Szegő Kernel Asymptotics on Complete Strictly Pseudoconvex CR Manifolds

We prove a Bochner–Kodaira–Nakano formula and establish Szegő kernel expansions on complete strictly pseudoconvex CR manifolds with transversal CR \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}$$\end{document}R-action under certain natural geometric conditions. As a consequence we show that such manifolds are locally CR embeddable.


Introduction
The first goal of this paper is to develop a differential geometric formalism on strictly pseudoconvex CR manifolds with R-action, analogous to the Kähler identities and Bochner-Kodaira-Nakano formula for Hermitian manifolds. We refine in this way Tanaka's formulas in the spirit of Demailly's general version of the latter formulas. This formalism leads to vanishing theorems and L 2 -estimates for the ∂ b -operator for complete CR manifolds.
The second goal is to generalize the result of Boutet de Monvel-Sjöstrand about the singularities of the Szegő kernel for complete strictly pseudoconvex CR manifolds with R-action. This entails global and local embeddability theorems for CR manifolds with R-action, including Sasakian manifolds. Moreover, by applying our result for the Grauert tube of a positive line bundle we obtain a new result about the expansion of the Bergman kernel on complete Kähler manifolds.
Let (X , T (1,0) X ) be a CR manifold of dimension 2n + 1, n ≥ 1. The orthogonal projection S (q) : L 2 0,q (X ) → ker b is called the Szegő projection, while its distribution kernel S (q) (x, y) is called the Szegő kernel, where (q) b denotes the Kohn Laplacian acting on (0, q)-forms. The study of the Szegő kernel is a classical subject in several complex variables and CR geometry. If X is compact strictly pseudoconvex and (0) b has closed range, Boutet de Monvel-Sjöstrand [5] showed that S (0) (x, y) is a complex Fourier integral operator. The Boutet de Monvel-Sjöstrand description of the Szegő kernel had a profound impact in several complex variables, symplectic and contact geometry, geometric quantization, and Kähler geometry. These ideas also partly motivated the introduction of the recent direct approaches and their various extensions, see [23,24].
However, almost all the results on Szegő kernel assumed that X is compact, while for non-compact complex manifolds the Bergman kernel asymptotics was comprehensively studied [18,19,[23][24][25], and used in the several applications mentioned above. Note that for CR manifolds, besides the global embeddability question [4,26], there is an important delicate specific issue, namely the local embeddability [1,20,22,27], which will be treated here by the analysis of the Szegő kernel.
The Szegő kernel was used by Boutet de Monvel-Guillemin [6] to introduce the Toeplitz quantization on compact contact manifolds. In the same vein, the question of "quantization commutes with reduction" was studied on CR manifolds in the recent paper [17]. It is natural to extend these results to complete Sasakian manifolds.
Let us see some simple examples. Consider the hypersurface Y := {z = (z 1 , . . . , z n ) ∈ C n ; Im z n = f (z 1 , . . . , z n−1 )}, where f ∈ C ∞ (C n−1 , R). Then Y is a non-compact CR manifold carrying many smooth CR functions, but even in this simple example we do not know the behavior of the associated Szegő kernel. Another example is the Heisenberg manifold H = C n × R with CR structure T (1,0) H := span ∂ ∂z j + i ∂φ ∂z j (z) ∂ ∂ x 2n+1 : 1 ≤ j ≤ n , where φ ∈ C ∞ (C n , R). Then, H is also a non-compact CR manifold and the Szegő kernel has been studied when φ is quadratic (see [14]). However, for general φ there are fewer results. Both Y and H are non-compact CR manifolds with transversal CR R-action. Therefore, we think that the study of the Szegő kernels on non-compact CR manifolds with transversal CR R-action is a very natural and interesting question.
In [15], the first author obtained the Szegő kernel asymptotic expansion on the nondegenerate part of the Levi form under the assumption that Kohn Laplacian has closed range in L 2 . The method in [15] works well for non-compact setting, but for general non-compact CR manifolds, the closed range property is not a natural assumption. In the Heisenberg case mentioned above, even for φ quadratic, (0) b does not have closed range; however, the Szegő kernel still has an asymptotic expansion.
In this paper, we show that (0) b has local closed range with respect to a spectral projection Q λ (see Definition 4.12) under certain geometric conditions. Furthermore, combining this local closed range property with a detailed analysis, we establish Szegő kernel asymptotic expansions on non-compact strictly pseudoconvex complete CR manifolds with transversal CR R-action under certain natural geometric conditions. To study the local closed range property, we establish a CR Bochner-Kodaira-Nakano formula analog to [9], see Theorem 3.3, which has its own interest. This is also a refinement of Tanaka's basic identities [28,Theorems 5.1,5.2] in our context. We remark that the results in this paper hold both for transversal CR R-action and S 1action.
We will work in the following setting. Let X be a connected smooth paracompact manifold of dimension 2n + 1, H X be a smooth sub-bundle of T X of rank 2n, and J be a smooth complex structure on the fibers of H X. Let T (1,0) X be the complex subbundle of the complexification CH X of H X, which corresponds to the i eigenspace of J , that is, T (1,0) We say that X is a CR manifold (of hypersurface type) if the formal integrability condition C ∞ (X , T (1,0) X ), C ∞ (X , T (1,0) X ) ⊂ C ∞ (X , T (1,0) X ).
(1. 1) holds. The sub-bundle H X is called Levi distribution and the annihilator (H X) 0 ⊂ T * X of H X is called the characteristic bundle of the CR manifold X . We will assume in the sequel that X is orientable. Since H X is oriented by its complex structure, it follows that (H X) 0 is a real orientable line bundle, thus trivial. A global frame of (H X) 0 , that is, a real non-vanishing 1-form ω 0 ∈ C ∞ (X , T * X ) such that (H X) 0 = Rω 0 , is called characteristic 1-form.
Given a characteristic 1-form ω 0 on X the Levi form L ω 0 is defined by We say that (X , H X, J ) is strictly pseudoconvex if there exists a characteristic 1-form ω 0 the Levi form L ω 0 x is positive definite at every point x ∈ X . If L ω 0 is positive definite, then dω 0 is symplectic on H X, thus ω 0 is a contact form and H X is a contact structure. Associated with a contact form ω 0 one has the Reeb vector field T = T ω 0 , uniquely defined by the equations ω 0 (T ) = 1, dω 0 (T , ·) = 0 on X .
More generally, we consider an arbitrary R-invariant Hermitian metric g = g X = · | · g = · | · on CT X such that (1.5) holds. Given such a metric we will denote by X its fundamental (1, 1)-form given by X (a, b) = √ −1 a | b g for a, b ∈ T (1,0) X . Let dv X be the volume form induced by the R-invariant metric g X as in (1.5). Let ( · | · ) be the L 2 inner product on the space of smooth compactly supported functions C ∞ c (X ) with respect to dv X . We denote by L 2 (X , dv X ) the completion of C ∞ c (X ) with respect to ( · | · ).
We denote by ∂ b the tangential Cauchy-Riemann operator (see Definition 2.3). The Szegő projection is the orthogonal projection with respect to ( · | · ), Then the Szegő projection is a Fourier integral operator with complex phase, that is, for any local coordinate patch (D, x = (x 1 , . . . , x 2n+1 )) with D X , we have e iϕ(x,y)t s(x, y, t)dt ∈ C ∞ (D × D), (1.8) where the phase function ϕ ∈ C ∞ (D × D) satisfies ϕ(x, y) = −ϕ(y, x), (1.9) and s(x, y, t) ∈ S n cl D × D × R + is a symbol of order n with asymptotic expansion s(x, y, t) = ∞ j=0 s j (x, y)t n− j whose leading term s 0 (x, y) satisfies s 0 (x, x) = 1 2 π −n−1 |det L x |, for all x ∈ D, (1.10) where det L x is the determinant of L x with respect to g X , cf. (4.73).
We will show in Lemma 2.7 that R is the pseudohermitian Ricci form with respect to the pseudohermitian structure ω 0 (see (2.10)). We refer to Definition 2.1 for the definition of the symbol space S n cl D × D × R + and to [19,Theorems 3.3,4.4] for more properties for the phase ϕ in (1.8).
Examples for the situation described in Theorem 1.2 are given by Galois coverings of compact strictly pseudoconvex CR manifolds (Examples 4.5, 5.3), circle bundles of positive line bundles over complete Kähler manifolds (Example 4.6), and, as mentioned before, the Heisenberg group (Sect. 5).
If we work with (n, 0)-forms we can drop some of the hypotheses of Theorem 1.2.

