# On the singularities of the Szegő kernels on CR orbifolds

Andrea Galasso and Chin-Yu Hsiao111Address: Room 407, Chee-Chun Leung
Cosmology Hall, National Taiwan University; ORCID iD: 0000-0002-5792-1674;
e-mail<EMAIL_ADDRESS>Address: Institute of Mathematics,
Academia Sinica, 6F, Astronomy-Mathematics Building, No.1, Sec.4, Roosevel:
ORCHID iD: 0000-0002-1781-0013; email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper we study the microlocal properties of the Szegő kernel of a
given compact connected orientable CR orbifold whose Kohn Laplacian has closed
range. This last assumption is satisfied if certain geometric conditions hold
true, as in the smooth case. As applications, we give a pure analytic proof of
Kodaira-Bailey theorem and explain how to generalize a CR version of
quantization commutes with reduction to orbifolds.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Orbifolds
    2. 2.2 CR orbifolds
    3. 2.3 Group actions on orbifolds
    4. 2.4 Operators on orbifolds
    5. 2.5 Microlocal Szegő kernel
    6. 2.6 Geometric conditions for the closed range property
  3. 3 Proof of Theorem 1.1
  4. 4 Applications
    1. 4.1 Proof of Theorem 1.3
    2. 4.2 Proof of Theorem 1.4
    3. 4.3 Quantization commutes with reduction

Keywords: CR orbifolds

Mathematics Subject Classification: 32A2

##  1 Introduction

The study of Szegő kernels on CR manifolds play an important role in
CR/Complex analysis and geometry. Boutet de Monvel and Sjöstrand [BS] showed
that the Szegő kernel on a strongly pseudoconex CR manifold is a complex
Fourier integral operator. Boutet de Monvel and Sjöstrand’s result has a
profound impact in CR/Complex analysis and geometry. When $X$ is a CR
orbifold, the study of the associated Szegő kernel on $X$ is a very natural
question and could have some application in CR/Complex geometry and geometric
quantization theory. For example, in [HMM] and [HH], the authors studied
quantization commutes with reduction for stronly speudococonex CR manifolds
under the assumption that the group action $G$ is free near $\mu^{-1}(0)$,
$\mu$ is the associated momentum map. When the group action $G$ is locally
free near $\mu^{-1}(0)$, we need to understand the behaviour of the Szegő
kernels on CR orbifolds. The main goal of this paper is to study the
associated Szegő kernel on a CR orbifold $X$ with non-degenerate Levi form. As
applications, we give a pure analytic proof of Kodaira-Bailey theorem and
explain how to generalize a CR version of quantization commutes with reduction
to orbifolds.

We now formulate our main results. We refer the reader to Section 2 for some
notations and terminology used here. Let $(X,\,T^{1,0}X)$ be a compact
orientable connected CR orbifold of dimension $2n+1,\,n\geq 1$. In Section 2
we recall the basic material concerning orbifolds, we give definitions for the
spaces of $(0,q)$-forms with $L^{2}(X)$ coefficients and for the Kohn
Laplacian $\square_{b}^{(q)}$ on the orbifold $X$. We can then define
abstractly the Szegő projector for $X$ to be the projector

$\Pi^{(q)}\,:\,L^{2}_{(0,q)}(X)\rightarrow\operatorname{Ker}\square_{b}^{(q)}\,.$

The main aim of this paper is to prove the following result, which is a
generalization of Theorem $1.2$ in [Hs].

######  Theorem 1.1.

Assume that the Levi form is non-degenerate of constant signature
$(n_{-},n_{+})$ on $X$. Suppose $\square_{b}^{(q)}$ has closed range. Then

$\Pi^{(q)}\equiv\Pi_{-}+\Pi_{+}$

where $\Pi_{\pm}\equiv 0$ if $q\neq n_{\pm}$. Let $q=n_{-}$. We have
$\mathrm{WF}^{\prime}(\Pi_{-})=\mathrm{diag}(\Sigma^{-}\times\Sigma^{-})$,
$\mathrm{WF}^{\prime}(\Pi_{+})=\mathrm{diag}(\Sigma^{+}\times\Sigma^{+})$ if
$q=n_{-}=n_{+}$,

$\Sigma^{\pm}=\left\\{(x,\,\lambda\,\omega_{0}(x))\in
T^{*}X\,:\,\lambda\,_{<}^{>}\,0\right\\}\,,$

$\mathrm{WF}^{\prime}(\Pi_{\pm})=\left\\{(x,\xi,y,-\eta)\in T^{*}X\times
T^{*}X;\,(x,\xi,y,\eta)\in\mathrm{WF}(\Pi_{\pm})\right\\}$,
$\mathrm{WF}(\Pi_{\pm})$ denotes the wave front set of $\Pi_{\pm}$ in the
sense of Hörmander. Moreover, $\Pi_{\pm}$ has the following microlocal
expression: Let $\Pi_{\pm}(x,y)\in\mathcal{D}^{\prime}(X\times
X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ be the distribution kernel of
$\Pi_{\pm}$. Consider an open set $U\subset X$ and an orbifold chart
$(\widetilde{U},G_{U})\rightarrow U$. We have

$\Pi_{\pm}(x,y)=\sum_{g\in
G_{U}}\Bigr{(}\int_{0}^{+\infty}e^{it\,\varphi_{\pm}(\widetilde{x},g\cdot\widetilde{y})}s_{\pm}(\widetilde{x},g\cdot\widetilde{y},t)\,\mathrm{d}t+F_{\pm}(\widetilde{x},g\cdot\widetilde{y})\Bigr{)}\
\ \mbox{on $U\times U$}\,,$

where $\pi(\widetilde{x})=x$, $\pi(\widetilde{y})=y$,
$\pi:\widetilde{U}\rightarrow U$ is the natural projection, $F_{\pm}$ are
smoothing operators on $\widetilde{U}$ and

$s_{\pm}(\widetilde{x},\widetilde{y},t)\sim\sum^{\infty}_{j=0}s^{j}_{\pm}(\widetilde{x},\widetilde{y})t^{n-j}$

in
$S^{n}_{1,0}(\widetilde{U}\times\widetilde{U}\times\mathbb{R}_{+}\,,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$,

$s_{+}(\widetilde{x},\widetilde{y},t)=0$ if $q\neq n_{+}$

and

$s^{0}_{-}(\widetilde{x}_{0},\widetilde{x}_{0})=\frac{1}{2}\pi^{-n-1}\lvert\det\mathcal{L}_{\widetilde{x}_{0}}\rvert\tau_{\widetilde{x}_{0},n_{-}}\,,\widetilde{x}_{0}\in\widetilde{U},$

if $q=n_{-}=n_{+}$,

$s^{0}_{+}(\widetilde{x}_{0},\widetilde{x}_{0})=\frac{1}{2}\pi^{-n-1}\lvert\det\mathcal{L}_{\widetilde{x}_{0}}\rvert\tau_{\widetilde{x}_{0},n_{+}}\,,\widetilde{x}_{0}\in\widetilde{U},$

and the phase functions $\varphi_{-}$, $\varphi_{+}$ satisfy

$\begin{array}[]{ll}&\varphi_{+},\varphi_{-}\in\mathcal{C}^{\infty}(\widetilde{U}\times\widetilde{U}),\
\ {\rm Im\,}\varphi_{\pm}(\widetilde{x},\widetilde{y})\geq 0,\\\
&\varphi_{-}(\widetilde{x},\widetilde{x})=0,\ \
\varphi_{-}(\widetilde{x},\widetilde{y})\neq 0\ \ \mbox{if}\ \
\widetilde{x}\neq\widetilde{y},\\\
&\mathrm{d}_{x}\varphi_{-}(\widetilde{x},\widetilde{y})\big{|}_{\widetilde{x}=\widetilde{y}}=-\omega_{0}(\widetilde{x}),\
\
\mathrm{d}_{\widetilde{y}}\varphi_{-}(\widetilde{x},\widetilde{y})\big{|}_{\widetilde{x}=\widetilde{y}}=\omega_{0}(\widetilde{x}),\\\
&-\overline{\varphi}_{+}(\widetilde{x},\widetilde{y})=\varphi_{-}(\widetilde{x},\widetilde{y}).\end{array}$

Moreover, we have

$\begin{split}&\sum_{g\in
G_{U}}\Bigr{(}\int_{0}^{+\infty}e^{it\,\varphi_{\pm}(\widetilde{x},g\cdot\widetilde{y})}s_{\pm}(\widetilde{x},g\cdot\widetilde{y},t)\,\mathrm{d}t+F_{\pm}(\widetilde{x},g\cdot\widetilde{y})\Bigr{)}\\\
&=\sum_{h,g\in
G_{U}}\frac{1}{\left|G_{U}\right|}\Bigr{(}\int_{0}^{+\infty}e^{it\,\varphi_{\pm}(h\circ\widetilde{x},g\cdot\widetilde{y})}s_{\pm}(h\cdot\widetilde{x},g\cdot\widetilde{y},t)\,\mathrm{d}t+F_{\pm}(h\circ\widetilde{x},g\cdot\widetilde{y})\Bigr{)},\end{split}$
(1)

where $\mathcal{L}_{\widetilde{x}_{0}}$ denotes the Levi form of $X$ at
$\widetilde{x}_{0}$, ${\rm
det\,}\mathcal{L}_{\widetilde{x}_{0}}=\lambda_{1}(\widetilde{x}_{0})\cdots\lambda_{n}(\widetilde{x}_{0})$,
$\lambda_{j}(\widetilde{x}_{0})$, $j=1,\ldots,n$, are the eigenvalues of
$\mathcal{L}_{\widetilde{x}_{0}}$ with respect to the given Hermitian metric
on $\mathbb{C}TX$ and $\tau_{\widetilde{x}_{0},n_{\pm}}$ are given by (17).

The main idea of the proof is the following: On $\widetilde{U}$, by using
approximate Szegő kernel $S$ constructed in [Hs] to define a kernel

$S^{(q)}_{U}(x,y):=\sum_{g,h\in
G_{U}}\frac{1}{\left|G_{U}\right|}S(h\cdot\widetilde{x},g\cdot\widetilde{y})$

which is micro-locally given by the formulas above. We then show that
$S^{(q)}_{U}$ is microlocally self-adjoint and it behaves as the identity when
restricted to $\operatorname{Ker}\square_{b}^{(q)}$, up to smoothing
contributions. Moreover, by using the result in [HM] about Szegő kernel
asymptotiocs for lower energy forms, we can show that $S^{(q)}_{U}$ satisfies
(1). Then using $S^{(q)}_{U}$ one can define globally an operator $S^{(q)}$ on
$X$ which we prove to be the projector $\Pi^{(q)}$ up to some smoothing
operator, if $\square_{b}^{(q)}$ has closed range. This is a consequence of
the properties of $S^{(q)}_{U}$ just mentioned and a theorem of functional
analysis. The property (1) is important for further study about the algebra of
Toeplitz operators on CR orbifolds.

Now, assume that $X$ admits a transversal and CR locally free $S^{1}$ action
$e^{i\theta}$. We take the Reeb vector field $R$ to be the vector field on $X$
induced by the $S^{1}$ action. For every $\ell\in\mathbb{N}$, put

$X_{\ell}:=\left\\{x\in X;\,e^{i\theta}x\neq
x,\theta\in\left[0,\frac{2\pi}{\ell}\right[,e^{i\frac{2\pi}{\ell}}x=x\right\\}.$

For every $k\in\mathbb{Z}$, set

${\rm Ker\,}\Box^{(q)}_{b,k}:=\left\\{u\in{\rm
Ker\,}\Box^{(q)}_{b};\,(e^{i\theta})^{*}u=e^{ik\theta}u\right\\}.$

Let

$\Pi^{(q)}_{k}:L^{2}_{(0,q)}(X)\rightarrow{\rm Ker\,}\Box^{(q)}_{b,k}$

be the orthogonal projection and let
$\Pi^{(q)}_{k}(x,y)\in\mathcal{C}^{\infty}(X\times
X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ be the distribution kernel of
$\Pi^{(q)}_{k}$. From Theorem 1.1, we can repeat the procedure in [GH2] and
get

######  Theorem 1.2.

With the assumptions and notations used above, assume that the Levi form is
non-degenerate of constant signature $(n_{-},n_{+})$ on $X$ and let $q=n_{-}$.
Fix $p\in X$ and assume that $p\in X_{\ell}$, for some $\ell\in\mathbb{N}$.
Consider an open set $U\subset X$, $p\in U$, and an orbifold chart
$(\widetilde{U},G_{U})\rightarrow U$. We have as $k\rightarrow+\infty$,

$\Pi^{(q)}_{k}(x,y)=\sum_{g\in G_{U}}\sum^{\ell-1}_{j=0}e^{\frac{2\pi
kj}{\ell}}e^{ik\,\Psi(\widetilde{x},e^{i\frac{2\pi j}{\ell}}\cdot
g\cdot\widetilde{y})}b(\widetilde{x},e^{i\frac{2\pi j}{\ell}}\cdot
g\cdot\widetilde{y},k)+\sum_{g\in
G_{U}}\sum^{\ell-1}_{j=0}F_{k}(\widetilde{x},e^{i\frac{2\pi j}{\ell}}\cdot
g\cdot\widetilde{y})$ (2)

where

$\begin{split}&\Psi\in\mathcal{C}^{\infty}(\widetilde{U}\times\widetilde{U}),\\\
&\Psi(\widetilde{x},\widetilde{x})=0,\ \ \mbox{for all
$\widetilde{x}\in\widetilde{U}$},\\\ &\frac{1}{C}\inf_{e^{i\theta}\in
S^{1}}\left\\{d^{2}(\widetilde{x},e^{i\theta}\widetilde{y})\right\\}\leq{\rm
Im\,}\Psi(\widetilde{x},\widetilde{y})\leq C\inf_{e^{i\theta}\in
S^{1}}\left\\{d^{2}(\widetilde{x},e^{i\theta}\widetilde{y})\right\\},\\\
&\mbox{$\forall(\widetilde{x},\widetilde{y})\in\widetilde{U}\times\widetilde{U}$,
$C>1$ is a constant},\end{split}$ (3)
$\begin{split}&\mbox{$b(\widetilde{x},\widetilde{y},k)\sim\sum^{+\infty}_{j=0}b_{j}(\widetilde{x},\widetilde{y})k^{n-j}$
in
$S^{n}(1;\widetilde{U}\times\widetilde{U},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$},\\\
&b_{j}(\widetilde{x},\widetilde{y})\in\mathcal{C}^{\infty}(\widetilde{U}\times\widetilde{U},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}),\
\ j=0,1,\ldots,\\\
&b_{0}(\widetilde{x},\widetilde{x})=\frac{1}{2}\pi^{-n-1}\lvert\det\mathcal{L}_{\widetilde{x}_{0}}\rvert\tau_{\widetilde{x}_{0},n_{-}}\,,\widetilde{x}_{0}\in\widetilde{U},\end{split}$
(4)

$F_{k}\in\mathcal{C}^{\infty}(\widetilde{U}\times\widetilde{U},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$
satisfies the following: for every $N_{1},N\in\mathbb{N}$, there is a constant
$C_{N_{1},N}$ independent of $k$ such that
$\left\|F_{k}(\widetilde{x},\widetilde{y})\right\|_{\mathcal{C}^{N_{1}}(\widetilde{U}\times\widetilde{U}}\leq
C_{N_{1},N}k^{-N}$, for every $k\gg 1$.

We refer the reader to [HM, Definition 8.1] for the meaning of semi-classical
symbol spaces
$S^{n}(1;\widetilde{U}\times\widetilde{U},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$.

