Abstract
We prove a Bochner–Kodaira–Nakano formula and establish Szegő kernel expansions on complete strictly pseudoconvex CR manifolds with transversal CR \(\mathbb {R}\)-action under certain natural geometric conditions. As a consequence we show that such manifolds are locally CR embeddable.
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1 Introduction
The first goal of this paper is to develop a differential geometric formalism on strictly pseudoconvex CR manifolds with \(\mathbb {R}\)-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 \(\overline{\partial }_b\)-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 \(\mathbb {R}\)-action. This entails global and local embeddability theorems for CR manifolds with \(\mathbb {R}\)-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 \(2n+1\), \(n\ge 1\). The orthogonal projection \(S^{(q)}:L^2_{0,q}(X)\rightarrow \ker \Box ^{(q)}_b\) onto \(\ker \Box ^{(q)}_b\) is called the Szegő projection, while its distribution kernel \(S^{(q)}(x,y)\) is called the Szegő kernel, where \(\Box ^{(q)}_b\) 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 \(\Box ^{(0)}_b\) 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 \(Y:=\{z=(z_1,\ldots ,z_n)\in {\mathbb {C}}^n;\, \mathrm{Im\,}z_n=f(z_1,\ldots ,z_{n-1})\}\), where \(f\in \mathscr {C}^\infty ({\mathbb {C}}^{n-1},{\mathbb {R}})\). 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 \(\mathbb {H}=\mathbb {C}^n\times \mathbb {R}\) with CR structure \(T^{(1,0)}\mathbb {H}:=\mathrm{span\,}\left\{ \frac{\partial }{\partial z_j}+ i\frac{\partial \phi }{\partial z_j}(z)\frac{\partial }{\partial x_{2n+1}}:1\le j\le n\right\} \), where \(\phi \in \mathscr {C}^\infty (\mathbb {C}^n,\mathbb {R})\). Then, \(\mathbb {H}\) is also a non-compact CR manifold and the Szegő kernel has been studied when \(\phi \) is quadratic (see [14]). However, for general \(\phi \) there are fewer results. Both Y and \(\mathbb {H}\) are non-compact CR manifolds with transversal CR \({\mathbb {R}}\)-action. Therefore, we think that the study of the Szegő kernels on non-compact CR manifolds with transversal CR \({\mathbb {R}}\)-action is a very natural and interesting question.
In [15], 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 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 \(\phi \) quadratic, \(\Box ^{(0)}_b\) does not have closed range; however, the Szegő kernel still has an asymptotic expansion.
In this paper, we show that \(\Box ^{(0)}_b\) has local closed range with respect to a spectral projection \(Q_\lambda \) (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 \(\mathbb {R}\)-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 \(\mathbb {R}\)-action and \(S^1\)-action.
We will work in the following setting. Let X be a connected smooth paracompact manifold of dimension \(2n+1\), HX be a smooth sub-bundle of TX of rank 2n, and J be a smooth complex structure on the fibers of HX. Let \(T^{(1,0)}X\) be the complex sub-bundle of the complexification \(\mathbb {C}HX\) of HX, which corresponds to the i eigenspace of J, that is, \(T^{(1,0)}X=\{v-iJv:v\in HX\}\). We say that X is a CR manifold (of hypersurface type) if the formal integrability condition
holds. The sub-bundle HX is called Levi distribution and the annihilator \((HX)^0\subset T^*X\) of HX is called the characteristic bundle of the CR manifold X.
We will assume in the sequel that X is orientable. Since HX is oriented by its complex structure, it follows that \((HX)^0\) is a real orientable line bundle, thus trivial. A global frame of \((HX)^0\), that is, a real non-vanishing 1-form \(\omega _0\in \mathscr {C}^\infty (X,T^*X)\) such that \((HX)^0=\mathbb {R}\omega _0\), is called characteristic 1-form.
Given a characteristic 1-form \(\omega _0\) on X the Levi form \(\mathscr {L}^{\omega _0}\) is defined by
We say that (X, HX, J) is strictly pseudoconvex if there exists a characteristic 1-form \(\omega _0\) the Levi form \(\mathscr {L}^{\omega _0}_x\) is positive definite at every point \(x\in X\). If \(\mathscr {L}^{\omega _0}\) is positive definite, then \(d\omega _0\) is symplectic on HX, thus \(\omega _0\) is a contact form and HX is a contact structure. Associated with a contact form \(\omega _0\) one has the Reeb vector field \(T=T^{\omega _0}\), uniquely defined by the equations
Assumption 1.1
\((X,HX,J,\omega _0)\) is an orientable strictly pseudoconvex CR manifold of dimension \(2n+1\), \(n\ge 1\), where HX is the Levi distribution, J is the complex structure, and \(\omega _0\) is a contact form. We assume that X is endowed with a smooth locally free \(\mathbb {R}\)-action preserving \(\omega _0\) and J such that the infinitesimal generator of the \(\mathbb {R}\)-action is a Reeb vector field, denoted T.
We denote by \(T^{(1,0)}X\) and \(T^{(0,1)}X\) the bundles of tangent vectors of type (1, 0) and (0, 1), respectively. By Assumption 1.1 the \(\mathbb {R}\)-action is CR and transversal (see (2.12) and (2.13)), hence we have a decomposition \(\mathbb {C}TX = T^{(1,0)}X \oplus T^{(0,1)}X \oplus \mathbb {C}T\). The Levi form (1.2) induces a Hermitian metric, called the Levi (or Webster) metric,
on TX and by extension on \(\mathbb {C}TX\), with the following properties:
In Sect. 2.2, we observe that Assumption 1.1 implies that the contact metric manifold \((X,\omega _0,T,J,g_{\mathscr {L}})\) is a Sasakian manifold. Conversely, every compact Sasakian manifold admits an \(\mathbb {R}\)-action as in Assumption 1.1.
Let \(K^*_X:=\det (T^{(1,0)}X)\) and let \(R^{K^*_X}_{\mathscr {L}}\) be the curvature of \(K^*_X\) induced by \(\langle \,\cdot \,|\,\cdot \,\rangle _{\mathscr {L}}\) (see (2.26) and (4.31)).
More generally, we consider an arbitrary \(\mathbb {R}\)-invariant Hermitian metric \(g=g_X=\langle \,\cdot \,|\,\cdot \,\rangle _g= \langle \,\cdot \,|\,\cdot \,\rangle \) on \(\mathbb {C}TX\) such that (1.5) holds. Given such a metric we will denote by \(\Theta _X\) its fundamental (1, 1)-form given by \(\Theta _X(a,\overline{b})=\sqrt{-1}\langle a\,|\,b\rangle _g\) for \(a,b\in T^{(1,0)}X\). Let \(dv_X\) be the volume form induced by the \(\mathbb {R}\)-invariant metric \(g_X\) as in (1.5). Let \((\,\cdot \,|\,\cdot \,)\) be the \(L^2\) inner product on the space of smooth compactly supported functions \(\mathscr {C}_c^\infty (X)\) with respect to \(dv_X\). We denote by \(L^2(X,dv_X)\) the completion of \(\mathscr {C}_c^\infty (X)\) with respect to \((\,\cdot \,|\,\cdot \,)\).
We denote by \(\overline{\partial }_b\) the tangential Cauchy–Riemann operator (see Definition 2.3). The Szegő projection is the orthogonal projection with respect to \((\,\cdot \,|\,\cdot \,)\),
on the space of square-integrable CR functions on X. The distribution kernel \(S^{(0)}(x,y)\in {\mathscr {D}}'(X\times X)\) of the Szegő projection is called the Szegő kernel. The main result of this article is as follows:
Theorem 1.2
Let \((X,HX,J,\omega _0)\) be an orientable strictly pseudoconvex CR manifold of dimension \(2n+1\), \(n\ge 1\), with an \(\mathbb {R}\)-action on X as in Assumption 1.1. Let \(g_X\) be an \(\mathbb {R}\)-invariant metric as in (1.5) and let \(\Theta _X\) be its fundamental form. Assume that the Levi metric \(g_\mathscr {L}\) is complete and there is \(C>0\) such that
Then the Szegő projection is a Fourier integral operator with complex phase, that is, for any local coordinate patch \((D,x=(x_1,\ldots ,x_{2n+1}))\) with \(D\Subset X\), we have
where the phase function \(\varphi \in \mathscr {C}^\infty (D\times D)\) satisfies
and \(s(x, y, t)\in S^{n}_{\mathrm{cl\,}} \big (D\times D\times {\mathbb {R}}_+\big )\) is a symbol of order n with asymptotic expansion \(s(x,y,t)=\sum _{j=0}^\infty s_j(x,y)t^{n-j}\) whose leading term \(s_0(x,y)\) satisfies
where \(\det {\mathscr {L}}_x\) is the determinant of \(\mathscr {L}_x\) with respect to \(g_X\), cf. (4.73).
We will show in Lemma 2.7 that \(R^{K^*_X}_{\mathscr {L}}={\text {Ric}}_{\mathscr {L}}\) where \({\text {Ric}}_{\mathscr {L}}\in \Omega ^{1,1}(X)\) is the pseudohermitian Ricci form with respect to the pseudohermitian structure \(\omega _0\) (see (2.10)). We refer to Definition 2.1 for the definition of the symbol space \(S^{n}_{\mathrm{cl\,}}\big (D\times D\times {\mathbb {R}}_+\big )\) and to [19, Theorems 3.3, 4.4] for more properties for the phase \(\varphi \) 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 \((X,HX,J,\omega _0)\) be an orientable strictly pseudoconvex CR manifold of dimension \(2n+1\), \(n\ge 1\), with an \(\mathbb {R}\)-action on X as in Assumption 1.1. Assume that the Levi metric \(g_{\mathscr {L}}\) is complete. Then the Szegő projection \(S^{(n,0)}:L^2_{n,0}(X)\rightarrow \ker \overline{\partial }_b \subset L^2_{n,0}(X)\) is a Fourier integral operator with complex phase, that is, for any local coordinate patch \((D,x=(x_1,\ldots ,x_{2n+1}))\) with \(D\Subset X\), the Szegő kernel has the form (1.8) with respect to the trivialization of \(K_X\) given by \(dz_1\wedge \ldots \wedge dz_n\).
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 \(K\subset X\) there exists a positive integer N and CR functions \(f_1,\ldots ,f_N\in L^2(X)\cap \mathscr {C}^\infty (X)\) such that \((f_1,\ldots ,f_N)\) is an embedding of K in \(\mathbb {C}^N\).
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 \(g_{\mathscr {L}}\) 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 \(\mathbb {R}\)-action). An analytic property that we use is that the spectral projections \(Q_\lambda \) of the operator \(\sqrt{-1}T\) (see 4.7) commute to \(\overline{\partial }_b\). 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 \((M,\Theta _M)\), where \(h^L\) denotes a Hermitian metric on L and \(\Theta _M\) is a positive (1, 1) form on M. For every \(k\in {\mathbb {N}}\), let \((L^k,h^{L^k})\) be the k-th power of \((L,h^L)\). The positive (1, 1) form \(\Theta _M\) and \(h^{L^k}\) induces a \(L^2\) inner product \((\,\cdot \,|\,\cdot \,)_{\Theta _M}\) on \(\Omega ^{0,q}_c(M,L^k))\). Let \(L^2_{0,q}(M,L^k)\) be the completion of \(\Omega ^{0,q}_c(M,L^k)\) with respect to \((\,\cdot \,|\,\cdot \,)_{\Theta _M}\). We write \(L^2(M,L^k):=L^2_{0,0}(M,L^k)\). Let
be the space of holomorphic square-integrable sections of \(L^k\). Let \(\{f_j^k\}_{j=1}^{{d_k}}\) be an orthonormal basis for \(H^0_{(2)}(M,L^k)\) with respect to \((\,\cdot \,|\,\cdot \,)_{\Theta _M}\), where \(d_k\in {\mathbb {N}}\cup \{\infty \}\). The Bergman kernel of \(L^k\) is
Let s be a local holomorphic frame of L defined on an open set \(D\Subset M\), \(|s|^2_{h^L}=e^{-2\phi }\), \(\phi \in \mathscr {C}^\infty (D,{\mathbb {R}})\). On D, we write \(f_j^k={\tilde{f}}_j^ks^{\otimes k}\), \({\tilde{f}}_j^k\in \mathscr {C}^\infty (D)\), \(j=1,\ldots ,d_k\). The localized Bergman kernel on D is given by
Let \(R^L\) be the Chern curvature of L induced by \(h^L\). Assume that \(\omega =\sqrt{-1}R^L\) is positive. Let \(K^*_M:=\det (T^{(1,0)}M)\) and let \(R^{K^*_M}_{\omega }\) be the curvature of \(K^*_M\) induced by \(\omega \). Applying Theorem 1.2 to the circle bundle of \((L,h^L)\), we get the following:
Theorem 1.6
Let \((L,h^L)\) be a Hermitian holomorphic line bundle over a Hermitian manifold \((M,\Theta _M)\) of dimension n. We assume that \(\omega =\sqrt{-1}R^L\) defines a complete Kähler metric on M. We assume moreover that there is \(C>0\) such that
Let s be a local holomorphic frame of L defined on an open set \(D\Subset M\). Then,
where \(\Phi \in \mathscr {C}^\infty (D\times D)\), \(\mathrm{Im\,}\Phi (x, y)\ge C|x-y|^2\), \(C>0\), \(\Phi (x,x)=0\), for every \(x\in D\),
\(b_j(x,y)\in \mathscr {C}^\infty (D\times D)\), \(j=0,1,\ldots \) , and
In particular, there exist coefficients \(\varvec{b}_r\in \mathscr {C}^\infty (X)\), \(r\in \mathbb {N}_0\), such that for any open set U of X with \({\overline{U}}\) compact, every \(\ell \in {\mathbb {N}}_0\) and every \(m\in {\mathbb {N}}\), there is a \(C_{U,\ell ,m}>0\) independent of k such that
We refer the reader to Sect. 2 for the precise meaning of the notation \(A_k\equiv B_k\mod O(k^{-\infty })\) on D in (1.14), \(S^n_{\mathrm{loc\,}}(1;D\times D)\) 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 \(\ell =4\). For general m, \(\ell \), 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 \(\partial \Theta _M\) are bounded below. Our conditions (1.13) are different from [23, Theorem 6.1.1], we replace the condition on \(\partial \Theta _M\) 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 \(\mathbb {R}\)-actions. In Sect. 3, we prove the Bochner–Kodaira formula on CR manifolds with \(\mathbb {R}\)-action. Section 4 is devoted to the proof of the asymptotics of the Szegő kernel. In Sect. 5, we examine the Heisenberg group.