Theorem 1.3
Let (X , H X, J , ω 0 ) be an orientable strictly pseudoconvex CR manifold of dimension 2n + 1, n ≥ 1, with an R-action on X as in Assumption 1.1. Assume that the Levi metric g L is complete. Then the Szegő projection S (n,0) : n,0 (X ) is a Fourier integral operator with complex phase, that is, for any local coordinate patch (D, x = (x 1 , . . . , x 2n+1 )) with D X, the Szegő kernel has the form (1.8) with respect to the trivialization of K X given by dz 1 ∧ . . . ∧ dz n .
The equivariant Kodaira embedding theorems for Sasakian manifolds were obtained in [13,16]. From Theorem 1.2, we obtain a Boutet de Monvel type embedding theorem [4] for complete Sasakian manifolds as follows, which is a generalization of the embedding theorem for compact Sasakian manifolds [26].

Corollary 1.4
In the situation of Theorem 1.2 the space of L 2 CR functions separate points and give local coordinates on X . In particular, for any compact set of K ⊂ X there exists a positive integer N and CR functions f 1 , . . . , As a consequence of Theorem 1.3 we obtain the following: Corollary 1.5 In the situation of Theorem 1.3 the space of L 2 CR (n, 0)-forms separate points and give local coordinates on X . Thus, X is locally CR embeddable in an Euclidean space. In particular, every Sasakian manifold with complete Levi metric g L is locally CR embeddable by global CR (n, 0)-forms.
The question arises if one can extend these results for general strictly pseudoconvex CR manifolds (without Assumption 1.1 about the existence of an R-action). An analytic property that we use is that the spectral projections Q λ of the operator √ −1T (see 4.7) commute to ∂ b . Beyond that it is not clear what would be general geometric or analytic conditions that would imply that the Szegő projector is a Fourier integral operator.
We now apply our main result to complex manifolds. Let (L, h L ) be a holomorphic line bundle over a Hermitian manifold (M, M ), where h L denotes a Hermitian metric on L and M is a positive (1, 1) form on M. For every k ∈ N, let (L k , h L k ) be the k-th power of (L, h L ). The positive (1, 1) form M and h L k induces a L 2 inner product Let s be a local holomorphic frame of L defined on an open set D M, The localized Bergman kernel on D is given by , for every x ∈ D.
In particular, there exist coefficients b r ∈ C ∞ (X ), r ∈ N 0 , such that for any open set U of X with U compact, every ∈ N 0 and every m ∈ N, there is a C U , ,m > 0 independent of k such that We refer the reader to Sect. 2 for the precise meaning of the notation A k ≡ B k mod O(k −∞ ) on D in (1.14), S n loc (1; D × D) and the asymptotic sums in (1.15) and (1.16).
For compact or certain complete Kähler-Einstein manifolds, the expansion (1.16) was obtained by Tian [29] for m = 0 and = 4. For general m, , and compact manifolds, the existence of the expansion was first obtained in [8,31]. In [23, Theorem 6.1.1] the expansion was generalized for complete Hermitian manifolds such that R K * M and ∂ M are bounded below. Our conditions (1.13) are different from [23, Theorem 6.1.1], we replace the condition on ∂ M by a condition on the volume form. The reason is that we use a local closed range condition instead of standard closed range or spectral gap condition. This paper is organized as follows. In Sect. 2, we recall necessary notions of microlocal analysis, pseudohermitian geometry, and strictly pseudoconvex CR manifolds with transversal CR R-actions. In Sect. 3, we prove the Bochner-Kodaira formula on CR manifolds with R-action. Section 4 is devoted to the proof of the asymptotics of the Szegő kernel. In Sect. 5, we examine the Heisenberg group.

