Weighted Composition Operators from Logarithmic Bloch Type Spaces to Bloch Type Spaces

Volume 2009, Article ID 964814,21pages doi:10.1155/2009/964814

Research Article

Weighted Composition Operators from Logarithmic

Bloch-Type Spaces to Bloch-Type Spaces

Stevo Stevi ´c

_{and Ravi P. Agarwal}

1_{Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11000 Beograd, Serbia} 2_{Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901-6988, USA}

Correspondence should be addressed to Stevo Stevi´c,[email protected]

Received 13 April 2009; Accepted 3 July 2009

Recommended by Radu Precup

The boundedness and compactness of the weighted composition operators from logarithmic Bloch-type spaces to Bloch-type spaces are studied here.

Copyrightq2009 S. Stevi´c and R. P. Agarwal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

LetDbe the unit disc in the complex planeC,dmzthe normalized Lebesgue area measure

onD,HDthe class of all holomorphic functions on D, andH∞Dthe space of bounded

holomorphic functions onDwith the normf_∞sup_z∈D|fz|.

The logarithmic Bloch-type space Bα

logβ Bα_logβD, α > 0, β ≥ 0, was recently

introduced in1. The space consists of allf ∈HDsuch that

bα,β

f:sup

z∈D1− |z|

α

ln e

β/α

1− |z|

β

fz<∞. 1.1

The norm onBα

logβ is introduced as follows: _f

Bα

logβ

f0bα,β

f. 1.2

When β 0,Bα

logβ becomes theα-Bloch spaceBα. For α-Bloch and other Bloch-type

spaces, see, for example,1–9, as well as the related references therein. Forα β 1,Bα

logβ

is the logarithmic Bloch space, which appeared in characterizing the multipliers of the Bloch

The little logarithmic Bloch-type spaceBα

logβ,0 B α

logβ,0D,α >0, β ≥0,consists of all

f∈ Bα

logβsuch that

lim

|z| →1−01− |z| α

ln e

β/α

1− |z|

β

fz0. 1.3

The following theorem summarizes the basic properties of the logarithmic Bloch-type spaces. Here, as usual, for fixedr∈0,1, frz frz, z∈D.

Theorem Asee1. The following statements are true. aThe logarithmic Bloch-type spaceBα

logβ is Banach with the norm given in1.2.

bBα

logβ_,0is a closed subset ofB α logβ.

cAssumef ∈ Bα

logβ.Thenf∈ B α

logβ,0if and only iflimr→1−f−frBαlogβ 0. dThe set of all polynomials is dense inBα

logβ,0.

eAssumef ∈ Bα

logβ,then for eachr ∈0,1,fr ∈ Bα_logβ_,0. Moreover fr_Bα

logβ

≤f_Bα

logβ

. _1.4

A positive continuous functionμonDis calledweight.

The Bloch-type spaceBμBμDconsists of allf∈HDsuch that

Bμ

fsup

z∈Dμz

fz<∞, 1.5

whereμis a weight. With the norm

f_B

μ f0Bμ

f, _1.6

the Bloch-type space becomes a Banach space.

The little Bloch-type spaceBμ,0 Bμ,0Dis a subspace ofBμconsisting of allf such

that

lim

|z| →1μz

_fz0.

1.7

Letϕbe a holomorphic self-map ofDandu∈HD. Forf ∈HDthe corresponding

weighted composition operator is defined by

uCϕ

fz uzfϕz, z∈D. 1.8

It is of interest to provide function-theoretic characterizations for when ϕ and u induce

For some classical results mostly on composition operators, see, for example,10. For some

recent related results, mostly inCn_{or related to Bloch-type or weighted-type spaces, see, for}

example,4,10–46and the references therein.

Here we study the boundedness and compactness of the weighted composition operator from the logarithmic Bloch-type space and the little logarithmic Bloch-type space to the Bloch-type or the little Bloch-type space.

In this paper, constants are denoted byC, they are positive and may differ from one

occurrence to the other. The notationabmeans that there is a positive constantCsuch that

a≤Cb. We say thata b, if bothabandbahold.

2. Auxiliary Results

In this section we quote several auxiliary results which will be used in the proofs of the main results.

Lemma 2.1. Assumeα >0,β≥0,then the following statements are true.

aAssumeγ≥β/αln 2,then the function

hx xα

lne

γ

x

β

2.1

is increasing on the interval0,2.

bThe function

h1x xα

lne

β/α

x

β

2.2

is increasing on the interval0,1.

Proof. aWe have

hx xα−1

lne

γ

x

β−1

αlne

γ

x −β . 2.3

Sincexα−1lneγ_/xβ−1 _{> 0,whenx ∈ 0,2andγ ≥ β/αln 2, and the functionHx}

αlneγ_{/x−βis decreasing on the interval0,2, we have}

αlne

γ

x −β > αln eγ

2 −βα

γ−ln 2−β

α ≥0, x∈0,2, 2.4

from which this statement follows.

The next lemma regarding the point evaluation functional onBα

logβ follows from 1,

Lemma 3and some elementary asymptotic relationship, such as

1− |z|α−1

ln e

β/α

1− |z|

β

1− |z|2α−1

ln e

β/α

1− |z|2

β

, α >1, β≥0. 2.5

Lemma 2.2. Letf ∈ Bα

logβD.Then

_fz≤C ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ _f Bα logβ

, α∈0,1orα1, β >1,

f0f_Bα

logβ

ln ln e

β/α

1− |z|2, αβ1,

f0f_Bα

logβ

ln e

β/α

1− |z|2

1−β

, α1, β∈0,1,

f0

f_Bα

logβ

1− |z|2α−1lneβ/α_{/1− |z|}2β

, α >1, β≥0,

2.6

for someC >0independent off.

