CR Paneitz operator on non-embeddable CR manifolds

The CR Paneitz operator plays a crucial role in CR geometry of dimension theree. For example, this operator appears in the transformation law of the logarithmic singularity of the Szegő kernel [Hirachi1993], which is also known as the CR $Q$-curvature [Fefferman-Hirachi2003]. Moreover, the non-negativity of the CR Paneitz operator is deeply connected to global embeddability [Chanillo-Chiu-Yang2012, Takeuchi2020-Paneitz] and the CR positive mass theorem [Cheng-Malchiodi-Yang2017], which has an application to the CR Yamabe problem.

Let $(M,T^{1,0}M,\theta)$ be a closed pseudo-Hermitian manifold of dimension three. We consider the CR Paneitz operator $P$ on $M$ as an unbounded operator on $L^{2}(M)$ with domain $C^{\infty}(M)$, which is closable. In the embeddable case, Hsiao [Hsiao2015] has studied analytic properties of this operator; see also [Takeuchi2023-GJMS] for some improvements and generalizations to higher dimensions. The CR Paneitz operator on an embeddable CR manifold is essentially self-adjoint and has closed range. Moreover, its spectrum is a discrete subset of $\mathbb{R}$ and consists only of eigenvalues. Furthermore, the eigenspace associated with each non-zero eigenvalue is a finite-dimensional subspace of $C^{\infty}(M)$.

It is natural to ask what happens in the non-embeddable case. We will first prove that $P$ is essentially self-adjoint even in the non-embeddable case.

The CR Paneitz operator $P$ is essentially self-adjoint; equivalently, the maximal closed extension of $P$ is self-adjoint.

We use the same letter $P$ for the maximal closed extension of the CR Paneitz operator by abuse of notation. Let $E$ be the resolution of the identity for $P$, and set

for $\lambda\geq 0$. Note that $\pi_{\lambda}$ is an orthogonal projection of $L^{2}(M)$. We will show that if $\lambda>0$, then $P\pi_{\lambda}$ is a smoothing operator (section 4). As an application, we study the spectrum $\operatorname{Spec}P$ of $P$. As opposed to the embeddable case, we can only show that $\operatorname{Spec}P$ is discrete except zero.

The set $\operatorname{Spec}P\setminus\{0\}$ is a discrete subset of $\mathbb{R}\setminus\{0\}$ and consists only of eigenvalues of finite multiplicity. Moreover, any eigenfunction of $P$ corresponding to each non-zero eigenvalue is smooth.

We will also show that $\pi_{\lambda}$ is a Heisenberg pseudodifferential operator (section 4). In particular, the singular support of its Schwartz kernel is contained in the diagonal $\Delta_{M}\subset M\times M$. Our proof is inspired by that of [Hsiao-Marinescu2017]. Remark that $\pi_{0}$ is not necessarily a Heisenberg pseudodifferential operator; see section 5.3.

We also study analytic properties of the CR Paneitz operator on the Rossi sphere, a homogeneous non-embeddable strictly pseudoconvex CR manifold; see [Chen-Shaw2001] for example. The unit sphere

has the canonical CR structure $T^{1,0}S^{3}$. This CR structure is spanned by

A canonical contact form $\theta$ on $S^{3}$ is given by

For a real number $0<\lvert t\rvert<1$, the Rossi sphere $(S^{3}_{t},T^{1,0}S^{3}_{t})$ is defined by

Denote by $P(t)$ the CR Paneitz operator with respect to $(S^{3}_{t},T^{1,0}S^{3}_{t},\theta)$. We can apply the theory of spherical harmonics since $P(t)$ is invariant under $U(2)$-action. Similar to the method in [Abbas-Brown-Ramasami-Zeytuncu2019], we can prove the following

The CR Paneitz operator $P(t)$ on the Rossi sphere $(S^{3}_{t},T^{1,0}S^{3}_{t},\theta)$ has infinitely many negative eigenvalues counted without multiplicity.

This paper is organized as follows. In section 2, we recall basic facts on CR manifolds and the definition of the CR Paneitz operator. section 3 gives a brief exposition of the Heisenberg calculus, which is a main tool for studying the CR Paneitz operator. section 4 is devoted to proofs of sections 1 and 1 as applications of Heisenberg calculus. In section 5, we study the CR Paneitz operator on the Rossi sphere via spherical harmonics. In section 6, we propose some related problems and give some observations.

