# Inequalities Involving Essential Norm Estimates of Product-Type Operators

Authors: Manisha Devi; Ajay K. Sharma; Kuldip Raj
Journal: Journal of Mathematics
 (2021)

---

## Abstract

Consider an open unit diskD=z∈ℂ:z<1 in the complex plane ℂ, ξ a holomorphic function on D, and ψ a holomorphic self-map of D. For an analytic function f, the weighted composition operator is denoted and defined as follows: Wξ,ψfz=ξzfψz. We estimate the essential norm of this operator from Dirichlet-type spaces to Bers-type spaces and Bloch-type spaces.


---

## Body

## 1. Introduction and Preliminaries

Consider an open unit diskD=z∈ℂ:z<1 in the complex plane ℂ. Let HD denote the class of all analytic functions on D, SD be the class of all holomorphic self-maps of D, and H∞ be the space of all bounded holomorphic functions on D. Let ξ∈HD and ψ be a holomorphic self-map of D. For z∈D, the composition operator and multiplication operator are, respectively, defined by(1)Cψfz=fψz,Mξfz=ξzfz,f∈HD.The weighted composition operator is denoted and defined as(2)Wξ,ψfz=ξzfψz,f∈HD,where Wξ,ψ is a product-type operator as Wξ,ψ=MξCψ. Clearly, this operator can be seen as a generalization of the composition operator and multiplication operator. It can be easily seen that, for ξ≡1, the operator reduced to Cψ. If ψz=z, the operator gets reduced to Mξ. This operator is basically a linear transformation of HD defined by Wξ,ψfz=ξzfψz=MξCψfz, for f in HD and z in D. The basic aim is to give the operator-theoretic characterization of these operators in terms of function-theoretic characterization of their including functions. Various holomorphic function spaces on various domains have been studied for the boundedness and compactness of weighted composition operators acting on them. Moreover, a number of papers have been studied on these operators acting on different spaces of holomorphic functions on various domains. For more details, see [1–14] and the references therein. We say that a linear operator is bounded if the image of a bounded set is a bounded set. Moreover, a linear operator is said to be compact if it maps the bounded sets to those sets whose closure is compact. For each α>0, the weighted Bloch space B is defined as follows:(3)B=f∈HD:supz∈D1−z2αf′z<∞.In this expression, seminormed is defined. This space forms a Banach space with the natural norm defined by(4)fB=f0+supz∈D1−z2αf′z.Forα=1, this space gets reduced to classical Bloch space. A function ω:D⟶0,∞ is said to be a weight if it is continuous. For z∈D, the weight ω is said to be radial if ωz=ωz. A weight ω is said to be a standard weight if limz⟶1−1ωz=0. For a weight function ω, the Bloch-type space Bω is defined by(5)Bω=f∈HD:supz∈Dωzf′z<∞.The little Bloch-type spaceBω,0 is the closure of the set of polynomials in Bω and is defined as follows:(6)Bω,0=f∈Bω:limz⟶1ωzf′z=0.BothBω and Bω,0 form a Banach space with the following norm:(7)fBω=f0+supz∈Dωzf′z.For more information about these spaces, one may refer [1–3, 5, 6, 15, 16] and the references therein. Likewise, for weight ω, the Bers-type space Aω is defined as follows:(8)Aω=f∈HD:fAω≔supz∈Dωzfz<∞.It is a nonseparable Banach space with the norm·Aω. The closure of the set of