berezin's operator calculus and higher order schwarz–pick lemma

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Berezin's operator calculus and higherorder Schwarz–Pick lemmaBo Li a & Yifei Pan ba Department of Mathematics, The University of Toledo, Toledo,OH 43606, USAb Department of Mathematical Sciences, Indiana University-Purdue University Fort Wayne, Fort Wayne, IN 46805, USAVersion of record first published: 25 Feb 2011.

To cite this article: Bo Li & Yifei Pan (2012): Berezin's operator calculus and higher orderSchwarz–Pick lemma, Complex Variables and Elliptic Equations: An International Journal, 57:5,523-538

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Complex Variables and Elliptic EquationsVol. 57, No. 5, May 2012, 523–538

Berezin’s operator calculus and higher order Schwarz–Pick lemma

Bo Lia* and Yifei Panb

aDepartment of Mathematics, The University of Toledo, Toledo, OH 43606, USA;bDepartment of Mathematical Sciences, Indiana University-Purdue University Fort

Wayne, Fort Wayne, IN 46805, USA

Communicated by Massimo Lanza de Cristoforis

(Received 23 December 2009; final version received 12 May 2010)

We provide a new and simple proof to the result in our study of Berezin’soperator calculus [B. Li, The Berezin transform and mth order Bergmanmetric, Trans. Amer. Math. Soc. (to appear)] that the mth order Bergmanmetric (Bm@v)(z) is a constant multiple of {(B1@v)(z)}

m on the unit ball,where (B1@v)(z) is the classical Bergman metric. Based on the reproducing-kernel theory, an approximation approach is developed to treat (Bm@v)(z)on the unit ball and C

n in a uniform way. Secondly, we discuss the interplaybetween our analysis in Berezin’s operator calculus and the higher orderSchwarz–Pick lemma in Dai et al. [S. Dai, H. Chen, and Y. Pan, TheSchwarz–Pick lemma of high order in several variables, Michigan Math. J.(to appear)]. As a consequence, the mth order Caratheodory–Reiffen metric(Cm@v)(z) is shown to be a constant multiple of {(C1@v)(z)}

m also on the unitball, where (C1@v)(z) is the classical infinitesimal Caratheodory–Reiffenmetric.

Keywords: Berezin transform; Schwarz–Pick lemma

AMS Subject Classifications: Primary 32F45; Secondary 47B32; 32A36

1. Introduction

Let � be a domain in Cn and H a Hilbert space of holomorphic functions on �. H is

called a holomorphic Hilbert space if the evaluation functional at each point z2� iscontinuous on H [1]. In this case, there is a unique function Kz2H such thatf(z)¼h f,Kzi for every f2H. The function K(w, z)¼Kz(w) on �� is called thereproducing kernel of H and holomorphic in the first variable and conjugateholomorphic in the second one. We will assume throughout this article thatkKzk

2¼K(z, z)40 for all z2�, and the set P of holomorphic polynomials is

included in H.For a fixed point a2�, let V ¼

Pnj¼1vj ðzÞ

@@zj¼Pn

j¼1vj ðzÞ@j be a nonzeroholomorphic vector field on � with v 0j s holomorphic, and for m� 1 define

SmðV, aÞ ¼ f f2H : k f k � 1, ðVjf ÞðaÞ ¼ 0, j ¼ 0, 1, . . . ,m� 1g, ð1Þ

*Corresponding author. Email: [email protected]

ISSN 1747–6933 print/ISSN 1747–6941 online

� 2012 Taylor & Francis

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where V 0¼ I is the identity operator. We denote by S0(V, a)¼ { f2H : k fk� 1}

the closed unit ball of H. It is clear that Sm(V, a) is nonempty since it contains

functions of the form cðz� aÞ� ¼ cQn

j¼1ðzj � aj Þ�j with j�j ¼

Pnj¼1 �j � m, where

c¼k(z� a)�k�1. Let

ðRmV ÞðaÞ ¼ supf2SmðV,aÞ

jðVmf ÞðaÞj2 ð2Þ

and write

fðBmV ÞðaÞg2 ¼ Kða, aÞ�1ðRmV ÞðaÞ: ð3Þ

It is standard that (R0V )(a)¼K(a, a) and (B0V )(a)¼ 1 [2].Note that for m� 1, the iteration Vm of a holomorphic vector field

V ¼Pn

j¼1 vj ðzÞ@j can be considered as a holomorphic differential operator of the

form Vm ¼Pmj�j¼0 f�ðzÞ@

� for some holomorphic functions f�’s on �. For fixed a2�

and each m� 1, the linear functional Vma : f! ðVmf ÞðaÞ defined on H is bounded by

Cauchy estimates and the fact that the norm of f2H dominates its supremum norm

over any compact set by the reproducing property f(z)¼h f,Kzi. So there exists a

unique element KVm,a2H such that

ðVmf ÞðaÞ ¼ h f,KVm,ai ð4Þ

and

KVm,aðzÞ ¼ hKVm,a,Kzi ¼ hKz,KVm,ai ¼ ðVmKzÞðaÞ:

