via(5.23) Π b,m ( x, x ′ ) := d m X j =1 ψ mj ( x ) ψ mj ( x ′ ) of ker (cid:0) (cid:3) b,m (cid:1) . The singularity of the Szegő kernel Theorem 1 corresponds to the on-diagonalasymptotics of the Fourier component Π b,m ( x, x ) as m → ∞ in the circle invariant case and wewish to describe it in this subsection.We shall first localize the problem. We use the local description (5.4) of X as the unit circlebundle of X = S L π −→ Y of a Hermitian holomorphic line bundle (cid:0) L, ∇ L , h L (cid:1) over a complexHermitian manifold Y . Finally and as partly noted before under the identification (5.14) onehas the m th Fourier component of the Kohn Laplacian(5.24) (cid:3) b,m = (cid:3) m := 12 D m is locally given in terms of the Kodaira Laplacian on tensor powers C ∞ ( Y ; L m ) . We now definea modify the frame { e , e } , used in the expressions (5.6), and define the frame { ˜ e , ˜ e } on R which agrees with { e , e } on B ̺ ( y ) and with { ∂ x , ∂ x } outside B ̺ ( x ) . Also define themodified metric ˜ g T Y and almost complex structure ˜ J on R to be standard in this frame andhence agreeing with g T Y , J on B ̺ ( x ) . The Christoffel symbol of the corresponding modifiedinduced connection on Λ , ∗ now satisfies ˜ a Λ , ∗ = 0 outside B ̺ ( x ) . Being identified with the Levi form, the curvature R L is semi-positive by assumption with itsorder of vanishing ord x (cid:0) R L (cid:1) = r x − ∈ N being given in terms of the type of the point x .We may then Taylor expand the curvature R L = X | α | = r x − R Lα y α dy dy | {z } = R L + O (cid:0) y r x − (cid:1) with(5.25) iR L ( e , e ) ≥ . (5.26)Now define the modified connection on L via ˜ ∇ L = d + Z dρ ρy k (cid:16) ˜ R L (cid:17) jk ( ρy ) | {z } =˜ a Lj dy j , where ˜ R L = χ (cid:18) | y | ̺ (cid:19) R L + (cid:20) − χ (cid:18) | y | ̺ (cid:19)(cid:21) R L . (5.27) ZEGŐ KERNEL 33 which agrees with ∇ L on B ̺ ( y ) . Note that the curvature ˜ R L of ˜ ∇ L above is also semi-positive bydefinition. Furthermore one also has ˜ R L = R L + O ( ̺ r y − ) and that the ( r x − -th derivative/jetof ˜ R L is non-vanishing at all points on R for(5.28) < ̺ < c (cid:12)(cid:12) j r y − R L ( y ) (cid:12)(cid:12) . Here c is a uniform constant on the manifold. We then define the modified Kodaira Diracoperator on R by the similar formula(5.29) ˜ D m = c ◦ ˜ ∇ Λ , ∗ ⊗ L m which agrees with D m on B ̺ ( y ) . The above satisfies a similar Lichnerowicz formula for thecorresponding Kodaira Laplacian (cid:3) m := ˜ D m = (cid:16) ˜ ∇ Λ , ∗ ⊗ L m (cid:17) ∗ ˜ ∇ Λ , ∗ ⊗ L m + m ˜ R L ( w, ¯ w ) [2 ¯ wi ¯ w −
1] + 12 ˜ r X ¯ wi ¯ w (5.30)where w = √ (˜ e − i ˜ e ) , ˜ r X := ˜ R T , X ( w, ¯ w ) , the adjoint being taken with respect to the metric ˜ g T Y and corresponding volume form. The above (5.30) again agrees with(5.31) ˜ (cid:3) m = (cid:3) m on B ̺ ( y ) while the endomorphism ˜ r X vanishes outside B ̺ ( y ) . Being semi-bounded below (5.30) is es-sentially self-adjoint. A similar argument as Corollary 23 gives a spectral gap(5.32) Spec (cid:0) ˜ (cid:3) m (cid:1) ⊂ { } ∪ (cid:2) c m /r x − c , ∞ (cid:1) . Thus for m ≫ , the resolvent (cid:0) ˜ (cid:3) m − z (cid:1) − is well-defined in a neighborhood of the origin inthe complex plane. From local elliptic regularity, the Bergman projector(5.33) ˜ B m : L (cid:0) R ; L ⊗ m (cid:1) → ker (cid:0) ˜ (cid:3) m (cid:1) then has a smooth Schwartz kernel with respect to the Riemannian volume of ˜ g T Y .Now we choose a set of such BRT trivializations n U j = (cid:16) − πs xj , πs xj (cid:17) × B ̺ ( x j ) o Nj =1 (5.8)centered at { x j ∈ X } Nj =1 with corresponding modified Laplacians (cid:8) ˜ (cid:3) m,j (cid:9) Nj =1 and Bergman pro-jectors n ˜ B m,j o Nj =1 such that n U j := (cid:16) − π s xj , π s xj (cid:17) × B ̺ ( x j ) o Nj =1 cover X . Choose a partitionof unity (cid:8) χ j ∈ C ∞ c (cid:0) U j ; [0 , (cid:1)(cid:9) Nj =1 subordinate to the the BRT cover, j = 1 , . . . , N . Furtherchoose ψ j ∈ C ∞ c ( U j ; [0 , such that ψ j = 1 on spt ( χ j ) and(5.34) σ j ∈ C ∞ c (cid:18) − πs x j , πs x j (cid:19) θ with Z σ j dθ = 1 in each such trivialization. We note that a finite propagation argument as in [29, Sec. 1.6] gives(5.35) χ j ˜ B m,j ψ j = χ j ˜ B m,j mod O (cid:0) m −∞ (cid:1) thus the right hand side above maybe assumed to be properly supported mod O ( m −∞ ) .Now define the approximate Szegő kernel via ˜Π m := ∞ X j =1 Z dy ′ dθ ′ χ j ( y, θ ) e imθ ˜ B m,j ( y, y ′ ) e − imθ ′ ψ j ( y ′ , θ ′ ) σ j ( θ ′ ) ! (5.36) ˜Π b,m := ˜Π m ◦ P m . (5.37)We now have the following localization lemma. ZEGŐ KERNEL 34
Lemma 24.