Let $M$ be a smooth three-dimensional manifold without boundary. A CR structure is a complex line subbundle $T^{1,0}M$ of the complexified tangent bundle $TM\otimes\mathbb{C}$ such that

where $T^{0,1}M$ is the complex conjugate of $T^{1,0}M$ in $TM\otimes\mathbb{C}$. Introduce an operator $\overline{\partial}_{b}\colon C^{\infty}(M)\to\Gamma((T^{0,1}M)^{\ast})$ by

A CR manifold $(M,T^{1,0}M)$ is said to be embeddable if there exists a smooth embedding $F$ from $M$ into some $\mathbb{C}^{N}$ such that $F_{\ast}T^{1,0}M\subset T^{1,0}\mathbb{C}^{N}$.

A CR structure $T^{1,0}M$ is said to be strictly pseudoconvex if there exists a nowhere-vanishing real one-form $\theta$ on $M$ such that $\theta$ annihilates $T^{1,0}M$ and

we call such a one-form a contact form. The triple $(M,T^{1,0}M,\theta)$ is called a pseudo-Hermitian manifold. Denote by $T$ the Reeb vector field with respect to $\theta$; that is, the unique vector field satisfying

Let $Z_{1}$ be a local frame of $T^{1,0}M$, and set $Z_{\overline{1}}=\overline{Z_{1}}$. Then $(T,Z_{1},Z_{\overline{1}})$ gives a local frame of $TM\otimes\mathbb{C}$, called an admissible frame. Its dual frame $(\theta,\theta^{1},\theta^{\overline{1}})$ is called an admissible coframe. The two-form $d\theta$ is written as

where $l_{1\overline{1}}$ is a positive function. We use $l_{1\overline{1}}$ and its multiplicative inverse $l^{1\overline{1}}$ to raise and lower indices.

A contact form $\theta$ induces a canonical connection $\nabla$, called the Tanaka-Webster connection with respect to $\theta$. It is defined by

with the following structure equations:

The tensor $A_{11}=\overline{A_{\overline{1}\overline{1}}}$ is called the Tanaka-Webster torsion. The curvature form ${\Omega_{1}}^{1}=d{\omega_{1}}^{1}$ of the Tanaka-Webster connection satisfies

where $\mathrm{Scal}$ is the Tanaka-Webster scalar curvature. We denote the components of a successive covariant derivative of a tensor by subscripts preceded by a comma, for example, $K_{1\overline{1},1}$; we omit the comma if the derivatives are applied to a function. We use the index $0$ for the component $T$ or $\theta$ in our index notation. In this notation, the operator $\overline{\partial}_{b}$ is given by

The commutators of the second derivatives for $u\in C^{\infty}(M)$ are given by

see [Lee1988]*(2.14). Define the Kohn Laplacian $\Box_{b}$ and the sub-Laplacian $\Delta_{b}$ by

for $u\in C^{\infty}(M)$. It follows from eq. 2.11 that

It is known that $(M,T^{1,0}M)$ is embeddable if and only if the Kohn Laplacian $\Box_{b}$ has closed range [Boutet_de_Monvel1975, Kohn1986].

The CR Paneitz operator $P$ is the fourth-order differential operator given by

where

This operator is real and formally self-adjoint; see [Gover-Graham2005]*Proposition 5.1 for example. Note that our $P$ is just the operator $P_{0,0}$ in this paper. Define an operator $\mathcal{C}\colon\Gamma((T^{0,1}M)^{\ast})\to C^{\infty}(M)$ by

It follows from eqs. 2.10 and 2.12 that $P=\mathcal{C}\overline{\partial}_{b}$. It is known that the CR Paneitz operator is non-negative if $(M,T^{1,0}M)$ is embeddable [Takeuchi2020-Paneitz]*Theorem 1.1. Conversely, $(M,T^{1,0}M)$ is embeddable if the CR Paneitz operator is non-negative and the Tanaka-Webster scalar curvature is positive [Chanillo-Chiu-Yang2012]*Theorem 1.4(a).

In this section, we recall basic properties of Heisenberg pseudodifferential operators; see [Beals-Greiner1988, Ponge2008-Book] for a comprehensive introduction to the Heisenberg calculus.

