Bergman kernels, TYZ expansions and Hankel operators on the Kepler manifold

0.26 MB·

by  Hélène Bommier-Hato

BERGMAN KERNELS, TYZ EXPANSIONS AND HANKEL
OPERATORS ON THE KEPLER MANIFOLD
6
1
HE´LE`NEBOMMIER-HATO,MIROSLAVENGLISˇ,ANDEL-HASSANYOUSSFI
0
2
n Abstract. ForaclassofO(n+1,R)invariantmeasuresontheKeplermani-
a foldpossessingfinitemomentsofallorders,wedescribethereproducingkernels
J of the associated Bergman spaces, discuss the corresponding asymptotic ex-
pansions ofTian-Yau-Zelditch, andstudy the relevantHankel operators with
4
conjugate holomorphic symbols. Related reproducing kernels on the minimal
1
ball are also discussed. Finally, we observe that the Kepler manifold either

does notadmitbalancedmetrics,orsuchmetricsarenotunique.
]
V
C
.
h
t 1. Introduction
a
m Let n 2 and consider the Kepler manifold in Cn+1 defined by
≥
[
H:= z Cn+1 :z z =0, z =0 ,
1 { ∈ • 6 }
v where z w := z w + + z w . This is the orbit of the vector e =
1 1 n+1 n+1
1 (1,i,0,..•.,0) under the O··(·n + 1,C)-action on Cn+1; it is also well-known that
2 H canbe identified with the cotangentbundle ofthe unit sphere Sn in Rn+1 minus
7
its zero section. The unit ball of H,
3
0
M:= z H: z 2 =z z <1
1. { ∈ | | • }
0 as well as its boundary ∂M = z H : z = 1 are invariant under O(n+1,C)
6 U(n +1) = O(n +1,R), and{in∈fact ∂|M| is t}he orbit of e under O(n + 1,R).
1 ∩Inparticular,thereisauniqueO(n+1,R)-invariantprobabilitymeasuredµon∂M,
:
v coming from the Haar measure on the (compact) group O(n+1,R). Explicitly,
i denoting
X
r ( 1)j 1
a α:=(n+1) − − dz dz dz on z =0
1 j n+1 j
z ∧···∧ ∧···∧ 6
j
(thisis,uptoconstantfactor,theuniqueSOc(n+1,C)-invariantholomorphicn-form
on H, see [21]) and defining a (2n 1)-form ω on ∂M by
−
ω(z)(V ,...,V ):=α(z) α(z)(z,V ,...,V ), V ,...,V T(∂M),
1 2n 1 1 2n 1 1 2n 1
− ∧ − − ∈
we then have dµ = ω/ω(∂M) (where, abusing notation, we denote by ω also the
measure induced by ω on ∂M). It follows, in particular, that dµ is also invariant
under complex rotations
z ǫz, ǫ T= z C: z =1 .
7→ ∈ { ∈ | | }
ResearchsupportedbyGACˇRgrantno.201/12/0426 andRVOfundingforICˇO67985840.
1
2 HE´LE`NEBOMMIER-HATO,MIROSLAVENGLISˇ,ANDEL-HASSANYOUSSFI
For a finite (nonnegative Borel) measure dρ on (0, ), we can therefore define a
rotation-invariantmeasure d(ρ µ) on H by ∞
⊗
∞
fd(ρ µ):= f(tζ)dµ(ζ)dρ(t),
ZH ⊗ Z0 Z∂M
and the (weighted) Bergman space
A2 := f L2(d(ρ µ)):f is holomorphic on RM ,
ρ⊗µ { ∈ ⊗ }
where R := sup t > 0 : t suppρ = sup z : z suppρ µ (with the under-
standingthatRM{ =HifR∈=+ ).}Itissta{n|da|rdth∈atA2 ⊗isa}reproducingkernel
Hilbert space, that is, there exi∞sts a function K (x,y)ρ⊗µK(x,y) on RM RM
ρ µ
⊗ ≡ ×
for which K(,y) A2 for each y, K(y,x)=K(x,y), and
· ∈ ρ⊗µ
f(z)= f(w)K(z,w)d(ρ µ)(w) f A2 .
