Abstract and Applied Analysis
Volume 2009 (2009), Article ID 246521, 9 pages
doi:10.1155/2009/246521
Research Article

Extended Cesàro Operators from Logarithmic-Type Spaces to Bloch-Type Spaces

Dinggui Gu

Department of Mathematics, JiaYing University, Meizhou, Guangdong 514015, China

Received 27 March 2009; Accepted 12 April 2009

Academic Editor: Stevo Stević

Copyright © 2009 Dinggui Gu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The boundedness and compactness of the extended Cesàro operator from logarithmic-type spaces to Bloch-type spaces on the unit ball are completely characterized in this paper.

1. Introduction

Let 𝔹 𝑛 be the unit ball of ℂ 𝑛 , 𝜕 𝔹 𝑛 the unit sphere of ℂ 𝑛 , 𝐻 ( 𝔹 𝑛 ) the space of all holomorphic functions in 𝔹 𝑛 , and 𝐻 ∞ the space of all bounded holomorphic functions on 𝔹 𝑛 . For 𝑓 ∈ 𝐻 ( 𝔹 𝑛 ) , let ℜ 𝑓 ( 𝑧 ) = 𝑛 𝑗 = 1 0 𝑥 0 2 0 0 𝑑 𝑧 𝑗 𝜕 𝑓 𝜕 𝑧 𝑗 ( 𝑧 ) ( 1 . 1 ) denote the radial derivative of 𝑓 ∈ 𝐻 ( 𝔹 𝑛 ) .

A positive continuous function 𝜇 on the interval [ 0 , 1 ) is called normal if there is 𝛿 ∈ [ 0 , 1 ) and 𝑠 and 𝑡 , 0 < 𝑠 < 𝑡 such that 𝜇 ( 𝑟 ) ( 1 − 𝑟 ) 𝑠 [ i s d e c r e a s i n g o n 𝛿 , 1 ) , l i m 𝑟 → 1 𝜇 ( 𝑟 ) ( 1 − 𝑟 ) 𝑠 = 0 , 𝜇 ( 𝑟 ) ( 1 − 𝑟 ) 𝑡 [ i s i n c r e a s i n g o n 𝛿 , 1 ) , l i m 𝑟 → 1 𝜇 ( 𝑟 ) ( 1 − 𝑟 ) 𝑡 = ∞ . ( 1 . 2 ) If we say that 𝜇 ∶ 𝔹 𝑛 → [ 0 , ∞ ) is normal we will also assume that 𝜇 ( 𝑧 ) = 𝜇 ( | 𝑧 | ) , 𝑧 ∈ 𝔹 𝑛 .

Let 𝜇 ∶ 𝔹 𝑛 → [ 0 , ∞ ) be normal. The Bloch-type space ℬ 𝜇 = ℬ 𝜇 ( 𝔹 𝑛 ) is the space of all functions 𝑓 ∈ 𝐻 ( 𝔹 𝑛 ) such that 𝑏 𝜇 ( 𝑓 ) = s u p 𝑧 ∈ 𝔹 𝑛 | | | | 𝜇 ( 𝑧 ) ℜ 𝑓 ( 𝑧 ) < ∞ . ( 1 . 3 ) ℬ 𝜇 is Banach space with the norm ‖ 𝑓 ‖ ℬ 𝜇 = | 𝑓 ( 0 ) | + 𝑏 𝜇 ( 𝑓 ) . The little Bloch-type space ℬ 𝜇 , 0 = ℬ 𝜇 , 0 ( 𝔹 𝑛 ) consists of all 𝑓 ∈ 𝐻 ( 𝔹 𝑛 ) such that l i m | 𝑧 | → 1 | | | | 𝜇 ( 𝑧 ) ℜ 𝑓 ( 𝑧 ) = 0 . ( 1 . 4 ) It is easy to see that ℬ 𝜇 , 0 is a closed subspace of ℬ 𝜇 . When 𝜇 ( 𝑟 ) = ( 1 − 𝑟 2 ) 𝛼 , 𝛼 ∈ ( 0 , ∞ ) , we obtain so-called 𝛼 -Bloch spaces and little 𝛼 -Bloch spaces, respectively, which for 𝛼 = 1 are reduced to classical Bloch spaces (see, e.g., [1–4] and the references therein). When 𝜇 ( 𝑟 ) = ( 1 − 𝑟 2 ) l n ( 𝑒 / ( 1 − 𝑟 2 ) ) , we obtain the logarithmic Bloch space ℒ ℬ = ℒ ℬ ( 𝔹 𝑛 ) and the little logarithmic Bloch space ℒ ℬ 0 = ℒ ℬ 0 ( 𝔹 𝑛 ) , respectively, (see [5]). It was shown that 𝑓 is a multiplier of ℬ if and only if 𝑓 ∈ 𝐻 ∞ and 𝑓 ∈ ℒ ℬ in [6].

