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Journal of Function Spaces and Applications
Volumeย 2012, Article IDย 343194, 21 pages
http://dx.doi.org/10.1155/2012/343194
Research Article

The Multiplication Operator from ๐น(๐‘,๐‘ž,๐‘ ) Spaces to ๐‘›th Weighted-Type Spaces on the Unit Disk

1Department of Mathematics, Jiangsu Normal University, Xuzhou 221116, China
2School of Mathematics and Physics Science, Xuzhou Institute of Technology, Xuzhou 221111, China

Received 2 February 2012; Revised 11 April 2012; Accepted 30 April 2012

Academic Editor: Amolย Sasane

Copyright ยฉ 2012 Yongmin Liu and Yanyan Yu. 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

Let ๐ป(๐”ป) be the space of analytic functions on ๐”ป and ๐‘ขโˆˆ๐ป(๐”ป). The boundedness and compactness of the multiplication operator ๐‘€๐‘ข from ๐น(๐‘,๐‘ž,๐‘ ),(or๐น0(๐‘,๐‘ž,๐‘ )) spaces to ๐‘›th weighted-type spaces on the unit disk are investigated in this paper.

1. Introduction

Let ๐ป(๐”ป) denote the space of all analytic functions in the open unit disc ๐”ป of the finite-complex plane โ„‚, ๐œ•๐”ป the boundary of ๐”ป, โ„•0 the set of all nonnegative integers and โ„• the set of all positive integers. Let ๐œ‡(๐‘ง) be a positive continuous function on ๐”ป (weight) such that ๐œ‡(๐‘ง)=๐œ‡(|๐‘ง|) and ๐‘›โˆˆโ„•0. The ๐‘›th weighted-type spaces on the unit disk ๐”ป, denoted by ๐’ฒ๐œ‡(๐‘›)(๐”ป) which were introduced in [1], consist of all ๐‘“โˆˆ๐ป(๐”ป) such that๐‘๐’ฒ๐œ‡(๐‘›)(๐”ป)(๐‘“)=sup๐‘งโˆˆ๐”ป๐œ‡||๐‘“(๐‘ง)(๐‘›)||(๐‘ง)<โˆž.(1.1) For ๐‘›=0, the space becomes the weighted-type space ๐ปโˆž๐œ‡(๐”ป) (see, e.g., [2โ€“4]), for ๐‘›=1 the Bloch-type space โ„ฌ๐œ‡(๐”ป) and for ๐‘›=2 the Zygmund-type space ๐’ต๐œ‡(๐”ป). For ๐œ‡(๐‘ง)=1โˆ’|๐‘ง|2, we obtain correspondingly the classical weighted-type space, the Bloch space โ„ฌ(๐”ป)=โ„ฌ, and the Zygmund space ๐’ต(๐”ป)=๐’ต. Some information on Zygmund-type spaces on the unit disk and some operators on them, for example, in [5โ€“9] and on the unit ball, can be found, for example, in [10, 11]. From now on, we will assume that ๐‘›โˆˆโ„•. Setโ€–๐‘“โ€–๐’ฒ๐œ‡(๐‘›)(๐”ป)=๐‘›โˆ’1๎“๐‘—=0||๐‘“(๐‘—)||(0)+๐‘๐’ฒ๐œ‡(๐‘›)(๐”ป)(๐‘“).(1.2) With this norm, the ๐‘›th weighted-type space becomes a Banach space.

The little ๐‘›th weighted-type space, denoted by ๐’ฒ(๐‘›)๐œ‡,0(๐”ป), is a closed subspace of ๐’ฒ๐œ‡(๐‘›)(๐”ป) consisting of those ๐‘“ for whichlim|๐‘ง|โ†’1๐œ‡||๐‘“(๐‘ง)(๐‘›)||(๐‘ง)=0.(1.3)

A positive continuous function ๐œ™ on [0,1) is called a normal if there exist positive numbers ๐‘Ž,๐‘,0<๐‘Ž<๐‘, and ๐‘ก0โˆˆ[0,1), such that๐œ™(๐‘ก)๎€ท1โˆ’๐‘ก2๎€ธ๐‘Ždecreasesfor๐‘ก0โ‰ค๐‘ก<1andlim๐‘กโ†’1โˆ’๐œ™(๐‘ก)๎€ท1โˆ’๐‘ก2๎€ธ๐‘Ž=0,๐œ™(๐‘ก)๎€ท1โˆ’๐‘ก2๎€ธ๐‘increasesfor๐‘ก0โ‰ค๐‘ก<1andlim๐‘กโ†’1โˆ’๐œ™(๐‘ก)๎€ท1โˆ’๐‘ก2๎€ธ๐‘=โˆž(1.4) (see [12]).

For 0<๐‘,๐‘ <โˆž,โˆ’2<๐‘ž<โˆž, a function ๐‘“โˆˆ๐ป(๐”ป) is said to belong to the general function space ๐น(๐‘,๐‘ž,๐‘ )=๐น(๐‘,๐‘ž,๐‘ )(๐”ป) ifโ€–๐‘“โ€–๐‘๐น(๐‘,๐‘ž,๐‘ )=||||๐‘“(0)๐‘+sup๐‘งโˆˆ๐”ป๎€œ๐”ป||๐‘“๎…ž(||๐‘ง)๐‘๎€ท1โˆ’|๐‘ง|2๎€ธ๐‘ž๎‚€||๐œ‘1โˆ’๐‘Ž(||๐‘ง)2๎‚๐‘ ๐‘‘๐ด(๐‘ง)<โˆž,(1.5) where ๐œ‘๐‘Ž(๐‘ง)=(๐‘Žโˆ’๐‘ง)/(1โˆ’๐‘Ž๐‘ง),๐‘Žโˆˆ๐”ป. An ๐‘“โˆˆ๐ป(๐”ป) is said to belong to ๐น0(๐‘,๐‘ž,๐‘ )=๐น0(๐‘,๐‘ž,๐‘ )(๐”ป) iflim|๐‘Ž|โ†’1๎€œ๐”ป||๐‘“๎…ž(||๐‘ง)๐‘๎€ท1โˆ’|๐‘ง|2๎€ธ๐‘ž๎‚€||๐œ‘1โˆ’๐‘Ž(||๐‘ง)2๎‚๐‘ ๐‘‘๐ด(๐‘ง)=0.(1.6) The space ๐น(๐‘,๐‘ž,๐‘ ) was introduced by Zhao in [13]. We can get many function spaces if we take some specific parameters of ๐‘,๐‘ž,๐‘ ; for example (see [13]), ๐น(๐‘,๐‘ž,๐‘ )=โ„ฌ(2+๐‘ž)/๐‘ and ๐น0(๐‘,๐‘ž,๐‘ )=โ„ฌ0(2+๐‘ž)/๐‘ for ๐‘ >1; ๐น(๐‘,๐‘ž,๐‘ )โŠ‚โ„ฌ(2+๐‘ž)/๐‘ and ๐น0(๐‘,๐‘ž,๐‘ )โŠ‚โ„ฌ0(2+๐‘ž)/๐‘ for 0<๐‘ โ‰ค1; ๐น(2,0,๐‘ )=๐‘„๐‘  and ๐น0(2,0,๐‘ )=๐‘„๐‘ ,0; ๐น(2,0,1)=BMOA and ๐น0(2,0,1)=VMOA. Since for ๐‘ž+๐‘ โ‰คโˆ’1, ๐น(๐‘,๐‘ž,๐‘ ) is the space of constant functions, we assume that ๐‘ž+๐‘ >โˆ’1.

The multiplication operator ๐‘€๐‘ข is defined by ๐‘€๐‘ข๐‘“=๐‘ข๐‘“. It is interesting to provide a function theoretic characterization of when ๐‘ข induces a bounded or compact composition operator on various spaces (see, e.g., [14โ€“20]). Yu and Liu in [21] studied the boundedness and compactness of the operator ๐ท๐‘€๐‘ข from mixed-norm spaces to the Bloch-type space. Steviฤ‡ in [22] studied the boundedness and compactness of the product of the differentiation and composition operator from the space of bounded analytic functions, the Bloch space and the little Bloch space to ๐‘›th weighted-type spaces on the unit disk. Zhang and Xiao in [23] studied the bounded and compact-weighted composition operator ๐‘ข๐ถ๐œ‘ from the ๐น(๐‘,๐‘ž,๐‘ ) space to the Bloch-type space in the unit disc. Zhang and Zeng in [24] studied the boundedness and compactness of weighted differentiation composition operators from weighted bergman space to ๐‘›th weighted-type spaces on the unit disk. Ye in [25] studied the boundedness and compactness of the weighted composition operator ๐‘ข๐ถ๐œ‘ from ๐น(๐‘,๐‘ž,๐‘ ) into the logarithmic Bloch space โ„ฌlog on the unit disk. Yang in [26] studied the boundedness and compactness of weighted differentiation composition operators from the ๐น(๐‘,๐‘ž,๐‘ ) space to the Bloch-type space. Yang in [27] studied the boundedness and compactness of the composition operator from the ๐น(๐‘,๐‘ž,๐‘ ) space to ๐‘›th weighted-type spaces on the unit disk. Zhou and Chen in [28] studied the weighted composition operator from the ๐น(๐‘,๐‘ž,๐‘ ) space to the Bloch-type space in the unit ball. Zhu in [29] studied the weighted composition operator from the ๐น(๐‘,๐‘ž,๐‘ ) space to ๐นโˆž๐œ‡ space in the unit ball. Steviฤ‡ in [30, 31] studied the boundedness and compactness of the integral operators between ๐น(๐‘,๐‘ž,๐‘ ) spaces and Bloch-type spaces in the unit ball. This paper focuses on the boundedness and compactness of the operators ๐‘€๐‘ข from ๐น(๐‘,๐‘ž,๐‘ )(or๐น0(๐‘,๐‘ž,๐‘ )) to ๐‘›th weighted-type spaces on the unit disk.

From now on, we will always assume that 0<๐‘, ๐‘ <โˆž, โˆ’2<๐‘ž<โˆž, ๐‘ž+๐‘ >โˆ’1, ๐œ‡(๐‘ง)=๐œ‡(|๐‘ง|) is normal and ๐‘›โˆˆโ„•. Further, for the sake of simplicity, ๐ถ will always denote an independent constant, which can be different from one display to another.

2. Auxiliary Results

In this section we formulate some auxiliary results which will be used in the proofs of the main results.