ZRM ⊗ ∀ ∈ ρ⊗µ
Our goal in this paper is to give a description of these reproducing kernels,
establishtheirasymptoticsasρvariesinacertainway(so-calledTian-Yau-Zelditch,
or TYZ, expansion), and study the Hankel operators on A2 . We also give an
ρ µ
analogous description for the reproducing kernels on the mini⊗mal ball
B:= z Cn : z 2+ z z <1 ,
{ ∈ | | | • | }
whichistheimageofMunderthe2-sheetedproperholomorphicmappinggivenby
the projection onto the first n coordinates.
In more detail, let
q := ∞tkdρ(t)
k
Z0
be the moments of the measure dρ (the values q = + being also allowed if the
k
∞
integral diverges). Our starting point is the following formula for the reproducing
kernels (whose proof goes by arguments which are already quite standard).
Theorem 1. For z,w RM with R as above,
∈
∞ (z w)l
K (z,w)= • ,
ρ µ
⊗ dl
l=0
X
with
q
2l
(1) d :=
l
N(l)
where
l+n 1 l+n 2 (2l+n 1)(l+n 2)!
N(l):= − + − = − − .
n 1 n 1 l!(n 1)!
(cid:18) − (cid:19) (cid:18) − (cid:19) −
Next, recall that, quite generally, for an n-dimensional complex manifold M
and a holomorphic line bundle L over M equipped with a Hermitian metric, the
so-called Kempf distortion functions ǫ , l =0,1,2,..., are defined by
l
ǫ (z):= h(s (z),s (z)),
l j j
j
X
where s is anorthonormalbasis of the Hilbert space L2 (L l,ωn) ofholomor-
{ j}j hol ⊗
phic sections of the l-th tensor power L l of L square-integrable with respect to
⊗
KEPLER MANIFOLD 3
the volume element ωn on M, where ω = curvh (which is assumed to be posi-
−
tive); see Kempf [16], Rawnsley [24], Ji [15] and Zhang [29]. These functions are
ofimportance in the study of projective embeddings and constantscalarcurvature
metrics (Donaldson [8]), where a prominent role is played, in particular, by their
asymptotic behaviour as l tends to infinity: namely, one has
∞
ǫl(z) ln aj(z)l−j as l +
≈ → ∞
j=0
X
intheC -sense,withsomesmoothcoefficientfunctionsa (z),anda (z)=1. This
∞ j 0
has been established in various contexts by Berezin [5] (for bounded symmetric
domains), Tian [26] and Ruan [25] (answering a conjecture of Yau) and Catlin [7]
andZelditch[28]forM compact,Engliˇs[9](for M astrictly pseudoconvexdomain
in Cn with smooth boundary and h subject to some technical hypotheses), etc.
If M is a domain in Cn and the line bundle is trivial (which certainly happens if
M is simply connected; in this case one need not restrict to integer l, but may
allow it to be any positive number), one can identify (holomorphic) sections of
L with (holomorphic) functions on M, h with a positive smooth weight on M,
L2 (L l,ωn) with A2 =A2 , and
hol ⊗ hlωn hldet[∂∂log 1]
h
(2) ǫ (z)=h(z)lK (z,z),
l hldet[∂∂log1]
h
where we (momentarily) denote by K the weighted Bergman kernel with respect
w
to a weightw onM (and similarly for A2). In the contextof our Kepler manifold,
w
this has been studied by Gramchev and Loi [11] for L = H C (i.e. the trivial
×
bundle) and h(z) = e z (so ω = i∂∂ z ; this turns out to be the symplectic
form inherited from the−|is|omorphism2H ∼=| T| ∗Sn\{zero section} mentioned in the
beginning of this paper [24]), who showed that
n 2
(3) ǫl(z)=ln+ (n−22)z(n−1)ln−1+ − zbkkln−k+Rl(z),
| | k=2 | |
X
with some constants b independent of z and remainder term R(z) = O(e clz )
k l − | |
for some c>0 (i.e. exponentially small).
Our second main result is the following.
