Compact differences of weighted composition operators on the weighted Bergman spaces

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R E S E A R C H

Open Access

Compact differences of weighted

composition operators on the weighted

Bergman spaces

Maocai Wang

1*

_{, Xingxing Yao}

2

_{and Fen Chen}

2

*_{Correspondence:}

[email protected]

1_{School of Computer, China}

University of Geosciences, Wuhan, 430074, China

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

Abstract

In this paper, we consider the compact differences of weighted composition operators on the standard weighted Bergman spaces. Some necessary and sufficient conditions for the differences of weighted composition operators to be compact are given, which extends Moorhouse’s results in (J. Funct. Anal. 219:70-92, 2005).

MSC: Primary 47B33; secondary 30D55; 46E15

Keywords: weighted Bergman space; weighted composition operator; compact diﬀerence

1 Introduction

Let D be the open unit disk in the complex plane C and T the boundary of D. Denote by

H(D) the space of all holomorphic functions on D and by S(D) the set of all holomorphic self-maps of D. Then, foru∈H(D) andϕ∈S(D), the weighted composition operatoruCϕ

induced byuandϕis given by

uCϕ(f) =u·f ◦ϕ, f ∈H(D).

Whenu≡,uCϕis the composition operatorCϕ, in other words,Cϕ(f) =f◦ϕ,f ∈H(D);

whenϕ(z) =z,uCϕis the multiplication operatorMu,i.e.,Mu(f) =u·f,f ∈H(D). Broadly,

one is interested in extracting properties ofuCϕacting on a given Banach space of

holo-morphic functions on D (boundedness, compactness, spectral properties,etc.) from func-tion theoretic properties ofuandϕand vice versa. In the past several decades, weighted composition operators on various spaces of holomorphic functions have been studied ex-tensively,e.g., [–].

As is well known, an early result of Shapiro and Taylor [] in  showed the non-existence of the angular derivative of the inducing map at any point of the boundary of the unit disk is a necessary condition for the compactness of the composition operator on the Hardy spaceH(D). Later, MacCluer and Shapiro [] proved that this condition is a necessary and suﬃcient condition for the compactness of composition operators on the weighted Bergman spacesApα(D) (α> –). Using the Nevanlinna counting function,

Shapiro [] completely characterized thoseϕwhich induce compact composition oper-ators on the Hardy spaceH_{(D). With the basic questions such as compactness settled,}

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it is natural to look at the topological structure of composition operators in the operator norm topology and this topic is of continuing interests in the theory of composition oper-ators. Berkson [] focused attention on the topological structure with his isolation result onHp(D) in , which was reﬁned later by Shapiro and Sundberg [], and MacCluer []. In [], Shapiro and Sundberg posed a question: Do the composition operators on

H_{(D) that diﬀer from}_C

ϕby a compact operator form the component ofCϕin the

oper-ator norm topology? While the same question was answered positively on the weighted Bergman spaces [], this turned out to be not true on the Hardy space []. Some other results on differences of weighted composition operators on spaces of holomorphic func-tions can be found, for example, in [–]. In relation with the study of the topological structures, the difference or the linear sum of composition operators on various settings has been a very active topic [, , –]. Recently, Moorhouse [] characterized com-pletely the compactness for the difference of two composition operators on the Bergman space over the unit disk, and Al-Rawashdeh and Narayan [] gave a sufficient condition for the same problem on the Hardy space. Here we continue this line to study compact differences of weighted composition operators acting on the standard weighted Bergman spaces.

The standard weighted Bergman spaceA

α(α> –) is deﬁned as follows:

Aα:=

f ∈H(D) :f_A_α=

D

f(z)dλα(z) <∞

,

wheredλα(z) =_π(α+ )( –|z|)αdA(z) anddAis the area measure on D. As is well known,

the Bergman spaceA

α is a reproducing kernel Hilbert space, the reproducing kernel at

z∈DisKz(w) =_(–_zw₎α+ and_K_z Aα

Kz→ weakly as|z| →.

In Section  we recall some related facts and results which are needed in the sequel, and then we prove our main results in Section . Section  deals with the compact perturba-tions of ﬁnite summaperturba-tions of a given weight composition operator.

