The Fundamental Solution to \Box_b on Quadric Manifolds -- Part 4. Nonzero Eigenvalues
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 4. NONZERO EIGENVALUES
ALBERT BOGGESS AND ANDREW RAICH
Abstract.
This paper is the fourth of a multi-part series in which we study the geometricand analytic properties of the Kohn Laplacian and its inverse on general quadric subman-ifolds of C n × C m . The goal of this article is explore the complex Green operator in thecase that the eigenvalues of the directional Levi forms are nonvanishing. We 1) investigatethe geometric conditions on M which the eigenvalue condition forces, 2) establish optimalpointwise upper bounds on complex Green operator and its derivatives, 3) explore the L p and L p -Sobolev mapping properties of the associated kernels, and 4) provide examples. Introduction
In this paper, we investigate the complex Green operator N on quadric submanifolds M ⊂ C n × C m for which all the eigenvalues of the directional Levi forms are nonzero. Thecomplex Green operator is the (relative) inverse to the Kohn Laplacian (cid:3) b . By definition, a quadric submanifold is defined as(1) M = { ( z, w ) ∈ C n × C m : Im w = φ ( z, z ) } where φ : C n × C n → C m is a sesquilinear vector-valued quadratic form. The Levi form inthe direction of ν ∈ S m − , the unit sphere in R m , is defined as φ ν ( z, z ) = φ ( z, z ) · ν . The Kohn Laplacian is defined as (cid:3) b = ¯ ∂ b ¯ ∂ ∗ b + ¯ ∂ ∗ b ¯ ∂ b where ¯ ∂ b is the usual tangential Cauchy-Riemann operator and ¯ ∂ ∗ b is its L adjoint. The (relative) inverse to (cid:3) b on ( p, q )-forms,when it exists, is called the complex Green operator and denoted by N p,q . The existenceof the complex Green operator produces the L -minimizing solution operator to the ¯ ∂ b -equation, ¯ ∂ ∗ b N p,q , in a canonical fashion. For background on the ¯ ∂ b and (cid:3) b -operators, pleasesee [Bog91, CS01, BS17].In this paper, our main interest is the class of quadrics with codimension m ≥ φ ν ( z, z ) has only nonzero eigenvalues for each ν ∈ S m − . We show, that the nonvanishing eigenvalue condition forces n to be even (soreplace n with 2 n ) with exactly half of the eigenvalues to be positive and half negative. For0 ≤ q ≤ n , we establish sharp upper bounds on the size of N ,q and its derivatives in termsof the control geometry on M that are analogous to the classical estimates on N for theHeisenberg group or the finite type hypersurface type case (that is, m = 1) in C [NRSW89,Chr91a, Chr91b, FK88]. This allows us to invoke the theory of homogeneous groups to prove L p and L p -Sobolev mapping properties for appropriate derivatives of N . When q = n , (cid:3) b isnot solvable by [PR03], but we can still estimate the canonical relative fundamental solution Mathematics Subject Classification.
Key words and phrases. quadric submanifolds, higher codimension, nonzero eigenvalues, complex Greenoperator, hypoellipticity, L p regularity.This work was supported by a grant from the Simons Foundation (707123, ASR). or (cid:3) b given by R ∞ e − s (cid:3) b ( I − S n ) ds where S n is the orthogonal projection onto ker (cid:3) b . Wealso provide several examples, illustrating our estimates.More generally, when the eigenvalues are not bounded away from zero, the control distancefails to govern estimates on N ,q . This failure is apparent in some general hypersurface typeCR manifolds as well as some simple higher codimension quadrics [Mac88, NS06, BRb]. Inhigher codimension, the correct geometry is far from understood as the singularities of N occur both on and off of the diagonal.For a bit more background and history, the tangential Cauchy-Riemann operator, or ¯ ∂ b ,and the associated Kohn Laplacian (cid:3) b are arguably the most important operators in severalcomplex variables because they are intrinsically intertwined with the complex geometry,topology, and analysis of CR manifolds. Solving the (cid:3) b -equation is often a product ofhard analysis and sophisticated functional analysis, and the solution produced by thesetechniques may have excellent function theoretic properties but is not constructive (e.g.,[Sha85, Koh86, HR11, HR15, CR]). Often, this approach is not (yet) sufficient to producethe estimates we seek on the solution in the higher codimension setting. Hence we restrict tothe class of quadrics, which have a Lie group structure which helps provide a more explicitformula for the solution that is suitable to estimate.In our opinion, one of the most beautiful results is the computation of N p,q on the Heisen-berg group by Folland and Stein [FS74]. The problem, though, is that their techniquedoes not easily generalize, especially to higher codimension. Consequently, one of main ap-proaches to the (cid:3) b -problem on these manifolds is through the (cid:3) b -heat equation. The firstresults in this direction were for the sub-Laplacian on the Heisenberg group by Hulanicki[Hul76] and Gaveau [Gav77]. More results followed for (cid:3) b on quadrics of increasing gen-erality [BR09, YZ08, CCT06, BGG96, BGG00, Eld09] culminating (so far) with our paper[BR11] where we compute the (cid:3) b -heat kernel on a general quadric. Virtually all of theseresults rely on the fact that we can identify M with its tangent space at the orgin, C n × R m ,and push the problem forward onto C n × R m . The problem with these papers (ours included)is that if we put coordinates ( z, t ) on M , the solution is only given up to a partial Fouriertransform in t . Given that [FS74] is the gold standard (for us), we are taking the formulafrom [BR11] and trying to undo the Fourier transform and integrate out the time variable.This allows us to recover to both the projection onto ker (cid:3) b as well as N ,q . In the earlierparts of the series, [BR13, BRa, BR20, BRb], we started with the formula for (cid:3) b -heat kerneland generated an integral formula for both diagonal part of the complex Green operator aswell as the projection onto ker (cid:3) b . We also categorized the class of quadrics of codimension2 in C into three (cid:3) b -invariant groups and computed 0th order asymptotics for the kernelsfor each of these groups. We noticed that in one case, where the directional Leviform hasnonvanishing eigenvalues, the complex Green operator was both solvable and hypoelliptic.Additionally, the estimates were particularly good, allowing us to prove continuity results in L p -Sobolev spaces, 1 < p < ∞ . In many respects, the current paper is a generalization ofthis case.In addition to our series of papers, Mendoza proves the following: Let M be a CR manifoldof CR codimension > (cid:3) b computedwith respect to any Hermitian metric is hypoelliptic in all degrees except those correspondingto the number of positive or negative eigenvalues of the Levi form [Men]. Additionally, in thespecial case that φ ( z, z ) is a sum of squares, Nagel, Ricci, and Stein [NRS01] proved pointwise pper bounds on both the complex Green operator and the projections onto ker (cid:3) b , and theyestablished the L p theory in addition.The outline of the paper is as follows. In the next section, we state our main results,primarily Theorem 2.1. We continue in Section 3 where we define our notation and explorethe geometric consequences of our hypotheses. The proof of Theorem 2.1 for q = n is spreadover Sections 4 - 11. In Section 12, we discuss the adjustments to adapt the argument forthe q = n case. We conclude the paper with several new examples in Section 13.2. Main Results
Define the projection π : C n × C m → C n × R m by π ( z, t + is ) = ( z, t ). For each quadric M ⊂ C n × C m , the projection π induces a CR structure and Lie group structure on C n × R m ,and we call this Lie group G (or G M ). The projection is therefore a CR isomorphism andwe use the same notation for objects on M and their pushfowards/pullbacks on G .We introduce only the notation necessary to state the main results. Define the normfunction ρ : C n × R m → [0 , ∞ ) by ρ ( z, t ) = max {| z | , | t | / } ≈ | z | + | t | / . For a multiindex I = ( I , I ) ∈ N n + m , the multiindex I ∈ N n records the differentiationin the z and ¯ z -variables, and I ∈ N m records the t -derivatives. Given such a multiindex I ,define the weighted order of I by h I i = | I | + 2 | I | and the order of I by | I | = | I | + | I | . Theorem 2.1.
Let M ⊂ C n × C m be a quadric submanifold defined by (1) with associatedprojection G , and assume that eigenvalues of the directional Levi forms are nonzero. Let ≤ q ≤ n and N = N ,q . For any multiindex I ∈ N n + m , there exists a constant C I > sothat | D I N ( z, t ) | ≤ C I ρ ( z, t ) n + m − h I i . Remark .
1. The homogeneous dimension of M is 2(2 n + m ), and we are inverting anorder two operator (with respect to ρ ). This explains the power of ρ in the denominatorof Theorem 2.1.2. The case q = n is special because ker (cid:3) b = 0. The relative fundamental solution that weestimate is R ∞ e − s (cid:3) b ( I − S n ) ds where S n : L ,n ( M ) → ker (cid:3) b ∩ L ,n ( M ) is the orthogonalprojection.Let W k,p ( M ) denote the Sobolev space of forms on M with z , ¯ z and t derivatives of order k are in L p ( M ). Theorem 2.3.
Let M ⊂ C n × C m be a quadric submanifold defined by (1) with associatedprojection G , and assume that eigenvalues of the directional Levi forms are nonzero. Let ≤ q ≤ n and N = N ,q . Given a multiindex I ∈ N n + m so that h I i = 2 , the operator D I N ,q is exactly regular on W k,p ( M ) for all k ≥ and all < p < ∞ . In other words, D I N ,q extends to a bounded operator on W k,p ( M ) . In particular, D I N ,q is a hypoellipticoperator.Proof. The proof follows easily following the approach of [BR20, Section 3]. Identifying M with C n × R m , we can view M as a homogeneous group with norm function ρ ( z, t ).From Theorem 2.1, it follows that the integration kernel of D I N ,q and its derivatives have he appropriate pointwise decay. A second consequence of Theorem 2.1 is that D I N ,q is atempered distribution, and combining this fact with the natural dilation structure and that D I N ,q is a convolution operator shows that D I N ,q is uniformly bounded on normalizedbump functions. This is exactly what is required to establish the L p boundedness, 1 < p < ∞ .From the fact that D I N ,q is a convolution operator, boundedness on W k,p ( C n × R m ) followsimmediately. (cid:3) Notation and Hypotheses
Suppose that M is the quadric submanifold M = { ( z, w ) ∈ C n × C m : Im w = φ ( z, z ) } . Recall that for ν ∈ S m − , φ ν ( z, z ) = φ ( z, z ) · ν = z ∗ A ν z where A ν is a Hermitian symmetricmatrix. Proposition 3.1. If m ≥ and if the eigenvalues of A ν are all nonzero for each ν ∈ S m − ,then n must be even. Furthermore for each ν ∈ S m − , half of the eigenvalues of A ν arepositive and half of the eigenvalues are negative, counting multiplicity.Proof. Note that if λ is an eigenvalue for A ν , then − λ is an eigenvalue for A − ν . If n isodd, then det A − ν = − det A ν . If m ≥
2, this change of sign in the determinant means thatdet A ν ′ = 0 for some other ν ′ ∈ S m − . Therefore, the assumption that all of eigenvalues arenonzero for each ν ∈ S m − implies that n must be even.Also note that all the eigenvalues of A ν are real. Let p ν ( λ ) = det( A ν − λI ) be thecharacteristic polynomial for A ν . Let P ν be the set of the positive roots of p ν . We areassuming that P ν is bounded away from zero for all ν ∈ S m − . Let K be a compact disc inthe open right half plane which contains P ν in its interior for all ν ∈ S m − . The number ofroots in P ν is given by the Argument Principle:Number of positive roots of p ν = 12 πi I ∂K p ′ ν ( λ ) dλp ν ( λ ) . This is clearly a continuous integer-valued function of ν ∈ S m − which is a connected set for m ≥
2. Therefore, the number of positive roots of p ν is constant for all ν ∈ S m − . Since n is even and A − ν = − A ν , we see that p − ν ( − λ ) = p ν ( λ ). Therefore if the number of positiveroots of p ν is k , then the number of negative roots of p − ν ( · ) is also k , which in turn impliesthat the number of positive roots of p − ν is n − k . Since the number of positive roots isconstant in ν , we conclude that k = n − k , and hence k = n/ (cid:3) The complex Green operator.
