# New Characterizations for the Essential Norms of Generalized Weighted Composition Operators between Zygmund-Type Spaces

Authors: Mostafa Hassanlou; Amir H. Sanatpour
Journal: Abstract and Applied Analysis
 (2021)

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## Abstract

We give different types of new characterizations for the boundedness and essential norms of generalized weighted composition operators between Zygmund-type spaces. Consequently, we obtain new characterizations for the compactness of such operators.


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## Body

## 1. Introduction

LetD denote the open unit ball of the complex plane ℂ and HD denote the space of all complex-valued analytic functions on D. By a weightν, we mean a strictly positive bounded function ν:D→ℝ+. The weighted-type spaceHν∞ consists of all functions f∈HD such that
(1)∥f∥ν=supz∈Dνz∣fz∣<∞.For a weightν, the associated weightν~ is defined by
(2)ν~z=sup∣fz∣:f∈Hν∞,∥f∥ν≤1−1.It is known that for thestandard weightsναz=1−z2α, 0<α<∞, and for the logarithmic weightνlogz=log2/1−z2−1, the associated weights and weights are the same.For each0<α<∞, the Bloch-type spaceBα consists of all functions f∈HD for which
(3)∥f∥sBα=supz∈D1−z2α∣f′z∣<∞.The spaceBα is a Banach space equipped with the norm
(4)∥f∥Bα=∣f0∣+∥f∥sBα,for each f∈Bα. The little Bloch-type spaceBα,0 is the closed subspace of Bα consisting of those functions f∈Bα satisfying
(5)lim∣z∣→11−z2α∣f′z∣=0.The classicZygmund spaceZ consists of all functions f∈HD which are continuous on the closed unit ball D¯ and
(6)sup∣feiθ+h+feiθ−h−2feiθ∣h<∞,where the supremum is taken over all θ∈ℝ and h>0. By [1] (Theorem 5.3), an analytic function f belongs to Z if and only if supz∈D1−z2∣f″z∣<∞. Motivated by this, for each 0<α<∞, the Zygmund-type spaceZα is defined to be the space of all functions f∈HD for which
(7)∥f∥sZα=supz∈D1−z2α∣f″z∣<∞.The spaceZα is a Banach space equipped with the norm
(8)∥f∥Zα=∣f0∣+∣f′0∣+∥f∥sZα,for each f∈Zα. The little Zygmund-type spaceZα,0 is the closed subspace of Zα consisting of those functions f∈Zα satisfying
(9)lim∣z∣→11−z2α∣f″z∣=0.Recall that for the Banach spacesX and Y, the space of all bounded operators T:X→Y is denoted by BX,Y, and the operator norm of T∈BX,Y is denoted by ∥T∥X→Y. The closed subspace of BX,Y containing all compact operators T:X→Y is denoted by KX,Y. The essential norm of T∈BX,Y, denoted by ∥T∥e,X→Y, is defined as the distance from T to KX,Y, that is,
(10)∥T∥e,X→Y=inf∥T−K∥X→Y:K∈KX,Y.Clearly, an operatorT∈BX,Y is compact if and only if ∥T∥e,X→Y=0. Therefore, essential norm estimates of bounded operators result in necessary and/or sufficient conditions for the compactness of such operators. Essential norm estimates of different types of operators between various classes of Banach spaces have been studied by many authors (see, for example, [2–7], and references therein).Letu and φ be analytic functions on D such that φD⊆D. The weighted composition operatoruCφ is defined by uCφf=u·f∘φ for all f∈HD. When u=1, we get the well-known composition operatorCφ given by Cφf=f∘φ for all f∈HD. Weighted composition operators appear in the study of dynamical systems, and also, it is known that isometries on many analytic function spaces are of the canonical forms of weighted composition operators. Operator theoretic properties of (weighted) composition operators have been studied by many authors between different classes of analytic function spaces (see, for example, [4, 5, 8, 9], and the references therein).For each nonnegative integerk, the generalized weighted composition operatorDφ,uk is defined by
(11)Dφ,ukfz=uzfkφz,for each f∈HD and z∈D. The class of generalized weighted composition operators include weighted composition operators uCφ=Dφ,u0, composition operators followed by differentiationDCφ=Dφ,φ′1 and composition operators proceeded by differentiationCφD=Dφ,11 [10]. Also, weighted types of operators DCφ and CφD are of the form Dφ,uk, that is, uDCφ=Dφ,uφ′1 and uCφD=Dφ,u1 [11].Boundedness and compactness of generalized weighted composition operators have been studied between Bloch-type spaces and Zygmund-type spaces in [2, 12] and between Bloch-type spaces and weighted-type spaces in [13, 14]. Essential norms of generalized weighted composition operators in these cases have been studied in [2, 3, 6, 15]. In [16], different characterizations for the boundedness and compactness of these operators between Bloch-type spaces are given, and also, their essential norms are investigated in [7]. The essential norm of some extensions of the generalized composition operators between kth weighted-type spaces on the unit disk has been calculated by Stević in [17]. In this paper, we first study boundedness of generalized weighted composition operators between Zygmund-type spaces and give new characterizations for the boundedness of these operators. Then, we find estimates for the essential norms of such operators in terms of the new characterizations. Consequently, we obtain different types of characterizations for the compactness of such operators.The following lemma, which will be used in the next chapters, collects some useful estimates for the functions in Zygmund-type spaces (see, for example, [6], and references therein).Lemma 1.