Lemma 2.1 (see [13, 27]). Assume that ๐‘“โˆˆ๐น(๐‘,๐‘ž,๐‘ ). Then, for each ๐‘›โˆˆโ„•, there is a positive constant ๐ถ, independent of ๐‘“ such that โ€–๐‘“โ€–โ„ฌ(2+๐‘ž)/๐‘โ‰ค๐ถโ€–๐‘“โ€–๐น(๐‘,๐‘ž,๐‘ ) and ||๐‘“(๐‘›)||โ‰ค(๐‘ง)๐ถโ€–๐‘“โ€–๐น(๐‘,๐‘ž,๐‘ )๎€ท1โˆ’|๐‘ง|2๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›,๐‘งโˆˆ๐”ป.(2.1) Moreover, if ๐‘“โˆˆ๐น0(๐‘,๐‘ž,๐‘ ), then ๐‘“โˆˆโ„ฌ0(2+๐‘ž)/๐‘.

Lemma 2.2 (see [32, 33]). Let ๐›ผ>0 and ๐‘“โˆˆโ„ฌ๐›ผ. Then, ||||โ‰คโŽงโŽชโŽชโŽจโŽชโŽชโŽฉ๐‘“(๐‘ง)๐ถโ€–๐‘“โ€–โ„ฌ๐›ผ,if20<๐›ผ<1,๐ถlog1โˆ’|๐‘ง|2โ€–๐‘“โ€–โ„ฌ๐›ผ,if๐ถ๐›ผ=1,(๎€ท๐›ผโˆ’1)1โˆ’|๐‘ง|2๎€ธ๐›ผโˆ’1โ€–๐‘“โ€–โ„ฌ๐›ผ,if๐›ผ>1.(2.2)

Lemma 2.3 (see [1]). Assume ๐‘Ž>0 and ๐ท๐‘›||||||||||||||||(๐‘Ž)=11โ‹ฏ1๐‘Ž(๐‘Ž+1)โ‹ฏ(๐‘Ž+๐‘›โˆ’1)๐‘Ž(๐‘Ž+1)(๐‘Ž+1)(๐‘Ž+2)โ‹ฏ(๐‘Ž+๐‘›โˆ’1)(๐‘Ž+๐‘›)โ‹ฎโ‹ฎโ‹ฑโ‹ฎ๐‘›โˆ’2๎‘๐‘—=0(๐‘Ž+๐‘—)๐‘›โˆ’2๎‘๐‘—=0(๐‘Ž+๐‘—+1)โ‹ฏ๐‘›โˆ’2๎‘๐‘—=0||||||||||||||||.(๐‘Ž+๐‘—+๐‘›โˆ’1)(2.3) Then, ๐ท๐‘›โˆ(๐‘Ž)=๐‘›โˆ’1๐‘—=0๐‘—!.

By standard arguments (see, e.g., [34] or Lemmaโ€‰โ€‰3 in [35]) the following lemma follows.

Lemma 2.4. Assume that ๐‘ขโˆˆ๐ป(๐”ป). Then, ๐‘€๐‘ขโˆถ๐น(๐‘,๐‘ž,๐‘ )(๐‘œ๐‘Ÿ๐น0(๐‘,๐‘ž,๐‘ ))โ†’๐’ฒ๐œ‡(๐‘›)(๐”ป) is compact if and only if ๐‘€๐‘ขโˆถ๐น(๐‘,๐‘ž,๐‘ )(๐‘œ๐‘Ÿ๐น0(๐‘,๐‘ž,๐‘ ))โ†’๐’ฒ๐œ‡(๐‘›)(๐”ป) is bounded and for any bounded sequence {๐‘“๐‘˜} in ๐น(๐‘,๐‘ž,๐‘ )(๐‘œ๐‘Ÿ๐น0(๐‘,๐‘ž,๐‘ )) which converges to zero uniformly on compact subsets of ๐”ป as ๐‘˜โ†’โˆž, one has โ€–๐‘€๐‘ข๐‘“๐‘˜โ€–๐’ฒ๐œ‡(๐‘›)(๐”ป)โ†’0 as ๐‘˜โ†’โˆž.

3. The Boundedness and Compactness of ๐‘€๐‘ข from ๐น(๐‘,๐‘ž,๐‘ ) to ๐’ฒ๐œ‡(๐‘›)(๐”ป)

In this section, we characterize the boundedness and compactness of ๐‘€๐‘ขโˆถ๐น(๐‘,๐‘ž,๐‘ )โ†’๐’ฒ๐œ‡(๐‘›)(๐”ป).

Theorem 3.1. Assume that ๐‘ขโˆˆ๐ป(๐”ป) and ๐œ‡ is normal.(a)If 2+๐‘ž>๐‘, then ๐‘€๐‘ขโˆถ๐น(๐‘,๐‘ž,๐‘ )โ†’๐’ฒ๐œ‡(๐‘›)(๐”ป) is bounded if and only if sup๐‘งโˆˆ๐”ป||||๐œ‡(|๐‘ง|)๐‘ข(๐‘ง)๎€ท1โˆ’|๐‘ง|2๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›<โˆž.(3.1)(b)If 2+๐‘ž<๐‘, then ๐‘€๐‘ขโˆถ๐น(๐‘,๐‘ž,๐‘ )โ†’๐’ฒ๐œ‡(๐‘›)(๐”ป) is bounded if and only if (3.1) holds and ๐‘ขโˆˆ๐’ฒ๐œ‡(๐‘›)(๐”ป).(c)If 2+๐‘ž=๐‘,๐‘ >1, then ๐‘€๐‘ขโˆถ๐น(๐‘,๐‘ž,๐‘ )โ†’๐’ฒ๐œ‡(๐‘›)(๐”ป) is bounded if and only if (3.1) holds and sup๐‘งโˆˆ๐”ป||๐‘ข๐œ‡(|๐‘ง|)(๐‘›)||2(๐‘ง)log1โˆ’|๐‘ง|2<โˆž.(3.2)