Theorem 2. Let K =K for
s ρ µ
⊗
(4) dρ(t)=2cme−st2mt2mn−1dt,
where c,m are fixed positive constants and s>0. Then as s + ,
→ ∞
(5) e sz2mK (z,z)= 2mn snn−1 bj +R (z),
− | | s (n 1)!c sj z 2mj s
− j=0 | |
X
where b are constants depending on m and n only,
j
(1 n)(mn n+1)
b =1, b = − − ,
0 1
2m
and R (z)=O(e δsz2m) with some δ >0.
s − | |
The result of Gramchev and Loi corresponds to m = 1 and c = 21−n , so it is
2 (n 1)!
−
recoveredasaspecialcase. Ourmethodofproofisagooddealsimplerthanin[11],
4 HE´LE`NEBOMMIER-HATO,MIROSLAVENGLISˇ,ANDEL-HASSANYOUSSFI
covers all m > 0, and is also extendible to other situations. We further note that
d(ρ µ) with the ρ from (4) actually coincides (up to a constant factor) with ωn
⊗
for ω = i∂∂(sz 2m), in full accordance with (2) (taking h(z) = e z2m, and with
2 | | −| |
l=1,2,... replacedbythecontinousparameters>0asalreadyremarkedabove).
ThereisalsoananalogousresultfortheBergman-typeweightsonMcorrespond-
ing to dρ(t)=χ (t)(1 t2)st2n 1dt, s> 1.
[0,1] −
− −
As for our last topic, recall that the Hankel operator H , g A2 , is the
operator from A2 into L2(ρ µ) defined by g ∈ ρ⊗µ
ρ⊗µ ⊗
H f :=(I P)(gf),
g
−
whereP :L2(ρ µ) A2 istheorthogonalprojection. Thisisadenselydefined
operator, which⊗is (e→xtenρd⊗sµto be) bounded e.g. whenever g is bounded. For the
(analogouslydefined) Hankel operators on the unit disc D in C or the unit ball Bn
of Cn, n 2, criteria for the membership of H in the Schatten classes p, p> 0,
g
≥ S
were given in the classical papers by Arazy, Fisher and Peetre [2] [1]: it turns out
that for p 1 there are no nonzero H in p on D, while for p > 1, H p
g g
if and only≤if g Bp(D), the p-th order BesSov space on D; while on Bn, n∈ S2,
∈ ≥
there are no nonzero H in p if p 2n, while for p > 2n, again H p
g g
if and only if g Bp(Bn). OSne says t≤hat there is a cut-off at p = 1 or p =∈ 2Sn,
respectively. The∈result remains in force also for D and Bn replaced any bounded
strictly-pseudoconvexdomaininCn,n 1,withsmoothboundary(Luecking[17]).
Our third main result shows that fo≥r the Bergman space A2(M) := A2 for
ρ µ
dρ(t)=χ (t)t2n 1dt on M, there is also a cut-off at p=2n. ⊗
[0,1] −
Theorem 3. Let p 1. Then the following are equivalent.
≥
(i) There exists nonconstant g A2(M) with H p.
g
∈ ∈S
(ii) There exists a nonzero homogeneous polynomial g of degree m 1 such
≥
that H p.
g
∈S
(iii) There exists m 1 such that H p for all homogeneous polynomials g
g
≥ ∈ S
of degree m.
(iv) p>2n.
(v) H p for any polynomial g.
g
∈S
We remark that the proofs in [2] and [1] relied on the homogeneity of Bn un-
der biholomorphic self-maps, and thus are not directly applicable for M with its
muchsmallerautomorphismgroup. The proofin[17]reliedon∂-techniques,which
probably could be adapted to our case of M i.e. of a smoothly bounded strictly
pseudoconvex domain not in Cn but in a complex manifold, and furthermore hav-
ing a singularity in the interior (cf. Ruppenthal [23]). Our method of proof of
Theorem 3, which is close in spirit to those of [2] and [1], is, however, much more
elementary.
The paper is organized as follows. The proof of Theorem 1, together with mis-
cellaneous necessary prerequisites and the results for the minimal ball, occupies
Section 2. Applications to the TYZ expansion appear in Section 3, and the re-
sults on Hankel operators in Section 4. The final section, Section 5, concludes by
a small observation concerning the so-called balanced metrics on H in the sense of
Donaldson [8].