For fixed vector field V, we write ea, j(z)¼KVj,a(z) for j¼ 0, 1, . . . ,m. In turn, (1)

and (2) can be rewritten as

SmðV, aÞ ¼ f f2S0ðV, aÞ : h f, ea, ji ¼ 0, j ¼ 0, 1, . . . ,m� 1g

and

ðRmV ÞðaÞ ¼ supf2SmðV,aÞ

jh f, ea,mij2: ð5Þ

Let Ha,m be the closed subspace spanned by fea, jgm�1j¼0 and PH?a,m be the orthogonal

projection onto H?a,m, the orthogonal complement space of Ha,m in H. Then

(RmV )(a) is exactly the square of the operator norm of the linear functional Vma

restricted to H?a,m by (2), and

ðRmV ÞðaÞ ¼ kPH?a,mea,mk2 ð6Þ

by (5) in the point of view of Hilbert space geometry. We remark that it is convenient

to view Bm in (3) as a mapping that associates a holomorphic vector field with a

function on �.Now suppose that H1 on �1 and H2 on �2 are two holomorphic Hilbert spaces

normed by k f k2H1¼R

�1j f ðzÞj2 d�1ðzÞ and k gk

2H2¼R

�2j gðwÞj2 d�2ðwÞ, where d�1

and d�2 are positive Borel measures on �1 and �2, respectively. Let be a

biholomorphic mapping of �1 onto �2 and assume that there is a nonvanishing

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holomorphic function j on �1 such that

�2ð ðN ÞÞ ¼

ZN

j j ðzÞj2 d�1ðzÞ ð7Þ

for each Borel subset N of �1. Then the change of variable formula (7) implies that

the operator

ðU f ÞðzÞ ¼ ð f � ÞðzÞ j ðzÞ ð8Þ

is an isometry of H2 onto H1, which in turn gives rise to

Kð�1Þðz,wÞ ¼ Kð�2Þð ðzÞ, ðwÞÞ j ðzÞj ðwÞ: ð9Þ

Consequently, using the unitary operator U in (8) and the identity (9), as in the

proof of Proposition 3.1 in [3], it is easy to show that for such , Bm enjoys the

transformation property as follows:

ðBmV Þð�1ÞðzÞ ¼ ðBm �ðV ÞÞ

ð�2Þð ðzÞÞ, ð10Þ

where �(V ) is the push-forward of the vector field V under .In particular, we specialize the vector field V to be the constant vector field

@v ¼Pn

j¼1 vj@j with v¼ (v1, . . . , vn)2Cn\{0}. In this case, we write ej ð�Þ ¼ ea, jð�Þ ¼

K@ jv ,að�Þ ¼ ð �@ j

vKÞð�, aÞ ¼ ð�@ jvKaÞð�Þ if no confusion arises. Then the set fejg

mj¼0 is linearly

independent in H since the set of holomorphic polynomials PH by our

assumption. An application of the Gram–Schmidt orthogonalization process gives

an orthonormal basis f’igmi¼0 for Ha,mþ1, where [4]

’0 ¼ J�1

2

0 e0,

’i ¼ fJi�1Jig�1

2

he0, e0i � � � he0, ei�1i e0

..

. . .. ..

. ...

hei, e0i � � � hei, ei�1i ei

��

for i¼ 1, . . . ,m, and

Ji ¼ Jið@v, aÞ ¼

he0, e0i � � � he0, eii

..

. . .. ..

.

hei, e0i � � � hei, eii

��

is the determinant of the Gram-matrix of the system fejgij¼0 for i¼ 0, 1, . . . ,m.

Note that Ji40 due to the linear independence of fejgij¼0 [4] for i¼ 0, 1, . . . ,m, and

(Bm@v)(a) can be expressed in a closed form in terms of the kernel function, i.e. [4]

ðRm@vÞðaÞ ¼ J�1m�1Jm, fðBm@vÞðaÞg2 ¼ fJ0Jm�1g

�1Jm, m � 1: ð11Þ

As remarked in [3], (Bm@v)(a) is homogeneous in v of degree m, and is NOT, strictly

speaking, an infinitesimal metric as m41. Care must be taken in considering

‘‘transformation laws’’ (10) for (Bm@v)(a) for m41. We also remark that for fixed

domain �, the quantity (Bm@v)(a) relies on the specific Hilbert space structure of H

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on �. The notation (Bm@v)(a) will be abused without indicating the holomorphic

Hilbert space H which is to be clear from the context.The prototypes of the spaces H are the Bergman spaces A2(�, d�) of all

holomorphic functions in L2(�, d�) on a bounded domain �Cn with the

normalized volume measure d�(z)¼V(�)�1dV(z) where dV is the Lebesgue measure

and V(�) is the volume of � with respect to dV, or the Segal–Bargmann space

H2(Cn, d�) of all entire functions in L2(Cn, d�) for the Gaussian measure

d�ðzÞ ¼ ð2�Þ�ne�jzj2

2 dVðzÞ:

For H¼A2(�, d�), the reproducing kernel K(z,w) is then just the original kernel

function of Bergman [4], and (Bm@v)(a) is exactly the mth order Bergman metric as

introduced by Burbea in [5]. In particular, (B1@v)(a) is the classical Bergman metric.

For H¼H2(Cn, d�), the kernel function Kðz,wÞ ¼ ehz,wi2 and (B1@v)(a) coincides (up to

a constant factor) with the Euclidean metric. As a Hermitian metric, the classical

Bergman metric can also be defined as

B1ða, vÞ ¼ ðB1@vÞðaÞ ¼

�Xnj,k¼1

@2

@zj@ �zklogKða, aÞvj �vk

�12

ð12Þ

for v2Tz(�)ffiCn the tangent space at z2� [6].

