# Essential Norms of Volterra Type Operators betweenZygmund Type Spaces

Authors: Shanli Ye; Caishu Lin
Journal: Journal of Function Spaces
 (2017)

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## Abstract

We investigate the boundedness of some Volterra type operators betweenZygmund type spaces. Then, we give the essential norms of such operators in terms of g,φ, their derivatives, and thenth power φn of φ.


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## Body

## 1. Introduction

LetD={z:z<1} be the open unit disk in the complex plane C and let ∂D={z:z=1} be its boundary, and H(D) denote the set of all analytic functions on D.For every0<α<∞, we denote by Bα the Bloch type space of all functions f∈H(D) satisfying (1)bαf=supz∈D⁡1-z2αf′z<∞endowed with the norm fBα=f0+bα(f). The little Bloch type space B0α consists of all f∈Bα satisfying limz→1-1-z2αf′z=0, and B0α is obviously the closed subspace of Bα. When α=1, we get the classical Bloch space B1=B and little Bloch space B01=B0. It is well known that, for 0<α<1, Bα is a subspace of H∞, the Banach space of bounded analytic functions on D. Some sources for results and references about the Bloch type functions are the papers of Yoneda [1], Stevic [2, 3], and the first author [4–7].For0<α<∞ we denote by Zα theZygmund type space of those functions f∈H(D) satisfying (2)supz∈D⁡1-z2αf′′z<∞,and the little Zygmund type space Z0α consists of all f∈Zα satisfying limz→1-1-z2αf′′z=0, and Z0α is obviously the closed subspace of Zα. It can easily be proved that Zα is a Banach space under the norm (3)fZα=f0+f′0+supz∈D⁡1-z2αf′′zand that Z0α is a closed subspace of Zα. When α=1, we get the classical Zygmund space Z1=Z and the little Zygmund space Z01=Z0. It is clear that f∈Z if and only if f′∈B1.We consider the weighted Banach spaces of analytic functions(4)Hv∞=f∈HD:Fv=supz∈D⁡vzfz<∞endowed with norm ·v, where the weight v:D→R+ is a continuous, strictly positive, and bounded function. The weight v is called radial, if v(z)=v(z) for all z∈D. For a weight v the associated weight v~ is defined by (5)v~z=sup⁡fz:f∈Hv∞,fv≤1-1,z∈D.We notice the standard weights vα(z)=1-z2α, where 0<α<∞, and it is well known that v~α=vα. We also consider the logarithmic weight (6)vlog=log⁡21-z2-1,z∈D.It is straightforward to show that v~log=vlog.For an analytic self-mapφ of D and a function u∈H(D), we define the weighted composition operator as uCφf=u·(f∘φ) for f∈H(D). Weighted composition operators have been extensively studied recently. It is interesting to provide a function theoretic characterization when φ and u induce a bounded or compact composition operator on various function spaces. Some results on the boundedness and compactness of concrete operators between some spaces of analytic functions one of which is of Zygmund type can be found, for example, in [8–19].Suppose thatg:D→C is an analytic map. Let Tg and Ig denote the Volterra type operators with the analytic symbol g on D, respectively:(7)Tgfz=∫0zfξg′ξdξ,z∈D,Igfz=∫0zf′ξgξdξ,z∈D.Ifg(z)=z, then Tg is an integral operator. While g(z)=ln⁡(1/1-z), then Tg is Cesàro operator. Pommerenke introduced the Volterra type operator Tg and characterized the boundedness of Tg between H2 spaces in [20]. More recently, boundedness and compactness of Volterra type operators between several spaces of analytic functions have been studied by many authors; one may see [21, 22].In this paper, we consider the following integral type operators, which were introduced by Li and Stevic (see, e.g., [10, 23]); they can be defined by (8)CφTgfz=∫0φzfξg′ξdξ,CφIgfz=∫0φzf′ξgξdξ,TgCφfz=∫0zfφξg′ξdξ,IgCφfz=∫0zf′φξgξdξ.We will characterize the boundedness of those integral type operators between Zygmund type spaces and also estimate their essential norms. The boundedness and compactness of these operators on the logarithmic Bloch space have been characterized in [22].Recall that essential normTe,X→Y of a bounded linear operator T:X→Y is defined as the distance from T to K(X,Y), the space of compact operators from X to Y, namely, (9)Te,X→Y=inf⁡T+KX→Y:K:X→Yis  compact.It provides a measure of noncompactness of T. Clearly, T is compact if and only if Te,X→Y=0.Throughout this paper, constants are denoted byC, they are positive and may differ from one occurrence to the other. The notation a≍b means that there are positive constants C1,C2 such that C1a≤b≤C2a.


## 2. Boundedness

In order to prove the main results of this paper. We need some auxiliary results.Lemma 1 (see [8, 13]).
For0<α<2 and let {fn} be a bounded sequence in Zα which converges to 0 uniformly on compact subsets of D. Then limn→∞⁡supz∈D⁡fnz=0.Lemma 2 (see [8, 13]).