Proof. Let 0<๐‘, ๐‘ <โˆž,โˆ’2<๐‘ž<โˆž, ๐‘ž+๐‘ >โˆ’1. Assume that conditions (3.1) holds. Then for all ๐‘งโˆˆ๐”ป||||๎€ท๐‘ข(๐‘ง)โ‰ค๐ถ1โˆ’|๐‘ง|2๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›.๐œ‡(|๐‘ง|)(3.3) Since 1๎€œ2๐œ‹02๐œ‹1||๐‘’๐‘–๐œƒ||โˆ’๐‘ง21๐‘‘๐œƒ=1โˆ’|๐‘ง|2,โˆ€๐‘งโˆˆ๐”ป,(3.4) let ๐›ฟ๐‘ง=(1+|๐‘ง|)/2, then we have |๐‘ง/๐›ฟ๐‘ง|=2|๐‘ง|/(1+|๐‘ง|)<1, so 1๎€œ2๐œ‹02๐œ‹1||๐›ฟ๐‘ง๐‘’๐‘–๐œƒ||โˆ’๐‘ง21๐‘‘๐œƒ=๐›ฟ2๐‘งโˆ’|๐‘ง|2.(3.5) By the Cauchy integral formula and (3.3), we obtain ||๐‘ข๎…ž||=1(๐‘ง)|||||๎€œ2๐œ‹02๐œ‹๐‘ข๎€ท๐›ฟ๐‘ง๐‘’๐‘–๐œƒ๎€ธ๎€ท๐›ฟ๐‘ง๐‘’๐‘–๐œƒ๎€ธโˆ’๐‘ง2๐›ฟ๐‘ง๐‘’๐‘–๐œƒ|||||๎€ท๐‘‘๐œƒโ‰ค๐ถ1โˆ’๐›ฟ2๐‘ง๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›๐œ‡๎€ท๐›ฟ๐‘ง๎€ธ1๎€œ2๐œ‹02๐œ‹๐›ฟ๐‘ง||๐›ฟ๐‘ง๐‘’๐‘–๐œƒ||โˆ’๐‘ง2๎€ท๐‘‘๐œƒ=๐ถ1โˆ’๐›ฟ2๐‘ง๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›๐œ‡๎€ท๐›ฟ๐‘ง๎€ธ๐›ฟ๐‘ง๐›ฟ2๐‘งโˆ’|๐‘ง|2๎€ทโ‰ค๐ถ1โˆ’๐›ฟ2๐‘ง๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›๐œ‡๎€ท๐›ฟ๐‘ง๎€ธ11โˆ’๐›ฟ๐‘ง.(3.6) Note that 12(1โˆ’|๐‘ง|)โ‰ค1โˆ’๐›ฟ2๐‘ง=๎€ท1+๐›ฟ๐‘ง๎€ธ๎€ท1โˆ’๐›ฟ๐‘ง๎€ธโ‰ค(1โˆ’|๐‘ง|),(3.7) and ๐œ‡ are normal, we have ||๐‘ข๎…ž||๎€ท(๐‘ง)โ‰ค๐ถ1โˆ’|๐‘ง|2๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›โˆ’1,๐œ‡(|๐‘ง|)(3.8) hence sup๐‘งโˆˆ๐”ป||||๐œ‡(|๐‘ง|)๐‘ขโ€ฒ(๐‘ง)๎€ท1โˆ’|๐‘ง|2๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›โˆ’1<โˆž.(3.9) Similarly, for ๐‘—โˆˆ{2,3,โ€ฆ,๐‘›}, we have that sup๐‘งโˆˆ๐”ป||๐‘ข๐œ‡(|๐‘ง|)(๐‘—)||(๐‘ง)๎€ท1โˆ’|๐‘ง|2๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›โˆ’๐‘—<โˆž.(3.10) Hence, if 2+๐‘ž>๐‘, by Lemmas 2.1 and 2.2, the Leibnitz formula, (3.1), (3.9), and (3.10), we have that |||๎€ท๐‘€๐œ‡(|๐‘ง|)๐‘ข๐‘“๎€ธ(๐‘›)|||||(๐‘ง)=๐œ‡(|๐‘ง|)(๐‘ข(๐‘ง)๐‘“(๐‘ง))(๐‘›)|||||||=๐œ‡(|๐‘ง|)๐‘›๎“๐‘—=0๐ถ๐‘—๐‘›๐‘ข(๐‘—)(๐‘ง)๐‘“(๐‘›โˆ’๐‘—)|||||||๐‘ข(๐‘ง)โ‰ค๐œ‡(|๐‘ง|)(๐‘›)||(๐‘ง)๐‘“(๐‘ง)+๐ถ๐‘›โˆ’1๎“๐‘—=0||๐ถ๐‘—๐‘›๐œ‡(|๐‘ง|)๐‘ข(๐‘—)||(๐‘ง)๎€ท1โˆ’|๐‘ง|2๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›โˆ’๐‘—โ€–๐‘“โ€–๐น(๐‘,๐‘ž,๐‘ )||๐‘ขโ‰ค๐ถ๐œ‡(|๐‘ง|)(๐‘›)||(๐‘ง)๎€ท1โˆ’|๐‘ง|2๎€ธ(2+๐‘žโˆ’๐‘)/๐‘โ€–๐‘“โ€–โ„ฌ(2+๐‘ž)/๐‘+๐ถ๐‘›โˆ’1๎“๐‘—=0||๐ถ๐‘—๐‘›๐œ‡(|๐‘ง|)๐‘ข(๐‘—)(||๐‘ง)๎€ท1โˆ’|๐‘ง|2๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›โˆ’๐‘—โ€–๐‘“โ€–๐น(๐‘,๐‘ž,๐‘ )โ‰ค๐ถโ€–๐‘“โ€–๐น(๐‘,๐‘ž,๐‘ ),(3.11) for every ๐‘งโˆˆ๐”ป and ๐‘“โˆˆ๐น(๐‘,๐‘ž,๐‘ ), if 2+๐‘ž<๐‘, using ๐‘ขโˆˆ๐’ฒ๐œ‡(๐‘›)(๐”ป), we have that |||๎€ท๐‘€๐œ‡(|๐‘ง|)๐‘ข๐‘“๎€ธ(๐‘›)|||||๐‘ข(๐‘ง)โ‰ค๐ถ๐œ‡(|๐‘ง|)(๐‘›)||(๐‘ง)โ€–๐‘“โ€–โ„ฌ(2+๐‘ž)/๐‘+๐ถ๐‘›โˆ’1๎“๐‘—=0||๐ถ๐‘—๐‘›๐œ‡(|๐‘ง|)๐‘ข(๐‘—)||(๐‘ง)๎€ท1โˆ’|๐‘ง|2๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›โˆ’๐‘—โ€–๐‘“โ€–๐น(๐‘,๐‘ž,๐‘ )โ‰ค๐ถโ€–๐‘ขโ€–๐’ฒ๐œ‡(๐‘›)(๐”ป)โ€–๐‘“โ€–โ„ฌ(2+๐‘ž)/๐‘+๐ถ๐‘›โˆ’1๎“๐‘—=0||๐ถ๐‘—๐‘›๐œ‡(|๐‘ง|)๐‘ข(๐‘—)(||๐‘ง)๎€ท1โˆ’|๐‘ง|2๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›โˆ’๐‘—โ€–๐‘“โ€–๐น(๐‘,๐‘ž,๐‘ )โ‰ค๐ถโ€–๐‘“โ€–๐น(๐‘,๐‘ž,๐‘ ),(3.12) for every ๐‘งโˆˆ๐”ป and ๐‘“โˆˆ๐น(๐‘,๐‘ž,๐‘ ), if 2+๐‘ž=๐‘, by (3.2), we have that |||๎€ท๐‘€๐œ‡(|๐‘ง|)๐‘ข๐‘“๎€ธ(๐‘›)|||||๐‘ข(๐‘ง)โ‰ค๐ถ๐œ‡(|๐‘ง|)(๐‘›)||2(๐‘ง)log1โˆ’|๐‘ง|2โ€–๐‘“โ€–โ„ฌ+๐ถ๐‘›โˆ’1๎“๐‘—=0||๐ถ๐‘—๐‘›๐œ‡(|๐‘ง|)๐‘ข(๐‘—)||(๐‘ง)๎€ท1โˆ’|๐‘ง|2๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›โˆ’๐‘—โ€–๐‘“โ€–๐น(๐‘,๐‘ž,๐‘ )โ‰ค๐ถโ€–๐‘“โ€–โ„ฌ+๐ถ๐‘›โˆ’1๎“๐‘—=0||๐ถ๐‘—๐‘›๐œ‡(|๐‘ง|)๐‘ข(๐‘—)(||๐‘ง)๎€ท1โˆ’|๐‘ง|2๎€ธ๐‘›โˆ’๐‘—โ€–๐‘“โ€–๐น(๐‘,๐‘ž,๐‘ )โ‰ค๐ถโ€–๐‘“โ€–๐น(๐‘,๐‘ž,๐‘ ),(3.13) for every ๐‘งโˆˆ๐”ป and ๐‘“โˆˆ๐น(๐‘,๐‘ž,๐‘ ). We also have that ||๎€ท๐‘€๐‘ข๐‘“๎€ธ||=||๐‘ข||(0)(0)๐‘“(0)โ‰ค๐ถโ€–๐‘“โ€–๐น(๐‘,๐‘ž,๐‘ ),(3.14) and for each ๐‘ โˆˆ{1,2,โ€ฆ,๐‘›โˆ’1}, |||๎€ท๐‘€๐‘ข๐‘“๎€ธ(๐‘ )|||=|||||(0)๐‘ ๎“๐‘—=0๐ถ๐‘—๐‘ ๐‘ข(๐‘—)(0)๐‘“(๐‘ โˆ’๐‘—)|||||(0)โ‰ค๐ถ๐‘ ๎“๐‘—=0||๐ถ๐‘—๐‘ ๐‘ข(๐‘—)||(0)โ€–๐‘“โ€–๐น(๐‘,๐‘ž,๐‘ ).(3.15) Using (3.11), (3.12), (3.13), (3.14), and (3.15) it follows that the operator ๐‘€๐‘ขโˆถ๐น(๐‘,๐‘ž,๐‘ )โ†’๐’ฒ๐œ‡(๐‘›)(๐”ป) is bounded.
On the other hand, suppose that ๐‘€๐‘ขโˆถ๐น(๐‘,๐‘ž,๐‘ )โ†’๐’ฒ๐œ‡(๐‘›)(๐”ป) is bounded, that is, there exists a constant ๐ถ such thatโ€–โ€–๐‘€๐‘ข๐‘“โ€–โ€–๐’ฒ๐œ‡(๐‘›)(๐”ป)โ‰ค๐ถโ€–๐‘“โ€–๐น(๐‘,๐‘ž,๐‘ )(3.16) for all ๐‘“โˆˆ๐น(๐‘,๐‘ž,๐‘ ). Then, we can easily obtain ๐‘ขโˆˆ๐’ฒ๐œ‡(๐‘›)(๐”ป) by taking ๐‘“(๐‘ง)=1. For a fixed ๐‘คโˆˆ๐”ป, and constants ๐‘1,๐‘2,โ€ฆ,๐‘๐‘›, set ๐‘“๐‘ค(๐‘ง)=๐‘›๎“๐‘—=1๐‘๐‘—๎€ท1โˆ’|๐‘ค|2๎€ธ๐‘—+1๎€ท1โˆ’๎€ธ๐‘ค๐‘ง๐›ผ+๐‘—,(3.17) where ๐›ผ=(2+๐‘ž)/๐‘. A straightforward calculation shows that, for ๐‘™โˆˆ{1,โ€ฆ,๐‘›}, ๐‘“๐‘ค(๐‘™)(๐‘ง)=๐‘›๎“๐‘—=1๐‘๐‘—๐‘™โˆ’1๎‘๐‘˜=0๎€ท(๐›ผ+๐‘—+๐‘˜)๐‘ค๎€ธ๐‘™๎€ท1โˆ’|๐‘ค|2๎€ธ๐‘—+1๎€ท1โˆ’๎€ธ๐‘ค๐‘ง๐›ผ+๐‘—+๐‘™,(3.18) so ๐‘“๐‘ค(๐‘™)๎€ท(๐‘ค)=๐‘ค๎€ธ๐‘™๎€ท1โˆ’|๐‘ค|2๎€ธ๐‘›๐›ผโˆ’1+๐‘™๎“๐‘—=1๐‘๐‘—๐‘™โˆ’1๎‘๐‘˜=0(๐›ผ+๐‘—+๐‘˜).