Preliminaries
We use the following notations through this article: N = {1, 2, . . .} is the set of natural numbers, N 0 = N {0}, R is the set of real numbers, . . , n, be coordinates of C n . We write

Notions of Microlocal Analysis
Let X be a C ∞ paracompact manifold. We let T X and T * X denote the tangent bundle of X and the cotangent bundle of X , respectively. The complexified tangent bundle of X and the complexified cotangent bundle of X are denoted by CT X and CT * X , respectively. Write · , · to denote the pointwise duality between T X and T * X . We extend · , · bilinearly to CT X × CT * X . Let D ⊂ X be an open set . The spaces of distributions of D and smooth functions of D will be denoted by D (D) and C ∞ (D), respectively. Let E (D) be the subspace of D (D) whose elements have compact support in D.
be a continuous map. We write A(x, y) to denote the distribution kernel of A. In this work, we will identify A with A(x, y). The following two statements are equivalent: If A satisfies (I) or (II), we say that A is smoothing on D. Let A, B : C ∞ c (D) → D (D) be continuous operators. We write Let D be an open coordinate patch of X with local coordinates x. We recall the following Hörmander symbol space.
If a and a j have the properties above, we write We explain now for the precise meaning of A k ≡ B k mod O(k −∞ ) on D in (1.14), S n loc (1; D × D) and the asymptotic sum in (1.15) (see also [18,Sect. 3.3]). A k-dependent smoothing operator A k : uniformly on every compact set in D × D, for all multi-indices α, β, and all N ∈ N. Let C k : We recall the definition of semi-classical Hörmander symbol spaces: 1) for every N 0 . From this, we form S m loc (1; Y , E) in the natural way, where Y is a smooth paracompact manifold and E is a vector bundle over Y .
Let X be an orientable paracompact smooth manifold of dimension 2n + 1 with n ≥ 1. The Levi form (1.2) of X at x ∈ X induces a Hermitian quadratic form on T Let g CT X be a Hermitian metric on CT X such that the decomposition CT X = T (1,0) X ⊕ T (0,1) X ⊕ CT is orthogonal. For u, v ∈ CT X we denote by u|v = u|v g the inner product given by g CT X and for u ∈ CT X, we write |u| 2 g := u|u g . Given such a metric we will denote by X its fundamental (1, 1)-form given by For u ∈ CT X and φ ∈ CT * X , the pointwise duality is defined by u, φ := φ(u). Let T * (1,0) X ⊂ CT * X be the dual bundle of T (1,0) X and T * (0,1) X ⊂ CT * X be the dual bundle of T (0,1) X . For p, q ∈ N 0 , the bundle of ( p, q) forms is denoted by T * ( p,q) X := ( p T * (1,0) X ) ∧ ( q T * (0,1) X ) and let T * •,• X := ⊕ p,q∈N 0 T * ( p,q) X . The induced Hermitian inner product on T •,• X and T * •,• X by · | · are still denoted by ·|· . The Hermitian norms are still denoted by | · |. Let p,q (X ) := C ∞ (X , T * ( p,q) X ) be the space of smooth ( p, q)-forms on X and •,• (X ) := p,q∈N 0 p,q (X ). Let C ∞ (X ) := 0,0 (X ).
be the anti-Kohn Laplacian on •,• (X ). We still denoted by ∂ b the maximal extension and by ∂ * b the Hilbert space adjoint with respect to the L 2 -inner product on X . We also denote by the Gaffney extension of the Kohn Laplacian with the domain By a result of Gaffney, b is a self-adjoint operator (see e.g., [23, Proposition 3.1.2]).

Pseudohermitian Geometry
The following is well known:  H X). Moreover, ∇ J = 0 and ∇dω 0 = 0 imply that the Tanaka-Webster connection is compatible with the Levi metric. By definition, the torsion of ∇ is given by In the following, we will use the Einstein summation convention. Let {Z α } n α=1 be a local frame of T (1,0) X and {θ α } n α=1 be the dual frame of {Z α } n α=1 . We use the notations Z α := Z α and θ α = θ α . Write We call ω β α the connection 1-form of Tanaka-Webster connection with respect to the frame {Z α } n α=1 . We denote by β α the Tanaka-Webster curvature 2-form. Then, By direct computation, we also have where C 0 is a 1-form. The term R β α jk is called the pseudohermitian curvature tensor and the form is called pseudohermitian Ricci form.