An 𝑓 ∈ 𝐻 ( 𝔹 𝑛 ) is said to belong to the logarithmic-type space 𝐻 ∞ l o g , if ‖ 𝑓 ‖ 𝐻 ∞ l o g = s u p 𝑧 ∈ 𝔹 𝑛 | | | | 𝑓 ( 𝑧 ) l n 𝑒 / 1 − | 𝑧 | 2 < ∞ . ( 1 . 5 ) It is easy to see that 𝐻 ∞ l o g becomes a Banach space under the norm ‖ ⋅ ‖ 𝐻 ∞ l o g , and that the inclusions 𝐻 ∞ ⊂ ℬ ⊂ 𝐻 ∞ l o g , hold. For some information of the space 𝐻 ∞ l o g see [7, 8].

Let 𝑔 ∈ 𝐻 ( 𝔹 𝑛 ) . The extended Cesàro operator on 𝐻 ( 𝔹 𝑛 ) is defined by 𝑇 𝑔 𝑓 ( 𝑧 ) = 1 0 𝑓 ( 𝑡 𝑧 ) ℜ 𝑔 ( 𝑡 𝑧 ) 𝑑 𝑡 𝑡 𝔹 0 𝑥 0 2 0 0 𝑑 , 𝑓 ∈ 𝐻 𝑛 , 𝑧 ∈ 𝔹 𝑛 . ( 1 . 6 ) This operator is a natural extension of a one-dimensional operator defined in [9]. Some other results on the one-dimensional operator can be found, for example, in [10, 11] (see also the references therein). For some extensions of operator (1.6) on the unit disk see [12–16]. On related operators on the unit polydisk see, for example, [17–21] and references therein. The boundedness and compactness of operator (1.6) between various spaces of holomorphic functions has been extensively studied recently, see, [17, 22–36]. For some integral operators on spaces of harmonic functions see, for example, [37] as well as the references therein. A new extension of operator (1.6) in the unit ball case have been recently introduced by Stević in [38] (see also [5, 39]).

In this paper, we study the extended Cesàro operator from 𝐻 ∞ l o g to Bloch-type spaces ℬ 𝜇 and ℬ 𝜇 , 0 . Sufficient and necessary conditions for the extended Cesàro operator 𝑇 𝑔 to be bounded and compact are given.

Throughout the paper, constants are denoted by 𝐶 , they are positive and may not be the same in every occurrence.

2. Main Results and Proofs

In this section, we give our main results and their proofs. Before stating these results, we need some auxiliary results, which are incorporated in the lemmas which follows.

Lemma 2.1. Assume that 𝑔 ∈ 𝐻 ( 𝔹 𝑛 ) and 𝜇 ∶ 𝔹 𝑛 → [ 0 , ∞ ) are normal. Then 𝑇 𝑔 ∶ 𝐻 ∞ l o g → ℬ 𝜇 is compact if and only if 𝑇 𝑔 ∶ 𝐻 ∞ l o g → ℬ 𝜇 is bounded and for any bounded sequence ( 𝑓 𝑘 ) 𝑘 ∈ ℕ in 𝐻 ∞ l o g which converges to zero uniformly on compact subsets of 𝔹 𝑛 as 𝑘 → ∞ , one has ‖ 𝑇 𝑔 𝑓 𝑘 ‖ ℬ 𝜇 → 0 as 𝑘 → ∞ .

The proof of Lemma 2.1 follows by standard arguments (see, e.g., Lemmas 3 in [20, 21, 29]). Hence, we omit the details.

Lemma 2.2. Assume that 𝜇 ∶ 𝔹 𝑛 → [ 0 , ∞ ) is normal. A closed set 𝐾 in ℬ 𝜇 , 0 is compact if and only if it is bounded and satisfies l i m | 𝑧 | → 1 s u p 𝑓 ∈ 𝐾 | | | | 𝜇 ( 𝑧 ) ℜ 𝑓 ( 𝑧 ) = 0 . ( 2 . 1 )

This lemma can be found in [5], and its proof is similar to the proof of Lemma 1 in [40]. Hence, it will be omitted.

The following result was proved in [8].

Lemma 2.3. There exist two functions 𝑓 1 , 𝑓 2 ∈ 𝐻 ∞ l o g ( 𝔹 1 ) such that | | 𝑓 1 | | + | | 𝑓 ( 𝑧 ) 2 | | 1 ( 𝑧 ) ≥ l n 1 − | 𝑧 | , 𝑧 ∈ 𝔹 1 . ( 2 . 2 )

Now we are in a position to state and prove our main results.

