# Properties of Toeplitz Operators on Some Holomorphic Banach Function Spaces

Authors: Ahmed El-Sayed Ahmed; Mahmoud Ali Bakhit
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

We characterize complex measuresμ on the unit ball of ℂn, for which the general Toeplitz operator Tμα is bounded or compact on the analytic Besov spaces Bp𝔹n, also on the minimal Möbius invariant Banach spaces B1𝔹n in the unit ball 𝔹n.


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## Body

## 1. Introduction

Let𝔹n 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 ℋ(𝔹n). The ball centered at  z  with radius  r  will be denoted by  B(z,r).  For α>-1, let  dνα(z)=cα(1-|z|2)αdν, where  dν  is the normalized Lebesgue volume measure on  𝔹n  and cα=Γ(n+α+1)/n!Γ(α+1) (where Γ denotes the Gamma function) so that να(𝔹n)≡1.For anyz=(z1,z2,…,zn), w=(w1,w2,…,wn)∈ℂn, the inner product is defined by  〈z,w〉=∑k=1nzkw¯k.  For  f∈ℋ(𝔹n),  we write(1.1)∇f(z)=(∂f(z)∂z1,∂f(z)∂z2,…,∂f(z)∂zn),Rf(z)=〈∇f,z¯〉=∑j=1nzj∂f(z)∂zj.
For  f∈ℋ(𝔹n)  and  z∈𝔹n,  set(1.2)Qf(z)=sup⁡⁡w∈Cn∖{0}|〈∇f(z),w̅〉|Hz(w,w),
where  Hz(w,w)  is the Bergman metric on  𝔹n, that is,(1.3)Hz(w,w)=(n+12)(1-|z|2)|w|2+|〈w,z〉|2(1-|z|2)2.
For   1<p<∞,  the Besov spaces Bp(𝔹n)  consists of all functions  f∈ℋ(𝔹n)  for which (see [1])(1.4)‖f‖Bp(Bn)p:=∫BnQf(z)pdν(z)<∞.
From [1], we know that for n≥2,  the Besov space is nontrivial if and only if  p>2n.The analytic Besov space is the minimal Möbius invariant Banach spaceB1(𝔹n) (see [2]) defined by(1.5)‖f‖B1(Bn):=∑|m|=n+1sup⁡⁡z∈Bn∫Bn|∂mf(z)∂zm|dν(z)<∞.
For α≥0,  a function  f∈ℋ(𝔹n)  is said to belong to the  α-Bloch spaces ℬα(𝔹n) if (see [3])(1.6)bα=sup⁡⁡z∈Bn|∇f(z)|(1-|z|2)α<∞.
The little Bloch space  ℬ0α(𝔹n)  consists of all  f∈ℬα(𝔹n)  such that(1.7)lim⁡|z|→1⁡|∇f(z)|(1-|z|2)α=0.
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 ℬ01 become the Bloch and the little Bloch space (see, e.g., [2]).For every pointa∈𝔹n,  the Möbius transformation  φa:𝔹n→𝔹n  is defined by(1.8)φa(z)=a-Pa(z)-SaQa(z)1-〈z,a〉,z∈Bn,
where  Sa=1-|a|2,Pa(z)=a〈z,a〉/|a|2,P0=0  and Qa=I-Pa  (see, e.g., [2] or [4]). The map  φa  has the following properties that  φa(0)=a, φa(a)=0, φa=φa-1 and(1.9)1-〈φa(z),φa(w)〉=(1-|a|2)(1-〈z,w〉)(1-〈z,a〉)(1-〈a,w〉),
where z  and w are arbitrary points in  𝔹n.  In particular,(1.10)1-|φa(z)|2=(1-|a|2)(1-|z|2)|1-〈z,a〉|2.The following result can be found in [3].Proposition 1.1.
Letf∈ℋ(𝔹n),2n<p<∞.  Then  f∈Bp(𝔹n)  if and only if
(1.11)∬Bn(|f(w)-f(z)||1-〈z,w〉|)p(1-|z|2)p/2(1-|w|2)p/2dμ(w)dν(z)<∞.Forα>-1  and  0<p<∞,  the weighted Bergman spaces  Aαp(𝔹n)  consists of all functions  f∈ℋ(𝔹n)   for which(1.12)‖f‖Aαpp:=∫Bn|f(z)|pdνα(z)<∞.
