# Szegö Kernels and Asymptotic Expansions for Legendre Polynomials

Authors: Roberto Paoletti
Journal: Journal of Complex Analysis
 (2017)

---

## Abstract

We present a geometric approach to the asymptotics of the Legendre polynomialsPk,n+1, based on the Szegö kernel of the Fermat quadric hypersurface, leading to complete asymptotic expansions holding on expanding subintervals of [-1,1].


---

## Body

## 1. Introduction

The search for asymptotic expansions and approximations of special functions is a very classical vein of research and is of great relevance in pure mathematics, in numerical analysis, mathematical physics, and the applied sciences (see, for instance, of course with no pretence of completion [1–4]).The goal of this paper is to develop a geometric approach to the asymptotics of the Legendre polynomialPk,n+1(t) for k→+∞, with t=cos(ϑ)∈[-1,1] and n≥1 fixed; as is well-known, Pk,n+1(t) is the restriction to Sn of the Legendre harmonic, expressed in polar coordinates on the sphere. For thorough discussions and terminology, see, for instance, [1, 3, 5–7]. We obtain an asymptotic expansion holding on expanding subintervals of [-1,1], rather than on fixed subintervals of the form [-1+δ,1-δ] for some given δ>0, as one typically finds in the literature.However, the actual point of this work is neither to present essentially new results nor to give an especially economic proof of Legendre asymptotics (the use of Szegö kernel machinery is arguably not more elementary than the traditional approaches). Rather, it is motivated by the following considerations. On the one hand, there is a conceptually very appealing view on spherical harmonics, due to Lebeau and Guillemin, based on the Szegö kernel of the Fermat quadric. On the other hand, in recent years, a considerable amount of work and attention has been devoted to algebro-geometric Szegö kernel asymptotics, which have played a fundamental role in complex geometry. Therefore, it seemsper se very natural and interesting to illustrate the important conceptual juncture between spherical harmonics and Szegö kernels, by revisiting classical results on Legendre asymptotics in view of these recent developments. In a broader perspective, the application of Szegö kernels to spherical harmonics seems a very promising area; a revisitation of this kind is also instrumental to the development of computational techniques that might be useful in future developments in this direction.Let us come to a closer description of the content of this paper. There is a tight relation betweenPk,n+1(t) and the orthogonal projector(1)Pk,n:L2Sn⟶Vk,n,where Vk,n is the space of level-k spherical harmonics on Sn; equivalently, Vk,n is the eigenspace of the (positive) Laplace-Beltrami operator on functions on Sn, corresponding to its kth eigenvalue λk,n=k(k+n-1).Namely, for any choice of an orthonormal basis(ϱknj)j=1Nk,n of Vk,n the distributional kernel Pk,n(·,·)∈C∞Sn×Sn satisfies(2)Pk,nq,q′=∑j=1Nk,nϱknjq·ϱknjq′¯,where q·q′=qtq′ (we think of q and q′ as columns vectors), and Nk,n is the dimension of Vk,n. By symmetry considerations, Pk,nq,q′ only depends on q·q′. In fact, with the normalization Pk,n+1(1)=1, (3)Pk,nq,q′=Nk,nvolSnPk,n+1q·q′.Thus, it is equivalent to give asymptotic expansions for Pk,n+1(cos(ϑ)) and for Pk,nq,q′ with q·q′=cos(ϑ).Since for any(q,q′)∈Sn×Sn we have(4)Pk,nq,q=Nk,nvolSn,Pk,nq,-q′=-1kPk,nq,q′,we may assume q≠±q′. Then there is a unique great circle parametrized by arc length going from q to q′ in a time ϑ∈(0,π) and qtq′=cos(ϑ).Our geometric approach uses, on the one hand, the specific relation between spherical harmonics onSn and the Hardy space of the Fermat quadric hypersurface in Pn [8, 9] and, on the other hand, the off-diagonal scaling asymptotics of the level-k Szegö kernel of polarized projective manifold [10, 11].The following asymptotic expansions involve a sequence of constantsCk,n>0 with a precise geometric meaning [9]. There is a natural algebraic isomorphism between the level-k Szegö kernel of the Fermat quadric Fn⊂Pn and Vk,n, given by a push-forward operation; this isomorphism is however unitary only up to an appropriate rescaling, and Ck,n is the corresponding scaling factor.An asymptotic expansion forCk,n is discussed in [9], building on the theory of [8]; an alternative derivation is given in Proposition 2 (with an explicit computation of the leading order term).In the following, the symbol~ stands for “has the same asymptotics as.”Theorem 1.
There exist smooth functionsAnl and Bnl  (l=1,2,…) on [0,π] such that the following holds. Let us fix C>0 and δ∈[0,1/6). Then, uniformly in (q,q′)∈Sn×Sn satisfying qtq′=cos(ϑ) with (5)Ck-δ<ϑ<π-Ck-δ,we have for k→+∞ an asymptotic expansion of the form (6)Pk,nq,q′=2n/2Ck,n21sin⁡ϑn-1/2·cos⁡αk,nϑ·Anϑ,k+sin⁡αk,nϑ·Bnϑ,k,where(7)αk,nϑ≕kϑ+ϑ2-π4n-1,Anϑ,k~1+∑l=1+∞k-lAnlϑsin⁡ϑ6l,Bnϑ,k~∑l=1+∞k-lBnlϑsin⁡ϑ6l.At thelth step, we have for some constant Cl>0(8)Anlϑsinϑ6l,Bnlϑsin⁡ϑ6l≤Clk-l1-6δ,and a similar estimate holds for the error term. Hence, the previous is an asymptotic expansion for δ∈[0,1/6).As mentioned, the same techniques yield an asymptotic expansion forCn,k (see (6.18) in [9]).Proposition 2.
Fork→+∞, we have an asymptotic expansion of the form: (9)Ck,n~n-1!22·volSnvolSn-11/2πk-n-1/4·1+∑j≥1k-jaj.If we insert the latter expansion in the one provided by Theorem1, we obtain the following.Corollary 3.
With the assumptions and notation of Theorem1, for k→+∞, there is an asymptotic expansion(10)Pk,nq,q′=2n+3/2n-1!1volSnvolSn-1πksin⁡ϑn-1/2·cos⁡αk,nϑ·Cnϑ,k+sin⁡αk,nϑ·Dnϑ,k,where Cn(ϑ,k) and Dn(ϑ,k) admit asymptotic expansions similar to those of An(ϑ,k) and Bn(ϑ,k), respectively (of course, with different functions Cnl and Dnl, l≥1).Pairing Corollary3 with (3), we obtain the following.Corollary 4.
In the same situation as in Theorem1, for k→+∞, there is an asymptotic expansion(11)Pk,n+1cos⁡ϑ=2n+1/2volSn-1πsinϑkn-1/2·cos⁡αk,nϑ·Enϑ,k+sin⁡αk,nϑ·Fnϑ,k,where again En(ϑ,k) and Fn(ϑ,k) admit asymptotic expansions similar to those of An(ϑ,k) and Bn(ϑ,k), respectively.Let us verify that Corollary4 fits with the classical asymptotics. For example, when n=1, we obtain(12)Pk,2cos⁡ϑ~cos⁡kϑ+⋯,so that the leading order term is the kth Chebyshev polynomial. Since it is known that in this case the Legendre polynomialis the Chebyshev polynomial ([6], page 11), this is in fact the only term of the expansion.Forn=2, we obtain the formula of Laplace (cfr [12], Section 4.6; [4], (8.01) of Ch. 4; [13], Theorem 8.21.2), but as a full asymptotic expansion holding uniformly on expanding subintervals converging to [-1,1] at a controlled rate, as above: (13)Pk,3cos⁡ϑ~2πksin⁡ϑcos⁡k+12ϑ-π4+Ok-3/2+6δ.For arbitraryn, Pk,n+1 is a multiple of a Gegenbauer polynomial ([2]; [6], page 16):(14)Pkn/2-1,n/2-1cos⁡ϑ=rk,nPk,n+1cos⁡ϑ.Given the standardization for Pk(n/2-1,n/2-1) ([14], section 10.8)(15)rk,n=Pkn/2-1,n/2-11=k+n2-1k=n/2kk!=Γk+n/2k!Γn/2,where Γ is of course the Gamma function. By (35.31) in [3], for k→+∞, we have(16)Γk+n2~kn/2Γk=kn/2k-1!.Therefore,(17)rk,n~kn/2k-1!k!Γn/2=kn/2-1Γn/2.If we use the well-known formula (see, e.g., (2) of [6])(18)volSn-1=2πn/2Γn/2,we obtain for Pk(n/2-1,n/2-1)(cos(ϑ)) as asymptotic expansion with leading order term(19)2n+1/2kn/2-1Γn2Γn/22πn/2πsin⁡ϑkn-1/2cos⁡αk,nϑ=1πk1cos⁡ϑ/2n-1/2sin⁡ϑ/2n-1/2cos⁡αk,nϑ,in agreement with (10) on page 198 of [14].


