The Szegö kernel on a class of noncompact CR manifolds of high codimension
aa r X i v : . [ m a t h . C V ] A ug THE SZEG ¨O KERNEL ON A CLASS OF NONCOMPACT CRMANIFOLDS OF HIGH CODIMENSION
ANDREW RAICH AND MICHAEL TINKER
Abstract.
We generalize Nagel’s formula for the Szeg¨o kernel and use it to compute theSzeg¨o kernel on a class of noncompact CR manifolds whose tangent space decomposes intoone complex direction and several totally real directions. We also discuss the control metricon these manifolds and relate it to the size of the Szeg¨o kernel. Introduction
The goal of this note is to derive a formula for the Szeg¨o kernel for a class of polynomialmodels that are CR manifolds whose maximal complex tangent space is one (complex)dimensional and totally real tangent space is n (real) dimensional. We also discuss the sizeof the Szeg¨o kernel in relation to the control metric.When a CR manifold M has a one (complex) dimensional maximal complex tangent space,then it is standard practice to identify ¯ ∂ b with a vector field ¯ Z that is antiholomorphicand tangential. The Szeg¨o kernel is then the orthogonal projection S M of L ( M ) onto L ( M ) ∩ ker ¯ Z . In complex analysis, the Szeg¨o kernel is a fundamental object of study, yetvery little is known about the Szeg¨o kernel when the tangent space to M has at least twototally real directions. In the case that M is a quadric submanifold (with no hypothesis onthe dimensionality of the maximal complex tangent space), then researchers have computedthe partial Fourier transform of the (cid:3) b -heat kernel, from which the partial Fourier transformof the Szeg¨o kernel can be obtained [BR11, CCM09, CCFI11]. This article represents thefirst time that a formula/estimate for the Szeg¨o kernel has been obtained for any exampleoutside of quadrics when M is not of hypersurface type.In an interesting twist, we will see in Section 3 that the control metric on S M is finite onthe manifold where the Szeg¨o kernel in nonzero. This behavior may provide a clue as to thebehavior of the Szeg¨o kernel in higher codimensions when for every point, the span of theantiholomorphic vector fields is a strictly smaller dimension than the the dimension of thetangent space.Let p , . . . , p n : R → R be a collection of n functions and P = ( p , . . . , p n ). The functions p j will typically be convex polynomials. Our model M P will be a subset of C × C n , and wedenote coordinates on C × C n by ( z, w ) where z = x + iy ∈ C and w = t + is ∈ C n . Set ∂∂ ¯ w = (cid:18) ∂∂ ¯ w , . . . , ∂∂ ¯ w n (cid:19) . Mathematics Subject Classification.
Key words and phrases.
Szeg¨o kernel, high codimension, control metric, polynomial model.This work was partially supported by a grant from the Simons Foundation ( n = 2 case in this paper was part of Tinker’s Ph.D. thesis which he completed under Raich’ssupervision. efine(1) M P = (cid:8) ( z, w ) ∈ C × C n : Im w = P ( x ) } . The maximal complex tangent space is spanned by the vector¯ Z = ∂∂ ¯ z − iP ′ ( x ) · ∂∂ ¯ w . Since the maximal complex tangent space has one dimension, the Szeg¨o kernel on M is theorthogonal projection S : L ( M ) → L ( M ) ∩ ker ¯ Z .We may identify M with C × R n under the identification (cid:0) z, t + iP ( x ) (cid:1) ←→ ( z, t ) . Under this identification, the vector field 2 ¯ Z pushes forward to the vector field¯ L = ∂∂x + i (cid:16) ∂∂y − P ′ ( x ) · ∂∂t (cid:17) . We have a choice of measure to put on M (and consequently on C × R n ). If n = 1 and P ( x ) = x , then M is the Heisenberg group H and Haar measure on M corresponds toLebesgue measure on C × R . Following precedent [Nag86, Chr91, Rai06b, Rai06a, Rai07,Rai12, BR13b, BR11, BR09, NS06, NRSW89, Has94, Str09], we use Lebesgue measure on C × R n .We can then identify the Szeg¨o projection S P on L ( M ) with a projection that (by anabuse of notation) we also call the Szeg¨o projection and denote by S P ; namely, the orthogonalprojection of L ( C × R n ) onto L ( C × R n ) ∩ ker ¯ L . By standard Hilbert space theory, thisSzeg¨o projection S P is given by integration against a kernel S P (cid:0) ( x, y, t ) , ( x ′ , y ′ , t ′ ) (cid:1) , that is, S P f ( x, y, t ) = Z C × R n S P (cid:0) ( x, y, t ) , ( x ′ , y ′ , t ′ ) (cid:1) f ( x ′ , y ′ , t ′ ) dx ′ dy ′ dt ′ . The first goal of this paper is to find a tractable expression for S P (cid:0) ( x, y, t ) , ( x ′ , y ′ , t ′ ) (cid:1) . Theorem 1.1.
