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R E S E A R C H

Open Access

General Toeplitz operators on weighted

Bloch-type spaces in the unit ball of

C

n

A El-Sayed Ahmed

*

*Correspondence: ahsayed80@hotmail.com Mathematics Department, Faculty of Science, Sohag University, Sohag, 82524, Egypt

Current address: Mathematics Department, Faculty of Science, Taif University, El-Hawiah, Box 888, Taif, Saudi Arabia

Abstract

In this paper, we consider the weighted Bloch-type spacesBαω,β(Bn) with

α

> 0 and

β

≥0 in the unit ball ofCn. We present some basic properties of the spacesBα,β ω (Bn),

then we consider the Toeplitz operatorTμα,β;ωacting betweenBαω,β(Bn) spaces, where

μ

is a positive Borel measure in the unit ballBn. Moreover, we characterize complex

measures

μ

for which the Toeplitz operatorTμα,β;ωis bounded or compact on

,β ω (Bn).

MSC: 47B35; 32A18

Keywords: Toeplitz operators; weighted Bloch-type spaces; weighted Bergman spaces

1 Introduction

We start here with some terminology, notations and definitions of various classes of ana-lytic functions defined on the unit ball ofCn.

LetBnbe the unit ball of then-dimensional complex Euclidean spaceCn. The boundary

ofBnis denoted bySnand is called the unit sphere inCn. Occasionally, we will also need

the closed unit ballBn. We denote the class of all holomorphic functions on the unit ball

BnbyH(Bn). The ball centered atz∈Cnwith radiusris denoted byB(z,r). Forα> –, letdνα(z) =( –|z|)αdν, whereis the normalized Lebesgue volume measure onBn

and=n!(n+(αα+)+) (wheredenotes the gamma function) so thatνα(Bn)≡. The surface

measure onSnis denoted by. Once again, we normalizeσso thatσα(Sn)≡. For any

z= (z,z, . . . ,zn),w= (w,w, . . . ,wn)∈Cn, the inner product is defined by

z,w=

n

k= zkwk.

For every pointa∈Bn, the Möbius transformationϕa:Bn→Bnis defined by

ϕa(z) =

aPa(z) –SaQa(z)

 –z,a , z∈Bn,

whereSa=

 –|a|,Pa(z) =az,a

|a| ,P=  andQa=IPa. The mapϕahas the following

properties thatϕa() =a,ϕa(a) = ,ϕa=ϕa–and

 –ϕa(z),ϕa(w)

= ( –|a|

)( –z,w)

( –z,a)( –a,w),

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wherezandware arbitrary points inBn. In particular,

 –ϕa(z) 

=( –|a|

)( –|z|)

| –z,a| .

ForfH(Bn), the holomorphic gradient off atzis defined by

f(z) =

∂f ∂z

(z), ∂f

∂z

(z), . . . , ∂f

∂zn

(z)

and the radial derivative off atzis defined by

Rf(z) =∇f,z=

n

j= zj

∂f(z)

∂zj

.

Similarly, the Möbius invariant complex gradient off atzis defined by

f(z) =∇(fϕz)().

Forα> , a functionfH(Bn) is said to belong to theα-Bloch spaces(Bn) if (see [])

(f)(Bn) =sup

z∈Bn

f(z) –|z|α<∞.

The little Bloch space

(Bn) consists of allf(Bn) such that lim

|z|→–∇f(z) –|z| α= .

With the normfBα=|f()|+bα(f)(Bn), we know that(Bn) becomes a Banach space

and

(Bn) is its closed subspace (see []). Forα= , the spacesB(Bn) andB(Bn) become

the Bloch and the little Bloch space, respectively (see, for example, [–]). Zhu in [] says that the normfB(Bn)is equivalent to

f()+sup z∈Bn

Rf(z) –|z|.

Forα> – and  <p<∞, the weighted Bergman spaceApα(Bn) consists of holomorphic

functionsfLp(Bn,

α) such that

fp Apα(Bn):=

Bn

f(z)pdνα(z) <∞,

that is,Apα(Bn) =Lp(Bn,dνα)∩H(Bn). When the weightα= , we simply writeAp(Bn) for

Ap(Bn). In the special case whenp= ,Aα(Bn) is a Hilbert space. It is well known that for

α> –, the Bergman kernel ofA

α(Bn) is given by

(z,w) = 

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Forα> –, a complex measureμis such that

B

n

 –|w|αdμ(w)=

Bn

dμα(w)

<∞.

The general Bergman projectionis the orthogonal projection of the measureμfrom

L(Bn,

α) intoAα(Bn) defined by

(μ)(z) =

Bn

( –|w|)α

( –z,w)n+α+(w) =

Bn

dμα(w)

( –z,w)n+α+.

The general Bergman projection of the functionf is

Pαf(z) =

Bn

f(w)( –|w|)α

( –z,w)n+α+(w) =

Bn

f(w)dνα(w)

( –z,w)n+α+.

