Weighted composition operators from weighted Bergman spaces with Békollé weights to Bloch type spaces

(1)

R E S E A R C H

Open Access

Weighted composition operators from

weighted Bergman spaces with Békollé

weights to Bloch-type spaces

Stevo Stevi´c

1,2*

_{and Ajay K Sharma}

3

*_{Correspondence: sstevic@ptt.rs} 1_{Mathematical Institute of the}

Serbian Academy of Sciences, Knez Mihailova 36/III, Beograd, 11000, Serbia

2_{Operator Theory and Applications}

Research Group, Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

Full list of author information is available at the end of the article

Abstract

Let

σ

be a Békollé weight function and

ν

be a weight function. In this paper, we characterize the boundedness and compactness of weighted composition operators acting from Bergman-type spacesAp_(σ_{) to Bloch-type spaces}_B

νandBν,0,

considerably extending some results in the literature.

MSC: Primary 47B33; 46E10; secondary 30D55

Keywords: weighted composition operator; Bergman-type spaces; Békollé weight; normal weight; Bloch-type spaces

1 Introduction and preliminaries

LetDdenote the open unit disk in the complex planeC,H(D) the space of all holomorphic functions onD, andS(D) the class of all holomorphic self-maps ofD. Letψ∈H(D) and

ϕ∈S(D), theweighted composition operator Wψ,ϕ is a linear operator onH(D) deﬁned

by

Wψ,ϕ(f)(z) =ψ(z)f

ϕ(z), z∈D

forf ∈H(D). It is of interest to provide function-theoretic characterizations when

sym-bolsϕ andψ induce a bounded or compact weighted composition operator between diﬀerent function spaces. Recently, numerous authors have studied the boundedness and compactness of weighted composition operators on spaces of analytic functions on var-ious domains (see, for example, [–] and the related references therein) as well as of some related operators involving composition ones on these spaces [–]. A joint fea-ture for the majority of these papers is that they study the operators from or to Bloch-type or Bergman-type spaces. Paper [] is one of the older papers that studied composition operators (caseψ(z)≡) on Bloch-type spaces and served as a motivation to several au-thors.

A continuous functionν:D→(,∞) is calledweight. Ifν(z) =ν(|z|),z∈D, the weight

is calledradial. If a weightνis such thatlim_|_z_|→_–ν(z) = , we will call it astandard weight.

Weightν(r),r=|z|, is callednormalif there exist positive numbersηandτ,  <η<τ, and

(2)

δ∈[, ) such that

ν(r)

( –r)η is decreasing on [δ, ) andlim_r_→_ ν(r) ( –r)η = ; ν(r)

( –r)τ is increasing on [δ, ) and_rlim_→_–

ν(r)

( –r)τ = +∞.

The classical weightsνα(r) = ( –r)α_,_α_{> –, are obviously normal.}

LetdA(z) =dx dy/π=r dr dθ/πstand for the normalized area measure inD. As usual, a measurable functiongonDis called Lebesgue integrable if

D

g(z)dA(z) <∞,

and it is writteng∈L₍_D_{). If}_g_∈_L₍_D_{) is nonnegative, we will write}_g_∈_L +(D).

For  <p<∞andσ a nonnegative Lebesgue integrable function onD, we denote by

Ap₍_σ_{) the Bergman-type space consisting of all functions}_f _∈_H₍_D_{) such that}

fp_Ap₍_σ₎=

D

f(z)pσ(z)dA(z) <∞.

Forσ(z) = ( –|z|₎α_,_α_{> –, the space becomes the (standard) weighted Bergman space}

Apα(D) =Apα.

The weights considered here are the so-calledBékollé weights[], which are Bergman spaces analogues of the Muckenhoupt classes used in harmonic analysis. Forp>  and α> –, letdAα(z) = (α+ )( –|z|)αdA(z),Bp(α) be the class consisting ofσ∈L



+(D) with

the property that there exists a constantC>  such that

S(θ,h)

σ(z)dAα(z)

S(θ,h) σ–

p_

p₍_z₎_dA_α₍_z₎

p

p_

≤CAα

S(θ,h) p

for any Carleson square

S(θ,h) =z=reiφ:  –h<r< ,|θ–φ|<h/, θ∈[, π],h∈(, ),

where /p+ /p= . It is not diﬃcult to see that all normal weights are the Békollé

weights.

It is well known [] that the following inclusions hold:

Bp(α)⊂Bp(α) if –  <α<α

and

Bp(α)⊂Bq(α) if  <p<q.