Strictly Pseudoconvex CR Manifolds with R-Action
be an R-action on X , see [13]. Let T be the infinitesimal generator of the R-action: (2.12) The R-action is transversal : (2.13) Note that (2.12) implies that L T preserves H X and [L T , J ] = 0. Since H X = ker ω 0 we have for U ∈ C ∞ (X , H X), .
We can therefore assume up to rescaling ω 0 by a smooth function that the infinitesimal generator of the R-action is a Reeb vector field T = T . This motivates the equality of the infinitesimal generator to the Reeb field in Assumption 1.1. By [28, Lemma 3.2 (3)]) we have 2J τ U = (L T J )U for any U ∈ H X, hence the pseudohermitian torsion τ vanishes, which means that the contact metric manifold (X , ω 0 , T , J , g L ) is a Sasakian manifold. Conversely, there exists a natural transversal CR R-action on any compact Sasakian manifold. Recall that compact Sasakian manifolds can be classified in three categories based on the properties of the Reeb foliation consisting of the orbits of the Reeb field (see [7,Definition 6.1.25]). If the orbits of the Reeb field are all closed, then the Reeb field T generates a locally free, isometric S 1 -action thus also an R-action on (X , g L ). In this case the Reeb foliation is called quasi-regular (and regular if the action is free). If the Reeb foliation is not quasi-regular, it is said to be irregular. In this case, T generates a transversal CR R-action on X .

14)
and there exists φ ∈ C ∞ (U , R) independent of x 2n+1 satisfying that is a frame of T (1,0) D, and dz j n j=1 ⊂ T * (1,0) D is the dual frame.
Let D = U × I be a BRT chart. Let f ∈ C ∞ (D) and u ∈ p,q (D) with u = I ,J u I J dz I ∧ dz J with ordered sets I , J and u I J ∈ C ∞ (D), for all I , J . We have The Levi form L in a BRT chart D ⊂ X has the form Indeed, the characteristic 1-form ω 0 and dω 0 on D are given by Note that it is independent of x 2n+1 . More precisely, We deduce that is independent of the choice of BRT coordinates, i.e., = U = U . Until further notice, we work on a BRT chart D = U × I. For p, q ∈ N 0 , let T * ( p,q) U be the bundle of ( p, q) forms on U and let T * •,• U := ⊕ p,q∈N 0 T * ( p,q) U . For p, q ∈ N 0 , let T ( p,q) U be the bundle of ( p, q) vector fields on U and let T •,• U := ⊕ p,q∈N 0 T ( p,q) U . The (1, 1) form induces Hermitian metrics on T •,• U and T * •,• U . We shall use · , · h to denote all the induced Hermitian metrics. The volume form on U induced by is given by dλ(z) := n /n!. Thus, the volume form dv X can be represented by The L 2 -inner product on •,• c (U ) with respect to is given by Let t ∈ R be fixed. The L 2 -inner product on •,• c (U ) with respect to and e −2tφ(z) is given by The Chern curvature of K * U := det(T (1,0) U ) with respect to is given by It is easy to see that R K * X is independent of the choice of BRT coordinates and hence We can check that {w j := n k=1 c k j ∂ ∂z k ; j = 1, . . . , n} and {w j := n k=1 c k j ∂ ∂z k ; j = 1, . . . , n} are orthonormal frames for T (1,0) U and T (0,1) U with respect to , respectively, and {e j ; j = 1, . . . , n}, {e j ; j = 1, . . . , n} are dual frames for {w j ; j = 1, . . . , n} and {w j ; j = 1, . . . , n}, respectively. We also write w j and w j to denote e j and e j , respectively, j = 1, . . . , n.
In the following, we will use Einstein summation convention. Write It is clear that {dz j } n j=1 and {dz j } n j=1 are the dual frames of {Z j } n j=1 and {Z j } n j=1 , respectively. Denote and we can check that the Hence, the pseudohermitian Ricci curvature tensor at origin is We get that On the other hand, by directed computation, we can check that (2.37) From (2.36) and (2.37), the lemma follows.

Bochner-Kodaira Formula on CR Manifolds with R-Action
In this section, we will prove the Bochner-Kodaira-Nakano for CR manifolds with transversal CR R-action. They are refinements of Tanaka's basic identities [28, Theorems 5.1, 5.2] in our context. Namely, Tanaka's formulas hold for any strictly pseudoconvex manifold endowed with the Levi metric, while our formulas are specific to CR manifolds with R-action endowed with arbitrary Hermitian metric X .

The Fourier Transform on BRT Charts
and we always assume that the summation is performed only over increasingly ordered The Fourier transform of the form u = I ,J u I J dz I ∧ dz J ∈ p,q c (D) with respect to x 2n+1 , denoted by u, is defined by for every z ∈ U . By using integration by parts, we have for u ∈ p,q be the Hermitian metric on the trivial line bundle U × C over U . The Chern connection of (U × C, e −2tφ ) is given by We can identify ∂∂φ with Levi form L and write R (U ×C,e −2tφ ) = 2tL . Moreover, we will identify •, where ∂ * , ∇ 1,0 * are the formal adjoints of ∂, ∇ 1,0 with respect to · , · L 2 (U ,e −2tφ ) , respectively, and ∂ * b , ∂ * b are the formal adjoints of ∂ b , ∂ b with respect to ( · | · ), respectively. (3.11) Thus the first equality holds. From Parseval's formula, Thus the second equality holds. The proofs of the third and the fourth equalities are similar.

CR Bochner-Kodaira-Nakano Formula I
Analog to [23, (1.4.32)], we define the Lefschetz operator X ∧ · on •,• (T * X ) and its adjoint = i( X ) with respect to the Hermitian inner product ·|· associated with X . The Hermitian torsion of X is defined by (3.14) (1,0) U be as in the discussion after (2.26). We can check that Note that i L j and i L j are the adjoints of e j ∧ and e j ∧, respectively. Since , which is independent of x 2n+1 . We remark that T is a differential operator of order zero. With respect to the Hermitian inner product ·|· associated with X , we have the adjoint operator T * , the conjugate operator T and the adjoint of the conjugate operator T * for T .