Letω: (, ]→(,∞) be a right-continuous and nondecreasing function. For a complex measureμ,α> –,β≥, andfL(Bn,

α+β), define weighted general Toeplitz operator

as follows:

Tμα,β;ωf(z) =

+β

( –|ϕz(w)|)βω( –|w|)

Bn

( –|w|)α+βf(w)

( –z,w)n+α+β+(w)

= +β

( –|ϕz(w)|)βω( –|w|)

Bn

f(w)dμα+β(w)

( –z,w)n+α+β+.

Thus+β;ω(μ)(z) =Tμα,β;ω()(z), where  stands for a constant function.

Toeplitz operators have been studied extensively on the Bergman spaces by many au-thors. For references, see [] and []. Boundedness and compactness of the general Toeplitz operators

μ on theα-Bloch(D) spaces have been investigated in [] on the

unit diskDfor  <α<∞. Also, in [], the authors extended the general Toeplitz operator

μto(Bn) with ≤α< . Recently, in [], the general Toeplitz operatorsTμαon the

analytic BesovBp(D) spaces with ≤p<∞have been investigated. Under a prerequisite

condition, the authors characterized a complex measureμon the unit disk for which μis

bounded or compact on the Besov spaceBp(D). For more studies on the Toeplitz operator,

we refer to [–].

In this paper, we consider the weighted Bloch-type spaces,β

ω (Bn) withα>  andβ≥

in the unit ball ofCn. We prove a certain integral representation theorem that is used to

determine the degree of growth of the functions in the space,β

ω (Bn). It is also proved

that the space,β

ω (Bn) is a Banach space for each weightα> ,β≥, and the Banach

dual of the Bergman spaceA(Bn) is,β

ω (Bn) for eachα≥,β≥. Further, we extend the

Toeplitz operator,β;ω

μ toBωα,β(Bn) in the unit ball ofCnand completely characterize the

positive Borel measureμsuch that,β;ω

μ is bounded or compact inBωα,β(Bn) spaces with

α+β≥.

Throughout the paper, we say that the expressionsAandBare equivalent, and writeABwhenever there exist positive constantsCandCsuch thatCABCA. As usual,

the letterCdenotes a positive constant, possibly different on each occurrence. Hereafter,

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Theorem .(see [, Theorem .]) Suppose b is real and s> –.Then the integrals

Ib(z) =

Sn

(ζ)

| –z,ζ|n+b, z∈Bn

and

Jb,s(z) =

Bn

( –|w|)sdν(w)

| –z,w|n++s+b, z∈Bn,

have the following asymptotic properties.

() Ifb< ,thenIb(z)andJb,s(z)are both bounded inBn.

() Ifb= ,then

Ib(z)≈Ib,s(z)≈log

 –|z| as|z| → –.

() Ifb> ,then

Ib(z)≈Jb,s(z)≈

 –|z|–b as|z| →–.

Lemma .(see [, Lemma .]) Supposeγ is a real constant and gL(Bn,).If

u(z) = –ϕa(z) β

Bn

g(w)(w)

( –z,w)γ, z∈Bn,

then

u(z)≤√|γ| –|z|  

Bn

g(w)(w)

| –z,w|γ+

, ∀z∈Bn.

Let β(·,·) be the Bergman metric on Bn. Denote the Bergman metric ball at w(j) by B(w(j),r) ={z∈Bn:β(w(j),z) <r}, wherew(j)∈Bnandr> .

Lemma .(see [, Theorem .]) For fixed r> ,there is a sequence{w(j)} ∈Bnsuch that:

• ∞j=B(w(j),r) =Bn;

there is a positive integerNsuch that eachz∈Bnis contained in at mostNof the sets B(w(j), r).

The following characterization of Carleson measures can be found in [], or in []. A positive Borel measureμon the unit ballBnis said to be a Carleson measure for the Bergman spaceApα(Bn) if

Bn

f(z)pdνα(z)≤Cfp

Apα(Bn), ∀fA p

α(Bn).

It is well known that a positive Borel measureμis a Carleson measure if and only if there is a positive constantCsuch that

sup w(j)B

n

μ(B(w(j),r))

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where{w(j)}is the sequence in Lemma .. Ifμsatisfies that

lim j→∞

μ(B(w(j),r)) ν(B(w(j),r)) = ,

thenμis called a vanishing Carleson measure.

For a given reasonable functionω: (, ]→(,∞), the weighted Bloch spaceof

sev-eral complex variables is defined as the set of all analytic functionsf onBnsatisfying

 –|z|αf(z)≤ –|z|, z∈Bn, whereα∈(,∞),

for some fixedC=Cf> . In the special case whereω≡,reduces to the classical Bloch

spaceBinCn. This class of functions extends and generalizes the well known Bloch space.

Now, we define the space,β;ω(Bn) in the unit ballBn. Forα>  andβ≥, a function

fH(Bn) is said to belong to the (α,β;ω)-Bloch space,β;ω(Bn) if

,β;ω(f)(Bn) = sup

a,z∈Bn

( –|z|)α+β

( –|ϕa(z)|)βω( –|z|)

f(z)<∞.

The little (α,β;ω)-Bloch space,β;ω,(Bn) is a subspace of,β;ω(Bn) consisting of allf

,β;ω(Bn) such that

lim

|a|→–|zlim|→

( –|z|)α+β

( –|ϕa(z)|)βω( –|z|)

f(z)= .

