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

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 ℜ𝑓(𝑧)=^𝑛_𝑗=1𝑧_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)^𝑠isdecreasingon[𝛿,1),lim_𝑟→1𝜇(𝑟)(1−𝑟)^𝑠=0,𝜇(𝑟)(1−𝑟)^𝑡isincreasingon[𝛿,1),lim_𝑟→1𝜇(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)=sup_{𝑧∈𝔹𝑛}𝜇(𝑧)|ℜ𝑓(𝑧)|<∞. (1.3)ℬ_𝜇 is Banach space with the norm ‖𝑓‖_ℬ𝜇=|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 lim_|𝑧|→1𝜇(𝑧)|ℜ𝑓(𝑧)|=0. (1.4) It is easy to see that ℬ_𝜇,0 is a closed subspace of ℬ_𝜇. When 𝜇(𝑟)=(1−𝑟²)^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−𝑟²)ln(𝑒/(1−𝑟²)), we obtain the logarithmic Bloch space ℒℬ=ℒℬ(𝔹_𝑛) and the little logarithmic Bloch space ℒℬ₀=ℒℬ₀(𝔹_𝑛), respectively, (see [5]). It was shown that 𝑓 is a multiplier of ℬ if and only if 𝑓∈𝐻^∞ and 𝑓∈ℒℬ in [6].

An 𝑓∈𝐻(𝔹_𝑛) is said to belong to the logarithmic-type space 𝐻^∞_log, if ‖𝑓‖_𝐻∞log=sup_{𝑧∈𝔹𝑛}|𝑓(𝑧)|ln(𝑒/(1−|𝑧|²))<∞. (1.5) It is easy to see that 𝐻^∞_log becomes a Banach space under the norm ‖⋅‖_𝐻^∞_log, and that the inclusions 𝐻^∞⊂ℬ⊂𝐻^∞_log, hold. For some information of the space 𝐻^∞_log see [7, 8].

Let 𝑔∈𝐻(𝔹_𝑛). The extended Cesàro operator on 𝐻(𝔹_𝑛) is defined by 𝑇_𝑔𝑓(𝑧)=¹₀𝑓(𝑡𝑧)ℜ𝑔(𝑡𝑧)𝑑𝑡𝑡,𝑓∈𝐻(𝔹_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^∞_log 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^∞_log→ℬ_𝜇 is compact if and only if 𝑇_𝑔∶𝐻^∞_log→ℬ_𝜇 is bounded and for any bounded sequence (𝑓_𝑘)_𝑘∈ℕ in 𝐻^∞_log which converges to zero uniformly on compact subsets of 𝔹_𝑛 as 𝑘→∞, one has ‖𝑇_𝑔𝑓_𝑘‖_ℬ𝜇→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 lim_|𝑧|→1sup_𝑓∈𝐾𝜇(𝑧)|ℜ𝑓(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 𝑓₁,𝑓₂∈𝐻^∞_log(𝔹₁) such that |𝑓₁(𝑧)|+|𝑓₂1(𝑧)|≥ln1−|𝑧|,𝑧∈𝔹₁. (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^∞_log→ℬ_𝜇 is bounded if and only if 𝑀=sup_{𝑧∈𝔹𝑛}𝑒𝜇(𝑧)|ℜ𝑔(𝑧)|ln1−|𝑧|²<∞. (2.3) Moreover, if 𝑇_𝑔∶𝐻^∞_log→ℬ_𝜇 is bounded then the following asymptotic relation holds ‖𝑇_𝑔‖_{𝐻∞log→ℬ𝜇}≍sup_{𝑧∈𝔹𝑛}𝑒𝜇(𝑧)|ℜ𝑔(𝑧)|ln1−|𝑧|²<∞. (2.4)

Proof. Assume that (2.3) holds. Then, for any 𝑓∈𝐻^∞_log, we have 𝜇(𝑧)|ℜ(𝑇_𝑔𝑒𝑓)(𝑧)|=𝜇(𝑧)|ℜ𝑔(𝑧)||𝑓(𝑧)|≤𝜇(𝑧)|ℜ𝑔(𝑧)|(ln1−|𝑧|²)‖𝑓‖_𝐻∞log. (2.5) In addition, it is easy to see that 𝑇_𝑔𝑓(0)=0. Therefore we have ‖𝑇_𝑔𝑓‖_ℬ𝜇=sup_{𝑧∈𝔹𝑛}𝜇(𝑧)|ℜ(𝑇_𝑔𝑓)(𝑧)|≤𝑀‖𝑓‖_𝐻∞log (2.6) as desired.
Conversely, assume that 𝑇_𝑔∶𝐻^∞_log→ℬ_𝜇 is bounded. For 𝑎∈𝔹_𝑛, set 𝑓_𝑎𝑒(𝑧)=ln1−⟨𝑧,𝑎⟩. (2.7) It is easy to see that 𝑓_𝑎∈𝐻^∞_log and sup_{𝑎∈𝔹𝑛}‖𝑓_𝑎‖_𝐻∞log<∞.
For any 𝑏∈𝔹_𝑛, we have ∞>‖𝑇_𝑔𝑓_𝑏‖_ℬ𝜇=sup_{𝑧∈𝔹𝑛}𝜇(𝑧)|ℜ(𝑇_𝑔𝑓_𝑏)(𝑧)|=sup_{𝑧∈𝔹𝑛}𝜇(𝑧)|ℜ𝑔(𝑧)||𝑓_𝑏𝑒(𝑧)|≥𝜇(𝑏)|ℜ𝑔(𝑏)|ln1−|𝑏|², (2.8) from which (2.3) follows, moreover sup_{𝑧∈𝔹𝑛}𝑒𝜇(𝑧)|ℜ𝑔(𝑧)|ln1−|𝑧|²≤𝐶‖𝑇_𝑔‖_{𝐻∞log→ℬ𝜇}. (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^∞_log→ℬ_𝜇 is compact if and only if lim_|𝑧|→1𝑒𝜇(𝑧)|ℜ𝑔(𝑧)|ln1−|𝑧|²=0. (2.10)