A functionσ∈L

+(D) belongs to classCp,p> , if there is a constantC>  such that

Dλ(r)

σ(z)dA(z)

Dλ(r)

σ– p_

p₍_z₎_dA₍_z₎

p_ p_

(3)

for every disk Dλ(r) ={z∈D:|z–λ|<r( –|λ|)}. Herer∈(, ) is ﬁxed, but the class

Cpis actually independent ofr∈(, ). Moreover,Bp(α)⊂Cpfor everyα> – and the

inclusion is strict. For more about the classesBp(α) andCpand the properties satisﬁed

by the weights in these classes, we refer to [] and [] and the references therein. Throughout this paper constants are denoted byC, they are positive and not necessarily the same at each occurrence. The notation ABmeans thatA≤CBfor someC>  independent of the variables involved into these quantities. IfABandBA, then we writeAB.

For functions inAp₍_σ_{), when the weight}_σ _{is in}_C

p, we have the following estimate,

which easily follows by using a standard procedure, namely, from the Cauchy inequality applied to functionf(k)_{, the subharmonicity of}_|_f_|p_,_p_{> , the integral Hölder inequality}

and the deﬁnition of classCp.

Lemma  Let r∈(, ),σ∈L₊(D)∩Cp,p> ,p> and k∈N.Then there is a constant

C> independent of z such that

_f(k)₍_z₎_≤_C(

Dz(r)σ(ζ)dA(ζ)) –/p

( –|z|₎k fAp(σ)

for every f ∈Ap₍_σ_).

The next lemma provides an asymptotic estimate for the norm of weighted Bergman kernel (see []).

Lemma  Let r∈(, )be ﬁxed,p> ,p> andη> –.Assume that p≥p,σ∈L+(D)is

such thatσ(z)/( –|z|₎α_{belongs to B}

p(α)andγ ≥(η+ )p/p– .Let

K_λγ(z) = 

( –λ¯z)γ+, λ∈D

(the reproducing kernel of the weighted Bergman space Apγ).Then Kλγ_Ap₍_σ₎

(_D

λ(r)σdA)

/p

( –|λ|₎γ+ .

Lemma  Let r∈(, )be ﬁxed,p> ,p> andη> –.Assume that p≥p,σ∈L+(D)is

such thatσ(z)/( –|z|₎α_{belongs to B}

p(α).Then

fλ(z) =

( –|λ|)(η+)p/p+

(_D

λ(r)σdA)

/p_{( –}_λ¯_z₎(η+)p/p+ ()

is in Ap₍_σ_)._Moreover_,

sup

λ∈D

fλAp₍_σ₎, ()

(4)

Proof Letγ = (η+ )p/p– . Then by Lemma  we have fλ(z)

|Kλγ+(z)|

K_λγ+Ap₍_σ₎

,

from which () follows. By Lemma  withk=  and Lemma , we have that

 =K_λγ()≤C

D(/)

σdA

–/p

K_λγ_Ap₍_σ₎

(_D

λ(r)σdA)

/p

( –|λ|₎γ+ .

Thus

( –|λ|₎(η+)p/p

(_D

λ(r)σdA)

/p =

( –|λ|₎γ+

(_D

λ(r)σdA)

/p . ()

Using () in () it is easy to see thatfλconverges to zero uniformly on compact subsets of

D, as|λ| →–_.

For a weightν, theBloch-type spaceBνonDis the space of all holomorphic functionsf

onDsuch that

sup

z∈D

ν(z)f(z)<∞.

Thelittle Bloch-type spaceBν,consists of allf ∈Bνsuch that

lim |z|→ν(z)f

₍_z₎_{= .}

Both spacesBνandBν,are Banach spaces with the norm

fBν =f()+sup

z∈D

ν(z)f(z),

andBν,is a closed subspace ofBν.

Depending on the weightν, various Bloch-type spaces are obtained. Forν(z) = ( –|z|₎α_, α> , the spaces are reduced to the so-calledα-Bloch, that is, the littleα-Bloch space, which is forα =  reduced to the classical Bloch space. The reader could see that pa-pers [–, , –, , , –] consider concrete operators on or to various Bloch-type spaces. Nowadays we know that if the image space is a Bloch-type, then usually it does not aﬀect much on the boundedness and compactness, so we will here consider the spaces with as much as general weights.

The compactness of a closed subsetL⊂Bν,can be characterized as follows.

Lemma  Letνbe a standard weight.A closed set L inBν,is compact if and only if it is

bounded with respect to the norm · Bν and satisﬁes lim

|z|→sup_f_∈_Lν(z)

_f₍_z₎_{= .}

This result for the caseν(z) =  –|z|_{was proved by Madigan and Matheson in []. By}

(5)

Motivated by [], as well as by [], here we characterize the boundedness and compact-ness of the weighted composition operators acting from the Bergman-type spacesAp₍_σ₎

to Bloch-type spacesBνandBν,. Our results extend some in [].

The following criterion for compactness follows by standard arguments which appeared for the ﬁrst time in [].

Lemma  Letσ∈L

+(D)be such thatσ(z)/( –|z|)αbelongs to Bp(α),νbe a standard

weight and Wψ,ϕ:Ap(σ)→Bν be bounded.Then Wψ,ϕ:Ap(σ)→Bν is compact if and

only if for any bounded sequence (fn)n∈N in Ap(σ)which converges to zero uniformly on

compact subsets ofD,we have

lim

n→∞Wψ,ϕfnBν= .