For everyf∈Zα, where α>0, one has(i)
f′z≤(2/1-α)fZα and fz≤(2/1-α)fZα for every 0<α<1;(ii)
f′z≤2log⁡(2/(1-z))fZα and fz≤fZα for α=1;(iii)
f′z≤(2/1-α)(fZα/(1-z)α-1), for every α>1;(iv)
fz≤(2/(α-1)(2-α))fZα, for every 1<α<2;(v)
fz≤2log⁡(2/1-z)fZα, for every α=2;(vi)
fz≤(2/α-1α-2)(fZα/(1-z)α-2), for every α>2.Lemma 3 (see [8]).
Let0<α<∞ and v a radial, nonincreasing weight tending to 0 at boundary of D, and let the weighted composition operator uCφ:Zα→Hv∞ be bounded.(i)
If0<α<2, then uCφ is a compact operator.(ii)
(10)uCφe,Z2→Hv∞≍limsupn→∞⁡log⁡nuφnv≍limsupφz→1⁡vzuzlog⁡21-φz2.(iii)
Ifα>2, then(11)uCφe,Zα→Hv∞≍limsupn→∞⁡n+1α-2uφnv≍limsupφz→1⁡vzuz1-φz2α-2.The following lemma is due to [24, 25].Lemma 4.
Letv and w be radial, nonincreasing weights tending to zero at the boundary of D. Then(i)
The weighted composition operatoruCφ maps Hv∞ into Hw∞ if and only if (12)supn≥0⁡uφnwznv≍supz∈D⁡wzv~φzuz<∞,with norm comparable to the above supermum.(ii)
(13)uCφe,Hv∞→Hw∞=limsupn→∞⁡uφnwznv=limsupφz→1⁡wzv~φzuz.Lemma 5 (see [26]).
For every0<α<∞, one has(14)limn→∞⁡n+1αznvα=2αeα,(15)limn→∞⁡log⁡nznvlog=1.Theorem 6.
Letφ be an analytic self-map of D and g∈H(D).(i)
If0<α<1, then IgCφ:Zα→Zβ is a bounded operator if and only if g′∈Hvβ∞ and (16)supn≥0⁡n+1αgφ′φnvβ≍supz∈D⁡1-z2β1-φz2αgzφ′z<∞.(ii)
Ifα=1, then IgCφ:Zα→Zβ is a bounded operator if and only if (17)supn≥0⁡n+1gφ′φnvβ≍supz∈D⁡1-z2β1-φz2gzφ′z<∞,(18)supn≥0⁡log⁡ng′φnvβ≍supz∈D⁡1-z2βlog21-φz2g′z<∞.(iii)
Ifα>1, then IgCφ:Zα→Zβ is a bounded operator if and only if (19)supn≥0⁡n+1αgφ′φnvβ≍supz∈D⁡1-z2β1-φz2αgzφ′z<∞,(20)supn≥0⁡n+1α-1g′φnvβ≍supz∈D⁡1-z2β1-φz2α-1g′z<∞.Proof.
Suppose thatIgCφ is bounded from Zα to Zβ. Using the test functions f(z)=z and f(z)=z2, we have (21)1-z2βIgCφz′′=1-z2βg′z<∞,(22)1-z2βIgCφz2′′=1-z2β2φ′zgz+2φzg′z<∞.Since φ is a self-map, we get that g′∈Hvβ∞, φ′g∈Hvβ∞.
For every0<α<∞ and given nonzero a∈D, we take the test functions (23)faz=1a2¯1-a221-a¯zα-1-a21-a¯zα-1,(24)haz=1a¯∫0z1-a21-a¯wαdw,(25)gaz=faz-haz,for every z∈D. One can show that fa,ha, and ga are in Z0α, sup1/2<a<1faZα<∞, and sup1/2<a<1haZα<∞. Since ga′(a)=0,ga′′(a)=α/(1-a2)α, it follows that for all z∈D with φz>1/2, we have (26)+∞>CgaZα≥IgCφgφzZβ≥1-z2βφ′zgzgφz′′φz-1-z2βg′zgφz′φz=1-z2βφ′zgzα1-φz2α.Then (27)supz∈D⁡φ′zgz1-z2β1-φz2α≤supφz≤1/2⁡φ′zgz1-z2β1-φz2α+supφz>1/2⁡φ′zgz1-z2β1-φz2α≤43αφ′gvβ+CgaZα<∞.
Now we use (14) and Lemma 4 to conclude that (28)supn≥0⁡n+1αgφ′φnvβ≍supz∈D⁡1-z2β1-φz2αgzφ′z<∞,which shows that (16) is necessary for all case.
Conversely, suppose thatg′∈Hvβ∞ and (16) holds. Assume that f∈Zα. From Lemma 2, it follows that(29)1-z2βIgCφf′′z=1-z2βφ′zgzf′′φz+g′zf′φz≤1-z2βφ′zgzf′′φz+1-z2βg′zf′φz≤1-z2β1-φz2αφ′zgzfZα+C1-z2βg′zfZα≤CfZα,IgCφf0=0,IgCφf′0=f′φ0g0≤fZαgφ0,which implies that IgCφ is bounded. This completes the proof of (i).