The motivation for us to study the mapping Bm originates from the recent

intensive research on Berezin’s operator calculus [7–12]. The Berezin transform,

introduced by Berezin in his quantization programme [13,14], provides a general

symbol calculus for linear operators on any holomorphic Hilbert space. More

specifically, for H a holomorphic Hilbert space on � and X2Op(H) the algebra of

bounded linear operators on H, the Berezin transform (or symbol) of X is given byeXðzÞ ¼ hXkz, kzi ¼ trðXAðzÞÞ, ð13Þ

where kzð�Þ ¼ Kð�, zÞKðz, zÞ�12 is the normalized kernel function at z, and

AðzÞ ¼ kz � kz ¼ h�, kzikz ð14Þ

is the projection onto the span of kz. It is well-known that eX is real analytic with

keXk1 � kXk, and X is uniquely determined by eX.The study on the interaction between the mth order Bergman metric and

Berezin’s operator calculus was initiated in [3] most recently, where the relationship

between (Bm@v)(a) and (B1@v)(a) has also been investigated. In particular, it was

shown there that for H¼A2(Bn, d�) on the open unit ball,

ðBm@vÞðaÞ ¼

�m!ðmþ nÞ!

n!ðnþ 1Þm

�12

fðB1@vÞðaÞgm: ð15Þ

In the same paper, it was also proved separately that an identity similar to (15) holds

with different constant factor on Cn for H¼H2(Cn, d�) (see Theorem 3.14 and

Theorem 3.18 of [3]).In view of the simple intuition that C

n could be considered as a ball with infinite

radius, and the fact that (Bm@v)(a) is ‘‘intrinsic’’ as interpreted in (10), it would be

of interest to find an approach that could deal with both cases uniformly in this


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Bergman-type spaces setting. It turns out that this goal can be achieved by

approximating the scaled Segal–Bargmann space H2(Cn, d�p) by a class of

holomorphic Hilbert spaces in an appropriate way, where d�pðzÞ ¼pn

�n e�pjzj2dVðzÞ.

This uniform treatment will be developed in Section 2 and Section 3, while a different

proof of (15) will be provided in the setting ofH¼A2(Bn, d��) the weighted Bergman

space in Section 2.Motivated by the definition of mth order Bergman metric, we introduce another

mapping Dm from the space of holomorphic vector fields to the set of functions on �

in Section 5, and provide an estimates for (Dm@v)(z) on the unit ball using our

analysis of Berezin’s operator calculus. It turns out that this estimate coincides with

the other one which is a direct consequence of the higher order Schwarz-Pick lemma

in [15] except a different constant factor. In our argument, we need an iteration

formula in [3] which will be refined by determining all coefficients completely in

Section 4.In Section 6, using the refined formula in Section 4 and the higher order Schwarz-

Pick lemma in [15], we are able to show that on the unit ball Bn,

ðCm@vÞðzÞ ¼ m!fðC1@vÞðzÞgm,

where (Cm@v)(z) is the mth order Caratheodory-Reiffen metric [5], and (C1@v)(z) is theclassical infinitesimal Caratheodory-Reiffen metric.

We conclude by pointing out that by using the refined formula in Section 4 and

some result in Section 6, our estimate for (Dm@v)(z) in Section 5 can be improved to

be the same as the other one directly from the higher order Schwarz-Pick lemma.

This perfect coincidence discloses some intimate relation between our analysis in

Berezin’s operator calculus and the higher order Schwarz-Pick lemma.

2. A new proof

In this section, we extend (15) to the setting of weighted Bergman space A2(Bn, d��)by a different proof, which provides the foundation for our method of approxima-

tion in Section 3.For �4�1, let d��(z)¼ c� (1� jzj2)�d�(z) be the probability measure on B

n

where c� ¼�ðnþ�þ1Þn!�ð�þ1Þ . For H¼A2(d��)¼A2(Bn, d��) the weighted Bergman space on

the unit ball Bn, the reproducing kernel is given by [16]

K�ðz,wÞ ¼1

ð1� hz,wiÞnþ1þ�¼ Kðz,wÞ

nþ1þ�nþ1 , ð16Þ

where K(z,w) is the kernel function of the standard Bergman space A2(Bn, d�) as�¼ 0. It is easy to check that the Bergman metric (12) induced by K�(z,w) on B

n is

fðB1@vÞðaÞg2 ¼ðnþ 1þ �Þfð1� jaj2Þjvj2 þ jha, vij2g

ð1� jaj2Þ2ð17Þ

by (16) and Proposition 1.4.22 in [17].It is standard that for each a in B

n, there is a a2Aut(Bn), the group of all

biholomorphic self-mapping on Bn, with the properties a(a)¼ 0 and a � a¼ I, the


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identity map. More precisely,

aðzÞ ¼a� Paz�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� jaj2

pQaz

1� hz, ai, ð18Þ

where P0¼ 0, Pa ¼1jaj2

a� a for a 6¼ 0, and Qa¼ I�Pa [18].

LEMMA 2.1 For a, z2Bn, v2C

n and a2Aut(Bn),

hz, við1� h aðzÞ, aiÞ ¼ ha, vi � h aðzÞ, �i,

where � ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� jaj2

pvþ hv, ai

1þffiffiffiffiffiffiffiffiffiffi1�jaj2p a with j�j2¼ (1� jaj2)jvj2þ jha, vij2.

Proof Without loss of generality, suppose that a 6¼ 0. By the involutive property

of a, it suffices to show that

h aðzÞ, við1� hz, aiÞ ¼ ha, vi � hz, �i:

By (18),

h aðzÞ, við1� hz, aiÞ ¼Da� Paz�


qQaz, v

E¼ ha, vi �

1

jaj2hz, aiha, vi �


qhz, vi þ


pjaj2

hz, aiha, vi

¼ ha, vi �Dz,


qvE�

1

1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� jaj2

p hz, aiha, vi

¼ ha, vi �

*z,


qvþ

hv, ai

1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� jaj2

p a

+:

Let � ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� jaj2

pvþ hv, ai

1þffiffiffiffiffiffiffiffiffiffi1�jaj2p a, and a direct calculation shows that j�j2¼

(1� jaj2)jvj2þ jha, vij2, which finishes the proof. g

In the setting of weighted Bergman space A2(d��), the key point of our new

method is to express the basis fej ¼ �@ jvK

�að�Þg

mj¼0 of Ha,mþ1 in terms of a set of

orthogonal elements.

