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Discrete Dynamics in Nature and Society
VolumeΒ 2008Β (2008), Article IDΒ 154263, 14 pages
http://dx.doi.org/10.1155/2008/154263
Research Article

On a New Integral-Type Operator from the Weighted Bergman Space to the Bloch-Type Space on the Unit Ball

Mathematical Institute of the Serbian Academy of Science, Knez Mihailova 36/III, 11000 Beograd, Serbia

Received 1 May 2008; Accepted 20 July 2008

Academic Editor: LeonidΒ Berezansky

Copyright © 2008 Stevo Stević. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce an integral-type operator, denoted by π‘ƒπ‘”πœ‘, on the space of holomorphic functions on the unit ball π”ΉβŠ‚β„‚π‘›, which is an extension of the product of composition and integral operators on the unit disk. The operator norm of π‘ƒπ‘”πœ‘ from the weighted Bergman space 𝐴𝑝𝛼(𝔹) to the Bloch-type space β„¬πœ‡(𝔹) or the little Bloch-type space β„¬πœ‡,0(𝔹) is calculated. The compactness of the operator is characterized in terms of inducing functions 𝑔 and πœ‘. Upper and lower bounds for the essential norm of the operator π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Ό(𝔹)β†’β„¬πœ‡(𝔹), when 𝑝>1, are also given.

1. Introduction

Let 𝔹 be the open unit ball in the complex vector space ℂ𝑛,𝑆=πœ•π”Ή its boundary, 𝔻 the open unit disk in the complex plane β„‚,𝑑𝑉(𝑧) the Lebesgue measure on 𝔹,𝑑𝑉𝛼(𝑧)=𝑐𝛼(1βˆ’|𝑧|2)𝛼𝑑𝑉(𝑧), where 𝛼>βˆ’1 and where the constant 𝑐𝛼 is chosen such that 𝑉𝛼(𝔹)=1,π‘‘πœŽ the normalized rotation invariant measure on 𝑆 (that is, 𝜎(𝑆)=1), 𝐻(𝔹) the class of all holomorphic functions on the unit ball and 𝐻∞=𝐻∞(𝔹) the space of all bounded holomorphic functions on 𝔹 with the normβ€–π‘“β€–βˆž=supπ‘§βˆˆπ”Ή||||.𝑓(𝑧)(1.1)

Let 𝑧=(𝑧1,…,𝑧𝑛) and 𝑀=(𝑀1,…,𝑀𝑛) be points in ℂ𝑛,βŸ¨π‘§,π‘€βŸ©=π‘›ξ“π‘˜=1π‘§π‘˜π‘€π‘˜(1.2)and √|𝑧|=βŸ¨π‘§,π‘§βŸ©.

For π‘“βˆˆπ»(𝔹) with the Taylor expansion βˆ‘π‘“(𝑧)=|𝛽|β‰₯0π‘Žπ›½π‘§π›½, letξ“β„œπ‘“(𝑧)=|𝛽|β‰₯0|𝛽|π‘Žπ›½π‘§π›½(1.3)be the radial derivative of 𝑓, where 𝛽=(𝛽1,𝛽2,…,𝛽𝑛) is a multi-index, |𝛽|=𝛽1+β‹―+𝛽𝑛 and 𝑧𝛽=𝑧𝛽11⋯𝑧𝛽𝑛𝑛. It is well known (see, e.g., [1]) thatβ„œπ‘“(𝑧)=𝑛𝑗=1π‘§π‘—πœ•π‘“πœ•π‘§π‘—(𝑧).(1.4)

For 𝑝>0 the Hardy space 𝐻𝑝=𝐻𝑝(𝔹) consists of all π‘“βˆˆπ»(𝔹) such that‖𝑓‖𝑝𝑝=sup0<π‘Ÿ<1ξ€œπ‘†||||𝑓(π‘Ÿπœ)π‘π‘‘πœŽ(𝜁)<∞.(1.5)It is well known that for every π‘“βˆˆπ»π‘ the radial limitπ‘“βˆ—(𝜁)∢=limπ‘Ÿβ†’1𝑓(π‘Ÿπœ)(1.6)exists almost everywhere on πœβˆˆπ‘†.

The weighted Bergman space 𝐴𝑝𝛼=𝐴𝑝𝛼(𝔹),𝑝>0,𝛼>βˆ’1, consists of all π‘“βˆˆπ»(𝔹) such that‖𝑓‖𝑝𝐴𝑝𝛼=ξ€œπ”Ή||||𝑓(𝑧)𝑝𝑑𝑉𝛼(𝑧)<∞.(1.7)When 𝑝β‰₯1, the weighted Bergman space with the norm ‖⋅‖𝐴𝑝𝛼 becomes a Banach space. If π‘βˆˆ(0,1), it is a Frechet space with the translation invariant metric𝑑(𝑓,𝑔)=β€–π‘“βˆ’π‘”β€–π‘π΄π‘π›Ό.(1.8)

Since for every π‘“βˆˆπ»π‘limπ›Όβ†’βˆ’1+0ξ€œπ”Ή||||𝑓(𝑧)π‘π‘‘π‘‰π›Όξ€œ(𝑧)=𝑆||π‘“βˆ—||(𝜁)π‘π‘‘πœŽ(𝜁),(1.9)we will also use the notation π΄π‘βˆ’1 for the Hardy space 𝐻𝑝.

A positive continuous function πœ™ on [0,1) is called normal (see [2]) if there is π›Ώβˆˆ[0,1) and π‘Ž and 𝑏,0<π‘Ž<𝑏 such thatπœ™(π‘Ÿ)(1βˆ’π‘Ÿ)π‘Žisdecreasingon[𝛿,1),limπ‘Ÿβ†’1πœ™(π‘Ÿ)(1βˆ’π‘Ÿ)π‘Ž=0;πœ™(π‘Ÿ)(1βˆ’π‘Ÿ)𝑏isincreasingon[𝛿,1),limπ‘Ÿβ†’1πœ™(π‘Ÿ)(1βˆ’π‘Ÿ)𝑏=∞.(1.10)From now on if we say that a function πœ‡βˆΆπ”Ήβ†’[0,∞) is normal, we will also assume that it is radial, that is, πœ‡(𝑧)=πœ‡(|𝑧|),π‘§βˆˆπ”Ή.