Throughout this section, we fix a closed pseudo-Hermitian manifold $(M,T^{1,0}M,\theta)$ of dimension three. Set

The Reeb vector field $T$ defines a nowhere-vanishing section $[T]$ of $TM/HM$. For sections $X_{0}$ and $Y_{0}$ of $TM/HM$ and $X^{\prime}$ and $Y^{\prime}$ of $HM$, the Lie bracket $[X_{0}+X^{\prime},Y_{0}+Y^{\prime}]$ is defined by

This bracket makes $\mathfrak{g}M$ a bundle of two-step nilpotent Lie algebras. The dilation $\delta_{r}$ on $\mathfrak{g}M$ is defined by

For $m\in\mathbb{Z}$, the space $S_{H}^{m}(M)$ consists of functions in $C^{\infty}((\mathfrak{g}M)^{\ast}\setminus\{0\})$ that are homogeneous of degree $m$ on each fiber. This space has a bilinear product

For $m\in\mathbb{Z}$, denote by $\Psi_{H}^{m}(M)$ the space of Heisenberg pseudodifferential operators $A\colon C^{\infty}(M)\to C^{\infty}(M)$ of order $m$. This space is closed under sum, complex conjugate, transpose, and formal adjoint [Ponge2008-Book]*Proposition 3.1.23. In particular, any $A\in\Psi_{H}^{m}(M)$ extends to a linear operator

where $\mathscr{D}^{\prime}(M)$ is the space of distributions on $M$. For example, $Z\in\Gamma(T^{1,0}M)$ is an element of $\Psi_{H}^{1}(M)$ and $T\in\Psi_{H}^{2}(M)$. Note that $\Psi_{H}^{-\infty}(M)\coloneqq\bigcap_{m\in\mathbb{Z}}\Psi_{H}^{m}(M)$ coincides with the space of smoothing operators on $M$. As in the usual pseudodifferential calculus, there exists the Heisenberg principal symbol, which has some good properties:

[[Ponge2008-Book]*Propositions 3.2.6 and 3.2.9] (i) The Heisenberg principal symbol $\sigma_{m}$ gives the following exact sequence:

(ii) For $A_{1}\in\Psi_{H}^{m_{1}}(M)$ and $A_{2}\in\Psi_{H}^{m_{2}}(M)$, the operator $A_{1}A_{2}$ is a Heisenberg pseudodifferential operator of order $m_{1}+m_{2}$, and

On the other hand, there exists a crucial difference between the usual pseudodifferential calculus and the Heisenberg one. Since the product of Heisenberg principal symbol is non-commutative, the commutator $[A_{1},A_{2}]$ of $A_{1}\in\Psi_{H}^{m_{1}}(M)$ and $A_{2}\in\Psi_{H}^{m_{2}}(M)$ is not an element of $\Psi_{H}^{m_{1}+m_{2}-1}(M)$ in general. However, we have the following

[[Takeuchi2023-GJMS]*Lemma 4.2] If $A\in\Psi_{H}^{m}(M)$, then $[T,A]\in\Psi_{H}^{m+1}(M)$.

Next we consider approximate inverses of Heisenberg pseudodifferential operators. We write $A\sim B$ if $A-B$ is a smoothing operator. Let $A\in\Psi_{H}^{m}(M)$. An operator $B\in\Psi_{H}^{-m}(M)$ is called a parametrix of $A$ if $AB\sim I$ and $BA\sim I$. The existence of a parametrix of a Heisenberg pseudodifferential operator is determined only by its Heisenberg principal symbol.

[[Ponge2008-Book]*Proposition 3.3.1] Let $A\in\Psi_{H}^{m}(M)$ with Heisenberg principal symbol $a\in S_{H}^{m}(M)$. Then the following are equivalent:

1. (1)

   $A$ has a parametrix;

2. (2)

   there exists $B\in\Psi_{H}^{-m}(M)$ such that $AB-I,BA-I\in\Psi_{H}^{-1}(M)$;

3. (3)

   there exists $b\in S_{H}^{-m}(M)$ such that $a\ast b=b\ast a=1$.

Now consider the Heisenberg differential operator $\Delta_{b}+I$ of order $2$. It is known that this operator has a parametrix; see the proof of [Ponge2008-Book]*Proposition 3.5.7 for example. Since $\Delta_{b}+I$ is positive and self-adjoint, the $k/2$-th power $(\Delta_{b}+I)^{k/2}$ of $\Delta_{b}+I$, $k\in\mathbb{Z}$, is a Heisenberg pseudodifferential operator of order $k$ [Ponge2008-Book]*Theorems 5.3.1 and 5.4.10. Using this operator, we define

This space is a Hilbert space with the inner product

The space $C^{\infty}(M)$ is dense in $W_{H}^{k}(M)$ for any $k\in\mathbb{Z}$, and we have

as topological vector spaces [Ponge2008-Book]*Proposition 5.5.3. Remark that the Hilbert space $W_{H}^{k}(M)$ coincides with the Folland-Stein space $S^{k,2}(M)$ as a topological vector space [Ponge2008-Book]*Proposition 5.5.5. Heisenberg pseudodifferential operators act on these Hilbert spaces as follows:

[[Ponge2008-Book]*Propositions 5.5.8 and [Takeuchi2023-GJMS]*Proposition 4.6] Any $A\in\Psi_{H}^{m}(M)$ extends to a continuous linear operator

for every $k\in\mathbb{Z}$. In particular if $m<0$, the operator $A\colon L^{2}(M)\to L^{2}(M)$ is compact.

