aa r X i v : . [ m a t h . DG ] S e p ANALYSIS OF THE CRITICAL CR GJMS OPERATOR
YUYA TAKEUCHI
Abstract.
The critical CR GJMS operator on a strictly pseudoconvex CRmanifold is a non-hypoelliptic CR invariant differential operator. We provethat, under the embeddability assumption, it is essentially self-adjoint and hasclosed range. Moreover, its spectrum is discrete, and the eigenspace corre-sponding to each non-zero eigenvalue is a finite-dimensional subspace of thespace of smooth functions. As an application, we obtain a necessary and suffi-cient condition for the existence of a contact form with zero CR Q -curvature. Contents
1. Introduction 12. CR manifolds 33. Model operators on the Heisenberg group 44. Heisenberg calculus 75. Proofs of the main results 10Acknowledgements 15References 151.
Introduction
It is one of the most important topics in both conformal and CR geometriesto study invariant differential operators. Analytic properties of such operators aredeeply connected to geometric problems, such as the Yamabe problem and theconstant Q -curvature problem.In conformal geometry, Graham, Jenne, Mason, and Sparling [GJMS92] haveconstructed a family of conformally invariant differential operators, called GJMSoperators. Let ( N, g ) be a Riemannian manifold of dimension n . For k ∈ N and k ≤ n/ n is even, the k -th GJMS operator P k is a differential operator acting on C ∞ ( N ) such that its principal part coincides with the k -th power of the Laplacian,and it has the following transformation law under the conformal change ˆ g = e g : e ( n/ k )Υ b P k = P k e ( n/ − k )Υ , where b P k is defined in terms of ˆ g . Analytic properties of P k on closed manifoldsare quite simple. From standard elliptic theory, it follows that P k is essentially self-adjoint and has closed range. Moreover, its spectrum is a discrete subset of R , and Mathematics Subject Classification.
Key words and phrases. critical CR GJMS operator, CR Q-curvature, Szegő projection, CRpluriharmonic function, Heisenberg calculus.This work was supported by JSPS Research Fellowship for Young Scientists and JSPS KAK-ENHI Grant Numbers JP16J04653 JP19J00063. the eigenspace corresponding to each eigenvalue is a finite-dimensional subspace of C ∞ ( N ).In CR geometry, Gover and Graham [GG05] have introduced a family of CRinvariant differential operators, called CR GJMS operators, via Fefferman con-struction. Let ( M, T , M, θ ) be a (2 n + 1)-dimensional pseudo-Hermitian manifoldand k ∈ N with k ≤ n + 1. The k -th CR GJMS operator P k is a differentialoperator acting on C ∞ ( M ) such that its principal part is the k -th power of thesub-Laplacian, and its transformation rule under the conformal change ˆ θ = e Υ θ isgiven by e ( n +1+ k )Υ / b P k = P k e ( n +1 − k )Υ / , where b P k is defined in terms of ˆ θ . Although P k is not elliptic, it is known to be subelliptic for 1 ≤ k ≤ n [Pon08]; in particular, the same statements as in theprevious paragraph also hold for P k on closed manifolds. However, the critical CRGJMS operator P n +1 is not even hypoelliptic. In fact, its kernel contains the spaceof CR pluriharmonic functions, which is infinite-dimensional on closed embeddableCR manifolds. In this paper, nevertheless, we will prove that similar results tothe above are true for P n +1 on the orthogonal complement of Ker P n +1 . In whatfollows, we simply write P for the critical CR GJMS operator.In the remainder of this section, let ( M, T , M, θ ) be a closed embeddablepseudo-Hermitian manifold of dimension 2 n + 1. Note that the embeddabilityautomatically holds if n ≥ Theorem 1.1.
The maximal closed extension of P is self-adjoint and has closedrange. We use the same letter P for the maximal closed extension of the critical CRGJMS operator by abuse of notation. Moreover, we obtain the following theoremon the spectrum of P : Theorem 1.2.
The spectrum of P is a discrete subset in R and consists only ofeigenvalues. Moreover, the eigenspace corresponding to each non-zero eigenvalueof P is a finite-dimensional subspace of C ∞ ( M ) . Furthermore, Ker P ∩ C ∞ ( M ) isdense in Ker P . In dimension three, Hsiao [Hsi15] has shown Theorems 1.1 and 1.2 by usingFourier integral operators with complex phase. Our proofs are similar to Hsiao’sones, but based on the
Heisenberg calculus , the theory of Heisenberg pseudodiffer-ential operators. The use of these operators simplifies some proofs and gives moreprecise regularity results.We will also give some applications of these theorems and their proofs. Let P and P be the space of CR pluriharmonic functions and its L -closure respectively.Then Ker P contains P , and the supplementary space W is defined by W := Ker P ∩ P ⊥ . Proposition 1.3.
