COMPOSITION OPERATORS BETWEEN GENERALLY WEIGHTED BLOCH SPACES OF POLYDISK

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Composition Operators Haiying Li, Peide Liu and Maofa Wang

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COMPOSITION OPERATORS BETWEEN GENERALLY WEIGHTED BLOCH SPACES OF

POLYDISK

HAIYING LI, PEIDE LIU AND MAOFA WANG

School of Mathematics and Statistics Wuhan University

Wuhan 430072, P.R. China EMail:tslhy2001@yahoo.com.cn

Received: 23 March, 2007

Accepted: 26 June, 2007

Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: Primary 47B38.

Key words: Holomorphic self-map; Composition operator; Bloch space; Generally weighted Bloch space.

Abstract: Let φ be a holomorphic self-map of the open unit polydisk Uⁿ in Cⁿ and p, q > 0.In this paper, the generally weighted Bloch spacesB_log^p (Uⁿ) are introduced, and the boundedness and compactness of composition operatorCφ

fromB_log^p (Uⁿ)toB_log^q (Uⁿ)are investigated.

Acknowledgements: Supported by the Natural Science Foundation of China (No. 10671147, 10401027).

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1. Introduction

Suppose thatDis a domain inCⁿ andφa holomorphic self-map ofD. We denote byH(D)the space of all holomorphic functions on D and define the composition operatorC_φonH(D)byC_φf =f ◦φ.

The theory of composition operators on various classical spaces, such as Hardy and Bergman spaces on the unit diskU in the finite complex planeChas been studied. However, the multivariable situation remains mysterious. It is well known in [3]

and [5] that the restriction of C_φto Hardy or standard weighted Bergman spaces on U is always bounded by the Littlewood subordination principle. At the same time, Cima, Stanton and Wogen confirmed in [1] that the multivariable situation is much different from the classical case (i.e., the composition operators on the Hardy space of holomorphic functions on the open unit ball ofC²as well as on many other spaces of holomorphic functions over a domain ofCⁿcan be unbounded, even whenn= 1 in [6]). Therefore, it would be of interest to pursue the function-theoretical or geo- metrical characterizations of those mapsφwhich induce bounded or compact composition operators. In this paper, we will pursue the function-theoretic conditions of those holomorphic self-mapsφ ofUⁿwhich induce bounded or compact composition operators from a generally weightedp−Bloch space to a q−Bloch space with p, q >0.

Forn ∈N, we denote byUⁿthe open unit polydisk inCⁿ :

Uⁿ={z = (z₁, z₂, . . . , z_n) :|z_j|<1, j = 1,2, . . . , n}, and

hz, wi=

n

X

j=1

z_jw_j, |z|=p hz, zi

for anyz = (z₁, z₂, . . . , z_n), w = (w₁, w₂, . . . , w_n)inCⁿ. α = (α₁, α₂, . . . , α_n)is

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said to be an n multi-index ifα_i ∈ N, written byα ∈ Nⁿ. Forα ∈ Nⁿ, we write z^α = z^α₁¹z^α₂²· · ·z^α_nⁿ andz_i⁰ = 1,1 ≤ i ≤ n for convenience. For z, w ∈ Cⁿ, we denote[z, w]_j =zwhenj = 0,[z, w]_j =wwhenj =n, and

[z, w]_j = (z₁, z₂, . . . , zn−j, wn−j+1, . . . , w_n)

whenj ∈ {1,2, . . . , n−1}. Then[z, w]n−j =wwhenj = 1, and[z, w]n−j+1 = w whenj =n+ 1, forj = 2,3, . . . , n,

[z, w]n−j+1 = (z₁, z₂, . . . , zj−1, w_j, . . . , w_n).

For anya∈Candz_j⁰ = (z1, z2, . . . , zj−1, zj+1, . . . , zn),we write (a, z_j⁰) = (z₁, z₂, . . . , z_j−1, a, z_j+1, . . . , z_n).