Constants.Throughout the paper we use the lettersCandcto denote various positive constants which may change at each occurrence. Variables indicating the dependency of constantsCandcwill be often speciﬁed in the parentheses. We use the notationXYor

YXfor non-negative quantitiesXandYto meanX≤CYfor some inessential constant

C> . Similarly, we use the notationX≈Yif bothXYandYXhold.

2 Preliminaries

For  <t<∞andξ∈T, lett,ξ be a non-tangential approach region atξdeﬁned by

t,ξ:=

z∈D:|z–ξ| ≤t –|z|

andt,ξ the boundary curve oft,ξ. Clearlyt,ξ has a corner atξ with angle less thanπ.

A functionf is said to have a non-tangential limit atξ, iflim_z_→ξf(z) exists in each

non-tangential regiont,ξ.

Letϕ be a holomorphic self-map of D. We say thatϕhas a finite angular derivative at ξ ∈T, if there exists a pointη∈T, such that the non-tangential limit asz→ξ of the difference quotientη–_ξϕ_–(_zz) exists as a finite complex value. Write

ϕ(ξ) :=∠lim

z→ξ

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DenoteF(ϕ) :={ξ∈T:|ϕ(ξ)|<∞}. Forξ ∈F(ϕ), by the Julia-Carathéodory theorem in [], we have

ϕ(ξ)= lim

z→ξ

z∈t,ξ

 –|ϕ(z)|

 –|z| (.)

for anyt> .

For any z∈D, letσz be the involutive automorphism of D which exchanges  toz,

namely,

σz(w) = z–w

 –zw, w∈D.

The pseudo-hyperbolic distance on D is deﬁned by

ρ(z,w) =σz(w)=

_{ –}z–_zww, z,w∈D.

Then, for anyz,w∈D, it is easy to see that

 –z–w  –zw

=( –|z|

_{)( –}_|_w_|₎

| –zw|

and

 –ρ(z,w)≤  –|z|



| –zw|≤ +ρ(z,w). (.)

Moreover, for anyz∈Dand  <r< , let

Er(z) :=

w∈D:ρ(z,w) <r

be the pseudo-hyperbolic disk with ‘center’zand ‘radius’r. It is well known that, for given  <r< ,

 –ρ(z,w)  +ρ(z,w)≤

 –|z|

 –|w| ≤

 +ρ(z,w)

 –ρ(z,w), w∈Er(z), (.)

and

λα

Er(z)

≈ –|z|+α, z∈D, (.)

where the constants in the estimate above depend only onrandα. In the sequel, we set ρ(z) :=ρ(ϕ(z),ϕ(z)) for the pseudo-hyperbolic distance ofϕ(z) andϕ(z).

The following lemma is cited from [].

Lemma . For α > –, let ϕ be a holomorphic self-map of D and u a non-negative,

bounded,and measurable function onD.Deﬁne the measure uλαby uλα(E) :=

Eu(z)dλα(z)

on all Borel subsets E⊆D.If

lim

|z|→u(z)

 –|z|

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then uλα◦ϕ–is a compactα-Carleson measure and the inclusion map Iα:Aα→L(uλα◦

ϕ–)is compact.

For more details as regards Carleson measures, see Section . in [].

3 Compact difference

Letϕ∈S(D) andu∈H(D). If the weighted composition operatoruCϕis bounded onAα

(α> –), then the adjoint (uCϕ)∗ofuCϕsatisﬁes

(uCϕ)∗Kz(w) =u(z)Kϕ(z)(w), z,w∈D.

Forϕ∈S(D), by the Schwarz-Pick theorem in [],

 –|z|

 –|ϕ(z)|<

 +|ϕ()|

 –|ϕ()|<∞ (.)

for anyz∈D.

The following lemma can be obtained by modifying Lemma . in [] (e.g., at the third line on p. in []). See also Proposition . in [] in a diﬀerent form for the unit ball case. Here, we give a more elementary proof for convenience.

Lemma . Letϕ andϕ be holomorphic self-maps ofD. Then,for any ξ ∈F(ϕ),the following holds:

lim

t→∞zlim→ξ

z∈t,ξ

 –|ϕ(z)|

 –ϕ(z)ϕ(z)

=

, ifϕ(ξ) =ϕ(ξ)andϕ(ξ) =ϕ(ξ),

, otherwise.

