Norm estimates for the Bergman and Cauchy-Szeg\"o projections over the Siegel upper half-space

0.26 MB·

by  Congwen Liu

NORM ESTIMATES FOR THE BERGMAN AND
CAUCHY-SZEGO¨ PROJECTIONS OVER THE SIEGEL UPPER
7 HALF-SPACE
1
0
2 CONGWENLIU

n
a Abstract. We obtain several estimates for the Lp operator norms of the
J Bergman and Cauchy-Szego¨ projections over the the Siegel upper half-space.

5 Asaby-product,wealsodeterminetheprecisevalueoftheLp operatornorm
1 ofafamilyofintegraloperatorsovertheSiegelupperhalf-space.

]
V
C
1. Introduction
.
h Let n be the Siegel upper half-space (or the generalized half-plane, following
t U
a the terminology of Kor´anyi [11, 12, 13, 14])
m
n := z Cn+1 :Imz > z 2
n+1 ′
[ U ∈ | |
and let b n be its boundary in(cid:8)Cn+1. Here and throughou(cid:9)t, we use the notation
2 U
v z =(z ,z ), where z =(z ,...,z ) Cn and z C1.
′ n+1 ′ 1 n n+1
9 ∈ ∈
Note that 0 =C := z C:Imz >0 , the classical upper half-plane. n is bi-
7 +
9 holomorphUicallyequival{ent∈totheunitba}llBn+1 inCn+1,viatheCayleytrUansform
1 Φ:B n given by
n+1
0 →U
z 1 z
. ′ n+1
1 (z′,zn+1) ,i − ,
7−→ 1+z 1+z
0 (cid:18) n+1 n+1(cid:19)
7 and so it is also referred to as the unbounded realization of the unit ball in Cn+1.
1 As usual, for p>0, the space Lp( n) consists of all Lebesgue measurable func-
: U
v tions f on n for which
i U
X 1/p
f := f(z)pdV(z)
r k kp | |
a (cid:26)Z (cid:27)
n
U
is finite, where dV = dm is the Lebesgue measure on Cn+1. The Bergman
2n+2
space Ap( n) is the closedsubspace ofLp( n) consisting of holomorphicfunctions
U U
on n. TheorthogonalprojectionfromL2( n)ontoA2( n),knownastheBergman
U U U
projection, can be expressed as an integral operator:
(P nf)(z)= K n(z,w)f(w)dV(w),
U U
Zn
U
2010 Mathematics Subject Classification. Primary32A35,47G10; Secondary32A26,30E20.
Key words and phrases. Siegel upper half-space; Bergman projection; Cauchy-Szego¨ projec-
tion;normestimates.
This work was supported by the National Natural Science Foundation of China grants
11571333, 11471301.
1
2 CONGWENLIU
with the Bergman kernel
n 2
(n+1)! i − −
(1.1) KUn(z,w)= 4πn+1 2(w¯n+1−zn+1)−z′·w′ .
(cid:20) (cid:21)
See [7, p.56, Lemma 5.1]. In the sequel, we shall use the notation
i
ρ(z,w) := (w¯ z ) z w .
2 n+1− n+1 − ′· ′
It has been long known that the Bergman projection P n extends to a bounded
operatorfromLp( n)toAp( n),for1<p< . See,forUinstance,[2,Lemma2.8].
U U ∞
In this paper, we are concerned with estimates of the operator norm of P n on
Lp( n). Our first main result is the following. U
U
Theorem 1. For 1<p< , we have
∞
Γ n+2 Γ n+2
p q (n+1)! π
(1.2) (cid:16) Γ2(cid:17)n+(cid:16)2 (cid:17) ≤ kPUnkp→p ≤ Γ2 n+2 sinπ,
2 2 p
where q := p is the(cid:0)conju(cid:1)gate exponent of p and (cid:0) (cid:1)
p 1
−
kPUnkp→p :=kPUnkLp(Un)→Ap(Un) =sup(cid:26)kPkUfnkfpkp :f ∈Lp(Un),f 6=0(cid:27).