2 Preliminaries
We use the following notations through this article: \({\mathbb {N}}=\{1,2,\ldots \}\) is the set of natural numbers, \({\mathbb {N}}_0={\mathbb {N}}\bigcup \{0\}\), \({\mathbb {R}}\) is the set of real numbers, \(\overline{{\mathbb {R}}}_+=\{x\in {\mathbb {R}};\, x\ge 0\}\). For \(m\in \mathbb {N}\), let \(x=(x_1,\ldots ,x_m)\) be coordinates of \(\mathbb {R}^m\). For \(n\in \mathbb {N}\), let \(z=(z_1,\ldots ,z_n)\), \(z_j=x_{2j-1}+\sqrt{-1}x_{2j}\), \(j=1,\ldots ,n\), be coordinates of \(\mathbb {C}^n\). We write
2.1 Notions of Microlocal Analysis
Let X be a \(\mathscr {C}^\infty \) paracompact manifold. We let TX 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 \(\mathbb CTX\) and \({\mathbb {C}} T^*X\), respectively. Write \(\langle \,\cdot \,,\cdot \,\rangle \) to denote the pointwise duality between TX and \(T^*X\). We extend \(\langle \,\cdot \,,\cdot \,\rangle \) bilinearly to \(\mathbb CTX\times {\mathbb {C}} T^*X\).
Let \(D\subset X\) be an open set . The spaces of distributions of D and smooth functions of D will be denoted by \({\mathscr {D}}'(D)\) and \(\mathscr {C}^\infty (D)\), respectively. Let \({\mathscr {E}}'(D)\) be the subspace of \({\mathscr {D}}'(D)\) whose elements have compact support in D. Let \(\mathscr {C}^\infty _c(D)\) be the subspace of \(\mathscr {C}^\infty (D)\) whose elements have compact support in D. Let \(A: \mathscr {C}^\infty _c(D)\rightarrow {\mathscr {D}}'(D)\) 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:
-
(I)
A is continuous: \({\mathscr {E}}'(D)\rightarrow \mathscr {C}^\infty (D)\),
-
(II)
\(A(x,y)\in \mathscr {C}^\infty (D\times D)\).
If A satisfies (I) or (II), we say that A is smoothing on D. Let \(A,B: \mathscr {C}^\infty _c(D)\rightarrow {\mathscr {D}}'(D)\) be continuous operators. We write
if \(A-B\) is a smoothing operator. We say that A is properly supported if the restrictions of the two projections \((x,y)\rightarrow x\), \((x,y)\rightarrow y\) to \(\mathrm{Supp\,}(A(x,y))\) are proper.
For \(m\in {\mathbb {R}}\), let \(H^m(D)\) denote the Sobolev space of order m on D. Put
Let D be an open coordinate patch of X with local coordinates x. We recall the following Hörmander symbol space.
Definition 2.1
For \(m\in {\mathbb {R}}\), \(S^m_{1,0}(D\times D\times {\mathbb {R}}_+)\) is the space of all \(a(x,y,t)\in \mathscr {C}^\infty (D\times D\times {\mathbb {R}}_+)\) such that for all compact \(K\Subset D\times D\) and all \(\alpha , \beta \in {\mathbb {N}}^{2n+1}_0\), \(\gamma \in {\mathbb {N}}_0\), there is a constant \(C_{\alpha ,\beta ,\gamma }>0\) such that
Put
Let \(a_j\in S^{m_j}_{1,0}(D\times D\times {\mathbb {R}}_+)\), \(j\in \mathbb {N}_0\), with \(m_j\searrow -\infty \), \(j\rightarrow \infty \). Then there exists \(a\in S^{m_0}_{1,0}(D\times D\times {\mathbb {R}}_+)\), unique modulo \(S^{-\infty }\), such that \(a-\sum ^{k-1}_{j=0}a_j\in S^{m_k}_{1,0}(D\times D\times {\mathbb {R}}_+)\) for \(k\in \mathbb {N}\). If a and \(a_j\) have the properties above, we write
The space \(S^{m}_{\mathrm{cl\,}}(D\times D\times {\mathbb {R}}_+)\) of classical symbols of order m is defined as the space of symbols \(s(x,y,t)\in S^{m}(D\times D\times {\mathbb {R}}_+)\) satisfying
We explain now for the precise meaning of \(A_k\equiv B_k\mod O(k^{-\infty })\) on D in (1.14), \(S^n_{\mathrm{loc\,}}(1;D\times D)\) and the asymptotic sum in (1.15) (see also [18, Sect. 3.3]). A k-dependent smoothing operator \(A_k:\Omega ^{0,q}_0(D)\rightarrow \Omega ^{0,q}(D)\) is called k-negligible if the kernel \(A_k(x, y)\) of \(A_k\) satisfies \(|\partial ^\alpha _x\partial ^\beta _yA_k(x, y)|=O(k^{-N})\) uniformly on every compact set in \(D\times D\), for all multi-indices \(\alpha \), \(\beta \), and all \(N\in {\mathbb {N}}\). Let \(C_k:\Omega ^{0,q}_0(D)\rightarrow \Omega ^{0,q}(D)\) be another k-dependent smoothing operator. We write \(A_k\equiv C_k\mod O(k^{-\infty })\) or \(A_k(x,y)\equiv C_k(x,y)\mod O(k^{-\infty })\) if \(A_k-C_k\) is k-negligible.
We recall the definition of semi-classical Hörmander symbol spaces:
Definition 2.2
Let U be an open set in \(\mathbb {R}^N\). Let \(S(1;U)=S(1)\) be the set of \(a\in \mathscr {C}^\infty (U)\) such that for every \(\alpha \in {\mathbb {N}}^N_0\), there exists \(C_\alpha >0\), such that \(|\partial ^\alpha _xa(x)|\le C_\alpha \) on U. If \(a=a(x,k)\) depends on \(k\in (1,\infty )\), we say that \(a(x,k)\in S_{\mathrm{loc\,}}(1)\) if \(\chi (x)a(x,k)\) is uniformly bounded in S(1) when k varies in \((1,\infty )\), for any \(\chi \in \mathscr {C}^\infty _0(U)\). For \(m\in \mathbb {R}\), we put \(S^m_{\mathrm{loc}}(1)=k^mS_{\mathrm{loc\,}}(1)\). If \(a_j\in S^{m_j}_{\mathrm{loc\,}}(1)\), \(m_j\searrow -\infty \), we say that \(a\sim \sum ^\infty _{j=0}a_j\) in \(S^{m_0}_{\mathrm{loc\,}}(1)\) if \(a-\sum ^{N_0}_{j=0}a_j\in S^{m_{N_0+1}}_{\mathrm{loc\,}}(1)\) for every \(N_0\). From this, we form \(S^m_{\mathrm{loc\,}}(1;Y, E)\) 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 \(n\ge 1\). The Levi form (1.2) of X at \(x\in X\) induces a Hermitian quadratic form on \(T^{(1,0)}_xX\) by
Let \(g^{\mathbb {C}TX}\) be a Hermitian metric on \(\mathbb {C}TX\) such that the decomposition \(\mathbb {C}TX = T^{(1,0)}X \oplus T^{(0,1)}X \oplus \mathbb {C}T\) is orthogonal. For \(u,v\in \mathbb {C}TX\) we denote by \(\langle u|v\rangle =\langle u|v\rangle _g\) the inner product given by \(g^{\mathbb {C}TX}\) and for \(u \in \mathbb {C}TX\), we write \(|u|^2_g := \langle u| u \rangle _g\). Given such a metric we will denote by \(\Theta _X\) its fundamental (1, 1)-form given by \(\Theta _X(a,\overline{b})=\sqrt{-1}\langle a\,|\,b\rangle _g\) for \(a,b\in T^{(1,0)}X\).
For \(p, q\in {\mathbb {N}}_0\), define \(T^{(p,q)}X :=(\Lambda ^pT^{(1,0)}X)\wedge (\Lambda ^qT^{(0,1)}X)\) and let \(T^{\bullet ,\bullet }X=\bigoplus _{p,q\in \mathbb {N}_0}T^{(p,q)}X\). For \(u\in \mathbb {C}TX\) and \(\phi \in \mathbb {C}T^*X\), the pointwise duality is defined by \(\langle u,\phi \rangle :=\phi (u).\) Let \(T^{*(1,0)}X\subset \mathbb {C}T^*X\) be the dual bundle of \(T^{(1,0)}X\) and \(T^{*(0,1)}X\subset \mathbb {C}T^*X\) be the dual bundle of \(T^{(0,1)}X\). For \(p, q\in {\mathbb {N}}_0\), the bundle of (p, q) forms is denoted by \(T^{*(p,q)}X:=(\Lambda ^pT^{*(1,0)}X)\wedge (\Lambda ^qT^{*(0,1)}X)\) and let \(T^{*\bullet ,\bullet }X:=\oplus _{p,q\in {\mathbb {N}}_0}T^{*(p,q)}X\). The induced Hermitian inner product on \(T^{\bullet ,\bullet }X\) and \(T^{*\bullet ,\bullet }X\) by \(\langle \,\cdot \,|\,\cdot \,\rangle \) are still denoted by \(\langle \cdot |\cdot \rangle \). The Hermitian norms are still denoted by \(|\cdot |\). Let \(\Omega ^{p,q}(X):=\mathscr {C}^{\infty }(X,T^{*(p,q)}X)\) be the space of smooth (p, q)-forms on X and \(\Omega ^{\bullet ,\bullet }(X):=\bigoplus _{p,q\in \mathbb {N}_0}\Omega ^{p,q}(X)\). Let \(\mathscr {C}^\infty (X):=\Omega ^{0,0}(X)\).
Definition 2.3
Let \(\pi ^{p,q}: \Lambda ^{p+q} \mathbb {C}T^*X\longrightarrow T^{*(p,q)}X\) be the natural projection for \(p,q\in \mathbb {N}_0\), \(p+q\ge 1\). The tangential (resp. anti-tangential) Cauchy–Riemann operator is given by
Let \(D\subset X\) be an open set. Let \(\Omega ^{p,q}_c(D)\) be the space of smooth (p, q)-forms on D with compact support in D. Let \(\Omega ^{\bullet ,\bullet }_c(D):= \bigoplus _{p,q\in \mathbb {N}_0}\Omega ^{p,q}_c(D)\). We write \(\mathscr {C}^\infty _c(D):=\Omega ^{0,0}_c(D)\). Let \((\,\cdot \,|\,\cdot \,)\) be the \(L^2\) inner product on \(\Omega ^{\bullet ,\bullet }_c(X)\) induced by \(\langle \,\cdot \,|\,\cdot \,\rangle \). Note that
where \(dv_X:=(\Theta _X^n/n!)\wedge \omega _0\) is the volume form induced by the Hermitian metric \(\Theta _X\) on X. Let \(L^2_{p,q}(X)\) be the completion of \(\Omega ^{p,q}_c(X)\) with respect to \((\,\cdot \,|\,\cdot \,)\). Let \(L^2_{\bullet ,\bullet }(X):=\bigoplus _{p,q\in \mathbb {N}_0} L^2_{p,q}(X)\). We write \(L^2(X):=L^2_{0,0}(X)\). We denote by \(\Vert u\Vert ^2:=(u|u)\) the \(L^2\)-norm on X. Let \(\overline{\partial }_b^*\) and \(\partial _b^*\) be the formal adjoints of \(\overline{\partial }_b\) and \(\partial _b\) with respect to \((\,\cdot \,|\,\cdot \,)\), respectively. Let \(\square _b:=\overline{\partial }_b\overline{\partial }_b^*+\overline{\partial }_b^* \overline{\partial }_b\) be the Kohn Laplacian on \(\Omega ^{\bullet ,\bullet }(X)\). Let \(\overline{\square }_b:=\partial _b\partial _b^*+\partial _b^* \partial _b\) be the anti-Kohn Laplacian on \(\Omega ^{\bullet ,\bullet }(X)\). We still denoted by \(\overline{\partial }_b\) the maximal extension and by \(\overline{\partial }_b^*\) 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, \(\square _b\) is a self-adjoint operator (see e.g., [23, Proposition 3.1.2]).
2.2 Pseudohermitian Geometry
The following is well known:
Proposition 2.4
[28, Proposition 3.1] Let \((X,HX,J,\omega _0)\) be an orientable strictly pseudoconvex CR manifold. Then there exists a unique affine connection, called Tanaka–Webster connection,
such that
-
(I)
\(\nabla _U \mathscr {C}^\infty (X,HX)\subset \mathscr {C}^\infty (X,HX)\) for \(U\in \mathscr {C}^\infty (X,TX)\).
-
(II)
\(\nabla T=\nabla J=\nabla d\omega _0=0\).
-
(III)
The torsion \(T_\nabla \) of \(\nabla \) satisfies: \(T_\nabla (U, V)=d\omega _0(U, V)T\), \(T_\nabla (T, JU)=-JT_\nabla (T, U)\), \(U, V\in \mathscr {C}^\infty (X, HX).\)
Recall that \(\nabla J\in \mathscr {C}^\infty (X, T^* X\otimes \mathscr {L}(HX,HX))\), \(\nabla d\omega _0\in \mathscr {C}^\infty (T^*X\otimes \Lambda ^2({\mathbb {C}} T^* X))\) are defined by \((\nabla _UJ)W=\nabla _U(JW)-J\nabla _UW\) and \(\nabla _Ud\omega _0(W, V)=Ud\omega _0(W, V)-d\omega _0(\nabla _UW, V)- d\omega _0(W, \nabla _UV)\) for \(U\in \mathscr {C}^\infty (X,TX), W, V\in \mathscr {C}^\infty (X,HX)\). Moreover, \(\nabla J=0\) and \(\nabla d\omega _0=0\) imply that the Tanaka–Webster connection is compatible with the Levi metric. By definition, the torsion of \(\nabla \) is given by \(T_\nabla (W, U)=\nabla _WU-\nabla _UW-[W, U]\) for \(U, V\in \mathscr {C}^\infty (X,TX)\) and \(\tau (T, U)\) for \(U\in \mathscr {C}^\infty (X,HX)\) is called pseudohermitian torsion.
In the following, we will use the Einstein summation convention. Let \(\{Z_\alpha \}_{\alpha =1}^n\) be a local frame of \(T^{(1,0)}X\) and \(\{\theta ^\alpha \}_{\alpha =1}^n\) be the dual frame of \(\{Z_\alpha \}_{\alpha =1}^n\). We use the notations \(Z_{\overline{\alpha }} :=\overline{Z_\alpha }\) and \(\theta ^{\overline{\alpha }} =\overline{\theta ^\alpha }\). Write
We call \(\omega _\alpha ^\beta \) the connection 1-form of Tanaka–Webster connection with respect to the frame \(\{Z_\alpha \}_{\alpha =1}^n\). We denote by \(\Theta _\alpha ^\beta \) the Tanaka–Webster curvature 2-form. Then,
By direct computation, we also have
where \(C_0\) is a 1-form. The term \(R_{\alpha j\overline{k}}^\beta \) is called the pseudohermitian curvature tensor and the form
is called pseudohermitian Ricci form.
2.3 Strictly Pseudoconvex CR Manifolds with \(\mathbb {R}\)-Action
Let \((X,T^{(1,0)}X)\) be a CR manifold of \(\dim X=2n+1\). Let \(r: {\mathbb {R}}\times X\rightarrow X\), \(r(x)=r\circ x\) for \(r\in \mathbb {R}\), be an \(\mathbb {R}\)-action on X, see [13]. Let \(\widehat{T}\) be the infinitesimal generator of the \(\mathbb {R}\)-action:
Definition 2.5
The \(\mathbb {R}\)-action is called locally free if \(\widehat{T}(x)\ne 0\) at every \(x\in X\).