Next we will prove (ii). The necessity in condition (17) has been proved above. Fixing a∈D with a>1/2, we take the function(30)kaz=pa¯za¯log⁡11-a-1,for z∈D, where (31)pz=z-11+log⁡11-z2+1.Then we have sup1/2<a<1kaZα≤C by [11]. Let a=φ(z). It follows that (32)IgCφkφzZβ≥1-z2βg′zkφz′φz-1-z2βφ′zgzkφz′′φz=1-z2βg′zlog⁡11-φz2-1-z2βφ′zgzα1-φz2α.Since (17) holds and IgCφ is bounded, we obtain that (33)supφz>1/2⁡1-z2βlog⁡11-φz2g′z≤supφz>1/2⁡1-z2βφ′zgzα1-φz2α+supφz>1/2⁡IgCφkφzZβ<∞.Noting g′∈Hvβ∞ and together with (15) and Lemma 4, we conclude that (18) holds.
The converse implication can be shown as in the proof of (i).
Finally we will prove (iii). We have proved that (19) holds above. To prove (20), we take function fφ(z) defined in (23) for every z∈D with φz>1/2 and obtain that (34)IgCφfφzZβ≥1-z2βg′zfφz′φz-1-z2βφ′zgzfφz′′φz=1-z2βg′z1φz¯1-φz2α-1-1-z2βφ′zgz2φz1-φz2α.Since IgCφ is bounded and (19) holds, we obtain that (35)supφz>1/2⁡1-z2β1-φz2α-1g′z≤2supφz>1/2⁡1-z2βφ′zgz21-φz2α+2supφz>1/2⁡IgCφfφzZβ<∞;therefore, we deduce that (20) holds by (14) and Lemma 4.
The converse implication can be shown as in the proof of (i).Theorem 7.
Letφ be an analytic self-map of D and g∈H(D).(i)
If0<α<1, then CφIg:Zα→Zβ is a bounded operator if and only if ((g∘φ)(φ′′)+g′∘φ(φ′)2)∈Hvβ∞ and (36)supn≥0⁡n+1αg∘φφ′2φnvβ≍supz∈D⁡1-z2β1-φz2αgφzφ′z2<∞.(ii)
Ifα=1, then CφIg:Zα→Zβ is a bounded operator if and only if (37)supn≥0⁡n+1g∘φφ′2φnvβ≍supz∈D⁡1-z2β1-φz2gφzφ′z2<∞,supn≥0⁡log⁡ng′∘φφ′2+g∘φφ′′φnvβ≍supz∈D⁡1-z2βlog21-φz2g′φzφ′z2+gφzφ′′z<∞.(iii)
Ifα>1, then CφIg:Zα→Zβ is a bounded operator if and only if (36) holds and (38)supn≥0⁡n+1α-1g′∘φφ′2+g∘φφ′′φnvβ≍supz∈D⁡1-z2β1-φz2α-1g′φzφ′z2+gφzφ′′z<∞.The proof is similar to that of Theorem6, and the details are omitted.Theorem 8.
Letφ be an analytic self-map of D and g∈H(D).(i)
If0<α<1, then CφTg:Zα→Zβ is a bounded operator if and only if ((g′∘φ)(φ′′)+(g′′∘φ)(φ′)2)∈Hvβ∞ and (g′∘φ)(φ′)2∈Hvβ∞.(ii)
Ifα=1, then CφTg:Zα→Zβ is a bounded operator if and only if ((g′∘φ)(φ′′)+(g′′∘φ)(φ′)2)∈Hvβ∞ and (39)supn≥0⁡log⁡ng′∘φφ′2φnvβ≍supz∈D⁡1-z2βlog⁡21-φz2g′φzφ′z2<∞;(iii)
If1<α<2, then CφTg:Zα→Zβ is a bounded operator if and only if ((g′∘φ)(φ′′)+(g′′∘φ)(φ′)2)∈Hvβ∞ and (40)supn≥0⁡n+1α-1g′∘φφ′2φnvβ≍supz∈D⁡1-z2β1-φz2α-1g′φzφ′z2<∞.(iv)
Ifα=2, then CφTg:Zα→Zβ is a bounded operator if and only if (41)supn≥0⁡n+1g′∘φφ′2φnvβ≍supz∈D⁡1-z2β1-φz2g′φzφ′z2<∞,(42)supn≥0⁡log⁡ng′′∘φφ′2+g′∘φφ′′φnvβ≍supz∈D⁡1-z2βlog⁡21-φz2g′′φzφ′z2+g′φzφ′′z<∞.(v)
Ifα>2, then CφTg:Zα→Zβ is a bounded operator if and only if (40) holds and (43)supn≥0⁡n+1α-2g′′∘φφ′2+g′∘φφ′′φnvβ≍supz∈D⁡1-z2β1-φz2α-2g′′φzφ′z2+g′φzφ′′z<∞.Proof.