(3.19) It is easy to see that ๐‘“๐‘คโˆˆ๐น(๐‘,๐‘ž,๐‘ ) for each ๐‘คโˆˆ๐”ป and โ€–๐‘“๐‘คโ€–๐น(๐‘,๐‘ž,๐‘ )โ‰ค๐ถ by using the same methods in [23]. By Lemma 2.3, using the same method in [22, 36], we can choose ๐‘1,๐‘2,โ€ฆ,๐‘๐‘›, the corresponding function is denoted by ๐‘“๐‘ค, such that ๐‘“๐‘ค(๐‘›)(๐‘ค)=๐‘›๎“๐‘—=1๐‘๐‘—๐‘™โˆ’1๎‘๐‘˜=0๎€ท(๐›ผ+๐‘—+๐‘˜)๐‘ค๎€ธ๐‘›๎€ท1โˆ’|๐‘ค|2๎€ธ๐›ผโˆ’1+๐‘›,๐‘“๐‘ค(๐‘™)(๐‘ค)=0,๐‘™โˆˆ{0,1,โ€ฆ,๐‘›โˆ’1}.(3.20) Therefore, โ€–โ€–๐‘€๐ถโ‰ฅ๐‘ข๐‘“๐‘คโ€–โ€–๐’ฒ๐œ‡(๐‘›)(๐”ป)||โ‰ฅ๐œ‡(|๐‘ค|)๐‘ข(๐‘ค)๐‘“๐‘ค(๐‘›)||=(๐‘ค)๐‘›๎“๐‘—=1๐‘๐‘—๐‘™โˆ’1๎‘๐‘˜=0(๐›ผ+๐‘—+๐‘˜)๐œ‡(|๐‘ค|)|๐‘ค|๐‘›||||๐‘ข(๐‘ค)๎€ท1โˆ’|๐‘ค|2๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›.(3.21) From this, we obtain sup|๐‘ค|>1/2||||๐œ‡(|๐‘ค|)๐‘ข(๐‘ค)๎€ท1โˆ’|๐‘ค|2๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›โ‰ค๐ถ.(3.22) Since ๐œ‡ is normal and ๐‘ขโˆˆ๐ป(๐”ป), we get sup|๐‘ค|โ‰ค1/2||||๐œ‡(|๐‘ค|)๐‘ข(๐‘ค)๎€ท1โˆ’|๐‘ค|2๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›โ‰ค๐ถsup|๐‘ค|โ‰ค1/2||||๐œ‡(|๐‘ค|)๐‘ข(๐‘ค)โ‰ค๐ถ,(3.23) which along with (3.22) implies that (3.1) is necessary for all case. Let 2+๐‘ž=๐‘,๐‘ >1, and ๐‘€๐‘ขโˆถ๐น(๐‘,๐‘ž,๐‘ )โ†’๐’ฒ๐œ‡(๐‘›)(๐”ป) be bounded. To prove (3.2), we set ๐‘”๐‘ค2(๐‘ง)=๐ดlog1โˆ’๎€ท๐‘ค๐‘ง+๐ตlog(2/(1โˆ’๎€ธ๐‘ค๐‘ง))2๎€ท๎€ทlog2/1โˆ’|๐‘ค|2๎€ธ๎€ธ,๐‘ง,๐‘คโˆˆ๐”ป.(3.24) By a direct calculation, we get ||๐‘”๎…ž๐‘ค||โ‰ค||๐ด||(๐‘ง)||1โˆ’||+2||๎€ท๎€ท๐‘ค๐‘ง๐ตlog2/1โˆ’||๐‘ค๐‘ง๎€ธ๎€ธ||๎€ท๎€ทlog2/1โˆ’|๐‘ค|2||||๎€ธ๎€ธ1โˆ’||โ‰ค๐ถ๐‘ค๐‘ง||1โˆ’||,๐‘ค๐‘ง(3.25) thus we have ๐‘”๐‘คโˆˆ๐น(๐‘,๐‘ž,๐‘ ) and sup๐‘คโˆˆ๐”ปโ€–๐‘”๐‘คโ€–๐น(๐‘,๐‘ž,๐‘ )โ‰ค๐ถ<โˆž (see [19, 30]). On the other hand, we have that ๐‘”๎…ž๐‘ค(๐‘ง)=๐ด๐‘ค1โˆ’๐‘ค๐‘ง+2๐ต๎€ท๎€ท๐‘คlog2/1โˆ’๐‘ค๐‘ง๎€ธ๎€ธ๎€ท๎€ท๎€ทlog2/1โˆ’|๐‘ค|2๎€ธ๎€ธ๎€ธ๎€ท1โˆ’๎€ธ,๐‘”๐‘ค๐‘ง๐‘ค๎…ž๎…ž๎€ท(๐‘ง)=๐ด๐‘ค๎€ธ2๎€ท1โˆ’๎€ธ๐‘ค๐‘ง2๎€ท+2๐ต๐‘ค๎€ธ2๎€ท๎€ท๎€ทlog2/1โˆ’|๐‘ค|2๎€ธ๎€ธ๎€ธ๎€ท1โˆ’๎€ธ๐‘ค๐‘ง2๎€ท+2๐ต๐‘ค๎€ธ2๎€ท๎€ทlog2/1โˆ’๐‘ค๐‘ง๎€ธ๎€ธ๎€ท๎€ท๎€ทlog2/1โˆ’|๐‘ค|2๎€ธ๎€ธ๎€ธ๎€ท1โˆ’๎€ธ๐‘ค๐‘ง2,๐‘”๐‘ค๎…ž๎…ž๎…ž๎€ท(๐‘ง)=2๐ด๐‘ค๎€ธ3๎€ท1โˆ’๎€ธ๐‘ค๐‘ง3๎€ท+6๐ต๐‘ค๎€ธ3๎€ท๎€ท๎€ทlog2/1โˆ’|๐‘ค|2๎€ธ๎€ธ๎€ธ๎€ท1โˆ’๎€ธ๐‘ค๐‘ง3๎€ท+2โ‹…2๐ต๐‘ค๎€ธ3๎€ท๎€ทlog2/1โˆ’๐‘ค๐‘ง๎€ธ๎€ธ๎€ท๎€ท๎€ทlog2/1โˆ’|๐‘ค|2๎€ธ๎€ธ๎€ธ๎€ท1โˆ’๎€ธ๐‘ค๐‘ง3,๐‘”๐‘ค(4)๎€ท(๐‘ง)=3โ‹…2๐ด๐‘ค๎€ธ4๎€ท1โˆ’๎€ธ๐‘ค๐‘ง4๎€ท+22๐ต๐‘ค๎€ธ4๎€ท๎€ท๎€ทlog2/1โˆ’|๐‘ค|2๎€ธ๎€ธ๎€ธ๎€ท1โˆ’๎€ธ๐‘ค๐‘ง4๎€ท+3!2๐ต๐‘ค๎€ธ4๎€ท๎€ทlog2/1โˆ’๐‘ค๐‘ง๎€ธ๎€ธ๎€ท๎€ท๎€ทlog2/1โˆ’|๐‘ค|2๎€ธ๎€ธ๎€ธ๎€ท1โˆ’๎€ธ๐‘ค๐‘ง4,๐‘”๐‘ค(5)๎€ท(๐‘ง)=4!๐ด๐‘ค๎€ธ5๎€ท1โˆ’๎€ธ๐‘ค๐‘ง5๎€ท+100๐ต๐‘ค๎€ธ5๎€ท๎€ท๎€ทlog2/1โˆ’|๐‘ค|2๎€ธ๎€ธ๎€ธ๎€ท1โˆ’๎€ธ๐‘ค๐‘ง5๎€ท+4!2๐ต๐‘ค๎€ธ5๎€ท๎€ทlog2/1โˆ’๐‘ค๐‘ง๎€ธ๎€ธ๎€ท๎€ท๎€ทlog2/1โˆ’|๐‘ค|2๎€ธ๎€ธ๎€ธ๎€ท1โˆ’๎€ธ๐‘ค๐‘ง5,โ‹ฎ๐‘”๐‘ค(๐‘š)(๎€ท๐‘ง)=(๐‘šโˆ’1)!๐ด๐‘ค๎€ธ๐‘š๎€ท1โˆ’๎€ธ๐‘ค๐‘ง๐‘š+๐ด๐‘š๐ต๎€ท๐‘ค๎€ธ๐‘š๎€ท๎€ท๎€ทlog2/1โˆ’|๐‘ค|2๎€ธ๎€ธ๎€ธ๎€ท1โˆ’๎€ธ๐‘ค๐‘ง๐‘š๎€ท+2(๐‘šโˆ’1)!๐ต๐‘ค๎€ธ๐‘š๎€ท๎€ทlog2/1โˆ’๐‘ค๐‘ง๎€ธ๎€ธ๎€ท๎€ท๎€ทlog2/1โˆ’|๐‘ค|2๎€ธ๎€ธ๎€ธ๎€ท1โˆ’๎€ธ๐‘ค๐‘ง๐‘š.(3.26) Moreover, we have that ๐‘”๐‘ค2(๐‘ค)=(๐ด+๐ต)log1โˆ’|๐‘ค|2,๐‘”๎…ž๐‘ค(๐‘ค)=(๐ด+2๐ต)๐‘ค1โˆ’|๐‘ค|2,๐‘”๐‘ค๎…ž๎…ž๎€ท(๐‘ค)=(๐ด+2๐ต)๐‘ค๎€ธ2๎€ท1โˆ’|๐‘ค|2๎€ธ2๎€ท+2๐ต๐‘ค๎€ธ2๎€ท๎€ท๎€ทlog2/1โˆ’|๐‘ค|2๎€ธ๎€ธ๎€ธ๎€ท1โˆ’|๐‘ค|2๎€ธ2,๐‘”๐‘ค๎…ž๎…ž๎…ž๎€ท(๐‘ค)=2(๐ด+2๐ต)๐‘ค๎€ธ3๎€ท1โˆ’|๐‘ค|2๎€ธ3๎€ท+6๐ต๐‘ค๎€ธ3๎€ท๎€ท๎€ทlog2/1โˆ’|๐‘ค|2๎€ธ๎€ธ๎€ธ๎€ท1โˆ’|๐‘ค|2๎€ธ3,๐‘”๐‘ค(4)(๎€ท๐‘ค)=6(๐ด+2๐ต)๐‘ค๎€ธ4๎€ท1โˆ’|๐‘ค|2๎€ธ4๎€ท+22๐ต๐‘ค๎€ธ4๎€ท๎€ท๎€ทlog2/1โˆ’|๐‘ค|2๎€ธ๎€ธ๎€ธ๎€ท1โˆ’|๐‘ค|2๎€ธ4,๐‘”๐‘ค(5)๎€ท(๐‘ค)=24(๐ด+2๐ต)๐‘ค๎€ธ5๎€ท1โˆ’|๐‘ค|2๎€ธ5๎€ท+100๐ต๐‘ค๎€ธ5๎€ท๎€ท๎€ทlog2/1โˆ’|๐‘ค|2๎€ธ๎€ธ๎€ธ๎€ท1โˆ’|๐‘ค|2๎€ธ5,โ‹ฎ๐‘”๐‘ค(๐‘š)๎€ท(๐‘ค)=(๐‘šโˆ’1)!(๐ด+2๐ต)๐‘ค๎€ธ๐‘š๎€ท1โˆ’|๐‘ค|2๎€ธ๐‘š+๐ด๐‘š๐ต๎€ท๐‘ค๎€ธ๐‘š๎€ท๎€ท๎€ทlog2/1โˆ’|๐‘ค|2๎€ธ๎€ธ๎€ธ๎€ท1โˆ’|๐‘ค|2๎€ธ๐‘š.(3.27) Taking ๐ด=2, ๐ต=โˆ’1, if ๐‘›=1, we have ๐œ‡||||๐‘ข(|๐‘ค|)๎…ž2(๐‘ค)log1โˆ’|๐‘ค|2||||โ‰คโ€–โ€–๐‘€๐‘ข๐‘”๐‘คโ€–โ€–๐’ฒ๐œ‡(๐‘›)(๐”ป)โ€–โ€–๐‘€โ‰ค๐ถ๐‘ขโ€–โ€–<โˆž,(3.28) if ๐‘›โ‰ 1, we have โ€–โ€–๐‘€๐‘ข๐‘”๐‘คโ€–โ€–๐’ฒ๐œ‡(๐‘›)(๐”ป)|||||โ‰ฅ๐œ‡(|๐‘ค|)๐‘›๎“๐‘—=0๐ถ๐‘—๐‘›๐‘ข(๐‘—)(๐‘ค)๐‘”๐‘ค(๐‘›โˆ’๐‘—)|||||||๐‘ข(๐‘ค)โ‰ฅ๐œ‡(|๐‘ค|)(๐‘›)(๐‘ค)๐‘”๐‘ค|||||||(๐‘ค)โˆ’๐œ‡(|๐‘ค|)๐‘›โˆ’1๎“๐‘—=0๐ถ๐‘—๐‘›๐‘ข(๐‘—)(๐‘ค)๐‘”๐‘ค(๐‘›โˆ’๐‘—)|||||||๐‘ข(๐‘ค)โ‰ฅ๐œ‡(|๐‘ค|)(๐‘›)(๐‘ค)๐‘”๐‘ค||ร—|||||(๐‘ค)โˆ’๐œ‡(|๐‘ค|)๐‘›โˆ’1๎“๐‘—=0๐ถ๐‘—๐‘›๐‘ข(๐‘—)(๐‘ค)๐ด๐‘›โˆ’๐‘—๐ต๎€ท๐‘ค๎€ธ๐‘›โˆ’๐‘—๎€ท๎€ท๎€ทlog2/1โˆ’|๐‘ค|2๎€ธ๎€ธ๎€ธ๎€ท1โˆ’|๐‘ค|2๎€ธ๐‘›โˆ’๐‘—|||||.(3.29) From (3.29), (3.1), (3.9), and (3.10) we have ๐œ‡||||๐‘ข(|๐‘ค|)(๐‘›)2(๐‘ค)log1โˆ’|๐‘ค|2||||โ‰คโ€–โ€–๐‘€๐‘ข๐‘”๐‘คโ€–โ€–๐’ฒ๐œ‡(๐‘›)(๐”ป)|||||+๐ถ๐‘›โˆ’1๎“๐‘—=0๐œ‡(|๐‘ค|)๐‘ข(๐‘—)(๐‘ค)๎€ท1โˆ’|๐‘ค|2๎€ธ๐‘›โˆ’๐‘—|||||โ€–โ€–๐‘€โ‰ค๐ถ๐‘ขโ€–โ€–+๐ถ<โˆž.(3.30) Using (3.28) and (3.30), it is easy to get that (3.2) holds, finishing the proof of the theorem.

