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Journal of Function Spaces and Applications
VolumeΒ 2012Β (2012), Article IDΒ 517689, 18 pages
http://dx.doi.org/10.1155/2012/517689
Research Article

Properties of Toeplitz Operators on Some Holomorphic Banach Function Spaces

1Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt
2Department of Mathematics, Faculty of Science, Taif University, P.O. Box 888, El-Hawiyah, El-Taif 5700, Saudi Arabia
3Department of Mathematics, Faculty of Science, Assiut Branch, Al-Azhar University, Assiut 32861, Egypt
4Department of Mathematics, Faculty of Science, Jazan University, P.O. Box 114, Jazan 45142, Saudi Arabia

Received 5 March 2012; Accepted 18 April 2012

Academic Editor: DashanΒ Fan

Copyright Β© 2012 Ahmed El-Sayed Ahmed and Mahmoud Ali Bakhit. 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 characterize complex measures πœ‡ on the unit ball of ℂ𝑛, for which the general Toeplitz operator π‘‡π›Όπœ‡ is bounded or compact on the analytic Besov spaces 𝐡𝑝(𝔹𝑛), also on the minimal MΓΆbius invariant Banach spaces 𝐡1(𝔹𝑛) in the unit ball 𝔹𝑛.

1. Introduction

Let 𝔹𝑛 be the unit ball of the 𝑛-dimensional complex Euclidean space ℂ𝑛. We denote the class of all holomorphic functions on the unit ball 𝔹𝑛 by β„‹(𝔹𝑛). The ball centered at 𝐳 with radius π‘Ÿ will be denoted by 𝐡(𝐳,π‘Ÿ). For 𝛼>βˆ’1, let π‘‘πœˆπ›Ό(𝐳)=𝑐𝛼(1βˆ’|𝐳|2)π›Όπ‘‘πœˆ, where π‘‘πœˆ is the normalized Lebesgue volume measure on 𝔹𝑛 and 𝑐𝛼=Ξ“(𝑛+𝛼+1)/𝑛!Ξ“(𝛼+1) (where Ξ“ denotes the Gamma function) so that πœˆπ›Ό(𝔹𝑛)≑1.

For any 𝐳=(𝑧1,𝑧2,…,𝑧𝑛), 𝐰=(𝑀1,𝑀2,…,𝑀𝑛)βˆˆβ„‚π‘›, the inner product is defined by βˆ‘βŸ¨π³,𝐰⟩=π‘›π‘˜=1π‘§π‘˜π‘€π‘˜. For π‘“βˆˆβ„‹(𝔹𝑛), we writeξ‚΅βˆ‡π‘“(𝐳)=πœ•π‘“(𝐳)πœ•π‘§1,πœ•π‘“(𝐳)πœ•π‘§2,…,πœ•π‘“(𝐳)πœ•π‘§π‘›ξ‚Ά,ξ«β„œπ‘“(𝐳)=βˆ‡π‘“,𝐳=𝑛𝑗=1π‘§π‘—πœ•π‘“(𝐳)πœ•z𝑗.(1.1) For π‘“βˆˆβ„‹(𝔹𝑛) and π³βˆˆπ”Ήπ‘›, set𝑄𝑓(𝐳)=supπ°βˆˆβ„‚π‘›β§΅{0}||ξ«βˆ‡π‘“(𝐳),𝐰||√𝐻𝐳(𝐰,𝐰),(1.2) where 𝐻𝐳(𝐰,𝐰) is the Bergman metric on 𝔹𝑛, that is,𝐻𝐳(𝐰,𝐰)=𝑛+121βˆ’|𝐳|2ξ€Έ|𝐰|2+||||⟨𝐰,𝐳⟩2ξ€·1βˆ’|𝐳|2ξ€Έ2.(1.3) For 1<𝑝<∞, the Besov spaces 𝐡𝑝(𝔹𝑛) consists of all functions π‘“βˆˆβ„‹(𝔹𝑛) for which (see [1])β€–π‘“β€–π‘π΅π‘ξ€·π”Ήπ‘›ξ€Έξ€œβˆΆ=𝔹𝑛𝑄𝑝𝑓(𝐳)π‘‘πœˆ(𝐳)<∞.(1.4) From [1], we know that for 𝑛β‰₯2, the Besov space is nontrivial if and only if 𝑝>2𝑛.

The analytic Besov space is the minimal MΓΆbius invariant Banach space 𝐡1(𝔹𝑛) (see [2]) defined by‖𝑓‖𝐡1(𝔹𝑛)ξ“βˆΆ=|π‘š|=𝑛+1supπ³βˆˆπ”Ήπ‘›ξ€œπ”Ήπ‘›||||πœ•π‘šπ‘“(𝐳)πœ•π³π‘š||||π‘‘πœˆ(𝐳)<∞.(1.5) For 𝛼β‰₯0, a function π‘“βˆˆβ„‹(𝔹𝑛) is said to belong to the 𝛼-Bloch spaces ℬ𝛼(𝔹𝑛) if (see [3])𝑏𝛼=supπ³βˆˆπ”Ήπ‘›||||ξ€·βˆ‡π‘“(𝐳)1βˆ’|𝐳|2𝛼<∞.(1.6) The little Bloch space ℬ𝛼0(𝔹𝑛) consists of all π‘“βˆˆβ„¬π›Ό(𝔹𝑛) such thatlim|𝐳|β†’1||||ξ€·βˆ‡π‘“(𝐳)1βˆ’|𝐳|2𝛼=0.(1.7) With the norm ‖𝑓‖ℬ𝛼(𝔹𝑛)=|𝑓(0)|+𝑏𝛼, we know that ℬ𝛼(𝔹𝑛) becomes a Banach space. For 𝛼=1, the spaces ℬ1 and ℬ10 become the Bloch and the little Bloch space (see, e.g., [2]).

For every point πšβˆˆπ”Ήπ‘›, the MΓΆbius transformation πœ‘πšβˆΆπ”Ήπ‘›β†’π”Ήπ‘› is defined byπœ‘πš(𝐳)=πšβˆ’π‘ƒπš(𝐳)βˆ’π‘†πšπ‘„πš(𝐳)1βˆ’βŸ¨π³,𝐚⟩,π³βˆˆπ”Ήπ‘›,(1.8) where π‘†πš=√1βˆ’|𝐚|2,π‘ƒπš(𝐳)=𝐚⟨𝐳,𝐚⟩/|𝐚|2,𝑃0=0 and π‘„πš=πΌβˆ’π‘ƒπš (see, e.g., [2] or [4]). The map πœ‘πš has the following properties that πœ‘πš(0)=𝐚, πœ‘πš(𝐚)=0, πœ‘πš=πœ‘πšβˆ’1 and1βˆ’βŸ¨πœ‘πš(𝐳),πœ‘πš(𝐰)⟩=1βˆ’|𝐚|2ξ€Έ(1βˆ’βŸ¨π³,𝐰⟩)(1βˆ’βŸ¨π³,𝐚⟩)(1βˆ’βŸ¨πš,𝐰⟩),(1.9) where 𝐳 and 𝐰 are arbitrary points in 𝔹𝑛. In particular,||πœ‘1βˆ’πš(||𝐳)2=ξ€·1βˆ’|𝐚|2ξ€Έξ€·1βˆ’|𝐳|2ξ€Έ||||1βˆ’βŸ¨π³,𝐚⟩2.(1.10)

The following result can be found in [3].

Proposition 1.1. Let π‘“βˆˆβ„‹(𝔹𝑛),2𝑛<𝑝<∞. Then π‘“βˆˆπ΅π‘(𝔹𝑛) if and only if 𝔹𝑛||||𝑓(𝐰)βˆ’π‘“(𝐳)||||ξ‚Ά1βˆ’βŸ¨π³,π°βŸ©π‘ξ€·1βˆ’|𝐳|2𝑝/2ξ€·1βˆ’|𝐰|2𝑝/2π‘‘πœ‡(𝐰)π‘‘πœˆ(𝐳)<∞.(1.11)

For 𝛼>βˆ’1 and 0<𝑝<∞, the weighted Bergman spaces 𝐴𝑝𝛼(𝔹𝑛) consists of all functions π‘“βˆˆβ„‹(𝔹𝑛) for whichβ€–π‘“β€–π‘π΄π‘π›Όξ€œβˆΆ=𝔹𝑛||||𝑓(𝐳)π‘π‘‘πœˆπ›Ό(𝐳)<∞.(1.12) It is clear that 𝐴𝑝𝛼=𝐿𝑝(𝔹𝑛,π‘‘πœˆπ›Ό)β‹‚β„‹(𝔹𝑛) and 𝐴𝑝𝛼 is a linear subspace of 𝐿𝑝(𝔹𝑛,π‘‘πœˆπ›Ό). When 𝛼=0, we simply write 𝐴𝑝(𝔹𝑛) for 𝐴𝑝0(𝔹𝑛). In the special case when 𝑝=2,𝐴2𝛼(𝔹𝑛) is a Hilbert space. It is well known that for 𝛼>βˆ’1 the Bergman kernel of 𝐴2𝛼(𝔹𝑛) is given by𝐾𝛼1(𝐳,𝐰)=(1βˆ’βŸ¨π³,𝐰⟩)𝑛+1+𝛼,𝐳,π°βˆˆπ”Ήπ‘›.(1.13)

