Integral Superposition-Type Operators on Some Analytic Function Spaces

All entire functions which transform a class of holomorphic Zygmund-type spaces into weighted analytic Bloch space using the socalled n-generalized superposition operator are characterized in this paper. Moreover, certain specific properties such as boundedness and compactness of the newly defined class of generalized integral superposition operators are discussed and established by using the concerned entire functions.


Fundamental Materials
Let D = fz : jzj < 1g be the known concerned open unit disk in ℂ. Assume that Ξ and Ξ 1 are two concerned metric spaces of holomorphic-type functions both defined on the complex unit disk D. Let ϕ be a concerned complex-valued function defined on D. The known superposition operator S ϕ on the concerned metric space Ξ is here defined by When the operator S ϕ f ∈ Ξ 1 for f ∈ Ξ 1 , we call that ϕ is induced by the concerned superposition operator from Ξ into Ξ 1 . When the concerned metric space Ξ contains specific concerned linear functions, thus, ϕ must be a concerned entire-type function. The symbol HðDÞ is supposed to be the class of all concerned holomorphic functions on D. For various emerging research studies on the superpositiontype operators, we refer to [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and others.
The motivation of this current study arises from the recent research of the theory of operators on the theory of complex-type function spaces.
The nonlinear n-generalized superposition operator can be defined in this current paper by The operator S g,n ϕ will be called the n-generalized superposition operator, where g n−1 ð Þ z ð Þ = d n−1 g z ð Þ dz n−1 , with } n − 1 } -order derivatives, n ∈ ℕ: ð3Þ Remark 1. When n = 1 and g = f ′ , the superposition operator is obtained because the difference S g,1 ϕ − S ϕ is a positive constant. Therefore, S g,n ϕ is a generalization of the superpositiontype operator. For good knowledge, the new operator S g,n ϕ is defined and discussed in the present concerned manuscript for the first time.
Remark 2. It should be noted that the graph of the n-generalized superposition operator S g,n ϕ is almost closed, but because the new operator is a nonlinear operator, then there is no guarantee to establish its boundedness. Nonetheless, there are some spaces Ξ and Ξ 1 , for instance, holomorphictype Bergman, holomorphic Bloch, and holomorphic-type Dirichlet spaces, which implies that the specific function ϕ must belong to a specific concerned class of entire-type functions, resulting in the boundedness-type property.
The following new definition can also be introduced.
Definition 5. Let h ∈ HðDÞ; then, h is said to be in the weighted little ν-type concerned Bloch space B ν,0 when Due to the result of Zygmund in [15] as well as the closed graph theorem, we can deduce that It is obvious to see that the space Z is a Banach-type space with the specific norm k•k Z , where Here, the concerned Zygmund-type space on D, denoted by Z 0 , is a concerned closed subspace of the holomorphictype space Z that consists of all holomorphic-type functions h, for which Using (7), it is very obvious to deduce that Some important and essential auxiliary results shall be proved in the next lemmas.

Lemma 6.
Let h ∈ B ν and w ∈ D; then, the following inequality can be obtained: for some λ which is independent of h, with Proof. Let jwj > 1/2, w = r ξ, and ξ ∈ ∂D. Then, we infer that Additionally, we deduce Let jwj ≤ 1/2; then, applying the known mean value theorem for the function hðwÞ − hð0Þ (cf. [16]) and considering Jensen's inequality, we deduce that where the homogeneous representation expansion of the function h results in the second inequality. Therefore, Combining (13) and (15), we deduce The proof is therefore established completely.
Hereafter, the letter Γ can be used to denote a concerned planar domain and ∂Γ defines its boundary.
In view of the known Riemann mapping theorem [16], for any Γ, there exists a holomorphic function h (that can be called a concerned Riemann mapping) which maps D onto Γ and the concerned origin to a prescribed concerned specific point. The known Euclidean metric of the specific point w to the concerned boundary Γ is denoted by dðw, ∂ ΓÞ; also, the concerned Riemann-type mapping h has the next interesting concerned property: which is comparatively similar to the concerned constructions of the concerned connected domains as the concerned images for specific classes of functions in different analytictype function spaces can be obtained in [4].

Auxiliary Results
The next lemmas establish some elementary properties of the ν-Bloch-type spaces of analytic functions.

Lemma 7.
Let h ∈ HðDÞ, and assume that (11) holds. Then, for the ν-Bloch space B ν , we have the following: (i) Every bounded concerned sequence ðh n Þ ∈ B ν is uniformly bounded on concerned compact sets (ii) For any concerned sequence ðh n Þ on B ν with kh n k B ν ⟶ 0, then h n − h n ð0Þ ⟶ 0 and the convergence is of uniform type on concerned compact sets Proof. If w ∈ Uð0, rÞ, r ∈ ð0, 1Þ; thus, the following inequalities can be deduced Hence, the result follows.