Moreover, we adopt the notation(z^[j⁰^])_j⁰_∈_Nfor an arbitrary subsequence of(z^[j])_j∈_N. Recall that the Bloch spaceB(Uⁿ)is the vector space of allf ∈H(Uⁿ)satisfying

b₁(f) = sup

z∈Uⁿ

Q_f(z)<∞, where

Qf(z) = sup

u∈Cⁿ\{0}

|h∇f(z),ui|¯

pH(z, u) , ∇f(z) = ∂f

∂z₁(z), . . . , ∂f

∂z_n(z)

and the Bergman metricH :Uⁿ×Cⁿ→[0,∞)onUⁿis H(z, u) =

n

X

k=1

|u_k|² 1− |z_k|²

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(for example see [9], [15]). It is easy to verify that both|f(0)|+b₁(f)and kfkB=|f(0)|+ sup

z∈Uⁿ n

X

k=1

∂f

∂z_k(z)

(1− |zk|²)

are equivalent norms onB(Uⁿ). In [10], [12] and [8], some characterizations of the Bloch spaceB(Uⁿ)have been given.

In a recent paper [2], a generalized Bloch space has been introduced, thep−Bloch space: forp > 0, a function f ∈ H(Uⁿ)belongs to the p−Bloch spaceB^p(Uⁿ)if there is someM ∈[0,∞)such that

n

X

k=1

∂f

∂z_k(z)

(1− |z_k|²)^p ≤M, ∀z ∈Uⁿ.

The references [13] to [7] studied these spaces and the operators in them.

Dana D. Clahane et al. in [2] proved the following two results:

Theorem A. Let φ be a holomorphic self-map of Uⁿ and p, q > 0. The following statements are equivalent:

(a) C_φis a bounded operator fromB^p(Uⁿ)toB^q(Uⁿ);

(b) There isM ≥0such that (1.1)

n

X

k, l=1

∂φ_l

∂zk

(z)

(1− |z_k|²)^q

(1− |φl(z)|²)^p ≤M, ∀z ∈Uⁿ.

Theorem B. Letφbe a holomorphic self-map ofUⁿandp, q >0. If condition (1.1) and

(1.2) lim

φ(z)→∂Uⁿ n

X

k, l=1

∂φ_l

∂z_k(z)

(1− |z_k|²)^q (1− |φ_l(z)|²)^p = 0

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hold, thenC_φis a compact operator fromB^p(Uⁿ)toB^q(Uⁿ).

Now we introduce the generally weighted Bloch spaceB^p_log(Uⁿ).

For p > 0, a function f ∈ H(U) belongs to the generally weighted p−Bloch spaceB_log^p (Uⁿ)if there is someM ∈[0,∞)such that

n

X

k=1

∂f

∂zk

(z)

(1− |z_k|²)^plog 2

1− |zk|² ≤M, ∀z ∈Uⁿ. Its norm inB_log^p (Uⁿ)is defined by

kfk_B^p

log =|f(0)|+ sup

z∈Uⁿ n

X

k=1

∂f

∂z_k(z)

(1− |z_k|²)^plog 2 1− |z_k|².

In this paper, we mainly characterize the boundedness and compactness of the composition operators between B_log^p (Uⁿ) and B_log^q (Uⁿ), and extend some corre- sponding results in [2] and [11] in several ways.

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2. Main Results and their Proofs

First, we have the following lemma:

Lemma 2.1. Letf ∈B_log^p (Uⁿ)andz ∈Uⁿ,then:

(a) |f(z)| ≤

1 + (1−p) log 2ⁿ

kfk_B^p

log,when0< p <1;

(b) |f(z)| ≤

1

2 log 2 +_{2nlog 2}¹ Pn

k=1log_1−|z⁴

k|²kfk_B^p

log,whenp= 1;

(c) |f(z)| ≤

1

n+ (p−1) log 2²^p−1

Pn k=1

1

(1−|z_k|²)^p−1kfk_B^p

log,whenp > 1.

Proof. Letp >0, z∈Uⁿ,from the definition ofk · k_B^p

log we have|f(0)| ≤ kfk_B^p

log

and (2.1)

∂f

∂z_k(z)

≤ kfk_B^p

log

(1− |z_k|²)^plog_1−|z²

k|²

≤ kfk_B^p

log

(1− |z_k|²)^plog 2 for everyz ∈Uⁿandk∈ {1,2, . . . , n}. Notice that

f(z)−f(0) =

n

X

k=1

f([0, z]n−k+1)−f([0, z]n−k)

=

n

X

k=1

z_k Z 1

0

∂f([0,(tz_k, z_k⁰)]n−k+1)

∂z_k dt

and then from the inequality (2.1), it follows that

|f(z)| ≤ |f(0)|+

n

X

k=1

|z_k| log 2

Z 1

0

kfk_B^p

log

(1− |tz_k|²)^pdt

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≤ kfk_B^p

log +kfk_B^p

log

log 2

n

X

k=1

Z |z_k|

0

1 (1−t²)^pdt.