The proof of the following lemma is similar to25, Lemma 2.1, so we omit it.

Lemma 2.3. Assumeμis a weight. A closed setKinBμ,0is compact if and only if it is bounded and

lim

|z| →1sup_f∈Kμz

fz0. _2.7

Remark 2.4. If inLemma 2.3we assume thatK is not closed, then the wordcompactcan be

replaced byrelatively compact.

The next characterization of compactness is proved in a standard waysee, e.g., the

proofs of the corresponding lemmas in10,30,47–49. Hence we omit it.

Lemma 2.5. Assume that u ∈ HD, ϕ is a holomorphic self-map of D, and μ is a weight. Let

X be one of the following spacesBα

logβ,Bα_logβ_,0,and Y one of the spacesBμ,Bμ,0. Then the operator

uCϕ :X → Y is compact if and only ifuCϕ :X → Y is bounded and for every bounded sequence

fkk∈N⊂Xconverging to0uniformly on compacts ofDone has

lim

Some concrete examples of the functions belonging to logarithmic Bloch-type spaces can be found in the next lemma.

Lemma 2.6. The following statements are true.

aAssume thatα /1andβ≥0,then

fwz 1

1−zwα−1lneγ_/1−zwβ, w∈D, 2.9

whereγ ≥β/αln 2andfw0 1/γβis a nonconstant function belonging toBα_logβ.

bAssume thatα1andβ∈0,∞\ {1},then

fw1z

ln e

γ

1−zw

1−β

, w∈D, 2.10

whereγ ≥βln 2andfw10 γ1−βis a nonconstant function belonging toBα_logβ.

cAssume thatαβ1,then

fw2z ln ln

eγ

1−zw, w∈D, 2.11

whereγ ≥1ln 2andfw20 lnγis a nonconstant function belonging toBα_logβ.

Moreover, for eachw∈D, it holds thatfw, fw1, fw2belong to the correspondingBα_logβ_,0space,

and for fixedαandβ

sup

w∈D

fw_Bα

logβ

≤C, sup

w∈D

fw1

B1 logβ

≤C, sup

w∈D

fw2

B1 log1

Proof. aLetw∈Dbe fixed. Then we have

1− |z|α

ln e

β/α

1− |z|

β _f

wz

1− |z|α

ln e

β/α

1− |z|

β

× α−1w

1−zwαlneγ_/1−zwβ −

βw

1−zwαlneγ_/1−zwβ1

≤ |α−1|1− |z|

α_lneβ/α_{/1− |z|}β

|1−zw|αlneγ_/|1−zw|β β

1− |z|αlneβ/α_{/1− |z|}β

|1−zw|αlneγ_/|1−zw|β1

≤

|α−1| β lneγ_/2

1− |z|αlneγ_{/1− |z|}β

|1−zw|αlneγ_/|1−zw|β

2.13

≤ |α−1| β

lneγ_/2, 2.14

where in 2.13 we have used that γ > β/α and in 2.14 we have used the fact that the

function in2.1is increasing on the interval0,2.

From2.13, since 1− |w| ≤ |1−zw|, z, w∈D, and byLemma 2.1a, we have that

1− |z|α

ln e

β/α

1− |z|

β f_wz

≤

|α−1| β lneγ_/2

1− |z|αlneγ_{/1− |z|}β

1− |w|αlneγ_{/1− |w|}β −→0,

2.15

as|z| → 1−0, from which it follows thatfw∈ Bα_logβ_,0,as desired.

bFor fixedw∈D, we have

1− |z|

ln e

β

1− |z|

β fw1

z 1− |z|

ln e

β

1− |z|

β

1−βw

1−zwlneγ_/1−zwβ

≤β−11− |z|lne

γ_{/1− |z|}β

|1−zw|lneγ_/|1−zw|β

2.16

≤β−1, 2.17

where in2.16we have used the assumptionγ > β, while in2.17, as ina, we have used

From2.16, and byLemma 2.1a, we obtain

1− |z|

ln e

β

1− |z|

β fw1

z≤β−1 1− |z|lne

γ_{/1− |z|}β

1− |w|lneγ_{/1− |w|}β −→0, 2.18

as|z| → 1−0. Hencefw1∈ B1_logβ_,0,finishing the proof of this statement.

cWe have

1− |z|

ln e

1− |z|

fw2

z 1− |z|

ln e

1− |z|

w

1−zwlneγ_/1−zw

≤ 1− |z|lne/1− |z| |1−zw|lneγ_/|1−zw|

2.19

≤ 1− |z|lneγ/1− |z|

1− |z|lneγ_{/1− |z|}≤1, 2.20

where we have used the assumptionγ > 1 and the fact that function2.1is increasing on

0,2.

From2.19,Lemma 2.1a, and sinceγ >1 we obtain

1− |z|

ln e

1− |z| f

wz≤

1− |z|lneγ_{/1− |z|}

1− |w|lneγ_{/1− |w|} −→0, 2.21

as|z| → 1−,that is,fw2∈ B1_log1_,0.