Theorem 2.4. Assume that 𝑔 ∈ 𝐻 ( 𝔹 𝑛 ) and 𝜇 ∶ 𝔹 𝑛 → [ 0 , ∞ ) is normal. Then 𝑇 𝑔 ∶ 𝐻 ∞ l o g → ℬ 𝜇 is bounded if and only if 𝑀 = s u p 𝑧 ∈ 𝔹 𝑛 𝜇 | | | | 𝑒 ( 𝑧 ) ℜ 𝑔 ( 𝑧 ) l n 1 − | 𝑧 | 2 < ∞ . ( 2 . 3 ) Moreover, if 𝑇 𝑔 ∶ 𝐻 ∞ l o g → ℬ 𝜇 is bounded then the following asymptotic relation holds ‖ ‖ 𝑇 𝑔 ‖ ‖ 𝐻 ∞ l o g → ℬ 𝜇 ≍ s u p 𝑧 ∈ 𝔹 𝑛 𝜇 | | | | 𝑒 ( 𝑧 ) ℜ 𝑔 ( 𝑧 ) l n 1 − | 𝑧 | 2 < ∞ . ( 2 . 4 )

Proof. Assume that (2.3) holds. Then, for any 𝑓 ∈ 𝐻 ∞ l o g , we have | | ℜ 𝑇 𝜇 ( 𝑧 ) 𝑔 𝑓 | | | | | | | | | | | | | | 𝑒 ( 𝑧 ) = 𝜇 ( 𝑧 ) ℜ 𝑔 ( 𝑧 ) 𝑓 ( 𝑧 ) ≤ 𝜇 ( 𝑧 ) ℜ 𝑔 ( 𝑧 ) l n 1 − | 𝑧 | 2 ‖ 𝑓 ‖ 𝐻 ∞ l o g . ( 2 . 5 ) In addition, it is easy to see that 𝑇 𝑔 𝑓 ( 0 ) = 0 . Therefore we have ‖ ‖ 𝑇 𝑔 𝑓 ‖ ‖ ℬ 𝜇 = s u p 𝑧 ∈ 𝔹 𝑛 𝜇 | | ℜ 𝑇 ( 𝑧 ) 𝑔 𝑓 | | ( 𝑧 ) ≤ 𝑀 ‖ 𝑓 ‖ 𝐻 ∞ l o g ( 2 . 6 ) as desired.
Conversely, assume that 𝑇 𝑔 ∶ 𝐻 ∞ l o g → ℬ 𝜇 is bounded. For 𝑎 ∈ 𝔹 𝑛 , set 𝑓 𝑎 𝑒 ( 𝑧 ) = l n 1 − ⟨ 𝑧 , 𝑎 ⟩ . ( 2 . 7 ) It is easy to see that 𝑓 𝑎 ∈ 𝐻 ∞ l o g and s u p 𝑎 ∈ 𝔹 𝑛 ‖ 𝑓 𝑎 ‖ 𝐻 ∞ l o g < ∞ .
For any 𝑏 ∈ 𝔹 𝑛 , we have ‖ ‖ 𝑇 ∞ > 𝑔 𝑓 𝑏 ‖ ‖ ℬ 𝜇 = s u p 𝑧 ∈ 𝔹 𝑛 | | ℜ 𝑇 𝜇 ( 𝑧 ) 𝑔 𝑓 𝑏 ( | | 𝑧 ) = s u p 𝑧 ∈ 𝔹 𝑛 | | | | | | 𝑓 𝜇 ( 𝑧 ) ℜ 𝑔 ( 𝑧 ) 𝑏 | | | | | | 𝑒 ( 𝑧 ) ≥ 𝜇 ( 𝑏 ) ℜ 𝑔 ( 𝑏 ) l n | | 𝑏 | | 1 − 2 , ( 2 . 8 ) from which (2.3) follows, moreover s u p 𝑧 ∈ 𝔹 𝑛 𝜇 | | | | 𝑒 ( 𝑧 ) ℜ 𝑔 ( 𝑧 ) l n 1 − | 𝑧 | 2 ‖ ‖ 𝑇 ≤ 𝐶 𝑔 ‖ ‖ 𝐻 ∞ l o g → ℬ 𝜇 . ( 2 . 9 ) From (2.6) and (2.9), we see that (2.4) holds. The proof is completed.