The supplementary space W is a finite dimensional subspace of C ∞ ( M ) . In dimension three, Proposition 1.3 has been already proved by Hsiao [Hsi15].However, in this case, the author [Tak19] has shown that W is equal to zero. Onthe other hand, for each n ≥
2, there exists a closed pseudo-Hermitian manifold
NALYSIS OF THE CRITICAL CR GJMS OPERATOR 3 ( M, T , M, θ ) of dimension 2 n + 1 such that W = 0; see the proof of [Tak18,Theorem 1.6].We will also tackle the zero CR Q -curvature problem. The CR Q -curvature Q ,introduced by Fefferman and Hirachi [FH03], is a smooth function on M such thatit transforms as follows under the conformal change ˆ θ = e Υ θ :(1.1) b Q = e − ( n +1)Υ ( Q + P Υ) , where b Q is defined in terms of ˆ θ . Marugame [Mar18] has proved that the total CR Q -curvature Q := Z M Qθ ∧ ( dθ ) n is always equal to zero. Moreover, the CR Q -curvature itself is identically zerofor pseudo-Einstein contact forms [FH03]. Hence it is natural to ask whether( M, T , M ) admits a contact form whose CR Q -curvature vanishes identically; thisis the zero CR Q -curvature problem . This problem has been solved affirmativelyfor embeddable CR three-manifolds by the author [Tak19]. However, it is still openin general. By the transformation law (1.1), it is necessary that Z M f Qθ ∧ ( dθ ) n = 0holds for any f ∈ Ker P ∩ C ∞ ( M ). Note that this condition is independent of thechoice of θ . The following proposition states that it is also a sufficient condition forembeddable CR manifolds: Proposition 1.4.
There exists a contact form ˆ θ on M such that the CR Q -curvature b Q vanishes identically if and only if Q ⊥ (Ker P ∩ C ∞ ( M )) . This paper is organized as follows. In Section 2, we recall basic facts on CR man-ifolds. Section 3 deals with convolution operators on the Heisenberg group, whichis a “model” of the Heisenberg calculus. In Section 4, we give a brief exposition ofthe Heisenberg calculus. Section 5 is devoted to proofs of the main results in thispaper. 2.
CR manifolds
Let M be an orientable smooth (2 n +1)-dimensional manifold without boundary.A CR structure is a rank n complex subbundle T , M of the complexified tangentbundle T M ⊗ C such that T , M ∩ T , M = 0 , (cid:2) Γ( T , M ) , Γ( T , M ) (cid:3) ⊂ Γ( T , M ) , where T , M is the complex conjugate of T , M in T M ⊗ C . Define a hyperplanebundle HM of T M by HM := Re T , M . A typical example of CR manifolds isa real hypersurface M in an ( n + 1)-dimensional complex manifold X ; this M hasthe canonical CR structure T , M := T , X | M ∩ ( T M ⊗ C ) . Take a nowhere-vanishing real one-form θ on M such that θ annihilates T , M .The Levi form L θ with respect to θ is the Hermitian form on T , M defined by L θ ( Z, W ) := −√− dθ ( Z, W ) , Z, W ∈ T , M. A CR structure T , M is said to be strictly pseudoconvex if the Levi form is positivedefinite for some θ ; such a θ is called a contact form . The triple ( M, T , M, θ ) is
YUYA TAKEUCHI called a pseudo-Hermitian manifold . Denote by T the Reeb vector field with respectto θ ; that is, the unique vector field satisfying θ ( T ) = 1 , T y dθ = 0 . Define an operator ∂ b : C ∞ ( M ) → Γ(( T , M ) ∗ ) by ∂ b f := df | T , M . A smooth function f is called a CR holomorphic function if ∂ b f = 0. A CRpluriharmonic function is a real-valued smooth function that is locally the real partof a CR holomorphic function. We denote by P the space of CR pluriharmonicfunctions.By using the Levi form and the volume form θ ∧ ( dθ ) n , we obtain the formaladjoint ∂ ∗ b : Γ(( T , M ) ∗ ) → C ∞ ( M ) of ∂ b . The Kohn Laplacian (cid:3) b and the sub-Laplacian ∆ b are defined by (cid:3) b := ∂ ∗ b ∂ b , ∆ b := (cid:3) b + (cid:3) b . Note that (cid:3) b = 12 ∆ b + √− nT ;see [Lee86, Theorem 2.3] for example. The Gaffney extension of the Kohn Lapla-cian, also denoted by (cid:3) b , is a self-adjoint operator on L ( M ). The kernel Ker (cid:3) b is the space of L CR holomorphic functions.The critical CR GJMS operator P is a differential operator of order 2 n + 2acting on C ∞ ( M ). It is known to be formally self-adjoint [GG05, Proposition 5.1].Moreover, it annihilates CR pluriharmonic functions [Hir14, Section 3.2].A CR manifold ( M, T , M ) is said to be embeddable if there exists a smoothembedding of M to some C N such that T , M = T , C N | M ∩ ( T M ⊗ C ). It isknown that a closed strictly pseudoconvex CR manifold ( M, T , M ) is embeddableif and only if (cid:3) b has closed range [BdM75, Koh86].3. Model operators on the Heisenberg group
