Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2008 / Article

Research Article | Open Access

Volume 2008 |Article ID 154263 | https://doi.org/10.1155/2008/154263

Stevo Stević, "On a New Integral-Type Operator from the Weighted Bergman Space to the Bloch-Type Space on the Unit Ball", Discrete Dynamics in Nature and Society, vol. 2008, Article ID 154263, 14 pages, 2008. https://doi.org/10.1155/2008/154263

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

Academic Editor: Leonid Berezansky
Received01 May 2008
Accepted20 Jul 2008
Published14 Sep 2008

Abstract

We introduce an integral-type operator, denoted by š‘ƒš‘”šœ‘, on the space of holomorphic functions on the unit ball š”¹āŠ‚ā„‚š‘›, which is an extension of the product of composition and integral operators on the unit disk. The operator norm of š‘ƒš‘”šœ‘ from the weighted Bergman space š“š‘š›¼(š”¹) to the Bloch-type space ā„¬šœ‡(š”¹) or the little Bloch-type space ā„¬šœ‡,0(š”¹) is calculated. The compactness of the operator is characterized in terms of inducing functions š‘” and šœ‘. Upper and lower bounds for the essential norm of the operator š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼(š”¹)ā†’ā„¬šœ‡(š”¹), when š‘>1, are also given.

1. Introduction

Let š”¹ be the open unit ball in the complex vector space ā„‚š‘›,š‘†=šœ•š”¹ its boundary, š”» the open unit disk in the complex plane ā„‚,š‘‘š‘‰(š‘§) the Lebesgue measure on š”¹,š‘‘š‘‰š›¼(š‘§)=š‘š›¼(1āˆ’|š‘§|2)š›¼š‘‘š‘‰(š‘§), where š›¼>āˆ’1 and where the constant š‘š›¼ is chosen such that š‘‰š›¼(š”¹)=1,š‘‘šœŽ the normalized rotation invariant measure on š‘† (that is, šœŽ(š‘†)=1), š»(š”¹) the class of all holomorphic functions on the unit ball and š»āˆž=š»āˆž(š”¹) the space of all bounded holomorphic functions on š”¹ with the normā€–š‘“ā€–āˆž=supš‘§āˆˆš”¹||||.š‘“(š‘§)(1.1)

Let š‘§=(š‘§1,ā€¦,š‘§š‘›) and š‘¤=(š‘¤1,ā€¦,š‘¤š‘›) be points in ā„‚š‘›,āŸØš‘§,š‘¤āŸ©=š‘›ī“š‘˜=1š‘§š‘˜š‘¤š‘˜(1.2)and āˆš|š‘§|=āŸØš‘§,š‘§āŸ©.

For š‘“āˆˆš»(š”¹) with the Taylor expansion āˆ‘š‘“(š‘§)=|š›½|ā‰„0š‘Žš›½š‘§š›½, letī“ā„œš‘“(š‘§)=|š›½|ā‰„0|š›½|š‘Žš›½š‘§š›½(1.3)be the radial derivative of š‘“, where š›½=(š›½1,š›½2,ā€¦,š›½š‘›) is a multi-index, |š›½|=š›½1+ā‹Æ+š›½š‘› and š‘§š›½=š‘§š›½11ā‹Æš‘§š›½š‘›š‘›. It is well known (see, e.g., [1]) thatā„œš‘“(š‘§)=š‘›ī“š‘—=1š‘§š‘—šœ•š‘“šœ•š‘§š‘—(š‘§).(1.4)

For š‘>0 the Hardy space š»š‘=š»š‘(š”¹) consists of all š‘“āˆˆš»(š”¹) such thatā€–š‘“ā€–š‘š‘=sup0<š‘Ÿ<1ī€œš‘†||||š‘“(š‘Ÿšœ)š‘š‘‘šœŽ(šœ)<āˆž.(1.5)It is well known that for every š‘“āˆˆš»š‘ the radial limitš‘“āˆ—(šœ)āˆ¶=limš‘Ÿā†’1š‘“(š‘Ÿšœ)(1.6)exists almost everywhere on šœāˆˆš‘†.

The weighted Bergman space š“š‘š›¼=š“š‘š›¼(š”¹),š‘>0,š›¼>āˆ’1, consists of all š‘“āˆˆš»(š”¹) such thatā€–š‘“ā€–š‘š“š‘š›¼=ī€œš”¹||||š‘“(š‘§)š‘š‘‘š‘‰š›¼(š‘§)<āˆž.(1.7)When š‘ā‰„1, the weighted Bergman space with the norm ā€–ā‹…ā€–š“š‘š›¼ becomes a Banach space. If š‘āˆˆ(0,1), it is a Frechet space with the translation invariant metricš‘‘(š‘“,š‘”)=ā€–š‘“āˆ’š‘”ā€–š‘š“š‘š›¼.(1.8)

Since for every š‘“āˆˆš»š‘limš›¼ā†’āˆ’1+0ī€œš”¹||||š‘“(š‘§)š‘š‘‘š‘‰š›¼ī€œ(š‘§)=š‘†||š‘“āˆ—||(šœ)š‘š‘‘šœŽ(šœ),(1.9)we will also use the notation š“š‘āˆ’1 for the Hardy space š»š‘.