As a consequence of the above discussion, we assumethe following: • For each ν ∈ S m − , there are n positive eigenvalues µ νj for j in some index set P ν ofcardinality n from the set { , , . . . , n } and n negative eigenvalues µ νk for k ∈ ( P ν ) c ,the complement of P ν . Remark . Given that our eigenvalues stay bounded away from 0 independently of ν ∈ S m − , we may arrange the indices so that P ν = P is independent of ν . enote the set of increasing q -tuples by I q = { K = ( k , . . . , k q ) ∈ N q : 1 ≤ k < k < · · · < k q ≤ n } . To write the fundamental solution for (cid:3) b [BRa] applied to a (0 , q )-form ofthe form f K d ¯ z K for a fixed K ∈ I q , we need to establish some notation. Fix λ ∈ R m − \ { } and set ν = λ | λ | ∈ S m − . We write z ∈ C n in terms of the unit eigenvectors of φ λ whichmeans that z λj = z νj is given by z ν = Z ( ν, z ) = U ( ν ) ∗ · z where U ( ν ) is the matrix whose columns are the eigenvectors, v νk , 1 ≤ k ≤ n of thedirectional Levi form φ ν , and · represents matrix multiplication with z written as a columnvector. Note that the corresponding orthonormal basis of (0 , d ¯ Z j ( ν, z ) , ≤ j ≤ n, where d ¯ Z ( ν, z ) = U ( ν ) T · d ¯ z where d ¯ z is written as a column vector of (0 , T stands fortranspose. Note that z ν = Z ( ν, z ) depends smoothly on z ∈ C n but only locally integrable as a function of ν ∈ S m − [Rai11].For each K ∈ I q , we will need to express d ¯ z K , in terms of d ¯ Z ( ν, z ) L for L ∈ I q . We have(2) d ¯ z K = X L ∈I q det( ¯ U ( ν ) K,L ) d ¯ Z ( ν, z ) L where ¯ U ( ν ) K,L is the q × q minor ¯ U ( ν ) comprised of elements in the rows K and columns L . Note that if q = 2 n , then the above sum only has one term and det( ¯ U ( ν ) K,K ) = 1. Inaddition, I = ∅ , so the sum (2) does not appear.Until Section 12, we work under the assumption that 0 ≤ q ≤ n is fixed and q = n .From [BRa], the the fundamental solution to (cid:3) b on (0 , q )-forms spanned by d ¯ z K is given byconvolution with the kernel N K ( z, t ) = K n,m X L ∈I q Z ν ∈ S m − det( ¯ U ( ν ) K,L ) d ¯ Z ( ν, z ) L (3) × Z r =0 (cid:18) Y j ∈ Lc ∩ Pj ∈ L ∩ Pc r | µ νj | | µ νj | (1 − r | µ νj | ) Y k ∈ L ∩ Pk ∈ Lc ∩ Pc | µ νk | (1 − r | µ νk | ) (cid:19) A ( r, ν, z ) − iν · t ) n + m − dr dνr where dν is surface measure on the unit sphere S m − , the dimensional constant(4) K n,m = 4 n (2 n + m − π ) m +2 n , and A ( r, ν, z ) = n X j =1 | µ νj | r | µ νj | − r | µ νj | ! | z νj | . Taking derivatives in z k or t ℓ is relatively straight forward because z only appears in A ( r, ν, z ) and t only appears in the ν · t term. In particular, we compute that for 1 ≤ k ≤ n ,(5) ∂∂z k A ( r, ν, z ) = n X j =1 | µ νj | r | µ νj | − r | µ νj | ! U ( ν ) ∗ j,k · Z j ( ν, z ) . imilarly, ∂ A ( r,ν,z ) ∂z k ∂z k = 0 as are all third (and higher) order derivatives. Also,(6) ∂∂z k ∂ ¯ z k A ( r, ν, z ) = n X j =1 | µ νj | r | µ νj | − r | µ νj | ! U ( ν ) ∗ j,k · U ( ν ) ∗ j,k . A key fact which will be used later is the following: If P ( u ) is a polynomial in u ∈ C , then(7) n X j =1 P ( µ νj ) | Z j ( ν, z ) | = z ∗ · U ( ν ) · P ( D ν ) · U ( ν ) ∗ · z = z ∗ · P ( A ν ) · z where D ν is the diagonal matrix with the eigenvalues of A ν as its diagonal entries. Theimportance of this equation is as follows. The right side is a quadratic expression in z and ¯ z with coefficients that are polynomials in the coordinates of ν (since A ν depends linearly on ν ).3.2. Derivative Notation.
We define a multiindex I = ( I , I ) ∈ N n + m where I ∈ N n is multiindex that records the z and ¯ z -derivatives and I ∈ N m records the t -derivatives.Recall that the weighted order of I is h I i = | I | + 2 | I | and the order of I is | I | = | I | + | I | .Each derivative in a t -variable introduces a component of ν into the numerator and increasesthe power of ( A ( r, ν, z ) − iν · t ) in the denominator by 1. A derivative in a z -variable ismore complicated to write down – either the power of ( A ( r, ν, z ) − iν · t ) increases by onein the denominator and a component of ∇ z A ( r, ν, z ) is introduced in the numerator or thedenominator remains unchanged and a term in the numerator changes from (5) to (6). Wewill not need a precise accounting of the constants but only the number of first and secondderivatives of A ( r, ν, z ) that appear. We denote ∇ z, ¯ z A to be the vector of first derivativeswith respect to both the z and ¯ z derivatives and ∇ z, ¯ z to denote all of the second orderderivatives of A . By an abuse of notation, we write D I n A ( r, ν, z ) − iν · t ) n + m − o = c n,m, | I | D I n ν I ( A ( r, ν, z ) − iν · t ) n + m − | I | o = X ( I ′ ,I ′′ | I ′ | +2 | I ′′ | = | I | c n,m,I ′ ,I ′′ , | I | ν I ( ∇ z, ¯ z A ( r, ν, z )) I ′ ( ∇ z, ¯ z A ( r, ν, z )) I ′′ ( A ( r, ν, z ) − iν · t ) n + m − | I ′ | + | I ′′ | + | I | . where | I ′ | is the number of first order derivatives in z or ¯ z and where | I ′′ | is the number ofsecond order derivatives in z and ¯ z . Note that | I ′ | + 2 | I ′′ | = | I | and not | I ′ | + | I ′′ | . Forexample, suppose that I = (2 , , , . . . , , z factors and one ¯ z factor. Then( ∇ z, ¯ z A ( r, ν, z )) I = (cid:16) ∂∂z A ( r, ν, z ) (cid:17) (cid:16) ∂∂ ¯ z A ( r, ν, z ) (cid:17) . nd | I ′ | = 1, | I ′′ | = 1 and | I | = 3. We analyze each piece of D I N separately and conse-quently, the integral to estimate is(8) N I ′ ,I ′′ ,I ( z, t ) = X L ∈I q Z ν ∈ S m − det( ¯ U ( ν ) K,L ) d ¯ Z ( ν, z ) L Z r =0 (cid:18) Y j ∈ Lc ∩ Pj ∈ L ∩ Pc r | µ νj | | µ νj | (1 − r | µ νj | ) Y k ∈ L ∩ Pk ∈ Lc ∩ Pc | µ νk | (1 − r | µ νk | ) (cid:19) × ν I ( ∇ z, ¯ z A ( r, ν, z )) I ′ ( ∇ z, ¯ z A ( r, ν, z )) I ′′ ( A ( r, ν, z ) − iν · t ) n + m − | I ′ | + | I ′′ | + | I | dν drr . The Case when | t | ≥ | z | , q = n The tricky case is | t | > | z | and so we will factor out a | t | n + m − | I ′ | + | I ′′ | + | I | from thedenominator and we will rotate ν coordinates via an orthogonal matrix M t chosen so that M t ( t/ | t | ) is the unit vector in the ν direction (so in the new coordinates, ν · t = ν ). Wealso set ν t = M − t ν andˆ q = z | t | / ∈ C n , and Q ( ν t , ˆ q ) = Z ( ν t , z ) | t | / = U ( ν t ) ∗ · z | t | / . Note that | Q ( ν t , ˆ q ) | = | ˆ q | since U ν t is unitary.Since ( ∇ z, ¯ z A ( r, ν t , z )) I ′ contains a monomial in z, ¯ z of degree I ′ , we obtain N I ′ ,I ′′ ,I ( z, t ) = | t | − (2 n + m − | I ′ | + | I ′′ | + | I | ) N I ′ ,I ′′ ,I ( q )= | t | − (2 n + m − h I i ) N I ′ ,I ′′ ,I (ˆ q )where N I ′ ,I ′′ ,I (ˆ q )(9)= X L ∈I q Z ν t ∈ S m − Z r =0 det( ¯ U ( ν t ) K,L ) d ¯ Z ( ν t , z ) L B L ( r, ν t ) ( ν t ) I ( ∇ z, ¯ z A ( r, ν t , ˆ q )) I ′ ( ∇ z, ¯ z A ( r, ν t , ˆ q )) I ′′ ( A ( r, ν t , ˆ q ) − iν ) n + m − | I ′ | + | I ′′ | + | I | dν drr and B L ( r, ν ) = Y j ∈ Lc ∩ Pj ∈ L ∩ Pc r | µ νj | | µ νj | − r | µ νj | Y k ∈ L ∩ Pk ∈ Lc ∩ Pc | µ νk | − r | µ νk | (10) A ( r, ν, ˆ q ) = n X j =1 | µ νj | r | µ νj | − r | µ νj | ! | Q j ( ν, ˆ q ) | . (11)To prove Theorem 2.1 in the case that | t | ≥ | z | and q = n , it suffices to prove the followingtheorem. Theorem 4.1.
There is a uniform constant
C > so that | N I ′ ,I ′′ ,I (ˆ q ) | ≤ C for all ˆ q ∈ C n . There are two primary terms which need to be analyzed: B L ( r, ν ), and A ( r, ν, ˆ q ). We firstconcentrate on the singularity at r = 1. The singularity at r = 0 is easier and is handled inSection 10. . Analysis of B L ( r, ν ) in the case r > / , q = n It turns out that the key to analyzing B L ( r, ν ) is B ∅ ( r, ν ). To this end, for 0 < r < u ∈ R , let(12) f ( r, u ) = ur u (1 − r u ) g ( r, u ) = f ( r, u ) + u = u (1 − r u ) . Note that g ( r, u ) = f ( r, − u ). Since µ νj > j ∈ P and µ νk < k ∈ P c , we can write B ( r, ν ) = B ∅ ( r, ν ) = Y j ∈ P r | µ νj | | µ νj | (1 − r | µ νj | ) Y k ∈ P c | µ νk | (1 − r | µ νk | )then B ( r, ν ) drr = Y j ∈ P f ( r, µ νj ) Y k ∈ P c g ( r, − µ νk ) drr (13) = n Y j =1 f ( r, µ νj ) drr . (14)Both descriptions of this term are useful. Note that the eigenvalues µ νj are not necessarilysmooth in ν ∈ S m − (though they are continuous). However as the next lemma shows, B ( r, ν ) is real analytic in both 0 < r < ν ∈ S m − and this uses the fact that theeigenvalues are bounded away from zero. Lemma 5.1.