Estimations2.12follow from2.14,2.17,2.20and by using the following facts

fw0

1

γβ, α /1, β≥1,

fw10 γ1−β, α1, β∈0,1,

fw20 lnγ, αβ1,

2.22

we finish the proof of the lemma.

Remark 2.7. Note that from Lemmas2.2and2.6the functionsfw, fw1, fw2defined in2.9–

3. Boundedness and Compactness of the Operator

uC

:

B

D or

B

D

→ B

D

This section studies the boundedness and compactness of the weighted composition operator

uCϕ :Bα_logβD orBα_logβ_,0D → BμD.

Case 1. α >1,β≥0.

Theorem 3.1. Assumeα >1,β≥0,ϕis an analytic self-map of the unit disk,u∈HD, andμis a weight. Then the operatoruCϕ:Bα_logβ orBα_logβ_,0 → Bμis bounded if and only if

sup

z∈Dμz

_uz ⎛ ⎜

⎝1 1

1−ϕz2α−1lneβ/α_/1−ϕz2β ⎞ ⎟

⎠<∞, 3.1

sup

z∈D

μz|uz|ϕz

1−ϕz2αlneβ/α_/1−ϕz2β

<∞. _3.2

Proof. First assume that3.1and3.2hold. Then, byLemma 2.2and the definition ofBα

logβ,

we have

uCϕf_Bμ u0f

ϕ0sup

z∈Dμz

uzfϕzuzfϕzϕz 3.3

≤C|u0|f_Bα

logβ

⎛ ⎜

⎝1 1

1−ϕ02α−1lneβ/α_/1−ϕ02β ⎞ ⎟ ⎠

Cf_Bα

logβ sup

z∈D

⎛ ⎜

⎝μzuz μz|u

_z|

1−ϕz2α−1lneβ/α_/1−ϕz2β ⎞ ⎟ ⎠

fBα

logβsup_z∈D

μz|uz|ϕz

1−ϕz2αlneβ/α_/1−ϕz2β

.

3.4

Applying3.1and3.2in3.4, the boundedness ofuCϕ:Bα_logβ orBα_logβ_,0 → Bμfollows.

Now assume the operatoruCϕ :Bα_logβ orBα_logβ_,0 → Bμis bounded. By taking the test

functionsfz≡1 andfz≡zwhich obviously belong toBα

logβ_,0, we obtain

sup

z∈Dμz

_uz<∞,

3.5

sup

z∈Dμz

_{uzϕz uzϕz<∞.}

From3.5and3.6, and since the functionϕis bounded, it follows that

sup

z∈Dμz

uzϕz<∞. 3.7

Forw∈D,set

gwz

1− |w|2

1−wzαlneβ/α_/1−wzβ−

1− |w|22

1−wzα1lneβ/α_/1−wzβ, z∈D. 3.8

We have thatgww 0,

g_w w − w

1− |w|2αlneβ/α_{/1− |w|}2β

, _3.9

and as an easy consequence ofLemma 2.6a, sup_w∈DgwBα

logβ ≤ Candgw ∈ B

α

logβ,0for each

w∈D.

Using these facts and the boundedness ofuCϕ : Bα_logβ orBα_logβ_,0 → Bμ, for the test

functionsgϕw, wherew∈Dandϕw/0, we get

μwuwϕwϕw

1−ϕw2αlneβ/α_/1−ϕw2β

≤uCϕgϕw_Bμ ≤CuCϕ_Bα

logβ→ Bμ

. 3.10

From3.10it follows that

sup

|ϕw|>1/2

μwuwϕw

1−ϕw2αlneβ/α_/1−ϕw2β

≤2CuCϕ_Bα

logβ→ Bμ

. 3.11

On the other hand, by using3.7andLemma 2.1b, we have

sup

|ϕw|≤1/2

μwuwϕw

1−ϕw2αlneβ/α_/1−ϕw2β

<sup

|w|<1

μw|uw|ϕw

3/4αlnβ4eβ/α_/3 <∞. 3.12

Hence,3.11and3.12imply3.2.

Let

Fwz

α11− |w|2

1−wzαlneβ/α_/1−wzβ −

α1− |w|22

Then

Fww

1

1− |w|2α−1lneβ/α_{/1− |w|}2β

,

F_w w − βw

1− |w|2αlneβ/α_{/1− |w|}2β1

,

3.14

and byLemma 2.6awe get sup_w∈DFwBα

logβ ≤C, andFw∈ B

α

logβ,0for everyw∈D. Using the

boundedness ofuCϕ :Bα_logβ orBα_logβ_,0 → Bμ, the test functionsFϕw, and equalities3.14

we get

μw|uw|

1−ϕw2α−1lneβ/α_/1−ϕw2β

≤uCϕFϕw_Bμ βμw|uw|

ϕwϕw

1−ϕw2αlneβ/α_/1−ϕw2β1

3.15

for eachϕw/0,w∈D.

From3.2,3.5,3.15, and using the fact that

sup

x∈0,1

ln e

β/α

1−x2 −1

≤ α

β, whenβ >0, 3.16

condition3.1follows.