A positive continuous function šœ™ on [0,1) is called normal (see [2]) if there is š›æāˆˆ[0,1) and š‘Ž and š‘,0<š‘Ž<š‘ such thatšœ™(š‘Ÿ)(1āˆ’š‘Ÿ)š‘Žisdecreasingon[š›æ,1),limš‘Ÿā†’1šœ™(š‘Ÿ)(1āˆ’š‘Ÿ)š‘Ž=0;šœ™(š‘Ÿ)(1āˆ’š‘Ÿ)š‘isincreasingon[š›æ,1),limš‘Ÿā†’1šœ™(š‘Ÿ)(1āˆ’š‘Ÿ)š‘=āˆž.(1.10)From now on if we say that a function šœ‡āˆ¶š”¹ā†’[0,āˆž) is normal, we will also assume that it is radial, that is, šœ‡(š‘§)=šœ‡(|š‘§|),š‘§āˆˆš”¹.

The weighted space š»āˆžšœ‡=š»āˆžšœ‡(š”¹) consists of all š‘“āˆˆš»(š”¹) such thatā€–š‘“ā€–š»āˆžšœ‡=supš‘§āˆˆš”¹||||šœ‡(š‘§)š‘“(š‘§)<āˆž,(1.11)where šœ‡ is normal. For šœ‡(š‘§)=(1āˆ’|š‘§|2)š›½, š›½>0, we obtain the weighted space š»āˆžš›½=š»āˆžš›½(š”¹) (see, e.g., [3ā€“5]).

The little weighted space š»āˆžšœ‡,0=š»āˆžšœ‡,0(š”¹) is a subspace of š»āˆžšœ‡ consisting of all š‘“āˆˆš»(š”¹) such thatlim|š‘§|ā†’1||||šœ‡(š‘§)š‘“(š‘§)=0.(1.12)

The class of all š‘“āˆˆš»(š”¹) such thatšµšœ‡(š‘“)=supš‘§āˆˆš”¹||||šœ‡(š‘§)ā„œš‘“(š‘§)<āˆž,(1.13)where šœ‡ is normal, is called the Bloch-type space, and is denoted by ā„¬šœ‡=ā„¬šœ‡(š”¹). With the normā€–š‘“ā€–ā„¬šœ‡=||||š‘“(0)+šµšœ‡(š‘“),(1.14)the Bloch-type space becomes a Banach space.

The little Bloch-type space ā„¬šœ‡,0 is a subspace of ā„¬šœ‡ consisting of those š‘“āˆˆā„¬šœ‡ such thatlim|š‘§|ā†’1||||šœ‡(š‘§)ā„œš‘“(š‘§)=0.(1.15)The š›¼-Bloch space ā„¬š›¼ is obtained for šœ‡(š‘§)=(1āˆ’|š‘§|2)š›¼,š›¼āˆˆ(0,āˆž) (see, e.g., [6ā€“11]). For š›¼=1 the space ā„¬1=ā„¬ becomes the classical Bloch space.

Let šœ‘ be a holomorphic self-map of š”¹. For any š‘“āˆˆš»(š”¹), the composition operator is defined byš¶šœ‘ī€·ī€øš‘“(š‘§)=š‘“šœ‘(š‘§),š‘§āˆˆš”¹.(1.16)It is of interest to provide function theoretic characterizations when š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, [3ā€“5, 7, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23] and the references therein.

Let š‘”āˆˆš»(š”») and šœ‘ be a holomorphic self-map of š”». For š‘“āˆˆš»(š”»), products of integral-type and composition operator are defined as follows:š¶šœ‘š½š‘”ī€œš‘“(š‘§)=0šœ‘(š‘§)š‘“(šœ)š‘”ī…ž(šœ)š‘‘šœ,š½š‘”š¶šœ‘ī€œš‘“(š‘§)=š‘§0š‘“ī€·ī€øš‘”šœ‘(šœ)ī…ž(šœ)š‘‘šœ.(1.17)

When šœ‘(š‘§)=š‘§, 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 ā„‚š‘› can be found, for example, in [27ā€“41] (see also the references therein).

In [42], among other results, we proved the following theorem regarding the boundedness of the operator š½š‘”š¶šœ‘āˆ¶š“š‘š›¼(š”»)ā†’ā„¬šœ‡(š”»).

Theorem 1.1. Assume that š‘>0,š›¼>āˆ’1,š‘”āˆˆš»(š”»),šœ‡ is normal, and šœ‘ is a holomorphic self-map of š”». Then š½š‘”š¶šœ‘āˆ¶š“š‘š›¼(š”»)ā†’ā„¬šœ‡(š”») is bounded if and only ifsupš‘§āˆˆš”»||š‘”šœ‡(š‘§)ī…ž||(š‘§)ī€·||||1āˆ’šœ‘(š‘§)2ī€ø(š›¼+2)/š‘<āˆž.(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 š‘”āˆˆš»(š”¹),š‘”(0)=0, and šœ‘ is a holomorphic self-map of š”¹. We introduce the following important integral-type operator on the space of holomorphic functions on š”¹: š‘ƒš‘”šœ‘ī€œ(š‘“)(š‘§)=10š‘“ī€·ī€øšœ‘(š‘”š‘§)š‘”(š‘”š‘§)š‘‘š‘”š‘”,š‘“āˆˆš»(š”¹),š‘§āˆˆš”¹.(1.19)

First note that when š‘›=1, the operator is reduced to an operator of the form as the second operator in (1.17). Indeed, since š‘”āˆˆš»(š”») and š‘”(0)=0, it follows that š‘”(š‘§)=š‘§š‘”0(š‘§),š‘§āˆˆš”» for some š‘”0āˆˆš»(š”»). By using this fact and the change of variables šœ=š‘”š‘§, we obtainš‘ƒš‘”šœ‘ī€œš‘“(š‘§)=10š‘“ī€·ī€øšœ‘(š‘”š‘§)š‘”š‘§š‘”0(š‘”š‘§)š‘‘š‘”š‘”=ī€œš‘§0š‘“ī€·ī€øš‘”šœ‘(šœ)0(šœ)š‘‘š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 š‘ƒš‘”šœ‘.