Let M P be a polynomial model defined by (1) . Then the Szeg¨o kernel for M P is given by the formula S P (cid:0) ( x, y, t ) , ( x ′ , y ′ , t ′ ) (cid:1) = Z Σ P C η,τ e πη (( x + x ′ )+ i ( y − y ′ )) e − πτ · ( P ( x )+ P ( x ′ ) − i ( t − t ′ )) dη dτ where C η,τ = Z R e π ( x ′ η − P ( x ′ ) · τ ) dx ′ and Σ P = { ( η, τ ) ∈ R × R n : C η,τ < ∞} . Theorem 1.1 generalizes the Szeg¨o kernel formula of Nagel [Nag86, p.302]. In [Nag86],Nagel investigates the case M p = { ( z, w ) ∈ C : Im w = p ( w ) } where p is a convex polyno-mial. If C η,r = R R e π ( ηx − rp ( x )) dx and Σ p = { ( η, r ) : C η,r < ∞} , then Nagel proves that(2) S p (cid:0) ( x, y, t ) , ( x ′ , y ′ , t ′ ) (cid:1) = Z Σ p C η,τ e πη (( x + x ′ )+ i ( y − y ′ )) e − πτ · ( p ( x )+ p ( x ′ ) − i ( t − t ′ )) dη dτ. We now explore several consequences of Theorem 1.1. heorem 1.2. Let M P be a model defined by (1) , and assume that p j ( x ) = a j p ( x ) for ≤ j ≤ n where a n = 1 and p ( x ) is a smooth function satisfying lim | x |→∞ p ( x ) | x | = ∞ . If welet t = ( s, t n ) and a = ( b, , then S P (cid:0) ( x, y, t ) , ( x ′ , y ′ , t ′ ) (cid:1) = δ [( s − s ′ ) − b ( t n − t ′ n )] S p (cid:0) ( x, y, t n ) , ( x ′ , y ′ , t ′ n ) (cid:1) where δ is the Dirac- δ in R n − . The size of the Szeg¨o kernel when M p is a model of three real dimensions and p is a convexpolynomial is well understood [Nag86, NRSW89, Chr91, Rai06a]. In particular if d ( · , · ) isthe control metric generated by the vector fields X = Re ¯ L and X = Im ¯ L , and B CC ( α, δ )is the control ball of radius δ , then if X J is a multiindex of operators X , X acting in either α = ( x, y, t ) or β = ( x ′ , y ′ , t ′ ), then | X J S P ( α, β ) | . | d ( α, β ) | −| J | | B CC ( α, d ( α, β )) | − . Thisyields an immediate corollary. Corollary 1.3.
Let M P be a model as in Theorem 1.2 where p is a convex polynomial. If X J is a multiindex of operators X , X acting in either α = ( x, y, t ) or β = ( x ′ , y ′ , t ′ ) , thenthere exists a constant C | J | > so that on supp δ [( s − s ′ ) − b ( t n − t ′ n )] | X J S M ( α, β ) | ≤ C | J | | d ( α, β ) | −| J | | B CC ( α, d ( α, β )) | . The proof is immediate, given the fact that X and X are tangential on the manifoldwhere s − s ′ = b ( t n − t ′ n ).If p j ( x ) = a j x , then M is an example of quadric submanifold. Quadrics have been studiedextensively [BR11, BR13a, CT00, CCT06, BGG96, BGG00, Gav77, Hul76] and [CCM09], inparticular for a more extensive background. In this case, we can compute all of the integralsexplicitly and prove the following theorem. Theorem 1.4.
