Weighted Composition Operators from Dirichlet-Zygmund Spaces into Zygmund-Type Spaces and Bloch-Type Spaces

where dAðzÞ = ð1/πÞdxdy is the normalized Lebesgue area measure. Dp−1 is a Banach space under the norm k·kDpp−1 . If f ′ ∈Dp−1, we say that f belongs to the Dirichlet-Zygmund space, denoted by Zp−1. To the best of our knowledge, this is the first work to study the Dirichlet-Zygmund space. Recall that the space B1, called the minimal Möbius invariant space, is the space of all f ∈HðDÞ that admit the representation f ðzÞ =∑j=1bjσt jðzÞ for some sequence fbjg in l1 and t j ∈D. The norm on f ∈ B1 is defined by


Introduction
Let HðDÞ denote the space of all analytic functions in the open unit disk D. For 1 ≤ p < ∞, the Dirichlet type space D p p−1 is the set of all f ∈ HðDÞ such that where dAðzÞ = ð1/πÞdxdy is the normalized Lebesgue area measure. D To the best of our knowledge, this is the first work to study the Dirichlet-Zygmund space.
Recall that the space B 1 , called the minimal Möbius invariant space, is the space of all f ∈ HðDÞ that admit the representation f ðzÞ = ∑ ∞ j=1 b j σ t j ðzÞ for some sequence fb j g in l 1 and t j ∈ D. The norm on f ∈ B 1 is defined by Here, σ a ðzÞ = ða − zÞ/ð1 − azÞ: For any f ∈ B 1 , the authors in [1] showed that there exists a constant C > 0 such that Therefore, Z 1 0 is in fact the space B 1 . We call v : D ⟶ ℝ + a weight, if v is a continuous, strictly positive and bounded function. v is called radial, if vðzÞ = vðjzjÞ for all z ∈ D. Let v be a radial weight. Recall that the Zygmund-type space Z v is the space that consists of all f ∈ HðDÞ such that Z v is a Banach space under the norm k·k Z v . We say that f belongs to the Bloch-type space When vðzÞ = 1 − jzj 2 , Z v = Z is called the Zygmund space, and B v = B is called the Bloch space, respectively. In particular, Z v is just the Bloch space when vðzÞ = ð1 − jzj 2 Þ 2 .
The weighted space, denoted by H ∞ v , is the set of all f ∈ HðDÞ such that We denote by SðDÞ the set of all analytic self-maps of D for simplicity. Let φ ∈ SðDÞ and ψ ∈ HðDÞ. The weighted composition operator ψC φ is defined as follows.
When ψ = 1, ψC φ is called the composition operator, denoted by C φ . See [2,3] for more results about the theory of composition operators and weighted composition operators.
In this paper, we follow the methods of [17] and give some characterizations for the boundedness, compactness, and essential norm of the operator ψC φ : We denoted by C a positive constant which may differ from one occurrence to the next. In addition, we will use the following notations throughout this paper:

Main Results and Proofs
In this section, we formulate and prove our main results in this paper.
Proof. Suppose r > 0 and g ∈ HðDÞ. Then, there exists a constant C > 0 such that which implies that The inequalities in (8) hold. Here, Dðz, rÞ is the hyperbolic disk (see [3]). From (8), we see that Z p p−1 are contained in the disk algebra for p > 1. Proof as desired.
Using Lemma 2 and similarly to the proof of Lemma 7 in [18], we get the following lemma.
Lemma 4 (see [5]). 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 For every sequence f f n g in the unit ball of X that exists an f ∈ X and a subsequence f f n j g such that Journal of Function Spaces (iii) The operator T : X ⟶ Y is continuous if X has the supremum norm and Y is given by the topology of uniform convergence on compact sets Then, T is a compact operator if and only if, given a bounded sequence f f n g in X such that f n ⟶ 0 uniformly on D, then the sequence kT f n k Y ⟶ 0 as n ⟶ ∞.
The following result is a direct consequence of Lemmas 3 and 4.
bounded in norm which converge to 0 uniformly in D.

Theorem 6.
Let v be a radial, nonincreasing weight tending to zero at the boundary of D. Let 1 < p < ∞, ψ ∈ HðDÞ, and φ ∈ SðDÞ. Then, the following statements are equivalent. and Proof. ðiiÞ ⇒ ðiÞ. For any z ∈ D and f ∈ Z p p−1 , by Lemma 1, we have Hence, Therefore, ψC φ : Z p p−1 ⟶ Z v is bounded. ðiÞ ⇒ ðiiÞ. Applying the operator ψC φ to z j with j = 0, 1 , 2 and using the boundedness of ψC φ , we get that ψ ∈ Z v , ψφ ∈ Z v , and ψφ 2 ∈ Z v . Hence, we obtain For any a ∈ D, set It is easy to check that Therefore, by the boundedness of ψC φ : Z p p−1 ⟶ Z v and arbitrary of a ∈ D, we get 3

Journal of Function Spaces
For w ∈ D, we get From (21) and (22), we obtain From (24), we get On one hand, from (25), we obtain On the other hand, from the fact that ψ, ψφ ∈ Z v , we get From (26) and (27), we see that P is finite. Using similar arguments, we see that Q is also finite.
ðiiÞ ⇔ ðiiiÞ. From [19], we see that the inequality in is equivalent to the operator ð2ψ By [20], the boundedness of ð2ψ′φ′ + ψφ′ ′ÞC φ is equivalent to Similarly, the inequality in is equivalent to The proof is complete.
Next, we consider the essential norm of ψC φ : Z Recall that the essential norm of T : X ⟶ Y is its distance to the set of compact operators K : X ⟶ Y, that is, Here, X, Y are Banach spaces, and T is a bounded linear operator. 4 Journal of Function Spaces Theorem 7. Let v be a radial, nonincreasing weight tending to zero at the boundary of D. Let 1 < p < ∞, ψ ∈ HðDÞ, and φ ∈ SðDÞ. Suppose that ψC φ : Here, Proof. First we show that kψC φ k e,Z p p−1 ⟶Z v ≳ max fE, Gg: Let fz j g j∈ℕ be a sequence in the unit disk such that jφðz j Þj ⟶ 1 as j ⟶ ∞. Define After a calculation, we get all k j and m j belong to Z p p−1 and k j φ z j À Á À Á = 0, k ′′ j φ z j À Á À Á = 0, k ′ j φ z j À Á À Á Moreover, k j and m j converge to 0 uniformly on D as j ⟶ ∞. Hence, for any compact operator K : Z p p−1 ⟶ Z v , by Lemma 5, we get