Weighted Differentiation Composition Operators to Bloch-Type Spaces

and Applied Analysis 3 (d) Dn φ,ψ 󸀠 : B0 → A α ∞ and Dn+1 φ,ψφ 󸀠 : B0 → A α ∞ are bounded. (e) Dn φ,ψ : B → Bα is bounded. (f) Dn φ,ψ 󸀠 : B → A α ∞ and Dn+1 φ,ψφ 󸀠 : B → A α ∞ are bounded. (g) sup z∈D((1 − |z| 2 ) α /(1 − |φ(z)| 2 ) n )|ψ 󸀠 (z)| < ∞ and sup z∈D((1 − |z| 2 ) α /(1 − |φ(z)| 2 ) n+1 )|ψ(z)φ 󸀠 (z)| < ∞. (h) sup k∈N‖D n φ,ψ 󸀠(z k )‖ Aα ∞ < ∞ and sup k∈N‖D n+1 φ,ψφ 󸀠(z k )‖ Aα ∞ < ∞. Proof. It is obvious that (f) ⇒ (b), (f) ⇒ (d), (e) ⇒ (c), and (e) ⇒ (a). Thus, we will prove the theorem according to the following steps. (I): (a) ⇒ (g), (c) ⇒ (g). (II): (b) ⇒ (g), (d) ⇒ (g). (III): (g) ⇒ (e), (g) ⇒ (f). (IV): (f) ⇔ (h). (I): (a) ⇒ (g), (c) ⇒ (g). Suppose that (a) or (c) holds. We choose the test function g 1 (z) = z . By Lemma 2, we get 󵄩󵄩󵄩󵄩g1 󵄩󵄩󵄩󵄩B ≤ 󵄩󵄩󵄩󵄩g1 󵄩󵄩󵄩󵄩BMOA ≲ 󵄩󵄩󵄩󵄩g1 󵄩󵄩󵄩󵄩∞ = 1. (19) So sup z∈D (1 − |z| 2 ) α 󵄨󵄨󵄨󵄨 ψ 󸀠 (z) 󵄨󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩󵄩 D n φ,ψ g 1 󵄩󵄩󵄩󵄩Bα < ∞. (20) Taking g 2 (z) = z n+1 and using the fact that |φ(z)| < 1, we have sup z∈D (1 − |z| 2 ) α 󵄨󵄨󵄨󵄨 ψ (z) φ 󸀠 (z) 󵄨󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩󵄩 D n φ,ψ g 2 󵄩󵄩󵄩󵄩Bα + 󵄩󵄩󵄩󵄩 D n φ,ψ g 1 󵄩󵄩󵄩󵄩Bα < ∞. (21) We now consider the function f λ (z) = (n + 1) 1 − 󵄨󵄨󵄨󵄨φ(λ) 󵄨󵄨󵄨󵄨 2 1 − φ(λ)z − (1 − |φ(λ)| 2 ) 2 (1 − φ(λ)z) 2 , λ ∈ D. (22) It is easy to check that f λ ∈ B 0 ∩ BMOA and ‖f λ ‖BMOA ≲ ‖f λ ‖ ∞ ≲ 1. Moreover,


Introduction
Let D be the open unit disk in the complex plane C, (D) the space of all functions holomorphic on D, ( ) = (1/ ) the normalized area measure on D, and ∞ the space of all bounded holomorphic functions with the norm ‖ ‖ ∞ = sup ∈D | ( )|.
Let > 0. The -Bloch space B on D is the space of all holomorphic functions on D such that The little -Bloch space B 0 consists of all ∈ B such that lim | | → 1 Both spaces B and B 0 are Banach spaces with the norm and B 0 is a closed subspace of B . If = 1, they become the classical Bloch space B and little Bloch space B 0 , respectively. For any > 0, the space A ∞ consists of functions ∈ (D) such that For information of such spaces, see, for example, [1][2][3][4].
For ∈ D, let ( ) = ( − )/(1 − ) be the automorphism of D that interchanges 0 and . Let the Green function in D with logarithmic singularity at be given by The space BMOA consists of all in the Hardy space 2 such that BMOA is a Banach space under following norm (see, e.g., [5]): Let and be holomorphic maps on the open unit disk D such that (D) ⊂ D. For a nonnegative integer , we define a linear operator , as follows: We call it weighted differentiation composition operators, which was defined in [6,7]. If = 0 and ≡ 1, , becomes induced by , defined as = ∘ , ∈ (D). If = 1 and ( ) = , then , is the differentiation operator defined as = ( ) . If = 0, then we get the weighted 2 Abstract and Applied Analysis composition operator defined as = ⋅ ( ∘ ). If = 1 and ( ) = ( ), then , reduces to . When ≡ 1, then , reduces to differentiation composition operator (also named as product of differentiation and composition operator). If we put ( ) = , then , = , the product of multiplication and differentiation operator.
Boundedness, compactness, and essential norm of weighted composition operator between Bloch-type spaces have been studied in [22][23][24]. Recently, Manhas and Zhao [25] and Hyvärinen and Lindström [26] gave a new characterization of boundedness and compactness of in terms of the norm of (for the compactness of composition operator, see [27,28]).
Throughout this paper, constants are denoted by ; they are positive and not necessarily the same at each occurrence. The notation ≲ means that there is a positive constant such that ≤ . When ≲ and ≲ , we write ≈ .