PROPOSITION 2.2 For j� 0,

�@ jvK

�aðwÞ ¼

�ð jþ nþ 1þ �Þ

�ðnþ 1þ �Þð1� jaj2Þ�ð

nþ1þ�2 þj Þ

Xj�¼0

j

�

� �ð�1Þ�ha, vij��h aðwÞ, �i

�k�aðwÞ,

where � ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� jaj2

pvþ hv, ai

1þffiffiffiffiffiffiffiffiffiffi1�jaj2p a with j�j2¼ (1� jaj2)jvj2þ jha, vij2, and the set

fh aðwÞ, �i�k�aðwÞg

j�¼0 consists of orthogonal elements in A2(d��).

Proof For v2Cn\{0}, we write �@ j

vK�ðw, uÞju¼a ¼ �@ j

vK�aðwÞ. By (16), it is easy to see that

�@ jvK

�aðwÞ ¼


�ðnþ 1þ �Þhw, vijð1� hw, aiÞ�ð jþnþ1þ�Þ:

Since the normalized kernel function k�a at a is

k�aðwÞ ¼K�ðw, aÞ

K�ða, aÞ12

¼ð1� jaj2Þ

nþ1þ�2

ð1� hw, aiÞnþ1þ�


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and [18]

1� h aðwÞ, ai ¼1� jaj2

1� hw, ai,

it follows that

hw, vijð1� hw, aiÞ�ð jþnþ1þ�Þ ¼ ð1� jaj2Þ�ðnþ1þ�

2 þj Þk�aðwÞ hw, við1� h aðwÞ, aiÞ� j

¼ ð1� jaj2Þ�ðnþ1þ�

2 þj Þk�aðwÞ ha, vi � h aðwÞ, �i� j

,

where the last equality follows from Lemma 2.1. Therefore,

�@ jvK

�aðwÞ ¼



nþ1þ�2 þj Þk�aðwÞ½ha, vi � h aðwÞ, �i�

j

¼�ð jþ nþ 1þ �Þ


nþ1þ�2 þj Þ

Xj�¼0

j

�

� �ð�1Þ�ha, vij��h aðwÞ, �i

�k�aðwÞ:

Next we need to show the orthogonality of elements in the set

fh aðwÞ, �i�k�aðwÞg

j�¼0. For this purpose, we consider the operator Ua associated

with a on A2(d��), defined by

ðUa f ÞðwÞ ¼ ð f � aÞðwÞk�aðwÞ:

Then it is easy to see that Ua is unitary on A2(d��) by Proposition 1.13 in [16]. Since

hw, �i�� j

�¼0is an orthogonal set in A2(d��) by (1.21) in [16], and

Uafhw, �i�g ¼ h aðwÞ, �i

�k�aðwÞ,

our assertion follows easily and the proof is completed. g

The next result gives the estimate for the greatest possible rate of growth of

higher order directional derivatives of Bergman space functions on the unit ball.

COROLLARY 2.3 For f2A2(d��) and m� 0,

maxk f k�1

jð@mv f ÞðaÞj2 ¼

½�ðmþ nþ 1þ �Þ�2

�ðnþ 1þ �Þðnþ 1þ �ÞmK�ða, aÞfðB1@vÞðaÞg

2m

�Xm�¼0

ðm!Þ2

�!½ðm� �Þ!�2�ð�þ nþ 1þ �Þ

jha, vij2

j�j2

� �m��

:

In particular, for �¼ 0,

maxk f k�1

jð@mv f ÞðaÞj2 ¼ðmþ nÞ!m!

n!ðnþ 1ÞmKða, aÞfðB1@vÞðaÞg

2mXm�¼0

m

�

� �mþ n

�þ n

� �jha, vij2

j�j2

� �m��

�½ðmþ nÞ!�2m!

ðn!Þ2ðnþ 1ÞmKða, aÞfðB1@vÞðaÞg

2m 1þjha, vij2

j�j2

� �m

: ð19Þ


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Proof By (4) with V¼ @v and Proposition 2.2, we know that

maxk f k�1

jð@mv f ÞðaÞj2 ¼ k �@mv K

�ak

2

¼�ðmþ nþ 1þ �Þ


nþ1þ�2 þmÞ

Xm�¼0

m

�

� �ð�1Þ�ha, vim��h aðwÞ, �i

�k�aðwÞ

2



nþ1þ�2 þmÞ

�� 2Xm�¼0

m

�

� �ha, vim��

�� 2 h aðwÞ, �i�k�aðwÞ

2:The rest of the proof follows from (17) and the fact that

h aðwÞ, �i�k�aðwÞ

2¼ khw, �i�k2 ¼ �ðnþ �þ 1Þ�!

�ð�þ nþ 1þ �Þj�j2�

by Lemma 1.11 of [16]. g

Now we are ready to state the main result of this section.

THEOREM 2.4 On A2(d��),

ðBm@vÞðaÞ ¼�ðmþ nþ 1þ �Þm!

�ðnþ 1þ �Þðnþ 1þ �Þm

� �12

fðB1@vÞðaÞgm: ð20Þ

Proof By Proposition 2.2, we know

ðPH?a,memÞðwÞ ¼ ½PH?a,mð�@mv K

�aÞ�ðwÞ


�ðnþ 1þ �Þð�1Þmð1� jaj2Þ�ð

nþ1þ�2 þmÞh aðwÞ, �i

mk�aðwÞ:

The same proof as in Corollary 2.3 shows that

kPH?a,memk2 ¼

�ðmþ nþ 1þ �Þm!