The function B ( r, ν ) = Q j ∈ P f ( r, µ νj ) Q k ∈ P c g ( r, − µ νk ) is real analytic in both < r < and in ν ∈ S m − .Proof. Using (13), write B ( r, ν ) = B + ( r, ν ) · B − ( r, ν ) where B + ( r, ν ) = Y j ∈ P f ( r, µ νj ); B − ( r, ν ) = Y k ∈ P c g ( r, − µ νk ) . It suffices to show that ln B + ( r, ν ) and ln B − ( r, ν ) are real analytic in 0 < r < ν ∈ S m − . We have ln B + ( r, ν ) = X j ∈ P ln ˜ f ( r, µ νj )where ˜ f ( r, z ) = zr z (1 − r z ) for z = u + iv . Since ˜ f ( r, z ) > z = u >
0, ln( ˜ f ( r, z )) is realanalytic in 0 < r < z = u + iv in a neighborhood, U ⊂ C containing the set { u + i u > } . Note that by hypothesis, there is a compact set K ⊂ { u + i u > } which contains all the positive eigenvalues µ νj for j ∈ P and ν ∈ S m − .Let γ ∈ U be a smooth simple closed curve which contains K . Let D ( ν, z ) = det( A ν − zI )where recall that A ν is the Hermitian matrix for φ ν ( z, z ). The eigenvalues µ νj , j ∈ P are theroots of the analytic function z → D ( ν, z ) that lie inside γ . By standard Residue theory, wehave ln B + ( r, ν ) = X j ∈ P ln ˜ f ( r, µ νj ) = 12 πi I z ∈ γ ln ˜ f ( r, z ) D ′ ( ν, z ) dzD ( ν, z ) here D ′ ( ν, z ) refers to the z -derivative of D ( ν, z ). Now observe that the right side is realanalytic in ν ∈ S m − since ν → A ν is real analytic in ν (and D ( ν, z ) = 0 for z ∈ γ ). Theproof of the analyticity of ln B − ( r, ν ) is similar. This completes the proof of the lemma. (cid:3) We observe that B L ( r, ν ) = B ( r, ν ) Y j ∈ L ∩ P c f ( r, − µ νj ) f ( r, µ νj ) Y k ∈ L ∩ P g ( r, µ νk ) g ( r, − µ νk ) = B ( r, ν ) Y j ∈ L r − µ νj . We need the following piece of notation for the next lemma. For J ∈ I q and ( ℓ , . . . , ℓ q ) ∈ N q , set ǫ ( ℓ ,...,ℓ q ) J = ( − | σ | if { ℓ , . . . , ℓ q } = J as sets and | σ | is the length of the permutationthat takes ( j , . . . , j q ) to J . Set ǫ ( ℓ ,...,ℓ q ) J = 0 otherwise.It may be the case the B L ( r, ν ) is not analytic, however, we have the following lemma.We also use the notation that if M is a matrix and J, L ∈ I q , the M J,L is the q × q minor of M with entries M jℓ , j ∈ J , ℓ ∈ L . Lemma 5.2.
The function ν X L ∈I q det( ¯ U ( ν ) K,L ) d ¯ Z ( ν, z ) L Y j ∈ L r − µ νj is real analytic in both < r < and in ν ∈ S m − . Moreover, (15) X L ∈I q det( ¯ U ( ν ) K,L ) d ¯ Z L ( ν, z ) Y j ∈ L r − µ νj = X J ∈I q det([ r − ¯ A ν ] K,J ) d ¯ z J . Remark . In view of the above expression for B L ( r, ν ), we record the following equationfor future reference(16) X L ∈I q det( ¯ U ( ν ) K,L ) d ¯ Z L ( ν, z ) B L ( r, ν ) = X J ∈I q det([ r − ¯ A ν ] K,J ) B ( r, ν ) d ¯ z J . which is real analytic in 0 < r < ν ∈ S m − in view of Lemma 5.1. Proof.
Once we show (15), the analyticity statement follows immediately from the fact that¯ A ν depends analytically on ν and therefore the matrix r − ¯ A ν will also depend analytically on ν . First, we record two basic equations. Suppose M is a N × N matrix with complex entriesand consider w = M z , where w, z ∈ C N . If 1 ≤ q ≤ N and K ∈ I q , then(17) d ¯ w K = X J ∈ I q det( ¯ M K,J ) d ¯ z J This is easily established using standard multilinear algebra.Second, conjugation by U ( ν ) diagonalizes the matrix A ν , and diagonalizes r − A ν . In par-ticular,(18) R − µ ν = U ( ν ) T r − ¯ A ν ¯ U ( ν )where R − µ ν is the (2 n ) × (2 n ) matrix with real entries, r − µ νj , on the diagonal and zeros offof the diagonal. ow we start with the left side of (15): X L ∈I q det( ¯ U ( ν ) K,L ) d ¯ Z L ( ν, z ) Y j ∈ L r − µ νj = X L ∈I q det( ¯ U ( ν ) K,L ) det( R − µ ν L,L ) d ¯ Z L ( ν, z )= X L ∈I q det([ ¯ U ( ν ) R − µ ν ] K,L ) d ¯ Z L ( ν, z )where the second equation uses the fact that R − µ ν is a diagonal matrix. Now use (18) andthe fact that ¯ U ( ν ) U ( ν ) T = I to conclude thatLeft side of (15) = X L ∈I q det[ r − ¯ A ν ¯ U ( ν )] K,L d ¯ Z L ( ν, z ) = d ( r − ¯ A ν ¯ z ) K where the last equality uses the equation z = U ( ν ) Z ( ν, z ) as well as (17) with w = r − A ν z .Now (15) follows by using (17) to expand out the right side of the above equation in termsof d ¯ z J . (cid:3) We make the following change of variables for s > r = r ( s ) = s − s + 1 or equivalently s = r + 11 − r with drr = 2 ds ( s − . Note that 1 / ≤ r < s ≥ Proposition 5.4.
1. The expansion of B ( r ( s ) ,ν ) r ′ ( s ) r ( s ) around s = ∞ is B ( r ( s ) , ν ) r ′ ( s ) r ( s ) = 22 n (1 − /s ) " n − X ℓ =0 P ℓ ( ν ) s n − ℓ − + O ( s, ν ) s (20) Typical Monomial in P ℓ ( ν ) = ν ℓ − e ; where e is even with ≤ e ≤ ℓ. (21) Here, P ℓ ( ν ) is a polynomial in ν = ( ν , . . . ν m ) ∈ S m − of total degree ℓ . By an abuse ofnotation, the term, ν ℓ − e , in (21) stands for a monomial in the coordinates of ν of totaldegree ℓ − e .Additionally, the (Taylor) remainder O ( s, ν ) is real analytic in s > and ν ∈ S m − .Furthermore O ( s, ν ) is bounded in s > .2. Modulo coefficients (that are computable but not relevant to the estimate), the expansionof det([ r ( s ) − ¯ A ν ] K,J ) around s = ∞ is comprised of a sums of terms ν ℓ ′ − e ′ s ℓ ′ where ℓ ′ ≥ , e ′ is an even integer with ≤ e ′ ≤ ℓ ′ , and (22) ν ℓ ′ − e ′ is a monomial of degree ℓ ′ − e ′ in the coordinates of ν ∈ S m − . To start the proof of Proposition 5.4, let F ( s, u ) = f ( r ( s ) , u ) , G ( s, u ) = g ( r ( s ) , u ) . Using (14), we obtain(23) B ( r ( s ) , ν ) r ′ ( s ) r ( s ) = 2 n Y j =1 F ( s, µ νj ) 1( s − . e will need to Taylor expand B ( r ( s ) , ν ) in s about s = ∞ , which is equivalent to letting s = 1 /w and expanding about w = 0. To this end, let(24) ˜ F ( w, u ) = w [ F (1 /w, u ) + u/
2] = w (cid:20) g (cid:18) − w w , u (cid:19) − u (cid:21) Lemma 5.5. ˜ F ( w, u ) is a real analytic function of w and u for − < w < and u ∈ R . Inaddition,1. For each fixed u , the function w → ˜ F ( w, u ) is an even function of w ;2. For each fixed w , the function u → ˜ F ( w, u ) is an even function of u ;3. The coefficients in the Taylor series expansions of ˜ F ( w, u ) in w about w = 0 are of theform: j th coefficient = (cid:26) if j is odd P j ( u ) if j is evenwhere P j ( u ) is a polynomial of degree j in u that involves only even powers of u .Proof. We have ˜ F ( w, u ) = wu − (cid:0) − w w (cid:1) u − wu wu − e u ln ( − w w ) − wu . (25)Since 1 − e z vanishes to first order in z at the origin, the ( u, w ) power series expansion of thedenominator has a factor of uw , which cancels with the uw in the numerator. The resultingterm is analytic and nonvanishing in a neighborhood of the origin. Hence ˜ F is real analytic.Part (2) follows easily from (24). Parts (1) follows by a calculation (Maple helps). For Part(3), we expand the exponential term appearing in (25) and cancel the common factor of uw to obtain ˜ F ( w, u ) = " L ( w ) + uw L ( w ) + ( uw ) L ( w ) + . . . − wu L ( w ) = w − ln (cid:0) − w w (cid:1) is analytic on − < w <
1. From repeated w -differentiations of˜ F , one can see that the j th w -derivative of ˜ F at w = 0 is a polynomial expression in u ofdegree j . In view of Part (2), this expression is zero if j is odd and only involves even powersof u when j is even as stated in Part (3). This concludes the proof of the lemma. (cid:3) We let w = 1 /s and unravel this lemma to imply the following expansions for F ( s, u ). F ( s, u ) = s − u u − s − u − u + 490 s + ∞ X j =3 p j ( u ) s j − (26)where p j ( u ) is a polynomial in u of degree 2 j with only even powers of u . The above seriesconverges uniformly on any closed subset of { s > } . Note that F has the linear term u/ u .Our next task is to use (26) to expand the expression B ( r ( s ) , ν ) given in (23) in powersof 1 /s (about s = ∞ ). To get started, here are the first few terms (in order of decreasing owers of s ): B ( r ( s ) , ν ) r ′ ( s ) r ( s ) = 2( s − n Y j =1 F ( s, µ νj )(27) = 2 s n n ( s − n Y j =1 " − µ νj s + ( µ νj ) − s + ∞ X k =2 p k ( µ νj ) s k (28)where p k ( u ) is a polynomial of degree 2 k with only even powers of u .Now, we expand the product on the right (denoted by Product) in terms of symmetricpolynomials in the variables µ ν , . . . , µ ν n . First, a definition. Definition 5.6. A symmetric polynomial of degree m on R N is a polynomial P of degree m in the variables ( u , . . . u N ) ∈ R N such that P ( u , . . . u N ) = P ( u σ (1) , . . . u σ ( N ) ) for allpermutations σ on { , , . . . , N } .An allowable multiindex α = ( α , . . . , α N (cid:1) is a nonincreasing N -tuple of nonnegativeintegers, that is, integers α j , 1 ≤ j ≤ N , satisfying α ≥ α ≥ · · · ≥ α N ≥
0. Let | α | = α + · · · + α N and define S α ( u , . . . , u N ) = ′ X i ,...,i N u α i . . . u α N i N where the sum is taken over all distinct indices i , . . . , i N each ranging from 1 to N .Note the prime over the sum emphasizes that the indices i j are distinct. Also for clarity, ifthe 2 n -tuple α ends with multiple zeros, we stop writing after the first zero. For example, wewrite S , ( µ ν , . . . , µ ν n ) for S , ,..., ( µ ν , . . . , µ ν n ). Clearly each S α ( u ) is a symmetric polynomialof degree | α | . For a fixed m >
0, the collection of S α ( u ) over all allowable multiindices α with | α | = m forms a basis of the space of symmetric polynomials of degree m on R N .From an examination of the product in (28) and using the fact that p k ( u ) is a polynomialof degree 2 k with only even powers of u , we obtain the following lemma. Lemma 5.7.