2 Boundedness and compactness ofWψ,ϕ:Ap(

σ

)→Bν(Bν,0)

In this section we formulate and prove the main results in this paper.

Theorem  Let r∈(, )be ﬁxed,p> ,p> ,α> –,νbe a weight,ψ∈H(D)andϕ∈

S(D).Assume that p≥p andσ ∈L+(D)is such thatσ(z)/( –|z|)α belongs to Bp(α).

Then Wψ,ϕ:Ap(σ)→Bνis bounded if and only if the following conditions are satisﬁed:

(i) M=supz∈Dν(z)|ψ(z)|(

Dϕ(z)(r)σdA)

–/p_<_∞_;

(ii) M=supz∈D

ν(z)|ψ(z)ϕ(z)| –|ϕ(z)| (

Dϕ(z)(r)σdA)

–/p_<_∞_.

Moreover,if Wψ,ϕ:Ap(σ)→Bνis bounded,then

Wψ,ϕAp₍_σ₎_→_B_ν_/_CM_+M_. ()

Proof First suppose that conditions (i) and (ii) hold. Then by Lemma  we have

ν(z)(Wψ,ϕf)(z)≤ν(z)ψ(z)f

ϕ(z)+ν(z)ψ(z)ϕ(z)fϕ(z)

ν(z)ψ(z)

Dϕ(z)(r)

σdA

–/p

+ν(z)|ψ(z)ϕ ₍_z₎_|  –|ϕ(z)|

Dϕ(z)(r)

σdA

–/p

fAp₍_σ₎. ()

Furthermore,

Wψ,ϕ(f)()=ψ()f

ϕ()ψ()

Dϕ()(r)

σdA

–/p

fAp₍_σ₎. ()

Using (i), (ii), () and (), we see that

Wψ,ϕfBν=ψ()f

ϕ()+sup

z∈D

ν(z)(Wψ,ϕf)(z)

ψ()

Dϕ()(r)

σdA

–/p

+M+M

fAp₍_σ₎.

So, we have thatWψ,ϕ:Ap(σ)→Bνis bounded and

Wψ,ϕAp₍_σ₎_→_B

(6)

Conversely, suppose thatWψ,ϕ :Ap(σ)→Bν is bounded. Then, by taking f(z)≡∈

Ap₍_σ_{), we have that}

sup

z∈D

ν(z)ψ(z)=Wψ,ϕ()_B

ν Wψ,ϕAp(σ)→Bν. ()

By takingf(z) =z∈Ap₍_σ_{), using the boundedness of}_ϕ _{and asymptotic estimate (), we}

easily get

sup

z∈D

ν(z)ψ(z)ϕ(z)Wψ,ϕAp₍_σ₎_→_B

ν. ()

Letλ=ϕ(ζ),ζ ∈D, andgλ(z) =τλ(z)fλ(z), wherefλis deﬁned in () andτλis deﬁned as

τλ(z) =  – –|λ|



 –λ¯z . ()

Thenτλ∈H∞as

sup

z∈D

τλ(z)≤sup

z∈D

 +  –|λ|



 –|λ||z|

≤.

Thusgλ∈Ap(σ) andsup_z_∈_DgλAp₍_σ₎. Moreover,

τλ(λ) = , τ_λ(z) = –λ¯  –|λ| 

( –λ¯z) and τ

λ(λ) =

–λ¯

 –|λ|. ()

We also have

fλ(λ) =

Dλ(r)

σdA

–/p

, ()

f_λ(z) =λ¯

(η+ )p

p + 

( –|λ|₎(η+)p/p+

(_D

λ(r)σdA)

/p_{( –}_λ¯_z₎(η+)p/p+ ()

and

fλ(λ) =λ¯

(η+ )p

p + 

₍

Dλ(r)σdA)

–/p

 –|λ| . ()

Thereforegλ(λ) =  and, from () and (), we have

gλ(λ) =τλ(λ)fλ(λ) +τλ(λ)fλ(λ) = –λ¯

(_D

λ(r)σdA)

–/p

 –|λ| .