�ðnþ 1þ �ÞK�ða, aÞ

j�j2

ð1� jaj2Þ2

� �m

:

Then by (3) and (6),

fðBm@vÞðaÞg2 ¼ K�ða, aÞ�1kPH?a,memk

2

¼�ðmþ nþ 1þ �Þm!

�ðnþ 1þ �Þ

j�j2

ð1� jaj2Þ2

� �m


�ðnþ 1þ �Þðnþ 1þ �ÞmfðB1@vÞðaÞg

2m,

as desired. g

Remark 2.5 When �¼ 0 in Theorem 2.4, (15) will be recovered simply, while it was

proved differently by using the transformation formula (10) and establishing the

iteration formula (24) of certain vector fields at the origin (see Section 4).


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3. A uniform approach

The objective of this section is to develop an approach which will deal with (Bm@v)(a)on balls of any radius and C

n uniformly.For 05r51 and �4�1, we denote Br¼ {z2C

n : jzj5r} and d��,rðzÞ ¼c�,rð1�

jzj2

r2Þ�d�ðzÞ with c�,r ¼

�ðnþ�þ1Þn!r2n�ð�þ1Þ

, then ��,r(Br)¼ 1, B1¼Bn the open unit

ball and d��,1(z)¼ d��(z) accordingly. It is clear that B1 and Br are biholomorphi-cally equivalent, and any biholomorphic mapping of B1 onto Br is of the form r�where � 2Aut(B1). In particular, for w ¼ ðzÞ ¼ z

r : Br ! B1, it is easy to see that �ð@vÞ ¼ @vr and

��ð ðN ÞÞ ¼

ZN

d��,rðzÞ

for each Borel subset NBr. Consequently, the kernel function of the weightedBergman spaces A2(Br, d��,r) is K�r ðz,wÞ ¼ ð1�

hz,wir2Þ�ðnþ1þ�Þ by (9). It also follows

from the transformation formula (10) and Theorem 2.4 that

ðBm@vÞðBrÞðaÞ ¼ ðBm@vrÞ

ðB1Þ

�a

r

�¼

�ðmþ nþ 1þ �Þm!

�ðnþ 1þ �Þðnþ 1þ �Þm

� �12�ðB1@vrÞ

ðB1Þ

�a

r

��m


�ðnþ 1þ �Þ

� �12�ðr2 � jaj2Þjvj2 þ jha, vij2

ðr2 � jaj2Þ2

�m2

: ð21Þ

LEMMA 3.1 For p40, the functions frðzÞ ¼ f ðz, rÞ ¼ ð1� jzj2

r2Þ�pr2 converge to epjzj

2

uniformly on the compacta of Cn as r!þ1.

Proof Using the limit definition of the natural base e, it is easy to see that

limr!þ1

frðzÞ ¼ epjzj2

for each z. We only need to show that the pointwise convergence is actually uniformon each compact subset M of C

n. For such an M, we can choose R40 such thatMBR. Note that

@

@rf ðz, rÞ ¼ �2pr 1�

jzj2

r2

� ��pr2ln

�1�jzj2

r2

�þ

jzj2

r2

1� jzj2

r2

( ):

For r4R and z2M, we claim that lnð1� jzj2

r2Þ þ

jzj2

r2

1�jzj2

r2

� 0. Let x ¼ jzj2

r2, then 0� x51

and we define gðxÞ ¼ lnð1� xÞ þ x1�x. So g(0)¼ 0 and g 0ðxÞ ¼ x

ð1�xÞ2� 0. Thus g(x)� 0

for 0� x51, which implies that @@r f ðz, rÞ � 0, and the functions fr(z) is decreasing in ras r4R for z2M. So the functions fr(z) are convergent to epjzj

2

uniformly on M asr!þ1 by Dini’s theorem. g

THEOREM 3.2 For fixed p40 and pr20 4 n for some r040, let �¼ pr2� n� 1 forr� r0. Then

limr!þ1

K�r ðz, zÞ ¼ epjzj2

uniformly on the compacta of Cn. Consequently, for H¼H2(Cn, d�p),

ðBm@vÞðC

nÞðaÞ ¼ lim

r!þ1ðBm@vÞ

ðBrÞðaÞ ¼ fm!pmjvj2mg12:


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Proof By our assumption, �¼ pr2� n� 14�1 and K�r ðz, zÞ ¼ ð1�jzj2

r2Þ�pr2 . Then by

Lemma 3.1, we know

limr!þ1

K�r ðz, zÞ ¼ epjzj2

ð22Þ

uniformly on the compacta of Cn.

On the other hand, by (11) we know that (Bm@v)(Br)(a) is a closed expression of

fhej, eii ¼ ð@iv�@ jvK

�r Þða, aÞg

mi, j¼0:

So the interchange of taking limit and differentiation is assured by (22) so that

limr!1ðBm@vÞ

ðBrÞðaÞ ¼ ðBm@vÞðC

nÞðaÞ

for some holomorphic Hilbert spaceH on Cn with kernel function K(p)(z, z)¼ epjzj

2

onthe diagonal of C

n�C

n. Since the function K(p)(z,w)¼ ephz,wi, as the kernel functionofH2(Cn, d�p), is uniquely determined by K(p)(z, z)¼ epjzj

2

by the standard uniquenesstheorem, and there is a bijective correspondence between Hilbert function spaces onCn and reproducing kernels on C

n [1, p. 19], we know that H must be represented byH2(Cn, d�p). Finally, with (21), it is easy to see that

ðBm@vÞðC

nÞðaÞ ¼ lim

r!1ðBm@vÞ

ðBrÞðaÞ

¼ limr!1

�ðmþ pr2Þm!