The Heisenberg group G is the Lie group with the underlying manifold R × C n and the multiplication( t, z ) · ( t ′ , z ′ ) := ( t + t ′ + 2 Im( z · z ′ ) , z + z ′ ) . The left translation by ( t, z ) and the inversion on G are denoted by l ( t,z ) and ι respectively.For α = 1 , . . . , n , we introduce a left-invariant complex vector field Z α by Z α := ∂∂z α + √− z α ∂∂t . The canonical CR structure T , G is spanned by Z , . . . , Z n . Define a left-invariantone-form θ on G by θ := dt + √− n X α =1 ( z α dz α − z α dz α ) . Then θ annihilates T , G and the Levi form L θ satisfies L θ ( Z α , Z β ) = 2 δ αβ ; inparticular, θ is a contact form on G . The Reeb vector field T coincides with ∂/∂t . NALYSIS OF THE CRITICAL CR GJMS OPERATOR 5
The Lie algebra g of G is isomorphic to R × C n as a linear space via g → R × C n ; tT + 2 n X α =1 Re( z α Z α ) ( t, z ) . Under this identification, the Lie bracket on g is given by[( t, z ) , ( t ′ , z ′ )] = (4 Im( z · z ′ ) , . Moreover, the exponential map g → G coincides with the identity map on R × C n .Furthermore, the dual g ∗ of g is also canonically isomorphic to R × C n as a linearspace. We write this linear coordinate as ( τ, ζ ).For r ∈ R + , the parabolic dilation δ r on R × C n is defined by δ r ( t, z ) = ( r t, rz ) . This dilation defines automorphisms on G , g , and g ∗ , for which we will use thesame letter δ r by abuse of notation. In what follows, the term “homogeneous” isdefined in terms of δ r . We will sometime write v for a point of G . Denote by dv the Lebesgue measure on G , which is a Haar measure on G .Let S ( G ) (resp. S ( g ∗ )) be the space of rapidly decreasing functions on G (resp. g ∗ ), and S ′ ( G ) (resp. S ′ ( g ∗ )) be that of tempered distributions on G (resp. g ∗ ).The coupling of f ∈ S ( G ) and k ∈ S ′ ( G ) is written as h k, f i . The pull-back by δ r induces endomorphisms on S ( G ) and S ( g ∗ ), and these extend to those on S ′ ( G )and S ′ ( g ∗ ). The Fourier transform F defines isomorphisms S ( G ) ∼ = −→ S ( g ∗ ) , S ′ ( G ) ∼ = −→ S ′ ( g ∗ );in our convention, the Fourier transform F ( f ) of f ∈ S ( G ) is defined by F ( f )( τ, ζ ) := Z G e −√− tτ +Re( z · ζ )) f ( t, z ) dv. Now we consider “model operators” of the Heisenberg calculus. For m ∈ R , setΣ mH := { a ∈ C ∞ ( g ∗ \ { } ) | δ ∗ r a = r m a } , which is the space of Heisenberg symbols of order m . Let G m be the space of g ∈ S ′ ( g ∗ ) such that g is smooth on g ∗ \ { } and satisfies δ ∗ r g = r m g + ( r m log r ) h, where h ∈ S ′ ( g ∗ ) with supp h ⊂ { } and δ ∗ r h = r m h . The restriction map G m → Σ mH is known to be surjective [BG88, Proposition 15.8]. Moreover, the inverseFourier transform gives an isomorphism F − : G m ∼ = −→ K − m − n − , where K l is the space of k ∈ S ′ ( G ) such that k is smooth on G \ { } and satisfies δ ∗ r k = r l k + ( r l log r ) ψ for a homogeneous polynomial ψ of degree l [BG88, Proposition 15.24]. We alsointroduce a function space on which Heisenberg symbols act. Let S ( G ) be thespace of f ∈ S ( G ) such that Z G ψ ( v ) f ( v ) dv = 0for any polynomial ψ on G . This condition is equivalent to that F ( f ) ∈ S ( g ∗ )vanishes to infinite order at the origin. YUYA TAKEUCHI
We denote by Ψ mH the space of endomorphisms A on S ( G ) commuting with lefttranslation and admitting its formal adjoint A ∗ of homogeneous degree m ; that is, A ∗ ◦ δ ∗ r = r m δ ∗ r ◦ A ∗ . We would like to define a canonical isomorphism between Σ mH and Ψ mH . Proposition 3.1.
Let a ∈ Σ mH and take g ∈ G m with g | g ∗ \{ } = a . Then theconvolution operator (3.1) f [ F − ( g ) ∗ f ]( v ) := (cid:10) F − ( g ) , f ◦ l v ◦ ι (cid:11) defines an endomorphism on S ( G ) and is independent of the choice of g . More-over, this operator commutes with left translation and is homogeneous of degree m .Furthermore, it is equal to zero if and only if a = 0 . Definition 3.2.
For a ∈ Σ mH , an operator O ( a ) : S ( G ) → S ( G ) is defined by(3.1). Proof of Proposition 3.1.