## 2. Preliminaries

### 2.1. The Geometric Picture

For the following, see [8, 9].LetS1n⊂Rn+1 be the unit sphere, and let us identify the tangent and cotangent bundles of S1n by means of the standard Riemannian metric. The unit (co)sphere bundle of S1n is given by the incidence correspondence(20)S∗S1n≅SS1n=q,p∈S1n×S1n:qtp=0.The Fermat quadric hypersurface in complex projective space is(21)Fn≔z∈Pn:ztz=0;let A be the restriction to Fn of the hyperplane line bundle. Given the standard Hermitian product on Cn+1, A is naturally a positive Hermitian line bundle, Fn inherits a Kähler structure ωFn (the restriction of the Fubini-Study metric), and the spaces of global holomorphic sections of higher powers of A, H0Fn,A⊗k have an induced Hermitian structure.The affine cone overFn is Cn={ztz=0}⊂Cn+1; the intersection X1≔Cn∩S12n+1 may be viewed as the unit circle bundle in the dual line bundle A∨. More generally, for any r>0, the intersection(22)Xr≔Cn∩Sr2n+1with the sphere of radius r is naturally identified with the circle bundle of radius r in A∨. In particular, (23)X2=q+ip:q2=p2=1,qtp=0is diffeomorphic to S∗(Sn) by the map β:(q,p)↦q+ip; furthermore, β is equivariant for the natural actions of O(n+1) on S∗(Sn) and X2 defined by, respectively,(24)B·q,p≕Bq,Bp,B·q+ip=Bq+iBpB∈On+1.We shall identifyS∗(Sn) and X2 and denote the projection by(25)ν:S∗Sn≅X2⟶Sn,q+ip⟼q.There is also a standard structure action ofS1 on X2, induced by fibrewise scalar multiplication in A∨, or equivalently in Cn+1. The latter action is intertwined by β with the “reverse” geodesic flow on S∗(Sn)≅S(Sn). The S1-orbits are the fibers of the circle bundle projection(26)π2:q+ip∈X2⟼q+ip∈Fn.This holds for anyr>0; we shall denote by πr:Xr→Fn the projection for general r>0.


### 2.2. The Metric onXr

Let us dwell on the metric aspect of (22); there are two natural choices of a Riemannian metric on Xr, hence of a Riemannian density, and we need to clarify the relation between the two.There is an obvious choice of a Riemannian metricgr′ on Xr, induced by the standard Euclidean product on Cn+1. With respect to gr′, the S1 orbits on Xr have length of 2πr. Clearly, gr′ is homogeneous of degree 2 with respect to the dilation μr:x∈X↦rx∈Xr, and therefore the corresponding volume form Υ′Xr on Xr is homogeneous of degree dim⁡X=2n-1. That is,(27)μr∗Υ′Xr=r2n-1Υ′X.An alternative and common choice of a Riemannian structureg1 on X1 comes from its structure of a unit circle bundle over Fn. Let α∈Ω1(X1) be the connection 1-form associated with the unique compatible covariant derivative on A, so that dα=2π1∗(ωFn). Also, let (28)HX1/Fn=ker⁡α,VX1/Fn=kerdπ1⊆TXdenote the horizontal and vertical tangent bundles for π1, respectively. There is a unique Riemannian metric g1 on X1 such that π1 a Riemannian submersion and the S1-orbits on X1 have unit length. The corresponding volume form on X1 is given by(29)ΥX1=1n-1!π1∗ωFn∧n-1∧12πα=12ππ1∗ΥFn∧α,where ΥFn=ωFn∧(n-1)/(n-1)! is the symplectic volume form on Fn.We wish to compare the two Riemannian metricsg1 and g1′, the corresponding volume forms, Υ′X1 and ΥX1, and densities, dVX and d′VX.Lemma 5.
Υ
X
1
=
1
/
2
π
Υ
′
X
1 and dVX1=1/2πd′VX1.Proof of Lemma5.
The connection 1-form for the Hopf mapS2n+1→Pn is (30)θ=i2ztdz¯t-z¯tdz;thus, α is the restriction of θ to X1. Let ω0 be the standard symplectic structure on Cn+1. Since θz(w)=ω0(z,w), we have ker⁡(θz)=z⊥ω0 (symplectic annihilator). In other words, (31)ker⁡θz=spanRz⊕z⊥h0∩TzS12n+1=z⊥h0,where z⊥h0 is the Hermitian orthocomplement of z for the standard Hermitian product.
Thus, ifz∈X1, then(32)HzX1/Fn=kerαz=z⊥h0∩TzCn=z⊥h0∩z¯⊥h0.On the other hand, Vz(X1/Fn)=spanR(iz). Thus, V(X1/Fn) and H(X1/Fn) are orthogonal with respect to both g1 (by construction) and g1′ (by the previous considerations). Hence, we may compare g1 and g1′ separately on H(X1/Fn) and V(X1/Fn).
On the complex vector bundleH(X1/Fn), g1′ and g1 are, respectively, the Euclidean scalar products associated with the restrictions of the (1,1)-forms (33)ω0=i2∂∂¯z2,ω1=i2∂∂¯ln⁡z2.Given that ω0 and ω1 agree on TS12n+2, g1=g1′ on H(X1/Fn).
On the other hand, bothg1 and g1′ are S1-invariant, but S1-orbits on X1 have length 2π for g1′ and 1 for g1. Thus, g1=g1′/2π on V(X1/Fn).
The claim follows directly from this.