This is motivated by recent work of Zhu [30], Dostani´c [4] and the author of
the present paper [18], in which sharp estimates for the norm of the Bergman
projection over the unit ball of Cn were obtained. It is also worth mentioning
that, in the recent years, there has been increasing interest in the study of the
size of Bergman projection in various context other than the Bergman space. See
[9, 10, 22, 23, 24, 28].
In the course of proving Theorem 1, we will precisely evaluate the Lp operator
norm of a family of integral operators as follows. For α> 1, we define
−
(n+1)! ρ(w,w)α
(T f)(z):= f(w)dV(w),
α 4πn+1 ρ(z,w)n+2+α
Zn | |
U
for,say,continuousf ofcompactsupport. It is a bounded mapofLp( n) to itself,
U
as [2, Lemma 2.8] shown. Our second main result is the following.
Theorem 2. If 1 p< and p(1+α)>1 then
≤ ∞
(n+1)!Γ 1+α 1 Γ 1
− p p
(1.3) T = .
k αkp→p Γ(cid:16)2 n+2+α (cid:17) (cid:16) (cid:17)
2
Notethat P n p p T0 p p,andhenc(cid:0)ethese(cid:1)condinequalityin(1.2)follows
k U k → ≤k k →
immediately from (1.3), together with the well-known formula
π
Γ(x)Γ(1 x)= .
− sin(πx)
We also remark that when n = 0, Theorem 2 gives an affirmative answer to a
conjecture ofDostanic in [5] (a partialanswer to this conjecture was givenin [21]).
THE BERGMAN AND CAUCHY-SZEGO¨ PROJECTIONS 3
Recall that the Berezin transform over n is defined by
U
K n(z,w)2
(B nf)(z) := | U | f(w)dV(w)
U K n(z,z)
Zn U
U
(n+1)! ρ(z,z)n+2
= f(w)dV(w),
4πn+1 ρ(z,w)2(n+2)
Zn | |
U
which plays important roles in Berezin’s theory of quantization as well as in the
theoryof Toeplitz operators. Note that B n =Tn∗+2, the adjointof Tn+2. Another
U
immediate consequence of Theorem 2 is the following.
Corollary 3. If 1<p then
≤∞
n+1
π 1
kBUnkp→p = psinπ 1+ kp .
p k=1(cid:18) (cid:19)
Y
When p= , the quantity on the right hand side should be interpreted as 1.
∞
This is an analogue of the main results in [3] and [20].
Our third main result concerns the Cauchy-Szeg¨o projection over b n. For f
U
holomorphic on n, we define
U
1
p
f Hp :=sup f(u+ti)pdβ(u) ,
k k t>0( | | )
Z
b n
U
where i=(0,i) Cn+1 and the measure dβ on b n is defined by the formula
′
∈ U
fdβ = f(z ,t+iz 2)dz dt,
′ ′ ′
| |
bZn CnZR
U ×
for (say) continuous f of compact support. See Section 2.1 below. Then we set
Hp( n):= f :f is holomorphic on n, f Hp < ,
U { U k k ∞}
which is the analogue for n of the classicalHardy space Hp of holomorphic func-
U
tions in the upper half-plane.
The space H2( n) can be identified with the closed subspace of L2(b n) con-
U U
sisting of functions fb that are boundary values of functions f H2( n), so
{ } ∈ U
there exists an orthogonal projection from L2(b n) onto H2( n). We denote this
U U
projection by C n and call it the Cauchy-Szeg¨o projection. It may also be written
U
as the Cauchy-Szeg¨o integral
(C nf)(z)= S n(z,u)f(u)dβ(u), z n,
U U ∈U
Z
b n
U
where
n!
(1.4) S n(z,w)= ρ(z,w)−n−1.