Some Lemmas
It is well known that ∞ ⊂ BMOA ⊂ B. From the definition of the norm, we know Indeed, Girela proved that in Corollary 5.2 of [5]. The following lemma is from Lemma 5 in [29] (see also Lemma 4.12 of [4]).

Lemma 1.
If ∈ (D), then The following lemma may be known, but we fail to find its reference; so we give a proof for the completeness of the paper.

Lemma 2.
Let ∈ (D). Then, Proof. Applying Littlewood-Paley identity and Lemma 1, we have It follows from the definitions of Bloch space and BMOA space that By Theorem 6.2 of [5] and the proof of Theorem 1 of [30], we have the following lemma.

Lemma 3. Let be a fixed positive integer and ∈ B with
then ‖ ‖ BMOA ≲ 1.

Lemma 4.
Suppose that is a fixed positive integer. Let ∈ N + , 0 ≤ ≤ 1, and If ≥ , then there are two positive constants and , depending only on , such that Proof. The proof is similar to that of Lemma 2.2 of [13] and is so omitted.

Boundedness of ,
In this section, we characterize the boundedness of , from BMOA and the Bloch space to Bloch-type spaces.
Theorem 5. Let > 0, ∈ (D), ∈ N + , and a holomorphic self-map of D. Then, the following statements are equivalent: Abstract and Applied Analysis and (e) ⇒ (a). Thus, we will prove the theorem according to the following steps.
From now on, we assume that ‖ ‖ ∞ = 1. For any integer ≥ , let Let with ≥ be the smallest positive integer such that Δ ̸ = ⌀. Since Δ is not empty for every integer ≥ and D = ∪ ∞ = Δ . By Lemma 4, for ∈ B, Thus, +1 , : B → A ∞ is bounded. Theorem 5 is proved.

Compactness of ,
The following criterion for the compactness is a useful tool and it follows from standard arguments, for example, Proposition 3.11 of [32] or Lemma 2.10 of [33]. We now give the compactness of , from BMOA and the Bloch space to Bloch-type spaces. : Proof. The proof is a modification of that of Theorem 5; so we give a sketch of the proof. We will prove the theorem according to the following steps.
That is, ∈ B , ∈ A ∞ . Let { } be a sequence in D such that | ( )| → 1 as → ∞. Now, we consider the function Simple computation shows that ∈ B 0 ∩ BMOA and It is also easy to check that → 0 uniformly on compact subsets of D as → ∞. Moreover, ] . (62) We have By Lemma 6, we get We next consider the function Similarly, we get ∈ B 0 ∩ BMOA and It is easy to see that converges to zero uniformly on compact subsets of D as → ∞ and ] . (67) Thus, Applying Lemma 6 again, we have Since ∈ D is arbitrary, we proved that (g) is true.
And let with ≥ be the smallest positive integer such that Δ ̸ = ⌀. For given > 0, there exists a large enough integer 1 with 1 > such that whenever > 1 . Let { } be a norm bounded sequence in B that converges to zero uniformly on compact subsets of D as → ∞. Denote = sup ‖ ‖ B < ∞. We get Then, where  B → A ∞ is compact. Similar as above, we can prove that +1 , : B → A ∞ is compact. The proof is complete.