The weighted space π»βˆžπœ‡=π»βˆžπœ‡(𝔹) consists of all π‘“βˆˆπ»(𝔹) such thatβ€–π‘“β€–π»βˆžπœ‡=supπ‘§βˆˆπ”Ή||||πœ‡(𝑧)𝑓(𝑧)<∞,(1.11)where πœ‡ is normal. For πœ‡(𝑧)=(1βˆ’|𝑧|2)𝛽, 𝛽>0, we obtain the weighted space π»βˆžπ›½=π»βˆžπ›½(𝔹) (see, e.g., [3–5]).

The little weighted space π»βˆžπœ‡,0=π»βˆžπœ‡,0(𝔹) is a subspace of π»βˆžπœ‡ consisting of all π‘“βˆˆπ»(𝔹) such thatlim|𝑧|β†’1||||πœ‡(𝑧)𝑓(𝑧)=0.(1.12)

The class of all π‘“βˆˆπ»(𝔹) such thatπ΅πœ‡(𝑓)=supπ‘§βˆˆπ”Ή||||πœ‡(𝑧)β„œπ‘“(𝑧)<∞,(1.13)where πœ‡ is normal, is called the Bloch-type space, and is denoted by β„¬πœ‡=β„¬πœ‡(𝔹). With the normβ€–π‘“β€–β„¬πœ‡=||||𝑓(0)+π΅πœ‡(𝑓),(1.14)the Bloch-type space becomes a Banach space.

The little Bloch-type space β„¬πœ‡,0 is a subspace of β„¬πœ‡ consisting of those π‘“βˆˆβ„¬πœ‡ such thatlim|𝑧|β†’1||||πœ‡(𝑧)β„œπ‘“(𝑧)=0.(1.15)The 𝛼-Bloch space ℬ𝛼 is obtained for πœ‡(𝑧)=(1βˆ’|𝑧|2)𝛼,π›Όβˆˆ(0,∞) (see, e.g., [6–11]). For 𝛼=1 the space ℬ1=ℬ becomes the classical Bloch space.

Let πœ‘ be a holomorphic self-map of 𝔹. For any π‘“βˆˆπ»(𝔹), the composition operator is defined byπΆπœ‘ξ€·ξ€Έπ‘“(𝑧)=π‘“πœ‘(𝑧),π‘§βˆˆπ”Ή.(1.16)It is of interest to provide function theoretic characterizations when πœ‘ induces bounded or compact composition operators on spaces of holomorphic functions. For some classical results in the topic (see, e.g., [12]). For some recent results see, for example, [3–5, 7, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23] and the references therein.

Let π‘”βˆˆπ»(𝔻) and πœ‘ be a holomorphic self-map of 𝔻. For π‘“βˆˆπ»(𝔻), products of integral-type and composition operator are defined as follows:πΆπœ‘π½π‘”ξ€œπ‘“(𝑧)=0πœ‘(𝑧)𝑓(𝜁)π‘”ξ…ž(𝜁)π‘‘πœ,π½π‘”πΆπœ‘ξ€œπ‘“(𝑧)=𝑧0π‘“ξ€·ξ€Έπ‘”πœ‘(𝜁)ξ…ž(𝜁)π‘‘πœ.(1.17)

When πœ‘(𝑧)=𝑧, operators in (1.17) are reduced to the integral operator introduced in [24]. For some other results on the operator; see, for example, [25, 26], and related references therein. Some results on related integral-type operators on spaces of holomorphic functions in ℂ𝑛 can be found, for example, in [27–41] (see also the references therein).

In [42], among other results, we proved the following theorem regarding the boundedness of the operator π½π‘”πΆπœ‘βˆΆπ΄π‘π›Ό(𝔻)β†’β„¬πœ‡(𝔻).

Theorem 1.1. Assume that 𝑝>0,𝛼>βˆ’1,π‘”βˆˆπ»(𝔻),πœ‡ is normal, and πœ‘ is a holomorphic self-map of 𝔻. Then π½π‘”πΆπœ‘βˆΆπ΄π‘π›Ό(𝔻)β†’β„¬πœ‡(𝔻) is bounded if and only ifsupπ‘§βˆˆπ”»||π‘”πœ‡(𝑧)ξ…ž||(𝑧)ξ€·||||1βˆ’πœ‘(𝑧)2ξ€Έ(𝛼+2)/𝑝<∞.(1.18)

One of the interesting questions is to extend operators in (1.17) in the unit ball settings and to study their function theoretic properties on spaces of holomorphic functions on the unit ball in terms of inducing functions.

Assume that π‘”βˆˆπ»(𝔹),𝑔(0)=0, and πœ‘ is a holomorphic self-map of 𝔹. We introduce the following important integral-type operator on the space of holomorphic functions on 𝔹: π‘ƒπ‘”πœ‘ξ€œ(𝑓)(𝑧)=10π‘“ξ€·ξ€Έπœ‘(𝑑𝑧)𝑔(𝑑𝑧)𝑑𝑑𝑑,π‘“βˆˆπ»(𝔹),π‘§βˆˆπ”Ή.(1.19)

First note that when 𝑛=1, the operator is reduced to an operator of the form as the second operator in (1.17). Indeed, since π‘”βˆˆπ»(𝔻) and 𝑔(0)=0, it follows that 𝑔(𝑧)=𝑧𝑔0(𝑧),π‘§βˆˆπ”» for some 𝑔0∈𝐻(𝔻). By using this fact and the change of variables 𝜁=𝑑𝑧, we obtainπ‘ƒπ‘”πœ‘ξ€œπ‘“(𝑧)=10π‘“ξ€·ξ€Έπœ‘(𝑑𝑧)𝑑𝑧𝑔0(𝑑𝑧)𝑑𝑑𝑑=ξ€œπ‘§0π‘“ξ€·ξ€Έπ‘”πœ‘(𝜁)0(𝜁)π‘‘πœ.(1.20)Hence operator (1.19) is a natural extension of the second operator in (1.17).