The approximate Szegő kernel (5.36) satisfies (5.38) ˜Π b,m − Π b,m = O (cid:0) m −∞ (cid:1) in the C ∞ norm on the product X × X .Proof. We first show that by direct computation that(5.39) ˜Π b,m Π b,m = Π b,m mod O (cid:0) m −∞ (cid:1) . To this end, let f ∈ C ∞ ( X ) and g := Π b,m f . Then g ∈ ker (cid:0) (cid:3) b,m (cid:1) and thus g (cid:0) ye iθ (cid:1) = g ( y ) e imθ with ˜ (cid:3) m,j g = 0 on each B ̺ ( x j ) by (5.24), (5.31). With(5.40) ˜ B m,j g = g mod O (cid:0) m −∞ (cid:1) on B ̺ ( x j ) following from a finite propagation argument, we may calculate ˜Π b,m g = ˜Π m g = ∞ X j =1 Z dy ′ dθ ′ χ j ( x ) e imθ ˜ B m,j ( y, y ′ ) e − imθ ′ σ j ( θ ′ ) g ( y ′ ) e imθ ′ = ∞ X j =1 χ j ( x ) g mod O (cid:0) m −∞ (cid:1) = g mod O (cid:0) m −∞ (cid:1) using (5.34), (5.36) and (5.40) showing (5.39).In similar vein, with P m f = g ∈ C ∞ m ( X ) satisfying g (cid:0) ye iθ (cid:1) = g ( y ) e imθ on each B ̺ ( x j ) asbefore we calculate ˜Π b,m (cid:3) b f = ˜Π m ◦ P m ◦ (cid:3) b f = ˜Π m ◦ (cid:3) b,m g = ∞ X j =1 Z dy ′ dθ ′ χ j ( x ) e imθ ˜ B m,j ( y, y ′ ) e − imθ ′ σ j ( θ ′ ) ˜ (cid:3) m,j g ( y ′ ) e imθ ′ mod O (cid:0) m −∞ (cid:1) = 0 mod O (cid:0) m −∞ (cid:1) (5.41)using (5.24), (5.31) and another finite propagation argument.Finally letting N m : L m ( X ) → Dom (cid:0) (cid:3) b,m (cid:1) ,N m f = ( f ∈ ker (cid:0) (cid:3) b,m (cid:1) , (cid:3) − b,m f ; f ∈ ker (cid:0) (cid:3) b,m (cid:1) ⊥ (5.42)denote the partial inverse of (cid:3) b,m we calculate ˜Π ∗ b,m = P m ˜Π ∗ b,m = ( N m (cid:3) b P m + Π b,m ) ˜Π ∗ b,m = Π b,m ˜Π ∗ b,m mod O (cid:0) m −∞ (cid:1) = Π b,m mod O (cid:0) m −∞ (cid:1) following (5.39), (5.41) and proving the proposition on account of the self-adjointness of Π b,m . (cid:3) ZEGŐ KERNEL 35
We note that the on-diagonal asymptotic expansion for the local Bergman kernel ˜ B m ( y, y ) follows in a similar fashion as Theorem 14. A slight difference here is that ˜ B m ( y, y ) is definedwith respect to a more general metric while the metric in Theorem 14 is flat. This howevermakes little difference to the argument and gives the following (cf. [31, Thm. 3]). Theorem 25.
For any differential operator P of order l , the derivative of the local Bergmankernel has the pointwise expansion on diagonal (5.43) P Π ˜ (cid:3) m ( y, y ) = m (2+ l ) /r y " N X j =0 c j ( P, y ) m − j/r y + O (cid:0) m l − N +1 (cid:1) , ∀ N ∈ N . The leading term, for P = 1 , is given c , (1 , y ) = Π g HXx ,j rx − L ,J HX (0 , > in termsof the Bergman kernel of the model Kodaira Laplacian on HX (see [31, Sec. A] ). Following this and the localization property of the Szegő kernel just proved now impliesTheorem 21 as below.
