# On Stević-Sharma Operators from General Class of Analytic Function Spaces into Zygmund-Type Spaces

Authors: M. A. Bakhit; A. Kamal
Journal: Journal of Function Spaces
 (2022)

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## Abstract

A Stevic′-Sharma operator denoted by Tψ1,ψ2,φ is a generalization product of multiplication, differentiation, and composition operators. In this paper, we characterize the bounded and compact Stevic′-Sharma operator Tψ1,ψ2,φ from a general class X of Banach function spaces into Zygmund-type spaces with some of the most convenient test functions on the open unit disk. Using several restrictive terms, we show that all bounded operators Tψ1,ψ2,φ from X into the little Zygmund-type spaces are compact. As an application, we show that our results hold up for some other domain spaces of Tψ1,ψ2,φ, such as the Hardy space and the weighted Bergman space.


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## Body

## 1. Introduction

Meanwhile, in this work,D=ζ∈ℂ:ζ<1 is the open unit disk of the complex plane ℂ. We denote by HOD the family of analytic functions in D; also, we denote by ASD the family of analytic self-maps of D and AutD the set of conformal automorphisms of the disk D. As is usual, the Banach space H∞ is the family of bounded functions h∈HOD, defined by the norm h∞=supζ∈Dhζ<∞.For any weighted functionμ, the weighted Banach space Hμ∞ is the space of functions h∈HOD such that
hμ∞=supζ∈Dhζμζ<∞.Moreover, the little weighted Banach spaceHμ,0∞ is the space of functions h∈HOD such that
limζ⟶1−hζμζ=0.The Bloch-type spaceBμ is the Banach space of functions h∈HOD such that h′∈Hμ∞, with the norm hBμ=h0+h′μ∞. The little Bloch-type space Bμ,0 is the closed subspace of Bμ consisting of functions h∈HOD such that h′∈Hμ,0∞.The space of all functionsh∈HOD is said to be a weighted Zygmund-type space Zμ, if all functions h are such that h″∈Hμ,0∞. The space Zμ becomes a Banach space with the norm
hZμ=h0+h′0+supζ∈Dh″ζμζ.The little weighted Zygmund-type spaceZμ,0 is the closed subspace of Zμ and consists of functions h∈HOD, such that limζ⟶1−h″ζμζ=0.Exactly as in the aforementioned spaces, whenμζ=1−ζ2d, we get the Zygmund-type space Zd. In addition, the Zd space is a Banach space with the norm h0+h′0+hZd, where hZd=supζ∈Dh″ζ1−ζ2d<∞. The Zygmund spaces also satisfy Zd⊊Zl whenever 0<d<l. When d=1, we get the classical Zygmund space denoted by Z. In fact, the motivation for the name and for studying the Zygmund-type spaces comes from the Zygmund class; see, for example, Chapter 5 of Duren’s book [1]. The Zygmund space and its subspace Z0 play an important role in connection to the theory of the Hardy spaces Hq,0<q<1. Indeed, in [2], it was shown that Z can be viewed as the dual of the Hardy space Hq,q=1/2. More generally, the spaces defined by replacing the function h in the definition of Z with its nth derivative, under the assumption h,h′,⋯,hn∈HOD∩CD for n≥1, can be viewed as duals of the Hardy spaces Hq,q=1/n+2. A recent study of several operators on Zygmund-type spaces has attracted significant research attention; see, for example, [3–5].For anyΨ∈HOD and φ∈ASD, we have the following linear operators:
(1)
Dhζ=h′ζ: the differentiation operator(2)
MΨhζ=Ψζhζ: the multiplication operator(3)
Cφhζ=h∘φζ=hφζ: the composition operator(4)
WΨ,φhζ=Ψζ·hφζ: the weighted composition operatorFor a unified manner of treatment of these operators, many researchers seek to present the various products of multiplication, composition, and differentiation operators; see, for example, [3, 4, 6–8]. Stevic′ et al. was the first to introduce the operator Tψ1,ψ2,φ in [7]. So, the operator Tψ1,ψ2,φ is called the Stevic′-Sharma operator, which is defined as
(1)Tψ1,ψ2,φhζ=ψ1ζhφζ+ψ2ζh′φζ,h∈HOD.Over the past 10 years, the boundedness and compactness of this operator have been studied extensively in the most well-known spaces of analytic functions; for example, see [5, 9].This paper proceeds as follows. In Section2, we present basic facts and prerequisites required for the Banach spaces. During Section 3, we characterize the boundedness of the Tψ1,ψ2,φ operators from a large class X of Banach spaces into the Zygmund-type space Zμ, also we give their norm estimates. Throughout Section 4, we study the compactness of Tψ1,ψ2,φ from a large class X of Banach spaces into the Zygmund-type space Zμ, under specific conditions on class X. The boundedness and compactness of Tψ1,ψ2,φ from X to Zμ,0 are investigated in Section 5. Finally, in Section 6 we apply our results to the Hardy spaces Hq,1≤q≤∞, and the weighted Bergman space Aβq, (β>−1 and 1≤q<∞).


### 1.1. Suppositions

In this paper, always, we will suppose the following:(i)
φ∈ASD and ψ1,ψ2∈HOD, unless otherwise stated(ii)
For anyh∈HOD, we will do the following differentiation over and over again:
(2)Tψ1,ψ2,φh″w=ψ1″whφw+2ψ1′wφ′w+ψ1wφ″w+ψ2″wh′φw+ψ1wφ′2w+2ψ2′wφ′w+ψ2wφ″wh″φw+ψ2wφ′2wh3φw.So, to simplify the formulation of our results, we will define the following quantities:(3)D0w=ψ1″wD1w=2ψ1′wφ′w+ψ1wφ″w+ψ2″wD2w=ψ1wφ′2w+2ψ2′wφ′w+ψ2wφ″wD3w=ψ2wφ′2w.(iii)
For any two quantitiesQ1h and Q2h that are jointly dependent on h∈HOD, we stipulate that Q1h≲Q2h, meaning that there is a positive constant C that fulfills Q1h≤CQ2h. Thus, when Q2h≲Q1h≲Q2h, we hold that Q1h≈Q2h and the quantities Q1h and Q2h are said to be equivalent. If Q1h≈Q2h, then Q2h<∞ if and only if Q1h<∞


