Composition Operators From Harmonic Lipschitz Space Into Weighted Harmonic Zygmund Space

. The paper investigate a necessary and suﬃcient condition for the composition operator from harmonic Lipschitz spaces Lip αH , (0 < α < 1) into weighted harmonic Zygmund spaces Z βH , (0 < β < ∞ ) to be bounded and compact on the open unit disk. As an application, it estimates the essential norms of such an operator from Lip αH into Z βH spaces.


Introduction
The operator theory has been characterized for spaces of analytic functions with different settings, and a significant number of related papers have appeared in the literature (see, for example, [7], [8], [10], [13], [17], and [22]).However, a similar investigation of the harmonic setting remains limited, see [2] and [14].
In [1], we have examined numerous characterizations of the weighted Bloch spaces and closed separable subspaces of harmonic mappings.We then presented the relationships between the weighted harmonic Bloch space and its Carleson measure.In [3], Aljuaid and Colonna studied the weighted Bloch space as the Banach space for harmonic mappings on an open unit disk.They then showed that the mappings in weighted Bloch space can be characterized in terms of a Lipschitz condition relative to the metric and can also be thought of as the harmonic growth space.Besides, in [5] they studied the harmonic Zygmund spaces and their closed separable subspace called the little harmonic Zygmund space.In [12], Colonna introduced and studied Bloch harmonic mappings on D as Lipschitz maps from the hyperbolic disk into C.In [19], Lusky investigated weighted spaces of harmonic functions on D and, in [20], isomorphism classes of weighted spaces of holomorphic and harmonic functions with a radial weight on C and on D. In [21], Yoneda studied harmonic Bloch spaces and harmonic Besov spaces.
Characterizations of the isometries between weighted spaces of harmonic functions were provided by Boyd and Rueda in [9].In [16], Jordá and Zarco studied Banach spaces of harmonic functions and composition operators between weighted Banach spaces of pluriharmonic functions.Isomorphisms on weighted Banach spaces of harmonic and holomorphic functions were treated in [15].
Lately, studies on operator theory acting on spaces of harmonic mappings on the unit disk have been conducted.In [4], the composition operators were studied on the Banach spaces of harmonic mappings on D, including the weighted Bloch spaces, the growth spaces, the Zygmund space, the analytic Besov spaces, and the space BMOA.Chao et al. in [11] studied composition operators in the space of bounded harmonic functions D, and then provided criteria for determining the essential norm of the difference between two composition operators.In [18], Laitila and Tylli characterized the weak compactness of the composition operators on vector-valued harmonic Hardy spaces and on the spaces of vector-valued Cauchy transforms for reflexive Banach spaces.
A harmonic mapping with domain D is a complex-valued function u such that: In this paper, let H(D) denote the space consisting of analytic functions on the unit disk D := {ζ ∈ C : |ζ| < 1}, Har (D) denote the space consisting all harmonic mappings.The harmonic mapping u is always a representation of the form h + f , where h and f are analytic functions.Up to the additive constants, this representation is unique.Therefore, u ∈ Har (D) if and only if u = h + f where h, f ∈ H(D) and f (0) = 0.For a general reference on the theory of harmonic functions, see [6].
Let S(D) be the set of all analytic or conjugate analytic self-maps of D. The composition operator C ϕ induced by ϕ ∈ S(D) is defined as the operator for all u ∈ Har (D).Surely, such an operator preserves harmonicity.
Recall that, for any two normed linear spaces X and Y , the linear operator T : X −→ Y is said to be bounded if there exists C > 0 such that T u Y ≤ C u X , ∀u ∈ X.Furthermore, a linear operator T : X −→ Y is said to be compact if it maps every bounded set in X to a relatively compact set in Y (i.e., a set whose closure is compact).
We start with several preliminaries that will be used to get the main results in this work, then we focus on the boundedness and the compactness of the composition operators from the harmonic Lipschitz spaces Li p α H , (0 < α < 1) into the harmonic weighted Zygmund spaces Z β H , (0 < β < ∞).We conclude by approximating the essential norm.