Ifβ= ,ω( –|z|) = , then we get theα-Bloch space(Bn) and the littleα-Bloch space

(Bn). Ifω( –|z|) = ,α=  andβ= , then we get the classical Bloch spaceB(Bn) and B(Bn). These classes extend the weighted Bloch spaces defined in [] to the setting of several complex variables.

The logarithmic (α,β;ω)-Bloch spaceLBαω,β(Bn) is the space of holomorphic functions

f such that

sup a,z∈Bn

( –|z|)α+β

( –|ϕa(z)|)βω( –|z|)

ln 

 –|z|

f(z)<∞.

Correspondingly, the little logarithmic (α,β;ω)-Bloch spaceLBαω,;β(Bn) is a subspace of

LBα,β

ω (Bn) consisting of all functionsf such that

lim

|a|→–|zlim|→

( –|z|)α+β

( –|ϕa(z)|)βω( –|z|)

ln 

 –|z|

f(z)= .

Ifω( –|z|) =  andβ= , then we get the logarithmicα-Bloch spaceLBα(Bn) and the little logarithmicα-Bloch spaceLBα(Bn). Ifω( –|z|) = ,α=  andβ= , then we get the logarithmic Bloch spaceLB(Bn) andLB(Bn) (see []).

2 Holomorphic(

α

,

β

;

ω

)-Bloch space in the unit ball In this section, we study the general (α,β;ω)-Bloch space,β

ω (Bn) in the unit ball ofCnby

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Lemma . Letα,β∈(,∞)and f,β

ω (Bn).Suppose that

ω( –t|z|)|z|dt

( –t|z|)α+β <∞. ()

Then

f(z)≤f()+fBα,β ω (Bn).

Proof Letz∈Bn, ≤t<  andf,β

ω (Bn). By the definition ofB α,β

ω (Bn) and|z|>

 , we

have that

f(z) –f

z

=

 

f(tz),zdt

≤ 

 

Rf(tz)dt

t

,β;ω(f)

( –|ϕa(tz)|)βω( –t|z|)

( –t|z|)α+β |z|dt

,β;ω(f)

( –|a|)βω( –t|z|)

| –tz,a|β( –t|z|)α|z|dt.

Since ( –|a|)≤ | –tz,a|and ( –t|z|)≤ | –tz,a|,a,z∈Bn, we get

f(z) –f

z

,β;ω(f)

( –|a|)βω( –|z|)

( –|a|)β( –t|z|)β( –t|z|)α|z|dt

≤βb α,β(f)

ω( –t|z|)|z|dt

( –t|z|)α+β ,

from which the result follows.

Theorem . For each <α,β<∞,γ > –and fH(Bn).Then the following conditions are equivalent:

(i) f,β ω (Bn);

(ii) The function (–|ϕ(–|z|)α+β

z(w)|)βω(–|w|)|Rf(z)|is bounded inBn;

(iii) There exists a functiongL∞(Bn)such that

f(z) = –ϕz(w) β

ω –|w|

Bn

g(w)dνγ(w)

( –z,w)n+α+β+γ, z∈Bn.

Proof By the Cauchy-Schwarz inequality inCn, we have

Rf(z)≤ |z|∇f(z)≤∇f(z).

This proves that (i)⇒(ii). If (ii) holds, then the function

g(z) =+β+γ

( –|z|)α+β

( –|ϕz(w)|)βω( –|w|)

f(z) + Rf(z)

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is bounded inBn. Forz∈Bnconsider the holomorphic function

F(z) =

Bn

g(w)( –|ϕz(w)|)βω( –|w|)dνγ(w)

( –z,w)n+α+β+γ

=

Bn

ω( –|w|) ( –z,w)n+α+β+γ

f(w) + Rf(w)

n+α+β+γ

dνα+β+γ(w).

As in the proof of Theorem . in [], we haveF=f.

This shows that (ii) implies (iii). That (iii) implies (i) follows from differentiating under the integral sign and then applying Theorem . in [].

Theorem . For eachα> ,β≥,α+β> and s=α+β– .If s> –,then the Banach dual of A(Bn)can be identified withBα,β

ω (Bn) (with equivalent norms)under the following

integral pairing:

f,gs=

Bn

f(z)g(z)dνs(z), fA(Bn),gBαω,β(Bn). ()

Proof It is easy to see that  – (α+β) +s> –. Ifg,β

ω (Bn), then by Theorem ., there

exists a functionhL∞(Bn) such that

g(z) = 

( –|ϕz(w)|)βω( –|w|)

Bn

h(w)–(α+β)+s(w)

( –z,w)n++s , z,w∈Bn,

andh∞≤CgBα,β

ω (Bn), whereCis a positive constant independent ofg. By Fubini’s

theorem,

f,gs=

Bn

f(z)h(z) –|z|–(α+β)+s(z)

=c–(α+β)+s

Bn

f(z)h(z)(z).

Applying Lemma . in [] for allfA(Bn), we have

Bn

f(z)(z)≤ fA(B

n).