KEPLER MANIFOLD 5
Largepartofthisworkwasdonewhilethesecondnamedauthorwasvisitingthe
othertwo;thehospitalityofUniversiteAixMarseilleinthisconnectionisgratefully
acknowledged.
2. Reproducing kernels
Denote by , k =0,1,2,..., the space of (restrictions to H of) polynomials on
k
Cn+1 homogePneous of degree k. Clearly, is isomorphic to the quotient of the
k
analogousspaceof k-homogeneouspolynomPialson allofCn by the k-homogeneous
component of the ideal generated by z z, and
•
k+n 1 k+n 2
dim = − + − =:N(k).
k
P n 1 n 1
(cid:18) − (cid:19) (cid:18) − (cid:19)
For z M and f L2(Sn,dσ), where σ stands for the normalized surface measure
on Sn,∈set ∈
(6) fˆ(z):= f(ζ)e z,ζ dσ(ζ),
h i
ZSn
and let k k(Sn) denote the subspace in L2(Sn,dσ) consisting of spherical
H ≡ H
harmonics of degree k. It is then known [14] [27] that the functions x (x z)k,
z H, span k, the functions z (z ζ)k, ζ Sn, span , and th7→e map•ping
k
∈ H 7→ • ∈ P
f fˆisanisomorphismof k onto ,foreachk. UsingtheFunke-Hecketheorem
k
7→ H P
[19, p. 20], it then follows that [21] [20]
(z ξ)k
(7) (z w)k(ξ w)ldµ(w)=δ •
kl
Z∂M • • N(k)
for all z H, ξ Cn+1; and, consequently, and are orthogonal in L2(dµ) if
k l
∈ ∈ P P
k =l, while
6
f(z)
(8) f(w)(z w)kdµ(w)=
Z∂M • N(k)
for all z H and f .
k
If f is∈a function∈hPolomorphic on RM (for some 0 < R ), then it has a
≤ ∞
unique decomposition of the form
∞
(9) f = f , f ,
k k k
∈P
k=0
X
withthe sum convergingabsolutely anduniformly oncompactsubsets ofRM ([20,
Lemma 3.1]). Let s = (s ,s ,...) be an arbitrary sequence of positive numbers.
0 1
We denote by A2 the space of all functions f holomorphic in some RM, R > 0,
s
for which,
f 2:= s f 2 < ,
k ks kk kkL2(∂M,dµ) ∞
equipped with the natural innerXproduct
∞
f,g s := sk fk,gk L2(∂M,dµ)
h i h i
k=0
X
forf = f ,g = g asin(9). ItisimmediatethatA2isaHilbertspacewhich
k k k k s
containseach , andthe linearspanofthe latter(i.e. the spaceofallpolynomials
k
on H) isPdensePin AP2s.
6 HE´LE`NEBOMMIER-HATO,MIROSLAVENGLISˇ,ANDEL-HASSANYOUSSFI
Proposition 4. Assume that
N(k) 1/2k
Rs :=liminf − >0
k sk
→∞ (cid:12) (cid:12)
(the value Rs = being also allowed).(cid:12)(cid:12) Then(cid:12)(cid:12) A2s is a reproducing kernel Hilbert
space of holomorp∞hic functions on RsM, with reproducing kernel
∞ N(k)
(10) Ks(x,y)= (x y)k.
s •
k
k=0
X
Proof. By the definition of Rs, the series (10) converges pointwise and locally uni-
formlyfor x,y ∈RsM. Moreover,for g =Ks(·,y) weplainly havegk = Ns(kk)(·•y)k
and by (7), g 2 = N(k)(y y)k, so
k kkL2(∂M,dµ) s2k •
∞ N(k)
g 2s = (y y)k =Ks(y,y)< .