polynomials in Aω forms a separable Banach space. This set is denoted by Aω,0 and is defined as(9)Aω,0=f∈Aω:limz⟶1−ωzfz=0.These spaces and their properties are discussed in many papers; some of these are [3, 15, 16] and the references therein. The Dirichlet space is defined as follows:(10)D=f∈HD:∫Df′z2dAz<∞,where dAz denotes the normalized Lebesgue area measure on D. With the following norm, it is a Hilbert space:(11)fD2=f02+∫Df′z2dAz.Consider a functionK:0,∞⟶0,∞ which is right continuous and increasing. In this paper, we consider function K as a weight function. With a weight function K, the Dirichlet type space DK is given as follows:(12)DK=f∈HD:∫Df′z2K1−z2dAz<∞.Clearly, spaceDK forms a Hilbert space with the norm ⋅DK defined by(13)fDK2=f02+∫Df′z2K1−z2dAz.Here, we haveKt=tp, 0≤p<∞, and DK gives Dp, that is, the usual Dirichlet-type space. This gives a classical Dirichlet space D for a case when p=0 and for p=1, and we gain the Hardy space H2. These spaces have been studied widely in various papers. For example, Aleman, in [17], obtained that each element of DK can be written as a quotient of two bounded functions in DK. Kerman and Sawyer [18], by taking some conditions on weight function K, characterized Carleson measures and multipliers of DK in terms of maximal operator.The Möbius invariant space generated byDK is denoted by QK. The space QK contains those functions f∈HD which satisfy the following:(14)supa∈Df°σa−faDK<∞,where σaz=a−z/1−a¯z is the Möbius transformation of D. Wulan and Zhu, in [19], characterized Lacunary series in the QK space under some conditions on weight function K. Furthermore, Wulan and Zhou [20] characterized space QK in terms of fractional-order derivatives of function. They also established a relationship between Morrey type spaces and QK space in terms of fractional order derivatives. In the study of QK spaces, the following two conditions play a very important role:(15)∫01φKssds<∞,(16)∫1∞φKss2ds<∞,where(17)φKs=sup0≤t≤1KstKt,0<s<∞.LetMDK be the class of multipliers of DK, that is,(18)MDK=f∈HD:fg∈DK,∀g∈DK.Bao et al., in [21], characterized the interpolating sequences for MDK of space DK, under certain conditions of weight function K. They also obtained corona theorem, ∂¯- equation, and corona-type decomposition theorem on MDK. For more details, see [9, 15, 21–25] and the references therein.From [26], one can see that if K satisfies (1), then(19)K1t=∫0tKsdss≈Kt,0<t<1.IfK satisfies (16), then(20)K2t=t∫t∞Ksdss2≈Kt,t>0.From condition (16), we get that K2t≈Kt for 0<t<1. Also, there exist C>0 sufficiently small for which t−CK1t is increasing and K2ttC−1 is decreasing. For more information about weight function K, one can refer [19–21].The criterion of boundedness as well as compactness has been discussed in many papers. Recently, Gürbüz, in [27], studied the boundedness of generalized commutators of rough fractional maximal and integral operators on generalized weighted Morrey spaces, respectively, and, in [28], he investigated the generalized weighted Morrey estimates for the boundedness of Marcinkiewicz integrals with rough kernel associated with Schrödinger operator and their commutators. Furthermore, in [29], Gürbuüz studied the behavior of multi-sublinear fractional maximal operators and rough multilinear fractional integral both on product Lp and weighted Lp spaces and, in [30], he obtained the boundedness of the variation and oscillation operators for the family of multilinear integrals with Lipschitz functions on weighted Morrey spaces. Among others in [3], we obtained the following results about boundedness and compactness of Wξ,ψ given as follows.Theorem 1.