By Assumption 1.1 we have
Note that (2.12) implies that \(L_{\widehat{T}}\) preserves HX and \([L_{\widehat{T}},J]=0\). Since \(HX=\ker \omega _0\) we have for \(U\in \mathscr {C}^{\infty }(X,HX)\),
We pose \(f=\omega _0(\widehat{T})\) and \(\omega _1=f^{-1}\omega _0\). Then \(L_{\widehat{T}}\omega _1=0\) and \(\omega _1(\widehat{T})=1\) since \((L_{\widehat{T}}\omega _1)(U)=0\) and \((L_{\widehat{T}}\omega _1)(\widehat{T})=\widehat{T}(\omega _1(\widehat{T}))=\widehat{T}(1)=0\). This also implies \(\iota _{\widehat{T}}d\omega _1=0\). We have thus
We can therefore assume up to rescaling \(\omega _0\) by a smooth function that the infinitesimal generator of the \(\mathbb {R}\)-action is a Reeb vector field \(T=\widehat{T}\). This motivates the equality of the infinitesimal generator to the Reeb field in Assumption 1.1.
By [28, Lemma 3.2 (3)]) we have \(2J\tau U=(L_T J )U\) for any \(U\in HX\), hence the pseudohermitian torsion \(\tau \) vanishes, which means that the contact metric manifold \((X,\omega _0,T,J,g_{\mathscr {L}})\) is a Sasakian manifold. Conversely, there exists a natural transversal CR \(\mathbb {R}\)-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 \(\mathbb {R}\)-action on \((X,g_{\mathscr {L}})\). 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 \(\mathbb {R}\)-action on X.
We use the local coordinates of Baouendi–Rothschild–Trevès (BRT charts) [3, Sect. 1], [19, Theorem 6.5] extensively as follows:
Theorem 2.6
(BRT charts) For each point \(x\in X\), there exists a coordinate neighborhood \(D=U\times \mathcal {I}\) with coordinates \(x=(x_1,\ldots ,x_{2n+1})\) centered at 0, where \(U=\{z=(z_1,\ldots ,z_n)\in \mathbb {C}^n:|z|<\epsilon \}\) and \({\mathcal {I}}=\{x_{2n+1}\in \mathbb {R}: |x_{2n+1}|<\epsilon _0 \}\), \(\epsilon , \epsilon _0>0\), \(z=(z_1,\ldots ,z_n)\) and \( z_j=x_{2j-1}+\sqrt{-1}x_{2j}, j=1,\ldots ,n\), such that
and there exists \(\phi \in \mathscr {C}^\infty (U,\mathbb {R})\) independent of \(x_{2n+1}\) satisfying that
is a frame of \(T^{(1,0)}D\), and \(\left\{ dz_j\right\} _{j=1}^n\subset T^{*(1,0)}D\) is the dual frame.
Let \(D=U\times {\mathcal {I}}\) be a BRT chart. Let \(f\in \mathscr {C}^\infty (D)\) and \(u\in \Omega ^{p,q}(D)\) with \(u=\sum _{I,J}u_{IJ}dz_I\wedge d\overline{z}_J\) with ordered sets I, J and \(u_{IJ}\in \mathscr {C}^\infty (D)\), for all I, J. We have
For \(u\in \Omega ^{p,q}(X)\), let \(\mathcal {L}_T u\) be the Lie derivative of u in the direction of T. For simplicity, we write Tu to denote \(\mathcal {L}_T u\). Since the \(\mathbb {R}\)-action is CR, \(Tu\in \Omega ^{p,q}(X)\). On a BRT chart D, for \(u\in \Omega ^{p,q}(D)\), \(u=\sum _{I,J}u_{IJ}dz_I\wedge d{\overline{z}}_J\), we have \(Tu=\sum _{I,J}(Tu_{IJ})\wedge dz_I\wedge d\overline{z}_J\) on D.
The Levi form \(\mathscr {L}\) in a BRT chart \(D\subset X\) has the form
Indeed, the characteristic 1-form \(\omega _0\) and \(d\omega _0\) on D are given by
From now on, we assume that \(\Theta _X\) is \({\mathbb {R}}\)-invariant. Let \(D=U\times {\mathcal {I}}\) be a BRT chart. The (1, 1) form \(\Theta =\Theta _U\) on U is defined by, for \(x=(z,x_{2n+1})\in D\),
Note that it is independent of \(x_{2n+1}\). More precisely,
Note that for another BRT coordinates \(D=\widetilde{U}\times \widetilde{{\mathcal {I}}}\), \(y=(w,y_{2n+1})\), there exist biholomorphic map \(H\in \mathscr {C}^\infty (U,{\tilde{U}})\) and \(G\in \mathscr {C}^\infty (U,{\mathbb {R}})\) such that \(H(z)=w\), for all \(z\in U\), \(y_{2n+1}=x_{2n+1}+G(z)\), for all \((z,x_{2n+1})\in U\times {\mathcal {I}}\) and \({\tilde{U}}=H(U)\), \(\tilde{{\mathcal {I}}}={\mathcal {I}}+G(U)\). We deduce that \(\Theta \) is independent of the choice of BRT coordinates, i.e., \(\Theta =\Theta _U=\Theta _{\widetilde{U}}\).
Until further notice, we work on a BRT chart \(D=U\times {\mathcal {I}}\). For \(p, q\in {\mathbb {N}}_0\), let \(T^{*(p,q)}U\) be the bundle of (p, q) forms on U and let \(T^{*\bullet ,\bullet }U:=\oplus _{p,q\in {\mathbb {N}}_0}T^{*(p,q)}U\). For \(p, q\in {\mathbb {N}}_0\), let \(T^{(p,q)}U\) be the bundle of (p, q) vector fields on U and let \(T^{\bullet ,\bullet }U:=\oplus _{p,q\in {\mathbb {N}}_0}T^{(p,q)}U\). The (1, 1) form \(\Theta \) induces Hermitian metrics on \(T^{\bullet ,\bullet }U\) and \(T^{*\bullet ,\bullet }U\). We shall use \(\langle \,\cdot \,,\,\cdot \,\rangle _h\) to denote all the induced Hermitian metrics. The volume form on U induced by \(\Theta \) is given by \(d\lambda (z):=\Theta ^n/n!\). Thus, the volume form \(dv_X\) can be represented by
The \(L^2\)-inner product on \(\Omega ^{\bullet ,\bullet }_c(U)\) with respect to \(\Theta \) is given by
Let \(t\in \mathbb {R}\) be fixed. The \(L^2\)-inner product on \(\Omega ^{\bullet ,\bullet }_c(U)\) with respect to \(\Theta \) and \(e^{-2t\phi (z)}\) is given by
The Chern curvature of \(K_U^*:=\det (T^{(1,0)}U)\) with respect to \(\Theta \) is given by
On X, define \(K^*_X:=\det (T^{(1,0)}X)\). Then, \(K^*_X\) is a CR line bundle over X. The Chern curvature \(R^{K^*_X}\) of \(K^*_X\) with respect to \(\Theta _X\) is defined as follows: On a BRT chart D, let
It is easy to see that \(R^{K_X^*}\) is independent of the choice of BRT coordinates and hence \(R^{K_X^*}\) is globally defined, i.e., \(R^{K^*_X}\in \Omega ^{1,1}(X)\).
Let \(\{ L_j\}_{j=1}^n\) be an \({\mathbb {R}}\)-invariant orthonormal frame of \(T^{(0,1)}D\) with the dual (orthonormal) frame \(\{ e_j \}_{j=1}^n\). Then \(\{ \overline{L}_j \}_{j=1}^n \) is an \({\mathbb {R}}\)-invariant orthonormal frame of \(T^{(1,0)}D \) with the dual (orthonormal) frame \(\{ \overline{e}_j \}_{j=1}^n\). Since \(\Theta _X\) is \({\mathbb {R}}\)-invariant, there exist \(c_j^k=c_j^k(z), w_j^k=w_j^k(z)\in \mathscr {C}^\infty (U)\), \(j,k=1,\ldots ,n\), satisfying \(\sum _{k=1}^n c_j^kw_k^l=\delta _j^l\), for all \(j, l=1,\ldots ,n\), such that for \(j=1,\ldots ,n\),
We can check that \(\{w_j:=\sum _{k=1}^n \overline{c}_j^k \frac{\partial }{\partial z_k};\, j=1,\ldots ,n\}\) and \(\{\overline{w}_j:=\sum _{k=1}^n c_j^k \frac{\partial }{\partial \overline{z}_k};\, j=1,\ldots ,n\}\) are orthonormal frames for \(T^{(1,0)}U\) and \(T^{(0,1)}U\) with respect to \(\Theta \), respectively, and \(\{{\overline{e}}_j;\, j=1,\ldots ,n\}\), \(\{e_j;\, j=1,\ldots ,n\}\) are dual frames for \(\{w_j;\, j=1,\ldots ,n\}\) and \(\{\overline{w}_j;\, j=1,\ldots ,n\}\), respectively. We also write \(w^j\) and \({\overline{w}}^j\) to denote \({\overline{e}}_j\) and \(e_j\), respectively, \(j=1,\ldots ,n\).
Lemma 2.7
We have
on X.
Proof
Fix \(p\in D\) and let \(x=(x_1,\ldots ,x_{2n+1})\) be BRT local coordinates defined on an open set D of p with \(x(p)=0\). We take \(x=(x_1,\ldots ,x_{2n+1})=(z_1,\ldots ,z_n,x_{2n+1})\), \(z_j=x_{2j-1}+ix_{2j}\), \(j=1,\ldots ,n\), so that
where \(\phi \in \mathscr {C}^\infty (D)\) is as in (2.15).
In the following, we will use Einstein summation convention. Write \(\nabla _{Z_i}Z_j=\Gamma ^l_{ij}Z_l\), where \(\nabla \) denotes the Tanaka–Webster connection (see Proposition 2.4). From [28, Lemma 3.2],
Directly,
and hence
for all \(i, j, l, k=1,\ldots ,n\). Accordingly, by (2.29) and (2.33), we get that for all \(i,j,k=1,\cdots ,n\),
Moreover, by taking \(\frac{\partial }{\partial \overline{z}_h}\) both sides in (2.33), from (2.29) and (2.33), it is not difficult to check that
It is clear that \(\{dz_j\}_{j=1}^n\) and \(\{d{\overline{z}}_j\}_{j=1}^n\) are the dual frames of \(\{Z_j\}_{j=1}^n\) and \(\{\overline{Z}_j\}_{j=1}^n\), respectively. Denote
and we can check that the (1, 1) part of \(d\omega _\alpha ^\beta \) is
and the (1, 1) part of \(\Theta _\alpha ^\beta =d\omega _\alpha ^\beta -\omega _\alpha ^\gamma \wedge \omega _\gamma ^\beta \) denoted by
equals the (1, 1) part of \(d\omega _\alpha ^\beta \). Hence, the pseudohermitian Ricci curvature tensor at origin is
We get that
On the other hand, by directed computation, we can check that
3 Bochner–Kodaira Formula on CR Manifolds with \(\mathbb {R}\)-Action
In this section, we will prove the Bochner–Kodaira–Nakano for CR manifolds with transversal CR \(\mathbb {R}\)-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 \(\mathbb {R}\)-action endowed with arbitrary Hermitian metric \(\Theta _X\).
3.1 The Fourier Transform on BRT Charts
Let \(D=U\times {\mathcal {I}}\) be a BRT chart. Let \(f\in \mathscr {C}^\infty _c(D)\). We write \(f=f(x)=f(z,x_{2n+1})\). For each fixed \(x_{2n+1}\in I\), \(f(\cdot ,x_{2n+1})\in \mathscr {C}_c^\infty (U)\). For each fixed \(z\in U\), \(f(z,\cdot )\in \mathscr {C}_c^\infty ({\mathcal {I}})\). Let \(p,q\in \mathbb {N}_0\), \(u\in \Omega _c^{p,q}(D)\). We write \(u=\sum _{I,J} u_{IJ}dz_I\wedge d\overline{z}_J\in \Omega ^{p,q}_c(D)\) and we always assume that the summation is performed only over increasingly ordered indices \(I=i_1<i_2<\ldots<i_p, J=j_1<j_2<\ldots <j_q\), and \(u_{IJ}\in \mathscr {C}^\infty _c(D)\), for all \(\{I, J\}\). For each fixed \(z\in U\), \(u_{IJ}(z,\cdot )\in \mathscr {C}_c^\infty ({\mathcal {I}})\).
Definition 3.1
The Fourier transform of the function \(f\in \mathscr {C}^\infty _c(D)\) with respect to \(x_{2n+1}\), denoted by \(\widehat{f}\), is defined by
The Fourier transform of the form \(u= \sum _{I,J} u_{IJ}dz_I\wedge d\overline{z}_J\in \Omega ^{p,q}_c(D)\) with respect to \(x_{2n+1}\), denoted by \(\widehat{u}\), is defined by
Note that \(\widehat{f}\in \mathscr {C}^\infty (U\times \mathbb {R})\) and \(\widehat{f}(\cdot ,t)\in \mathscr {C}_c^\infty (U)\) for every \(t\in \mathbb {R}\). Similarly, \(\widehat{u}\in \Omega ^{p,q}(U\times \mathbb {R})\) and \(\widehat{u}(\cdot ,t)\in \Omega ^{p,q}_c(U)\) for every \(t\in \mathbb {R}\). From Parseval’s formula, we have for \(u, v\in \Omega _c^{p,q}(D)\),
for every \(z\in U\). By using integration by parts, we have for \(u\in \Omega ^{p,q}_c(D)\),
Let \(t\in \mathbb {R}\) be fixed. Let \(|(z,1)|^2_h:=e^{-2t\phi (z)}\) be the Hermitian metric on the trivial line bundle \(U\times \mathbb {C}\) over U. The Chern connection of \((U\times \mathbb {C},e^{-2t\phi })\) is given by
Indeed, \(\nabla ^{(U\times \mathbb {C},e^{-2t\phi })}= d+h^{-1}\partial h=d+e^{2t\phi }\partial (e^{-2t\phi })\). The curvature of \((U\times \mathbb {C},e^{-2t\phi })\) is
We can identify \(\partial \overline{\partial }\phi \) with Levi form \(\mathscr {L}\) and write \(R^{(U\times \mathbb {C},e^{-2t\phi })}=2t\mathscr {L}\). Moreover, we will identify \(\Omega ^{\bullet ,\bullet }(U)\) and \(\Omega ^{\bullet ,\bullet }_c(U)\) with \(\Omega ^{\bullet ,\bullet }(U, U\times {\mathbb {C}})\) and \(\Omega ^{\bullet ,\bullet }_c(U,U\times {\mathbb {C}})\), respectively.