As an easy consequence of the asymptotic expansion of the Szegő kernel along
the diagonal we also have the following theorem, see also Corollary $1.1$ in
[P].

######  Theorem 1.3.

Under the same assumptions and notations as in Theorem 1.2 above, assume that
$X=X_{\ell_{0}}\cup X_{\ell_{1}}\cup\cdots\cup X_{\ell_{u}}$,
$1\leq\ell_{0}<\ell_{1}<\cdots<\ell_{u}$, $\ell_{j}\in\mathbb{N}$,
$j=0,1,\ldots,u$, $u\in\mathbb{N}$. Denote by $p$ the least common multiple of
$l_{0},l_{1},\dots,l_{n}$. We have

$\lim_{k\rightarrow+\infty}\frac{1}{(kp)^{n}}{\rm dim\,}{\rm
Ker\,}\Box^{(q)}_{b,kp}=\frac{\ell_{0}}{2}\pi^{-n-1}\int_{X}\left|{\rm
det\,}\mathcal{L}_{x}\right|\mathrm{dV}_{X}.$

We work with the same assumptions and notations as in Theorem 1.2 above and
assume that $X$ is strongly pseudoconvex. Let $M$ be the Hodge orbifold
obtained by $M=X/S^{1}$. There exists a complex proper line orbibundle
$(L,h^{L})$ such that $X\rightarrow M$ is the circle orbibundle $X\subset
L^{*}$. For $k\in\mathbb{N}$, let $H^{0}(M,L^{k})$ be the space of global
holomorphic sections of $M$ with values in $L^{k}$. The space ${\rm
Ker\,}\Box^{(0)}_{b,k}$ is isomorphic to $H^{0}(M,L^{k})$. Let
$\left\\{f_{j}\right\\}^{d_{k}}_{j=1}\subset H^{0}(M,L^{k})$ be an orthonormal
basis. The Kodaira-Bailey map is given by

$\begin{split}\Phi_{k}:M&\rightarrow\mathbb{C}\mathbb{P}^{d_{k}-1},\\\
x&\rightarrow[f_{1}(x),\ldots,f_{d_{k}}(x)].\end{split}$ (5)

As an application of Theorem 1.2, we will give an analytic proof of the
following Kodaira-Bailey embedding theorem

######  Theorem 1.4.

With the same assumptions and notations as in Theorem 1.2 and suppose that $X$
is strongly pseudoconvex. Then, the Kodaira-Bailey map
$\Phi_{kp}:M\hookrightarrow\mathbb{C}\mathbb{P}^{d_{kp}-1}$ is an embedding
for $k$ sufficiently large, where $p$ is the least common multiple of
$l_{0},l_{1},\dots,l_{n}$.

It should be mention that Ma and Marinescu [MM] also gave a proof of Theorem
1.4.

We assume that $X$ admits a CR compact Lie group action $G$, ${\rm dim\,}G=d$.
We do not assume that $X$ admits a transversal and $S^{1}$ action. We will
work with Assumption 2.1 below and suppose that $X$ is strongly pseudoconvex.
We refer the reader to Section 2.3 and Section 4.3 for the notations and
terminology used here. We will assume that $0$ is a regular value of the
moment map $\mu$ and thus $\mu^{-1}(0)$ is a suborbifold of $X$. Put
$X_{G}:=\mu^{-1}(0)/G$. Note that both $X$ and $X_{G}$ are orbifolds. Let
$\Box^{(0)}_{b,X_{G}}$ be the Kohn Laplacian on the orbifold $X_{G}$. Let

$({\rm Ker\,}\Box^{(0)}_{b})^{G}:=\left\\{u\in{\rm
Ker\,}\Box^{(0)}_{b};\,g^{*}u=u,\forall g\in G\right\\}\,.$

As an application of Theorem 1.1, we generalize a CR version of quantization
commutes with reduction to orbifolds.

######  Theorem 1.5.

There is a map $\sigma:({\rm Ker\,}\Box^{(0)}_{b})^{G}\subset
L^{2}(X)\rightarrow{\rm Ker\,}\Box^{(0)}_{b,X_{G}}\subset L^{2}(X_{G})$ such
that $\sigma$ is Fredholm, that is,

  * (i)

${\rm Ker\,}\sigma\cap({\rm Ker\,}\Box^{(0)}_{b})^{G}$ is a finite dimensional
subspace of $\mathcal{C}^{\infty}(X)$,

  * (ii)

${\rm Coker\,}\sigma\cap{\rm Ker\,}\Box^{(0)}_{b,X_{G}}$ is a finite
dimensional subspace of $\mathcal{C}^{\infty}(X_{G})$.

We refer the reader to (47) for the explicit formula for $\sigma$.

Let us recall prior literature. First there is the paper [S], which deals with
the Szegő kernel on an orbifold circle bundle. The semi-classical behavior of
Bergman kernels on complex orbifolds was studied by Dai, Liu and Ma [DLM]. Our
approach is closer in spirit to [DLM] where an explicit expression for an
operator on a local chart $(\tilde{U},G_{U})\rightarrow U$ is given. This is
also motivated by previous results concerning $G$-invariant Szegő kernel for
locally free $G$-action on CR manifolds we obtained in [GH2], in this case the
CR reduction is an orbifold.

It is known that a compact connected strongly pseudoconvex CR orbifold $X$
with a transversal CR $S^{1}$ action can be identified with the circle
orbibundle of a positive holomorphic orbifold line bundle $(L,h)\rightarrow M$
over a compact complex Hodge orbifold $M$. By this identification the $k$-th
Fourier component of the Szegő kernel function and the Bergman kernel function
for the $k$-th tensor power of the line bundle $L$ are equal up to a constant
factor of $2\pi$. Hence in this paper we obtained a phase function version of
the Bergman kernels expansions on orbifolds (see [MM, Theorems 5.4.10, 5.4.11]
or [DLM, Theorem 1.4] for another version).

##  2 Preliminaries

###  2.1 Orbifolds

We recall basic definitions we need about orbifolds, for a more precise
discussion see [MM] and [ALR] and references therein. Define $\mathcal{M}_{s}$
to be the category whose objects are pairs $(M,G)$, where $M$ is a connected
smooth manifold and $G$ is a finite group acting smoothly on $M$, and whose
morphisms $\Phi\,:\,(M,G)\rightarrow(M^{\prime},G^{\prime})$ are families of
open embeddings $\varphi:\,M\rightarrow M^{\prime}$ such that for each
$\varphi\in\Phi$ there exists an injective group homomorphism
$\lambda_{\varphi}:G\rightarrow G^{\prime}$ satisfying

$\varphi(g\cdot x)=\lambda_{\varphi}(g)\cdot\varphi(x),\qquad x\in M,\,g\in
G\,.$

Let $X$ be a paracompact Hausdorff space and let $\mathcal{U}$ be a covering
of $X$ consisting of connected open subsets. Then we say that the topological
space $X$ has an orbifold structure if

  * i)

for each $U\in\mathcal{U}$ there exists
$(\widetilde{U},G_{U})\in\mathcal{M}_{s}$ and a ramified covering
$(\widetilde{U},G_{U})\rightarrow U$ such that $U\cong\widetilde{U}/G_{U}$;

  * ii)

for any $U,\,V\in\mathcal{U}$ there exists a morphism
$\Phi_{V,U}:(\widetilde{U},G_{U})\rightarrow(\widetilde{V},G_{V})$ which
covers the inclusion $U\subset V$ and satisfies
$\varphi_{WU}=\varphi_{WV}\circ\varphi_{VU}$ for any $U,\,V,\,W\in\mathcal{U}$
such that $U\subset V\subset W$.

We recall that for each $x\in X$ one can always find local coordinates
$(\widetilde{U},G_{U})$, $\widetilde{U}\subseteq\mathbb{R}^{n}$, such that $x$
is a fixed point of $G_{U}$ and $G_{U}$ acts linearly on $\mathbb{R}^{n}$,
$x\equiv 0\in\mathbb{R}^{n}$. If $|G_{U}\rvert>1$ then we say that $x$ is a
singular point, otherwise we say that $x$ is regular. There exists an open and
dense subset in $X$ on which all the points have conjugated stabilizers (see
[DK] p. 116), thus if the action of $G$ on $M$ in the definition of
$\mathcal{M}_{s}$ is assumed to be effective as in [MM], then set of regular
points is open and dense in $X$.

A vector orbibundle $(E,p)$ on an orbifold $X$ is an orbifold $E$ together
with a continuous projection $p:E\rightarrow X$ such that for
$U\in\mathcal{U}$ there exists a $G_{U}$-equivariant lift
$\widetilde{p}\,:\widetilde{E}_{U}\rightarrow\widetilde{U}$ defining a trivial
vector bundle such that the diagram

$\displaystyle(\widetilde{E}_{U},$ $\displaystyle
G^{E}_{U})\xrightarrow{\widetilde{p}}(\widetilde{U},G_{U})$
$\displaystyle\downarrow$ $\displaystyle\qquad\qquad\downarrow$ $\displaystyle
E_{U}$ $\displaystyle\quad\,\,\,\,\xrightarrow{p}\quad U$

commutes. As before, when we say that
$\widetilde{p}\,:\widetilde{E}_{U}\rightarrow\widetilde{U}$ is
$G_{U}$-equivariant we mean that there exists an injective group homomorphism
$\lambda_{p}:G^{E}_{U}\rightarrow G_{U}$ such that

$\widetilde{p}(g\cdot e)=\lambda_{p}(g)\cdot\widetilde{p}(e)\qquad
e\in\widetilde{E}_{U},\,g\in G_{U}^{E}\,;$

if $\lambda_{p}$ is a group isomorphism, we say that the vector orbibundle is
proper.

A smooth section of a vector orbibundle $(E,p)$ over $X$ is a smooth map
$s:X\rightarrow E$ such that $p\circ s=1_{X}$, we denote the space of smooth
sections as $\mathcal{C}^{\infty}(X,E)$. We recall that a smooth map
$f:X\rightarrow Y$ between orbifolds is a continuous map between the
underlying topological spaces such that for each $U\in\mathcal{U}$ there
exists a local lift $(\widetilde{f}_{U},\overline{f}_{U})$ such that
$\widetilde{f}_{U}:\widetilde{U}\rightarrow\widetilde{V}$ is smooth,
$\overline{f}_{U}:G_{U}\rightarrow G_{V}$ is a group homomorphism, the diagram

$\displaystyle(\widetilde{U},$ $\displaystyle
G_{U})\xrightarrow{\widetilde{f}}(\widetilde{V},G_{V})$
$\displaystyle\downarrow$ $\displaystyle\qquad\qquad\downarrow$ $\displaystyle
U$ $\displaystyle\quad\,\,\,\,\xrightarrow{f}\quad V$

commutes and $\widetilde{f}_{U}$ is $\overline{f}_{U}$-equivariant:

$\widetilde{f}_{U}(g\cdot x)=\overline{f}_{U}(g)\widetilde{f}_{U}(x)\,,\qquad
g\in G_{U},\,x\in\widetilde{U}\,.$

######  Proposition 2.1.

The tangent orbibundle $TX$ is defined by gluing together the bundles defined
over the charts

$T_{\widetilde{x}}\widetilde{U}\rightarrow
T_{x}U:=T_{\widetilde{x}}\widetilde{U}/G_{U}\,,$

where the action of $G_{U}$ is induced by differentiating the action on
$\widetilde{U}$, $\mathrm{d}g$. It has a natural structure of proper vector
orbibundle. The smooth sections of it are called vector fields.

We can then define the differential of a smooth map $f:X\rightarrow Y$ to be
the smooth map $\mathrm{d}f:TX\rightarrow TY$ such that locally for a given
$x\in U$, $f(x)\in V$ we have a commutative diagram

$\displaystyle(T_{\widetilde{x}}\widetilde{U},$ $\displaystyle
G_{U})\xrightarrow{\mathrm{d}_{\widetilde{x}}\widetilde{f}}(T_{\widetilde{f}(\widetilde{x})}\widetilde{V},G_{V})$
$\displaystyle\downarrow$ $\displaystyle\qquad\qquad\,\,\,\downarrow$
$\displaystyle T_{x}U$
$\displaystyle\quad\,\,\,\,\xrightarrow{\mathrm{d}_{x}f}\,\,T_{f(x)}V$

where $\mathrm{d}_{\widetilde{x}}\widetilde{f}$ is
$\overline{f}_{U}$-equivariant. We say that $f$ is an immersion at $x\in X$ if
$\mathrm{d}f$ is injective, so this means that there is a local lift
$\widetilde{f}_{U}:\widetilde{U}\rightarrow\widetilde{V}$ such that both
$\mathrm{d}_{\widetilde{x}}\widetilde{f}$ and $\overline{f}_{U}$ are
injective.

There are different ways to construct new vector bundles out of old, see
[MiSt] p. $29$ and also [ALR] p. $14$. Given any continuous functor
$\mathcal{F}$ from vector spaces to vector spaces we obtain a new orbibundle
$\mathcal{F}(TX)\rightarrow X$ with fibers
$\mathcal{F}(T_{\widetilde{x}}\widetilde{U})/G_{U}$. In particular, this
allows us to construct the cotangent orbibundle $T^{*}X$ and exterior powers
$\wedge^{q}T^{*}X$. We recall the definition of sub-orbibundle that we will
require later to define $T^{1,0}X$. A sub-orbibundle $(F,p_{|F})$ of a vector
orbibundle $(E,p)$ is a suborbifold of $E$ being a vector bundle on $X$. We
recall that a orbifold $Y$ is a suborbifold of an orbifold $X$ if there exists
an immersion $\iota:Y\rightarrow X$, we identify $Y$ with its image in $X$.

######  Example 2.1.

Consider an action of a finite group $G$ on $\mathbb{R}^{2n+1}$. A form on the
orbifold $\mathbb{R}^{2n+1}/G$ is a $G$-invariant form on $\mathbb{R}^{2n+1}$.

At this point we can give the definition of exterior derivative. Consider a
chart $(\widetilde{U},G_{U})\rightarrow U$ of an orbifold $X$, a $k$-form
$\omega$ defines a $G_{U}$-invariant $k$-form $\omega_{U}$ on $\widetilde{U}$,
locally this is given by

$\omega_{U}:=f_{U}\,\mathrm{d}x^{j_{1}}\wedge\mathrm{d}x^{j_{2}}\wedge\dots\wedge\mathrm{d}x^{j_{k}}\qquad\text{
on }\widetilde{U}\subseteq\mathbb{R}^{n}\,,$

where $f_{U}$ is a $G_{U}$-invariant function on $\widetilde{U}$. The exterior
derivative of $\omega$ is given locally by

$\mathrm{d}\omega_{U}:=\sum_{i=1}^{n}\frac{\partial f_{U}}{\partial
x^{i}}\mathrm{d}x^{i}\wedge\mathrm{d}x^{J}\text{ on
}\widetilde{U}\subseteq\mathbb{R}^{n}\,,$

where $J$ is the multi-index $J=(j_{1},\dots j_{n})$. It is easy to check that
it defines a $(k+1)$-form which is invariant under the action of $G_{U}$.

