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: [email protected] 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 andβ
≥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 complexmeasures
μ
for which the Toeplitz operatorTμα,β;ωis bounded or compact onBα,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) =cα( –|z|)αdν, wheredνis the normalized Lebesgue volume measure onBn
andcα=n!(n+(αα+)+) (wheredenotes the gamma function) so thatνα(Bn)≡. The surface
measure onSnis denoted bydσ. 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) =
a–Pa(z) –SaQa(z)
–z,a , z∈Bn,
whereSa=
–|a|,Pa(z) =az,a
|a| ,P= andQa=I–Pa. The mapϕahas the following
properties thatϕa() =a,ϕa(a) = ,ϕa=ϕa–and
–ϕa(z),ϕa(w)
= ( –|a|
)( –z,w)
( –z,a)( –a,w),
wherezandware arbitrary points inBn. In particular,
–ϕa(z)
=( –|a|
)( –|z|)
| –z,a| .
Forf∈H(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) =∇(f ◦ϕz)().
Forα> , a functionf ∈H(Bn) is said to belong to theα-Bloch spacesBα(Bn) if (see [])
bα(f)(Bn) =sup
z∈Bn
∇f(z) –|z|α<∞.
The little Bloch spaceBα
(Bn) consists of allf ∈Bα(Bn) such that lim
|z|→–∇f(z) –|z| α= .
With the normfBα=|f()|+bα(f)(Bn), we know thatBα(Bn) becomes a Banach space
andBα
(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
functionsf∈Lp(Bn,dν
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
Kα(z,w) =
Forα> –, a complex measureμis such that
B
n
–|w|αdμ(w)=
Bn
dμα(w)
<∞.
The general Bergman projectionPαis the orthogonal projection of the measureμfrom
L(Bn,dν
α) intoAα(Bn) defined by
Pα(μ)(z) =cα
Bn
( –|w|)α
( –z,w)n+α+dμ(w) =cα
Bn
dμα(w)
( –z,w)n+α+.
The general Bergman projection of the functionf is
Pαf(z) =cα
Bn
f(w)( –|w|)α
( –z,w)n+α+dν(w) =cα
Bn
f(w)dνα(w)
( –z,w)n+α+.
Letω: (, ]→(,∞) be a right-continuous and nondecreasing function. For a complex measureμ,α> –,β≥, andf ∈L(Bn,dν
α+β), define weighted general Toeplitz operator
as follows:
Tμα,β;ωf(z) =
cα+β
( –|ϕz(w)|)βω( –|w|)
Bn
( –|w|)α+βf(w)
( –z,w)n+α+β+dμ(w)
= cα+β
( –|ϕz(w)|)βω( –|w|)
Bn
f(w)dμα+β(w)
( –z,w)n+α+β+.
ThusPα+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 operatorsTα
μ on theα-BlochBα(D) spaces have been investigated in [] on the
unit diskDfor <α<∞. Also, in [], the authors extended the general Toeplitz operator
Tα
μtoBα(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 whichTα 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 spacesBα,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 spaceBα,β
ω (Bn). It is also proved
that the spaceBα,β
ω (Bn) is a Banach space for each weightα> ,β≥, and the Banach
dual of the Bergman spaceA(Bn) isBα,β
ω (Bn) for eachα≥,β≥. Further, we extend the
Toeplitz operatorTα,β;ω
μ toBωα,β(Bn) in the unit ball ofCnand completely characterize the
positive Borel measureμsuch thatTα,β;ω
μ is bounded or compact inBωα,β(Bn) spaces with
α+β≥.