Combining this, we see that

f,gs≤ hfA(B

n)≤CgBωα,β(Bn)fA(Bn).

Conversely, ifFis a bounded linear functional onA(Bn) andfA(Bn), then

fr(z) =

( –|ϕz(w)|)βω( –|w|)

Bn

fr(w)dνs(w)

( –z,w)n++s for  <r< .

It is easy to verify (using the homogeneous expansion of the kernel function) that

F(fr) =

Bn

fr(w)Fz

( –z,w)n++s( –|ϕz(w)|)βω( –|w|)

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Define a functiongonBnby

g(w) =Fz

( –z,w)n++s( –|ϕz(w)|)βω( –|w|)

.

Then

F(fr) =

Bn

fr(w)g(w)dνs(w) =f,gs.

It remains for us to show thatg,β ω .

We interchange differentiation and the application ofF, which can be justified by using the homogeneous expansion of the kernel. The result is

Rg(w) = (n+  +s)Fz

 ( –z,w)n++s( –|ϕ

z(w)|)βω( –|w|)

.

SinceFis bounded onA(Bn), we have

Rg(w)≤ CF

( –|ϕz(w)|)βω( –|w|)

Bn

(w)

| –z,w|n++s.

An application of Theorem . fors+  =α+βthen shows that

Rg(w)≤ CF ( –|z|)α+β( –|ϕ

z(w)|)βω( –|w|)

.

This shows thatg,β

ω (Bn) and completes the proof of the theorem.

Lemma . If n> ,α+β>,then f,β

ω (Bn)if and only if the function

( –|z|)α+β–

( –|ϕw(z)|)βω( –|z|) ∇

f(z)

is bounded inBn.

Proof Recall from Lemma . in [] that

 –|z|∇f(z)≤ ∇f(z), z∈Bn.

So, the boundedness of

( –|z|)α+β–

( –|ϕw(z)|)βω( –|z|) ∇

f(z)

implies that of

( –|z|)α+β

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On the other hand, iff,β

ω (Bn), then by Theorem .,

f(z) = –ϕw(z) β

ω –|z|

Bn

g(w)(w)

( –z,w)n+α+β, z∈Bn,

wheregis a function inL∞(Bn). Now we letf(z) =h(z)u(z), whereh(z) = ( –|ϕw(z)|)β×

ω( –|z|) and

u(z) =

Bn

g(w)(w) ( –z,w)n+α+β.

An application of Lemma . gives

u(z)≤ |n+α+β|√ –|z|

Bn

g(w)(w)

| –z,w|n+α+β+

, ∀z∈Bn.

Sinceg(z) is bounded, by Theorem . we have

Bn

g(w)(w)

| –z,w|n+α+β+ ≈

 –|z|–(α+β).

So,

u(z)≤C –|z|–(α+β).

It is easy to check that∇h(z) =∇(hϕz)() = .

Using the product rule, we have

f(z)≤ ∇h(z)u(z)+h(z) ∇u(z)≤ ∇h(z)u(z)

and we have

f(z)≤ |n+α+β|√ –|z|   –ϕ

a(z) β

Bn

g(w)(w)

| –z,w|n+α+β+

for allz∈Bn.

Hence,

( –|z|)α+β–

( –|ϕw(z)|)βω( –|z|) ∇

f(z)

is bounded inBn. This completes the proof.

Lemma . Let <α+β≤.Letλbe any real number satisfying the following properties: • ≤λα+βif <α+β< ;

•  <λ< ifα+β= ;

α+β– ≤λ≤if <α+β≤.

Then,for allz,w∈Bna holomorphic function f,β

ω (Bn)if and only if

sup z=w

( –|z|)λ( –|w|)α+βλ

( –|ϕw(z)|)βω( –|z|)

|f(z) –f(w)|

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Proof Letf,β

ω (Bn). By a similar proof to the one for Theorem . in [], we have

f(z) –f(w)=√n|zw| 

ftz– ( –t)wdt

for anyz,w∈Bnwithz=w. We know that

fBα,β

ω (Bn)≈wsup

,z∈Bn

( –|z|)α+β

( –|ϕw(z)|)βω( –|z|) f(z).

Thus, there is a constantC>  such that

|f(z) –f(w)|

|zw| ≤CfBαω,β(Bn)

( –|ϕa(tz– ( –t)a)|)βω( –|tz– ( –t)w|)

( –|tz– ( –t)w|)α+β dt.

Since ( –|z|)≤ | –w,z|and

 –tz+ ( –t)w≥ –tz+ ( –t)w≥ –|w|+|w|–|z|t,

we get

 –ϕw

tz– ( –t)wβ = ( –|w|

)β( –|tz– ( –t)w|)β

| –tz– ( –t)w,w|β

≤( –|w|)β( –|tz– ( –t)w|)β

( –|w|)β

≤( +|w|)β( –|tz– ( –t)w|)β

( –|w|)β

≤( –|tz– ( –t)w|)β

( –|w|)β .

