1 Introduction

Suppose that is the set of all analytic self-maps of the unit disk (the analytic maps from to itself). Let be, as usual, the collection of all analytic functions on . The composition operator, induced by an analytic self-map of the unit disk φ, could be defined as For , the weighted composition operator could be defined as

The study of composition operators and weighted composition operators has been established for over 6 decades. Many mathematicians aimed at the research of (weighted) composition operators on spaces of analytic functions on or on some high-dimension domains (e.g., the unit ball of , the unit polydisk of ). We refer to reference [1] for studying the history of composition operators acting on different spaces of analytic functions.

As one of the classical space of analytic functions on , the Bloch space, is defined as

Importantly, is maximal among all Möbius-invariant spaces of analytic functions and turns to a complete normed linear space endowed with the norm

The logarithmic Bloch space is defined as It is a Banach space endowed with the norm .

The μ–Bloch space is defined as where the weight function μ(z) is a positive continuous function in . Then is a Banach space endowed with the norm .

Recall that a Young’s function φ: [0, ∞) → [0, ∞) is a strictly increasing convex function such that φ(0) = 0 and .

In the recent years, the boundedness and compactness of composition operators among different Bloch(-type) spaces were studied (see, e.g., [2–17] and the references therein). Moreover, as the study of Hardy-Orlicz space and Bergman-Orlicz space (see, e.g., [8–14]), the Bloch-Orlicz space was defined as a generalization of .

It was firstly defined in [15], for some λ > 0 depending of f, where φ is the Young’s function. Basic properties of the convex function imply and (1.1)

We can assume without loss of generality that φ⁻¹ is differentiable (as the authors did in [15]). Suppose that then defines a semi-norm for . In this way, is a Banach space endowed with the norm

In this paper, we generalize the Bloch-Orlicz space to the logarithmic Bloch-Orlicz space. Note that it is a generalization of the logarithmic Bloch space . The logarithmic Bloch-Orlicz space is defined as follows, for some λ > 0 depending of f, where φ is the Young’s function and φ⁻¹ is differentiable. Moreover, defines a semi-norm for , where Thus, becomes a Banach space endowed with the norm

This paper is organized as follows: in Section 2 we recall some basic facts on the logarithmic Bloch-Orlicz space. In Section 3 we investigate the boundedness of weighted composition operators on the logarithmic Bloch-Orlicz space. Moreover, in Section 4 we investigate the compactness of weighted composition operators on the logarithmic Bloch-Orlicz space.

We say that if there exists a constant M_φ > 1 such that which also implies that and

2 Preliminaries

In this section, we present several basic conclusions for the study of .

Proposition 2.1 [16] holds for each .

Proof. The proof is similar with Lemma 2 in [15].

For each , a decreasing sequence {λ_n,log}_n of positive numbers can be chosen, satisfying and For any positive integer , let . Observe that {S_n} is increasing and bounded and hence there exists a real number satisfying It follows that Observe that

Thus we have S′ ≤ S. Moreover, holds for each and . Taking limit as n → ∞, the above inequality becomes for each , which is equivalent to say that This completes the proof.

Remark 2.2 The inequality (2.1) holds for all and by Lemma 2.1, where M_φ > 1 is a constant only dependent of φ. In fact, a simple estimation shows that where more details can be found in [17]. The inequality above also implies that the evaluation functional is continuous on , where is fixed.

The proposition below shows that the logarithmic Bloch-Orlicz space is isometrically equal to a μ–Bloch space.

Proposition 2.3 The logarithmic Bloch-Orlicz space is isometrically equal to a μ_log–Bloch space, where

In other words, holds for each .

Proof. Deducted by Proposition 2.1, for each and , which implies that μ_log(z)|f>′(z)| ≤ ‖f‖_φ,log holds for all . Therefore, with . Conversely, for each , holds for all , which is equivalent with It follow that Therefore, we obtain that with . Combining what we have observed above, we complete the proof.

Corollary 2.4 The equivalent condition (2.4) holds for each .

Proof. The sufficiency part is obvious. The necessity is deducted by Proposition 2.1 and the estimation .

For two real numbers A and B, we say A ≲ B if there exists a constant C ≠ 0 such that A ≤ CB, by which the complexity of all constants appearing is simplified.

3 Boundedness of weighted composition operator on

In this section we investigate the boundedness of weighted composition operators on the logarithmic Bloch-Orlicz space under the condition (this is an unexpected hypothesis). However, for those Young’s funtion , the boundedness of weighted composition operators on the logarithmic Bloch-Orlicz space remains to be an open question.

The first lemma contains some trivial but complicated calculations, which will be used in the proof in this section.

Lemma 3.1 Let where t ∈ [0, 1] and , then |g_t(z)| < 2.

