# Upper and Lower Bounds for Essential Norm of Weighted Composition Operators from Bergman Spaces with Békollé Weights

Authors: Elina Subhadarsini; Ajay K. Sharma
Journal: Journal of Function Spaces
 (2020)

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## Abstract

Letσ be a weight function such that σ/1−z2α is in the class Bp0α of Békollé weights, μ a normal weight function, ψ a holomorphic map on D, and φ a holomorphic self-map on D. In this paper, we give upper and lower bounds for essential norm of weighted composition operator Wψ,φ acting from weighted Bergman spaces Apσ to Bloch-type spaces Bμ.


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## Body

## 1. Introduction and Preliminaries

LetD be the open unit disk in the complex plane ℂ and HD the space of all holomorphic functions on D. For a ψ∈HD and φ a holomorphic self-map of D, the weighted composition operatorWψ,φ is a linear operator on HD defined by Wψ,φf=ψf∘φ, f∈HD. Several authors have studied these weighted composition operators on different spaces of analytic functions, see for example, [1–12] and the related references therein. Recently, Stevic and Sharma [12] characterized boundness and compactness of Wψ,φ acting from weighted Bergman spaces Apσ to Bloch-type spaces Bμ. Motivated by results in [12], in this paper, we give upper and lower bounds for essential norm of a weighted composition operator acting between these spaces.A continuous functionσ:0,1⟶0,∞ is called a weight or a weight function. We extend it on D by defining σz=σ∣z∣ for all z∈D. For 0<p<∞ and σ a weight, denoted by Apσ the weighted Bergman space consisting of holomorphic functions f on D such that
(1)fApσp=∫DfzpσzdAz<∞,where dA is the normalized area measure in D. If σz=σγz=1−z2γγ>−1, then Apσ is the well-known weighted Bergman space Aγp.Forp0>1 and α>−1, the class Bp0α of Békollé weights consists of weights σ with the property that there exists a constant C>0 such that
(2)∫Sθ,hσdAα≤CAαSθ,hp0∫Sθ,hσp′0/p0dAα−p0/p′0.Here,α>−1,dAαz=α+11−z2αdAz is the probability measure on D,Sθ,h=z=reiϕ:1−h<r<1,∣θ−ϕ∣<h/2,θ∈0,2π,h∈0,1 is the Carleson square in D, and p′0 is the conjugate exponent of p0, that is, 1/p0+1/p′0=1. Recall that a weightμ is normal if there exist positive numbers η and τ, 0<η<τ, and δ∈0,1 such that
(3)μr1−rηisdecreasingonδ,1andlimr→1μr1−rη=0,μr1−rτisincreasingonδ,1andlimr→1μr1−rτ=∞.It is well known that classical weightsσαz=1−z2α,α>−1 are normal weights.For a normal weightμ, the weighted Bloch-type spaceBμ on D is the space of all functions f in HD such that supz∈Dμzf′z<∞. The space Bμ is a Banach space with the norm
(4)fBμ=f0+supz∈Dμzf′z.Throughout this paper,r∈0,1 is fixed, p>0, p0>1, and η>−1. We also assume that p0≥p, σ a weight function such that σ/1−z2α belongs to Bp0α, γ≥η+2p0/p−2, and Kλγ=1/1−λ¯zγ+2 be the reproducing kernel of the Bergman space Apσγ. Constants are denoted by C; they are positive and not necessarily the same at each occurrence. The notation A≲B means that A is less than or equal to a constant multiple of B, and D≳E means that a constant multiple of D is greater than or equal to E. When A≲B as well as A≳B, then we write A≍B.