Throughout the paper, we say that the expressionsAandBare equivalent, and writeA≈ Bwhenever there exist positive constantsCandCsuch thatCA≤B≤CA. As usual,
the letterCdenotes a positive constant, possibly different on each occurrence. Hereafter,
Theorem .(see [, Theorem .]) Suppose b is real and s> –.Then the integrals
Ib(z) =
Sn
dσ(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 g∈L(Bn,dν).If
u(z) = –ϕa(z) β
Bn
g(w)dν(w)
( –z,w)γ, z∈Bn,
then
∇u(z)≤√|γ| –|z|
Bn
g(w)dν(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), ∀f ∈A 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))
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 spaceBωof
sev-eral complex variables is defined as the set of all analytic functionsf onBnsatisfying
–|z|α∇f(z)≤Cω –|z|, z∈Bn, whereα∈(,∞),
for some fixedC=Cf> . In the special case whereω≡,Bωreduces to the classical Bloch
spaceBinCn. This class of functions extends and generalizes the well known Bloch space.
Now, we define the spaceBα,β;ω(Bn) in the unit ballBn. Forα> andβ≥, a function
f ∈H(Bn) is said to belong to the (α,β;ω)-Bloch spaceBα,β;ω(Bn) if
bα,β;ω(f)(Bn) = sup
a,z∈Bn
( –|z|)α+β
( –|ϕa(z)|)βω( –|z|)
∇f(z)<∞.
The little (α,β;ω)-Bloch spaceBα,β;ω,(Bn) is a subspace ofBα,β;ω(Bn) consisting of allf ∈
Bα,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 spaceBα(Bn) and the littleα-Bloch space
Bα
(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 spaceBα,βω (Bn) in the unit ball ofCnby
Lemma . Letα,β∈(,∞)and f ∈Bα,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 ∈Bα,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
≤bα,β;ω(f)
( –|ϕa(tz)|)βω( –t|z|)
( –t|z|)α+β |z|dt
≤bα,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
≤bα,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 f ∈H(Bn).Then the following conditions are equivalent:
(i) f∈Bα,β ω (Bn);
(ii) The function (–|ϕ(–|z|)α+β
z(w)|)βω(–|w|)|Rf(z)|is bounded inBn;
(iii) There exists a functiong∈L∞(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) =cα+β+γ
cγ
( –|z|)α+β
( –|ϕz(w)|)βω( –|w|)
f(z) + Rf(z)
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), f∈A(Bn),g∈Bαω,β(Bn). ()
Proof It is easy to see that – (α+β) +s> –. Ifg∈Bα,β
ω (Bn), then by Theorem ., there
exists a functionh∈L∞(Bn) such that
g(z) =
( –|ϕz(w)|)βω( –|w|)
Bn
h(w)dν–(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|dν–(α+β)+s(z)
=c–(α+β)+s
Bn
f(z)h(z)dν(z).
Applying Lemma . in [] for allf∈A(Bn), we have
Bn
f(z)dν(z)≤ fA(B
n).
Combining this, we see that
f,gs≤ h∞fA(B
n)≤CgBωα,β(Bn)fA(Bn).
Conversely, ifFis a bounded linear functional onA(Bn) andf ∈A(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|)
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∈Bα,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
dν(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∈Bα,β
ω (Bn) and completes the proof of the theorem.
Lemma . If n> ,α+β>,then f ∈Bα,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
On the other hand, iff ∈Bα,β
ω (Bn), then by Theorem .,
f(z) = –ϕw(z) β
ω –|z|
Bn
g(w)dν(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)dν(w) ( –z,w)n+α+β.
An application of Lemma . gives
∇u(z)≤ |n+α+β|√ –|z|
Bn
g(w)dν(w)
| –z,w|n+α+β+
, ∀z∈Bn.
Sinceg(z) is bounded, by Theorem . we have
Bn
g(w)dν(w)
| –z,w|n+α+β+ ≈
–|z|–(α+β).
So,
∇u(z)≤C –|z|–(α+β).
It is easy to check that∇h(z) =∇(h◦ϕz)() = .
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)dν(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: • ≤λ≤α+βif <α+β< ;
• <λ< ifα+β= ;
• α+β– ≤λ≤if <α+β≤.