Proposition 3.2
Let \(u,v\in \Omega ^{\bullet ,\bullet }_c(D)\). We have
where \(\overline{\partial }^*,\nabla ^{1,0*}\) are the formal adjoints of \(\overline{\partial },\nabla ^{1,0}\) with respect to \(\langle \,\cdot \,,\,\cdot \,\rangle _{L^2(U,e^{-2t\phi })}\), respectively, and \(\overline{\partial }_b^*\), \(\partial ^*_b\) are the formal adjoints of \(\overline{\partial }_b\), \(\partial _b\) with respect to \((\,\cdot \,|\,\cdot \,)\), respectively.
Proof
Let \(u=\sum _{I,J}u_{IJ}dz_I\wedge d\overline{z}_J\). By \(\overline{\partial }_b u=\sum _{I,J} \sum _{j=1}^n\left( \frac{\partial u_{IJ}}{\partial \overline{z}_j}- i\frac{\partial \phi }{\partial \overline{z}_j}\frac{\partial u_{IJ}}{\partial x_{2n+1}}\right) d\overline{z}_j\wedge dz_I\wedge d\overline{z}_J\),
Thus the first equality holds. From Parseval’s formula,
Meanwhile, we have
Thus the second equality holds. The proofs of the third and the fourth equalities are similar. \(\square \)
3.2 CR Bochner–Kodaira–Nakano Formula I
Analog to [23, (1.4.32)], we define the Lefschetz operator \(\Theta _X\wedge \cdot \) on \(\bigwedge ^{\bullet ,\bullet }(T^*X)\) and its adjoint \(\Lambda =i(\Theta _X)\) with respect to the Hermitian inner product \(\langle \cdot |\cdot \rangle \) associated with \(\Theta _X\). The Hermitian torsion of \(\Theta _X\) is defined by
Let \(D=U\times {\mathcal {I}}\) be a BRT chart and let \(\{{\overline{L}}_j\}^n_{j=1}\subset T^{(1,0)}D\), \(\{{\overline{e}}_j\}^n_{j=1}\subset T^{*(1,0)}D\), \(\{w_j\}^n_{j=1}\subset T^{(1,0)}U\) be as in the discussion after (2.26). We can check that
Note that \(i_{L_j}\) and \(i_{\overline{L}_j}\) are the adjoints of \(e_j\wedge \) and \(\overline{e}_j\wedge \), respectively.
Since \(\partial \Theta (z)=\partial _b\Theta _X(x)\) on D, and \(\Theta \wedge \cdot =\sqrt{-1}\overline{e}_j\wedge e_j\wedge \cdot \), \(\Lambda =-\sqrt{-1}i_{\overline{w}_j}i_{w_j}\) on U, see [23, 1.4.32], we have \(\mathcal {T}=[\Lambda ,\partial _b\Theta ]=[\Lambda ,\partial \Theta ]\) on \(\Omega ^{\bullet ,\bullet }(D)\), which is independent of \(x_{2n+1}\). We remark that \(\mathcal {T}\) is a differential operator of order zero. With respect to the Hermitian inner product \(\langle \cdot |\cdot \rangle \) associated with \(\Theta _X\), we have the adjoint operator \(\mathcal {T}^*\), the conjugate operator \(\overline{\mathcal {T}}\) and the adjoint of the conjugate operator \(\overline{\mathcal {T}}^*\) for \(\mathcal {T}\).
Theorem 3.3
With the notations used above, we have on \(\Omega ^{\bullet ,\bullet }(X)\),
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 \(D=U\times {\mathcal {I}}\). We will use the same notations as before. Let
where \(\nabla ^{1,0}\) is given by (3.5), \(\overline{\partial }^*,\nabla ^{1,0*}\) are the formal adjoints of \(\overline{\partial },\nabla ^{1,0}\) with respect to \(\langle \,\cdot \,,\,\cdot \,\rangle _{L^2(U,e^{-2t\phi })}\), respectively. From [23, (1.4.44)],
Let \(u, v\in \Omega ^{\bullet ,\bullet }_c(D)\). Let \(s_1(z):=e^{t\phi (z)}\widehat{u}(z,t)\in \Omega ^{\bullet ,\bullet }(U\times {\mathbb {R}})\), \(s_2(z):=e^{t\phi (z)}\widehat{v}(z,t)\in \Omega ^{\bullet ,\bullet }(U\times {\mathbb {R}})\). Firstly, we have
In fact, from Proposition 3.2,
Similarly, we have
Thirdly, we have
In fact, it follows from
Fourthly, we consider the rest terms
By Proposition 3.2, we have
Thus we obtain
Similarly, we obtain
From (3.17), (3.18), (3.19), (3.22), and (3.23), we get that for \(u, v\in \Omega ^{\bullet ,\bullet }_c(D)\),
The theorem follows. \(\square \)
Corollary 3.4
(CR Nakano’s inequality I) With the notations used above, for any \(u\in \Omega _c^{\bullet ,\bullet }(X)\),
If \((X,T^{(1,0)}X)\) is Kähler, i.e., \(d\Theta _X=0\), then
Proof
By the Cauchy–Schwarz inequality, Theorem 3.3 and since \(\mathcal {T}=0\), \(\mathcal {T}^*=0\) if \(d\Theta _X=0\), we get the corollary. \(\square \)
The following follows from straightforward calculation, we omit the proof.
Proposition 3.5
For a real (1, 1)-form \(\sqrt{-1}\alpha \in \Omega ^{1,1}(D)\), if we choose local orthonormal frame \(\{{\overline{L}}_j\}_{j=1}^n\) of \(T^{(1,0)}D\) with the dual frame \(\{{\overline{e}}_j\}_{j=1}^n\) of \(T^{*(1,0)}D\) such that \(\sqrt{-1}\alpha =\sqrt{-1}\lambda _j(x){\overline{e}}_j\wedge e_j\) at a given point \(x\in D\), then for any \(f=\sum _{I,J}f_{IJ}(x){\overline{e}}^I \wedge e^J \in \Omega ^{\bullet ,\bullet }(D)\), we have
Corollary 3.6
With the notations used above, let \(\Theta _X\) be a Hermitian metric on X such that
Then for any \(u\in \Omega _c^{n,q}(X)\) with \(1\le q\le n\),
Proof
By applying (3.26) for \(\sqrt{-1}\alpha :=2\sqrt{-1}\mathscr {L}\), \(\lambda _j=1\) for all j, we have
By \(d\Theta _X=d(2\sqrt{-1}\mathscr {L})=0\) and Corollary 3.4, we obtain
\(\square \)
Let E be a CR line bundle over X (see Definition 2.4 in [13]. We say that E is a \({\mathbb {R}}\)-equivariant CR line bundle over X if the \({\mathbb {R}}\)-action on X can be CR lifted to E and for every point \(x\in X\), 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 \({\mathbb {R}}\)-action on E. From now on, we assume that E is a \({\mathbb {R}}\)-equivariant CR line bundle over X with a \({\mathbb {R}}\)-invariant Hermitian metric \(h^E\) on E. For \(p, q\in {\mathbb {N}}_0\), let \(\Omega ^{p,q}(X,E)\) be the space of smooth (p, q)-forms of X with values in E and let \(\Omega ^{\bullet ,\bullet }(X,E):= \oplus _{p,q\in {\mathbb {N}}_0}\Omega ^{p,q}(X,E)\). Let \(\Omega ^{p,q}_c(X,E)\) be the subspace of \(\Omega ^{p,q}(X,E)\) whose elements have compact support in X and let \(\Omega ^{\bullet ,\bullet }_c(X,E):= \oplus _{p,q\in {\mathbb {N}}_0}\Omega ^{p,q}_c(X,E)\). For \(p, q\in {\mathbb {N}}_0\), let
be the tangential Cauchy–Riemann operator with values in E. Let \((\,\cdot \,|\,\cdot \,)_E\) be the \(L^2\) inner product on \(\Omega ^{\bullet ,\bullet }_c(X,E)\) induced by \(\langle \,\cdot \,|\,\cdot \,\rangle \) and \(h^E\). Let
be the formal adjoint of \(\overline{\partial }_{b,E}\) with respect to \((\,\cdot \,|\,\cdot \,)_E\). Put
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
Let
where \(((\nabla ^E)^{1,0})^*\) is the adjoint of \((\nabla ^E)^{1,0}\) with respect to \((\,\cdot \,|\,\cdot \,)_E\). Let \(R^E\in \Omega ^{1,1}(X)\) be the curvature of E induced by \(h^E\) given by \(R^E:=-2\overline{\partial }_b\partial _b\Phi \) on D, where \(\Phi \) is as in (3.32). It is easy to check that \(R^E\) is globally defined. Let \(D=U\times {\mathcal {I}}\) be a BRT chart. Since E is \({\mathbb {R}}\)-equivariant, on D, E is a holomorphic line bundle over U. We can repeat the proof of Theorem 3.3 with minor changes and conclude the following:
Theorem 3.7
Let E be a \({\mathbb {R}}\)-equivariant CR line bundle over X with a \({\mathbb {R}}\)-invariant Hermitian metric \(h^E\). With the notations used above, we have on \(\Omega ^{\bullet ,\bullet }(X,E)\),
where \(R^E\in \Omega ^{1,1}(X)\) is the curvature of E induced by \(h^E\).
3.3 CR Bochner–Kodaira–Nakano Formula II
The bundle \(K^*_X:=\det (T^{(1,0)}X)\) is a \({\mathbb {R}}\)-equivariant CR line bundle over X. The (1, 1) form \(\Theta _X\) induces a \({\mathbb {R}}\)-invariant Hermitian metric \(h^{K^*_X}\) on \(K^*_X\). Let \(R^{K^*_X}\) be the curvature of \(K^*_X\) induced by \(h^{K^*_X}\). Let
be the natural isometry defined as follows: Let \(D=U\times {\mathcal {I}}\) be a BRT chart. Let \(\{{\overline{L}}_j\}^n_{j=1}\subset T^{(1,0)}D\), \(\{{\overline{e}}_j\}^n_{j=1}\subset T^{*(1,0)}D\) be as in the discussion after (2.26). Then,
It is easy to see that the definition above is independent of the choices of \({\mathbb {R}}\)-invariant orthonormal frame \(\{{\overline{L}}_j\}^n_{j=1}\subset T^{(1,0)}D\) 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 \(\Omega ^{0,\bullet }(X)\),
where \(\{{\overline{L}}_j\}_{j=1}^n\) is a local \({\mathbb {R}}\)-invariant orthonormal frame of \(T^{(1,0)}X\) with dual frame \(\{{\overline{e}}_j \}_{j=1}^n\subset T^{*(1,0)}X\).
Proof
Let \(u\in \Omega ^{0,q}(X)\). From (3.37) and (3.36), we have
It is straightforward to check that
From (3.39), (3.40) and noting that \(({\overline{e}}_j\wedge i_{{\overline{L}}_k}-i_{L_j}e_k\wedge )v= e_k\wedge i_{L_j}v\), \(\mathcal {T}^*(\nabla ^{K^*_X})^{1,0}v=0\), for every \(v\in \Omega ^{n,q}(X,K^*_X)\) and T commutes with \(\Psi \), we get (3.38). \(\square \)
Corollary 3.9
With the notations used above, assume that \(2\sqrt{-1}\mathscr {L}=\Theta _X\) and there is \(C>0\) such that
Then, for any \(u\in \Omega _c^{0,q}(X)\) with \(1\le q\le n\), we have
Proof
Since \(2\sqrt{-1}\mathscr {L}=\Theta _X\), we can choose \({\mathbb {R}}\)-invariant orthonormal frame \(\{{\overline{L}}_j\}^n_{j=1}\) such that \(2\mathscr {L}(\overline{L}_j,L_k)=\delta _{jk}\), for every j, k. We write \(u=\sum _{J}u_Je_J\) on D with \(u_J\in \mathscr {C}^\infty (D)\) and \(e_J=e_{j_1}\wedge \ldots \wedge e_{j_q}\), \(j_1<\ldots <j_q\). We have
Since \(\sqrt{-1}R^{K_X^*}\ge -C\Theta _X\), as (3.42), we can check that
Since \(d\Theta _X=d(2\sqrt{-1}\mathscr {L})=0\), we have \(\mathcal {T}=[\Lambda ,\partial _b\Theta _X]=0\). From this observation, (3.38), (3.42), and (3.43), we obtain
holds for every \(u\in \Omega _c^{0,q}(X)\) with \(1\le q\le n\). \(\square \)
4 Szegő Kernel Asymptotics
In this section, we will establish Szegő kernel asymptotic expansions on X under certain curvature assumptions.
4.1 Complete CR Manifolds
Let X be a CR manifold as in Assumption 1.1. Let \(g_X\) be the \(\mathbb {R}\)-invariant Hermitian metric as in (1.5). We will assume in the following that the Riemannian metric induced by \(g_X\) on TX is complete and study the extension \(\overline{\partial }_b\), \(\overline{\partial }_b^*\), and T. We denote by the same symbols the maximal weak extensions in \(L^2\) of these differential operators.
Since \(g_X\) is complete we know by [10, Lemma 2.4, p. 366] that there exists a sequence \(\{\chi _k\}^{\infty }_{k=1}\subset \mathscr {C}^\infty _c(X)\) such that \(0\le \chi _k\le 1\), \(\chi _{k+1}=1\) on \(\mathrm{supp\,}\chi _k\), \(|d\chi _k|_g\le \frac{1}{2^k}\), for every \(k\in \mathbb {N}\), and \(\bigcup ^{\infty }_{k=1}\mathrm{supp\,}\chi _k=X\). By using this sequence as in the Andreotti–Vesentini lemma on complex Hermitian manifolds (cf. [10, Theorem 3.2, p. 368], [23, Lemma 3.3.1]) and the classical Friedrichs lemma, we obtain the following.
Lemma 4.1
Assume that \((X,g_X)\) is complete. Then \(\Omega _c^{p,q}(X)\) is dense in \({{\,\mathrm{Dom}\,}}\overline{\partial }_b\), \({{\,\mathrm{Dom}\,}}\overline{\partial }_b^*\), \({{\,\mathrm{Dom}\,}}T\), \({{\,\mathrm{Dom}\,}}\overline{\partial }_b\cap {{\,\mathrm{Dom}\,}}\overline{\partial }_b^*\), and \({{\,\mathrm{Dom}\,}}T\cap {{\,\mathrm{Dom}\,}}\overline{\partial }_b\cap {{\,\mathrm{Dom}\,}}\overline{\partial }_b^*\) with respect to the graph norms of \(\overline{\partial }_b\), \(\overline{\partial }_b^*\), T, \(\overline{\partial }_b+\overline{\partial }_b^*\), and \(\overline{\partial }_b+\overline{\partial }_b^*+T\). Here the graph norm of a linear operator R is defined by \(\Vert u\Vert +\Vert Ru\Vert \) for \(u\in {{\,\mathrm{Dom}\,}}R\).
As a consequence, analog to [23, Corollary 3.3.3], we obtain the following:
Corollary 4.2
If \((X,g_X)\) is complete, then the maximal extension of the formal adjoint of \(\overline{\partial }_b\) and T coincide with their Hilbert space adjoint, respectively.
Corollary 4.3
If \((X,g_X)\) is complete, then \(\sqrt{-1}T: {{\,\mathrm{Dom}\,}}\sqrt{-1}T\subset L^2_{\bullet ,\bullet }(X)\rightarrow L^2_{\bullet ,\bullet }(X)\) is self-adjoint, that is, \((\sqrt{-1}T)^*=\sqrt{-1}T\).