Letω and K be two weight functions, ξ∈HD, and ψ be a self-holomorphic map on D. Then, the operator Wξ,ψ:DK⟶Bω is bounded if and only if the following conditions hold:(i)
M1=supz∈Dωzξ′z/K1−ψz21−ψz2<∞(ii)
M2=supz∈Dωzξzψ′z/K1−ψz21−ψz22<∞Furthermore, if the operatorWξ,ψ:DK⟶Bω is bounded, then(21)M1+M2≲Wξ,ψDK⟶Bω≲1+M1+M2.Theorem 2.
Letω be a standard weight, ξ∈HD, and ψ be a self-holomorphic map on D. Let K be a weight function. Assume that Wξ,ψ:DK⟶Bω is bounded. Then, the operator Wξ,ψ:DK⟶Bω is compact if and only if the following conditions hold:(i)
limψz⟶1ωzξ′z/K1−ψz21−ψz2=0(ii)
limψz⟶1ωzξzψ′z/K1−ψz21−ψz22=0Theorem 3.
Letω be a weight and K be a weight function, ξ∈HD, and ψ be a self-holomorphic map on D. Then, the operator Wξ,ψ:DK⟶Aω is bounded if and only if the following condition holds:(22)l1=supz∈DωzξzK1−ψz21−ψz2<∞.Theorem 4.
Letω be a standard weight, ξ∈HD, and ψ be a self-analytic map on D. Let K be a weight function. Assume that the operator Wξ,ψ:DK⟶Aω is bounded. Then, Wξ,ψ:DK⟶Aω is compact if and only if the following condition is satisfied:(23)limψz⟶1−ωzξzK1−ψz21−ψz2=0.The aim of this paper is to provide some estimates of essential norm of the operatorWξ,ψ:DK⟶Bω as well as of Wξ,ψ:DK⟶Aω.Assume thatT:X1⟶X2 is a bounded linear operator for Banach spaces X1 and X2. The essential norm of operator T is denoted and defined as follows:(24)Te,X1⟶X2=infT−EX1⟶X2:EmappingX1intoX2is compact,where ·X1⟶X2 is the operator norm. In other words, the essential norm is the distance from compact operators E mapping X1 into X2 to the bounded linear operator T:X1⟶X2. If X1=X2, that is, the two Banach spaces are same, then the norm is simply denoted by ·e. For unbounded linear operator T:X1⟶X2, we have Te,X1⟶X2=∞. As the class of all compact operators is contained in the class of all bounded operators, in fact, this subset is closed, which implies that the operator T is compact if and only if Te,X1⟶X2=0. Thus, the estimate of essential norm leads to the compactness of the operator. Various results on the essential norm of different operators such as multiplication, composition, differentiation, weighted composition, generalized weighted composition, and their different combinations are studied in numerous research papers, and some of the references are [31–37].This study is formulated in a systematic way. Introduction and literature part is kept in Section1. In Section 2, we estimated the essential norm of operator Wξ,ψ:DK⟶Bω. Finally, in Section 3, we estimated the essential norm of operator Wξ,ψ:DK⟶Aω. Throughout the paper, the notation a≲b, for any two positive quantities a and b, which means that a≤Cb, where C is some positive constant. The value of constant C may change from one place to the other. We write a≈b if a≲b and b≲a.