k k s • ∞
k
k=0
X
Thus Ks(,y) A2s for y RsM. Furthermore, for f A2s, by (8)
· ∈ ∈ ∈
f (y) N(k) f (w)(y w)kdµ(w)
k k
Xk | |≤Xk (cid:12) Z∂M • (cid:12)
= (cid:12)(cid:12)s f ,N(k)( y)k (cid:12)(cid:12)
k k
sk ·• L2(∂M,dµ)
Xk (cid:12)D E (cid:12)
(cid:12) (cid:12)
≤ sk(cid:12)kfkkL2(∂M,dµ) Ns(kk)(·•y)k (cid:12)L2(∂M,dµ)
Xk (cid:13) (cid:13)
s f 2 (cid:13)(cid:13) 1/2 N(cid:13)(cid:13)(k)( y)k 2 1/2
≤ kk kkL2(∂M,dµ) sk ·• L2(∂M,dµ)
(cid:16)Xk (cid:17) (cid:16)Xk (cid:13) (cid:13) (cid:17)
= f sKs(y,y)1/2, (cid:13)(cid:13) (cid:13)(cid:13)
k k
implying that the series kfk converges locally uniformly on RsM and that the
point evaluation f f(y) is continuous on A2s for all y RsM. Finally, removing
7→ P ∈
absolute values in the last computation shows that f(y) = f,Ks(,y) s, proving
that Ks is the reproducing kernel for A2s. h · i (cid:3)
Recall now that a sequence s=(sk)k N is called a Stieltjes moment sequence if
∈
it has the form
s =s (ν):= ∞rkdν(r)
k k
Z0
for some nonnegative measure ν on [0,+ ), called a representing measure for s;
∞
or, alternatively,
s = ∞t2kdρ(t)
k
Z0
for the nonnegative measure dρ(t)=dν(t2). These sequences have been character-
ized by Stieltjes in terms of a positive definiteness conditions. It follows from the
above integral representation that each Stieltjes moment sequence is either non-
vanishing, that is, s > 0 for all k, or else s = cδ for all k for some c 0.
k k 0k
≥
Fix a nonvanishing Stieltjes moment sequence s = (s ). By the Cauchy-Schwarz
k
KEPLER MANIFOLD 7
inequality we see that the sequence sk+1 is nondecreasing and hence converges as
sk
k + to the radius of convergence R2 of the series
s
→ ∞
∞ N(k)
zk, z C.
s ∈
k
k=0
X
We can now prove our first theorem from the Introduction.
ProofofTheorem1. RecallthatwedenotedR=sup t>0:t suppρ . IfR= ,
{ ∈ } ∞
then for any r > 0 we have q2k ≥ r∞t2kdρ(t) ≥ r2kρ((r,∞)) with ρ((r,∞)) > 0,
1/2k 1/2k
so liminf q r; thus q + . If R < , then the same argument
k→∞ 2k ≥ 1/2k 2k R → ∞ ∞
shows that liminf q r for any r <R, while from
k→∞ 2k ≥
R
q R2k dρ=R2kρ([0,R])
2k
≤
Z0
1/2k
with ρ([0,R]) < we get limsup q R. Setting s = q we thus get in
either case ∞ k→∞ 2k ≤ k 2k
R=Rs.
Now for any r<R and f A2 , we have by the uniform convergence of (9)
∈ ρ⊗µ
f 2d(ρ µ)= f f d(ρ µ)
j k
ZrM| | ⊗ j,k ZrM ⊗
X
r
= tj+kf (ζ)f (ζ)dµ(ζ)dρ(t)
j k
j,k Z0 Z∂M
X
r
= t2kdρ(t) f 2
k kkL2(∂M,dµ)
Xk (cid:16)Z0 (cid:17)
by the orthogonality of and for j =k. Letting r R, we thus get
k l
P P 6 ր
kfk2ρ⊗µ = q2kkfkk2L2(∂M,dµ) =kfk2s.
k
X
Hence A2 A2, with equal norms. Since clearly A2 for each l and the
fsoprananoyffPρ⊗l µisA⊂d2enss.eTinheAc2sla,iimt fnoollwowfsoltlohwatsafrcotmualtlhyeAla2ρPs⊗tlµp∈=ropAρo⊗2ssµiatniodn.kfkρ⊗µ = kfk(cid:3)s
∈ ρ⊗µ
Asanexample,letφbeanonnegativeintegrablefunctionon(0, ),andconsider
∞
the volume element
α(z) α(z)
α (z):=φ(z 2) ∧ .
φ | | ( 1)n(n+1)/2(2i)n
−
Let A2 and K be the correspondingweightedBergmanspace andits reproducing
φ φ
kernel, respectively.