Proof First we notice that

 –|ϕ(z)|

 –ϕ(z)ϕ(z)

= –|ϕ(z)|



 –|z| ·

 –|z|

 –ϕ(z)ϕ(z)

= –|ϕ(z)|



 –|z| ·



–|ϕ(z)|

–|z| +

ϕ(z)(ϕ(z)–ϕ(z))

–|z|

= –|ϕ(z)|



 –|z| ·



–|ϕ(z)|

–|z| +(z) ,

where(z) =ϕ(z)(ϕ(z)–ϕ(z))

–|z| , and

lim inf

z→ξ

(z)≥lim inf

z→ξ

 

 –|ϕ(z)|

 –|z| . (.)

Ifϕhas no ﬁnite angular derivative atξ, namely,

lim inf

z→ξ

 –|ϕ(z)|

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then, for anyt> , by (.), we have

lim

z→ξ

z∈t,ξ

₍_z₎₌_∞_.

Ifϕhas ﬁnite angular derivative atξandϕ(ξ)=ϕ(ξ), then it follows clearly that

lim

z→ξ

z∈t,ξ

(z)=∞.

Ifϕhas ﬁnite angular derivative atξ andϕ(ξ) =ϕ(ξ), then

lim

z→ξ

z∈t,ξ

(z)= lim

z→ξ

z∈t,ξ

ξ–z

 –|z|ϕ(z)

ϕ(ξ) –ϕ(z)

ξ–z –

ϕ(ξ) –ϕ(z)

ξ–z

= t ϕ

(ξ) –ϕ(ξ).

Thus ifϕ(ξ) =ϕ(ξ) andϕ(ξ) =ϕ(ξ), we have

lim

t→∞zlim→ξ

z∈t,ξ

(z) = .

Otherwise ifϕ(ξ) =ϕ(ξ) andϕ(ξ)=ϕ(ξ), then

lim

t→∞zlim→ξ

z∈t,ξ

(z)=∞.

Consequently, we get the desired result.

To further study compact diﬀerences of weighted composition operators onA

α, we

de-ﬁneFu(ϕ) as

Fu(ϕ) :=

ξ∈T:lim sup

z→ξ

u(z)  –|z|



 –|ϕ(z)| = 

.

It is easy to check thatFu(ϕ)⊆F(ϕ) ifuis bounded. To avoid the trivial case, in the sequel

we assumeFui(ϕi)=∅,i= , ,i.e., neitheruCϕnoruCϕis compact onA



α.

In the following we take the test functions

gw(z) :=

( –|w|)

( –wz)α+

, w,z∈D.

First note that{gw}is bounded inAα. Indeed, note that

Ic(w) =

D

( –|z|₎α

| –wz|α++cdA(z)≈



( –|w|₎c, |w| →

forc>  by Lemma .. in [], and then

gw_A α=

α+  π

D

( –|w|_{)( –}_|_z_|₎α

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Again it is well known thatgw(z)→ uniformly on any compact subset of D, and hence

thatgw→ weakly as|w| →.

We now give some necessary conditions for the diﬀerence of weighted composition op-erators to be compact.

Theorem . Letϕ,ϕbe holomorphic self-maps ofDand let u,ube bounded holomor-phic functions onDsuch that neither uCϕnor uCϕ is compact on Aα.If uCϕ–uCϕ

is compact on Aα,then the following statements are true:

() Fu(ϕ) =Fu(ϕ).

() ∠limz→ξ|u(z) –u(z)|= for anyξ∈Fu(ϕ). () lim_|_z_|→_ρ(z)(|u(z)| –|z|



–|ϕ(z)| +|u(z)|

 –|z|

–|ϕ(z)|) = .

Proof DenoteT :=uCϕ–uCϕ for short, and assume thatT is compact on Aα. For

ξ∈Fu(ϕ), it is easy to see that

lim

w→ξ

w∈t,ξ

Tgϕ(w)



Aα=  (.)

for anyt> . Using the submean value type inequality in [] and equation (.), then, for a given  <r< ,

Tgϕ(w)



A α≥

Er(w)

Tgϕ(w)(z)



dλα(z)

 –|w|α+Tgϕ(w)(w)



.