Suppose thatCφTg is bounded from Zα to Zβ space.
(i) Case
0
<
α
<
1. Using functions f=1∈Zα and f=z∈Zα, we obtain (44)supz∈D⁡1-z2βg′′φzφ′z2+g′φzφ′′z<∞,supz∈D⁡1-z2βg′φzφ′z2<∞.Then we obtain that ((g′∘φ)(φ′′)+(g′′∘φ)(φ′)2)∈Hvβ∞ and (g′∘φ)(φ′)2∈Hvβ∞ are necessary for all case.
For the converse implication, suppose that((g′∘φ)(φ′′)+(g′′∘φ)(φ′)2)∈Hvβ∞ and (g′∘φ)(φ′)2∈Hvβ∞. For f∈Zα, it follows from Lemma 2 that (45)1-z2βCφTgf′′z=1-z2βφ′z2g′φzf′φz+g′′φzφ′z2+g′φzφ′′zfφz≤1-z2βφ′z2g′φzf′φz+1-z2βg′′φzφ′z2+g′φzφ′′zfφz≤1-z2βφ′z2g′φzfZα+1-z2βg′′φzφ′z2+g′φzφ′′zfZα≤CfZα,CφTgf0=∫0φ0fζg′ζdζ≤maxζ≤φ0⁡fζmaxζ≤φ0⁡g′ζ≤21-αfZαmaxζ≤φ0⁡g′ζ,CφTgf′0=fφ0φ′0g′φ0≤fZαφ′0g′φ0.Then CφTg is bounded. This completes the proof of (i).
(ii) Case
α
=
1. We consider the test function kφ(z)(z) defined in (30) for every z∈D with φz>1/2. It follows that (46)CφTgkφzZβ≥log⁡11-φz21-z2βφ′z2g′φz-1-z2βφ′′zg′φz+g′′φzφz2kφzφz.Since ((g′∘φ)(φ′′)+(g′′∘φ)(φ′)2)∈Hvβ∞ and supφz>1/2CφTgkφ(z)Zβ≤C, we get (47)supφz>1/2⁡log⁡11-φz21-z2βφ′z2g′φz≤supφz>1/2⁡1-z2βφ′′zg′φz+g′′φzφz2kφzφz+CφTgkφzZβ<∞.Then we have (48)supz∈D⁡log⁡11-φz21-z2βφ′z2g′φz≤supφz≤1/2⁡log⁡11-φz21-z2βφ′z2g′φz+supφz>1/2⁡log⁡11-φz21-z2βφ′z2g′φz≤C+log43g′∘φφ′2vβ<∞.On the other hand, from (15) and Lemma 4, we have (49)supn≥0⁡log⁡ng′∘φφ′2φnvβ≍supz∈D⁡1-z2βlog⁡21-φz2g′φzφ′z2.Hence (39) holds.
The converse implication can be shown as in the proof of (i).
(iii) Case
1
<
α
<
2. ((g′∘φ)(φ′′)+(g′′∘φ)(φ′)2)∈Hvβ∞ has been proved above. We take the test function fφ(z) in (23) for every z∈D with φz>1/2; by the same way as (ii), we can obtain that (40) holds.
The converse implication can be shown as in the proof of (i).
(iv) Case
α
=
2. We have proved that (41) holds above. To prove (42), we consider another test function ta(z)=log⁡(2/1-a¯z⁡). Clearly ta∈Z2 and sup1/2<a<1taZ2<∞. For every z∈D with φz>1/2, it follows that (50)supz∈D⁡1-z2βCφTgta′′z≥supφz>1/2⁡log⁡11-φz21-z2βφ′z2g′′φz+g′φzφ′′z-supφz>1/2⁡1-z2β1-φz2g′φzφz2.Applying (41) we get (51)supφz>1/2⁡log⁡11-φz21-z2βφ′z2g′′φz+g′φzφ′′z≤supφz>1/2⁡1-z2β1-φz2g′φzφz2+CφTgtaZβ<∞.
Noting((g′∘φ)(φ′′)+(g′′∘φ)(φ′)2)∈Hvβ∞ and using Lemma 4 and (15), we conclude that (42) holds.