(2.2)

Forp= 1, we have:

(2.3)

n

X

k=1

Z |z_k|

0

1

(1−t²)^pdt =

n

X

k=1

1

2log1 +|z_k| 1− |zk| ≤

n

X

k=1

1

2log 4 1− |zk|². Ifp > 0andp6= 1, then

(2.4)

n

X

k=1

Z |z_k|

0

1

(1−t²)^pdt≤

n

X

k=1

Z |z_k|

0

1

(1−t)^pdt =

n

X

k=1

1−(1− |z_k|)^1−p

1−p .

Now for (a), from (2.4), (2.5)

n

X

k=1

Z |zk|

0

1

(1−t²)^pdt≤ 1 1−p. From (2.2) and (2.5), it follows that

|f(z)| ≤

1 + n

(1−p) log 2

kfk_B^p

log. For (b), Sincelog _1−|z⁴

k|² >log 4 = 2 log 2for eachk∈ {1,2, . . . , n},then

(2.6) 1< 1

2nlog 2

n

X

k=1

log 4

1− |z_k|². Combining (2.2), (2.3) and (2.6) we get

|f(z)| ≤ 1

2 log 2 + 1 2nlog 2

ⁿ X

k=1

log 4

1− |z_k|²kfk_B^p

log.

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For (c), from (2.4) we have

n

X

k=1

Z |z_k|

0

1

(1−t²)^pdt ≤

n

X

k=1

1−(1− |z_k|)^p−1 (p−1)(1− |zk|)^p−1 (2.7)

≤

n

X

k=1

2^p−1

(p−1)(1− |z_k|²)^p−1. By (2.2) and (2.7), we obtain

|f(z)| ≤ kfk_B^p

log + 2^p−1

(p−1) log 2

n

X

k=1

1

(1− |z_k|²)^p−1kfk_B^p

log

≤ 1

n + 2^p−1 (p−1) log 2

n

X

k=1

1

(1− |z_k|²)^p−1kfk_B^p

log.

Lemma 2.2. Forp >0, l∈ {1,2, . . . , n}andw∈U, the functionf_w^l :Uⁿ→C, f_w^l(z) =

Z z_l

0

1

(1−wt)^plog _1−wt² dt belongs toB_log^p (Uⁿ).

Proof. Letk, l∈ {1,2, . . . , n}andw∈U, then

(2.8) ∂f_w^l

∂z_k = 0, ∀z ∈Uⁿ, k6=l and

(2.9) ∂f_w^l

∂z_l(z) = 1

(1−wz_l)^plog _1−wz²

l

, ∀z ∈Uⁿ.

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An easy estimate shows that there is0< M <+∞such that (1− |wz|)^plog _1−|wz|²

|1−wz|^plog _|1−wz|² ≤M, ∀z, w ∈U.

Therefore, by (2.8) and (2.9), we have

|f_w^l(0)|+

n

X

k=1

∂f_w^l

∂z_k(z)

(1− |z_k|²)^plog 2 1− |z_k|²

= (1− |zl|²)^plog_1−|z²

l|²

|1−wzl|^p|log_1−wz²

l|

≤ (1− |zl|²)^plog_1−|z²

l|²

(1− |wzl|)^plog _1−|wz²

l|

· (1− |wzl|)^plog _1−|wz²

l|

|1−wzl|^plog _|1−wz²

l|

≤ 2^p

pelog 2 ·M <+∞

and thus{f_w^l :w∈U, l ∈ {1,2, . . . , n}} ⊂B_log^p (Uⁿ).

Theorem 2.3. Let φ be a holomorphic self-map of the open unit polydisk Uⁿ and p, q >0, then the following statements are equivalent:

(a) C_φis a bounded operator fromB^p_log(Uⁿ)andB_log^q (Uⁿ);

(b) There is anM >0such that (2.10)

n

X

k,l=1

∂φ_l

∂z_k(z)

· (1− |z_k|²)^q

(1− |φ_l(z)|²)^p · log_1−|z²

k|²

log _1−|φ²

l(z)|²

≤M, ∀z ∈Uⁿ.

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Proof. Firstly, assume that (b) is true. By Lemma2.1, there is aC >0such that for allf ∈B_log^p (Uⁿ),

(2.11) |f(φ(0))| ≤Ckfk_B^p

log. Then for allz ∈Uⁿ,

n

X

k=1

∂(C_φf)

∂z_k (z)

(1− |z_k|²)^qlog 2 1− |z_k|²

≤

n

X

l=1

∂f

∂ξ_l(φ(z))

(1− |φ_l(z)|²)^plog 2 1− |φ_l(z)|²

×

n

X

k=1

∂φ_l

∂z_k(z)

(1− |z_k|²)^q

(1− |φ_l(z)|²)^p · log_1−|z²

k|²

log _1−|φ²

l(z)|²

≤Mkfk_B^p

log, and (a) is obtained.

Conversely, letl ∈ {1,2, . . . , n},if (a) is true, i.e. there is aC ≥0such that (2.12) kC_φfk ≤Ckfk_B^p

log, ∀f ∈B^p_log(Uⁿ), then, by Lemma2.2and (2.12), there is aQ >0such that

n

X

k=1

n

X

l=1

∂f_w^l

∂ξ_l(φ(z))· ∂φ_l

∂z_k(z)

(1−|z_k|²)^qlog 2

1− |z_k|² ≤CQ, ∀w∈U, z ∈Uⁿ. Lettingw=φ(z),and using (2.8) and (2.9), we have

n

X

l,k=1

∂φ_l

∂zk

(z)

· (1− |z_k|²)^q

(1− |φl(z)|²)^p · log_1−|z²

k|²

log _1−|φ²

l(z)|²

≤CQ.

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Lemma 2.4. Let φ : Uⁿ → Uⁿ be holomorphic and p, q > 0, then C_φ is com- pact fromB_log^p (Uⁿ)toB_log^q (Uⁿ)if and only if for any bounded sequence(f_j)j∈N in B_log^p (Uⁿ), when f_j → 0 uniformly on compacta in Uⁿ, then kC_φf_jk_B^q

log → 0 as

j → ∞.

Proof. Assume thatC_φis compact and (f_j)j∈N is a bounded sequence inB_log^p (Uⁿ) withf_j → 0uniformly on compacta in Uⁿ. If the contrary is true, then there is a subsequence(f_jm)m∈Nand aδ > 0such thatkC_φf_jmk_B^q

log ≥ δfor allm ∈ N. Due to the compactness ofCφ, we choose a subsequence(fjml◦φ)l∈Nof(Cφfjm)m∈N= (f_jm◦φ)_m∈_Nand someg ∈B_log^p (Uⁿ),such that

(2.13) lim

l→∞kf_jml◦φ−gk_B^q

log = 0.

Since Lemma2.1 implies that for any compact subsetK ⊂ Uⁿ, there is aC_k ≥ 0 such that

(2.14) |f_jml(φ(z))−g(z)| ≤C_kkf_jml◦φ−gk_B^q

log, ∀l ∈N, z∈K.

By (2.13),f_jml◦φ−g →0uniformly on compact subset inUⁿ. Sincef_jmlφ(z)→0 asl → ∞for eachz ∈Uⁿ, and by (2.14), theng = 0; (2.13) shows

l→∞lim kC_φ(f_jml)k_B^q

log = 0, it gives a contradiction.

Conversely, assume that (gj)j∈Nis a sequence in B_log^p (Uⁿ)such thatkgjk_B^p

log ≤ M for all j ∈ N. Lemma 2.1 implies that if(g_j)j∈N is uniformly bounded on any compact subset inUⁿand normal by Montel’s theorem, then there is a subsequence (g_jm)m∈N of (g_j)j∈N which converges uniformly on compacta in Uⁿ to some g ∈ H(Uⁿ). It follows that ^∂g_∂z^jm

l → _∂z^∂g

l uniformly on compacta in Uⁿ for each l ∈

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{1,2, . . . , n}. Thus g ∈ B_log^p (Uⁿ) withkg_jm −gk_B^p

log ≤ M +kgk_B^p

log < ∞ and g_jm−g converges to0on compacta inUⁿ, so by the hypotheses,g_jm◦φ →g◦φin B_log^q (Uⁿ). ThereforeC_φis a compact operator fromB_log^p (Uⁿ)toB_log^q (Uⁿ).

Lemma 2.5. If for every f ∈ B_log^p (Uⁿ), C_φf belongs to B_log^q (Uⁿ), then φ^α ∈ B_log^q (Uⁿ)for eachn-multi-indexα.

Proof. As is well known, every polynomialpα : Cⁿ →Cdefined bypα(z) =z^α is inB^p_log(Uⁿ). Thus, by the assumptionC_φ(z^α) = φ^α ∈B_log^q (Uⁿ).