Research Project 1. Let š‘‹ and š‘Œ be two Banach spaces of holomorphic functions on the unit ball in ā„‚š‘› (e.g., the weighted Bergman space š“š‘š›¼, the Bloch-type space ā„¬šœ‡, the Hardy space š»š‘ space, the weighted space š»āˆžšœ‡, the Besov space šµš‘, BMOA etc.) Characterize the boundedness, compactness, essential norms, and other operator theoretic properties of the operator š‘ƒš‘”šœ‘āˆ¶š‘‹ā†’š‘Œ in terms of function theoretic properties of inducing functions šœ‘ and š‘”.

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 š‘¢š¶šœ‘ mapping the Bloch space ā„¬ to the weighted space š»āˆžš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 š‘‹ and š‘Œ be two Banach spaces of holomorphic functions as in Research project 1. Calculate the operator norm of š‘ƒš‘”šœ‘āˆ¶š‘‹ā†’š‘Œ in terms of inducing functions šœ‘ and š‘”.

In this paper, among other results, we will calculate the operator norm of š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼(š”¹)ā†’ā„¬šœ‡(š”¹). 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, š¶ denotes a positive constant not necessarily the same at each occurrence. The notation š“ā‰šµ means that there is a positive constant š¶ such that š“/š¶ā‰¤šµā‰¤š¶š“.

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 š‘āˆˆ(0,āˆž) and š›¼ā‰„āˆ’1. Then for all š‘“āˆˆš“š‘š›¼(š”¹) and š‘§āˆˆš”¹, the following inequality holds:||||ā‰¤š‘“(š‘§)ā€–š‘“ā€–š“š‘š›¼ī€·1āˆ’|š‘§|2ī€ø(š‘›+1+š›¼)/š‘.(2.1)

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

Lemma 2.2. Suppose that 0<š‘<āˆž,š›¼ā‰„āˆ’1,š‘”āˆˆš»(š”¹),šœ‡ is normal, and šœ‘ is a holomorphic self-map of š”¹. Then the operator š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡ is compact if and only if š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡ is bounded and for every bounded sequence (š‘“š‘˜)š‘˜āˆˆā„• in š“š‘š›¼ converging to zero uniformly on compacts of š”¹, one has ā€–š‘ƒš‘”šœ‘š‘“š‘˜ā€–ā„¬šœ‡ā†’0 as š‘˜ā†’āˆž.

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

Lemma 2.3. Suppose that 0<š‘<āˆž,š›¼>āˆ’1, thenā€–š‘“ā€–š‘š“š‘š›¼ā‰||||š‘“(0)š‘+ī€œš”¹||||āˆ‡š‘“(š‘§)š‘ī€·1āˆ’|š‘§|2ī€øš‘+š›¼š‘‘š‘‰(š‘§),(2.2)for every š‘“āˆˆš“š‘š›¼ (here āˆ‡š‘“=((šœ•š‘“/šœ•š‘§1),ā€¦,(šœ•š‘“/šœ•š‘§š‘›))).

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

Lemma 2.4. Suppose that šœ‡ is normal. A closed set š¾ in ā„¬šœ‡,0 is compact if and only if it is bounded andlim|š‘§|ā†’1supš‘“āˆˆš¾||||šœ‡(š‘§)ā„œš‘“(š‘§)=0.(2.3)

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

Lemma 2.5. Assume that š‘“,š‘”āˆˆš»(š”¹) and š‘”(0)=0. Thenā„œš‘ƒš‘”šœ‘(š‘“)(š‘§)=š‘“(šœ‘(š‘§))š‘”(š‘§).(2.4)

Proof. Since the function š‘“(šœ‘(š‘§))š‘”(š‘§) is holomorphic and š‘”(0)=0, it has the Taylor expansion in the following form āˆ‘š›¼ā‰ 0š‘Žš›¼š‘§š›¼. Thenā„œī€ŗš‘ƒš‘”šœ‘ī€»ī€œ(š‘“)(š‘§)=ā„œ10ī“š›¼ā‰ 0š‘Žš›¼(š‘”š‘§)š›¼š‘‘š‘”š‘”ī‚µī“=ā„œš›¼ā‰ 0š‘Žš›¼š‘§|š›¼|š›¼ī‚¶=ī“š›¼ā‰ 0š‘Žš›¼š‘§š›¼,(2.5)as claimed.

3. The Norm of the Operator š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡

In this section, we calculate the norm ā€–š‘ƒš‘”šœ‘ā€–š“š‘š›¼ā†’ā„¬šœ‡.

Theorem 3.1. Assume that š‘>0,š›¼ā‰„āˆ’1,š‘”āˆˆš»(š”¹),šœ‡ is normal, šœ‘ is a holomorphic self-map of š”¹, and š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼(š”¹)ā†’ā„¬šœ‡(š”¹) is bounded. Thenā€–š‘ƒš‘”šœ‘ā€–š“š‘š›¼ā†’ā„¬šœ‡=ā€–š‘ƒš‘”šœ‘ā€–š“š‘š›¼ā†’ā„¬šœ‡,0=supš‘§āˆˆš”¹||||šœ‡(š‘§)š‘”(š‘§)ī€·||||1āˆ’šœ‘(š‘§)2ī€ø(š‘›+1+š›¼)/š‘=āˆ¶š‘€.(3.1)