Thus

|f(z) –f(w)|

|zw| ≤CfBαω,β(Bn)

( –|w|+ (|w|–|z|)t)α( –|w|)βdt. ()

If|z|=|a|, then

|f(z) –f(w)|

|zw| ≤CfBαω,β(Bn) 

( –|w|)α+βdt

CfBα,β ω (Bn)

( –|z|)λ( –|w|)α+βλ. ()

Now suppose|z| =|w|as in [], there is a constantC>  such that this integral in () is dominated by

C

( –|z|)λ( –|w|)α+βλ.

Combining with (), we get that wheneverz=w,

|f(z) –f(w)|

|zw| ≤

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This proves the necessity. The proof of the sufficiency condition is much akin to the cor-responding one in [], so the proof is omitted.

Proposition . Suppose f,β

ω (Bn)and≤α+β≤.Letλbe any real number

satis-fying:

•  <λ< ifα+β= ;

α+β– ≤λ≤if <α+β≤.

Then

sup z,w∈Bn

( –|z|)α+βλ–( –|w|)α+βλ|f(z) –f(w)|

( –|ϕw(z)|)βω( –|z|)| –w,z|(α+β)–(λ+)|zPz(w) –SzQz(w)| ≤CfBα,β

ω (Bn). ()

Proof Letz=  in (), then we have

 –|w|α+βλ|f() –f(w)|

|w| ≤Cbα,β;ω(f)(Bn), w∈Bn\{}.

Now, replacingf byfϕw, we get

 –|u|α+βλ|fϕw() –fϕw(u)|

|u| ≤Cbα,β;ω(fϕw)(Bn), u∈Bn\{}. ()

Since

∇(fϕw)(z)= ∇f

ϕw(z),

by Lemma ., we obtain that

,β;ω(fϕw)(Bn)≈ sup z,w∈Bn

( –|w|)α+β–

( –|ϕw(z)|)βω( –|z|) ∇

(fϕw)(z)

= sup z,w∈Bn

( –|w|)α+β–

( –|ϕw(z)|)βω( –|z|) ∇

w(z)

= sup z,w∈Bn

( –|w|)α+β–( –|ϕ

w(z)|)α+β–

( –|ϕw(z)|)α+β–( –|ϕw(z)|)βω( –|z|) ∇

w(z).

Then

,β;ω(fϕz)(Bn)≤CfBα,β ω (Bn)

( –|w|)α+β–

( –|ϕz(w)|)α+β–ω( –|w|) ≤ CfBωα,β(Bn)

( –|z|)α+β–.

Lettingu=ϕw(z) andw=zin (), we obtain

|f(z) –f(w)|

|ϕw(z)|ω( –|z|)( –|ϕw(z)|)α+βλ

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Since

 –ϕz(w)=

( –|z|)( –|w|) | –w,z|

and

ϕz(w) =

zPz(w) –SzQz(w)

 –w,z .

Consequently,

( –|z|)α+βλ–( –|w|)α+βλ|f(z) –f(w)|

ω( –|z|)( –|ϕz(w)|)α+βλ| –w,z|(α+β)–(λ+)|zPz(w) –SzQz(w)| ≤CfBα,β

ω (Bn).

3 Boundedness of general Toeplitz operators

In this section, we study the boundedness of general Toeplitz operators acting on the weighted Bloch-type spaces,β

ω (Bn) in the unit ball ofCn.

Theorem . Letμbe a positive Borel measure onBn.Then we have

() ifα+β= ,thenTα,β;ω

μ is bounded onBωα,β(Bn)if and only ifPα+β–;ω(μ)∈LBω(Bn)

andμis a Carleson measure; () ifα=β= ,thenTα,β;ω

μ is bounded onBαω,β(Bn)if and only if

+β–;ω(μ)∈(Bn)∩LBω(Bn)andμis a Carleson measure;

() ifα> ,β> ,thenTα,β;ω

μ is bounded onBαω,β(Bn)if and only if

+β–;ω(μ)∈Bωα,β(Bn)andμis a Carleson measure.

Proof Since the Banach dual of A(Bn) is,β

ω (Bn) under the pairing (), to prove the

boundedness of,β;ω

μ , it suffices to show that

f,Tμα,β;ω(g)

αCfA(Bn)gBωα,β(Bn) for allfA(Bn) andgBα,β

ω (Bn), whereCis a positive constant that does not depend on

f org.

Fors=α+β– , by Fubini’s theorem, we get

f,Tμα,β;ωg

s=

Bn

f(z)Tμα,β;ωg(z)dνs(z)

=+β–

Bn

f(z)g(z) –|z|α+β–(z).

Using the operator+β;ω, we have

f,Tμα,β;ωg

s=+β–

Bn

(Iz,w;ω+β;ω)(f g)(z)

 –|z|α+β–(z)

++β–

Bn

+β;ω(f g)(z)

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whereIz,w;ω=(–|ϕ I

z(w)|)βω(–|w|), andIis the identity operator. Also, (Iz,w;ω+β;ω)(f g)(z)

= f(z)g(z) ( –|ϕz(w)|)βω( –|w|)

+β

( –|ϕz(w)|)βω( –|w|)

Bn

f(w)g(w)( –|w|)α+β

( –z,w)n+α+β+ (w)

= +β

( –|ϕz(w)|)βω( –|w|) ×

Bn

(g(z) –g(w))f(w)( –|w|)α+β

( –z,w)n+α+β+( –|ϕ

z(w)|)βω( –|w|)

(z).