Theorem 3.2. Assumeα > 1,β ≥ 0,ϕis an analytic self-map of the unit disk,u ∈ HD, and

μis a weight. Then the operator uCϕ : Bα_logβ orBα_logβ_,0 → Bμ is compact if and only if uCϕ :

Bα

logβ or B α

logβ_,0 → Bμis bounded

lim

|ϕz|→1

μzuz

⎛ ⎜

⎝1 1

1−ϕz2α−1lneβ/α_/1−ϕz2β ⎞ ⎟

⎠0, 3.17

lim

|ϕz| →1

μz|uz|ϕz

1−ϕz2αlneβ/α_/1−ϕz2β

0. 3.18

Proof. Suppose that uCϕ : Bα_logβ orBα_logβ_,0 → Bμ is compact. Then it is clear that uCϕ :

Bα

logβ orB α

Hence assume thatϕ∞1. Letzm_m∈Nbe a sequence inDsuch that|ϕzm| → 1 asm →

∞, andgmz gϕzmz, m ∈N,wheregwis defined in3.8. Then supm∈NgmBα

logβ < ∞,

gm → 0 uniformly on compacts ofDasm → ∞,gmϕzm 0,and

g_m ϕzm

− ϕzm

1− |ϕzm|2 α

lneβ/α_/1−ϕz m2

β. 3.19

Hence from3.10andLemma 2.5we have that

μzmuzmϕzmϕzm

1− |ϕzm|2 α

lneβ/α_/1−ϕz m2

β ≤uCϕgmBμ −→0 asm−→ ∞ 3.20

from which3.18follows.

LetFm Fϕzm, m∈NwhereFwis defined in3.13. Then supm∈NFmBα

logβ <∞and

Fm → 0 uniformly on compact subsets ofDasm → ∞. SinceuCϕ:Bα_logβ orBα_logβ_,0 → Bμis

compact, we see that

lim

m→ ∞uCϕFmBμ 0. 3.21

From3.15we have

μzm|uzm|

1− |ϕzm|2 α−1

lneβ/α_/1−ϕz m2

β

≤ uCϕFm_Bμ

βμzmuzmϕzmϕzm

1− |ϕzm|2 α

lneβ/α_/1−ϕz m2

β1,

3.22

which along with3.16,3.18, and3.21implies

lim

|ϕz| →1

μz|uz|

1− |ϕz|2α−1lneβ/α_/1−ϕz2β

0. 3.23

On the other hand, we have

μz|uz|

1− |ϕz|2α−1lneβ/α_/1−ϕz2β

≥Cμzuz, 3.24

Conversely, assume thatuCϕ :Bα_logβ orBα_logβ_,0 → Bμis bounded and3.17and3.18

hold. From the proof ofTheorem 3.1we know that

Bμu sup z∈Dμz

_{uz<∞, K} 2 sup

z∈Dμz

_{ϕz|uz|<∞. 3.25}

On the other hand, from3.17and3.18we have that, for everyε >0, there is aδ ∈0,1,

such that

μzuz

⎛ ⎜

⎝1 1

1−ϕz2α−1lneβ/α_/1−ϕz2β ⎞ ⎟ ⎠< ε,

μz|uz|ϕz

1− |ϕz|2αlneβ/α_/1−ϕz2β

< ε

3.26

wheneverδ <|ϕz|<1.

Assumefmm∈Nis a sequence inBα_logβorBα_logβ_,0such that supm∈NfmBα

logβ ≤Landfm

converges to 0 uniformly on compact subsets ofDasm → ∞.LetK {z∈D:|ϕz| ≤δ}.

Then from3.25,3.26, and byLemma 2.2, it follows that

sup

z∈Dμz

uCϕfm

z

≤sup

z∈Kμz

ϕz|uz|f_mϕzsup

z∈Kμz

uzfm

ϕz

sup

z∈D\Kμz

ϕz|uz|f_m ϕz sup

z∈D\Kμz

uzfm

ϕz

≤K2sup |w|≤δ

f_m wCsup

z∈D\K

μzuz

×

⎛ ⎜

⎝1 1

1−ϕz2α−1lneβ/α_/1−ϕz2β ⎞ ⎟ ⎠fm_Bα

logβ

Bμusup

|w|≤δ

fmwCsup z∈D\K

μz|uz|ϕz

1− |ϕz|2αlneβ/α_/1−ϕz2β

fm_Bα

logβ

≤K2sup |w|≤δ

f_m wBμusup |w|≤δ

fmw2Cεfm_Bα

logβ

.

Therefore

uCϕfm_Bμ fm

ϕ0|u0|sup

z∈Dμz

uCϕfm

z

≤K2sup |w|≤δ

_f

mwBμusup |w|≤δ

_fmw2CLεfm

ϕ0|u0|.

3.28

Sincefmm∈N converges to zero on compact subsets ofDas m → ∞, by the Weierstrass

theorem it follows that the sequencef_mm∈Nalso converges to zero on compact subsets ofD

asm → ∞, in particular limm→ ∞sup|w|≤δ|fmw|0 and limm→ ∞|fmϕ0|0. Using these

facts and lettingm → ∞in the last inequality, we obtain that

lim sup

m→ ∞

uCϕfm_Bμ ≤2CLε. 3.29

Sinceεis an arbitrary positive number it follows that the last limit is equal to zero. Applying

Lemma 2.5, the implication follows.