Theorem 2.5. Assume that 𝑔 ∈ 𝐻 ( 𝔹 𝑛 ) and 𝜇 ∶ 𝔹 𝑛 → [ 0 , ∞ ) is normal. Then 𝑇 𝑔 ∶ 𝐻 ∞ l o g → ℬ 𝜇 is compact if and only if l i m | 𝑧 | → 1 𝜇 | | | | 𝑒 ( 𝑧 ) ℜ 𝑔 ( 𝑧 ) l n 1 − | 𝑧 | 2 = 0 . ( 2 . 1 0 )

Proof. Suppose that 𝑇 𝑔 ∶ 𝐻 ∞ l o g → ℬ 𝜇 is compact. Let ( 𝑧 𝑘 ) 𝑘 ∈ ℕ be a sequence in 𝔹 𝑛 such that l i m 𝑘 ⇀ ∞ | 𝑧 𝑘 | = 1 . Set 𝑓 𝑘 𝑒 ( 𝑧 ) = l n 1 − ⟨ 𝑧 , 𝑧 𝑘 ⟩ 2 𝑒 l n | | 𝑧 1 − 𝑘 | | 2 − 1 , 𝑘 ∈ ℕ . ( 2 . 1 1 ) It is easy to see that s u p 𝑘 ∈ ℕ ‖ 𝑓 𝑘 ‖ 𝐻 ∞ l o g < ∞ . Moreover 𝑓 𝑘 → 0 uniformly on compact subsets of 𝔹 𝑛 as 𝑘 → ∞ . By Lemma 2.1, l i m 𝑘 → ∞ ‖ ‖ 𝑇 𝑔 𝑓 𝑘 ‖ ‖ ℬ 𝜇 = 0 . ( 2 . 1 2 ) In addition, ‖ ‖ 𝑇 𝑔 𝑓 𝑘 ‖ ‖ ℬ 𝜇 = s u p 𝑧 ∈ 𝔹 𝑛 𝜇 | | ( 𝑧 ) ℜ 𝑔 ( 𝑧 ) 𝑓 𝑘 | | 𝑧 ( 𝑧 ) ≥ 𝜇 𝑘 | | 𝑧 ℜ 𝑔 𝑘 | | 𝑒 l n | | 𝑧 1 − 𝑘 | | 2 , ( 2 . 1 3 ) which together with (2.12) implies that l i m 𝑘 → ∞ 𝜇 𝑧 𝑘 | | 𝑧 ℜ 𝑔 𝑘 | | 𝑒 l n | | 𝑧 1 − 𝑘 | | 2 = 0 . ( 2 . 1 4 ) From the above inequality we see that (2.10) holds.
Conversely, assume that (2.10) holds. From Theorem 2.4 we see that 𝑇 𝑔 ∶ 𝐻 ∞ l o g → ℬ 𝜇 is bounded. In order to prove that 𝑇 𝑔 ∶ 𝐻 ∞ l o g → ℬ 𝜇 is compact, according to Lemma 2.1, it suffices to show that if ( 𝑓 𝑘 ) 𝑘 ∈ ℕ is a bounded sequence in 𝐻 ∞ l o g converging to 0 uniformly on compact subsets of 𝔹 𝑛 , then l i m 𝑘 → ∞ ‖ ‖ 𝑇 𝑔 𝑓 𝑘 ‖ ‖ ℬ 𝜇 = 0 . ( 2 . 1 5 )