For 𝛼>βˆ’1, a complex measure πœ‡ such that||||ξ€œπ”Ήπ‘›ξ€·1βˆ’|𝐰|2𝛼||||=||||ξ€œπ‘‘πœ‡(𝐰)π”Ήπ‘›π‘‘πœ‡π›Ό||||(𝐰)<∞(1.14) define a Toeplitz operator as follows:π‘‡π›Όπœ‡π‘“(𝐳)=π‘π›Όξ€œπ”Ήπ‘›ξ€·1βˆ’|𝐰|2𝛼𝑓(𝐰)(1βˆ’βŸ¨π³,𝐰⟩)𝑛+𝛼+1ξ€œπ‘‘πœ‡(𝐰)=𝔹𝑛𝑓(𝐰)π‘‘πœ‡π›Ό(𝐰)(1βˆ’βŸ¨π³,𝐰⟩)𝑛+𝛼+1,(1.15) where π³βˆˆπ”Ήπ‘› and π‘“βˆˆπΏ1(𝔹𝑛,(1βˆ’|𝐳|2)π›Όπ‘‘πœ‡).

For 𝛼,𝛽>βˆ’1, define the function 𝑃𝛼,𝛽(𝑓)(𝐳), for π³βˆˆπ”Ήπ‘› by:𝑃𝛼,𝛽𝑓(𝐳)=π‘π›Όξ€œπ”Ήπ‘›ξ€·π‘“(𝐰)1βˆ’|𝐰|2𝛼(1βˆ’βŸ¨π³,𝐰⟩)𝑛+𝛽+1ξ€œπ‘‘πœˆ(𝐰)=𝔹𝑛𝑓(𝐰)π‘‘πœˆπ›Ό(𝐰)(1βˆ’βŸ¨π³,𝐰⟩)𝑛+𝛽+1.(1.16) In case 𝛽=𝛼 we write 𝑃𝛼 instead of 𝑃𝛼,𝛼 and we have that 𝑃𝛼(πœ‡)(𝐳)=π‘‡π›Όπœ‡(1)(𝐳), where 1 stands for the constant function. For 𝛽>𝛼 the function 𝑃𝛼,𝛽(πœ‡) is equivalent to the (π›½βˆ’π›Ό) fractional derivative of 𝑃𝛼(πœ‡). The Bergman projection 𝑃𝛼 is the orthogonal form 𝐿2(𝔹𝑛,π‘‘πœˆπ›Ό) onto 𝐴2𝛼(𝔹𝑛) defined by:𝑃𝛼𝑓(𝐳)=π‘π›Όξ€œπ”Ήπ‘›πΎπ›Ό(𝐳,𝐰)𝑓(𝐰)π‘‘πœˆπ›Ό(𝐰).(1.17)

The Bergman projection 𝑃𝛼 naturally extends to an integral operator on 𝐿1(𝔹𝑛,π‘‘πœˆπ›Ό).

Toeplitz operators have been studied extensively on the Bergman spaces by many authors. For references, see [5, 6]. Boundedness and compactness of general Toeplitz operators π‘‡π›Όπœ‡ on the 𝛼-Bloch ℬ𝛼(𝔻) spaces have been investigated in [7] on the unit disk 𝔻 for 0<𝛼<∞. Also in [8], the authors extend the Toeplitz operator π‘‡π›Όπœ‡ to ℬ𝛼(𝔹𝑛) in the unit ball of ℂ𝑛 and completely characterize the positive Borel measure πœ‡ such that π‘‡π›Όπœ‡ is bounded or compact on ℬ𝛼(𝔹𝑛) with 1≀𝛼<2. Recently, in [9], general Toeplitz operators π‘‡π›Όπœ‡ on the analytic Besov 𝐡𝑝(𝔻) spaces with 1≀𝑝<∞ have been investigated. Under a prerequisite condition, the authors characterized complex measure πœ‡ on the unit disk 𝔻 for which π‘‡π›Όπœ‡ is bounded or compact on Besov space 𝐡𝑝(𝔻). For more details on several studies of different classes of Toeplitz operators we refer to [6, 10–16] and others.

In the present paper, we will extend the general Toeplitz operators π‘‡π›Όπœ‡ to 𝐡𝑝(𝔹𝑛) in the unit ball o f ℂ𝑛 and completely characterize the positive Borel measure πœ‡ such that π‘‡π›Όπœ‡ is bounded or compact on the 𝐡𝑝(𝔹𝑛) spaces with 2𝑛<𝑝<∞. The extension requires some different techniques from those used in [9].

Let 𝛽(β‹…,β‹…) be the Bergman metric on 𝔹𝑛. Denote the Bergman metric ball at 𝐚 by 𝐡(𝐚,π‘Ÿ)={π³βˆˆπ”Ήπ‘›βˆΆπ›½(𝐚,𝐳)<π‘Ÿ, where πšβˆˆπ”Ήπ‘› and π‘Ÿ>0}.

Lemma 1.2 (see [2, Theorem  2.23]). For fixed π‘Ÿ>0, there is a sequence {𝐰(𝑗)}βˆˆπ”Ήπ‘› such that (i)β‹ƒβˆžπ‘—=1𝐡(𝐰(𝑗),π‘Ÿ)=𝔹𝑛; (ii)there is a positive integer 𝑁 such that each π³βˆˆπ”Ήπ‘› is contained in at most 𝑁 of the sets 𝐡(𝐰(𝑗),2π‘Ÿ).

A positive Borel measure πœ‡ on the unit ball 𝔹𝑛 is said to be a Carleson measure for 𝐡𝑝(𝔹𝑛) if there exists 𝐢>0 such thatξ€œπ”Ήπ‘›||||𝑓(𝐳)π‘π‘‘πœ‡(𝐳)≀𝐢‖𝑓‖𝐡𝑝(𝔹𝑛),βˆ€π‘“βˆˆπ΅π‘ξ€·π”Ήπ‘›ξ€Έ.(1.18)

The following characterization of Carleson measures can be found in [2] or in [5]. A positive Borel measure πœ‡ on the unit ball 𝔹𝑛 is said to be a Carleson measure for the Bergman space 𝐴𝑝𝛼(𝔹𝑛) ifξ€œπ”Ήπ‘›||||𝑓(𝐳)π‘π‘‘πœˆπ›Ό(𝐳)≀𝐢‖𝑓‖𝑝𝐴𝑝𝛼𝔹𝑛,βˆ€π‘“βˆˆπ΄π‘π›Όξ€·π”Ήπ‘›ξ€Έ.(1.19) It is well known that a positive Borel measure πœ‡ is a (𝐴𝑝(𝔹𝑛),𝑝)-Carleson measure if and only if sup𝐰(𝑗)βˆˆπ”Ήπ‘›πœ‡ξ€·π΅ξ€·π°(𝑗),π‘Ÿξ€Έξ€Έπœˆξ€·π΅ξ€·π°(𝑗),π‘Ÿξ€Έξ€Έ<∞,(1.20)

where {𝐰(𝑗)} is the sequence in Lemma 1.2. If πœ‡ satisfies thatlimπ‘—β†’βˆžπœ‡ξ€·π΅ξ€·π°(𝑗),π‘Ÿξ€Έξ€Έπœˆξ€·π΅ξ€·π°(𝑗),π‘Ÿξ€Έξ€Έ=0,(1.21)

then πœ‡ is called vanishing Carleson measure for 𝐴𝑝(𝔹𝑛).

These two are special cases of a more general notion of Carleson measures on normed spaces of analytic functions.