Now we formulate the following big research project related to the operator π‘ƒπ‘”πœ‘.

Research Project 1. Let 𝑋 and π‘Œ be two Banach spaces of holomorphic functions on the unit ball in ℂ𝑛 (e.g., the weighted Bergman space 𝐴𝑝𝛼, the Bloch-type space β„¬πœ‡, the Hardy space 𝐻𝑝 space, the weighted space π»βˆžπœ‡, the Besov space 𝐡𝑝, BMOA etc.) Characterize the boundedness, compactness, essential norms, and other operator theoretic properties of the operator π‘ƒπ‘”πœ‘βˆΆπ‘‹β†’π‘Œ in terms of function theoretic properties of inducing functions πœ‘ and 𝑔.

Another interesting question is to find the exact value of the norm of operators on spaces of holomorphic functions. Majority of papers in the area only find asymptotics of the operator norm of certain linear operators on some spaces of holomorphic functions. There are a few papers which calculate the operator norm of these operators. Recently in [4] we calculated operator norm of the weighted composition operator π‘’πΆπœ‘ mapping the Bloch space ℬ to the weighted space π»βˆžπœ‡, which motivates us to find the norms of weighted composition and other closely related operators between various spaces of holomorphic functions.

Research Project 2. Let 𝑋 and π‘Œ be two Banach spaces of holomorphic functions as in Research project 1. Calculate the operator norm of π‘ƒπ‘”πœ‘βˆΆπ‘‹β†’π‘Œ in terms of inducing functions πœ‘ and 𝑔.

In this paper, among other results, we will calculate the operator norm of π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Ό(𝔹)β†’β„¬πœ‡(𝔹). We will also characterize the boundedness, compactness, and the essential norm of the operator. These results partially solve problems posed in the above research projects.

Throughout the paper, 𝐢 denotes a positive constant not necessarily the same at each occurrence. The notation 𝐴≍𝐡 means that there is a positive constant 𝐢 such that 𝐴/𝐢≀𝐡≀𝐢𝐴.

2. Auxiliary Results

In this section, we give several auxiliary results, which are used in the proofs of the main results.

Lemma 2.1 (see [43, Corollary 3.5]). Suppose that π‘βˆˆ(0,∞) and 𝛼β‰₯βˆ’1. Then for all π‘“βˆˆπ΄π‘π›Ό(𝔹) and π‘§βˆˆπ”Ή, the following inequality holds:||||≀𝑓(𝑧)‖𝑓‖𝐴𝑝𝛼1βˆ’|𝑧|2ξ€Έ(𝑛+1+𝛼)/𝑝.(2.1)

The following criterion for the compactness follows by standard arguments (see, e.g., [12, 20, 34–36]). Hence, we omit its proof.

Lemma 2.2. Suppose that 0<𝑝<∞,𝛼β‰₯βˆ’1,π‘”βˆˆπ»(𝔹),πœ‡ is normal, and πœ‘ is a holomorphic self-map of 𝔹. Then the operator π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡ is compact if and only if π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡ is bounded and for every bounded sequence (π‘“π‘˜)π‘˜βˆˆβ„• in 𝐴𝑝𝛼 converging to zero uniformly on compacts of 𝔹, one has β€–π‘ƒπ‘”πœ‘π‘“π‘˜β€–β„¬πœ‡β†’0 as π‘˜β†’βˆž.

The following result can be found in [44]. For closely related results, see also [11, 45–52] and the references therein.

Lemma 2.3. Suppose that 0<𝑝<∞,𝛼>βˆ’1, then‖𝑓‖𝑝𝐴𝑝𝛼≍||||𝑓(0)𝑝+ξ€œπ”Ή||||βˆ‡π‘“(𝑧)𝑝1βˆ’|𝑧|2𝑝+𝛼𝑑𝑉(𝑧),(2.2)for every π‘“βˆˆπ΄π‘π›Ό (here βˆ‡π‘“=((πœ•π‘“/πœ•π‘§1),…,(πœ•π‘“/πœ•π‘§π‘›))).

The following lemma can be proved similar to [53, Lemma 1].

Lemma 2.4. Suppose that πœ‡ is normal. A closed set 𝐾 in β„¬πœ‡,0 is compact if and only if it is bounded andlim|𝑧|β†’1supπ‘“βˆˆπΎ||||πœ‡(𝑧)β„œπ‘“(𝑧)=0.(2.3)

The following lemma is related to [32, Lemma 1] and [34, Lemma 2].

Lemma 2.5. Assume that 𝑓,π‘”βˆˆπ»(𝔹) and 𝑔(0)=0. Thenβ„œπ‘ƒπ‘”πœ‘(𝑓)(𝑧)=𝑓(πœ‘(𝑧))𝑔(𝑧).(2.4)

Proof. Since the function 𝑓(πœ‘(𝑧))𝑔(𝑧) is holomorphic and 𝑔(0)=0, it has the Taylor expansion in the following form βˆ‘π›Όβ‰ 0π‘Žπ›Όπ‘§π›Ό. Thenβ„œξ€Ίπ‘ƒπ‘”πœ‘ξ€»ξ€œ(𝑓)(𝑧)=β„œ10𝛼≠0π‘Žπ›Ό(𝑑𝑧)𝛼𝑑𝑑𝑑=β„œπ›Όβ‰ 0π‘Žπ›Όπ‘§|𝛼|𝛼=𝛼≠0π‘Žπ›Όπ‘§π›Ό,(2.5)as claimed.

3. The Norm of the Operator π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡

In this section, we calculate the norm β€–π‘ƒπ‘”πœ‘β€–π΄π‘π›Όβ†’β„¬πœ‡.