Let M P be the quadric submanifold defined by M a | x | = { ( z, w ) ∈ C × C n : Im w = x a } where a = ( a , . . . , a n ) ∈ R n and a n > . Then S a | x | (cid:0) ( x, y, t ) , ( x ′ , y ′ , t ′ ) (cid:1) = S a n | x | (cid:0) ( x, y, t n ) , ( x ′ , y ′ , t ′ n ) (cid:1) δ [ a n ( s − s ′ ) − b ( t n − t ′ n )]= 2 a n δ [ a n ( s − s ′ ) − b ( t n − t ′ n )] (cid:0) πa n [( x − x ′ ) + ( y − y ′ ) ] − πi [( t n − t ′ n ) + a n ( x + x ′ )( y − y ′ )] (cid:1) Remark . The condition that a n > a n = 0, butwe keep a n > S λx (( x, y, t ) , ( x ′ , y ′ , t ′ )) where λ >
0, and from that expression, we can see there is nothingdistinguished about the n th coordinate, except the fact that a n = 0.The outline of the remainder of the paper consists of the proofs of the main theorems inSection 2 and a discussion of the control geometry in Section 3.2. Proofs of the Main Theorems
Proof of the Szeg¨o kernel formula. roof of Theorem 1.1. The proof of Theorem 1.1 follows from two observations. The firstis that ¯ L is translation invariant in y and t . This means we can take the partial Fouriertransform in y and t . Given a function f ( x, y, t ), we define the partial Fourier transform of f to be F f ( x, η, τ ) = ˆ f ( x, η, τ ) = Z Z R × R n e − πi ( y,t ) · ( η,τ ) f ( x, y, t ) dy dt. Under F , with ( η, τ ) as the transform variables of ( y, t ), the operator¯ L ˆ¯ L = ∂∂x − πη + 2 πP ′ ( x ) · τ = e π ( xη − P ( x ) · τ ) ∂∂x e − π ( xη − P ( x ) · τ ) . Set Ψ( x, η, τ ) = e − π ( xη − P ( x ) · τ ) and M Ψ : L ( R , dx ) → L ( R , e π ( xη − P ( x ) · τ ) ) to be the isometrydefined by f Ψ f . Since M Ψ and F are isometries, ¯ Lf = 0 if and only if ddx { M Ψ F f } = 0.The second observation is that ker ddx are constant functions. The function f = 1 is in L ( R , e π ( xη − P ( x ) · τ ) ) exactly when C η,τ < ∞ . Assuming C η,τ < ∞ , then the projection of g onto ker ddx is the operator P η,τ given by P η,τ g ( x ) = P η,τ g = 1 C η,τ Z R g ( x ′ ) e π ( x ′ η − P ( x ) · τ ) dx ′ . If the operator P = P η,τ on L ( R , e π ( xη − P ( x ) · τ ) ) with the understanding that P η,τ = 0 when( η, τ ) Σ. Consequently, S = F − M Ψ − P M Ψ F . Expanding the right-hand side yields the desired formula. (cid:3)
Proof of the Szeg¨o kernel formula when P = ap . Proof of Theorem 1.2.
We use the following notation: a = ( a , . . . , a n ), b = ( a , . . . , a n − ), τ = ( σ, τ n ). Also, a · τ = τ n + b · σ . Since lim | x |→∞ p ( x ) | x | = ∞ , C η,τ < ∞ if and only if a · τ > τ n > − b · σ ), and this condition is independent of η . We usea superscript to denote which model to which various expressions refer. Also, P ( x ) = p ( x ) a ,so C apη,τ = Z R e π ( xη − p ( x ) a · τ ) dx = C pη,τ n + σ · b . Consequently, we use Theorem 1.1 and compute that S ap (cid:0) ( x, y, t ) , ( x ′ , y ′ , t ′ ) (cid:1) = Z Σ ap C apη,τ e πη ( x + x ′ + i ( y − y ′ )) e − πτ · ( a ( p ( x )+ p ( x ′ )) − i ( t − t ′ )) dτ dη = Z R n − Z R Z ∞ τ n = − b · σ C pη,τ n + σ · b e πη ( x + x ′ + i ( y − y ′ )) e − π ( τ n + b · σ )( p ( x )+ p ( x ′ ) − i ( t n − t ′ n )) e πiσ · [( s − s ′ ) − b ( t n − t ′ n )] dτ n dη dσ where the last line uses the equality τ · ( t − t ′ ) = σ · ( s − s ′ ) + τ n ( t n − t ′ n ) and the fact that a n = 1. Shifting the variable τ + b · σ τ , comparing the resulting formula to (2), andrecognizing that resulting integration in σ results in a δ [( s − s ′ ) − b ( t n − t ′ n )] finishes theproof. (cid:3) .3. The quadric case.
Proof of Theorem 1.4.