If $X$ is an oriented $n$ dimensional orbifold, we define the integral of a
smooth section $\omega$ of the orbibundle $\Lambda^{n}T^{*}X$ as follows.
Suppose that $\mathrm{supp}(\omega)$ is compactly supported on
$U\in\mathcal{U}$, we define

$\int_{U}\omega:=\frac{1}{\lvert
G_{U}\rvert}\int_{\widetilde{U}}\widetilde{\omega}_{U}\,.$

In the general case, for every open cover of $X$ by positively oriented
charts, $(U_{i},G_{U_{i}})$, we have a partition of unity, $(\rho_{i})_{i\in
I}$ subordinate to this cover. We define,

$\int_{X}\omega:=\sum_{i\in I}\frac{1}{\lvert
G_{U_{i}}\rvert}\int_{\widetilde{U}_{i}}\widetilde{\rho}_{i}\widetilde{\omega}_{U_{i}}\,.$
(6)

It remains to show that it does not depend on the choice of open cover and on
the choice of partition of unity. Now, given another open cover
$((V_{j},G_{V_{j}}))_{j\in J}$ and an associated partition of unity
$(\sigma_{j})_{j\in J}$, since
$\mathrm{supp}(\rho_{i}\sigma_{j}\omega)\subseteq U_{i}\cap V_{j}$ we have

$\frac{1}{\lvert
G_{U_{i}}\rvert}\int_{\widetilde{U}_{i}}\widetilde{\rho}_{i}\widetilde{\sigma}_{j,i}\widetilde{\omega}_{U_{i}}=\frac{1}{\lvert
G_{V_{j}}\rvert}\int_{\widetilde{V}_{j}}\widetilde{\rho}_{i,j}\widetilde{\sigma}_{j}\omega_{V_{j}}$

where $\widetilde{\sigma}_{j,i}$ is defined on $\widetilde{U}_{i}$ as the
composition between $\sigma_{j}:U_{i}\rightarrow\mathbb{R}$ and the covering
$\widetilde{U}_{i}\rightarrow U_{i}$, in a similar way
$\widetilde{\rho}_{i,j}$ is defined. Now,

$\displaystyle\sum_{i\in I}\frac{1}{\lvert
G_{U_{i}}\rvert}\int_{\widetilde{U}_{i}}\widetilde{\rho}_{i}\widetilde{\omega}_{U_{i}}$
$\displaystyle=\sum_{i\in I}\sum_{j\in J}\frac{1}{\lvert
G_{U_{i}}\rvert}\int_{\widetilde{U}_{i}}\widetilde{\rho}_{i}\widetilde{\sigma}_{j,i}\widetilde{\omega}_{U_{i}}$
$\displaystyle=\sum_{i\in I}\sum_{j\in J}\frac{1}{\lvert
G_{V_{j}}\rvert}\int_{\widetilde{V}_{j}}\widetilde{\rho}_{i,j}\widetilde{\sigma}_{j}\omega_{V_{j}}$
$\displaystyle=\sum_{j\in J}\frac{1}{\lvert
G_{V_{j}}\rvert}\int_{\widetilde{V}_{j}}\widetilde{\sigma}_{j}\widetilde{\omega}_{V_{j}}\,,$

where we use the fact that

$\sum_{j\in J}\widetilde{\sigma}_{j,i}=1,\quad\sum_{i\in
I}\widetilde{\rho}_{i,j}=1\,.$

###  2.2 CR orbifolds

In this section, we recall the definition of CR orbifolds. We also refer the
reader to [KHH, Section 5]. Let $X$ be a compact, connected and orientable
orbifold of dimension $2n+1$, $n\geq 1$, a CR structure on $X$ is a sub-
orbibundle $T^{1,0}X$ of rank $n$ of the complexified orbifold tangent bundle
$TX\otimes\mathbb{C}$ (which is defined locally for each $x\in X$ by taking
$T_{\tilde{x}}\widetilde{U}\otimes\mathbb{C}$), satisfying $T^{1,0}X\cap
T^{0,1}X=\\{0\\}$, where $T^{0,1}X=\overline{T^{1,0}X}$, and
$[\mathcal{V},\mathcal{V}]\subset\mathcal{V}$, where $\mathcal{V}$ is the
space of smooth sections of the orbibundle $T^{0,1}X$. We also fix a Hermitian
metric $\langle\cdot|\cdot\rangle$ on $TX\otimes\mathbb{C}$ so that
$T^{1,0}X\perp T^{0,1}X$, as before we mean that $\langle\cdot|\cdot\rangle$
is defined locally on each local chart $U$ as $G_{U}$-invariant Hermitian
metric on $T_{\tilde{x}}\widetilde{U}\otimes\mathbb{C}$; from now on when it
is clear how to define a geometric structure on an orbibundle we will not
always recall the definition. Furthermore, notice that every Hermitian metric
$h^{\prime}$ on $\widetilde{U}$ gives rise to a $G_{U}$-invariant metric
$h_{U}$ on $\widetilde{U}$ by averaging over the group:

$h_{{U}}(v,\,w):=\sum_{g\in
G_{U}}h^{\prime}(\mathrm{d}_{\widetilde{x}}g(v),\,\mathrm{d}_{\widetilde{x}}g(w)),\qquad
v,w\in T_{\widetilde{x}}\widetilde{U}\,.$

For $u\in TX\otimes\mathbb{C}$, we write $|u|^{2}:=\langle\,u\,|\,u\,\rangle$.

There is a unique sub-orbibundle $HX$ of $TX$ such that
$HX\otimes\mathbb{C}=T^{1,0}X\oplus T^{0,1}X$, i.e. $HX$ is the real part of
$T^{1,0}X\oplus T^{0,1}X$. Let $J:HX\rightarrow HX$ be the complex structure
map given by $J(u+\overline{u})=\imath u-\imath\overline{u}$, for every $u\in
T^{1,0}X$. By complex linear extension of $J$ to $TX\otimes\mathbb{C}$, the
$\imath$-eigenspace of $J$ is $T^{1,0}X\,=\,\left\\{V\in
HX\otimes\mathbb{C}\,;\,JV\,=\,\imath V\right\\}.$ We shall also write
$(X,HX,J)$ to denote a CR orbifold.

We fix a real non-vanishing $1$-form
$\omega_{0}\in\mathcal{C}^{\infty}(X,T^{*}X)$ so that
$\langle\,\omega_{0}(x)\,,\,u\,\rangle=0$, for every $u\in H_{x}X$, for every
$x\in X$. For each $x\in X$, we define a quadratic form on $HX$ by

$\mathcal{L}_{x}(U,V)=\frac{1}{2}\mathrm{d}\omega_{0}(JU,V),\qquad\forall\
U,V\in H_{x}X.$

We extend $\mathcal{L}$ to $HX\otimes\mathbb{C}$ by complex linear extension.
Then, for $U,V\in T^{1,0}_{x}X$,

$\mathcal{L}_{x}(U,\overline{V})=\frac{1}{2}\mathrm{d}\omega_{0}(JU,\overline{V})=-\frac{1}{2i}\mathrm{d}\omega_{0}(U,\overline{V}).$
(7)

The Hermitian quadratic form $\mathcal{L}_{x}$ on $T^{1,0}_{x}X$ is called
Levi form at $x$. This means that for each local coordinate patch
$(\widetilde{U},G_{U})\rightarrow U$ there exists a $G_{U}$-invariant
Hermitian quadratic form on $(T_{\widetilde{x}}\widetilde{U},G_{U})$. In this
paper, we always assume that the Levi form $\mathcal{L}$ on $T^{1,0}X$ is non-
degenerate of constant signature $(n_{-},n_{+})$ on $X$, where $n_{-}$ denotes
the number of negative eigenvalues of the Levi form and $n_{+}$ denotes the
number of positive eigenvalues of the Levi form. Let
$R\in\mathcal{C}^{\infty}(X,TX)$ be the non-vanishing vector field determined
by

$\begin{split}\omega_{0}(R)=-1,\quad\mathrm{d}\omega_{0}(R,\cdot)\equiv 0\ \
\mbox{on $TX$}.\end{split}$ (8)

Note that $X$ is a contact orbifold with contact form $\omega_{0}$, contact
plane $HX$ and $R$ is the Reeb vector field. We have
$\langle\,R\,|\,R\,\rangle=1$ and $R$ is orthogonal to $T^{1,0}X\oplus
T^{0,1}X$.

Denote by $T^{*1,0}X$ and $T^{*0,1}X$ the dual bundles of $T^{1,0}X$ and
$T^{0,1}X$, respectively. They can be identified with sub-orbibundles of the
complexified cotangent orbibundle $T^{*}X\otimes\mathbb{C}$. Define the vector
orbibundle of $(0,q)$-forms by $T^{*0,q}X:=\wedge^{q}T^{*0,1}X$. The Hermitian
metric $\langle\,\cdot\,|\,\cdot\,\rangle$ on $TX\otimes\mathbb{C}$ induces,
by duality, a Hermitian metric on $T^{*}X\otimes\mathbb{C}$ and also on the
orbibundles of $(0,q)$ forms $T^{*0,q}X,q=0,1,\cdots,n$. We shall also denote
all these induced metrics by $\langle\,\cdot\,|\,\cdot\,\rangle$. For $u\in
T^{*0,q}X$, we write $\left|u\right|^{2}:=\langle\,u\,|\,u\,\rangle$. Note
that we have the point-wise orthogonal decompositions:

$\begin{array}[]{c}T^{*}X\otimes\mathbb{C}=T^{*1,0}X\oplus
T^{*0,1}X\oplus\left\\{\lambda\omega_{0};\,\lambda\in\mathbb{C}\right\\},\\\
TX\otimes\mathbb{C}=T^{1,0}X\oplus T^{0,1}X\oplus\left\\{\lambda
R;\,\lambda\in\mathbb{C}\right\\}.\end{array}$

Let $U$ be an open set of $X$. Let $\Omega^{0,q}(U)$ denote the space of
smooth sections of $T^{*0,q}X$ over $D$ and let $\Omega^{0,q}_{c}(U)$ be the
subspace of $\Omega^{0,q}(U)$ whose elements have compact support in $U$; more
precisely an element $u$ in $\Omega^{0,q}_{c}(U)$ has a lift
$\tilde{u}\in\Omega^{0,q}_{c}(\tilde{U})$ which is $G_{U}$-invariant. The
metric on $TX\otimes\mathbb{C}$ induces a metric on
$\Lambda^{q}(T^{*}X\otimes\mathbb{C})$, so we have a well-defined orthogonal
projector

$\pi^{0,q}\,:\,\Lambda^{q}(T^{*}X\otimes\mathbb{C})\rightarrow T^{*0,q}X\,.$

We can define the tangential Cauchy-Riemann operator $\overline{\partial}_{b}$
by composing with the exterior differential

$\overline{\partial}_{b}=\pi^{0,q+1}\circ\mathrm{d}\,:\,\Omega^{0,q}(X)\rightarrow\Omega^{0,q+1}(X)\,,$

thus, on a local chart $(\widetilde{U},G_{U})\rightarrow U$,
$\overline{\partial}_{b}$ acts on $G_{U}$-invariant functions on
$\widetilde{U}$ and commutes with the action of $G_{U}$ on $\tilde{U}$. Let
$\mathrm{dV}_{X}(x)$ be the volume form on $X$ induced by the Hermitian metric
$\langle\,\cdot\,|\,\cdot\,\rangle$. For a local chart
$(\widetilde{U},G_{U})\rightarrow U$, let $\mathrm{dV}_{\widetilde{U}}(x)$ be
the volume form on $\widetilde{U}$ induced by the Hermitian metric
$\langle\,\cdot\,|\,\cdot\,\rangle$. The natural global $L^{2}$ inner product
$(\,\cdot\,|\,\cdot\,)$ on $\Omega^{0,q}(X)$ induced by $\mathrm{dV}_{X}(x)$
and $\langle\,\cdot\,|\,\cdot\,\rangle$ is given by

$(\,u\,|\,v\,):=\int_{X}\langle\,u(x)\,|\,v(x)\,\rangle\,\mathrm{dV}_{X}(x)\,,\quad
u,v\in\Omega^{0,q}(X)\,.$

We denote by $L^{2}_{(0,q)}(X)$ the completion of $\Omega^{0,q}(X)$ with
respect to $(\,\cdot\,|\,\cdot\,)$. It should be clear that the elements in
$L^{2}_{(0,q)}(X)$ are $L^{2}$ sections of the orbibundle
$\Lambda^{q}(T^{*}X\otimes\mathbb{C})$, the integral appearing in the
definition of the $L^{2}$ inner product $(\,\cdot\,|\,\cdot\,)$ is meant to be
interpreted as in the previous section, see (6). We extend
$(\,\cdot\,|\,\cdot\,)$ to $L^{2}_{(0,q)}(X)$ in the standard way. For $f\in
L^{2}_{(0,q)}(X)$, we denote $\left\|f\right\|^{2}:=(\,f\,|\,f\,)$. We extend
$\overline{\partial}_{b}$ to $L^{2}_{(0,r)}(X)$, $r=0,1,\ldots,n$, by

$\overline{\partial}_{b}:{\rm Dom\,}\overline{\partial}_{b}\subset
L^{2}_{(0,r)}(X)\rightarrow L^{2}_{(0,r+1)}(X)\,,$

where ${\rm Dom\,}\overline{\partial}_{b}:=\\{u\in
L^{2}_{(0,r)}(X);\,\overline{\partial}_{b}u\in L^{2}_{(0,r+1)}(X)\\}$ and, for
any $u\in L^{2}_{(0,r)}(X)$, $\overline{\partial}_{b}u$ is defined in the
sense of distributions. We also write

$\overline{\partial}^{*}_{b}:{\rm Dom\,}\overline{\partial}^{*}_{b}\subset
L^{2}_{(0,r+1)}(X)\rightarrow L^{2}_{(0,r)}(X)$

to denote the Hilbert space adjoint of $\overline{\partial}_{b}$ in the
$L^{2}$ space with respect to $(\,\cdot\,|\,\cdot\,)$. Let $\Box^{(q)}_{b}$
denote the (Gaffney extension) of the Kohn Laplacian given by

$\begin{array}[]{c}{\rm Dom\,}\Box^{(q)}_{b}=\Big{\\{}s\in
L^{2}_{(0,q)}(X);\,s\in{\rm Dom\,}\overline{\partial}_{b}\cap{\rm
Dom\,}\overline{\partial}^{*}_{b},\,\overline{\partial}_{b}s\in{\rm
Dom\,}\overline{\partial}^{*}_{b},\,\overline{\partial}^{*}_{b}s\in{\rm
Dom\,}\overline{\partial}_{b}\Big{\\}}\,,\\\
\Box^{(q)}_{b}s=\overline{\partial}_{b}\overline{\partial}^{*}_{b}s+\overline{\partial}^{*}_{b}\overline{\partial}_{b}s\>\>\text{for
$s\in{\rm Dom\,}\Box^{(q)}_{b}$}\,.\end{array}$ (9)

###  2.3 Group actions on orbifolds

Consider a compact connected Lie group action $G$. A smooth action of $G$ on
an orbifold $X$ is a continuous action $\cdot$ between the underlying
topological spaces such that for each $g\in G$ the map
$g\cdot\,:\,X\rightarrow X$ is smooth by mean of definition in Section 2.1:
this means that locally $(G_{U},\widetilde{U})\rightarrow U$, for each $g\in
G$, there exists a local lift $(\widetilde{g\cdot}_{U},\overline{g\cdot}_{U})$
such that $\widetilde{g\cdot}_{U}:\widetilde{U}\rightarrow\widetilde{V}$ is
smooth, $\overline{g\cdot}_{U}:G_{U}\rightarrow G_{V}$ is a group
homomorphism, the diagram

$\displaystyle(\widetilde{U},$ $\displaystyle
G_{U})\xrightarrow{\widetilde{g\cdot}}(\widetilde{V},G_{V})$
$\displaystyle\downarrow$ $\displaystyle\qquad\qquad\downarrow$ $\displaystyle
U$ $\displaystyle\quad\,\,\,\,\xrightarrow{g\cdot}\quad V$

commutes and $\widetilde{g\cdot}_{U}$ is $\overline{g\cdot}_{U}$-equivariant:

$\widetilde{g\cdot}_{U}(g\cdot
x)=\overline{g\cdot}_{U}(g)\widetilde{g\cdot}_{U}(x)\,,\qquad g\in
G_{U},\,x\in\widehat{U}\,.$

In this work, when $X$ admits a compact Lie group action $G$, we suppose that

######  Assumption 2.1.

We assume $g^{\ast}\omega_{0}=\omega_{0}$ on $X$ and $g_{\ast}J=Jg_{\ast}$ on
$HX$, for every $g\in G$, where $g^{*}$ and $g_{*}$ denote the pull-back map
and push-forward map of $G$, respectively.