Theorem 3.1. Assume that 𝑝>0,𝛼β‰₯βˆ’1,π‘”βˆˆπ»(𝔹),πœ‡ is normal, πœ‘ is a holomorphic self-map of 𝔹, and π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Ό(𝔹)β†’β„¬πœ‡(𝔹) is bounded. Thenβ€–π‘ƒπ‘”πœ‘β€–π΄π‘π›Όβ†’β„¬πœ‡=β€–π‘ƒπ‘”πœ‘β€–π΄π‘π›Όβ†’β„¬πœ‡,0=supπ‘§βˆˆπ”Ή||||πœ‡(𝑧)𝑔(𝑧)ξ€·||||1βˆ’πœ‘(𝑧)2ξ€Έ(𝑛+1+𝛼)/𝑝=βˆΆπ‘€.(3.1)

Proof. If π‘“βˆˆπ΄π‘π›Ό, then by Lemmas 2.5 and 2.1 we obtainβ€–π‘ƒπ‘”πœ‘π‘“β€–β„¬πœ‡=supπ‘§βˆˆπ”Ή||ξ€·ξ€Έ||πœ‡(𝑧)𝑔(𝑧)π‘“πœ‘(𝑧)≀‖𝑓‖𝐴𝑝𝛼supπ‘§βˆˆπ”Ή||||πœ‡(𝑧)𝑔(𝑧)||||(1βˆ’πœ‘(𝑧)2)(𝑛+1+𝛼)/𝑝,(3.2)from which it follows thatβ€–π‘ƒπ‘”πœ‘β€–π΄π‘π›Όβ†’β„¬πœ‡β‰€π‘€.(3.3)
Now we prove the reverse inequality. For π‘€βˆˆπ”Ή fixed, set𝑓𝑀(𝑧)=1βˆ’|𝑀|2ξ€Έ(𝑛+1+𝛼)/𝑝1βˆ’βŸ¨π‘§,π‘€βŸ©2(𝑛+1+𝛼)/𝑝,π‘§βˆˆπ”Ή.(3.4)We have that ‖𝑓𝑀‖𝐴𝑝𝛼=1, for each π‘€βˆˆπ”Ή. For 𝛼>βˆ’1 this fact is well known. The proof for the case 𝛼=βˆ’1 could be less known, and we give a proof of it for the lack of a specific reference and for the benefit of the reader. Let 𝑧=π‘Ÿπœ, πœβˆˆπ‘†, then we have ‖𝑓𝑀‖𝑝𝑝=sup0<π‘Ÿ<1ξ€œπ‘†(1βˆ’|𝑀|2𝑛||||1βˆ’βŸ¨π‘§,π‘€βŸ©2𝑛=ξ€·π‘‘πœŽ(𝜁)1βˆ’|𝑀|2𝑛sup0<π‘Ÿ<1ξ€œπ‘†||ξ€·ξ€Έ1βˆ’βŸ¨π‘§,π‘€βŸ©βˆ’π‘›||2=ξ€·π‘‘πœŽ(𝜁)1βˆ’|𝑀|2𝑛sup0<π‘Ÿ<1ξ€œπ‘†||||βˆžξ“π‘˜=0Ξ“(𝑛+π‘˜)π‘ŸΞ“(π‘˜+1)Ξ“(𝑛)π‘˜βŸ¨πœ,π‘€βŸ©π‘˜||||2=ξ€·π‘‘πœŽ(𝜁)1βˆ’|𝑀|2𝑛sup0<π‘Ÿ<1ξ€œπ‘†βˆžξ“π‘˜=0ξ‚€Ξ“(𝑛+π‘˜)Γ(π‘˜+1)Ξ“(𝑛)2π‘Ÿ2π‘˜||||⟨𝜁,π‘€βŸ©2π‘˜=ξ€·π‘‘πœŽ(𝜁)1βˆ’|𝑀|2𝑛sup∞0<π‘Ÿ<1ξ“π‘˜=0Ξ“(𝑛+π‘˜)π‘ŸΞ“(π‘˜+1)Ξ“(𝑛)2π‘˜|𝑀|2π‘˜=ξ€·1βˆ’|𝑀|2𝑛sup0<π‘Ÿ<11ξ€·1βˆ’π‘Ÿ2|𝑀|2𝑛=1,(3.5)where we have used the following formula (see, e.g., [1])ξ€œπ‘†||||⟨𝜁,π‘€βŸ©2π‘˜π‘‘πœŽ(𝜁)=Ξ“(π‘˜+1)Ξ“(𝑛)Ξ“(𝑛+π‘˜)|𝑀|2π‘˜.(3.6)
From this and the boundedness of π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡, we haveβ€–β€–π‘ƒπ‘”πœ‘β€–β€–π΄π‘π›Όβ†’β„¬πœ‡=β€–β€–π‘“πœ‘(𝑀)β€–β€–π΄π‘π›Όβ€–β€–π‘ƒπ‘”πœ‘β€–β€–π΄π‘π›Όβ†’β„¬πœ‡β‰₯β€–β€–π‘ƒπ‘”πœ‘ξ€·π‘“πœ‘(𝑀)ξ€Έβ€–β€–β„¬πœ‡=supπ‘§βˆˆπ”Ή||||||π‘“πœ‡(𝑧)𝑔(𝑧)πœ‘(𝑀)ξ€·ξ€Έ||||||||π‘“πœ‘(𝑧)β‰₯πœ‡(𝑀)𝑔(𝑀)πœ‘(𝑀)ξ€·ξ€Έ||=||||πœ‘(𝑀)πœ‡(𝑀)𝑔(𝑀)ξ€·||||1βˆ’πœ‘(𝑀)2ξ€Έ(𝑛+1+𝛼)/𝑝.(3.7)Taking the supremum in (3.7) over π‘€βˆˆπ”Ή, we obtainβ€–β€–π‘ƒπ‘”πœ‘β€–β€–π΄π‘π›Όβ†’β„¬πœ‡β‰₯𝑀.(3.8)From (3.3) and (3.8), it follows that β€–π‘ƒπ‘”πœ‘β€–π΄π‘π›Όβ†’β„¬πœ‡=𝑀.
Sinceβ€–β€–π‘ƒπ‘”πœ‘β€–β€–π΄π‘π›Όβ†’β„¬πœ‡,0β‰€β€–β€–π‘ƒπ‘”πœ‘β€–β€–π΄π‘π›Όβ†’β„¬πœ‡(3.9)and the proof of (3.8) does not depend on the space β„¬πœ‡ (we may replace it by β„¬πœ‡,0) the second equality in (3.1) also holds.