For ℓ ≥ , the coefficient of s ℓ in the Product on the right side of (28) is alinear combination of S α ( µ ν , . . . , µ ν n ) , with | α | = ℓ, ℓ − , ℓ − , . . . , ℓ − e where e is the largest even integer which is less than or equal to ℓ . As an illustration of this lemma, we write out the first few terms of the Product on theright side of (28)Product = 1 − s − n X k =1 µ νk + s − (1 / n X k =1 [( µ νk ) −
1] + X j = k µ νj µ νk ! + . . . = 1 − s S , ( µ ν ) + 1 s (cid:16) (1 / S , ( µ ν ) − n ) + S , , ( µ ν ) (cid:17) + . . . . Now we need transform the S α ( µ ν ) into a more useful basis involving elementary symmetricfunctions. efinition 5.8. For 0 ≤ ℓ ≤ N , the elementary symmetric function of degree ℓ in R N is(29) E ℓ ( u ) = X ( j ,...,j ℓ ) ∈I ℓ u j · · · u j ℓ . With N = 2 n , the key fact about the E ℓ ( µ ν ) is that they appear as coefficients in thecharacteristic polynomial for A ν :(30) det( A ν − λI ) = λ n + n X ℓ =1 ( − ℓ E ℓ ( µ ν ) λ n − ℓ . Note that each row of A ν depends linearly and homogeneously on ν and thus the coefficientof λ n − ℓ , i.e., E ℓ ( µ ν ), is a homogenous polynomial of degree ℓ in the coordinates of ν =( ν , . . . , ν m ) ∈ S m − . As a consequence, we have Lemma 5.9. E ℓ ( µ ν ) , is a homogenous polynomial of degree ℓ in the coordinates of ν =( ν , . . . , ν m ) ∈ S m − . In particular, E ℓ ( µ ν ) is analytic in ν even though the eigenvalues µ νj are not necessarilydifferentiable in ν . Definition 5.10.
Suppose L = ( ℓ , . . . , ℓ j , . . . ) is a multiindex (of indeterminate length)with ℓ j ≥ ℓ j +1 and only a finite number of the ℓ j are nonzero. For u = ( u , . . . , u N ), define E L ( u ) = E ℓ ( u ) · E ℓ ( u ) · · · E ℓ N ( u ) ,E L ( u ) is a symmetric polynomial of degree | L | = ℓ + · · · + ℓ j + . . . The next theorem is [Sta99, Theorem 7.4.4].
Theorem 5.11.
For a given integer, m ≥ , the collection of { E L ( u ); | L | = m ; u ∈ R N } is a basis for the space of symmetric polynomials of degree m on R N . The following corollary follows from this theorem and Lemma 5.7.
Corollary 5.12.
In the expansion of B ( r ( s ) , ν ) r ′ ( s ) r ( s ) given in (28), the coefficient of s n − − ℓ is expressible as a linear combination of E L ( µ ν ) = E k ( µ ν ) n k · · · E ( µ ν ) n E ( µ ν ) n . . . , k ≥ where L = ( n k , . . . , n ) with | L | = n + 2 n + . . . kn k = ℓ − e , where e is an even integer with ≤ e ≤ ℓ . Moreover, this coefficient is a linear combination of monomials in the componentsof ν = ( ν , . . . , ν m ) ∈ S m − each having degree ℓ − e . We will not need to know the exact values of the coefficients in this expansion. Rather,the key phrase is the last sentence in the above corollary: the coefficient of s n − − ℓ is a linearcombination of monomials in the components of ν = ( ν , . . . , ν m ) ∈ S m − each having degree ℓ − e . roof of Proposition 5.4. In view of Corollary 5.12 and (28), equations (20) and (21) bothhold. Additionally, the real analyticity of the Taylor remainder term O ( s, ν ) for s > ν ∈ S m − is assured from Lemma 5.1, the (Taylor) remainder O ( s, ν ). Furthermore O ( s, ν )is bounded in s > r ( s ) − u about s = ∞ yields r ( s ) − u = (cid:16) s − s + 1 (cid:17) − u = 1 − us + 2 u s − u (1 + 2 u ) s + ∞ X k =4 ˜ p k ( u ) s k where ˜ p k ( u ) is a a polynomial that has only odd powers of u if k is odd and even powers of u if k is even (this fact can be proven by setting w = s , and Taylor expansion around w = 0,and an induction argument on the form of the derivatives). This means r ( s ) − A ν = 1 − A ν s + 2 A ν s − A ν (1 + 2 A ν ) s + ∞ X k =4 ˜ p k ( A ν ) s k . Equation (22) now follows from expanding the appropriate q × q minor determinant. (cid:3) Expansion of A in Denominator in the case / ≤ r < , q = n The formula for A ( r, ν, ˆ q ) is given in (11). Using (12), the coefficient function in front of | Q j ( ν, ˆ q ) | is f ( r, µ νj ) + g ( r, µ νj ) = 2 f ( r, µ νj ) + µ νj which in the s variables (where r = r ( s ) = s − s +1 ), using (26), this becomes(31) Coefficient of | Q j | = 2 F ( s, µ νj ) + µ νj = s + ∞ X k =1 p k ( µ νj ) s k − where p k ( u ) is a polynomial of degree 2 k with only even powers of u . From (26), the firsttwo terms are p ( u ) = u −
13 ; p ( u ) = − u − u + 445 etc.Now using (11), (31), and (7), we obtain A ( r ( s ) , ν, ˆ q ) − iν = s | ˆ q | + n X j =1 ∞ X k =1 p k ( µ νj ) s k − | Q j ( ν, ˆ q ) | − iν = ( s | ˆ q | − iν ) + ∞ X k =1 ˆ q ∗ · p k ( A ν ) · ˆ qs k − . (32)We denote by e j ∈ C n the j th unit vector e j = (0 , . . . , , . . . ,
0) (1 in the j th position). Weobserve that(33) ∂A∂z j (cid:12)(cid:12)(cid:12) ( r ( s ) ,ν, ˆ q ) = s ˆ q ∗ · e j + ∞ X k =1 ˆ q ∗ · p k ( A ν ) · e j s k − and(34) ∂ A∂z j ∂ ¯ z j (cid:12)(cid:12)(cid:12) ( r ( s ) ,ν, ˆ q ) = se ∗ j · e j + ∞ X k =1 e ∗ j · p k ( A ν ) · e j s k − ince the substitution of ˆ q for z comes after the differentiation.Note the coefficients ˆ q ∗ · p k ( A ν ) · ˆ q consist of quadratic terms in ˆ q and ¯ˆ q together with alinear combination of monomial terms in the coordinates of ν of degree 2 k − e where e iseven with 0 ≤ e ≤ k .7. Expanding the Kernel for N in the case / ≤ r < , q = n From (9) and (16), to estimate N I ′ ,I ′′ ,I (ˆ q ), we must investigate the integrands(35) N K,J (ˆ q, s, ν ) = det([ r ( s ) − ¯ A ν ] K,J ) B ( r ( s ) , ν t ) r ′ ( s ) r ( s ) ( ν t ) I ( ∇ z, ¯ z A ( r ( s ) , ν t , ˆ q )) I ′ ( ∇ z, ¯ z A ( r ( s ) , ν t , ˆ q )) I ′′ ( A ( r ( s ) , ν t , ˆ q ) − iν ) n + m − | I ′ | + | I ′′ | + | I | . For nonzero V ∈ C , consider the Taylor expansion1( V + ζ ) n + m − | I ′ | + | I ′′ | + | I | = 1 V n + m − | I ′ | + | I ′′ | + | I | + ∞ X j =1 α j ζ j V n + m − | I ′ | + | I ′′ | + | I | + j which converges uniformly for | ζ | ≤ | V | / α j are unimportant). We make useof the following expansions: From (32) with V = s | q | − iν and ζ = P ∞ k =1 ˆ q ∗ · p k ( A νt ) · ˆ qs k − wehave ( ν t ) I ( ∇ z, ¯ z A ( r ( s ) , ν t , ˆ q )) I ′ ( ∇ z, ¯ z A ( r ( s ) , ν t , ˆ q )) I ′′ ( A ( r ( s ) , ν t , ˆ q ) − iν ) n + m − | I ′ | + | I ′′ | + | I | = ( ν t ) I (cid:20) s | ˆ q | − iν ) n + m − | I ′ | + | I ′′ | + | I | + ∞ X j =1 α j hP ∞ k =1 ˆ q ∗ · p k ( A νt ) · ˆ qs k − i j ( s | ˆ q | − iν ) n + m − j + | I ′ | + | I ′′ | + | I | (cid:21) (36) × ( ∇ z, ¯ z A ( r ( s ) , ν t , ˆ q )) I ′ ( ∇ z, ¯ z A ( r ( s ) , ν t , ˆ q )) I ′′ . Carefully writing out ( ∇ z, ¯ z A ( r ( s ) , ν t , ˆ q )) I ′ and ( ∇ z, ¯ z A ( r ( s ) , ν t , ˆ q )) I ′′ would be more confusingthan useful, as we only need the lead term and the generic expression for the higher orderterms. Using (33), we write( ∇ z, ¯ z A ( r ( s ) , ν t , ˆ q )) I ′ = s | I ′ | C ,I ′ (cid:0) (ˆ q, ¯ˆ q ) | I ′ | (cid:1) + ∞ X K =1 X k ··· + k | I ′ | = Kkj ≥ , all j s | I ′ |− K C k ,...,k I ′ ,I ′ (cid:0) (ˆ q, ¯ˆ q ) | I ′ | (cid:1) p k ,I ′ ( ν t ) · · · p k | I ′ | ,I ′ ( ν t )(37)and using (34), we have( ∇ z, ¯ z A ( r ( s ) , ν t , ˆ q )) I ′′ = s | I ′′ | C ,I ′′ + ∞ X K =1 X k ··· + k | I ′′ | = Kkj ≥ , all j s | I ′′ |− K C k ,...,k I ′′ ,I ′′ p k ,I ′′ ( ν t ) · · · p k | I ′′ | ,I ′′ ( ν t ) . (38)Here, C ,I ′ (cid:0) (ˆ q, ¯ˆ q ) | I ′ | (cid:1) and C k ,...,k I ′ ,I ′ (cid:0) (ˆ q, ¯ˆ q ) | I ′ | (cid:1) denote polynomial expressions involving ˆ q and¯ˆ q of degree | I ′ | and C k ,...,k I ′′ ,I ′′ are constants (independent of ˆ q ). rom the derivative products (37) and (38), a typical term in ( ∇ z, ¯ z A ) I ′ ( ∇ z, ¯ z A ) I ′′ is of theform(39) s | I ′ | + | I ′′ |− K C K (cid:0) (ˆ q, ¯ˆ q ) | I | (cid:1) p k ( ν t ) · · · p k | I | ( ν t )where C K (cid:0) (ˆ q, ¯ˆ q ) | I | (cid:1) is a polynomial in ˆ q and ¯ˆ q of degree at most | I | and k + · · · + k | I ′ | + | I ′′ | = K ≥ k j is a nonnegative integer.The main term is the lowest degree term in s and is given by( ν t ) I s | I ′ | + | I ′′ | C ((ˆ q, ¯ˆ q ) | I ′ | )( s | ˆ q | − iν ) n + m − | I ′ | + | I ′′ | + | I | where C ((ˆ q, ¯ˆ q ) | I ′ | ) is a monomial in terms in the coordinates for (ˆ q, ¯ˆ q ) of degree | I ′ | . Its exactexpression is possible to compute but not relevant for this calculation.Letting K j,I ′ ,I ′′ = k + · · · + k j + k I + · · · + k | I ′ | + | I ′′ | ≥ | I ′ | + | I ′′ | , a typical term from the expansion of (36) is(40) Typical Term in (36)= ( ν t ) I s | I ′ | + | I ′′ |− K j,I ′ ,I ′′ C ˜ K (cid:0) (ˆ q, ¯ˆ q ) | I ′ | (cid:1) p k ( ν t ) · · · p k | I ′ | + | I ′′ | ( ν t ) × C (ˆ q j , ¯ˆ q j )˜ p k − e ( ν t ) . . . ˜ p k j − e j ( ν t )( s | ˆ q | − iν ) n + m − j + | I ′ | + | I ′′ | + | I | s (2 k − ··· +(2 k j − where k ℓ ≥ j ≥ j = 0) C (ˆ q j , ¯ˆ q j ) stands for monomial terms inthe coordinates for ˆ q of degree j and ¯ˆ q , of degree j , and where each ˜ p k a − e a ( ν ), 1 ≤ a ≤ j is a monomial in the coordinates of ν of degree 2 k a − e a . Here, e a is an even integer with0 ≤ e a ≤ k a . Set E j = e + · · · + e j and incorporate the matrix M − t into the C (ˆ q j , ¯ˆ q j ) termto obtain(41) Typical Term in (40) = C t ((ˆ q, ¯ˆ q ) j + | I ′ | ) ν K j,I ′ ,I ′′ − E j ν I ( s | ˆ q | − iν ) n + m − j + | I ′ | + | I ′′ | + | I | s K j,I ′ ,I ′′ − ( j + | I ′ | + | I ′′ | ) and E j is a an even integer with 0 ≤ E j ≤ k + · · · + k j ). Note that we have used thesame abuse of notation with ν K j − E j as we did in (21) and the dependence on t is a (possiblynonsmooth but certainly bounded) dependence on t/ | t | . We will not need all the terms inthe expansion - just up through j + | I ′ | + | I ′′ | = 2 n − j + | I ′ | + | I ′′ | = 2 n (and therefore K j,I ′ ,I ′′ := K n ≥ n ). In particular, using (37) and (38),(42) Typical