## 1.1. Suppositions

In this paper, always, we will suppose the following:(i)
φ∈ASD and ψ1,ψ2∈HOD, unless otherwise stated(ii)
For anyh∈HOD, we will do the following differentiation over and over again:
(2)Tψ1,ψ2,φh″w=ψ1″whφw+2ψ1′wφ′w+ψ1wφ″w+ψ2″wh′φw+ψ1wφ′2w+2ψ2′wφ′w+ψ2wφ″wh″φw+ψ2wφ′2wh3φw.So, to simplify the formulation of our results, we will define the following quantities:(3)D0w=ψ1″wD1w=2ψ1′wφ′w+ψ1wφ″w+ψ2″wD2w=ψ1wφ′2w+2ψ2′wφ′w+ψ2wφ″wD3w=ψ2wφ′2w.(iii)
For any two quantitiesQ1h and Q2h that are jointly dependent on h∈HOD, we stipulate that Q1h≲Q2h, meaning that there is a positive constant C that fulfills Q1h≤CQ2h. Thus, when Q2h≲Q1h≲Q2h, we hold that Q1h≈Q2h and the quantities Q1h and Q2h are said to be equivalent. If Q1h≈Q2h, then Q2h<∞ if and only if Q1h<∞


## 2. Some of the Prerequisites Required for Banach Spaces

Following popular terminology in functional analysis, we stipulate a Banach spaceX whose elements are functions h∈HOD, with norm hX and whose functionals of point-evaluation are bounded. Always we will suppose through this paper that X is a Banach space of functions h∈HOD.LetEζ indicate the functional of point-evaluation at ζ. So, for any ζ∈D, we can define
(4)ΛXζ=suphζ:h∈X,hX≤1=EζX.Thus, for eachh∈X and ζ∈D, we obtain
(5)hζ≤hXΛXζ.Now, under appropriate modifications, we will list the specific conditions given in [10], including the conditions used to formulate the results of the present work.
(I)
For allζ∈D and s∈0,1, we get ΛXsζ≲ΛXζ(II)
With respect to the uniform convergence topology on compact subsets ofD, the unit ball BX of a Banach space X is comparatively compact(III)
By a positive constantδ in compact subsets of D, the norm ΛXζ is bounded below and limζ⟶1ΛXζ=∞(IV)
Fors∈0,1, let fsζ=fsζ,ζ∈D where f∈X. Then, the linear map Ts:f⟶fs is a compact map on X, also sup0<s<1TsX<∞(V)
For allh∈X and S∈AutD, we have Skh∈X and SkhX≲hX, where k=1,2,3(VI)
For allζ∈D, let h∈X; then 1−ζ2khkζ≲hXΛXζ, where k=1,2,3The following proposition also is proved in [10].Proposition 1.
Forζ∈D, the map ζ⟶ΛXζ is bounded on compact subsets of D. Further, if a Banach space X is reflexive, then BX is compact and shows relatively of uniform convergence, with respect to the topology on a compact subset of D.The Lemma2 helps to distinguish the properties of the operator Tψ1,ψ2,φ concerned with studying in this article.Lemma 2.
Let there be a Banach spaceX satisfying the above conditions (V) and (VI). Suppose that φ∈ASD, then for any w∈D and h∈X, such that hX≤1, there is a set of functions Fw,k defined in X, for k=1,2,3, such that supw∈DFw,kX<∞.
(6)Fw,kjφw=0,ifk>j,Fw,kjφw=k!−1khφw1−φw2j,ifk=j.
Moreover, the sequencesFw,k converge to 0 uniformly on compact subsets of D, when w⟶1.Proof.
Letρbζ=b−ζ/1−b¯ζ, which is the automorphism of D that changes zero and b, that is, ρb∈AutD, for any ζ∈D. Note that
(7)ρb′b=−11−b2,ρb″b=−2b¯1−b22,ρb3b=4b¯21−b23.
Now, fixh∈X, such that hX≤1, and consider the functions Fw,kw=ρφwwkhw, for all k∈1,2,3 and w∈D. Let S=ρφw in condition (V); then for all k=1,2,3, we have Fw,k∈X and
(8)Fw,kX=SkhX≲hX≲1.
Also, it is not difficult to prove that(9)Fw,1φw=0,Fw,1′φw=−hφw1−φw2,Fw,2φw=Fw,2′φw=0,F″w,2φw=2hφw1−φw22,Fw,3φw=Fw,3′φw=F″w,3φw=0,Fw,33φw=−6hφw1−φw23.
Finally, the uniform convergence to0 of the sequences Fw,k is self-evident, when w⟶1.