Theorem 3.2. Assume that ๐‘ขโˆˆ๐ป(๐”ป) and ๐œ‡ is normal.(a)If 2+๐‘ž>๐‘, then ๐‘€๐‘ขโˆถ๐น(๐‘,๐‘ž,๐‘ )โ†’๐’ฒ๐œ‡(๐‘›)(๐”ป) is compact if and only if lim|๐‘ง|โ†’1||||๐œ‡(|๐‘ง|)๐‘ข(๐‘ง)๎€ท1โˆ’|๐‘ง|2๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›=0.(3.31)(b)If 2+๐‘ž<๐‘, then ๐‘€๐‘ขโˆถ๐น(๐‘,๐‘ž,๐‘ )โ†’๐’ฒ๐œ‡(๐‘›)(๐”ป) is compact if and only if (3.31) holds and ๐‘ขโˆˆ๐’ฒ๐œ‡(๐‘›)(๐”ป).(c)If 2+๐‘ž=๐‘,๐‘ >1, then ๐‘€๐‘ขโˆถ๐น(๐‘,๐‘ž,๐‘ )โ†’๐’ฒ๐œ‡(๐‘›)(๐”ป) is compact if and only if (3.31) holds and lim|๐‘ง|โ†’1||๐‘ข๐œ‡(|๐‘ง|)(๐‘›)||2(๐‘ง)log1โˆ’|๐‘ง|2=0.(3.32)