Remainder Term in (36) = O t (1) O ( ν, s )( s | ˆ q | − iν ) n + m − s K n − n where O t (1) is a real analytic function of the coordinates of ˆ q and ¯ˆ q that may depend on t .Also, O ( ν, s ) stands for a real analytic function in ν ∈ S m − and s > s .Note that the power of s in the denominator is at least 2 n since K n ≥ n , as mentionedabove.In the expansions of B ( r ( s ) , ν ) r ′ ( s ) r ( s ) and det([ r − ¯ A ν ] K,J ) given in (20) and (22), respectively,writing (1 − /s ) − = P ∞ j ′ =0 s − j ′ . Therefore, by (21), we see that up to the coefficients of ome polynomials, a typical term in the expansion of det([ r − ¯ A ν ] K,J ) B ( r ( s ) ,ν ) r ′ ( s ) r ( s ) is(43) Typical Term of det([ r − ¯ A ν ] K,J ) B ( r ( s ) , ν ) r ′ ( s ) r ( s ) = s n − j ′ − − ℓ ′ ν ℓ ′ − e ′ together with a remainder of the form O ( ν,s ) s j ′ . Note e ′ is even and 0 ≤ e ′ ≤ ℓ ′ .Now the typical term of N K,J is the product of a term in (41) with a term in (43). ThereforeTypical Term in N K,J = C (ˆ q, ¯ˆ q ) j + | I ′ | s N − − ℓ ν ℓ − e ( ν t ) I ( s | ˆ q | − iν ) N + m − | I | where(44) N = 2 n + j + | I ′ | + | I ′′ | , ℓ = ℓ ′ + 2 j ′ + 2 K j + | I ′ | + | I ′′ | , e = e ′ + E j + 2 j ′ . (45)Note that e is even with 0 ≤ e ≤ ℓ , due to the constraints listed in on the indices in (41)and (43).The remainder term for N K,J is the product of the remainders given in (42) and theremainder given just after (43): a typical term comprising the remainder is(46) Typical Remainder Term for N K,J = O (ˆ q n ) O ( ν, s )( s | ˆ q | − iν ) n + m − s K n − n +2+2 j ′ where O ( ν, s ) is real analytic function in ν ∈ S m − and s ≥ s . Note thatthe exponent in s in the denominator is at least 2 since K j ≥ j, j ≥
1. We will now showthat the integral (over ν ∈ S m − , and s ≥
1) of the typical term in (44) is bounded in ˆ q . Wewill also show the same for a remainder term in (46).As to the first task, let ˆ r = | ˆ q | > H N,ℓ,m,e,I (ˆ r, s, ν ) = s N − − ℓ ν ℓ + I − e ( s ˆ r − iν ) N + m − | I | . To establish Theorem 4.1 over the region 1 / ≤ r <
1, we need to show that for each ℓ ≥
0, there is a uniform constant C such that(47) (cid:12)(cid:12)(cid:12) Z ν ∈ S m − Z ∞ s =3 H N,ℓ,m,e,I (ˆ r, s, ν ) dν (cid:12)(cid:12)(cid:12) ≤ C for all ˆ r > Unit Sphere Integrals.
To compute the integral of H N,ℓ,m,e,I (ˆ r, s, ν ) over the unit sphere, S m − in R m , we needto use some easy facts about spherical integrals:(1) Let ν = ( ν , ν ′ ) ∈ S m − , then ν ′ belongs to a m − R m − ofradius | ν ′ | = p − ν .(2) Let θ be the “angle” between ν and the ν ′ plane; note that ν = sin( θ ), − π/ ≤ θ ≤ π/
2; and | ν ′ | = cos θ .(3) Surface measure on the unit sphere in R m is dν = (cos θ ) m − dθ dν ′ where dν ′ issurface measure on, S m − , the unit sphere in R m − .(4) The integral of any odd function of ν ′ over S m − , the unit sphere in R m − , will bezero. sing the last fact, we claim that we can assume the monomial ν ℓ + I − e depends on ν only. To see this, write ν ℓ + I − e = ( ν ′ ) a ν b with | a | + b = | ℓ | + | I | − | e | . By (4), if | a | wereodd, then the ν ′ -integral would be zero. Thus we can assume a = e ′ where | e ′ | is even, whichimplies | b | = | ℓ | + | I | − ( | e | + | e ′ | ) = | ℓ | + | I | − | E | with | E | even. We can then factor outthe ( ν ′ ) a from the ν integral and we are left with ν ℓ + I − E within the ν integral.We now change variables and let x = ν = sin θ , − ≤ x ≤
1. Note that cos θ = √ − x and dθ = dx √ − x . Therefore(48) dν = (1 − x ) ( m − / dx dν ′ where dν ′ is surface measure on S m − . The desired estimate in (47) will follow from thefollowing lemma: Lemma 8.1.
For any nonnegative integers N , m and ℓ with m ≥ and any even integer E with ≤ E ≤ | ℓ | + | I | , let A ℓ,E,I N,m (ˆ r ) = Z x = − Z ∞ s =3 (1 − x ) ( m − / s N − − ℓ x ℓ − E + | I | ds dx ( s ˆ r − ix ) N + m − | I | then A ℓ,E,I N,m,I (ˆ r ) is a smooth function of ˆ r > up to ˆ r = 0 . As shown in the proof, the lemma is not true if E is odd. Proof of Lemma 8.1.
First write A ℓ,E,I N,m (ˆ r ) = C N,ℓ,I D N − (2+ ℓ )ˆ r n B ℓ,Em,I (ˆ r ) o where C N,ℓ is a constant and B ℓ,Em,I (ˆ r ) = Z x = − Z ∞ s =3 (1 − x ) ( m − / x ℓ − E + | I | ds dx ( s ˆ r − ix ) m + ℓ +1+ | I | . Here, D j ˆ r indicates the j th derivative with respect to ˆ r . The index j is allowed to be negativein which case this means the | j | th anti-derivative with respect to ˆ r (with a particular initialcondition specified at a fixed value of ˆ r = ˆ r > B ℓ,Em,I (ˆ r ) is smooth for ˆ r > r = 0. The s -integral can be computed to give:(49) B ℓ,Em,I (ˆ r ) = 1ˆ r ( m + ℓ + | I | ) b ℓ,Em,I (ˆ r )where b ℓ,Em,I (ˆ r ) = Z x = − (1 − x ) ( m − / x ℓ − E + | I | dx (3ˆ r − ix ) m + ℓ + | I | . We need to show b ℓ,Em,I (ˆ r ) is smooth in ˆ r > r = 0 and that(50) b ℓ,Em,I (ˆ r = 0) = 0for then (49) will imply that B ℓ,Em,I (ˆ r ) is smooth in ˆ r > r = 0. To this end, we notethat for ˆ r >
0, the integrand of b ℓ,Em,I (ˆ r ) has an analytic extension in x to the upper half of he complex plane. So we can deform the integral using Cauchy to the top half of the unitcircle, denoted by C + from z = − z = +1 to obtain b ℓ,Em,I (ˆ r ) = Z z ∈ C + (1 − z ) ( m − / z ℓ − E + | I | dz (3ˆ r − iz ) m + ℓ + | I | . This expression shows that b ℓ,Em,I (ˆ r ) extends smoothly (in fact, analytically) in ˆ r to a neigh-borhood of ˆ r = 0. All that remains to show is that b ℓ,Em,I (ˆ r = 0) = 0. We have(51) ( − i ) m + ℓ + | I | b ℓ,Em,I (ˆ r = 0) = Z z ∈ C + (1 − z ) ( m − / dzz m + E . If m = 3, then this integral is R z ∈ C + dzz E = 0 since e is even. If m is odd and greater than3, then this integral can be reduced using integration by parts with dv = 1 /z m + E dz and u = (1 − z ) ( m − / (note there are no boundary terms at z = ±
1) to obtain b ℓ,Em (ˆ r = 0) = c m,ℓ Z z ∈ C + (1 − z ) ( m − / dzz m + E − . One can continue integrating by parts this until the power of (1 − z ) is zero to obtain(52) b ℓ,Em,I (ˆ r = 0) = ˜ c m,ℓ,I Z z ∈ C + dzz E = 0 . This establishes (50) for m odd. (Note clearly, the above integral is not zero if E is odd,which is why this assumption is so necessary).If m ≥ b ℓ,Em,I (ˆ r = 0) = ˜ c m,ℓ,I Z z ∈ C + √ − z dzz E . Since E is even, let E = 2 k for a nonnegative integer k . Amazingly, there is a closed-formantiderivative:(53) Z √ − z z E dz = − k X j =0 (1 − z ) j +3 / z j +3 (cid:18) kj (cid:19) j + 3) . Clearly this antiderivative vanishes at both z = ±
1. If m = 2, then one can integrate byparts in (51) with dv = z dz √ − z and reduce to this integral to (53). Thus, Lemma 8.1, andhence (47) are proved. (cid:3) The Remainder Term, q = n To restate the remainder in (46)Remainder = O ( ν, s )( s | ˆ q | − iν ) n + m − s J , with J ≥ . We use the facts (1)-(3) about spherical integrals in the previous section with x = ν . Since s − J is integrable over { s ≥ } and since O ( ν ′ , ν , x ) is real analytic (and hence uniformlybounded) in ν ′ ∈ ( √ − x ) S m − ( m − √ − x ), it suffices toprove the following lemma, which will finish the proof of Theorem 4.1 for the integral overthe region 1 / ≤ r < emma 9.1. For m ≥ , let R ( s, ˆ r, ν ′ ) = Z x = − (1 − x ) m − O ( ν ′ , x, s ) dx ( s ˆ r − ix ) n + m − . Then R ( s, ˆ r, ν ′ ) is uniformly bounded for s ≥ , ˆ r ≥ , and ν ′ ∈ ( √ − x ) S m − .Proof. Divide up the interval − ≤ x ≤ {| x | ≥ / } and {| x | ≤ / } . The denomi-nator is bounded below on {| x | ≥ / } . The numerator is also bounded except in the case m = 2 in which case (1 − x ) m − has an integrable singularity at x = ± {| x | ≤ / } , we replace x by z ∈ C and note that the integrand can beextended to analytic function z in a complex neighborhood of the interval − / ≤ x ≤ / C be a path in this neighborhood and in the upper half plane which connects z = − / z = 1 / Z / x = − / (1 − x ) m − O ( ν ′ , x, s ) dx ( sr − ix ) n + m − = Z z ∈ C (1 − z ) m − O ( ν ′ , z, s ) dz ( sr − iz ) n + m − . Since the denominator is uniformly bounded away from zero, for z ∈ C , s ≥ r = | ˆ q | >
0, the integral on the right is uniformly bounded in ν ′ , r , and s . This completes theproof. (cid:3) The case ≤ r ≤ / , q = n Our starting point is (9) which equates to (35) but we wish to remain in the r variable.We fix K , restrict the r integral to 0 ≤ r ≤ / N AI ′ ,I ′′ ,I (ˆ q )= X L ∈I q Z ν ∈ S m − Z r =0 det( ¯ U ( ν ) K,L ) d ¯ Z ( z, ν t ) L B L ( r, ν t ) ( ν t ) I ( ∇ z, ¯ z A ( r, ν t , ˆ q )) I ′ ( ∇ z, ¯ z A ( r, ν t , ˆ q )) I ′′ ( A ( r, ν t , ˆ q ) − iν ) n + m − | I ′ | + | I ′′ | + | I | dν drr (54)= X J ∈I q d ¯ z J (cid:20) Z ν ∈ S m − Z r =0 det([ r − ¯ A ν ] K,J ) B ( r, ν t ) ( ν t ) I ( ∇ z, ¯ z A ( r, ν t , ˆ q )) I ′ ( ∇ z, ¯ z A ( r, ν t , ˆ q )) I ′′ ( A ( r, ν t , ˆ q ) − iν ) n + m − | I ′ | + | I ′′ | + | I | (cid:21) dν drr . (55)We denote by N JI ′ ,I ′′ ,I (ˆ q ) the d ¯ z J coefficient of N AI ′ ,I ′′ ,I (ˆ q ).We devote the remainder of this section to the proof of the following lemma, which willestablish Theorem 4.1 for the integral over the region 0 < r < / Lemma 10.1. (56) | N JI ′ ,I ′′ ,I (ˆ q ) | ≤ C for all ˆ q = z | t | / ∈ C n where C is a uniform constant. Recall from (10) that B L ( r, ν ) = Y j ∈ Lc ∩ Pj ∈ L ∩ Pc r | µ νj | | µ νj | − r | µ νj | Y k ∈ L ∩ Pk ∈ Lc ∩ Pc | µ νk | − r | µ νk | . ince L ∈ I q and q = n , at least one of L c ∩ P or L ∩ P c is nonempty. This means thereexist constants C > c = min n X j ∈ Lc ∩ Pj ∈ L ∩ Pc | µ νj | : ν ∈ S m − and L ∈ I q o so that(57) (cid:12)(cid:12)(cid:12) B ( r, ν ) r (cid:12)(cid:12)(cid:12) ≤ Cr c − for 0 < r < / . From this estimate, it follows that the integrals in (54) and therefore in (55) over { ≤ r ≤ / } × {| ν | ≥ / } are uniformly bounded for ˆ q ∈ C n . Moreover, we know from (16) andthe accompanying remark that the integrand of N JI ′ ,I ′′ ,I (ˆ q ) is real analytic in ν ∈ S m − and0 < r ≤ / ν -integral over | ν | ≤ /
2. We have A ( r, ν, ˆ q ) = n X j =1 | µ νj | r | µ νj | − r | µ νj | ! | Q j ( ν, ˆ q ) | with Q ( ν, ˆ q ) = | t | − / Z ( ν, z ) = U ( ν ) ∗ · ˆ q where U ( ν ) is the unitary matrix which diagonalizes the scalar valued Levi form in thenormal direction ν . For u ∈ R , let(58) Λ( u ) = | u | (cid:18) r | u | − r | u | (cid:19) . As a generalization of (7), we have(59) A ( r, ν, ˆ q ) = n X j =1 Λ( µ j ) | Q j ( ν, ˆ q ) | = ˆ q ∗ · Λ( A ν ) · ˆ q where A ν is the Hessian matrix of Φ( z, z ) · ν . Here Λ( A ν ) is computed by replacing | u | by p A ν in (58) and where (cid:16) I − r √ A ν (cid:17) − is the matrix inverse of I − r √ A ν . Furthermore, r √ A ν is defined as exp (cid:16) ln r p A ν (cid:17) . Note that since all the eigenvalues of A ν are real and boundedaway from zero, the (operator) norm of the matrix r √ A ν , for 0 ≤ r ≤ /
2, is less than onesince ln r <
0, guaranteeing the existence of the inverse of I − r √ A ν . For this analysis towork, we need to know the map ν → p A ν is analytic in ν , established in the followinglemma. Lemma 10.2.
The map ν → p A ν is analytic for ν ∈ S m − .Proof. Observe that the matrix X = A ν is a Hermitian symmetric matrix with positiveeigenvalues which are contained in a compact interval, say [ c , R ] ⊂ R with R > c >
0, forall ν ∈ S m − . So consider the power series for √ X about X = RI : √ X = ∞ X n =0 a n [ X − RI ] n hich has radius of convergence R . Since the open disc centered at x = R of radius R contains all the eigenvalues of A ν in its interior, the following series converges uniformly in ν : p A ν = ∞ X n =0 a n [ A ν − RI ] n . This series is clearly analytic in ν ∈ S m − . (cid:3) Proof of Lemma 10.1.
The expression for A given in (59) and the discussion following showsthat X ( r, ν, ˆ q ) := (cid:20) ∂∂ν { A ( r, ν t , ˆ q ) } − iI (cid:21) − is a smooth matrix on { < r ≤ / } × { ν ∈ S m − } with X ( r, ν, ˆ q ) → r → r in the denominator). Moreover, since dr u du = r u ln r , differentiation of A ( r, ν t , ˆ q ), X ( r, ν, ˆ q ), B ( r, ν t ), or det([ r − ¯ A ν ] K,J ) produces a term of the same size with a possible ad-ditional ln r term. However, from (57), there is always a r c − term, and r c − | ln r | N isintegrable at 0 for any power N since c > { ν ∈ S m − ; | ν | ≤ / } as the graph over the set { ν = ( ν , ν ′ ); | ν | ≤ / ν ′ ∈ S m − } . We use integration by parts for the integral over | ν | ≤ / Z ν =1 / ν = − / ( ν t ) I ( ∇ z, ¯ z A ( r, ν t , ˆ q )) I ′ ( ∇ z, ¯ z A ( r, ν t , ˆ q )) I ′′ ( A ( r, ν t , ˆ q ) − iν ) n + m − | I ′ | + | I ′′ | + | I | det([ r − ¯ A ν ] K,J ) B ( r, ν t ) r dν = Z ν =1 / ν = − / ∂∂ν (cid:26) − (2 n + m + | I ′ | + | I ′′ | + | I | − − ( A ( r, ν t , ˆ q ) − iν ) n + m + | I ′ | + | I ′′ | + | I |− (cid:27) × ( ν t ) I ( ∇ z, ¯ z A ( r, ν t , ˆ q )) I ′ ( ∇ z, ¯ z A ( r, ν t , ˆ q )) I ′′ X ( r, ν t , ˆ q ) det([ r − ¯ A ν ] K,J ) B ( r, ν t ) r dν = − Z ν =1 / ν = − / − (2 n + m + | I ′ | + | I ′′ | + | I | − − ( A ( r, ν t , ˆ q ) − iν ) n + m + | I ′ | + | I ′′ | + | I |− × ∂∂ν ( ( ν t ) I ( ∇ z, ¯ z A ( r, ν t , ˆ q )) I ′ ( ∇ z, ¯ z A ( r, ν t , ˆ q )) I ′′ X ( r, ν t , ˆ q ) det([ r − ¯ A ν ] K,J ) B ( r, ν t ) r ) dν + Boundary Terms at | ν | = 1 / . The power of ( A ( r, ν t , ˆ q ) − iν ) in the denominator has been reduced by one. As discussedabove and analogous to (57), we have (cid:12)(cid:12)(cid:12)(cid:12) ∂∂ν ( X ( r, ν t , ˆ q ) det([ r − ¯ A ν ] K,J ) B ( r, ν t )( ν t ) I ( ∇ z, ¯ z A ( r, ν t , ˆ q )) I ′ ( ∇ z, ¯ z A ( r, ν t , ˆ q )) I ′′ r ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | ln r | r c − which is integrable in 0 ≤ r ≤ /
2. The boundary terms are also controlled by a similarestimate. e continue integrating by parts in ν until we reduce the fractional expression involving( A ( r, ν t , ˆ q ) − iν ) to a log-term, to obtain: Z ν =1 / ν = − / ( ν t ) I ( ∇ z, ¯ z A ( r, ν t , ˆ q )) I ′ ( ∇ z, ¯ z A ( r, ν t , ˆ q )) I ′′ ( A ( r, ν t , ˆ q ) − iν ) n + m − | I ′ | + | I ′′ | + | I | det([ r − ¯ A ν ] K,J ) B ( r, ν t ) r dν = c n,m,I ′ ,I ′′ ,I Z ν =1 / ν = − / ln (cid:2) A ( r, ν t , ˆ q ) − iν (cid:3) E [ X ( r, ν t , ˆ q ) , det([ r − ¯ A ν ] K,J ) , B ( r, ν t ) , A ( r, ν t , ˆ q ) , I ′ , I ′′ , I ] r dν + Boundary Terms at | ν | = 1 / c n,m,I ′ ,I ′′ ,I is a constant depending only on n, m, I ′ , I ′′ , I ; ln is the principal branchof the logarithm defined on the right half-plane (note A ( r, ν t , ˆ q ) > E [ X ( r, ν t , ˆ q ) , det([ r − ¯ A ν ] K,J ) , B ( r, ν t ) , A ( r, ν t , ˆ q ) , I ′ , I ′′ , I ] is an expression involve a sum ofproducts of ν -derivatives of X ( r, ν t , ˆ q ), det([ r − ¯ A ν ] K,J ), and B ( r, ν t ) where the total numberof derivatives is 2 n + m + | I ′ | + | I ′′ | + | I | −
1. Note that | ln [ A ( r, ν t , ˆ q ) − iν ] | is integrablein ν uniformly in the other variables ν ′ ∈ S m − and 0 ≤ r ≤ /
2. In addition (cid:12)(cid:12)(cid:12) E [ X ( r, ν t , ˆ q ) , B ( r, ν t )] r (cid:12)(cid:12)(cid:12) ≤ C | ln r | n + m + | I ′ | + | I ′′ | + | I |− r c − which is also integrable ν ′ ∈ S m − and 0 ≤ r ≤ /
2. Similar estimates hold for the boundaryterms. This establishes (56) and completes the proof of Lemma 10.1. This also concludesthe proof of Theorem 4.1 and hence establishes Theorem 2.1 when | t | ≥ | z | . (cid:3) The | z | ≥ | t | case, q = n To complete the proof of Theorem 2.1, we have left to check the case when | z | ≥ | t | . Asbefore, we break the integral up into two cases: 0 < r ≤ and < r ≤ The case < r ≤ . As before, we start with the harder case. Fortunately, though,the bulk of the preliminary computations still hold. We take (41) as our starting point. Thedifferences between the | t | and | z | dominant cases, though, is that in the manipulationsleading to (41) we do not want to factor | t | out of the integral and replace z by ˆ q . We alsoworry about the C t ((ˆ q, ¯ˆ q ) j + | I ′ | term which is now a C ( z, ¯ z ) j + | I ′ | term. We will use sizeestimates and ignore completely the (uniformly bounded) ν terms. Thus, (41) simplifies to(60) | Typical Term | ≤ C ( z, ¯ z ) j + | I ′ | ( s | z | ) n + m − j + | I ′ | + | I ′′ | + | I | s K j,I ′ ,I ′′ − ( j + | I ′ | + | I ′′ | ) and we estimate Z ∞ s =3 Z ν ∈ S m − C ( z, ¯ z ) j + | I ′ | ( s | z | ) n + m − j + | I ′ | + | I ′′ | + | I | s K j,I ′ ,I ′′ − ( j + | I ′ | + | I ′′ | ) dν ds ≤ C j,I ′ ,I ′′ ,I | z | j + | I ′ | | z | n + m − j + | I ′ | + | I ′′ | + | I | ) = C j,I ′ ,I ′′ ,I | z | n + m − h I i ) . Since h I i = | I ′ | + 2 | I ′′ | + 2 | I | , this establishes the estimate in Theorem 2.1 for this term.The remainder term (46) has a similarly straightforward adaptation and estimate. The case < r ≤ . The estimates in this case will also follow from size estimates.We established the key estimate on B ( r, ν ) in (57). Moreover, since the eigenvalues for ν ∈ S m − are bounded away from zero (say by c ), we have11 − r u ≤ − (1 / c ≤ C. It therefore follows that for j = 0 , , |∇ jz, ¯ z A ( r, ν, z ) | ∼ C | z | − j . Consequently, we ignore the t -term and estimate (8) directly by | N I ′ ,I ′′ ,I ( z, t ) | ≤ C I ′ ,I ′′ ,I Z r =0 Z ν ∈ S m − r c − | z | | I ′ | dν dr | z | n + m − | I ′ | + | I ′′ | + | I | ) = C I ′ ,I ′′ ,I | z | n + m − h I i ) . This establishes the desired estimate in the case when | z | ≥ | t | and hence concludes theproof of Theorem 2.1. 12. The case q = n The techniques that prove the estimates in the q = n case are robust enough to work inthe q = n case, as well. However, the non-triviality of ker (cid:3) b changes for the formula for N K ( z, t ), and in this section, we sketch the argument for the I = ∅ case, which is when thereare no derivatives. We also assume, without loss of generality, that the set of positive indices P = { , , . . . , n } .We computed the relative solution to (cid:3) b in the case q = n given by R ∞ e − s (cid:3) b ( I − S n ) ds in [BRa]. Following the notation of [BRa], for each q -tuple L ∈ I q , we setΓ L = { α ∈ S m − : µ αℓ > ℓ ∈ L and µ αℓ < ℓ L } . If L ∈ I n , then Γ L = ∅ , unless L = P , in which case Γ P = S m − . Therefore, from [BRa,Theorem 2.2, Part 3], if K ∈ I n , then N K ( z, t ) = K n,m (cid:20) X L ∈I n ,L = P Z ν ∈ S m − det( ¯ U ( ν ) K,L ) d ¯ Z ( z, ν ) L Z r =0 B L ( r, ν )( A ( r, ν, z ) − iν · t ) n + m − dr dνr (61)+ Z ν ∈ S m − det( ¯ U ( ν ) K,P ) d ¯ Z ( z, ν ) P Z r =0 (cid:20) B P ( r, ν )( A ( r, ν, z ) − iν · t ) n + m − − | det A ν | ( A (0 , ν, z ) − iν · t ) n + m − (cid:21) dr dνr (cid:21) (62)12.1. The case / < r < . As above, we split up the r -integral into 0 < r ≤ / / < r <
1. The challenge is in the region when 1 / < r < L ∈ I n in (61). After factoring out | t | nd rotating coordinates so that ν · t/ | t | = ν , we set ˆ q = z | t | / and rewrite the integral over1 / ≤ r ≤ | t | n + m − N K = X L ∈I n Z ν ∈ S m − det( ¯ U ( ν t ) K,L ) d ¯ Z (ˆ q, ν t ) L Z r =1 / B L ( r, ν t )( A ( r, ν t , ˆ q ) − iν ) n + m − dr dνr (63) − Z ν ∈ S m − det( ¯ U ( ν t ) K,P ) d ¯ Z (ˆ q, ν t ) P Z r =1 / | det A ν t | ( A (0 , ν t , ˆ q ) − iν ) n + m − dr dνr (64)and where (as above) ν t = M − t ( ν ) and M t is an orthogonal transformation on R m with M t ( t/ | t | ) = e . The analysis of (63) is precisely the same as we carried out in Sections 3-11.Thus we focus on (64). We show the following Proposition 12.1.