Corollary 3.2. Assume that 𝑝>0,𝛼β‰₯βˆ’1,π‘”βˆˆπ»(𝔹),πœ‡ is normal, and πœ‘ is a holomorphic self-map of 𝔹. Then π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡ is bounded if and only ifsupπ‘§βˆˆπ”Ή||||πœ‡(𝑧)𝑔(𝑧)ξ€·||||1βˆ’πœ‘(𝑧)2ξ€Έ(𝑛+1+𝛼)/𝑝<∞.(3.10)

Proof. If π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡ is bounded, then (3.10) follows from Theorem 3.1. If (3.10) holds, then the boundedness of π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡ follows from (3.3).

4. The Boundedness of the Operator π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡,0

Here we characterize the boundedness of the operator π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡,0.

Theorem 4.1. Assume that 𝑝>0,𝛼β‰₯βˆ’1,π‘”βˆˆπ»(𝔹),πœ‡ is normal, and πœ‘ is a holomorphic self-map of 𝔹. Then π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡,0 is bounded if and only if π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡ is bounded and π‘”βˆˆπ»βˆžπœ‡,0.

Proof. Assume that π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡ is bounded and π‘”βˆˆπ»βˆžπœ‡,0. Then, for each polynomial 𝑝, we have||πœ‡(𝑧)β„œπ‘ƒπ‘”πœ‘||||ξ€·ξ€Έ||||||𝑝(𝑧)=πœ‡(𝑧)𝑔(𝑧)π‘πœ‘(𝑧)β‰€πœ‡(𝑧)𝑔(𝑧)β€–π‘β€–βˆžβŸΆ0,as|𝑧|⟢1(4.1)from which it follows that π‘ƒπ‘”πœ‘(𝑝)βˆˆβ„¬πœ‡,0.
Since the set of all polynomials is dense in 𝐴𝑝𝛼, we have that for every π‘“βˆˆπ΄π‘π›Ό there is a sequence of polynomials (π‘π‘˜)π‘˜βˆˆβ„• such thatlimπ‘˜β†’βˆžβ€–β€–π‘“βˆ’π‘π‘˜β€–β€–π΄π‘π›Ό=0.(4.2)From this and since the operator π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡ is bounded, it follows thatβ€–β€–π‘ƒπ‘”πœ‘π‘“βˆ’π‘ƒπ‘”πœ‘π‘π‘˜β€–β€–β„¬πœ‡β‰€β€–β€–π‘ƒπ‘”πœ‘β€–β€–π΄π‘π›Όβ†’β„¬πœ‡β€–β€–π‘“βˆ’π‘π‘˜β€–β€–π΄π‘π›ΌβŸΆ0,(4.3)as π‘˜β†’βˆž. Hence π‘ƒπ‘”πœ‘(𝐴𝑝𝛼)βŠ‚β„¬πœ‡,0. Since β„¬πœ‡,0 is a closed subset of β„¬πœ‡, the boundedness of π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡,0 follows.
Now assume that π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡,0 is bounded. Then clearly π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡ is bounded. Taking the test function 𝑓(𝑧)=1βˆˆπ΄π‘π›Ό, we obtain π‘”βˆˆπ»βˆžπœ‡,0.

5. Compactness of the Operator π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡

This section is devoted to studying of the compactness of the operator π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡. We prove the following result.

Theorem 5.1. Assume that 𝑝>0,𝛼β‰₯βˆ’1,π‘”βˆˆπ»(𝔹),πœ‡ is normal, πœ‘ is a holomorphic self-map of 𝔹, and the operator π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡ is bounded. Then the operator π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡ is compact if and only iflim|πœ‘(𝑧)|β†’1||||πœ‡(𝑧)𝑔(𝑧)ξ€·||||1βˆ’πœ‘(𝑧)2ξ€Έ(𝑛+1+𝛼)/𝑝=0.(5.1)