So by (.)

lim

w→ξ

w∈t,ξ

 –|w|

 –|ϕ(w)| α+

u(w) –u(w)

 –|ϕ(w)|

 –ϕ(w)ϕ(w) α+



= 

for allt> . Sinceu,uare bounded holomorphic functions on D andξ∈F(ϕ), then it

follows from our assumptionξ∈Fu(ϕ) that

lim

t→∞wlim→ξ

w∈t,ξ

 –|ϕ(w)|

 –ϕ(w)ϕ(w)

= ,

and therefore

lim

t→∞wlim→ξ

w∈t,ξ

_u_(w) –u(w)= .

So () is obtained, and thusFu(ϕ)⊆Fu(ϕ) by Lemma .. Similarly, we haveFu(ϕ)⊆

Fu(ϕ). Thus the proof for () is complete.

To prove (), we assume that there exists a sequence{zn}with|zn| → such that

lim

n→∞ρ(zn)

u(zn)

  –|zn|

 –|ϕ(zn)|

+u(zn)

  –|zn|

 –|ϕ(zn)|

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Without loss of generality, we may further assume that

_u_₍_z_n₎  –|zn|

 –|ϕ(zn)|≥

_u_₍_z_n₎  –|zn|

 –|ϕ(zn)|

for alln. Then

lim sup

n→∞ ρ(zn)

u(zn)

  –|zn|

 –|ϕ(zn)|

> . (.)

Due to (.) and the boundedness ofuon D, by passing to a subsequence if necessary, we

can suppose that

lim

n→∞ρ(zn) =a, nlim→∞u(zn)



=a and lim

n→∞

 –|zn|

 –|ϕ(zn)|

=a

for some constantsa∈(, ],a> ,a> . Thenlimn→∞|u(zn)|=aby the obtained

facts () and (). Actually, we may further assume that

lim

n→∞

 –|zn|

 –|ϕ(zn)|

=a

for somea> . We putfn:=Kzn/KznAα, whereKznis the reproducing kernel function at

zn∈DinAαfor eachn≥. Sofn→ weakly asn→ ∞. We will arrive at a contradiction

to the compactness ofuCϕ–uCϕ by showing (uCϕ–uCϕ)∗fn (n→ ∞) inAα.

In fact, notice that

(uCϕ–uCϕ)

∗_f

n



Aα

= 

Kzn_A α

D

u(zn)Kϕ(zn)–u(zn)Kϕ(zn) 

dλα

u(zn)

  –|zn|

 –|ϕ(zn)|

α+

+u(zn)

  –|zn|

 –|ϕ(zn)|

α+

– u(zn)u(zn) –ρ(zn)

α+

 ( –|zn|

₎α+

( –|ϕ(zn)|)

α+

 _{( –}|_ϕ_₍_z_n₎|₎α+

≥ – –ρ(zn)

α+  _u

(zn)u(zn)

( –|zn|)α+

( –|ϕ(zn)|)

α+

 ( –|_ϕ_(z_n)|)α+

for alln. Then

lim inf

n→∞(uCϕ–uCϕ)

∗_f

n



Aα≥

 – –a_

α+  _a

(aa)

α+  _{> .}

The contradiction implies (), which completes the proof.

To give a suﬃcient condition for the compact diﬀerence of weighted composition op-erators, we need the following fact from [], pp.-: for anyε>  small enough and

f ∈A

α,

{z∈D:ρ(z)≤ε}

f(ϕ) –f(ϕ)dλαεf_A

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To simplify our suﬃcient condition, we use the following simple lemma.

Lemma . Letϕ,ϕbe holomorphic self-maps ofDand let u,ube bounded holomor-phic functions onD.If

lim

z→ξu(z) –u(z)=  for anyξ∈Fu(ϕ)∪Fu(ϕ)

and

lim

|z|→ρ(z)

u(z)

  –|z|

 –|ϕ(z)|

+u(z)

  –|z|

 –|ϕ(z)|

= ,

then Fu(ϕ) =Fu(ϕ).