Let $(M,T^{1,0}M,\theta)$ be a closed strictly pseudoconvex CR manifold of dimension three. If $(M,T^{1,0}M)$ is embeddable, then there exist the orthogonal projection to $\operatorname{Ker}\Box_{b}$, known as the Szegő projection, and the partial inverse of $\Box_{b}$ [Beals-Greiner1988]*Theorem 25.20. If $(M,T^{1,0}M)$ is non-embeddable, then we can not use the partial inverse of $\Box_{b}$. However, we have an approximate Szegő projection and partial inverse of $\Box_{b}$.

[[Beals-Greiner1988]*Proposition 25.4 and Corollaries 25.64 and 25.67] There exist $S\in\Psi_{H}^{0}(M)$ and $N\in\Psi_{H}^{-2}(M)$ such that

We first show some commutation relations of $S$ and $\overline{S}$.

One has $[S,\overline{\Box}_{b}],[\overline{S},\Box_{b}]\in\Psi_{H}^{1}(M)$.

Next we consider the composition $S$ and $\overline{S}$.

One has $S\overline{S},\overline{S}S\in\Psi_{H}^{-1}(M)$.

If $M$ is embeddable and $S$ is the Szegő projection, then we have $S\overline{S}\sim\overline{S}S\sim 0$; see [Hsiao2015]*Lemma 4.2.

Let $P$ be the CR Paneitz operator on $(M,T^{1,0}M,\theta)$. Since $P=\mathcal{C}\overline{\partial}_{b}$, we have $PS\sim 0$. Taking the complex conjugate yields $P\overline{S}\sim 0$. Hence $\Pi_{0}\coloneqq S+\overline{S}\in\Psi_{H}^{0}(M)$ satisfies $P\Pi_{0}\sim 0$. Set $G_{0}\coloneqq N\overline{N}\in\Psi_{H}^{-4}(M)$. It follows from sections 4 and 4 that

modulo $\Psi_{H}^{-1}(M)$; note that $\mathcal{Q}\in\Psi_{H}^{2}(M)$. Thus we have

The operator $I+R_{0}\in\Psi_{H}^{0}(M)$ has a parametrix $A_{0}\in\Psi_{H}^{0}(M)$. Moreover, $A_{0}$ satisfies $A_{0}-I\in\Psi_{H}^{-1}(M)$.

The proof of the following proposition is inspired by that of [Beals-Greiner1988]*Proposition 25.4.

There exist $\Pi_{\infty}\in\Psi_{H}^{0}(M)$ and $G_{\infty}\in\Psi_{H}^{-4}(M)$ such that

Consider $P$ as an unbounded closed operator acting on $L^{2}(M)$ by the maximal closed extension. The domain $\operatorname{Dom}P$ contains $W_{H}^{4}(M)$ by section 3. Conversely, any $u\in\operatorname{Dom}P$ is an element of $W_{H}^{4}(M)$ modulo $\operatorname{Ran}\Pi_{\infty}$ by the lemma below.

For $u\in\operatorname{Dom}P$, one has $u-\Pi_{\infty}u\in W_{H}^{4}(M)$. In particular, $\operatorname{Dom}P=\operatorname{Ran}\Pi_{\infty}+W_{H}^{4}(M)$.

Let $E$ be the resolution of the identity for $P$ and fix $\lambda\geq 0$. Set

This is an orthogonal projection of $L^{2}(M)$ and satisfies $\pi_{\lambda}P=P\pi_{\lambda}$ on $\operatorname{Dom}P$. If $\lambda>0$,

is a continuous self-adjoint operator and satisfies

If $\lambda>0$, then $P\pi_{\lambda}\sim 0$.

If $\lambda>0$, then $\pi_{\lambda}$ and $N_{\lambda}$ are Heisenberg pseudodifferential operators of order $0$ and $-4$ respectively. Moreover, $\pi_{\lambda}$ and $N_{\lambda}$ coincide with $\Pi_{\infty}$ and $G_{\infty}$ respectively modulo $\Psi_{H}^{-\infty}(M)$.