Proof. Suppose that 𝑇_𝑔∶𝐻^∞_log→ℬ_𝜇 is compact. Let (𝑧_𝑘)_𝑘∈ℕ be a sequence in 𝔹_𝑛 such that lim_𝑘⇀∞|𝑧_𝑘|=1. Set 𝑓_𝑘𝑒(𝑧)=(ln1−⟨𝑧,𝑧_𝑘⟩)²𝑒(ln1−|𝑧_𝑘|²)⁻¹,𝑘∈ℕ. (2.11) It is easy to see that sup_𝑘∈ℕ‖𝑓_𝑘‖_𝐻∞log<∞. Moreover 𝑓_𝑘→0 uniformly on compact subsets of 𝔹_𝑛 as 𝑘→∞. By Lemma 2.1, lim_𝑘→∞‖𝑇_𝑔𝑓_𝑘‖_ℬ𝜇=0. (2.12) In addition, ‖𝑇_𝑔𝑓_𝑘‖_ℬ𝜇=sup_{𝑧∈𝔹𝑛}𝜇(𝑧)|ℜ𝑔(𝑧)𝑓_𝑘(𝑧)|≥𝜇(𝑧_𝑘)|ℜ𝑔(𝑧_𝑘𝑒)|ln1−|𝑧_𝑘|², (2.13) which together with (2.12) implies that lim_𝑘→∞𝜇(𝑧_𝑘)|ℜ𝑔(𝑧_𝑘𝑒)|ln1−|𝑧_𝑘|²=0. (2.14) 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^∞_log→ℬ_𝜇 is bounded. In order to prove that 𝑇_𝑔∶𝐻^∞_log→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^∞_log converging to 0 uniformly on compact subsets of 𝔹_𝑛, then lim_𝑘→∞‖𝑇_𝑔𝑓_𝑘‖_ℬ𝜇=0. (2.15)
Let (𝑓_𝑘)_𝑘∈ℕ be a bounded sequence in 𝐻^∞_log 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 𝑒𝜇(𝑧)|ℜ𝑔(𝑧)|ln1−|𝑧_𝑘|²<𝜀 (2.16) 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_𝑔𝑓_𝑘‖_ℬ𝜇=sup_{𝑧∈𝔹𝑛}𝜇(𝑧)|ℜ(𝑇_𝑔𝑓_𝑘)(𝑧)|=sup_{𝑧∈𝔹𝑛}𝜇(𝑧)|ℜ𝑔(𝑧)𝑓_𝑘(𝑧)|≤(sup_{{𝑧∈𝔹𝑛∶|𝑧|≤𝛿}}+sup_{{𝑧∈𝔹𝑛∶𝛿≤|𝑧|<1}})𝜇(𝑧)|ℜ𝑔(𝑧)||𝑓_𝑘(𝑧)|≤‖𝑔‖_ℬ𝜇sup_𝑧∈𝐾|𝑓_𝑘(𝑧)|+sup_{{𝑧∈𝔹𝑛∶𝛿≤|𝑧|<1}}𝑒𝜇(𝑧)|𝑔(𝑧)|(ln1−|𝑧|²)‖𝑓_𝑘‖_𝐻∞log≤‖𝑔‖_ℬ𝜇sup_𝑧∈𝐾|𝑓_𝑘(𝑧)|+𝐶𝜀. (2.17) Observe that 𝐾 is a compact subset of 𝔹_𝑛, so that lim_𝑘→∞sup_𝑧∈𝐾|𝑓_𝑘(𝑧)|=0. (2.18) Therefore limsup_𝑘→∞‖𝑇_𝑔𝑓_𝑘‖_ℬ𝜇≤𝐶𝜀. (2.19) Since 𝜀>0 is an arbitrary positive number it follows that the last limit is equal to zero. Therefore, 𝑇_𝑔∶𝐻^∞_log→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^∞_log→ℬ is bounded if and only if 𝑇_𝑔∶ℬ→ℬ is bounded; 𝑇_𝑔∶𝐻^∞_log→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^∞_log→ℬ_𝜇,0 is bounded;(b)𝑇_𝑔∶𝐻^∞_log→ℬ_𝜇,0 is compact;(c)lim_|𝑧|→1𝑒𝜇(𝑧)|ℜ𝑔(𝑧)|ln1−|𝑧|²=0. (2.20)