Proof IfFu(ϕ)= Fu(ϕ), we may assume thatξ∈Fu(ϕ) butξ∈/Fu(ϕ). Thenξ∈F(ϕ), and ξ ∈/ F(ϕ) by the assumption limz→ξ|u(z) –u(z)|=  and ξ ∈Fu(ϕ). Hence by Lemma .,

lim

t→∞zlim→ξ

z∈t,ξ

 –|ϕ(z)|

 –ϕ(z)ϕ(z)

= .

Note that

 –ρ(z) =( –|ϕ(z)|

_{)( –}_|_ϕ (z)|) | –ϕ(z)ϕ(z)|

,

and (.) implies

 –|ϕ(z)| | –ϕ(z)ϕ(z)|

≤.

So

lim

t→∞zlim→ξ

z∈t,ξ

ρ(z) = ,

and then

lim

t→∞zlim→ξ

z∈t,ξ

ρ(z)u(z)

 –|z|

 –|ϕ(z)|

+u(z)

 –|z|

 –|ϕ(z)|

= .

This leads to a contradiction to the assumption. ThusFu(ϕ) =Fu(ϕ).

We are now ready to give our suﬃciency theorem.

Theorem . Letϕ,ϕbe holomorphic self-maps ofDand let u,u be bounded holo-morphic functions onD.If the following hold:

() limz→ξ|u(z) –u(z)|= for anyξ∈Fu(ϕ)∪Fu(ϕ)and () lim_|z|→ρ(z)(|u(z)| –|z|



–|ϕ(z)| +|u(z)|

 –|z|

–|ϕ(z)|) = ,

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Proof Assume that{fn}is any bounded sequence inAα such thatfn→ (n→ ∞)

uni-formly on each compact subsets of D. Givenε> , we put

Q:=z∈D:ρ(z)≤ε , Q:= D\Q.

Now we can write

(uCϕ–uCϕ)fn



A α=

D

|uCϕfn–uCϕfn|

_d_λ

α= Q + Q (.)

for eachn.

LetχQbe the characteristic function ofQ, then by the assumption (),

lim

|z|→χQ

u(z)

  –|z|

 –|ϕ(z)|

+u(z)

  –|z|

 –|ϕ(z)|

= .

So by Lemma ., for the second term of the right-hand side of (.),

Q

ufn(ϕ) –ufn(ϕ) 

dλα

D

χQuCϕ(fn)

 dλα+

D

χQuCϕ(fn)



dλα→

asn→ ∞. For anyξ∈Fu(ϕ), by the assumption (), there existsδ(ξ) >  such that|u(z) –

u(z)|<εwhenever|z–ξ|<δ(ξ). We decomposeQinto two parts,Q:=H+H, where H:=Q∩(

ξ∈Fu_(ϕ){z∈D:|z–ξ|<δ(ξ)}) andH:=Q\H. Also, for the ﬁrst term of the right-hand side of (.), we have

Q

ufn(ϕ) –ufn(ϕ) 

dλα

Q

|u|fn(ϕ) –fn(ϕ) 

dλα+

Q

|u–u|fn(ϕ) 

dλα

Q

fn(ϕ) –fn(ϕ)dλα+

Q

|u–u|fn(ϕ)dλα

Q

fn(ϕ) –fn(ϕ) 

dλα+

i=

i=

Hi

|u–u|fn(ϕ) 

dλα

Q

fn(ϕ) –fn(ϕ)dλα+

H

|u–u|fn(ϕ)dλα

+ i= i= H

|ui|fn(ϕ) 

dλα. (.)

Note that by (.)

Q

fn(ϕ) –fn(ϕ) 

dλαε

for alln. Also, by the deﬁnition ofH, we can easily get

H

|u–u|fn(ϕ) 

dλα≤εCϕfn_A αε

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for allnand

lim

|z|→χHu(z)

  –|z|

 –|ϕ(z)|

= . (.)

We now claim that

lim

|z|→χH u(z)

  –|z|

 –|ϕ(z)|

=  (.)

and

lim

|z|→χH

_u_₍_z₎  –|z|

 –|ϕ(z)|

= . (.)

Indeed, if either (.) or (.) fails, then we will arrive at a contradiction to (.), and thus the desired is obtained. To this end, we assume that there exist someη∈Tand a sequence

zn∈Hsatisfyingzn→ηsuch that

lim

n→∞u(zn)

  –|zn|

 –|ϕ(zn)|

> , (.)

or

lim

n→∞u(zn)

  –|zn|

 –|ϕ(zn)|

> . (.)

