Szeg\H{o} kernel asymptotics on some non-compact complete CR manifolds

We establish Szeg\H{o} kernel asymptotic expansions on non-compact strictly pseudoconvex complete CR manifolds with transversal CR $\mathbb{R}$-action under certain natural geometric conditions.


INTRODUCTION
Let ( Ì 1 0 ) be a CR manifold of dimension 2Ò + 1, Ò 1. The orthogonal projection Ë (Õ) : Ä 2 0 Õ ( ) Ker (Õ) onto Ker (Õ) is called the Szegő projection, while its distribution kernel Ë (Õ) (Ü Ý) is called the Szegő kernel, where (Õ) denote the Kohn Laplacian acting on (0 Õ) forms. The study of the Szegő kernel is a classical subject in several complex variables and CR geometry. When is strictly pseudoconvex, compact and (0) has closed range, Boutet de Monvel-Sjöstrand [BS76] showed that Ë (0) (Ü Ý) 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 [MM07,MM08].
However, almost all the results on Szegő kernel assumed that is compact, while for non-compact complex manifolds the Bergman kernel asymptotics was comprehensively studied [MM07,MM08,MM15]. Therefore, it is interesting to look for counterparts for the Szegő kernel on non-compact CR manifolds. Let us see some simple examples and describe our motivation briefly. On C Ò , consider the hypersurface := Þ = (Þ 1 Þ Ò ) ¾ C Ò ; Im Þ Ò = (Þ 1 Þ Ò 1 ) where ¾ C ½ (C Ò 1 R). Then, is a non-compact CR manifold. There are many smooth CR functions on . Even in this simple example, we do not know the behavior of the associated Szegő kernel. Let us see another example. Consider H = C Ò ¢ R with CR structure where ¾ C ½ (C Ò R). Then, H is also a non-compact CR manifold and the Szegő kernel has been studied when is quadratic (see [HHL18]) but for general , there are fewer results. Both 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 [Hsi10], the first author obtained the Szegő kernel asymptotic expansion on the non-degenerate part of the Levi form under the assumption that Kohn Laplacian has closed range. It should be mentioned that the method in [Hsi10] works well for noncompact setting. But for general non-compact CR manifolds, closed range property is not a natural assumption since in the Heisenberg case mentioned above, even for quadratic, (0) does not have closed range but we still have Szegő kernel asymptotic expansion. In this paper, we show that (0) has local closed range with respect to some operator É (see Definition 4.12) under certain geometric conditions. Furthermore, combining this local closed range property and by more 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 have local closed range property, we established the CR Bochner-Kodaira-Nakano formula analogue to [Dem85], see Theorem 3.1, which has its own interest. This is also a refinement of Tanaka's basic identities [Tan75, Theorems 5.1, 5.2] in our context. We remark that the results in this paper hold both for transversal CR R-action and Ë 1 -action.
We now formulate our main result. We will work in the following setting.
Assumption 1.1. ( À Â 0 ) is an orientable strictly pseudoconvex CR manifold of dimension 2Ò + 1, Ò 1, where À is the Levi distribution, Â is the complex structure and 0 is a contact form, endowed with a smooth transversal CR R-action on preserving 0 and Â.
Remark 1.4. Our conditions (1.10) are different from [MM07, Theorem 6.1.1]. In particular, we do not need that the uniform bounded condition for Θ Å . The reason is that we can get asymptotic expansion under local closed range condition instead of standard closed range or spectral gap condition.

Let
be an open set . The spaces of distributions of and smooth functions of will be denoted by D ¼ ( ) and C ½ ( ) respectively. Let E ¼ ( ) be the subspace of D ¼ ( ) whose elements have compact support in . Let C ½ ( ) be the subspace of C ½ ( ) whose elements have compact support in . Let : C ½ ( ) D ¼ ( ) be a continuous map. We write (Ü Ý) to denote the distribution kernel of . In this work, we will identify with (Ü Ý). The following two statements are equivalent (I) is continuous: If satisfies (a) or (b), we say that is smoothing on . Let if is a smoothing operator. We say that is properly supported if the restrictions of the two projections (Ü Ý) Ü, (Ü Ý) Ý to Supp Ã are proper.

Let
:= £ + £ be the Kohn Laplacian on Ω¯ ¯( ). Let := £ + £ be the anti-Kohn Laplacian on Ω¯ ¯( ). We still denoted by the maximal extension and by £ the Hilbert space adjoint with respect to the Ä 2 -inner product on . We also denote by 2.1. The R-action on . Let ( Ì 1 0 ) be a CR manifold of dim = 2Ò + 1. Let Ö : R ¢ , Ö(Ü) = Ö AE Ü for Ö ¾ R, be a R-action on , see [HHL17]. Let Ì be the global real vector field on given by From now on, assume that admits a transversal and CR R-action. We take the Hermitian metric ¡ ¡ so that Ì = Ì .
is a frame of Ì 1 0 , and is the dual frame.
Thus the first equality holds. From Parserval's formula, Thus the second equality holds. The proofs of the third and the fourth equalities are similar.

