# Integral Superposition-Type Operators on Some Analytic Function Spaces

Authors: A. El-Sayed Ahmed; S. Omran
Journal: Journal of Function Spaces
 (2021)

---

## Abstract

All entire functions which transform a class of holomorphic Zygmund-type spaces into weighted analytic Bloch space using the so-calledn-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.


---

## Body

## 1. Fundamental Materials

LetD=z:z<1 be the known concerned open unit disk in ℂ. Assume that Ξ and Ξ1 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 Sϕ on the concerned metric space Ξ is here defined by
(1)Sϕf=ϕ∘f,f∈Ξ.When the operatorSϕf∈Ξ1 for f∈Ξ1, we call that ϕ is induced by the concerned superposition operator from Ξ into Ξ1. When the concerned metric space Ξ contains specific concerned linear functions, thus, ϕ must be a concerned entire-type function. The symbol HD is supposed to be the class of all concerned holomorphic functions on D. For various emerging research studies on the superposition-type operators, we refer to [1–14] 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 nonlinearn-generalized superposition operator can be defined in this current paper by
(2)Sϕg,nfn,z=∫0zϕ′fξgn−1ξdξ,whereg∈HDandn∈ℕ.The operatorSϕg,n will be called the n-generalized superposition operator, where
(3)gn−1z=dn−1gzdzn−1,with"n−1"‐orderderivatives,n∈ℕ.Remark 1.
Whenn=1 and g=f′, the superposition operator is obtained because the difference Sϕg,1−Sϕ is a positive constant. Therefore, Sϕg,n is a generalization of the superposition-type operator. For good knowledge, the new operator Sϕg,n is defined and discussed in the present concerned manuscript for the first time.Remark 2.
It should be noted that the graph of then-generalized superposition operator Sϕg,n 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 Ξ1, for instance, holomorphic-type 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.Definition 3 [9].
For a given nondecreasing bounded and continuous functionν:0,1⟶0,∞, the concerned function h∈HD is said to be in the weighted concerned space Bν when
(4)hBν≔supz∈Dνzh′z<∞.Remark 4.
In the above definition, suppose thatνt=1−tα,0<t<∞; then, we obtain the definition of α-Bloch space, where 0<α<∞.The following new definition can also be introduced.Definition 5.
Leth∈HD; then, h is said to be in the weighted little ν-type concerned Bloch space Bν,0 when
(5)limz⟶1−νzh′z=0.Due to the result of Zygmund in [15] as well as the closed graph theorem, we can deduce that
(6)h∈Z⟺supz∈D1−z2h′′z<∞.It is obvious to see that the spaceZ is a Banach-type space with the specific norm ∙Z, where
(7)hZ=h0+h′0+supz∈D1−z2h′′z.Here, the concerned Zygmund-type space onD, denoted by Z0, is a concerned closed subspace of the holomorphic-type space Z that consists of all holomorphic-type functions h, for which
(8)limz⟶11−z2h′′z=0.Using (7), it is very obvious to deduce that
(9)h′z−h′0≤ChZlog11−z2.Some important and essential auxiliary results shall be proved in the next lemmas.Lemma 6.
Leth∈Bν and w∈D; then, the following inequality can be obtained:
(10)hz≤h0+λhBν,for some λ which is independent of h, with
(11)Iν=∫01rdrνr<∞.Proof.
Letw>1/2, w=rξ, and ξ∈∂D. Then, we infer that
(12)hw−hrξ2=∫1/21wh′twdt≤∫1/21wh′twdt≤2hBν∫01νtwwdt≤λhBν.
Additionally, we deduce(13)hw≤maxw≤1/2hw+ChBν.
Letw≤1/2; then, applying the known mean value theorem for the function hw−h0 (cf. [16]) and considering Jensen’s inequality, we deduce that
(14)maxw≤1/2hw−h0≤4n∫h≤3/4hw−h0dAw≤4n∫w≤3/4h′w12dAw1≤3nmaxw≤3/4h′w12,where the homogeneous representation expansion of the function h results in the second inequality.
Therefore,(15)max∣w∣≤1/2hw≤h0+3nmaxw≤3/4h′w≤h0+3nν34hBν.
Combining (13) and (15), we deduce
(16)hw≤h0+3nν34hBν+λhBν≤f0+3nν34+λhBν.
The proof is therefore established completely.Hereafter, the letterΓ 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 dw,∂Γ; also, the concerned Riemann-type mapping h has the next interesting concerned property:
(17)14νwh′w≤dhw,∂Γ≤νzh′w,forallw∈D, which is comparatively similar to the concerned constructions of the concerned connected domains as the concerned images for specific classes of functions in different analytic-type function spaces can be obtained in [4].