In general, let πœ‡ be a positive measure on 𝔹𝑛 and 𝑋 a MΓΆbius invariant space. For 0<𝑝<1; then πœ‡ is an (𝑋,𝑝)-Carleson measure if there is a constant 𝐢>0 so that (see [2])ξ€œπ”Ήπ‘›||||𝑓(𝐳)π‘π‘‘πœ‡(𝐳)≀𝐢‖𝑓‖𝑝𝑋,βˆ€π‘“βˆˆπ‘‹.(1.22)

Also, defineβ€–πœˆβ€–π‘‹,𝑝=supπ‘“βˆˆπ‘‹,‖𝑓‖𝑋≀1ξ€œπ”Ήπ‘›||||𝑓(𝐳)π‘π‘‘πœ‡(𝐳).(1.23)

We say that πœ‡ is vanishing (𝑋,𝑝)-Carleson measure if for any sequence {𝑓𝑛}βˆˆπ‘‹ with ‖𝑓𝑛‖𝑋≀1 and such that 𝑓𝑛→0 uniformly on compact subset of 𝔹𝑛, we have thatlimπ‘›β†’βˆžξ€œπ”Ήπ‘›||𝑓𝑛(||𝐳)π‘π‘‘πœ‡(𝐳)=0.(1.24)

Throughout the paper, we will say that the expressions 𝐴 and 𝐡 are equivalent, and write π΄β‰ˆπ΅, whenever there exist positive constants 𝐢1 and 𝐢2 such that 𝐢1𝐴≀𝐡≀𝐢2𝐴. As usual, the letter 𝐢 will denote a positive constant, possibly different on each occurrence.

2. Bounded Toeplitz Operators on 𝐡𝑝(𝔹𝑛) Spaces

We are going to work with Toeplitz operators acting on Besov spaces 𝐡𝑝(𝔹𝑛) in the unit ball of ℂ𝑛.

We start with the following lemma.

Lemma 2.1. Let 0<𝑝<∞, βˆ’1<𝛼, 𝑑<∞. If 𝑃0,π›Όξ€œπ‘“(𝐳)=𝔹𝑛𝑓(𝐳)(1βˆ’βŸ¨π³,𝐰⟩)𝑛+𝛼+1π‘‘πœˆ(𝐳),(2.1)then 𝑃0,𝛼 is a bounded operator from 𝐿𝑝(𝔹𝑛,π‘‘πœˆπ‘‘) into 𝐴𝑝𝑑+𝑝𝛼(𝔹𝑛) if and only if βˆ’π‘π›Ό<𝑑+1<𝑝.

Proof. Let 𝑇𝑓(𝐳)=1βˆ’|𝐰|2𝛼𝑃0,𝛼=𝑓(𝐳)1βˆ’|𝐰|2ξ€Έπ›Όξ€œπ”Ήπ‘›π‘“(𝐳)(1βˆ’βŸ¨π³,𝐰⟩)𝑛+𝛼+1π‘‘πœˆ(𝐳).(2.2) By Theorem  2.10 in [2], we know that 𝑇 is bounded on 𝐿𝑝(𝔹𝑛,π‘‘πœˆπ‘‘) if and only if βˆ’π‘π›Ό<𝑑+1<𝑝. However, it is obvious that 𝑃0,𝛼 is bounded from 𝐿𝑝(𝔹𝑛,π‘‘πœˆπ‘‘) into 𝐴𝑝𝑑+𝑝𝛼(𝔹𝑛) if and only if 𝑇 is bounded on 𝐿𝑝(𝔹𝑛,π‘‘πœˆπ‘‘).

Theorem 2.2. Let 2𝑛<𝑝<∞,𝛼>βˆ’1 and let πœ‡ be a positive Borel measure on 𝔹𝑛. If πœ‡ is a (𝐴𝑝(𝔹𝑛),𝑝)-Carleson measure, then the Toeplitz operator π‘‡π›Όπœ‡ is bounded on 𝐡𝑝(𝔹𝑛) spaces if and only if 𝑃𝛼(πœ‡)(𝐰) is a (𝐡𝑝(𝔹𝑛),𝑝)-Carleson measure.