For everyf∈Zα, we have
(i)
∣f′z∣≤2/1−α∥f∥Zα and ∣fz∣≤2/1−α∥f∥Zα for 0<α<1(ii)
∣f′z∣≤2∥f∥Zlog2/1−∣z∣ and ∣fz∣≤∥f∥Z for α=1(iii)
∣f′z∣≤2/α−1∥f∥Zα/1−∣z∣α−1, for 1<α<∞(iv)
∣fz∣≤2/α−12−α∥f∥Zα, for 1<α<2(v)
∣fz∣≤2∥f∥Z2log2/1−∣z∣, for α=2(vi)
∣fz∣≤2/α−1α−2∥f∥Zα/1−∣z∣α−2, for 2<α<∞It is known that for eachn≥2 and 0<α<∞, we have
(12)∣fnz∣≤∥f∥Bα1−z2α+n−1,for all f∈Bα and z∈D (see [18]). Therefore, for each n≥2 and 0<α<∞, we have
(13)∣fn+1z∣≤∥f∥Zα1−z2α+n−1,for all f∈Zα and z∈D. Note that, by the definition of Zygmund-type spaces, it is clear that (13) also holds in the case of n=1.In this paper, for real scalarsA and B, the notation A≲B means A≤cB for some positive constant c. Also, the notation A≈B means A≲B and B≲A.


## 2. Boundedness

For eacha∈D, the following test functions in HD will be used in our proofs:
(14)faz=1−a221−a¯zα,gaz=1−a231−a¯zα+1,haz=1−a241−a¯zα+2.Moreover, in order to simplify the notations, we define(15)Au,φ,α,β,n=supz∈D1−z2β1−φz2α+n−2u″z,Bu,φ,α,β,n=supz∈D1−z2β1−φz2α+n−12u′zφ′z+uzφ″z,Cu,φ,α,β,n=supz∈D1−z2β1−φz2α+nuzφ′2z.In the next theorem, we give three different characterizations for the boundedness ofDφ,un:Zα→Zβ.Theorem 2.
Letu∈HD, φ be an analytic self-map of D and n,α≠1,1. Then, the following statements are equivalent for each 0<β<∞.
(i)
Dφ,un:Zα→Zβ is bounded(ii)
supj≥1jα−2∥Dφ,unIj+1∥Zβ<∞, where Ijz=zj for each j≥1 and z∈D(iii)
u∈Zβ,(16)supz∈D1−z2β∣uzφ′2z∣<∞,supz∈D1−z2β∣2u′zφ′z+uzφ″z∣<∞,supa∈D∥Dφ,unfa∥Zβ<∞,supa∈D∥Dφ,unga∥Zβ<∞,supa∈D∥Dφ,unha∥Zβ<∞(iv)
maxAu,φ,α,β,n,Bu,φ,α,β,n,Cu,φ,α,β,n<∞Moreover, this is also equivalent to(17)max∥u∥Zβ,Bu,φ,α,β,1,Cu,φ,α,β,1<∞,in the special case of n=1 and 0<α<1.Proof.