## 2. Essential Norm ofWψ,φ:Apσ⟶Bμ

In this section, we give upper and lower bounds for the essential norm of weighted composition operatorWψ,φ:Apσ⟶Bμ.Recall that ifX and Y are two Banach spaces, then the essential norm ∥T∥e,X→Y of a bounded linear operator T:X⟶Y is defined as
(5)∥T∥e,X→Y=inf∥T−K∥:KiscompactfromXtoY,where ∥T∥ denotes the usual operator norm. Clearly, T is compact if and only if ∥T∥e,X→Y=0.Theorem 1.
Let∈1,∞,ψ∈HD, and φ be a holomorphic self-map of D such that ∥φ∥∞=1. Assume that Wψ,φ:Apσ⟶Bμ is bounded. Then,
(6)∥Wψ,φ∥e,Apσ→Bμ≍limsup∣φz∣→1μz∣ψ′z∣∫DφzrσdA−1/p+limsup∣φz∣→1μz∣ψzφ′z∣1−φz2∫DφzrσdA−1/p,
whereDφzr=w∈D:∣w−φz∣<r1−∣φz∣.To prove the main result of this paper, we need the following lemmas. The next two lemmas can be found in [12].Lemma 2.
The following estimates hold:(1)
For eachz∈D, we have that(7)fkz≤C∫DzrσdA−1/p1−z2kfApσforallf∈Aσp(2)
For eachλ∈D, we have that(8)∥Kλγ∥Apσ≍∫DλrσdA1/p1−λ2γ+2,where Dλr=z∈D:∣z−λ∣<r1−∣λ∣Lemma 3.
For eachλ∈D, the function fλ defined as
(9)fλz=1−λ21+η+2p0/p1−λ¯z1+η+2p0/p∫DλrσdA−1/p
is inApσ. Moreover, supλ∈D∥fλ∥Apσ≍1 and fλ converges to zero, uniformly on compact subsets of D as ∣λ∣⟶1.The next lemma can be found in [8].Lemma 4.
Letp∈1,∞. If a bounded sequence fkk∈ℕ in Apσ converges to 0 uniformly on compact subsets of D, then fkk∈ℕ also converges to 0 weakly in Apσ.Proof of Theorem 1.
Lower bound. Letζjj∈ℕ be a sequence in D such that ∣φζj∣⟶1 as j⟶∞ and
(10)limsupj→∞μζjψ′ζj∫DφζjrσdA−1/p=limsup∣φz∣→1μz∣ψ′z∣∫DφzrσdA−1/p.
For eachj∈ℕ, let γj be defined as γjz=1−1−φζj21−φζj¯z−1. Then, γj∈H∞ and supj∈ℕ∣γjz∣≤3. Consider the family of functions defined as gjz=γjzfφζjz, where fφζj is defined as in (9). Also, by Lemma 3, supj∥gj∥Apσ≲1 and gjn∈ℕ converges to zero uniformly on compact subsets of D as j⟶∞. By Lemma 4, gj converges to zero weakly in Apσ. Thus, for any compact operator K:Apσ⟶Bμ, we have that ∥Kgj∥Bμ⟶0 as j⟶∞. Moreover,
(11)γjφζj=0,γj′z=−φζj¯1−φζj21−φζj¯z2,γ′jφζj=−φζj¯1−φζj2.
Also,gjφζj=0 and
(12)fφζjφζj=∫DφζjrσdA−1/p.