Theorem 3.3. Assumeα >0, β≥0, ϕis an analytic self-map of the unit disk,u∈HD, andμis a weight. ThenuCϕ :Bα_logβ_,0 → Bμ,0is bounded if and only ifuCϕ:Bα_logβ_,0 → Bμis bounded

lim

|z| →1μz

_uz0,

3.30

lim

|z| →1μz|uz|

ϕz0. _3.31

Proof. First assume thatuCϕ :Bα_logβ_,0 → Bμ,0is bounded. Then, it is clear thatuCϕ :Bα_logβ_,0 →

Bμis bounded, and as usual by taking the test functionsfz≡1 andfz≡z,and using the

factϕ_∞≤1, we obtain3.30and3.31.

Conversely, assume that the operatoruCϕ : Bα_logβ_,0 → Bμ is bounded,u ∈ Bμ,0,and

condition3.31holds.

Then, for each polynomialp, we have

μzuCϕp

z≤μzuzpϕzμzuzϕzpϕz

≤μzuzp_∞μzuzϕzp_∞,

3.32

from which along with conditions 3.30 and 3.31 it follows that uCϕp ∈ Bμ,0. Since

according to Theorem A the set of all polynomials is dense inBα

logβ_,0,we see that for every

f∈ Bα

logβ,0there is a sequence of polynomialspnn∈Nsuch that

lim

n→ ∞f−pnBαlogβ

From this and by the boundedness of the operatoruCϕ:Bα_logβ_,0 → Bμwe have that

_{uCϕf−uCϕpn} Bμ ≤

_uCϕ Bα

logβ ,0→ Bμ

_f−pn Bα

logβ ,0

−→0 _3.34

asn → ∞.HenceuCϕBα_logβ_,0⊆ Bμ,0,and consequentlyuCϕ :Bα_logβ_,0 → Bμ,0is bounded.

Remark 3.4. Note thatTheorem 3.3holds for allα >0 andβ≥0.

Theorem 3.5. Assumeα >1, β≥0, ϕis an analytic self-map of the unit disk,u∈HD, andμis a weight. Then the operatoruCϕ:Bα_logβ or Bα_logβ_,0 → Bμ,0is compact if and only if

lim

|z| →1

μz|uz|

1− |ϕz|2α−1lneβ/α_/1−ϕz2β

0, _3.35

lim

|z| →1

μz|uz|ϕz

1− |ϕz|2αlneβ/α_/1−ϕz2β

0. _3.36

Proof. IfuCϕ:Bα_logβ orBα_logβ_,0 → Bμ,0is compact, then it is bounded so that conditions3.30

and3.31hold. On the other hand,uCϕ :Bα_logβ orBα_logβ_,0 → Bμis compact, which implies

that3.17and3.18hold.

By3.18we have that, for everyε >0, there exists anr∈0,1such that

μz|uz|ϕz

1− |ϕz|2αlneβ/α_/1−ϕz2β

< ε, _3.37

whenr <|ϕz|<1. From3.31, there exists aρ∈0,1such that

μz|uz|ϕz< εh1

1−r2 3.38

whenρ <|z|<1, and whereh1is the function inLemma 2.1b.

Therefore, whenρ <|z|<1 andr <|ϕz|<1, we have that

μz|uz|ϕz

1− |ϕz|2αlneβ/α_/1−ϕz2β

On the other hand, ifρ <|z|<1 and|ϕz| ≤r, from3.38andLemma 2.1bwe have

μz|uz|ϕz

1− |ϕz|2αlneβ/α_/1−ϕz2β

≤ μz|uz|ϕz

h11−r2

< ε. 3.40

Combining3.39and3.40, we obtain3.36. Similarly, from3.17and3.30is obtained

3.35, as claimed.

Conversely, assume that3.35 and3.36 hold. First note that3.35implies3.30.

Indeed if3.30did not hold then there would be a sequenceznn∈Nand aδ >0 such that

inf

n∈Nμznu

_z

n≥δ 3.41

and limn→ ∞|ϕzn|L∈D.From this and the continuity of the function

h2x 1

1−x2α−1_lneβ/α_/1−x2β, x∈0,1, 3.42

we would have that

inf

n∈N

μzn|uzn|

1− |ϕzn|2 α−1

lneβ/α_/1−ϕz n2

β ≥δinf_0,1h2x>0, 3.43

which is a contradiction with3.35.

For anyf ∈ Bα

logβ, we have

μzuCϕf

z≤Cf_Bα

logβ

⎛ ⎜

⎝μzuz μz|u

_z|

1− |ϕz|2α−1lneβ/α_/1−ϕz2β

μz|uz|ϕz

1− |ϕz|2αlneβ/α_/1−ϕz2β

⎞ ⎟ ⎠.

3.44

Using conditions3.30,3.35, and3.36in3.44, it follows thatuCϕf ∈ Bμ,0for eachf ∈

Bα

logβ, moreover the set

uCϕ

f ∈ Bα

logβ

or Bα

logβ,0

:f_Bα

logβ

≤1 3.45

Taking the supremum in3.44 over the unit ball of the spaceBα

logβ orB α

logβ,0,then

letting|z| → 1 and using conditions3.30,3.35, and3.36, we obtain

lim

|z| →1_fsup

Bα

logβ≤1

μzuCϕf

z0, 3.46

from which along withLemma 2.3the compactness of the operatoruCϕ :Bα_logβ orBα_logβ_,0 →

Bμ,0follows.

Case 2. α1, β∈0,1.