U 4πn+1
See [7, p.61, Proposition 5.3] or [27, p.536, Theorem 1]. A classical theorem by
Kor´anyi and Va´gi [13, Theorem 6.1] asserts that the Cauchy-Szeg¨o projection has
an extension to a bounded operator from Lp(b n) onto Hp( n), for 1<p< .
U U ∞
Our third main result gives a lower bound for the operator norm of C n.
U
4 CONGWENLIU
Theorem 4. For all 1<p< ,
∞
Γ n+1 Γ n+1
p q
(1.5) kCUnkLp(bUn)→Hp(Un) ≥ (cid:16) Γ2(cid:17)n+(cid:16)1 (cid:17),
2
where q := p is the conjugate exponent of p. (cid:0) (cid:1)
p 1
−
We shall deduce Theorem 4 from Theorem 1, with the help of the following
inequality, which makes a connection between the norms of the two operators and
might be of independent interest.
Theorem 5. For all 1<p< ,
∞
(1.6) kCUnkLp(bUn)→Hp(Un) ≥ kPUn−1kLp(Un−1)→Ap(Un−1).
The above results suggest the following.
Conjecture 6. For all 1<p< , we have
∞
Γ n+2 Γ n+2
p q
(1.7) kPUnkLp(Un)→Ap(Un) = (cid:16) Γ2(cid:17)n+(cid:16)2 (cid:17)
2
and (cid:0) (cid:1)
Γ n+1 Γ n+1
p q
(1.8) kCUnkLp(bUn)→Hp(Un) = (cid:16) Γ2(cid:17)n+(cid:16)1 (cid:17),
2
where q := p is the conjugate exponent of p. (cid:0) (cid:1)
p 1
−
Note that when n=0, the conjectured (1.8) reads
1
(1.9) kCC+kLp(R)→Hp(C+) = sinπ.
p
ThiscoincideswithavariantoftheGohberg-Krupnikconjecture,whichwasproved
by Hollenbeck and Verbitsky [8] only in 2000. For the proof of (1.9), see [17,
p.373]. This provides a support for our conjecture. It is also noteworthy that the
conjectured(1.8)wouldimply (1.7),inviewofTheorems1and5. Seealso[18]and
[19] for their counterparts in the setting of the unit ball.
The rest of the paper is organized as follows: In Section 2 we review some
definitions and basic facts, and in Section 3 we establish several technical lemmas,
someofthemmightbeofindependentinterest. Sections4isdevotedtotheproofof
Theorem2. Ourfirstmainresult,Theorem1willbeprovedinSections5. Sections
6 is devoted to the proof of Theorem 5.
Acknowledgements. The author is indebted to Lifang Zhou for correcting two
errors in an earlier version of the paper, and to Guangbin Ren and Xieping Wang
for many helpful comments.
2. Preliminaries
2.1. Heisenberg group. We recall the definition of the Heisenberg group and
some basic facts which can be found in [27, Chapter XII] and [16, Chapter 9].
We denote by Hn the Heisenberg group, that is, the set
Cn R= [ζ,t]:ζ Cn,t R
× { ∈ ∈ }
THE BERGMAN AND CAUCHY-SZEGO¨ PROJECTIONS 5
endowed with the Heisenberg group operation
[ζ,t] [η,s]=[ζ+η,t+s+2Im(ζ η¯)].
· ·
Here we shall use square brackets [ ] for elements of the Heisenberg group to
distinguish them from points in Cn+1, for which parentheses () are used. The
identity element is [0,0] and the inverse of [ζ,t] is [ζ,t] 1 = [ ζ, t]. The Haar
−
measure dh on Hn is the usual Lebesgue measure dζdt on Cn −R−, here we write
h = [ζ,t], ζ Cn,t R. (To be more precise, dζ = dηdξ i×f ζ = η +iξ with
η,ξ Rn.) ∈ ∈
T∈o eachelement h=[ζ,t] ofHn, we associatethe following (holomorphic)affine
self-mapping of n:
U
(2.1) h: (z ,z ) (z +ζ,z +t+2iz ζ¯+iζ 2).