Proof. Let 2𝑛<𝑝,π‘ž<∞ where 1/𝑝+1/π‘ž=1 and let 𝛼>βˆ’1. We know that the dual spaces of 𝐡𝑝(𝔹𝑛) are π΅π‘ž(𝔹𝑛) under the paring βŸ¨π‘“,π‘”βŸ©=𝑓(0)ξ€œπ‘”(0)+π”Ήπ‘›β„œπ‘“(𝐳)β„œπ‘”(𝐳)π‘‘πœˆ(𝐳),π‘“βˆˆπ΅π‘ξ€·π”Ήπ‘›ξ€Έ,π‘”βˆˆπ΅π‘žξ€·π”Ήπ‘›ξ€Έ.(2.3) To prove the boundedness of π‘‡π›Όπœ‡, it suffices to show that ||ξ«π‘‡π›Όπœ‡ξ¬||(𝑓),𝑔≀𝐢‖𝑓‖𝐡𝑝(𝔹𝑛)β€–π‘”β€–π΅π‘ž(𝔹𝑛),(2.4) for all π‘“βˆˆπ΅π‘(𝔹𝑛) and π‘”βˆˆπ΅π‘ž(𝔹𝑛), where 𝐢 is a positive constant that does not depend on 𝑓 or 𝑔.
Now we define 𝐺(𝐰) by the following: 𝐺(𝐰)=𝐰𝑃0,𝛼+1ξ€œβ„œπ‘”(𝐰)=π°π”Ήπ‘›β„œπ‘”(𝐳)(1βˆ’βŸ¨π³,𝐰⟩)𝑛+𝛼+2π‘‘πœˆ(𝐳).(2.5) Then ξ«π‘‡π›Όπœ‡ξ¬π‘“,𝑔=π‘‡π›Όπœ‡π‘“(0)ξ€œπ‘”(0)+π”Ήπ‘›π‘‡π›Όπœ‡(β„œπ‘“)(𝐳)β„œπ‘”(𝐳)π‘‘πœˆ(𝐳)=π‘‡π›Όπœ‡π‘“(0)𝑔(0)+π‘π›Όξ€œπ”Ήπ‘›ξƒ©ξ€œπ”Ήπ‘›ξ€·1βˆ’|𝐰|2ξ€Έπ›Όβ„œπ‘“(𝐰)(1βˆ’βŸ¨π³,𝐰⟩)𝑛+𝛼+1ξƒͺπ‘‘πœ‡(𝐰)β„œπ‘”(𝐳)π‘‘πœˆ(𝐳).(2.6) Since 𝑓(𝑀)βˆ’π‘“(0)=𝑓(𝐰)βˆ’π‘ƒπ›Όξ‚€π‘“ξ‚(𝐰)=𝑓(𝐰)βˆ’π‘π›Όξ€œπ”Ήπ‘›ξ€·π‘“(𝐳)1βˆ’|𝐳|2𝛼(1βˆ’βŸ¨π³,𝐰⟩)𝑛+𝛼+1π‘‘πœˆ(𝐳)=π‘π›Όξ€œπ”Ήπ‘›ξ‚€π‘“(𝐰)βˆ’ξ‚π‘“(𝐳)(1βˆ’βŸ¨π³,𝐰⟩)𝑛+𝛼+1π‘‘πœˆπ›Ό(𝐳),(2.7) we have π‘‡π›Όπœ‡ξ€œπ‘“(0)=𝔹𝑛𝑓(𝐰)π‘‘πœ‡π›Ό(ξ€œπ°)=𝑓(0)π”Ήπ‘›π‘‘πœ‡π›Ό(𝐰)+𝑐2𝛼𝔹𝑛𝑓(𝐰)βˆ’ξ‚π‘“(𝐳)(1βˆ’βŸ¨π³,𝐰⟩)𝑛+𝛼+1π‘‘πœˆπ›Ό(𝐳)π‘‘πœ‡π›Ό(𝐰).(2.8) This implies ||π‘‡π›Όπœ‡||||||𝑓(0)≀𝐢𝑓(0)+𝑐2𝛼𝔹𝑛||||𝑓(𝐰)βˆ’π‘“(𝐳)||||1βˆ’βŸ¨π³,π°βŸ©π‘›+𝛼+1π‘‘πœˆπ›Ό(𝐳)π‘‘πœ‡π›Ό(𝐰).(2.9) By Proposition 1.1, we have 𝔹𝑛||||𝑓(𝐰)βˆ’π‘“(𝐳)||||1βˆ’βŸ¨π³,π°βŸ©π‘›+𝛼+1π‘‘πœˆπ›Ό(𝐳)π‘‘πœ‡π›Ό=ξƒ©ξ€œ(𝐰)𝔹𝑛1βˆ’|𝐳|2ξ€Έπ‘π›Όβˆ’π‘/2ξ€œπ”Ήπ‘›||||𝑓(𝐰)βˆ’π‘“(𝐳)𝑝1βˆ’|𝐳|2𝑝/2ξ€·1βˆ’|𝐰|2𝑝/2||||1βˆ’βŸ¨π³,π°βŸ©π‘Γ—ξ€·1βˆ’|𝐰|2ξ€Έπ‘π›Όβˆ’π‘/2(1βˆ’βŸ¨π³,𝐰⟩)𝑝(𝑛+𝛼)ξƒͺπ‘‘πœ‡(𝐰)π‘‘πœˆ(𝐳)1/𝑝≀𝐢‖𝑓‖𝐡𝑝(𝔹𝑛)ξ€œπ”Ήπ‘›ξ€·1βˆ’|𝐳|2ξ€Έπ›Όβˆ’(1/2)ξ€œπ”Ήπ‘›ξ€·1βˆ’|𝐰|2ξ€Έπ›Όβˆ’(1/2)||||1βˆ’βŸ¨π³,π°βŸ©π‘›+π›Όπ‘‘πœ‡(𝐰)π‘‘πœˆ(𝐳).(2.10) Since πœ‡ is a (𝐴𝑝(𝔹𝑛),𝑝)-Carleson measure, taking π›Όβˆ’1/2>βˆ’1, then as in [8] (see also Proposition  1.4.10 of [4]), we get ξ€·1βˆ’|𝐳|2ξ€Έβˆ’1/2ξ€œπ”Ήπ‘›ξ€·1βˆ’|𝐰|2ξ€Έπ›Όβˆ’1/2||||1βˆ’βŸ¨π³,π°βŸ©π‘›+π›Όπ‘‘πœ‡(𝐰)≀𝐢.(2.11) Then, ||π‘‡π›Όπœ‡||||||𝑓(0)≀𝐢𝑓(0)+𝐢‖𝑓‖𝐡𝑝(𝔹𝑛)ξ€œπ”Ήπ‘›π‘‘πœˆπ›Ό(𝐳)≀𝐢‖𝑓‖𝐡𝑝(𝔹𝑛).(2.12) Therefore ||π‘‡π›Όπœ‡||𝑓(0)𝑔(0)β‰€πΆβ€–π‘“β€–π΅π‘ž(𝔹𝑛)β€–π‘”β€–π΅π‘ž(𝔹𝑛).(2.13)
By Fubini’s Theorem we haveξ«π‘‡π›Όπœ‡ξ¬=ξ€œπ‘“,π‘”π”Ήπ‘›π‘‡π›Όπœ‡(β„œπ‘“)(𝐳)β„œπ‘”(𝐳)π‘‘πœˆ(𝐳)=π‘π›Όξ€œπ”Ήπ‘›ξƒ©ξ€œπ”Ήπ‘›ξ€·1βˆ’|𝐰|2ξ€Έπ›Όβ„œπ‘“(𝐰)(1βˆ’βŸ¨π³,𝐰⟩)𝑛+𝛼+1ξƒͺπ‘‘πœ‡(𝐰)β„œπ‘”(𝐳)π‘‘πœˆ(𝐳)=π‘π›Όξ€œπ”Ήπ‘›ξ€·π‘“(𝐰)1βˆ’|𝐰|2ξ€Έπ›Όξƒ©π°ξ€œπ”Ήπ‘›β„œπ‘”(𝐳)π‘‘πœˆ(𝐳)(1βˆ’βŸ¨π³,𝐰⟩)𝑛+𝛼+2ξƒͺπ‘‘πœ‡(𝐰)=π‘π›Όξ€œπ”Ήπ‘›π‘“(𝐰)𝐺(𝐰)1βˆ’|𝐰|2ξ€Έπ›Όπ‘‘πœ‡(𝐰).(2.14) Using the operator 𝑃𝛼, divide the integral ξ€œπ”Ήπ‘›π‘“(𝐰)𝐺(𝐰)1βˆ’|𝐰|2ξ€Έπ›Όξ€œπ‘‘πœ‡(𝐰)=𝔹𝑛𝑓(𝐰)𝐺(𝐰)1βˆ’|𝐰|2ξ€Έπ›Όπ‘‘πœ‡(𝐰),(2.15)we have ξ«π‘‡π›Όπœ‡ξ¬π‘“,𝑔=π‘π›Όξ€œπ”Ήπ‘›ξ‚ƒξ€·πΌβˆ’π‘ƒπ›Όξ€Έξ‚€π‘“πΊ(𝐰)1βˆ’|𝐰|2ξ€Έπ›Όπ‘‘πœ‡(𝐰)+π‘π›Όξ€œπ”Ήπ‘›π‘ƒπ›Όξ‚€π‘“πΊξ‚ξ€·(𝐰)1βˆ’|𝐰|2ξ€Έπ›Όπ‘‘πœ‡(𝐰)=𝐼1+𝐼2,(2.16)where 𝐼 is the identity operator, and ξ€·πΌβˆ’π‘ƒπ›Όξ€Έξ‚€π‘“πΊξ‚(𝐰)=𝑓(𝐰)𝐺(𝐰)βˆ’π‘π›Όξ€œπ”Ήπ‘›π‘“(𝐳)𝐺(𝐳)1βˆ’|𝐳|2𝛼(1βˆ’βŸ¨π³,𝐰⟩)𝑛+𝛼+1π‘‘πœˆ(𝐳)=π‘π›Όξ€œπ”Ήπ‘›(𝑓(𝐰)βˆ’π‘“(𝐳))𝐺(𝐳)1βˆ’|𝐳|2𝛼(1βˆ’βŸ¨π³,𝐰⟩)𝑛+𝛼+1π‘‘πœˆ(𝐳).(2.17)