If (.) holds, thenη∈Fu(ϕ). Thusη∈Fu(ϕ) due to the fact thatFu(ϕ) =Fu(ϕ) by Lemma .. If (.) holds, then

lim

n→∞u(zn)

  –|zn|

 –|ϕ(zn)|

> ,

because

 –|zn|

 –|ϕ(zn)|≥

 –ε  +ε

 –|zn|

 –|ϕ(zn)|

byzn∈Hand (.). Thus we also haveη∈Fu(ϕ). This leads to a contradiction to (.). So our claim holds. Thus by Lemma ., we have

i=

i=

H

|ui|fn(ϕ)dλα=

i=

i=

D

|χHui|

_f

n(ϕ)dλα→

asn→ ∞. Therefore the proof is complete.

The following, given in [] and [], respectively, are immediate consequences of Theo-rems . and ..

Corollary . Letϕ,ϕ be holomorphic self-maps ofDand a,b non-zero constants.If F(ϕ)=∅and F(ϕ)=∅,then aCϕ+bCϕ is compact on A



αif and only if a+b= and

lim_|z|→ρ(z)( –|z|



–|ϕ(z)| +

–|z|

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Corollary . Letϕ,ϕbe holomorphic self-maps ofD,then Cϕ–Cϕis compact on Aα

if and only iflim_|z|→ρ(z)( –|z|



–|ϕ(z)| +

–|z|

–|ϕ(z)|) = .

Corollary . Let u,ube bounded holomorphic functions onD,then Mu–Muis

com-pact on A

αif and only if u=u. 4 Compact perturbation

In the ﬁnal section, we consider the compact perturbation of ﬁnite summations of a given weighted composition operator.

Theorem . For i= , , . . . ,N,letϕ,ϕi∈S(D)and u,uibounded holomorphic functions onD.Suppose that Fui(ϕi)=∅for each i,Fui(ϕi)∩Fuj(ϕj) =∅(i=j),Fu(ϕ) =

N

i=Fui(ϕi).

Deﬁneρi(z) :=|_–ϕ(_ϕz)–ϕi(z)

i(z)ϕ(z)|.If () limz→ξρi(z)(|u(z)| –|z|



–|ϕ(z)| +|ui(z)| –|z|



–|ϕi(z)|) = ,and () limz→ξ|u(z) –ui(z)|= for anyξ∈Fui(ϕi)

for every i= , , . . . ,N,then uCϕ–

N

i=uiCϕiis compact on A



α.

Proof Deﬁne Di := {z ∈D : |ui(z)| –|z|



–|ϕi(z)| ≥ |uj(z)|

 –|z|

–|ϕj(z)|, for allj=i} for each i= , , . . . ,N. Fixε>  and denoteEi:={z∈Di,ρi(z)≤ε}andEi:= Di\Ei. To end the proof,

we assume that{fn}is any bounded sequence inAα such thatfn→ (n→ ∞) uniformly

on each compact subset of D. For ≤i≤N,

Di

ufn(ϕ) –

N

k=

ukfn(ϕk)



dλα

Di

ufn(ϕ) –uifn(ϕi)

 dλα+

k=i

Di

ukfn(ϕk)

 dλα

Ei

_uf_n(ϕ) –uifn(ϕi)

 dλα+

E_i

_uf_n(ϕ)dλα

+

E_i

uifn(ϕi)dλα+

k=i

Di

ukfn(ϕk)dλα. (.)

LetχD_iandχ_E

ibe the characteristic functions of DiandE

i, respectively, then it is obvious

from the assumption () that

lim

|z|→χEi

ui(z)

  –|z|

 –|ϕi(z)|

+u(z)  –|z|



 –|ϕ(z)|

= . (.)