### 2.3. The Szegö Kernel onXr

The following analysis is based on the equivariant asymptotics of the Szegö kernel ofX2 [11]. We refer the reader to [10, 11, 15] for a thorough discussion of Szegö kernels in the algebro-geometric context and to [16, 17] for the basic microlocal theory that underlies the subject (see also the neat discussion of Hardy spaces in [9]). To put things into perspective, however, let us recall that if Y is the boundary of a pseudoconvex domain, its Hardy space H(Y)⊂L2(Y) is the Hilbert space of square summable boundary values of holomorphic functions. The Szegö projector is then the orthogonal projector Π:L2(Y)→H(Y); with some abuse of notation, the Szegö kernel Π∈D′(Y×Y) is the corresponding distributional kernel. A description of Π as a Fourier integral operator was given in [16].In the special algebro-geometric case, whereY is the dual unit circle bundle of a positive line bundle on a complex projective manifold, H(Y) is the orthogonal direct sum of its isotypical components Hk(Y),k=0,1,2,…, under the S1-action; correspondingly, Π=⨁kΠk, where Πk is the orthogonal projector onto Hk(Y); since Hk(Y) is finite-dimensional, Πk is a smoothing operator; therefore, its Schwartz kernel is a C∞ function on Y×Y. Many local asymptotic properties of Πk, for k→+∞, were first discovered in [10, 11, 15] building on the theory of [16]. We shall recall what is needed here shortly.Let us now come to the specific case in point. For everyr>0, Xr is the boundary of a strictly pseudoconvex domain, and as such it carries a CR structure, a Hardy space H(Xr), and a Szegö projector Πr:L2(Xr)→H(Xr). We aim to relate the various Πr’s.LetO(Cn∖{0}) be the ring of holomorphic functions on the conic complex manifold Cn∖{0}. Let Ok(Cn∖{0})⊂O(Cn∖{0}) be the subspace of holomorphic functions of degree of homogeneity k.For everyr>0 and k=0,1,2,… let Hk(Xr)⊂H(Xr) be the finite-dimensional kth isotypical component of H(Xr) with respect to the standard S1-action. Restriction induces an algebraic isomorphism Ok(Cn∖{0})→Hk(Xr); with a slight abuse of language, we shall denote by the same symbol an element of Hk(Xr) and the corresponding element of Ok(Cn∖{0}).Suppose that(skj)j=0Nk⊆Ok(Cn∖{0}) yields by restriction an orthonormal basis of Hk(X1):(34)∫X1skjxsklx¯dVX1x=δjl.Setting y=rx and using (27) together with Lemma 5, we get(35)∫Xrskjyskly¯12πd′VXry=r2n+2k-1∫X1skjxsklx¯12πd′VX1x=r2n+2k-1δjl.Therefore, we have the following.Lemma 6.
If(skj)j=0Nk⊆Ok(Cn∖{0}) yields by restriction an orthonormal basis of Hk(X1) with respect to d′VX1, then for every r>0(36)r-k+n-1/2skjj=0Nkyields by restriction an orthonormal basis of Hk(Xr), with respect to d′VXr/2π.Let nowΠr,k be the level-k Szegö kernel on Xr, that is, the orthogonal projector(37)Πr,k:L2Xr,d′VXr2π⟶HkXr.By Lemma 6, its Schwartz kernel Πr,k∈C∞(Xr×Xr) is given by(38)Πr,ky,y′=r-2k+2n-1∑j=0Nkskjy·skjy′¯y,y′∈Xr.When pulled back toX1, this is (here x,x′∈X1)(39)Πr,krx,rx′=r-2k+2n-1∑j=0Nkskjrx·skjrx′¯=r-2n-1∑j=0Nks^kjx·s^kjx′¯=r1-2nΠ1,kx,x′.In particular,(40)Π2,k2x,2x′=22nΠ1,kx,x′.We shall make repeated use of the following asymptotic property ofΠ1,k, which follows from the microlocal description of Π as an FIO (explicit exponential estimates are discussed in [18]).Theorem 7.
LetdistFn be the distance function on Fn associated with the Kähler metric. Given any C,ϵ>0, uniformly for x,x′∈X satisfying(41)distFnπx,πx′≥Ckϵ-1/2,we have(42)Π1,kx,x′=Ok-∞when k→+∞.


### 2.4. Heisenberg Local Coordinates

There are two unit circle bundles in our picture: the Hopf fibrationπ:S12n+1→Pn, and π1:X1→Fn. Clearly, π1 is the pull-back of π under the inclusion Fn↪Pn. Both S12n+1 and X1 are boundaries of strictly pseudoconvex domains and carry a CR structure.On bothS12n+1 and X1, we may consider privileged systems of coordinates calledHeisenberg local coordinates (HLC). In these coordinates, Szegö kernel asymptotics exhibit a “universal” structure [11]; we refer toibidem for a detailed discussion.Givenz0∈X1, a HLC system on X1 centered at z0 will be denoted in additive notation:(43)θ,v∈-π,π×B2n-20,δ⟼z0+θ,v∈X1.Here θ∈(-π,π) is an “angular” coordinate measuring displacement along the S1-orbit through z0 (the fiber through z0 of π1:X1→Fn); instead, v∈B2n-2(0,δ)⊆R2n-2≅Cn-1 descends to a local coordinate on Fn centered at m0=π(x0), inducing a unitary isomorphism T[z0]Fn≅Cn-1. We may thus think of v as a tangent vector in T[z0]Fn.Here this additive notation might be misleading, sinceX1⊂Cn+1. Therefore, we shall write z0+X1(θ,v) for HLC on X1 centered at z0. We shall generally abridge notation by writing z0+X1v for z0+X1(0,v).Similarly,(θ,w)∈(-π,π)×B2n(0,δ)↦z0+S12n+1(θ,v) will denote a system of Heisenberg local coordinates on S12n+1 centered at z0. There is in fact a natural choice of HLC on S12n+1 centered at any z0∈S12n+1.Namely, let(a1,…,an) be an orthonormal basis of the Hermitian orthocomplement z0⊥h⊆Cn+1, and for w=(wj)∈Cn let us set(44)z0+S12n+1θ,w≔eiθ1+w2z0+∑j=1nwjaj.Since there is a canonical unitary identification z0⊥h≅T[z0]Pn, we shall also write this as z0+S12n+1(θ,v) with (θ,v)∈(-π,π)×T[z0]Pn.Ifz0∈X1, HLC on X1 centered at z0 can be chosen so that they agree to second order with the former HLC on S12n+1. More precisely, we may assume that for any v∈T[z0]Fn⊂T[z0]Pn we have(45)z0+Xθ,w=z0+S12n+1θ,v+R2v,where R2 is a function vanishing to second order at the origin.Givenv,w∈Cn+1≅R2n+2, let us define(46)ψ2v,w≕-iω0v,w-12v-w2;here ω0 is the standard symplectic structure, and · is the standard Euclidean norm. We shall make use of the following asymptotic expansion, for which we refer again to [11].Theorem 8.
Let us fixC>0 and ϵ∈(0,1/6). Then for any z∈X1, and for any choice of HLC on X1 centered at z, there exist polynomials Pj of degree ≤3j and parity j on T[z]Fn×T[z]Fn≅R2n-2×R2n-2, such that the following holds. Uniformly in v1,v2∈T[z]Fn with vj≤Ckϵ for j=1,2, and θ1,θ2∈(-π,π), one has for k→+∞ the following asymptotic expansion:(47)Π1,kz+θ1,v1k,z+θ2,v2k~kπn-1eikθ1-θ2+ψ2v1,v21+∑j=1+∞k-j/2Pjv1,v2.In the given range, the above is an asymptotic expansion, since(48)k-j/2Pjv1,v2≤Cjk-j/21-6ϵ.


### 2.5.Pk and Π2,k

As discussed in [9], the push-forward operator ν∗:C∞(X2)→C∞(Sn) yields by restriction an algebraic isomorphism (49)C∞X2∩HX2⟶C∞Sn,for every k, and (49) yields by restriction an isomorphism(50)HkX2⟶Vk,which is unitary up to a dilation by a constant factor Ck,n>0. Thus, we have (51)ν∗sL2Sn=Ck,nsHX2s∈HkX2.Therefore, if(σkj)j=0Nk is an orthonormal basis of Hk(X2), then(52)Ck,n-1·ν∗σkjj=0Nkis an orthonormal basis of Vk. It follows that Pk,n in (2) is given by(53)Pk,n=1Ck,n2ν×ν∗Π2,k,where ν×ν:X2×X2→Sn×Sn is the product projection.More explicitly, forq∈Sn let S(q⊥)≅Sn-1 be the unit sphere centered at the origin in the orthocomplement q⊥, and let dVS(q⊥) be the Riemannian density on S(q⊥); then(54)Pk,nq0,q1=1Ck,n2∫Sq0⊥∫Sq1⊥Π2,kq0+ip,q1+ip′dVSq0⊥pdVSq1⊥p′.


### 2.6.Πr,k and Conjugation

Conjugationσ:z↦z¯ in Cn+1 leaves invariant the affine cone Cn and every Xr. Furthermore, it yields a Riemannian isometry of Xr into itself. For f∈OCn∖{0}, let us set(55)fσz≕fz¯¯.If f∈OkCn∖{0}, then fσ∈OkCn∖{0}.Hence, if(skj)j⊆OkCn∖{0} yields by restriction an orthonormal basis of Hk(Xr), then so does (skjσ)j. Thus, for any z0,z1∈Xr, we have(56)Πr,kz¯0,z¯1=∑jskjz¯0·skjz¯1¯=∑jskjσz0¯·skjσz1=Πrkz1,z0=Πrkz0,z1¯.