Let ( 𝑓 𝑘 ) 𝑘 ∈ ℕ be a bounded sequence in 𝐻 ∞ l o g such that 𝑓 𝑘 → 0 uniformly on compact subsets of 𝔹 𝑛 as 𝑘 → ∞ . By (2.10) we have that for any 𝜀 > 0 , there is a constant 𝛿 ∈ ( 0 , 1 ) , such that 𝜇 | | | | 𝑒 ( 𝑧 ) ℜ 𝑔 ( 𝑧 ) l n | | 𝑧 1 − 𝑘 | | 2 < 𝜀 ( 2 . 1 6 ) whenever 𝛿 < | 𝑧 | < 1 . Let 𝐾 = { 𝑧 ∈ 𝔹 𝑛 ∶ | 𝑧 | ≤ 𝛿 } . From (2.10) we see that 𝑔 ∈ ℬ 𝜇 . Equality (2.16) along with the fact that 𝑔 ∈ ℬ 𝜇 implies ‖ ‖ 𝑇 𝑔 𝑓 𝑘 ‖ ‖ ℬ 𝜇 = s u p 𝑧 ∈ 𝔹 𝑛 𝜇 | | ℜ 𝑇 ( 𝑧 ) 𝑔 𝑓 𝑘 | | ( 𝑧 ) = s u p 𝑧 ∈ 𝔹 𝑛 | | 𝜇 ( 𝑧 ) ℜ 𝑔 ( 𝑧 ) 𝑓 𝑘 | | ≤ ( 𝑧 ) s u p { 𝑧 ∈ 𝔹 𝑛 ∶ | 𝑧 | ≤ 𝛿 } + s u p { 𝑧 ∈ 𝔹 𝑛 ∶ 𝛿 ≤ | 𝑧 | < 1 } | | | | | | 𝑓 𝜇 ( 𝑧 ) ℜ 𝑔 ( 𝑧 ) 𝑘 | | ( 𝑧 ) ≤ ‖ 𝑔 ‖ ℬ 𝜇 s u p 𝑧 ∈ 𝐾 | | 𝑓 𝑘 | | ( 𝑧 ) + s u p { 𝑧 ∈ 𝔹 𝑛 ∶ 𝛿 ≤ | 𝑧 | < 1 } | | | | 𝑒 𝜇 ( 𝑧 ) 𝑔 ( 𝑧 ) l n 1 − | 𝑧 | 2 ‖ ‖ 𝑓 𝑘 ‖ ‖ 𝐻 ∞ l o g ≤ ‖ 𝑔 ‖ ℬ 𝜇 s u p 𝑧 ∈ 𝐾 | | 𝑓 𝑘 | | ( 𝑧 ) + 𝐶 𝜀 . ( 2 . 1 7 ) Observe that 𝐾 is a compact subset of 𝔹 𝑛 , so that l i m 𝑘 → ∞ s u p 𝑧 ∈ 𝐾 | | 𝑓 𝑘 | | ( 𝑧 ) = 0 . ( 2 . 1 8 ) Therefore l i m s u p 𝑘 → ∞ ‖ ‖ 𝑇 𝑔 𝑓 𝑘 ‖ ‖ ℬ 𝜇 ≤ 𝐶 𝜀 . ( 2 . 1 9 ) Since 𝜀 > 0 is an arbitrary positive number it follows that the last limit is equal to zero. Therefore, 𝑇 𝑔 ∶ 𝐻 ∞ l o g → ℬ 𝜇 is compact. The proof is completed.