## 2. Auxiliary Results

The next lemmas establish some elementary properties of theν-Bloch-type spaces of analytic functions.Lemma 7.
Leth∈HD, and assume that (11) holds. Then, for the ν-Bloch space Bν, we have the following:
(i)
Every bounded concerned sequencehn∈Bν is uniformly bounded on concerned compact sets(ii)
For any concerned sequencehn on Bν with hnBν⟶0, then hn−hn0⟶0 and the convergence is of uniform type on concerned compact setsProof.
Ifw∈U0,r, r∈0,1; thus, the following inequalities can be deduced
(18)hnw−hn0=∫01hn′twwdt≤hnBν∫01twdtνtw≤ChnBν.Hence, the result follows.Lemma 8.
Assume thath,g∈HD and ϕ:D⟶D. Thus, for the n-generalized superposition operator Sϕg,n:Bν⟶Bν which is a concerned compact operator and ⟺Sϕg,n:Bν⟶Bν which is actually bounded for any bounded concerned sequence hnn∈N∈Bν with hn⟶0 uniformly bounded on concerned compact sets if n⟶∞, we obtain Sϕg,nhnBν⟶0 as n⟶∞.Proof.
Now, the concerned assertions (7), (9), and (11) of Lemma 3.7 in [17] can be satisfied for the analytic-type Bν 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 hn can be taken as a concerned sequence in the specific closed unit ball of Bν. Therefore, using Lemma 7 implies that hn can be seen as uniformly bounded on concerned compact sets. Hence, applying the theorem of Montel [18], we can get a concerned subsequence hnkn1<n2<⋯, for which hnk⟶h which indeed is uniformly bounded on concerned compact sets, for some concerned functions h∗∈HD. Then, it is enough to clear that h∗∈Bν. From Fatou’s theorem [16] and the concerned assumptions, we deduce that
(19)νwh∗′w=limk⟶∞νwhnk′w≤limk⟶∞hnkBν<∞.
Therefore, in view of Lemma 3.7 in [17], we infer that the n-generalized superposition operator Sϕg,n:Bν⟶Bν is compact ⇔ for any bounded concerned sequence hn∈Bν with hn⟶0 uniformly bounded on compact sets as n⟶∞, hnϕ0+Sϕg,nhnBν⟶0, as n⟶∞. Thus, the proof of Lemma 8 is clearly obtained.Lemma 9.
Leth∈HD. The concerned closed set K∈Bν,0 is compact iff it is bounded and satisfies the next concerned condition:
(20)limw⟶1suph∈Kνwh′w=0.Proof.
Assume thatK 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 h1,h2,⋯,hn∈K, for which the analytic function h∈K yields that h−hjBν<ε/2 for some 1≤j≤n; hence,
(21)νwh′w≤νwfj′w+ε2.
Now, for eachj, we can find rj∈0,1, for which
(22)νwhj′w<ε2,wheneverrj<w<1.
By lettingr=maxr1,⋯,rn, we obtain
(23)νzh′w≤ε,wheneverr<w<1andh∈K.
This establishes that (20) holds.
On the other hand, assume thatK⊂Bν,0 is closed and bounded and satisfies (20).
Then, the setK can be a normal family. When hn is a concerned sequence in the set K, then passing to a specific subsequence, we can suppose that hn⟶h with the concerned uniformly converging on the concerned compact subsets of D. Hence, we can clear that hn⟶h in Bν,0. Then, for a given ε>0, using (20), we can find r∈0,1, for which
(24)νzh′w≤ε2,forallr<w<1andallh∈K.
Since the convergencehn′⟶h′ is uniformly bounded on the concerned compact subsets of the disk D, it yields that the type of convergence for hn′⟶h′ is pointwise on D. Additionally, we deduce that
(25)νwh′w≤ε2,forallr<z<1.
Therefore,(26)νwhn′w−h′w≤ε,∀r<w<1.
Because the convergencehn′⟶h′ is uniformly bounded on rD¯, then we can find a specific positive integer N, for which
(27)hn′w−h′w≤ε,forall∣w∣≤randn≥N.
Hence,(28)νwhn′w−h′w≤ε,∀w∈Dandalln≥N.
Hence,hn⟶h in Bν. 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.


