Norms of Composition Operators from Weighted Harmonic Bloch Spaces into Weighted Harmonic Zygmund Spaces

This article examines the norms of composition operators from the weighted harmonic Bloch space B λ H , 0 < λ < ∞ to the weighted harmonic Zygmund space Z β H , 0 < β < ∞ . The critical norm is on the open unit disk. We ﬁ rst give necessary and su ﬃ cient conditions where the composition operator between B λ H and Z β H is bounded. Secondly, we will study the compactness case of the composition operator between B λ H and Z β H . Finally, we will estimate the essential norms of the composition operator between B λ H and Z β H .


Introduction
Operator theory for spaces of analytic functions has been described in various settings, and there is a rich volume of studies in the academic literature that focus on the operator theory of spaces related to analytic functions on the unit disk.These studies delve into diverse environments, and the references will be highlighted below.
In [1], the second author discusses the essential norms of Stević-Sharma operators from general Banach spaces into Zygmund-type spaces, and in [2], the authors characterize the bounded and compact Stević-Sharma operator from a general class X of Banach function spaces into Zygmundtype space.In [3], the authors show a new essential norm estimate of composition operators from weighted Bloch space into μ-Bloch spaces.Cowen and MacCluer in [4] investigated composition operators on spaces of analytic functions.In [5], the necessary and sufficient conditions for the compactness and boundedness of product operator from H ∞ to Zygmund spaces were characterized.
Yet, there is a noticeable lack of investigations that offer a comprehensive look into the harmonic setting.We would like to highlight several of these references below.Characterization composition operators on some Banach spaces of harmonic mappings were discussed in [6].Colonna in [7] discussed the Bloch constant of bounded harmonic mappings.Lusky studied the weighted spaces of harmonic and holomorphic functions in [8] and then in [9] determined the isomorphism classes of weighted spaces of harmonic and analytic functions.Characterization of the harmonic Bloch space and the harmonic Besov spaces by an oscillation in [10].Jordá and Zarco studied the weighted Banach spaces of harmonic functions and the isomorphisms on weighted Banach spaces of harmonic and holomorphic functions in [11,12].
This paper is part of a series of works that address several different properties of composition operators between weighted Banach spaces of harmonic mappings.We discussed the boundedness, compactness, and the essential norm of composition operators from the space of bounded harmonic mappings H ∞ into the harmonic Zygmund space Z H in [13].Bakhit et al. in [14] discussed the boundedness, compactness, and the essential norm of composition operators from harmonic Lipschitz space into Z H .A harmonic mapping is a complex-valued function f with simply connected domain Ω such that Here, let Hol U be the space containing all analytic functions on the unit disk U and H U be the space of harmonic mappings, while G be a compact subset of the unit disk U. Further, let H ∞ H U be the space of all bounded mappings f ∈ H U equipped with the norm The harmonic mapping f always can invariably be represented in the form g + h, where both g, h ∈ Hol U .Up to an additive constant, this representation attains uniqueness.For the scope of our study, we will focus on harmonic mappings with the domain U. Therefore, See [15], as an excellent reference on the harmonic function theory.
The composition operator C ψ induced by analytic selfmaps ψ U ⟶ U (or conjugate analytic self-maps) can be expressed as Surely, this operator preserves harmonicity (see [6]).
In this work, we begin with some preliminaries that we use to derive the main results.We continue our research in [13,14] by focusing on the boundedness and the compactness of C ψ from harmonic λ-Bloch space B λ H into the weighted harmonic Zygmund space Z β H .We conclude by estimating the essential norm from B λ H into Z β H . Let Q and W be two normed linear spaces.Then, the linear operator T Q ⟶ W is bounded if there exists a positive constant C such that Further, the operator T Q ⟶ W is compact if every bounded set in Q whose closure is compact, while the essential norm T e of T Q ⟶ W is its distance from the compact operators in the operator norm.Then, the essential norm of T Q ⟶ W is given by T e,Q⟶W = inf f can be characterized as The quantity gives a Banach space structure on B λ H (see [16]).The little harmonic λ-Bloch space B λ H,0 is considered as the subspace of B λ H consisting of f ∈ H U such that Obviously, H is with the harmonic Zygmund space Z H (see [13]).
The following lemma will help to characterize the boundedness of Its proof is the proof of Theorem 19 in [16].
Let x ∈ U be a fixed point, and let α ∈ 1, 2, 3 .For any ξ ∈ U, we consider the test functions F λ x,α defined as Moreover, it is evident that lim x ⟶1 F λ x,α = 0 uniformly on G .Recall that the power series representation of F λ x,α is For all n ∈ ℕ and α ∈ 1, 2, 3 , by direct calculations, we know that Then, we have As before, for all ξ ∈ U, Then, we have Throughout this article, the notation X⪯Y means that X ≤ CY, where C > 0 is a constant.Therefore, the notation X ≈ Y means that X and Y are equivalent, when Y⪯X⪯Y.