Remark 2.6. From [24] and Theorems 2.4 and 2.5, we see that 𝑇 𝑔 ∶ 𝐻 ∞ l o g → ℬ is bounded if and only if 𝑇 𝑔 ∶ ℬ → ℬ is bounded; 𝑇 𝑔 ∶ 𝐻 ∞ l o g → ℬ is compact if and only if 𝑇 𝑔 ∶ ℬ → ℬ is compact.

Theorem 2.7. Assume that 𝑔 ∈ 𝐻 ( 𝔹 𝑛 ) and 𝜇 ∶ 𝔹 𝑛 → [ 0 , ∞ ) is normal. Then the following statements are equivalent: (a) 𝑇 𝑔 ∶ 𝐻 ∞ l o g → ℬ 𝜇 , 0 is bounded;(b) 𝑇 𝑔 ∶ 𝐻 ∞ l o g → ℬ 𝜇 , 0 is compact;(c) l i m | 𝑧 | → 1 𝜇 | | | | 𝑒 ( 𝑧 ) ℜ 𝑔 ( 𝑧 ) l n 1 − | 𝑧 | 2 = 0 . ( 2 . 2 0 )

Proof. (b) ⇒ (a). This implication is obvious.
(a) ⇒ (c). Assume that 𝑇 𝑔 ∶ 𝐻 ∞ l o g → ℬ 𝜇 , 0 is bounded. Now we prove that (2.20) holds. Note that (2.20) is equivalent with l i m | 𝑧 | → 1 | | | | 1 𝜇 ( 𝑧 ) ℜ 𝑔 ( 𝑧 ) l n 1 − | 𝑧 | = 0 . ( 2 . 2 1 ) Hence we only need to show that (2.21) holds. This can be done by contradiction. Now assume that the condition (2.21) does not hold. If it was, then it would exist 𝜀 0 > 0 and a sequence ( 𝑧 ( 𝑗 ) ) 𝑗 ∈ ℕ ∈ 𝔹 𝑛 , such that 𝑧 ( 𝑗 ) → 𝜕 𝔹 𝑛 , and 𝜇 𝑧 ( 𝑗 ) | | 𝑧 ℜ 𝑔 ( 𝑗 ) | | 1 l n | | 𝑧 1 − ( 𝑗 ) | | ≥ 𝜀 0 > 0 ( 2 . 2 2 ) for sufficiently large 𝑗 . We may assume that l i m 𝑗 → ∞ 𝑧 ( 𝑗 ) = ( 1 , 0 , … , 0 ) and also | | 𝑧 1 − ( 𝑗 ) | | > | | 𝑧 1 − 1 ( 𝑗 ) | | 2 , 𝑗 ∈ ℕ . ( 2 . 2 3 ) According to Lemma 2.3 we know that there exist two functions ℎ 1 , ℎ 2 ∈ 𝐻 ∞ l o g ( 𝔹 1 ) such that | | ℎ 1 𝑧 1 | | + | | ℎ 2 𝑧 1 | | 1 ≥ l n | | 𝑧 1 − 1 | | , 𝑧 1 ∈ 𝔹 1 . ( 2 . 2 4 ) Let 𝐹 1 ( 𝑧 ) = ℎ 1 𝑧 1 , 𝐹 2 ( 𝑧 ) = ℎ 2 𝑧 1 𝑧 , 𝑧 = 1 , … , 𝑧 𝑛 ∈ 𝔹 𝑛 . ( 2 . 2 5 ) Then clearly 𝐹 1 , 𝐹 2 ∈ 𝐻 ∞ l o g . By the boundedness of 𝑇 𝑔 ∶ 𝐻 ∞ l o g → ℬ 𝜇 , 0 we have 𝑇 𝑔 𝐹 1 , 𝑇 𝑔 𝐹 2 ∈ ℬ 𝜇 , 0 . ( 2 . 2 6 ) On the other hand, 𝜇 𝑧 ( 𝑗 ) | | ℜ 𝑇 𝑔 𝐹 1 𝑧 ( 𝑗 ) | | + | | ℜ 𝑇 𝑔 𝐹 2 𝑧 ( 𝑗 ) | | 𝑧 = 𝜇 ( 𝑗 ) | | 𝑧 ℜ 𝑔 ( 𝑗 ) | | | | 𝐹 1 𝑧 ( 𝑗 ) | | + | | 𝐹 2 𝑧 ( 𝑗 ) | | 𝑧 = 𝜇 ( 𝑗 ) | | 𝑧 ℜ 𝑔 ( 𝑗 ) | | | | | ℎ 1 𝑧 1 ( 𝑗 ) | | | + | | | ℎ 2 𝑧 1 ( 𝑗 ) | | | 𝑧 ≥ 𝜇 ( 𝑗 ) | | 𝑧 ℜ 𝑔 ( 𝑗 ) | | 1 l n | | 𝑧 1 − 1 ( 𝑗 ) | | 𝑧 ≥ 𝐶 𝜇 ( 𝑗 ) | | 𝑧 ℜ 𝑔 ( 𝑗 ) | | 1 l n | | 𝑧 1 − ( 𝑗 ) | | ≥ 𝐶 𝜀 0 > 0 ( 2 . 2 7 ) for sufficiently large 𝑗 . Since 𝑧 ( 𝑗 ) → 𝜕 𝔹 𝑛 , from the above inequality we obtain that 𝑇 𝑔 𝐹 1 , 𝑇 𝑔 𝐹 2 ∉ ℬ 𝜇 , 0 , which is a contradiction.
(c) ⇒ (b). From (1.5) we have that | | ℜ 𝑇 𝜇 ( 𝑧 ) 𝑔 𝑓 | | | | | | 𝑒 ( 𝑧 ) ≤ 𝜇 ( 𝑧 ) ℜ 𝑔 ( 𝑧 ) l n 1 − | 𝑧 | 2 ‖ 𝑓 ‖ 𝐻 ∞ l o g . ( 2 . 2 8 ) Taking the supremum in the above inequality over all 𝑓 ∈ 𝐻 ∞ l o g such that ‖ 𝑓 ‖ 𝐻 ∞ l o g ≤ 1 , then letting | 𝑧 | → 1 , by (2.20) we arrive at l i m | 𝑧 | → 1 s u p ‖ 𝑓 ‖ 𝐻 ∞ l o g ≤ 1 𝜇 | | ℜ 𝑇 ( 𝑧 ) 𝑔 | | ( 𝑓 ) ( 𝑧 ) = 0 . ( 2 . 2 9 ) From this and by employing Lemma 2.2, we see that 𝑇 𝑔 ∶ 𝐻 ∞ l o g → ℬ 𝜇 , 0 is compact. The proof is completed.

From Theorems 2.4, 2.5, and 2.7, we have the following corollary.