i⇒ii Suppose that n≥1 and Dφ,un:Zα→Zβ is a bounded operator. Since jα−1Ij is a bounded sequence in Bα (see, for example, [19]), one can see that jα−2Ij+1 is a bounded sequence in Zα. Therefore, boundedness of Dφ,un:Zα→Zβ implies that supj≥1jα−2∥Dφ,unIj+1∥Zβ<∞
ii⇒iii Let M=supj≥1jα−2∥Dφ,unIj+1∥Zβ<∞. Recalling the definition of fa, ga, and ha, one can see that
(18)faz=1−a22∑j=0∞Γj+αj!Γαa¯jzj,gaz=1−a23∑j=0∞Γj+α+1j!Γα+1a¯jzj,haz=1−a24∑j=0∞Γj+α+2j!Γα+2a¯jzj,for each a,z∈D. Note that
(19)faz=1−a22+1−a22∑j=1∞Γj+αj!Γαa¯jzj=1−a22+1−a22∑j=0∞Γj+1+αj+1!Γαa¯j+1zj+1=1−a22+1−a22∑j=0∞j+αj+1Γj+αj!Γαa¯j+1zj+1.Also, by Stirling’s formula, we know thatΓj+α/j!Γα≈jα−1 as j→∞ (see [2]). Consequently,
(20)∥Dφ,unfa∥Zβ≲1−a22∥Dφ,un1∥Zβ+1−a22∑j=0∞j+αj+1jα−1aj+1∥Dφ,unIj+1∥Zβ=1−a22∑j=0∞j+αj+1jaj+1jα−2∥Dφ,unIj+1∥Zβ≤1−a22∑j=0∞j+αaj+1jα−2∥Dφ,unIj+1∥Zβ≲M,and sincea∈D was arbitrary, we get supa∈D∥Dφ,unfa∥Zβ<∞. Applying a similar argument to the functions ga and ha implies iii.iii⇒iv For each a,z∈D define
(21)kaz=α+n1−a221−a¯zα−2α1−a231−a¯zα+1+αα+1α+n+11−a241−a¯zα+2.Then, one can see thatka∈Zα, supa∈D∥ka∥Zα<∞, kana=kan+1a=0 and
(22)kφan+2φa=2αα+1⋯α+nφa¯n+21−φa2α+n.Moreover,(23)supa∈D∥Dφ,unka∥Zβ≤α+nsupa∈D∥Dφ,unfa∥Zβ+2αsupa∈D∥Dφ,unga∥Zβ+αα+1α+n+1supa∈D∥Dφ,unha∥Zβ<∞,and by the definition of∥·∥Zβ, we have
(24)∞>∥Dφ,unkφa∥Zβ≥supz∈D1−z2β∣ukφan∘φ″z∣≥1−a2β∣ukφan∘φ″a∣=2αα+1⋯α+n1−a2βuaφ′2aφan+21−φa2α+n.Therefore,(25)sup∣φa∣>1/21−a2β1−φa2α+n∣uaφ′2a∣<∞.On the other hand,(26)sup∣φa∣≤1/21−a2β1−φa2α+n∣uaφ′2a∣≲sup∣φa∣≤1/21−a2β∣uaφ′2a∣<∞.Now, (25) and (26) imply that
(27)Cu,φ,α,β,n=supz∈D1−z2β1−φz2α+n∣uzφ′2z∣<∞.In order to prove thatBu,φ,α,β,n<∞, the similar approach can be applied using the test functions la defined as
(28)laz=α+nα+n+21−a221−a¯zα−α2α+2n+31−a231−a¯zα+1+αα+11−a241−a¯zα+2.Also, for the proof ofAu,φ,α,β,n<∞, the argument is similar, using the test functions
(29)maz=α+n+11−a221−a¯zα−2α1−a231−a¯zα+1+αα+1α+n+21−a241−a¯zα+2.The special case ofn=1 and 0<α<1 can be proved by a similar argument and applying Lemma 1.iv⇒i By applying (13), for every f∈Zα and n≥2, we get
(30)∣Dφ,unf″z∣=∣u″zfnφz+2u′zφ′z+uzφ″zfn+1z+uzφ′2zfn+2φz∣≤∣u″zfnφz∣+∣2u′zφ′z+uzφ″zfn+1z∣+∣uzφ′2zfn+2φz∣≤∣u″z∣1−φz2α+n−2∥f∥Zα+∣2u′zφ′z+uzφ″z∣1−φz2α+n−1∥f∥Zα+∣uzφ′2z∣1−φz2α+n∥f∥Zα.Multiplying both sides by1−z2β and taking supremum over z∈D, one can get (i).In the case ofn=1 and α=1, we have the following result.Theorem 3.