## 2.1. The Geometric Picture

For the following, see [8, 9].LetS1n⊂Rn+1 be the unit sphere, and let us identify the tangent and cotangent bundles of S1n by means of the standard Riemannian metric. The unit (co)sphere bundle of S1n is given by the incidence correspondence(20)S∗S1n≅SS1n=q,p∈S1n×S1n:qtp=0.The Fermat quadric hypersurface in complex projective space is(21)Fn≔z∈Pn:ztz=0;let A be the restriction to Fn of the hyperplane line bundle. Given the standard Hermitian product on Cn+1, A is naturally a positive Hermitian line bundle, Fn inherits a Kähler structure ωFn (the restriction of the Fubini-Study metric), and the spaces of global holomorphic sections of higher powers of A, H0Fn,A⊗k have an induced Hermitian structure.The affine cone overFn is Cn={ztz=0}⊂Cn+1; the intersection X1≔Cn∩S12n+1 may be viewed as the unit circle bundle in the dual line bundle A∨. More generally, for any r>0, the intersection(22)Xr≔Cn∩Sr2n+1with the sphere of radius r is naturally identified with the circle bundle of radius r in A∨. In particular, (23)X2=q+ip:q2=p2=1,qtp=0is diffeomorphic to S∗(Sn) by the map β:(q,p)↦q+ip; furthermore, β is equivariant for the natural actions of O(n+1) on S∗(Sn) and X2 defined by, respectively,(24)B·q,p≕Bq,Bp,B·q+ip=Bq+iBpB∈On+1.We shall identifyS∗(Sn) and X2 and denote the projection by(25)ν:S∗Sn≅X2⟶Sn,q+ip⟼q.There is also a standard structure action ofS1 on X2, induced by fibrewise scalar multiplication in A∨, or equivalently in Cn+1. The latter action is intertwined by β with the “reverse” geodesic flow on S∗(Sn)≅S(Sn). The S1-orbits are the fibers of the circle bundle projection(26)π2:q+ip∈X2⟼q+ip∈Fn.This holds for anyr>0; we shall denote by πr:Xr→Fn the projection for general r>0.


## 2.2. The Metric onXr

Let us dwell on the metric aspect of (22); there are two natural choices of a Riemannian metric on Xr, hence of a Riemannian density, and we need to clarify the relation between the two.There is an obvious choice of a Riemannian metricgr′ on Xr, induced by the standard Euclidean product on Cn+1. With respect to gr′, the S1 orbits on Xr have length of 2πr. Clearly, gr′ is homogeneous of degree 2 with respect to the dilation μr:x∈X↦rx∈Xr, and therefore the corresponding volume form Υ′Xr on Xr is homogeneous of degree dim⁡X=2n-1. That is,(27)μr∗Υ′Xr=r2n-1Υ′X.An alternative and common choice of a Riemannian structureg1 on X1 comes from its structure of a unit circle bundle over Fn. Let α∈Ω1(X1) be the connection 1-form associated with the unique compatible covariant derivative on A, so that dα=2π1∗(ωFn). Also, let (28)HX1/Fn=ker⁡α,VX1/Fn=kerdπ1⊆TXdenote the horizontal and vertical tangent bundles for π1, respectively. There is a unique Riemannian metric g1 on X1 such that π1 a Riemannian submersion and the S1-orbits on X1 have unit length. The corresponding volume form on X1 is given by(29)ΥX1=1n-1!π1∗ωFn∧n-1∧12πα=12ππ1∗ΥFn∧α,where ΥFn=ωFn∧(n-1)/(n-1)! is the symplectic volume form on Fn.We wish to compare the two Riemannian metricsg1 and g1′, the corresponding volume forms, Υ′X1 and ΥX1, and densities, dVX and d′VX.Lemma 5.
Υ
X
1
=
1
/
2
π
Υ
′
X
1 and dVX1=1/2πd′VX1.Proof of Lemma5.
The connection 1-form for the Hopf mapS2n+1→Pn is (30)θ=i2ztdz¯t-z¯tdz;thus, α is the restriction of θ to X1. Let ω0 be the standard symplectic structure on Cn+1. Since θz(w)=ω0(z,w), we have ker⁡(θz)=z⊥ω0 (symplectic annihilator). In other words, (31)ker⁡θz=spanRz⊕z⊥h0∩TzS12n+1=z⊥h0,where z⊥h0 is the Hermitian orthocomplement of z for the standard Hermitian product.
Thus, ifz∈X1, then(32)HzX1/Fn=kerαz=z⊥h0∩TzCn=z⊥h0∩z¯⊥h0.On the other hand, Vz(X1/Fn)=spanR(iz). Thus, V(X1/Fn) and H(X1/Fn) are orthogonal with respect to both g1 (by construction) and g1′ (by the previous considerations). Hence, we may compare g1 and g1′ separately on H(X1/Fn) and V(X1/Fn).
On the complex vector bundleH(X1/Fn), g1′ and g1 are, respectively, the Euclidean scalar products associated with the restrictions of the (1,1)-forms (33)ω0=i2∂∂¯z2,ω1=i2∂∂¯ln⁡z2.Given that ω0 and ω1 agree on TS12n+2, g1=g1′ on H(X1/Fn).
On the other hand, bothg1 and g1′ are S1-invariant, but S1-orbits on X1 have length 2π for g1′ and 1 for g1. Thus, g1=g1′/2π on V(X1/Fn).
The claim follows directly from this.


## 2.3. The Szegö Kernel onXr

The following analysis is based on the equivariant asymptotics of the Szegö kernel ofX2 [11]. We refer the reader to [10, 11, 15] for a thorough discussion of Szegö kernels in the algebro-geometric context and to [16, 17] for the basic microlocal theory that underlies the subject (see also the neat discussion of Hardy spaces in [9]). To put things into perspective, however, let us recall that if Y is the boundary of a pseudoconvex domain, its Hardy space H(Y)⊂L2(Y) is the Hilbert space of square summable boundary values of holomorphic functions. The Szegö projector is then the orthogonal projector Π:L2(Y)→H(Y); with some abuse of notation, the Szegö kernel Π∈D′(Y×Y) is the corresponding distributional kernel. A description of Π as a Fourier integral operator was given in [16].In the special algebro-geometric case, whereY is the dual unit circle bundle of a positive line bundle on a complex projective manifold, H(Y) is the orthogonal direct sum of its isotypical components Hk(Y),k=0,1,2,…, under the S1-action; correspondingly, Π=⨁kΠk, where Πk is the orthogonal projector onto Hk(Y); since Hk(Y) is finite-dimensional, Πk is a smoothing operator; therefore, its Schwartz kernel is a C∞ function on Y×Y. Many local asymptotic properties of Πk, for k→+∞, were first discovered in [10, 11, 15] building on the theory of [16]. We shall recall what is needed here shortly.Let us now come to the specific case in point. For everyr>0, Xr is the boundary of a strictly pseudoconvex domain, and as such it carries a CR structure, a Hardy space H(Xr), and a Szegö projector Πr:L2(Xr)→H(Xr). We aim to relate the various Πr’s.LetO(Cn∖{0}) be the ring of holomorphic functions on the conic complex manifold Cn∖{0}. Let Ok(Cn∖{0})⊂O(Cn∖{0}) be the subspace of holomorphic functions of degree of homogeneity k.For everyr>0 and k=0,1,2,… let Hk(Xr)⊂H(Xr) be the finite-dimensional kth isotypical component of H(Xr) with respect to the standard S1-action. Restriction induces an algebraic isomorphism Ok(Cn∖{0})→Hk(Xr); with a slight abuse of language, we shall denote by the same symbol an element of Hk(Xr) and the corresponding element of Ok(Cn∖{0}).Suppose that(skj)j=0Nk⊆Ok(Cn∖{0}) yields by restriction an orthonormal basis of Hk(X1):(34)∫X1skjxsklx¯dVX1x=δjl.Setting y=rx and using (27) together with Lemma 5, we get(35)∫Xrskjyskly¯12πd′VXry=r2n+2k-1∫X1skjxsklx¯12πd′VX1x=r2n+2k-1δjl.Therefore, we have the following.Lemma 6.
If(skj)j=0Nk⊆Ok(Cn∖{0}) yields by restriction an orthonormal basis of Hk(X1) with respect to d′VX1, then for every r>0(36)r-k+n-1/2skjj=0Nkyields by restriction an orthonormal basis of Hk(Xr), with respect to d′VXr/2π.Let nowΠr,k be the level-k Szegö kernel on Xr, that is, the orthogonal projector(37)Πr,k:L2Xr,d′VXr2π⟶HkXr.By Lemma 6, its Schwartz kernel Πr,k∈C∞(Xr×Xr) is given by(38)Πr,ky,y′=r-2k+2n-1∑j=0Nkskjy·skjy′¯y,y′∈Xr.When pulled back toX1, this is (here x,x′∈X1)(39)Πr,krx,rx′=r-2k+2n-1∑j=0Nkskjrx·skjrx′¯=r-2n-1∑j=0Nks^kjx·s^kjx′¯=r1-2nΠ1,kx,x′.In particular,(40)Π2,k2x,2x′=22nΠ1,kx,x′.We shall make repeated use of the following asymptotic property ofΠ1,k, which follows from the microlocal description of Π as an FIO (explicit exponential estimates are discussed in [18]).Theorem 7.
LetdistFn be the distance function on Fn associated with the Kähler metric. Given any C,ϵ>0, uniformly for x,x′∈X satisfying(41)distFnπx,πx′≥Ckϵ-1/2,we have(42)Π1,kx,x′=Ok-∞when k→+∞.