Moreover,

lim

|z|→χDi uk(z)

 –|z|

 –|ϕk(z)|

=  (.)

for ﬁxediand eachk=i. Indeed, if (.) fails for somek=i, then there existξ ∈Tand

zn∈Disatisfyingzn→ξsuch that

lim

n→∞uk(zn)

  –|zn|

 –|ϕk(zn)|

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Thenξ∈Fuk(ϕk) and

lim

n→∞ui(zn)

  –|zn|

 –|ϕi(zn)|

> 

by the deﬁnition of Di, which impliesξ∈Fui(ϕi). SoFui(ϕi)∩Fuk(ϕk)=∅wheni=k, which contradicts our assumption. Then the last three terms of (.) tend to  asn→ ∞by Lemma .. In the following, we consider the ﬁrst term of (.) by a similar argument to the proof of Theorem .. For anyξ∈Fui(ϕi), there existsδ(ξ) >  such that

u(z) –ui(z)<ε,

whenever|z–ξ|<δ(ξ). We decomposeEiinto two parts asEi=Hi+Hi, where

Hi:=Ei∩

ξ∈Fu(ϕ)

z∈D:|z–ξ|<δ(ξ)

andHi:=Ei\Hi. Note that

Ei

ufn(ϕ) –uifn(ϕi)

 dλα

Ei

fn(ϕ) –fn(ϕi)dλα+

Ei

|u–ui|fn(ϕi)dλα

Ei

fn(ϕ) –fn(ϕi)

 dλα+

j=

j=

Hij

|u–ui|fn(ϕi)

 dλα

Ei

fn(ϕ) –fn(ϕi)

 dλα+

Hi

|u–ui|fn(ϕi)

 dλα

+

Hi

|u|fn(ϕi)

 dλα+

Hi

|ui|fn(ϕi)



dλα. (.)

Clearly, by (.)

Ei

fn(ϕ) –fn(ϕi)

 dλα≤

{z∈D:ρi(z)≤ε}

fn(ϕ) –fn(ϕi)



dλαε

for alln. Also, by the deﬁnition ofHi, we can easily get

Hi

|u–ui|fn(ϕi)dλα≤εCϕifn



A αε



for alln. Moreover,

lim

|z|→χHi

u(z)  –|z|



 –|ϕ(z)|= . (.)

Indeed, if (.) fails, there exist someζ∈Tand a sequence{zn} ⊆Hisatisfyingzn→ζ

such that

lim

|z|→

u(z)  –|z|



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thenζ∈Fu(ϕ) =

N

k=Fuk(ϕk). Since thisζ ∈/Fuk(ϕk) whenk=iby (.), thenζ ∈Fui(ϕi), which contradicts the deﬁnition ofHi. So (.) holds. We now claim that

lim

|z|→χHi

_u_i(z)  –|z|



 –|ϕi(z)|

=  (.)

and

lim

|z|→χHi

u(z)  –|z|



 –|ϕi(z)|

= . (.)

Indeed, the argument for (.) is similar to (.) and we omit it. To prove that (.) holds, we assume that there exist someη∈Tand a sequence{zn} ⊆Hisuch thatzn→ηand

lim

n→∞u(zn)

  –|zn|

 –|ϕi(zn)|

> .

Note that

 –|zn|

 –|ϕ(zn)| ≥

 –ε  +ε

 –|zn|

 –|ϕi(zn)|

,

because of{zn} ⊆Hi and (.). Thenη∈Fu(ϕ), which contradicts (.). Thus again by

Lemma ., we have

Hi

|ui|fn(ϕi)

 dλα+

Hi

|u|fn(ϕi)



dλα→

asn→ ∞. So

Di

ufn(ϕ) –

N

k=

ukfn(ϕk)



dλα→,

and then

uCϕ–

N

i= uiCϕi

fn



A α

N

i=

Di

ufn(ϕ) –

N

k=

ukfn(ϕk)



dλα→

asn→ ∞, which completes the proof.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and signiﬁcantly in writing this article. All authors read and approved the ﬁnal manuscript.

Author details

1_{School of Computer, China University of Geosciences, Wuhan, 430074, China.}2_{School of Mathematics and Statistics,}

Wuhan University, Wuhan, 430072, China.

Acknowledgements

The authors would like to thank the anonymous referees for their comments and suggestions, which improved our ﬁnal manuscript. The authors also thank Prof. K Zhu and Z Wu for their very helpful comments. In addition, the ﬁrst author is supported by Natural Science Foundation of China (41571403), and the second and third authors are supported by Natural Science Foundation of China (11271293).

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