It follows from [CGGP92, Proposition 2.2] that (3.1) de-fines an endomorphism on S ( G ) commuting with left translation and homoge-neous of degree m . Assume that g ′ also satisfies g ′ | g ∗ \{ } = a . Then the supportof g ′ − g is contained in { } ⊂ g ∗ . Hence F − ( g ′ − g ) is a polynomial on G , and so F − ( g ′ − g ) ∗ f = 0 for any f ∈ S ( G ). This implies the independence of the choiceof g . Next, suppose that the operator (3.1) is equal to zero. For any f ∈ S ( G ),we have (cid:10) F − ( g ) , f ◦ ι (cid:11) = 0. Hence g annihilates F ( S ( G )). Since C ∞ c ( g ∗ \ { } )is a subspace of F ( S ( G )), the support of g is contained in { } ⊂ g ∗ . Therefore a = g | g ∗ \{ } = 0. (cid:3) The operator O ( a ) is well-behaved under formal adjoint and composition. Theorem 3.3. (i) The formal adjoint of O ( a ) , a ∈ Σ mH , is given by O ( a ) . Inparticular, O ( a ) is formally self-adjoint if and only if a is real-valued.(ii) There exists a bilinear product ∗ : Σ m H × Σ m H → Σ m + m H such that O ( a ) O ( a ) = O ( a ∗ a ) for any a ∈ Σ m H and a ∈ Σ m H .Proof. (i) Take g ∈ G m with g | g ∗ \{ } = a The formal adjoint of O ( a ) is given bythe convolution with respect to F − ( g ) ◦ ι = F − ( g );see [CGGP92, Section 3]. Thus we have ( O ( a )) ∗ = O ( a ).(ii) See [Pon08, Proposition 3.1.3(2)]. (cid:3) In particular, O defines an injective map from Σ mH to Ψ mH . In fact, this is anisomorphism. Proposition 3.4.
For any A ∈ Ψ mH , there exists the unique a ∈ Σ mH such that A = O ( a ) .Proof. Let A ∈ Ψ mH . By [CGGP92, Proposition 3.2], we have k ∈ K − m − n − suchthat Af = k ∗ f for any f ∈ S ( G ). If we define a ∈ Σ mH by a := ( F k ) | g ∗ \{ } , then O ( a ) coincides with A by definition. (cid:3) NALYSIS OF THE CRITICAL CR GJMS OPERATOR 7
Definition 3.5.
The
Heisenberg symbol σ m : Ψ mH → Σ mH is defined by the inverse map of O .It follows from Theorem 3.3 that σ m ( A ∗ ) = σ m ( A ) , σ m + m ( A A ) = σ m ( A ) ∗ σ m ( A )for A ∈ Ψ mH , A ∈ Ψ m H , and A ∈ Ψ m H . In particular, A is formally self-adjoint ifand only if σ m ( A ) is real-valued.Before the end of this section, we note a relation between the Reeb vector fieldand Ψ mH . Lemma 3.6.
The Reeb vector field T commutes with any A ∈ Ψ mH .Proof. The vector field T generates the flow l ( t, . Since A ∈ Ψ mH commutes withleft translation, we have (cid:2) T , A (cid:3) = 0. (cid:3) Heisenberg calculus
In this section, we recall basic properties of Heisenberg pseudodifferential opera-tors; see [BG88,Pon08] for a comprehensive introduction to the Heisenberg calculus.Throughout this section, we fix a closed pseudo-Hermitian manifold (
M, T , M, θ )of dimension 2 n + 1. Let g M := ( T M/HM ) ⊕ HM.
The Reeb vector field T defines a nowhere-vanishing section [ T ] of T M/HM .For sections X and Y of T M/HM and X ′ and Y ′ of HM , the Lie bracket[ X + X ′ , Y + Y ′ ] is defined by[ X + X ′ , Y + Y ′ ] := − dθ ( X ′ , Y ′ )[ T ] . This bracket makes g M a bundle of two-step nilpotent Lie algebras. The dilation δ r on g M is defined by δ r | T M/HM := r , δ r | HM := r. It follow from the definition of the Lie bracket that δ r is a fiberwise Lie algebraisomorphism. Set GM := g M as a smooth fiber bundle with the fiberwise groupstructure defined via the Baker-Campbell-Hausdorff formula. The dilation δ r on g M induces that on GM , which we write as δ r for abbreviation.Take a local frame ( Z α ) of T , M on an open set U ⊂ M such that L θ ( Z α , Z β ) = 2 δ αβ . Then the map(4.1) g M | U → U × g ; p, t [ T ] + 2 Re n X α =1 z α Z α ! ( p, t, z )gives an isomorphism between fiber bundles of Lie algebras. This isomorphism iscompatible with the dilation. The identification (4.1) induces those on GM andthe dual bundle g ∗ M := ( g M ) ∗ of g M :(4.2) GM | U → U × G, g ∗ M | U → U × g ∗ . These are also compatible with the dilation. Let ( Z ′ α ) be another local frameof T , M on U satisfying L θ ( Z ′ α , Z ′ β ) = 2 δ αβ . This gives another identification YUYA TAKEUCHI g M | U → U × g . These two identifications relate with each other by