Let $\mathfrak{g}$ denote the Lie algebra of $G$ and $e$ denotes the identity
element in $G$. For a moment denote by $f:G\times X\rightarrow X$ the action,
so that $f(g,x)=g\cdot x$. Fix $x\in X$, define

$\xi_{X}(x)=\mathrm{d}_{e}f(\cdot,x)\left(\xi\right)\,.$

For any $\xi\in\mathfrak{g}$, $\xi_{X}$ is the so called vector field on $X$
induced by $\xi$.

By the motivation explained in the introduction, in Theorem 1.2, we assume
that $X$ admits a CR and transversal $S^{1}$-action which is locally free,
$T\in\mathcal{C}^{\infty}(X,\,TX)$ denotes the global real vector field given
by the infinitesimal circle action. In this case, we will take $T$ to be our
Reeb vector field $R$. Note that in Theorem 1.1, we do not assume that $X$
admits a group action.

###  2.4 Operators on orbifolds

We first introduce some notations. Let $U\subset X$ be an open set and let $E$
be a vector orbibundle over $U$. As smooth case, let
$\mathcal{D}^{\prime}(U,E)$ denote the space of distribution sections over $U$
and let $\mathcal{E}^{\prime}(U,E)$ denote the space of distribution sections
over $U$ whose elements have compact support in $U$. Let
$\mathcal{C}^{\infty}(U,E)$ denote the space of smooth sections over $U$ and
let $\mathcal{C}^{\infty}_{c}(U,E)$ denote the space of smooth sections over
$U$ whose elements have compact support in $U$. For $s\in\mathbb{R}$, let
$H^{s}_{{\rm comp\,}}(U,E)$ denote the Sobolev space of sections over $U$ of
order $s$ whose elements have compact support in $U$. Put

$H^{s}_{{\rm loc\,}}(U,E):=\left\\{u\in\mathcal{D}^{\prime}(U,E);\,\chi u\in
H^{s}_{{\rm comp\,}}(U,E),\forall\chi\in\mathcal{C}^{\infty}_{c}(U)\right\\}.$

Let $H^{s}(X,E)$ denote the Sobolev space of sections over $X$ of order $s$.
Let $F$ be anther vector orbibundle over $U$. We write $E\boxtimes F^{*}$ to
denote the vector orbibundle over $U\times U$ with fiber over $U\times U$
consisting of the linear maps from $F_{y}$ to $E_{x}$.

Let $H:\mathcal{C}^{\infty}_{c}(U,E)\rightarrow\mathcal{D}^{\prime}(U,E)$ be a
continuous operator and let $H(x,y)\in\mathcal{D}^{\prime}(U\times
U,E\boxtimes E^{*})$ be the distribution kernel of $H$. We say that $H$ is
smoothing on $U$ if $H(x,y)\in\mathcal{C}^{\infty}(U\times U,E\boxtimes
E^{*})$. When $H$ is smoothing on $U$, we write $H\equiv 0$ (on $U$) or
$H(x,y)\equiv 0$ (on $U$). More precisely, let
$(\widetilde{U},G_{U})\rightarrow U$ be an orbifold chart. $H\equiv 0$ means
that we can find a
$\widetilde{H}(x,y)\in\mathcal{C}^{\infty}(\widetilde{U}\times\widetilde{U},\widetilde{E}\boxtimes\widetilde{E}^{*})$
such that

$H(x,y)=\sum_{h,g\in
G_{U}}\frac{1}{\left|G_{U}\right|}\widetilde{H}(h\cdot\widetilde{x},g\cdot\widetilde{y}),$

for all $(x,y)\in U\times U$, where $\pi(\tilde{x})=x$, $\pi(\tilde{y})=y$,
$\pi:\tilde{U}\rightarrow U$ is the natural projection and $\widetilde{E}$ is
the lifting of $E$. Let $G(x,y)\in\mathcal{D}^{\prime}(U\times U,E\boxtimes
E^{*})$. We will write $G$ to denote the continuous operator
$G:\mathcal{C}^{\infty}_{c}(U,E)\rightarrow\mathcal{D}^{\prime}(U,E)$ with
distribution kernel $G(x,y)$.

Let
$H:\mathcal{C}^{\infty}(X,T^{*0,q}X)\rightarrow\mathcal{D}^{\prime}(X,T^{*0,q}X)$
be a continuous operator. We say tat $H$ is a complex Fourier integral
operator if let $(\widetilde{U},G_{U})\rightarrow U$ be any orbifold chart of
$X$, there is a complex Fourier integral operator
$\widetilde{H}_{U}:\Omega^{0,q}(\tilde{U})\rightarrow\Omega^{0,q}(\tilde{U})$
has a distribution kernel
$\widetilde{H}_{U}(\widetilde{x},\widetilde{y})\in\mathcal{D}^{\prime}(\widetilde{U}\times\widetilde{U},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$
such that

${H}(x,y)\equiv\frac{1}{\left|G_{U}\right|}\sum_{g,h\in
G_{U}}\widetilde{H}_{{U}}(h\cdot\widetilde{x},g\cdot\widetilde{y})$

for $(x,y)\in U\times U$, where $\pi(\tilde{x})=x$, $\pi(\tilde{y})=y$,
$\pi:\tilde{U}\rightarrow U$ is the natural projection.

We recall Hörmander symbol space. Since we work on a given
$\widetilde{U}\subset\mathbb{R}^{2n+1}$, we use notation $x$ for an element
$x\in\widetilde{U}$ instead of $\tilde{x}$. Let
$\widetilde{U}\subset\mathbb{R}^{2n+1}$ be a local coordinate patch with local
coordinates $x=(x_{1},\ldots,x_{2n+1})$. We recall the following definition.

######  Definition 2.1.

For $m\in\mathbb{R}$,
$S^{m}_{1,0}(\widetilde{U}\times\widetilde{U}\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$
is the space of all smooth functions
$a(x,y,t)\in\mathcal{C}^{\infty}(\widetilde{U}\times\widetilde{U}\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$
such that, for all compact $K\Subset\widetilde{U}\times\widetilde{U}$ 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

$\left|\partial^{\alpha}_{x}\partial^{\beta}_{y}\partial^{\gamma}_{t}a(x,y,t)\right|\leq
C_{\alpha,\beta,\gamma}(1+\left|t\right|)^{m-\gamma},\ \ \mbox{for every
$(x,y,t)\in K\times\mathbb{R}_{+},t\geq 1$}.$

Put

$S^{-\infty}(\widetilde{U}\times\widetilde{U}\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}):=\bigcap_{m\in\mathbb{R}}S^{m}_{1,0}(\widetilde{U}\times\widetilde{U}\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}).$

Let $a_{j}\in
S^{m_{j}}_{1,0}(\widetilde{U}\times\widetilde{U}\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$,
$j=0,1,2,\ldots$ with $m_{j}\rightarrow-\infty$, as $j\rightarrow\infty$. Then
there exists $a\in
S^{m_{0}}_{1,0}(\widetilde{U}\times\widetilde{U}\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$
unique modulo $S^{-\infty}$, such that $a-\sum^{k-1}_{j=0}a_{j}\in
S^{m_{k}}_{1,0}(\widetilde{U}\times\widetilde{U}\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}\big{)}$
for $k=0,1,2,\ldots$.

If $a$ and $a_{j}$ have the properties listed above, then we write
$a\sim\sum^{\infty}_{j=0}a_{j}$ (in the space
$S^{m_{0}}_{1,0}\big{(}\tilde{U}\times\tilde{U}\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}\big{)}$
). Furthermore, we write

$s(x,y,t)\in S^{m}_{{\rm
cl\,}}(\widetilde{U}\times\widetilde{U}\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$

if $s(x,y,t)\in
S^{m}_{1,0}(\widetilde{U}\times\widetilde{U}\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$
and

$\begin{array}[]{lc}&s(x,y,t)\sim\sum^{\infty}_{j=0}s^{j}(x,y)t^{m-j}\text{ in
}S^{m}_{1,0}(\widetilde{U}\times\widetilde{U}\times\mathbb{R}_{+}\,,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})\,,\\\
&s^{j}(x,y)\in\mathcal{C}^{\infty}(\widetilde{U}\times\widetilde{U},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}),\
j\in\mathbb{N}_{0}.\end{array}$

We simply write $S^{m}_{1,0}$ to denote
$S^{m}_{1,0}(\widetilde{U}\times\widetilde{U}\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$,
$m\in\mathbb{R}\cup\left\\{-\infty\right\\}$.

Let $\widetilde{E}$ be a vector bundle over $\widetilde{U}$. Let
$m\in\mathbb{R}$, $0\leq\rho,\delta\leq 1$. Let
$S^{m}_{\rho,\delta}(T^{*}\widetilde{U},\widetilde{E})$ denote the Hörmander
symbol space on $T^{*}\widetilde{U}$ with values in $\widetilde{E}$ of order
$m$ type $(\rho,\delta)$ and let $S^{m}_{{\rm
cl\,}}(T^{*}\widetilde{U},\widetilde{E})$ denote the space of classical
symbols on $T^{*}\widetilde{U}$ with values in $\widetilde{E}$ of order $m$.
Let
$L^{m}_{\rho,\delta}(\widetilde{U},\widetilde{E}\boxtimes\widetilde{E}^{*})$
($L^{m}_{{\rm cl\,}}(\widetilde{U},\widetilde{E}\boxtimes\widetilde{E}^{*})$)
be the space of pseudodifferential operators on $\widetilde{U}$ of order $m$
from sections of $E$ to sections of $E$ with full symbol $a\in
S^{m}_{\rho,\delta}(T^{*}\widetilde{U},\widetilde{E})$ ($a\in S^{m}_{{\rm
cl\,}}(T^{*}\widetilde{U},\widetilde{E})$). Let $E$ be a vector orbibundle
over $X$. Let
$A:\mathcal{C}^{\infty}(X,E)\rightarrow\mathcal{D}^{\prime}(X,E)$ be a
continuous operator with distribution kernel
$A(x,y)\in\mathcal{D}^{\prime}(X\times X,E\boxtimes E^{*})$. We write $A\in
L^{m}_{\rho,\delta}(X,E\otimes E^{*})$ ($A\in L^{m}_{{\rm cl\,}}(X,E\otimes
E^{*})$) if for any orbifold chart $(\widetilde{U},G_{U})\rightarrow U$, there
is a $\widetilde{A}_{U}\in
L^{m}_{\rho,\delta}(\widetilde{U},\widetilde{E}\boxtimes\widetilde{E}^{*})$
($\widetilde{A}_{U}\in L^{m}_{{\rm
cl\,}}(\widetilde{U},\widetilde{E}\boxtimes\widetilde{E}^{*})$) such that

$A(x,y)\equiv\frac{1}{\left|G_{U}\right|}\sum_{g,h\in
G_{U}}\widetilde{A}_{{U}}(h\cdot\widetilde{x},g\cdot\widetilde{y})$

for $(x,y)\in U\times U$, where $\pi(\tilde{x})=x$, $\pi(\tilde{y})=y$,
$\pi:\tilde{U}\rightarrow U$ is the natural projection and $\widetilde{E}$ is
the lifting of $E$.

###  2.5 Microlocal Szegő kernel

On each open set $U\subseteq X$, we define operators $S^{(q)}_{U}$ having the
same microlocal structure described in our main results. We call this
operators approximate Szegő kernels. We shall review some results in [Hs]
concerning the existence of a microlocal Hodge decomposition of the Kohn
Laplacian on an open set of a CR manifold where the Levi form is non-
degenerate. Since the proofs are local we can carry out the same analysis on
the chart $(G_{U},\widetilde{U})$ for a given open set $U$. Since we work on a
fix $\widetilde{U}\subset\mathbb{R}^{2n+1}$, we use notation $x$ for an
element $x\in\widetilde{U}$. For the following theorems we refer to Chapter 6,
Chapter 7 and Chapter 8 of Part I in [Hs].

######  Theorem 2.1.

We assume that the $G_{U}$-invariant Levi form is non-degenerate of constant
signature $(n_{-},n_{+})$ at each point of $\widetilde{U}$. Let $q\neq
n_{-},n_{+}$. Then, there is a properly supported operator
$A_{\widetilde{U}}\in
L^{-1}_{\frac{1}{2},\frac{1}{2}}(\widetilde{U},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$
such that $\Box^{(q)}_{b}A_{\widetilde{U}}=I+F_{\widetilde{U}}$ on
$\widetilde{U}$, $F_{\widetilde{U}}$ is a smoothing operator on
$\widetilde{U}$.

Until further notice, we assume that the Levi form is non-degenerate of
constant signature $(n_{-},n_{+})$ on $X$. Let $p_{0}(x,\xi)\in
C^{\infty}(T^{*}X)$ be the principal symbol of $\Box^{(q)}_{b}$. Note that
$p_{0}(x,\xi)$ is a $G_{U}$-invariant polynomial of degree $2$ in $\xi$ when
written locally on $\widetilde{U}$. Recall that the characteristic manifold of
$\Box^{(q)}_{b}$ is given by $\Sigma=\Sigma^{+}\cup\Sigma^{-}$, where
$\Sigma^{+}$ and $\Sigma^{-}$ are given by

$\Sigma^{+}=\left\\{(x,\lambda\,\omega_{0}(x))\in
T^{*}X;\,\lambda>0\right\\},\>\>\Sigma^{-}=\left\\{(x,\lambda\,\omega_{0}(x))\in
T^{*}X;\,\lambda<0\right\\},$ (10)

where $\omega_{0}\in\mathcal{C}^{\infty}(X,T^{*}X)$ is the uniquely determined
global contact $1$-form.