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 \(2\sqrt{-1}\mathscr {L}=\Theta _X\), \(g_X\) is complete and there is \(C>0\) such that
Then, for any \(u\in L^2_{0,q}(X)\), \(1\le q\le n\), \(u\in {{\,\mathrm{Dom}\,}}\overline{\partial }_b\bigcap \mathrm{Dom}\overline{\partial }^*_b\bigcap {{\,\mathrm{Dom}\,}}(\sqrt{-1}T)\), we have
Proof
Let \(u\in L^2_{0,q}(X)\), \(1\le q\le n\), \(u\in {{\,\mathrm{Dom}\,}}\overline{\partial }_b\bigcap {{\,\mathrm{Dom}\,}}\overline{\partial }^*_b\bigcap {{\,\mathrm{Dom}\,}}(\sqrt{-1}T)\). From Lemma 4.1, we can find \(\{u_j\}^{\infty }_{j=1}\subset \Omega ^{\bullet ,\bullet }_c(X)\) such that
From (3.41), we have for every \(j=1,2,\ldots \),
Taking \(j\rightarrow \infty \) in (4.3) and using (4.2), we get (4.1). \(\square \)
Let us describe two examples of CR manifolds with complete \(\mathbb {R}\)-invariant metric \(g_X\).
Example 4.5
Let \((X,HX,J,\omega _0)\) be a compact strictly pseudoconvex CR manifold as in Assumption 1.1 and let \(g_X\) be a \(\mathbb {R}\)-invariant metric as in (1.5). Let \(\pi :\widetilde{X}\rightarrow X\) be a Galois covering of X, that is, there exists a discrete, proper action \(\Gamma \) such that \(X=\widetilde{X}/\Gamma \). By pulling back the objects from X by the projection \(\pi \) we obtain a strictly pseudoconvex CR manifold \((\widetilde{X},H\widetilde{X},\widetilde{J},\widetilde{\omega }_0)\) satisfying Assumption 1.1. Moreover, the metric \(\widetilde{g}_X=\pi ^*g_X\) is a complete \(\mathbb {R}\)-invariant metric satisfying (1.5).
Example 4.6
Let us consider now the case of a circle bundle associated to a Hermitian holomorphic line bundle. Let \((M,J,\Theta _M)\) be a complete Hermitian manifold. Let \((L,h^L)\rightarrow M\) be a Hermitian holomorphic line bundle over M. Let \(h^{L^*}\) be the Hermitian metric on \(L^*\) induced by \(h^L\). Let
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, HX, J) inherited from the complex structure of \(L^*\) by setting \(T^{(1,0)}X= TX\cap T^{(1,0)} L^{*}\).
In this situation, \(S^{1}\) acts on X by fiberwise multiplication, denoted \((x,e^{i\theta })\mapsto xe^{i\theta }\). A point \(x\in X\) is a pair \(x=(p,\lambda )\), where \(\lambda \) is a linear functional on \(L_p\), the \(S^1\) action is \(xe^{i\theta }=(p,\lambda )e^{i\theta }=(p,e^{i\theta }\lambda )\).
Let \(\omega _0\) be the connection 1-form on X associated to the Chern connection \(\nabla ^L\). Then
where \(R^L\) is the curvature of \(\nabla ^L\). Assume \(R^L\) is positive, hence X is a strictly pseudoconvex CR manifold. Hence, \((X,HX,J,\omega _0)\) fulfills Assumption 1.1. We denote by \(\partial _\theta \) the infinitesimal generator of the \(S^1\) action on X. The span of \(\partial _\theta \) defines a rank one sub-bundle \(T^VX\cong TS^1\subset TX\), the vertical sub-bundle of the fibration \(\pi :X\rightarrow M\). Moreover (1.3) holds for \(T=\partial _\theta \).
We construct now a Riemannian metric on X. Let \(g_M\) be a J-invariant metric on TM associated to \(\Theta _M\). The Chern connection \(\nabla ^L\) on L induces a connection on the \(S^1\)-principal bundle \(\pi :X\rightarrow M\), and let \(T^H X \subset TX\) be the corresponding horizontal bundle. Let \(g_X= \pi ^*g_M\oplus d\theta ^2/4\pi ^2\) be the metric on \(TX= T^HX\oplus T S^1\), with \(d\theta ^2\) the standard metric on \(S^1= \mathbb {R}/2\pi \mathbb {Z}\). Then \(g_X\) is a \(\mathbb {R}\)-invariant Hermitian metric on X satisfying (1.5). Since \(g_M\) is complete it is easy to see that \(g_X\) is also complete.
4.2 The Operators \(Q_\lambda \), \(Q_{[\lambda _1,\lambda ]}\), \(Q_{\tau }\)
From now on, we assume that X is a CR manifold satisfying Assumption 1.1 and \((X,g_X)\) is complete. Let \({\mathbb {S}}\) denote the spectrum of \(\sqrt{-1}T\). By the spectral theorem, there exists a finite measure \(\mu \) on \({\mathbb {S}}\times {\mathbb {N}}\) and a unitary operator
with the following properties: If \(h: \mathbb S\times {\mathbb {N}}\rightarrow {\mathbb {R}}\) is the function \(h(s,n)=s\), then the element \(\xi \) of \(L^2_{\bullet ,\bullet }(X)\) lies in \({{\,\mathrm{Dom}\,}}(\sqrt{-1}T)\) if and only if \(hU(\xi )\in L^2(\mathbb S\times {\mathbb {N}}, d\mu )\). We have
Let \(\lambda _1, \lambda \in {\mathbb {R}}\), \(\lambda _1<\lambda \), and let \(\tau (t)\in \mathscr {C}^\infty ({\mathbb {R}},[0,1])\). Put
where \(\mathbb {1}_A\) denotes the characteristic function of the set A. Let
be the orthogonal projections with respect to \((\,\cdot \,|\,\cdot \,)\).
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:
When X is non-compact, the \({\mathbb {R}}\)-action does not always come from a CR torus action. For example, when X is the Heisenberg group (see Sect. 5), the standard \({\mathbb {R}}\)-action on X is free and does not come from any CR torus action.
Assume that the \({\mathbb {R}}\)-action is free. Let \(D=U\times {\mathcal {I}}\) be a BRT chart with BRT coordinates \(x=(x_1,\ldots ,x_{2n+1})\). Since the \({\mathbb {R}}\)-action is free, we can extend \(x=(x_1,\ldots ,x_{2n+1})\) to \({\hat{D}}:=U\times {\mathbb {R}}\). We identify \({\hat{D}}\) with an open set in X.
Lemma 4.7
Assume that the \({\mathbb {R}}\)-action is free. Let \(D=U\times {\mathcal {I}}\) be a BRT chart with BRT coordinates \(x=(x_1,\ldots ,x_{2n+1})\). Let \(\lambda _1, \lambda \in {\mathbb {R}}\), \(\lambda _1<\lambda \). For \(u\in \Omega ^{\bullet ,\bullet }_c(D)\), we have
and \(\mathrm{supp\,}Q_\lambda u\subset {\hat{D}}\), \(\mathrm{supp\,} Q_{[\lambda _1,\lambda ]}u\subset {\hat{D}}\), \(\mathrm{supp\,} Q_{\tau }u\subset {\hat{D}}\), where \({\hat{D}}\) is as in the discussion after (4.8).
Proof
Let \(\chi \in \mathscr {C}^\infty _c({\mathbb {R}})\), \(\chi =1\) on \([-1,1]\), \(\chi =0\) outside \([-2,2]\). For every \(M>0\), put \(\tau _{M}(t):=\chi (\frac{t}{M})\tau (t)\). Then,
From the Helffer–Sjöstrand formula [12, Proposition 7.2], we see that
where \(\tilde{\tau }_{M}\in \mathscr {C}^\infty _c({\mathbb {C}})\) is an extension of \(\tau _{M}\) with \(\frac{\partial \tilde{\tau }_{M}}{\partial {\overline{z}}}=0\) on \({\mathbb {R}}\). It is not difficult to see that for \(u\in \Omega ^{\bullet ,\bullet }_c(D)\),
and \(\mathrm{supp\,}(z-\sqrt{-1}T)^{-1}u\subset {\hat{D}}\). From (4.13) and (4.14), we have
and \(\mathrm{supp\,}Q_{\tau _{M}}u\subset {\hat{D}}\), for every \(u\in \Omega ^{\bullet ,\bullet }_c(D)\). By Cauchy integral formula, we see that
From this observation and (4.15), we deduce that
and \(\mathrm{supp\,}Q_{\tau _{M}}u\subset {\hat{D}}\), for every \(u\in \Omega ^{\bullet ,\bullet }_c(D)\). From (4.12) and (4.16), we get (4.11).
Let \(\gamma _\varepsilon \in \mathscr {C}^\infty _c({\mathbb {R}})\), \(\lim _{\varepsilon \rightarrow 0}\gamma _\varepsilon (t)= \mathbb {1}{(-\infty ,\lambda ]}(t)\), for every \(t\in {\mathbb {R}}\). We can repeat the proof above and get that
and \(\mathrm{supp\,}Q_{\lambda }u\subset {\hat{D}}\), for every \(u\in \Omega ^{\bullet ,\bullet }_c(D)\). We obtain (4.9). The proof of (4.10) is similar. \(\square \)
We now assume that the \({\mathbb {R}}\)-action is not free. From (4.8), we know that the \({\mathbb {R}}\)-action comes from a CR torus action \(\mathbb T^d=(e^{i\theta _1},\ldots ,e^{i\theta _d})\) on X and \(\omega _0\), \(\Theta _X\) are \({\mathbb {T}}^d\) invariant. Since the \({\mathbb {R}}\)-action comes from the \({\mathbb {T}}^d\)-action, there exist \(\beta _1,\ldots ,\beta _d\in {\mathbb {R}}\), such that
where \(T_j\) is the vector field on X given by \(T_ju:= \frac{\partial }{\partial \theta _j}((1,\ldots ,1,e^{i\theta _j},1,\ldots ,1)^*u)\) \(|_{\theta _j=0}\), \(u\in \Omega ^{\bullet ,\bullet }(X)\), \(j=1,\ldots ,d\). For \((m_1,\ldots ,m_d)\in {\mathbb {Z}}^d\), put
and let
be the orthogonal projection. It is not difficult to see that for every \(u\in L^2_{\bullet ,\bullet }(X)\), we have
From Lemma 4.7 and (4.19), we conclude that
Proposition 4.8
Let \(\lambda _1, \lambda \in {\mathbb {R}}\), \(\lambda _1<\lambda \). For \(u\in {{\,\mathrm{Dom}\,}}\overline{\partial }_b\), we have \(Q_\lambda u, Q_{[\lambda _1,\lambda ]}u, Q_\tau u\in {{\,\mathrm{Dom}\,}}\overline{\partial }_b\) and \(\overline{\partial }_bQ_\lambda u=Q_\lambda \overline{\partial }_bu\), \(\overline{\partial }_bQ_{[\lambda _1,\lambda ]} u=Q_{[\lambda _1,\lambda ]}\overline{\partial }_bu\), \(\overline{\partial }_bQ_\tau u=Q_\tau \overline{\partial }_bu\). Similarly, for \(u\in {{\,\mathrm{Dom}\,}}\overline{\partial }^*_b\), we have \(Q_\lambda u, Q_{[\lambda _1,\lambda ]}u, Q_\tau u\in {{\,\mathrm{Dom}\,}}\overline{\partial }^*_b\) and \(\overline{\partial }^*_bQ_\lambda u= Q_\lambda \overline{\partial }^*_bu\), \(\overline{\partial }^*_bQ_{[\lambda _1,\lambda ]} u= Q_{[\lambda _1,\lambda ]}\overline{\partial }^*_bu\), \(\overline{\partial }^*_bQ_\tau u= Q_\tau \overline{\partial }^*_bu\).
For \(\lambda \in {\mathbb {R}}\), define
where \(\Box _{b}\) is defined in (2.8), (2.9). From Proposition 4.8, we see that
From now on, we write \(\Box ^{(q)}_b\) and \(\Box ^{(q)}_{b,\lambda }\) to denote \(\Box _b\) and \(\Box _{b,\lambda }\) acting on (0, q) forms, respectively.
4.3 Local Closed Range for \(\Box ^{(0)}_b\)
In this section, we will establish the local closed range property for \(\Box ^{(0)}_b\) under appropriate curvature assumptions. We first need the following.
Lemma 4.9
Assume that \(2\sqrt{-1}\mathscr {L}=\Theta _X\), \(g_X\) is complete and there is \(C>0\) such that
Then, for any \(u\in L^2_{0,q}(X)\), \(1\le q\le n\), \(u\in {{\,\mathrm{Dom}\,}}\overline{\partial }_b\cap {{\,\mathrm{Dom}\,}}\overline{\partial }^*_b\cap \mathscr {E}(\lambda ,\sqrt{-1}T)\), \(\lambda \in {\mathbb {R}}\), \(\lambda <-C\), we have
Proof
Let \(\lambda <-C\) and let \(u\in {{\,\mathrm{Dom}\,}}\overline{\partial }_b\cap {{\,\mathrm{Dom}\,}}\overline{\partial }^*_b\cap \mathscr {E}(\lambda ,\sqrt{-1}T)\), \(u\in L^2_{0,q}(X)\), \(1\le q\le n\). Let \(M\gg 1\) and let \(u_M:=Q_{[-M,\lambda ]}u\). By Proposition 4.8, we see that
From this observation and (4.1), we have
Since \(\lambda <-C\) we deduce from (4.23) that
Letting \(M\rightarrow \infty \) in (4.24) we get (4.22). \(\square \)
For every \(q=0,1,\ldots ,n\), put \(\mathscr {E}^{(q)}(\lambda ,\sqrt{-1}T):= \mathscr {E}(\lambda ,\sqrt{-1}T)\bigcap L^2_{0,q}(X)\). We prove now a vanishing theorem for harmonic forms which are eigenforms of \(\sqrt{-1}T\).
Theorem 4.10
Assume that \(2\sqrt{-1}\mathscr {L}=\Theta _X\), \(g_X\) is complete and there is \(C>0\) such that
Let \(q\in \{1,\ldots ,n\}\). Let \(\lambda \in {\mathbb {R}}\), \(\lambda <-C\). The operator
has closed range and \(\ker \Box ^{(q)}_{b,\lambda }=\{0\}\). Hence, there is a bounded operator
such that
Proof
Let \(u\in {{\,\mathrm{Dom}\,}}\Box ^{(q)}_{b,\lambda }\). From (4.22), we have
Hence,
From (4.26), the theorem follows. \(\square \)
We now consider (0, 1)-forms. Let \(G^{(1)}_\lambda \) be as in (4.25). Since \(G^{(1)}_\lambda \) is \(L^2\) bounded, there is \(C_0>0\) such that
We use the previous result to solve the \(\overline{\partial }_b\)-equation for eigenforms of \(\sqrt{-1}T\).
