Weighted Composition Operators from Besov Zygmund-Type Spaces into Zygmund-Type Spaces

The boundedness, compactness, and essential norm of weighted composition operators from Besov Zygmund-type spaces into Zygmund-type spaces are investigated in this paper.


Introduction
Let D denote the open unit disk in the complex plane ℂ and HðDÞ the space of all analytic functions in D. For an analytic self-map φ of D and u ∈ HðDÞ, the weighted composition operator uC φ is defined as follows: When u = 1, uC φ is just the composition operator, denoted by C φ . In the past several decades, composition operators and weighted composition operators have received much attention and appear in various settings in the literature (see, for example, [2-5, 8, 10, 13, 15, 16, 19]).
Let α ∈ ð0,∞Þ. The Bloch type space B α consists of those functions f ∈ HðDÞ for which B α is a Banach space under the above norm. It is known that when α = 1, B 1 = B is the classical Bloch space.
For 0 < β < ∞, an f ∈ HðDÞ is said to be in the Zygmundtype space Z β , if It is easy to check that Z β is a Banach space under the norm ∥·∥ Z β . When β = 1, Z 1 = Z is the Zygmund space. When β > 1, Z β is just the Bloch type space B β−1 . In particular, when β = 2, Z β is just the Bloch space B. Hence, the Zygmund space is the space of all f ∈ HðDÞ such that f ′ ∈ B with norm Let dA be the normalized area measure on D. For 1 < p < ∞, the Besov space, denoted by B p , is the space of all f ∈ HðDÞ such that space. Besov spaces are Möbius invariant in the sense that b p ðf ∘ ψÞ = b p ð f Þ for all f ∈ B p and ψ ∈ AutðDÞ, the set of all Möbius maps of D (see [1,19]). In [4], Colonna and Tjani introduced a new class type space Z p p−2 , called the Besov Zygmund-type space, which consists of all f ∈ HðDÞ such that f ′ ∈ B p . Since the Besov space is contained in the Bloch space, it follows that the Besov Zygmund-type space is a subset of the Zygmund space, and hence, it is contained in the disk algebra.
In this work, we give some characterizations for the boundedness, compactness, and the essential norm of the operator uC φ : Z p p−2 ⟶ Z β . Throughout the paper, we denote by C a positive constant which may differ from one occurrence to the next. In addition, we say that

Main Results and Proofs
In this section, we formulate and prove our main results in this paper. For this purpose, we need the following lemmas.
Proof. For f ∈ B p , it is well known that Then, the inequalities in (6) follow from the definition of the Besov Zygmund-type space. Since the Zygmund space is continuously embedded into H ∞ , as shown in Lemma 2.1 of The proof is complete.
Lemma 2 (see [4]). Let 1 < p < ∞. Every sequence in Z p p−2 bounded in norm has a subsequence which converges uniformly in D to a function in Z p p−2 .
Lemma 3 (see [4]). Let X be a Banach space that is continuously contained in the disk algebra, and let Y be any Banach space of analytic functions on D. Suppose that (i) the point evaluation functionals on Y are continuous (ii) for every sequence f f n g in the unit ball of X that exists f ∈ X and a subsequence f f n j g such that The following result is a direct consequence of Lemmas 2 and 3.
The following estimates are fundamental in operator theory and function spaces on the unit disk (see ( [19], Lemma 3.10)).
Lemma 5 (see [19]). Suppose that z ∈ D,c is real, t > −1, and (i) If c < 0, then as a function of z,I c,t is bounded on D Now, we are in a position to give the following characterization of bound composition operators from Z p p−2 to Z β . Theorem 6. Let 1 < p < ∞, 0 < β < ∞, u ∈ HðDÞ, and φ be an analytic self-map of D. Then, uC φ : Journal of Function Spaces Proof. First, suppose that u ∈ Z β , (11) and (12) hold. For arbitrary z ∈ D and f ∈ Z p p−2 , by Lemma 1, we have Therefore, uC φ : Z p p−2 ⟶ Z β is bounded. Conversely, suppose that uC φ : Z p p−2 ⟶ Z β is bounded. Applying the operator uC φ to z j with j = 0, 1, 2, and using the boundedness of uC φ , we get that u ∈ Z β , uφ ∈ Z β , and uφ 2 ∈ Z β . Hence, For a ∈ D such that |a | > ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 − 2/e p , set Then, Thus, for and by Lemma 5, So, Therefore, by the boundedness of uC φ : Z p p−2 ⟶ Z β , we get Since f a ″ ðaÞ = 0, |f a ′ ðaÞ | = ðlog ð2/ð1 − jaj 2 ÞÞÞ 1−ð1/pÞ , for any w ∈ D such that ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 − 2/e p < |φðwÞ | <1, we have which implies that

Journal of Function Spaces
By (14), we get From (23) and (24), we see that (11) holds. For a ∈ D, define So, By Lemma 5, we see that g a ∈ Z p p−2 and sup a∈D ∥g a ∥ Z p p−2 < ∞. By the boundedness of uC φ : Z p p−2 ⟶ Z β , we get sup a∈D ∥uC φ g a ∥ Z β < ∞: After a calculation, g a a ð Þ = 0, Hence, for any w ∈ D, On the one hand, from (28), we obtain On the other hand, by (15), we get From (29) and (30), we see that (12) holds. The proof is complete.
Next, we estimate the essential norm of uC φ : Z p p−2 ⟶ Z β . Recall that the essential norm of uC φ : Z p p−2 ⟶ Z β is defined as the distance from uC φ to the set of compact operators K : Z p p−2 ⟶ Z β , that is, Theorem 7. Let 1 < p < ∞, 0 < β < ∞, u ∈ HðDÞ, and φ be an analytic self-map of D such that uC φ : Z p p−2 ⟶ Z β is bounded. Then, Here, Proof. First we prove that ∥uC φ ∥ e,Z p p−2 →Z β ≳ max E, G f g: ð34Þ Let fz j g j∈ℕ be a sequence in D such that |φðz j Þ | ⟶1 as j ⟶ ∞. Define