It is clear that   Aαp=Lp(𝔹n,dνα)⋂ℋ(𝔹n)  and  Aαp  is a linear subspace of Lp(𝔹n,dνα).  When α=0,  we simply write  Ap(𝔹n)  for  A0p(𝔹n).  In the special case when p=2,Aα2(𝔹n) is a Hilbert space. It is well known that for  α>-1  the Bergman kernel of Aα2(𝔹n) is given by(1.13)Kα(z,w)=1(1-〈z,w〉)n+1+α,z,w∈Bn.Forα>-1,  a complex measure μ  such that(1.14)|∫Bn(1-|w|2)αdμ(w)|=|∫Bndμα(w)|<∞
define a Toeplitz operator as follows:(1.15)Tμαf(z)=cα∫Bn(1-|w|2)αf(w)(1-〈z,w〉)n+α+1dμ(w)=∫Bnf(w)dμα(w)(1-〈z,w〉)n+α+1,
where z∈𝔹n  and  f∈L1(𝔹n,(1-|z|2)αdμ).Forα,β>-1,  define the function  Pα,β(f)(z),  for  z∈𝔹n  by:(1.16)Pα,βf(z)=cα∫Bnf(w)(1-|w|2)α(1-〈z,w〉)n+β+1dν(w)=∫Bnf(w)dνα(w)(1-〈z,w〉)n+β+1.
In case  β=α  we write Pα  instead of Pα,α  and we have that Pα(μ)(z)=Tμα(1)(z), where  1  stands for the constant function. For β>α  the function Pα,β(μ)  is equivalent to the (β-α)  fractional derivative of Pα(μ). The Bergman projection Pα  is the orthogonal form L2(𝔹n,dνα)  onto  Aα2(𝔹n)  defined by:(1.17)Pαf(z)=cα∫BnKα(z,w)f(w)dνα(w).The Bergman projectionPα  naturally extends to an integral operator on  L1(𝔹n,dνα).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  Tμα  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 Tμα to ℬα(𝔹n)  in the unit ball of ℂn  and completely characterize the positive Borel measure μ  such that  Tμα  is bounded or compact on ℬα(𝔹n) with 1≤α<2. Recently, in [9], general Toeplitz operators  Tμα  on the analytic Besov Bp(𝔻)  spaces with 1≤p<∞  have been investigated. Under a prerequisite condition, the authors characterized complex measure μ on the unit disk  𝔻  for which Tμα is bounded or compact on Besov space Bp(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 operatorsTμα to  Bp(𝔹n)  in the unit ball o f ℂn  and completely characterize the positive Borel measure μ such that  Tμα  is bounded or compact on the  Bp(𝔹n)  spaces with  2n<p<∞.  The extension requires some different techniques from those used in [9].Letβ(·,·) be the Bergman metric on  𝔹n. Denote the Bergman metric ball at  a by  B(a,r)={z∈𝔹n:β(a,z)<r,   where a∈𝔹n   and   r>0}.Lemma 1.2 (see [2, Theorem  2.23]).
For fixedr>0,  there is a sequence  {w(j)}∈𝔹n  such that (i)
⋃j=1∞B(w(j),r)=𝔹n;(ii)
there is a positive integerN such that each  z∈𝔹n  is contained in at most  N  of the sets B(w(j),2r).A positive Borel measureμ on the unit ball  𝔹n  is said to be a Carleson measure for Bp(𝔹n) if there exists  C>0  such that(1.18)∫Bn|f(z)|pdμ(z)≤C‖f‖Bp(Bn),∀f∈Bp(Bn).The following characterization of Carleson measures can be found in [2] or in [5]. A positive Borel measure μ on the unit ball  𝔹n  is said to be a Carleson measure for the Bergman space  Aαp(𝔹n)  if(1.19)∫Bn|f(z)|pdνα(z)≤C‖f‖Aαp(Bn)p,∀f∈Aαp(Bn).