′ n+1 ′ n+1 ′
7−→ · | |
The mappings (2.1) are simply transitive on the boundary b n of n, so we can
U U
identify the Heisenberg group with b n via its action on the origin
U
Hn [ζ,t] (ζ,t+iζ 2) b n.
∋ 7−→ | | ∈ U
This identificationallowsus to transportthe Haarmeasure dh onHn to a measure
dβ on b n; that is, we have the integration formula
U
(2.2) fdβ = f(z ,t+iz 2)dz dt,
′ ′ ′
| |
bZn CnZR
U ×
for (say) continuous f of compact support. The measure dβ is invariant under the
action of Hn, that is, dβ(h(z))=dβ(z) for each h Hn.
∈
We shall make frequent use of the following Fubini’s theorem on n:
U
∞
(2.3) fdV = f(u+ti)dβ(u)dt,
Zn Z0 bZn
U U
whichisvalidfor,say,continuousf ofcompactsupport. Thiscanbeeasilyverified
by substituting (2.2) into the right hand side of (2.3)
Finally, it is easy to verify that
(2.4) ρ(h(z),h(w))=ρ(z,w),
for each h Hn, and for all z n, w b n.
∈ ∈U ∈ U
2.2. Cayleytransform. RecallthattheCayleytransformΦ:B nisgiven
n+1
→U
by
z 1 z
′ n+1
(z′,zn+1) ,i − .
7−→ 1+z 1+z
(cid:18) n+1 (cid:18) n+1(cid:19)(cid:19)
It is immediate to calculate that
2iz i z
Ψ:=Φ−1 :(z′,zn+1) ′ , − n+1 .
7−→ i+z i+z
(cid:18) n+1 n+1(cid:19)
Again, we refer to [27, Chapter XII] and [16, Chapter 9] for the properties of
thesetwomappings. Fortheconvenienceoflaterreference,werecordthe following
lemma.
Lemma 7. The mappings Φ and Ψ have the following elementary properties:
6 CONGWENLIU
(i) The identity
1 ξ η
(2.5) ρ(Φ(ξ),Φ(η))= − ·
(1+ξ )(1+η )
n+1 n+1
holds for all ξ,η B .
n+1
(ii) The real Jacobian∈of Φ at ξ B is
n+1
∈
4
(2.6) (J Φ)(ξ)= .
R 1+ξ 2(n+2)
n+1
| |
(iii) The identity
ρ(z,w)
(2.7) 1 Ψ(z) Ψ(w)=
− · ρ(z,i)ρ(i,w)
holds for all z,w n, where i=(0,i).
′
∈U
(iv) The identity
ρ(z,z)
(2.8) Ψ(z)2 =1
| | − ρ(z,i)2
| |
holds for all z n.
∈U
(v) The real Jacobian of Ψ at z n is
∈U
1
(2.9) (J Ψ)(z)= .
R 4ρ(z,i)2(n+2)
| |
Note that the mappings Φ and Ψ extend also to the boundaries of the domains
B and n. Thus, b n correspondsviaΨtothe unitsphereS ,exceptforthe
n+1 n+1
U U
“south pole” (0, 1). It is easy to check that the identity
′
−
1 Ψ(z) Ψ(w)
(2.10) ρ(z,w)= − ·
(1+[Ψ(z)] )(1+[Ψ(w)] )
n+1 n+1
holds for all z n and w b n.
Finally, writ∈inUg dσ for th∈e nUormalized surface measure on the unit sphere S ,
n+1
one has the following change of variables formula (see [27, p.575, 7.2(b)]):
4πn+1 f(Φ(ξ))
(2.11) fdβ = dσ(ξ).
n! 1+ξ 2(n+1)
bUZn SnZ+1 | n+1|
2.3. Mo¨bius transformations. The group of all one-to-one holomorphic map-
pingsofB ontoB (the so-calledautomorphismsofB )willbe denotedby
n+1 n+1 n+1
Aut(B ). It is generatedby the unitary transformationsonCn+1 along with the
n+1
Mo¨bius transformations ϕ given by
ξ
ξ Pξη (1 ξ 2)21Qξη
ϕ (η):= − − −| | ,
ξ
1 η ξ
− ·
where ξ B , P is the orthogonal projection onto the space spanned by ξ, and
n+1 ξ
∈
Q η =η P η.