## 2. Essential Norm of Weighted Composition Operator from Dirichlet-Type Space to Bloch-Type Space

Theorem 5.
Letω be a standard weight, ξ∈HD, and ψ be a self-analytic map on D. Let K be a weight function. Assume that Wξ,ψ:DK⟶Bω is bounded. Then,(25)Wξ,ψe,DK⟶Bω≈maxR,S,where(26)R≔limsupψz⟶1ωzξ′zK1−ψz21−ψz2,S≔limsupψz⟶1ωzξzψ′zK1−ψz21−ψz22.Proof.
At first, we show that(27)Wξ,ψe,DK⟶Bω≳maxR,S.
Forz∈D, define a function(28)hzw=a01−z2ε/2K1−z21−z¯w1+ε/2+a11−|z|21+ε/2K1−z21−z¯w2+ε/2,where a0=2+ε/2 and a1=−1+ε/2. It can be easily checked that hz∈DK and, for all z∈D, hzDK≲1. On calculation, we have hz′z=0 and hzz=1/K1−z21−z2. Furthermore, on compact subsets of D, hz converges to zero as z⟶1. Hence, for any compact operator E:DK⟶Bω and any ζnn∈D such that ψζn⟶1, we obtain(29)Wξ,ψ−EDK⟶Bω≳Wξ,ψ−EhψζnBω≥Wξ,ψhψζnBω−EhψζnBω.
In the above inequality, takelimsupψζn⟶1 on both sides, and we obtain(30)Wξ,ψ−EDK⟶Bω≳limsupψζn⟶1Wξ,ψhψζnBω≳limsupψζn⟶1ωζnξ′ζnK1−ψζn21−ψζn2=R.
Again, forz∈D, define another function:(31)kzw=b01−z2ε/2K1−z21−z¯w1+ε/2+b11−z21+ε/2K1−z21−z¯w2+ε/2,where b0=1 and b1=−1. In the similar manner, we can check that kz∈DK and, for all z∈D, kzDK≲1. On calculation, we have kzz=0 and kz′z=−z¯/K1−z21−z22. Furthermore, on compact subsets of D, kz converges to zero as z⟶1. Thus, for any compact operator E:DK⟶Bω and any ζnn∈D such that ψζn⟶1, we obtain(32)Wξ,ψ−EDK⟶Bω≳Wξ,ψ−EkψζnBω≥Wξ,ψkψζnBω−EkψζnBω.
By takinglimsupψζn⟶1 on both sides of the above inequality, we obtain(33)Wξ,ψ−EDK⟶Bω≳limsupψζn⟶1Wξ,ψkψζnBω≳limsupψζn⟶1ωζnξζnψ′ζnK1−ψζn21−ψζn22=S.
On applying the definition of essential norm, we find that(34)Wξ,ψe,DK⟶Bω=infEWξ,ψ−EDK⟶Bω≳maxR,S.
Next, we prove that(35)Wξ,ψe,DK⟶Bω≲maxR,S.
Forδ∈0,1, consider Eδ:HD⟶HD, defined as follows:(36)Eδfz=fδz=fδz,f∈HD.
Clearly,Eδ is compact on DK and EδDK⟶DK≤1. Consider a sequence δn⊂0,1 satisfying δn⟶1 as n⟶∞. Then, for all n∈ℕ, operator Wξ,ψEδn:DK⟶Bω is compact. By using the definition of essential norm, we obtain(37)Wξ,ψe,DK⟶Bω≤limsupn⟶∞Wξ,ψ−Wξ,ψEδnDK⟶Bω.
Therefore, we only have to prove that(38)limsupn⟶∞Wξ,ψ−Wξ,ψEδnDK⟶Bω≲maxR,S.
Letf be a function in DK satisfying fDK≤1; then, we have(39)Wξ,ψ−Wξ,ψEδnfDK⟶Bω=ξ0fψ0−ξ0fδnψ0+ξf−fδn°ψBω.