By Proposition 1.1, we have ||𝐼1||=𝑐𝛼||||ξ€œπ”Ήπ‘›ξ‚ƒξ€·πΌβˆ’π‘ƒπ›Όξ€Έξ‚€π‘“πΊξ€·ξ‚ξ‚„(𝐰)1βˆ’|𝐰|2𝛼||||π‘‘πœ‡(𝐰)=𝑐2𝛼|||||𝔹𝑛(𝑓(𝐰)βˆ’π‘“(𝐳))𝐺(𝐳)1βˆ’|𝐳|2𝛼1βˆ’|𝐰|2𝛼(1βˆ’βŸ¨π³,𝐰⟩)𝑛+𝛼+1|||||π‘‘πœˆ(𝐳)π‘‘πœ‡(𝐰)=𝑐2π›Όξ€œπ”Ήπ‘›||||𝐺(𝐳)1βˆ’|𝐳|2ξ€Έπ›Όξ€œπ”Ήπ‘›||𝑓||ξ€·(𝐰)βˆ’π‘“(𝐳)1βˆ’|𝐰|2𝛼||||1βˆ’βŸ¨π³,π°βŸ©π‘›+𝛼+1π‘‘πœ‡(𝐰)π‘‘πœˆ(𝐳)=𝑐2π›Όξƒ©ξ€œπ”Ήπ‘›||||𝐺(𝐳)𝑝1βˆ’|𝐳|2ξ€Έπ‘π›Όβˆ’π‘/2Γ—ξ€œπ”Ήπ‘›||||𝑓(𝐰)βˆ’π‘“(𝐳)𝑝1βˆ’|𝐳|2𝑝/2ξ€·1βˆ’|𝐰|2𝑝/2||||1βˆ’βŸ¨π³,π°βŸ©π‘β‹…ξ€·1βˆ’|𝐰|2ξ€Έπ‘π›Όβˆ’π‘/2||||1βˆ’βŸ¨π³,π°βŸ©π‘(𝑛+𝛼)ξƒͺπ‘‘πœ‡(𝐰)π‘‘πœˆ(𝐳)1/𝑝≀𝐢‖𝑓‖𝐡𝑝(𝔹𝑛)ξ€œπ”Ήπ‘›||𝐺||ξ€·(𝐳)1βˆ’|𝐳|2ξ€Έπ›Όβˆ’1/2ξ€œπ”Ήπ‘›ξ€·1βˆ’|𝐰|2ξ€Έπ›Όβˆ’1/2||||1βˆ’βŸ¨π³,π°βŸ©π‘›+π›Όπ‘‘πœ‡(𝐰)π‘‘πœˆ(𝐳).(2.18)
By Lemma 2.1, the operator 𝑃0,𝛼 is bounded from 𝐿𝑝(𝔹𝑛,π‘‘πœˆπ‘‘) into 𝐴𝑝𝑑+𝑝𝛼(𝔹𝑛) whenever βˆ’π‘π›Ό<𝑑+1<𝑝. Since π‘”βˆˆπ΅π‘ž(𝔹𝑛) if and only if β„œπ‘”βˆˆπ΄π‘žπ‘žβˆ’2(𝔹𝑛), and we have from above, 𝑃0,𝛼+1 maps π΄π‘žπ‘žβˆ’2(𝔹𝑛) boundedly into π΄π‘ž(π‘žβˆ’2)+π‘ž(𝛼+1)(𝔹𝑛), whenever βˆ’π‘ž(𝛼+1)<(π‘žβˆ’2)+1<π‘ž, or π‘ž>1/(𝛼+2), which is always true if 𝛼>βˆ’1. Thus 𝐺(𝐰)βˆˆπ΄π‘žπ‘ž(𝛼+2)βˆ’2(𝔹𝑛). It can easily seen that 𝐺∈𝐴1𝛼(𝔹𝑛) and that ‖𝐺‖𝐴1𝛼(𝔹𝑛)β‰€πΆβ€–πΊβ€–π΄π‘žπ‘ž(𝛼+2)βˆ’2(𝔹𝑛)β‰€πΆβ€–π‘”β€–π΅π‘ž(𝔹𝑛). Thus ||𝐼1||≀𝐢‖𝑓‖𝐡𝑝(𝔹𝑛)ξ€œπ”Ήπ‘›||||𝐺(𝐳)1βˆ’|𝐳|2𝛼1βˆ’|𝐳|2ξ€Έβˆ’1/2ξ€œπ”Ήπ‘›ξ€·1βˆ’|𝐰|2ξ€Έπ›Όβˆ’1/2||||1βˆ’βŸ¨π³,π°βŸ©π‘›+𝛼ξƒͺπ‘‘πœ‡(𝐰)π‘‘πœˆ(𝐳)≀𝐢‖𝑓‖𝐡𝑝(𝔹𝑛)ξ€œπ”Ήπ‘›β€–πΊβ€–π΄1𝛼1βˆ’|𝐳|2ξ€Έβˆ’1/2ξ€œπ”Ήπ‘›ξ€·1βˆ’|𝐰|2ξ€Έπ›Όβˆ’1/2||||1βˆ’βŸ¨π³,π°βŸ©π‘›+π›Όπ‘‘πœ‡(𝐰)π‘‘πœˆ(𝐳)≀𝐢‖𝑓‖𝐡𝑝(𝔹𝑛)ξ€œπ”Ήπ‘›β€–π‘”β€–π΅π‘ž(𝔹𝑛)ξ€·1βˆ’|𝐳|2ξ€Έβˆ’1/2ξ€œπ”Ήπ‘›ξ€·1βˆ’|𝐰|2ξ€Έπ›Όβˆ’1/2||||1βˆ’βŸ¨π³,π°βŸ©π‘›+π›Όπ‘‘πœ‡(𝐰)π‘‘πœˆ(𝐳).(2.19)
By (2.11), we get ||𝐼1||≀𝐢‖𝑓‖𝐡𝑝(𝔹𝑛)β€–π‘”β€–π΅π‘ž(𝔹𝑛).(2.20)
Next consider 𝐼2, we have ||𝐼2||=𝑐𝛼||||ξ€œπ”Ήπ‘›π‘ƒπ›Όξ‚€π‘“πΊξ‚(𝐳)π‘‘πœ‡π›Ό||||(𝐳)=𝑐2𝛼|||||𝔹𝑛𝑓(𝐰)𝐺(𝐰)1βˆ’|𝐰|2𝛼||||1βˆ’βŸ¨π³,π°βŸ©π‘›+𝛼+1π‘‘πœˆ(𝐰)π‘‘πœ‡π›Ό|||||(𝐳)=π‘π›Όξ€œπ”Ήπ‘›||||||||𝑓(𝐰)𝐺(𝐰)1βˆ’|𝐰|2ξ€Έπ›Όπ‘π›Όξ€œπ”Ήπ‘›ξ€·1βˆ’|𝐳|2ξ€Έπ›Όπ‘‘πœ‡(𝐳)||||1βˆ’βŸ¨π³,π°βŸ©π‘›+𝛼+1ξ€œπ‘‘πœˆ(𝐰)≀𝐢𝔹𝑛‖𝐺‖𝐴1𝛼(𝔹𝑛)||𝑓||𝑃(𝐰)π›Όξ€œ(πœ‡)(𝐰)π‘‘πœˆ(𝐰)β‰€πΆπ”Ήπ‘›β€–π‘”β€–π΅π‘ž(𝔹𝑛)||||𝑃𝑓(𝐰)𝛼(πœ‡)(𝐰)π‘‘πœˆ(𝐰).(2.21) Therefore, π‘‡π›Όπœ‡ is bounded on 𝐡𝑝(𝔹𝑛) if and only if ξ€œπ”Ήπ‘›||||𝑃𝑓(𝐰)𝛼(πœ‡)(𝐰)π‘‘πœˆ(𝐰)≀𝐢‖𝑓‖𝐡𝑝(𝔹𝑛)(2.22) if and only if the measure 𝑃𝛼(πœ‡)(𝐰) is a (𝐡𝑝(𝔹𝑛),𝑝)-Carleson measure.