Proof. If š‘“āˆˆš“š‘š›¼, then by Lemmas 2.5 and 2.1 we obtainā€–š‘ƒš‘”šœ‘š‘“ā€–ā„¬šœ‡=supš‘§āˆˆš”¹||ī€·ī€ø||šœ‡(š‘§)š‘”(š‘§)š‘“šœ‘(š‘§)ā‰¤ā€–š‘“ā€–š“š‘š›¼supš‘§āˆˆš”¹||||šœ‡(š‘§)š‘”(š‘§)||||(1āˆ’šœ‘(š‘§)2)(š‘›+1+š›¼)/š‘,(3.2)from which it follows thatā€–š‘ƒš‘”šœ‘ā€–š“š‘š›¼ā†’ā„¬šœ‡ā‰¤š‘€.(3.3)
Now we prove the reverse inequality. For š‘¤āˆˆš”¹ fixed, setš‘“š‘¤ī€·(š‘§)=1āˆ’|š‘¤|2ī€ø(š‘›+1+š›¼)/š‘ī€·ī€ø1āˆ’āŸØš‘§,š‘¤āŸ©2(š‘›+1+š›¼)/š‘,š‘§āˆˆš”¹.(3.4)We have that ā€–š‘“š‘¤ā€–š“š‘š›¼=1, for each š‘¤āˆˆš”¹. For š›¼>āˆ’1 this fact is well known. The proof for the case š›¼=āˆ’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 š‘§=š‘Ÿšœ, šœāˆˆš‘†, then we have ā€–š‘“š‘¤ā€–š‘š‘=sup0<š‘Ÿ<1ī€œš‘†(1āˆ’|š‘¤|2ī€øš‘›||||1āˆ’āŸØš‘§,š‘¤āŸ©2š‘›=ī€·š‘‘šœŽ(šœ)1āˆ’|š‘¤|2ī€øš‘›sup0<š‘Ÿ<1ī€œš‘†||ī€·ī€ø1āˆ’āŸØš‘§,š‘¤āŸ©āˆ’š‘›||2=ī€·š‘‘šœŽ(šœ)1āˆ’|š‘¤|2ī€øš‘›sup0<š‘Ÿ<1ī€œš‘†||||āˆžī“š‘˜=0Ī“(š‘›+š‘˜)š‘ŸĪ“(š‘˜+1)Ī“(š‘›)š‘˜āŸØšœ,š‘¤āŸ©š‘˜||||2=ī€·š‘‘šœŽ(šœ)1āˆ’|š‘¤|2ī€øš‘›sup0<š‘Ÿ<1ī€œš‘†āˆžī“š‘˜=0ī‚€Ī“(š‘›+š‘˜)ī‚Ī“(š‘˜+1)Ī“(š‘›)2š‘Ÿ2š‘˜||||āŸØšœ,š‘¤āŸ©2š‘˜=ī€·š‘‘šœŽ(šœ)1āˆ’|š‘¤|2ī€øš‘›supāˆž0<š‘Ÿ<1ī“š‘˜=0Ī“(š‘›+š‘˜)š‘ŸĪ“(š‘˜+1)Ī“(š‘›)2š‘˜|š‘¤|2š‘˜=ī€·1āˆ’|š‘¤|2ī€øš‘›sup0<š‘Ÿ<11ī€·1āˆ’š‘Ÿ2|š‘¤|2ī€øš‘›=1,(3.5)where we have used the following formula (see, e.g., [1])ī€œš‘†||||āŸØšœ,š‘¤āŸ©2š‘˜š‘‘šœŽ(šœ)=Ī“(š‘˜+1)Ī“(š‘›)Ī“(š‘›+š‘˜)|š‘¤|2š‘˜.(3.6)
From this and the boundedness of š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡, we haveā€–ā€–š‘ƒš‘”šœ‘ā€–ā€–š“š‘š›¼ā†’ā„¬šœ‡=ā€–ā€–š‘“šœ‘(š‘¤)ā€–ā€–š“š‘š›¼ā€–ā€–š‘ƒš‘”šœ‘ā€–ā€–š“š‘š›¼ā†’ā„¬šœ‡ā‰„ā€–ā€–š‘ƒš‘”šœ‘ī€·š‘“šœ‘(š‘¤)ī€øā€–ā€–ā„¬šœ‡=supš‘§āˆˆš”¹||||||š‘“šœ‡(š‘§)š‘”(š‘§)šœ‘(š‘¤)ī€·ī€ø||||||||š‘“šœ‘(š‘§)ā‰„šœ‡(š‘¤)š‘”(š‘¤)šœ‘(š‘¤)ī€·ī€ø||=||||šœ‘(š‘¤)šœ‡(š‘¤)š‘”(š‘¤)ī€·||||1āˆ’šœ‘(š‘¤)2ī€ø(š‘›+1+š›¼)/š‘.(3.7)Taking the supremum in (3.7) over š‘¤āˆˆš”¹, we obtainā€–ā€–š‘ƒš‘”šœ‘ā€–ā€–š“š‘š›¼ā†’ā„¬šœ‡ā‰„š‘€.(3.8)From (3.3) and (3.8), it follows that ā€–š‘ƒš‘”šœ‘ā€–š“š‘š›¼ā†’ā„¬šœ‡=š‘€.
Sinceā€–ā€–š‘ƒš‘”šœ‘ā€–ā€–š“š‘š›¼ā†’ā„¬šœ‡,0ā‰¤ā€–ā€–š‘ƒš‘”šœ‘ā€–ā€–š“š‘š›¼ā†’ā„¬šœ‡(3.9)and the proof of (3.8) does not depend on the space ā„¬šœ‡ (we may replace it by ā„¬šœ‡,0) the second equality in (3.1) also holds.