It is well known that a positive Borel measure μ is a  (Ap(𝔹n),p)-Carleson measure if and only if (1.20)sup⁡⁡w(j)∈Bnμ(B(w(j),r))ν(B(w(j),r))<∞,where{w(j)}  is the sequence in Lemma 1.2. If μ  satisfies that(1.21)lim⁡j→∞⁡μ(B(w(j),r))ν(B(w(j),r))=0,thenμ  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, letμ  be a positive measure on  𝔹n  and  X  a Möbius invariant space. For  0<p<1;  then μ is an (X,p)-Carleson measure if there is a constant  C>0  so that (see [2])(1.22)∫Bn|f(z)|pdμ(z)≤C‖f‖Xp,∀f∈X.Also, define(1.23)‖ν‖X,p=sup⁡⁡f∈X,‖f‖X≤1∫Bn|f(z)|pdμ(z).We say thatμ  is vanishing  (X,p)-Carleson measure if for any sequence  {fn}∈X  with ∥fn∥X≤1  and such that  fn→0  uniformly on compact subset of  𝔹n,  we have that(1.24)lim⁡n→∞⁡∫Bn|fn(z)|pdμ(z)=0.Throughout the paper, we will say that the expressionsA  and  B  are equivalent, and write   A≈B,   whenever there exist positive constants  C1 and  C2 such that  C1A≤B≤C2A.  As usual, the letter  C  will denote a positive constant, possibly different on each occurrence.


## 2. Bounded Toeplitz Operators onBp(𝔹n) Spaces

We are going to work with Toeplitz operators acting on Besov spacesBp(𝔹n)  in the unit ball of  ℂn.We start with the following lemma.Lemma 2.1.
Let0<p<∞, -1<α, t<∞.  If
(2.1)P0,αf(z)=∫Bnf(z)(1-〈z,w〉)n+α+1dν(z),then  P0,α   is a bounded operator from  Lp(𝔹n,dνt)   into  At+pαp(𝔹n)  if and only if  -pα<t+1<p.Proof.
Let(2.2)Tf(z)=(1-|w|2)αP0,αf(z)=(1-|w|2)α∫Bnf(z)(1-〈z,w〉)n+α+1dν(z).
By Theorem  2.10 in [2], we know that  T  is bounded on  Lp(𝔹n,dνt)   if and only if -pα<t+1<p.  However, it is obvious that  P0,α  is bounded from Lp(𝔹n,dνt) into At+pαp(𝔹n)  if and only if T is bounded on Lp(𝔹n,dνt).Theorem 2.2.
Let2n<p<∞,α>-1 and let  μ be a positive Borel measure on   𝔹n.  If   μ  is a  (Ap(𝔹n),p)-Carleson measure, then the Toeplitz operator  Tμα is bounded on  Bp(𝔹n) spaces if and only if   Pα(μ)(w) is a   (Bp(𝔹n),p)-Carleson measure.Proof.
Let2n<p,q<∞ where 1/p+1/q=1 and let  α>-1. We know that the dual spaces of  Bp(𝔹n) are Bq(𝔹n) under the paring
(2.3)〈f,g〉=f(0)g(0)¯+∫BnRf(z)Rg(z)¯dν(z),f∈Bp(Bn),g∈Bq(Bn).
To prove the boundedness of Tμα, it suffices to show that
(2.4)|〈Tμα(f),g〉|≤C‖f‖Bp(Bn)‖g‖Bq(Bn),
for all f∈Bp(𝔹n)  and  g∈Bq(𝔹n), where C is a positive constant that does not depend on  f or  g.
Now we defineG(w)  by the following:
(2.5)G(w)=wP0,α+1Rg(w)=w∫BnRg(z)(1-〈z,w〉)n+α+2dν(z).
Then
(2.6)〈Tμαf,g〉=Tμαf(0)g(0)¯+∫BnTμα(Rf)(z)Rg(z)¯dν(z)=Tμαf(0)g(0)¯+cα∫Bn(∫Bn(1-|w|2)αRf(w)(1-〈z,w〉)n+α+1dμ(w))Rg(z)¯dν(z).
Since
(2.7)f(w)¯-f(0)¯=f(w)¯-Pα(f¯)(w)=f(w)¯-cα∫Bnf(z)¯(1-|z|2)α(1-〈z,w〉)n+α+1dν(z)=cα∫Bn(f(w)¯-f(z)¯)(1-〈z,w〉)n+α+1dνα(z),
we have
(2.8)Tμαf(0)=∫Bnf(w)dμα(w)=f(0)∫Bndμα(w)+cα2∬Bn(f(w)¯-f(z)¯)(1-〈z,w〉)n+α+1dνα(z)dμα(w).
This implies
(2.9)|Tμαf(0)|≤C|f(0)|+cα2∬Bn|f(w)-f(z)||1-〈z,w〉|n+α+1dνα(z)dμα(w).