## 2.4. Heisenberg Local Coordinates

There are two unit circle bundles in our picture: the Hopf fibrationπ:S12n+1→Pn, and π1:X1→Fn. Clearly, π1 is the pull-back of π under the inclusion Fn↪Pn. Both S12n+1 and X1 are boundaries of strictly pseudoconvex domains and carry a CR structure.On bothS12n+1 and X1, we may consider privileged systems of coordinates calledHeisenberg local coordinates (HLC). In these coordinates, Szegö kernel asymptotics exhibit a “universal” structure [11]; we refer toibidem for a detailed discussion.Givenz0∈X1, a HLC system on X1 centered at z0 will be denoted in additive notation:(43)θ,v∈-π,π×B2n-20,δ⟼z0+θ,v∈X1.Here θ∈(-π,π) is an “angular” coordinate measuring displacement along the S1-orbit through z0 (the fiber through z0 of π1:X1→Fn); instead, v∈B2n-2(0,δ)⊆R2n-2≅Cn-1 descends to a local coordinate on Fn centered at m0=π(x0), inducing a unitary isomorphism T[z0]Fn≅Cn-1. We may thus think of v as a tangent vector in T[z0]Fn.Here this additive notation might be misleading, sinceX1⊂Cn+1. Therefore, we shall write z0+X1(θ,v) for HLC on X1 centered at z0. We shall generally abridge notation by writing z0+X1v for z0+X1(0,v).Similarly,(θ,w)∈(-π,π)×B2n(0,δ)↦z0+S12n+1(θ,v) will denote a system of Heisenberg local coordinates on S12n+1 centered at z0. There is in fact a natural choice of HLC on S12n+1 centered at any z0∈S12n+1.Namely, let(a1,…,an) be an orthonormal basis of the Hermitian orthocomplement z0⊥h⊆Cn+1, and for w=(wj)∈Cn let us set(44)z0+S12n+1θ,w≔eiθ1+w2z0+∑j=1nwjaj.Since there is a canonical unitary identification z0⊥h≅T[z0]Pn, we shall also write this as z0+S12n+1(θ,v) with (θ,v)∈(-π,π)×T[z0]Pn.Ifz0∈X1, HLC on X1 centered at z0 can be chosen so that they agree to second order with the former HLC on S12n+1. More precisely, we may assume that for any v∈T[z0]Fn⊂T[z0]Pn we have(45)z0+Xθ,w=z0+S12n+1θ,v+R2v,where R2 is a function vanishing to second order at the origin.Givenv,w∈Cn+1≅R2n+2, let us define(46)ψ2v,w≕-iω0v,w-12v-w2;here ω0 is the standard symplectic structure, and · is the standard Euclidean norm. We shall make use of the following asymptotic expansion, for which we refer again to [11].Theorem 8.
Let us fixC>0 and ϵ∈(0,1/6). Then for any z∈X1, and for any choice of HLC on X1 centered at z, there exist polynomials Pj of degree ≤3j and parity j on T[z]Fn×T[z]Fn≅R2n-2×R2n-2, such that the following holds. Uniformly in v1,v2∈T[z]Fn with vj≤Ckϵ for j=1,2, and θ1,θ2∈(-π,π), one has for k→+∞ the following asymptotic expansion:(47)Π1,kz+θ1,v1k,z+θ2,v2k~kπn-1eikθ1-θ2+ψ2v1,v21+∑j=1+∞k-j/2Pjv1,v2.In the given range, the above is an asymptotic expansion, since(48)k-j/2Pjv1,v2≤Cjk-j/21-6ϵ.


## 2.5.Pk and Π2,k

As discussed in [9], the push-forward operator ν∗:C∞(X2)→C∞(Sn) yields by restriction an algebraic isomorphism (49)C∞X2∩HX2⟶C∞Sn,for every k, and (49) yields by restriction an isomorphism(50)HkX2⟶Vk,which is unitary up to a dilation by a constant factor Ck,n>0. Thus, we have (51)ν∗sL2Sn=Ck,nsHX2s∈HkX2.Therefore, if(σkj)j=0Nk is an orthonormal basis of Hk(X2), then(52)Ck,n-1·ν∗σkjj=0Nkis an orthonormal basis of Vk. It follows that Pk,n in (2) is given by(53)Pk,n=1Ck,n2ν×ν∗Π2,k,where ν×ν:X2×X2→Sn×Sn is the product projection.More explicitly, forq∈Sn let S(q⊥)≅Sn-1 be the unit sphere centered at the origin in the orthocomplement q⊥, and let dVS(q⊥) be the Riemannian density on S(q⊥); then(54)Pk,nq0,q1=1Ck,n2∫Sq0⊥∫Sq1⊥Π2,kq0+ip,q1+ip′dVSq0⊥pdVSq1⊥p′.


## 2.6.Πr,k and Conjugation

Conjugationσ:z↦z¯ in Cn+1 leaves invariant the affine cone Cn and every Xr. Furthermore, it yields a Riemannian isometry of Xr into itself. For f∈OCn∖{0}, let us set(55)fσz≕fz¯¯.If f∈OkCn∖{0}, then fσ∈OkCn∖{0}.Hence, if(skj)j⊆OkCn∖{0} yields by restriction an orthonormal basis of Hk(Xr), then so does (skjσ)j. Thus, for any z0,z1∈Xr, we have(56)Πr,kz¯0,z¯1=∑jskjz¯0·skjz¯1¯=∑jskjσz0¯·skjσz1=Πrkz1,z0=Πrkz0,z1¯.