Basic properties of the auxiliary functions entailed to the proof of the boundedness of weighted composition operator ψC_ϕ are described in the next lemma.

Lemma 3.2 For , and , suppose that and where . Then the auxiliary functions p_a and q_a have the properties as follows:

(i) the auxiliary function p_a belongs to the logarithmic Bloch-Orlicz space with .

(ii) the auxiliary function q_a belongs to the logarithmic Bloch-Orlicz space with .

Proof. As easy calculation shows, and Observe that Then we conclude that Further observe that, by Lemma 3.1 and Then we conclude that .

Though the approach we use in the proof of the boundedness is standard, it is not trivial since the collection contains not only (almost) linear functions, it also contains the convex functions which line between the the line whose tangent is 1 and a line whose tangent is M_φ, a positive number depending on φ.

Theorem 3.3 For , the weighted composition operator ψC_ϕ is bounded on if and only if and hold.

Proof. Suppose that M₁ < ∞, M₂ < ∞. For each , observe that where C is chosen such that M_φφ⁻¹(1)M₁ + M₂ ≤ C and the second inequality is deducted by (2.1). Then we conclude that ψC_ϕ is bounded on by and the estimation (2.1) by taking z = ϕ(0).

Conversely, if ψC_ϕ is bounded on , then there exists a constant C ≥ 0 such that for each . By Proposition 2.1 and the condition above, holds for each , which is equivalent with (3.1)

Taking , employing (3.1) we obtain that (3.2)

Further taking , employing (3.1) again we obtain that holds for each . It follows that (3.3)

On the one hand, for each , suppose that

Obviously, and

By Lemma 3.2, we have that It follows by the boundedness of ψC_ϕ that and hence by L₁ < ∞ we conclude that

On the other hand, for each , we define

By Lemma 3.2, Obviously,

For each , we have

Then we conclude that Combining what we have observed above, we complete the proof.

4 Compactness of weighted composition operator on

In this section we investigate the compactness of weighted composition operators on the logarithmic Bloch-Orlicz space, where the approach we use in the proof within is standard (see, e.g., [16]).

The first lemma contains some trivial but complicated calculations, which will be used in the proof of the compactness of weighted composition operator on . Moreover, it can be proved in a similar way with Lemma 3.2.

Lemma 4.1 For , and , suppose that where . Then the auxiliary function belongs to the logarithmic Bloch-Orlicz space with .

Theorem 4.2 For , the weighted composition operator ψC_ϕ is compact on if and only if ψC_ϕ is bounded on , (4.1) and (4.2)

Proof. Suppose that the weighted composition operator ψC_ϕ is bounded on and (4.1) (4.2) hold. Note that L₁ < ∞ and L₂ < ∞ defined in the proof of Theorem 3.3 (see, (3.2) and (3.3), respectively) by the boundedness of ψC_ϕ. For every ϵ > 0, there exists an 0 < r < 1 such that for |ϕ(z)| > r, and hold. For a chosen sequence that satisfy and {f_n} converges to zero uniformly on any compact subsets of the unit disk as n → ∞, where K is a fixed constant. It is sufficient to show that by the compactness of ψC_ϕ. Note that and converges to zero uniformly on any compact subsets of the unit disk. It follows by Proposition 2.3 that Then we conclude that ψC_ϕ is compact on by the arbitrariness of ϵ > 0.

Conversely, suppose that ψC_ϕ is compact on and hence ψC_ϕ is bounded on . We prove (4.1) and (4.2) hold as follows. Let {z_n,log}_n be a sequence in the unit disk satisfying If such sequence does not exist, then the proof is completed.

On the one hand, for each , we consider the function , where is constructed in Lemma 4.1. Then we have and Note that is bounded uniformly in and uniformly converges to zero on any compact subset of the unit disk as n → ∞ by (1.1). Thus we have It follows that and hence

On the other hand, we consider the function , where q_a is constructed in Lemma 3.2. Then we have and Note that is bounded uniformly in and uniformly converges to zero on any compact subset of the unit disk as n → ∞ since Thus we have It follows that By the same arguments shown in the proof of the boundedness of ψC_ϕ on , we conclude that Combining what we have observed above, we complete the proof.

5 Boundedness and compactness of weighted composition operators on

Recall that it is proved in Proposition 2.3 that the logarithmic Bloch-Orlicz space is isometrically equal to a μ_log–Bloch space, where Hence, the boundedness and compactness of weighted composition operators on the μ_log–Bloch space could be naturally given.

Theorem 5.1 For , the weighted composition operator ψC_ϕ is bounded on if and only if and hold.

Theorem 5.2 For , the weighted composition operator ψC_ϕ is compact on if and only if ψC_ϕ is bounded on , and

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