Now, we will characterize boundedness of Toeplitz operators on the minimal MΓΆbius invariant Banach spaces of holomorphic functions 𝐡1(𝔹𝑛) in the unit ball of ℂ𝑛.

Theorem 2.3. Let πœ‡ be a positive Borel measure on 𝔹𝑛. If πœ‡ is a (𝐴𝑝(𝔹𝑛),𝑝)-Carleson measure, then the Toeplitz operator π‘‡π›Όπœ‡ is bounded on 𝐡1(𝔹𝑛) spaces if and only if |π‘š|=𝑛+1|||πœ•π‘šπœ•π°π‘šπ‘ƒπ›Ό|||(πœ‡)(𝐰)π‘‘πœˆ(𝐰)(2.23) is a (𝐡1(𝔹𝑛),1)-Carleson measure.

Proof. We will use the fact that the dual spaces of 𝐡1(𝔹𝑛) are the Bloch space ℬ(𝔹𝑛) under the paring ξ€œβŸ¨π‘“,π‘”βŸ©=π”Ήπ‘›β„œπ‘“(𝐳)β„œπ‘”(𝐳)π‘‘πœˆ(𝐳),π‘“βˆˆπ΅1𝔹𝑛𝔹,π‘”βˆˆβ„¬π‘›ξ€Έ.(2.24) Similarly, as in the proof of Theorem 2.2, by duality, we have that π‘‡π›Όπœ‡ is bounded on 𝐡1(𝔹𝑛) spaces if and only if ||ξ«π‘‡π›Όπœ‡ξ¬||(𝑓),𝑔=𝑐𝛼||||ξ€œπ”Ήπ‘›π‘“(𝐰)𝐺(𝐰)1βˆ’|𝐰|2𝛼||||π‘‘πœ‡(𝐰)≀𝐢‖𝑓‖𝐡1(𝔹𝑛)‖𝑔‖ℬ(𝔹𝑛),(2.25) for all π‘“βˆˆπ΅1(𝔹𝑛) and π‘”βˆˆβ„¬(𝔹𝑛), where 𝐺(𝐰)=𝐰𝑃0,𝛼+1ξ€œβ„œπ‘”(𝐰)=π°π”Ήπ‘›β„œπ‘”(𝐳)(1βˆ’βŸ¨π³,𝐰⟩)𝑛+𝛼+2π‘‘πœˆ(𝐳).(2.26)
Using the fact that |||||ξ€œπ”Ήπ‘›β„œπ‘”(𝐳)||||1βˆ’βŸ¨π³,π°βŸ©π‘›+𝛼+2|||||β‰ˆ||||ξ€œπ‘‘πœˆ(𝐳)𝔹𝑛1βˆ’|𝐳|2ξ€Έβ„œπ‘”(𝐳)(1βˆ’βŸ¨π³,𝐰⟩)𝑛+𝛼+3||||π‘‘πœˆ(𝐳),(2.27) for π‘”βˆˆβ„¬(𝔹𝑛), we have that |𝐺(𝐰)|(1βˆ’|𝐰|2)𝛼+1<∞, which means that πΊβˆˆβ„¬π›Ό+2(𝔹𝑛). Now using the operator 𝑃𝛼+1, we have ξ«π‘‡π›Όπœ‡ξ¬π‘“,𝑔=π‘π›Όξ€œπ”Ήπ‘›ξ‚ƒξ€·πΌβˆ’π‘ƒπ›Ό+1𝑓𝐺(𝐰)1βˆ’|𝐰|2ξ€Έπ›Όπ‘‘πœ‡(𝐰)+π‘π›Όξ€œπ”Ήπ‘›π‘ƒπ›Ό+1𝑓𝐺(𝐰)1βˆ’|𝐰|2ξ€Έπ›Όπ‘‘πœ‡(𝐰)=𝐼1+𝐼2,ξ€·πΌβˆ’π‘ƒπ›Ό+1𝑓𝐺(𝐰)=𝑐𝛼+2ξ€œπ”Ήπ‘›(𝑓(𝐰)βˆ’π‘“(𝐳))𝐺(𝐳)1βˆ’|𝐳|2𝛼+1(1βˆ’βŸ¨π³,𝐰⟩)𝑛+𝛼+2π‘‘πœˆ(𝐳).(2.28)
By Proposition 1.1, we have ||𝐼1||=𝑐𝛼||||ξ€œπ”Ήπ‘›ξ‚ƒξ€·πΌβˆ’π‘ƒπ›Ό+1𝑓𝐺(𝐰)1βˆ’|𝐰|2𝛼||||π‘‘πœ‡(𝐰)=𝑐𝛼𝑐𝛼+1|||||𝔹𝑛(𝑓(𝐰)βˆ’π‘“(𝐳))𝐺(𝐳)1βˆ’|𝐳|2𝛼+2ξ€·1βˆ’|𝐰|2𝛼||||1βˆ’βŸ¨π³,π°βŸ©π‘›+𝛼+2|||||π‘‘πœˆ(𝐳)π‘‘πœ‡(𝐰)=𝑐𝛼𝑐𝛼+1ξ€œπ”Ήπ‘›||||𝐺(𝐳)1βˆ’|𝐳|2𝛼+1ξ€œπ”Ήπ‘›||𝑓||ξ€·(𝐰)βˆ’π‘“(𝐳)1βˆ’|𝐰|2𝛼||||1βˆ’βŸ¨π³,π°βŸ©π‘›+𝛼+2ξ€œπ‘‘πœ‡(𝐰)π‘‘πœˆ(𝐳)≀𝐢𝔹𝑛‖𝑓‖𝐡𝑝(𝔹𝑛)||||𝐺(𝐳)1βˆ’|𝐳|2𝛼+1/2ξ€œπ”Ήπ‘›ξ€·1βˆ’|𝐰|2ξ€Έπ›Όβˆ’1/2||||1βˆ’βŸ¨π³,π°βŸ©π‘›+𝛼+2ξ€œπ‘‘πœ‡(𝐰)π‘‘πœˆ(𝐳)≀𝐢𝔹𝑛‖𝑓‖𝐡𝑝(𝔹𝑛)‖𝐺‖𝐴1𝛼+1(𝔹𝑛)ξ€·1βˆ’|𝐳|2ξ€Έβˆ’1/2ξ€œπ”Ήπ‘›ξ€·1βˆ’|𝐰|2ξ€Έπ›Όβˆ’1/2||||1βˆ’βŸ¨π³,π°βŸ©π‘›+𝛼+1π‘‘πœ‡(𝐰)π‘‘πœˆ(𝐳)≀𝐢‖𝑓‖𝐡1(𝔹𝑛)‖𝑔‖ℬ(𝔹𝑛).(2.29)
Next consider 𝐼2, notice first that 𝑃𝛼(πœ‡)(𝐰)=π‘π›Όξ€œπ”Ήπ‘›π‘‘πœ‡π›Ό(𝐳)(1βˆ’βŸ¨π³,𝐰⟩)𝑛+𝛼+1;|π‘š|=𝑛+1|||πœ•π‘šπœ•π°π‘šπ‘ƒπ›Ό|||β‰ˆ|||||ξ€œ(πœ‡)(𝐰)π”Ήπ‘›ξ€·π³ξ€Έπ‘šπ‘‘πœ‡π›Ό(𝐳)||||1βˆ’βŸ¨π³,π°βŸ©π‘›+𝛼+2|||||.(2.30) Thus, ||𝐼2||=𝑐𝛼||||ξ€œπ”Ήπ‘›(𝐳)π‘šπ‘ƒπ›Ό+1𝑓𝐺(𝐳)π‘‘πœ‡π›Ό||||(𝐳)=𝑐𝛼𝑐𝛼+1|||||ξ€œπ”Ήπ‘›ξ€·π³ξ€Έπ‘šξ€œπ”Ήπ‘›π‘“(𝐰)𝐺(𝐰)1βˆ’|𝐰|2𝛼+1(1βˆ’βŸ¨π³,𝐰⟩)𝑛+𝛼+2π‘‘πœˆ(𝐰)π‘‘πœ‡π›Ό|||||(𝐳)=π‘π›Όξ€œπ”Ήπ‘›||||||||𝑓(𝐰)𝐺(𝐰)1βˆ’|𝐰|2𝛼+1𝑐𝛼+1ξ€œπ”Ήπ‘›ξ€·π³ξ€Έπ‘šξ€·1βˆ’|𝐳|2ξ€Έπ›Όπ‘‘πœ‡(𝐳)||||1βˆ’βŸ¨π³,π°βŸ©π‘›+𝛼+2ξƒͺπ‘‘πœˆ(𝐰)=π‘π›Όξ€œπ”Ήπ‘›||||||||𝑓(𝐰)𝐺(𝐰)1βˆ’|𝐰|2𝛼+1|π‘š|=𝑛+1|||πœ•π‘šπœ•π°π‘šπ‘ƒπ›Ό|||(πœ‡)(𝐰)π‘‘πœˆ(𝐰).(2.31) It is known that (𝐴1(𝔹𝑛))βˆ—=ℬ𝛽+1(𝔹𝑛) under the paring⟨𝐹,π»βŸ©π›½=π‘π›½ξ€œπ”Ήπ‘›πΉ(𝐰)𝐻(𝐰)1βˆ’|𝐰|2ξ€Έπ›½π‘‘πœ‡(𝐰),𝐹∈𝐴1𝔹𝑛,π»βˆˆβ„¬π›½+1𝔹𝑛.(2.32)
Since πΊβˆˆβ„¬π›Ό+2(𝔹𝑛),π‘”βˆˆβ„¬(𝔹𝑛) for by the above duality we get sup‖𝑔‖𝑛)ℬ(𝔹≀1||𝐼2||β‰ˆπΆsup‖𝑔‖𝑛)ℬ(𝔹≀1ξ€œπ”Ήπ‘›β€–πΊβ€–π΄1𝛼+1(𝔹𝑛)||||𝑓(𝐰)|π‘š|=𝑛+1|||πœ•π‘šπœ•π°π‘šπ‘ƒπ›Ό|||(πœ‡)(𝐰)π‘‘πœˆ(𝐰)≀𝐢sup‖𝑔‖𝑛)ℬ(𝔹≀1ξ€œπ”Ήπ‘›||𝑓||(𝐰)|π‘š|=𝑛+1|||πœ•π‘šπœ•π°π‘šπ‘ƒπ›Ό|||(πœ‡)(𝐰)π‘‘πœˆ(𝐰).(2.33)Therefore, π‘‡π›Όπœ‡ is bounded on 𝐡𝑝(𝔹𝑛) if and only if ξ€œπ”Ήπ‘›||||𝑓(𝐰)|π‘š|=𝑛+1|||πœ•π‘šπœ•π°π‘šπ‘ƒπ›Ό|||(πœ‡)(𝐰)π‘‘πœˆ(𝐰)≀𝐢‖𝑓‖𝐡𝑝(𝔹𝑛)(2.34) if and only if the measure βˆ‘|π‘š|=𝑛+1|(πœ•π‘š/πœ•π°π‘š)𝑃𝛼(πœ‡)(𝐰)|π‘‘πœˆ(𝐰) is a (𝐡𝑝(𝔹𝑛),𝑝)-Carleson measure.

3. Compact Toeplitz Operators on 𝐡𝑝(𝔹𝑛) Spaces

In this section we will characterize compact Toeplitz operators on 𝐡𝑝(𝔹𝑛) spaces in the unit ball of ℂ𝑛. We need the following lemma.

Lemma 3.1. Let 0<𝑝<∞, βˆ’1<𝛼 and π‘‡π›Όπœ‡ be bounded linear operator from 𝐡𝑝(𝔹𝑛) into 𝐡𝑝(𝔹𝑛) in the unit ball. Then π‘‡π›Όπœ‡ is compact on 𝐡𝑝(𝔹𝑛) spaces if and only if β€–π‘‡π›Όπœ‡π‘“π‘—β€–π΅π‘(𝔹𝑛)β†’0 as π‘—β†’βˆž wheneve r {𝑓𝑗} is a bounded sequence in 𝐡𝑝(𝔹𝑛) that converges to 0 uniformly on 𝔹𝑛.

Proof. This lemma can be proved by Montel’s Theorem.

Theorem 3.2. Let 2𝑛<𝑝<∞,𝛼>βˆ’1 and let πœ‡ be a positive Borel measure on 𝔹𝑛. If πœ‡ is a vanishing (𝐴𝑝(𝔹𝑛),𝑝)-Carleson measure, then the Toeplitz operator π‘‡π›Όπœ‡ is compact on 𝐡𝑝(𝔹𝑛) spaces if and only if 𝑃𝛼(πœ‡)(𝐰) is a vanishing (𝐡𝑝(𝔹𝑛),𝑝)-Carleson measure.