For each0<β<∞, Dφ,u1:Z→Zβ is bounded if and only if
(31)supz∈D1−z2β∣u″z∣log11−φz2<∞,supz∈D1−z2β1−φz2∣2u′zφ′z+uzφ″z∣<∞,supz∈D1−z2β1−φz22∣uzφ′2z∣<∞.Proof.
The sufficiency part is a consequence of Lemma1(ii) and definition of the norm in Zygmund-type spaces. Now, suppose that Dφ,u1:Z→Zβ is bounded. Then, by applying Dφ,u1z,Dφ,u1z2,Dφ,u1z3∈Zβ, we get u,u′φ′+uφ″,uφ′2∈Zβ. Also, by defining
(32)kaz=31−a221−a¯z−31−a231−a¯z2+1−a241−a¯z3,for each a,z∈D, one can see that ka∈Z, supa∈D∥ka∥Z<∞, kφa′φa=0, kφa″φa=0 and kφa‴φa=16φa¯3/1−φa22. Therefore,
(33)∥Dφ,u1kφa∥Zβ≥1−a2βuaφ′2af‴φa=161−a2β1−φa23uaφ′2aφa3,and consequently,
(34)sup∣φa∣>1/21−a2β1−φa22∣uaφ′2a∣≲sup∣φa∣>1/2∥Dφ,u1kφa∥Zβ<∞.On the other hand,uφ′2∈Zβ implies that
(35)sup∣φa∣≤1/21−a2β1−φa22∣uaφ′2a∣<∞,which completes the proof of (iii). In order to prove (ii), for eacha,z∈D, define the test functions
(36)laz=81−a221−a¯z−71−a231−a¯z2+21−a241−a¯z3.Then, one can prove (ii) by a similar approach as in (iii) and using the factsla∈Z, supa∈D∥la∥Z<∞, lφa′φa=0, lφa‴φa=0 and lφa″φa=−2φa¯2/1−φa2α.In order to prove (i), consider the test functions(37)taz=hφa¯zφa¯log11−φa2−1,for each a,z∈D, where hz=z−11+log1/1−z2+1. Then, one can see that ta∈Z, supa∈D∥ta∥Z<∞, ta′φa=log1/1−φa2 and
(38)ta″φa=2φa¯1−φa2,ta‴φa=2φa¯21−φa221+log11−φa2−1.Since the operatorDφ,u1:Z→Zβ is bounded, we get
(39)Dφ,u1taZβ≥1−a2βu″ata′φa−1−a2β2u′aφ′a+uaφ″ata″φa−uaφ′2ata‴φa=1−a2βu″alog11−φa2−1−a2β2u′aφ′a+uaφ″a2φa¯1−φa2−1−a2βuaφ′2a2φa¯21−φa221+log11−φa2−1.Therefore,(40)supφa>121−a2βu″alog11−φa2≤supφa>1/2Dφ,u1taZβ+supφa>1/21−a2β2u′aφ′a+uaφ″a2φa¯1−φa2+supφa>1/21−a2βuaφ′2a2φa¯21−φa221+log11−φa2−1<∞.On the other hand, sinceu∈Zβ, we have
(41)sup∣φa∣≤1/21−a2β∣u″a∣log21−φa2<∞,which completes the proof.


## 3. Essential Norms

In this section, we investigate essential norm estimates ofDφ,un:Zα→Zβ in different cases of α, β, and n. In order to simplify the notations, we define
(42)E=limsup∣a∣→1∥Dφ,unfa∥Zβ,F=limsupa→1∥Dφ,unga∥Zβ,G=limsup∣a∣→1∥Dφ,unha∥Zβ,A=limsup∣φz∣→11−z2β1−φz2α+n−2u″z,B=limsup∣φz∣→11−z2β1−φz2α+n−12u′zφ′z+uzφ″z,C=limsup∣φz∣→11−z2β1−φz2α+nuzφ′2z.Theorem 4.