CR BOCHNER FORMULA WITH R-ACTION
Recall that we work with the assumption that admits a transversal CR R-action on . We will prove Bochner-Kodaira-Nakano formulas in the CR setting. They are refinements of Tanaka's basic identities [Tan75, 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 Θ .
3.1. CR Bochner-Kodaira-Nakano formula I. Analogue to [MM07,(1.4.32)], we define the Lefschetz operator Θ ¡ on Î¯ ¯( Ì £ ) and its adjoint Λ = (Θ ) with respect to the Hermitian inner product ¡ ¡ associated with Θ . The Hermitian torsion of Θ is defined by be as in the discussion after (2.32). We can check that Note that Ä and Ä are the adjoints of and respectively. Since We remark that Ì is a differential operator of order zero. With respect to the Hermitian inner product ¡ ¡ associated with Θ , we have the adjoint operator Ì £ , the conjugate operator Ì and the adjoint of the conjugate operator Ì £ for Ì .
Theorem 3.1. With the notations used above, we have on Ω¯ ¯( ), Proof. Since the both side of (3.3) are globally defined, we can check (3.3) on a BRT chart. Now, we work on a BRT chart = Í ¢ Á. We will use the same notations as before. Let In fact, from Proposition 2.7, Fourthly, we consider the rest terms By Proposition 2.7, we have Similarly, we obtain (3.10) From (3.4), (3.5), (3.6), (3.9) and (3.10), we get that for Ù Ú ¾ Ω¯ ( ), The theorem follows.
The following follows from straightforward calculation, we omit the proof  [HML17]). Here we also use Ì to denote the vector field acting on sections of induced by the R-action on Ψ : Ì £0 Õ Ì £Ò Õ ª Ã £ be the natural isometry defined as follows: Let = Í ¢ Á be a BRT chart. Let Ä Ò =1 Ì 1 0 , Ò =1 Ì £1 0 be as in the discussion after (2.32). Then, It is easy to see that the definition above is independent of the choices of R-invariant orthonormal frame Ä Ò =1 Ì 1 0 and hence is globally defined. We have the isometry: Moreover, it is straightforward to see that (3.25) We can now prove Theorem 3.6. With the notations used above, we have on Ω 0 ¯( ), Proof. Let Ù ¾ Ω 0 Õ ( ). From (3.25) and (3.24), we have It is straightforward to check that Proof. Since 2 Ô 1L = Θ , we can choose R-invariant orthonormal frame Ä Ò =1 such that 2L (Ä Ä ) = AE , for every . We write Ù = È Â Ù Â Â on with Ù Â ¾ C ½ ( ) and

SZEGŐ KERNEL ASYMPTOTICS
In this section, we will establish Szegő kernel asymptotic expansions on under certain curvature assumptions. 4.1. Complete CR manifolds. Let be a CR manifold as in Assumption 1.1. Let be the R-invariant Hermitian metric as in (1.3). We will assume in the following that the Riemannian metric induced by on Ì is complete and study the extension , £ and Ì . We denote by the same symbols the maximal weak extensions in Ä 2 of these differentials operator. Since Using these results and extend the estimates from Corollary 3.7 as follows.  Galois covering of , that is, there exists a discrete, proper action Γ such that = Γ.
Since is a hypersurface in the complex manifold Ä £ , it a has a CR structure ( À Â) inherited from the complex structure of Ä £ by setting Ì 1 0 = Ì Ì 1 0 Ä £ .
In this situation, Ë 1 acts on by fiberwise multiplication, denoted (Ü ) Ü . A point Ü ¾ is a pair Ü = (Ô ), where is a linear functional on Ä Ô , the Ë 1 action is Let 0 be the connection 1-form on associated to the Chern connection Ö Ä . Then
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 = ( ½ ) on and 0 , Θ are T invariant.
Since the R-action comes from the T -action, there exist ¬ 1 ¬ ¾ R, such that where 0 0 is a constant as in (4.27).
Note that Ä 2 0 Õ ( ) Ñ = E (Õ) ( Ñ Ô 1 ) and the Ä 2 -Dolbeault complex (Ä 2 0 ¯( Å Ä Ñ ) ) is isomorphic to the -complex (Ä 2 0 ¯( ) Ñ ). Hence the assertion follows from Theo- For a given point Ü 0 ¾ , let Ï Ò =1 be an orthonormal frame of Ì 1 0 with respect to ¡ ¡ near Ü 0 , for which the Levi form is diagonal at Ü 0 . Put    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 = ( ½ ) on and 0 , Θ are T invariant. We will use the same notations as in the discussion before Proposition 4.8. We need Proof. We also write Ý = ( Note that the following formula holds for every « ¾ R, where the integral is defined as an oscillatory integral and AE « is the Dirac measure at «. Using (4.67), (4.71) and the Fourier inversion formula, (4.70) becomes From (4.72), the claim (4.68) follows.
With this , 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.
Example 5.3. Let ( Ì 1 0 ) be a strictly pseudoconvex, CR manifold of dimension 2Ò + 1, Ò 1, with a discrete, proper, CR action Γ such that the quotient Γ is compact. Assume admits a transversal CR R-action on and let Θ be a Γ-invariant, R-invariant, Hermitian metric on . Then the conclusion of Theorem 1.2 holds. In fact, Γ-covering manifold is complete and we can find the desired constant depending on the fundamental domain Í ⋐ given by the Γ-action such that (1.5) fulfilled. As a consequence, if we consider the circle bundle case in which Ê Ä = 2L , we could obtain the Bergman kernel expansion for covering manifold [MM07, 6.1.2].