By Proposition ., we have

|I|=+β– B

n

(Iz,w;ω+β;ω)(f g)(z)

 –|z|α+β–(z)

=+β–+β

B

n

Bn

(g(z) –g(w))f(w)( –|w|)α+β( –|z|)α+β–

( –z,w)n+α+β+( –|ϕ

z(w)|)βω( –|w|)

(w)(z)

=+β–+β

B

n

f(w) –|w|α+β

×

Bn

(g(z) –g(w))( –|z|)α+β–

( –z,w)n+α+β+( –|ϕz(w)|)βω( –|w|)(z)(w)

+β–+β

Bn

f(w) –|w|λ

×

Bn

( –|z|)(α+β)–λ–( –|w|)α+βλ|f(z) –f(w)|

| –w,z|(α+β)–(λ+)|zP

z(w) –SzQz(w)|

× |zPz(w) –SzQz(w)|( –|z|)λ–(α+β) | –z,w|n–(α+β)+λ+( –|ϕ

z(w)|)βω( –|w|)

(z)(w)

C

Bn

gBα,β

ω (Bn)f(w) –|w| λ

Bn

( –|z|)λ–(α+β)

| –z,w|n–(α+β)+λ+(z)(w).

Sinceμis a Carleson measure, takingλ– (α+β) > –, then as in [] or in [, Proposi-tion ..], for fixedr> , we get

Bn

( –|z|)λ–(α+β)

| –z,w|n–(α+β)+λ+(z) ≤

j=

μ(B(z(j),r))

ν(B(z(j),r))

B(z(j),r)

( –|z|)λ–(α+β)

| –z,w|n–(α+β)+λ+(z)≤C.

Therefore,

|I| ≤CfA(B

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Next consideringI, we have

|I|=+β– B

n

+β;ω(f g)(z)

 –|z|α+β–(z)

=+β–+β

B

n

Bn

f(w)g(w)( –|w|)α+β( –|z|)α+β–

( –z,w)n+α+β+( –|ϕz(w)|)βω( –|w|)(w)(z)

+β

Bn

f(w) –|w|α+βg(w)

×

+β–

( –|ϕz(w)|)βω( –|w|)

Bn

( –|z|)α+β–(z) | –z,w|n+α+β+

(w)

C

Bn

fA(Bn)

 –|w|α+βg(w),β;ω

μ (w)(w),

where

Qαμ,β;ω(w) =

+β–

( –|ϕz(w)|)βω( –|w|)

Bn

( –|z|)α+β–(z) | –z,w|n+α+β+ .

As in [], by simple calculation, we have

Qαμ,β;ω(w) =+β–;ω(μ)(w) +

n+α+βR+β–;ω(μ)(w). ()

It is easy to see that

() ifα+β= and+α–;ω(μ)∈LBαω,β(Bn), then

 –|w|Qαμ,β;ω(w)

ln 

 –|w|

L∞(Bn);

() ifα=β= ,+α–;ω(μ)∈(Bn)∩LBω(Bn), then

 –|w|,β;ω

μ (w)∈L∞(Bn)

and

 –|w|,β;ω μ (w)

ln 

 –|w|

L∞(Bn);

() ifα> ,β> , and+α–;ω(μ)∈Bωα,β(Bn), then

 –|w|α+βQμα,β;ω(w)∈L∞(Bn).

This implies that |I| ≤CfA(B

n)gBαω,β(Bn). Hence,T

α,β;ω

μ is a bounded operator on

,β

ω (Bn) withα> ,β≥.

Conversely, suppose that,β;ω

μ is a bounded operator onBαω,β(Bn). Take

fw(z) =

( –|w|)t

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It is clear thatfwA(B

n)≤C. On the other hand, take

gw(z) =

( –|w|)n++t–(α+β)

( –z,w)n+t+ ; ϕw(z)≡ and ω

 –|z|≡ fort> .

Then, we havegwBα,β

ω (Bn)≤C. Therefore

f, μg

s=+β–

 –|w|n++t–(α+β)

Bn

( –|z|)α+β–(z) | –z,w|(n+t+) ≤CTμα,β;ωfwA(Bn)gwBα,β

ω (Bn)≤C.

Thus,

 –|w|n++t–(α+β)

B(w,r)

( –|z|)α+β–(z) | –z,w|(n+t+) ≤C

for everyw∈Bn. This implies that

sup w∈Bn

μ(B(w,r))

ν(B(w,r))<∞.

Henceμis a Carleson measure onBn.

From the proof of the sufficient condition, we find that there exists a constantCsuch that

|I|=+β

B

n

f(w) –|w|α+βg(w)Qαμ,β;ω(w)(w)

CfA(Bn)g

,β ω (Bn).

This implies that

g(w),β;ω

μ (w) –|w|

α+β

CgBα,β ω (Bn).