Theorem 3.6. Assume thatϕis an analytic self-map of the unit disk,u∈HD, andμis a weight. Then the operatoruCϕ :B1_logβ orB1_logβ_,0 → Bμis bounded if and only if

sup

z∈Dμz

_uz ⎛ ⎝₁

ln e

β

1−ϕz2

1−β⎞ ⎠_<∞,

sup

z∈D

μz|uz|ϕz

1− |ϕz|2lneβ_/1−ϕz2β

<∞.

3.47

Proof. The proof of the theorem is similar to the proof ofTheorem 3.1. The sufficiency follows

by using the triangle inequality in3.3and then the third inequality inLemma 2.2and the

definition of the spaceB1 logβ.

For the necessity it is enough to follow the lines of the corresponding part of the proof ofTheorem 3.1and use the test functionsfz≡1, fz≡z,

fwz

fw1z 2

fw1w

−

fw1z 3

fw1w

2, w∈D, 3.48

gwz 3

fw1z 2

fw1w

−2

fw1z 3

fw1w

2, w∈D, 3.49

which belong toB1

logβfor the functions in3.48and3.49it easily follows byLemma 2.6b,

wherefw1zis the function in2.10. We omit the details.

The proofs of the following two theorems are similar to the proofs of Theorems3.2and

3.5, where the test functions in3.48and3.49are used as well as the lemmas inSection 2.

Theorem 3.7. Assume thatϕis an analytic self-map of the unit disk,u∈HD,andμis a weight. Then the operatoruCϕ:B1_logβ or B1_logβ_,0 → Bμis compact if and only ifuCϕ:B1_logβ orB1_logβ_,0 →

Bμis bounded

lim

|ϕz| →1μz _uz

⎛ ⎝₁

ln e

β

1−ϕz2

1−β⎞ ⎠_0,

lim

|ϕz| →1

μz|uz|ϕz

1−ϕz2lneβ_/1−ϕz2β

0.

3.50

Theorem 3.8. Assume thatϕis an analytic self-map of the unit disk,u∈HD, andμis a weight. Then the operatoruCϕ :B1_logβ orB1_logβ_,0 → Bμ,0is compact if and only if

lim

|z| →1μzu _z

⎛ ⎝₁

ln e

β

1−ϕz2

_1−β⎞ ⎠_0,

lim

|z| →1

μz|uz|ϕz

1−ϕz2lneβ_/1−ϕz2β

0.

3.51

Case 3. αβ1.

The following results were proved in15. Hence we quote them for the benefit of the

reader, and without any proof.

Theorem 3.9. Assume thatϕis an analytic self-map of the unit disk,u∈HD, andμis a weight. Then the operatoruCϕ :B1_log1 orB1_log1_,0 → Bμis bounded if and only if

sup

z∈Dμz|u

_z|max

1,ln ln e

1−ϕz2

<∞,

sup

z∈D

μz|uz|ϕz

1− |ϕz|2lne/1− |ϕz|2 <∞.

3.52

Theorem 3.10. Assume thatϕis an analytic self-map of the unit disk,u∈HD, andμis a weight. Then the operatoruCϕ:B1_log1 or B1_log1_,0 → Bμis compact if and only ifuCϕ :B1_log1 orB1_log1_,0 →

Bμis bounded

lim

|ϕz| →1μz

uzmax

1,ln ln e

1−ϕz2

0,

lim

|ϕz| →1

μz|uz|ϕz

1−ϕz2lne/1− |ϕz|20.

Theorem 3.11. Assume thatϕis an analytic self-map of the unit disk,u∈HD, andμis a weight. Then the operatoruCϕ :B1_log1 orB1_log1_,0 → Bμ,0is compact if and only if

lim

|z| →1μz|u

_z|max

1,ln ln e

1−ϕz2

0,

lim

|z| →1

μz|uz|ϕz

1−ϕz2lne/1− |ϕz|2

0.

3.54

Case 4. α∈0,1, orα1 andβ >1.

Here we consider the casesα∈0,1, orα1 andβ >1.

Theorem 3.12. Assume thatα∈0,1, orα1andβ > 1, u∈HD, μis a weight, andϕis a holomorphic self-map ofD.ThenuCgϕ:Bα_logβ orBα_logβ_,0 → Bμis bounded if and only ifu∈ Bμand

condition3.2holds.

Proof. The sufficiency follows by using the first inequality inLemma 2.2and the definition of the spaceBα

logβ in3.3.

For the necessity, by using the test functionsfz≡1, fz≡zwe first get conditions

3.5and3.7. To get3.2for the caseα1 andβ >1 we use the test functions

fwz 21− |w| 2

1−zwf

1

w z−

1− |w|22

1−zw2 f

1

w z, w∈D. 3.55

Note thatfww fw1w,

f_ww

1−βw

1− |w|2lneγ_{/1− |w|}2β

, _3.56

and similar toLemma 2.6b, sup_w∈DfwBα

logβ ≤Candfw∈ B

α

logβ_,0for eachw∈D.

Hence for the familyfϕww∈D,we get

1−βμw|uw|ϕwϕw

1−ϕw2lneβ/α_/1−ϕw2β

≤CuCϕfϕw_Bμ μw|u

_w|

lneβ/α_/1−ϕw2β−1

,

3.57

from which along with3.5and the assumptionβ >1, easily follows3.2in this case.