Proof. First assume that the operator π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡ is compact. If β€–πœ‘β€–βˆž<1, then condition (5.1) is vacuously satisfied. Hence, assume that β€–πœ‘β€–βˆž=1 and assume to the contrary that (5.1) does not hold. Then there is a sequence (π‘§π‘˜)π‘˜βˆˆβ„• satisfying the condition |πœ‘(π‘§π‘˜)|β†’1 as π‘˜β†’βˆž and 𝛿>0 such thatπœ‡(π‘§π‘˜)||𝑔(π‘§π‘˜)||ξ€·||1βˆ’πœ‘(π‘§π‘˜)||2ξ€Έ(𝑛+1+𝛼)/𝑝β‰₯𝛿,π‘˜βˆˆβ„•.(5.2)For π‘€βˆˆπ”Ή fixed, setπΉπ‘˜(𝑧)=π‘“πœ‘(π‘§π‘˜)(𝑧),π‘˜βˆˆβ„•,(5.3)where 𝑓𝑀 is defined in (3.4). Recall that ‖𝑓𝑀‖𝐴𝑝𝛼=1, for each π‘€βˆˆπ”Ή. Then β€–πΉπ‘˜β€–π΄π‘π›Ό=1,π‘˜βˆˆβ„• and it is easy to see that πΉπ‘˜β†’0 uniformly on compacts of 𝔹 as π‘˜β†’βˆž. Hence, by Lemma 2.2, it follows thatlimπ‘˜β†’βˆžβ€–β€–π‘ƒπ‘”πœ‘πΉπ‘˜β€–β€–β„¬πœ‡=0.(5.4)
On the other hand, by Lemma 2.5 and (5.2), we obtainβ€–β€–π‘ƒπ‘”πœ‘πΉπ‘˜β€–β€–β„¬πœ‡=supπ‘§βˆˆπ”Ή||||||πΉπœ‡(𝑧)𝑔(𝑧)π‘˜ξ€·ξ€Έ||β‰₯πœ‘(𝑧)πœ‡(π‘§π‘˜)||π‘”ξ€·π‘§π‘˜ξ€Έ||ξ€·||πœ‘ξ€·π‘§1βˆ’π‘˜ξ€Έ||2ξ€Έ(𝑛+1+𝛼)/𝑝β‰₯𝛿>0,(5.5)for every π‘˜βˆˆβ„•, which contradicts with (5.4).
Now assume that (5.1) holds. Then for every πœ€>0 there is an π‘Ÿβˆˆ(0,1) such that when π‘Ÿ<|πœ‘(𝑧)|<1,||||πœ‡(𝑧)𝑔(𝑧)ξ€·||||1βˆ’πœ‘(𝑧)2ξ€Έ(𝑛+1+𝛼)/𝑝<πœ€.(5.6)
On the other hand, since the operator π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡ is bounded, for 𝑓≑1βˆˆπ΄π‘π›Ό, we obtain β€–π‘”β€–π»βˆžπœ‡<∞.
Assume that (β„Žπ‘˜)π‘˜βˆˆβ„• is a bounded sequence in 𝐴𝑝𝛼 converging to zero uniformly on compacts of 𝔹 as π‘˜β†’βˆž. Let supπ‘˜βˆˆβ„•β€–β„Žπ‘˜β€–π΄π‘π›Ό=𝑀1. Then by Lemma 2.1 and (5.6), for π‘Ÿ<|πœ‘(𝑧)|<1, we obtain||||||β„Žπœ‡(𝑧)𝑔(𝑧)π‘˜ξ€·ξ€Έ||πœ‘(𝑧)≀supπ‘˜βˆˆβ„•β€–β„Žπ‘˜β€–π΄π‘π›Ό||||πœ‡(𝑧)𝑔(𝑧)ξ€·||||1βˆ’πœ‘(𝑧)2ξ€Έ(𝑛+1+𝛼)/𝑝<𝑀1πœ€.(5.7)If |πœ‘(𝑧)|β‰€π‘Ÿ, we have||||||β„Žπœ‡(𝑧)𝑔(𝑧)π‘˜ξ€·ξ€Έ||πœ‘(𝑧)β‰€β€–π‘”β€–π»βˆžπœ‡sup|𝑀|β‰€π‘Ÿ||β„Žπ‘˜||(𝑀)⟢0,asπ‘˜βŸΆβˆž.(5.8)From (5.7) and (5.8), it follows that β€–π‘ƒπ‘”πœ‘β„Žπ‘˜β€–β„¬πœ‡β†’0 as π‘˜β†’βˆž, from which the compactness of the operator π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡ follows.

6. Compactness of the Operator π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡,0

This section characterizes the compactness of the operator π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡,0.

Theorem 6.1. Assume 𝑝>0,𝛼β‰₯βˆ’1,π‘”βˆˆπ»(𝔹),πœ‡ is normal, πœ‘ is a holomorphic self-map of 𝔹, and the operator π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡,0 is bounded. Then the operator π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡,0 is compact if and only iflim|𝑧|β†’1||||πœ‡(𝑧)𝑔(𝑧)ξ€·||||1βˆ’πœ‘(𝑧)2ξ€Έ(𝑛+1+𝛼)/𝑝=0.(6.1)

Proof. Assume π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡,0 is compact. Then clearly π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡,0 is bounded and as in Theorem 4.1 we have that π‘”βˆˆπ»βˆžπœ‡,0.
Hence if β€–πœ‘β€–βˆž<1, thenlim|𝑧|β†’1||||πœ‡(𝑧)𝑔(𝑧)ξ€·||||1βˆ’πœ‘(𝑧)2ξ€Έ(𝑛+1+𝛼)/𝑝≀lim|𝑧|β†’1||||πœ‡(𝑧)𝑔(𝑧)ξ€·1βˆ’β€–πœ‘β€–2βˆžξ€Έ(𝑛+1+𝛼)/𝑝=0,(6.2)from which the result follows in this case.
Now assume β€–πœ‘β€–βˆž=1. By using the test functions πΉπ‘˜(𝑧)=π‘“πœ‘(π‘§π‘˜)(𝑧), π‘˜βˆˆβ„•, defined in (5.3) we obtain that condition (5.1) holds, which implies that for every πœ€>0, there is an π‘Ÿβˆˆ(0,1) such that for π‘Ÿ<|πœ‘(𝑧)|<1, condition (5.6) holds.
Since π‘”βˆˆπ»βˆžπœ‡,0, there is 𝜎∈(0,1) such that for 𝜎<|𝑧|<1||||ξ€·πœ‡(𝑧)𝑔(𝑧)<πœ€1βˆ’π‘Ÿ2ξ€Έ(𝑛+1+𝛼)/𝑝.(6.3)
Hence, if |πœ‘(𝑧)|β‰€π‘Ÿ and 𝜎<|𝑧|<1, we have||||πœ‡(𝑧)𝑔(𝑧)ξ€·||||1βˆ’πœ‘(𝑧)2ξ€Έ(𝑛+1+𝛼)/𝑝≀||||πœ‡(𝑧)𝑔(𝑧)ξ€·1βˆ’π‘Ÿ2ξ€Έ(𝑛+1+𝛼)/𝑝<πœ€.(6.4)From (5.6) and (6.4), condition (6.1) follows.
Now assume that condition (6.1) holds. Then the quantity 𝑀 in Theorem 3.1 is finite. Using this fact and the following inequality||πœ‡(𝑧)β„œπ‘ƒπ‘”πœ‘||||ξ€·ξ€Έ||𝑓(𝑧)β‰€πœ‡(𝑧)𝑔(𝑧)π‘“πœ‘(𝑧)≀‖𝑓‖𝐴𝑝𝛼||||πœ‡(𝑧)𝑔(𝑧)ξ€·||||1βˆ’πœ‘(𝑧)2ξ€Έ(𝑛+1+𝛼)/𝑝,(6.5)it follows that the set π‘ƒπ‘”πœ‘({π‘“βˆΆβ€–π‘“β€–π΄π‘π›Όβ‰€1}) is bounded in β„¬πœ‡, moreover, in view of (6.1), it is bounded in β„¬πœ‡,0. Taking the supremum in the last inequality over the unit ball in 𝐴𝑝𝛼, then letting |𝑧|β†’1, using condition (6.1) and employing Lemma 2.4, we obtain the compactness of the operator π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡,0, as desired.