�ð pr2Þ

� �12 ðr2 � jaj2Þjvj2 þ jha, vij2

ðr2 � jaj2Þ2

� �m2

¼ fm!pmjvj2mg12,

where the last equality follows from Stirling’s formula. g

4. Revisiting to an iteration formula

For �¼Bn and w¼ a(z)2Aut(Bn) of the form (18), we define gðwÞ ¼ ð aÞ�ðvÞ ¼

0aðzÞv ¼ ð g1ðwÞ, . . . , gnðwÞÞ, then gð0Þ ¼ 0aðaÞv and [3, Lemma 3.7]

j gð0Þj ¼ðB1@vÞðaÞffiffiffiffiffiffiffiffiffiffiffi

nþ 1p , ð23Þ

where (B1@v)(a) is the classical Bergman metric associated with the (unweighted)Bergman space A2(Bn, d�). Moreover, let @mg stand for the operator f

Pnj¼1 gj ðwÞ@wj

gm

and @mgð0Þ ¼ @m 0aðaÞv

for the operator fPn

j¼1 gj ð0Þ@wjgm. Let X be a Banach space and

C1(Bn,X) be the set of smooth mappings from Bn into X possessing strong

derivatives of all orders (see e.g. [11]). An iteration formula for the action of @mg onC1(Bn,X) at w¼ 0 has been established in [3] as follows: for f2C1(Bn,X)and m� 1,

ð@mg f ÞðwÞjw¼0 ¼Xmj¼1

CðmÞj hða, vÞm�jð@ j

gð0Þ f ÞðwÞjw¼0, ð24Þ

where h(a, v)¼ (1� jaj2)�1hv, ai, and CðmÞj ’s are constants depending on m and j

with CðmÞm ¼ 1.


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Moreover, a careful examination of the proof of (24) in [3] indicates that CðmÞj ’s

are determined by the recurrence relation

CðmÞj ¼ C

ðm�1Þj�1 þ 2jC

ðm�1Þj þ j ð jþ 1ÞC

ðm�1Þjþ1 , 1 � j � m, ð25Þ

where CðmÞm ¼ 1 for m� 1. We denote Cð0Þ0 ¼ 1 and C

ðmÞ0 ¼ 0 for m40, C

ðmÞj ¼ 0 for

j4m or j50. Actually, we can determine these coefficients explicitly.

PROPOSITION 4.1 For m� 1 and 1� j�m,

CðmÞj ¼

ðm� 1Þ!m!

ð j� 1Þ!j!ðm� j Þ!: ð26Þ

Proof We proceed by induction on m and j. It is clear that for m¼ 1 and j¼ 1,

Cð1Þ1 ¼ 1. So (26) holds. Now we assume that (26) is true for some m¼ k� 1 and for

all 1� j� k. Then for m¼ kþ 1 and j¼ 1, by (25) and our hypotheses for m¼ k,

we have

Cðkþ1Þ1 ¼ C

ðkÞ0 þ 2C

ðkÞ1 þ 2C

ðkÞ2 ¼ 2C

ðkÞ1 þ 2C

ðkÞ2 ¼ 2k!þ ðk� 1Þk! ¼ ðkþ 1Þ!,

which shows that (26) is true for m¼ kþ 1 and j¼ 1.Now for m¼ kþ 1 and 2� j� kþ 1, by (25) and our hypotheses for m¼ k again,

we have

Cðkþ1Þj ¼ C

ðkÞj�1þ 2jC

ðkÞj þ j ð jþ 1ÞC

ðkÞjþ1

¼ðk� 1Þ!k!

ð j� 2Þ!ð j� 1Þ!ðkþ 1� j Þ!þ 2j

ðk� 1Þ!k!

ð j� 1Þ!j!ðk� j Þ!þ j ð jþ 1Þ

ðk� 1Þ!k!

j!ð jþ 1Þ!ðk� 1� j Þ!

¼ðk� 1Þ!k!

ð j� 1Þ!j!ðkþ 1� j Þ!fð j� 1Þ jþ 2j ðkþ 1� j Þ þ ðk� j Þðkþ 1� j Þg

¼ðk� 1Þ!k!

ð j� 1Þ!j!ðkþ 1� j Þ!fkðkþ 1Þg

¼k!ðkþ 1Þ!

ð j� 1Þ!j!ðkþ 1� j Þ!,

which shows that (26) is also true for m¼ kþ 1 and 2� j� kþ 1. Therefore, we can

conclude that (26) holds for all m� 1 and 1� j�m. g

5. Connections to higher order Schwarz–Pick lemma

Most recently, a higher order Schwarz-Pick lemma formulated in terms of the

classical Bergman metric has been proved in [15] for the Frechet derivative of

f¼ ( f1, . . . , fk)2H(Bn,Bk) the class of holomorphic mappings from Bn into B

k (even

in the setting of complex Hilbert balls [19]). For v2Cn\{0}, we observe that the

Frechet derivative Dm( f, a, v) of f at point a2Bn of order m defined in [15] can be

expressed in our notations as

Dmð f, a, vÞ ¼ ð@mv f ÞðaÞ ¼ ðð@

mv f1ÞðaÞ, . . . , ð@mv fkÞðaÞÞ 2C

k,


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and the higher order Schwarz–Pick lemma in [15] can be rewritten as

ðB1@ð@mv f ÞðaÞÞð f ðaÞÞ � m!1ffiffiffiffiffiffiffiffiffiffiffinþ 1p 1þ

jha, vij

j�j

� �� m�1

ðB1@vÞðaÞ� m

: ð27Þ

Note that the Bergman metric used in (27) differs from the one used in [15] by a

factorffiffiffiffiffiffiffiffiffiffiffinþ 1p

. One immediate consequence of (27) combining with (17) (�¼ 0) isthat for a2B

n and v2Cn,

supf2HðBn,BkÞ

jð@mv f ÞðaÞj �m!