We use Theorem 1.2. In the case that p ( x ) = x and n = 1, we seethat if λ >
0, then C λx η,τ = e πη λτ √ λτ , so applying Nagel’s formula yields S λx (( x, y, t ) , ( x ′ , y ′ , t ′ )) = Z ∞ Z R √ τ λe − πη λτ e − πτ [ λ ( x + x ′ ) − i ( t − t ′ )] e πη [( x + x ′ )+ i ( y − y ′ )] dη dτ = 2 λ (cid:0) πλ [( x − x ′ ) + ( y − y ′ ) ] − πi [( t − t ′ ) + λ ( x + x ′ )( y − y ′ )] (cid:1) We are not assuming that a n = 1, and we could use a change of variables to reduce to thiscase, but it is simpler to make the change of variables ˜ σ = σ , and ˜ τ n = a · τ . Next, an easycomputation establishes that Σ ax = { ( η, τ ) : a · τ > } = { ( η, ˜ τ ) : ˜ τ n > } and C ax η,τ = e πη a · τ √ a · τ Consequently, we compute that τ n = (˜ τ n − σ · b ) /a n and S ax (cid:0) ( x, y, t ) , ( x ′ , y ′ , t ′ ) (cid:1) = Z Σ ax √ a · τ e − πη a · τ e πη [( x + x ′ )+ i ( y − y ′ )] e − πτ · [ a ( x + x ′ ) − i ( t − t ′ )] dη dτ = Z R Z ∞ Z R n − p ˜ τ n e − πη τn e πη [( x + x ′ )+ i ( y − y ′ )] e − π ˜ τ n ( x + x ′ ) e πiσ · ( s − s ′ ) e πi (˜ τ n − σ · b )( t n − t ′ n ) /a n dσ d ˜ τ n dη = 1 a n Z R Z ∞ Z R n − p ˜ τ n e − πη τn e πη [( x + x ′ )+ i ( y − y ′ )] e − π ˜ τ n [( x + x ′ ) − i ( t n − t ′ n ) /a n ] e πiσ · [( s − s ′ ) − ban ( t n − t ′ n )] dσ dη d ˜ τ n = 1 a n δ [ a n ( s − s ′ ) − b ( t n − t ′ n )] (cid:0) π [( x − x ′ ) + ( y − y ′ ) ] + 2 πi [( t n − t ′ n ) /a n + ( x + x ′ )( y − y ′ )] (cid:1) = S a n | x | (cid:0) ( x, y, t n ) , ( x ′ , y ′ , t ′ n ) (cid:1) δ [ a n ( s − s ′ ) − b ( t n − t ′ n )] . (cid:3) Connection to the control geometry
Since on a finite type domain boundary in C the Szeg¨o kernel is governed by the controlmetric, we may naturally ask whether the same holds on a codimension CR manifold withat least two totally real directions. It is easy to extend the notion of finite commutator type;we simply require that the real and imaginary parts of ¯ L , X and X , along with a finitenumber m of their iterated commutators, span the real tangent space at every point of M P . Definition 3.1.
With notation as above, let { Y , . . . , Y q } be some enumeration of the vectorfields X , X , and all their iterated commutators of length less than or equal m . Define the“degree” of each vector field Y j byd( Y j ) = length of the iterated commutator that forms Y j Now let the distance between p, q ∈ M P be the infimum of δ > γ : [0 , M P with γ (0) = p , γ (1) = q so that for almost all r ∈ (0 , γ ′ ( r ) = q X j =1 c j ( t ) Y j ( γ ( r )) , | c j ( r ) | < δ d( Y j ) nder this condition, the control distance yields a metric, but one which currently defiesany tractable description. (Indeed, even on a domain boundary M p , a serious amount of workis required to prove the equivalence of the control metric to the pseudometrics investigatedby Nagel et al. [NSW85].) Although M ap ( x ) is not of finite type, there is a submanifold of M ap ( x ) on which the control distance is finite and a direct connection to the Szeg¨o kernel on M ap ( x ) . Here X = ∂∂x and X = ∂∂y − p ′ ( x ) a · ∂∂t , so every potentially non-zero commutatoris of the form Y k = [ k − z }| { X , [ X , · · · , [ X , X ] · · · ]] (2 ≤ k ≤ m )That is, Y k = − p ( k ) ( x ) a · ∂∂t . This forces { X , X , Y k , Y k } to span only the subspace generated by { ∂∂x , ∂∂y , a · ∂∂t } . Thusthe real tangent space is never spanned, at any point of M ap ( x ) , and { X , X } do not generatea finite control metric.We may still consider control distance on M ap ( x ) . This distance is finite and less than some δ > γ : [0 , M ap ( x ) such that γ ′ ( ς ) = c ( ς ) X ( γ ( ς )) + c ( ς ) X ( γ ( ς )) + m X k =2 c k ( ς ) Y k ( γ ( ς ))with | c ( ς ) | , | c ( ς ) | < δ and | a k ( ς ) | < δ k for almost all ς ∈ (0 , γ ( ς ) = ( γ ( ς ) , γ ( ς ) , γ ( ς ) , . . . , γ n +2 ( ς )) can only exist if ( γ ′ , . . . , γ ′ n +2 )is parallel to a for almost all ς ∈ (0 , M ap ( x ) is asingular distribution supported on exactly the subspace where the control distance on M ap ( x ) is finite. On this subspace, the control ball is well-defined and exactly determines the sizeof the Szeg¨o kernel on M ap ( x ) , treating the subspace as an R and applying the Nagel et. al.machinery. References [BGG96] R. Beals, B. Gaveau, and P.C. Greiner. The Green function of model step two hypoelliptic oper-ators and the analysis of certain tangential Cauchy Riemann complexes.
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