## 3. Boundedness ofTψ1,ψ2,φ:X⟶Zμ

In this section, we characterize the boundedness of theTψ1,ψ2,φ operators from a large class X of Banach spaces into the Zygmund-type space Zμ; also we give their norm estimates.First of all, to simplify the formulation of the next results, forΛXw as in (4), we will define the following:
(10)K0~=supw∈DμwD0wΛXφw,K1~=supw∈DμwD1w1−φw2ΛXφw,K2~=supw∈DμwD2w1−φw22ΛXφw,K3~=supw∈DμwD3w1−φw23ΛXφw.Now, we introduce the main results of this section.Theorem 3.
Let there be a Banach spaceX satisfying the above conditions (V) and (VI). Then, Tψ1,ψ2,φ:X⟶Zμ is a bounded operator if and only if all the quantities Kj~ are finite, for j=0,1,2,3. Moreover, in which case
(11)Tψ1,ψ2,φX⟶Zμ≈∑j=03Kj~.Proof.
First, suppose thatTψ1,ψ2,φ:X⟶Zμ is bounded. Then,
(12)Tψ1,ψ2,φhZμ≲hX,forallh∈X.
Setk=3 in Lemma 2, we observe that
(13)Fw,3φw=Fw,3′φw=F″w,3φw=0,Fw,33φw=−6hφw1−φw23.
Also, we haveFw,3∈X and Fw,3X≲hX≲1.
Then, we obtain(14)supw∈DμwTψ1,ψ2,φFw,3″w=supw∈DμwD3wFw,33φw=supw∈Dμw6D3whφw1−φw23≤Tψ1,ψ2,φXhX≲Tψ1,ψ2,φX⟶Zμ.
For anyh∈X with hX≤1, by (5), we have
(15)K3~=supw∈DμwD3w1−φw23ΛXφw≲Tψ1,ψ2,φX⟶Zμ<∞.
Next, we prove thatK2~ is finite. So, set k=2 in Lemma 2; we observe that Fw,2φw=Fw,2′φw=0, but
(16)F″w,2φw=2hφw1−φw22.
Also, we haveFw,2∈X and Fw,2X≲hX≲1.
Then, we obtain(17)supw∈DμwTψ1,ψ2,φFw,2″w=supw∈DμwD2wF″w,2φw+D3wFw,23φw≤Tψ1,ψ2,φX⟶ZμhX≲Tψ1,ψ2,φX⟶Zμ.
Using the triangle inequality, we get(18)supw∈DμwD2wF″w,2φw≲Tψ1,ψ2,φX⟶Zμ+supw∈DμwD3wFw,23φw.
Condition (VI) fork=3 shows that
(19)1−φw23Fw,23φw≲hXΛXφw.
Thus, by (16), we have
(20)supw∈DμwD2w2hφw1−φw22≲Tψ1,ψ2,φX⟶Zμ+supw∈Dμw1−φw231−φw23D3wFw,23φw≲Tψ1,ψ2,φX⟶Zμ+supw∈DμwD3w1−φw23ΛXφw.
Using (15), we obtain
(21)K2~=supw∈DμwD2w1−φw22ΛXφw≲Tψ1,ψ2,φX⟶Zμ+K3~,(22)≲Tψ1,ψ2,φX⟶Zμ<∞.
For the third time, we go back to Lemma2 with k=1; we know Fw,1φb=0, also Fw,1∈X, and Fw,1X=ShX≲hX≲1.
Then,(23)supw∈DμwTψ1,ψ2,φFw,1″w=supw∈DμwD1wFw,1′φw+D2wF″w,1φw+D3wFw,13φw≤hXTψ1,ψ2,φX⟶Zμ≲Tψ1,ψ2,φX⟶Zμ.
We repeat using the triangle inequality; we get(24)supw∈DμwD1wFw,1′φw≲Tψ1,ψ2,φX⟶Zμ+supw∈DμwD3wFw,13φw+supw∈DμwD2wF″w,1φw.
Repeat condition (VI) fork=2,3 and since Fw,1′φb=−hφb/1−φb2, we obtain
(25)supw∈DμwD1wFw,1′φw−hφw1−φw2≲Tψ1,ψ2,φX⟶Zμ+supw∈DμwD3w1−φw23ΛXφw+supw∈DμwD2w1−φw22ΛXφw.
Using (15) and (21), we obtain
(26)K1~=supw∈DμwD1w1−φw2ΛXφw≲Tψ1,ψ2,φX⟶Zμ+K2~+K3~≲hX<∞.
Finally, we go to prove thatK0~ is finite. For w∈D fix h∈X, such that hX≤1. Then, we have
(27)supw∈Dμwψ1″whw=supw∈DμwTψ1,ψ2,φh″w−D1wh′φw−D2wh″φw−D3wh3φw.
SinceTψ1,ψ2,φ:X⟶Zμ is bounded, by condition (VI) for k=1,2,3, we observe that
(28)supw∈Dμwψ1″whw≲Tψ1,ψ2,φX⟶Zμ+supw∈DμwD3w1−φw23ΛXφw+supw∈DμwD2w1−φw22ΛXφw+supw∈DμwD1w1−φw2ΛXφw.
Now, using (15), (21), and (26), we have
(29)K0~=supw∈Dμwψ1″wΛXφw≲Tψ1,ψ2,φX⟶Zμ+K1~+K2~+K3~≲Tψ1,ψ2,φX⟶Zμ<∞.
Because of this, all the quantitiesKj~ are finite, where j=0,1,2,3, and
(30)Tψ1,ψ2,φX⟶Zμ≳∑j=03Kj~.
On the other hand, suppose all the quantitiesKj~ are finite, where j=0,1,2,3. For fix w∈D we let h∈X, such that hX≤1. For k=1,2,3, by conditions (VI) and (4), we obtain
(31)supw∈DμwTψ1,ψ2,φh″w≤supw∈DμwD0wΛXφw+supw∈DμwD1w1−φw23ΛXφw+supw∈DμwD2w1−φw22ΛXφw+supw∈DμwD3w1−φw2ΛXφw≤C∑j=03Kj~.
In addition to that,(32)Tψ1,ψ2,φh0+Tψ1,ψ2,φh′0≤ψ10+ψ201−φ02ΛXφ0+ψ1′0+ψ10φ′0+ψ2′01−φ02ΛXφ0+ψ20φ′01−φw22ΛXφ0≤C.
We can choose a sufficiently largeC, the highest estimate
(33)Tψ1,ψ2,φX⟶Zμ≲∑j=03Kj~<∞.
Moreover, from (1) and (33), we have
(34)Tψ1,ψ2,φX⟶Zμ≈∑j=03Kj~.
That ends the proof of Theorem3.In the case ofψ1w=ψw and ψ2w=0 in Theorem 3, we define the following quantities:
(35)E0μ,ψ,φ=μwψ″wΛXφw;E1μ,ψ,φ=μw2ψ′wφ′w+ψwφ″w1−φw2ΛXφw;E2μ,ψ,φ=μwψwφ′2w1−φw22ΛXφw.Now we deduce the following result as a corollary of Theorem3.Corollary 4 ([4], Theorem 3.1).
Let a Banach spaceX be satisfying the above conditions (V) and (VI). Then, Tψ1,0,φ:X⟶Zμ is a bounded operator if and only if all the quantities Kj are finite, for j=0,1,2. Moreover, in which case Tψ1,0,φX≈∑j=02‍Kj, where
(36)K0=supw∈DE0μ,ψ,φ,K1=supw∈DE1μ,ψ,φ,K2=supw∈DE2μ,ψ,φ.


