The Fundamental Solution to \Box_b on Quadric Manifolds -- Part 1. General Formulas
aa r X i v : . [ m a t h . C V ] J a n THE FUNDAMENTAL SOLUTION TO (cid:3) b ON QUADRIC MANIFOLDS –PART 1. GENERAL FORMULAS
ALBERT BOGGESS AND ANDREW RAICH
Abstract.
This paper is the first of a three part series in which we explore geometric and analyticproperties of the Kohn Laplacian and its inverse on general quadric submanifolds of C n × C m . Inthis paper, we present a streamlined calculation for a general integral formula for the complex Greenoperator N and the projection onto the nullspace of (cid:3) b . The main application of our formulas isthe critical case of codimension two quadrics in C where we discuss the known solvability andhypoellipticity criteria of Peloso and Ricci [PR03]. We also provide examples to show that ourformulas yield explicit calculations in some well-known cases: the Heisenberg group and a Cartesianproduct of Heisenberg groups. Introduction
The goal of this paper is to present an explicit integral formula for the complex Green operatorand the projection onto the null space of the Kohn Laplacian on quadric submanifolds of C n × C m .Our result generalizes the formula of [BR13] from the specific case of codimension 2 quadrics in C to the general case of arbitrary n and m , and we also prove a formula for the complex Green operatorwhen it is only a relative inverse of the Kohn Laplacian and not a full inverse. Additionally, thenew proof is significantly simpler and uses our calculation of the (cid:3) b -heat kernel on general quadrics[BR11]. We then provide several applications of our formula to the case of codimension 2 quadricsin C and provide context for how this case fits into the solvability/hypoellipticity framework ofPeloso and Ricci [PR03]. We also provide a calculation of the complex Green operator in severalinstances when it is only a relative fundamental solution: (cid:3) b on the Heisenberg group on functionsand (0 , n )-forms, and on the Cartesian product of Heisenberg groups at the top degree. We concludewith a computation of the Szeg¨o projection on the Cartesian product of Heisenberg groups.This paper is the first of a series where we explore the geometry and analysis of the Kohn Lapla-cian (cid:3) b and its (relative) inverse, the complex Green operator, on quadric submanifolds in C n × C m .The (cid:3) b -equation, (cid:3) b u = f , governs the behavior of boundary values of holomorphic functions, andthe (cid:3) b -operator is a naturally occurring, non-constant coefficient, non-elliptic operator. Solvingthe equation has been tantalizing mathematicians for the better part of fifty years, and while muchis known for solvability/regularity in L p and other spaces (especially L ) for hypersurface typeCR manifolds, there has been much less work done to determine the structure of the complexGreen operator, denoted by N , especially on non-hypersurface type CR manifolds. The problemis that the techniques used to solve the equation are functional analytic in nature and thereforenon-constructive. Consequently, to have any hope of finding an explicit solution, we need additionalstructure on the CR manifold. From our perspective, the gold standard for results in this area isthe calculation by Folland and Stein [FS74] in which they find a beautiful, closed form expressionfor N on the Heisenberg group. Mathematics Subject Classification.
Key words and phrases.
Quadric submanifolds, tangential Cauchy-Riemann operator, ¯ ∂ b , complex Green operator,Szeg¨o kernel, Szeg¨o projection, fundamental solution, Heisenberg group.This work was supported by a grant from the Simons Foundation (707123, ASR). n the decades following [FS74], mathematicians developed machinery to solve the (cid:3) b -heat equa-tion (or the heat equation associated to the sub-Laplacian) on certain Lie groups. From theformulas for the heat equations, in principle, it is only a matter of integrating the time variableout to recover the formula for N . The first results for the heat equation were for the sub-Laplacianon the Heisenberg group by Hulanicki [Hul76] and Gaveau [Gav77]. More results followed in asimilar vein, and the vast majority rely on the group Fourier transform and Hermite functions[PR03, BR09, BR11, YZ08, CCT06, BGG96, BGG00, Eld09]. The problem with these techniquesis that the formulas that they generate for the heat kernel are only given up to a partial Fouriertransform that is uncomputable in practice. Consequently, any information giving precise sizeestimates or asymptotics, let alone a formula in the spirit of Folland and Stein, is absent.A quadric submanifold M ⊂ C n × C m is a CR manifold of the form(1) M = { ( z, w ) ∈ C n × C m : Im w = φ ( z, z ) } where φ : C n × C n C m is a sesquilinear form (i.e., φ ( z, z ′ ) = φ ( z ′ , z )). The fundamental solutionto (cid:3) b or the sub-Laplacian on quadrics has been studied many authors, including [BG88, BGG96,BR09, BR13, CCT06, FS74, PR03]. See also Part II of the series, [BR20], in which we find useablesufficient conditions for a map T between quadrics to be a (cid:3) b -preserving Lie group isomorphism aswell as establish a framework for which appropriate derivatives of the complex Green operator arecontinuous in L p and L p -Sobolev spaces (and hence are hypoelliptic). We apply the general resultsto codimension 2 quadrics in C .There are two higher codimension papers that need mentioning. First, in [NRS01], Nagel et. al.analyze (cid:3) b and its geometry in the special class of decoupled quadrics where Im w j = P nk =1 a jk | z k | .However, many of the interesting cases do not fall into this category. Second, Raich and Tinkercompute the the Szeg¨o kernel for the polynomial model M = { ( z, w ) ∈ C × C n : Im w = ( a , . . . , a n − , p (Re z ) (cid:9) where p : R → R is a smooth function satisfying lim | x |→∞ p ( x ) | x | = ∞ , and the constants a , . . . , a n − are nonzero [RT15]. The authors write an explicit formula for the Szeg¨o kernel based on an integralformula of Nagel [Nag86] and show that there are significant blowups off of the diagonal. Raichand Tinker evaluate all of the integrals in the case p ( x ) = x . That is, the CR manifold M isa quadric, but this case is very special because the tangent space (at each point) has only onecomplex direction, so every degree is either top or bottom. These are often the exceptional casesand can give misleading intuition.Associated to each quadric is the Levi form φ ( z, z ′ ) and for each λ ∈ R m , the directional Leviform in the direction λ is φ λ ( z, z ′ ) = φ ( z, z ′ ) · λ where · is the usual dot product without conjugation.For each λ , there is an n × n matrix A λ so that φ λ ( z, z ′ ) = z ∗ A λ z ′ and we identify the eigenvalues of φ λ with the eigenvalues of the matrix A λ . Here, the ∗ designatesthe Hermitian transpose.The outline of the remainder of the paper is as follows: In Section 2, we record the main resultfor general quadrics and provide additional context. In Section 3, we discuss the CR geometry andLie group structure of a general quadric. In Section 4, we prove the main result, and we devoteSection 5 to explicit examples.1.1. Acknowledgements.