Corollary 2.8. Let 𝑔 ∈ 𝐻 ( 𝔹 𝑛 ) . Then (1) 𝑇 𝑔 ∶ 𝐻 ∞ l o g → ℒ ℬ is bounded if and only if s u p 𝑧 ∈ 𝔹 𝑛 1 − | 𝑧 | 2 | | | | 𝑒 ℜ 𝑔 ( 𝑧 ) l n 1 − | 𝑧 | 2 2 < ∞ . ( 2 . 3 0 ) (2) 𝑇 𝑔 ∶ 𝐻 ∞ l o g → ℒ ℬ is compact if and only if 𝑇 𝑔 ∶ 𝐻 ∞ l o g → ℒ ℬ 0 is bounded if and only if 𝑇 𝑔 ∶ 𝐻 ∞ l o g → ℒ ℬ 0 is compact if and only if l i m | 𝑧 | → 1 1 − | 𝑧 | 2 | | | | 𝑒 ℜ 𝑔 ( 𝑧 ) l n 1 − | 𝑧 | 2 2 = 0 . ( 2 . 3 1 )

Acknowledgment

The author is supported partly by the Natural Science Foundation of Guangdong Province (no. 07006700).

References

  1. D. D. Clahane and S. Stević, “Norm equivalence and composition operators between Bloch/Lipschitz spaces of the ball,” Journal of Inequalities and Applications, vol. 2006, Article ID 61018, 11 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. S. Li and S. Stević, “Some characterizations of the Besov space and the α-Bloch space,” Journal of Mathematical Analysis and Applications, vol. 346, no. 1, pp. 262–273, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  3. S. Li and H. Wulan, “Characterizations of α-Bloch spaces on the unit ball,” Journal of Mathematical Analysis and Applications, vol. 343, no. 1, pp. 58–63, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  4. K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, vol. 226 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2005. View at MathSciNet
  5. S. Stević, “On a new operator from the logarithmic Bloch space to the Bloch-type space on the unit ball,” Applied Mathematics and Computation, vol. 206, no. 1, pp. 313–320, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. K. H. Zhu, “Multipliers of BMO in the Bergman metric with applications to Toeplitz operators,” Journal of Functional Analysis, vol. 87, no. 1, pp. 31–50, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. X. Fu and X. Zhu, “Weighted composition operators on some weighted spaces in the unit ball,” Abstract and Applied Analysis, vol. 2008, Article ID 605807, 8 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. D. Girela, J. Á. Peláez, F. Pérez-González, and J. Rättyä, “Carleson measures for the Bloch space,” Integral Equations and Operator Theory, vol. 61, no. 4, pp. 511–547, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  9. Ch. Pommerenke, “Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation,” Commentarii Mathematici Helvetici, vol. 52, no. 4, pp. 591–602, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. A. Aleman and J. A. Cima, “An integral operator on Hp and Hardy's inequality,” Journal d'Analyse Mathématique, vol. 85, pp. 157–176, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. A. G. Siskakis and R. Zhao, “A Volterra type operator on spaces of analytic functions,” in Function Spaces, vol. 232 of Contemporary Mathematics, pp. 299–311, American Mathematical Society, Providence, RI, USA, 1999. View at Zentralblatt MATH · View at MathSciNet
  12. S. Li and S. Stević, “Generalized composition operators on Zygmund spaces and Bloch type spaces,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1282–1295, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. S. Li and S. Stević, “Products of composition and integral type operators from H∞ to the Bloch space,” Complex Variables and Elliptic Equations, vol. 53, no. 5, pp. 463–474, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. S. Li and S. Stević, “Products of Volterra type operator and composition operator from H∞ and Bloch spaces to Zygmund spaces,” Journal of Mathematical Analysis and Applications, vol. 345, no. 1, pp. 40–52, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. S. Li and S. Stević, “Products of integral-type operators and composition operators between Bloch-type spaces,” Journal of Mathematical Analysis and Applications, vol. 349, no. 2, pp. 596–610, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. S. Stević, “Norms of some operators from Bergman spaces to weighted and Bloch-type spaces,” Utilitas Mathematica, vol. 76, pp. 59–64, 2008. View at Zentralblatt MATH · View at MathSciNet
  17. D.-C. Chang, S. Li, and S. Stević, “On some integral operators on the unit polydisk and the unit ball,” Taiwanese Journal of Mathematics, vol. 11, no. 5, pp. 1251–1285, 2007. View at Zentralblatt MATH · View at MathSciNet
  18. D.-C. Chang and S. Stević, “Estimates of an integral operator on function spaces,” Taiwanese Journal of Mathematics, vol. 7, no. 3, pp. 423–432, 2003. View at Zentralblatt MATH · View at MathSciNet