Theorem 4.11
Assume that \(2\sqrt{-1}\mathscr {L}=\Theta _X\), \(g_X\) is complete and there is \(C>0\) such that
Let \(\lambda \in {\mathbb {R}}\), \(\lambda <-C\). For every \(v\in \mathscr {E}^{(1)}(\lambda ,\sqrt{-1}T)\) with \(\overline{\partial }_bv=0\), we can find \(u\in {{\,\mathrm{Dom}\,}}\overline{\partial }_b\bigcap \mathscr {E}^{(0)}(\lambda ,\sqrt{-1}T)\) such that
where \(C_0>0\) is a constant as in (4.27).
Proof
Let \(v\in \mathscr {E}^{(1)}(\lambda ,\sqrt{-1}T)\) with \(\overline{\partial }_bv=0\). From (4.25), we have
Since \(\overline{\partial }_b\Bigr (\overline{\partial }^*_b\overline{\partial }_bG^{(1)}_\lambda v\Bigr )= \overline{\partial }_bv-\overline{\partial }^2_b\,\overline{\partial }^*_bG^{(1)}_\lambda v=0\), \(\overline{\partial }^*_b\Bigr (\overline{\partial }^*_b\overline{\partial }_bG^{(1)}_\lambda v\Bigr )=0\), \(\overline{\partial }^*_b\overline{\partial }_bG^{(1)}_\lambda v\in \ker \Box ^{(1)}_{b,\lambda }\). From Theorem 4.10, we see that \(\overline{\partial }^*_b\overline{\partial }_bG^{(1)}_\lambda v=0\). From this observation and (4.29), we get \(v=\overline{\partial }_bu\), \(u=\overline{\partial }^*_bG^{(1)}_\lambda v\). Now,
where \(C_0>0\) is as in (4.27). The theorem follows. \(\square \)
Fix \(q\in \{0,1,\ldots ,n\}\). Let \(S^{(q)}: L^2_{0,q}(X)\rightarrow {{\,\mathrm{Ker}\,}}\square _b^{(q)}\) be the orthogonal projection with respect to \((\,\cdot \,|\,\cdot \,)\). From Proposition 4.8, we can check that
We recall the following notion introduced in [19, Definition 1.8].
Definition 4.12
Fix \(q\in \{0,1,2,\ldots ,n\}\). Let \(Q:L^2_{0,q}(X)\rightarrow L^2_{0,q}(X)\) be a continuous operator. We say that \(\Box ^{(q)}_b\) has local \(L^2\) closed range on an open set \(D\subset X\) with respect to Q if for every \(D'\Subset D\), there exist constants \(C_{D'}>0\) and \(p\in {\mathbb {N}}\), such that
We remind that we do not assume that \(\Theta _X=2\sqrt{-1}\mathscr {L}\). The Levi form \(2\sqrt{-1}\mathscr {L}\) induces a Hermitian metric \(\langle \,\cdot \,|\,\cdot \,\rangle _{\mathscr {L}}\) on \(\mathbb CTX\) and \(\langle \,\cdot \,|\,\cdot \,\rangle _{\mathscr {L}}\) induces a Hermitian metric \(\langle \,\cdot \,|\,\cdot \,\rangle _{\mathscr {L}}\) on \(T^{*\bullet ,\bullet }X\). More precisely, if X is strictly pseudoconvex, i.e., \(2\sqrt{-1}\mathscr {L}\in \Omega ^{1,1}(X)\) is positive definite, then we can construct a Hermitian metric \(\langle \,\cdot \,|\,\cdot \,\rangle _{\mathscr {L}}\) on \(\mathbb CTX=T^{(1,0)}X\oplus T^{(0,1)}X\oplus \mathbb {C}\{T\}\) in the following way: For arbitrary \(a,b\in T^{(1,0)}X\), \(\langle a| b\rangle _\mathscr {L}:=2\mathscr {L}(a,\overline{b})\), \(\langle \overline{a}| \overline{b}\rangle _\mathscr {L}:=\langle b| a\rangle _\mathscr {L}\), \(\langle a| \overline{b}\rangle _\mathscr {L}:=0\) and \(\langle T| T\rangle _\mathscr {L}:=1\). We simply use \(2\sqrt{-1}\mathscr {L}\) to represent \(\langle \,\cdot \,|\,\cdot \,\rangle _{\mathscr {L}}\). Let \((\,\cdot \,|\,\cdot \,)_{\mathscr {L}}\) be the \(L^2\) inner product on \(\Omega ^{\bullet ,\bullet }_c(X)\) induced by \(\langle \,\cdot \,|\,\cdot \,\rangle _{\mathscr {L}}\) and let \(L^2_{\bullet ,\bullet }(X,\mathscr {L})\) be the completion of \(\Omega ^{\bullet ,\bullet }_c(X)\) with respect to \((\,\cdot \,|\,\cdot \,)_{\mathscr {L}}\). We write \(L^2(X,\mathscr {L}):=L^2_{0,0}(X,\mathscr {L})\). For \(f\in L^2_{\bullet ,\bullet }(X,\mathscr {L})\), we write \(\Vert f\Vert ^2_{\mathscr {L}}:=(\,f\,|\,f\,)_{\mathscr {L}}\).
Let \(R^{K_X^*}_{\mathscr {L}}\) be the Chern curvature of \(K^*_X\) with respect to the Hermitian metric \(\langle ,\rangle _{\mathscr {L}}\) on X, see (2.26). Locally it can be represented by
where \(\{Z_j\}^n_{j=1}\subset T^{(1,0)}X\) is as in (2.15).
For \(u\in \Omega ^{\bullet ,\bullet }_c(X)\), from Lemma 4.7 and (4.19), we see that \(Q_\lambda u\), \(Q_{[\lambda _1,\lambda ]}u\), \(Q_\tau u\) are independent of the choices of \({\mathbb {R}}\)-invariant Hermitian metrics on X.
Theorem 4.13
Assume that \(g_\mathscr {L}\) is complete and there is \(C>0\) such that
Let \(D\Subset X\) be an open set. Let \(\lambda \in {\mathbb {R}}\), \(\lambda <-C\). Then, \(\square _b^{(0)}\) has local closed range on D with respect to \(Q_\lambda \).
Proof
Let \(u\in \mathscr {C}^\infty _c(D)\). Let \(v:=\overline{\partial }_bQ_\lambda u =Q_\lambda \overline{\partial }_bu\). Since \(\overline{\partial }_bu\in \Omega ^{0,1}_c(D)\),
From Theorem 4.11, there exists \(g\in L^2(X,\mathscr {L})\) with
such that
where \(C_0>0\) is a constant as in (4.28). Since \(\overline{\partial }_b(I-S^{(0)})Q_\lambda u=\overline{\partial }_bQ_\lambda u\) and \((I-S^{(0)})Q_\lambda u\perp \ker \overline{\partial }_b\), we have
where \(C>0\) is a constant as in (4.32). From (4.32), (4.33), and (4.35), we have
Since \(\overline{\partial }_bu\) has compact support in D, there exists \(C_1>0\) independent of u such that
(4.36) and (4.37), the theorem follows. \(\square \)
For \(\lambda \in {\mathbb {R}}\), \(\lambda \le 0\), let \(\tau _\lambda \in \mathscr {C}^\infty ({\mathbb {R}},[0,1])\), \(\tau _\lambda =1\) on \(]-\infty , 2\lambda ]\), \(\tau _\lambda =0\) outside \((-\infty ,\lambda ]\). It is clear that \(\Vert Q_{\tau _\lambda }(I-S^{(0)})u\Vert \le \Vert Q_{\lambda }(I-S^{(0)})u\Vert \), for every \(u\in L^2(X)\). From this observation and Theorem 4.13, we deduce that
Theorem 4.14
Assume that \(g_\mathscr {L}\) is complete and there is \(C>0\) such that
Let \(D\Subset X\) be an open set. Let \(\lambda \in {\mathbb {R}}\), \(\lambda <-C\). Then, \(\square _b^{(0)}\) has local closed range on D with respect to \(Q_{\tau _\lambda }\).
4.4 Local Closed Range for \(\square _{b}^{(n,0)}\)
In this section, we will establish the local closed range property for \(\Box ^{(n,0)}_b\) 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 \(\square _{b}\) be the Gaffney extension of the usual Kohn Laplacian. Let \(\square _{b}^{(n,q)}\) be the restriction of \(\square _{b}\) acting on (n, q)-forms. Set
Let \(S^{(n,q)}: L^2_{n,q}(X)\rightarrow {{\,\mathrm{Ker}\,}}\square _{b}^{(n,q)}\) be orthogonal projection. It is known that \(Q_\tau , Q_\lambda ,Q_{[\lambda _1,\lambda ]}\) 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 \(\mathbb {R}\)-action. Let \(\Theta _X\) be an \(\mathbb {R}\)-invariant Hermitian metric on X. Assume that \(g_\mathscr {L}\) is complete. Let \(D\Subset X\) be an open set. Let \(\lambda \in {\mathbb {R}}\), \(\lambda <0\). Then, \(\square _b^{(n,0)}\) has local closed range on D with respect to \(Q_\lambda \), i.e., there exists \(C>0\) such that for all \(u\in \Omega _c^{n,0}(D)\),
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 \(\Theta _X\) be a Hermitian metric on X such that
Then for any \(u\in L^2_{n,q}(X)\cap {{\,\mathrm{Dom}\,}}\overline{\partial }_b\cap {{\,\mathrm{Dom}\,}}\overline{\partial }^*_b \cap {{\,\mathrm{Dom}\,}}(\sqrt{-1}T)\), \(1\le q \le n\), we have
Lemma 4.17
Assume that \(2\sqrt{-1}\mathscr {L}=\Theta _X\) and \(g_X\) is complete. Then, for any \(u\in L^2_{n,q}(X)\cap {{\,\mathrm{Dom}\,}}\overline{\partial }_b\cap {{\,\mathrm{Dom}\,}}\overline{\partial }^*_b\cap \mathscr {E}(\lambda ,\sqrt{-1}T)\), \(\lambda <0\), and \(1\le q\le n\), we have
Proof
Let \(u\in L^2_{n,q}(X)\cap {{\,\mathrm{Dom}\,}}\overline{\partial }_b\cap {{\,\mathrm{Dom}\,}}\overline{\partial }^*_b \cap \mathscr {E}(\lambda ,\sqrt{-1}T)\), \(\lambda <0\), and \(1\le q\le n\). Let \(M\gg 1\) be a sufficiently large positive real number such that \(-M<\lambda \) and \(u_M:=Q_{[-M,\lambda ]}u\). By Proposition 4.8, we see that
Then
By letting \(M\rightarrow +\infty \) we complete the proof. \(\square \)
Moreover, we have the following analog of Theorem 4.10.
Theorem 4.18
Let X be a CR manifold with a transversal CR \(\mathbb {R}\)-action. Let \(\Theta _X\) be an \(\mathbb {R}\)-invariant Hermitian metric on X. Assume that \(2\sqrt{-1}\mathscr {L}=\Theta _X\) and \(g_X\) is complete. Let \(1\le q\le n\) and \(\lambda <0\). Then the operator
has closed range, and
Hence, there exists a bounded operator
such that
Therefore, we have \(C_0>0\) such that, for all \(v\in \mathscr {E}^{(n,1)}(\lambda ,\sqrt{-1}T)\),
Secondly, we solve the \(\overline{\partial }_b\)-equation as follows.
Theorem 4.19
Let X be a CR manifold with a transversal CR \(\mathbb {R}\)-action. Let \(\Theta _X\) be an \(\mathbb {R}\)-invariant Hermitian metric on X. Assume that \(2\sqrt{-1}\mathscr {L}=\Theta _X\) and \(g_X\) is complete. Then for every \(\lambda <0\) and every \(v\in \mathscr {E}^{(n,1)}(\lambda ,\sqrt{-1}T)\) with \(\overline{\partial }_b v=0\), there exists \(u\in {{\,\mathrm{Dom}\,}}\overline{\partial }_b\cap \mathscr {E}^{(n,0)}(\lambda ,\sqrt{-1}T)\) such that
Proof
Let \(\lambda <0\). Let \(v\in \mathscr {E}^{(n,1)}(\lambda ,\sqrt{-1}T)\) with \(\overline{\partial }_bv=0\). We have
Since \(\overline{\partial }_b\Bigr (\overline{\partial }^*_b\overline{\partial }_bG^{(n,1)}_\lambda v\Bigr )= \overline{\partial }_bv-\overline{\partial }^2_b\,\overline{\partial }^*_bG^{(n,1)}_\lambda v=0\), \(\overline{\partial }^*_b\Bigr (\overline{\partial }^*_b\overline{\partial }_bG^{(n,1)}_\lambda v\Bigr )=0\),
Thus, we see that \(\overline{\partial }^*_b\overline{\partial }_bG^{(n,1)}_\lambda v=0\) and \(v=\overline{\partial }_bu\) with \(u=\overline{\partial }^*_bG^{(n,1)}_\lambda v\). Thus obtain
The proof is complete. \(\square \)
Proof of Theorem 4.15
Note that \(\Theta _X\) is not necessarily equal to \(2\sqrt{-1}\mathscr {L}\) so we have to deduce the general case to the particular case considered in Theorems 4.16–4.19. Let \(\lambda <0\). Let \(v:=\overline{\partial }_b Q_\lambda u=Q_\lambda \overline{\partial }_b u\). Note that \(Q_\lambda \) is independent of the choice of \(\Theta _X\). Then \(v\in L^2_{n,1}(X,\mathscr {L})\cap L^2_{n,1}(X)\). From the above theorem, we can find \(g\in L^2_{n,0}(X,\mathscr {L})\) such that \(\overline{\partial }_b g=v\) and
where the first inequality in (4.55) follows from Theorem 4.19. We claim that
and thus \(g\in L^2_{n,0}(X,\mathscr {L})\cap L^2_{n,0}(X)\). In fact, we write locally
With respect to \(\Theta _X=\sqrt{-1}\langle Z_i|Z_j \rangle dz_i\wedge dz_j\) and \(2\sqrt{-1}\mathscr {L}=\sqrt{-1}\langle Z_i|Z_j \rangle _\mathscr {L}dz_i\wedge dz_j\),
and, respectively, the volume forms are given by
Thus, the claim follows from
Since \(\overline{\partial }_b(I-S^{(n,0)})Q_\lambda u=\overline{\partial }_b Q_\lambda u=v\) and \((I-S^{(n,0)})Q_\lambda u \perp {{\,\mathrm{Ker}\,}}\overline{\partial }_b\), we have \(\overline{\partial }_b(I-S^{(n,0)})Q_\lambda u\) is the solution of minimal norm with respect to \(\Theta _X\), i.e.,
by \({{\,\mathrm{supp}\,}}(u)\Subset D\). \(\square \)
4.5 \(L^2\) Estimates for \(\overline{\partial }_{b,E}\)
In this section, we prove an analog for the \(\overline{\partial }_{b,E}\)-operator of the \(L^2\)-estimates of the Hörmander–Andreotti–Vesentini estimates for \(\overline{\partial }\). As in the case of complex manifolds, we use the Bochner–Kodaira–Nakano formula in the present form (3.36). In order to eliminate the first-order error term \([(\nabla ^E)^{1,0},\mathcal {T}^*]-[\overline{\partial }_{b,E},\overline{\mathcal {T}}^*]\) in (3.36), we reformulate (3.36) can be reformulated as in [10, VII.1]. Under the hypothesis of Theorem 3.7, we have on \(\Omega ^{\bullet ,\bullet }(X,E)\),
where \(\overline{\square }_{b,\mathcal {T}}^E:=[(\nabla ^E)^{1,0}+\mathcal {T}, (\nabla ^E)^{1,0*}+\mathcal {T}^*]\) is a positive formally self-adjoint operator, and
is an operator of order zero depending only on the torsion of Hermitian metric \(\Theta _X\).