######  Theorem 2.2.

With the same assumptions as in the previous theorem, let $q=n_{-}$ or
$n_{+}$. Then, by adopting notation as in Section 2.4, there exist properly
supported continuous operators $A\in
L^{-1}_{\frac{1}{2},\frac{1}{2}}(\widetilde{U},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$,
and operators $S_{-},S_{+}\in
L^{0}_{\frac{1}{2},\frac{1}{2}}(\tilde{U},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$,
such that

$\begin{split}&\mbox{$\Box^{(q)}_{b}A+S_{-}+S_{+}=I$ on $\widetilde{U}$},\\\
&\Box^{(q)}_{b}S_{-}\equiv 0\ \ \mbox{on $\widetilde{U}$},\ \
\Box^{(q)}_{b}S_{+}\equiv 0\ \ \mbox{on $\tilde{U}$},\\\ &A\equiv A^{*}\ \
\mbox{on $\widetilde{U}$},\ \ S_{-}A\equiv 0\ \ \mbox{on $\widetilde{U}$},\ \
S_{+}A\equiv 0\ \ \mbox{on $\widetilde{U}$},\\\ &S_{-}\equiv S_{-}^{*}\equiv
S_{-}^{2}\ \ \mbox{on $\widetilde{U}$},\\\ &S_{+}\equiv S_{+}^{*}\equiv
S_{+}^{2}\ \ \mbox{on $\widetilde{U}$},\\\ &S_{-}S_{+}\equiv S_{+}S_{-}\equiv
0\ \ \mbox{on $\widetilde{U}$},\end{split}$ (11)

where $A^{*}$, $S_{-}^{*}$ and $S_{+}^{*}$ are the formal adjoints of $A$,
$S_{-}$ and $S_{+}$ with respect to $(\,\cdot\,|\,\cdot\,)$ respectively and
$S_{-}(x,y)$ satisfies

$S_{-}(x,y)\equiv\int^{\infty}_{0}e^{i\varphi_{-}(x,y)t}s_{-}(x,y,t)dt\ \
\mbox{on $\widetilde{U}$}$

with a symbol $s_{-}(x,y,t)\in S^{n}_{{\rm
cl\,}}\big{(}\widetilde{U}\times\widetilde{U}\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}\big{)}$
such that

$\begin{split}&s_{-}(x,y,t)\sim\sum^{\infty}_{j=0}s^{j}_{-}(x,y)t^{n-j}\quad\text{
in
}S^{n}_{1,0}\big{(}\widetilde{U}\times\widetilde{U}\times\mathbb{R}_{+}\,,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}\big{)}\,,\\\
&s^{j}_{-}(x,y)\in
C^{\infty}\big{(}\widetilde{U}\times\widetilde{U},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}\big{)},\
\ j\in\mathbb{N}_{0},\end{split}$ (12)

and phase function $\varphi_{-}$ such that $\varphi=\varphi_{-}$ satisfies

$\begin{split}&\varphi\in\mathcal{C}^{\infty}(\widetilde{U}\times\widetilde{U}),\
\ {\rm Im\,}\varphi(x,y)\geq 0,\\\ &\varphi(x,x)=0,\ \ \varphi(x,y)\neq 0\ \
\mbox{if}\ \ x\neq y,\\\ &d_{x}\varphi(x,y)\big{|}_{x=y}=-\omega_{0}(x),\ \
d_{y}\varphi(x,y)\big{|}_{x=y}=\omega_{0}(x),\\\
&\varphi(x,y)=-\overline{\varphi}(y,x).\end{split}$ (13)

Moreover, there is a function
$f\in\mathcal{C}^{\infty}(\widetilde{U}\times\widetilde{U})$ such that

$p_{0}(x,\varphi^{\prime}_{x}(x,y))-f(x,y)\varphi(x,y)$ (14)

vanishes to infinite order at $x=y$. Similarly,

$S_{+}(x,y)\equiv\int^{\infty}_{0}\\!\\!e^{i\varphi_{+}(x,y)t}s_{+}(x,y,t)dt\
\ \mbox{on $\widetilde{U}$}$

with $s_{+}(x,y,t)\in S^{n}_{{\rm
cl\,}}\big{(}\widetilde{U}\times\widetilde{U}\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}\big{)}$
satisfying

$s_{+}(x,y,t)\sim\sum^{\infty}_{j=0}s^{j}_{+}(x,y)t^{n-j}$ (15)

in
$S^{n}_{1,0}\big{(}\widetilde{U}\times\widetilde{U}\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}\big{)}$,

$s^{j}_{+}(x,y)\in\mathcal{C}^{\infty}\big{(}\widetilde{U}\times\widetilde{U},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}\big{)},\
\ j\in\mathbb{N}_{0},$

and $-\overline{\varphi}_{+}(x,y)$ satisfies (13) and (14). Moreover, if
$q\neq n_{+}$, then $s_{+}(x,y,t)$ vanishes to infinite order at $x=y$. If
$q\neq n_{-}$, then $s_{-}(x,y,t)$ vanishes to infinite order at $x=y$.

The operators $S_{+}$, $S_{-}$ are called _approximate Szegő kernels_.

######  Remark 2.1.

With the notations and assumptions used in Theorem 2.2, furthermore suppose
that $q=n_{-}\neq n_{+}$. Since $s_{+}(x,y,t)$ vanishes to infinite order at
$x=y$, we have $S_{+}\equiv 0$ on $\widetilde{U}$. Similarly, if $q=n_{+}\neq
n_{-}$, then $S_{-}\equiv 0$ on $\widetilde{U}$.

We pause and introduce some notations. For a given point $p\in\widetilde{U}$,
let $\\{W_{j}\\}_{j=1}^{n}$ be an orthonormal frame of
$(T^{1,0}\widetilde{U},\langle\,\cdot\,|\,\cdot\,\rangle)$ near $p$, for which
the Levi form is diagonal at $p$. Put

$\mathcal{L}_{p}(W_{j},\overline{W}_{\ell})=\mu_{j}(p)\delta_{j,\ell}\,,\;\;j,\ell=1,\ldots,n\,.$

We will denote by

$\det\mathcal{L}_{p}=\prod_{j=1}^{n}\mu_{j}(p)\,.$ (16)

Let $\\{e_{j}\\}_{j=1}^{n}$ denote the basis of $T^{*0,1}\widetilde{U}$, dual
to $\\{\overline{W}_{j}\\}^{n}_{j=1}$. We assume that $\mu_{j}(p)<0$ if $1\leq
j\leq n_{-}$ and $\mu_{j}(p)>0$ if $n_{-}+1\leq j\leq n$. Put

$\begin{array}[]{ll}&\mathcal{N}(p,n_{-}):=\left\\{ce_{1}(p)\wedge\ldots\wedge
e_{n_{-}}(p);\,c\in\mathbb{C}\right\\},\\\
&\mathcal{N}(p,n_{+}):=\left\\{ce_{n_{-}+1}(p)\wedge\ldots\wedge
e_{n}(p);\,c\in\mathbb{C}\right\\}\end{array}$

and let

$\tau_{p,n_{-}}:T^{*0,q}_{p}\widetilde{U}\rightarrow\mathcal{N}(p,n_{-})\,,\quad\tau_{p,n_{+}}:T^{*0,n-q}_{p}\widetilde{U}\rightarrow\mathcal{N}(p,n_{+})\,,$
(17)

is be the orthogonal projections onto $\mathcal{N}(p,n_{-})$ and
$\mathcal{N}(p,n_{+})$ with respect to $\langle\,\cdot\,|\,\cdot\,\rangle$,
respectively. For $J=(j_{1},\ldots,j_{q})$, $1\leq j_{1}<\cdots<j_{q}\leq n$,
let $e_{J}:=e_{j_{1}}\wedge\cdots\wedge e_{j_{q}}$. For
$\left|I\right|=\left|J\right|=q$, $I$, $J$ are strictly increasing, let
$e_{I}\otimes(e_{J})^{*}$ be the linear transformation from $T^{*0,q}X$ to
$T^{*0,q}X$ given by

$(e_{I}\otimes(e_{J})^{*})(e_{K})=\delta_{J,K}e_{I},$

for every $\left|K\right|=q$, $K$ is strictly increasing, where
$\delta_{J,K}=1$ if $J=K$, $\delta_{J,K}=0$ if $J\neq K$. For any $f\in
T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}$, we have

$f=\sideset{}{{}^{\prime}}{\sum}_{\left|I\right|=\left|J\right|=q}c_{I,J}e_{I}\otimes(e_{J})^{*},$

$c_{I,J}\in\mathbb{C}$, for all $\left|I\right|=\left|J\right|=q$, $I$, $J$
are strictly increasing, where $\sum^{\prime}$ means that the summation is
performed only over strictly increasing multi-indices. We call
$c_{I,J}e_{I}\otimes(e_{J})^{*}$ the component of $f$ in the direction
$e_{I}\otimes(e_{J})^{*}$. Let $I_{0}=(1,2,\ldots,q)$. We can check that

$\tau_{p,n_{-}}=e_{I_{0}}(p)\otimes(e_{I_{0}}(p))^{*}.$

The following formula for the leading term $s^{0}_{-}$ on the diagonal follows
from [Hs, §9]. The formula for the leading term $s^{0}_{+}$ on the diagonal
follows similarly.

######  Theorem 2.3.

Let $q=n_{-}$. For the leading term $s^{0}_{-}(x,y)$ of the expansion (12) of
$s_{-}(x,y,t)$, we have

$s^{0}_{-}(x_{0},x_{0})=\frac{1}{2}\pi^{-n-1}\left|\det\mathcal{L}_{x_{0}}\right|\tau_{x_{0},n_{-}}\,,\>\>x_{0}\in\widetilde{U}.$

Let $q=n_{+}$. For the leading term $s^{0}_{+}(x,y)$ of the expansion (15) of
$s_{+}(x,y,t)$, we have

$s^{0}_{+}(x_{0},x_{0})=\frac{1}{2}\pi^{-n-1}\left|\det\mathcal{L}_{x_{0}}\right|\tau_{x_{0},n_{-}}\,,\>\>x_{0}\in\widetilde{U}.$

###  2.6 Geometric conditions for the closed range property

We recall that, given $q\in\\{0,\ldots,n\\}$, the Levi form is said to satisfy
condition $Y(q)$ at $p\in X$, if $\mathcal{L}_{p}$ has at least either
$\min{(q+1,n-q)}$ pairs of eigenvalues with opposite signs or
$\max{(q+1,n-q)}$ eigenvalues of the same sign.

######  Remark 2.2.

Assume that the Levi form is non degenerate of constant signature
$(n_{-},n_{+})$. Given $q\in\\{0,\,\dots,\,n\\}$, then $Y(q)$ holds if
$q\notin\\{n_{-},\,n_{+}\\}$. Furthermore, notice that if $q=n_{-}=n_{+}$,
$q-1\notin\\{n_{-},n_{+}\\}$ and $q+1\notin\\{n_{-},n_{+}\\}$, then $Y(q-1)$
and $Y(q+1)$ hold. Eventually, we remark that if $q=n_{-}$ and $\lvert
n_{-}-n_{+}\rvert>1$, then then $Y(q-1)$ and $Y(q+1)$ hold. In particular if
$X$ is a strongly pseudoconvex CR orbifold of dimension greater than $5$, then
$Y(1)$ holds.

We can repeat the same proof as in manifold case with small modification and
get the following:

######  Theorem 2.4.

Let $X$ be a compact CR orbifold and suppose that $Y(q)$ holds. Then,
$\Box^{(q)}_{b}$ has closed range.

Furthermore, suppose that $Y(q)$ fails but $Y(q-1)$ and $Y(q+1)$ hold, then
$\Box^{(q)}_{b}$ has closed range.

In the following, we will give a proof of the first pat of Theorem 2.4 under
the assumption that the Levi form is non-degenerate of constant signature
$(n_{-},n_{+})$. Since the proof is related to the construction of
parametrices for Kohn Lapalcians on orbifolds, therefore we think it is worth
to give a proof under the assumption that the Levi form is non-degenerate of
constant signature $(n_{-},n_{+})$. We suppose that $Y(q)$ holds. Then,
$q\notin\left\\{n_{-},n_{+}\right\\}$. We shall prove:

$\exists\,C>0\quad\text{such that}\quad\lVert\square^{(q)}_{b}u\rVert\geq
C\,\lVert u\rVert\quad\forall u\perp\operatorname{Ker}\square^{(q)}_{b}\,;$
(18)

which is equivalent to prove that $\square^{(q)}_{b}$ has closed range, see
Lemma C.1.1 in [MM]. We need the following lemma.

######  Lemma 2.1.

There exists a continuous operator

$A:H^{s}(X,T^{*0,q}X)\rightarrow H^{s+1}(X,T^{*0,q}X),\ \ \forall
s\in\mathbb{R},$

such that $\square^{(q)}_{b}A=I+F$ where $F$ is smoothing on $X$.

###### Proof of Lemma 2.1.

Fix an orbifold chart $(G_{U},\tilde{U})\rightarrow U$. By Theorem 2.1, we can
define on $\widetilde{U}$,

$\hat{A}_{U}(x,y)=\sum_{g,h\in G_{U}}\frac{1}{\lvert
G_{U}\rvert}A_{\widetilde{U}}(h\cdot x,g\cdot y)\quad\text{ and
}\quad\hat{F}_{U}(x,y)=\sum_{g,h\in G_{U}}\frac{1}{\lvert
G_{U}\rvert}F_{\widetilde{U}}(h\cdot x,g\cdot y),$

so that $\hat{A}_{U}:H^{s}_{{\rm comp\,}}(\widetilde{U},T^{*0,q}X)\rightarrow
H^{s+1}_{{\rm loc\,}}(\widetilde{U},T^{*0,q}X)$ is continuous, for all
$s\in\mathbb{R}$, $\hat{F}_{U}$ is smoothing and they satisfy the following

$\Box^{(q)}_{b}\hat{A}_{U}=I+\hat{F}_{U}\quad\text{ on }\quad U\,,$

where $\hat{A}_{U}$ and $\hat{F}_{U}$ are as in Theorem 2.1. Now, since $X$ is
compact we can consider an open cover $\\{U_{j}\\}_{j\in J}$,
$(\widetilde{U}_{j},G_{U_{j}})\rightarrow U_{j}$ orbifold chart, for every
$j$, and a partition of unity $\chi_{j}\in\mathcal{C}_{c}^{\infty}(U_{j})$ so
that we have

$\Box^{(q)}_{b}\hat{A}_{U_{j}}\chi_{j}=\chi_{j}+\hat{F}_{U_{j}}\chi_{j}\quad\text{
on }\quad U_{j}\,.$

Eventually, set $\hat{A}=\sum_{j}\hat{A}_{j}\chi_{j}$ and
$\hat{F}=\sum_{j}\hat{F}_{j}\chi_{j}$, we can check

$\Box^{(q)}_{b}\hat{A}=I+\hat{F}\,,$

globally on $X$ and we get the lemma. ∎

We are ready to prove (18). Suppose by the contrary that (18) does not hold.
Then there exists a sequence of $(0,q)$ forms such that $u_{j}\in
L^{2}_{(0,q)}(X)$, $j=1,2,\ldots$, with $\lVert u_{j}\rVert=1$ and
$u_{j}\perp\operatorname{Ker}\square^{(q)}_{b}$ such that
$\lVert\square^{(q)}_{b}u_{j}\rVert<\frac{1}{j}\lVert u_{j}\rVert$ for each
$j=1,2\dots$. By Lemma 2.1, we have

$\hat{A}^{*}\square_{b}^{(q)}u_{j}=u_{j}+\hat{F}^{*}u_{j},\ \ j=1,2,\ldots,$

where $\hat{A}^{*}$ and $\hat{F}^{*}$ are adjoints of $\hat{A}$ and $\hat{F}$
respectively. Since $\hat{A}^{*}\,:\,L^{2}_{(0,q)}(X)\rightarrow
H^{1}(X,T^{*0,1}X)$ is continuous, we get

$\lVert
u_{j}\rVert_{1}-\lVert\hat{F}^{*}u_{j}\rVert_{1}\leq\lVert\hat{A}^{*}\square^{(q)}_{b}u_{j}\rVert_{1}\leq
C\lVert\square_{b}^{(q)}u_{j}\rVert\leq\frac{C}{j}\lVert u_{j}\rVert\,,$

for all $j=1,2,\ldots$, where $\left\|\cdot\right\|_{1}$ denotes the Sobolev
norm of order one and $C>0$ is a constant independent of $u_{j}$. Since
$\hat{F}:L^{2}_{(0,q)}(X)\rightarrow H^{1}(X,T^{*0,q}X)$ is continuous, we get

$\lVert u_{j}\rVert_{1}\leq C^{\prime},\ \ \mbox{for all $j=1,2,\ldots$},$

where $C^{\prime}>0$ is a constant independent of $j$. By Rellich’s Lemma,
there is a subsequence $1\leq j_{1}<j_{2}<\dots$ such that
$u_{j_{s}}\rightarrow u$ in $L^{2}_{(0,q)}(X)$ as $s\rightarrow+\infty$. Since
$u_{j}\perp{\rm Ker\,}\square^{(q)}_{b}$ for each $j$, then $u\perp{\rm
Ker\,}\square^{(q)}_{b}$; but
$\square^{(q)}_{b}u=\lim_{s}\square^{(q)}_{b}u_{j_{s}}=0$ and thus we get a
contradiction.