Theorem 5. We have
1
Kφ(z,w)= 2tF(n−1)(t)+(n 1)F(n−2)(t) ,
(n 1)!cM − t=z w
− h i •
where
α(z) α(z)
cM =(n 1) ∧
− M ( 1)n(n+1)/2(2i)n
Z −
8 HE´LE`NEBOMMIER-HATO,MIROSLAVENGLISˇ,ANDEL-HASSANYOUSSFI
and
(11) F(t)= ∞ tk, where c := ∞tkφ(t)dt.
k
c
k=0 k Z0
X
Proof. It was shown in [20, Lemma 2.1] that for any measurable function f on H,
f(z) α(z)∧α(z) =2cM ∞ f(tζ)t2n−3dµ(ζ)dt,
ZH (−1)n(n+1)/2(2i)n Z0 Z∂M
with cM as above. Thus αφ =ρ µ for
⊗
dρ(t)=2cMt2n−3φ(t2)dt,
and by the last theorem, K is given by (10) with
φ
sk = ∞t2kdρ(t)=cM ∞tk+n−2φ(t)dt=cMck+n 2.
Z0 Z0 −
Now by an elementary manipulation,
k+n 1 tk 1 d n 1tk+n 1
− = − −
n 1 c (n 1)! dt c
Xk (cid:18) − (cid:19) k+n−2 − Xk (cid:16) (cid:17) k+n−2
1 d n 1
= − (tF(t))
(n 1)! dt
− (cid:16) (cid:17)
1
= tF(n 1)(t)+(n 1)F(n 2)(t) ,
− −
(n 1)! −
− h i
and similarly
k+n 2 tk t d n 1tk+n 2
− = − −
n 1 c (n 1)! dt c
Xk (cid:18) − (cid:19) k+n−2 − Xk (cid:16) (cid:17) k+n−2
t
= F(n 1)(t).
−
(n 1)!
−
Thus
N(k)tk 1
= 2tF(n 1)(t)+(n 1)F(n 2)(t)
− −
c (n 1)! −
k+n 2
Xk − − h i
and the assertion follows. (cid:3)
Example. Take φ(r) = (1 r)m for r [0,1] and φ(r) = 0 for r > 1, where
− ∈
m> 1. Then c = k!Γ(m+1) and we recover the formula from [20]
− k Γ(m+k+2)
Γ(n+m) (n 1)+(n+1+2m)z w
K (z,w)= − •
φ (n 1)!cMΓ(m+1) (1 z w)n+m+1
− − •
for the “standard” Bergman kernels on M (with respect to α α).
∧
Further applications of Theorem 1 will occur later in the sequel.
We conclude this section by considering the unit ball B in Cn with respect to
the “minimal norm” given by
N (z)= z 2+ z z .
∗ | | | • |
This norm was shown to be of interestpin the study of several problems related to
proper holomorphic mappings and the Bergmankernel, see [13], [22], [21], [20] and
[18].
KEPLER MANIFOLD 9
Suppose that φ is as before and consider the measure dV on Cn with density
φ
φ(N2) with respect to the Lebesgue measure. Namely,
∗
dV (z):=φ(z 2+ z z )dV(z)
φ
| | | • |
where dV(z) denotes the Lebesgue measure on Cn normalized so that the volume
of the minimal ball is equal to one. We denote by A2(Cn) the Bergman-Fock type
φ
space on Cn with respect to dV , consisting of all measurable functions f on Cn
φ
which are holomorphic in the ball
B := z Cn : z 2+ z z <R
φ φ
{ ∈ | | | • | }
and satisfy p
(12) f 2 := f(z)2dV (z)<+ .
k kφ ZCn| | φ ∞
Here R is the square root of the radius of convergence of the series (11). We let
φ
L2(Cn)denotethespaceofallmeasurablefunctionsf inCn verifying(12). Finally,
φ
wedefine the operators∆ , j =0,1,actingonpowerseriesin z by their actionson
j
the monomials zm as follows
[m−1]
2 m
(∆ zm)(x,y):=2 xm 1 2kyk,
0 − −
2k+1
k=0 (cid:18) (cid:19)
X
[m]
2 m
(∆1zm)(x,y):=2 xm−2kyk, x,y C.