Theorem 3.3 With the notations used above, we have on
Proof Since the both side of (3.16) are globally defined, we can check (3.16) on a BRT chart. Now, we work on a BRT chart D = U × I. We will use the same notations as before. Let where ∇ 1,0 is given by (3.5), ∂ * , ∇ 1,0 * are the formal adjoints of ∂, ∇ 1,0 with respect to · , · L 2 (U ,e −2tφ ) , respectively. From [23, (1.4.44)], Similarly, we have Thirdly, we have Fourthly, we consider the rest terms .
Thus we obtain Similarly, we obtain The theorem follows.

Corollary 3.4 (CR Nakano's inequality I) With the notations used above, for any u
If (X , T (1,0) X ) is Kähler, i.e., d X = 0, then Proof By the Cauchy-Schwarz inequality, Theorem 3.3 and since T = 0, T * = 0 if d X = 0, we get the corollary.
The following follows from straightforward calculation, we omit the proof.

Proposition 3.5 For a real
(3.26) Corollary 3.6 With the notations used above, let X be a Hermitian metric on X such that

27)
Then for any u ∈ n,q By d X = d(2 √ −1L ) = 0 and Corollary 3.4, we obtain Let E be a CR line bundle over X (see Definition 2.4 in [13]. We say that E is a R-equivariant CR line bundle over X if the R-action on X can be CR lifted to E and for every point x ∈ X , we can find a T -invariant local CR frame of E defined near x (see [16,Definitions 2.6,2.9]). Here, we also use T to denote the vector field acting on sections of E induced by the R-action on E. From now on, we assume that E is a R-equivariant CR line bundle over X with a R-invariant Hermitian metric h E on E. For p, q ∈ N 0 , let p,q (X , E) be the space of smooth ( p, q)-forms of be the tangential Cauchy-Riemann operator with values in E. Let ( · | · ) E be the L 2 inner product on •,• c (X , E) induced by · | · and h E . Let be the connection on E induced by h E given as follows: Let s be a T -invariant local CR frame of E on an open set D of X , Then, It is straightforward to check that (3.33) is independent of the choices of T -invariant local CR trivializing sections s and hence is globally defined. Put

36)
where R E ∈ 1,1 (X ) is the curvature of E induced by h E .

CR Bochner-Kodaira-Nakano Formula II
The bundle K * be the natural isometry defined as follows: It is easy to see that the definition above is independent of the choices of R-invariant orthonormal frame {L j } n j=1 ⊂ T (1,0) D and hence is globally defined. We have the isometry: Moreover, it is straightforward to see that We can now prove Theorem 3.8 With the notations used above, we have on 0,• (X ), Proof Let u ∈ 0,q (X ). From (3.37) and (3.36), we have (3.39) It is straightforward to check that (3.40) From (3.39), (3.40) and noting that (e j ∧i L k −i L j e k ∧)v = e k ∧i L j v, T * (∇ K * X ) 1,0 v = 0, for every v ∈ n,q (X , K * X ) and T commutes with , we get (3.38).

Corollary 3.9
With the notations used above, assume that 2 √ −1L = X and there is C > 0 such that Then, for any u ∈ 0,q such that 2L (L j , L k ) = δ jk , for every j, k. We write u = J u J e J on D with u J ∈ C ∞ (D) and e J = e j 1 ∧ . . . ∧ e j q , j 1 < . . . < j q . We have

Szegő Kernel Asymptotics
In this section, we will establish Szegő kernel asymptotic expansions on X under certain curvature assumptions.

Complete CR Manifolds
Let X be a CR manifold as in Assumption 1.1. Let g X be the R-invariant Hermitian metric as in (1.5). We will assume in the following that the Riemannian metric induced by g X on T X is complete and study the extension ∂ b , ∂ * b , and T . We denote by the same symbols the maximal weak extensions in L 2 of these differential operators.

Lemma 4.1 Assume that (X , g X ) is complete. Then
Here the graph norm of a linear operator R is defined by u + Ru for u ∈ Dom R.
As a consequence, analog to [23, Corollary 3.3.3], we obtain the following:

Corollary 4.3 If (X , g X ) is complete, then
Using these results, we extend the estimates from Corollary 3.9 as follows: Theorem 4.4 Let X be a CR manifold as in Assumption 1.1. Assume that 2 √ −1L = X , g X is complete and there is C > 0 such that Then, for any u ∈ L 2 0,q (X ), From (3.41), we have for every j = 1, 2, . . ., Taking j → ∞ in (4.3) and using (4.2), we get (4.1).
Let us describe two examples of CR manifolds with complete R-invariant metric g X .