## 4. Compactness ofTψ1,ψ2,φ:X⟶Zμ

Here, we study the compactness ofTψ1,ψ2,φ from a large class X of Banach spaces into the Zygmund-type space Zμ, under specific conditions on class X. Before that, we mention the Lemma 5, which can be proved in a standard way; for example, see [6, 10].Lemma 5.
Let a Banach spaceX be satisfying condition (II) and whose the functionals of point-evaluation are continuous. Then, the bounded operator T from X into Zygmund-type space Zμ is compact if and only if for any bounded sequence hm in X which converges uniformly to 0 as m⟶∞ on compact subsets of D, limn⟶∞ThmZμ=0.
Moreover, to simplify the formulation of the main results in this section, forΛXw as in (4) and δ>0, we set
(37)L0~=limδ⟶1supφw>δμwD0wΛXφw,L1~=limδ⟶1supφw>δμwD1w1−φw2ΛXφw,L2~=limδ⟶1supφw>δμwD2w1−φw22ΛXφw,L3~=limδ⟶1supφw>δμwD3w1−φw23ΛXφw.Now, we introduce the main results of this section.Theorem 6.
Let a Banach spaceX be satisfy the above conditions (II), (V), and (VI). Suppose that Tψ1,ψ2,φ:X⟶Zμ is a bounded operator, and let
(38)γ≔infw∈DΛXφw>0.
Then,Tψ1,ψ2,φ:X⟶Zμ is a compact operator if and only if all the quantities Lj~=0, for j=0,1,2,3.Proof.
It is clear that condition (38) holds if the constants contain in X, that is ΛXw≥1X−1. First, assume that Tψ1,ψ2,φ:X⟶Zμ is a compact operator. Obviously, the result is intuitive if φ∞<1. So from now, we suppose φ∞=1. Also from now to the end of the proof, we let a sequence bm in D, be such that limm⟶∞φbm=1, for φbm>1/2,
(39)L0~=limm⟶∞μbmD0bmΛXφbm,L1~=limm⟶∞μbmD1bm1−φbm2ΛXφbm,L2~=limm⟶∞μbmD2bm1−φbm22ΛXφbm,L3~=limm⟶∞μbmD3bm1−φbm23ΛXφbm.
Sincehw≤hXΛXw for each h∈X and w∈D, then, for each m∈ℕ and fix M>0, there exists hm∈X such that hmX≤1 and
(40)hmφbm>ΛXφbm−M.
Clearly,hm is a norm-bounded sequence; also w⟶ΛXw is a bounded map on compact sets in D. Using the fact hmw≤hmXΛXw, then hm is a uniformly bounded sequence on compact sets. For all w∈D and m∈ℕ, set the functions
(41)fmw=1−φbm2w1−φbm¯whmw.
Then, we see that(42)fmw=1−φbm21−wΛXw,(43)fmφbm=φbmhmφbm.
Clearly, alsofm converges uniformly to 0 on compact subsets of D. By the condition (V), we obtain fm∈X and fmX≲hmX≲1.
Sincefm∈X, by the functions defined in Lemma 2 for k=1,2,3, we can consider the functions
(44)Fm,kbm=ρφbmbmkfmbm.
By Lemma2, we observe that
(45)Fm,kjφbm=0,ifk>j,(46)Fm,kjφbm=k!−1kΛXφbm1−φbm2j,ifk=j.
Also, we haveFm,k∈X and Fm,kX≲fmX≲1, for all k=1,2,3.
Using the hypothesis that condition (II) holds, by applying Lemma5, for all k=1,2,3 we obtain
(47)limm⟶∞Tψ1,ψ2,φFm,kZμ=0.
In the case ofk=3 in (45) and (47), we get
(48)Fm,3φbm=Fbm,3′φbm=F″m,3φbm=0,Fm,33φbm=−6fmφbm1−φbm23,limm⟶∞μbmTψ1,ψ2,φFm,3″bm=0.
Thence,(49)limm⟶∞μbm6D3bmfmφbm1−φbm23=0.
From (40) and (43), we have
(50)μbm6D3bmφbm1−φbm23ΛXφbm−M≤μbm6D3bm1−φbm23φbmhmφbm=μbm6D3bm1−φbm23fmφbm.
Thus, by (38), we have
(51)μbm6D3bm1−φbm23≤supw∈Dμw6D3wφw1−φw23≤6γsupw∈DμwD3wφw1−φw23ΛXφw(52)=6γK3~.
We know from Theorem3 and the boundedness of the operator Tψ1,ψ2,φ, that all the quantities Kj~ are finite, where j=0,1,2,3. Therefore, we obtain from (50) and (51):
(53)μbm6D3bmφbm1−φbm23ΛXφbm≤μbm6D3bm1−φbm23fmφbm+μbm6D3bmφbm1−φbm23M≤μbm6D3bm1−φbm23fmφbm+6MγK3~.
SinceM is an arbitrary constant, we conclude that
(54)L3~=limm⟶∞μbmD3bm1−φbm23ΛXφbm=limm⟶∞6MγK3~=0.
Next, we will show thatL2~=0. In the case of k=2 in (45) and (47), we observe that
(55)Fm,2φbm=Fm,2′φbm=0;F″m,2φbm=2fmφbm1−φbm22.
Also(56)limm⟶∞μbmTψ1,ψ2,φFm,2″bm=0.
Then, there is an integerN that corresponds with the number M fixed above, whenever m>N such that
(57)μbmTψ1,ψ2,φFm,2″bm=μbmD2bmF″m,2φbm+D3bmFm,23φbm<M2.
SinceFm,2∈X and Fm,2X≲fmX≲1, by condition (VI) for k=3, using the triangle inequality, for every m>N, we obtain
(58)μbmD2bmF″m,2φbm<M2+μbmD3bmFm,23φbm≲M2+μbmD3bm1−φbm23ΛXφbm.
By (55), we conclude that
(59)limm⟶∞μbm2D2bmfmφbm1−φbm22≲M+limm⟶∞μbmD3bm1−φbm23ΛXφbm≲M+L3~=M.
Again from (40) and (43), for every m sufficiently large and φbm>1/2, we have