By Proposition 1.1, we have
(2.10)∬Bn|f(w)-f(z)||1-〈z,w〉|n+α+1dνα(z)dμα(w)=(∫Bn(1-|z|2)pα-p/2∫Bn|f(w)-f(z)|p(1-|z|2)p/2(1-|w|2)p/2|1-〈z,w〉|p×(1-|w|2)pα-p/2(1-〈z,w〉)p(n+α)dμ(w)dν(z))1/p≤C‖f‖Bp(Bn)∫Bn(1-|z|2)α-(1/2)∫Bn(1-|w|2)α-(1/2)|1-〈z,w〉|n+αdμ(w)dν(z).
Since  μ  is a  (Ap(𝔹n),p)-Carleson measure, taking  α-1/2>-1,  then as in [8] (see also Proposition  1.4.10 of [4]), we get
(2.11)(1-|z|2)-1/2∫Bn(1-|w|2)α-1/2|1-〈z,w〉|n+αdμ(w)≤C.
Then,
(2.12)|Tμαf(0)|≤C|f(0)|+C‖f‖Bp(Bn)∫Bndνα(z)≤C‖f‖Bp(Bn).
Therefore
(2.13)|Tμαf(0)g(0)|≤C‖f‖Bq(Bn)‖g‖Bq(Bn).
By Fubini’s Theorem we have(2.14)〈Tμαf,g〉=∫BnTμα(Rf)(z)Rg(z)¯dν(z)=cα∫Bn(∫Bn(1-|w|2)αRf(w)(1-〈z,w〉)n+α+1dμ(w))Rg(z)¯dν(z)=cα∫Bnf(w)(1-|w|2)α(w¯∫BnRg(z)¯dν(z)(1-〈z,w〉)n+α+2)dμ(w)=cα∫Bnf(w)G(w)¯(1-|w|2)αdμ(w).
Using the operator  Pα,  divide the integral
(2.15)∫Bnf(w)G(w)¯(1-|w|2)αdμ(w)=∫Bnf(w)¯G(w)(1-|w|2)αdμ¯(w),we have
(2.16)〈Tμαf,g〉=cα∫Bn[(I-Pα)(fG¯)](w)(1-|w|2)αdμ(w)+cα∫BnPα(fG¯)(w)(1-|w|2)αdμ(w)=I1+I2,where  I  is the identity operator, and
(2.17)(I-Pα)(fG¯)(w)=f(w)G(w)¯-cα∫Bnf(z)G(z)¯(1-|z|2)α(1-〈z,w〉)n+α+1dν(z)=cα∫Bn(f(w)-f(z))G(z)¯(1-|z|2)α(1-〈z,w〉)n+α+1dν(z).
By Proposition1.1, we have
(2.18)|I1|=cα|∫Bn[(I-Pα)(fG¯)](w)(1-|w|2)αdμ(w)|=cα2|∬Bn(f(w)-f(z))G(z)¯(1-|z|2)α(1-|w|2)α(1-〈z,w〉)n+α+1dν(z)dμ(w)|=cα2∫Bn|G(z)|(1-|z|2)α∫Bn|f(w)-f(z)|(1-|w|2)α|1-〈z,w〉|n+α+1dμ(w)dν(z)=cα2((1-|w|2)pα-p/2|1-〈z,w〉|p(n+α)∫Bn|G(z)|p(1-|z|2)pα-p/2×∫Bn|f(w)-f(z)|p(1-|z|2)p/2(1-|w|2)p/2|1-〈z,w〉|p⋅(1-|w|2)pα-p/2|1-〈z,w〉|p(n+α)dμ(w)dν(z))1/p≤C‖f‖Bp(Bn)∫Bn|G(z)|(1-|z|2)α-1/2∫Bn(1-|w|2)α-1/2|1-〈z,w〉|n+αdμ(w)dν(z).