Proof. Assume that conditions (3.31) hold. By Theorem 3.1, ๐‘€๐‘ขโˆถ๐น(๐‘,๐‘ž,๐‘ )โ†’๐’ฒ๐œ‡(๐‘›)(๐”ป) is bounded. For any bounded sequence {๐‘“๐‘˜} in ๐น(๐‘,๐‘ž,๐‘ ) with ๐‘“๐‘˜โ†’0 uniformly on compact subsets of ๐”ป establish the assertion, it suffices, in view of Lemma 2.4, to show that โ€–โ€–๐‘€๐‘ข๐‘“๐‘˜โ€–โ€–๐’ฒ๐œ‡(๐‘›)(๐”ป)โŸถ0as๐‘˜โŸถโˆž.(3.33) We assume that โ€–๐‘“๐‘˜โ€–๐น(๐‘,๐‘ž,๐‘ )โ‰ค1. From (3.31) and (3.32), given ๐œ–>0, there exists a ๐›ฟโˆˆ(0,1), when ๐›ฟ<|๐‘ง|<1, we have ||||๐œ‡(|๐‘ง|)๐‘ข(๐‘ง)๎€ท1โˆ’|๐‘ง|2๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›<๐œ–,(3.34)๐œ‡||๐‘ข(|๐‘ง|)(๐‘›)||2(๐‘ง)log1โˆ’|๐‘ง|2<๐œ–.(3.35) By (3.34) and the Cauchy integral formula, we obtain ||๐‘ข๎…ž||=1(๐‘ง)|||||๎€œ2๐œ‹02๐œ‹๐‘ข๎€ท๐›ฟ๐‘ง๐‘’๐‘–๐œƒ๎€ธ๎€ท๐›ฟ๐‘ง๐‘’๐‘–๐œƒ๎€ธโˆ’๐‘ง2๐›ฟ๐‘ง๐‘’๐‘–๐œƒ|||||๎€ท๐‘‘๐œƒ<๐œ–1โˆ’๐›ฟ2๐‘ง๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›๐œ‡๎€ท๐›ฟ๐‘ง๎€ธ1๎€œ2๐œ‹02๐œ‹๐›ฟ๐‘ง||๐›ฟ๐‘ง๐‘’๐‘–๐œƒ||โˆ’๐‘ง2๎€ท๐‘‘๐œƒ=๐œ–1โˆ’๐›ฟ2๐‘ง๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›๐œ‡๎€ท๐›ฟ๐‘ง๎€ธ๐›ฟ๐‘ง๐›ฟ2๐‘งโˆ’|๐‘ง|2๎€ทโ‰ค๐œ–๐ถ1โˆ’|๐‘ง|2๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›โˆ’1,๐œ‡(|๐‘ง|)(3.36) when ๐›ฟ<|๐‘ง|<1. Hence, ||๐‘ข๐œ‡(|๐‘ง|)๎…ž||(๐‘ง)๎€ท1โˆ’|๐‘ง|2๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›โˆ’1<๐ถ๐œ–,(3.37) when ๐œ<|๐‘ง|<1. Similarly, for ๐‘—โˆˆ{2,โ€ฆ,๐‘›}, we have that ||๐‘ข๐œ‡(|๐‘ง|)(๐‘—)||(๐‘ง)๎€ท1โˆ’|๐‘ง|2๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›โˆ’๐‘—<๐ถ๐œ–,(3.38) when ๐›ฟ<|๐‘ง|<1. Since ๐‘“๐‘˜โ†’0 uniformly on compact subsets of ๐”ป, Cauchyโ€™s estimate gives that ๐‘“๐‘˜(๐‘—) converges to 0 uniformly on compact subsets of ๐”ป for each ๐‘—โˆˆ{1,โ€ฆ,๐‘›}, there exists a ๐พ0โˆˆโ„• such that ๐‘˜>๐พ0 implies that ๐‘›โˆ’1๎“๐‘—=0๐‘—โˆ’1๎“๐‘š=0๐ถ๐‘š๐‘—||๐‘ข(๐‘š)(0)๐‘“๐‘˜(๐‘—โˆ’๐‘š)||(0)+sup|๐‘ง|โ‰ค๐›ฟ|||||๐œ‡(|๐‘ง|)๐‘›๎“๐‘—=0๐ถ๐‘—๐‘›๐‘ข(๐‘—)(๐‘ง)๐‘“๐‘˜(๐‘›โˆ’๐‘—)|||||(๐‘ง)<๐ถ๐œ–.(3.39) If 2+๐‘ž>๐‘, from (3.34), (3.38), (3.39), and Lemmas 2.1 and 2.2, we have โ€–โ€–๐‘€๐‘ข๐‘“๐‘˜โ€–โ€–๐’ฒ๐œ‡(๐‘›)(๐”ป)=๐‘›โˆ’1๎“๐‘—=0|||๎€ท๐‘ข๐‘“๐‘˜๎€ธ(๐‘—)|||(0)+sup๐‘งโˆˆ๐”ป๐œ‡|||๎€ท(|๐‘ง|)๐‘ข๐‘“๐‘˜๎€ธ(๐‘›)|||โ‰ค(๐‘ง)๐‘›โˆ’1๎“๐‘—=0๐‘—โˆ’1๎“๐‘š=0๐ถ๐‘š๐‘—||๐‘ข(๐‘š)(0)๐‘“๐‘˜(๐‘—โˆ’๐‘š)||(0)+sup|๐‘ง|โ‰ค๐›ฟ|||||๐œ‡(|๐‘ง|)๐‘›๎“๐‘—=0๐ถ๐‘—๐‘›๐‘ข(๐‘—)(๐‘ง)๐‘“๐‘˜(๐‘›โˆ’๐‘—)|||||(๐‘ง)+sup|๐‘ง|>๐›ฟ|||||๐œ‡(|๐‘ง|)๐‘›๎“๐‘—=0๐ถ๐‘—๐‘›๐‘ข(๐‘—)(๐‘ง)๐‘“๐‘˜(๐‘›โˆ’๐‘—)|||||(๐‘ง)<๐ถ๐œ–+๐ถsup|๐‘ง|>๐›ฟ||๐œ‡(|๐‘ง|)๐‘ข(๐‘›)(๐‘ง)๐‘“๐‘˜||(๐‘ง)+๐ถsup|๐‘ง|>๐›ฟ๐‘›โˆ’1๎“๐‘—=0|||||๐ถ๐‘—๐‘›๐œ‡(|๐‘ง|)๐‘ข(๐‘—)(๐‘ง)๎€ท1โˆ’|๐‘ง|2๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›โˆ’๐‘—|||||โ€–โ€–๐‘“๐‘˜โ€–โ€–๐น(๐‘,๐‘ž,๐‘ )โ‰ค๐ถ๐œ–+๐ถsup|๐‘ง|>๐›ฟ|||||๐œ‡(|๐‘ง|)๐‘ข(๐‘›)(๐‘ง)๎€ท1โˆ’|๐‘ง|2๎€ธ(2+๐‘žโˆ’๐‘)/๐‘|||||โ€–โ€–๐‘“๐‘˜โ€–โ€–โ„ฌ(2+๐‘ž)/๐‘+๐ถsup|๐‘ง|>๐›ฟ๐‘›โˆ’1๎“๐‘—=0|||||๐ถ๐‘—๐‘›๐œ‡(|๐‘ง|)๐‘ข(๐‘—)(๐‘ง)๎€ท1โˆ’|๐‘ง|2๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›โˆ’๐‘—|||||โ‰ค๐ถ๐œ–+๐ถsup|๐‘ง|>๐›ฟ|||||๐œ‡(|๐‘ง|)๐‘ข(๐‘›)(๐‘ง)๎€ท1โˆ’|๐‘ง|2๎€ธ(2+๐‘žโˆ’๐‘)/๐‘|||||+๐ถsup|๐‘ง|>๐›ฟ๐‘›โˆ’1๎“๐‘—=0|||||๐ถ๐‘—๐‘›๐œ‡(|๐‘ง|)๐‘ข(๐‘—)(๐‘ง)๎€ท1โˆ’|๐‘ง|2๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›โˆ’๐‘—|||||<๎ƒฉ2+๐ถ๐‘›๎“๐‘—=0๐ถ๐‘—๐‘›๎ƒช๐ถ๐œ–,(3.40) when ๐‘˜>๐พ0. If 2+๐‘ž<๐‘, then ๐‘“๐‘˜โˆˆ๐น(๐‘,๐‘ž,๐‘ )โŠ‚โ„ฌ(2+๐‘ž)/๐‘,๐‘˜โˆˆโ„•. By [37, Lemmaโ€‰3.2], we have lim๐‘˜โ†’โˆžsup๐‘งโˆˆ๐”ป||๐‘“๐‘˜||(๐‘ง)=0.(3.41) By (3.38), (3.39), Lemma 2.1, and ๐‘ขโˆˆ๐’ฒ๐œ‡(๐‘›)(๐”ป), we get โ€–โ€–๐‘€๐‘ข๐‘“๐‘˜โ€–โ€–๐’ฒ๐œ‡(๐‘›)(๐”ป)=๐‘›โˆ’1๎“๐‘—=0|||๎€ท๐‘ข๐‘“๐‘˜๎€ธ(๐‘—)|||(0)+sup๐‘งโˆˆ๐”ป๐œ‡|||๎€ท(|๐‘ง|)๐‘ข๐‘“๐‘˜๎€ธ(๐‘›)|||(๐‘ง)<๐ถ๐œ–+๐ถsup|๐‘ง|>๐›ฟ||๐œ‡(|๐‘ง|)๐‘ข(๐‘›)(๐‘ง)๐‘“๐‘˜||(๐‘ง)+๐ถsup|๐‘ง|>๐›ฟ๐‘›โˆ’1๎“๐‘—=0|||||๐ถ๐‘—๐‘›๐œ‡(|๐‘ง|)๐‘ข(๐‘—)(๐‘ง)๎€ท1โˆ’|๐‘ง|2๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›โˆ’๐‘—|||||โ€–โ€–๐‘“๐‘˜โ€–โ€–๐น(๐‘,๐‘ž,๐‘ )โ‰ค๐ถ๐œ–+๐ถsup๐‘งโˆˆ๐”ป||๐‘“๐‘˜||(๐‘ง)+๐ถsup|๐‘ง|>๐›ฟ๐‘›โˆ’1๎“๐‘—=0|||||๐ถ๐‘—๐‘›๐œ‡(|๐‘ง|)๐‘ข(๐‘—)(๐‘ง)๎€ท1โˆ’|๐‘ง|2๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›โˆ’๐‘—|||||โ‰ค๐ถ๐œ–+๐ถ๐œ–+๐ถsup|๐‘ง|>๐›ฟ๐‘›โˆ’1๎“๐‘—=0|||||๐ถ๐‘—๐‘›๐œ‡(|๐‘ง|)๐‘ข(๐‘—)(๐‘ง)๎€ท1โˆ’|๐‘ง|2๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›โˆ’๐‘—|||||<๎ƒฉ2+๐ถ๐‘›๎“๐‘—=0๐ถ๐‘—๐‘›๎ƒช๐ถ๐œ–,(3.42) when ๐‘˜>๐พ0. If 2+๐‘ž=๐‘, from (3.35), (3.38), (3.39), and Lemmas 2.1 and 2.2, we get โ€–โ€–๐‘€๐‘ข๐‘“๐‘˜โ€–โ€–๐’ฒ๐œ‡(๐‘›)(๐”ป)=๐‘›โˆ’1๎“๐‘—=0|||๎€ท๐‘ข๐‘“๐‘˜๎€ธ(๐‘—)|||(0)+sup๐‘งโˆˆ๐”ป๐œ‡|||๎€ท(|๐‘ง|)๐‘ข๐‘“๐‘˜๎€ธ(๐‘›)|||(๐‘ง)<๐ถ๐œ–+๐ถsup|๐‘ง|>๐›ฟ||๐œ‡(|๐‘ง|)๐‘ข(๐‘›)(๐‘ง)๐‘“๐‘˜||(๐‘ง)+๐ถsup|๐‘ง|>๐›ฟ๐‘›โˆ’1๎“๐‘—=0|||||๐ถ๐‘—๐‘›๐œ‡(|๐‘ง|)๐‘ข(๐‘—)(๐‘ง)๎€ท1โˆ’|๐‘ง|2๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›โˆ’๐‘—|||||โ€–โ€–๐‘“๐‘˜โ€–โ€–๐น(๐‘,๐‘ž,๐‘ )<๐ถ๐œ–+๐ถsup|๐‘ง|>๐›ฟ||๐œ‡(|๐‘ง|)๐‘ข(๐‘›)||2(๐‘ง)log1โˆ’|๐‘ง|2โ€–โ€–๐‘“๐‘˜โ€–โ€–โ„ฌ+๐ถsup|๐‘ง|>๐›ฟ๐‘›โˆ’1๎“๐‘—=0|||||๐ถ๐‘—๐‘›๐œ‡(|๐‘ง|)๐‘ข(๐‘—)(๐‘ง)๎€ท1โˆ’|๐‘ง|2๎€ธ((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›โˆ’๐‘—|||||โ‰ค๐ถ๐œ–+๐ถsup|๐‘ง|>๐›ฟ||๐œ‡(|๐‘ง|)๐‘ข(๐‘›)||2(๐‘ง)log1โˆ’|๐‘ง|2โ€–โ€–๐‘“๐‘˜โ€–โ€–๐น(๐‘,๐‘ž,๐‘ )+๐ถsup|๐‘ง|>๐›ฟ๐‘›โˆ’1๎“๐‘—=0|||||๐ถ๐‘—๐‘›๐œ‡(|๐‘ง|)๐‘ข(๐‘—)(๐‘ง)๎€ท1โˆ’|๐‘ง|2๎€ธ๐‘›โˆ’๐‘—|||||<๎ƒฉ2+๐ถ๐‘›๎“๐‘—=0๐ถ๐‘—๐‘›๎ƒช๐ถ๐œ–,(3.43) when ๐‘˜>๐พ0. It follows that the operator ๐‘€๐‘ขโˆถ๐น(๐‘,๐‘ž,๐‘ )โ†’๐’ฒ๐œ‡(๐‘›)(๐”ป) is compact.