(60)μbmD2bm1−φbm22ΛXφbm−M≤μbmD2bm1−φbm222fmφbm<2M.
Once secondly by (38), we obtain
(61)supw∈DμbmD2bm1−φbm22≤supw∈DμwD2w1−φw22≤1γsupw∈DμwD2w1−φw22ΛXφw=1γK2~.
Therefore, from (59) and (61), we get
(62)L2~=limm⟶∞μbmD2bm1−φbm22ΛXφbm≤MγK2~+2MM.
SinceM is an arbitrary constant, we conclude that L2~=0.
We go to what is needed to prove thatL1~=0. In the case of k=1 in (45) and (47), we observe that
(63)Fm,1φbm=0,Fm,1′φbm=−fmφbm1−φbm2.
Also,(64)limm⟶∞μbmTψ1,ψ2,φFm,1″bm=0.
Then, there is an integerN that corresponds with the number M fixed above, whenever m>N such that
(65)μbmTψ1,ψ2,φFm,1″bm=μbmD1bmFm,1′φbm+D2bmFm,12φbm+D3bmFm,13φbm<M.
Repeat condition (VI), fork=2,3, and using (55), since L3~=L2~=0, we get
(66)μbmD1bmFm,1′φbm<M+μbmD3bmFm,13φbm+μbmD2bmF″m,1φbm≲M+μbmD3bm1−φbm23ΛXφbm+μbmD2bm1−φbm22ΛXφbm<M.
By (40) and (43), for every m sufficiently large and φbm>1/2, we have
(67)μbmD1bm1−φbm2ΛXφbm−M≤μbmD1bm1−φbm22fmφbm<2M.
By (38), we obtain
(68)μbmD1bm1−φbm2≤1γsupw∈DμwD1w1−φw2ΛXφw≤1γK1~.
Therefore, by (60) and (61), we have
(69)L1~=limm⟶∞supw∈DμwD1w1−φw2ΛXφw≤MγK1~+2M.
SinceM is an arbitrary constant, we conclude that L1~=0.
Finally, we go to prove thatL0~=0. For each m∈ℕ and fix M>0, let hm and fm be instituted as in (41) in terms of the sequence bm and M. Using the hypothesis that condition (II) holds, with apply Lemma 5, we guarantee that
(70)limm⟶∞μbmTψ1,ψ2,φfm″bm=0.
Corresponding to the numberM, there is an integer N, whenever m≥N, such that
(71)μbmTψ1,ψ2,φfm″bm=μbmD0bmfmφbm+D1bmfm′φbm+D2bmf″mφbm+D3bmfm3φbm<M.
SinceL3~=L2~=L1~=0, by condition (VI), we observe that
(72)μbmD0bmfmφbm<M+μbmD1bmfm′φbm+μbmD2bmf″mφbm+μbmD3bmfm3φbm≲M+μbmD1bm1−φbm2ΛXφbm+μbmD2bm1−φbm22ΛXφbm+μbmD3bm1−φbm23ΛXφbm<M.
Thus, for everym sufficiently large,
(73)μbmD0bmΛXφbm−M≲μbmD0bm2fmφbm<M.
By (38), we deduce
(74)μbmD0bm≤1γμbmD0wΛXφbm.
Hence, for everym sufficiently large, we have
(75)μbmD0bmΛXφbm≲Mγsupw∈DμwD0wΛXφw+M=Mγ+M,which means L0~=limm⟶∞μwD0wΛXφw=0.
Conversely, assume that the quantitiesLj~=0, for all j=0,1,2,3. By applying Lemma 5, to prove that Tψ1,ψ2,φ:X⟶Zμ is a compact operator, it is enough to find a sequence hm in X converges uniformly to 0, on compact subsets of D and such that limn⟶∞Tψ1,ψ2,φhmX=0.
Now, fixM>0, for each m∈ℕ, suppose hm is a sequence in X. Let 0<δ<1, whenever φw>δ, such that
(76)μwD0wΛXφw<M4,(77)μwD1w1−φw2ΛXφw<M4,(78)μwD2w1−φw22ΛXφw<M4,(79)μwD3w1−φw23ΛXφw<M4.
Since a sequencehm converges uniformly to 0 on w≤δ, it does so all sequences hmk, for k=1,2,3. Thus, there is an integer N>0, for all m>N, such that hmk<δ, for k=0,1,2,3 and all w≤δ. Hence, as in Theorem 3 if m>N and φw≤δ, using the quantities Kj~, for j=0,1,2,3, then
(80)μwD0wΛXφw≲MK0~,μwD1w1−φw2ΛXφw≲MK1~,μwD2w1−φw22ΛXφw≲MK2~,μwD3w1−φw23ΛXφw≲MK3~.
Therefore, one last time by (38) for φw≤δ, we obtain
(81)μwTψ1,ψ2,φhm″w≤1γμwD0whmφwΛXφw+μwD1w1−φw2hm′φwΛXφw+μwD2w1−φw22h″mφwΛXφw+μwD3w1−φw23hm3φwΛXφw<Mγ∑j=03Kj~.
Letφw>δ, and let m be large sufficient. Then, by (38), condition (VI) for k=1,2,3, and the hypothesis (76), we have
(82)μwTψ1,ψ2,φhm″w≲μwD0wΛXφw+μwD1w1−φw2ΛXφw+μwD2w1−φw22ΛXφw+μwD3w1−φw23ΛXφw<M.
The last step, since the sequenceshm and h′m are converged uniformly to 0 on compact subsets of D, then, they converge pointwise. Thus,
(83)limn⟶∞Tψ1,ψ2,φh0+Tψ1,ψ2,φh′0=0.
By this with the last two inequalities, we see thatlimn⟶∞Tψ1,ψ2,φhmX=0. According to Lemma 5, we already got the compactness of Tψ1,ψ2,φ:X⟶Zμ.
Now, in the case ofψ1w=ψw and ψ2w=0 in Theorem 6, by using the quantities in (35), we deduce the following result as a corollary, see Theorem 3.2 in [4].Corollary 7.
Let a Banach spaceX be satisfying the above conditions (II), (V), and (VI). Suppose that Tψ,φ:X⟶Zμ is a bounded operator and suppose that (38) holds. Then, Tψ,φ:X⟶Zμ is a compact operator if and only if all the quantities Lj=0, for j=0,1,2, where
(84)L0=limδ⟶1supφw>δE0μ,ψ,φ,L1=limδ⟶1supφw>δE1μ,ψ,φ,L2=limδ⟶1supφw>δE2μ,ψ,φ.