Theorem 4.20
(\(L^2\)-estimates for \(\overline{\partial }_b\)) Let X be a CR manifold with a smooth locally free CR \(\mathbb {R}\)-action. Let \(\Theta _X\) be an \(\mathbb {R}\)-invariant Hermitian metric on X. Assume \(g_X\) is complete. Let E be a \({\mathbb {R}}\)-equivariant CR line bundle over X with a \({\mathbb {R}}\)-invariant Hermitian metric \(h^E\). Assume that for some (r, q), \(q\ge 1\), there exists a function \(\psi : X\rightarrow [0,\infty )\) such that, for all \(s\in \Omega _c^{r,q}(X,E)\), pointwisely
Then, for any \(f\in L^2_{r,q}(X,E)\) satisfying \(\overline{\partial }_{b,E} f=0\) and \(\int _X \psi ^{-1}|f|^2dv_X< \infty \), there exists \(g\in L^2_{r,q-1}(X,E)\) such that \(\overline{\partial }_{b,E} g=f\) and \(\Vert g\Vert ^2\le \int _X \psi ^{-1}|f|^2dv_X\).
Proof
Consider the complex of closed densely defined operators
where T and S are maximal extensions of \(\overline{\partial }_{b,E}\). We apply (4.61) and obtain for all \(s\in \Omega _c^{r,q}(X,E)\), it follows that
By Cauchy–Schwarz inequality,
Since \(g_X\) is complete, the above inequality still holds for all \(s\in {{\,\mathrm{Dom}\,}}(S)\cap {{\,\mathrm{Dom}\,}}(T^*)\) by the density Lemma 4.1. Consider now \(s\in {{\,\mathrm{Dom}\,}}(T^*)\) and write the orthogonal decomposition \(s=s_1+s_2\) with \(s_1\in {{\,\mathrm{Ker}\,}}(S)\) and \(s_2\in {{\,\mathrm{Ker}\,}}(S)^\perp \subset [\mathrm{Im}(S^*)]\subset {{\,\mathrm{Ker}\,}}T^*\). So \(s_1=s-s_2\in {{\,\mathrm{Ker}\,}}(S)\cap {{\,\mathrm{Dom}\,}}(T^*)\). Recall \(f\in {{\,\mathrm{Ker}\,}}(S)\),
We consider \(\lambda :\mathrm{Im}(T^*)\rightarrow \mathbb {C}\) defined by \(\lambda (T^* s)=(f|s)\) for \(s\in {{\,\mathrm{Dom}\,}}(T^*)\). We see that \(\lambda \) is \(\mathbb {C}\)-antilinear map and \(|\lambda (T^*s)|\le (\int _X\psi ^{-1}|f|^2dv_X)^{1/2}\Vert T^*s\Vert \), i.e., \(\lambda \) is bounded with norm \(\Vert \lambda \Vert \le (\int _X\psi ^{-1}|f|^2dv_X)^{1/2}\). By the complex Hahn–Banach theorem we can extend \(\lambda \) to \(L^2_{r,q-1}(X,E)\) with the same norm \(\Vert \lambda \Vert \le (\int _X\psi ^{-1}|f|^2dv_X)^{1/2}\).
By the Riesz representation theorem, there exists \(g\in L^2_{r,q-1}(X,E)\) such that \(\lambda (\alpha )=(g|\alpha )\) for \(\alpha \in L^2_{r,q-1}(X,E)\) with \(\Vert g\Vert ^2= \Vert \lambda \Vert ^2\le \int _X\psi ^{-1}|f|^2dv_X\). We set \(\alpha =T^*s\), \(s\in {{\,\mathrm{Dom}\,}}(T^*)\). Thus \((g|T^*s)=\lambda (T^*s)=(f|s)\) and \(g\in {{\,\mathrm{Dom}\,}}(T^{**})={{\,\mathrm{Dom}\,}}(T)\) satisfying \(Tg=f\). \(\square \)
For certain complete CR manifold endowed with a Nakano q-semipositive line bundle, the \(L^2\) method applies to solve the \(\overline{\partial }_{b,E}\)-equation for (n, q)-forms as follows. For the cohomology aspect of Nakano q-semipositive line bundles on complex manifolds, see [30].
Corollary 4.21
Let X be a CR manifold with a smooth locally free CR \(\mathbb {R}\)-action. Let \(\Theta _X\) be an \(\mathbb {R}\)-invariant Hermitian metric on X. Assume \(g_X\) is complete. Assume \(\mathscr {L}=0\) and \(d\Theta _X=0\). Let E be a \({\mathbb {R}}\)-equivariant CR line bundle over X with a \({\mathbb {R}}\)-invariant Hermitian metric \(h^E\). Let \(\lambda _1\le \cdots \le \lambda _n\) be eigenvalues of \(R^E\) with respect to \(\Theta _X\). Assume \((E,h^E)\) is Nakano q-semipositive with respect to \(\Theta _X\), i.e., \(\lambda _1+\cdots +\lambda _q\ge 0\). Then, for any \(f\in L^2_{n,q}(X,E)\) satisfying \(\overline{\partial }_{b,E} f=0\) and \(\int _X (\lambda _1+\cdots +\lambda _q)^{-1}|f|^2dv_X< \infty \), there exists \(g\in L^2_{n,q-1}(X,E)\) such that \(\overline{\partial }_{b,E} g=f\) and \(\Vert g\Vert ^2\le \int _X (\lambda _1+\cdots +\lambda _q)^{-1}|f|^2dv_X\).
4.6 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 \(2\sqrt{-1}\mathscr {L}=\Theta _X\), \(g_X\) is complete and let \(\lambda <0\) and \(1\le q\le n\). 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 \(2\sqrt{-1}\mathscr {L}=\Theta _X\), \(g_X\) is complete and let \(C>0\) such that
Let \(\lambda <-C\) and \(1\le q\le n\). 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.
Corollary 4.24
(Andreotti–Vesentini) Let \((M,\omega )\) be a complete Kähler manifold of dimension n and let \((L,h^L)\rightarrow M\) be a Hermitian holomorphic line bundle such that \(\sqrt{-1}R^L=\omega \) and there is \(C>0\) such that \(\sqrt{-1}R^{K^*_M}_{\omega }\ge -C\omega \) on M. Then there exists \(m_0\in \mathbb {N}\) such that for every \(m\ge m_0\) we have \(H^q_{(2)}(M,L^m)=0\) for \(q\ge 1\), where \(H^q_{(2)}(M,L^m)\) denotes the \(L^2\) q-th Dolbeault cohomology group with respect to the metric \((h^L)^m\) and volume form \(\omega ^n/n!\) with values in \(L^m\).
Proof
We apply the previous results for the CR manifold X constructed in Example 4.6. In this case \(T=\partial _\theta \). For \(m\in \mathbb {Z}\), the space \(L^2_{0,q}(M,L^m)\) is isometric to the space of m-equivariant \(L^2\) forms on X, \(L^2_{0,q}(X)_m=\{u\in L^2_{0,q}(X): (e^{i\theta })^*u=e^{im\theta }u, \text {for any}\, e^{i\theta }\in S^{1}\}\). Note that \(L^2_{0,q}(X)_m=\mathscr {E}^{(q)}(-m,\sqrt{-1}\partial _\theta )\) and the \(L^2\)-Dolbeault complex \((L^2_{0,\bullet }(M,L^m),\overline{\partial })\) is isomorphic to the \(\overline{\partial }_b\)-complex \((L^2_{0,\bullet }(X)_m,\overline{\partial }_b)\). Hence, the assertion follows from Theorem 4.10. \(\square \)
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.
Corollary 4.25
(Andreotti–Vesentini) Let \((M,\omega )\) be a complete Kähler manifold of dimension n and let \((L,h^L)\rightarrow M\) be a Hermitian holomorphic line bundle such that \(\sqrt{-1}R^L=\omega \). Then \(H^{n,q}_{(2)}(M,L)=0\) for \(q\ge 1\), where \(H^{n,q}_{(2)}(M,L)\) denotes the \(L^2\) q-th Dolbeault cohomology group with respect to the metric \(h^L\) and volume form \(\omega ^n/n!\) with values in \(K_X\otimes L\).
4.7 Szegö Kernel Asymptotic Expansions
In this section, we prove Theorem 1.2 and Corollary 1.4. We first introduced some notations. Let \(D\subset X\) be an open coordinate patch with local coordinates \(x=(x_1,\ldots ,x_{2n+1})\). Let \(m\in {\mathbb {R}}\), \(0\le \rho ,\delta \le 1\). Let \(S^m_{\rho ,\delta }(T^*D)\) denote the Hörmander symbol space on \(T^*D\) of order m type \((\rho ,\delta )\) and let \(S^m_{\mathrm{cl\,}}(T^*D)\) denote the space of classical symbols on \(T^*D\) of order m, see Grigis–Sjöstrand [11, Definition 1.1 and p. 35] and Definition 2.1. Let \(L^m_{\rho ,\delta }(D)\) and \(L^m_{\mathrm{cl\,}}(D)\) denote the space of pseudodifferential operators on D of order m type \((\rho ,\delta )\) and the space of classical pseudodifferential operators on D of order m, respectively.
Let \(\Sigma \) be the characteristic manifold of \(\Box _b\). We have
We recall the following definition introduced in [19, Definition 2.4].
Definition 4.26
Let \(Q:L^2(X)\rightarrow L^2(X)\) be a continuous operator. Let \(D\Subset X\) be an open local coordinate patch of X with local coordinates \(x=(x_1,\ldots ,x_{2n+1})\) and let \(\eta =(\eta _1,\ldots ,\eta _{2n+1})\) be the dual variables of x. We write
if for every \(D'\Subset D\),
where \(q(x,\eta )\in S^0_{1,0}(T^*D')\) and there exist \(M>0\) and a conic open neighborhood \(\Lambda _-\) of \(\Sigma ^-\) such that for every \((x,\eta )\in T^*D'\cap \Lambda _-\) with \(|\eta |\ge M\), we have \(q(x,\eta )=0\).
For a given point \(x_0\in D\), let \(\{W_j\}_{j=1}^{n}\) be an orthonormal frame of \(T^{(1,0)}X\) with respect to \(\langle \,\cdot \,|\,\cdot \,\rangle \) near \(x_0\), for which the Levi form is diagonal at \(x_0\). Put
We will denote by
We recall the following results in [19, Theorems 1.9, 5.1].
Theorem 4.27
Let \(D\Subset X\) be an open coordinate patch with local coordinates \(x=(x_1,\ldots ,x_{2n+1})\). Let \(Q:L^2(X)\rightarrow L^2(X)\) be a continuous operator and let \(Q^*\) be the \(L^2\) adjoint of Q with respect to \((\,\cdot \,|\,\cdot \,)\). Suppose that \(\Box ^{(0)}_b\) has local \(L^2\) closed range on D with respect to Q and \(QS^{(0)}=S^{(0)}Q\) on \(L^2(X)\) and
where \(Q_0\in L^0_{\mathrm{cl\,}}(D)\). Then,
where
\(a(x, y, t)\in S^{n}_{\mathrm{cl\,}}\big (D\times D\times {\mathbb {R}}_+\big )\) and the leading term \(a_0(x,y)\) of the expansion (2.4) of a(x, y, t) satisfies
where \(\det {\mathcal {L}}_x\) is the determinant of the Levi form defined in (4.73), \(q(x,\eta )\in \mathscr {C}^\infty (T^*D)\) is the principal symbol of Q.
We refer the reader to [19, Theorems 3.3, 4.4] for more properties for the phase \(\varphi \) in (4.75). Let \(D=U\times {\mathcal {I}}\) be a BRT chart with BRT coordinates \(x=(x_1,\ldots ,x_{2n+1})\). For \(\lambda \in {\mathbb {R}}\), put
where \(\tau _\lambda \in \mathscr {C}^\infty ({\mathbb {R}})\) is as in the discussion before Theorem 4.14. It is not difficult to see that
Assume that the \({\mathbb {R}}\)-action is free. From (4.11), we see that \(Q_{\tau _\lambda } ={\hat{Q}}_{\tau _\lambda }\) on D. From this observation, Theorems 4.14, 4.27, (4.78), and noticing that \(Q^*_{\tau _\lambda }S^{(0)}Q_{\tau _\lambda }= Q_{\tau ^2_\lambda }S^{(0)}\), where \(Q^*_{\tau _\lambda }\) is the \(L^2\) adjoint of \(Q_{\tau _\lambda }\) with respect to \((\,\cdot \,|\,\cdot \,)\), we get
Theorem 4.28
Suppose that the \({\mathbb {R}}\)-action is free. Assume that \(g_\mathscr {L}\) is complete and there is \(C>0\) such that
Let \(D=U\times {\mathcal {I}}\Subset X\) be a BRT chart with BRT coordinates \(x=(x_1,\ldots ,x_{2n+1})\). Let \(\lambda \in {\mathbb {R}}\), \(\lambda <-C\). Then,
where \(\varphi \in \mathscr {C}^\infty (D\times D)\) is as in (4.74), \(s(x, y, t)\in S^{n}_{\mathrm{cl\,}}\big (D\times D\times {\mathbb {R}}_+\big )\) and the leading term \(s_0(x,y)\) of the expansion (2.4) of s(x, y, t) satisfies
We now assume that the \({\mathbb {R}}\)-action is not free. From (4.8), we know that the \({\mathbb {R}}\)-action comes from a CR torus action \(\mathbb T^d=(e^{i\theta _1},\ldots ,e^{i\theta _d})\) on X and \(\omega _0\), \(\Theta _X\) are \({\mathbb {T}}^d\) invariant. We will use the same notations as in the discussion before Proposition 4.8. We need
Lemma 4.29
Suppose that the \({\mathbb {R}}\)-action is not free. With the notations and assumptions used above, let \(D=U\times {\mathcal {I}}\Subset X\) be a BRT chart with BRT coordinates \(x=(x',x_{2n+1})\), \(x'=(x_1,\ldots ,x_{2n})\). Fix \(D_0\Subset D\) and \(\lambda \in {\mathbb {R}}\). For \(u\in \mathscr {C}^\infty _c(D_0)\), we have
where \(\chi \in \mathscr {C}^\infty _c(I)\), \(\chi (x_{2n+1})=1\) for every \((x',x_{2n+1})\in D_0\) and \(\beta _1\in {\mathbb {R}},\ldots ,\beta _d\in {\mathbb {R}}\) are as in (4.17).
