Weighted Composition Operators from Derivative Hardy Spaces into n-th Weighted-Type Spaces

When p � 2, S2 is a Hilbert space. In [4], Roan started the study of composition operators on the space S. In [5], MacCluer investigated composition operators on the space S in terms of Carleson measure. +e boundedness and compactness of weighted composition operators onS were studied in [6]. See references [4–7] and references therein for more study of composition operators and weighted composition operators on the space S. If μ is a radial, positive, and continuous function on D, then μ is called a radial weight. Let μ be a radial weight and n ∈ N, the set of all positive integers. LetWnμ denote the n-th weighted space, which consists of all f ∈ H(D) such that


Introduction
Let H(D) denote the space of analytic functions on the open unit disk D. Let S(D) denote the set of all analytic self-maps of D. Let φ ∈ S(D). e composition operator C φ with the symbol φ is defined by Let φ ∈ S(D) and ψ ∈ H(D). e weighted composition operator ψC φ is defined on H(D) by It is important to give function theoretic descriptions when ψ and φ induce a bounded or compact weighted composition operator on various function spaces. See references [1,2] for more information of this research field.
For 0 < p < ∞, the Hardy space, denoted by H p , consists of all functions f ∈ H(D) such that (see [3]) As usual, H ∞ denotes the space of bounded analytic functions in D. If f ′ ∈ H p , we say that f belongs to the derivative Hardy space, denoted by S p . When p > 1, the space S p is a Banach space under the norm defined by When p � 2, S 2 is a Hilbert space. In [4], Roan started the study of composition operators on the space S p . In [5], MacCluer investigated composition operators on the space S p in terms of Carleson measure. e boundedness and compactness of weighted composition operators on S p were studied in [6]. See references [4][5][6][7] and references therein for more study of composition operators and weighted composition operators on the space S p .
If μ is a radial, positive, and continuous function on D, then μ is called a radial weight. Let μ be a radial weight and n ∈ N, the set of all positive integers. Let W n μ denote the n-th weighted space, which consists of all f ∈ H(D) such that It is a Banach space with the norm ‖ · ‖ W n μ . When n � 1 and n � 2, W n μ becomes the Bloch-type space B μ and the Zygmund-type space Z μ , respectively. Furthermore, when μ(z) � (1 − |z| 2 ), B μ � B is the Bloch space and Z μ � Z is the Zygmund space. For some results on the space W n μ , see references [2,[8][9][10][11][12][13].
Let n, k ∈ N 0 with k ≤ n. Recall that the partial Bell polynomials are defined as follows: where the sum taken over all sequences j 1 , j 2 , . . . , j n− k+1 of nonnegative integers such that the following two conditions hold: See reference [14] for more information about Bell polynomials.
In [15], Stević studied the boundedness and compactness of the composition operator from A p α to W n μ on the unit disk. In [12], Stević studied the boundedness and compactness of weighted composition operators from H ∞ and the Bloch space to W n μ . See references [8][9][10] for more characterizations for weighted composition operators from H ∞ and the Bloch space to W n μ . In [16], Zhu and Du studied the boundedness, compactness, and essential norm of weighted composition operators from weighted Bergman spaces with doubling weight A p ω to W n μ . Recall that the essential norm of a bounded linear operator T: X ⟶ Y is its distance to the set of compact operators K mapping X into Y, that is, Here, X and Y are Banach spaces and ‖ · ‖ X⟶Y denotes the operator norm. In [17], Colonna and Tjani studied the boundedness and compactness of weighted composition operators from derivative Hardy spaces S p to Bloch-type space B μ .
In this paper, we use Bell polynomials to study the boundedness and compactness of weighted composition operators from derivative Hardy spaces S p to n-th weightedtype spaces W n μ . Moreover, we give an estimate for the essential norm of weighted composition operators from S p to W n μ . roughout the paper, we denote by C a positive constant which may differ from one occurrence to the next. In addition, we say that A≲B if there exists a constant C such that A ≤ CB. e symbol A ≈ B means that A≲B≲A.