We can repeat the proof of [HM, Theoredm 6.24] with minor change and get the
following

######  Theorem 2.5.

Assume that the Levi form is non-degenerate of constant signature
$(n_{-},n_{+})$ on $X$. Suppose that $X$ admits a transversal and CR
$\mathbb{R}$-action. For any $q\in\left\\{0,1,\ldots,n\right\\}$,

$\Box^{(q)}_{b}:{\rm Dom\,}\Box^{(q)}_{b}\subset L^{2}_{(0,q)}(X)\rightarrow
L^{2}_{(0,q)}(X)$

has closed range.

##  3 Proof of Theorem 1.1

Let

$\Pi^{(q)}:L^{2}_{(0,q)}(X)\rightarrow{\rm Ker\,}\Box^{(q)}_{b}$

be the orthogonal projection (Szegő projection). Let us recall the following
known global result.

######  Theorem 3.1.

Suppose that the Kohn Laplacian $\square_{b}^{(q)}:{\rm
Dom\,}\square_{b}^{(q)}\subset L^{2}_{(0,q)}(X)\rightarrow L^{2}_{(0,q)}(X)$
has closed range, then there exits a bounded operator
$N:L^{2}_{(0,q)}(X)\rightarrow{\rm Dom\,}\square_{b}^{(q)}$ such that

$N\square_{b}^{(q)}+\Pi^{(q)}=I\quad\text{on }{\rm
Dom\,}\square_{b}^{(q)}\quad\text{ and
}\quad\square_{b}^{(q)}N+\Pi^{(q)}=I\quad\text{on }L^{2}_{(0,q)}(X)\,.$

Consider an orbifold chart $(G_{U},\widetilde{U})\rightarrow U$, let

$S_{-},S_{+},A:\Omega^{0,q}_{c}(\widetilde{U})\rightarrow\Omega_{c}^{0,q}(\widetilde{U})$

be as in Theorem 2.2. Recall that $S_{-},S_{+}$ and $A$ are properly supported
on $\widetilde{U}$. Define on $\widetilde{U}$ the following $G_{U}$-invariant
smooth kernels

$\begin{split}&\hat{S}_{-}(x,y)=\sum_{h,g\in
G_{U}}\frac{1}{\left|G_{U}\right|}S_{-}(h\cdot x,g\cdot
y),\quad\hat{S}_{+}(x,y)=\sum_{h,g\in
G_{U}}\frac{1}{\left|G_{U}\right|}S_{+}(h\cdot x,g\cdot y),\\\
&\hat{A}(x,y)=\sum_{h,g\in G_{U}}\frac{1}{\left|G_{U}\right|}A(h\cdot x,g\cdot
y)\,.\end{split}$

It is not difficult to see that $\hat{S}_{-}$, $\hat{S}_{+}$ and $A$ are
properly supported. For every $u\in\Omega^{0,q}_{c}(U)$, we have

$\hat{A}u=\sum_{h,g\in
G_{U}}\frac{1}{\left|G_{U}\right|}\int_{\widetilde{U}}A(h\cdot x,g\cdot
y)u(y)\mathrm{dV}_{\widetilde{U}}(y)=\sum_{h,g\in
G_{U}}\frac{1}{\left|G_{U}\right|}\int_{\widetilde{U}}A(h\cdot
x,y)u(g^{-1}\cdot y)\mathrm{dV}_{\widetilde{U}}(y),$

which is in $\Omega^{0,q}_{c}(U)$ because $A$ is proper and $u\circ g^{-1}$ is
compactly supported. Similarly one can deal with $\hat{S}_{-}$ and
$\hat{S}_{+}$. Thus, we have
$\hat{S}_{-},\hat{S}_{+},\hat{A}\,:\,\Omega^{0,q}_{c}(U)\rightarrow\Omega_{c}^{0,q}(U)$.

By the properties of $S_{-},S_{+}$ and $A$, we have

$\square_{b}^{(q)}\hat{A}+\hat{S}_{-}+\hat{S}_{+}=I\quad\text{on }\tilde{U}$
(19)

and

$\square_{b}^{(q)}\hat{S}_{-}\equiv 0\quad\text{on }\tilde{U}\,.$ (20)

Now, apply $\Pi^{(q)}$ to (19), for each $u\in\Omega_{c}^{0,q}(U)$ we have

$\Pi^{(q)}(\square_{b}^{(q)}\hat{A}+\hat{S}_{-}+\hat{S}_{+})u=\Pi^{(q)}u\,,$
(21)

where $\Pi^{(q)}u\in L^{2}_{(0,q)}(X)$ and the left hand side is well-defined
since we have already checked that $\hat{S}_{-},\hat{S}_{+},\hat{A}$ are
properly supported. Now, notice that for each $u,v\in\Omega_{c}^{0,q}(U)$ we
have
$(\,\Pi^{(q)}\square_{b}^{(q)}\hat{A}u\,|\,v\,)=(\,\hat{A}u\,|\,\square_{b}^{(q)}\Pi
v\,)=0$ which implies $\Pi^{(q)}\square_{b}^{(q)}\hat{A}=0$ on
$\Omega_{c}^{0,q}(U)$. Thus, by (21), we also get the following desired
property

$\Pi^{(q)}(\hat{S}_{-}+\hat{S}_{+})=\Pi^{(q)}\quad\text{on
}\Omega_{c}^{0,q)}(U)\,.$ (22)

Recall from Theorem 3.1 the existence of the operator $N$; since $\hat{S}_{-}$
and $\hat{S}_{+}$ are properly supported we have

$N\square_{b}^{(q)}(\hat{S}_{-}+\hat{S}_{+})+\Pi^{(q)}(\hat{S}_{-}+\hat{S}_{+})=\hat{S}_{-}+\hat{S}_{+}\quad\text{
on }\Omega_{c}^{0,q}(U)\,.$

Now, we get that $\square_{b}^{(q)}(\hat{S}_{-}+\hat{S}_{+})$ is a smoothing
operator on $U$, which we will denote by $F$. Thus, by (22) we write

$NF+\Pi^{(q)}=\hat{S}_{-}+\hat{S}_{+}=:\hat{S}\quad\text{ on
}\Omega_{c}^{0,q}(U)\,.$ (23)

We recall that since $F$ is smoothing and properly supported on $U$, it maps
$\mathcal{E}^{\prime}(U,T^{*0,q}X)$ to compactly supported smooth sections, so
$F:\mathcal{E}^{\prime}(U,T^{*0,q}X)\rightarrow\Omega_{c}^{0,q}(U)$. Thus, by
composition we get in (23) an operator
$NF:\mathcal{E}^{\prime}(U,T^{*0,q}X)\rightarrow L^{2}_{(0,q)}(X)$. Taking the
adjoint in (23), we get $\hat{S}^{*}-\Pi^{(q)}=F^{*}N$ on
$\Omega_{c}^{0,q}(U)$ and thus by making use again of (23) we obtain

$(\hat{S}^{*}-\Pi^{(q)})(\hat{S}-\Pi^{(q)})=F^{*}N^{2}F\quad\text{ on
}\Omega_{c}^{0,q}(U)$ (24)

where it is easy to see that the right hand side is smoothing. Thus, the left
hand side of (24) is also smoothing and we have

$\hat{S}^{*}\hat{S}-\hat{S}^{*}\Pi^{(q)}-\Pi^{(q)}\hat{S}+\Pi^{2}\equiv 0\,.$

Eventually by (22) we have

$\Pi^{(q)}\equiv\hat{S}^{*}\hat{S}.$ (25)

From (19) and $\hat{S}^{*}\Box^{(q)}_{b}\equiv 0$, we get

$\hat{S}^{*}\hat{S}\equiv\hat{S}.$ (26)

From (25) and (26),we get

$\Pi^{(q)}\equiv\hat{S}_{-}+\hat{S}_{+}.$ (27)

Define on $\widetilde{U}$ the following kernels

$\tilde{S}_{-}(x,y)=\sum_{g\in G_{U}}S_{-}(x,g\cdot
y),\quad\tilde{S}_{+}(x,y)=\sum_{g\in G_{U}}S_{+}(x,g\cdot y).$

Note that $\tilde{S}_{-}(x,y)$ and $\tilde{S}_{+}(x,y)$ are not
$G_{U}$-invariant. Put $\tilde{S}:=\tilde{S}_{-}+\tilde{S}_{+}$.

######  Lemma 3.1.

With the notations used above, we have

$\hat{S}\equiv\tilde{S}$ on $\tilde{U}$. (28)

###### Proof.

The open set $\widetilde{U}$ is a non-compact CR manifold. Let

$\Box^{(q)}_{b,\widetilde{U}}:{\rm Dom\,}\Box^{(q)}_{b,\widetilde{U}}\subset
L^{2}_{(0,q)}(\widetilde{U})\rightarrow L^{2}_{(0,q)}(\widetilde{U})$

be the Gaffney extension of Kohn Laplacian on $\widetilde{U}$ with respect to
$\langle\,\cdot\,|\,\cdot\,\rangle$ and $\mathrm{dV}_{\widetilde{U}}$.
$\Box^{(q)}_{b,\widetilde{U}}$ is self-adjoint. Fix $\lambda>0$ and set
$\Pi^{(q)}_{\leq\lambda,\widetilde{U}}:=1_{[0,\lambda]}(\Box^{(q)}_{b,\widetilde{U}})$,
where $1_{[0,\lambda]}(\Box^{(q)}_{b,\widetilde{U}})$ denotes the functional
calculus of $\Box^{(q)}_{b,\widetilde{U}}$ with respect to $1_{[0,\lambda]}$.
From [HM, Theorem 4.7], we have

$\Pi^{(q)}_{\leq\lambda,\widetilde{U}}\equiv S_{-}+S_{+}\ \ \mbox{on
$\widetilde{U}$}.$ (29)

Let $L^{2}_{(0,q)}(\widetilde{U})^{G_{U}}$ be the space of $G_{U}$-invariant
$L^{2}$ $(0,q)$ forms. Let

$Q:L^{2}_{(0,q)}(\widetilde{U})\rightarrow
L^{2}_{(0,q)}(\widetilde{U})^{G_{U}}$

be the orthogonal projection. We claim that

$\mbox{$Q\circ\Pi^{(q)}_{\leq\lambda,\widetilde{U}}=\Pi^{(q)}_{\leq\lambda,\widetilde{U}}\circ
Q$ on $L^{2}_{(0,q)}(\widetilde{U})$}.$ (30)

Let $\tau\in\mathcal{C}^{\infty}_{c}(]0,\lambda[)$. Since $Q$ commutes with
$\Box^{(q)}_{b,\widetilde{U}}$, we have

$Q(z-\Box^{(q)}_{b,\widetilde{U}})^{-1}=(z-\Box^{(q)}_{b,\widetilde{U}})^{-1}Q\
\ \mbox{on $L^{2}_{(0,q)}(\widetilde{U})$},$ (31)

for every $z\in\mathbb{C}$ with ${\rm Im\,}z\neq 0$. From (31) and Helffer-
Sjöstrand formula, we have

$\begin{split}&\tau(\Box^{(q)}_{b,\widetilde{U}})Q=\frac{1}{2\pi
i}\int_{\mathbb{C}}\frac{\partial\tilde{\tau}}{\partial\overline{z}}(z-\Box^{(q)}_{b,\widetilde{U}})^{-1}Qdz\wedge
d\overline{z}\\\ &=\frac{1}{2\pi
i}\int_{\mathbb{C}}Q\frac{\partial\tilde{\tau}}{\partial\overline{z}}(z-\Box^{(q)}_{b,\widetilde{U}})^{-1}dz\wedge
d\overline{z}=Q\tau(\Box^{(q)}_{b,\widetilde{U}}),\end{split}$ (32)

where $\tilde{\tau}$ is an almost analytic extension of $\tau$. From (32), we
get the claim (30).

From (29) and (30), we get

$Q\circ(S_{-}+S_{+})\equiv(S_{-}+S_{+})Q\equiv Q(S_{-}+S_{+})Q.$ (33)

Note that $\hat{S}=\left|G_{U}\right|Q(S_{-}+S_{+})Q$,
$\tilde{S}=\left|G_{U}\right|(S_{-}+S_{+})Q$. From this observation and (33),
we get (28). ∎

From (27) and (28), Theorem 1.1 follows.

##  4 Applications

###  4.1 Proof of Theorem 1.3

We have the following formula

$\dim{\rm Ker\,}\Box^{(q)}_{b,kp}=\int_{X}{\rm
Tr\,}\Pi^{(q)}_{kp}(x,x)\mathrm{dV}_{X}(x).$

As explained in the introduction, $X$ can be written as a disjoint union based
on the orbit type induced by the circle action and $X_{\ell_{0}}$ is an open
and dense subset of $X$. Moreover, from (2), we see that there is a constant
$C>0$ independent of $k$ such that

${\rm Tr\,}\Pi^{(q)}_{kp}(x,x)\leq Ck^{n},$ (34)

for all $x\in X$. From (34), we can apply Lebesgue’s dominated convergence
theorem and get

$\lim_{k\rightarrow+\infty}k^{-n}\dim{\rm
Ker,}\Box^{(q)}_{b,kp}=\int_{X_{\ell_{0}}}\lim_{k\rightarrow+\infty}k^{-n}{\rm
Tr\,}\Pi^{(q)}_{kp}(x,x)\mathrm{dV}_{X}(x).$ (35)

From (2) and (35), Theorem 1.3 follows.