Let0<α,β<∞ and n≥1 with n,α≠1,1. If Dφ,un:Zα→Zβ is a bounded operator, then
(43)∥Dφ,un∥e,Zα→Zβ≈limsupj→∞jα−2∥Dφ,unIj+1∥Zβ≈maxE,F,G≈maxA,B,C.Proof.
First, note that, as mentioned in the proof of Theorem2, jα−2Ij+1 is a bounded sequence in Zα. Since polynomials are dense in Zα,0, by [20] (Proposition 2.1), jα−2Ij+1 converges weakly to 0 in Zα,0 and hence in Zα. Therefore, for each compact operator K:Zα→Zβ, we have limj→∞∥Kjα−2Ij+1∥Zβ=0, and consequently,
(44)∥Dφ,un∥e,Zα→Zβ=infK∥Dφ,un−K∥Zα→Zβ≲infKlimsupj→∞∥Dφ,un−Kjα−2Ij+1∥Zβ≲limsupj→∞jα−2∥Dφ,unIj+1∥Zβ.Next, we prove that(45)∥Dφ,un∥e,Zα→Zβ≈maxE,F,G.One can see thatfa,ga,ha∈Zα,0 having uniformly bounded ∥·∥Zα norm. Also, these functions converge to 0 uniformly on compact subsets of D, as ∣a∣→1, which implies their weak convergence to 0 in Zα. Therefore, for each compact operator K:Zα→Zβ, we have
(46)∥Dφ,un−K∥Zα→Zβ≲limsup∣a∣→1∥Dφ,un−Kfa∥Zβ≥limsup∣a∣→1∥Dφ,unfa∥Zβ−limsup∣a∣→1∥Kfa∥Zβ=limsup∣a∣→1∥Dφ,unfa∥Zβ=E.The same argument is valid forga and ha, that is,
(47)∥Dφ,un−K∥Zα→Zβ≲F,∥Dφ,un−K∥Zα→Zβ≲G.SinceK:Zα→Zβ was an arbitrary compact operator, we get
(48)∥Dφ,un∥e,Zα→Zβ=infK∥Dφ,un−K∥Zα→Zβ≲maxE,F,G.For each0<r<1 and f∈HD, the function fr∈HD is defined by frz=frz for all z∈D. Note that fr→f uniformly on compact subsets of D when r→1. Also, the operator Kr:Zα→Zα given by Krf=fr is a compact operator for each 0<r<1. Let rj be a sequence in 0,1 converging to 1 as j→∞. Then, Dφ,unKrj:Zα→Zβ is a compact operator for each j≥1. Hence,
(49)∥Dφ,un∥e,Zα→Zβ≤limsupj→∞∥Dφ,un−Dφ,unKrj∥Zα→Zβ.Therefore, it is enough to prove that(50)limsupj→∞∥Dφ,un−Dφ,unKrj∥Zα→Zβ≲maxE,F,G.Note that for everyf∈Zα with ∥f∥Zα≤1, we have
(51)∥Dφ,un−Dφ,unKrjf∥Zβ=∣u0fnφ0−rjnu0fnrjφ0∣+∣u′0f−frjnφ0+u0f−frjn+1φ0φ′0∣+∥u·f−frjn∘φ∥sZβ.Sincerj→1, we have
(52)limsupj→∞∣u0fnφ0−rjnu0fnrjφ0∣=0,limsupj→∞∣u0f−frjnφ0+u0f−frjn+1φ0φ′0∣=0.Therefore, for eachN≥1, we get