ξ ξ
−
It is easily shown that the mapping ϕ satisfies
ξ
ϕ (0)=ξ, ϕ (ξ)=0, ϕ (ϕ (η))=η.
ξ ξ ξ ξ
THE BERGMAN AND CAUCHY-SZEGO¨ PROJECTIONS 7
Furthermore, for all η,ω B ,
n+1
∈
(1 ξ 2)(1 η ω)
(2.12) 1 ϕ (η) ϕ (ω) = −| | − · ,
ξ ξ
− · (1 η ξ)(1 ξ ω)
− · − ·
and in particular,
1 ξ 2
(2.13) 1 ϕ (η) ξ = −| | .
ξ
− · 1 η ξ
− ·
Finally, an easy computation shows that
(1 η ϕ (ω))(1 ξ ω)
ξ
(2.14) 1 ϕ (η) ω = − · − ·
ξ
− · 1 η ξ
− ·
holds for all η,ω B .
n+1
∈
The best general reference here is [26, Chapter 2].
2.4. Hypergeometric functions. We use the classical notation
a, b ∞ (a)k(b)k λk
F ; λ :=
2 1 c (c) k!
(cid:20) (cid:21) k=0 k
X
with c=0, 1, 2,..., where
6 − −
(a) =1, (a) =a(a+1)...(a+k 1) for k 1.
0 k
− ≥
denotes the Pochhammer symbol of a. This series gives an analytic function for
λ <1, called the Gauss hypergeometric function associated to (a,b,c).
| |
We refer to [1, Chapter 2] for the properties of these functions. Here, we only
record three formulas for later reference.
a, b Γ(c)Γ(c a b)
(2.15) F ; 1 = − − , Re(c a b)>0.
2 1 c Γ(c a)Γ(c b) − −
(cid:20) (cid:21) − −
a, b c a, c b
(2.16) 2F1 c ; λ = (1−λ)c−a−b2F1 − c − ; λ .
(cid:20) (cid:21) (cid:20) (cid:21)
a, b Γ(c) 1
F ; λ = tb 1(1 t)c b 1(1 tλ) adt,
2 1 c Γ(b)Γ(c b) − − − − − −
(cid:20) (cid:21) − Z0
(2.17) Rec>Reb>0; arg(1 λ) <π.
| − |
2.5. Schur’s test. The following lemma, usually called Schur’s test, is one of the
most commonly used results for proving the Lp-boundedness of integraloperators.
See, for example, [31].
Lemma 8. Suppose that (X,µ) is a σ-finite measure space and Q(x,y) is a non-
negative measurable function on X X and T is the associated integral operator
×
Tf(x)= Q(x,y)f(y)dµ(y).
ZX
Let 1 < p < and 1 + 1 = 1. If there exist a positive constant C and a positive
∞ p q
measurable function g on X such that
Q(x,y)g(y)qdµ(y) Cg(x)q
≤
ZX
8 CONGWENLIU
for almost every x in X and
Q(x,y)g(x)pdµ(x) Cg(y)p
≤
ZX
for almost every y in X, then T is bounded on Lp(X,µ) with T C.
k k≤
3. Technical lemmas
3.1. An elementary inequality.
Lemma 9. Let a,b R and c>max a+b+1,a+1,b . Then
∈ { }
a, b a, b
(3.1) F ; λ F ; 1 C(a,b,c) 1 λ
2 1 c − 2 1 c ≤ ·| − |
(cid:12) (cid:20) (cid:21) (cid:20) (cid:21)(cid:12)
for all λ D,(cid:12)(cid:12)where (cid:12)(cid:12)
∈ (cid:12) (cid:12)
Γ(c)Γ(c a b 1)
C(a,b,c):= a max 2−a−1, − − − .