(v) Case
α
>
2
. We have proved that (40) holds above. Applying test function fφ(z) in (23) for every z∈D with φz>1/2, we have (52)S1=supφz>1/2⁡1-z2βφ′z2g′φzφz¯1-φz2α-1=supφz>1/2⁡1-z2βφ′z2g′φzfφz′φz≤supφz>1/2⁡CφTgfφzZβ<∞.With the same calculation for test function tφ(z)(φ(z))=(1-φz2)2/(1-φ(z)¯z)α with φz>1/2, then supφz>1/2tφ(z)Zα≤C, and we have that (53)S2=supφz>1/2⁡1-z2βφ′z2g′′φz+g′φzφ′′z1-φz2α-2+αφz¯φ′z2g′φz1-φz2α-1=supφz>1/2⁡1-z2βCφTgtφz′′φz≤CφTgZα→Zβsupφz>1/2⁡tφzZα<∞.Therefore, (54)supφz>1/2⁡1-z2βφ′z2g′′φz+g′φzφ′′z1-φz2α-2≤S2+αsupφz>1/2⁡φ′z2g′φz1-φz2α-1≤S2+αsupφz>1/2⁡φ′z2g′φzφz¯1-φz2α-1≤S2+αS1<∞.Since ((g′∘φ)(φ′′)+(g′′∘φ)(φ′)2)∈Hvβ∞, we conclude that (43) holds.Theorem 9.
Letφ be an analytic self-map of D and g∈H(D).(i)
If0<α<1, then TgCφ:Zα→Zβ is a bounded operator if and only if g′φ′∈Hvβ∞ and g′′∈Hvβ∞.(ii)
Ifα=1, then TgCφ:Zα→Zβ is a bounded operator if and only if g′′∈Hvβ∞ and (55)supn≥0⁡log⁡ng′φ′φnvβ≍supz∈D⁡1-z2βlog⁡21-φz2g′zφ′z<∞.(iii)
If1<α<2, then TgCφ:Zα→Zβ is a bounded operator if and only if g′′∈Hvβ∞ and (56)supn≥0⁡n+1α-1g′φ′φnvβ≍supz∈D⁡1-z2β1-φz2α-1g′zφ′z<∞.(iv)
Ifα=2, then TgCφ:Zα→Zβ is a bounded operator if and only if(57)supn≥0⁡n+1g′φ′φnvβ≍supz∈D⁡1-z2β1-φz2g′zφ′z<∞,supn≥0⁡log⁡ng′′φnvβ≍supz∈D⁡1-z2βlog⁡21-φz2g′′z<∞.(v)
Ifα>2, then TgCφ:Zα→Zβ is a bounded operator if and only if (56) holds and (58)supn≥0⁡n+1α-2g′′φnvβ≍supz∈D⁡1-z2β1-φz2α-2g′′z<∞.The proof is similar to that of Theorem8, and the details are omitted.


## 3. Essential Norms

In this section we estimate the essential norms of these integral type operators onZygmund type spaces in terms of g,φ, their derivatives, and thenth power φn of φ.LetZα~={f∈Zα:f(0)=f′(0)=0} and Bα~={f∈Bα:f(0)=0}. We note that every compact operator T∈K(Zα~,Zβ) can be extended to a compact operator K∈K(Zα,Zβ). In fact, for every f∈Zα,f-f(0)-f′(0)z∈Zα~, and we can define K(f)=T(f-f(0)-f′(0)z)+f(0)+f′(0)z.Forr∈(0,1), we consider the compact operator Kr:Zα→Zβ defined by Krf(z)=f(rz).Lemma 10.
IfX(IgCφ,CφIg,CφTg,TgCφ) is a bounded operator from Zα to Zβ space, then (59)Xe,Zα→Zβ=Xe,Zα~→Zβ.Proof.
ClearlyXe,Zα→Zβ≥Xe,Zα~→Zβ. Then we prove Xe,Zα→Zβ≤Xe,Zα~→Zβ.
LetT∈K(Zα,Zβ) be given. Let {rn} be an increasing sequence in (0,1) converging to 1 and A={h∣h=a+bz}, the closed subspace of Zα. Then (60)X-TZα→Zβ=supf∈Zα,fZα≤1⁡Xf-TfZβ≤supf∈Zα,fZα≤1⁡Xf-f0-f′0z-Tf-f0-f′0zZβ+supf∈Zα,fZα≤1⁡Xf0+f′0z-Tf0+f′0zZβ≤supg∈Zα~,gZα≤1⁡Xg-TZα~gZβ+suph∈A,hZα≤1⁡Xh-TαhZβ.Hence (61)infT∈KZα,Zβ⁡X-TZα→Zβ≤infT∈KZα,Zβ⁡X-TZα~Zα→Zβ+infT∈KZα,Zβ⁡X-TαZα→Zβ≤Xe,Zα~→Zβ+limn→∞⁡XI-KrnZα→Zβ.Since limn→∞⁡X(I-Krn)Zα→Zβ=0, we have Xe,Zα→Zβ≤Xe,Zα~→Zβ, and the proof is finished.LetDα:Zα→Bα and Sα:Bα→Hvα∞ be the derivative operators. Then clearly Dα and Sα are linear isometries on Zα~ and Bα~, respectively, and (62)SβDβIgCφDα-1Sα-1=gφ′Cφ+g′CφSα-1.Therefore (63)IgCφe,Zα~→Zβ≤g′Cφe,Bα~→Hvβ∞+gφ′Cφe,Hvα∞→Hvβ∞.Similarly,(64)SβDβCφIgDα-1Sα-1=g∘φφ′2Cφ+g′∘φφ′2+g∘φφ′′CφSα-1,(65)CφIge,Zα~→Zβ≤g′∘φφ′2Cφe,Hvα∞→Hvβ∞+g′∘φφ′2+g∘φφ′′Cφe,Bα~→Hvβ∞,(66)SβDβCφTgDα-1Sα-1=g′′∘φφ′2+g′∘φφ′′CφDα-1Sα-1+g′∘φφ′2CφSα-1,(67)CφTge,Zα~→Zβ≤g′∘φφ′2Cφe,Bα~→Hvβ∞+g′′∘φφ′2+g′∘φφ′′Cφe,Zα~→Hvβ∞,(68)SβDβTgCφDα-1Sα-1=g′′CφDα-1Sα-1+g′φ′CφSα-1,(69)TgCφe,Zα~→Zβ≤g′φ′e,Bα~→Hvβ∞+g′′Cφe,Zα~→Hvβ∞.Theorem 11.