Ifα+β= , we have

g(w),β;ω

μ (w) –|w|

Cg

,β ω (Bn).

Takegw(z) =ln–z,w;ϕw(z)≡ andω( –|z|)≡. It is clear thatgwLBω(Bn)≤C. Taking z=w, then

Qαμ,β;ω(w) –|w|

ln 

 –|w|

C.

From () we have+β–(μ)∈LBω(Bn). Letα=β= , we have

g(w)Qμα,β;ω(w) –|w|

CgBα,β ω (Bn).

Takegw(z) =–z,w+ln–z,w;ϕw(z)≡ andω( –|z|)≡. It is clear that

gwBω(Bn)LB

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Takingz=w, then

g(w)Qμα,β;ω(w) –|w|

=

  –|w| +ln

  –|w|

,β

μ (w) –|w|



Qαμ,β;ω(w) –|w|

+Qαμ,β;ω(w) –|w|

 ln 

 –|w|

C.

By (), then+β–;ω(μ)∈LBω(Bn) and+β–;ω(μ)∈(Bn).

Whenα,β> , takinggw(z) = ( –z,w)–(α+β), we havegwBα,β

ω (Bn)≤C.

From Lemma ., we get

Qαμ,β;ω(w) –|w|

α+β

C forw∈Bn.

By () it is obvious that+β–;ω(μ)∈Bωα,β(Bn).

This completes the proof of Theorem ..

4 Compactness of general Toeplitz operators

In this section, we study the compactness of Toeplitz operators on the weighted Bloch-type spaces,β

ω (Bn) in the unit ball ofCn. We need the following lemma.

Lemma . Let  <α <∞, ≤β<∞and Tα,β;ω

μ be a bounded linear operator from

,β

ω (Bn)intoBωα,β(Bn).When <α< , ≤β< andα+β< ,then Tμα,β;ωis compact

if and only if

lim j→∞

Tμα,β;ωfjBα,β ω (Bn)= ,

whenever(fj)is a bounded sequence inBωα,β(Bn)that converges touniformly onBn.

Proof This lemma can be proved by Montel’s theorem and Lemma ..

Theorem . Letμbe a positive Borel measure onBn.We have the following: () ifα+β= ,thenTα,β;ω

μ is compact onBωα,β(Bn)if and only ifPα+β–;ω(μ)∈LBω;(Bn) andμis a vanishing Carleson measure;

() ifα=β= ,thenTα,β;ω

μ is compact onB α,β

ω (Bn)if and only if

+β–;ω(μ)∈;(Bn)∩LBω;(Bn)andμis a vanishing Carleson measure;

() ifα> ,β> ,thenTα,β;ω

μ is compact onBωα,β(Bn)if and only ifPα+β–(μ)∈Bαω,;β(Bn) andμis a vanishing Carleson measure.

Proof Forα+β≥, let (gj) be a sequence inBαω,β(Bn) satisfyinggjBαω,β(Bn)≤ andgj converges to  uniformly asj→ ∞onBn. SupposefA(Bn). By duality, we have that ,β;ω

μ is compact onB α,β

ω (Bn) if and only if

lim j→∞fsup

A(Bn)≤

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Similarly, as in the proof of Theorem . fors=α+β– , we have

f,Tμα,β;ωgj

s=+β–

Bn

f(z)gj(z)

 –|z|α+β–(z)

=+β–

Bn

(Iz,w;ω+β;ω)(f gj)

(z) –|z|α+β–(z)

++β–

Bn

+β;ω(f gj)(z)

 –|z|α+β–(z) =J+J.

For fixed  <ε< , sinceμis a vanishing Carleson measure, there exists  <η<  such that

 –|z|λ

Bn\ηBn

( –|w|)λ–(α+β)

| –z,w|n+λ+–(α+β)(w) <ε,

whereηBn={z∈Cn,|z|<η}andλ– (α+β) > –. For a positive constant  <δ< , as in

the proof of Theorem ., by Proposition ., we obtain

|J|=+β–+β

B

n

Bn

(gj(z) –gj(w))f(w)( –|w|)α+β( –|z|)α+β–

( –z,w)n+α+( –|ϕ

w(z)|)βω( –|z|)

(w)(z)

=+β–+β

B

n

f(w) –|w|α+β

×

Bn

(gj(z) –gj(w))( –|z|)α+β–

( –z,w)n+α+( –|ϕw(z)|)βω( –|z|)(z)(w)

+β–+β

Bn\δBn

f(w) –|w|α+β

×

Bn

|gj(z) –gj(w)|( –|z|)α+β– | –z,w|n+α+( –|ϕ

w(z)|)βω( –|z|)

(z)(w)

++β–+β

δBn

f(w) –|w|α+β

×

Bn

|gj(z) –gj(w)|( –|z|)α+β– | –z,w|n+α+( –|ϕ

w(z)|)βω( –|z|)

(z)(w)

=L+L.

Sincegj→ asj→ ∞on compact subsets ofBn, we can choosejbig enough so that

f(w) –|w|α+β<ε.