## 5. Boundedness and Compactness ofTψ1,ψ2,φ:X⟶Zμ,0

In this part, we will focus on the operatorsTψ1,ψ2,φ mapping a large class X of a Banach spaces into the little Zygmund-type space Zμ,0. In particular, especially, we develop the compactness and the boundedness conditions of Tψ1,ψ2,φ:X⟶Zμ,0.As is the case in the previous two sections, to simplify the formulation of the results, we define the following limits forΛXw as in (4).
(85)W0~=limw⟶1μwD0wΛXφw,W1~=limw⟶1μwD1w1−φw2ΛXφw,W2~=limw⟶1μwD2w1−φw22ΛXφw,W3~=limw⟶1μwD3w1−φw23ΛXφw.Now, we go to the main results of this section.Theorem 8.
Let a Banach spaceX be satisfying the above conditions (II), (V), and (VI) and suppose that (38) holds. Then, the equivalence of the following conditions is certain.
(i)
Tψ1,ψ2,φ:X⟶Zμ,0 is a compact operator(ii)
Tψ1,ψ2,φ:X⟶Zμ,0 is a bounded operator(iii)
The quantitiesWj~=0, for j=0,1,2,3Proof.
i⇒ii is evident.
To prove (ii)⇒(iii), assume that Tψ1,ψ2,φ:X⟶Zμ,0 is a bounded operator. Then,
(86)Tψ1,ψ2,φhZμhX,forallh∈X.Firstly, we let a sequencebm in D, be such that limm⟶∞bm=1. Fix h∈X with hX≤1, by the functions defined in Lemma 2 for k=1,2,3, we can consider the functions
(87)Fm,kw=ρφbmbmkhw.By Lemma2, we inferred that
(88)Fm,kjφbm=0,ifk>j,(89)Fm,kjφbm=k!−1kΛXφbm1−φbm2j,ifk=j.Also by using condition (V), we have that all the sequencesFm,k are bounded in X. Now, set k=3 in (88), we observe that
(90)Fm,3φbm=Fm,3′φbm=F″m,3φbm=0,Fm,33φbm=−6hφbm1−φbm23.SinceTψ1,ψ2,φFm,3∈Zμ,0, then, we obtain
(91)limm⟶∞μbm6D3bmhφbm1−φbm23=6limm⟶∞μbmTψ1,ψ2,φFm,3″bm=0.Over all,h∈X with hX≤1, by taking the supremum and by (5), we have
(92)W3~=limm⟶∞μbmD3bm1−φbm23ΛXφbm=0.Similarly, we prove thatW2~=0. So, set k=2 in (88); we observe that
(93)Fm,2φbm=Fm,2′φbm=0,F″m,2φbm=2hφbm1−φbm22.SinceTψ1,ψ2,φFm,2∈Zμ,0, then, we obtain
(94)limm⟶∞μbmTψ1,ψ2,φFm,2″bm=limm⟶∞μbmD2bmF″m,2φbm+D3bmFm,23φbm=0.Then, by condition (VI) fork=3 with (92), we have
(95)limm⟶∞μbmD2bmF″m,2φbm≲limm⟶∞μbmD3bm1−φbm23ΛXφbm=0.Again, over allh∈X with hX≤1, by taking the supremum and by (5) we have
(96)W2~=limm⟶∞μbmD2bm1−φbm22ΛXφbm=0.Third likewise, we prove thatW1~=0. Set k=1 in (88), we observe that
(97)Fm,1φbm=0;Fm,1′φbm=−fmφbm1−φbm2.Also, sinceTψ1,ψ2,φFm,1∈Zμ,0, by condition (VI) for k=2,3 with (92) and (96), we obtain
(98)limm⟶∞μbmD1bmFm,1′φbmlimm⟶∞μbmD2bm1−φbm22ΛXφbm+limm⟶∞μbmD3bm1−φbm23ΛXφbm=0.Again, over allh∈X with hX≤1, by taking the supremum and by (5), we have
(99)W1~=limm⟶∞μbmD1bm1−φbm2ΛXφbm=0.Finally, we will prove thatW0~=0. For each h∈X as above. Using the fact that Tψ1,ψ2,φh∈Zμ,0, we get
(100)limm⟶∞μbmTψ1,ψ2,φh″bm=limm⟶∞μbmD0bmhφbm+D1bmh′φbm+D2bmh″φbm+D3bmh3φbm=0.SinceW3~=W2~=W1~=0, by condition (VI), we observe that
(101)limm⟶∞μbmD0bmhφbmlimm⟶∞μbmD1bm1−φbm2ΛXφbm+limm⟶∞μbmD2bm1−φbm22ΛXφbm+limm⟶∞μbmD3bm1−φbm23ΛXφbm=0.Thus, over allh∈X with hX≤1, by taking the supremum and by (5) we have
(102)limm⟶∞μbmD0bmΛXφbm=0,which means W0~=0.Conversely, we prove (iii)⇒(i), assume that all the quantities Wj~=0, for all j=0,1,2,3. By applying Theorem 6, we have the compactness of the operator Tψ1,ψ2,φ:X⟶Zμ. Thus, to prove that Tψ1,ψ2,φ:X⟶Zμ,0 is a compact operator, it is enough to show for all h∈X that Tψ1,ψ2,φh∈Zμ,0. So, let h∈X, then by condition (VI) for k=1,2,3, and (5), we observe that
(103)limw⟶1μwTψ1,ψ2,φh″w≲hXlimw⟶1μwψ1″wΛXφw+limw⟶1μwD1w1−φw2ΛXφw+limw⟶1μwD2w1−φw22ΛXφw+limw⟶1μwD3w1−φw23ΛXφw=0.Thus, as desirable,Tψ1,ψ2,φh∈Zμ,0.In the caseψ1w=ψw and ψ2w=0 in Theorem 8, by using the quantities in (35), we deduce the following result as a corollary; see Theorem 3.2 in [4].Corollary 9.
Let a Banach spaceX be satisfying the above conditions (II), (V), and (VI) and suppose that (38) holds. Then, the equivalence of the following conditions is certain:
(i)
Tψ,φ:X⟶Zμ,0 is a compact operator(ii)
Tψ,φ:X⟶Zμ,0 is a bounded operator(iii)
The quantitiesWj=0, for j=0,1,2, where(104)W0=limw⟶1E0μ,ψ,φ,W1=limw⟶1E1μ,ψ,φ,W2=limw⟶1E2μ,ψ,φ.