Example 4.5
Let (X , H X, J , ω 0 ) be a compact strictly pseudoconvex CR manifold as in Assumption 1.1 and let g X be a R-invariant metric as in (1.5). Let π : X → X be a Galois covering of X , that is, there exists a discrete, proper action such that X = X / . By pulling back the objects from X by the projection π we obtain a strictly pseudoconvex CR manifold ( X , H X , J , ω 0 ) satisfying Assumption 1.1. Moreover, the metric g X = π * g X is a complete R-invariant metric satisfying (1.5). be the circle bundle of L * ; it is isomorphic to the S 1 principal bundle associated to L. Since X is a hypersurface in the complex manifold L * , it has a CR structure (X , H X, J ) inherited from the complex structure of L * by setting T (1,0) X = T X ∩ T (1,0) L * .
In this situation, S 1 acts on X by fiberwise multiplication, denoted (x, e iθ ) → xe iθ . A point x ∈ X is a pair x = ( p, λ), where λ is a linear functional on L p , the S 1 action is xe iθ = ( p, λ)e iθ = ( p, e iθ λ).
Let ω 0 be the connection 1-form on X associated to the Chern connection ∇ L . Then where R L is the curvature of ∇ L . Assume R L is positive, hence X is a strictly pseudoconvex CR manifold. Hence, (X , H X, J , ω 0 ) fulfills Assumption 1.1. We denote by ∂ θ the infinitesimal generator of the S 1 action on X . The span of ∂ θ defines a rank one sub-bundle T V X ∼ = T S 1 ⊂ T X, the vertical sub-bundle of the fibration π : X → M.
We construct now a Riemannian metric on X . Let g M be a J -invariant metric on T M associated to M . The Chern connection ∇ L on L induces a connection on the S 1 -principal bundle π : X → M, and let T H X ⊂ T X be the corresponding horizontal bundle. Let g X = π * g M ⊕ dθ 2 /4π 2 be the metric on T X = T H X ⊕ T S 1 , with dθ 2 the standard metric on S 1 = R/2π Z. Then g X is a R-invariant Hermitian metric on X satisfying (1.5). Since g M is complete it is easy to see that g X is also complete.

The Operators Q , Q [ 1 , ] , Q
From now on, we assume that X is a CR manifold satisfying Assumption 1.
(4.6) where 1 A denotes the characteristic function of the set A. Let be the orthogonal projections with respect to ( · | · ). Since X is strictly pseudoconvex, from [21, Lemma 3.4 (3), p. 239], [13, Theorem 3.5], we have one of the following two cases: (a) The R -action is free, (b) The R -action comes from a CR torus actionT d on X and ω 0 and X are T d invariant.
(4.8) When X is non-compact, the R-action does not always come from a CR torus action. For example, when X is the Heisenberg group (see Sect. 5), the standard R-action on X is free and does not come from any CR torus action.
, for every t ∈ R. We can repeat the proof above and get that and supp Q λ u ⊂D, for every u ∈ •,• c (D). We obtain (4.9). The proof of (4.10) is similar.
We now assume that the R-action is not free. From (4.8), we know that the R-action comes from a CR torus action T d = (e iθ 1 , . . . , e iθ d ) on X and ω 0 , X are T d invariant.
Since the R-action comes from the T d -action, there exist β 1 , . . . , β d ∈ R, such that where T j is the vector field on X given by T j u := ∂ ∂θ j ((1, . . . , 1, e iθ j , 1, . . . , 1 j = 1, . . . , d. For (m 1 , . . . , m d  be the orthogonal projection. It is not difficult to see that for every u ∈ L 2 •,• (X ), we have From Lemma 4.7 and (4.19), we conclude that (2.9). From Proposition 4.8, we see that From now on, we write b,λ to denote b and b,λ acting on (0, q) forms, respectively.

Local Closed Range for (0) b
In this section, we will establish the local closed range property for (0) b under appropriate curvature assumptions. We first need the following.
Theorem 4.10 Assume that 2 √ −1L = X , g X is complete and there is C > 0 such that has closed range and ker Hence, (4.26) From (4.26), the theorem follows.
We now consider (0, 1)-forms. Let G (1) λ be as in (4.25). Since G We use the previous result to solve the ∂ b -equation for eigenforms of √ −1T .

Theorem 4.11
Assume that 2 √ −1L = X , g X is complete and there is C > 0 such that where C 0 > 0 is a constant as in (4.27).
where C 0 > 0 is as in (4.27). The theorem follows.
b be the orthogonal projection with respect to ( · | · ). From Proposition 4.8, we can check that (4.30) We recall the following notion introduced in [19, Definition 1.8].

Theorem 4.13
Assume that g L is complete and there is C > 0 such that Let D X be an open set. Let λ ∈ R, λ < −C. Then, (0) b has local closed range on D with respect to Q λ .
From Theorem 4.11, there exists g ∈ L 2 (X , L ) with where C 0 > 0 is a constant as in (4.28).
It is clear that Q τ λ (I − S (0) )u ≤ Q λ (I − S (0) )u , for every u ∈ L 2 (X ). From this observation and Theorem 4.13, we deduce that Theorem 4.14 Assume that g L is complete and there is C > 0 such that Let D X be an open set. Let λ ∈ R, λ < −C. Then, b has local closed range on D with respect to Q τ λ .

Local Closed Range for (n,0) b
In this section, we will establish the local closed range property for (n,0) b under appropriate curvature assumptions. We observe that the condition (4.32) can be removed, if we consider (n, 0)-forms instead of smooth function. We will adopt the same notation as before.
Let b be the Gaffney extension of the usual Kohn Laplacian. Let be orthogonal projection. It is known that Q τ , Q λ , Q [λ 1 ,λ] commutes with S (n,q) on L 2 n,q (X ). Now, we present the main result of this section as follows: Theorem 4.15 Let X be a CR manifold with a transversal CR R-action. Let X be an R-invariant Hermitian metric on X . Assume that g L is complete. Let D X be an open set. Let λ ∈ R, λ < 0. Then, (n,0) b has local closed range on D with respect to Q λ , i.e., there exists C > 0 such that for all u ∈ n,0 c (D), This result is very natural in view of the Kodaira vanishing theorem, in the same way as Theorem 4.13 is parallel to the Kodaira-Serre type vanishing theorem. The proof is analog to the proof of Theorem 4.13.
Firstly, from Corollary 3.6 and the density Lemma 4.1, we obtain the following:

Lemma 4.16
With the notations used above, let X be a Hermitian metric on X such that
Moreover, we have the following analog of Theorem 4.10.