## 3. Superpositions on Bloch and Zygmund Spaces

In this section, some interesting and important properties of the new class of then-generalized superposition Sϕg,n acting from the analytic-type Zygmund classes to the analytic ν-Bloch classes are clearly established.Theorem 10.
Leth,g∈HD and ϕ:D⟶D. Then, the n-generalized superposition operator Sϕg,n:Z⟶Bν is bounded ⟺(29)F≔supw∈Dνwgn−1wlog11−hw2<∞.Proof.
Assume that (29) holds. Hence, for arbitrary w∈D and h∈Z, we deduce that
(30)νwSϕg,nh′w=νwgn−1wϕ′hw≤CϕZνwgn−1wlog11−hw2.
From the above inequality (29) and because Sϕg,nh0=0, we deduce that the operator
(31)Sϕg,n:Z⟶Bν,is a bounded operator.
On the other hand, suppose thatSϕg,n:Z⟶Bν is bounded. Set
(32)fw=w−11+log11−w2+1,and also set the function
(33)hbw=fb¯wb¯log11−b2−1,where b∈D with 1/2<b<1. Then, the easy calculation gives
(34)hb′w=log11−b¯w2log11−b2−1,h′′bw=2b¯1−b¯wlog11−b¯w2log11−b2−1,which yields that
(35)h′′bw=21−wC+log11−b2log11−b2−1≤C1−w,with 1/2<b<1 and sup1/2<b<1hbZ<∞. Hence, the following inequality can be deduced:
(36)Sϕg,nhϕbBν=νwSϕghϕb′w∣≥νbgn−1blog11−hb2.
Hence, by using the maximum modulus principle, we obtain (29). The proof is therefore completely established.For the next theorem, the analytics-Bloch-type space is considered as follows:
(37)hwBs=supw∈D1−w2sh′w<∞,where0<s<1.Theorem 11.
Assume thatνw=1−w2s, where 0<s<1. Suppose that h,g∈HD and ϕ:D⟶D. Then, the operator Sϕg,n:Z⟶Bs is a compact operator iff Sϕg,n:Z⟶Bs is bounded, and the following condition holds:
(38)lim∣hw∣⟶11−w2sgn−1wlog11−hw2=0.Proof.
Assume the boundedness for the operatorSϕg,n:Z⟶Bs holds as well as condition (38) holds too. Therefore, since the operator Sϕg,n is bounded with ϕw=w, we deduce that
(39)M1≔1−w2sgn−1w<∞.
Lethkk∈ℕ be a concerned sequence in the Zygmund space Z, for which
(40)supk∈ℕhkZ≤η,whereηisapositiveconstant, and hk⟶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∈0,1, such that δ<hw<1, which implies that
(41)1−w2sgn−1wlog11−hz2<εη.
Now, setk=w1∈D:w1<δ. By (9), we deduce that
(42)Sϕg,nhkBs=supw∈D1−w2sϕk′hwgwn−1≤suphw≤δ1−w2sϕk′hwgn−1w+supδ<hw≤11−w2sϕk′hwgn−1w≤M1supw1∈kϕk′w1+CϕkZsupδ<hw≤11−w2sgn−1wlog11−hw2≤M1supw1∈kϕk′w1+εC.
Applying the known Cauchy’s estimate, when we have the specific sequencehkk∈ℕ that has a uniform type that converges to zero on a specific compact concerned subset of D, we obtain that the specific sequence ϕ′kk∈ℕ converges to zero on a specific compact concerned subset of D as k⟶∞. Because k is actually compact, it yields that
(43)limk⟶∞supw1∈kϕk′w1=0.
Lettingk⟶∞ in the last inequality, we deduce that
(44)limk⟶∞supSϕg,nhkBs≤εC.
Because the numberε is an arbitrary and positive, this yields that
(45)limk⟶∞supSϕg,nhkBs=0.
The concerned specific implication is therefore proved.
On the other hand, assume thatSϕg,n:Z⟶Bs is a compact operator. Remarking that the function hb 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
(46)ϕb′b=log11−b2,foreachb∈D\0.
Letwkk∈ℕ be a concerned specific sequence in the disk D, for which hwk⟶1 when k⟶∞. The test function ϕkk∈ℕ can be defined by
(47)ϕkw=h¯wkw−1h¯wk1+log11−h¯wkz2+1log11−hwk2−1.
From the proof steps in Theorem10, we deduce that supk∈ℕϕkZ≤C. Furthermore, the specific function ϕk converges to zero uniformly on a concerned compact subset of the disk D. Then, using Lemma 7, we obtain that the operator ∥Sϕg,nhk∥Bs⟶0, when k⟶∞. Moreover,
(48)Sϕg,nhkBs=supw∈D1−wk2sϕk′hwkgwkn−1≥1−wk2sϕk′hwkgn−1wk≥1−w2sgn−1wlog11−hw2.
Therefore, we can obtain(49)limk⟶∞1−wk2sgn−1wklog11−hwk2=0.
Thus, the concerned proof of our result is obtained clearly.Now, we give the following characterizations concerning the boundedness of then-generalized superposition operator Sϕg,n:Z⟶Bν,0 on the boundary of the unit disk.Theorem 12.
Assume thatνw=1−w2s, where 0<s<1. Suppose that h,g∈HD and ϕ:D⟶D. Then, the operator Sϕg,n:Z⟶B0s is a bounded operator iff condition (38) holds as well as
(50)limw⟶11−w2αgn−1w=0.Proof.
First, assume that conditions (38) and (50) hold. Using condition (38), thus, for every constant ε>0, we can get r∈0,1, for which
(51)1−w2sgn−1wlog11−hw2<ε,with r<hw<1. By (38), we can get ρ∈0,1, for which
(52)1−w2sgn−1w<εlog1/1−r2,when ρ<w<1.