## 6. Implementation of the Results on Other Domains

In this section, we will show that our estimations hold up for any domain space choice provided that the domain is reflexive. Note that all spaces handled below contain the constant functions, so the following condition holds for all:(105)γ≔infw∈DΛXφw>0.


### 6.1. The Banach SpaceH∞

Observe that the mapζ⟶Λζ is a constant 1 mapping. Clearly for the H∞ space, all the conditions (II)-(V) hold. By Theorem 5.4 in [11], for any integer k, we know that the space H∞ and the Bloch space B are connected by the condition
(106)1−ζkhkζ≲hB,forζ∈D,h∈B.Then, theH∞ space satisfies also the condition (VI). So, we deduce that all the results in Section 3–5 are applicable. We set the following quantities:
(107)Q0=μwD0w,Q1=μwD1w1−φw2,Q2=μwD2w1−φw22,Q3=μwD3w1−φw23.The theory below summarizes the results are applicable for the domain spaceH∞.Theorem 10.
The operatorTψ1,ψ2,φ:H∞⟶Zμ is a bounded operator if and only if all the quantities supw∈DQj are finite, for j=0,1,2,3, in which case Tψ1,ψ2,φH∞⟶Zμ≈∑j=13supw∈DQj.
Moreover, assume thatTψ1,ψ2,φ:H∞⟶Zμ is a bounded operator; then, it is a compact operator if and only if limδ⟶1supφw>δQj=0, for all j=0,1,2,3.Theorem 11.
The following conditions are equivalent:(i)
Tψ1,ψ2,φ:H∞⟶Zμ,0 is a compact operator(ii)
Tψ1,ψ2,φ:H∞⟶Zμ,0 is a bounded operator(iii)
The quantitieslimw⟶1Qj=0, for j=1,2,3 and ψ1∈Zμ,0


### 6.2. The Weighted Bergman SpaceAβq

For allβ>−1 and 1≤q<∞, the weighted Bergman space Aβq is the set of all functions h∈HOD, such that
(108)hAβqq=∫DhζqdAβζ<∞,where dAβζ=β+11−ζ2βdAζ is the weighted Lebesgue measure. When β>−1, the positive measure dAβ was normalized to be a probability measure. The weighted Bergman-type space Aβq is a Banach space when 1≤q<∞; also, it is a complete metric space when 0<q<1. Several properties of Aβq spaces are discussed in [12, 13].Lemma12 was proved in [12] and Lemma 5.3 in [10].Lemma 12.
For a positive integern and ζ∈D, let β>−1 and 1≤q<∞ and let h∈Aβq. Then,
(109)ΛAβqζ=11−ζ2β+2/q,hnζ≲hAβq1−ζ2n+β+2/q.
Set the following quantities:(110)R0=μwD0w1−φw2β+2/q,R1=μwD1w1−φw2β+2+q/q,R2=μwD2w1−φw2β+2+2q/q,R3=μwD3w1−φw2β+2+3/q.
Now, we summarize the results are applicable for the domain spaceAβq.Theorem 13.
The operatorTψ1,ψ2,φ:Aβq⟶Zμ is a bounded operator if and only if all the quantities supw∈DRj are finite, for j=0,1,2,3, in which case Tψ1,ψ2,φAβq⟶Zμ≈∑j=03supw∈DRj.
Moreover, assume thatTψ1,ψ2,φ:Aβq⟶Zμ is a bounded operator; then, it is a compact operator if and only if limδ⟶1supφw>δRj=0, for all j=0,1,2,3.Proof.
From Lemma12, we have that the conditions (V) and (VI) hold for all k=1,2,3. Then, the norm estimate of the operator is an outcome instant from Theorem 3.
The results forTψ1,ψ2,φ:Aβq⟶Zμ,0 are an outcome instant from Theorem 8.Theorem 14.
The following conditions are equivalent:(i)
Tψ1,ψ2,φ:Aβq⟶Zμ,0 is a compact operator(ii)
Tψ1,ψ2,φ:Aβq⟶Zμ,0 is a bounded operator(iii)
The quantitieslimw⟶1Rj=0, for j=0,1,2,3Proof.
From Lemma12, we have that the conditions (V) and (VI) hold for all k=1,2,3. Then, the results are an outcome instant from Theorem 8.


### 6.3. The Hardy SpaceHq

Forq≥1 and h∈HOD, the Hardy space HqD=Hq is the Banach space with the norm
(111)hHqq=sup0<s<1∫02πhseiθqdθ2π.We know that the explicit formula for Hardy spaceHq is given by
(112)ΛHqζ=11−ζ21/q.Moreover, for a positive integern,ζ∈D and 1≤q<∞ if h∈Hq, then,
(113)hnζ≲hHq1−ζ2n+1/q.From (112) and (113), also, we have that the conditions (V) and (VI) hold for all k=1,2,3 on Hardy space Hq. Then, the norm estimate of the operator is an outcome instant from Theorem 3.The results forTψ1,ψ2,φ:∈Hq⟶Zμ,0 are an instant outcome from Theorem 8. So, we can summarize the results as follows.Theorem 15.
The operatorTψ1,ψ2,φ:Hq⟶Zμ is a bounded operator if and only if all the quantities supw∈DRj^ are finite, for j=0,1,2,3, in which case Tψ1,ψ2,φHq⟶Zμ≈∑j=03supw∈DRj^.
Assume thatTψ1,ψ2,φ:Aβq⟶Zμ is a bounded operator, then it is a compact operator if and only if limδ⟶1supφw>δRj^=0, for all j=0,1,2,3, where
(114)R0^=μwψ1″w1−φw21/q,R1^=μwD1w1−φw21+q/q,R2^=μwD2w1−φw21+2q/q,R3^=μwD3w1−φw21+3q/q.Theorem 16.
The following conditions are equivalent:(i)
Tψ1,ψ2,φ:Hq⟶Zμ,0 is a compact operator(ii)
Tψ1,ψ2,φ:Hq⟶Zμ,0 is a bounded operator(iii)
The quantitieslimw⟶1Rj^=0, for j=0,1,2,3