7. Essential Norm of π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡

Let 𝑋 and π‘Œ be Banach spaces, and let πΏβˆΆπ‘‹β†’π‘Œ be a bounded linear operator. The essential norm of the operator πΏβˆΆπ‘‹β†’π‘Œ, denoted by ‖𝐿‖𝑒,π‘‹β†’π‘Œ, is defined as follows:‖𝐿‖𝑒,π‘‹β†’π‘Œξ€½=inf‖𝐿+πΎβ€–π‘‹β†’π‘Œξ€ΎβˆΆπΎiscompactfromπ‘‹π‘‘π‘œπ‘Œ,(7.1)where β€–β‹…β€–π‘‹β†’π‘Œ denote the operator norm.

From this definition and since the set of all compact operators is a closed subset of the set of bounded operators, it follows that operator 𝐿 is compact if and only if ‖𝐿‖𝑒,π‘‹β†’π‘Œ=0.

In this section, we study the essential norm of the operator π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡ for the case 𝑝>1.

Theorem 7.1. Assume that π‘βˆˆ(1,∞),𝛼β‰₯βˆ’1,π‘”βˆˆπ»(𝔹),𝑔(0)=0,πœ‘ is a holomorphic self-map of 𝔹, and π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡ is bounded. Then the following inequalities hold:limsup|πœ‘(𝑧)|β†’1||||πœ‡(𝑧)𝑔(𝑧)ξ€·||||1βˆ’πœ‘(𝑧)2ξ€Έ(𝑛+1+𝛼)/π‘β‰€β€–β€–π‘ƒπ‘”πœ‘β€–β€–π‘’,π΄π‘π›Όβ†’β„¬πœ‡β‰€2limsup|πœ‘(𝑧)|β†’1||||πœ‡(𝑧)𝑔(𝑧)ξ€·||||1βˆ’πœ‘(𝑧)2ξ€Έ(𝑛+1+𝛼)/𝑝.(7.2)