a smoothfamily ( U ( p )) p ∈ U of unitary matrices; that is, U × g → U × g ; ( p, t, z ) ( p, t, U ( p ) · z ) . The same is true for GM and g ∗ M .For m ∈ R , the space Σ mH ( M ) consists of functions in C ∞ ( g ∗ M \ { } ) that arehomogeneous of degree m on each fiber. Under the identification (4.2), the fiberwiseproduct ∗ induces a well-defined bilinear product ∗ : Σ m H ( M ) × Σ m H ( M ) → Σ m + m H ( M ) . Now we consider Heisenberg pseudodifferential operators. For m ∈ R , denote byΨ mH ( M ) the space of Heisenberg pseudodifferential operators A : C ∞ ( M ) → C ∞ ( M ) of order m . This space is closed under complex conjugate, transpose, and formaladjoint [Pon08, Proposition 3.1.23]. In particular, any A ∈ Ψ mH extends to a linearoperator A : D ′ ( M ) → D ′ ( M ) , where D ′ ( M ) is the space of distributions on M . For example, V ∈ Γ( HM ) is an el-ement of Ψ H ( M ) and T ∈ Ψ H ( M ). Note that Ψ −∞ H ( M ) := T m ∈ R Ψ mH ( M ) coincideswith the space of smoothing operators on M . As in the usual pseudodifferentialcalculus, there exists the Heisenberg principal symbol σ m : Ψ mH ( M ) → Σ mH ( M ) , which has the following properties: Proposition 4.1 ([Pon08, Propositions 3.2.6 and 3.2.9]) . (i) The Heisenberg prin-cipal symbol σ m gives the following exact sequence: → Ψ m − H ( M ) → Ψ mH ( M ) σ m −−→ Σ mH ( M ) → . (ii) For A ∈ Ψ m H ( M ) and A ∈ Ψ m H ( M ) , the operator A A is a Heisenbergpseudodifferential operator of order m + m , and σ m + m ( A A ) = σ m ( A ) ∗ σ m ( A ) . On the other hand, there exists a crucial difference between the usual pseudodif-ferential calculus and the Heisenberg one. Since the product ∗ is non-commutative,the commutator [ A , A ] of A ∈ Ψ m H ( M ) and A ∈ Ψ m H ( M ) is not an element ofΨ m + m − H ( M ) in general. However, we have the following Lemma 4.2.
Let A ∈ Ψ mH ( M ) . Then [ T, A ] ∈ Ψ m +1 H ( M ) .Proof. It is enough to show that σ m +2 ([ T, A ]) = 0, or equivalently, σ ( T ) ∗ σ m ( A ) = σ m ( A ) ∗ σ ( T ) . Fix an identification (4.2). Then σ ( T ) ∈ Σ H ( M ) is given by σ ( T )( p, τ, ζ ) = √− τ = σ ( T )( τ, ζ );see [Pon08, Example 3.25]. Hence it suffices to prove that σ ( T ) ∗ a = a ∗ σ ( T )holds for any a ∈ Σ mH . From Lemma 3.6, we obtain O ( σ ( T ) ∗ a ) = T O ( a ) = O ( a ) T = O ( a ∗ σ ( T )) , which is equivalent to σ ( T ) ∗ a = a ∗ σ ( T ). (cid:3) NALYSIS OF THE CRITICAL CR GJMS OPERATOR 9
Next, consider approximate inverses of Heisenberg pseudodifferential operators.We write A ∼ B if A − B is a smoothing operator. Definition 4.3.
Let A ∈ Ψ mH ( M ). An operator B ∈ Ψ − mH ( M ) is called a parametrix of A if AB ∼ I and BA ∼ I .The existence of a parametrix of a Heisenberg pseudodifferential operator isdetermined only by its Heisenberg principal symbol. Proposition 4.4 ([Pon08, Proposition 3.3.1]) . Let A ∈ Ψ mH ( M ) with Heisenbergprincipal symbol a ∈ Σ mH ( M ) . Then the following are equivalent: (1) A has a parametrix; (2) there exists B ∈ Ψ − mH ( M ) such that AB − I, BA − I ∈ Ψ − H ( M ) ; (3) there exists b ∈ Σ − mH ( M ) such that a ∗ b = b ∗ a = 1 . Now consider the Heisenberg differential operator ∆ b + 1 of order 2. It is knownthat this operator has a parametrix; see the proof of [Pon08, Proposition 3.5.7] forexample. Since ∆ b +1 is positive and self-adjoint, the s -th power (∆ b +1) s of ∆ b +1, s ∈ R , is a Heisenberg pseudodifferential operator of order 2 s [Pon08, Theorems5.3.1 and 5.4.10]. Using this operator, we define W sH ( M ) := n u ∈ D ′ ( M ) | (∆ b + 1) s/ u ∈ L ( M ) o . This space is a Hilbert space with the inner product( u, v ) s = (cid:16) (∆ b + 1) s/ u, (∆ b + 1) s/ v (cid:17) L ( M ) ;write k·k s for the norm determined by ( · , · ) s . The space C ∞ ( M ) is dense in W sH ( M ),and C ∞ ( M ) = T s ∈ R W sH ( M ) [Pon08, Proposition 5.5.3]. Note that, for k ∈ N ,the Hilbert space W kH ( M ) coincides with the Folland-Stein space S k, ( M ) as atopological vector space [Pon08, Proposition 5.5.5]. Similar to the usual L -Sobolevspace theory, we obtain the following Lemma 4.5.