The authors would like to express their sincere appreciation for thedetailed review from the referee and the excellent editorial comments from Harold Boas. . Results and Discussion
Under the projection π : C n × C m → C n × R m given by π ( z, t + is ) = ( z, t ), we may identify aquadric M with C n × R m . The projection induces both a CR structure and Lie group structure on C n × R m , and we denote this Lie group by G (or G M ). Thus the projection is a CR isomorphismand we refer to the pushforwards and pullbacks of objects from M to G without changing thenotation.2.1. The Main Result – General Formulas for the Solution of (cid:3) b on Quadrics. To statethe main result, we need to introduce some notation. For each λ ∈ R m \ { } , let µ λ , . . . , µ λn bethe eigenvalues of φ λ (or equivalently, the Hermitian symmetric matrix, A λ ) and v λ , . . . , v λn be anorthonormal set of eigenvectors. This means(2) φ λ ( v λj , v λk ) = δ jk µ λj . Let α = λ/ | λ | , then µ λj = | λ | µ αj and v λj = v αj . If z ∈ C n is expressed in terms of the unit eigenvectorsof φ λ , then z λj = z αj is given by z α = Z ( z, α ) = ( U α ) ∗ · z where U α is the matrix whose columns are the eigenvectors, v αj , 1 ≤ j ≤ n , and · represents matrixmultiplication with z written as a column vector. Note that the corresponding orthonormal basisof (0 , d ¯ Z j ( z, α ) , ≤ j ≤ n, where d ¯ Z ( z, α ) = ( U α ) T · d ¯ z where d ¯ z is written as a column vector of (0 , T stands for transpose.It is a fact that the eigenvalues, eigenvectors and hence z α = Z ( z, α ) depend smoothly on z ∈ C n . However, while the dependence of the eigenvalues is continuous (in fact Lipschitz) in α , theeigenvectors may only be functions of bounded variation (SBV) in α [Rai11, Theorem 9.6].Let I q = { L = ( ℓ , . . . , ℓ q ) : 1 ≤ ℓ < ℓ < · · · < ℓ q ≤ n } . For each K ∈ I q , we will need toexpress d ¯ z K , in terms of d ¯ Z ( z, α ) L for L ∈ I q . We have(3) d ¯ z K = X L ∈I q C K,L ( α ) d ¯ Z ( z, α ) L where C K,L ( α ) are the appropriate q × q minor determinants of ¯ U α . Note that if q = n , then theabove sum only has one term and C K,K ( α ) = 1. Additionally, when q = 0, I = ∅ and the sum (3)does not appear.Denote by ν ( λ ) = ν ( α ) the number of nonzero eigenvalues of φ λ . For each q -tuple L ∈ I q , setΓ L = { α ∈ S m − : µ αℓ > ℓ ∈ L and µ αℓ < ℓ L } and ε αj,L = ( sgn( µ λj ) if j ∈ L − sgn( µ λj ) if j L. Remark . If ν = max λ ∈ R m ν ( λ ), then { λ ∈ R m \ { } : ν ( λ ) = ν } is a Zariski open set andhence carries full Lebesgue measure. In particular, if one of the sets Γ L is nonempty, then ν = n .When ν < n , we arrange our eigenvalues so that µ λν +1 = · · · µ λn = 0 and write z ′ = ( z , . . . , z ν ) and z ′′ = ( z ν +1 , . . . , z n ). Definition 2.2.
Given an index in K ∈ I q , we say that a current N K = P K ′ ∈I q ˜ N K ′ ( z, t ) d ¯ z K ′ isa fundamental solution to (cid:3) b on forms spanned by d ¯ z K if (cid:3) b N K = δ ( z, t ) d ¯ z K . K acts on smooth forms with compact support by componentwise convolution with respect tothe group structure on G , that is, if f = f d ¯ z K , then N K ∗ f = P K ′ ∈I q ˜ N K ′ ∗ f d ¯ z K ′ . Thus if f = f d ¯ z K is a smooth form with compact support, then (cid:3) b { N K ∗ f } = f . In cases where (cid:3) b hasa nontrivial kernel, we let S K be the projection (Szeg¨o) operator onto this kernel and we say that N K is a relative fundamental solution if (cid:3) b { N K ∗ f } = f − S K ( f ) holds for all compactly supportedforms spanned by d ¯ z K . On quadrics, (cid:3) b never has closed range in L so the complex Green operatorcannot be continuous in L . As a consequence, we can only discuss a relative inverse and not the relative inverse. However, a relative inverse is called canonical if its output is orthogonal to ker (cid:3) b whenever it belongs to L .We can now state our main result. Theorem 2.3.