Spaces treated in this paper
Most of the research on harmonic mappings in the last two decades has been conducted by analyzing the function theoretic aspects.
Firstly, let H ∞ H = H ∞ H (D) denote the space of all bounded harmonic mappings u on D equipped with the norm The harmonic weighted Bloch space B α H .For α ∈ (0, ∞), the harmonic α-Bloch space B α H contains all u ∈ Har (D) is defined such that , the harmonic α-Bloch seminorm β α u can be characterized as The quantity yields a Banach space structure on B α H ; see [3].The harmonic little α-Bloch space B α H,0 is defined as the subspace of B α H consisting of the mappings u ∈ Har (D) such that In [12], Bloch harmonic mappings were introduced and the connection between the Lipschitz constant of a bounded harmonic mapping and its supremum norm was studied.
The harmonic Lipschitz space.For α ∈ (0, 1), Li p α H consists of all complex-valued harmonic mappings u on D satisfying the condition: there exists a constant C > 0 such that The norm of the harmonic Lipschitz space Li p α H is defined by the quantity |w −z| α : w = z}.Therefore, for w ∈ D, we have Then, taking the supremum over all w ∈ D, we get The elements of Li p α H are characterized by the following harmonic Bloch condition: (2. 2) The weighted harmonic Zygmund space Z β H .For β ∈ (0, ∞), Z β H consists of all complex-valued harmonic mappings u ∈ Har (D) such that H is with the harmonic Zygmund space Z H ; see [5].
Remark 2.1.When u ∈ H(D), the mapping ∂u ∂ζ reduces to u and ∂u ∂ζ = ∂ 2 u ∂ζ 2 = 0. Thus, for all 0 < β < ∞, the collection of analytic functions in the space Z β H is the classical weighted Zygmund space Z β and both norms are identical.
By Theorem 19 provided in [3], we have the following characterization of the harmonic Bloch-type mappings.Given 0 < α < 1 and let u ∈ Har (D), then (2.4) Let b ∈ D be fixed, and let k ∈ {1, 2, 3}.Then, for any ζ ∈ D, we consider a set of three functions F α b,k as follows: Moreover, it is evident that lim |b|→1 F α b,k = 0 uniformly on compact subsets D ⊂ D. Recall the power series representations of F α b,k are given as By direct calculation, we know that, for all n ∈ N and k ∈ {1, 2, 3}, Then, we obtain As before, for all ζ ∈ D, we have Then, we have Thus, for every k ∈ N, it can be demonstrated that Throughout this paper, we use the notation A B, which implies that there is a constant C > 0 such that A ≤ CB.Therefore, when B A B, we use the notation A ≈ B, meaning that A and B are equivalent.
For the second part L 2 , we similarly let 0 < s < 1. (3.12) Therefore, the quantity L 2 is finite.
Noting that for any ζ ∈ D and u ∈ Har Therefore, because |ϕ(0)| < 1, we note that On the other hand, for any ζ ∈ D and u ∈ Har (D), Since u ∈ Har (D), by (2.4), we obtain Finally, by taking the supremum over all ζ ∈ D the boundedness of C ϕ : Li p α H → Z β H follows from above.The proof of Theorem 3.1 is complete.

Compactness
In this section, we shift our attention to discussing the compactness of C ϕ : Li p α H → Z β H .The following criterion lemma for the compactness is similar to the case of Banach spaces of analytic functions (the analytic case), see for example Proposition 3.11 of [13].The following result indicates that the compactness of the composition operators can be characterized in terms of the sequence C ϕ p j Z β H , where p j (w ) = j −α (w j + w j ), for w ∈ D and when j ≥ 0 is an integer.Theorem 4.1.Let ϕ ∈ S(D), 0 < α < 1 and 0 < β < ∞ and assume that the operator H is bounded.Then the following are equivalent: (1) The composition operator  .Then, for ε > 0, there is N ∈ N such that By using the test function (2.6), for k = 1, 2, 3 and ζ ∈ D, we have Moreover, On the other hand, for any ζ ∈ D let 0 < s < 1 sufficiently close to 1 such that |ϕ(ζ)| > s, thus Since ε is arbitrary, for k = 1, 2, 3, it follows that Going back to the proof of Theorem 3.1, from (3.7) we know Moreover, from (3.8) we know Using (4.2), we have Moreover, by (4.3), we have (3) =⇒ (1).Suppose that (3) holds.By Theorem 3.1 and the boundedness of C ϕ : Li p α H → Z β H , we see that For any ε > 0, from (3), there is 0 < s < 1, where Now we let a sequence {h m } in the harmonic space Li p α H with and h m → 0 uniformly on compact subsets G ⊂ D, as m → ∞.To prove the compactness of Thus, using (4.4), for |ϕ(w )| > s we have Once again, going back to the proof of Theorem 3.1, from (3.1) and (3.2), we know which implies that The proof of the main theorem of this section is complete.Our next goal of this paper is to provide an approximation of the essential norm.