Then,for allz,w∈Bna holomorphic function f ∈Bα,β
ω (Bn)if and only if
sup z=w
( –|z|)λ( –|w|)α+β–λ
( –|ϕw(z)|)βω( –|z|)
|f(z) –f(w)|
Proof Letf ∈Bα,β
ω (Bn). By a similar proof to the one for Theorem . in [], we have
f(z) –f(w)=√n|z–w|
∇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)|
|z–w| ≤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)|
|z–w| ≤CfBαω,β(Bn)
( –|w|+ (|w|–|z|)t)α( –|w|)βdt. ()
If|z|=|a|, then
|f(z) –f(w)|
|z–w| ≤CfBαω,β(Bn)
( –|w|)α+βdt≤
CfBα,β ω (Bn)
( –|z|)λ( –|w|)α+β–λ. ()
Now suppose|z| =|w|as in [], there is a constantC> such that this integral in () is dominated by
C
( –|z|)λ( –|w|)α+β–λ.
Combining with (), we get that wheneverz=w,
|f(z) –f(w)|
|z–w| ≤
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∈Bα,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+β–λ–( –|w|)α+β–λ|f(z) –f(w)|
( –|ϕw(z)|)βω( –|z|)| –w,z|(α+β)–(λ+)|z–Pz(w) –SzQz(w)| ≤CfBα,β
ω (Bn). ()
Proof Letz= in (), then we have
–|w|α+β–λ|f() –f(w)|
|w| ≤Cbα,β;ω(f)(Bn), w∈Bn\{}.
Now, replacingf byf ◦ϕw, we get
–|u|α+β–λ|f◦ϕw() –f◦ϕw(u)|
|u| ≤Cbα,β;ω(f ◦ϕw)(Bn), u∈Bn\{}. ()
Since
∇(f ◦ϕw)(z)= ∇f
ϕw(z),
by Lemma ., we obtain that
bα,β;ω(f◦ϕw)(Bn)≈ sup z,w∈Bn
( –|w|)α+β–
( –|ϕw(z)|)βω( –|z|) ∇
(f ◦ϕw)(z)
= sup z,w∈Bn
( –|w|)α+β–
( –|ϕw(z)|)βω( –|z|) ∇
fϕw(z)
= sup z,w∈Bn
( –|w|)α+β–( –|ϕ
w(z)|)α+β–
( –|ϕw(z)|)α+β–( –|ϕw(z)|)βω( –|z|) ∇
fϕw(z).
Then
bα,β;ω(f◦ϕz)(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+β–λ
Since
–ϕz(w)=
( –|z|)( –|w|) | –w,z|
and
ϕz(w) =
z–Pz(w) –SzQz(w)
–w,z .
Consequently,
( –|z|)α+β–λ–( –|w|)α+β–λ|f(z) –f(w)|
ω( –|z|)( –|ϕz(w)|)α+β–λ| –w,z|(α+β)–(λ+)|z–Pz(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 spacesBα,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
Pα+β–;ω(μ)∈Bω(Bn)∩LBω(Bn)andμis a Carleson measure;
() ifα> ,β> ,thenTα,β;ω
μ is bounded onBαω,β(Bn)if and only if
Pα+β–;ω(μ)∈Bωα,β(Bn)andμis a Carleson measure.
Proof Since the Banach dual of A(Bn) isBα,β
ω (Bn) under the pairing (), to prove the
boundedness ofTα,β;ω
μ , it suffices to show that
f,Tμα,β;ω(g)
α≤CfA(Bn)gBωα,β(Bn) for allf ∈A(Bn) andg∈Bα,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)
=cα+β–
Bn
f(z)g(z) –|z|α+β–dμ(z).
Using the operatorPα+β;ω, we have
f,Tμα,β;ωg
s=cα+β–
Bn
(Iz,w;ω–Pα+β;ω)(f g)(z)
–|z|α+β–dμ(z)
+cα+β–
Bn
Pα+β;ω(f g)(z)
whereIz,w;ω=(–|ϕ I
z(w)|)βω(–|w|), andIis the identity operator. Also, (Iz,w;ω–Pα+β;ω)(f g)(z)
= f(z)g(z) ( –|ϕz(w)|)βω( –|w|)
– cα+β
( –|ϕz(w)|)βω( –|w|)
Bn
f(w)g(w)( –|w|)α+β
( –z,w)n+α+β+ dν(w)
= cα+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|)
dν(z).