###  4.2 Proof of Theorem 1.4

We will identify $H^{0}(M,L^{kp})$ with ${\rm Ker\,}\Box^{(0)}_{b,kp}$ and
since we work locally on an orbifold chart $(\widetilde{U},G_{U})\rightarrow
U$ we always write $x$ for local coordinates on $\tilde{U}$. Let us first
proof that the differential of the map $\Phi_{kp}$ is injective, if $k\gg 1$.
Consider an open set $U$ around a point $x_{0}\in X_{\ell}$ and an orbifold
chart $(\widetilde{U},G_{U})\rightarrow U$ such that $x_{0}$ is a fix point of
$G_{U}$. Let $x=(x_{1},\ldots,x_{2n+1})$ be local coordinates of
$\widetilde{U}$ with $x(x_{0})=0$ and $T={\partial}/{\partial x_{2n+1}}$. Let
$z=(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}$, $z_{j}=x_{2j-1}+ix_{2j}$,
$j=1,\ldots,n$. Let $\chi\in\mathcal{C}^{\infty}_{c}(\widetilde{U})$ with
$\int_{X}\chi\,\mathrm{dV}_{X}=1$. For $k\in\mathbb{N}$, put

$\chi_{k}(x)=\chi(k^{\frac{1}{2}+\varepsilon}z,k^{1+\varepsilon}x_{2n+1})\quad\text{
and }\quad\hat{\chi}_{k}(x):=\sum_{g\in G_{U}}\chi_{k}(g\cdot x)\,,$

where $\varepsilon>0$ is a small constant. Let us define

$u_{k}:=k^{1+(2n+1)\varepsilon}\Pi^{(0)}_{kp}(\hat{\chi}_{k})\in{\rm
Ker\,}\Box^{(0)}_{b,kp}.$

From (2), we can check that

$\lim_{k\rightarrow+\infty}u_{k}(x_{0})=p^{n}\frac{\ell}{2}\pi^{-n-1}\left|\det\mathcal{L}_{x_{0}}\right|.$
(36)

For each $j=1,\dots n$, it is easy to see that the function

$\tilde{f}_{j}:\widetilde{U}\rightarrow\mathbb{C}\,,\qquad z\mapsto
f_{j}(z)=\sum_{g\in G_{U}}g\cdot z_{j}$

is $G_{U}$-invariant and smooth and thus descends naturally to a continuous
function $f_{j}:U\rightarrow\mathbb{C}$ on $U$. Let us also set

$u_{k}^{(j)}=k^{1+(2n+1)\varepsilon}\Pi^{(0)}_{kp}(f_{j}\hat{\chi}_{k})\in{\rm
Ker\,}\Box^{(0)}_{b,k}\,.$

From $\Psi(x,x)=0$ and by using integration by parts, we get

$\begin{split}&\lim_{k\rightarrow+\infty}(\frac{\partial}{\partial
z_{q}}u^{(j)}_{k})(x_{0})=\delta_{j,q}p^{n}\frac{\ell}{2}\pi^{-n-1}\left|\det\mathcal{L}_{x_{0}}\right|,\\\
&\lim_{k\rightarrow+\infty}u^{(j)}_{k}(x_{0})=0,\end{split}$ (37)

for every $j,q=1,\ldots,n$. Now, consider sections
$g_{1},\dots,g_{d_{pk}-n-1}$ such that

$\\{u_{k},u_{k}^{(1)},\dots,u_{k}^{(n)},g_{1},\dots,g_{d_{kp}-n-1}\\}$

is a basis of $H^{0}(M,L^{kp})$. As a consequence of (36) and (37), it is easy
to see that the map

$x\mapsto\left(\frac{u^{1}_{k}}{u_{k}}(x),\dots,\frac{u^{n}_{k}}{u_{k}}(x),\frac{g_{1}}{u_{k}}(x),\dots,\frac{g_{d_{k}-n-1}}{u_{k}}(x)\right)$
(38)

is injective if $k$ is sufficiently large. The Kodaira embedding map is
constructed by using an orthonormal basis of $H^{0}(M,L^{kp})$; the basis used
to define the map (38) is not orthonormal in general. By considering the
change of basis matrix we get that $\mathrm{d}\Phi_{kp}$ is injective if $k$
is sufficiently large.

We now need to prove that $\Phi_{kp}$ is globally injective. By absurd, up to
passing to a subsequence, suppose that there are $x_{k},\,y_{k}\in M$ with
$x_{k}\neq y_{k}$ such that $\Phi_{kp}(x_{k})=\Phi_{kp}(y_{k})$, for each $k$.
We prove the theorem by showing that we get a contradiction.

Assume that $\lim_{k\rightarrow+\infty}x_{k}=x_{0}$,
$\lim_{k\rightarrow+\infty}y_{k}=y_{0}$. If $x_{0}\neq y_{0}$, we can repeat
the procedure in [Hs1] and find $u_{k},v_{k}\in H^{0}(M,L^{kp})$ such that
$\left|u_{k}(x_{k})\right|^{2}_{h^{L^{kp}}}\geq Ck^{n}$,
$\left|v_{k}(x_{k})\right|^{2}_{h^{L^{kp}}}\leq\frac{C}{2}k^{n}$,
$\left|u_{k}(y_{k})\right|^{2}_{h^{L^{kp}}}\leq\frac{C}{2}k^{n}$,
$\left|v_{k}(y_{k})\right|^{2}_{h^{L^{kp}}}\geq Ck^{n}$, for all $k\gg 1$,
where $C>0$ is a constant independent of $k$. Thus,
$\Phi_{kp}(x_{k})\neq\Phi_{kp}(y_{k})$, for $k$ large. We get a contradiction
and thus we must have $x_{0}=y_{0}$.

For every $k\in\mathbb{N}$, put

$A_{k}:=\left\\{(x,y)\in M\times M;\,\Phi_{k}(x)=\Phi_{k}(y)\right\\}.$

We need

######  Lemma 4.1.

There is a constant $k_{0}$ such that for every $j\in\mathbb{N}$ and every
$k\in\mathbb{N}$ with $k\geq k_{0}$, we have

$A_{j+k}\subset A_{j}.$

###### Proof.

From the proof of (36), we see that there is a $k_{0}\in\mathbb{N}$ such that
for every $x\in X$ and every $k\in\mathbb{N}$, $k\geq k_{0}$, we can find
$u_{k}\in{\rm Ker\,}\Box^{(0)}_{b,kp}$ such that
$\left|u_{k}(x)\right|\geq\frac{1}{2}$. Fix $j\in\mathbb{N}$, let
$k\in\mathbb{N}$ with $k\geq k_{0}$ and let $(x,y)\notin A_{j}$. We claim that
$(x,y)\notin A_{j+k}$. Since $(x,y)\notin A_{j}$, we can find $v_{j}\in{\rm
Ker\,}\Box^{(0)}_{b,jp}$ so that $v_{j}(x)=1$ and $v_{j}(y)=0$. Let
$u_{k}\in{\rm Ker\,}\Box^{(0)}_{b,kp}$ such that
$\left|u_{k}(x)\right|\geq\frac{1}{2}$. Then, $u_{k}v_{j}\in{\rm
Ker\,}\Box^{(0)}_{b,(j+k)p}$, $\left|(u_{k}v_{j})(x)\right|\geq\frac{1}{2}$,
$\left|(u_{k}v_{j})(y)\right|=0$. Thus, $(x,y)\notin A_{j+k}$. The lemma
follows. ∎

We now have
$\lim_{k\rightarrow+\infty}x_{k}=\lim_{k\rightarrow+\infty}y_{k}=0$. Consider
an open set $U$ around a given point $x_{0}\in X_{\ell}$ and an orbifold chart
$(\widetilde{U},G_{U})\rightarrow U$ such that $x_{0}$ is a fix point of
$G_{U}$. Let $x=(x_{1},\ldots,x_{2n+1})$ be local coordinates of
$\widetilde{U}$ with $x(x_{0})=0$. For every $j\in\mathbb{N}$, take
$k_{j}\in\mathbb{N}$ with $k_{j}\geq k_{0}$ such that

$\left|x_{j+k_{j}}\right|\leq\frac{1}{j^{3}}\quad\text{ and
}\quad\left|y_{j+k_{j}}\right|\leq\frac{1}{j^{3}}\,.$ (39)

From Lemma 4.1, we see that $(x_{j+k_{j}},y_{j+k_{j}})\subset A_{j}$. Thus,
for every $j$, we replace $(x_{j},y_{j})$ by $(x_{j+k_{j}},y_{j+k_{j}})$ and
conclude that there are $x_{k},\,y_{k}\in M$, $x_{k}\neq y_{k}$ such that
$\Phi_{kp}(x_{k})=\Phi_{kp}(y_{k})$, for each $k$ and

$\lim_{k\rightarrow+\infty}\left|k^{2}x_{k}\right|=\lim_{k\rightarrow+\infty}\left|k^{2}y_{k}\right|=0\,.$
(40)

Put

$f_{k}(t)=\frac{\lvert\Pi^{(0)}_{kp}(tx_{k}+(1-t)y_{k},y_{k})\rvert^{2}}{\Pi^{(0)}_{kp}(tx_{k}+(1-t)y_{k})\Pi^{(0)}_{kp}(y_{k})}$

where we write $\Pi^{(0)}_{kp}(x)=\Pi^{(0)}_{kp}(x,x)$, for any $x\in X$. From
(40), we can check that

$f_{k}(t)=\sum_{g\in
G_{U}}e^{-2kp\,\mathrm{Im}\Psi(t{x}_{k}+(1-t){y}_{k},g\cdot{y}_{k})}\cdot\left[1+\frac{1}{k}\tilde{R}_{k}(t)+\varepsilon_{k}(t)\right],$
(41)

where

$\left|\partial^{j}_{t}\widetilde{R}_{k}(t)\right|\leq
C_{j}\left|x_{k}-y_{k}\right|^{j}\quad\text{and}\quad\left|\partial^{j}_{t}\varepsilon_{k}(t)\right|\leq
C_{N,j}k^{-N}\left|x_{k}-y_{k}\right|^{j}\,,$

for all $j\in\mathbb{N}\cup\left\\{0\right\\}$ and every $N\in\mathbb{N}$,
where $C_{j},C_{N,j}>0$ are constants independent of $k$. For ease of
notation, pose
$F_{k}(t):=-2kp\,\mathrm{Im}\Psi(t{x}_{k}+(1-t){y}_{k},g\cdot{y}_{k})$ for
given $x_{k}$, $y_{k}$ and $g\in G_{U}$. From (40), we have

$\lvert
F^{\prime}_{k}(t)\rvert=\lvert\langle-2kp\,\mathrm{Im}\Psi^{\prime}_{x}(t{x}_{k}+(1-t){y}_{k},g\cdot{y}_{k}),{x}_{k}-{y}_{k}\rangle\rvert\leq\frac{1}{c_{0}k}\lvert{x}_{k}-{y}_{k}\rvert$

and

$F^{\prime\prime}_{k}(t)=\langle-2kp\,\mathrm{Im}\Psi^{\prime\prime}_{x}(t{x}_{k}+(1-t){y}_{k},g\cdot{y}_{k}),x_{k}-y_{k}\rangle<-c_{0}k\,\lvert
x_{k}-y_{k}\rvert^{2}$

for the computation of the second derivative of (41), here $c_{0}$ is a
positive constant. We get

$\limsup_{k}\frac{f^{\prime\prime}(t)}{k\,\lvert x_{k}-y_{k}\rvert^{2}}<0$
(42)

for each $t$.

By Cauchy-Schwartz inequality we have $0\leq f_{k}(t)\leq 1$, for any
$t\in[0,1]$. From $\Phi_{kp}(x_{k})=\Phi_{kp}(y_{k})$, for each $k$, we can
check that $f_{k}(0)=f_{k}(1)=1$. Thus, for each $k$, there is a
$t_{k}\in[0,1]$ such that $f^{\prime\prime}(t_{k})=0$. Hence,

$\limsup_{k}\frac{f^{\prime\prime}(t)}{k\,\lvert x_{k}-y_{k}\rvert^{2}}\geq 0$

which contradicts (42).

###  4.3 Quantization commutes with reduction

In this section, we assume that $X$ admits a CR compact Lie group action $G$,
${\rm dim\,}G=d$. We will work Assumption 2.1 and suppose that $X$ is strongly
pseudoconvex. We will use the same notations as in Section 2.3. The moment map
associated to $\omega_{0}$ is the map given by
$\mu:X\rightarrow\mathfrak{g}^{*}$ such that for all $x\in X$ and
$\xi\in\mathfrak{g}$, we have

$\langle\,\mu(x)\,,\,\xi\,\rangle=\omega_{0}(x)(\xi_{X}).$ (43)

Let us recall the following theorem which is well-known in the setting of
symplectic manifolds. Throughout this section we will always assume that $0$
is a regular value, and thus $\mu^{-1}(0)$ is a suborbifold of $X$.

######  Proposition 4.1.

Suppose that $0$ is a regular value, then the action of $G$ on $\mu^{-1}(0)$
is locally free.

###### Proof.

Since $0$ is a regular value then for each $p\in\mu^{-1}(0)$,
$\mathrm{d}_{p}\mu:T_{p}X\rightarrow\mathfrak{g}^{*}$ is surjective. Let us
identify $\mathfrak{g}\cong\mathfrak{g}^{*}$, then for every
$\xi\in\mathfrak{g}\setminus\\{0\\}$ there exists $V\in T_{p}X$ such that
$\mathrm{d}_{p}\mu(V)=\xi$. Thus, we have

$0\neq\langle\mathrm{d}_{p}\mu(V),\xi\rangle=\mathrm{d}_{p}\omega_{0}(\xi_{X}(p),V)=-b_{p}(\xi_{X}(p),J_{p}V)$

and we can conclude that $\xi_{X}(p)\neq 0$. ∎

######  Proposition 4.2.

Suppose $0$ is a regular value, then $X_{G}=\mu^{-1}(0)/G$ is an orbifold.

###### Proof.

By the previous proposition the action of $G$ on $\mu^{-1}(0)$ is locally
free. From the proof of Corollary B$.31$ in [GGK] we can conclude. ∎

Corollary B$.31$ in [GGK] is a consequence of the Slice Theorem which states
that there exists a $G$-equivariant diffeomorphism $\phi$ between
$G\times_{G_{x}}D$ and a tubular neighborhood of the orbit $G\cdot x$, we
refer to Appendix B in [GGK] for precise definitions. More precisely, $\phi$
is induced by the exponential map $E_{p}:U_{0}\rightarrow U\subseteq M$, where
$U_{0}$ is an open neighborhood of $0$ in $T_{x}M$. We have $G_{x}$
equivariant decomposition

$T_{x}X=T_{x}(G\cdot x)\oplus W$

where $W$ is the normal bundle to $G\cdot x$. As a consequence of this, it is
easy to see the the CR orbifold structure on $X$ descends naturally to a CR
orbifold structure on $X$ and so the other $G$-invariant structures defined on
$X$. Moreover, we can repeat the proof of [HMM, Theorem 2.5] and deduce that
$X_{G}$ is a strongly pseudoconvex CR orbifold.

We take the Hermitian metric $\langle\,\cdot\,|\,\cdot\,\rangle$ on
$\mathbb{C}TX$ so that $\langle\,\cdot\,|\,\cdot\,\rangle$ is $G$-invariant.
Let $\langle\,\cdot\,|\,\cdot\,\rangle_{X_{G}}$ be the Hermitian metric on
$\mathbb{C}TX_{G}$ induced by $\langle\,\cdot\,|\,\cdot\,\rangle$ and let
$(\,\cdot\,|\,\cdot\,)_{X_{G}}$ be the inner product on $L^{2}(X_{G})$ induced
by $\langle\,\cdot\,|\,\cdot\,\rangle_{X_{G}}$. Let $\Box^{(0)}_{b,X_{G}}$ be
the Kohn Laplacian on the orbifold $X_{G}$ with respect to
$(\,\cdot\,|\,\cdot\,)_{X_{G}}$. Let
$\Pi_{X_{G}}:L^{2}(X_{G})\rightarrow\Box^{(0)}_{b,X_{G}}$ be the Szegő
projection. Fix $p\in\mu^{-1}(0)$. Let $(\widetilde{U},G_{U})\rightarrow U$ be
an orbifold chart of $X$, $p\in U$. By the previous discussion we have that
$G\cdot p=[p]\in X_{G}$ and a sufficiently small neighborhood $U_{[p]}$ of
$[p]$ in $X_{G}$ has a orbifold local chart $(G_{p}\times
G_{U},\widetilde{W})\rightarrow U_{[p]}$, where
$\widetilde{W}/G_{p}=(\widetilde{U}\cap\mu^{-1}(0))/G_{p}$ and
$G_{p}:=\left\\{g\in G;\,gp=p\right\\}$. Thus, from Theorem 1.1, the Szegő
kernel for the orbifold $X_{G}$ has the following expression in the local
chart $(G_{p}\times G_{U},\widetilde{W})\rightarrow U_{[p]}$

$\Pi_{X_{G}}(x,y)\equiv\sum_{h\in G_{U},g\in
G_{p}}\int_{0}^{+\infty}e^{it\,\varphi_{-}^{X_{G}}(\widetilde{x},g\cdot
h\cdot\widetilde{y})}s_{-}^{X_{G}}(\widetilde{x},g\cdot
h\cdot\widetilde{y},t)\,\mathrm{d}t$ (44)

where $x,y\in U_{[p]}$ and $\varphi_{-}^{X_{G}}$ and $s_{-}^{X_{G}}$ are
respectively the complex phase function and the symbol of $S_{X_{G}}$ as
described in Theorem 1.1 for the CR orbifold $X_{G}$.