## 3. Proof of Theorem1

Proof of Theorem1.Given q0,q1∈Sn-1 with q1≠±q0, let γ+ be the unique unit speed geodesic on Sn such that γ+(0)=q0 and γ+(ϑ)=q′ for some ϑ∈(0,π). Then (57)p0≔γ˙+0∈Sn-1q0⊥,p1≔γ˙+ϑ∈Sn-1q1⊥.The reverse geodesic γ-(ϑ)≕γ(-ϑ) satisfies γ-(0)=q, γ˙-(0)=-p0 and γ-(ϑ′)=q′ for a unique ϑ′=-ϑ∈(-π,0).Although they project down to the same locus inSn, γ+ and γ- correspond to distinct fibers of the circle bundle projection π:X(2)→Fn. Let us express the (co)tangent lift γ~± of the geodesics γ± in complex coordinates, and set p1=γ˙+(ϑ). Then(58)γ~±θ=γ±θ+iγ˙±θ=e-iθq0±ip0=q1±ip1.In view of (26), we have (59)q0±ip0,q1±ip1∈π2-1q0±ip0.On the other hand, [q0+ip0]≠[q0-ip0]∈Fn, since q0+ip0 and q0-ip0 are linearly independent in Cn+1.Thus, we have the following.Lemma 9.
Supposeq0,q1∈Sn and q1≠±q0. Then the only points [z]∈Fn such that(60)ν-1q0∩π2-1z≠∅,ν-1q1∩π2-1z≠∅are(61)z+=q0+ip0,z-=q0-ip0.By Theorem7, for fixed p and p′ and k→+∞, we have(62)Π2,kq0+ip,q1+ip′=Ok-∞,unless p=±p0 and p′=±p1. Therefore, for a fixed ϑ∈(0,π) integration in (54) may be localized in a small neighborhood of (±p0,±p1), perhaps at the cost of disregarding a negligible contribution to the asymptotics.Since however we are allowingϑ to approach 0 or π at a controlled rate, we need to give a more precise quantitative estimate of how small the previous neighborhood may be chosen when k→+∞.To this end, let us introduce some further notation. Given linearly independenta,b∈Sn, let us set(63)Ra,b≕spanRa,b⊆Rn+1,Ra,bC≔Ra,b⊗C=spanCa,b⊆Cn+1.Furthermore, for v≤1, we shall set (64)S±v≕-1±1-v2.A straightforward computation yields the following.Lemma 10.
Assume thatq0+ip0∈X2 and q1+ip1=e-iϑ(q0+ip0) with ϑ∈(0,π). Then any p∈Sn-1(q0⊥) with p0tp≥0, respectively, p0tp≤0, may be written uniquely in the form (65)p=1+S+vp0+v,respectively (66)p=1+S-vp0+v,where v∈q0⊥∩p0⊥=R(q0,q1)⊥ (the Euclidean orthocomplement) has norm ≤1 and (67)S±v≕-1±1-v2.Proposition 11.
Let us fixC>0, δ∈(0,1/6) and ϵ>δ. Then there exist constants D,ϵ1>0 such that the following holds. Suppose that (1)
Ck-δ<ϑ<π-Ck-δ;(2)
qj+ipj∈X2  for j=0,1;(3)
q1+ip1=e-iϑ(q0+ip0);(4)
vj∈q0⊥∩q1⊥  for j=0,1;(5)
1≥max{v0,v1}≥Ckϵ-1/2;(6)
pj′=(1+Sj(vj))pj+vj∈Sn-1(qj⊥) for j=0,1, where Sj can be either one of S± (Lemma 10).
Then(68)distFnq0+ip0′,q1+ip1′≥Dkϵ1-1/2for every k≫0.In view of Theorem7, Proposition 11 implies the following.Corollary 12.
Uniformly in the range of Proposition11, we have(69)Π2,kq0+ip0′,q1+ip1′=Ok-∞.Proof of Proposition11.
Let us set forγ∈[-π,π](70)Φγ,p0′,p1′≕e-iγq0+ip0′-q1+ip1′.Let distFn be the restriction to Fn of the distance function on Pn. Then(71)distFnq0+ip0′,q1+ip1′=12min⁡Φγ,p0′,p1′:γ∈0,2π.The factor in front is needed because while the Hopf map S12n+1→Pn is a Riemannian submersion, the projection S22n+1→Pn is so only after a constant rescaling of the metric.
We are reduced to proving that in the given range there exist constantsD,ϵ1>0 such that for every k≫0 and γ∈[0,2π](72)Φγ,p0′,p1′≥Dkϵ1-1/2.
We have(73)Φγ,p0′,p1′=e-iγq0+ip0+iS0v0p0+iv0-e-iϑq0+ip0+iS1v1p1+iv1=Aq0+Bp0+ie-iγv0-v1,where (74)A≔e-iγ-e-iϑ+iS1v1sin⁡ϑ=cos⁡γ-cos⁡ϑ+i-sin⁡γ+sin⁡ϑ1+S1v1,(75)B≔ie-iγ-e-iϑ+ie-iγS0v0-S1v1cosϑ=sinγ1+S0v0-sinϑ+icosγS0v0-S1v1cosϑ+cosγ-cosϑ.
Regarding the two summands on the last line of (73), we have(76)Aq0+Bp0∈Rq0,q1C,ie-iγv0-v1∈Rq0,q1C⊥h,where ⊥h denotes the Hermitian orthocomplement. Hence,(77)Φγ,p0′,p1′2≥e-iγv0-v12≥1-cosγv02+v12.
Since1-cos⁡γ vanishes exactly to second order at γ=0,π,2π, there exists D>0 such that for γ∈[0,2π], we have(78)1-cosγ≥D2min⁡γ2,γ-π2,γ-2π2.Given this and (77), we conclude that, under the present hypothesis,(79)Φγ,p0′,p1′≥Dmin⁡γ,γ-π,2π-γmax⁡v,v′≥CDmin⁡γ,γ-π,2π-γkϵ-1/2.Let us now pick δ′ with ϵ>δ′>δ and assume(80)min⁡γ,γ-π,2π-γ≥k-δ′.
Then(81)Φγ,p,p′≥CDkϵ-δ′-1/2.This establishes (72) with ϵ1=ϵ-δ′, in the case where (80) holds. Thus, we are reduced to assuming (82)min⁡γ,γ-π,2π-γ≤k-δ′.Then we also have |sin(γ)|≤k-δ′. Let us then look at the first summand on the last line of (73). We have an Hermitian orthogonal direct sum(83)Rq0,q1C=Rq0,p0C=spanCq0⊕spanCp0.On the other hand, since sin(ϑ) vanishes exactly to first order at ϑ=0 and ϑ=π, there exists E>0 such that for ϑ∈(0,π) under the assumptions of the lemma we have(84)sin⁡ϑ≥Emin⁡ϑ,π-ϑ≥ECk-δ.Hence, in view of (75), we have for some D1>0 and k≫0(85)Φγ,p,p′≥Aq0+Bp0≥B≥RB=sin⁡γ1+S0v0-sinϑ≥sinϑ-k-δ′≥ECk-δ-k-δ′≥12ECk-δ≥12ECk-1/6,since δ′>δ and δ<1/6. This establishes (72) with ϵ1=1/3 when (82) holds.
The proof of Proposition11 is complete.Equations (65) and (66) parametrize neighborhoods of p0 and -p0, respectively. Therefore, Proposition 11 implies that in (54) only a negligible contribution to the asymptotics is lost, if integration in p and p′ yields by restriction the shrinking neighborhoods of ±p0 and ±p1, of radii Okϵ-1/2.This may be rephrased as follows. Letϱ∈C0∞Rn+1 be even, supported in a small neighborhood of the origin, and identically equal to one in a smaller neighborhood of the origin. Then the asymptotics of (54) are unchanged, if the integrand is multiplied by (86)ϱk1/2-ϵp-p0+ϱk1/2-ϵp+p0·ϱk1/2-ϵp′-p1+ϱk1/2-ϵp′+p1.In this way, the integrand splits into four summands. In fact, only two of these are nonnegligible fork→+∞. Namely, consider the summand containing the factor (87)ϱk1/2-ϵp-p0ϱk1/2-ϵp′+p1.On its support,p lies in a shrinking neighborhood of p0 and p′ in a shrinking neighborhood of -p1. Therefore, on the same support q0+ip lies in a shrinking neighborhood of q0+ip0, and q1-ip′ lies in a shrinking neighborhood of q1-ip1. Since(88)12q0+ip0∧12q1-ip1=12q0+ip0∧eiϑq0-ip0=-iq0∧p0has unit norm, on the support of (87) [q0+ip] and [q1+ip′] remain at a distance ≥2/3, say, in projective space. This implies that as k→+∞(89)Π2,kq0+ip,q1+ip′=Ok-∞uniformly in (p,p′) in the support of (87). A similar argument applies to the summand containing the factor(90)ϱk1/2-ϵp+p0ϱk1/2-ϵp′-p1.Thus, we may rewrite (54) as follows:(91)Pk,nq0,q1~Pk,nq0,q1++Pk,nq0,q1-,where(92)Pk,nq0,q1±≔1Ck,n2∫Sq0⊥∫Sq1⊥ϱk1/2-ϵp∓p0ϱk1/2-ϵp′∓p1·Π2,kq0+ip,q1+ip′dVSq0⊥pdVSq1⊥p′.As a further reduction, we need only deal with one ofPk,n(q0,q1)±.Lemma 13.
P
k
,
n
(
q
0
,
q
1
)
±
=
P
k
(
q
0
,
q
1
)
∓
¯.Proof of Lemma13.
Let us apply the change of integration variablep↦-p and p′↦-p′ and apply (56). Since ϱ is even, we get(93)Pk,nq0,q1-=1Ck,n2∫Sq0⊥∫Sq1⊥ϱk1/2-ϵp-p0ϱk1/2-ϵp′-p1·Π2,kq0+ip,q1+ip′¯dVSq0⊥pdVSq1⊥p′=Pkq0,q1+¯.Lemma13 and (91) imply(94)Pk,nq0,q1~2RPkq0,q1+.In the definition ofPk(q0,q1)+, integration is over a shrinking neighborhood of (p0,p1)∈S(q0⊥)×S(q1⊥). We can thus make use of the parametrization (65) and write in (92):(95)p=p0+Av0,p′=p1+Av1,where we have set(96)Avj≕vj+S+vjpj.It is also harmless to replace p-pj by vj in the rescaled cut-offs in (92). Let us also set zj=qj+ipj and recall that z1=e-iϑz0. We then obtain(97)Pk,nq0,q1+=1Ck,n2∫q0⊥∩q1⊥∫q0⊥∩q1⊥ϱk1/2-ϵv0ϱk1/2-ϵv1·Π2,kz0+iAv0,z1+iAv1·Vv0,v1dv0dv1,where V(0,0)=1.Let us pass to rescaled integration variablesvj↦vj/k in (97). Then(98)Pk,nq0,q1+=k1-nCk,n2∫q0⊥∩q1⊥∫q0⊥∩q1⊥ϱk-ϵv0ϱk-ϵv1·Π2,kz0+ikA0kv0,z1+ikA1kv1·Vv0k,v1kdv0dv1,with(99)Ajkv≔v+k·S+vkpj.Let us consider the Szegö term in the integrand. In view of (40), this is(100)Π2,kz0+ikA0kv0,z1+ikA1kv1=22neikϑΠ1,kz02+1kiA0kv02,z02+1kieiϑA1kv12.Now the sums in the previous expression are just algebraic sums inCn+1; in order to apply the scaling asymptotics of Theorem 8, we need to first express the argument of (100) in terms of local Heisenberg coordinates on X1 centered at z0/2.Lemma 14.