Suppose M is a quadric CR submanifold of C n + m given by (1) with associatedprojection G . Fix K ∈ I q .1. If | Γ L | = 0 for all L ∈ I q , then the fundamental solution to (cid:3) b on forms spanned by d ¯ z K is givenby N K ( z, t ) = 4 n π ) m + n X L ∈I q Z α ∈ S m − C K,L ( α ) d ¯ Z ( z, α ) L Z r =0 | log r | n − ν ( α ) ν ( α ) Y j =1 r (1 − ε αj,L ) | µ αj | | µ αj | − r | µ αj | ( n + m − A α ( r, z ) − iα · t ) n + m − dr dαr (4) where A α ( r, z ) = 2 | log r | | z ′′ α | + ν ( α ) X j =1 | µ αj | r | µ αj | − r | µ αj | ! | z αj | .
2. If | Γ L | > for at least one L ∈ I q , then orthogonal projection onto the ker (cid:3) b applied to formsspanned by d ¯ z K is given by convolution with the (0 , q ) -form: (5) S K ( z, t ) = 4 n ( n + m − π ) m + n X L ∈I q Z α ∈ Γ L C K,L ( α ) d ¯ Z ( z, α ) L Q nj =1 | µ αj | ( P nj =1 | µ αj || z αj | − iα · t ) n + m dα. In the case K = ∅ , S ∅ ( z, t ) is the Szeg¨o kernel.3. If | Γ L | > for at least one L ∈ I q , then the canonical relative fundamental solution to (cid:3) b givenby R ∞ e − s (cid:3) b ( I − S q ) ds applied to forms spanned by d ¯ z K is given by N K ( z, t ) =4 n ( n + m − π ) m + n X L ∈I q Z α Γ L C K,L ( α ) d ¯ Z ( z, α ) L Z r =0 n Y j =1 r (1 − ε αj,L ) | µ αj | | µ αj | − r | µ αj | A ( r, z ) − iα · t ) n + m − dr dαr + Z α ∈ Γ L C K,L ( α ) d ¯ Z ( z, α ) L Z r =0 h(cid:16) n Y j =1 | µ αj | − r | µ αj | (cid:17) A α ( r, z ) − iα · t ) n + m − − Q nj =1 | µ αj | ( A α (0 , z ) − iα · t ) n + m − i dr dαr (6) here A α ( r, z ) = n X j =1 | µ αj | r | µ αj | − r | µ αj | ! | z αj | . In all cases, the integrals converge absolutely.Remark . In many of the most important cases, the functions C K,L ( α ) = δ KL , and formulasfrom the theorem simplify. There are several cases when this simplifcation occurs. The first iswhen q = 0 or q = n . The second is when the orthonormal basis { v λj } is independent of λ . Thisindependence happens both when m = 1 (the hypersurface type case) or in the sum of squares caseconsidered by Nagel, Ricci, and Stein [NRS01], discussed in Section 1. Remark . It is a straightforward exercise to recover the classical complex Green operator on theHeisenberg group from (4) [BR13]. Additionally, [BR13, Theorem 2] is now a simple and immediateapplication of (4).2.2.
Solvability, hypoellipticity, and φ λ . In [PR03], Peloso and Ricci say that1. (cid:3) b is solvable if given any smooth (0 , q )-form ψ on G with compact support, there exists a(0 , q )-current u on G so that (cid:3) b u = ψ ;2. (cid:3) b is hypoelliptic if given any (0 , q )-current ψ on G , ψ is smooth on any open set on which (cid:3) b ψ is smooth.Peloso and Ricci are able to characterize solvability and hypoellipticity of (cid:3) b . Theorem 2.6 ([PR03]) . Let n + ( λ ) , resp., n − ( λ ) , be the number of positive, resp., negative eigen-values of φ λ . Then1. (cid:3) b is solvable on (0 , q ) -forms if and only if there does not exist λ ∈ R m \ { } for which n + ( λ ) = q and n − ( λ ) = n − q .2. (cid:3) b is hypoelliptic on (0 , q ) -forms if and only if there does not exist λ ∈ R m \ { } for which n + ( λ ) ≤ q and n − ( λ ) ≤ n − q .Remark . The condition | Γ L | > S K ( z, t ) and is easy tocheck. By combining the solvability criteria of [PR03] (Theorem 2.6) and the formula for the (cid:3) b -heat kernel from [BR11] (Theorem 3.1 below), it must be the case that ν = n (see (10)) as S L ( z, t ) = lim s →∞ H L ( s, z, t ) and solvability is equivalent S L ( z, t ) = 0. The latter statementfollows from the fact the condition in part 1 of Theorem 2.6 is an open condition, that is, whensolvability fails, Γ L is a (union of) cones, at least one of which will be open and hence has nonzeromeasure. 3. The Kohn Laplacian on Quadrics
For a discussion of the group theoretic properties of G , please see [BR11] or [PR03]. By definition,the operator (cid:3) b is defined on (0 , q ) forms as (cid:3) b = ¯ ∂ b ¯ ∂ ∗ b + ¯ ∂ ∗ b ¯ ∂ b without reference to any particularcoordinate system. However in order to do computations, we need formulas for (cid:3) b with respect tocarefully chosen coordinates.For v ∈ R n ≈ C n , let ∂ v be the real vector field given by the directional derivative in thedirection of v . Then the right invariant vector field at an arbitrary g = ( z, w ) ∈ M correspondingto v is given by X v ( g ) = ∂ v + 2 Im φ ( v, z ) · D t = ∂ v − φ ( z, v ) · D t . Let
J v be the vector in R n which corresponds to iv in C n (where i = √− G is then spanned by vectors of the form:(7) Z v ( g ) = (1 / X v − iX Jv ) = (1 / ∂ v − i∂ Jv ) − iφ ( z, v ) · D t nd(8) ¯ Z v ( g ) = (1 / X v + iX Jv ) = (1 / ∂ v + i∂ Jv ) + iφ ( z, v ) · D t . Let v , . . . , v n be any orthonormal basis for C n . Let X j = X v j , Y j = X Jv j , and let Z j =(1 / X j − iY j ), ¯ Z j = (1 / X j + iY j ) be the right invariant CR vector fields defined above (whichare also the left invariant vector fields for the group structure with φ replaced by − φ ). A (0 , q )-formcan be expressed as P K ∈I q φ K d ¯ z K . An explicit formula for (cid:3) b on quadrics is written down byPeloso and Ricci [PR03] (see also [BR11]) which takes the following form: if φ = P K φ K d ¯ z K is a(0 , q )-form, then(9) (cid:3) b ( X K ∈I q φ K d ¯ z K ) = X K,L ∈I q (cid:3) vLK φ K d ¯ z L where (cid:3) vLL = (1 / n X ℓ =1 (cid:0) Z ℓ ¯ Z ℓ + ¯ Z ℓ Z ℓ (cid:1) + (1 / X ℓ ∈ L [ Z ℓ , ¯ Z ℓ ] − X k L [ Z k , ¯ Z k ] If L = K , then (cid:3) vLK is zero unless | L ∩ K | = q −