Proof. Assume that (πœ‘(π‘§π‘˜))π‘˜βˆˆβ„• is a sequence in 𝔹 such that |πœ‘(π‘§π‘˜)|β†’1 as π‘˜β†’βˆž. Note that the sequence (π‘“πœ‘(π‘§π‘˜))π‘˜βˆˆβ„• (where 𝑓𝑀 is defined in (3.4)) is such that β€–π‘“πœ‘(π‘§π‘˜)‖𝐴𝑝𝛼=1 for each π‘˜βˆˆβ„• and it converges to zero uniformly on compacts of 𝔹. From this and by [11, Theorems 2.12 and 4.50], it follows that π‘“πœ‘(π‘§π‘˜)β†’0 weakly in 𝐴𝑝𝛼, as π‘˜β†’βˆž (here we use the condition 𝑝>1). Hence, for every compact operator πΎβˆΆπ΄π‘π›Όβ†’β„¬πœ‡, we have that β€–πΎπ‘“πœ‘(π‘§π‘˜)β€–β„¬πœ‡β†’0 as π‘˜β†’βˆž. Thus, for every such sequence and for every compact operator πΎβˆΆπ΄π‘π›Όβ†’β„¬πœ‡, we have thatβ€–β€–π‘ƒπ‘”πœ‘β€–β€–+πΎπ΄π‘π›Όβ†’β„¬πœ‡β‰₯limsupπ‘˜β†’βˆžβ€–β€–π‘ƒπ‘”πœ‘π‘“πœ‘(π‘§π‘˜)β€–β€–β„¬πœ‡βˆ’β€–β€–πΎπ‘“πœ‘(π‘§π‘˜)β€–β€–β„¬πœ‡β€–β€–π‘“πœ‘(π‘§π‘˜)‖‖𝐴𝑝𝛼=limsupπ‘˜β†’βˆžβ€–β€–π‘ƒπ‘”πœ‘π‘“πœ‘(π‘§π‘˜)β€–β€–β„¬πœ‡β‰₯limsupπ‘˜β†’βˆžπœ‡(π‘§π‘˜)||π‘”ξ€·π‘§π‘˜ξ€Έπ‘“πœ‘(π‘§π‘˜)ξ€·πœ‘ξ€·π‘§π‘˜||ξ€Έξ€Έ=limsupπ‘›β†’βˆžπœ‡ξ€·π‘§π‘˜ξ€Έ||π‘”ξ€·π‘§π‘˜ξ€Έ||ξ€·||πœ‘ξ€·π‘§1βˆ’π‘˜ξ€Έ||2ξ€Έ(𝑛+1+𝛼)/𝑝.(7.3)
Taking the infimum in (7.3) over the set of all compact operators πΎβˆΆπ΄π‘π›Όβ†’β„¬πœ‡, we obtainβ€–β€–π‘ƒπ‘”πœ‘β€–β€–π‘’,π΄π‘π›Όβ†’β„¬πœ‡β‰₯limsupπ‘›β†’βˆžπœ‡ξ€·π‘§π‘˜ξ€Έ||π‘”ξ€·π‘§π‘˜ξ€Έ||ξ€·||πœ‘ξ€·π‘§1βˆ’π‘˜ξ€Έ||2ξ€Έ(𝑛+1+𝛼)/𝑝,(7.4)from which the first inequality follows.
In the sequel we prove the second inequality. Assume that (π‘Ÿπ‘™)π‘™βˆˆβ„• is a sequence which increasingly converges to 1. Consider the operators defined byξ€·π‘ƒπ‘”π‘Ÿπ‘™πœ‘π‘“ξ€Έξ€œ(𝑧)=10ξ€·π‘Ÿπ‘”(𝑑𝑧)π‘“π‘™ξ€Έπœ‘(𝑑𝑧)𝑑𝑑𝑑,π‘™βˆˆβ„•.(7.5)We prove that these operators are compact. Indeed, since |π‘Ÿπ‘™πœ‘(𝑧)|β‰€π‘Ÿπ‘™<1, it follows that condition (5.1) in Theorem 5.1 is vacuously satisfied, from which the claim follows.
Recall that π‘”βˆˆπ»βˆžπœ‡. Let 𝜌∈(0,1) be fixed for a moment. Employing Lemma 2.1, and using the factβ€–β€–π‘“βˆ’π‘“π‘Ÿπ‘™β€–β€–π΄π‘π›Όβ‰€2‖𝑓‖𝐴𝑝𝛼,π‘™βˆˆβ„•,(7.6)which follows by using the triangle inequality for the norm, the monotonicity of the integral meansπ‘€π‘π‘ξ€œ(𝑓,π‘Ÿ)=𝑆||||𝑓(π‘Ÿπœ)π‘π‘‘πœŽ(𝜁)(7.7)and the polar coordinates, we have β€–β€–π‘ƒπ‘”πœ‘βˆ’π‘ƒπ‘”π‘Ÿπ‘™πœ‘β€–β€–π΄π‘π›Όβ†’β„¬πœ‡=sup‖𝑓‖𝐴𝑝𝛼≀1supπ‘§βˆˆπ”Ή||||||π‘“ξ€·ξ€Έξ€·π‘Ÿπœ‡(𝑧)𝑔(𝑧)πœ‘(𝑧)βˆ’π‘“π‘™ξ€Έ||πœ‘(𝑧)≀sup‖𝑓‖𝐴𝑝𝛼≀1sup|πœ‘(𝑧)|β‰€πœŒ||||||π‘“ξ€·ξ€Έξ€·π‘Ÿπœ‡(𝑧)𝑔(𝑧)πœ‘(𝑧)βˆ’π‘“π‘™ξ€Έ||πœ‘(𝑧)+sup‖𝑓‖𝐴𝑝𝛼≀1sup|πœ‘(𝑧)|>𝜌||||||π‘“ξ€·ξ€Έξ€·π‘Ÿπœ‡(𝑧)𝑔(𝑧)πœ‘(𝑧)βˆ’π‘“π‘™ξ€Έ||πœ‘(𝑧)β‰€β€–π‘”β€–π»βˆžπœ‡sup‖𝑓‖𝐴𝑝𝛼≀1sup|πœ‘(𝑧)|β‰€πœŒ||π‘“ξ€·ξ€Έξ€·π‘Ÿπœ‘(𝑧)βˆ’π‘“π‘™ξ€Έ||πœ‘(𝑧)+2sup|πœ‘(𝑧)|>𝜌||||πœ‡(𝑧)𝑔(𝑧)ξ€·||||1βˆ’πœ‘(𝑧)2ξ€Έ(𝑛+1+𝛼)/𝑝.(7.8)
LetπΌπ‘™βˆΆ=sup‖𝑓‖𝐴𝑝𝛼≀1sup|πœ‘(𝑧)|β‰€πœŒ||ξ€·π‘Ÿπ‘“(πœ‘(𝑧))βˆ’π‘“π‘™ξ€Έ||.πœ‘(𝑧)(7.9)If 𝛼>βˆ’1, then by using the mean value theorem, the subharmonicity of the partial derivatives of 𝑓 and Lemma 2.3, we have 𝐼𝑙≀sup‖𝑓‖𝐴𝑝𝛼≀1sup|πœ‘(𝑧)|β‰€πœŒξ€·1βˆ’π‘Ÿπ‘™ξ€Έ||||πœ‘(𝑧)sup|𝑀|β‰€πœŒ||||βˆ‡π‘“(𝑀)(7.10)β‰€πΆπœŒξ€·1βˆ’π‘Ÿπ‘™ξ€Έsup‖𝑓‖𝐴𝑝𝛼≀1ξ‚€ξ€œ|𝑀|≀(1+𝜌)/2||||βˆ‡π‘“(𝑀)𝑝1βˆ’|𝑀|2𝑝+𝛼𝑑𝑉(𝑀)1/π‘β‰€πΆπœŒξ€·1βˆ’π‘Ÿπ‘™ξ€Έsup‖𝑓‖𝐴𝑝𝛼≀1ξ‚€ξ€œπ”Ή||||𝑓(𝑀)𝑝1βˆ’|𝑀|2𝛼𝑑𝑉(𝑀)1/𝑝(1)β‰€πΆπœŒ(1βˆ’π‘Ÿπ‘™)⟢0asπ‘™βŸΆβˆž.(7.11)
If 𝛼=βˆ’1, then applying in (7.10) the known fact that for each compact πΎβŠ‚π”Ή,supπ‘€βˆˆπΎ||||βˆ‡π‘“(𝑀)≀𝐢‖𝑓‖𝑝,(7.12) for some 𝐢 independent of 𝑓 (see [11]), we obtain that (7.11) also holds in this case.
Letting π‘™β†’βˆž in (7.8), using (7.11), and then letting πœŒβ†’1, the second inequality in (7.2) follows, finishing the proof of the theorem.

Motivated by Theorem 7.1, we leave the following open problem.

Open Problem 1. Find the exact value of the essential norm of the operator π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›Όβ†’β„¬πœ‡.

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