Let use recall that

$({\rm Ker\,}\Box^{(0)}_{b})^{G}:=\left\\{u\in{\rm
Ker\,}\Box^{(0)}_{b};\,g^{*}u=u,\forall g\in G\right\\}\,.$

Let $\Pi_{G}:L^{2}(X)\rightarrow({\rm Ker\,}\Box^{(0)}_{b})^{G}$ be the
orthogonal projection ($G$-invariant Szegő projection). In [HH] and [GH2], we
studied the $G$-invariant Szegő kernel.

We pause and introduce some notations. Fix $x\in\mu^{-1}(0)$, consider the map

$\begin{split}R_{x}:\mathfrak{g}^{*}&\rightarrow\mathfrak{g}^{*},\\\
u&\rightarrow R_{x}u,\
\langle\,R_{x}u\,|\,v\,\rangle=\langle\,d\omega_{0}(x)\,,\,Ju\wedge
v\,\rangle,\ \ v\in\mathfrak{g}^{*},\end{split}$

where $J$ is the natural complex structure map on $HX:={\rm Re\,}T^{1,0}X$.
Let

${\rm det\,}R_{x}=\lambda_{1}(x)\cdots\lambda_{d}(x),$

$\lambda_{j}(x)$, $j=1,\ldots,d$, are the eigenvaules of $R_{x}$ with respect
to $\langle\,\cdot\,|\,\cdot\,\rangle$. Put

$Y_{x}:=\left\\{gx\in\mu^{-1}(0);\,g\in G\right\\}$

and let $\mathrm{dV}_{Y_{x}}$ be the volume form on $Y_{x}$ induced by the
given Hermitian metric $\langle\,\cdot\,|\,\cdot\,\rangle$. Put

$V_{{\rm eff\,}}:=\int_{Y_{x}}\mathrm{dV}_{Y_{x}}.$

We now come back to our situation. From Theorem 1.1, we can repeat the
procedure in [GH2] and deduce the following: Fix $p\in\mu^{-1}(0)$. Let
$(\widetilde{U},G_{U})\rightarrow U$ be an orbifold chart of $X$, $p\in U$.
Then, the distributional kernel of $\Pi_{G}$ satisfies

$\Pi_{G}(x,y)\equiv\sum_{h\in G_{U},g\in G_{p}}\int_{0}^{\infty}e^{\imath
t\,\Phi_{-}(\widetilde{x},g\cdot
h\cdot\widetilde{y})}a_{-}(\widetilde{x},\,g\cdot
h\cdot\widetilde{y},\,t)\,\mathrm{d}t\ \ \ \mbox{on $U\times U$},$ (45)

where

$a_{-}(\widetilde{x},\widetilde{y},\,t)\sim\sum_{j=0}^{+\infty}a_{-}^{j}(\widetilde{x},\widetilde{y})\,t^{n-d/2-j}\in
S^{n-d/2}_{1,0}(\widetilde{U}\times\widetilde{U}\times\mathbb{R})\,,$

$a_{-}^{j}\in\mathcal{C}^{\infty}(\widetilde{U}\times\widetilde{U}),\,j\in\mathbb{N}_{0}$,
and for every $\widetilde{x}\in\widetilde{\mu^{-1}(0)}$,
$\widetilde{\mu^{-1}(0)}\subset\widetilde{U}$ is the lifting of $\mu^{-1}(0)$
on $\widetilde{U}$, we have

$a_{-}^{0}(\widetilde{x},\,\widetilde{x})=2^{d-1}\frac{1}{V_{{\rm
eff\,}}(x)|G_{x}|}\pi^{-n-1+\frac{d}{2}}\left|\det
R_{\widetilde{x}}\right|^{-\frac{1}{2}}\left|\det\mathcal{L}_{\widetilde{x}}\right|\,.$
(46)

We refer to [HH] or [GH2] for a precise description of the complex phase
function $\Phi_{-}$ and the symbol $a_{-}$. Notice that $x$ and $y$ can be
assumed to be $x=\exp_{p}(v)$ and $y=\exp_{p}(w)$ for some $v,\,w\in W$ since
$S^{G}_{-}$ is $G$-invariant.

Now we can introduce explicitly a map $\sigma:({\rm
Ker\,}\Box^{(0)}_{b})^{G}\rightarrow{\rm Ker\,}\Box^{(0)}_{b,X_{G}}$. Let
$\mathcal{C}^{\infty}(\mu^{-1}(0))^{G}$ denote the set of all $G$-invariant
smooth functions on $\mu^{-1}(0)$. Let

$\iota_{G}:\mathcal{C}^{\infty}(\mu^{-1}(0))^{G}\rightarrow\mathcal{C}^{\infty}(X_{G})$

be the natural identification. Let $\iota:\mu^{-1}(0)\rightarrow X$ be the
natural inclusion and let
$\iota^{*}:\mathcal{C}^{\infty}(X)\rightarrow\mathcal{C}^{\infty}(\mu^{-1}(0))$
be the pull-back of $\iota$. Let

$f(x)=\sqrt{V_{{\rm eff\,}}(x)\left|G_{x}\right|}.$

Eventually, let $E\in L^{-\frac{d}{4}}_{{\rm cl\,}}(X_{G})$ be an elliptic
pseudodifferential operator with principal symbol
$\sigma^{0}_{E}(x,\xi)=\left|\xi\right|^{-\frac{d}{4}}$, we define

$\begin{split}\sigma:\mathcal{C}^{\infty}(X)&\rightarrow{\rm
Ker\,}\Box^{(0)}_{b,X_{G}},\\\ u&\mapsto{\Pi_{X_{G}}}\circ
E\circ\iota_{G}\circ f\circ\iota^{*}\circ\Pi_{G}u.\end{split}$ (47)

Let
$\sigma^{\,*}:\mathcal{C}^{\infty}(X_{G})\rightarrow\mathcal{D}^{\prime}(X)$
be the formal adjoint of $\sigma$. Following the same computations as in [HH]
with some modification, by an application of stationary phase lemma of [MeSj]
and by making use of (44), which is a consequence of Theorem 1.1 and by
equation (45), we obtain

######  Theorem 4.1.

Under the hypothesis of this section, suppose that $\Box^{(0)}_{b,X_{G}}$ has
closed range. Then, $\sigma^{\,*}$ maps $\mathcal{C}^{\infty}(X_{G})$
continuously to $\mathcal{C}^{\infty}(X)$. Hence,
$\sigma^{*}\,\sigma:\mathcal{C}^{\infty}(X)\rightarrow\mathcal{C}^{\infty}(X)$
is a well-defined continuous operator whose distribution kernel satisfies

$\sigma^{*}\,\sigma=\Pi_{G}+\Gamma,$ (48)

where $\Gamma:\mathcal{C}^{\infty}(X)\rightarrow\mathcal{C}^{\infty}(X)$ is a
continuous map and satisfies the following: there exists $\varepsilon>0$ such
that $\Gamma:H^{s}(X)\rightarrow H^{s+\varepsilon}(X)$ is continuous for every
$s\in\mathbb{\mathbb{missing}}R$, $H^{s}(X)$ is the Sobolev space of functions
over $X$ order $s$.

Moreover,

$\sigma\,\sigma^{*}=\Pi_{X_{G}}+\hat{\Gamma},$ (49)

where
$\hat{\Gamma}:\mathcal{C}^{\infty}(X_{G})\rightarrow\mathcal{C}^{\infty}(X_{G})$
is a continuous map and satisfies the following: there exists $\delta>0$ such
that $\hat{\Gamma}:H^{s}(X_{G})\rightarrow H^{s+\delta}(X_{G})$ is continuous
for every $s\in\mathbb{\mathbb{missing}}R$, $H^{s}(X_{G})$ is the Sobolev
space of functions over $X_{G}$ order $s$.

From Theorem 4.1, we see that $\sigma$ can be extended by density to

$\sigma:L^{2}(X)\rightarrow{\rm Ker\,}\Box^{(0)}_{b,X_{G}}.$ (50)

######  Corollary 4.1.

Under the preceding assumptions and notations, the map $\sigma$ given by (47)
and (50) has the following properties:

  * (i)

${\rm Ker\,}\sigma\cap({\rm Ker\,}\Box^{(0)}_{b})^{G}$ is a finite dimensional
subspace of $\mathcal{C}^{\infty}(X)$.

  * (ii)

${\rm Coker\,}\sigma\cap{\rm Ker\,}\Box^{(0)}_{b,X_{G}}$ is a finite
dimensional subspace of $\mathcal{C}^{\infty}(X_{G})$.

Moreover, we have $\sigma:({\rm Ker\,}\Box^{(0)}_{b})^{G}\rightarrow{\rm
Ker\,}\Box^{(0)}_{b,X_{G}}$ is a Fredholm map.

###### Proof.

Let $u\in{\rm Ker\,}\sigma\cap({\rm Ker\,}\Box^{(0)}_{b})^{G}$. From (48), we
have

$0=\sigma^{*}\,\sigma u=(\Pi_{G}+\Gamma)u=u+\Gamma u.$ (51)

From the previous theorem and (51), we deduce that $u\in H^{\varepsilon}(X)$.
By applying again the previous theorem and (51), we get $u\in
H^{2\varepsilon}(X)$. Continuing in this way, we conclude that
$u\in\mathcal{C}^{\infty}(X)$.

Suppose that ${\rm Ker\,}\sigma\cap({\rm Ker\,}\Box^{(0)}_{b})^{G}$ is not a
finite dimensional subspace of $\mathcal{C}^{\infty}(X)$. We can find
$u_{j}\in{\rm Ker\,}\sigma\cap({\rm Ker\,}\Box^{(0)}_{b})^{G}$,
$j=1,2,\ldots$, $(\,u_{j}\,|\,u_{k}\,)=\delta_{j,k}$, for every
$j,k=1,\ldots$. From the previous theorem and (51), we notice that
$\\{u_{j}\\}_{j\in\mathbb{N}}$ is a bounded set in $H^{\varepsilon}(X)$. By
Rellich’s lemma, there is a subsequence $1\leq j_{1}<j_{2}<\cdots$,
$\lim_{s\rightarrow+\infty}j_{s}=+\infty$, such that $u_{j_{s}}\rightarrow u$
in $L^{2}(X)$ as $s\rightarrow+\infty$, for some $u\in L^{2}(X)$. Since
$(\,u_{j}\,|\,u_{k}\,)=\delta_{j,k}$, for every $j,k=1,\ldots$, we have

$\lVert u_{j_{s}}-u_{k_{s}}\rVert_{L^{2}}=\sqrt{2}$

and we get a contradiction since $u_{j_{s}}$ and $u_{k_{s}}$ converge to $u$
as $k_{s},j_{s}$ goes to infinity. Thus, ${\rm Ker\,}\sigma\cap({\rm
Ker\,}\Box^{(0)}_{b})^{G}$ is a finite dimensional subspace of
$\mathcal{C}^{\infty}(X)$.

We can repeat the procedure above with minor change and deduce that ${\rm
Coker\,}\sigma\cap{\rm Ker\,}\Box^{(0)}_{b,X_{G}}$ is a finite dimensional
subspace of $\mathcal{C}^{\infty}(X_{G})$. ∎

Recall that when $X$ is a circle orbibundle, we have a natural circle action
which commutes with the action of $G$ on $X$ and it is transversal to the CR
structure. This induces a decomposition of $({\rm Ker\,}\Box^{(0)}_{b})^{G}$
into Fourier components $({\rm Ker\,}\Box^{(0)}_{b})^{G}_{k}$ and similarly
${\rm Ker\,}\Box^{(0)}_{b,X_{G}}$ splits as a direct sum $({\rm
Ker\,}\Box^{(0)}_{b,X_{G}})_{k}$, $k\in\mathbb{Z}$. Now, we can take the
limits as $k$ goes to infinity in the previous corollary and we get that
quantization commutes with reduction for $k$ large, $X$ and $X_{G}$ may be
orbifolds.

Acknowledgments: This project was started during the first author’s
postdoctoral fellowship at the National Center for Theoretical Sciences in
Taiwan; we thank the Center for the support. Andrea Galasso thanks Yi-Sheng
Wang for conversations about orbifolds. 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.

## References

  * [ALR] Adem, A., Leida, J., Ruan, Y., Orbifolds and Stringy Topology, Cambridge Tracts in Mathematics, Cambridge: Cambridge University Press (2007).
  * [BG] C. P. Boyer and K. Galicki, Sasakian geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008.
  * [BS] L. Boutet de Monvel, J. Sjöstrand, Sur la singularité des noyaux de Bergman et de Szegő, Astérisque 34-35 (1976), 123–164.
  * [DLM] X. Dai, K. Liu, and X. Ma, On the asymptotic expansion of Bergman kernel, J. Differential Geom. 72 (2006), no. 1, 1–41.
  * [DK] J.J. Duistermaat, Johan A.C. Kolk Lie Groups, Universitext, Springer-Verlag Berlin Heidelberg (2000), ISBN 978-3-642-56936-4.
  * [GGK] V. Ginzburg, Y. Karshon, V. Guillemin, Moment map, cobordisms and Hamiltonian group actions, Mathematical Surveys and Monographs Volume: 98; 2002; 350 pp.
  * [GH2] A. Galasso, C.-Y. Hsiao, Toeplitz operators on CR manifolds and group actions, arxiv.org/abs/2108.11061.
  * [GH2] A. Galasso, C.-Y. Hsiao, Functional calculus and quantization commutes with reduction for Toeplitz operators on CR manifolds, arXiv:2112.11257.
  * [KHH] K. Fritsch, H. Herrmann and C.-Y. Hsiao, $G$-equivariant embedding theorems for CR manifolds of high codimension, Michigan Math. J. Advance Publication 1–44 (2021), DOI: 10.1307/mmj/20205864.
  * [HH] C.-Y. Hsiao, R.-T. Huang, G-invariant Szegő kernel asymptotics and CR reduction, Calc. Var. Partial Differential Equations 60 (2021), no. 1, Paper No. 47, 48 pp.
  * [HMM] C.-Y. Hsiao, X. Ma and G. Marinescu,Geometric quantization on CR manifolds, arXv: 1906.05627.
  * [Hs] C.-Y. Hsiao, Projections in several complex variables, Mémoires de la Société Mathématique de France 123 (2010), 1-136.
  * [Hs1] C.-Y. Hsiao, Bergman Kernel Asymptotics and a Pure Analytic Proof of the Kodaira Embedding Theorem, Complex Analysis and Geometry, 2015, Springer Proceedings in Mathematics and Statistics.
  * [HM] C.-Y. Hsiao, G. Marinescu On the singularities of the Szegő projections on lower energy forms, J. Diff. Geom. 107 (2017), no. 1, pp 83–155.
  * [MM] G. Marinescu, X. Ma Holomorphic Morse inequalities and Bergman kernels, Progress in Mathematics, Series Volume $254$, 2007, Birkhäuser Basel.
  * [MeSj] A. Melin and J. Sjöstrand, Fourier integral operators with complex valued phase functions, Springer Lecture Notes in Math 459 (1074), pp. 120-223, Springer-Verlag, Berlin.
  * [MiSt] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press (1974), Princeton, New Jersey.
  * [P] R. Paoletti, Asymptotics of Szegő kernels under Hamiltonian torus actions, Israel Journal of Mathematics 191 (2012), no. 1, 363–403.
  * [S] J. Song, The Szegő kernel on an orbifold circle bundle, arXiv:math/0405071.