1, in which case (cid:3) vLK = ( − d kl [ Z k , ¯ Z ℓ ]where k ∈ K is the unique element not in L and ℓ ∈ L is the unique element not in K and d kl is thenumber of indices between k and ℓ . The notation (cid:3) vLK indicates the dependency of this differentialoperator on the particular orthonormal basis v = ( v , . . . , v n ) chosen and the resulting basis (i.e., Z , . . . , Z n ) and the associated dual basis of (0 , d ¯ z , . . . , d ¯ z n ).Note that if | L ∩ K | = q −
1, then (cid:3) vLK is quite simple since it is a linear combination over C of t j derivatives. In the next section, we will use the coordinates z α = Z ( z, α ) derived from the basis v α , . . . , v αn used in Section 2, and we will see that we can ignore the (cid:3) v α LK when L = K .3.1. Fourier Transform of (cid:3) b . Since the quadric defining equations are independent of t ∈ R m ,we can use the Fourier transform in the t -variables:ˆ f ( λ ) = 1(2 π ) m Z R m f ( t ) e − iλ · t dt. In the case that f is a function of ( z, t ), we use the notation f ( z, ˆ λ ) to denote the partial Fouriertransform of f in the t -variables. We transform (cid:3) b via the Fourier transform and consider thefundamental solution to the heat operator in the transformed variables. We then use the z α coordinates relative to the basis v αj chosen in Section 2 for the z -variable in f ( z α , ˆ λ ) with α = λ | λ | .Thus, λ plays two roles - first as the Fourier transform variable and second, as the label for thecoordinates relative to the basis v αj which diagonalizes φ λ . Also note that the operation of Fouriertransform in t and the operation of expressing z in terms of the z α coordinates are interchangeable(i.e., these operations commute).For a general orthonormal basis v = { v , . . . , v n } , let (cid:3) v, ˆ λLL be the partial Fourier transform in t of the sub-Laplacian (cid:3) vLL . When v = v α , we have (from [BR11]): (cid:3) v α , ˆ λLL = −
14 ∆ + 2 i n X k =1 µ λk Im { z αk ∂ z αk } + n X k =1 ( µ λk ) | z αk | − X k ∈ L µ λk − X k L µ λk where ∆ is the ordinary Laplacian in z = z α coordinates. lso note that (cid:3) v α LK = ( − d kℓ [ Z v αk , ¯ Z v αℓ ]= 2 i ( − d kℓ Re φ ( v αk , v αℓ ) · D t using (7) and (8) . Using (2), we conclude that the Fourier transform of (cid:3) v α LK is (cid:3) v α , ˆ λLK = − − d kℓ Re φ ( v αk , v αℓ ) · λ = 0when L = K (i.e., when ℓ = k ). The signficance of this calculation is that the partial Fouriertransform of (cid:3) LK (expressed in global coordinates) is incorporated into the operators (cid:3) v α , ˆ λLL .Next, we recall the heat kernel and Szeg¨o kernel for the (cid:3) v α , ˆ λLL heat equation. Let ˜ H L ( s, z α , ˆ λ )be the “heat kernel”, i.e., the solution to the following boundary value problem: (cid:20) ∂∂s + (cid:3) v α , ˆ λLL (cid:21) { ˜ H L ( s, z α , ˆ λ ) } = 0 for s > H L ( s = 0 , z α , ˆ λ ) = (2 π ) − m/ δ ( z α )= (2 π ) − m/ δ ( z )where δ is the Dirac-delta function centered at the origin in the z variables. Let ˜ S L ( z α , ˆ λ ) be theSzeg¨o kernel which represents orthogonal projection of L ( C n ) onto the kernel of (cid:3) v α , ˆ λLL . Note, thetilde over the ˜ H L and ˜ S L indicates that these terms are functions rather than differential forms.By contrast, N K and S K in Theorem 2.3 do not have tildes and they are differential (0 , q )-forms. Theorem 3.1. [BR11]
Let L ∈ I q be a given multiindex of length q and fix a nonzero λ ∈ R m and α = λ | λ | . Then(1) The heat kernel which solves the above boundary value problem is (10) ˜ H L ( s, z α , ˆ λ ) = 2 n − ν ( α ) (2 π ) m/ n s n − ν ( α ) e − | z ′′{ α }| s ν ( α ) Y j =1 e sε αj,L | µ λj | | µ λj | sinh( s | µ λj | ) e −| µ λj | coth( s | µ λj | ) | z αj | (2) If α ∈ Γ L , then the projection onto ker (cid:3) v α , ˆ λLL is given by (11) ˜ S L ( z α , ˆ λ ) = lim s →∞ ˜ H L ( s, z α , ˆ λ ) = 4 n (2 π ) n + m/ n Y j =1 | µ λj | e −| µ λj || z αj | , otherwise ˜ S L ( z α , ˆ λ ) = 0 .(3) The connection between the fundamental solution to the heat equation and the canonicalrelative fundamental solution to (cid:3) v α , ˆ λLL , denoted ˜ N L ( z α , ˆ λ ) , is given as follows: (12) ˜ N L ( z α , ˆ λ ) = Z ∞ h ˜ H L ( s, z α , ˆ λ ) − ˜ S L ( z α , ˆ λ ) i ds. In particular, (13) (cid:3) v α , ˆ λLL { ˜ N L ( z α , ˆ λ ) } = (2 π ) − m/ ( δ ( z ) − ˜ S L ( z α , ˆ λ ))Both of the kernels ˜ H L ( s, · , ˆ λ ) and ˜ S L ( · , ˆ λ ) act on L ( C n ) via a twisted convolution, ∗ λ , where( f ∗ λ g )( z ) = R w ∈ C n f ( w ) g ( z − w ) e − iλ · Im φ ( z,w ) dw , as defined in Section 5.4 of [BR11], but thisplays no role here. et F − λ denote the inverse Fourier transform in λ - that is, if ˜ f ( z, λ ) is an integrable functionof λ ∈ R m , then F − λ ( ˜ f ( z, λ ))( t ) := 12 m/ Z λ ∈ R m ˜ f ( z, λ ) e iλ · t dλ. Now we can formulate our relative solution to (cid:3) b and Szeg¨o kernel in terms of the inverse Fouriertransform. Proposition 3.2.
For a given index K ∈ I q , the relative fundamental solution to (cid:3) b applied to aform spanned by d ¯ z K given by R ∞ e − s (cid:3) b ( I − S K ) ds is (14) N K ( z, t ) = F − λ X L ∈I q C K,L ( α ) ˜ N L ( z α , ˆ λ ) d ¯ Z ( z, α ) L ( t ) Moreover, the orthogonal projection onto the ker (cid:3) b applied to forms spanned by d ¯ z K is given byconvolution with the (0 , q ) -form (15) S K ( z, t ) = F − λ (2 π ) − m/ X L ∈I q C K,L ( α ) ˜ S L ( z α , ˆ λ ) d ¯ Z ( z, α ) L ( t ) Proof.
With the definitions of N K and S K given by (14) and (15), respectively, we shall show (cid:3) b N K = I − S K . On the transform side, we have (cid:3) v α , ˆ λb n N K ( z α , ˆ λ ) o = X L ∈I q C K,L ( α ) (cid:3) v α , ˆ λb n ˜ N L ( z α , ˆ λ ) d ¯ Z ( z, α ) L o = X L ∈I q C K,L ( α ) (cid:3) ˆ λLL (cid:8) ˜ N L ( z α , ˆ λ ) (cid:9) d ¯ Z ( z, α ) L = (2 π ) − m/ X L ∈I q C K,L ( α ) (cid:2) δ ( z α ) − ˜ S L ( z α , ˆ λ ) (cid:3) d ¯ Z ( z, α ) L from (13)= (2 π ) − m/ δ ( z ) ⊗ λ d ¯ z K − (2 π ) − m/ X L ∈I q C K,L ( α ) ˜ S L ( z α , ˆ λ ) d ¯ Z ( z, α ) L from (3)where the function 1 λ is the constant function which is 1 in the λ coordinates. Now take the inverseFourier transform (in λ ) of both sides. The left side becomes (cid:3) b N K and then use the fact that F − λ { (2 π ) − m/ λ } ( t ) = δ ( t ) and we obtain (cid:3) b { N K ( z, t ) } = δ ( z ) δ ( t ) d ¯ z K − F − λ (cid:26) (2 π ) − m/ X L ∈I q C K,L ( α ) ˜ S L ( z α , ˆ λ ) d ¯ Z ( z, α ) L (cid:27) = δ ( z ) δ ( t ) d ¯ z K − S K ( z, t ) using (15)as desired. (cid:3) A New Derivation of the Integral Formula – Proof of Theorem 2.3
Proof of Theorem 2.3.
We first assume the Szeg¨o kernel is zero, that is, | Γ L | = ∅ for all L ∈I q . Consequently, it follows from (12) that ˜ N L ( z α , ˆ λ ) = R ∞ ˜ H L ( s, z α , ˆ λ ) ds . To prepare for thecalculation of ˜ N L ( z, t ), we use polar coordinates and write λ = ατ where α belongs to the unitsphere S m − and τ >
0. We observe2 e sε αj,L | µ λj | | µ λj | sinh( s | µ λj | ) = 4 e sτε αj,L | µ αj | | µ αj | τe s | µ αj | τ − e − s | µ αj | τ = 4 e sτ ( ε αj,L − | µ αj | | µ αj | τ − e − s | µ αj | τ nd coth( s | µ λj | ) = e sτ | µ αj | + e − sτ | µ αj | e sτ | µ αj | − e − sτ | µ αj | = 1 + e − sτ | µ αj | − e − sτ | µ αj | . We now recover N K ( z, t ) from (14) by computing the inverse Fourier transform using polarcoordinates ( λ = ατ, α ∈ S m − , τ > N K ( z, t ) = (2 π ) − m/ X L ∈I q Z ∞ s =0 Z λ ∈ R m C K,L ( α ) d ¯ Z ( z, α ) L ˜ H L ( s, z, ˆ λ ) e it · λ dλ ds = (2 π ) − m/ X L ∈I q Z ∞ s =0 Z ∞ τ =0 Z α ∈ S m − C K,L ( α ) d ¯ Z ( z, α ) L ˜ H L ( s, z, c ατ ) e it · ατ τ m − dα dτ ds where dα is the surface volume form on the unit sphere in R m .Let r = e − sτ in the s -integral and so ds = − dr/ (2 τ r ) and the oriented r -limits of integrationbecome 1 to 0. We obtain N K ( z, t ) = 4 n π ) m + n X L ∈I q Z r =0 Z α ∈ S m − C K,L ( α ) d ¯ Z ( z, α ) L Z ∞ τ =0 | log r | n − ν ( α ) ν ( α ) Y j =1 r (1 − ε αj,L ) | µ αj | | µ αj | − r | µ αj | e − τ ( A α ( r,z ) − it · α ) τ n + m − dτ dα drr where A α ( r, z ) = 2 | log r | | z ′′ α | + ν ( α ) X j =1 | µ αj | r | µ αj | − r | µ αj | ! | z αj | . We now perform the τ -integral by using the following formula: Z ∞ τ =0 τ p e − aτ dτ = p ! a p +1 for Re a > S K ( z, t ) = 4 n (2 π ) m + n X L ∈I q Z α ∈ Γ L C K,L ( α ) d ¯ Z ( z, α ) L Z ∞ (cid:16) n Y j =1 | µ αj | (cid:17) τ n + m − e − τ ( P nj =1 | µ αj || z αj | − iα · t ) dτ dα = 4 n ( n + m − π ) m + n X L ∈I q Z α ∈ Γ L C K,L ( α ) d ¯ Z ( z, α ) L Q nj =1 | µ αj | ( P nj =1 | µ αj || z αj | − iα · t ) n + m dα which concludes the proof for (5).Finally, if S L ( z, t ) = 0, then using (14) and (12) N K ( z, t )= 1(2 π ) m Z ∞ s =0 X L ∈I q (cid:18)Z α ∈ Γ L C K,L ( α ) d ¯ Z ( z, α ) L Z ∞ τ =0 (cid:16) ˜ H L ( s, z, c τ α ) − ˜ S L ( z, c τ α ) (cid:17) e iτ ( α · t ) τ m − dτ dα (cid:19) ds + 1(2 π ) m Z ∞ s =0 X L ∈I q (cid:18)Z α ∈ Γ L Z α Γ L C K,L ( α ) d ¯ Z ( z, α ) L Z ∞ τ =0 ˜ H L ( s, z, c τ α ) e iτ ( α · t ) τ m − dτ dα (cid:19) ds = I K + II K . he second set of integrals is virtually identical to what we computed earlier and we get II K =4 n ( n + m − π ) m + n Z r =0 X L ∈I q Z α Γ L C K,L ( α ) d ¯ Z ( z, α ) L n Y j =1 r (1 − ε αj,L ) | µ αj | | µ αj | − r | µ αj | dα ( A ( r, z ) − iα · t ) n + m − drr where A α ( r, z ) = n X j =1 | µ αj | r | µ αj | − r | µ αj | ! | z αj | . For the first set of integrals, we observe that I K = 4 n π ) m + n X L ∈I q (cid:18)Z r =0 Z α ∈ Γ L C K,L ( α ) d ¯ Z ( z, α ) L Z ∞ τ =0 h(cid:16) n Y j =1 | µ αj | − r | µ αj | (cid:17) e − τ ( A α ( r,z ) − it · α ) − n Y j =1 | µ αj | e τ ( A α (0 ,z ) − it · α ) i τ n + m − dτ dα drr = 4 n ( n + m − π ) m + n X L ∈I q (cid:18)Z r =0 Z α ∈ Γ L C K,L ( α ) d ¯ Z ( z, α ) L h(cid:16) n Y j =1 | µ αj | − r | µ αj | (cid:17) A α ( r, z ) − it · α ) n + m − − Q nj =1 | µ αj | ( A α (0 , z ) − it · α ) n + m − i dα drr . This completes the proof of (6).That the convergence of the resulting integrals is absolute follows from a straightforward Taylorexpansion argument around r = 0 and r = 1, the only possible points where the integrand appearsto blow up. (cid:3) Examples
We analyze three examples in this section, all of which fall into the cases discussed in Remark2.4 so the formulas from Theorem 2.3 are slightly simpler. We discuss codimension 2 quadrics in C when q = 0 ,
2, the Heisenberg group (so m = 1), and the product the Heisenberg groups (so wefall into the sum of squares case).5.1. Codimension quadrics in C . When n = m = 2, we wrote down the formulas for N inthe case of three canonical examples [BR13]: • M where φ ( z, z ) = ( | z | , | z | ) T • M where φ ( z, z ) = (2 Re( z ¯ z ) , | z | − | z | ) T • M where φ ( z, z ) = (2 | z | , z ¯ z )) T These examples are canonical in the sense that any quadric in C × C whose Levi form has imagewhich is not contained in a one-dimensional cone is biholomorphic to one of these three examples(see [Bog91]). Additionally, these three examples perfectly demonstrate the three possibilities forsolvability/hypoellipticity of (cid:3) b on quadrics.The quadric M is simply a Cartesian product of Heisenberg groups and both solvability andhypoellipticity are impossible for any degree. In this case, A λ = (cid:18) λ λ (cid:19) , o the eigenvalues of A λ are λ and λ , so { ( n + ( λ ) , n − ( λ )) } = { (2 , , (1 , , (1 , , (0 , , (0 , } .For M , it follows from Peloso and Ricci [PR03] that solvability and hypoellipticity occur for(0 , q ) forms if and only if q = 0 or q = 2. In this case, A λ = (cid:18) λ λ λ − λ (cid:19) which gives us eigenvalues ±| λ | , so that for all λ ∈ R \{ , } , ( n + ( λ ) , n − ( λ )) = (1 , N ( z, t ) = C ( | z | + | t | ) − / , where C is a constant [BR13, Theorem 3].For M , (cid:3) b is solvable if and only if q = 0 or 2 and is never hypoelliptic. In this case, A λ = (cid:18) λ λ λ (cid:19) so that the eigenvalues are λ ± | λ | . Thus { ( n + ( λ ) , n − ( λ )) } = { (1 , , (1 , , (0 , } with the degen-erate values occurring when λ = 0. In Corollary 5.1 below, we give a more useful formula for N on M [BR13]. The analysis of the operator is extremely complicated and delicate and is the subjectof a later work in the series [BR]. We must mention the paper of Nagel, Ricci, and Stein whichanalyzes L p estimates on a class of higher codimension quadrics in C n × C m which depend only on | z j | , 1 ≤ j ≤ m [NRS01]. However, their result applies to neither M nor M for these quadricscanot be described in this manner.5.2. Example M . Let q = 0. As defined in Section 1, M = { ( z, w ) ∈ C × C : Im w = 2 | z | , Im w = 2 Re( z ¯ z ) } . Here, m = n = 2, and for α = (cos θ, sin θ ), we easily compute µ α = 1 + cos θ , µ α = cos θ −
1. Thefunction φ satisfies both A and A , though we will concentrate on the case q = 0 (the case for q = 2 is similar). Since L = { } , ε αj = − sgn( µ αj ) and so ε α = − ε α = +1 (except when θ = 0or 1 which is a set of measure zero). We obtain N ( z, t ) = (2 π ) − Z πθ =0 Z r =0 r cos θ σ ( θ ) σ ( θ )(1 − r σ ( θ ) )(1 − r σ ( θ ) )(16) × dr dθ ( − iα ( θ ) · t + σ ( θ ) E ( r, θ ) | z θ | + σ ( θ ) E ( r, θ ) | z θ | ) where α ( θ ) = (cos θ, sin θ ) , σ ( θ ) = 1 + cos θ, σ ( θ ) = 1 − cos θ, E j ( r, θ ) = 1 + r σ j ( θ ) − r σ j ( θ ) . We first wrote this formula in [BR13]. We wish to express it in a more useful and computable formwhich we will use in [BR].We let t = ( t , t ) which gives α ( θ ) · t = t cos θ + t sin θ . We also let x = r σ , so dx = σ r σ − dr and σ = σ σ = 1 − cos θ θ , dθ = dσ ( σ + 1) √ σ and obtain cos θ = 1 − σ σ and sin θ = ± √ σ σ here ± is + for θ ∈ [0 , π ] and − for θ ∈ ( π, π ]. Also the interval 0 ≤ θ ≤ π corresponds to theoriented σ interval [0 , ∞ ) and the interval π ≤ θ ≤ π corresponds to ( ∞ , z is expressed in terms of the eigenvectors of φ λ . To this end, we set z {√ σ } = 1 √ σ (cid:0) z + √ σz (cid:1) z {√ σ } = − √ σ (cid:0) √ σz − z (cid:1) and ˜ z {√ σ } = − z {√ σ } = 1 √ σ (cid:0) − z + √ σz (cid:1) ˜ z {√ σ } = − z {√ σ } = − √ σ (cid:0) √ σz + z (cid:1) We then obtain the following corollary to Theorem 2.3:
Corollary 5.1.
The fundamental solution to (cid:3) b for M on functions is given by convolution withthe kernel N ( z, t )= 2(2 π ) − Z ∞ σ =0 Z x =0 √ σ ( σ + 1)(1 − x )(1 − x σ ) dx dσ h − i (cid:0) t − σ + t √ σ (cid:1) + (cid:16) x − x (cid:17) | z {√ σ }| + σ (cid:16) x σ − x σ (cid:17) | z {√ σ }| i + 2(2 π ) − Z ∞ σ =0 Z x =0 √ σ ( σ + 1)(1 − x )(1 − x σ ) dx dσ h − i (cid:0) t − σ − t √ σ (cid:1) + (cid:16) x − x (cid:17) | ˜ z {√ σ }| + σ (cid:16) x σ − x σ (cid:17) | ˜ z {√ σ }| i . This formula is the launching point for [BR].5.3.
The Heisenberg group.
Denote the Heisenberg group H n ∼ = R n × R . The Kohn Laplacian (cid:3) b has a nontrivial kernel in the case that L = ∅ or L = { , . . . , n } . The calculation for these twocases is identical and we prove the details in the case L = { , . . . , n } . A derivation of a relatedformula from the classical methods appears in [Ste93, pp.615-617]. We set(17) log (cid:16) | z | − it | z | + it (cid:17) = log( | z | − it ) − log( | z | + it )for all z ∈ C n and t ∈ R and assume that the logarithm is defined via the principal branch. Theorem 5.2.
On the Heisenberg group H n ,1. The relative fundamental solution e − s (cid:3) b ( I − S ) to (cid:3) b = ¯ ∂ ∗ b ¯ ∂ b on functions is given by theintegration kernel N ∅ ( z, t ) = 2 n − ( n − π n +1 | z | + it ) n h log (cid:16) | z | + it | z | − it (cid:17) − n − X j =1 j i .
2. The relative fundamental solution e − s (cid:3) b ( I − S n ) to (cid:3) b = ¯ ∂ b ¯ ∂ ∗ b on (0 , n ) -forms is given by theintegration kernel N { ,...,n } ( z, t ) = 2 n − ( n − π n +1 | z | − it ) n h log (cid:16) | z | − it | z | + it (cid:17) − n − X j =1 j i . emark . (1) Up to a function in ker (cid:3) b , our formula appears to be the complex conjugateof the formula in [Ste93, Chapter XIII, Equation (51)]. This is a consequence of the factthat our computations are taken with respect to right invariant vector fields and not leftinvariant vector fields.(2) For a discussion regarding the consequences of the existence of a relative fundamentalsolution, we again refer the reader to [Ste93, Chapter XIII, Section 4.2]. It is easy tosee that the convolution N with a Schwartz function will be an object in L and henceorthogonal to ker (cid:3) b . Proof.