Conversely, assume that ๐‘€๐‘ขโˆถ๐น(๐‘,๐‘ž,๐‘ )โ†’๐’ฒ๐œ‡(๐‘›)(๐”ป) is compact. Taking ๐‘“=1, we get ๐‘ขโˆˆ๐’ฒ๐œ‡(๐‘›)(๐”ป). Let {๐‘ง๐‘˜} be a sequence in ๐”ป such that |๐‘ง๐‘˜|โ†’1 as ๐‘˜โ†’โˆž. Taking the test functions โ„Ž๐‘ง๐‘˜, where โ„Ž๐‘ง๐‘˜ is defined by (3.17), we will write โ„Ž๐‘˜(๐‘ง)=๐‘“๐‘ง๐‘˜(๐‘ง).(3.44) We obtain that โ„Ž๐‘˜โˆˆ๐น(๐‘,๐‘ž,๐‘ ), sup๐‘˜โˆˆโ„•โ€–โ€–โ„Ž๐‘˜โ€–โ€–๐น(๐‘,๐‘ž,๐‘ )โ‰ค๐ถ,(3.45) and for ๐‘กโˆˆ{1,โ€ฆ,๐‘›โˆ’1}, โ„Ž๐‘˜(๐‘›)๎€ท๐‘ง๐‘˜๎€ธ=๐‘›๎“๐‘—=1๐‘๐‘—๐‘™โˆ’1๎‘๐‘˜=0๎€ท(๐›ผ+๐‘—+๐‘˜)๐‘ง๐‘˜๎€ธ๐‘›๎‚€||๐‘ง1โˆ’๐‘˜||2๎‚((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›,โ„Ž๐‘˜(๐‘ก)๎€ท๐‘ง๐‘˜๎€ธ=0.(3.46) For |๐‘ง|=๐‘Ÿ<1, we have ||โ„Ž๐‘˜||=|||||||(๐‘ง)๐‘›๎“๐‘—=1๐‘๐‘—๎‚€||๐‘ง1โˆ’๐‘˜||2๎‚๐‘—+1๎€ท1โˆ’๐‘ง๐‘˜๐‘ง๎€ธ๐›ผ+๐‘—|||||||โ‰ค๐‘›๎“๐‘—=1๐‘๐‘—๎‚€||๐‘ง1โˆ’๐‘˜||2๎‚๐‘—+1(1โˆ’๐‘Ÿ)๐›ผ+๐‘—โŸถ0as๐‘˜โŸถโˆž,(3.47) that is, โ„Ž๐‘˜ converges to 0 uniformly on compact subsets of ๐”ป, using the compactness of ๐‘€๐‘ขโˆถ๐น(๐‘,๐‘ž,๐‘ )โ†’๐’ฒ๐œ‡(๐‘›)(๐”ป), we obtain ๐œ‡๎€ท๐‘ง๐‘˜๎€ธ||๐‘ข๎€ท๐‘ง๐‘˜๎€ธ||||๐‘ง๐‘˜||๐‘›๎‚€||๐‘ง1โˆ’๐‘˜||2๎‚((2+๐‘žโˆ’๐‘)/๐‘)+๐‘›โ€–โ€–๐‘€โ‰ค๐ถ๐‘ขโ„Ž๐‘˜โ€–โ€–๐’ฒ๐œ‡(๐‘›)(๐”ป)โŸถ0as๐‘˜โŸถโˆž,(3.48) and consequently (3.31) holds.
Assume that 2+๐‘ž=๐‘,๐‘ >1 and ๐‘€๐‘ขโˆถ๐น(๐‘,๐‘ž,๐‘ )โ†’๐’ฒ๐œ‡(๐‘›)(๐”ป) is compact, we only need to prove (3.32) holds. To do this, let {๐‘ง๐‘˜} be a sequence in ๐”ป such that |๐‘ง๐‘˜|โ†’1 as ๐‘˜โ†’โˆž. Set ๐‘Ž๐‘˜2=log||๐‘ง1โˆ’๐‘˜||2,๐ฝ๐‘˜๐ธ(๐‘ง)=๐‘Ž๐‘˜๎‚ต2log1โˆ’๐‘ง๐‘˜๐‘ง๎‚ถ2+๐น๐‘Ž2๐‘˜๎‚ต2log1โˆ’๐‘ง๐‘˜๐‘ง๎‚ถ3.(3.49) Since ||๐ฝ๎…ž๐‘˜||โ‰ค||||(๐‘ง)2๐ธ๐‘Ž๐‘˜๎‚ต2log1โˆ’๐‘ง๐‘˜๐‘ง๎‚ถ๐‘ง๐‘˜1โˆ’๐‘ง๐‘˜๐‘ง||||+||||3๐น๐‘Ž2๐‘˜๎‚ต2log1โˆ’๐‘ง๐‘˜๐‘ง๎‚ถ2๐‘ง๐‘˜1โˆ’๐‘ง๐‘˜๐‘ง||||โ‰ค๐ถ||1โˆ’๐‘ง๐‘˜๐‘ง||,(3.50) we have ๐ฝ๐‘˜โˆˆ๐น(๐‘,๐‘ž,๐‘ ) and sup๐‘˜โˆˆโ„•โ€–๐ฝ๐‘˜โ€–๐น(๐‘,๐‘ž,๐‘ )โ‰ค๐ถ (see [19, 30]), and ๐ฝ๐‘˜โ†’0 uniformly on compact subsets of ๐”ป. Since ๐‘€๐‘ขโˆถ๐น(๐‘,๐‘ž,๐‘ )โ†’๐’ฒ๐œ‡(๐‘›)(๐”ป) is compact, we have โ€–โ€–๐‘€๐‘ข๐ฝ๐‘˜โ€–โ€–๐’ฒ๐œ‡(๐‘›)(๐”ป)โŸถ0as๐‘˜โŸถโˆž.(3.51) We also have that ๐ฝ๎…ž๐‘˜(๐‘ง)=2๐ธ๐‘Ž๐‘˜๎‚ต2log1โˆ’๐‘ง๐‘˜๐‘ง๎‚ถ๐‘ง๐‘˜1โˆ’๐‘ง๐‘˜๐‘ง+3๐น๐‘Ž2๐‘˜๎‚ต2log1โˆ’๐‘ง๐‘˜๐‘ง๎‚ถ2๐‘ง๐‘˜1โˆ’๐‘ง๐‘˜๐‘ง,๐ฝ๐‘˜๎…ž๎…ž(๐‘ง)=2๐ธ๐‘Ž๐‘˜๎€ท๐‘ง๐‘˜๎€ธ2๎€ท1โˆ’๐‘ง๐‘˜๐‘ง๎€ธ2+2๐ธ๐‘Ž๐‘˜๎‚ต2log1โˆ’๐‘ง๐‘˜๐‘ง๎‚ถ๎€ท๐‘ง๐‘˜๎€ธ2๎€ท1โˆ’๐‘ง๐‘˜๐‘ง๎€ธ2+3โ‹…2๐น๐‘Ž2๐‘˜๎‚ต2log1โˆ’๐‘ง๐‘˜๐‘ง๎‚ถ๎€ท๐‘ง๐‘˜๎€ธ2๎€ท1โˆ’๐‘ง๐‘˜๐‘ง๎€ธ2+3๐น๐‘Ž2๐‘˜๎‚ต2log1โˆ’๐‘ง๐‘˜๐‘ง๎‚ถ2๎€ท๐‘ง๐‘˜๎€ธ2๎€ท1โˆ’๐‘ง๐‘˜๐‘ง๎€ธ2,๐ฝ๐‘˜๎…ž๎…ž๎…ž(๐‘ง)=2โ‹…2๐ธ๐‘Ž๐‘˜๎€ท๐‘ง๐‘˜๎€ธ3๎€ท1โˆ’๐‘ง๐‘˜๐‘ง๎€ธ3+2๐ธ๐‘Ž๐‘˜๎€ท๐‘ง๐‘˜๎€ธ3๎€ท1โˆ’๐‘ง๐‘˜๐‘ง๎€ธ3+4๐ธ๐‘Ž๐‘˜๎‚ต2log1โˆ’๐‘ง๐‘˜๐‘ง๎‚ถ๎€ท๐‘ง๐‘˜๎€ธ3๎€ท1โˆ’๐‘ง๐‘˜๐‘ง๎€ธ3+3โ‹…2๐น๐‘Ž2๐‘˜๎€ท๐‘ง๐‘˜๎€ธ3๎€ท1โˆ’๐‘ง๐‘˜๐‘ง๎€ธ3+3โ‹…2โ‹…2๐น๐‘Ž2๐‘˜๎‚ต2log1โˆ’๐‘ง๐‘˜๐‘ง๎‚ถ๎€ท๐‘ง๐‘˜๎€ธ3๎€ท1โˆ’๐‘ง๐‘˜๐‘ง๎€ธ3+3โ‹…2๐น๐‘Ž2๐‘˜๎‚ต2log1โˆ’๐‘ง๐‘˜๐‘ง๎‚ถ๎€ท๐‘ง๐‘˜๎€ธ3๎€ท1โˆ’๐‘ง๐‘˜๐‘ง๎€ธ3+6๐น๐‘Ž2๐‘˜๎‚ต2log1โˆ’๐‘ง๐‘˜๐‘ง๎‚ถ2๎€ท๐‘ง๐‘˜๎€ธ3๎€ท1โˆ’๐‘ง๐‘˜๐‘ง๎€ธ3,=6๐ธ๐‘Ž๐‘˜๎€ท๐‘ง๐‘˜๎€ธ3๎€ท1โˆ’๐‘ง๐‘˜๐‘ง๎€ธ3+4๐ธ๐‘Ž๐‘˜๎‚ต2log1โˆ’๐‘ง๐‘˜๐‘ง๎‚ถ๎€ท๐‘ง๐‘˜๎€ธ3๎€ท1โˆ’๐‘ง๐‘˜๐‘ง๎€ธ3+6๐น๐‘Ž2๐‘˜๎€ท๐‘ง๐‘˜๎€ธ3๎€ท1โˆ’๐‘ง๐‘˜๐‘ง๎€ธ3+18๐น๐‘Ž2๐‘˜๎‚ต2log1โˆ’๐‘ง๐‘˜๐‘ง๎‚ถ๎€ท๐‘ง๐‘˜๎€ธ3๎€ท1โˆ’๐‘ง๐‘˜๐‘ง๎€ธ3+6๐น๐‘Ž2๐‘˜๎‚ต2log1โˆ’๐‘ง๐‘˜๐‘ง๎‚ถ2๎€ท๐‘ง๐‘˜๎€ธ3๎€ท1โˆ’๐‘ง๐‘˜๐‘ง๎€ธ3,๐ฝ๐‘˜(4)(๐‘ง)=22๐ธ๐‘Ž๐‘˜๎€ท๐‘ง๐‘˜๎€ธ4๎€ท1โˆ’๐‘ง๐‘˜๐‘ง๎€ธ4+3!2๐ธ๐‘Ž๐‘˜๎‚ต2log1โˆ’๐‘ง๐‘˜๐‘ง๎‚ถ๎€ท๐‘ง๐‘˜๎€ธ4๎€ท1โˆ’๐‘ง๐‘˜๐‘ง๎€ธ4+36๐น๐‘Ž2๐‘˜๎€ท๐‘ง๐‘˜๎€ธ4๎€ท1โˆ’๐‘ง๐‘˜๐‘ง๎€ธ4+66๐น๐‘Ž2๐‘˜๎‚ต2log1โˆ’๐‘ง๐‘˜๐‘ง๎‚ถ๎€ท๐‘ง๐‘˜๎€ธ4๎€ท1โˆ’๐‘ง๐‘˜๐‘ง๎€ธ4+3!3๐น๐‘Ž2๐‘˜๎‚ต2log1โˆ’๐‘ง๐‘˜๐‘ง๎‚ถ2๎€ท๐‘ง๐‘˜๎€ธ4๎€ท1โˆ’๐‘ง๐‘˜๐‘ง๎€ธ4,โ‹ฎ๐ฝ๐‘˜(๐‘š)๐‘(๐‘ง)=๐‘š๐ธ๐‘Ž๐‘˜๎€ท๐‘ง๐‘˜๎€ธ๐‘š๎€ท1โˆ’๐‘ง๐‘˜๐‘ง๎€ธ๐‘š+(๐‘šโˆ’1)!2๐ธ๐‘Ž๐‘˜๎‚ต2log1โˆ’๐‘ง๐‘˜๐‘ง๎‚ถ๎€ท๐‘ง๐‘˜๎€ธ๐‘š๎€ท1โˆ’๐‘ง๐‘˜๐‘ง๎€ธ๐‘š+๐‘‘๐‘š๐น๐‘Ž2๐‘˜๎€ท๐‘ง๐‘˜๎€ธ๐‘š๎€ท1โˆ’๐‘ง๐‘˜๐‘ง๎€ธ๐‘š+๐‘๐‘š๐น๐‘Ž2๐‘˜๎‚ต2log1โˆ’๐‘ง๐‘˜๐‘ง๎‚ถ๎€ท๐‘ง๐‘˜๎€ธ๐‘š๎€ท1โˆ’๐‘ง๐‘˜๐‘ง๎€ธ๐‘š+(๐‘šโˆ’1)!3๐น๐‘Ž2๐‘˜๎‚ต2log1โˆ’๐‘ง๐‘˜๐‘ง๎‚ถ2๎€ท๐‘ง๐‘˜๎€ธ๐‘š๎€ท1โˆ’๐‘ง๐‘˜๐‘ง๎€ธ๐‘š.(3.52) Therefore, ๐ฝ๐‘˜๎€ท๐‘ง๐‘˜๎€ธ=(๐ธ+๐น)๐‘Ž๐‘˜,๐ฝ๎…ž๐‘˜๎€ท๐‘ง๐‘˜๎€ธ=(2๐ธ+3๐น)๐‘ง๐‘˜||๐‘ง1โˆ’๐‘˜||2,๐ฝ๐‘˜๎…ž๎…ž๎€ท๐‘ง๐‘˜๎€ธ๎€ท=(2๐ธ+3๐น)๐‘ง๐‘˜๎€ธ2๎‚€||๐‘ง1โˆ’๐‘˜||2๎‚2+2(๐ธ+3๐น)๐‘Ž๐‘˜๎€ท๐‘ง๐‘˜๎€ธ2๎‚€||๐‘ง1โˆ’๐‘˜||2๎‚2,๐ฝ๐‘˜๎…ž๎…ž๎…ž๎€ท๐‘ง๐‘˜๎€ธ๎€ท=2(2๐ธ+3๐น)๐‘ง๐‘˜๎€ธ3๎‚€||๐‘ง1โˆ’๐‘˜||2๎‚3+6(๐ธ+3๐น)๐‘Ž๐‘˜๎€ท๐‘ง๐‘˜๎€ธ3๎‚€||๐‘ง1โˆ’๐‘˜||2๎‚3+6๐น๐‘Ž2๐‘˜๎€ท๐‘ง๐‘˜๎€ธ3๎‚€||๐‘ง1โˆ’๐‘˜||2๎‚3,๐ฝ๐‘˜(4)๎€ท๐‘ง๐‘˜๎€ธ๎€ท=3!(2๐ธ+3๐น)๐‘ง๐‘˜๎€ธ4๎‚€||๐‘ง1โˆ’๐‘˜||2๎‚4+22(๐ธ+3๐น)๐‘Ž๐‘˜๎€ท๐‘ง๐‘˜๎€ธ4๎‚€||๐‘ง1โˆ’๐‘˜||2๎‚4+36๐น๐‘Ž2๐‘˜๎€ท๐‘ง๐‘˜๎€ธ4๎‚€||๐‘ง1โˆ’๐‘˜||2๎‚4,โ‹ฎ๐ฝ๐‘˜(๐‘š)๎€ท๐‘ง๐‘˜๎€ธ๎€ท=(๐‘šโˆ’1)!(2๐ธ+3๐น)๐‘ง๐‘˜๎€ธ๐‘š๎‚€||๐‘ง1โˆ’๐‘˜||2๎‚๐‘š+๐‘๐‘š๐ธ+๐‘๐‘š๐น๐‘Ž๐‘˜๎€ท๐‘ง๐‘˜๎€ธ๐‘š๎‚€||๐‘ง1โˆ’๐‘˜||2๎‚๐‘š+๐‘‘๐‘š๐น๐‘Ž2๐‘˜๎€ท๐‘ง๐‘˜๎€ธ๐‘š๎‚€||๐‘ง1โˆ’๐‘˜||2๎‚๐‘š.(3.53) Taking ๐ธ=3 and ๐น=โˆ’2, if ๐‘›=1, we have ๐œ‡๎€ท||๐‘ง๐‘˜||๎€ธ|||||๐‘ข๎…ž๎€ท๐‘ง๐‘˜๎€ธ2log||๐‘ง1โˆ’๐‘˜||2|||||โ‰คโ€–โ€–๐‘€๐‘ข๐ฝ๐‘˜โ€–โ€–๐’ฒ๐œ‡(๐‘›)(๐”ป)โŸถ0as๐‘˜โŸถโˆž,(3.54) if ๐‘›โ‰ 1, we have โ€–โ€–๐‘€๐‘ข๐ฝ๐‘˜โ€–โ€–๐’ฒ๐œ‡(๐‘›)(๐”ป)๎€ท||๐‘งโ‰ฅ๐œ‡๐‘˜||๎€ธ|||||๐‘›๎“๐‘—=0๐ถ๐‘—๐‘›๐‘ข(๐‘—)๎€ท๐‘ง๐‘˜๎€ธ๐ฝ๐‘˜(๐‘›โˆ’๐‘—)๎€ท๐‘ง๐‘˜๎€ธ|||||๎€ท||๐‘งโ‰ฅ๐œ‡๐‘˜||๎€ธ||๐‘ข(๐‘›)๎€ท๐‘ง๐‘˜๎€ธ๐ฝ๐‘˜๎€ท๐‘ง๐‘˜๎€ธ||๎€ท||๐‘งโˆ’๐œ‡๐‘˜||๎€ธ|||||๐‘›โˆ’1๎“๐‘—=0๐ถ๐‘—๐‘›๐‘ข(๐‘—)๎€ท๐‘ง๐‘˜๎€ธ๐ฝ๐‘˜(๐‘›โˆ’๐‘—)๎€ท๐‘ง๐‘˜๎€ธ|||||๎€ท||๐‘งโ‰ฅ๐œ‡๐‘˜||๎€ธ|||||๐‘ข(๐‘›)๎€ท๐‘ง๐‘˜๎€ธ2log||๐‘ง1โˆ’๐‘˜||2|||||๎€ท||๐‘งโˆ’๐œ‡๐‘˜||๎€ธ|||||๐‘›โˆ’1๎“๐‘—=0๐ถ๐‘—๐‘›๐‘ข(๐‘—)๎€ท๐‘ง๐‘˜๎€ธ๐ฝ๐‘˜(๐‘›โˆ’๐‘—)๎€ท๐‘ง๐‘˜๎€ธ|||||.(3.55) From (3.31), (3.35), (3.37), and (3.38), we have that for sufficiently large ๐‘˜: ๐œ‡๎€ท||๐‘ง๐‘˜||๎€ธ|||||๐‘ข(๐‘›)๎€ท๐‘ง๐‘˜๎€ธ2log||๐‘ง1โˆ’๐‘˜||2|||||โ‰คโ€–โ€–๐‘€๐‘ข๐ฝ๐‘˜โ€–โ€–๐’ฒ๐œ‡(๐‘›)(๐”ป)๎€ท||๐‘ง+๐œ‡๐‘˜||๎€ธ|||||๐‘›โˆ’1๎“๐‘—=0๐ถ๐‘—๐‘›๐‘ข(๐‘—)๎€ท๐‘ง๐‘˜๎€ธ๐ฝ๐‘˜(๐‘›โˆ’๐‘—)๎€ท๐‘ง๐‘˜๎€ธ|||||โ‰คโ€–โ€–๐‘€๐‘ข๐ฝ๐‘˜โ€–โ€–๐’ฒ๐œ‡(๐‘›)(๐”ป)+๐ถ๐‘›โˆ’1๎“๐‘—=0โŽ›โŽœโŽœโŽœโŽ๐œ‡๎€ท||๐‘ง๐‘˜||๎€ธ||๐‘ข(๐‘—)๎€ท๐‘ง๐‘˜๎€ธ||๐‘Ž๐‘˜๎‚€||๐‘ง1โˆ’๐‘˜||2๎‚๐‘›โˆ’๐‘—+๐œ‡๎€ท||๐‘ง๐‘˜||๎€ธ||๐‘ข(๐‘—)๎€ท๐‘ง๐‘˜๎€ธ||๐‘Ž2๐‘˜๎‚€||๐‘ง1โˆ’๐‘˜||2๎‚๐‘›โˆ’๐‘—โŽžโŽŸโŽŸโŽŸโŽ โ‰คโ€–โ€–๐‘€๐‘ข๐ฝ๐‘˜โ€–โ€–๐’ฒ๐œ‡(๐‘›)(๐”ป)+๐ถ๐‘›โˆ’1๎“๐‘—=0๐œ‡๎€ท||๐‘ง๐‘˜||๎€ธ||๐‘ข(๐‘—)๎€ท๐‘ง๐‘˜๎€ธ||๎‚€||๐‘ง1โˆ’๐‘˜||2๎‚๐‘›โˆ’๐‘—<๐ถ๐œ–.(3.56) Using (3.54) and (3.56), it is easy to get that (3.32) holds. From which we obtain the desired result.