Proof. Let 2𝑛<𝑝,π‘ž<∞ where 1/𝑝+1/π‘ž=1 and let {𝑓𝑗} be a sequence in 𝐡𝑝(𝔹𝑛) satisfying ‖𝑓𝑗‖𝐡𝑝(𝔹𝑛)≀1 and such that 𝑓𝑗 converges to 0 uniformly as π‘—β†’βˆž on compact subsets of 𝔹𝑛, and let π‘”βˆˆπ΅π‘ž(𝔹𝑛). By duality, we have that π‘‡π›Όπœ‡ is compact on 𝐡𝑝(𝔹𝑛) if and only if limπ‘—β†’βˆžsup‖𝑔‖𝑛)π΅π‘ž(𝔹≀1||ξ«π‘‡π›Όπœ‡ξ€·π‘“π‘—ξ€Έξ¬||,𝑔=0.(3.1) As in the proof of Theorem 2.2, ξ«π‘‡π›Όπœ‡ξ€·π‘“π‘—ξ€Έξ¬,𝑔=π‘‡π›Όπœ‡π‘“π‘—(0)ξ€œπ‘”(0)+π”Ήπ‘›π‘‡π›Όπœ‡ξ€·β„œπ‘“π‘—ξ€Έ(𝐳)β„œπ‘”(𝐳)π‘‘πœˆ(𝐳)=π‘‡π›Όπœ‡π‘“π‘—(0)𝑔(0)+π‘π›Όξ€œπ”Ήπ‘›π‘“π‘—(𝐰)𝐺(𝐰)π‘‘πœ‡π›Ό(𝐰),(3.2) where 𝐺(𝐰)=𝐰𝑃0,𝛼+1ξ€œβ„œπ‘”(𝐰)=π°π”Ήπ‘›β„œπ‘”(𝐳)(1βˆ’βŸ¨π³,𝐰⟩)𝑛+𝛼+2π‘‘πœˆ(𝐳).(3.3) Also as in the proof of Theorem 2.2, ||π‘‡π›Όπœ‡||𝑓(0)≀𝐢‖𝑓‖𝐡𝑝(𝔹𝑛).(3.4)
Since |βˆ«π”Ήπ‘›π‘‘πœ‡π›Ό(𝐰)|<∞ and πœ‡ is a vanishing (𝐴𝑝(𝔹𝑛),𝑝)-Carleson measure, and 𝑓𝑗 converges to 0 uniformly as π‘—β†’βˆž on compact subsets of 𝔹𝑛, we get that π‘‡π›Όπœ‡π‘“(0)⟢0asπ‘—βŸΆβˆž.(3.5)
Thus π‘‡π›Όπœ‡ is compact on 𝐡𝑝(𝔹𝑛) if and only if limπ‘—β†’βˆžsup‖𝑔‖𝑛)π΅π‘ž(𝔹≀1||||ξ€œπ”Ήπ‘›π‘“π‘—(𝐰)𝐺(𝐰)π‘‘πœ‡π›Ό||||(𝐰)=0.(3.6) Using the operator 𝑃𝛼, we have thatξ€œπ”Ήπ‘›π‘“π‘—(𝐰)𝐺(𝐰)π‘‘πœ‡π›Ό(𝐰)=π‘π›Όξ€œπ”Ήπ‘›ξ‚ƒξ€·πΌβˆ’π‘ƒπ›Όξ€Έξ‚€π‘“π‘—πΊ(𝐳)π‘‘πœ‡π›Ό(𝐳)+π‘π›Όξ€œπ”Ήπ‘›π‘ƒπ›Όξ‚€π‘“π‘—πΊξ‚(𝐳)π‘‘πœ‡π›Ό(𝐳).=𝐽1+𝐽2.(3.7) For 0<π‘Ÿ<1 and π‘Ÿπ”Ήπ‘›={π³βˆˆβ„‚π‘›,|𝐳|β‰€π‘Ÿ}, we have ||𝐽1||=𝑐𝛼||||ξ€œπ”Ήπ‘›ξ‚ƒξ€·πΌβˆ’π‘ƒπ›Όξ€Έξ‚€π‘“π‘—πΊξ€·ξ‚ξ‚„(𝐰)1βˆ’|𝐰|2𝛼||||π‘‘πœ‡(𝐰)=𝑐2𝛼|||||𝔹𝑛𝑓𝑗(𝐰)βˆ’π‘“π‘—ξ€Έ(𝐳)𝐺(𝐳)1βˆ’|𝐳|2𝛼1βˆ’|𝐰|2𝛼(1βˆ’βŸ¨π³,𝐰⟩)𝑛+𝛼+1|||||π‘‘πœˆ(𝐳)π‘‘πœ‡(𝐰)=𝑐2π›Όξ‚΅ξ€œπ”Ήπ‘›β§΅π‘Ÿπ”Ήπ‘›+ξ€œπ‘Ÿπ”Ήπ‘›ξ‚Ά||||𝐺(𝐳)1βˆ’|𝐳|2ξ€Έπ›Όξ€œπ”Ήπ‘›||𝑓𝑗(𝐰)βˆ’π‘“π‘—||ξ€·(𝐳)1βˆ’|𝐰|2𝛼||||1βˆ’βŸ¨π³,π°βŸ©π‘›+𝛼+1π‘‘πœ‡(𝐰)π‘‘πœˆ(𝐳)=𝐿1+𝐿2.(3.8) For a fixed πœ€>0, since πœ‡ is a vanishing (𝐴𝑝(𝔹𝑛),𝑝)-Carleson measure, let π‘Ÿ sufficiently close to 1 so that ξ€·1βˆ’|𝐳|2ξ€Έβˆ’1/2ξ€œπ”Ήπ‘›β§΅π‘Ÿπ”Ήπ‘›ξ€·1βˆ’|𝐰|2ξ€Έπ›Όβˆ’1/2||||1βˆ’βŸ¨π³,π°βŸ©π‘›+π›Όπ‘‘πœ‡(𝐰)𝑑<πœ€.(3.9) Similarly, as in the proof of Theorem 2.2, by Proposition 1.1, 𝐿1=𝑐2π›Όξ€œπ”Ήπ‘›β§΅π‘Ÿπ”Ήπ‘›||||𝐺(𝐳)1βˆ’|𝐳|2ξ€Έπ›Όξ€œπ”Ήπ‘›||𝑓𝑗(𝐰)βˆ’π‘“π‘—(||𝐳)1βˆ’|𝐰|2𝛼||||1βˆ’βŸ¨π³,π°βŸ©π‘›+𝛼+1ξ€œπ‘‘πœ‡(𝐰)π‘‘πœˆ(𝐳)β‰€πΆπ”Ήπ‘›β§΅π‘Ÿπ”Ήπ‘›β€–β€–π‘“π‘—β€–β€–π΅π‘(𝔹𝑛)||||𝐺(𝐳)1βˆ’|𝐳|2ξ€Έπ›Όβˆ’1/2ξ€œπ”Ήπ‘›ξ€·1βˆ’|𝐰|2ξ€Έπ›Όβˆ’1/2||||1βˆ’βŸ¨π³,π°βŸ©π‘›+π›Όβ€–β€–π‘“π‘‘πœ‡(𝐰)π‘‘πœˆ(𝐳)β‰€πΆπœ€β€–π‘—β€–β€–π΅π‘(𝔹𝑛)‖𝐺‖𝐴1𝛼(𝔹𝑛)β€–β€–π‘“β‰€πΆπœ€π‘—β€–β€–π΅π‘(𝔹𝑛)β€–π‘”β€–π΅π‘ž(𝔹𝑛)β‰€πœ€.(3.10)
Since 𝑓𝑗→0 as π‘—β†’βˆž on compact subsets of 𝔹𝑛, we cane choose 𝑗 big enough so that ||||𝐺(𝐳)1βˆ’|𝐳|2𝛼<πœ€.(3.11) Therefore,𝐿2=𝑐2π›Όξ€œπ‘Ÿπ”Ήπ‘›||||𝐺(𝐳)1βˆ’|𝐳|2ξ€Έπ›Όξ€œπ”Ήπ‘›||𝑓𝑗(𝐰)βˆ’π‘“π‘—(||𝐳)1βˆ’|𝐰|2𝛼||||1βˆ’βŸ¨π³,π°βŸ©π‘›+𝛼+1ξ€œπ‘‘πœ‡(𝐰)π‘‘πœˆ(𝐳)β‰€πΆπ‘Ÿπ”Ήπ‘›β€–β€–π‘“π‘—β€–β€–π΅π‘ž(𝔹𝑛)||||𝐺(𝐳)1βˆ’|𝐳|2ξ€Έπ›Όβˆ’1/2ξ€œπ”Ήπ‘›ξ€·1βˆ’|𝐰|2ξ€Έπ›Όβˆ’(1/2)||||1βˆ’βŸ¨π³,π°βŸ©π‘›+π›Όπ‘‘πœ‡(𝐰)π‘‘πœˆ(𝐳)β‰€πΆπœ€β€–πΊβ€–π΄1𝛼(𝔹𝑛)β‰€πΆπœ€β€–π‘”β€–π΅π‘ž(𝔹𝑛).(3.12)
Hence |𝐽1|<πΆπœ€, where 𝐢 does not depend on 𝑔(𝐳), and so limπ‘—β†’βˆžsup‖𝑔‖𝑛)π΅π‘ž(𝔹≀1||𝐽1||=0.(3.13) Thus, π‘‡π›Όπœ‡ is compact on 𝐡𝑝(𝔹𝑛) if and only iflimπ‘—β†’βˆžsup‖𝑔‖𝑛)π΅π‘ž(𝔹≀1||𝐽2||=0.(3.14) Again, as in the proof of Theorem 2.2, we have ||𝐽2||=𝑐𝛼||||ξ€œπ”Ήπ‘›π‘ƒπ›Όξ‚€π‘“π‘—πΊξ‚(𝐳)π‘‘πœ‡π›Ό||||(𝐳)=𝑐2𝛼|||||𝔹𝑛𝑓𝑗(𝐰)𝐺(𝐰)1βˆ’|𝐰|2𝛼(1βˆ’βŸ¨π³,𝐰⟩)𝑛+𝛼+1π‘‘πœˆ(𝐰)π‘‘πœ‡π›Ό|||||(𝐳)=π‘π›Όξ€œπ”Ήπ‘›||𝑓𝑗||||||ξ€·(𝐰)𝐺(𝐰)1βˆ’|𝐰|2ξ€Έπ›Όπ‘π›Όξ€œπ”Ήπ‘›ξ€·1βˆ’|𝐳|2ξ€Έπ›Όπ‘‘πœ‡(𝐳)(1βˆ’βŸ¨π³,𝐰⟩)𝑛+𝛼+1ξ€œπ‘‘πœˆ(𝐰)≀𝐢𝔹𝑛‖𝐺‖𝐴1𝛼(𝔹𝑛)||𝑓𝑗||𝑃(𝐰)π›Όξ€œ(πœ‡)(𝐰)π‘‘πœˆ(𝐰)β‰€πΆπ”Ήπ‘›β€–π‘”β€–π΅π‘ž(𝔹𝑛)||𝑓𝑗||𝑃(𝐰)𝛼(πœ‡)(𝐰)π‘‘πœˆ(𝐰).(3.15) Therefore, π‘‡π›Όπœ‡ is compact on 𝐡𝑝(𝔹𝑛) if and only if limπ‘—β†’βˆžξ€œπ”Ήπ‘›||𝑓𝑗(||𝑃𝐰)𝛼(πœ‡)(𝐰)π‘‘πœˆ(𝐰)=0,(3.16) which is equivalent to say that 𝑃𝛼(πœ‡)(𝐰) is a vanishing (𝐡𝑝(𝔹𝑛),𝑝)-Carleson measure.