Now(13)fφζj′z=φζj¯η+2p0p+1∫DφζjrσdA−1/p1−φζj21+η+2p0/p1−φζj¯z2+η+2p0/pfφζj′φζj=φζj¯η+2p0p+1∫DφζjrσdA−1/p1−φζj2.
Therefore, from (12) and (13), we have that
(14)gj′φζj=γ′jφζjfφζjφζj+γjφζjfφζj′φζj=−φζj¯∫DφζjrσdA−1/p1−φζj2.
Using the facts that∥Wψ,φ∥e,Apσ→Bμ≳limsupj→∞∥Wψ,φgj−Kgj∥Bμand
limsupj→∞∥Kgj∥Bμ=0, we have that
(15)∥Wψ,φ∥e,Apσ→Bμ≳limsupj→∞μζj∣ψ′ζgjφζj+ψζjφ′ζjgj′φζj∣≥limsupj→∞μζj∣ψζjφ′ζj∣∫DφζjrσdA−1/p1−φζj2.
Again, letζj be a sequence in D such that ∣φζj∣⟶1 as j⟶∞ and
(16)limj→∞μζj∣ψζjφ′ζjφζj∣∫DφζjrσdA−1/p=limsup∣φz∣→1μz∣ψzφ′zφz∣∫DφzrσdA−1/p.
For eachj∈ℕ, let hj be defined as
(17)hjz=11+η+2p0/p1−φζj21+η+2p0/p1−φζj¯z1+η+2p0/p−12+η+2p0/p1−φζj22+η+2p0/p1−φζj¯z2+η+2p0/p∫DφζjrσdA−1/p.
Then, by Lemma3, supj∥hj∥Apσ≲1 and hjn∈ℕ converges to zero uniformly on compact subsets of D as j⟶∞. By Lemma 4, hj converges to zero weakly in Apσ. Thus, for any compact operator K:Apσ⟶Bμ, we have that ∥Khj∥Bμ⟶0 as j⟶∞. Moreover, hj′φζj=0 and
(18)hjφζj=11+η+2p0/p2+η+2p0/p∫DφζjrσdA−1/p.
Thus, using (18), we have that
(19)∥Wψ,φ∥e,Apσ→Bμ≳limsupj→∞μζj∣ψ′ζjhjφζj+ψζjφ′ζjhj′φζj∣≳limsupj→∞μζj∣ψ′ζj∣∫DφζjrσdA−1/p.
Combining (15) and (19), we have that
(20)∥Wψ,φ∥e,Apσ→Bμ≳limsup∣φz∣→1μz∣ψ′z∣∫DφzrσdA−1/p+limsup∣φz∣→1μz∣ψzφ′z∣1−φz2∫DφzrσdA−1/p.
Upper bound. Letf∈Apσ be such that ∥f∥Apσ≤1. Let
(21)Ljfz=fajz,
whereaj=j/j+1.
Then, by Theorem 6.1 in [3], we have that Lj:Apσ⟶Apσ is compact. Since Wψ,φ:Apσ⟶Bμ is bounded, so Wψ,φLj:Apσ⟶Bμ is compact. Thus, for fixed r in 0,1, we have that
(22)∥Wψ,φ∥e,Apσ→Bμ≤sup∥f∥Apσ≤1∥Wψ,φI−Ljf∥Bμ≤sup∥f∥Apσ≤1∣Wψ,φI−Ljf0∣+supz∈Dμz∣Wψ,φI−Ljf′z∣≤sup∥f∥Apσ≤1∣Wψ,φI−Ljf0∣+sup∣φz∣≤rμz∣Wψ,φI−Ljf′z∣+sup∣φz∣>rμz∣Wψ,φI−Ljf′z∣,where I is the identity operator on Apσ. Now
(23)Wψ,φI−Ljf′z≤ψ′zfφz−fajφz+ψzφ′zf′φz−ajf′ajφz≤ψ′zfφz−fajφz+∣ψzφ′z∣j+1f′ajφz+ψzφ′zf′φz−f′ajφz.
Let∣φz∣≤r and ζ=φz. Let Γ=ajζ,ζ be the line segment from ajζ to ζ. Then, Γ⊂D0,r, where D0,r=z:∣z∣≤r. Thus, by Lemma 2, we have that
(24)∣f′ζ−f′ajζ∣=∣∫Γf′′λdλ∣≤∣ζ∣j+1supλ∈D0,r∣f′′λ∣≲∣ζ∣j+1supλ∈D0,r∫DλrσdA−1/p1−λ22∥f∥Apσ.
Again, letγ0=η+2p0/p−2. Then, by Lemma 2, we have that