By Lemma2.1, the operator  P0,α  is bounded from  Lp(𝔹n,dνt)  into At+pαp(𝔹n) whenever  -pα<t+1<p.  Since  g∈Bq(𝔹n) if and only if  ℜg∈Aq-2q(𝔹n),  and we have from above,  P0,α+1  maps Aq-2q(𝔹n)  boundedly into  A(q-2)+q(α+1)q(𝔹n), whenever  -q(α+1)<(q-2)+1<q,  or q>1/(α+2),  which is always true if  α>-1.  Thus G(w)∈Aq(α+2)-2q(𝔹n). It can easily seen that  G∈Aα1(𝔹n)  and that ∥G∥Aα1(𝔹n)≤C∥G∥Aq(α+2)-2q(𝔹n)≤C∥g∥Bq(𝔹n). Thus
(2.19)|I1|≤C‖f‖Bp(Bn)∫Bn|G(z)|(1-|z|2)α((1-|z|2)-1/2∫Bn(1-|w|2)α-1/2|1-〈z,w〉|n+αdμ(w))dν(z)≤C‖f‖Bp(Bn)∫Bn‖G‖Aα1(1-|z|2)-1/2∫Bn(1-|w|2)α-1/2|1-〈z,w〉|n+αdμ(w)dν(z)≤C‖f‖Bp(Bn)∫Bn‖g‖Bq(Bn)(1-|z|2)-1/2∫Bn(1-|w|2)α-1/2|1-〈z,w〉|n+αdμ(w)dν(z).
By (2.11), we get
(2.20)|I1|≤C‖f‖Bp(Bn)‖g‖Bq(Bn).
Next considerI2, we have
(2.21)|I2|=cα|∫BnPα(fG¯)(z)dμα(z)|=cα2|∬Bnf(w)G(w)¯(1-|w|2)α|1-〈z,w〉|n+α+1dν(w)dμα(z)|=cα∫Bn|f(w)||G(w)|(1-|w|2)αcα∫Bn(1-|z|2)αdμ(z)|1-〈z,w〉|n+α+1dν(w)≤C∫Bn‖G‖Aα1(Bn)|f(w)|Pα(μ)(w)dν(w)≤C∫Bn‖g‖Bq(Bn)|f(w)|Pα(μ)(w)dν(w).
Therefore, Tμα  is bounded on  Bp(𝔹n)  if and only if
(2.22)∫Bn|f(w)|Pα(μ)(w)dν(w)≤C‖f‖Bp(Bn)
if and only if the measure Pα(μ)(w) is a (Bp(𝔹n),p)-Carleson measure.Now, we will characterize boundedness of Toeplitz operators on the minimal Möbius invariant Banach spaces of holomorphic functionsB1(𝔹n) in the unit ball of  ℂn.Theorem 2.3.
Letμ  be a positive Borel measure on  𝔹n. If   μ is a  (Ap(𝔹n),p)-Carleson measure, then the Toeplitz operator   Tμα  is bounded on  B1(𝔹n) spaces if and only if
(2.23)∑|m|=n+1|∂m∂wmPα(μ)(w)|dν(w)
is a (B1(𝔹n),1)-Carleson measure.Proof.
We will use the fact that the dual spaces ofB1(𝔹n) are the Bloch space  ℬ(𝔹n) under the paring
(2.24)〈f,g〉=∫BnRf(z)Rg(z)¯dν(z),f∈B1(Bn),g∈B(Bn).
Similarly, as in the proof of Theorem 2.2, by duality, we have that  Tμα  is bounded on B1(𝔹n) spaces if and only if
(2.25)|〈Tμα(f),g〉|=cα|∫Bnf(w)G(w)¯(1-|w|2)αdμ(w)|≤C‖f‖B1(Bn)‖g‖B(Bn),
for all  f∈B1(𝔹n)  and  g∈ℬ(𝔹n), where
(2.26)G(w)=wP0,α+1Rg(w)=w∫BnRg(z)(1-〈z,w〉)n+α+2dν(z).
Using the fact that(2.27)|∫BnRg(z)|1-〈z,w〉|n+α+2dν(z)|≈|∫Bn(1-|z|2)Rg(z)(1-〈z,w〉)n+α+3dν(z)|,
for  g∈ℬ(𝔹n),  we have that  |G(w)|(1-|w|2)α+1<∞,  which means that  G∈ℬα+2(𝔹n).  Now using the operator  Pα+1,  we have
(2.28)〈Tμαf,g〉=cα∫Bn[(I-Pα+1)(fG¯)](w)(1-|w|2)αdμ(w)+cα∫BnPα+1(fG¯)(w)(1-|w|2)αdμ(w)=I1+I2,(I-Pα+1)(fG¯)(w)=cα+2∫Bn(f(w)-f(z))G(z)¯(1-|z|2)α+1(1-〈z,w〉)n+α+2dν(z).