Boundedness
In this section, the boundedness of weighted composition operators from S p to W n μ is characterized.
for every f ∈ S p .
Proof. e first inequality follows from the fact that S p are contained in the disk algebra for p > 1. In addition, it is well known that for every f ∈ H p , there exists a positive constant C such that which implies the second inequality. For any a ∈ D, 1 < p < ∞, and j ∈ 1, 2, . . . , n + 1 After a calculation, for each j ∈ 1, 2, . . . , n + 1 Moreover, v i,a converges to 0 uniformly in D.

Journal of Mathematics
Proof.
e proof is similar to the proof of eorem 1 in [16]. Hence, we omit the details of the proof.
For the simplicity of this paper, we define From the definition, we see that, for example, is fact will be used in the proof of the following theorem. Now, we are in a position to state and prove the first result in this paper.
Now, we assume that for To get the desired result, we only need to show that Applying the operator ψC φ for h j (z) � z j , we obtain Hence, from the boundedness of φ and triangle inequality again, we get the desired result.
Let n � 2. We get the following result.

Essential Norm
In this section, we obtain some estimates for the essential norm of the weighted composition operator ψC φ : S p ⟶ W n μ , and we need the following lemma.
Lemma 3 (see [17]). Let X be a Banach space that is continuously contained in the disk algebra, and let Y be any Lemma 4 (see [17]). Let 1 < p < ∞. Every sequence in S p bounded in norm has a subsequence which converges uniformly in D to a function in S p .
The following result is a direct consequence of Lemma 3 and Lemma 4.

Lemma 5.
Let n ∈ N, 1 < p < ∞, and μ be a weight. If T is a bounded linear operator from S p into W n μ , then T is compact if and only if ‖Tf n ‖ W n μ ⟶ 0 as n ⟶ ∞ for any sequence f n in S p bounded in norm which convergences to 0 uniformly in D. ψ ∈ H(D), and μ be a weight such that ψC φ : S p ⟶ W n μ is bounded. en, If sup z∈D |φ(z)| < 1, there is a number δ ∈ (0, 1) such that sup z∈D |φ(z)| < δ. In this case, (32) is vacuously satisfied and the desired result follows.
Assume that sup z∈D |φ(z)| � 1. Let z j j∈N be a sequence in D such that |φ(z j )| ⟶ 1 as j ⟶ ∞. Since ψC φ : S p ⟶ W n μ is bounded, for any compact operator K: S p ⟶ W n μ and i ∈ 1, . . . , n { }, by using Lemma 2 and Lemma 5, we obtain Hence, as desired. Now, we show that Journal of Mathematics

(37)
Let r ∈ [0, 1) and define K r f(z) � f r (z) � f(rz). en, K r : S p ⟶ S p is compact and ‖K r ‖ S p ⟶ S p ≤ 1. It is clear that f r ⟶ f uniformly on compact subsets of D as r ⟶ 1. Let r j ⊂ (0, 1) be a sequence such that r j ⟶ 1 as en, for any positive integer j, the operator ψC φ K r j : S p ⟶ W n μ is compact. By the definition of the essential norm, we get Hence, it is sufficient to show that For any f ∈ S p such that ‖f‖ S p ≤ 1, where N ∈ N such that r j ≥ (2/3) for all j ≥ N. Since for any nonnegative integer s, (f − f r j ) (s) ⟶ 0 uniformly on compact subsets of D as j ⟶ ∞. It is clear that while For any i ∈ 1, . . . , n { }, by Lemma 1 and Lemma 2, Journal of Mathematics Taking the limit as N ⟶ ∞, by Lemma 1, we get By the same reason, we have Hence, by (40)-(43), (45), and (46), we obtain limsup |φ(z)|⟶1 μ(z) I n i (z) 1 − |φ(z)| 2 i+(1/p)− 1 .

(47)
Hence, from the last two inequalities, we obtain (39). e proof is complete.

(48)
Let n � 1. We get the following result, which was first obtained in [17].
Let n � 2. We get the following result.

Data Availability
No data were used to support this study.

Conflicts of Interest
e author declares that there are no conflicts of interest.
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