Corollary 3.2. Assume that š‘>0,š›¼ā‰„āˆ’1,š‘”āˆˆš»(š”¹),šœ‡ is normal, and šœ‘ is a holomorphic self-map of š”¹. Then š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡ is bounded if and only ifsupš‘§āˆˆš”¹||||šœ‡(š‘§)š‘”(š‘§)ī€·||||1āˆ’šœ‘(š‘§)2ī€ø(š‘›+1+š›¼)/š‘<āˆž.(3.10)

Proof. If š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡ is bounded, then (3.10) follows from Theorem 3.1. If (3.10) holds, then the boundedness of š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡ follows from (3.3).

4. The Boundedness of the Operator š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡,0

Here we characterize the boundedness of the operator š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡,0.

Theorem 4.1. Assume that š‘>0,š›¼ā‰„āˆ’1,š‘”āˆˆš»(š”¹),šœ‡ is normal, and šœ‘ is a holomorphic self-map of š”¹. Then š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡,0 is bounded if and only if š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡ is bounded and š‘”āˆˆš»āˆžšœ‡,0.

Proof. Assume that š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡ is bounded and š‘”āˆˆš»āˆžšœ‡,0. Then, for each polynomial š‘, we have||šœ‡(š‘§)ā„œš‘ƒš‘”šœ‘||||ī€·ī€ø||||||š‘(š‘§)=šœ‡(š‘§)š‘”(š‘§)š‘šœ‘(š‘§)ā‰¤šœ‡(š‘§)š‘”(š‘§)ā€–š‘ā€–āˆžāŸ¶0,as|š‘§|āŸ¶1(4.1)from which it follows that š‘ƒš‘”šœ‘(š‘)āˆˆā„¬šœ‡,0.
Since the set of all polynomials is dense in š“š‘š›¼, we have that for every š‘“āˆˆš“š‘š›¼ there is a sequence of polynomials (š‘š‘˜)š‘˜āˆˆā„• such thatlimš‘˜ā†’āˆžā€–ā€–š‘“āˆ’š‘š‘˜ā€–ā€–š“š‘š›¼=0.(4.2)From this and since the operator š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡ is bounded, it follows thatā€–ā€–š‘ƒš‘”šœ‘š‘“āˆ’š‘ƒš‘”šœ‘š‘š‘˜ā€–ā€–ā„¬šœ‡ā‰¤ā€–ā€–š‘ƒš‘”šœ‘ā€–ā€–š“š‘š›¼ā†’ā„¬šœ‡ā€–ā€–š‘“āˆ’š‘š‘˜ā€–ā€–š“š‘š›¼āŸ¶0,(4.3)as š‘˜ā†’āˆž. Hence š‘ƒš‘”šœ‘(š“š‘š›¼)āŠ‚ā„¬šœ‡,0. Since ā„¬šœ‡,0 is a closed subset of ā„¬šœ‡, the boundedness of š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡,0 follows.
Now assume that š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡,0 is bounded. Then clearly š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡ is bounded. Taking the test function š‘“(š‘§)=1āˆˆš“š‘š›¼, we obtain š‘”āˆˆš»āˆžšœ‡,0.

5. Compactness of the Operator š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡

This section is devoted to studying of the compactness of the operator š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡. We prove the following result.

Theorem 5.1. Assume that š‘>0,š›¼ā‰„āˆ’1,š‘”āˆˆš»(š”¹),šœ‡ is normal, šœ‘ is a holomorphic self-map of š”¹, and the operator š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡ is bounded. Then the operator š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡ is compact if and only iflim|šœ‘(š‘§)|ā†’1||||šœ‡(š‘§)š‘”(š‘§)ī€·||||1āˆ’šœ‘(š‘§)2ī€ø(š‘›+1+š›¼)/š‘=0.(5.1)