Theorem 3.3. Let πœ‡ be a positive Borel measure on 𝔹𝑛. If πœ‡ is a (𝐴𝑝(𝔹𝑛),𝑝)-Carleson measure, then the Toeplitz operator π‘‡π›Όπœ‡ is compact on 𝐡1(𝔹𝑛) spaces if and only if |π‘š|=𝑛+1||||πœ•π‘šπ‘ƒπ›Ό(πœ‡)πœ•π°π‘š||||(𝐰)π‘‘πœˆ(𝐰)(3.17) is a vanishing (𝐡1(𝔹𝑛),1)-Carleson measure.

Proof. Let {𝑓𝑗} be a sequence in 𝐡𝑝(𝔹𝑛) satisfying ‖𝑓𝑗‖𝐡1(𝔹𝑛)≀1 and such that 𝑓𝑗 converges to 0 uniformly as π‘—β†’βˆž on compact subsets of 𝔹𝑛, and let π‘”βˆˆβ„¬(𝔹𝑛). By duality, we have that π‘‡π›Όπœ‡ is compact on 𝐡1(𝔹𝑛) if and only if limπ‘—β†’βˆžsup‖𝑔‖𝑛)ℬ(𝔹≀1||ξ«π‘‡π›Όπœ‡ξ€·π‘“π‘—ξ€Έξ¬||,𝑔=0.(3.18) Thus, π‘‡π›Όπœ‡ is compact on 𝐡1(𝔹𝑛) if and only if limπ‘—β†’βˆžsup‖𝑔‖𝑛)ℬ(𝔹≀1||||ξ€œπ”Ήπ‘›π‘“π‘—(𝐰)𝐺(𝐰)π‘‘πœ‡π›Ό||||(𝐰)=0.(3.19) Using the operator 𝑃𝛼, we have that ξ€œπ”Ήπ‘›π‘“π‘—(𝐰)𝐺(𝐰)π‘‘πœ‡π›Ό(𝐰)=π‘π›Όξ€œπ”Ήπ‘›ξ‚ƒξ€·πΌβˆ’π‘ƒπ›Όξ€Έξ‚€π‘“π‘—πΊ(𝐳)π‘‘πœ‡π›Ό(𝐳)+π‘π›Όξ€œπ”Ήπ‘›π‘ƒπ›Όξ‚€π‘“π‘—β„Žξ‚(𝐳)π‘‘πœ‡π›Ό(𝐳).=𝐽1+𝐽2.(3.20) As in the proof of Theorem 2.3, we have ||𝐽1||ξ€œβ‰€πΆπ”Ήπ‘›β€–β€–π‘“π‘—β€–β€–π΅1(𝔹𝑛)‖𝐺‖𝐴1(𝔹)ξ€·1βˆ’|𝐳|2ξ€Έβˆ’1/2ξ€œπ”Ήπ‘›ξ€·1βˆ’|𝐰|2ξ€Έπ›Όβˆ’1/2||||1βˆ’βŸ¨π³,π°βŸ©π‘›+𝛼+1π‘‘πœ‡(𝐰)π‘‘πœˆ(𝐳).(3.21) Notice that ‖𝑓𝑗‖𝐡𝑝(𝔹𝑛) implies that ‖𝑓𝑗‖𝐡1(𝔹𝑛)≀𝐢. Since 𝑓𝑗 converges to 0 uniformly as π‘—β†’βˆž on compact subsets of 𝔹𝑛, and πœ‡ is a (𝐴𝑝(𝔹𝑛),𝑝)-Carleson measure, we get that πΊβˆˆβ„¬π›Ό+2(𝔹𝑛) and ‖𝐺‖ℬ𝛼+2(𝔹𝑛)≀𝐢‖𝑔‖ℬ(𝔹𝑛). Thus ||𝐽1||‖‖𝑓≀𝐢𝑗‖‖𝐡1(𝔹𝑛)‖𝑔‖ℬ(𝔹𝑛),limπ‘—β†’βˆžsup‖𝑔‖ℬ≀1||𝐽1||=0.(3.22) Therefore, π‘‡π›Όπœ‡ is compact on 𝐡1(𝔹𝑛) if and only if limπ‘—β†’βˆžsup‖𝑔‖𝑛)ℬ(𝔹≀1||𝐽2||=0.(3.23) We have shown in the proof of Theorem 2.3sup‖𝑔‖𝑛)ℬ(𝔹≀1||𝐽2||≀𝐢sup‖𝑔‖𝑛)ℬ(𝔹≀1ξ€œπ”Ήπ‘›||𝑓𝑗(||𝐰)β„œπ‘ƒπ›Ό(πœ‡)(𝐰)π‘‘πœˆ(𝐰).(3.24) Therefore, π‘‡π›Όπœ‡ is compact on 𝐡1(𝔹𝑛) if and only if limπ‘—β†’βˆžξ€œπ”Ήπ‘›||𝑓𝑗||(𝐰)|π‘š|=𝑛+1|||πœ•π‘šπœ•π°π‘šπ‘ƒπ›Ό|||(πœ‡)(𝐰)π‘‘πœˆ(𝐰)=0,(3.25) which is equivalent to saying that the measure βˆ‘|π‘š|=𝑛+1|(πœ•π‘š/πœ•π°π‘š)𝑃𝛼(πœ‡)(𝐰)|π‘‘πœˆ(𝐰) is a vanishing (𝐡1(𝔹𝑛),1)-Carleson measure.

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