2k ∈
k=0(cid:18) (cid:19)
X
If f(z)= c zk is a power series of radius of convergence R, then the series
k k
P (∆ f)(x,y):= c (∆ zk)(x,y2)
j k j
k
X
converges as long as x + y <R and we have
| | | |
f(x+y) f(x y)
(∆ f)(x,y)= − − , y =0,
0
y 6
(∆ f)(x,y)=f(x+y)+f(x y).
1
−
Using these notations, we then have the following.
Theorem 6. The space A2(Cn)coincides with theclosureof theholomorphic poly-
φ
nomials in L2(Cn) and its reproducing kernel is given by
φ
(n+1)2
Kφ,Cn(z,w)= 2(z w)∆0(F(n−1))(z w,z z w w)
(n 1)!cM • • • · •
− h
+∆1(F(n−1))(z w,z z w w)
• • · •
+(n 1)∆ (F(n 2))(z w,z z w w) ,
0 −
− • • · •
i
with F as in (11).
Proof. We will use the technique developed in [20]. Let Pr : Cn+1 Cn be the
→
projection onto the first n coordinates
Pr(z ,...,z ,z )=(z ,...,z )
1 n n+1 1 n
10 HE´LE`NEBOMMIER-HATO,MIROSLAVENGLISˇ,ANDEL-HASSANYOUSSFI
andι:=Pr H. Thenι:H Cn 0 isaproperholomorphicmappingofdegree2.
| → \{ }
The branching locus of ι consists of points with z = 0, and its image under ι
n+1
consists of all x Cn 0 with n x2 =0. The local inverses Φ and Ψ of ι are
∈ \{ } j=1 j
given for z Cn 0 by
∈ \{ } P
Φ(z)=(z,i√z z),
•
Ψ(z)=(z, i√z z).
− •
In view of Lemma 3.1 of [21], we see that
1+n
Φ (α )= φ(z)1/2( 1)ndz dz ,
∗ φ 1 n
i√z z − ∧···∧
•
1+n
Ψ (α )= φ(z)1/2( 1)ndz dz .
∗ φ 1 n
i√z z − ∧···∧
− •
Iff :Cn Cisameasurablefunctionandz H,weconsidertheoperatorU =U
φ
→ ∈
by setting
z
n+1
(Uf)(z):= (f ι)(z).
√2(n+1) ◦
Using the same arguments as in the proof of Lemma 4.1 in [20], it can be shown
that U is an isometry from L2(Cn) into L2(H). More precisely, we have
φ φ
Uf(z)2α (z)= f(z)2dV (z).
φ φ
ZH| | ZCn| |
In addition, the arguments used in the proof of part (2) of the latter lemma show
that the image (H) of A2(Cn) under U is a closed proper subspace of A2(H),
Eφ φ φ
and U is unitary from A2(Cn) onto (H). From the technique used in the proof
φ Eφ
of Lemma 2.4 in [20], we the get the following lemma.
Lemma 7. If Φ and Ψ are as before, then
K (z,Φ(w)) K (z,Ψ(w))
zn+1Kφ,Cn(ιz,w)=(n+1)2 φ + φ ,
" Φn+1(w) Ψn+1(w) #
for all z H, w Cn.
∈ ∈
The rest of the proof of the theorem now follows from the last lemma and the
identities used in the proof of Theorem A in [20]. (cid:3)
Example. Let φ(r)=e cr, c>0. Then s =k!/ck+1, F(t)=cect and we obtain
− k
2(n+1)2cn 2
−
Kφ,Cn(z,w)=
(n 1)!cM ×
−
ecz w 2c2(n 1+z w)S(c2z zw w)+cC(c2z zw w) ,
•
− • • • • •
where we wrote for brev(cid:2)ity S(t)= sinh√t, C(t)=cosh√t. (cid:3)
√t
3. TYZ expansions
Proof of Theorem 2. For the measure (4), we have by the change of variable x =
st2m,
q =2cm ∞t2ke st2mt2mn 1dt= c ∞ x 2k+2m2mn−1e xdx=cΓ(k+mmn).
2k Z0 − − sZ0 (cid:16)s(cid:17) − sk+mmn