Theorem 4.18 Let X be a CR manifold with a transversal CR R-action. Let X be an R-invariant
Hermitian metric on X . Assume that 2 √ −1L = X and g X is complete. Let 1 ≤ q ≤ n and λ < 0. Then the operator has closed range, and Ker : Secondly, we solve the ∂ b -equation as follows.

Theorem 4.19
Let X be a CR manifold with a transversal CR R-action. Let X be an R-invariant Hermitian metric on X . Assume that 2 √ −1L = X and g X is complete. Then for every λ < 0 and every v ∈ E (n,1) (λ, (4.54) The proof is complete.

Proof of Theorem 4.15
Note that X is not necessarily equal to 2 √ −1L so we have to deduce the general case to the particular case considered in Theorems 4. 16-4.19.
Note that Q λ is independent of the choice of X . Then v ∈ L 2 n,1 (X , L ) ∩ L 2 n,1 (X ). From the above theorem, we can find g ∈ L 2 n,0 (X , L ) such that ∂ b g = v and where the first inequality in (4.55) follows from Theorem 4.19. We claim that g = g L and thus g ∈ L 2 n,0 (X , L ) ∩ L 2 n,0 (X ). In fact, we write locally and, respectively, the volume forms are given by n X ∧ ω 0 = n!( (4.58) Thus, the claim follows from (4.59) S (n,0) )Q λ u is the solution of minimal norm with respect to X , i.e., by supp(u) D.

L 2 Estimates for @ b,E
In this section, we prove an analog for the ∂ b,E -operator of the L 2 -estimates of the Hörmander-Andreotti-Vesentini estimates for ∂. As in the case of complex manifolds, we use the Bochner-Kodaira-Nakano formula in the present form (3.36 is a positive formally self-adjoint operator, and is an operator of order zero depending only on the torsion of Hermitian metric X . Then, for any f ∈ L 2 r ,q (X , E) satisfying ∂ b,E f = 0 and X ψ −1 | f | 2 dv X < ∞, there exists g ∈ L 2 r ,q−1 (X , E) such that ∂ b,E g = f and g 2 ≤ X ψ −1 | f | 2 dv X .
Proof Consider the complex of closed densely defined operators where T and S are maximal extensions of ∂ b,E . We apply (4.61) and obtain for all s ∈ r ,q c (X , E), it follows that (4.65) By Cauchy-Schwarz inequality, Since g X is complete, the above inequality still holds for all s ∈ Dom(S) ∩ Dom(T * ) by the density Lemma 4.1. Consider now s ∈ Dom(T * ) and write the orthogonal decomposition s = s 1 +s 2 with s 1 ∈ Ker(S) and s 2 ∈ Ker(S) ⊥ ⊂ [Im(S * )] ⊂ Ker T * .

Corollary 4.21 Let X be a CR manifold with a smooth locally free CR R-action. Let
X be an R-invariant Hermitian metric on X . Assume g X is complete. Assume L = 0 and d X = 0. Let E be a R-equivariant CR line bundle over X with a R-invariant Hermitian metric h E . Let λ 1 ≤ · · · ≤ λ n be eigenvalues of R E with respect to X . Assume (E, h E ) is Nakano q-semipositive with respect to X , i.e., λ 1 + · · · + λ q ≥ 0. Then, for any f ∈ L 2 n,q (X , E) satisfying ∂ b,E f = 0 and X (λ 1 + · · · + λ q ) −1 | f | 2 dv X < ∞, there exists g ∈ L 2 n,q−1 (X , E) such that ∂ b,E g = f and g 2 ≤ X (λ 1 + · · · + λ q ) −1 | f | 2 dv X .

Vanishing Theorems
In this section, we present some vanishing theorems that follow from the previous L 2 estimates. We obtain first a CR counterpart of the Kodaira vanishing theorem [23,Theorem 1.5.4.(a)] as follows:

Corollary 4.22
Assume that 2 √ −1L = X , g X is complete and let λ < 0 and 1 ≤ q ≤ n. Then, we have This follows immediately from Theorem 4.18.
We obtain a CR counterpart of the Kodaira-Serre vanishing theorem [23, Theorem 1.5.6] as follows: Corollary 4.23 Assume that 2 √ −1L = X , g X is complete and let C > 0 such that Let λ < −C and 1 ≤ q ≤ n. Then, we have This follows immediately from Theorem 4.10. We note that the previous vanishing theorems on CR manifolds imply the following generalizations due to Andreotti-Vesentini [2] of the Kodaira-Serre and Kodaira vanishing theorems for complete Kähler manifolds. Proof We apply the previous results for the CR manifold X constructed in Example 4.6. In this case T = ∂ θ . For m ∈ Z, the space . Hence, the assertion follows from Theorem 4.10.
In the same vein recover from Theorem 4.18 the following vanishing theorem for the L 2 -cohomology of positive bundles twisted with the canonical bundle on complete Kähler manifolds.