ðnþ 1Þm2

1þjha, vij

j�j

� �m�1

ðB1@vÞðaÞ� m

: ð28Þ

We notice that (28) has the same flavour as (19) in Corollary 2.3, and believe that

there might have been certain connection between our study of higher order‘intrinsic’ metrics in Berezin’s operator calculus and the higher order Schwarz–Pick

lemma (27). Actually, our belief will be enhanced by the fact that we are able torecover (28) with less sharp constant factor based on our analysis in Berezin’s

operator calculus.First of all, motivated by (28) and the definition of the mth order Bergman

metric, it would be interesting to consider another mapping DðkÞm defined on the space

of holomorphic vector fields by

ðDðkÞm V ÞðaÞ ¼ supf2Hð�,BkÞ

jðVmf ÞðaÞj ¼ supf2Hð�,BkÞ

Xkj¼1

jðVmfj ÞðaÞj2

( )12

,

where � is a bounded domain, V is a holomorphic vector field on � and H(�, Bk) is

the class of holomorphic mappings from � into Bk. It is also easy to prove that

ðDðkÞm V Þð�1ÞðzÞ ¼ ðDðkÞm �ðV ÞÞð�2Þð ðzÞÞ ð29Þ

for :�1!�2 a biholomorphic mapping. We write ðDð1Þm V ÞðaÞ ¼ ðDmV ÞðaÞ. It turnsout that the definition of ðDðkÞm V ÞðaÞ is independent of k.

PROPOSITION 5.1 For any k� 1,

ðDðkÞm V ÞðaÞ ¼ ðDmV ÞðaÞ:

Proof For f2H(�, B1), we define f ¼ ð f, 0, . . . , 0Þ 2Hð�,Bk

Þ. Then

jðVmf ÞðaÞj ¼ jðVmf ÞðaÞj

for any holomorphic vector field V. It follows that ðDmV ÞðaÞ � ðDðkÞm V ÞðaÞ. For the

opposite inequality, we observe that for f2H(�, Bk),

jðVmf ÞðaÞj ¼ supfjhðVmf ÞðaÞ, ij : 2Ck, jj ¼ 1g

¼ sup

��Xkj¼1

�j ðVmfj ÞðaÞ

�� : 2Ck, jj ¼ 1

�

¼ sup

��ðVmXkj¼1

�j fj ÞðaÞ

�� : 2Ck, jj ¼ 1

�� supfjðVmhÞðaÞj : h2Hð�,B1

Þg

¼ ðDmV ÞðaÞ:

The proof is completed. g


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Recall that for the rank one self-adjoint operator A(z) in (14), the mapping Lm

defined on the space of holomorphic vector fields by

ðLmV ÞðaÞ ¼ kðVmAÞðaÞktr

was introduced in [3].

PROPOSITION 5.2 On any bounded domain �,

ðDmV ÞðaÞ � ðLmV ÞðaÞ:

Proof From the definition (13) of the Berezin transform, we know that for

X2Op(A2(�)),

jðVmeX ÞðaÞj ¼ jtr½XðVmAÞðaÞ�j � kXkkðVmAÞðaÞktr: ð30Þ

If we take X¼Mf the multiplication operator on the Bergman space A2(�) with the

bounded symbol f2H(�,B1) in (30), it is easy to check that eMf ¼ f and

kMfk¼k f k1� 1. Our assertion follows easily. g

In our terminology, (28) is the same as the statement that on the unit ball Bn,

ðDðkÞm @vÞðaÞ ¼ ðDm@vÞðaÞ �m!

ðnþ 1Þm2

1þjha, vij

j�j

� �m�1

ðB1@vÞðaÞ� m

, ð31Þ

which can be recovered with less sharp constant factor as follows:

PROPOSITION 5.3 For X2Op(A2(Bn)),

jð@mveX ÞðaÞj � kXk mþ n

n

� �12 m!

ðnþ 1Þm2

1þjha, vij

j�j

� �m�1

ðB1@vÞðaÞ� m

: ð32Þ

Consequently,

ðDm@vÞðaÞ �mþ n

n

� �12 m!

ðnþ 1Þm2

1þjha, vij

j�j

� �m�1

ðB1@vÞðaÞ� m

: ð33Þ

Proof It was obtained by the formula (24) in section 4.1 of [3] that on the unit ball,

ðLm@vÞðaÞ ¼Xmj¼1

jCðmÞj j

2 j!ð jþ nÞ!

n!jhða, vÞj2ðm�j Þj gð0Þj2j

( )12

¼ðm� 1Þ!m!fðB1@vÞðaÞg

mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin!ðnþ 1Þm

p Xmj¼1

ð jþ nÞ!

½ð j� 1Þ!�2j!½ðm� j Þ!�2jha, vij

j�j

� �2ðm�j Þ( )1

2

,

where (23) and (26) are used in the last equality. Thus,

fðLm@vÞðaÞg �m!fðB1@vÞðaÞg

mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðnþ 1Þm

p Xmj¼1

nþ j

n

� �12 m� 1

j� 1

� �jha, vij

j�j

� �m�j

�mþ n

n

� �12 m!