(53)limsupj→∞∥Dφ,un−Dφ,unKrjf∥Zβ≤limsupj→∞∥u·f−frjn∘φ∥sZβ≤limsupj→∞sup∣φz∣≤rN1−z2βf−frjnφzu″z+limsupj→∞sup∣φz∣>rN1−z2βf−frjnφzu″z+limsupj→∞sup∣φz∣≤rN1−z2βf−frjn+1φz2u′zφ′z+uzφ″z+limsupj→∞sup∣φz∣>rN1−z2βf−frjn+1φz2u′zφ′z+uzφ″z+limsupj→∞sup∣φz∣≤rN1−z2βf−frjn+2φzuzφ′2z+limsupj→∞sup∣φz∣>rN1−z2βf−frjn+2φzuzφ′2z≔S1+S2+S3+S4+S5+S6,respectively. Since the operatorDφ,un:Zα→Zβ is bounded, by Theorem 2, we have
(54)supz∈D1−z2β∣u″z∣<∞,supz∈D1−z2β∣2u′zφ′z+uzφ″z∣<∞,supz∈D1−z2β∣uzφ′2z∣<∞.Also, for eachk≥1, frjk→fk uniformly on compact subsets of D, since frj→f uniformly on compact subsets of D. These facts imply that
(55)S1=S3=S5=0,and hence, we just need to estimate S2, S4, and S6. By applying (13) and Lemma 1 and using the test functions ma defined in Theorem 2, we have
(56)S2=limsupj→∞sup∣φz∣>rN1−z2βf−frjnφzu″z≤limsupj→∞sup∣φz∣>rN1−z2β∥f−frj∥Zα1−φz2α+n−2u″z≲sup∣φz∣>rN1−z2βφznφzn1−φz2α+n−2u″z≲sup∣a∣>rNα+n+22αα+1⋯α+n−1∥Dφ,unma∥Zβ≲sup∣a∣>rN∥Dφ,unα+n+1fa−2αga+αα+1α+n+2ha∥Zβ≤α+n+1sup∣a∣>rN∥Dφ,unfa∥Zβ+2αsup∣a∣>rN∥Dφ,unga∥Zβ+αα+1α+n+2sup∣a∣>rN∥Dφ,unha∥Zβ.Therefore, asN→∞, we get
(57)S2≲limsup∣a∣→1∥Dφ,unfa∥Zβ+limsup∣a∣→1∥Dφ,unga∥Zβ+limsup∣a∣→1∥Dφ,unha∥Zβ≲maxE,F,G.Similarly, by applying the test functionska and la, one can prove
(58)S4≲maxE,F,G,S6≲maxE,F,G,and therefore,
(59)∥Dφ,un∥e,Zα→Zβ≤limsupj→∞∥Dφ,un−Dφ,unKrj∥Zα→Zβ≲maxE,F,G.By applying (56), we get
(60)S2≲limsup∣φz∣→11−z2β1−φz2α+n−2∣u″z∣=A,and similarly, one can see that S4≲B and S6≲C. Therefore,
(61)∥Dφ,un∥e,Zα→Zβ≲maxA,B,C.In order to prove the converse of (61), let zj be a sequence in D such that ∣φzj∣→1, as j→∞, and define the test functions
(62)kj=kφzj,lj=lφzj,mj=mφzj,where ka, la, and ma are defined in the proof of Theorem 2. Then, kj,lj, and mj are bounded sequences in Zα,0 which converge to zero on compact subsets of D. Also, for each j≥1, we have
(63)kjnφzj=kjn+1φzj=0,kjn+2φzj=2αα+1⋯α+nφzjn+21−φzj2α+n,ljnφzj=ljn+2φzj=0,ljn+1φzj=−αα+1⋯α+nφzjn+11−φzj2α+n−1,mjn+1φzj=mjn+2φzj=0,mjnφzj=α+n+22αα+1⋯α+n−1φzjn1−φzj2α+n−2.Thus, for any compact operatorK:Zα→Zβ, we get