Using this fact we obtain

νWψ,ϕgϕ(ζ)Bν ≥ν(ζ)ψ(ζ)gϕ(ζ)

ϕ(ζ)+ψ(ζ)ϕ(ζ)gϕ(ζ)

ϕ(ζ)

≥ν(ζ)ψ(ζ)ϕ(ζ)ϕ(ζ) (_D

ϕ(ζ)(r)σdA)

–/p

(7)

Thus, for a ﬁxedδ∈(, ), we obtain

sup |ϕ(ζ)|>δ

ν(ζ)ψ(ζ)ϕ(ζ) (_D

ϕ(ζ)(r)σdA)

–/p

 –|ϕ(ζ)| Wψ,ϕAp(σ)→Bν. ()

By using () and (), we have that

sup |ϕ(ζ)|≤δ

ϕ(ζ)(r)σdA)

–/p

 –|ϕ(ζ)|

= sup

|ϕ(ζ)|≤δ

ν(ζ)|ψ(ζ)ϕ(ζ)| ( –|ϕ(ζ)|₎(η+)p/p+

( –|ϕ(ζ)|₎(η+)p/p

(_ϕ₍_ζ₎₍_r₎σdA)/p



( –δ₎(η+)p/p+Wψ,ϕAp(σ)→Bν. ()

Hence from () and () we have

sup

ζ∈D

ϕ(ζ)(r)σdA)

–/p

 –|ϕ(ζ)| Wψ,ϕAp(σ)→Bν. ()

Letλ=ϕ(ζ),ζ ∈D, andfλbe deﬁned in (). Recall thatsupλ∈DfλAp₍_σ₎. Hence, by

() and () we get

ν Wψ,ϕfϕ(ζ)Bν≥ψ(ζ)fϕ(ζ)

ϕ(ζ)+ψ(ζ)ϕ(ζ)fϕ(ζ)

ϕ(ζ)

≥ν(ζ)ψ(ζ)

Dϕ(ζ)(r)

σdA

–/p

–

(η+ )p

p + 

ν(ζ)|ψ(ζ)||ϕ(ζ)|  –|ϕ(ζ)| ϕ(ζ)

Dϕ(ζ)(r)

σdA

–/p

,

from which, along with the boundedness ofϕ, it follows that

ν(ζ)ψ(ζ)

Dϕ(ζ)(r)

σdA

–/p

≤ Wψ,ϕAp₍_σ₎_→_B

ν+C

ν(ζ)|ψ(ζ)||ϕ(ζ)|  –|ϕ(ζ)|

Dϕ(ζ)(r)

σdA

–/p

. ()

Taking the supremum overζ ∈Din () and using (), we get

sup

ζ∈D

ν(ζ)ψ(ζ)

Dϕ(ζ)(r)

σdA

–/p

ν. ()

From () and () we have

M+MWψ,ϕAp₍_σ₎_→_B_ν_/_C. ()

(8)

Theorem  Let r∈(, )be ﬁxed,p> ,p> ,α> –,νbe a standard weight,ψ∈H(D)

andϕ∈S(D).Assume that p≥p,σ∈L+(D)is such thatσ(z)/( –|z|)αbelongs to Bp(α)

and Wψ,ϕ:Ap(σ)→Bνis bounded.Then Wψ,ϕ:Ap(σ)→Bνis compact if and only if the

following conditions are satisﬁed: (i) lim_|ϕ(z)|→ν(z)|ψ(z)|(

Dϕ(z)(r)σdA)

–/p_{= }_;

(ii) lim|ϕ(z)|→ν(z)|ψ(z)ϕ

₍_z₎_|

–|ϕ(z)| (

Dϕ(z)(r)σdA)

–/p_{= }_.

Proof First suppose that conditions (i) and (ii) hold. Then by Lemma  it is suﬃcient to show that if (fn)n∈Nis a bounded sequence inAp(σ) that converges to zero uniformly on compact subsets ofD, thenWψ,ϕfnBν → asn→ ∞. Let (fn)n∈N⊂Ap(σ) be such a

sequence. By conditions (i) and (ii), we have that for anyε> , there isδ∈(, ) such that

ν(z)ψ(z)

Dϕ(z)(r)

σdA

–/p

<ε ()

and

ν(z)|ψ(z)ϕ(z)|  –|ϕ(z)|

Dϕ(z)(r)

σdA

–/p

<ε, ()

wheneverδ<|ϕ(z)|< .

LetK={z∈D:|z| ≤δ}. Clearly,Kis a compact subset ofD. We have

Wψ,ϕfnBν =ψ()fn

ϕ()+sup

ζ∈D

ν(ζ)(Wψ,ϕfn)(ζ)

≤ψ()fn

ϕ()+sup

ζ∈D

ν(ζ)ψ(ζ)fn

ϕ(ζ)

+sup

ζ∈D

ν(ζ)ψ(ζ)ϕ(ζ)f_nϕ(ζ)

≤ψ()fn

ϕ()+ sup {ζ∈D:ϕ(ζ)∈K}ν(ζ)

ψ(ζ)fn

ϕ(ζ)

+ sup

{ζ∈D:δ<|ϕ(ζ)|<}

ν(ζ)ψ(ζ)fn

ϕ(ζ)

+ sup

{ζ∈D:ϕ(ζ)∈K}

+ sup

{ζ∈D:δ<|ϕ(ζ)|<}

≤ψ()fn

ϕ()+ψ_B_νsup

z∈K

fn(z)+Nsup z∈K

f_n(z)