Clearly,limn⟶∞ξ0fψ0−ξ0fδnψ0=0. Furthermore, consider a large enough N∈ℕ such that, for all n≥N, we have δn≥1/2. Thus, we obtain(40)limsupn⟶∞ξf−fδn∘ψBω≤llimsupn⟶∞supψz≤δNωzξzf−fδn′ψzψ′z+limsupn⟶∞supψz≤δNωzξzf−fδn′ψzψ′z+limsupn⟶∞supψz≤δNωzξ′zf−fδnψz+limsupn⟶∞supψz≤δNωzξ′zf−fδnψz=S1+S2+S3+S4,where(41)S1=limsupn⟶∞supψz≤δNωzξzf−fδn′ψzψ′z,S2=limsupn⟶∞supψz≤δNωzξzf−fδn′ψzψ′z,S3=limsupn⟶∞supψz≤δNωzξ′zf−fδnψz,S4=limsupn⟶∞supψz≤δNωzξ′zf−fδnψz.
Taking the operatorWξ,ψ to 1 and z and using its boundedness, it easily follows that ξ∈Bω and(42)E˜=supz∈Dωzψ′zξz<∞.
Also, on compact subsets ofD, δnfδn′ uniformly converges to f′ as n⟶∞; thus, we have(43)S1≤E˜limsupn⟶∞supw≤δNf′w−δnf′δnw=0.
Similarly, forξ∈Bω and the fact that fδn converges uniformly to f on compact subsets of D as n⟶∞, we obtain(44)S3≤ξBωlimsupn⟶∞supw≤δNfw−fδnw=0.
Now, considerS2. We have S2≤limsupn⟶∞P1+P2, where(45)P1=supψz>δNωzξzψ′zf′ψz,P2=supψz>δNδnωzξzψ′zf′δnψz.
First, we considerP1. As fDK≤1, we obtain(46)P1=supψz>δNωzξz‖ψ′z‖f′ψz≲1δNfDKsupψz>δNωzξzψ′zψzK1−ψz21−ψz22.
On taking limit asN⟶∞, we obtain(47)limsupn⟶∞P1≲limsupψz⟶1ωzξzψ′zK1−ψz21−ψz22=S.
In the similar manner, we obtain(48)limsupn⟶∞P2≲limsupψz⟶1ωzξzψ′zK1−ψz21−ψz22=S.
On combining the above two inequalities, we obtain(49)S2≤limsupn⟶∞P1+P2≲S.
Next, considerS4. We have S4≤limsupn⟶∞P3+P4, where(50)P3=supψz>δNωzξ′zfψz,P4=supψz>δNωzξ′zfδnψz.
By similar calculation, we obtain(51)P3=supψz>δNωzξ′zfψz≲1δNfDKsupψz>δNωzξ′z1K1−ψz21−ψz2.
On taking limitN⟶∞, we obtain(52)limsupn⟶∞P3≲limsupψz⟶1ωzξ′zK1−ψz21−ψz2=R.
In the similar manner, we obtain(53)limsupn⟶∞P4≲limsupψz⟶1ωzξ′zK1−ψz21−ψz2=R.
Combining the above two inequalities, we obtain(54)S4≤limsupn⟶∞P3+P4≲R.
On combining (40), (43), (44), (49), and (54), we obtain(55)limsupn⟶∞Wξ,ψ−Wξ,ψEδnDK⟶Bω=limsupn⟶∞supfDK≤1Wξ,ψ−Wξ,ψEδnfBω=limsupn⟶∞supfDK≤1ξf−fδn∘ψBω≲R+S.
Thus, inequalities (37) and (55) imply that(56)Wξ,ψe,DK⟶Bω≤R+S≲maxR,S.
Hence, inequalities (34) and (56) complete the theorem.
The following corollary can be easily obtained from Theorem5.Corollary 1.
Letω be a standard weight and ψ be a self-analytic map on D. Let K be a weight function. Assume that Cψ:DK⟶Bω is bounded. Then,(57)Cψe,DK⟶Bω≈limsupψz⟶1ωzψ′zK1−ψz21−ψz22.