| |· Γ(c a 1)Γ(c b)
(cid:26) − − − (cid:27)
Proof. Since c>b and c a b>0, by (2.17), we have
− −
a, b a, b
F ; λ F ; 1
2 1 c − 2 1 c
(cid:12) (cid:20) (cid:21) (cid:20) (cid:21)(cid:12)
(3.2) (cid:12)(cid:12)(cid:12) Γ(c) 1tb−1(1 t)c−(cid:12)(cid:12)(cid:12)b−1 (1 tλ)−a (1 t)−a dt
≤ Γ(b)Γ(c b) − − − −
− Z0
for all λ D. By the mean value theorem we hav(cid:12)(cid:12)e (cid:12)(cid:12)
∈
(1 tλ) a (1 t) a sup a(1 ϑ) a 1 1 λ
− − − −
− − − ≤ ϑ tD − ·| − |
(cid:12)(cid:12) (cid:12)(cid:12) a∈ m(cid:12)(cid:12) ax 2−a−1,(1(cid:12)(cid:12) t)−a−1 1 λ.
≤ | |· − | − |
Substituting this into (3.2) yields the desired(cid:8)inequality (3.1). (cid:9) (cid:3)
3.2. A Forelli-type formula. Thefollowinglemmadealswithintegrationonb n
U
of functions of fewer variables, which is in the same spirit as a result of Forelli [6,
p.383] (see also [26, p.14] or [29, p.10, Lemma 1.9]).
Suppose 0 k n and let k be the Siegel upper half-space in Ck+1. It is
≤ ≤ U
convenient to let
k
ρ (w) := Imw w 2
k k+1− | j|
j=1
X
We think of ρ as a “height function” in k. Note that
k U
ρ (z) = ρ(z,z) = Imz z 2.
n n+1−| ′|
Lemma 10. Suppose 0 k <n and f is a function on b n that depends only on
≤ U
z , ,z . Then f can be regarded as defined on k and
n k+1 n+1
− ··· U
πn k
(f Π )dβ = − ρ (w)n k 1f(w)dm (w),
◦ k (n k 1)! k − − 2k+2
bZn − − Zk
U U
where Π is the orthogonal projection of Cn+1 onto Ck+1 given by
k
(z , ,z ) (z , ,z ),
1 n+1 n k+1 n+1
··· 7−→ − ···
THE BERGMAN AND CAUCHY-SZEGO¨ PROJECTIONS 9
and
πn k
−
c := .
n,k
(n k 1)!
− −
In particular when k =n 1, this reads
−
(3.3) (f Π )dβ =π f(w)dm (w).
n 1 2n
◦ −
Z Z
b n n−1
U U
Proof. For convenience, we use the notation z =(z ,z ), where
† ‡
z =(z ,...,z ) Cn k and z =(z , ,z ) Ck+1.
† 1 n k − ‡ n k+1 n+1
− ∈ − ··· ∈
By an approximation argument, it suffices for us to prove the result when f is
continuous in Ck+1 and has support in z Ck+1 :ρ (z )>r for some r >0.
‡ ∈ k ‡ 0 0
Fix such an f and consider the integrals
(cid:8) (cid:9)
I(r)= (f Π )dV, 0<r< .
k
◦ ∞
Z
ρ (z)>r
{ n }
By Fubini’s theorem (2.3), we have
∞
I(r)= (f Π )(u+ti)dβ(u) dt.
k
( ◦ )
Z Z
r b n
U
We then differentiate this to obtain
(3.4) I (0)= (f Π )(u)dβ(u).