(25)1=∣Kζγ00∣≤C∫D01/2σdA−1/p∥Kζγ0∥Apσ≍∫DζrσdA1/p1−ζ2γ0+2.
Thus,(26)1−ζ2η+2p0/p∫DζrσdA1/p≤1−ζ2γ0+2∫DζrσdA1/p≲1.
Thus, from (24) and (26), we have that
(27)∣f′ζ−f′ajζ∣≲∣ζ∣j+1supλ∈D0,r11−λ24+η+2p0/p1−λ22+η+2p0/p∫DλrσdA1/p∥f∥Apσ≲∣ζ∣j+111−r24+η+2p0/p∥f∥Apσ.
Similarly, we can show that(28)∣fζ−fajζ∣≲∣ζ∣j+111−r23+η+2p0/p∥f∥Apσ.
Also, by Lemma2 and equation (26), we have that
(29)sup∣φz∣≤r∣f′ajζ∣≲sup∣φz∣≤r∫DajφzrσdA−1/p1−aj2φz2∥f∥Apσ≲11−aj2r21+η+2p0/p∥f∥Apσ.
SinceWψ,φ:Apσ⟶Bμ is bounded, so
(30)∥Wψ,φf∥Bμ≲∥f∥Apσfor each f∈Apσ. By taking, respectively, fz=1 and fz=z and using the fact that ∣φz∣<1, we have that
(31)M1=supz∈Dμz∣ψ′z∣<∞andM2=supz∈Dμz∣ψzφ′z∣<∞.
Combining (23) and (27)-(29), we have that
(32)sup∥f∥Apσ≤1sup∣φz∣≤rμz∣Wψ,φI−Ljf′z∣⟶0asj⟶∞.
Using (28), we have that
(33)∣Wψ,φI−Ljf0∣=∣ψ0fφ0−ψ0fajφ0∣≲∣ψ0φ0∣j+1∥f∥Apσ1−r23+η+2p0/p.
Using (31), (32), and (33), we have that
(34)sup∥f∥Apσ≤1sup∣φz∣≤r∣Wψ,φI−Ljf0∣+∣Wψ,φI−Ljf′z∣⟶0as j⟶∞. The last term in the right-hand side of (22) is dominated by
(35)sup∣φz∣>rμz∣ψ′z∣∣fφz∣+∣fajφz∣+sup∣φz∣>rμz∣ψzφ′z∣∣f′φz∣+aj∣f′ajφz∣,which is further dominated by a constant multiple of
(36)sup∣φz∣>rμz∣ψ′z∣∫DφzrσdA−1/p+∫DajφzrσdA−1/p∥f∥Apσ+sup∣φz∣>rμz∣ψzφ′z∣∫DφzrσdA−1/p1−φz2+∫DajφzrσdA−1/p1−aj2φz2∥f∥Apσ.
Lettingj⟶∞ in (36), we get
(37)limsupj→∞sup∥f∥Apσ≤1sup∣φz∣>rμz∣Wψ,φI−Lkf′z∣≲sup∣φz∣>rμz∣ψ′z∣∫DφzrσdA−1/p+sup∣φz∣>rμz∣ψzφ′z∣∫DφzrσdA−1/p1−φz2.
Using (34) and (37) in (22), we have that
(38)∥Wψ,φ∥e,Apσ→Bμ≲sup∣φz∣>rμz∣ψ′z∣∫DφzrσdA−1/p+sup∣φz∣>rμz∣ψzφ′z∣1−φz2∫DφzrσdA−1/p.
Finally, lettingr⟶1, then we get
(39)∥Wψ,φ∥e,Apσ→Bμ≲limsup∣φz∣→1μz∣ψ′z∣∫DφzrσdA−1/p+limsup∣φz∣→1μz∣ψzφ′z∣1−φz2∫DφzrσdA−1/p.
Combining (20) and (39), we get the desired result.Corollary 5.
Letp∈1,∞, ψ∈HD, and φ be a holomorphic self-map of D, such that ∥φ∥∞=1. Let Wψ,φ:Apσ⟶Bμ is bounded. Then, Wψ,φ:Apσ⟶Bμ is compact if and only if the following conditions are satisfied:
(1)
limsup∣φz∣→1μz∣ψ′z∣∫DφzrσdA−1/p=0(2)
limsupφz→1μz∣ψzφ′z∣/1−φz2∫DφzrσdA−1/p=0



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