By Proposition1.1, we have
(2.29)|I1|=cα|∫Bn[(I-Pα+1)(fG¯)](w)(1-|w|2)αdμ(w)|=cαcα+1|∬Bn(f(w)-f(z))G(z)¯(1-|z|2)α+2(1-|w|2)α|1-〈z,w〉|n+α+2dν(z)dμ(w)|=cαcα+1∫Bn|G(z)|(1-|z|2)α+1∫Bn|f(w)-f(z)|(1-|w|2)α|1-〈z,w〉|n+α+2dμ(w)dν(z)≤C∫Bn‖f‖Bp(Bn)|G(z)|(1-|z|2)α+1/2∫Bn(1-|w|2)α-1/2|1-〈z,w〉|n+α+2dμ(w)dν(z)≤C∫Bn‖f‖Bp(Bn)‖G‖Aα+11(Bn)(1-|z|2)-1/2∫Bn(1-|w|2)α-1/2|1-〈z,w〉|n+α+1dμ(w)dν(z)≤C‖f‖B1(Bn)‖g‖B(Bn).
Next considerI2,  notice first that
(2.30)Pα(μ)(w)=cα∫Bndμα(z)(1-〈z,w〉)n+α+1;∑|m|=n+1|∂m∂wmPα(μ)(w)|≈|∫Bn(z̅)mdμα(z)|1-〈z,w〉|n+α+2|.
Thus,
(2.31)|I2|=cα|∫Bn(z)mPα+1(fG¯)(z)¯dμα(z)|=cαcα+1|∫Bn(z̅)m∫Bnf(w)G(w)¯(1-|w|2)α+1(1-〈z,w〉)n+α+2dν(w)dμα(z)|=cα∫Bn|f(w)||G(w)|(1-|w|2)α+1(cα+1∫Bn(z̅)m(1-|z|2)αdμ(z)|1-〈z,w〉|n+α+2)dν(w)=cα∫Bn|f(w)||G(w)|(1-|w|2)α+1∑|m|=n+1|∂m∂wmPα(μ)(w)|dν(w).
It is known that (A1(𝔹n))*=ℬβ+1(𝔹n)  under the paring(2.32)〈F,H〉β=cβ∫BnF(w)H(w)¯(1-|w|2)βdμ(w),F∈A1(Bn),H∈Bβ+1(Bn).
SinceG∈ℬα+2(𝔹n),g∈ℬ(𝔹n)  for by the above duality we get
(2.33)sup⁡⁡‖g‖B(Bn)≤1|I2|≈Csup⁡⁡‖g‖B(Bn)≤1∫Bn‖G‖Aα+11(Bn)|f(w)|∑|m|=n+1|∂m∂wmPα(μ)(w)|dν(w)≤Csup⁡⁡‖g‖B(Bn)≤1∫Bn|f(w)|∑|m|=n+1|∂m∂wmPα(μ)(w)|dν(w).Therefore,  Tμα  is bounded on  Bp(𝔹n)  if and only if
(2.34)∫Bn∑|f(w)|∑|m|=n+1|∂m∂wmPα(μ)(w)|dν(w)≤C‖f‖Bp(Bn)
if and only if the measure ∑|m|=n+1|(∂m/∂wm)Pα(μ)(w)|dν(w) is a  (Bp(𝔹n),p)-Carleson measure.


## 3. Compact Toeplitz Operators onBp(𝔹n) Spaces

In this section we will characterize compact Toeplitz operators onBp(𝔹n) spaces in the unit ball of  ℂn. We need the following lemma.Lemma 3.1.
Let0<p<∞, -1<α  and   Tμα  be bounded linear operator from   Bp(𝔹n) into  Bp(𝔹n)  in the unit ball. Then  Tμα is compact on  Bp(𝔹n)  spaces if and only if                         ∥Tμαfj∥Bp(𝔹n)→0  as  j→∞  wheneve r  {fj}  is a bounded sequence in   Bp(𝔹n)  that converges to  0 uniformly on  𝔹n.Proof.
This lemma can be proved by Montel’s Theorem.Theorem 3.2.
Let2n<p<∞,α>-1  and let   μ be a positive Borel measure on   𝔹n.  If  μ is a vanishing   (Ap(𝔹n),p)-Carleson measure, then the Toeplitz operator  Tμα  is compact on  Bp(𝔹n)  spaces if and only if  Pα(μ)(w)  is a vanishing  (Bp(𝔹n),p)-Carleson measure.Proof.