Proof of Theorem 21.
By the localization property (5.38) it suffices to show the pointwise ex-pansion of the approximate Szegő kernel (5.37). In showing the expansion at x ∈ X , wemay further assume the BRT cover and partition of unity defining (5.37) is chosen so that χ j = ( , j = 10 , j > , near x . We then compute ˜Π b,m = Z π dθ ˜Π m (cid:0) x, xe iθ (cid:1) e imθ = s x − X l =0 Z πsx ( l +1) πsx l dθ ˜Π m (cid:0) x, xe iθ (cid:1) e imθ = s x − X l =0 e i πlmsx Z πsx l dθ ˜Π m (cid:0) x, xe iθ (cid:1) e imθ = s x − X l =0 e i πlmsx Z πsx l dθ ˜Π m (cid:0) x, xe iθ (cid:1) e imθ = s x − X l =0 e i πlmsx ! (cid:18)Z dθ σ ( θ ) (cid:19) ˜ B m ( y, y )= s x − X l =0 e i πlmsx ! m /r y " N X j =0 c j ( x ) m − j/r y + O (cid:0) m − (2 N +1) /r y (cid:1) , ∀ N ∈ N , (5.44)from (5.37), Theorem 25 to prove (5.3). (cid:3) As noted in [31, Rem. 24] the expansion for the local Bergman kernels ˜ B m ( y, y ) is the sameas the positive case on X ,s (the strongly pseudoconvex points) and furthermore uniform inany C l -topology on compact subsets of X ,s cf. [29, Theorem 4.1.1]. In particular the first twocoefficients for y ∈ Y are given by c ( y ) = Π g HXx ,j rx − L ,J HX (0 ,
0) = 12 π τ L c ( y ) = 116 π τ L (cid:2) κ − ∆ ln τ L (cid:3) . (5.45) ZEGŐ KERNEL 36
The derivative expansion on X ,s is also known to satisfy c = c = . . . = c [ l − ] = 0 (i.e. beginsat the same leading order m ).In the next section we shall also need uniform estimates on the local Bergman kernels asbelow. Theorem 26.
The local Bergman kernel satisfies the estimate (5.46) (cid:20) inf x ∈ X r, ≤ s Π g HXx ,j rx − L ,J HX (0 , (cid:21) [1 + o (1)] m /r ≤ ˜ B m ( y, y ) ≤ (cid:20) sup x ∈ X Π g HXx ,j rx − L ,J HX (0 , (cid:21) [1 + o (1)] m with the o (1) terms being uniform in x ∈ X .Furthermore, there exists constants C l , l = 0 , , . . . , uniform in y ∈ Y , such that for anydifferential operator P l of order l , the derivative of the local Bergman kernel satisfies the estimate (5.47) (cid:12)(cid:12)(cid:12) P l ˜ B m ( y, y ) (cid:12)(cid:12)(cid:12) ≤ m // C l ˜ B m ( y, y ) . Proof.
Note that theorem Theorem 25 already shows(5.48) Π m ( y, y ) ≥ C r y (cid:0)(cid:12)(cid:12) j r y − R L (cid:12)(cid:12) m (cid:1) /r y − c y ∀ y ∈ Y , with c y = c (cid:16)(cid:12)(cid:12) j r y − R L ( y ) (cid:12)(cid:12) − (cid:17) = O | j ry − R L ( y ) | − (1) being a ( y -dependent) constantgiven in terms of the norm of the first non-vanishing jet. The norm of this jet affects thechoice of ̺ needed for (5.28); which in turn affects the C ∞ -norms of the coefficients of (5.43)via (5.27). We first show that this estimate extends to a small ( (cid:12)(cid:12) j r y − R L ( y ) (cid:12)(cid:12) - dependent) sizeneighborhood of y . To this end, for any ε > there exists a uniform constant c ε dependingonly on ε and (cid:13)(cid:13) R L (cid:13)(cid:13) C r such that(5.49) (cid:12)(cid:12) j r y − R L ( y ) (cid:12)(cid:12) ≥ (1 − ε ) (cid:12)(cid:12) j r y − R L ( y ) (cid:12)(cid:12) , ∀ y ∈ B c ε | j ry − R L | ( y ) . We begin by rewriting the model Kodaira Laplacian ˜ (cid:3) m (5.30) near y in terms of geodesiccoordinates centered at y . In the region y ∈ B c ε | j ry − R L | ( y ) ∩ n C (cid:0)(cid:12)(cid:12) j R L ( y ) (cid:12)(cid:12) m (cid:1) ≥ m /r y Π g TYy ,j ry − y R L ,J TYy (0 , o a rescaling of ˜ (cid:3) m by δ m − / , now centered at y , shows Π m ( y , y ) = m Π g TY y ,j y R L ,J TY y (0 ,