Essential norm
In this section, we characterize the essential norms of the composition operators from Li p α H to Z H .We know that the essential norm T e of an operator T is its distance from the compact operators in the operator norm.Precisely, consider X and Y to be Banach spaces and let T : X → Y be a bounded linear operator, then the essential norm of T between X and Y is then given by First, we define Thus, Hence, we obtain For all ζ ∈ D, it can be proven that uniformly on compact subsets G ⊂ D.Moreover, by direct calculation we see that H is a compact operator, by Lemma 4.1 we have Similarly, we have Thus, and Hence, we obtain Secondly, we prove that For any 0 ≤ δ < 1, let the operator T δ : Har (D) → Har (D) such that Without a doubt, u δ → u uniformly on compact subsets of the unit disk as δ → 1.Moreover, T δ is a compact operator on Li p α H and T δ Lip α H →Lip α H ≤ 1.For {δ i } ⊂ (0, 1) a sequence such that δ i → 1 as i → ∞.Thus, for all positive integers i , we obtain is a compact operator.However, the definition of the essential norm, indicates that Thus, we only need to demonstrate that (5. 2) It is obvious that Now let N ∈ N be large enough and δ i ≥ 1 2 , for all i ≥ N. Then Since C ϕ : Li p α H → Z β H is bounded, from Theorem 3.1, we see that In addition, since the following limits are uniformly on compact subsets G ⊂ D, Then, we have Hence, by the above equations, we have Now we estimate the quantities R j , where j = 1, 2, 3. We define Consequently, Similarly, we see that Because u Lip α H ≤ 1, for all u ∈ Li p α H , we have (5.9) Thus, we obtain (5.12) Hence, by applying (5.5) and (5.12), we determine that Finally, we prove that According to the definition of the essential norm, we only need to prove that From (5.6), we see that Moreover, for (5.9), we see that ( Recall that, the sequence p j (w ) = j −α (w j + w j ), for w ∈ D and when j ≥ 0 is an integer.Then (5.18)By (5.17) and (5.18), we have achieved the desired result.

Conclusions
In this work, an interesting result in harmonic mappings about the operator-theoretic properties of composition operators between harmonic Lipschitz spaces Li p α H , (0 < α < 1) and weighted harmonic Zygmund spaces Z β H , (0 < β < ∞) has been obtained.It is well known that the existing similar results in spaces of analytic functions have been applied many times to the composition operators between Lip α H , (0 < α < 1) and weighted harmonic Zygmund spaces Z β H , (0 < β < ∞).We hope that this study can attract people's attention to the operator theory on harmonic mappings.

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Lemma 4 . 1 .
The bounded operator T : Li p α H → Z β H is compact if and only if T u m Z β H → 0 as m → ∞, for any bounded sequence {u m } m∈N in Li p α H converges to zero uniformly on compact subsets G ⊂ D.

Proof. ( 1 )
=⇒(2).As in the proof of Theorem 3.1, since the sequence {p j } is bounded in the harmonic space Li p α H and converges to zero uniformly on compact subsets G ⊂ D, if C ϕ : Li p α H → Z β H is compact, then by Lemma 4.1, lim j→∞ C ϕ p j Z β H = 0.

1 ,
we verify that C ϕ : Li p α H → Z β H is compact.

Theorem 5 . 1 .≈.
Let ϕ ∈ S(D) and consider the bounded operator C ϕ : Li p α H max{B 1 , B 2 }.Proof.By using the test function (2.5), for k = 1, 2, 3 and ζ ∈ D, we prove that max Fix b ∈ D, since for all 1 ≤ k ≤ 3, F α b,k ∈ Li p α H and F α b,k converges uniformly to 0 on compact subsets G ⊂ D.Then, for a compact operator T : Li p α H → Z β H , we have

H.
Next, to prove that C ϕ e,Lip α H →Z β H max{B 1 , B 2 }, we define the sequence {w i } such that lim i→∞ |ϕ(w i )| = 1, for w ∈ D.Moreover, we define