Whenα∈0,1, condition3.2follows as inTheorem 3.1, by using the test functions

Theorem 3.13. Assume that α ∈ 0,1, or α 1 and β > 1, u ∈ HD, μ is a weight, and ϕ is a holomorphic self-map of D, and uCϕ : Bα_logβ orBα_logβ_,0 → Bμ is bounded. Then

uCϕ :Bα_logβ orB α

logβ_,0 → Bμis compact if

lim

|ϕz| →1μz

_{uz0, 3.58}

and condition3.18holds.

Proof. The proof is similar to the corresponding parts of the proofs of Theorems3.2and3.7, so is omitted.

Remark 3.14. Note that if α ∈ 0,1, orα 1 andβ > 1 anduCϕ : Bα_logβ orBα_logβ_,0 → Bμ

is compact, then condition3.18is proved as in Theorems 3.2and 3.7, by using the test

functions in3.8and3.48. Ifϕ∞ < 1 then condition3.58is vacuously satisfied. At the

moment, we are not sure if the compactness implies condition3.58in the caseϕ_∞ 1.

Hence for the interested readers we leave this as an open problem.

The following theorem is proved as the corresponding part ofTheorem 3.5.

Theorem 3.15. Assume thatα∈0,1, orα1andβ > 1, u∈HD, μis a weight, andϕis a holomorphic self-map ofD. Then the operatoruCϕ :Bα_logβ orBα_logβ_,0 → Bμ,0is compact ifu∈ Bμ,0

and condition3.36holds.

Remark 3.16. Note that ifuCϕ :Bα_logβ orBα_logβ_,0 → Bμ,0is compact, then clearlyu∈ Bμ,0.

References

1 S. Stevi´c, “On new Bloch-type spaces,”Applied Mathematics and Computation, vol. 215, no. 2, pp. 841– 849, 2009.

2 J. M. Anderson, J. Clunie, and Ch. Pommerenke, “On Bloch functions and normal functions,”Journal f ¨ur die reine und angewandte Mathematik, vol. 270, pp. 12–37, 1974.

3 L. Brown and A. L. Shields, “Multipliers and cyclic vectors in the Bloch space,” The Michigan Mathematical Journal, vol. 38, no. 1, pp. 141–146, 1991.

4 D. D. Clahane and S. Stevi´c, “Norm equivalence and composition operators between Bloch/Lipschitz spaces of the ball,”Journal of Inequalities and Applications, vol. 2006, Article ID 61018, 11 pages, 2006.

5 S. Li and S. Stevi´c, “Weighted-Hardy functions with Hadamard gaps on the unit ball,” Applied Mathematics and Computation, vol. 212, no. 1, pp. 229–233, 2009.

6 S. Stevi´c, “On an integral operator on the unit ball inCn_{,”Journal of Inequalities and Applications, vol.}

2005, no. 1, pp. 81–88, 2005.

7 S. Stevi´c, “On Bloch-type functions with Hadamard gaps,”Abstract and Applied Analysis, vol. 2007, Article ID 39176, 8 pages, 2007.

8 S. Yamashita, “Gap series andα-Bloch functions,”Yokohama Mathematical Journal, vol. 28, no. 1-2, pp. 31–36, 1980.

9 K. Zhu,Spaces of Holomorphic Functions in the Unit Ball, vol. 226 ofGraduate Texts in Mathematics, Springer, New York, NY, USA, 2005.

10 C. C. Cowen and B. D. MacCluer,Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, Fla, USA, 1995.

12 X. Fu and X. Zhu, “Weighted composition operators on some weighted spaces in the unit ball,”

Abstract and Applied Analysis, vol. 2008, Article ID 605807, 8 pages, 2008.

13 P. Galanopoulos, “OnBlogtoQ_logp pullbacks,”Journal of Mathematical Analysis and Applications, vol.

337, no. 1, pp. 712–725, 2008.

14 D. Gu, “Weighted composition operators from generalized weighted Bergman spaces to weighted-type spaces,”Journal of Inequalities and Applications, vol. 2008, Article ID 619525, 14 pages, 2008.

15 S. G. Krantz and S. Stevi´c, “On the iterated logarithmic Bloch space on the unit ball,”Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 5-6, pp. 1772–1795, 2009.

16 S. Li and S. Stevi´c, “Composition followed by differentiation between Bloch type spaces,”Journal of Computational Analysis and Applications, vol. 9, no. 2, pp. 195–205, 2007.

17 S. Li and S. Stevi´c, “Weighted composition operators from Bergman-type spaces into Bloch spaces,”

Proceedings of Indian Academy of Sciences: Mathematical Sciences, vol. 117, no. 3, pp. 371–385, 2007.

18 S. Li and S. Stevi´c, “Weighted composition operators fromα-Bloch space toH∞on the polydisc,”

Numerical Functional Analysis and Optimization, vol. 28, no. 7-8, pp. 911–925, 2007.

19 S. Li and S. Stevi´c, “Weighted composition operators fromH∞to the Bloch space on the polydisc,”

Abstract and Applied Analysis, vol. 2007, Article ID 48478, 13 pages, 2007.

20 S. Li and S. Stevi´c, “Generalized composition operators on Zygmund spaces and Bloch type spaces,”

Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1282–1295, 2008.

21 S. Li and S. Stevi´c, “Weighted composition operators betweenH∞andα-Bloch spaces in the unit ball,”Taiwanese Journal of Mathematics, vol. 12, no. 7, pp. 1625–1639, 2008.