0) + O | j ry − R L ( y ) | − (1)= m (cid:12)(cid:12) j R L ( y ) (cid:12)(cid:12) Π g TY y , j y RL | j RL ( y ) | ,J TY y (0 ,
0) + O | j ry − R L ( y ) | − (1) ≥ m /r y Π g TYy ,j ry − y R L ,J TYy (0 ,
0) + O | j ry − R L ( y ) | − (1) (5.50)as in (5.48). Now, in the region y ∈ B c ε | j ry − R L | ( y ) ∩ n C (cid:0)(cid:12)(cid:12) j R L ( y ) /j R L ( y ) (cid:12)(cid:12) m (cid:1) / ≥ m /r y Π g TYy ,j ry − y R L ,J TYy (0 , ≥ C (cid:0)(cid:12)(cid:12) j R L ( y ) (cid:12)(cid:12) m (cid:1)o ZEGŐ KERNEL 37 a rescaling of ˜ (cid:3) m by δ m − / centered at y similarly shows Π m ( y , y ) = m / (cid:2) O (cid:0) m /r − / (cid:1)(cid:3) Π g TY y ,j y R L /j y R L ,J TY y (0 , O | j ry − R L ( y ) | − (1)= m / (cid:2) O (cid:0) m /r − / (cid:1)(cid:3) (cid:12)(cid:12) j y R L /j y R L (cid:12)(cid:12) / Π g TY y , j y RL/j y RL | j y RL/j y RL | ,J TY y (0 , O | j ry − R L ( y ) | − (1) (5.51) ≥ (1 − ε ) m /r y Π g TYy ,j ry − y R L ,J TYy (0 ,
0) + O | j ry − R L ( y ) | − (1) (5.52)Next, in the region y ∈ B c ε | j ry − R L | ( y ) ∩ n C (cid:0)(cid:12)(cid:12) j R L ( y ) /j R L ( y ) (cid:12)(cid:12) m (cid:1) / ≥ m /r y Π g TYy ,j ry − y R L ,J TYy (0 , ≥ max h C (cid:0)(cid:12)(cid:12) j R L ( y ) (cid:12)(cid:12) m (cid:1) , C (cid:0)(cid:12)(cid:12) j R L ( y ) /j R L ( y ) (cid:12)(cid:12) m (cid:1) / io a rescaling of ˜ (cid:3) m by δ m − / centered at y shows Π m ( y , y ) = m / (cid:2) O (cid:0) m /r − / (cid:1)(cid:3) Π g TY y ,j y R L /j y R L ,J TY y (0 ,
0) + O | j ry − R L ( y ) | − (1)= m / (cid:2) O (cid:0) m /r − / (cid:1)(cid:3) (cid:12)(cid:12) j y R L /j y R L (cid:12)(cid:12) / Π g TY y , j y RL/j y RL | j y RL/j y RL | ,J TY y (0 ,
0) + O | j ry − R L ( y ) | − (1) ≥ (1 − ε ) m /r y Π g TYy ,j ry − y R L ,J TYy (0 ,
0) + O | j ry − R L ( y ) | − (1) (5.53)Continuing in this fashion, we are finally left with the region y ∈ B c ε | j ry − R L | ( y ) ∩ n m /r y Π g TYy ,j ry − y R L ,J TYy (0 , ≥ max h C (cid:0)(cid:12)(cid:12) j R L ( y ) (cid:12)(cid:12) m (cid:1) , . . . , C r y − (cid:0)(cid:12)(cid:12) j r y − R L ( y ) /j r y − R L ( y ) (cid:12)(cid:12) m (cid:1) / ( r y − io . In this region we have (cid:12)(cid:12) j r y − R L ( y ) /j r y − R L ( y ) (cid:12)(cid:12) ≥ (1 − ε ) (cid:12)(cid:12) j r y − R L ( y ) (cid:12)(cid:12) + O (cid:0) m /r y − / ( r y − (cid:1) following (5.49) with the remainder being uniform. A rescaling by δ m − /ry then giving a similarestimate in this region, we have finally arrived at Π m ( y , y ) ≥ (1 − ε ) m /r y Π g TYy ,j ry − y R L ,J TYy (0 ,
0) + O | j ry − R L ( y ) | − (1) ∀ y ∈ B c ε | j ry − R L | ( y ) .Finally a compactness argument finds a finite set of points { y j } Nj =1 such that the correspond-ing B c ε (cid:12)(cid:12)(cid:12) j ryj − R L (cid:12)(cid:12)(cid:12) ( y j ) ’s cover Y . This gives a uniform constant c ,ε > such that Π m ( y, y ) ≥ (1 − ε ) (cid:20) inf y ∈ Y r Π g TYy ,j r − y R L ,J TYy (0 , (cid:21) m /r − c ,ε ∀ y ∈ Y , ε > proving the lower bound of (5.46). The argument for the upper bound is similar.The proof of the uniform estimate on the derivative (5.47) is similar. Given ε > we