+C sup

{ζ∈D:δ<|ϕ(ζ)|<}

ν(ζ)ψ(ζ)

Dϕ(ζ)(r)

σdA

–/p

fnAp₍_σ₎

+C sup

{ζ∈D:δ<|ϕ(ζ)|<}

ν(ζ)|ψ(ζ)ϕ(ζ)|  –|ϕ(ζ)|

Dϕ(ζ)(r)

σdA

–/p

fnAp₍_σ₎, ()

where we have used the fact thatψ∈BνandN=sup_ζ_∈_Dν(ζ)|ψ(ζ)ϕ(ζ)|<∞. Using () and () along with facts that

fn

ϕ()<ε, sup

z∈K

fn(z)<ε and sup z∈K

(9)

for someN∈Nand for alln≥N, in (), we haveWψ,ϕfnBν <Cεforn≥N. Since

ε>  is arbitrary, we have thatWψ,ϕfnBν→ asn→ ∞. HenceWψ,ϕ:Ap(σ)→Bνis

compact.

Conversely, suppose thatWψ,ϕ:Ap(σ)→Bνis compact. Let (ζn)n∈Nbe a sequence in Dsuch that |ϕ(ζn)| → asn→ ∞. If such a sequence does not exist, then (i) and (ii)

are vacuously satisﬁed. Letgn(z) =τϕ(ζn)(z)fϕ(ζn)(z), wherefλis deﬁned in () andτλis

de-ﬁned in (). Then as in Theorem ,τϕ(ζn)Ap₍_σ₎ and by Lemma , f_ϕ₍_ζ_n)_Ap₍_σ₎

and (fϕ(ζn))n∈N converges to zero uniformly on compact subsets ofDasn→ ∞. Thus

gnAp₍_σ₎ and (g_n)_n_∈_Nconverges to zero uniformly on compact subsets ofDasn→ ∞.

SinceWψ,ϕ:Ap(σ)→Bν is compact, we have thatWψ,ϕgnBν → asn→ ∞. On the

other hand, from () we have

Wψ,ϕgnBν≥

ν(z)|ψ(ζn)ϕ(ζn)||ϕ(ζn)|

 –|ϕ(ζn)|

Dϕ(ζn)(r)

σdA

–/p

.

Using these two facts we have that

lim |ϕ(ζn)|→

ν(z)|ψ(ζn)ϕ(ζn)|

 –|ϕ(ζn)|

Dϕ(ζn)(r)

σdA

–/p

= , ()

from which (i) follows.

Letfλbe deﬁned in (). Thensup_n_∈_Nfϕ(ζn)Ap₍_σ₎ andf_ϕ₍_ζ_n) converges to zero

uni-formly on compact subsets ofDasn→ ∞. SinceWψ,ϕ:Ap(σ)→Bνis compact, we have

that

lim

n→∞Wψ,ϕfϕ(ζn)Bν= . ()

From (), we have

ν(ζn)ψ(ζn)

Dϕ(ζn)(r)

σdA

–/p

≤ Wψ,ϕfϕ(ζn)Ap₍_σ₎_→_B

ν+C

ν(ζn)|ψ(ζn)||ϕ(ζn)|

 –|ϕ(ζn)|

Dϕ(ζn)(r)

σdA

–/p

,

which along with () and () implies that

lim |ϕ(ζn)|→ν(ζn)

ψ(ζn)

Dϕ(ζn)(r)

σdA

–/p

= ,

from which (ii) follows.

Next we characterize the boundedness and compactness ofWψ,ϕ:Ap(σ)→Bν,.

Lemma  Let r∈(, )be ﬁxed,p> ,p> ,α> –,νbe a standard weight,ψ∈H(D)

andϕ∈S(D).Assume that p≥p,σ∈L+(D)is such thatσ(z)/( –|z|)αbelongs to Bp(α).

Then

lim |z|→ν(z)

ψ₍_z₎

Dϕ(z)(r)

σdA

–/p

(10)

if and only ifψ∈Bν,and

lim |ϕ(z)|→ν(z)

ψ₍_z₎

Dϕ(z)(r)

σdA

–/p

= . ()

Proof Suppose that () holds. Then, sinceσ∈L(D), we have

ν(z)ψ(z)ν(z)ψ(z)

Dϕ(z)(r)

σdA

–/p

→

as|z| →. Henceψ ∈Bν,. On the other hand, since |ϕ(z)| → implies|z| →, ()

automatically holds.

Conversely, suppose thatψ∈Bν,and () hold. By (), for everyε> , there exists δ∈(, ) such that

ν(z)ψ(z)

Dϕ(z)(r)

σdA

–/p

<ε, ()

whenδ<|ϕ(z)|< .

On the other hand, sinceψ∈Bν,, there existsγ ∈(, ) such that

ν(z)ψ(z)≤ε –δ(η+)p/p,

wheneverγ <|z|< .