4. The Boundedness of ๐‘€๐‘ข from ๐น(๐‘,๐‘ž,๐‘ ) (or๐น0(๐‘,๐‘ž,๐‘ )) to ๐’ฒ(๐‘›)๐œ‡,0(๐”ป)

In this section, we characterize the boundedness of ๐‘€๐‘ขโˆถ๐น(๐‘,๐‘ž,๐‘ )(or๐น0(๐‘,๐‘ž,๐‘ ))โ†’๐’ฒ(๐‘›)๐œ‡,0(๐”ป).

Theorem 4.1. Assume that ๐‘ขโˆˆ๐ป(๐”ป) and ๐œ‡ is normal. Then,(1)๐‘€๐‘ขโˆถ๐น(๐‘,๐‘ž,๐‘ )โ†’๐’ฒ(๐‘›)๐œ‡,0(๐”ป) is bounded if and only if ๐‘€๐‘ขโˆถ๐น(๐‘,๐‘ž,๐‘ )โ†’๐’ฒ๐œ‡(๐‘›)(๐”ป) is bounded and lim|๐‘ง|โ†’1||๐‘ข๐œ‡(๐‘ง)(๐‘›)||(๐‘ง)=0.(4.1)(2)๐‘€๐‘ขโˆถ๐น0(๐‘,๐‘ž,๐‘ )โ†’๐’ฒ(๐‘›)๐œ‡,0(๐”ป) is bounded if and only if ๐‘€๐‘ขโˆถ๐น0(๐‘,๐‘ž,๐‘ )โ†’๐’ฒ๐œ‡(๐‘›)(๐”ป) is bounded and (4.1) holds.

Proof. (1) Assume that ๐‘€๐‘ขโˆถ๐น(๐‘,๐‘ž,๐‘ )โ†’๐’ฒ๐œ‡(๐‘›)(๐”ป) is bounded and condition (4.1) holds. Since ๎€œ๐‘ข(๐‘ง)โˆ’๐‘ข(0)=10๐‘ฅ๐‘ข๎…ž(๐‘ง๐‘ฅ)๐‘‘๐‘ฅ,(4.2) it follows that ๐œ‡||๐‘ข||||๐‘ข|