finda uniform c ε such that (5.49) holds for each y ∈ Y and y ∈ B c ε | j ry − R L | ( y ) . Then rewrite the ZEGŐ KERNEL 38 model Kodaira Laplacian ˜ (cid:3) m (5.30) near y in terms of geodesic coordinates centered at y . Inthe region y ∈ B c ε | j ry − R L | ( y ) ∩ n C (cid:0)(cid:12)(cid:12) j R L ( y ) (cid:12)(cid:12) m (cid:1) ≥ m /r y Π g TYy ,j ry − y R L ,J TYy (0 , o a rescaling of ˜ (cid:3) m by δ m − / , now centered at y , shows ∂ α Π m ( y , y ) = m π (cid:0) ∂ α τ L ( y ) (cid:1) + O | j ry − R L ( y ) | − (1) following (5.45) as r y = 2 . Diving the above by (5.50) gives | ∂ α Π m ( y , y ) | Π m ( y , y ) ≤ (cid:12)(cid:12) ∂ α τ L ( y ) (cid:12)(cid:12) τ L ( y ) + O | j ry − R L ( y ) | − (cid:0) m − (cid:1) ≤ m | α | / sup y ∈ Y (cid:12)(cid:12)(cid:12)h j | α | Π g TYy ,j y R L /j y R L ,J TYy i (0 , (cid:12)(cid:12)(cid:12) Π g TYy ,j y R L /j y R L ,J TYy (0 , Π m ( y , y )+ O | j ry − R L ( y ) | − (cid:0) m − (cid:1) Next, in the region y ∈ B c ε | j ry − R L | ( y ) ∩ n C (cid:0)(cid:12)(cid:12) j R L ( y ) /j R L ( y ) (cid:12)(cid:12) m (cid:1) / ≥ m /r y Π g TYy ,j ry − y R L ,J TYy (0 , ≥ C (cid:0)(cid:12)(cid:12) j R L ( y ) (cid:12)(cid:12) m (cid:1)o a rescaling of ˜ (cid:3) m by δ m − / centered at y similarly shows ∂ α Π m ( y , y ) = m (2+ | α | ) / (cid:2) O (cid:0) m /r − / (cid:1)(cid:3) h ∂ α Π g TY y ,j y R L /j y R L ,J TY y i (0 , O | j ry − R L ( y ) | − (cid:0) m (1+ | α | ) / (cid:1) . Dividing this by (5.52) gives | ∂ α Π m ( y , y ) | Π m ( y , y ) ≤ m | α | / (1 + ε ) (cid:12)(cid:12)(cid:12)h ∂ α Π g TY y ,j y R L /j y R L ,J TY y i (0 , (cid:12)(cid:12)(cid:12)(cid:2) Π g TY y ,j y R L /j y R L ,J TY y (cid:3) (0 , O | j ry − R L ( y ) | − (cid:0) m ( | α |− / (cid:1) ≤ m | α | / (1 + ε ) sup y ∈ Y (cid:12)(cid:12)(cid:12)h j | α | Π g TYy ,j y R L /j y R L ,J TYy i (0 , (cid:12)(cid:12)(cid:12) Π g TYy ,j y R L /j y R L ,J TYy (0 , + O | j ry − R L ( y ) | − (cid:0) m ( | α |− / (cid:1) . Continuing in this fashion as before eventually gives | ∂ α Π m ( y , y ) | Π m ( y , y ) ≤ m | α | / (1 + ε ) sup y ∈ Y (cid:12)(cid:12)(cid:12)h j | α | Π g TYy ,j y R L /j y R L ,J TYy i (0 , (cid:12)(cid:12)(cid:12) Π g TYy ,j y R L /j y R L ,J TYy (0 , + O | j ry − R L ( y ) | − (cid:0) m ( | α |− / (cid:1) ZEGŐ KERNEL 39 ∀ y ∈ Y , y ∈ B c ε | j ry − R L | ( y ) , ∀ α ∈ N . By compactness one again finds a uniform c ,ε such that | ∂ α Π m ( y, y ) | Π m ( y, y ) ≤ m | α | / (1 + ε ) sup y ∈ Y (cid:12)(cid:12)(cid:12)h j | α | Π g TYy ,j y R L /j y R L ,J TYy i (0 , (cid:12)(cid:12)(cid:12) Π g TYy ,j y R L /j y R L ,J TYy (0 , + c ,ε ∀ y ∈ Y , proving the lemma. (cid:3) Equivariant CR Embedding
In this section we construct the CR embedding for X required to prove Theorem 3. Firstly,setting m = p. ( s !) ∈ ( s !) . N in (5.18), (5.23), (5.46) and (5.44) the base locus(6.1) Bl p ( X ) := (cid:8) x ∈ X | s ( x ) = 0 , ∀ s ∈ H b,p. ( s !) ( X ) (cid:9) = ∅ is empty for p ≫ . Thus the subspace(6.2) Φ p,x := (cid:8) s ∈ H b,p. ( s !) ( X ) | s ( x ) = 0 (cid:9) ⊂ H b,p. ( s !) ( X ) is a hyperplane for each x ∈ X . Identifying the Grassmanian G (cid:16) d p. ( s !) − H b,p. ( s !) ( X ) (cid:17) , d p. ( s !) = dim H b,p. ( s !) ( X ) , with the projective space P h H b,p. ( s !) ( X ) ∗ i , by sending a non-zeroelement of H b,p. ( s !) ( X ) ∗ to its kernel, gives a well-defined Kodaira map(6.3) Φ p : X → P (cid:2) H b,p. ( s !) ( X ) ∗ (cid:3) for p ≫ .In terms of the basis n ψ p. ( s !)1 , . . . , ψ p. ( s !) d p. ( s !) o of ker (cid:16) (cid:3) b,p. ( s !) (cid:17) = H b,p. ( s !) ( X ) , with ¯ ∂ b ψ j = 0 , j = 1 , . . . , d p. ( s !) , and corresponding dual basis of H b,p. ( s !) ( X ) ∗ the map is written(6.4) Φ p ( x ) := (cid:16) ψ p. ( s !)1 ( x ) , . . . , ψ p. ( s !) d p. ( s !) ( x ) (cid:17) ∈ C d p. ( s !) and is seen to be CR.We now define the augmented Kodaira map Ψ p : X → C N , Ψ p := (cid:0) Φ p , Ψ p , . . . , Ψ sp (cid:1) , Ψ kp := ψ p.k , . . . , ψ p.kd p.k | {z } =:Ψ k, p ; ψ ( p +1) .k , . . . , ψ ( p +1) .kd ( p +1) .k | {z } =:Ψ k, p , ≤ k ≤ s,N := d p. ( s !) + s X k =1 (cid:0) d p.k + d ( p +1) .k (cid:1) , (6.5)which is again CR. We shall now show that the above augmented map is an embedding for p ≫ . We first show that it is an immersion, whereby it suffices to show that its firstcomponent Φ p (6.4) defines an immersion; the augmented components of (6.5) are required toseparate further points. Theorem 27.
The Kodaira map Φ p (6.3) is an immersion for p ≫ . ZEGŐ KERNEL 40
Proof.
We work in a BRT trivialization (5.8), (5.8) near x ∈ X . We choose χ ∈ C ∞ c (( − ε, ε ) ; [0 , , χ = 1 on (cid:0) − ε , ε (cid:1) , σ ( θ ) ∈ C ∞ c (cid:0) − πs , πs (cid:1) , R θσ ( θ ) dθ = 1 and set u = y χ (cid:0) m /r x y (cid:1) σ ( θ ) e imθ u = y χ (cid:0) m /r x y (cid:1) σ ( θ ) e imθ u = χ (cid:0) m /r x y (cid:1) ( mθ ) σ ( mθ ) e imθ and v j = Π m,b u j , j = 1 , , , (6.6)with m = p. ( s !) and x = ( y, θ ) being BRT coordinates.The equations ¯ ∂ b Π m,b ( ., x ) = 0 , ¯ ∂ ∗ b Π m,b ( ., x ) = 0 written in the BRT chart give ∂ y Π m,b ( x ′ , x ) = m r x y ′ X | α | = r x − R Lα y ′ α + O ( y ′ r x ) Π m,b ( x ′ , x ) ∂ y Π m,b ( x ′ , x ) = m − r x y ′ X | α | = r x − R Lα y ′ α + O ( y ′ r x ) Π m,b ( x ′ , x ) (6.7)from (5.8), (5.7). Further note that Π m,b u j ( x ) = Z dx ′ Π m,b ( x, x ′ ) u j ( x ′ )= Z dx ′ Π m,b ( x ′ , x ) u j ( x ′ )= s x Z dx ′ ˜Π m ( x ′ , x ) u j ( x ′ ) mod O (cid:0) m −∞ (cid:1) (6.8)from (5.38). We now estimate the derivative of the CR functions (6.6). Below c ij ( | j r x − L| ) , ≤ i, j ≤ , continuous positive functions of the norms of the jet of the Levi form. We thenhave ∂ y v = s x m Z dy ′ dθ r x ( y ′ ) X | α | = r x − R Lα y ′ α + O (cid:0) y ′ r x +1 (cid:1) × ˜Π m ( x ′ , x ) χ (cid:0) m /r x y ′ (cid:1) σ ( θ ) e imθ + O (cid:0) m −∞ (cid:1) = s x Z dy ′ dθ r x ( y ′ ) X | α | = r x − R Lα y ′ α + O (cid:0) m − /r x y ′ r x +1 (cid:1) × ˜Π m (cid:0) m − /r x x ′ , (cid:1) χ ( y ′ ) σ ( θ ) e imθ = s x Z dy ′ dθ r x ( y ′ ) X | α | = r x − R Lα y ′ α + O (cid:0) m − /r x y ′ r x +1 (cid:1) × h Π g HXx ,j rx − L ,J HX ( x ′ ,
0) + O (cid:0) m − /r x (cid:1)i χ ( y ′ ) σ ( θ ) e imθ ≥ ε r x +2 c (cid:0)(cid:12)(cid:12) j r x − L (cid:12)(cid:12)(cid:1) + O x (cid:0) m − /r x (cid:1) (6.9)using (6.7) and (6.8). ZEGŐ KERNEL 41