Thus if γ <|z|<  andδ<|ϕ(z)|< , we have that () holds, while ifγ <|z|<  and

|ϕ(z)| ≤δ, then from () we have

ν(z)ψ(z)

Dϕ(z)(r)

σdA

–/p

= ν(z)|ψ ₍_z₎_| ( –|ϕ(z)|₎(η+)p/p

( –|ϕ(z)|₎(η+)p/p

(_D

ϕ(z)(r)σdA)

–/p

ν(z)|ψ(z)|

( –δ₎(η+)p/p <ε. ()

Combining () and (), we easily obtain that () holds.

The following lemma is proved similarly. Hence we omit the proof.

Lemma  Let r∈(, )be ﬁxed,p> ,p> ,α> –,νbe a standard weight,ψ∈H(D)

andϕ∈S(D).Assume that p≥p,σ∈L+(D)is such thatσ(z)/( –|z|)αbelongs to Bp(α).

Then

lim |z|→

ν(z)|ψ(z)ϕ(z)|  –|ϕ(z)|

Dϕ(z)(r)

σdA

–/p

= 

if and only if

lim |z|→ν(z)

(11)

and

lim |ϕ(z)|→

ν(z)|ψ(z)ϕ(z)|  –|ϕ(z)|

Dϕ(z)(r)

σdA

–/p

= .

Theorem  Let r∈(, )be ﬁxed,p> ,p> ,α> –,νbe a standard weight,ψ∈H(D)

andϕ∈S(D).Assume that p≥p,σ∈L+(D)is radial and such thatσ(z)/( –|z|)α

be-longs to Bp(α).Then Wψ,ϕ:Ap(σ)→Bν,is bounded if and only if Wψ,ϕ:Ap(σ)→Bνis

bounded,ψ∈Bν,and

lim

|z|→ν(z)ψ(z)ϕ

₍_z₎_{= .} _()

Proof First suppose thatWψ,ϕ:Ap(σ)→Bν,is bounded. Then it is obvious thatWψ,ϕ:

Ap₍_σ₎_→_Bν_{is also bounded. By taking}_f₍_z₎_≡__∈_Ap₍_σ_{), we have that}_ψ_∈_Bν

,. By taking

f(z) =z∈Ap₍_σ_{) and using the fact that}_ψ_∈_Bν

,, we have that () holds.

Conversely, assume thatWψ,ϕ:Ap(σ)→Bνis bounded,ψ∈Bν,and () holds. Then,

for each polynomialp, we have that

ν(z)(Wψ,ϕp)(z)≤ν(z)ψ(z)p

ϕ(z)+ν(z)ψ(z)ϕ(z)pϕ(z),

from which, along withψ∈Bν,and (), it follows thatWψ,ϕp∈Bν,. Since the set of

all polynomials is dense inAp₍_σ_{), we have that for every}_f_∈_Ap₍_σ_{), there is a sequence of}

polynomials (pn)n∈Nsuch thatf –pnAp₍_σ₎→ asn→ ∞. Hence, by the boundedness

ofWψ,ϕ:Ap(σ)→Bν, we have

Wψ,ϕf–Wψ,ϕpnBν≤ Wψ,ϕAp(σ)→Bνf–pnAp(σ)→

asn→ ∞. SinceBν,is a closed subspace ofBν, we have thatWψ,ϕ(Ap(σ))⊆Bν,and so

Wψ,ϕ:Ap(σ)→Bν,is bounded.

Theorem  Let r∈(, )be ﬁxed,p> ,p> ,α> –,νbe a standard weight,ψ∈H(D)

andϕ∈S(D).Assume that p≥p,σ∈L+(D)is radial and such thatσ(z)/( –|z|)αbelongs

to Bp(α)and Wψ,ϕ:A

p₍_σ₎_→_Bν

,is bounded.Then Wψ,ϕ:Ap(σ)→Bν,is compact if and

only if the following conditions are satisﬁed: (i) lim|z|→ν(z)|ψ(z)|(

Dϕ(z)(r)σdA)

–/p_{= }_;

(ii) lim_|_z_|→_ν(z)|ψ(z)ϕ(z)|

–|ϕ(z)| (

Dϕ(z)(r)σdA)

–/p_{= }_.

Proof First suppose thatWψ,ϕ:Ap(σ)→Bν,is compact. By takingf(z)≡∈Ap(σ), we

have thatψ∈Bν,. By takingf(z) =z∈Ap(σ) and using the fact thatψ ∈Bν,, we have

that () holds. Thus ifϕ∞< , then from () and the fact thatψ∈Bν,we have

lim |z|→ν(z)

ψ(z)

Dϕ(z)(r)

σdA

–/p

= lim |z|→

ν(z)|ψ(z)| ( –|ϕ(z)|₎(η+)p/p

( –|ϕ(z)|)(η+)p/p

(_D

ϕ(z)(r)σdA)

/p

≤C lim |z|→

ν(z)|ψ(z)| ( –ϕ

(12)

From () and () we have

lim |z|→

ν(z)|ψ(z)ϕ(z)|  –|ϕ(z)|

Dϕ(z)(r)

σdA

–/p

= lim |z|→

ν(z)|ψ(z)ϕ(z)| ( –|ϕ(z)|₎(η+)p/p+

( –|ϕ(z)|₎(η+)p/p

(_D

ϕ(z)(r)σdA)

/p

≤C lim |z|→

ν(z)|ψ(z)ϕ(z)| ( –ϕ

∞)(η+)p/p+ = .