Therefore, whenρ<w<1 and r<hw<1, we deduce that
(53)1−w2sgn−1wlog11−hw2<ε.
Ifρ<w<1 and hz≤r, then we can infer that
(54)1−w2sgn−1wlog11−hw2<1−w2sgn−1wlog11−r2<ε.
From (53) and (54), we deduce
(55)limw⟶11−w2sgn−1wlog11−hw2=0.
From (55), using the known maximum modulus theorem as well as Theorem 10, the concerned property of boundedness for the concerned operator Sϕg,n:Z⟶Bs can be obtained.
Assumingf∈Z and using (9), we get
(56)1−w2sSϕg,n′w≤CϕZ1−w2sgn−1wlog11−hw2.
In view of (55), we can deduce that the operator Sϕg,n∈B0s, for each h∈Z. Because B0s is a concerned closed subset of the analytic space Bs, it follows therefore that
(57)Sϕg,nw⊆B0s.
Hence, the boundedness of the operatorSϕg,n:Z⟶B0s can be obtained immediately.
For the converse direction of the proof, assuming that the operatorSϕg,n:Z⟶B0s holds the boundedness property, then for ϕz=z, we infer that (50) holds. Now suppose that the concerned condition (38) does not hold. Therefore, we can have ε0>0 and a concerned sequence wkk∈ℕ∈D, for which limk⟶∞hwk=1. Also, we have
(58)1−wk2sgn−1wklog11−hwk2≥ε0>0.
Then, by choosing a sufficiently largek, the following specific inequalities can be directly obtained:
(59)1−hwk−12>1−hzk,k∈ℕ.
Then, for every nonnegative defined integerm, mostly we can find one wk, for which
(60)1−12m≤hwk<1−12m+1.
Then, there ism0∈ℕ, such that for any concerned Carleson window,
(61)Ω=reiθ:0<1−r<lΩ,θ−θ0<lΩ,and m∈ℕ; thus, we can almost find m0 elements in
(62)hwk∈Ω:2−m−1lΩ<1−hwk<2−mlΩ.
Hence,hwkk∈ℕ is an interpolating concerned specific sequence for the analytic space Bs.
Then, we may have some certain functionsPw∈Bs, for which
(63)Phwk=log11−hwk2,k∈ℕ.
Assume thatϕhw=∫0hwPξdξ. Therefore, in view of the definition of the analytic Bloch space Bs and analytic Zygmund functions, we deduce that h∈Z. Hence, we infer that
(64)1−wk2sSϕg,n′wk=1−wk2sgn−1wkϕ′hwk=1−wk2sgn−1wkPhwk=1−wk2sgn−1wklog11−hwk2≥ε0>0.
Becauselimk⟶∞hwk=1 which yields that limk⟶∞wk=1, therefore, using the above inequality, we deduce that Sϕg,n∈Bω,0s, which is a contradiction. The proof of Theorem 12 is completely finished.Theorem 13.
Suppose thath,g∈HD also assumes that ϕ defines a concerned entire-type function. Thus, the considered operator Sϕg,n:Bν⟶Bν is a compact operator iff
(65)Sϕg,nφbBν⟶0,asb⟶1.Proof.
Assuming that the compactness property for the operatorSϕg:Bν⟶Bν is satisfied. Therefore, φbw:b∈D can be viewed as a concerned bounded set in the analytic Bloch space Bν and the specific function φb−b⟶0, where the convergence is of uniform type on specific compact concerned sets when b⟶1.
Hence, applying Lemma8, we deduce that
(66)limb⟶1Sϕg,nφbBν=0.
On the other hand, assume that(67)limb⟶1Sϕg,nφbBν=0,asb⟶1.
Lethn be a concerned bounded sequence in the analytic-type space Bν, for which hn⟶0 is uniformly bounded on concerned specific compact sets, when n⟶∞. Then, we can clear that
(68)limn⟶0Sϕg,nhnBν=0.
Assume thatε>0 is given also fixing 0<δ<1, for which b>δ; hence, we obtain
(69)Sϕg,nφbBν<ε,which implies that for any w0∈D, ϕw0>δ and Sϕg,nφϕw0B<ε.
Particularly, we have(70)ϕ′φϕw0gn−1w0νw0<ε.
Therefore, consideringn∈ℕ and w0∈D, as well as ϕw0>δ, (70) gives
(71)Sϕg,nhnBν=supw∈Dϕ′hnw0gn−1w0νw0≤εconst.
The concerned setK1=z:z≤δ is a concerned compact subset of the disk D. Also, the convergence of ϕn′⟶0 is uniformly bounded on specific compact concerned sets:
(72)supz∈K1ϕn′z⟶0,whenn⟶∞.
Hence, we can findn0 which can be chosen to be large enough; then, ϕ′hn<ε, for any n>n0, with z∈D, for which hz≤δ. Hence,
(73)Sϕg,nhnBν=ϕ′hngn−1zνz<εgn−1zνz≤ε∥g∥Bν<εconst.
Then, wheren≥n0,
(74)SϕgfnBν<εconst.
Therefore, combining equations (71) and (74), we deduce that
(75)Sϕg,nhnBν<εconst.,forn≥n0.
Therefore, (75) gives Sϕg,nhnBν⟶0 as n⟶∞. Hence, in view of the concerned Lemma 8, we deduce that the operator Sϕg,n:Bν⟶Bν is actually a compact operator.Corollary 14.
Let0<s<1 and h,g∈HD and let ϕ be a holomorphic-type self-mapping on D. Thus, Sϕg,n:Bν⟶Bs is a compact operator iff
(76)Sϕg,nφbBν⟶0,asb⟶1.Proof.
The proof of this corollary can be obtained immediately from the proof of Theorem13 by setting νw=1−ws, 0<s<1, in the image of the operator Sϕg,n.