And similarly, ∂ y v = s x m Z dy ′ dθ r x y ′ y ′ X | α | = r x − R Lα y ′ α + O (cid:0) y ′ r x +1 (cid:1) × ˜Π m ( x ′ , χ (cid:0) m /r x y ′ (cid:1) σ ( θ ) e imθ + O (cid:0) m −∞ (cid:1) = s x Z dy ′ dθ r x y ′ y ′ X | α | = r x − R Lα y ′ α + O (cid:0) m − /r x y ′ r x +1 (cid:1) × ˜Π m (cid:0) m − /r y x ′ , (cid:1) χ ( y ′ ) σ ( θ ) e imθ = s x Z dy ′ dθ r x y ′ y ′ X | α | = r x − R Lα y ′ α + O (cid:0) m − /r x y ′ r x +1 (cid:1) × h Π g HXx ,j rx − L ,J HX ( x ′ ,
0) + O (cid:0) m − /r x (cid:1)i χ ( y ′ ) σ ( θ ) e imθ = s x Z dy ′ dθ (cid:2) O (cid:0) y ′ r x +1 (cid:1) + O (cid:0) m − /r x (cid:1)(cid:3) χ ( y ′ ) σ ( θ ) e imθ ≤ ε r y +3 c (cid:0)(cid:12)(cid:12) j r x − L (cid:12)(cid:12)(cid:1) + O x (cid:0) m − /r x (cid:1) (6.10)using a Taylor expansion Π g HXx ,j rx − L ,J HX ( y,
0) = Π + y Π + y Π ; Π = Π g HXx ,j rx − L ,J HX (0 , on spt ( χ ) . Finally we compute ∂ θ v = s x m Z dy ′ dθ ˜Π m ( x ′ , x ) y ′ χ (cid:0) m /r x y ′ (cid:1) σ ( θ ) e imθ = s x m − /r x Z dy ′ dθm − /r x ˜Π m (cid:0) m − /r y x ′ , (cid:1) y ′ χ ( y ′ ) σ ( θ ) e imθ = s x m − /r x Z dy ′ dθ h Π g HXx ,j rx − L ,J HX ( x ′ ,
0) + O (cid:0) m − /r x (cid:1)i y ′ χ ( y ′ ) σ ( θ ) e imθ ≤ m − /r x ε c (cid:0)(cid:12)(cid:12) j r x − L (cid:12)(cid:12)(cid:1) + O x (cid:0) m − /r x (cid:1) . We have similar estimates on derivatives of v ∂ y v ≤ ε r y +2 c (cid:0)(cid:12)(cid:12) j r y − R L (cid:12)(cid:12)(cid:1) + O x (cid:0) m − /r x (cid:1) ∂ y v ≥ ε r y +2 c (cid:0)(cid:12)(cid:12) j r y − R L (cid:12)(cid:12)(cid:1) + O x (cid:0) m − /r x (cid:1) ∂ θ v ≤ m − /r x ε c (cid:0)(cid:12)(cid:12) j r x − L (cid:12)(cid:12)(cid:1) + O x (cid:0) m − /r x (cid:1) . (6.11)for two further constants c (cid:0)(cid:12)(cid:12) j r y − R L (cid:12)(cid:12)(cid:1) , c C (cid:0)(cid:12)(cid:12) j r y − R L (cid:12)(cid:12)(cid:1) (cid:0)(cid:12)(cid:12) j r y − R L (cid:12)(cid:12)(cid:1) depending only on thenorm of jet of the Levi tensor at x .Finally, and in similar vein, we estimate the derivative of v ∂ θ v = m Z dzdθ ˜Π b,m ( z, χ (cid:0) m /r x y (cid:1) ( mθ ) σ ( mθ )= Z dzdθ h Π g HXx ,j rx − L ,J HX ( y,
0) + O (cid:0) m − /r x (cid:1)i χ ( y ) θσ ( θ ) ≥ ε c (cid:0)(cid:12)(cid:12) j r y − R L (cid:12)(cid:12)(cid:1) + O x (cid:0) m − /r x (cid:1) (6.12) ZEGŐ KERNEL 42 and ∂ y v = m Z dzdθ ˜Π b,m ( z, χ (cid:0) m /r x y (cid:1) ( mθ ) σ ( mθ )= m Z dzdθ r x y X | α | = r x − R Lα y α + O ( y r x ) × ˜Π b,m ( z, χ (cid:0) m /r x z (cid:1) ( mθ ) σ ( mθ )= Z dzdθ (cid:2) O (cid:0) m − m /r r y r x − (cid:1)(cid:3) ˜Π b,m (cid:0) m − /r y z, (cid:1) χ ( z ) θσ ( θ ) ≤ m − /r x ε r x +1 c (cid:0)(cid:12)(cid:12) j r y − R L (cid:12)(cid:12)(cid:1) + O x (cid:0) m − /r x (cid:1) . (6.13)and similarly for ∂ y v . Following these estimates there exists C (cid:0)(cid:12)(cid:12) j r y − R L (cid:12)(cid:12)(cid:1) such that the dif-ferential of x ( v , v , v ) , and thus of Φ p , m = p. ( s !) , is invertible at x for ε < C (cid:0)(cid:12)(cid:12) j r y − R L (cid:12)(cid:12)(cid:1) and p > C (cid:0)(cid:12)(cid:12) j r y − R L (cid:12)(cid:12)(cid:1) . Thus for some C (cid:0)(cid:12)(cid:12) j r y − R L (cid:12)(cid:12)(cid:1) the differential of Φ p is invertible on a C (cid:0)(cid:12)(cid:12) j r y − R L (cid:12)(cid:12)(cid:1) ball centered at x for p > C (cid:0)(cid:12)(cid:12) j r y − R L (cid:12)(cid:12)(cid:1) ; which completes the proof followinga compactness argument. (cid:3) Next to show the Kodaira map is injective, one needs the following definition.