Proof. First assume that the operator š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡ is compact. If ā€–šœ‘ā€–āˆž<1, then condition (5.1) is vacuously satisfied. Hence, assume that ā€–šœ‘ā€–āˆž=1 and assume to the contrary that (5.1) does not hold. Then there is a sequence (š‘§š‘˜)š‘˜āˆˆā„• satisfying the condition |šœ‘(š‘§š‘˜)|ā†’1 as š‘˜ā†’āˆž and š›æ>0 such thatšœ‡(š‘§š‘˜)||š‘”(š‘§š‘˜)||ī€·||1āˆ’šœ‘(š‘§š‘˜)||2ī€ø(š‘›+1+š›¼)/š‘ā‰„š›æ,š‘˜āˆˆā„•.(5.2)For š‘¤āˆˆš”¹ fixed, setš¹š‘˜(š‘§)=š‘“šœ‘(š‘§š‘˜)(š‘§),š‘˜āˆˆā„•,(5.3)where š‘“š‘¤ is defined in (3.4). Recall that ā€–š‘“š‘¤ā€–š“š‘š›¼=1, for each š‘¤āˆˆš”¹. Then ā€–š¹š‘˜ā€–š“š‘š›¼=1,š‘˜āˆˆā„• and it is easy to see that š¹š‘˜ā†’0 uniformly on compacts of š”¹ as š‘˜ā†’āˆž. Hence, by Lemma 2.2, it follows thatlimš‘˜ā†’āˆžā€–ā€–š‘ƒš‘”šœ‘š¹š‘˜ā€–ā€–ā„¬šœ‡=0.(5.4)
On the other hand, by Lemma 2.5 and (5.2), we obtainā€–ā€–š‘ƒš‘”šœ‘š¹š‘˜ā€–ā€–ā„¬šœ‡=supš‘§āˆˆš”¹||||||š¹šœ‡(š‘§)š‘”(š‘§)š‘˜ī€·ī€ø||ā‰„šœ‘(š‘§)šœ‡(š‘§š‘˜)||š‘”ī€·š‘§š‘˜ī€ø||ī€·||šœ‘ī€·š‘§1āˆ’š‘˜ī€ø||2ī€ø(š‘›+1+š›¼)/š‘ā‰„š›æ>0,(5.5)for every š‘˜āˆˆā„•, which contradicts with (5.4).
Now assume that (5.1) holds. Then for every šœ€>0 there is an š‘Ÿāˆˆ(0,1) such that when š‘Ÿ<|šœ‘(š‘§)|<1,||||šœ‡(š‘§)š‘”(š‘§)ī€·||||1āˆ’šœ‘(š‘§)2ī€ø(š‘›+1+š›¼)/š‘<šœ€.(5.6)
On the other hand, since the operator š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡ is bounded, for š‘“ā‰”1āˆˆš“š‘š›¼, we obtain ā€–š‘”ā€–š»āˆžšœ‡<āˆž.
Assume that (ā„Žš‘˜)š‘˜āˆˆā„• is a bounded sequence in š“š‘š›¼ converging to zero uniformly on compacts of š”¹ as š‘˜ā†’āˆž. Let supš‘˜āˆˆā„•ā€–ā„Žš‘˜ā€–š“š‘š›¼=š‘€1. Then by Lemma 2.1 and (5.6), for š‘Ÿ<|šœ‘(š‘§)|<1, we obtain||||||ā„Žšœ‡(š‘§)š‘”(š‘§)š‘˜ī€·ī€ø||šœ‘(š‘§)ā‰¤supš‘˜āˆˆā„•ā€–ā„Žš‘˜ā€–š“š‘š›¼||||šœ‡(š‘§)š‘”(š‘§)ī€·||||1āˆ’šœ‘(š‘§)2ī€ø(š‘›+1+š›¼)/š‘<š‘€1šœ€.(5.7)If |šœ‘(š‘§)|ā‰¤š‘Ÿ, we have||||||ā„Žšœ‡(š‘§)š‘”(š‘§)š‘˜ī€·ī€ø||šœ‘(š‘§)ā‰¤ā€–š‘”ā€–š»āˆžšœ‡sup|š‘¤|ā‰¤š‘Ÿ||ā„Žš‘˜||(š‘¤)āŸ¶0,asš‘˜āŸ¶āˆž.(5.8)From (5.7) and (5.8), it follows that ā€–š‘ƒš‘”šœ‘ā„Žš‘˜ā€–ā„¬šœ‡ā†’0 as š‘˜ā†’āˆž, from which the compactness of the operator š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡ follows.

6. Compactness of the Operator š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡,0

This section characterizes the compactness of the operator š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡,0.

Theorem 6.1. Assume š‘>0,š›¼ā‰„āˆ’1,š‘”āˆˆš»(š”¹),šœ‡ is normal, šœ‘ is a holomorphic self-map of š”¹, and the operator š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡,0 is bounded. Then the operator š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡,0 is compact if and only iflim|š‘§|ā†’1||||šœ‡(š‘§)š‘”(š‘§)ī€·||||1āˆ’šœ‘(š‘§)2ī€ø(š‘›+1+š›¼)/š‘=0.(6.1)

Proof. Assume š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡,0 is compact. Then clearly š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡,0 is bounded and as in Theorem 4.1 we have that š‘”āˆˆš»āˆžšœ‡,0.
Hence if ā€–šœ‘ā€–āˆž<1, thenlim|š‘§|ā†’1||||šœ‡(š‘§)š‘”(š‘§)ī€·||||1āˆ’šœ‘(š‘§)2ī€ø(š‘›+1+š›¼)/š‘ā‰¤lim|š‘§|ā†’1||||šœ‡(š‘§)š‘”(š‘§)ī€·1āˆ’ā€–šœ‘ā€–2āˆžī€ø(š‘›+1+š›¼)/š‘=0,(6.2)from which the result follows in this case.
Now assume ā€–šœ‘ā€–āˆž=1. By using the test functions š¹š‘˜(š‘§)=š‘“šœ‘(š‘§š‘˜)(š‘§), š‘˜āˆˆā„•, defined in (5.3) we obtain that condition (5.1) holds, which implies that for every šœ€>0, there is an š‘Ÿāˆˆ(0,1) such that for š‘Ÿ<|šœ‘(š‘§)|<1, condition (5.6) holds.
Since š‘”āˆˆš»āˆžšœ‡,0, there is šœŽāˆˆ(0,1) such that for šœŽ<|š‘§|<1||||ī€·šœ‡(š‘§)š‘”(š‘§)<šœ€1āˆ’š‘Ÿ2ī€ø(š‘›+1+š›¼)/š‘.(6.3)
Hence, if |šœ‘(š‘§)|ā‰¤š‘Ÿ and šœŽ<|š‘§|<1, we have||||šœ‡(š‘§)š‘”(š‘§)ī€·||||1āˆ’šœ‘(š‘§)2ī€ø(š‘›+1+š›¼)/š‘ā‰¤||||šœ‡(š‘§)š‘”(š‘§)ī€·1āˆ’š‘Ÿ2ī€ø(š‘›+1+š›¼)/š‘<šœ€.(6.4)From (5.6) and (6.4), condition (6.1) follows.
Now assume that condition (6.1) holds. Then the quantity š‘€ in Theorem 3.1 is finite. Using this fact and the following inequality||šœ‡(š‘§)ā„œš‘ƒš‘”šœ‘||||ī€·ī€ø||š‘“(š‘§)ā‰¤šœ‡(š‘§)š‘”(š‘§)š‘“šœ‘(š‘§)ā‰¤ā€–š‘“ā€–š“š‘š›¼||||šœ‡(š‘§)š‘”(š‘§)ī€·||||1āˆ’šœ‘(š‘§)2ī€ø(š‘›+1+š›¼)/š‘,(6.5)it follows that the set š‘ƒš‘”šœ‘({š‘“āˆ¶ā€–š‘“ā€–š“š‘š›¼ā‰¤1}) is bounded in ā„¬šœ‡, moreover, in view of (6.1), it is bounded in ā„¬šœ‡,0. Taking the supremum in the last inequality over the unit ball in š“š‘š›¼, then letting |š‘§|ā†’1, using condition (6.1) and employing Lemma 2.4, we obtain the compactness of the operator š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡,0, as desired.