Let2n<p,q<∞  where 1/p+1/q=1 and let  {fj}  be a sequence in  Bp(𝔹n)  satisfying  ∥fj∥Bp(𝔹n)≤1 and such that  fj  converges to  0  uniformly as j→∞ on compact subsets of  𝔹n, and let g∈Bq(𝔹n).  By duality, we have that  Tμα is compact on  Bp(𝔹n) if and only if
(3.1)lim⁡j→∞⁡sup⁡⁡‖g‖Bq(Bn)≤1|〈Tμα(fj),g〉|=0.
As in the proof of Theorem 2.2,
(3.2)〈Tμα(fj),g〉=Tμαfj(0)g(0)¯+∫BnTμα(Rfj)(z)Rg(z)¯dν(z)=Tμαfj(0)g(0)¯+cα∫Bnfj(w)G(w)¯dμα(w),
where
(3.3)G(w)=wP0,α+1Rg(w)=w∫BnRg(z)(1-〈z,w〉)n+α+2dν(z).
Also as in the proof of Theorem 2.2,
(3.4)|Tμαf(0)|≤C‖f‖Bp(Bn).
Since|∫𝔹ndμα(w)|<∞  and μ  is a vanishing (Ap(𝔹n),p)-Carleson measure, and  fj  converges to 0 uniformly  as j→∞  on compact subsets of  𝔹n, we get that
(3.5)Tμαf(0)⟶0asj⟶∞.
ThusTμα  is compact on Bp(𝔹n) if and only if
(3.6)lim⁡j→∞⁡sup⁡⁡‖g‖Bq(Bn)≤1|∫Bnfj(w)G(w)¯dμα(w)|=0.
Using the operator  Pα,  we have that(3.7)∫Bnfj(w)G(w)¯dμα(w)=cα∫Bn[(I-Pα)(fjG¯)](z)dμα(z)+cα∫BnPα(fjG¯)(z)dμα(z).=J1+J2.
For  0<r<1  and  r𝔹n={z∈ℂn,|z|≤r},  we have
(3.8)|J1|=cα|∫Bn[(I-Pα)(fjG¯)](w)(1-|w|2)αdμ(w)|=cα2|∬Bn(fj(w)-fj(z))G(z)¯(1-|z|2)α(1-|w|2)α(1-〈z,w〉)n+α+1dν(z)dμ(w)|=cα2(∫Bn∖rBn+∫rBn)|G(z)|(1-|z|2)α∫Bn|fj(w)-fj(z)|(1-|w|2)α|1-〈z,w〉|n+α+1dμ(w)dν(z)=L1+L2.
For a fixed  ε>0, since  μ is a vanishing  (Ap(𝔹n),p)-Carleson measure, let  r sufficiently close to 1  so that
(3.9)(1-|z|2)-1/2∫Bn∖rBn(1-|w|2)α-1/2|1-〈z,w〉|n+αdμ(w)d<ε.
Similarly, as in the proof of Theorem 2.2, by Proposition 1.1,
(3.10)L1=cα2∫Bn∖rBn|G(z)|(1-|z|2)α∫Bn|fj(w)-fj(z)|(1-|w|2)α|1-〈z,w〉|n+α+1dμ(w)dν(z)≤C∫Bn∖rBn‖fj‖Bp(Bn)|G(z)|(1-|z|2)α-1/2∫Bn(1-|w|2)α-1/2|1-〈z,w〉|n+αdμ(w)dν(z)≤Cε∥‖fj‖Bp(Bn)‖G‖Aα1(Bn)≤Cε‖fj‖Bp(Bn)‖g‖Bq(Bn)≤ε.
Sincefj→0 as j→∞  on compact subsets of  𝔹n,  we cane choose  j  big enough so that
(3.11)|G(z)|(1-|z|2)α<ε.
Therefore,(3.12)L2=cα2∫rBn|G(z)|(1-|z|2)α∫Bn|fj(w)-fj(z)|(1-|w|2)α|1-〈z,w〉|n+α+1dμ(w)dν(z)≤C∫rBn‖fj‖Bq(Bn)|G(z)|(1-|z|2)α-1/2∫Bn(1-|w|2)α-(1/2)|1-〈z,w〉|n+αdμ(w)dν(z)≤Cε‖G‖Aα1(Bn)≤Cε‖g‖Bq(Bn).
Hence|J1|<Cε,  where  C  does not depend on  g(z),  and so
(3.13)lim⁡j→∞⁡sup⁡⁡‖g‖Bq(Bn)≤1|J1|=0.