  19. S. Stević, “Cesàro averaging operators,” Mathematische Nachrichten, vol. 248-249, no. 1, pp. 185–189, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. S. Stević, “Boundedness and compactness of an integral operator on a weighted space on the polydisc,” Indian Journal of Pure and Applied Mathematics, vol. 37, no. 6, pp. 343–355, 2006. View at Zentralblatt MATH · View at MathSciNet
  21. S. Stević, “Boundedness and compactness of an integral operator in a mixed norm space on the polydisk,” Sibirskiĭ Matematicheskiĭ Zhurnal, vol. 48, no. 3, pp. 694–706, 2007. View at MathSciNet
  22. K. Avetisyan and S. Stević, “Extended Cesàro operators between different Hardy spaces,” Applied Mathematics and Computation, vol. 207, no. 2, pp. 346–350, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. Z. Hu, “Extended Cesàro operators on mixed norm spaces,” Proceedings of the American Mathematical Society, vol. 131, no. 7, pp. 2171–2179, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. Z. Hu, “Extended Cesáro operators on the Bloch space in the unit ball of ℂn,” Acta Mathematica Scientia. Series B, vol. 23, no. 4, pp. 561–566, 2003. View at Zentralblatt MATH · View at MathSciNet
  25. Z. Hu, “Extended Cesàro operators on Bergman spaces,” Journal of Mathematical Analysis and Applications, vol. 296, no. 2, pp. 435–454, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. S. Li, “Riemann-Stieltjes operators from F(p,q,s) spaces to α-Bloch spaces on the unit ball,” Journal of Inequalities and Applications, vol. 2006, Article ID 27874, 14 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. S. Li and S. Stević, “Integral type operators from mixed-norm spaces to α-Bloch spaces,” Integral Transforms and Special Functions, vol. 18, no. 7-8, pp. 485–493, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. S. Li and S. Stević, “Riemann-Stieltjes operators on Hardy spaces in the unit ball of ℂn,” Bulletin of the Belgian Mathematical Society. Simon Stevin, vol. 14, no. 4, pp. 621–628, 2007. View at Zentralblatt MATH · View at MathSciNet
  29. S. Li and S. Stević, “Riemann-Stieltjes-type integral operators on the unit ball in ℂn,” Complex Variables and Elliptic Equations, vol. 52, no. 6, pp. 495–517, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. S. Li and S. Stević, “Compactness of Riemann-Stieltjes operators between F(p,q,s) spaces and α-Bloch spaces,” Publicationes Mathematicae Debrecen, vol. 72, no. 1-2, pp. 111–128, 2008. View at MathSciNet
  31. S. Li and S. Stević, “Riemann-Stieltjes operators between different weighted Bergman spaces,” Bulletin of the Belgian Mathematical Society. Simon Stevin, vol. 15, no. 4, pp. 677–686, 2008. View at MathSciNet
  32. S. Li and S. Stević, “Riemann-Stieltjes operators between mixed norm spaces,” Indian Journal of Mathematics, vol. 50, no. 1, pp. 177–188, 2008. View at Zentralblatt MATH · View at MathSciNet
  33. S. Li and S. Stević, “Cesàro-type operators on some spaces of analytic functions on the unit ball,” Applied Mathematics and Computation, vol. 208, no. 2, pp. 378–388, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. S. Stević, “On an integral operator on the unit ball in ℂn,” Journal of Inequalities and Applications, vol. 2005, no. 1, pp. 81–88, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. X. Tang, “Extended Cesàro operators between Bloch-type spaces in the unit ball of ℂn,” Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 1199–1211, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. J. Xiao, “Riemann-Stieltjes operators on weighted Bloch and Bergman spaces of the unit ball,” Journal of the London Mathematical Society. Second Series, vol. 70, no. 1, pp. 199–214, 2004. View at Zentralblatt MATH · View at MathSciNet
  37. S. Stević, “Compactness of the Hardy-Littlewood operator on some harmonic function spaces,” Siberian Mathematical Journal, vol. 50, no. 1, pp. 167–180, 2009. View at Publisher · View at Google Scholar
  38. S. Stević, “On a new operator from H∞ to the Bloch-type space on the unit ball,” Utilitas Mathematica, vol. 77, pp. 257–263, 2008. View at Zentralblatt MATH · View at MathSciNet
  39. S. Stević, “On a new integral-type operator from the weighted Bergman space to the Bloch-type space on the unit ball,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 154263, 14 pages, 2008. View at Zentralblatt MATH · View at MathSciNet
  40. K. Madigan and A. Matheson, “Compact composition operators on the Bloch space,” Transactions of the American Mathematical Society, vol. 347, no. 7, pp. 2679–2687, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet