(1)COMPOSITION OPERATORS BETWEEN GENERALLY WEIGHTED BLOCH SPACES OF POLYDISK HAIYING LI, PEIDE LIU, AND MAOFA WANG SCHOOL OFMATHEMATICS ANDSTATISTICS WUHANUNIVERSITY WUHAN430072, P.R

COMPOSITION OPERATORS BETWEEN GENERALLY WEIGHTED BLOCH SPACES OF POLYDISK

HAIYING LI, PEIDE LIU, AND MAOFA WANG SCHOOL OFMATHEMATICS ANDSTATISTICS

WUHANUNIVERSITY

WUHAN430072, P.R. CHINA

tslhy2001@yahoo.com.cn

Received 23 March, 2007; accepted 26 June, 2007 Communicated by S.S. Dragomir

ABSTRACT. Letφbe a holomorphic self-map of the open unit polydiskUⁿinCⁿandp, q >0.

In this paper, the generally weighted Bloch spacesB_log^p (Uⁿ)are introduced, and the bounded- ness and compactness of composition operatorCφfromB^p_log(Uⁿ)toB_log^q (Uⁿ)are investigated.

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

2000 Mathematics Subject Classification. Primary 47B38.

1. INTRODUCTION

Suppose thatDis a domain inCⁿandφa holomorphic self-map ofD. We denote byH(D) the space of all holomorphic functions onDand 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 planeC has been studied. However, the mul- tivariable situation remains mysterious. It is well known in [3] and [5] that the restriction of C_φto Hardy or standard weighted Bergman spaces onU 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 ofC² as well as on many other spaces of holomorphic functions over a domain of Cⁿ can be unbounded, even when n = 1in [6]). Therefore, it would be of interest to pursue the function-theoretical or geometri- cal characterizations of those mapsφwhich induce bounded or compact composition operators.

In this paper, we will pursue the function-theoretic conditions of those holomorphic self-maps φ of Uⁿ which induce bounded or compact composition operators from a generally weighted p−Bloch space to aq−Bloch space withp, q >0.

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

091-07

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

zjwj, |z|=p hz, zi

for anyz = (z₁, z₂, . . . , z_n), w = (w₁, w₂, . . . , w_n)inCⁿ. α = (α₁, α₂, . . . , α_n)is said to be an n multi-index if α_i ∈ N, written by α ∈ Nⁿ. For α ∈ Nⁿ, we write z^α = z₁^α1z₂^α2· · ·z_n^αn andz_i⁰ = 1,1 ≤ i ≤ n for convenience. Forz, w ∈ Cⁿ, we denote [z, w]_j = z whenj = 0, [z, w]_j =wwhenj =n, and

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

when j ∈ {1,2, . . . , n−1}. Then [z, w]n−j = w when j = 1, and [z, w]n−j+1 = w when j =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, z_j+1, . . . , z_n),we write (a, z_j⁰) = (z₁, z₂, . . . , zj−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

Q_f(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

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

z∈Uⁿ n

X

k=1

∂f

∂z_k(z)

(1− |z_k|²)

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

In a recent paper [2], a generalized Bloch space has been introduced, the p−Bloch space:

for p > 0, a function f ∈ H(Uⁿ) belongs to the p−Bloch space B^p(Uⁿ) if there is some M ∈[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 φbe a holomorphic self-map of Uⁿ andp, 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

∂z_k(z)

(1− |z_k|²)^q

(1− |φ_l(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

∂φl

∂z_k(z)

(1− |zk|²)^q (1− |φ_l(z)|²)^p = 0 hold, thenC_φis a compact operator fromB^p(Uⁿ)toB^q(Uⁿ).

Now we introduce the generally weighted Bloch spaceB_log^p (Uⁿ).

Forp >0, a functionf ∈H(U)belongs to the generally weightedp−Bloch spaceB_log^p (Uⁿ) if there is someM ∈[0,∞)such that

n

X

k=1

∂f

∂z_k(z)

(1− |zk|²)^plog 2

1− |z_k|² ≤M, ∀z ∈Uⁿ. Its norm inB^p_log(Uⁿ)is defined by

kfk_B^p

log =|f(0)|+ sup

z∈Uⁿ n

X

k=1

∂f

∂zk

(z)

(1− |z_k|²)^plog 2 1− |zk|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 corresponding results in [2] and [11] in several ways.

2. MAINRESULTS AND THEIRPROOFS

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 2^2p−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

≤ kfk_B^p

log + kfk_B^p

log

log 2

n

X

k=1

Z |zk|

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

2log 1 +|z_k| 1− |z_k| ≤

n

X

k=1

1

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

(2.4)

n

X

k=1

Z |zk|

0

1

(1−t²)^pdt≤

n

X

k=1

Z |zk|

0

1

(1−t)^pdt=

n

X

k=1

1−(1− |zk|)^1−p

1−p .

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

n

X

k=1

Z |z_k|

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. For (c), from (2.4) we have

(2.7)

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− |z_k|)^p−1 ≤

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 zl

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

∂zl

(z) = 1

(1−wz_l)^plog _1−wz²

l

, ∀z ∈Uⁿ. 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− |z_l|²)^plog _1−|z²

l|²

|1−wz_l|^p|log _1−wz²

l|

≤ (1− |zl|²)^plog _1−|z²

l|²

(1− |wz_l|)^plog_1−|wz²

l|

·(1− |wzl|)^plog_1−|wz²

l|

|1−wz_l|^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. Letφbe a holomorphic self-map of the open unit polydiskUⁿandp, q >0, then the following statements are equivalent:

(a) C_φis a bounded operator fromB_log^p (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.

Proof. Firstly, assume that (b) is true. By Lemma 2.1, there is a C > 0 such that for all f ∈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

∂ξl

(φ(z))

(1− |φ_l(z)|²)^plog 2 1− |φl(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_log^p (Uⁿ), then, by Lemma 2.2 and (2.12), there is aQ >0such that

n

X

k=1

n

X

l=1

∂f_w^l

∂ξl

(φ(z))· ∂φ_l

∂zk

(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

∂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

≤CQ.

Lemma 2.4. Let φ : Uⁿ → Uⁿ be holomorphic and p, q > 0, then C_φ is compact from B_log^p (Uⁿ)toB_log^q (Uⁿ)if and only if for any bounded sequence(f_j)j∈NinB_log^p (Uⁿ),whenf_j →0 uniformly on compacta inUⁿ,thenkC_φf_jk_B^q

log →0asj → ∞.

Proof. Assume thatCφis compact and(fj)j∈Nis a bounded sequence inB_log^p (Uⁿ)withfj →0 uniformly on compacta inUⁿ. If the contrary is true, then there is a subsequence(f_jm)m∈Nand aδ >0such thatkC_φf_jmk_B^q

log ≥δ for allm∈ N. Due to the compactness ofC_φ, we choose a subsequence(f_jml◦φ)_l∈Nof(C_φf_jm)_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 Lemma 2.1 implies that for any compact subsetK ⊂Uⁿ, there is aC_k ≥0such 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)→0asl→ ∞ 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(g_j)_j∈N is a sequence inB_log^p (Uⁿ) such thatkg_jk_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 in Uⁿ and 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 ∈ {1,2, . . . , n}. Thus g ∈ B_log^p (Uⁿ) with kg_jm − gk_B^p

log ≤M+kgk_B^p

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

Lemma 2.5. If for everyf ∈B_log^p (Uⁿ), C_φfbelongs toB^q_log(Uⁿ), thenφ^α∈B_log^q (Uⁿ)for each n-multi-indexα.

Proof. As is well known, every polynomialp_α : Cⁿ→Cdefined byp_α(z) = z^αis inB_log^p (Uⁿ).

Thus, by the assumptionC_φ(z^α) = φ^α ∈B_log^q (Uⁿ).

Theorem 2.6. Suppose that p, q > 0, φ : Uⁿ → Uⁿ is a holomorphic self-map such that φ_k ∈B_log^q (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_log^p (Uⁿ)toB_log^q (Uⁿ).

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

(2.16) kf_jk_B^p

log ≤C, ∀j ∈N.

By Lemma 2.4, 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 ∈ {1,2, . . . , n}. Now let ε >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 all z ∈ Uⁿ satisfying d(φ(z), ∂Uⁿ) < r. By using a subsequence and the chain rule for derivatives, (2.16) and (2.18) guarantee that for all suchzandj ∈N,

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

∂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 case d(φ(z), ∂Uⁿ) ≥ r, let Er = {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

≤

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 havefj(φ(0)) → 0asj → ∞, andkC_φfjk_B^q

log → 0asj → ∞,

thusC_φis 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, then C_φ is bounded fromB_log^p (Uⁿ) toB^q_log(Uⁿ)by Theorem 2.3, and φ_k ∈B_log^q (Uⁿ)for eachk∈ {1,2, . . . , n}by Lemma 2.5. The proof follows on applying (2.15)

and Theorem 2.6.

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