Definition 28.
The peak function S px ∈ H b,p. ( s !) ( X ) at x ∈ X is the unit norm element ofthe orthogonal complement to Φ ⊥ p,x ⊂ H b,p. ( s !) ( X ) (6.2).Clearly, if the orthonormal basis n ψ , . . . , ψ d p. ( s !) o of ker (cid:16) (cid:3) b,p. ( s !) (cid:17) = H b,p. ( s !) ( X ) is chosenso that ψ j ( x ) = 0 , ≤ j ≤ d p. ( s !) − , one has S px = ψ d p. ( s !) . From (5.23) one then has (cid:12)(cid:12) S px ( x ) (cid:12)(cid:12) = Π b,p. ( s !) ( x , x ) S px ( x ) = 1Π b,p. ( s !) ( x , x ) Π b,p. ( s !) ( x, x ) .S px ( x ) . (6.14) Theorem 29.
The augmented Kodaira map Ψ p (6.3) is injective for p ≫ .Proof. We assume to the contrary that there are two sequences of points x p j , x p j , j = 1 , , . . . ,such that p j → ∞ as j → ∞ and x p j = x p j Ψ p j (cid:16) x p j (cid:17) = Ψ p j (cid:16) x p j (cid:17) , ∀ j. (6.15)By compactness we may further suppose x p j → x , x p j → x as j → ∞ .Case i: Suppose e iθ x = x , ∀ e iθ ∈ S , i.e. the limit points do not lie on the same S orbit.The equation (6.15) in particular implies Φ p j (cid:16) x p j (cid:17) = Φ p j (cid:16) x p j (cid:17) by definition (6.5). Thus (6.14)implies Π b,p j . ( s !) (cid:16) x p j , x p j (cid:17) Π b,p j . ( s !) (cid:16) x p j , x p j (cid:17) = (cid:12)(cid:12)(cid:12) Π b,p j . ( s !) (cid:16) x p j , x p j (cid:17)(cid:12)(cid:12)(cid:12) . The left hand side above is uniformly bounded below by cp /rj on account of (5.36), (5.38),(5.46). While the right hand side can be seen to be O (cid:0) p −∞ j (cid:1) on choosing the S orbits of theBRT charts containing x , x in defining (5.36) to be disjoint.Case ii: Suppose e iθ x = x , for some e iθ ∈ S . We now again consider a BRT chart (5.8)of the form U = (cid:16) − πs x , πs x (cid:17) × B ̺ ( x ) containing the point x . As before this is obtained as ZEGŐ KERNEL 43 the unit circle bundle S L → Y over a hypersurface Y ⊂ X containing the point x . For each j we denote by h x p j i , h x p j i ∈ Y the unique points satisfying x p j ∈ S . h x p j i , x p j ∈ S h x p j i . The inclusion further defines a local holomorphic map Φ p : Y → P h H b,p. ( s !) ( X ) ∗ i satisfying Φ p j (cid:16)h x p j i(cid:17) = Φ p j (cid:16)h x p j i(cid:17) , j = 1 , , . . . . Via the Noetherian property for analytic sets as in[29, Sec. 5.1] this gives h x p j i = h x p j i or x p j = e iθ x p j ∈ S .x p j for j ≫ . Thus x p j , x p j lie onthe same orbit and with k = s x pj = s x pj we have Ψ kp (cid:16) x p j (cid:17) = Ψ kp (cid:16) x p j (cid:17) k k (cid:16) Ψ k, p (cid:16) x p j (cid:17) , Ψ k, p (cid:16) x p j (cid:17)(cid:17) (cid:16) e ip.kθ Ψ k, p (cid:16) x p j (cid:17) , e i ( p +1) .kθ Ψ k, p (cid:16) x p j (cid:17)(cid:17) . It now follows that θ ∈ πk Z implying x p j = x p j and contradicting (6.15). (cid:3) We note that following the closed range property (5.21) for ¯ ∂ b of 5.2 our embedding theoremTheorem 3, aside from the equivariance, can be obtained from the main theorem of [11]. Acknowledgments.
The authors would like to thank J. Sjöstrand and G. Marinescu fortheir guidance and several discussions regarding Boutet de Monvel-Sjöstrand theory. The au-thors also thank professors L. Lempert, D. Phong and T. Ohsawa for their insightful commentsregarding this work.
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