Thus,  Tμα  is compact on  Bp(𝔹n) if and only if(3.14)lim⁡j→∞⁡sup⁡⁡‖g‖Bq(Bn)≤1|J2|=0.
Again, as in the proof of Theorem 2.2, we have
(3.15)|J2|=cα|∫BnPα(fjG¯)(z)dμα(z)|=cα2|∬Bnfj(w)G(w)¯(1-|w|2)α(1-〈z,w〉)n+α+1dν(w)dμα(z)|=cα∫Bn|fj(w)||G(w)|(1-|w|2)αcα∫Bn(1-|z|2)αdμ(z)(1-〈z,w〉)n+α+1dν(w)≤C∫Bn‖G‖Aα1(Bn)|fj(w)|Pα(μ)(w)dν(w)≤C∫Bn‖g‖Bq(Bn)|fj(w)|Pα(μ)(w)dν(w).
Therefore,  Tμα  is compact on Bp(𝔹n) if and only if
(3.16)lim⁡j→∞⁡∫Bn|fj(w)|Pα(μ)(w)dν(w)=0,
which is equivalent to say that Pα(μ)(w)  is a vanishing (Bp(𝔹n),p)-Carleson measure.Theorem 3.3.
Letμ  be a positive Borel measure on   𝔹n. If   μ is a  (Ap(𝔹n),p)-Carleson measure, then the Toeplitz operator  Tμα is compact on  B1(𝔹n) spaces if and only if
(3.17)∑|m|=n+1|∂mPα(μ)∂wm(w)|dν(w)
is a vanishing  (B1(𝔹n),1)-Carleson measure.Proof.
Let{fj}  be a sequence in  Bp(𝔹n)  satisfying  ∥fj∥B1(𝔹n)≤1 and such that  fj converges to 0 uniformly as  j→∞  on compact subsets of 𝔹n, and let  g∈ℬ(𝔹n).  By duality, we have that  Tμα is compact on  B1(𝔹n)  if and only if
(3.18)lim⁡j→∞⁡sup⁡⁡‖g‖B(Bn)≤1|〈Tμα(fj),g〉|=0.
Thus,  Tμα  is compact on B1(𝔹n) if and only if
(3.19)lim⁡j→∞⁡sup⁡⁡‖g‖B(Bn)≤1|∫Bnfj(w)G(w)¯dμα(w)|=0.
Using the operator  Pα, we have that
(3.20)∫Bnfj(w)G(w)¯dμα(w)=cα∫Bn[(I-Pα)(fjG¯)](z)dμα(z)+cα∫BnPα(fjh¯)(z)dμα(z).=J1+J2.
As in the proof of Theorem 2.3, we have
(3.21)|J1|≤C∫Bn‖fj‖B1(Bn)‖G‖A1(B)(1-|z|2)-1/2∫Bn(1-|w|2)α-1/2|1-〈z,w〉|n+α+1dμ(w)dν(z).
Notice that  ∥fj∥Bp(𝔹n)   implies that  ∥fj∥B1(𝔹n)≤C.  Since fj  converges to 0 uniformly as  j→∞  on compact subsets of  𝔹n, and  μ is a (Ap(𝔹n),p)-Carleson measure, we get that  G∈ℬα+2(𝔹n) and  ∥G∥ℬα+2(𝔹n)≤C∥g∥ℬ(𝔹n). Thus
(3.22)|J1|≤C‖fj‖B1(Bn)‖g‖B(Bn),lim⁡j→∞⁡sup⁡⁡‖g‖B≤1|J1|=0.
Therefore,  Tμα  is compact on  B1(𝔹n)  if and only if
(3.23)lim⁡j→∞⁡sup⁡⁡‖g‖B(Bn)≤1|J2|=0.
We have shown in the proof of Theorem 2.3(3.24)sup⁡⁡‖g‖B(Bn)≤1|J2|≤Csup⁡⁡‖g‖B(Bn)≤1∫Bn|fj(w)|RPα(μ)(w)dν(w).
Therefore, Tμα  is compact on B1(𝔹n) if and only if
(3.25)lim⁡j→∞⁡∫Bn|fj(w)|∑|m|=n+1|∂m∂wmPα(μ)(w)|dν(w)=0,
which is equivalent to saying that the measure  ∑|m|=n+1|(∂m/∂wm)Pα(μ)(w)|dν(w)  is a vanishing  (B1(𝔹n),1)-Carleson measure.



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*Source: 517689-2012-05-31.xml*