Therefore,

L≤εC

δBn

Bn

|gj(z) –gj(w)|( –|z|)α+β– | –z,w|n+α+( –|ϕ

w(z)|)βω( –|z|)

(z)(w)

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Now, takingδsuch that  – [ε( –η)n++λ]λδ< , then

L≤CgjBα,β ω (Bn)

Bn\δBn

f(w) –|w|λ

Bn

( –|z|)λ–(α+β)

| –z,w|n+λ+–(α+β)(z)(w) ≤C

Bn\δBn

f(w) –|w|λ

Bn\ηBn

( –|z|)λ–(α+β)

| –z,w|n+λ+–(α+β)(z)(w)

+C

Bn\δBn

f(w) –|w|λ

ηBn

( –|z|)λ–(α+β)

| –z,w|n+λ+–(α+β)(z)(w) ≤

Bn\δBn

f(w)(w) +C

Bn\δBn

f(w) ( –δ)

λ

( –η)n++λdν(w)

CεfA(B

n).

Hence|J|<, whereCdoes not depend onf(z), and so

lim j→∞fsup

A(Bn)≤ |J|= .

Thus,,β;ω

μ is compact onBαω,β(Bn) if and only if

lim j→∞fsup

A(Bn)≤ |J|= .

Again, as in the proof of Theorem ., we have

|J| ≤C

Bn

fA(B

n)

 –|w|α+βg(w)Qαμ,β;ω(w)(w).

From () it is easy to see that

() ifα+β= and+α–;ω(μ)∈LBαω,,β(Bn), then

lim

|w|→

 –|w|Qαμ,β;ω(w)

ln 

 –|w|

= ;

() ifα=β= ,+α–;ω(μ)∈;(Bn)∩LBω;(Bn), then

lim

|w|→

 –|w|Qαμ,β;ω(w) = 

and

lim

|w|→

 –|w|Qαμ,β;ω(w)

ln 

 –|w|

= ;

() ifα> ,β> , and+α–;ω(μ)∈Bωα,;β(Bn), then

lim

|w|→

(19)

Combined withgj→ asj→ ∞on compact subsets ofBn, we have

lim j→∞fsup

A(Bn)≤ |J|= .

Therefore,

lim j→∞T

α,β;ω μ gjBα,β

ω (Bn)= ,

which implies that,β;ω

μ is a compact operator.

Next assume that ,β;ω

μ is a compact operator onBωα,β(Bn). Again, as in the proof of

Theorem ., we take

fw(z) =

( –|w|)t

( –z,w)n+t+ fort> .

We know thatfwA(Bn)C. On the other hand, take

gw(z) =

( –|w|)n++t–(α+β)

( –z,w)n+t+ ; ϕw(z)≡ and ω

 –|z|≡ fort> .

ThengwBα,β

ω (Bn)≤Candgw→ uniformly on compact subsets ofBn, as|w| →,

f,,β;ω μ g

s

=+β–

 –|w|n++t–(α+β)

Bn

( –|z|)α+β–(z) | –z,w|(n+t+) ≤CfwA(B

n)T

α,β

μ gwBα,β(Bn).

From Lemma ., we have

lim

|w|→fwA (B

n)T

α,β;ω μ gwBα,β

ω (Bn)= , ∀w∈Bn.

This implies thatμis a vanishing Carleson measure onBn. Next let

fw(z) =

( –|w|)α+β

( –z,w)n+α+β+.

Then, we havefwA(B

n)≤C. Let{gj}be a bounded sequence inB

α,β

ω (Bn) that converges

to zero uniformly asj→ ∞onBn. By the compactness of,β;ω

μ , we have

 = lim

j→∞J=jlim→∞+β

Bn

fw(z)

 –|z|α+βgj(z)Q

α,β;ω

μ (z)(z)

= lim j→∞+β

 –|w|α+β

Bn

( –|z|)α+βg

j(z)Q

α,β;ω μ (z)

( –z,w)n+α+β+ (z)

= lim j→∞

(20)

Whenα+β= , taking

gw(z) =

ln 

 –z,w

ln 

 –|w| –

; ϕw(z)≡ and ω

 –|z|≡,

with|w| ≥ 

, we have+β–;ω(μ)∈LBω;(Bn).

Whenα=β= , taking

gw(z) =

 –|w|

( –z,w)+ln

 –z,w; ϕw(z)≡ and ω

 –|z|≡,

we have+β–;ω(μ)∈LBω(Bn)∩(Bn).

Finally, whenα,β> , take

gw(z) =

 –|w|

( –z,w)α+β; ϕw(z)≡ and ω

 –|z|≡.

Then, it is obvious that+β–;ω(μ)∈Bωα,β(Bn).

This completes the proof of Theorem ..

Remark . It is still an open problem to study the properties of radial Toeplitz operators on the studied spaces of this paper. For more information on radial Toeplitz operators, we refer to [, ].

Competing interests

The author declares that they have no competing interests.

Received: 17 October 2012 Accepted: 17 April 2013 Published: 9 May 2013

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doi:10.1186/1029-242X-2013-237

Cite this article as:El-Sayed Ahmed:General Toeplitz operators on weighted Bloch-type spaces in the unit ball ofCn.

References

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