LetIgCφ be a bounded operator from Zα to Zβ space.(i)
If0<α<1, then (70)IgCφe,Zα→Zβ≍limsupn→∞⁡n+1αgφ′φnvβ.(ii)
Ifα=1, then (71)IgCφe,Z→Zβ≍max⁡limsupn→∞⁡n+1gφ′φnvβ,limsupn→∞⁡log⁡ng′φnvβ.(iii)
Ifα>1, then (72)IgCφe,Zα→Zβ≍max⁡limsupn→∞⁡n+1αgφ′φnvβ,limsupn→∞⁡n+1α-1g′φnvβ.Proof.
(i) We start with the upper bound. First we show thatg′Cφ is a compact weighted composition operator for Bα into Hvβ∞. Suppose that {fn} is bounded sequence in Bα. From Lemma 3.6 in [27], {fn} has a subsequence {fnk} which converges uniformly on D to a function, which we can assume to be identically zero. Then it follows from Theorem 6 and Lemma 1 that (73)limk→∞⁡supz∈D⁡1-z2βg′zfnkφz≤Climk→∞⁡supz∈D⁡fnkz=0,which shows that g′Cφ:Bα→Hvβ∞ is a compact operator and g′Cφe,Bα→Hvβ∞=0. Applying (63), Lemmas 4, 5, and 10, we get that (74)IgCφe,Zα→Zβ≤gφ′Cφe,Hvα∞→Hvβ∞=limsupφz→1⁡1-z2β1-φz2αgzφ′z=limsupn→∞⁡gφ′φnvβznvα=e2ααlimsupn→∞⁡n+1αgφ′φnvβ.
For the lower bound, let{zn}⊆D with φzn>1/2 and φzn→1 as n→∞. Taking gn=gφ(zn) defined in (25), we obtain that {gn} is bounded sequence in Z0α converging to 0 uniformly on compact subset of D and supn∈NgnZα≤C. For every compact operator T:Zα→Zβ, (75)CIgCφ-TZα→Zβ≥limsupn→∞⁡IgCφgnZβ-limsupn→∞⁡TgnZβ≥αlimsupn→∞⁡1-zn2βgznφ′zn1-φzn2α.Now we use (14) and Lemma 4 to obtain that (76)IgCφe,Zα→Zβ≥IgCφ-TZα→Zβ≥αClimsupφzn→1⁡gznφ′zn1-zn2β1-φzn2α=αClimsupn→∞⁡gφ′φnvβznvα=Ce2ααlimsupn→∞⁡n+1αgφ′φnvβ.Hence (70) holds.
(ii) The boundedness ofIgCφ guarantees that gφ′Cφ:Hv1∞→Hvβ∞ and g′Cφ:B→Hvβ∞ are bounded weighted composition operators. Theorem 3.4 in [28] ensures that (77)g′Cφe,B→Hvβ∞≍limφz→1⁡1-z2βg′zlog⁡21-φz2.Now we use Lemmas 4, 5 and (63) to conclude that (78)IgCφe,Z→Zβ≤g′Cφe,B~→Hvβ∞+gφ′Cφe,Hv1∞→Hvβ∞≤Climsupn→∞⁡gφ′φnvβznv1+Climsupn→∞⁡g′φnvβznvlog=Climsupn→∞⁡log⁡ng′φnvβ+Ce2limsupn→∞⁡n+1gφ′φnvβ≤Cmax⁡limsupn→∞⁡n+1gφ′φnvβ,limsupn→∞⁡log⁡ng′φnvβ.