7. Essential Norm of š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡

Let š‘‹ and š‘Œ be Banach spaces, and let šæāˆ¶š‘‹ā†’š‘Œ be a bounded linear operator. The essential norm of the operator šæāˆ¶š‘‹ā†’š‘Œ, denoted by ā€–šæā€–š‘’,š‘‹ā†’š‘Œ, is defined as follows:ā€–šæā€–š‘’,š‘‹ā†’š‘Œī€½=infā€–šæ+š¾ā€–š‘‹ā†’š‘Œī€¾āˆ¶š¾iscompactfromš‘‹š‘”š‘œš‘Œ,(7.1)where ā€–ā‹…ā€–š‘‹ā†’š‘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 šæ is compact if and only if ā€–šæā€–š‘’,š‘‹ā†’š‘Œ=0.

In this section, we study the essential norm of the operator š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡ for the case š‘>1.

Theorem 7.1. Assume that š‘āˆˆ(1,āˆž),š›¼ā‰„āˆ’1,š‘”āˆˆš»(š”¹),š‘”(0)=0,šœ‘ is a holomorphic self-map of š”¹, and š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡ is bounded. Then the following inequalities hold:limsup|šœ‘(š‘§)|ā†’1||||šœ‡(š‘§)š‘”(š‘§)ī€·||||1āˆ’šœ‘(š‘§)2ī€ø(š‘›+1+š›¼)/š‘ā‰¤ā€–ā€–š‘ƒš‘”šœ‘ā€–ā€–š‘’,š“š‘š›¼ā†’ā„¬šœ‡ā‰¤2limsup|šœ‘(š‘§)|ā†’1||||šœ‡(š‘§)š‘”(š‘§)ī€·||||1āˆ’šœ‘(š‘§)2ī€ø(š‘›+1+š›¼)/š‘.(7.2)