(64)∥Dφ,un−K∥Zα→Zβ≲limsupj→∞∥Dφ,unkj∥Zβ−limsupj→∞∥Kkj∥Zβ≲limsupj→∞1−zj2βφzjn+21−φzj2α+nuzjφ′2zj=limsup∣φz∣→11−z2β1−φz2α+nuzφ′2z,∥Dφ,un−K∥Zα→Zβ≲limsupj→∞∥Dφ,unlj∥Zβ−limsupj→∞∥Klj∥Zβ≲limsupj→∞1−zj2βφzjn+11−φzj2α+n−12u′zjφ′zj+uzjφ″zj=limsup∣φz∣→11−z2β1−φz2α+n−12u′zφ′z+uzφ″z,∥Dφ,un−K∥Zα→Zβ≲limsupj→∞∥Dφ,unmj∥Zβ−limsupj→∞∥Kmj∥Zβ≲limsupj→∞1−zj2βφzjn1−φzj2α+n−2u″zj=limsup∣φz∣→11−z2β1−φz2α+n−2u″z.Hence,(65)∥Dφ,un∥e,Zα→Zβ=infK∥Dφ,un−K∥Zα→Zβ≲limsup∣φz∣→11−z2β1−φz2α+nuzφ′2z=C,and similarly,
(66)∥Dφ,un∥e,Zα→Zβ≲B,∥Dφ,un∥e,Zα→Zβ≲A,which implies that
(67)∥Dφ,un∥e,Zα→Zβ≲maxA,B,C.To prove the final case, first note that by Theorem2, we have
(68)M=supj≥1jα−2∥Dφ,unIj+1∥Zβ<∞.Recall that, for eacha,z∈D,
(69)faz=1−a221−a¯zα=1−a22∑j=0∞Γj+αj!Γαa¯jzj.For eachN≥1, as in the proof of Theorem 2, we have
(70)∥Dφ,unfa∥Zβ≲1−a22∑j=0∞j+αaj+1jα−2∥Dφ,unIj+1∥Zβ=1−a22∑j=0N−1j+αaj+1jα−2∥Dφ,unIj+1∥Zβ+∑j=N∞j+αaj+1jα−2∥Dφ,unIj+1∥Zβ≲1−a22∑j=0N−1j+αaj+1+supj≥Njα−2∥Dφ,unIj+1∥Zβ∑j=N∞j+αaj+1,which implies that
(71)limsup∣a∣→1∥Dφ,unfa∥Zβ≲supj≥Njα−2∥Dφ,unIj+1∥Zβ.Therefore, whenN→∞, we get
(72)E=limsup∣a∣→1∥Dφ,unfa∥Zβlimsupj→∞jα−2∥Dφ,unIj+1∥Zβ.A similar approach implies that(73)F≲limsupj→∞jα−2∥Dφ,unIj+1∥Zβ,G≲limsupj→∞jα−2∥Dφ,unIj+1∥Zβ,and, by applying (59), we conclude that
(74)∥Dφ,un∥e,Zα→Zβ≲maxE,F,G≲limsupj→∞jα−2∥Dφ,unIj+1∥Zβ.This, along with (44), completes the proof.Corollary 5.
Let0<α,β<∞ and n≥1 with n,α≠1,1. If Dφ,un:Zα→Zβ is a bounded operator, then the following statements are equivalent:
(i)
Dφ,un:Zα→Zβ is compact(ii)
limsupj→∞jα−2∥Dφ,unIj+1∥Zβ=0(iii)
limsup∣a∣→1∥Dφ,unfa∥Zβ=limsup∣a∣→1∥Dφ,unga∥Zβ=limsup∣a∣→1∥Dφ,unha∥Zβ=0(iv)
limsupφz→11−z2β/1−φz2α+n−2u″z=0,limsupφz→11−z2β1−φz2α+n−12u′zφ′z+uzφ″z=0,
limsup∣φz∣→11−z2β/1−φz2α+n∣uzφ′2z∣=0We next prove the result of Theorem4 in the case of n=1 and α=1.Theorem 6.
Let0<β<∞ and Dφ,u1:Z→Zβ be a bounded operator. Then,
(75)∥Dφ,u1∥e,Z→Zβ≈maxlimsup∣φz∣→11−z2β∣u″z∣log21−φz2α,limsup∣φz∣→11−z2β1−φz2∣2u′zφ′z+uzφ″z∣,limsup∣φz∣→11−z2β1−φz22∣uzφ′2z∣.Proof.