Thus in this case conditions (i) and (ii) follow.

Now assumeϕ_∞= . Let (zn)n∈N⊂Dbe a sequence such thatlimn→∞|ϕ(zn)|= . By

the proof of Theorem , we have

lim |ϕ(z)|→ν(z)

ψ(z)

Dϕ(z)(r)

σdA

–/p

=  ()

and

lim |ϕ(z)|→

ν(z)|ψ(z)ϕ(z)|  –|ϕ(z)|

Dϕ(z)(r)

σdA

–/p

= . ()

Using () and the fact thatψ∈Bν,, by Lemma , we get (i). Using () and the fact that

lim_|z|→ν(z)|ψ(z)ϕ(z)|= , by Lemma , we get (ii).

Conversely, by taking the supremum in () over allf ∈Ap₍_σ_{) such that}_f

Ap₍_σ₎≤ and

then letting|z| →, we obtain that

lim |z|→_fsup

Ap(σ)≤

ν(z)(Wψ,ϕf)(z)= .

Thus, by Lemma , we obtain thatWψ,ϕ:Ap(σ)→Bν,is compact.

Note that ifσis also a normal weight, then it is easy to see that for a ﬁxedr∈(, ), the following relationship holds:

σ|ζ|σ|z|, ζ∈Dz(r). ()

Thus from () we have

Dϕ(z)(r)

σdAσϕ(z) –ϕ(z). ()

Using () and Theorems -, we obtain the following corollaries.

Corollary  Let p> ,νandσ be normal weights,ψ∈H(D)andϕ∈S(D).Then Wψ,ϕ:

Ap(σ)→Bνis bounded if and only if the following conditions are satisﬁed: (i) M=supz∈D

ν(z)|ψ(z)|

σ(|ϕ(z)|)/p_(–_|_ϕ₍_z₎_|₎/p<∞;

(ii) M=supz∈D

ν(z)|ψ(z)ϕ(z)|

(13)

Moreover,if Wψ,ϕ:Ap(σ)→Bνis bounded,then

ν/CM+M.

Corollary  Let p> , ν and σ be normal weights, ψ ∈H(D), ϕ∈ S(D) and Wψ,ϕ :

Ap(σ)→Bν be bounded.Then Wψ,ϕ:Ap(σ)→Bνis compact if and only if the following

conditions are satisﬁed: (i) lim_|ϕ(z)|→ ν(z)|ψ

₍_z₎_|

σ(|ϕ(z)|)/p_(–_|_ϕ₍_z₎_|₎/p= ;

(ii) lim|ϕ(z)|→ ν(z)|ψ(z)ϕ

₍_z₎_|

σ(|ϕ(z)|)/p_(–_|_ϕ₍_z₎_|₎+/p= .

Corollary  Let p> ,νandσbe normal weights,ψ∈H(D)andϕ∈S(D).Then Wψ,ϕ:

Ap(σ)→Bν,is bounded if and only if Wψ,ϕ:Ap(σ)→Bνis bounded,ψ∈Bν,and

lim |z|→ν(z)

ψ(z)ϕ(z)= .

Corollary  Let p> , ν and σ be normal weights, ψ ∈H(D), ϕ∈ S(D)and Wψ,ϕ :

Ap₍_σ₎_→_Bν

,be bounded.Then Wψ,ϕ:Ap(σ)→Bν,is compact if and only if the following

conditions are satisﬁed: (i) lim_|_z_|→_ ν(z)|ψ(z)|

σ(|ϕ(z)|)/p_(–_|_ϕ₍_z₎_|₎/p= ;

(ii) lim_|_z_|→_ ν(z)|ψ(z)ϕ(z)|

σ(|ϕ(z)|)/p_(–_|_ϕ₍_z₎_|₎+/p= .

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally to the writing of this paper. They read and approved this version of the manuscript.