′ k
− ◦
Z
b n
U
On the other hand, if 0 < r < r , an application of the classical Fubini’s theorem
0
shows that
I(r) = (f Π )(z ,z )dm (z ) dm (z )
k † ‡ 2n 2k † 2k+2 ‡
( ◦ − )
Z Z
ρ (z‡)>r z†2<ρ (z‡) r
{ k } {| | k − }
= (nπn−kk)! ρk(z‡)−r n−kf(z‡)dm2k+2(z‡)
Z −
{ρk(z‡)>r} (cid:0) (cid:1)
= (nπn−kk)! ρk(z‡)−r n−kf(z‡)dm2k+2(z‡),
− Z
k (cid:0) (cid:1)
U
where the last equality follows from the assumption that f is supported in z
‡
Ck+1 :ρ (z )>r . Differentiation then gives { ∈
k ‡ 0}
πn k
(3.5) I′(0)=−(n k− 1)! ρk(z‡)n−k−1f(z‡)dm2k+2(z‡).
− − Z
k
U
Comparison of (3.4) and (3.5) gives (10). (cid:3)
Corollary11. Letf beafunctionofonecomplexvariable. Then, foranyz b n,
∈ U
we have
πn
(3.6) f(2iρ(w,z))dβ(w) = f(λ)(Imλ)n−1dm2(λ).
(n 1)!
bUZn − CZ+
10 CONGWENLIU
Proof. Let z b n be fixed. We put h := [ z , z + iz 2] Hn. Then
′ n+1 ′
∈ U − − | | ∈
h(z)=0. Hence, by (2.4), we see that
f(2iρ(w,z))dβ(w) = f(2iρ(h(w),0))dβ(w)
Z Z
b n b n
U U
= f(2iρ(u,0))dβ(u) = f(u )dβ(u),
n+1
Z Z
b n b n
U U
where the second equality follows from the Hn-invariance of dβ. Finally, an appli-
cation of Lemma 10 (with k =0) completes the proof. (cid:3)
Corollary 12. Suppose 1 k<n. Then for f L1(b n) we have
≤ ∈ U
fdβ =cn,k f( ρk(w)η,w)dσn k(η) ρk(w)n−k−1dm2k+2(w).
( − )
bUZn UZk SnZ−k p
In particular, when k =n 1,
−
2π
1
(3.7) fdβ =π f( ρ (w)eiθ,w)dθ dm (w).
(2π n−1 ) 2n
bZn nZ−1 Z0 q
U U
Proof. As in the proof of Lemma 10, we write z = (z ,z ), where z Cn k and
† ‡ † −
z Ck+1. Then ∈
‡
∈
fdβ = f(z†,z‡)dβ(z).
Z Z
b n b n
U U
Note that if z = (z ,z ) b n then z 2 = ρ (z ). Since dβ(g(z)) = dβ(z) for
every unitary linear†tra‡ns∈formUation g |on†|Cn, whkere‡ g((z ,z )):=(g(z ),z ),
′ n+1 ′ n+1
f(z†,z‡)dβ(z)= f(|z†|η,z‡)dβ(z)= f( ρk(z‡)η,z‡)dβ(z)
bZn bZn bZn q
U U U
for any fixed η S . Integrating over η S and applying Fubini’s theorem,
n k n k
∈ − ∈ −
we obtain
fdβ = f( ρk(z‡)η,z‡)dσn k(η) dβ(z).
( − )
bUZn bUZn SnZ−k q
The inner integral above defines a function that only depends on the last k +1
variables. Therefore, an application of Lemma 10 completes the proof. (cid:3)
3.3. Evaluation of some integrals.
Lemma 13. Let θ >0 and γ > 1. The identities
−
dβ(u) 4πn+1Γ(θ)
(3.8) = ρ(z,z) θ
−
ρ(z,u)n+1+θ Γ2 n+1+θ
bZn | | 2
U
and (cid:0) (cid:1)
ρ(w,w)γdV(w) 4πn+1Γ(1+γ)Γ(θ)
(3.9) = ρ(z,z) θ
−
Zn |ρ(z,w)|n+2+θ+γ Γ2 n+2+2θ+γ
U
hold for all z n. (cid:16) (cid:17)
∈U