Theorem 2.6. Suppose thatp, q >0, φ:Uⁿ →Uⁿis a holomorphic self-map such thatφ_k∈B^q_log(Uⁿ)for eachk∈ {1,2, . . . , n}and

(2.15) lim

φ(z)→∂Uⁿ n

X

k,l=1

∂φ_l

∂z_k(z)

· (1− |z_k|²)^q

(1− |φ_l(z)|²)^p · log_1−|z²

k|²

log _1−|φ²

l(z)|²

= 0,

thenC_φis a compact operator fromB^p_log(Uⁿ)toB_log^q (Uⁿ).

Proof. Let(f_j)j∈Nbe a sequence in B_log^p (Uⁿ)withf_j → 0uniformly on compacta inUⁿand

(2.16) kfjk_B^p

log ≤C, ∀j ∈N.

By Lemma2.4, it suffices to show that

(2.17) lim

j→∞kCφfjk_B^q

log = 0.

Notice that if kφ_mk_B^q

log = 0 for all m ∈ {1,2, . . . , n}, then φ = 0 and C_φ has finite rank. Therefore, we can assume C > 0 and kφ_mk_B^q

log > 0 for some m ∈

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{1,2, . . . , n}. Now letε >0, from (2.15), there is anr∈(0,1)such that (2.18)

n

X

k,l=1

∂φ_l

∂z_k(z)

· (1− |z_k|²)^q

(1− |φ_l(z)|²)^p · log _1−|z²

k|²

log_1−|φ²

l(z)|²

< ε 2C

for allz ∈ Uⁿ satisfyingd(φ(z), ∂Uⁿ) < r. By using a subsequence and the chain rule for derivatives, (2.16) and (2.18) guarantee that for all suchz andj ∈N,

n

X

k=1

∂(C_φf)

∂z_k (z)

(1− |zk|²)^qlog 2 1− |z_k|²

≤

n

X

l=1

∂f

∂ξ_l(φ(z))

(1− |φ_l(z)|²)^plog 2 1− |φ_l(z)|²

×

n

X

k=1

∂φl

∂z_k(z)

· (1− |zk|²)^q

(1− |φ_l(z)|²)^p · log_1−|z²

k|²

log _1−|φ²

l(z)|²

≤C· ε 2C = ε

2.

To obtain the same estimate in the cased(φ(z), ∂Uⁿ)≥r, letE_r={w: d(w, ∂Uⁿ)≥ r}. Since E_r is compact, by the hypothesis, (f_j)_j∈_N and the sequences of partial derivatives

∂fj

∂zl

j∈N

converge to0uniformly onEr,respectively. Then

n

X

k=1

∂(C_φf_j)

∂z_k (z)

(1− |z_k|²)^qlog 2 1− |z_k|²

≤

n

X

k=1

∂f_j

∂ξ_l(φ(z))

·

n

X

k=1

∂φ_l

∂z_k(z)

·(1− |z_k|²)^qlog 2 1− |z_k|²

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≤

n

X

l=1

∂f_j

∂ξ_l(φ(z))

· kφ_lk_B^q

log

≤

n

X

l=1

sup

w∈Er

∂f_j

∂ξ_l(w)

· kφ_lk_B^q

log ≤ ε

2 (asj →+∞).

Since{φ(0)}is compact, we havef_j(φ(0)) →0asj → ∞, andkC_φf_jk_B^q

log →0as j → ∞, thusCφis a compact operator fromB_log^p (Uⁿ)toB_log^q (Uⁿ).

Theorem 2.7. Letφ be a holomorphic self-map ofUⁿ andp, q > 0. If conditions (2.10) and (2.15) hold, thenC_φis a compact operator fromB_log^p (Uⁿ)toB_log^q (Uⁿ).

Proof. If (2.10) is true, thenC_φis bounded fromB_log^p (Uⁿ)toB_log^q (Uⁿ)by Theorem 2.3, andφ_k∈B_log^q (Uⁿ)for eachk ∈ {1,2, . . . , n}by Lemma2.5. The proof follows on applying (2.15) and Theorem2.6.

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References

[1] J.A. CIMA, C.S. STANTON,ANDW.R. WOGEN, On boundedness of composition operators onH(B₂), Proc. Amer. Math. Soc., 91 (1984), 217–222.

[2] D.D. CLAHANE, S. STEVIC AND ZEHUA ZHOU, Composition operators between generalized Bloch space of the ploydisk, to appear.

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