## 6.1. The Banach SpaceH∞

Observe that the mapζ⟶Λζ is a constant 1 mapping. Clearly for the H∞ space, all the conditions (II)-(V) hold. By Theorem 5.4 in [11], for any integer k, we know that the space H∞ and the Bloch space B are connected by the condition
(106)1−ζkhkζ≲hB,forζ∈D,h∈B.Then, theH∞ space satisfies also the condition (VI). So, we deduce that all the results in Section 3–5 are applicable. We set the following quantities:
(107)Q0=μwD0w,Q1=μwD1w1−φw2,Q2=μwD2w1−φw22,Q3=μwD3w1−φw23.The theory below summarizes the results are applicable for the domain spaceH∞.Theorem 10.
The operatorTψ1,ψ2,φ:H∞⟶Zμ is a bounded operator if and only if all the quantities supw∈DQj are finite, for j=0,1,2,3, in which case Tψ1,ψ2,φH∞⟶Zμ≈∑j=13supw∈DQj.
Moreover, assume thatTψ1,ψ2,φ:H∞⟶Zμ is a bounded operator; then, it is a compact operator if and only if limδ⟶1supφw>δQj=0, for all j=0,1,2,3.Theorem 11.
The following conditions are equivalent:(i)
Tψ1,ψ2,φ:H∞⟶Zμ,0 is a compact operator(ii)
Tψ1,ψ2,φ:H∞⟶Zμ,0 is a bounded operator(iii)
The quantitieslimw⟶1Qj=0, for j=1,2,3 and ψ1∈Zμ,0


## 6.2. The Weighted Bergman SpaceAβq

For allβ>−1 and 1≤q<∞, the weighted Bergman space Aβq is the set of all functions h∈HOD, such that
(108)hAβqq=∫DhζqdAβζ<∞,where dAβζ=β+11−ζ2βdAζ is the weighted Lebesgue measure. When β>−1, the positive measure dAβ was normalized to be a probability measure. The weighted Bergman-type space Aβq is a Banach space when 1≤q<∞; also, it is a complete metric space when 0<q<1. Several properties of Aβq spaces are discussed in [12, 13].Lemma12 was proved in [12] and Lemma 5.3 in [10].Lemma 12.
For a positive integern and ζ∈D, let β>−1 and 1≤q<∞ and let h∈Aβq. Then,
(109)ΛAβqζ=11−ζ2β+2/q,hnζ≲hAβq1−ζ2n+β+2/q.
Set the following quantities:(110)R0=μwD0w1−φw2β+2/q,R1=μwD1w1−φw2β+2+q/q,R2=μwD2w1−φw2β+2+2q/q,R3=μwD3w1−φw2β+2+3/q.
Now, we summarize the results are applicable for the domain spaceAβq.Theorem 13.
The operatorTψ1,ψ2,φ:Aβq⟶Zμ is a bounded operator if and only if all the quantities supw∈DRj are finite, for j=0,1,2,3, in which case Tψ1,ψ2,φAβq⟶Zμ≈∑j=03supw∈DRj.
Moreover, assume thatTψ1,ψ2,φ:Aβq⟶Zμ is a bounded operator; then, it is a compact operator if and only if limδ⟶1supφw>δRj=0, for all j=0,1,2,3.Proof.
From Lemma12, we have that the conditions (V) and (VI) hold for all k=1,2,3. Then, the norm estimate of the operator is an outcome instant from Theorem 3.
The results forTψ1,ψ2,φ:Aβq⟶Zμ,0 are an outcome instant from Theorem 8.Theorem 14.
The following conditions are equivalent:(i)
Tψ1,ψ2,φ:Aβq⟶Zμ,0 is a compact operator(ii)
Tψ1,ψ2,φ:Aβq⟶Zμ,0 is a bounded operator(iii)
The quantitieslimw⟶1Rj=0, for j=0,1,2,3Proof.
From Lemma12, we have that the conditions (V) and (VI) hold for all k=1,2,3. Then, the results are an outcome instant from Theorem 8.


## 6.3. The Hardy SpaceHq

Forq≥1 and h∈HOD, the Hardy space HqD=Hq is the Banach space with the norm
(111)hHqq=sup0<s<1∫02πhseiθqdθ2π.We know that the explicit formula for Hardy spaceHq is given by
(112)ΛHqζ=11−ζ21/q.Moreover, for a positive integern,ζ∈D and 1≤q<∞ if h∈Hq, then,
(113)hnζ≲hHq1−ζ2n+1/q.From (112) and (113), also, we have that the conditions (V) and (VI) hold for all k=1,2,3 on Hardy space Hq. Then, the norm estimate of the operator is an outcome instant from Theorem 3.The results forTψ1,ψ2,φ:∈Hq⟶Zμ,0 are an instant outcome from Theorem 8. So, we can summarize the results as follows.Theorem 15.
The operatorTψ1,ψ2,φ:Hq⟶Zμ is a bounded operator if and only if all the quantities supw∈DRj^ are finite, for j=0,1,2,3, in which case Tψ1,ψ2,φHq⟶Zμ≈∑j=03supw∈DRj^.
Assume thatTψ1,ψ2,φ:Aβq⟶Zμ is a bounded operator, then it is a compact operator if and only if limδ⟶1supφw>δRj^=0, for all j=0,1,2,3, where
(114)R0^=μwψ1″w1−φw21/q,R1^=μwD1w1−φw21+q/q,R2^=μwD2w1−φw21+2q/q,R3^=μwD3w1−φw21+3q/q.Theorem 16.
The following conditions are equivalent:(i)
Tψ1,ψ2,φ:Hq⟶Zμ,0 is a compact operator(ii)
Tψ1,ψ2,φ:Hq⟶Zμ,0 is a bounded operator(iii)
The quantitieslimw⟶1Rj^=0, for j=0,1,2,3



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