Proof. Assume that (šœ‘(š‘§š‘˜))š‘˜āˆˆā„• is a sequence in š”¹ such that |šœ‘(š‘§š‘˜)|ā†’1 as š‘˜ā†’āˆž. Note that the sequence (š‘“šœ‘(š‘§š‘˜))š‘˜āˆˆā„• (where š‘“š‘¤ is defined in (3.4)) is such that ā€–š‘“šœ‘(š‘§š‘˜)ā€–š“š‘š›¼=1 for each š‘˜āˆˆā„• and it converges to zero uniformly on compacts of š”¹. From this and by [11, Theorems 2.12 and 4.50], it follows that š‘“šœ‘(š‘§š‘˜)ā†’0 weakly in š“š‘š›¼, as š‘˜ā†’āˆž (here we use the condition š‘>1). Hence, for every compact operator š¾āˆ¶š“š‘š›¼ā†’ā„¬šœ‡, we have that ā€–š¾š‘“šœ‘(š‘§š‘˜)ā€–ā„¬šœ‡ā†’0 as š‘˜ā†’āˆž. Thus, for every such sequence and for every compact operator š¾āˆ¶š“š‘š›¼ā†’ā„¬šœ‡, we have thatā€–ā€–š‘ƒš‘”šœ‘ā€–ā€–+š¾š“š‘š›¼ā†’ā„¬šœ‡ā‰„limsupš‘˜ā†’āˆžā€–ā€–š‘ƒš‘”šœ‘š‘“šœ‘(š‘§š‘˜)ā€–ā€–ā„¬šœ‡āˆ’ā€–ā€–š¾š‘“šœ‘(š‘§š‘˜)ā€–ā€–ā„¬šœ‡ā€–ā€–š‘“šœ‘(š‘§š‘˜)ā€–ā€–š“š‘š›¼=limsupš‘˜ā†’āˆžā€–ā€–š‘ƒš‘”šœ‘š‘“šœ‘(š‘§š‘˜)ā€–ā€–ā„¬šœ‡ā‰„limsupš‘˜ā†’āˆžšœ‡(š‘§š‘˜)||š‘”ī€·š‘§š‘˜ī€øš‘“šœ‘(š‘§š‘˜)ī€·šœ‘ī€·š‘§š‘˜||ī€øī€ø=limsupš‘›ā†’āˆžšœ‡ī€·š‘§š‘˜ī€ø||š‘”ī€·š‘§š‘˜ī€ø||ī€·||šœ‘ī€·š‘§1āˆ’š‘˜ī€ø||2ī€ø(š‘›+1+š›¼)/š‘.(7.3)
Taking the infimum in (7.3) over the set of all compact operators š¾āˆ¶š“š‘š›¼ā†’ā„¬šœ‡, we obtainā€–ā€–š‘ƒš‘”šœ‘ā€–ā€–š‘’,š“š‘š›¼ā†’ā„¬šœ‡ā‰„limsupš‘›ā†’āˆžšœ‡ī€·š‘§š‘˜ī€ø||š‘”ī€·š‘§š‘˜ī€ø||ī€·||šœ‘ī€·š‘§1āˆ’š‘˜ī€ø||2ī€ø(š‘›+1+š›¼)/š‘,(7.4)from which the first inequality follows.
In the sequel we prove the second inequality. Assume that (š‘Ÿš‘™)š‘™āˆˆā„• is a sequence which increasingly converges to 1. Consider the operators defined byī€·š‘ƒš‘”š‘Ÿš‘™šœ‘š‘“ī€øī€œ(š‘§)=10ī€·š‘Ÿš‘”(š‘”š‘§)š‘“š‘™ī€øšœ‘(š‘”š‘§)š‘‘š‘”š‘”,š‘™āˆˆā„•.(7.5)We prove that these operators are compact. Indeed, since |š‘Ÿš‘™šœ‘(š‘§)|ā‰¤š‘Ÿš‘™<1, it follows that condition (5.1) in Theorem 5.1 is vacuously satisfied, from which the claim follows.
Recall that š‘”āˆˆš»āˆžšœ‡. Let šœŒāˆˆ(0,1) be fixed for a moment. Employing Lemma 2.1, and using the factā€–ā€–š‘“āˆ’š‘“š‘Ÿš‘™ā€–ā€–š“š‘š›¼ā‰¤2ā€–š‘“ā€–š“š‘š›¼,š‘™āˆˆā„•,(7.6)which follows by using the triangle inequality for the norm, the monotonicity of the integral meansš‘€š‘š‘ī€œ(š‘“,š‘Ÿ)=š‘†||||š‘“(š‘Ÿšœ)š‘š‘‘šœŽ(šœ)(7.7)and the polar coordinates, we have ā€–ā€–š‘ƒš‘”šœ‘āˆ’š‘ƒš‘”š‘Ÿš‘™šœ‘ā€–ā€–š“š‘š›¼ā†’ā„¬šœ‡=supā€–š‘“ā€–š“š‘š›¼ā‰¤1supš‘§āˆˆš”¹||||||š‘“ī€·ī€øī€·š‘Ÿšœ‡(š‘§)š‘”(š‘§)šœ‘(š‘§)āˆ’š‘“š‘™ī€ø||šœ‘(š‘§)ā‰¤supā€–š‘“ā€–š“š‘š›¼ā‰¤1sup|šœ‘(š‘§)|ā‰¤šœŒ||||||š‘“ī€·ī€øī€·š‘Ÿšœ‡(š‘§)š‘”(š‘§)šœ‘(š‘§)āˆ’š‘“š‘™ī€ø||šœ‘(š‘§)+supā€–š‘“ā€–š“š‘š›¼ā‰¤1sup|šœ‘(š‘§)|>šœŒ||||||š‘“ī€·ī€øī€·š‘Ÿšœ‡(š‘§)š‘”(š‘§)šœ‘(š‘§)āˆ’š‘“š‘™ī€ø||šœ‘(š‘§)ā‰¤ā€–š‘”ā€–š»āˆžšœ‡supā€–š‘“ā€–š“š‘š›¼ā‰¤1sup|šœ‘(š‘§)|ā‰¤šœŒ||š‘“ī€·ī€øī€·š‘Ÿšœ‘(š‘§)āˆ’š‘“š‘™ī€ø||šœ‘(š‘§)+2sup|šœ‘(š‘§)|>šœŒ||||šœ‡(š‘§)š‘”(š‘§)ī€·||||1āˆ’šœ‘(š‘§)2ī€ø(š‘›+1+š›¼)/š‘.(7.8)
Letš¼š‘™āˆ¶=supā€–š‘“ā€–š“š‘š›¼ā‰¤1sup|šœ‘(š‘§)|ā‰¤šœŒ||ī€·š‘Ÿš‘“(šœ‘(š‘§))āˆ’š‘“š‘™ī€ø||.šœ‘(š‘§)(7.9)If š›¼>āˆ’1, then by using the mean value theorem, the subharmonicity of the partial derivatives of š‘“ and Lemma 2.3, we have š¼š‘™ā‰¤supā€–š‘“ā€–š“š‘š›¼ā‰¤1sup|šœ‘(š‘§)|ā‰¤šœŒī€·1āˆ’š‘Ÿš‘™ī€ø||||šœ‘(š‘§)sup|š‘¤|ā‰¤šœŒ||||āˆ‡š‘“(š‘¤)(7.10)ā‰¤š¶šœŒī€·1āˆ’š‘Ÿš‘™ī€øsupā€–š‘“ā€–š“š‘š›¼ā‰¤1ī‚€ī€œ|š‘¤|ā‰¤(1+šœŒ)/2||||āˆ‡š‘“(š‘¤)š‘ī€·1āˆ’|š‘¤|2ī€øš‘+š›¼ī‚š‘‘š‘‰(š‘¤)1/š‘ā‰¤š¶šœŒī€·1āˆ’š‘Ÿš‘™ī€øsupā€–š‘“ā€–š“š‘š›¼ā‰¤1ī‚€ī€œš”¹||||š‘“(š‘¤)š‘ī€·1āˆ’|š‘¤|2ī€øš›¼ī‚š‘‘š‘‰(š‘¤)1/š‘(1)ā‰¤š¶šœŒ(1āˆ’š‘Ÿš‘™)āŸ¶0asš‘™āŸ¶āˆž.(7.11)
If š›¼=āˆ’1, then applying in (7.10) the known fact that for each compact š¾āŠ‚š”¹,supš‘¤āˆˆš¾||||āˆ‡š‘“(š‘¤)ā‰¤š¶ā€–š‘“ā€–š‘,(7.12) for some š¶ independent of š‘“ (see [11]), we obtain that (7.11) also holds in this case.
Letting š‘™ā†’āˆž in (7.8), using (7.11), and then letting šœŒā†’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 š‘ƒš‘”šœ‘āˆ¶š“š‘š›¼ā†’ā„¬šœ‡.

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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.


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