Letzj be a sequence in D such that ∣φzj∣→1, as j→∞. Consider the test functions kj=kφzj and lj=lφzj defined in the proof of Theorem 3. Then, kj and lj are bounded sequences in Z0 which converge to zero on compact subsets of D. Also,
(76)kj′φzj=kj″φzj=0,kj‴φzj=16φzj31−φzj22,lj′φzj=lj‴φzj=0,lj″φzj=−2φzj21−φzj2.Thus, for any compact operatorK:Z→Zβ, we have
(77)∥Dφ,u1−K∥Z→Zβ≲limsupj→∞∥Dφ,u1kj∥Zβ−limsupj→∞∥Kkj∥Zβ≲limsupj→∞1−zj2βφzj31−φzj22uzjφ′2zj=limsup∣φz∣→11−z2β1−φz22uzφ′2z,∥Dφ,u1−K∥Z→Zβ≲limsupj→∞∥Dφ,u1lj∥Zβ−limsupj→∞∥Klj∥Zβ≲limsupj→∞1−zj2βφzj21−φzj22u′zjφ′zj+uzjφ″zj=limsup∣φz∣→11−z2β1−φz22u′zφ′z+uzφ″z.Therefore,(78)∥Dφ,u1∥e,Z→Zβ=infK∥Dφ,u1−K∥Z→Zβ≲limsup∣φz∣→11−z2β1−φz22uzφ′2z,and also,
(79)∥Dφ,u1∥e,Z→Zβ≲limsup∣φz∣→11−z2β1−φz22u′zφ′z+uzφ″z.Now, consider the test functionstj=tφzj defined in the proof of Theorem 3. Then, tj is a bounded sequence in Z0 which converges to zero on compact subsets of D. Moreover,
(80)tj′φzj=log11−φzj2,tj″φzj=2φzj¯1−φzj2,tj‴φzj=2φzj¯21−φzj221+log11−φzj2−1.Consequently,(81)∥Dφ,u1∥e,Z→Zβ=infK∥Dφ,u1−K∥Z→Zβ≳limsupj→∞1−zj2β∣u″zj∣log11−φzj2−limsupj→∞1−zj2β∣2u′zjφ′zj+uzjφ″zj∣2∣φzj∣1−φzj2−limsupj→∞1−zj2β∣uzjφ′2zj∣2φzj21−φzj221+log11−φzj2−1=limsup∣φz∣→11−z2β∣u″z∣log11−φz2−limsup∣φz∣→11−z2β∣2u′zφ′z+uzφ″z∣21−φz2−limsup∣φz∣→11−z2β∣uzφ′2z∣21−φz221+log11−φz2−1≳limsup∣φz∣→11−z2β∣u″z∣log11−φz2−∥Dφ,u1∥e,Z→Zβ−∥Dφ,u1∥e,Z→Zβ,which implies that
(82)∥Dφ,u1∥e,Z→Zβ≲limsup∣φz∣→11−z2β∣u″z∣log11−φz2.The proof of upper estimate is similar to the proof of Theorem4 using the operators Kr and Lemma 1.Corollary 7.
Let0<β<∞ and Dφ,u1:Z→Zβ be a bounded operator. Then, Dφ,u1:Z→Zβ is compact if and only if
(83)limsup∣φz∣→11−z2β∣u″z∣log21−φz2α=0,limsup∣φz∣→11−z2β1−φz2∣2u′zφ′z+uzφ″z∣=0,limsup∣φz∣→11−z2β1−φz22∣uzφ′2z∣=0.Remark 8.
Montes-Rodríguez in [8] (Theorem 2.1), and also Hyva¨rinen et al. in [4] (Theorem 2.4), proved that ifνandωare radial and nonincreasing weights tending to zero at the boundary ofD, then(i)
the weighted composition operatoruCφ maps Hν∞ into Hω∞ if and only if(84)supn≥0∥uφn∥ω∥zn∥ν≍supz∈Dωzν~φz∣uz∣<∞,with norm comparable to the above supremum.
(ii)
∥uCφ∥e,Hν∞→Hω∞=limsupn→∞∥uφn∥ω/∥zn∥ν=limsup∣φz∣→1ωz/ν~φzuzAlso, by [5] (Lemma 2.1), we know that for each 0<α<∞,
(iii)
limsupn→∞n+1α∥zn∥να=2α/eα(iv)
limsupn→∞logn∥zn∥νlog=1By applying these facts, our results in this paper containing terms of the typeωz/ν~φz∣uz∣ can be restated in terms of u and φn (see, for example, [9], and references therein for these types of results).



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