## 4. Conclusions

An interesting approach for defining and discussing a new class of integral superposition-type operators is provided in this concerned manuscript. This class of operators is based on both an inducing entire function and a concerned analytic function of “n−1”-order derivatives, where n∈ℕ.Essential results of the nonlinearn-generalized superposition operator defined for the first time are introduced in the present manuscript. Using the new class of operators, certain classical and recent classes of analytic-type function spaces such as Zygmund and ν-Bloch classes are studied. Careful foundations for the concerned weighted analytic-type function class theoretic results are devoted to laying and discussing properties of the operator Sϕg,n such as the usual properties of boundedness and compactness. Both the new operator Sϕg,n and the concerned analytic function space ultraproduct constructions are studied in detail.For then-generalized superposition operator Sϕg,n, the next useful important remarks are stated as unsolved interesting future research.Remark 15.
Can one discuss some certain specific properties such as boundedness and compactness for the operatorSϕg,n on some other types of analytic function spaces such as Besov- and Q-type spaces?Remark 16.
How about estimating the essential norms of then-generalized superposition operator Sϕg,n?
In addition, there are some interesting recent studies on some types of Morrey spaces by applying some certain classes of operators (see [19, 20]). Is it possible to apply the n-generalized superposition operator Sϕg,n in lieu of the considered operators in such studies?Remark 17.
There are some recent interesting and important studies of some hypercomplex function spaces using quaternion techniques (see [21–24]). On the other hand, there is no specific study with the help of operators on such hypercomplex spaces.



---

*Source: 5531955-2021-03-30.xml*