On the other hand, let{zn} be a sequence in D such that φzn>1/2 and φzn→1 as n→∞. Given (79)hnz=hφzn¯zφzn¯log⁡21-φzn2-1-∫0zlog3⁡21-φzn¯ωdωlog⁡21-φzn2-2,where h(z)=(z-1)((1+log⁡2/1-z)2+1), from [11] we know that {hn} is a bounded sequence in Z10 which converges to zero uniformly on compact subsets of D, and (80)hn′′φzn=-φzn¯1-φzn2,hn′φzn=0,supn⁡hnZ<+∞.For every compact operator T:Z→Zβ, we have T(hn)Zβ→0 as n→∞. By Lemmas 4 and 5, we obtain that (81)CIgCφ-TZ→Zβ≥IgCφhnZ→Zβ≥limsupn→∞⁡1-zn2βgznφ′znφzn1-φzn2≥limsupn→∞⁡gφ′φnvβznv1=e2limsupn→∞⁡n+1gφ′φnvβ.Now we take another function (82)fn=hφzn¯zφzn¯log⁡21-φzn2-1.From [11] we know that {fn} is a bounded sequence in Z10 which converges to zero uniformly on compact subsets of D, and supn≥1fnZ<+∞. It follows from Lemmas 4 and 5 that (83)CIgCφe,Z→Zβ≥limn→∞sup⁡IgCφfnZβ≥limsupn→∞⁡1-zn2βg′znlog⁡21-φzn2-limsupn→∞⁡1-zn2βgznφ′znφzn1-φzn2.Noting that limsupn→∞⁡n+1⁡(gφ′)φnvβ≤2C/eIgCφe,Z→Zβ, we obtain(84)C+2CeIgCφe,Z→Zβ≥limsupφzn→1⁡1-zn2βg′znlog⁡21-φzn2=limsupn→∞⁡log⁡ng′φnvβ.Hence we have IgCφe,Z→Zβ≥Cmax⁡{limsupn→∞⁡(n+1)gφ′φnvβ,limsupn→∞⁡(log⁡n)g′φnvβ}.
(iii) Letα>1. The proof of the upper bound is similar to that of (ii). From the proof of (i), we get that, for some constant C,(85)CIgCφe,Zα→Zβ≥limsupφz→1⁡gzφ′z1-zn2β1-φz2α.
Now, let{zn} be as before and note that the function fn=fφ(zn) given in (23). Then {fn} is bounded sequence in Z0α converging to zero uniformly on compact subsets of D; therefore (86)CIgCφe,Zα→Zβ≥limn→∞⁡IgCφfnZβ≥2αlimsupn→∞⁡1-zn2β1-φzn2α-1g′zn-limsupn→∞⁡1-zn2βgznφ′znφzn1-φzn2α.By (85), we have (87)CIgCφe,Zα→Zβ≥limsupn→∞⁡1-zn2β1-φzn2α-1g′zn,and the rest of the proof is similar to that of the previous, and we omit it.Theorem 12.
Letφ be an analytic self-map of D and g∈H(D), and CφTg:Zα→Zβ is a bounded operator.(i)
If0<α<1, then (88)CφTge,Zα→Zβ=0.(ii)
Ifα=1, then (89)CφTge,Z→Zβ≍limn→∞⁡sup⁡log⁡ng′∘φφ′2φnvβ.(iii)
If1<α<2, then (90)CφTge,Zα→Zβ≍limn→∞⁡sup⁡n+1α-1g′∘φφ′2φnvβ.(iv)
Ifα=2, then (91)CφTge,Z2→Zβ≍max⁡limn→∞⁡sup⁡n+1g′∘φφ′2φnvβ,limn→∞⁡sup⁡log⁡ng′′∘φφ′2+g′∘φφ′′φnvβ.(v)
Ifα>2, then(92)CφTge,Zα→Zβ≍max⁡limn→∞⁡sup⁡n+1α-1g′∘φφ′2φnvβ,limn→∞⁡sup⁡n+1α-2g′′∘φφ′2+g′∘φφ′′φnvβ.Proof.
(i) For the compactness of(g′∘φ(φ′)2)Cφ, the argument is similar to the proof of Theorem 11(i); then we have ((g′∘φ)φ′2)Cφe,Bα→Hvβ∞=0. Hence by (67) and Lemma 3, we get that CφTge,Zα→Zβ=0.
Next we will prove (ii). The boundedness ofCφTg guarantees that (g′∘φ(φ′)2)Cφ:B→Hvβ∞ and (g′′∘φ(φ′)2+g′∘φ(φ′′))Cφ:Z→Hvβ∞ are bounded weighted composition operators. We know that if uCφ:Z→Hvβ∞ is a bounded operator, then uCφ is a compact operator by Lemma 3. Hence we consider the boundedness of CφTg:Z→Zβ and just consider that (g′∘φ(φ′)2)Cφ:B→Hvβ∞ is a bounded operator.
Theorem3.4 in [28] ensures that (93)g′∘φφ′2Cφe,B→Hvβ∞≍limφ→1⁡1-z2βg′φzφ′z2log⁡21-φz2.From (67) and Lemmas 4 and 5, we have (94)CφTge,Z→Zβ≤Climsupn→∞⁡g′∘φφ′2φnvβznvlog=Climsupn→∞⁡log⁡ng′∘φφ′2φnvβ.