Supposez=q+ip∈X1 and choose a system of HLC on X1 centered at z. Then for δp~0∈Rn+1 and eiϑ∈S1 such that z+ieiϑδp∈X1 we have(101)z+ieiϑδp=z+X10,ieiϑδp+R2θ;δp,for a suitable smooth function R2(θ;·) vanishing to second order at the origin (in v).Proof of Lemma14.
In view of (45), it suffices to prove the statement on S12n+1, working with the HLC (44). Since z,z+ieiϑδp∈Cn, we have(102)0=ztz+2ieiϑztδp-e2iϑδptδp=2ieiϑztδp-e2iϑδp2,so that iztδp=eiϑδp2/2.
Let us look forβ>0 and h∈z⊥h (Hermitian orthocomplement) such that(103)z+ieiϑδp=βz+h.If this is possible at all, then necessarily β=1/z+h, as z+ieiϑδp=1. Then (104)z+ieiϑδp=z+S12n+10,h.
Assuming that (103) may be solved, then, taking the Hermitian product with z on both sides of (103) and using (102), we get (105)β=ztz¯-ie-iϑδp=1-ie-iϑztδp=1-12δp2>0.
With this value ofβ, let us set (106)h≔1βz+ieiθδp-z,so that (103) is certainly satisfied. We need to verify that h∈z⊥h. Indeed we have(107)htz¯=1β1+ieiθδptz¯-1=1β1-12δp2-1=0.Since h=ieiθδp+R2(δp), the proof of the lemma is complete.Notice thath is given for δp~0 by an asymptotic expansion in homogeneous polynomials of increasing degree in δp of the form (108)h~ieiθδp+12δp2z+i2eiθδp2δp+14δp4z+⋯.This holds on S12n+1, but a similar expansion obviously holds on X1, possibly with modified terms in higher degree.Let us apply Lemma14 with z=z0/2 and δpj=eiθ(Ajk(vj)/2)/k (we will set θ=0 for j=0 and θ=ϑ for j=1). To this end, let us note that in view of (99) for k→+∞ there is an asymptotic expansion of the form(109)Ajkv~∑j≥01kl/2Pj,l+1v,where Pj,l is a homogeneous (vector valued) polynomial function of degree l and Pj1(v)=v. Hence, (110)δpj=eiθ2kAjkvj~eiθ2∑j≥01kl+1/2Pj,l+1vj=eiθkvj2+⋯.Making use of (110) in (108) we obtain(111)hkj~∑l≥11kl/2Qjlθ;v=1kieiθvj2+∑l≥11kl/2Qj,l+1θ;vj,where Qjl(θ;·) is a homogeneous polynomial function of degree l and we have emphasized the dependence on k.Thus, we obtain forj=0 (with θ=0) that(112)z02+1kiA0kv02=z02+X10,hk0,where(113)hk0~1kiv02+∑l≥11kl/2Q0,l+10;v0=1kak0,with ak0 defined by the latter equality. Similarly, for j=1 (with θ=ϑ), we have (114)z02+1kieiϑA1kv12=z02+X10,hk1,where (115)hk1~1kieiϑv12+∑l≥11kl/2Q1,l+1ϑ;v1=1kak1.Let us return to (100). In view of Theorem 8, we get(116)Π1,kz02+1kiA0kv02,z02+1kieiϑA1kv12=Π1,kz02+X11kak0,z02+X11kak1~kπn-1eψ2ak0,ak1·1+∑b=1+∞k-b/2Pbak0,ak1.We have(117)ψ2ak0,ak1~12ψ2v0,eiϑv1+∑l≥11kl/2Q~l+2ϑ;v0,v1,where Q~l(ϑ;·,·) is a homogeneous C-valued polynomial of degree l. For any r≥1 and l1,…,lr≥1, we have(118)∏j=1r1klj/2Q~lj+2ϑ;v0,v1=1k∑j=1rlj/2Q^∑j=1rlj+2rϑ;v0,v1,where Q^∑j=1rlj+2r(ϑ;·,·) is homogeneous of degree ∑j=1rlj+2r. Since lj≥1 for every j, we have ∑j=1rlj+2r≤3∑j=1rlj.One can see from this that(119)eψ2ak0,ak1~e1/2ψ2v0,eiϑv11+∑l≥11kl/2Blϑ;v0,v1,where Bl(ϑ;·,·) is a polynomial of degree ≤3l, having the same parity as l.Similarly, recalling thatPb has the same parity as b and degree ≤3b, each summand k-b/2Pb(ak0,ak1) in (116) gives rise to an asymptotic expansion in terms of the form(120)1kb/2∏a=1r1kla/2Rla+1ϑ;v0,v1=1kb+∑a=1rla/2R~∑a=1rla+rϑ;v0,v1,where Rl(ϑ;·,·) and R~l(ϑ;·,·) are homogeneous polynomials of the given degree, r≤3b, and b-r is even. Then 3(b+∑a=1rla)≥∑a=1rla+r, and (b+∑a=1rla)-(∑a=1rla+r)=b-r is also even. Hence, each summand k-b/2Pb(ak0,ak1) (b≥1) yields an asymptotic expansion of the form(121)∑l≥1k-l/2Tblϑ;v0,v1,where again each Tbl has the same parity as l and degree ≤3l.Putting this all together, we obtain an asymptotic expansion for the integrand in (98).Lemma 15.
Forl≥0, there exist polynomials Zl(ϑ;·,·) of degree ≤3l and parity (-1)l, with Z0(ϑ;·,·)=1, such that(122)Π2,kz0+ikA0kv0,z1+ikA1kv1·Vv0k,v1k~22neikϑkπn-1e1/2ψ2v0,eiϑv1∑l≥01kl/2Zlϑ;v0,v1.Proof of Lemma15.
The previous arguments yield an asymptotic expansion of the given form for the first factor. We need only multiply the latter expansion by the Taylor expansion of the second factor.Since integration in (98) takes place over a poly-ball or radius Okϵ in q0⊥∩q1⊥2, the expansion may be integrated term by term. In addition, given that the exponent and the cut-offs are even functions of (v0,v1), only terms of even parity yield a nonzero integral. Hence, we may discard the half-integer powers and obtain(123)Pkq0,q1+~k1-nCk,n222neikϑkπn-1∑l≥0k-lP^lϑ+,where(124)P^lϑ+≔∫q0⊥∩q1⊥∫q0⊥∩q1⊥ϱk-ϵv0ϱk-ϵv1·e1/2ψ2v0,eiϑv1Z2lϑ;v0,v1dv0dv1.We can slightly simplify the previous asymptotic expansion as follows. First, as emphasized the dependence on (q0,q1) is of course only through the angle ϑ. In particular, in (124) nothing is lost by assuming that q0 and q1 span the 2-plane {0}×R2⊆Rn+1, and therefore that q0⊥∩q1⊥=Rn-1×{0}.Furthermore, given (46), we have(125)ψ2v0,eiϑv1=-isin⁡ϑv0tv1-12v0-cos⁡ϑv12-12sin⁡ϑ2v12.With the change of variables (126)v0v1=2b0+cotϑb11sinϑb1we obtain (127)ψ2v0,eiϑv1=-12b02-ib0tb1-121+2icot⁡ϑb12.Since Z2l(ϑ,·,·) is even and has degree ≤6l, we can write(128)Z2lϑ;b0+cot⁡ϑb1,b1sin⁡ϑ=1sin⁡ϑ6lTlϑ;b0,b1,where Tl(ϑ;·,·) is an even polynomial of degree ≤6l, with smooth bounded coefficients for ϑ∈[0,π]. Thus,(129)P^lϑ+=2sin⁡ϑn-11sin⁡ϑ6l∫Rn-1∫Rn-1e-1/2b02-ib0tb1-1/21+2icot⁡ϑb12·ϱk-ϵ2b0+cot⁡ϑb1ϱk-ϵ2sin⁡ϑ-1b1Tlϑ;b0,b1·db0db1.There is a constantC>0 such that the support of(130)1-ϱk-ϵ2b0+cot⁡ϑb1ϱk-ϵ2sin⁡ϑ-1b1is contained in the locus, where b0,b1≥Ckϵsin(ϑ). Under the assumptions of the theorem, this implies, perhaps for a different constant C>0, that b0,b1≥Ckϵ-δ. On the other hand, the exponent in (129) satisfies(131)-12b02-ib0tb1-121+2icot⁡ϑb12≤-12b02+b12.Given that ϵ>δ (statement of Proposition 11), we conclude that only a negligible contribution to the asymptotics is lost, if the cut-off function is omitted and integration is now extended to all of Rn-1×Rn-1.We can thus rewrite (123) as follows:(132)Pkq0,q1+~1Ck,n222eikϑ1πsin⁡ϑn-1∑l≥0k-l1sin⁡ϑ6lP~lϑ+,where(133)P~lϑ+=∫Rn-1∫Rn-1e-1/2b02-ib0tb1-1/21+2icot⁡ϑb12Tlϑ;b0,b1db0db1.Let us setBϑ=(1+icot⁡(ϑ))In-1. The leading order coefficient is(134)P~0ϑ+=∫Rn-1e-1/21+2icot⁡ϑb12∫Rn-1e-ib0tb1-1/2b02db0db1=2πn-1/2∫Rn-1e-1/22+2icot⁡ϑb12db1=2n-1/2πn-11det⁡Bϑ=2πn-1sin⁡ϑn-1/2eiϑ/2-π/4n-1.Given (134), (132), and (94), Pk(q0,q1) has an asymptotic expansion for k→+∞ with leading order term(135)2n/2Ck,n21sin⁡ϑn-1/2coskϑ+ϑ2-π2n-1.For anyl, we can write(136)P~lϑ+=∫Rn-1e-1/21+2icot⁡ϑb12∫Rn-1e-ib0tb1-1/2b02Tlϑ;b0,b1db0db1=∫Rn-1e-1/22+2icot⁡ϑb12Tlϑ;b1db1,where Tl(ϑ;·) is an even polynomial of degree ≤6l.Let us introduce the Fourier transform(137)Fc=∫Rn-1e-1/22+2icot⁡ϑb12-ib1tcdb1=2πn-1/2sin⁡ϑn-1/2eiϑ/2-π/4n-1e-1/22+2icot⁡ϑ-1c2.Then (136) is the result of applying an even differential polynomial Pl(Dc) of degree ≤6l to F(c) and then evaluating the result at c=0.Given this and (135), we conclude that (138)Pkq0,q1=2n/2Ck,n21sinϑn-1/2·coskϑ+ϑ2-π4n-1·Aϑ+sinkϑ+ϑ2-π4n-1·Bϑ,where(139)Aϑ~1+∑l=1+∞k-lAlϑsin⁡ϑ6l,Bϑ~∑l=1+∞k-lBlϑsin⁡ϑ6l,with Al and Bl smooth functions of ϑ on [0,2π].Proof of Theorem1 is complete.