Proof
We also write \(y=(y',y_{2n+1})=(y_1,\ldots ,y_{2n+1})\), \(y'=(y_1,\ldots ,y_{2n})\), to denote the BRT coordinates x. Let \(u\in \mathscr {C}^\infty _c(D_0)\). From (4.19), it is easy to see that on D,
Now, we claim that
Let \(u\in \mathscr {C}^\infty _c(D_0)\). From Fourier inversion formula, it is straightforward to see that
From (4.84) and the definition of \({\hat{R}}_{\tau _\lambda }\), we have
Note that the following formula holds for every \(\alpha \in {\mathbb {R}}\),
where the integral is defined as an oscillatory integral and \(\delta _\alpha \) is the Dirac measure at \(\alpha \). Using (4.82), (4.86), and the Fourier inversion formula, (4.85) becomes
From (4.87), the claim (4.83) follows. \(\square \)
To study \(Q_{\tau ^2_\lambda }S^{(0)}\) when the \({\mathbb {R}}\) is not free, we also need the following two known results [19, Theorems 3.2, 5.2].
Theorem 4.30
We assume that the \({\mathbb {R}}\)-action is arbitrary. Let \(D\Subset X\) be a coordinate patch with local coordinates \(x=(x_1,\ldots ,x_{2n+1})\). Then there exist properly supported continuous operators \(A\in L^{-1}_{\frac{1}{2},\frac{1}{2}}(D)\) , \({\tilde{S}}\in L^{0}_{\frac{1}{2},\frac{1}{2}}(D)\), such that
where \(A^*\), \({\tilde{S}}^*\) are the formal adjoints of A, \({\tilde{S}}\) with respect to \((\,\cdot \,|\,\cdot \,)\), respectively, and \({\tilde{S}}(x,y)\) satisfies
where \(\varphi (x,y)\in \mathscr {C}^\infty (D\times D)\) and \(s(x, y, t)\in S^{n}_{\mathrm{cl\,}}(D\times D\times {\mathbb {R}}_+)\) are as in (4.79).
Theorem 4.31
Let us consider an arbitrary \({\mathbb {R}}\)-action and let \(Q:L^2(X)\rightarrow L^2(X)\) be a continuous operator and let \(Q^*\) be the \(L^2\) adjoint of Q with respect to \((\,\cdot \,|\,\cdot \,)\). Suppose that \(\Box ^{(0)}_b\) has local \(L^2\) closed range on D with respect to Q and \(QS^{(0)}=S^{(0)}Q\) on \(L^2(X)\). Let D be a coordinate patch with local coordinates \(x=(x_1,\ldots ,x_{2n+1})\). We have
where \({\tilde{S}}\) is as in Theorem 4.30.
For the proof we need the following.
Lemma 4.32
Suppose that the \({\mathbb {R}}\)-action is not free. Fix \(p\in X\). Let \(D=U\times {\mathcal {I}}\) be a BRT chart defined near p with BRT coordinates \(x=(x',x_{2n+1})\), \(x'=(x_1,\ldots ,x_{2n})\), \(x(p)=0\). Fix \(D_0\Subset D\), \(p\in D_0\), and \(\lambda \in {\mathbb {R}}\). Then,
where \({\hat{R}}_{\tau _\lambda }\) and \({\tilde{S}}\) are as in Lemma 4.29 and Theorem 4.30, respectively.
Proof
From (4.75), we may assume \(D_0\) is small so that
where \(C>0\) is a constant. Let \(g\in \mathscr {C}^\infty _c(D_0)\). From (4.81) and (4.89), we have
where we also write \(u=(u',u_{2n+1})\), \(u'=(u_1,\ldots ,u_{2n})\), to denote the BRT coordinates x. Since \(u_{2n+1}\ne y_{2n+1}\), for every \((u',u_{2n+1})\in D_0\), \(y_{2n+1}\in \mathrm{Supp\,}(1-\chi (y_{2n+1}))\), we can integrate by parts in \(\eta _{2n+1}\) and rewrite (4.92):
Let
By (4.91) there exists \(c>0\) such that
for every \((x,u)\in D_0\times D_0\). Hence, we can integrate by parts in \(u_{2n+1}\) and \(\eta _{2n+1}\) and deduce that
We have
where \(dv_X(u)=dv(u')du_{2n+1}\). From (4.94) and (4.95), we have
where \(\Vert \cdot \Vert _{D_0,s}\) denotes the standard Sobolev norm of order s on \(D_0\), \(C, {\hat{C}}, {\hat{C}}_0>0\) are constants, \(\tau \in \mathscr {C}^\infty _0(D)\), \(\tau =1\) near \(D_0\). From (4.96), we deduce that
Let \(\triangle _X: \mathscr {C}^\infty (X)\rightarrow \mathscr {C}^\infty (X)\) be the standard Laplacian on X induced by \(\langle \,\cdot \,|\,\cdot \,\rangle \). Since \(\langle \,\cdot \,|\,\cdot \,\rangle \) is \({\mathbb {T}}^d\) invariant, \(\triangle _X\) is \({\mathbb {T}}^d\) invariant. Fix \(s\in {\mathbb {N}}\). Let
be a parametrix of \(\triangle _X^s\) on \(D_0\) and \(G_s\) is properly supported on \(D_0\). Hence, there is a properly supported smoothing operator
such that
for all \(g\in \mathscr {C}^\infty _c(D_0)\). 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 \(G_s: H^{-2s}_{\mathrm{comp\,}}(D)\rightarrow H^0_{\mathrm{comp\,}}(D)\) is continuous, we can repeat the proof of (4.96) and conclude that
where \(C, C_1>0\) are constants. From (4.97), (4.98), (4.99), and (4.102), we get that
Hence, \({\tilde{S}}{\hat{R}}_{\tau _\lambda }\) is smoothing on \(D_0\). \(\square \)
Theorem 4.33
Suppose that the \({\mathbb {R}}\)-action is not free. Assume that \(g_\mathscr {L}\) is complete and there is \(C>0\) such that
Let \(D=U\times {\mathcal {I}}\Subset X\) be a BRT chart with BRT coordinates \(x=(x_1,\ldots ,x_{2n+1})\). Let \(\lambda \in {\mathbb {R}}\), \(\lambda <-C\). Then,
where \(\varphi \in \mathscr {C}^\infty (D\times D)\) and \(s(x, y, t)\in S^{n}_{\mathrm{cl\,}}\big (D\times D\times {\mathbb {R}}_+\big )\) are as in (4.79).
Proof
From (4.81), (4.90), and Lemma 4.32, we see that on D,
Using this observation, we can repeat the proof of [19, Theorem 5.8] and obtain the conclusion. \(\square \)
Theorem 4.34
Let us consider an arbitrary \({\mathbb {R}}\)-action. Assume that \(g_\mathscr {L}\) is complete and there is \(C>0\) such that
Let \(D=U\times I\) be a BRT chart with BRT coordinates \(x=(x_1,\ldots ,x_{2n+1})\). Let \(\lambda \in {\mathbb {R}}\), \(\lambda <-C\). Then,
Proof
From (4.88), we have
From Lemma 4.32, we have
Since \(\mathrm{WF\,}(I-{\hat{Q}}_{\tau ^2_\lambda })\bigcap \Sigma ^- =\emptyset \) and \(\mathrm{WF'\,}({\tilde{S}})=\mathrm{diag\,}(\Sigma ^-\times \Sigma ^-)\), we have
where \(\mathrm{WF\,}(I-{\hat{Q}}_{\tau ^2_\lambda })\) denotes the wave front set of \(I-{\hat{Q}}_{\tau ^2_\lambda }\) and
From (4.105) and (4.106), we get
and hence
From (4.107) and (4.108), we get
The theorem follows. \(\square \)
We can now prove the main result of this work.
Theorem 4.35
(=Theorem 1.2) Let the \({\mathbb {R}}\)-action be arbitrary. Assume that \(g_\mathscr {L}\) is complete and there is \(C>0\) such that
Let \(D\Subset X\) be a local coordinate patch with local coordinates \(x=(x_1,\ldots ,x_{2n+1})\). Then,
where \(\varphi \in \mathscr {C}^\infty (D\times D)\) satisfies (4.75), \(s(x, y, t)\in S^{n}_{\mathrm{cl\,}}\big (D\times D\times {\mathbb {R}}_+\big )\) and the leading term \(s_0(x,y)\) of the expansion (2.4) of s(x, y, t) satisfies
Proof
Let \(\lambda \in {\mathbb {R}}\), \(\lambda <-C\). From Theorem 4.34, we have
We can repeat the proofs of Theorems 4.28 and 4.33 and get
where \(\varphi \in \mathscr {C}^\infty (D\times D)\), \({\hat{s}}(x, y, t), {\tilde{s}}(x,y,t)\in S^{n}_{\mathrm{cl\,}} \big (D\times D\times {\mathbb {R}}_+\big )\) are as in (4.79). From (4.111) and (4.112), the theorem follows. \(\square \)
Proof of Theorem 1.3
The proof is analogous to the proof of Theorem 1.2 by using Theorem 4.15 instead of Theorem 4.13. \(\square \)
Proof of Corollary 1.4
Let K be a compact set of X. Fix \(x\in K\). 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 \(V_x\subset U_x\) of x and global smooth \(L^2\) CR functions \((f_{0,x}, f_{1,x},\cdots ,f_{N_x,x} )=F_x\) such that \(F_x: U_x\rightarrow \mathbb {C}^{N_x+1}\) is an embedding and \(\sup _{K\setminus U_x}|f_{0,x}|\le \frac{1}{2}\), \(\inf _{V_x}|f_{0,x}|\ge 1\). There exists \(x_1,x_2,\ldots ,x_m\in K\) such that \(K\subset V_{x_1}\cup V_{x_2}\cup V_{x_m}\subset U_{x_1}\cup U_{x_1}\cup U_{x_2}\cup U_{x_m}\). Then \(K\ni x\mapsto (F_{x_1},\cdots , F_{x_m})\) is an embedding. \(\square \)
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 \((x_1,\ldots ,x_{2n+1})\) and observing that in these coordinates a CR (n, 0)-form equals \(fdz_1\wedge \ldots \wedge dz_n\) with f a CR function on K. \(\square \)
5 Examples
We now consider Heisenberg group \(\mathbb {H}=\mathbb {C}^n\times \mathbb {R}\) with CR structure
where \(\phi \in \mathscr {C}^\infty (\mathbb {C}^n,\mathbb {R})\). Let \((\,\cdot \,|\,\cdot \,)_{\mathbb {H}}\) be the \(L^2\) inner product on \(\mathbb {H}\) induced by the Euclidean measure dx on \({\mathbb {R}}^{2n+1}\). Let
be the orthogonal projection with respect to \((\,\cdot \,|\,\cdot \,)_{\mathbb {H}}\) and let \(S_{\mathbb {H}}(x,y)\in {\mathscr {D}}'(\mathbb {H}\times \mathbb {H})\) be the distribution kernel of \(S_{\mathbb {H}}\). From Theorem 1.2, we deduce
Corollary 5.1
With the notations used above, assume that \(\left( \frac{\partial ^2\phi (z)}{\partial z_j\partial {\overline{z}}_k}\right) ^n_{j,k=1}\) is positive definite, at every \(z\in {\mathbb {C}}^n\). Let \(0<\lambda _1(z)\le \ldots \le \lambda _n(z)\) be the eigenvalues of \(\left( \frac{\partial ^2\phi (z)}{\partial z_j\partial {\overline{z}}_k}\right) ^n_{j,k=1}\), for every \(z\in {\mathbb {C}}^n\). Suppose that there is \(C>0\) such that
Let \(D\Subset \mathbb {H}\) be any open set. Then,
where \(\varphi \in \mathscr {C}^\infty (D\times D)\) and \(s(x, y, t)\in S^{n}_{\mathrm{cl\,}}\big (D\times D\times {\mathbb {R}}_+\big )\) are as in Theorem 1.2.
Example 5.2
With the notations used in Corollary 5.1, assume that
with \(r(z)\in \mathscr {C}^\infty _c(\mathbb {C}^n)\) and \(\sqrt{-1}\partial \overline{\partial }(|z|^2+r(z))>0\) on \(\mathbb {C}^n\). With this \(\phi \), we can check the conditions of Corollary 5.1 fulfilled as follows. In fact, in this case, we have
with some \(F(z)\in \mathscr {C}^\infty _c(\mathbb {C}^n)\). And we have
Since \(r(z)\in \mathscr {C}^\infty _c(\mathbb {C}^n)\) and \(\sqrt{-1}\partial \overline{\partial }\phi =\sqrt{-1}\partial \overline{\partial }(|z|^2+r(z))>0\), we have a uniform lower bound for the smallest eigenvalue, i.e., \(\lambda _1(z)>1/C_1\) for some \(C_1>0\). Moreover, we can choose \(C_2>0\) sufficiently large such that
since the first term \(\sqrt{-1}\partial \overline{\partial }\left( -\log (1+F(z))\right) \) is a real (1, 1)-form with compact support in \(\mathbb {C}^n\) and the second term \(\sqrt{-1}\partial \overline{\partial }\phi \) is a real positive (1, 1)-form with a uniformly positive lower bound for the smallest eigenvalue \(\lambda _1(z)>1/C_1\) on \(\mathbb {C}^n\). Finally, we obtain \(C:=\max \{C_1,C_2\}>0\) as desired in (5.2).
With this \(\phi \), 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 \((X,T^{(1,0)}X)\) be a strictly pseudoconvex, CR manifold of dimension \(2n+1\), \(n\ge 1\), with a discrete, proper, CR action \(\Gamma \) such that the quotient \(X/\Gamma \) is compact. Assume X admits a transversal CR \(\mathbb {R}\)-action on X and let \(\Theta _X\) be a \(\Gamma \)-invariant, \(\mathbb {R}\)-invariant, Hermitian metric on X. Then the conclusion of Theorem 1.2 holds. In fact, the \(\Gamma \)-covering manifold is complete and we can find the desired constant C depending on the fundamental domain \(U\Subset X\) given by the \(\Gamma \)-action such that (1.7) is fulfilled. As a consequence, if we consider the circle bundle case in which \(R^L=2\mathscr {L}\), we could obtain the Bergman kernel expansion for covering manifold [23, 6.1.2].
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Chin-Yu Hsiao was partially supported by Taiwan Ministry of Science and Technology projects 108-2115-M-001-012-MY5, 109-2923-M-001-010-MY4 and Academia Sinica Career Development Award.
George Marinescu partially supported by the DFG funded project SFB TRR 191 ‘Symplectic Structures in Geometry, Algebra and Dynamics’ (Project-ID 281071066–TRR 191).
Huan Wang thanks his friends and former colleagues for their help during the Coronavirus pandemic.
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Hsiao, CY., Marinescu, G. & Wang, H. Szegő Kernel Asymptotics on Complete Strictly Pseudoconvex CR Manifolds. J Geom Anal 32, 266 (2022). https://doi.org/10.1007/s12220-022-00990-4
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DOI: https://doi.org/10.1007/s12220-022-00990-4