22 S. Li and S. Stevi´c, “Products of integral-type operators and composition operators between Bloch-type spaces,”Journal of Mathematical Analysis and Applications, vol. 349, no. 2, pp. 596–610, 2009.

23 M. Lindstr ¨om and E. Wolf, “Essential norm of the difference of weighted composition operators,”

Monatshefte f ¨ur Mathematik, vol. 153, no. 2, pp. 133–143, 2008.

24 B. D. MacCluer and R. Zhao, “Essential norms of weighted composition operators between Bloch-type spaces,”The Rocky Mountain Journal of Mathematics, vol. 33, no. 4, pp. 1437–1458, 2003.

25 K. Madigan and A. Matheson, “Compact composition operators on the Bloch space,”Transactions of the American Mathematical Society, vol. 347, no. 7, pp. 2679–2687, 1995.

26 A. Montes-Rodr´ıguez, “Weighted composition operators on weighted Banach spaces of analytic functions,”Journal of the London Mathematical Society, vol. 61, no. 3, pp. 872–884, 2000.

27 S. Ohno, “Weighted composition operators betweenH∞and the Bloch space,”Taiwanese Journal of Mathematics, vol. 5, no. 3, pp. 555–563, 2001.

28 S. Ohno and R. Zhao, “Weighted composition operators on the Bloch space,”Bulletin of the Australian Mathematical Society, vol. 63, no. 2, pp. 177–185, 2001.

29 J. Shi and L. Luo, “Composition operators on the Bloch space of several complex variables,”Acta Mathematica Sinica, vol. 16, no. 1, pp. 85–98, 2000.

30 S. Stevi´c, “Composition operators betweenH∞andα-Bloch spaces on the polydisc,”Zeitschrift f ¨ur Analysis und ihre Anwendungen, vol. 25, no. 4, pp. 457–466, 2006.

31 S. Stevi´c, “Weighted composition operators between mixed norm spaces andHα∞spaces in the unit

ball,”Journal of Inequalities and Applications, vol. 2007, Article ID 28629, 9 pages, 2007.

32 S. Stevi´c, “Essential norms of weighted composition operators from theα-Bloch space to a weighted-type space on the unit ball,”Abstract and Applied Analysis, vol. 2008, Article ID 279691, 11 pages, 2008.

33 S. Stevi´c, “Norm of weighted composition operators from Bloch space toHμ∞on the unit ball,”Ars

Combinatoria, vol. 88, pp. 125–127, 2008.

34 S. Stevi´c, “On a new integral-type operator from the weighted Bergman space to the Bloch-type space on the unit ball,”Discrete Dynamics in Nature and Society, vol. 2008, Article ID 154263, 14 pages, 2008.

35 S. Stevi´c, “Essential norms of weighted composition operators from the Bergman space to weighted-type spaces on the unit ball,”Ars Combinatoria, vol. 91, pp. 391–400, 2009.

36 S. Stevi´c, “On a new integral-type operator from the Bloch space to Bloch-type spaces on the unit ball,”Journal of Mathematical Analysis and Applications, vol. 354, no. 2, pp. 426–434, 2009.

37 S. Stevi´c, “Weighted composition operators from weighted Bergman spaces to weighted-type spaces on the unit ball,”Applied Mathematics and Computation, vol. 212, no. 2, pp. 499–504, 2009.

38 S.-I. Ueki, “Composition operators on the Privalov spaces of the unit ball ofCn_{,”Journal of the Korean}

Mathematical Society, vol. 42, no. 1, pp. 111–127, 2005.

39 S.-I. Ueki and L. Luo, “Compact weighted composition operators and multiplication operators between Hardy spaces,”Abstract and Applied Analysis, vol. 2008, Article ID 196498, 12 pages, 2008.

41 E. Wolf, “Compact differences of composition operators,” Bulletin of the Australian Mathematical Society, vol. 77, no. 1, pp. 161–165, 2008.

42 E. Wolf, “Weighted composition operators between weighted Bergman spaces and weighted Bloch type spaces,”Journal of Computational Analysis and Applications, vol. 11, no. 2, pp. 317–321, 2009.

43 W. Yang, “Weighted composition operators from Bloch-type spaces to weighted-type spaces,” to appear inArs Combinatoria.

44 S. Ye, “Weighted composition operator between the littleα-Bloch spaces and the logarithmic Bloch,”

Journal of Computational Analysis and Applications, vol. 10, no. 2, pp. 243–252, 2008.

45 X. Zhu, “Generalized weighted composition operators from Bloch type spaces to weighted Bergman spaces,”Indian Journal of Mathematics, vol. 49, no. 2, pp. 139–150, 2007.

46 X. Zhu, “Weighted composition operators fromFp, q, sspaces toH∞μ spaces,”Abstract and Applied

Analysis, vol. 2009, Article ID 290978, 14 pages, 2009.

47 S. Li and S. Stevi´c, “Riemann-Stieltjes-type integral operators on the unit ball inCn_{,”Complex Variables}

and Elliptic Equations, vol. 52, no. 6, pp. 495–517, 2007.

48 S. Stevi´c, “Boundedness and compactness of an integral operator on a weighted space on the polydisc,”Indian Journal of Pure and Applied Mathematics, vol. 37, no. 6, pp. 343–355, 2006.