Proof. (b)⇒(a). This implication is obvious.
(a)⇒(c). Assume that 𝑇_𝑔∶𝐻^∞_log→ℬ_𝜇,0 is bounded. Now we prove that (2.20) holds. Note that (2.20) is equivalent with lim_|𝑧|→11𝜇(𝑧)|ℜ𝑔(𝑧)|ln1−|𝑧|=0. (2.21) 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 𝜀₀>0 and a sequence (𝑧^(𝑗))_𝑗∈ℕ∈𝔹_𝑛, such that 𝑧^(𝑗)→𝜕𝔹_𝑛, and 𝜇(𝑧^(𝑗))|ℜ𝑔(𝑧^(𝑗)1)|ln1−|𝑧^(𝑗)|≥𝜀₀>0 (2.22) for sufficiently large 𝑗. We may assume that lim_𝑗→∞𝑧^(𝑗)=(1,0,…,0) and also 1−|𝑧^(𝑗)|>1−|𝑧₁^(𝑗)|2,𝑗∈ℕ. (2.23) According to Lemma 2.3 we know that there exist two functions ℎ₁,ℎ₂∈𝐻^∞_log(𝔹₁) such that |ℎ₁(𝑧₁)|+|ℎ₂(𝑧₁1)|≥ln1−|𝑧₁|,𝑧₁∈𝔹₁. (2.24) Let 𝐹₁(𝑧)=ℎ₁(𝑧₁),𝐹₂(𝑧)=ℎ₂(𝑧₁),𝑧=(𝑧₁,…,𝑧_𝑛)∈𝔹_𝑛. (2.25) Then clearly 𝐹₁,𝐹₂∈𝐻^∞_log. By the boundedness of 𝑇_𝑔∶𝐻^∞_log→ℬ_𝜇,0 we have 𝑇_𝑔𝐹₁,𝑇_𝑔𝐹₂∈ℬ_𝜇,0. (2.26) On the other hand, 𝜇(𝑧^(𝑗))(|ℜ(𝑇_𝑔𝐹₁)(𝑧^(𝑗))|+|ℜ(𝑇_𝑔𝐹₂)(𝑧^(𝑗))|)=𝜇(𝑧^(𝑗))|ℜ𝑔(𝑧^(𝑗))|(|𝐹₁(𝑧^(𝑗))|+|𝐹₂(𝑧^(𝑗))|)=𝜇(𝑧^(𝑗))|ℜ𝑔(𝑧^(𝑗))|(|ℎ₁(𝑧₁^(𝑗))|+|ℎ₂(𝑧₁^(𝑗))|)≥𝜇(𝑧^(𝑗))|ℜ𝑔(𝑧^(𝑗)1)|ln1−|𝑧₁^(𝑗)|≥𝐶𝜇(𝑧^(𝑗))|ℜ𝑔(𝑧^(𝑗)1)|ln1−|𝑧^(𝑗)|≥𝐶𝜀₀>0 (2.27) for sufficiently large 𝑗. Since 𝑧^(𝑗)→𝜕𝔹_𝑛, from the above inequality we obtain that 𝑇_𝑔𝐹₁,𝑇_𝑔𝐹₂∉ℬ_𝜇,0, which is a contradiction.
(c)⇒(b). From (1.5) we have that 𝜇(𝑧)|ℜ(𝑇_𝑔𝑒𝑓)(𝑧)|≤𝜇(𝑧)|ℜ𝑔(𝑧)|(ln1−|𝑧|²)‖𝑓‖_𝐻∞log. (2.28) Taking the supremum in the above inequality over all 𝑓∈𝐻^∞_log such that ‖𝑓‖_𝐻∞log≤1, then letting |𝑧|→1, by (2.20) we arrive at lim_|𝑧|→1sup_{‖𝑓‖𝐻∞log≤1}𝜇(𝑧)|ℜ(𝑇_𝑔(𝑓))(𝑧)|=0. (2.29) From this and by employing Lemma 2.2, we see that 𝑇_𝑔∶𝐻^∞_log→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^∞_log→ℒℬ is bounded if and only if sup_{𝑧∈𝔹𝑛}(1−|𝑧|²𝑒)|ℜ𝑔(𝑧)|(ln1−|𝑧|²)²<∞. (2.30)(2)𝑇_𝑔∶𝐻^∞_log→ℒℬ is compact if and only if 𝑇_𝑔∶𝐻^∞_log→ℒℬ₀ is bounded if and only if 𝑇_𝑔∶𝐻^∞_log→ℒℬ₀ is compact if and only if lim_|𝑧|→1(1−|𝑧|²𝑒)|ℜ𝑔(𝑧)|(ln1−|𝑧|²)²=0. (2.31)

Acknowledgment

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