Let (65) N K (ˆ q, t ) = Z ν ∈ S m − det( ¯ U ( ν t ) K,P ) d ¯ Z (ˆ q, ν t ) P Z r =1 / | det A ν t | ( A (0 , ν t , ˆ q ) − iν ) n + m − dr dνr Then there are positive constants c and C such that | N K (ˆ q, t ) | ≤ C for all | ˆ q | ≤ c .Remark . Note that the case | ˆ q | > c falls into the | z | dominant case which is mucheasier to handle.We devote the remainder of this section to proving this proposition. To prove the propo-sition, we need the following analyticity lemma. Lemma 12.3.
The following functions are analytic as a function of ν ∈ S m − : • ν → | det( A ν ) |• ν → A (0 , ν, ˆ q ) = P nj =1 | µ νj || ˆ q νj | • ν → det( ¯ U ( ν ) K,P ) d ¯ Z (ˆ q, ν ) P = P J ∈I n det( ¯ U ( ν ) K,P ) det[ U ( ν ) P,J ] T d ¯ z J Proof.
For the first bullet, note that A ν has n positive and n negative eigenvalues and so | det A ν | = ( − n det A ν . So the expression in the first bullet is analytic since A ν is linear in ν . For the second bullet, note that A ν and | A ν | = p A ν have the same eigenvectors. Therefore n X j =1 | µ νj || ˆ q νj | = n X j =1 | µ νj || Q j ( ν, ˆ q ) | = ˆ q ∗ U ( ν ) · | D ν | · U ( ν ) ∗ ˆ q = ˆ q ∗ p A ν ˆ q (66)which is analytic in ν . Here, D ν is the diagonal matrix with the eigenvalues of A ν as itsdiagonal entriesl, and √ is the principal branch of the square root of a positive definiteHermitian symmetric matrix.Showing the expression in the third bullet is analytic in ν is equivalent to showing thatthe expression(67) det( ¯ U ( ν ) K,P ) det[ U ( ν ) P,J ] T is real analytic in ν ∈ S m − for each J, K in I n . e shall need the standard branch of the function arctan z , which is holomorphic on C \ { z = iy : x = 0 and | y | ≥ } . Let I n be the 2 n × n identity matrix and consider thesequence 1 π (cid:16) arctan( nA ν ) + π I n (cid:17) for j = 1 , , . . . . Since the eigenvalues of A ν are bounded away from zero, each of thesematrices in this sequence is analytic in ν ∈ S m − and is diagonalized by U ( ν ) and U ( ν ) ∗ .Furthermore, this sequence converges uniformly in ν as j → ∞ to a matrix A ν , which isanalytic in ν with n eigenvalues equal to 1 and n eigenvalues equal to −
1. Also A ν isdiagonalized by U ( ν ) and [ U ( ν )] ∗ .Now consider ˜ A ν = 12 ( A ν + I ) . ˜ A ν is analytic in ν and has n eigenvalues equal to 1 and n eigenvalues equal to zero. It isalso diagonalized by U ( ν ) and [ U ( ν )] ∗ . Therefore,[ U ( ν )] ∗ ˜ A ν U ( ν ) = D , where D = (cid:18) I n n n n (cid:19) and where I n is the n × n identity matrix and 0 n is the n × n zero matrix. Therefore˜ A ν = ¯ U ( ν ) D [ U ( ν )] T . Taking determinants, we havedet [ ˜ A ν ] KJ = X L,L ′ det( ¯ U ( ν ) K,L ) det( D L,L ′ ) det([ U ( ν ) L ′ ,J ] T ) . Given that D is diagonal, the only nonzero contributions to this sum occur when L = L ′ = P , which is the set of positive indices = 1 , . . . , n . We obtaindet [ ˜ A ν ] KJ = det( ¯ U ( ν ) K,P ) det([ U ( ν ) P,J ] T ) . Since the left side is analytic in ν , so is the right side and this establishes the analyticity of(67) and thus concludes the proof of the lemma. (cid:3) Proof of Proposition 12.1.
Note that the r -integral in (65) can be computed exactly (asln(2)), so we need only examine the ν -integral. Clearly, the integral over | ν | ≥ / q and t . So we restrict attention to the region { ν ∈ S m − ; | ν | ≤ / } .The key is to examine the integral over ν -slices of this region. Without loss of generality,let us assume we are on a region, V , of the sphere where ν m = h ( ν , ν ′ ) can be written asan analytic function of the other variables ( ν , ν ′ ) with ν ′ = ( ν , . . . , ν m − ). We may furtherassume that the “cap” V is large enough so that projection of V onto ν m = 0 contains thedisk { ( ν , ν ′ ) ∈ R m − : | ν | ≤ and | ν ′ | ≤ } . We also write dν = g ( ν , ν ′ ) dν dν ′ where g isan analytic function on V . Let G (ˆ q, t, ν , ν ′ ) = det( ¯ U ˜ ν t K,P ) d ¯ Z (ˆ q, ˜ ν t ) P | det A ˜ ν t | g ( ν , ν ′ ) where ˜ ν t = M − t ( ν , ν ′ , h ( ν , ν ′ )) . rom Lemma 12.3, G (ˆ q, t, ν , ν ′ ) is analytic in ν , ν ′ and uniformly bounded in ν ∈ V ⊂ S m − ,ˆ q , and t . We need to show that there are positive constants c and C so that(68) (cid:12)(cid:12)(cid:12) Z | ν |≤ / G (ˆ q, t, ν , ν ′ ) dν ( A (0 , ˜ ν t , ˆ q ) − iν ) n + m − (cid:12)(cid:12)(cid:12) ≤ C for all | ν ′ | ≤
12 and | ˆ q | ≤ c . We shall proceed by using Cauchy’s Theorem to bump the contour of integration aroundthe potential singularity at ν = 0. First, let δ > G (ˆ q, t, ν , ν ′ ) and A (0 , ˜ ν t , ˆ q ) analytically continue from { ν ∈ R ; | ν | ≤ / } to a neighbor-hood of the rectangle ˜ V = { ζ = ν + iη ∈ C ; | ν | ≤ / ≤ η ≤ δ } in the upper halfplane and for all ( ν , ν ′ , h ( ν , ν ′ )) ∈ V . Also note from (66) that A (0 , ν, ˆ q ) = ˆ q ∗ p A ν ˆ q ≥ ν ∈ V . In addition, the analytic extension of A (0 , ν, ˆ q ) to ˜ V is the function A (0 , ˜ ν t ( ζ ) , ˆ q ) := A (0 , M − t ( ζ , ν ′ , h ( ζ , ν t ) , ˆ q ) . Furthermore, its ζ derivative is uniformly bounded by ˜ C | ˆ q | for ζ ∈ ˜ V and ν ∈ V where˜ C > A (0 , ˜ ν t ( ζ ) , ˆ q ) ≥ − ˜ C | ˆ q | η for ζ = ν + iη ∈ ˜ V . This inequality implies | A (0 , ˜ ν t ( ζ ) , ˆ q ) − iζ | ≥ (1 − ˜ C | ˆ q | ) η for ζ = ν + iη ∈ ˜ V . Let γ be the upper three sides of the boundary of the rectangle of ˜ V , i.e. the union of thethree line segments, respectively, from − / − / iδ ; from − / iδ to 1 / iδ , andfrom 1 / iδ to 1 /
2. The above inequality shows that there is a constant c > | ˆ q | < c , then | A (0 , ˜ ν t ( ζ ) , ˆ q ) − iζ | > ζ inside ˜ V and | A (0 , ˜ ν t ( ζ ) , ˆ q ) − iζ | ≥ c for ζ ∈ γ . Now we can use Cauchy’s Theorem to deform the path of integration in (68) to γ and theproof of the estimate in (68) easily follows. This concludes the proof of the proposition. (cid:3) The cases < r < / and | z | > | t | . The estimates of N K for the interval 0
Here, we record four examples with complex tangent dimension 2 n ≥ m ≥ n = 2 and m = 2 originally computedin [BR13, BR20]: xample 13.1. n = 2, m = 2, and q = 0. Consider Φ( z, z ) = ( φ ( z, z ) , φ ( z, z )) where φ ( z, z ) = 2 Re( z ¯ z ) φ ( z, z ) = | z | − | z | . The eigenvalues of the A ν (the Hessian of Φ( z, z ) · ν ) are +1 and − N L with L = ∅ and so ε = − ε = +1. Since m = 2, S m − is just the unit circleparameterized by ν = (cos θ, sin θ ), 0 ≤ θ ≤ π and dν = dθ . We rotate θ coordinates sothat ν · t becomes | t | sin θ . From (3), we obtain N ( z, t ) = 4 π ) Z Z π r (1 − r ) dθ (cid:2)(cid:0) r − r (cid:1) | z | − i | t | sin θ (cid:3) drr . We let s = r +11 − r , ds = dr (1 − r ) to obtain N ( z, t ) = 4 π ) Z π Z ∞ ds dθ [ s | z | − i | t | sin θ ] = 4(2 π ) | z | | t | Z π dθ [ | ˆ q | − i sin θ ] with ˆ q = z/ | t | / = 12 π | z | + | t | ] / ≈ | z | + | t | / ) ≈ ρ ( z, t ) as indicated by Theorem 2.1. Example 13.2. n = 2 and m = 3. Consider Φ( z, z ) = ( φ ( z, z ) , φ ( z, z ) , φ ( z, z )) where φ ( z, z ) = 2 Re( z ¯ z ) φ ( z, z ) = | z | − | z | φ ( z, z ) = 2 Im( z ¯ z ) . Let ν = ( ν , ν , ν ) be a unit vector in R . Then A ν = (cid:18) ν ν − iν ν + iν − ν (cid:19) . The characteristic equation for the eigenvalues is det( A ν − λI ) = λ − | ν | = λ − λ = +1 , − n = 2 and m = 3, we obtain N ( z, t ) = 4 π ) Z Z ν ∈ S r (1 − r ) dν dr (cid:2)(cid:0) r − r (cid:1) | z | − i | t | ν (cid:3) drr . We now let s = r +1 r − as before and let x = ν . Using (48), we write dν = dx dφ where φ isthe angular measure of the S copy of the equator of S . Since φ does not appear in theintegrand, its integral provides a factor of 2 π . We obtain N ( z, t ) = 4 π ) Z ∞ Z x = − π dx ds [ s | z | − i | t | x ] = 2 π | z | + | t | ) ≈ | z | + | t | / ) ≈ ρ ( z, t ) s indicated by Theorem 2.1. Example 13.3. n = 4, m = 4, q = 0. Consider Φ( z, z ) = ( φ ( z, z ) , φ ( z, z ) , φ ( z, z ) , φ ( z, z ))where φ ( z, z ) = 2Re( z ¯ z ) + 2Re( z ¯ z ) φ ( z, z ) = 2Re( z ¯ z ) − z ¯ z ) φ ( z, z ) = 2Im( z ¯ z ) − z ¯ z ) φ ( z, z ) = − z ¯ z ) + 2Im( z ¯ z ) . Let ν = ( ν , ν , ν , ν ) be a unit vector in R . Then(69) A ν = ν − iν − ν − iν ν + iν ν + iν ν − iν ν + iν − ν + iν ν − iν . The characteristic polynomial (in λ ) is the quadratic polynomial ( λ − with eigenvalues+1 , +1 , − , − n = 4 and m = 4, we obtain N ( z, t ) = 4 π ) Z Z ν ∈ S r (1 − r ) dν (cid:2)(cid:0) r − r (cid:1) | z | − i | t | ν (cid:3) drr . We let ˆ q = z/ | t | / and s = r − r (as before) to obtain N ( z, t ) = 4 (2 π ) | t | Z ∞ Z ν ∈ S ( s − dν ds ( | ˆ q | s − iν ) . Now we let x = ν and use (48) with m = 4 to write dν = √ − x dx dν ′ where dν ′ is surfacemeasure on S (the equator of S ). When ν ′ is integrated out, this provides a factor of 4 π .We obtain N ( z, t ) = 4 π )2 (2 π ) | t | Z ∞ Z x = − ( s − √ − x dx ds ( | ˆ q | s − ix ) = 152 π | z | + | t | ) / ≈ | z | + | t | / ) ≈ ρ ( z, t ) as indicated by Theorem 2.1. Example 13.4.