ðnþ 1Þm2

1þjha, vij

j�j

� �m�1

ðB1@vÞðaÞ� m

:

Now (32) follows easily from (30) by taking V¼ @v, and (33) is from

Proposition 5.2. g


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6. mth order Caratheodory–Reiffen metric

In this section, we would like to derive an identity similar to (20) on the unit ball Bn

between the mth order Caratheodory-Reiffen metric and the classical infinitesimal

Caratheodory-Reiffen metric.For a bounded domain �, we replace the Hilbert space H by H(�,D) the class of

holomorphic functions f on � with values in the unit disc D, and the Hilbert space

norm by the supremum norm k�k1 in (1), and denote

TmðV, aÞ ¼ f f2Hð�,DÞ : ðVjf ÞðaÞ ¼ 0, j ¼ 0, 1, . . . ,m� 1g

and define a mapping on the space of holomorphic vector fields by

ðCmV ÞðaÞ ¼ supf2TmðV,aÞ

jðVmf ÞðaÞj:

Then it is clear that (CmV )(a)� (DmV )(a) and (C1V )(a)¼ (D1V )(a) on any bounded

domain. It is also easy to show that Cm satisfies the transformation formula

ðCmV Þð�1ÞðzÞ ¼ ðCm �ðV ÞÞ

ð�2Þð ðzÞÞ, ð34Þ

where :�1!�2 is biholomorphic and �(V ) is the push-forward of the vector

field V under . If the vector field V¼ @v for v2Cn\{0}, (Cm@v)(a) is exactly the mth

order Caratheodory-Reiffen metric introduced by Burbea in [5], and (C1@v)(a) is justthe classical infinitesimal Caratheodory-Reiffen metric [17].

LEMMA 6.1 For e1¼ (1, 0, . . . , 0)2Cn and m� 1,

ðCm@e1Þð0Þ ¼ ðDm@e1 Þð0Þ ¼ m!

on the unit ball Bn.

Proof It is clear that (Cm@e1)(0)� (Dm@e1)(0) by their definitions. For f ðzÞ ¼ zm1 , it is

easy to check that f2Tm(@e1, 0) and ð@me1 f Þð0Þ ¼ m!. So m!� (Cm@e1)(0), while

(Dm@e1)(0)�m! by (31). g

THEOREM 6.2 On the unit ball Bn,

ðCm@vÞðaÞ ¼ m!fðC1@vÞðaÞgm:

Proof Based on (24), the same argument as in the proof of Proposition 3.13 of [3]

yields that

ðCm@gÞð0Þ ¼ ðCm@gð0ÞÞð0Þ: ð35Þ

By (34) and (35), we have

ðCm@vÞðaÞ ¼ ðCm@gÞð0Þ ¼ ðCm@gð0ÞÞð0Þ: ð36Þ

We choose a unitary transformation U such that Uð gð0Þj gð0ÞjÞ ¼ e1, where e1¼

(1, 0, . . . , 0), then by (34) again,

ðCm@gð0ÞÞð0Þ ¼ j gð0Þjm�Cm@ gð0Þ

j gð0Þj

�ð0Þ ¼ j gð0Þjm

�Cm@Uð gð0Þ

j gð0ÞjÞ

�ð0Þ ¼ j gð0ÞjmðCm@e1Þð0Þ:

ð37Þ


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Since j gð0Þj ¼ j 0aðaÞvj ¼ ðC1@vÞðaÞ, (36), (37) and Lemma 6.1 imply that

ðCm@vÞðaÞ ¼ j gð0ÞjmðCm@e1Þð0Þ ¼ m!fðC1@vÞðaÞg

m:

g

Remark 6.3 We remark that (31) could be recovered exactly by an alternativeargument involving Lemma 6.1 in which a particular case of (31) has been used. Thatis, by (29) and (24),

ðDm@vÞðaÞ ¼ ðDm@gÞð0Þ �Xmj¼1

CðmÞj jhða, vÞj

m�jðDj@gð0ÞÞð0Þ

¼Xmj¼1

CðmÞj jhða, vÞj

m�jj gð0Þj j ðDj@e1Þð0Þ

¼m!fðB1@vÞðaÞg

mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðnþ 1Þm

p Xmj¼1

m� 1

j� 1

� �jha, vij

j�j

� �m�j

,

which is exactly (31) after reindexing.

Remark 6.4 Combining the results in [3,5] and this note, we see that

ðCm@vÞðaÞ � ðBm@vÞðaÞ, ðDm@vÞðaÞ � ðLm@vÞðaÞ

on any bounded domain, and they are equivalent to each other and comparable to{(B1@v)(a)}

m on the unit ball. We conjecture that they are also equivalent on anybounded symmetric domain in C

n.

Acknowledgements

The authors thank Prof Jingbo Xia for suggesting the essential idea of the new proof inSection 2, and they are also grateful to Dr Trieu Le for insightful discussions in the finding ofthe formula in Proposition 4.1.

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Soc. 134 (2006), pp. 2285–2294.[12] B. Li, The Berezin transform and Laplace–Beltrami operator, J. Math. Anal. Appl. 327

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[13] F.A. Berezin, Covariant and contravariant symbols of operators, Math. USSR Izv. 6(1972), pp. 1117–1151.

[14] F.A. Berezin, Quantization, Math. USSR Izv. 8 (1974), pp. 1109–1163.

[15] S. Dai, H. Chen, and Y. Pan, The Schwarz-Pick lemma of high order in several variables,Michigan Math. J. (to appear).

[16] K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Springer-Verlag, New York,2005.

[17] S. Krantz, Function Theory of Several Complex Variables, Wiley-Interscience, New York,1982.

[18] W. Rudin, Function Theory in the Unit Ball of Cn, Springer-Verlag, New York, Berlin,

1980.[19] S. Dai, H. Chen, and Y. Pan, The high order Schwarz-Pick lemma on complex Hilbert

balls, Sci. China (to appear).


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berezin's operator calculus and higher order schwarz–pick lemma

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