Author details

1_{Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, Beograd, 11000, Serbia.}2_Operator

Theory and Applications Research Group, Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia. 3_{School of Mathematics, Shri Mata Vaishno Devi University, Kakryal, Katra, JK 182320, India.}

Received: 10 August 2015 Accepted: 7 October 2015 References

1. Colonna, F: New criteria for boundedness and compactness of weighted composition operators mapping into the Bloch space. Cent. Eur. J. Math.11(1), 55-73 (2013)

2. Colonna, F, Li, S: Weighted composition operators from the minimal Möbius invariant space into the Bloch space. Mediterr. J. Math.10(1), 395-409 (2013)

3. Li, S, Stevi´c, S: Weighted composition operators from Bergman-type spaces into Bloch spaces. Proc. Indian Acad. Sci. Math. Sci.117(3), 371-385 (2007)

4. Li, S, Stevi´c, S: Weighted composition operators fromH∞to the Bloch space on the polydisc. Abstr. Appl. Anal.2007, Article ID 48478 (2007)

5. Li, S, Stevi´c, S: Weighted composition operators betweenH∞andα-Bloch spaces in the unit ball. Taiwan. J. Math.12, 1625-1639 (2008)

6. Li, S, Stevi´c, S: Weighted composition operators from Zygmund spaces into Bloch spaces. Appl. Math. Comput.

206(2), 825-831 (2008)

7. Luo, L, Ueki, S: Weighted composition operators between weighted Bergman and Hardy spaces on the unit ball ofCn_{. J. Math. Anal. Appl.}₃₂₆_{, 88-100 (2007)}

8. MacCluer, B, Zhao, R: Essential norms of weighted composition operators between Bloch-type spaces. Rocky Mt. J. Math.33(4), 1437-1458 (2003)

9. Sharma, AK, Ueki, S: Composition operators between weighted Bergman spaces with admissible Békollé weights. Banach J. Math. Anal.8, 64-88 (2014)

10. Stevi´c, S: Essential norms of weighted composition operators from theα-Bloch space to a weighted-type space on the unit ball. Abstr. Appl. Anal.2008, Article ID 279691 (2008)

11. Stevi´c, S: Norm of weighted composition operators from Bloch space toH∞μon the unit ball. Ars Comb.88, 125-127 (2008)

(14)

13. Stevi´c, S: Weighted composition operators from weighted Bergman spaces to weighted-type spaces on the unit ball. Appl. Math. Comput.212, 499-504 (2009)

14. Stevi´c, S, Ueki, SI: Weighted composition operators from the weighted Bergman space to the weighted Hardy space on the unit ball. Appl. Math. Comput.215, 3526-3533 (2010)

15. Stevi´c, S, Zhou, Z, Chen, R: Weighted composition operators between Bloch type spaces in the polydisc. Sb. Math.

201(1-2), 289-319 (2010)

16. Ueki, SI: Order bounded weighted composition operators mapping into the Bergman space. Complex Anal. Oper. Theory6(3), 549-560 (2012)

17. Ueki, SI, Luo, L: Essential norms of weighted composition operators between weighted Bergman spaces of the ball. Acta Sci. Math.74, 829-843 (2008)

18. Li, S, Stevi´c, S: Composition followed by diﬀerentiation between Bloch type spaces. J. Comput. Anal. Appl.9(2), 195-205 (2007)

19. Li, S, Stevi´c, S: Generalized composition operators on Zygmund spaces and Bloch type spaces. J. Math. Anal. Appl.

338, 1282-1295 (2008)

20. Sharma, AK: Products of multiplication, composition and diﬀerentiation between weighted Bergman-Nevanlinna and Bloch-type spaces. Turk. J. Math.35, 275-291 (2011)

21. Stevi´c, S: On a new integral-type operator from the Bloch space to Bloch-type spaces on the unit ball. J. Math. Anal. Appl.354, 426-434 (2009)

22. Stevi´c, S: On an integral-type operator from logarithmic Bloch-type and mixed-norm spaces to Bloch-type spaces. Nonlinear Anal. TMA71, 6323-6342 (2009)

23. Stevi´c, S: Products of integral-type operators and composition operators from the mixed norm space to Bloch-type spaces. Sib. Math. J.50(4), 726-736 (2009)

24. Stevi´c, S: On an integral operator between Bloch-type spaces on the unit ball. Bull. Sci. Math.134, 329-339 (2010) 25. Yang, W, Zhu, X: Generalized weighted composition operators from area Nevanlinna spaces to Bloch-type spaces.

Taiwan. J. Math.16(3), 869-883 (2012)

26. Zhu, X: Generalized weighted composition operators from Bloch-type spaces to weighted Bergman spaces. Indian J. Math.49(2), 139-149 (2007)

27. Madigan, K, Matheson, A: Compact composition operators on the Bloch space. Trans. Am. Math. Soc.347(7), 2679-2687 (1995)

28. Békollé, D: Inégalite á poids pour le projecteur de Bergman dans la boule unité de_Cn_{. Stud. Math.}₇₁_{, 305-323} (1981/82)

29. Constantin, O: Carleson embeddings and some classes of operators on weighted Bergman spaces. J. Math. Anal. Appl.365, 668-682 (2010)

30. Constantin, O: Discretizations of integral operators and atomic decomposition in vector-valued weighted Bergman spaces. Integral Equ. Oper. Theory59, 523-554 (2007)