For s < s , the embedding W s H ( M ) ֒ → W s H ( M ) is compact.Proof. The operator (∆ b + 1) s ′ / , s ′ ∈ R , gives an isometry W s + s ′ H ( M ) → W sH ( M ),and so we may assume that s = 0. From [Pon08, Proposition 5.5.7], we derivethat the embedding W s H ( M ) ֒ → W H ( M ) = L ( M ) is the composition of the twoembeddings W s H ( M ) ֒ → H s / ( M ) and H s / ( M ) ֒ → L ( M ), where H s ( M ) is theusual L -Sobolev space on M of order s . Thus the compactness of W s H ( M ) ֒ → L ( M ) follows from Rellich’s lemma. (cid:3) Heisenberg pseudodifferential operators act on these Hilbert spaces as follows:
Proposition 4.6.
Any A ∈ Ψ mH ( M ) extends to a continuous linear operator A : W s + mH ( M ) → W sH ( M ) for every s ∈ R . In particular if m < , the operator A : L ( M ) → L ( M ) iscompact.Proof. The former statement follows from [Pon08, Propositions 5.5.8]. The latterone is a consequence of the former one and Lemma 4.5. (cid:3) Proofs of the main results
In this section, we prove the main results in this paper. In what follows, we fix aclosed embeddable pseudo-Hermitian manifold (
M, T , M, θ ) of dimension 2 n + 1.For µ ∈ R , we define a formally self-adjoint Heisenberg differential operator L µ of order 2 by L µ := 12 ∆ b + √− µT. It is known that L µ has a parametrix N µ ∈ Ψ − H ( M ) if and only if µ / ∈ ± ( n + 2 N );see the proof of [Pon08, Proposition 3.5.7] for example. On the other hand, theembeddability of M implies that there exist the partial inverse N n ∈ Ψ − H ( M )of L n = (cid:3) b and the orthogonal projection S ∈ Ψ H ( M ) to Ker (cid:3) b , called the Szegő projection [BG88, Theorem 24.20 and Corollary 25.67]. Taking the complexconjugate gives the partial inverse N − n ∈ Ψ − H ( M ) of L − n = (cid:3) b and the orthogonalprojection S ∈ Ψ H ( M ) to Ker (cid:3) b Lemma 5.1.
For any µ ∈ R , one has [ L µ , S ] ∈ Ψ H ( M ) .Proof. We have[ L µ , S ] = [ L n , S ] + √−
12 ( µ − n )[ T, S ] = √−
12 ( µ − n )[ T, S ] ∈ Ψ H ( M )by Lemma 4.2. (cid:3) This lemma implies a property of S and S . Lemma 5.2.
One has
SS, SS ∈ Ψ − H ( M ) .Proof. Since SL n = L − n S = 0, we have L SS = [ L , S ] S + SL S = [ L , S ] S + 12 S ( L n + L − n ) S = [ L , S ] S. By Lemma 5.1, [ L , S ] S ∈ Ψ H ( M ). On the other hand, L has a parametrix N ∈ Ψ − H ( M ). Hence SS ∼ N [ L , S ] S ∈ Ψ − H ( M ) . Taking the complex conjugate yields SS ∈ Ψ − H ( M ). (cid:3) The critical CR GJMS operator P on ( M, T , M, θ ) coincides with L − n L − n +2 · · · L n − L n modulo Ψ n +1 H ( M ); see [Pon08, Proposition 3.5.7]. Set G := N n N n − · · · N − n +2 N − n ∈ Ψ − n − H ( M ) , Π := S + S ∈ Ψ H ( M ) NALYSIS OF THE CRITICAL CR GJMS OPERATOR 11
Then modulo Ψ − H ( M ), P G ≡ L − n L − n +2 · · · L n − L n N n N n − · · · N − n +2 N − n = L − n L − n +2 · · · L n − ( I − S ) N n − · · · N − n +2 N − n ≡ ( I − S ) L − n L − n +2 · · · L n − N n − · · · N − n +2 N − n ∼ ( I − S )( I − S )= I − S − S + SS ≡ I − Π . Thus we have R := P G + Π − I ∈ Ψ − H ( M ) . Lemma 5.3.
The operator I + R ∈ Ψ H ( M ) has a parametrix A ∈ Ψ H ( M ) .Moreover, A satisfies A − I ∈ Ψ − H ( M ) .Proof. Since I ( I + R ) − I = ( I + R ) I − I = R ∈ Ψ − H ( M ) ,I + R has a parametrix A by Proposition 4.4. From R ∈ Ψ − H ( M ) and Proposi-tion 4.1, we obtain σ ( A ) = σ (( I + R ) A ) = σ ( I ) , which means A − I ∈ Ψ − H ( M ). (cid:3) The proof of the following proposition is inspired by that of [BG88, Proposition25.4].
Proposition 5.4.
There exist G ∞ ∈ Ψ − n − H ( M ) and Π ∞ ∈ Ψ H ( M ) such that G ∗∞ ∼ G ∞ , Π ∗∞ ∼ Π ∞ ∼ Π ∞ ,P G ∞ + Π ∞ ∼ G ∞ P + Π ∞ ∼ I, Π ∞ P ∼ P Π ∞ = 0 , Π ∞ G ∞ ∼ G ∞ Π ∞ ∼ , Π ∞ − Π ∈ Ψ − H ( M ) . Proof.