Lemma 8.
Assume that h, g ∈ HðDÞ and ϕ : D ⟶ D. Thus, for the n-generalized superposition operator S g,n ϕ : B ν ⟶ B ν which is a concerned compact operator and ⟺S g,n ϕ : B ν ⟶ B ν which is actually bounded for any bounded concerned sequence ðh n Þ n∈N ∈ B ν with h n ⟶ 0 uniformly bounded on concerned compact sets if n ⟶ ∞, we obtain kS g,n Proof. Now, the concerned assertions (7), (9), and (11) of Lemma 3.7 in [17] can be satisfied for the analytic-type B ν space. In view of Lemma 7, thus, it is clear to show that the concerned assertions (i) and (ii) hold. To clear that (ii) holds, assume that the specific sequence ðh n Þ can be taken as a concerned sequence in the specific closed unit ball of B ν . Therefore, using Lemma 7 implies that ðh n Þ can be seen as uniformly bounded on concerned compact sets. Hence, applying the theorem of Montel [18], we can get a concerned subsequence ðh n k Þðn 1 < n 2 <⋯Þ, for which h n k ⟶ h which indeed is uniformly bounded on concerned compact sets, for some concerned functions h * ∈ HðDÞ. Then, it is enough to clear that h * ∈ B ν . From Fatou's theorem [16] and the concerned assumptions, we deduce that Therefore, in view of Lemma 3.7 in [17], we infer that the n-generalized superposition operator S g,n Thus, the proof of Lemma 8 is clearly obtained.

Lemma 9.
Let h ∈ HðDÞ. The concerned closed set K ∈ B ν,0 is compact iff it is bounded and satisfies the next concerned condition: Proof. Assume that K is compact, and let ε > 0; thus, the specific balls that centered at the specific elements of the concerned K with radii ε/2 cover K; then, by the help of compactness, we can find functions h 1 , h 2 , ⋯, h n ∈ K, for which the analytic function h ∈ K yields that kh − h j k B ν < ε/2 for some 1 ≤ j ≤ n; hence, Now, for each j, we can find r j ∈ ð0, 1Þ, for which By letting r = max fr 1 , ⋯, r n g, we obtain This establishes that (20) holds.
On the other hand, assume that K ⊂ B ν,0 is closed and bounded and satisfies (20).
Then, the set K can be a normal family. When ðh n Þ is a concerned sequence in the set K, then passing to a specific subsequence, we can suppose that h n ⟶ h with the concerned uniformly converging on the concerned compact 3 Journal of Function Spaces subsets of D. Hence, we can clear that h n ⟶ h in B ν,0 . Then, for a given ε > 0, using (20), we can find r ∈ ð0, 1Þ, for which , for all r < w j j < 1 and all h ∈ K: ð24Þ Since the convergence h n ′ ⟶ h ′ is uniformly bounded on the concerned compact subsets of the disk D, it yields that the type of convergence for h n ′ ⟶ h ′ is pointwise on D. Additionally, we deduce that Therefore, Because the convergence h n ′ ⟶ h ′ is uniformly bounded on r D, then we can find a specific positive integer N, for which Hence, Hence, h n ⟶ h in B ν . Because K is specifically closed, then we can infer that h ∈ K. Therefore, we deduce that the concerned set K is a compact set. The proof of Lemma 9 is therefore completely established.

Superpositions on Bloch and Zygmund Spaces
In this section, some interesting and important properties of the new class of the n-generalized superposition S g,n ϕ acting from the analytic-type Zygmund classes to the analytic ν -Bloch classes are clearly established.