By Proposition ., we have
|I|=cα+β– B
n
(Iz,w;ω–Pα+β;ω)(f g)(z)
–|z|α+β–dμ(z)
=cα+β–cα+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|)
dν(w)dμ(z)
=cα+β–cα+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|)dμ(z)dν(w)
≤cα+β–cα+β
Bn
f(w) –|w|λ
×
Bn
( –|z|)(α+β)–λ–( –|w|)α+β–λ|f(z) –f(w)|
| –w,z|(α+β)–(λ+)|z–P
z(w) –SzQz(w)|
× |z–Pz(w) –SzQz(w)|( –|z|)λ–(α+β) | –z,w|n–(α+β)+λ+( –|ϕ
z(w)|)βω( –|w|)
dμ(z)dν(w)
≤C
Bn
gBα,β
ω (Bn)f(w) –|w| λ
Bn
( –|z|)λ–(α+β)
| –z,w|n–(α+β)+λ+dμ(z)dν(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+dμ(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+dν(z)≤C.
Therefore,
|I| ≤CfA(B
Next consideringI, we have
|I|=cα+β– B
n
Pα+β;ω(f g)(z)
–|z|α+β–dμ(z)
=cα+β–cα+β
B
n
Bn
f(w)g(w)( –|w|)α+β( –|z|)α+β–
( –z,w)n+α+β+( –|ϕz(w)|)βω( –|w|)dν(w)dμ(z)
≤cα+β
Bn
f(w) –|w|α+βg(w)
×
cα+β–
( –|ϕz(w)|)βω( –|w|)
Bn
( –|z|)α+β–dμ(z) | –z,w|n+α+β+
dν(w)
≤C
Bn
fA(Bn)
–|w|α+βg(w)Qα,β;ω
μ (w)dν(w),
where
Qαμ,β;ω(w) =
cα+β–
( –|ϕz(w)|)βω( –|w|)
Bn
( –|z|)α+β–dμ(z) | –z,w|n+α+β+ .
As in [], by simple calculation, we have
Qαμ,β;ω(w) =Pα+β–;ω(μ)(w) +
n+α+βRPα+β–;ω(μ)(w). ()
It is easy to see that
() ifα+β= andPα+α–;ω(μ)∈LBαω,β(Bn), then
–|w|Qαμ,β;ω(w)
ln
–|w|
∈L∞(Bn);
() ifα=β= ,Pα+α–;ω(μ)∈Bω(Bn)∩LB–ω(Bn), then
–|w|Qα,β;ω
μ (w)∈L∞(Bn)
and
–|w|Qα,β;ω μ (w)
ln
–|w|
∈L∞(Bn);
() ifα> ,β> , andPα+α–;ω(μ)∈B–ωα,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
Bα,β
ω (Bn) withα> ,β≥.
Conversely, suppose thatTα,β;ω
μ is a bounded operator onBαω,β(Bn). Take
fw(z) =
( –|w|)t
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,Tα μg
s=cα+β–
–|w|n++t–(α+β)
Bn
( –|z|)α+β–dμ(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–dμ(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|=cα+β
B
n
f(w) –|w|α+βg(w)Qαμ,β;ω(w)dν(w)
≤CfA(Bn)g
Bα,β ω (Bn).
This implies that
g(w)Qα,β;ω
μ (w) –|w|
α+β
≤CgBα,β ω (Bn).
Ifα+β= , we have
g(w)Qα,β;ω
μ (w) –|w|
≤Cg
Bα,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 havePα+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
Takingz=w, then
g(w)Qμα,β;ω(w) –|w|
=
–|w| +ln
–|w|
Qα,β
μ (w) –|w|
≤Qαμ,β;ω(w) –|w|
+Qαμ,β;ω(w) –|w|
ln
–|w|
≤C.