Let A ∈ Ψ H ( M ) be a parametrix of I + R , and setΠ ∞ := Π A ∈ Ψ H ( M ) , G ∞ := ( I − Π ∞ ) G A ∈ Ψ − n − H ( M ) . Note that Π ∞ − Π = Π ( A − I ) ∈ Ψ − H ( M ) . Since P Π = 0, we have P Π ∞ = 0 and Π ∗∞ P = 0. Moreover, P G ∞ + Π ∞ = ( P G + Π ) A = ( I + R ) A ∼ I, G ∗∞ P + Π ∗∞ ∼ I. Hence Π ∗∞ ∼ Π ∗∞ ( P G ∞ + Π ∞ ) = Π ∗∞ Π ∞ = ( G ∗∞ P + Π ∗∞ )Π ∞ ∼ Π ∞ . We also have Π ∞ G ∞ = (Π ∞ − Π ∞ ) G A ∼ and G ∞ Π ∞ ∼ ( G ∗∞ P + Π ∗∞ ) G ∞ Π ∞ ∼ G ∗∞ ( I − Π ∞ )Π ∞ = G ∗∞ (Π ∞ − Π ∞ ) ∼ . Therefore G ∗∞ ∼ G ∗∞ ( I − Π ∞ ) ∼ G ∗∞ P G ∞ ∼ ( G ∗∞ P + Π ∞ ) G ∞ ∼ G ∞ , which completes the proof. (cid:3) Consider P as an unbounded closed operator on L ( M ) by the maximal closedextension. The domain Dom P contains W n +2 H ( M ) by Proposition 4.6. Conversely,any u ∈ Dom P is an element of W n +2 H ( M ) modulo Ker P by the lemma below. Lemma 5.5.
For u ∈ Dom P , one has u − Π ∞ u ∈ W n +2 H ( M ) . In particular, Dom P = Ker P + W n +2 H ( M ) .Proof. Set R ∞ := G ∞ P + Π ∞ − I ∈ Ψ −∞ H ( M ) . If v = P u ∈ L ( M ), then u − Π ∞ u = G ∞ v − R ∞ u ∈ W n +2 H ( M ) . In particular, u ∈ Ker P + W n +2 H ( M ) since Π ∞ u ∈ Ker P . (cid:3) Lemma 5.6.
The range
Ran P of P is orthogonal to Ran Π ∞ in L ( M ) .Proof. Assume u ∈ Dom P and v ∈ L ( M ). Take a sequence ( v j ) ∈ C ∞ ( M ) suchthat v j converges to v in L ( M ) as j → + ∞ . Since Π ∞ ∈ Ψ H ( M ), the functionΠ ∞ v j is smooth and converges to Π ∞ v in L ( M ) as j → + ∞ also. Hence( P u, Π ∞ v ) = lim j →∞ ( P u, Π ∞ v j ) = lim j →∞ ( u, P Π ∞ v j ) = 0 , which completes the proof. (cid:3) Proof of Theorem 1.1.
We first prove that P is self-adjoint. To this end, it is enoughto show that P is symmetric. Let u, v ∈ Dom P . From Lemma 5.5, it followsthat v ′ := v − Π ∞ v is in W n +2 H ( M ). Take a sequence ( v j ) in C ∞ ( M ) such that v j converges to v ′ in W n +2 H ( M ) as j → + ∞ . Then P v j converges to P v ′ = P v in L ( M ) as j → + ∞ by the continuity of P : W n +2 H ( M ) → L ( M ). From NALYSIS OF THE CRITICAL CR GJMS OPERATOR 13
Lemma 5.6, we derive (
P u, v ) = ( P u, v ′ ) + ( P u, Π ∞ v ) = lim j →∞ ( P u, v j ) = lim j →∞ ( u, P v j ) = ( u, P v ) , which means that P is symmetric.We next prove that P : Dom P → L ( M ) has closed range. It suffices to showthat there exists ǫ > k P u k ≥ ǫ k u k for any u ∈ Dom P ∩ (Ker P ) ⊥ . Note that (Ker P ) ⊥ ⊂ Ker Π ∗∞ since Ran Π ∞ ⊂ Ker P . Set(5.1) R ′∞ := P G ∞ + Π ∞ − I ∈ Ψ −∞ H ( M ) . Note that(5.2) G ∗∞ P + Π ∗∞ = I + ( R ′∞ ) ∗ . Suppose that we can take a sequence ( u j ) in Dom P ∩ (Ker P ) ⊥ such that k u j k = 1 , k P u j k ≤ j . From (5.2), it follows that u j = G ∗∞ ( P u j ) − ( R ′∞ ) ∗ u j is uniformly bounded in W n +2 H ( M ). By Lemma 4.5, we may assume that u j converges to some u ∈ L ( M ) as j → + ∞ . From the definition of u j , we derivethat u is in (Ker P ) ⊥ and k u k = 1. However, since k P u j k ≤ /j , we have u ∈ Dom P and P u = 0. This is a contradiction. (cid:3)
Since P is a range-closed operator, there exist the partial inverse G of P andthe orthogonal projection Π to Ker P . Next, we show that these operators areHeisenberg pseudodifferential operators. Theorem 5.7.