Theorem 10.
Let h, g ∈ HðDÞ and ϕ : D ⟶ D. Then, the n -generalized superposition operator S g,n Proof. Assume that (29) holds. Hence, for arbitrary w ∈ D and h ∈ Z, we deduce that From the above inequality (29) and because S g,n ϕ hð0Þ = 0, we deduce that the operator S g,n is a bounded operator. On the other hand, suppose that S g,n and also set the function where b ∈ D with 1/ ffiffi ffi 2 p < jbj < 1. Then, the easy calculation gives which yields that kh b k Z < ∞. Hence, the following inequality can be deduced: Hence, by using the maximum modulus principle, we obtain (29). The proof is therefore completely established.
For the next theorem, the analytic s-Bloch-type space is considered as follows: Proof. Assume the boundedness for the operator S g,n ϕ : Z ⟶ B s holds as well as condition (38) holds too. Therefore, since the operator S g,n ϕ is bounded with ϕðwÞ = w, we deduce that Let ðh k Þ k∈ℕ be a concerned sequence in the Zygmund space Z, for which and h k ⟶ 0 is uniformly bounded on a concerned compact subset of D as k ⟶ ∞. Using (38), we obtain that for every ε > 0, there is a constant δ ∈ ð0, 1Þ, such that δ < jhðwÞj < 1, which implies that Now, set k = fw 1 ∈ D : jw 1 j < δg. By (9), we deduce that Applying the known Cauchy's estimate, when we have the specific sequence ðh k Þ k∈ℕ that has a uniform type that converges to zero on a specific compact concerned subset of D, we obtain that the specific sequence ðϕ′ k Þ k∈ℕ converges to zero on a specific compact concerned subset of D as k ⟶ ∞. Because k is actually compact, it yields that Letting k ⟶ ∞ in the last inequality, we deduce that Because the number ε is an arbitrary and positive, this yields that The concerned specific implication is therefore proved.
On the other hand, assume that S g,n ϕ : Z ⟶ B s is a compact operator. Remarking that the function h b given by (33) has a uniform type that converges to zero on a specific compact concerned subset of D as |b | ⟶1. In addition, we have Let ðw k Þ k∈ℕ be a concerned specific sequence in the disk D, for which jhðw k Þj ⟶ 1 when k ⟶ ∞. The test function ðϕ k Þ k∈ℕ can be defined by From the proof steps in Theorem 10, we deduce that sup k∈ℕ kϕ k k Z ≤ C. Furthermore, the specific function ϕ k converges to zero uniformly on a concerned compact subset of the disk D. Then, using Lemma 7, we obtain that the operator ∥S g,n Therefore, we can obtain Thus, the concerned proof of our result is obtained clearly. Now, we give the following characterizations concerning the boundedness of the n-generalized superposition operator S g,n ϕ : Z ⟶ B ν,0 on the boundary of the unit disk.
Assuming f ∈ Z and using (9), we get In view of (55), we can deduce that the operator S g,n ϕ ∈ B s 0 , for each h ∈ Z. Because B s 0 is a concerned closed subset of the analytic space B s , it follows therefore that Then, by choosing a sufficiently large k, the following specific inequalities can be directly obtained: Then, for every nonnegative defined integer m, mostly we can find one w k , for which Then, there is m 0 ∈ ℕ, such that for any concerned Carleson window, and m ∈ ℕ; thus, we can almost find m 0 elements in Hence, ðhðw k ÞÞ k∈ℕ is an interpolating concerned specific sequence for the analytic space B s .
Then, we may have some certain functions PðwÞ ∈ B s , for which Assume that ϕðhðwÞÞ = Ð hðwÞ 0 PðξÞdξ. Therefore, in view of the definition of the analytic Bloch space B s and analytic Zygmund functions, we deduce that h ∈ Z. Hence, we infer that Because lim k⟶∞ jhðw k Þj = 1 which yields that lim k⟶∞ jw k j = 1, therefore, using the above inequality, we deduce that S g,n ϕ ∈B s ω,0 , which is a contradiction. The proof of Theorem 12 is completely finished.

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Journal of Function Spaces Theorem 13. Suppose that h, g ∈ HðDÞ also assumes that ϕ defines a concerned entire-type function. Thus, the considered operator S g,n ϕ : B ν ⟶ B ν is a compact operator iff S g,n Proof. Assuming that the compactness property for the operator S g ϕ : B ν ⟶ B ν is satisfied. Therefore, fφ b ðwÞ: b ∈ Dg can be viewed as a concerned bounded set in the analytic Bloch space B ν and the specific function φ b − b ⟶ 0, where the convergence is of uniform type on specific compact concerned sets when jbj ⟶ 1.
Hence, applying Lemma 8, we deduce that On the other hand, assume that Let ðh n Þ be a concerned bounded sequence in the analytic-type space B ν , for which h n ⟶ 0 is uniformly bounded on concerned specific compact sets, when n ⟶ ∞. Then, we can clear that Assume that ε > 0 is given also fixing 0 < δ < 1, for which jbj > δ; hence, we obtain S g,n which implies that for any w 0 ∈ D, jϕðw 0 Þj > δ and kS g,n ϕ φ ϕðw 0 Þ k B < ε. Particularly, we have Therefore, considering n ∈ ℕ and w 0 ∈ D, as well as jϕ ðw 0 Þj > δ, (70) gives