By (), thenPα+β–;ω(μ)∈LBω(Bn) andPα+β–;ω(μ)∈Bω(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 thatPα+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 spacesBα,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
Bα,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
Pα+β–;ω(μ)∈Bω;(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. Supposef ∈A(Bn). By duality, we have that Tα,β;ω
μ is compact onB α,β
ω (Bn) if and only if
lim j→∞fsup
A(Bn)≤
Similarly, as in the proof of Theorem . fors=α+β– , we have
f,Tμα,β;ωgj
s=cα+β–
Bn
f(z)gj(z)
–|z|α+β–dμ(z)
=cα+β–
Bn
(Iz,w;ω–Pα+β;ω)(f gj)
(z) –|z|α+β–dμ(z)
+cα+β–
Bn
Pα+β;ω(f gj)(z)
–|z|α+β–dμ(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)dμ(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|=cα+β–cα+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|)
dν(w)dμ(z)
=cα+β–cα+β
B
n
f(w) –|w|α+β
×
Bn
(gj(z) –gj(w))( –|z|)α+β–
( –z,w)n+α+( –|ϕw(z)|)βω( –|z|)dμ(z)dν(w)
≤cα+β–cα+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|)
dμ(z)dν(w)
+cα+β–cα+β
δBn
f(w) –|w|α+β
×
Bn
|gj(z) –gj(w)|( –|z|)α+β– | –z,w|n+α+( –|ϕ
w(z)|)βω( –|z|)
dμ(z)dν(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|)
dμ(z)dν(w)
Now, takingδsuch that – [ε( –η)n++λ]λ ≤δ< , then
L≤CgjBα,β ω (Bn)
Bn\δBn
f(w) –|w|λ
Bn
( –|z|)λ–(α+β)
| –z,w|n+λ+–(α+β)dμ(z)dν(w) ≤C
Bn\δBn
f(w) –|w|λ
Bn\ηBn
( –|z|)λ–(α+β)
| –z,w|n+λ+–(α+β)dμ(z)dν(w)
+C
Bn\δBn
f(w) –|w|λ
ηBn
( –|z|)λ–(α+β)
| –z,w|n+λ+–(α+β)dμ(z)dν(w) ≤Cε
Bn\δBn
f(w)dν(w) +C
Bn\δBn
f(w) ( –δ)
λ
( –η)n++λdν(w)
≤CεfA(B
n).
Hence|J|<Cε, whereCdoes not depend onf(z), and so
lim j→∞fsup
A(Bn)≤ |J|= .
Thus,Tα,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)dν(w).
From () it is easy to see that
() ifα+β= andPα+α–;ω(μ)∈LBαω,,β(Bn), then
lim
|w|→
–|w|Qαμ,β;ω(w)
ln
–|w|
= ;
() ifα=β= ,Pα+α–;ω(μ)∈Bω;(Bn)∩LBω;(Bn), then
lim
|w|→
–|w|Qαμ,β;ω(w) =
and
lim
|w|→
–|w|Qαμ,β;ω(w)
ln
–|w|
= ;
() ifα> ,β> , andPα+α–;ω(μ)∈Bωα,;β(Bn), then
lim
|w|→
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 thatTα,β;ω
μ is a compact operator.
Next assume that Tα,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,Tα,β;ω μ g
s
=cα+β–
–|w|n++t–(α+β)
Bn
( –|z|)α+β–dμ(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 ofTα,β;ω
μ , we have
= lim
j→∞J=jlim→∞cα+β
Bn
fw(z)
–|z|α+βgj(z)Q
α,β;ω
μ (z)dν(z)
= lim j→∞cα+β
–|w|α+β
Bn
( –|z|)α+βg
j(z)Q
α,β;ω μ (z)
( –z,w)n+α+β+ dν(z)
= lim j→∞
Whenα+β= , taking
gw(z) =
ln
–z,w
ln
–|w| –
; ϕw(z)≡ and ω
–|z|≡,
with|w| ≥
, we havePα+β–;ω(μ)∈LBω;(Bn).
Whenα=β= , taking
gw(z) =
–|w|
( –z,w)+ln
–z,w; ϕw(z)≡ and ω
–|z|≡,
we havePα+β–;ω(μ)∈LBω(Bn)∩Bω(Bn).