The operators G and Π are Heisenberg pseudodifferential operatorsof order − n − and respectively. Moreover, Π coincides with Π modulo Ψ − H ( M ) .Proof. First note that ΠΠ ∞ = Π ∞ , Π ∗∞ Π = Π ∗∞ . since Ran Π ∞ ⊂ Ker P . Composing Π to (5.1) from the left and taking its adjoint,we have Π ∞ = Π + Π R ′∞ , Π ∗∞ = Π + ( R ′∞ ) ∗ Π . Hence Π − Π ∞ = − Π R ′∞ = ( R ′∞ ) ∗ Π R ′∞ − Π ∗∞ R ′∞ , which is a smoothing operator. In particular, Π is a Heisenberg pseudodifferentialoperator of order 0. Moreover, Π − Π ∈ Ψ − H ( M ) since Π ∞ − Π ∈ Ψ − H ( M ). Nextconsider G . Set(5.3) R ′′∞ := P G ∞ + Π − I ∈ Ψ −∞ H ( M ) . Composing G to (5.3) and taking its adjoint give( I − Π) G ∞ = G + GR ′′∞ , ( G ∞ ) ∗ ( I − Π) = G + ( R ′′∞ ) ∗ G. Hence G − ( I − Π) G ∞ = − GR ′′∞ = ( R ′′∞ ) ∗ GR ′′∞ − ( G ∞ ) ∗ ( I − Π) R ′′∞ , which is a smoothing operator. Therefore G is a Heisenberg pseudodifferentialoperator of order − n − (cid:3) This theorem proves Theorem 1.2.
Proof of Theorem 1.2.
From Proposition 4.6 and Theorem 5.7, we derive that thepartial inverse G : L ( M ) → L ( M ) is a compact self-adjoint operator. Hencethe spectrum σ ( G ) of G is bounded and consists only of eigenvalues, and 0 is theonly accumulation point of σ ( G ). Moreover, for any non-zero eigenvalue λ , theeigenspace H λ := Ker( G − λ ) is finite-dimensional, and there exists the followingorthogonal decomposition: L ( M ) = Ker G ⊕ M λ ∈ σ ( G ) \{ } H λ . Furthermore, since G maps W sH ( M ) to W s +2 n +2 H ( M ), the eigenspace H λ is a linearsubspace of C ∞ ( M ). By the definition of the partial inverse, H λ is the eigenspaceof P with eigenvalue 1 /λ , and Ker G = Ker P . Hence the spectrum σ ( P ) is discreteand consists only of eigenvalues, and the eigenspace corresponding to each non-zero eigenvalue is a finite-dimensional subspace of C ∞ ( M ). Moreover, Ker P ∩ C ∞ ( M ) is dense in Ker P since the orthogonal projection Π to Ker P is a Heisenbergpseudodifferential operator of order 0. (cid:3) An argument similar to the proof of Theorem 5.7 also gives Proposition 1.3.
Proof of Proposition 1.3.
Let π be the orthogonal projection to P . Note that Π − π is the orthogonal projection to W . Hence it is enough to prove that Π − π is asmoothing operator. Since Π ∼ Π ∞ , it suffices to show that π − Π ∞ is a smoothingoperator. From the construction of Π ∞ , we deriveRan Π ∞ ⊂ Ran Π ⊂ Ran π. Hence π Π ∞ = Π ∞ , Π ∗∞ = Π ∗∞ π. It follows from (5.1) thatΠ ∞ = π + πR ′∞ , Π ∗∞ = π + ( R ′∞ ) ∗ π. Therefore we have π − Π ∞ = − πR ′∞ = ( R ′∞ ) ∗ πR ′∞ − Π ∗∞ R ′∞ , which is a smoothing operator. (cid:3) As an application of results in this section, we give a necessary and sufficientcondition for the zero CR Q -curvature problem. NALYSIS OF THE CRITICAL CR GJMS OPERATOR 15
Proof of Proposition 1.4.
As we saw in the introduction, Q ⊥ (Ker P ∩ C ∞ ( M ))if there exists a contact form with zero Q -curvature. Conversely, assume that Q is orthogonal to Ker P ∩ C ∞ ( M ). It follows from Theorem 1.2 that Q is in factorthogonal to Ker P . Then Υ := − GQ ∈ C ∞ ( M ) and P Υ = − Q . Hence ˆ θ := e Υ θ satisfies b Q = 0. (cid:3) Acknowledgements
The author is grateful to Charles Fefferman, Kengo Hirachi, and Paul Yangfor helpful comments. A part of this work was carried out during his visit toPrinceton University with the support from The University of Tokyo/PrincetonUniversity Strategic Partnership Teaching and Research Collaboration Grant, andthe Program for Leading Graduate Schools, MEXT, Japan. He would also like tothank Princeton University for its kind hospitality.
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Department of Mathematics, Graduate School of Science, Osaka University, 1-1Machikaneyama-cho, Toyonaka, Osaka 560-0043, Japan
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