## 4. Proof of Proposition2

Proof of Proposition2.
The diagonal restrictionPk,n(q,q) may be computed in two different ways. On the one hand, since Pk,n(q,q) is constant, we have (140)Pk,nq,q=Nk,nvolSn=2volSnkn-1n-1!+Okn-2.
On the other hand, (54) with q0=q1=q yields(141)Pk,nq,q=1Ck,n2∫Sq⊥∫Sq⊥Π2,kq+ip,q+ip′dVSq⊥pdVSq⊥p′=1Ck,n2∫Sq⊥Fkq,pdVSq⊥p,where (142)Fkq,p≕∫Sq⊥Π2,kq+ip,q+ip′dVSq⊥p′.
Again, integration indVSq⊥(p′) localizes in a shrinking neighborhood of p. Hence, we may let (143)p′=p+Av,Av=v+S+vp,where v∈q⊥∩p⊥, and introduce the cut-off ϱk1/2-ϵv. Passing to rescaled coordinates, and setting z=q+ip, we get(144)Fkq,p=1kn-1/2∫q⊥∩p⊥ϱk-ϵvΠ2,kz,z+ikAkvVvkdv,where(145)Akv=v+kS+vkp,V0=1.
By Lemma15 (with z=z0=z1, v0=0, v1=v, ϑ=0), we have(146)Π2,kz,z+ikAkv·Vvk~22nkπn-1e-1/4v2∑l≥01kl/2Zlv,for certain polynomials Zl of degree ≤3l and parity (-1)l, with Z0(·)=1.
As before, the expansion may be integrated term by term and, by parity, only the summands withl even yield a nonzero contribution. In addition, only a negligible contribution is lost if the cut-off is omitted and integration is extended to all of q⊥∩p⊥≅Rn-1. Therefore,(147)Fkq,p~22nkn-1/2πn-1∑l≥0k-l∫Rn-1e-1/4v2Z2lvdv=12kπn-1/2+⋯.
Inserting this in (141), we obtain an asymptotic expansion(148)Pk,nq,q~volSn-1Ck,n212kπn-1/2+⋯.
Comparing (140) and (148), we obtain an asymptotic expansion in descending powers of k, of the form(149)Ck,n~volSnvolSn-122·n-1!1/2πk-n-1/4+⋯.



---

*Source: 7823545-2017-10-31.xml*