This is the same example as Example 12.3, except with q = 1 and K = { } .The matrix A ν from (69) has associated eigensystem (with entries written as pairs { v, λ ) } where v is a unit eigenvector with eigenvalue λ is (cid:26) √ − − ν − iν ν − iν , (cid:27) , (cid:26) √ ν + iν ν − iν , (cid:27)(cid:26) √ − ν − iν ν − iν , − (cid:27)(cid:26) √ ν + iν − ν − iν , − (cid:27) . e have U ( ν ) = (cid:0) v ν v ν v ν v ν (cid:1) = 1 √ − − ν − iν ν + iν − ν − iν ν + iν − ν − iν ν − iν ν − iν ν − iν where v ν = 1 √ − − ν − iν ν − iν , v ν = 1 √ ν + iν ν − iν , v ν = 1 √ − ν − iν ν − iν , v ν = 1 √ ν + iν − ν − iν so that U ( ν ) ∗ A ν U ( ν ) = (cid:18) I − I (cid:19) where I is the 2 × | µ νj | = 1 for all j and ν ∈ S m − , A ν ( r, z ) = n X j =1 | µ νj | r | µ νj | − r | µ νj | ! | z νj | = X j =1 (cid:18) r − r (cid:19) | z νj | = 1 + r − r | z | . Next, n Y j =1 r (1 / − ε νj,L ) | µ νj | | µ νj | (1 − r | µ νj | ) = 1(1 − r ) ( r L ∈ { , } r L ∈ { , } Next, we compute d ¯ Z ( ν, z ) = U ( ν ) T · d ¯ z = 1 √ − − ν − iν ν − iν ν + iν ν − iν − ν − iν ν − iν ν + iν − ν − iν d ¯ z d ¯ z d ¯ z d ¯ z . Next, d ¯ Z ( ν, z ) = U ( ν ) T · d ¯ z , then multiplying both sides by ¯ U ( ν ) where U ( ν ) T is the transposeof U produces ¯ U ( ν ) · d ¯ Z ( z, ν ) = ¯ U ( ν ) · U ( ν ) T · d ¯ z = U ( ν ) · U ( ν ) ∗ · d ¯ z = d ¯ z. Also, det( ¯ U ( ν ) K ′ ,L ) = ¯ U ( ν ) k ′ ,ℓ . From (3) and (4) and the computations in this example, N ( z, t )= − K , Z ν ∈ S h − d ¯ z − ( ν + iν ) d ¯ z + ( ν − iν ) d ¯ z (cid:1)i Z r =0 r (1 − r ) dν ( r − r | z | − iν · t ) drr + K , Z ν ∈ S h d ¯ z − ( ν + iν ) d ¯ z + ( ν − iν ) d ¯ z i Z r =0 r (1 − r ) dν ( r − r | z | − iν · t ) drr here K , = (6!)2(2 π ) . Reorganizing, we have N ( z, t ) = (cid:20) K , Z ν ∈ S Z r =0 r (1 − r ) r − r | z | − iν · t ) dr dν (cid:21) d ¯ z (70) + (cid:20) K , Z ν ∈ S Z r =0 ( ν + iν )(1 − r )(1 − r ) r − r | z | − iν · t ) dr dν (cid:21) d ¯ z − (cid:20) K , Z ν ∈ S Z r =0 ( ν − iν )(1 − r )(1 − r ) r − r | z | − iν · t ) dr dν (cid:21) d ¯ z . We observe that with ˆ q = z/ | t | / , Z r (1 − r ) ( r − r | ˆ q | + ia ) dr = − a + 6 ia | ˆ q | + 25 | ˆ q | | ˆ q | ( ia + | ˆ q | ) and Z − r (1 − r ) ( r − r | ˆ q | + ia ) dr = ia + 6 | ˆ q | | ˆ q | ( ia + | ˆ q | ) . Let’s also observe the estimate in the special case that t = ( | t | , , . . . ,
0) and only the d ¯ z component (since K = { } ) and compute I = Z ν ∈ S Z r =0 r (1 − r ) r − r | z | − iν · t ) dr dν = 1 | t | Z ν ∈ S Z r =0 r (1 − r ) r − r | ˆ q | − iν ) dr dν = 1240 | ˆ q | | t | Z ν ∈ S − ν − iν | ˆ q | + 25 | ˆ q | ( | ˆ q | − iν ) dν. Integrating in spherical coordinates, we compute I = 1240 | t | | z | Z π Z π Z π − cos α − i cos α | ˆ q | + 25 | ˆ q | ( | ˆ q | − i cos α ) sin α sin α dα dα dα = π | t | | z | Z π − π − cos α − i cos α | ˆ q | + 25 | ˆ q | ( | ˆ q | − i cos α ) sin α dα . The last integral follows from the fact that cos α and sin α are even functions. If f (cos α , sin α )is the integrand, then Z π − π f (cos α , sin α ) dα = I | z | =1 f (cid:16) z + z , z − z i (cid:17) iz dz. If g ( z ) = f (cid:16) z + z , z − z i (cid:17) iz then g ( z ) = − iz ( − z ) ( − | ˆ q | z + (1 + z ) + 12 i | ˆ q | ( z + z ))(2 | ˆ q | z − i (1 + z )) as poles at z = i ( −| ˆ q | ± p | ˆ q | + 1). The pole at z = i ( −| ˆ q | + p | ˆ q | + 1) occurs insidethe unit disk and is easily computed using Mathematica. In fact,Res( g, i ( −| ˆ q | + p | ˆ q | + 1)) = − | ˆ q | i ( − | ˆ q | )2(1 + | ˆ q | ) . In summary, the d ¯ z component of N in (70) is I = − πi π | t | i ( − | ˆ q | )2(1 + | ˆ q | ) . Similar calculations with t = ( | t | , , ,
0) shows that the d ¯ z component of N in (70) is II = Z ν ∈ S Z r =0 ( ν + iν )(1 − r )(1 − r ) r − r | z | − iν · t ) dr dν = 2 π i | t | | ˆ q | | ˆ q | ) . and the d ¯ z component of N in (70) is III = Z ν ∈ S Z r =0 ( ν − iν )(1 − r )(1 − r ) r − r | z | − iν · t ) dr dν = 0 . In summary, we have computed that N (cid:0) z, ( | t | , , , (cid:1) = 15 π | t | + | z | ) (cid:20) − | t | + 5 | z | | t | + | z | ) d ¯ z + i | z | | t | ( | t | + | z | ) d ¯ z (cid:21) . This expression has norm ≈ ρ ( z, t ) − as indicated by Theorem 2.1. Example 13.5.
This is a modification of Example 12.3 where the eigenvalues of A ν do notdepend analytically on ν . Let Φ( z, z ) = ( φ ( z, z ) , φ ( z, z ) , φ ( z, z ) , φ ( z, z )) where φ ( z, z ) = 2Re( z ¯ z ) + 2Re( z ¯ z ) φ ( z, z ) = 2Re( z ¯ z ) − z ¯ z ) φ ( z, z ) = 2Im( z ¯ z ) − z ¯ z ) φ ( z, z ) = − z ¯ z ) + 2(1 + b )Im( z ¯ z )where b is a small real number. Let ν = ( ν , ν , ν , ν ) be a unit vector in R . Then, it iseasy to compute the complex Hessian of Φ ν = Φ( z, z ) · ν : A ν = HessianΦ ν = ν − iν − ν − i (1 + b ) ν ν + iν ν + iν ν − iν ν + iν − ν + i (1 + b ) ν ν − iν . Note that when b = 0, then this is Example 3.The characteristic polynomial (in λ ) turns out to be a quadratic polynomial in λ so theeigenvalues (though messy) can be computed as µ ν > , µ ν > , µ ν < , µ ν < µ ν = p Λ + , µ ν = p Λ − , µ ν = − p Λ + , µ ν = − p Λ − nd where Λ ± = ν + ν + ν + (1 / b + 2 b + 2) ν ± | ν || b | (cid:18) ( b + 2) ν ν + ν (cid:19) / . Note that when b = 0, µ ν = µ ν = 1 and µ ν = µ ν = − b nonzero,but small, these eigenvalues are not smooth at ν = 0 and ν , ν = 0 due to the presence of | ν | . 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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School of Mathematical and Statistical Sciences, Arizona State University, PhysicalSciences Building A-Wing Rm. 216, 901 S. Palm Walk, Tempe, AZ 85287-1804Department of Mathematical Sciences, 1 University of Arkansas, SCEN 327, Fayet-teville, AR 72701
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