Szegö Kernel Asymptotic Expansions
In this section, we prove Theorem 1. Let be the characteristic manifold of b . We have For a given point x 0 ∈ D, let {W j } n j=1 be an orthonormal frame of T (1,0) X with respect to · | · near x 0 , for which the Levi form is diagonal at x 0 . Put (4.72) We will denote by We recall the following results in [19, Theorems 1.9, 5.1]. (x 1 , . . . , x 2n+1 ). Let Q : L 2 (X ) → L 2 (X ) be a continuous operator and let Q * be the L 2 adjoint of Q with respect to ( · | · ). Suppose that where

Theorem 4.27 Let D X be an open coordinate patch with local coordinates x =
ϕ(x, y) = −ϕ(y, x), a(x, y, t) ∈ S n cl D × D × R + and the leading term a 0 (x, y) of the expansion (2.4) of a(x, y, t) satisfies where det L x is the determinant of the Levi form defined in (4.73), q(x, η) ∈ C ∞ (T * D) is the principal symbol of Q.
where τ λ ∈ C ∞ (R) is as in the discussion before Theorem 4.14. It is not difficult to see thatQ Assume that the R-action is free. From (4.11), we see that Q τ λ =Q τ λ on D. From this observation, Theorems 4.14, 4.27, (4.78), and noticing that Q * where Q * τ λ is the L 2 adjoint of Q τ λ with respect to ( · | · ), we get Theorem 4.28 Suppose that the R-action is free. Assume that g L is complete and there is C > 0 such that where the integral is defined as an oscillatory integral and δ α is the Dirac measure at α. Using (4.82), (4.86), and the Fourier inversion formula, (4.85) becomes (4.87) From (4.87), the claim (4.83) follows.
To study Q τ 2 λ S (0) when the R is not free, we also need the following two known results [19, Theorems 3.2, 5.2]. Theorem 4. 30 We assume that the R-action is arbitrary. Let D X be a coordinate patch with local coordinates x = (x 1 , . . . , x 2n+1 ). Then there exist properly supported continuous operators A ∈ L −1  where ϕ(x, y) ∈ C ∞ (D × D) and s(x, y, t) ∈ S n cl (D × D × R + ) are as in (4.79).

Theorem 4.31
Let us consider an arbitrary R-action and let Q : L 2 (X ) → L 2 (X ) be a continuous operator and let Q * be the L 2 adjoint of Q with respect to ( · | · ). Suppose that (0) b has local L 2 closed range on D with respect to Q and Q S (0) = S (0) Q on L 2 (X ). Let D be a coordinate patch with local coordinates x = (x 1 , . . . , x 2n+1 ). We have Q * S (0) Q ≡S * Q * QSon D, (4.90) whereS is as in Theorem 4.30.
be a parametrix of s X on D 0 and G s is properly supported on D 0 . Hence, there is a properly supported smoothing operator for all g ∈ C ∞ c (D 0 ). Now, on D 0 , Since F s is smoothing, we have where C > 0 is a constant. Now, we can integrate by parts and repeat the proof of (4.95) and show that where From (4.100), (4.101), and noticing that G s : is continuous, we can repeat the proof of (4.96) and conclude that Hence,SR τ λ is smoothing on D 0 .
Using this observation, we can repeat the proof of [19,Theorem 5.8] and obtain the conclusion.

Theorem 4.34
Let us consider an arbitrary R-action. Assume that g L is complete and there is C > 0 such that Let D = U × I be a BRT chart with BRT coordinates x = (x 1 , . . . , x 2n+1 ). Let λ ∈ R, λ < −C. Then, The theorem follows.
We can now prove the main result of this work. Proof of Corollary 1.4 Let K be a compact set of X . Fix x ∈ K . From Theorem 1.2 and the fact that the Szegő kernel is smoothing away the diagonal, we can repeat the proof of [19,Theorem 1.10] and deduce that there are open neighborhoods V x ⊂ U x of x and global smooth L 2 CR functions ( f 0,x , f 1,x , · · · , f N x ,x ) = F x such that F x : U x → C N x +1 is an embedding and sup K \U x | f 0,x | ≤ 1 2 , inf V x | f 0,x | ≥ 1. There exists x 1 , x 2 , . . . , Then K x → (F x 1 , · · · , F x m ) is an embedding.
Proof of Corollary 1. 5 We proceed as in the proof of Corollary 1.5 by working on a compact coordinate patch K with coordinates (x 1 , . . . , x 2n+1 ) and observing that in these coordinates a CR (n, 0)-form equals f dz 1 ∧ . . . ∧ dz n with f a CR function on K .

Examples
We now consider Heisenberg group H = C n × R with CR structure T (1,0) where φ ∈ C ∞ (C n , R). Let ( · | · ) H be the L 2 inner product on H induced by the Euclidean measure dx on R 2n+1 . Let , for every z ∈ C n . Suppose that there is C > 0 such that ≤ C, for every z ∈ C n . where ϕ ∈ C ∞ (D × D) and s(x, y, t) ∈ S n cl D × D × R + are as in Theorem 1.2. Example 5. 2 With the notations used in Corollary 5.1, assume that φ(z) = |z| 2 + r (z), (5.4) with r (z) ∈ C ∞ c (C n ) and √ −1∂∂(|z| 2 + r (z)) > 0 on C n . With this φ, we can check the conditions of Corollary 5.1 fulfilled as follows. In fact, in this case, we have det ∂ 2 φ ∂z j ∂z k n j,k=1 = det ∂ 2 (|z| 2 + r (z)) ∂z j ∂z k n j,k=1  F(z))) ∈ 1,1 c (C n ).
With this φ, it is easy to see that (5.2) hold. This example shows that, after small perturbation of the Levi form of Heisenberg group, we still can obtain the Szegő kernel expansion via Corollary 5.1. (X , T (1,0) X ) be a strictly pseudoconvex, CR manifold of dimension 2n + 1, n ≥ 1, with a discrete, proper, CR action such that the quotient X / is compact. Assume X admits a transversal CR R-action on X and let X be a -invariant, R-invariant, Hermitian metric on X . Then the conclusion of Theorem 1.2 holds. In fact, the -covering manifold is complete and we can find the desired constant C depending on the fundamental domain U X given by the -action such that (1.7) is fulfilled. As a consequence, if we consider the circle bundle case in which R L = 2L , we could obtain the Bergman kernel expansion for covering manifold [23, 6.1.2].

Example 5.3 Let
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Conflict of interest
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