## 3. Essential Norm of Weighted Composition Operator from Dirichlet-Type Space to Bers-Type Space

In this section, we consider the Bers-type spaces and estimated the essential norm of weighted composition operator fromDK to Aω.Theorem 6.
Letω be a standard weight, ξ∈HD, and ψ be a self-analytic map on D. Let K be a weight function. Assume that the operator Wξ,ψ:DK⟶Aω is bounded. Then,(58)Wξ,ψe,DK⟶Aω≈limsupψz⟶1−ωzξzK1−ψz21−ψz2.Proof.
Firstly, we prove that(59)Wξ,ψe,DK⟶Aω≳limsupψz⟶1−ωzξzK1−ψz21−ψz2.
Consider a functionfz∈DK such that fzDK≲1, and on compact subsets of D, fz converges to zero as z⟶1. Thus, for any compact operator E:DK⟶Aω and any ζnn∈D such that ψζn⟶1−, we obtain(60)Wξ,ψ−EDK⟶Aω≳Wξ,ψ−EfψζnAω≥Wξ,ψfψζnBω−EfψζnBω.
Takinglimsupψζna⟶1− on both sides, we obtain(61)Wξ,ψ−EDK⟶Aω≳limsupψζn⟶1−Wξ,ψfψζnAω≳limsupψζn⟶1−ωζnξζnK1−ψζn21−ψζn2.
On applying the definition of essential norm, we find that(62)Wξ,ψe,DK⟶Aω=infEWξ,ψ−EDK⟶Aω≳limsupψζn⟶1−ωζnξζnK1−ψζn21−ψζn2.
Finally, we prove that(63)Wξ,ψe,DK⟶Aω≲limsupψζn⟶1−ωζnξζnK1−ψζn21−ψζn2.
For this, considerEδ:HD⟶HD with δ∈0,1 and a sequence δn⊂0,1 satisfying δn⟶1 as n⟶∞ defined in Theorem 5. Then, for all n∈ℕ, the operator Wξ,ψEδn:DK⟶Aω is compact. By using the definition of essential norm, we obtain(64)Wξ,ψe,DK⟶Aω≤limsupn⟶∞Wξ,ψ−Wξ,ψEδnDK⟶Aω.
So, we only have to prove that(65)limsupn⟶∞Wξ,ψ−Wξ,ψEδnDK⟶Bω≲limsupψζn⟶1−ωζnξζnK1−ψζn21−ψζn2.
Letf be a function in DK satisfying fDK≤1; then, we have(66)Wξ,ψ−Wξ,ψEδnfDK⟶Aω=ξf−fδn∘ψAω.
Furthermore, consider a large enoughN∈ℕ such that, for all n≥N, we have δn≥1/2. Thus, we obtain(67)limsupn⟶∞ξf−fδn∘ψAω≤limsupn⟶∞supψz≤δNωzξzf−fδnψz+limsupn⟶∞supψz≤δNωzξzf−fδnψz=A1+A2,where(68)A1=limsupn⟶∞supψz≤δNωzξzf−fδnψz,A2=limsupn⟶∞supψz≤δNωzξzf−fδnψz.
Similar to Theorem5, for ξ∈Aω and the fact that fδn converges uniformly to f on compact subsets of D as n⟶∞, we obtain(69)A1≤ξAωlimsupn⟶∞supw≤δNfw−fδnw=0.
Next, we considerA2. We have A2≤limsupn⟶∞R1+R2, where(70)R1=supψz>δNωzξzfψz,R2=supψz>δNωzξzfδnψz.
On calculation, we obtain(71)R1=supψz>δNωzξzfψz≲1δNfDKsupψz>δNωzξz1K1−ψz21−ψz2.
TakingN⟶∞, we obtain(72)limsupn⟶∞R1≲limsupψz⟶1−ωzξzK1−ψz21−ψz2.
Similarly,(73)limsupn⟶∞R2≲limsupψz⟶1−ωzξzK1−ψz21−ψz2.
On combining the above two inequalities, we obtain(74)A2≤limsupn⟶∞R1+R2≲limsupψz⟶1−ωzξzK1−ψz21−ψz2.
From inequalities (67), (69), and (74), we obtain(75)limsupn⟶∞Wξ,ψ−Wξ,ψEδnDK⟶Aω=limsupn⟶∞supfDK≤1Wξ,ψ−Wξ,ψEδnfAω=limsupn⟶∞supfDK≤1ξf−fδn∘ψAω≲limsupψz⟶1−ωzξzK1−ψz21−ψz2.
Thus, inequalities (64) and (75) imply that(76)Wξ,ψe,DK⟶Aω≲limsupψz⟶1−ωzξzK1−ψz21−ψz2.
Hence, inequalities (62) and (76) complete the theorem.
The following corollary can be easily obtained from Theorem6.Corollary 2.
Letω be a standard weight and ψ be a self-analytic map on D. Let K be a weight function. Assume that the operator Cψ:DK⟶Aω is bounded. Then,(77)Cψe,DK⟶Aω≈limsupψz⟶1−ωzK1−ψz21−ψz2.



---

*Source: 8811309-2021-03-03.xml*