Since L = { , . . . , n } , the Szeg¨o kernel S ( z, ˆ λ ) = S L ( z, ˆ λ ) has support supp S L ( z, ˆ λ ) = [0 , ∞ )which means (suppressing L ) N ( z, t ) = 1 √ π Z ∞ Z ∞ (cid:0) H ( s, z, ˆ λ ) − S ( z, ˆ λ ) (cid:1) e itλ ds dλ + 1 √ π Z −∞ Z ∞ H ( s, z, ˆ λ ) e itλ ds dλ Equation (6) yields(18) N ( z, t ) = 4 n ( n − π ) n +1 Z r h − r ) n r − r | z | − it ) n − | z | − it ) n i + r n − (1 − r ) n r − r | z | + it ) n dr Set a = | z | − it | z | + it and for δ > a δ = | z | + δ − it | z | + δ + it (so | a | = | a δ | = 1). The reason that we introduce a δ isthat a logarithm appears in the integral, and log a is not well defined with the principal branch if | z | = 0. By introducing δ , it is immediate that for any a δ log a δ = log( | z | + δ − it ) − log( | z | + δ + it )and by sending δ →
0, we obtain log a as in (17). Ignoring the constants, we compute I δ = Z r h r )( | z | + δ ) − it (1 − r )) n − | z | + δ − it ) n i + r n − ((1 + r )( | z | + δ ) + it (1 − r )) n dr = Z r h | z | + δ + it ) r + | z | + δ − it ) n − | z | + δ − it ) n i + r n − (( | z | + δ − it ) r + ( | z | + δ + it )) n dr = 1( | z | + δ + it ) n (cid:20) Z (cid:16) r + a δ ) n − a nδ (cid:17) drr + Z r n − ( a δ r + 1) n dr (cid:21) For the second integral, we change variables r = s and compute Z s n − ( a δ s + 1) n ds = Z ∞ r + a δ ) n drr . Thus, ( | z | + δ + it ) n I δ = lim ǫ → (cid:20) Z ∞ ǫ r + a δ ) n drr + 1 a nδ log ǫ (cid:21) . A geometric series argument shows that1 r ( r + a δ ) n = 1 a nδ r − a nδ ( r + a δ ) − n − X j =1 a n − jδ ( r + a δ ) j +1 . herefore( | z | + δ + it ) n I δ = lim ǫ → (cid:20) Z ∞ ǫ a nδ r − a nδ ( r + a δ ) dr − n − X j =1 Z ∞ ǫ a n − jδ ( r + a δ ) j +1 dr + 1 a nδ log ǫ (cid:21) = lim ǫ → (cid:20) log( a δ + ǫ ) a nδ − n − X j =1 ja n − jδ ( a δ + ǫ ) j (cid:21) = 1 a nδ (cid:16) log a δ − n − X j =1 j (cid:17) . Thus, if we set N δ ( z, t ) to equal the right hand side of (18) except with | z | replaced by | z | + δ ,then | z | + δ − it | z | + δ − it stays away from the branch cut and N δ ( z, t ) = 2 n − ( n − π n +1 | z | + δ − it ) n h log (cid:16) | z | + δ − it | z | + δ + it (cid:17) − n − X j =1 j i = 2 n − ( n − π n +1 | z | + δ − it ) n h log( | z | + δ − it ) − log( | z | + δ + it ) − n − X j =1 j i . This function is continuous in δ , thus we may send δ → (cid:3) The Cartesian product of Heisenberg groups.
In contrast to the explicit computabilityof the Heisenberg group case, if M = { ( z, w ) ∈ C × C : Im w j = | z j | } ,L = { , } , and α = (cos θ, sin θ ), then Γ α { , } is the first quadrant and from Theorem 2.3, we have N { , } ( z, t ) = 1 π Z r =0 Z π π | cos θ sin θ | r | cos θ | (1 − r | cos θ | )(1 − r | sin θ | ) 1( A α ( r ) − i ( t cos θ + t sin θ )) n + m − dθ drr + 1 π Z r =0 Z π π | cos θ sin θ | r | cos θ +sin θ | (1 − r | cos θ | )(1 − r | sin θ | ) 1( A α ( r ) − i ( t cos θ + t sin θ )) n + m − dθ drr + 1 π Z r =0 Z π π | cos θ sin θ | r | sin θ | (1 − r | cos θ | )(1 − r | sin θ | ) 1( A α ( r ) − i ( t cos θ + t sin θ )) n + m − dθ drr + 1 π Z r =0 Z π cos θ sin θ h − r cos θ )(1 − r sin θ ) 1( A α ( r ) − i ( t cos θ + t sin θ )) n + m − − A α (0) − i ( t cos θ + t sin θ )) n + m − i dθ drr where A α ( r ) = | cos θ | r | cos θ | − r | cos θ | ! | z | + | sin θ | r | sin θ | − r | sin θ | ! | z | . On the other hand, using (5), we compute the Szeg¨o kernel S ∅ ( z, t ) = 6 π Z π π cos θ sin θ (( | z | + it ) cos θ + ( | z | + it ) sin θ ) dθ = 1 π ( | z | + it ) ( | z | + it ) . References [BG88] R. Beals and P. Greiner.
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School of Mathematical and Statistical Sciences, Arizona State University, Physical SciencesBuilding A-Wing Rm. 216, 901 S. Palm Walk, Tempe, AZ 85287-1804Department of Mathematical Sciences, 1 University of Arkansas, SCEN 327, Fayetteville, AR72701
Email address : [email protected], [email protected]@asu.edu, [email protected]