In order to proveCφTge,Z→Zβ≥limsupn→∞⁡(log⁡n)((g′∘φ)(φ′)2)φnvβ, we take the function (95)gnz=φzn¯z-1φzn¯1+log⁡11-φzn¯z2+1log⁡11-φzn2-1-an,where (96)an=φzn2-1φzn¯1+log⁡11-φzn22+1log⁡11-φzn2-1,and limn→∞⁡an=0. From [16] we obtain that {gn} is a bounded sequence in Z10 which converges to zero uniformly on compact subsets of D. By a direct calculation, we have (97)gnφzn=0,gn′φzn=log⁡11-φzn2.For every compact operator T:Z→Zβ, we have T(hn)Zβ→0 as n→∞. Let M=supn≥1gnZβ. It follows from Lemma 5 that (98)MCφTg-Te,Z→Zβ≥CφTgfne,Z→Zβ≥limsupn→∞⁡1-zn2βg′φznφ′zn2log⁡11-φzn2≥limsupn→∞⁡g′∘φφ′2φnvβznv≥limsupn→∞⁡log⁡ng′∘φφ′2φnvβ.This completes the proof.
The proof of (iii) is the same as that of Theorem11 (iii); we do not prove it again.
(iv) Letα=2. Applying Lemma 3 (ii) and Theorem 3.2 in [29], we get that(99)g′∘φφ′2Cφe,B2→Hvβ∞≍limsupn→∞⁡n+1g′∘φφ′2φnvβ,g′′∘φφ′2+g′∘φφ′′Cφe,Z2→Hvβ∞≍limsupn→∞⁡log⁡ng′′∘φφ′2+g′∘φφ′′φnvβ,which yields the upper bound by (67).
With the same arguments as in the proof of Theorems8 and 11, for some constant C, we have(100)CCφTge,Z2→Zβ≥limsupn→∞⁡n+1g′∘φφ′2φnvβ.
On the other hand, let{zn}⊆D with φzn>1/2 and φzn→1 as n→∞. Let the test function (101)Onz=1+log⁡21-φzn¯z2log⁡21-φzn2-1.From [8] we obtain that {On} is a bounded sequence in Z02 which converges to zero uniformly on compact subsets of D, and (102)limn→∞⁡1-zn2βg′φznφ′zn2On′φzn=2limn→∞⁡1-zn2β1-φzn2g′φznφ′zn2φzn≤CCφTge,Z2→Zβ.Applying Theorem 8 we get (103)CCφTge,Z2→Zβ≥CφTgOnZβ≥limn→∞⁡1-zn2βg′′φznφ′zn2+g′φznφ′′znOnφzn-limn→∞⁡1-zn2βg′φznφ′zn2On′φzn≥limn→∞⁡1-zn2βg′′φznφ′zn2+g′φznφ′′znlog⁡21-φzn2-2CCφTge,Z2→Zβ.Hence(104)limn→∞⁡1-zn2βg′′φznφ′zn2+g′φznφ′′znlog⁡21-φzn2≤CCφTge,Z2→Zβ.On the other hand, the lower bound can be easily proved by Lemmas 4 and 5.
Ifα>2, the proof is similar to that of (iv) except that we now choose the test function tn(z)=(1-φzn2)2/(1-φ(zn)¯z)α instead of On(z). This completes the proof of Theorem 12.Using the same methods of Theorems11 and 12, we can have the following results.Theorem 13.
LetCφIg be a bounded operator from Zα to Zβ space.(i)
If0<α<1, then (105)CφIge,Zα→Zβ≍limsupn→∞⁡n+1αg∘φφ′2φnvβ.(ii)
Ifα=1, then (106)CφIge,Z→Zβ≍max⁡limsupn→∞⁡n+1g∘φφ′2φnvβ,limsupn→∞⁡log⁡ng′∘φφ′2+g∘φφ′′φnvβ.(iii)
Ifα>1, then (107)CφIge,Zα→Zβ≍max⁡limsupn→∞⁡n+1αg∘φφ′2φnvβ,limsupn→∞⁡n+1α-1g′∘φφ′2+g∘φφ′′φnvβ.Theorem 14.
Letφ be an analytic self-map of D and g∈H(D), and TgCφ:Zα→Zβ is a bounded operator.(i)
If0<α<1, then (108)TgCφe,Zα→Zβ=0.(ii)
Ifα=1, then (109)TgCφe,Z→Zβ≍limsupn→∞⁡log⁡ng′φ′φnvβ.(iii)
If1<α<2, then (110)TgCφe,Zα→Zβ≍limsupn→∞⁡n+1α-1g′φ′φnvβ.(iv)
Ifα=2, then (111)TgCφe,Z2→Zβ≍max⁡limsupn→∞⁡n+1g′φ′φnvβ,limsupn→∞⁡log⁡ng′′φnvβ.(v)
Ifα>2, then (112)TgCφe,Zα→Zβ≍max⁡limsupn→∞⁡n+1α-1g′φ′φnvβ,limsupn→∞⁡n+1α-2g′′φnvβ.



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*Source: 1409642-2017-04-11.xml*