Boundedness
In this section, we work on the boundedness of the operator ⪯1 (the authors in Theorem 2.9 of [17] have demonstrated that Therefore, Conversely, suppose that (20) holds and set Now, multiplying the above expressions ( 26) and ( 27) by 1 − ξ 2 β , we have By Lemma 2, we know that where H is a bounded operator, it suffices to show that both quantities L 1 and L 2 are finite.For ξ ∈ U since C ψ p 1 = ψ + ψ, we have Then, Moreover, we know that By the linearity of the test functions F λ ψ ξ ,α in ( 14), for α = 1, 2, 3 and ξ ∈ U, we have Journal of Function Spaces From ( 16), for α = 1, 2, 3 and ξ ∈ U, we obtain Next, for α = 1, 2, 3, we let By equation (38), for α = 1, 2, 3, we obtain Thus, from (39), we obtain Moreover, from (40), we obtain

Journal of Function Spaces
If we let ψ ξ ≤ s in (32), we have Therefore, the quantity L 2 is finite, and the proof is complete.

Compactness
In this section, we focus on discussing the compactness of the operator The proof of the following lemma is a slight modification of the proof of Proposition 3.11 in [4] (the case of Banach spaces of analytic functions).
The following theorem shows that the compactness of Proof.First, we consider the sequence p i z = i λ−1 z i + z i , for z ∈ U and i ∈ ℕ 0 .Since the sequence p i is bounded in B λ H and converges to zero uniformly on H is a bounded operator, and (47) holds by Lemma 4.
On the other hand, assume that Now, we define a sequence h i in B λ H with M = sup j∈ℕ h j B λ H < ∞, and h i ⟶ 0 uniformly on G, as i ⟶ ∞.

Essential Norm
In this section, our emphasis shifts to a comprehensive discussion regarding the essential norms of the operator

Journal of Function Spaces
Proof.First, for α = 1, 2, 3 and ξ ∈ U, by the test function ( 13), we will prove that x,α ∈ B λ H and F λ x,α converges uniformly to 0 on G.Then, for a compact operator Thus, Hence, we obtain define the sequence w i such that lim i⟶∞ ψ w i = 1, for w ∈ U. We also define uniformly on G.Moreover, by simple calculation, we have H is a compact operator, by Lemma 4, we have 74 Similarly, we have 75 Thus, Hence, we obtain Journal of Function Spaces Secondly, we prove that 78 Now, we consider the operator T γ H U ⟶ H U , for any 0 ≤ γ < 1 such that For any sequence γ i ⊂ 0, 1 such that γ i ⟶ 1 as i ⟶ ∞, we obtain But the definition of the essential norm says 81 Thus, we only need to demonstrate that It is clear that On the other hand, we consider Now, we let N ∈ ℕ be large enough and γ i ≥ 1/2, for all i ≥ N.Then, Moreover, all the limits, Hence, by the above equations, we have Next, assume ψ ξ > γ N , and we have To find estimates of the quantities R 1 , R 2 , R 3 , and R 4 , we define and n ≥ 2 and by Lemma 2, we have Similarly, we see that 96 Thus, we obtain Journal of Function Spaces Similarly, we see that By the inequalities (94)-(98), we obtain Hence, by applying (90) and (99), we determine that Finally, we prove that Now, we only need to prove that From (93), we see that Similarly, Further, for (96), we see that The proof now is complete.
Now assume the test function F λ x,α with x ∈ U in (14), for α = 1, 2, 3.By linearity of the composition operator, for any fixed positive integer n ≥ 2, we obtain
Journal of Function Spaces are uniformly on G.Then, we have

Theorem 7 .
For ψ U ⟶ U, let C ψ B λ H ⟶ Z Recall that the sequence p i z = i λ−1 z i + z i , for z ∈ U and when i ∈ ℕ 0 .Then, p i B λ H ≈1, and p i converges uniformly to 0 on G. Therefore, by Lemma 4, we see that i⟶∞ C ψ p i Z β H = limsup i⟶∞ i λ−1 ψ i + ψ i β H is bounded, then by Theorem 2