Finally, whenα,β> , take
gw(z) =
–|w|
( –z,w)α+β; ϕw(z)≡ and ω
–|z|≡.
Then, it is obvious thatPα+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
References
1. Hahn, KT, Choi, KS: Weighted Bloch spaces inCn. J. Korean Math. Soc.35, 171-189 (1998)
2. Li, S, Wulan, H: Characterizations ofα-Bloch spaces on the ball. J. Math. Anal. Appl.343(1), 58-63 (2008)
3. Nowak, M: Bloch and Möbius invariant Besov spaces on the unit ball ofCn. Complex Var. Theory Appl.44, 1-12 (2001)
4. Zhu, K: Bloch-type spaces of analytic functions. Rocky Mt. J. Math.23, 1143-1177 (1993) 5. Zhu, K: Spaces of Holomorphic Functions in the Unit Ball. Springer, New York (2004)
6. Zhu, K: Positive Toeplitz operators on weighted Bergman spaces of bounded symmetric domains. J. Oper. Theory20, 329-357 (1988)
7. Zhu, K: Operator Theory in Function Spaces. Dekker, New York (1990)
8. Wu, Z, Zhao, R, Zorboska, N: Toeplitz operators on Bloch-type spaces. Proc. Am. Math. Soc.134, 3531-3542 (2006) 9. Wang, X, Liu, T: Toeplitz operators on Bloch-type spaces in the unit ball ofCn. J. Math. Anal. Appl.368, 727-735 (2010)
10. Wu, Z, Zhao, R, Zorboska, N: Toeplitz operators on analytic Besov spaces. Integral Equ. Oper. Theory60, 435-449 (2008)
11. El-Sayed Ahmed, A, Bakhit, MA: Properties of Toeplitz operators on some holomorphic Banach function spaces. J. Funct. Spaces Appl.2012, Article ID 517689 (2012)
12. Das, N, Sahoo, M: Positive Toeplitz operators on the Bergman space. Ann. Funct. Anal.4(2), 171-182 (2013) 13. Englis, M: Toeplitz operators and localization operators. Trans. Am. Math. Soc.361, 1039-1052 (2009) 14. Englis, M: Toeplitz operators and weighted Bergman kernels. J. Funct. Anal.255, 1419-1457 (2008)
15. Perälä, A: Toeplitz operators on Bloch-type spaces and classes of weighted Sobolev distributions. Integral Equ. Oper. Theory71(1), 113-128 (2011)
16. Sánchez-Nungaray, A, Vasilevski, N: Toeplitz operators on the Bergman spaces with pseudodifferential defining symbols. In: Karlovich, YI, Rodino, L, Silbermann, B, Spitkovsky, IM (eds.) Operator Theory, Pseudo-Differential Equations, and Mathematical Physics. Operator Theory: Advances and Applications, vol. 228, pp. 355-374. Birkhäuser, Basel (2013)
17. Vasilevski, NL: Commutative Algebras of Toeplitz Operators on the Bergman Space. Operator Theory: Advances and Applications, vol. 185, xxix. Birkhäuser, Basel (2008)
18. El-Sayed Ahmed, A: Criteria for functions to be weighted Bloch. J. Comput. Anal. Appl.11(2), 252-262 (2009) 19. Zhu, K: Multipliers ofBMOin the Bergman metric with applications to Toeplitz operators. J. Funct. Anal.87, 31-50
(1989)
21. Zhao, R: A characterization of Bloch-type spaces on the unit ball ofCn. J. Math. Anal. Appl.330, 291-297 (2007)
22. Rudin, W: Function Theory in the Unit Ball ofCn. Springer, New York (1980)
23. Grudsky, S, Maximenko, E, Vasilevski, NL: Radial Toeplitz operators on the unit ball and slowly oscillating sequences. Commun. Math. Anal.14(2), 77-94 (2013)
24. Zhou, ZH, Chen, WL, Dong, XT: The Berezin transform and radial operators on the Bergman space of the unit ball. Complex Anal. Oper. Theory7(1), 313-329 (2013)
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.
