Integration Operators On Weighted Bloch Spaces

Nihonkai Math. J.

Vo1.12(2001), 123-133

Integration Operators On Weighted Bloch Spaces

Rikio Yoneda

Abstract

Let 9 be an analytic function on the open unit disk D in the complex plane C. We shall study the following operator

Jg(f)(z):= l ^z f(()g'(()d(

on the Bloch space B. We show that the operator J g is bounded on B if and only if

sup(1-lzI ² ) (IOg~1 ^{1 2 ) Ig'(z)1 < +00,}

zED 1- z

and the operator J

is compact on B if and only if

lim (1 -Izf) (log ~I ^{1 2 ) 19'(z)/ = o.}

Izl-->l- 1 - z

And we shall also characterize the boundedness and compactness of J g on weighted Bloch spaces.

KeyWords and Phrases: integration operator, Bloch space, compactness, boundedness.

§1. Introduction

Let D = {z ^E C; Izi < I} denote the open unit disk in the complex plane C and let aD = {z E C; Izi = I} denote the unit circle. Let H(D) denote the space of analytic functions on D. Let dA(z) be the normalized area measure on D. Let 1 ~ p < +00. The Hardy space HP is defined to be the Banach space of analytic functions f on D with the norm

1 2~ ~

II f II p := (sup - r If(reiO)IPde) < +00.

O<r<l 27r Jo

D

D 1 (3( ) 1 1 l+1'I'z(w I h ( ) z-w

11 fr 1

ror z, wE

et z, W := '2 og ^{1- '1'%}

,were 'Pz W = 1-z.w· vve wi equent y use the following properties of 'Pz :

1 ^{-1 ()1 2 _} (1-lzI ² )(I-lwI ² )

'Pz ^{W -} 2 '

11-zwl

2000 Mathematics Subject Classification: Primary 30 D 45 .

epz(z) =0, epz(O) = z, epz ⁰ epz(w) = w.

For 0 < r < +00, let D(z) = D(z,r) = {w E D; (3(z,w) < r} denote the Bergman disk. Then D( z, r) is a Euclidean disk with Euclidean center C and radius R

e ^{r _} e- ^r

respectively, where t = E (0,1). We denote by ID(z, r)/ the normalized area of D(z, r).

e ^r + e- ^r

Then ID(z, r)1 is comparable to (1-lzI 2 ?

The Bloch space B is the space of functions 1 ^E H(D) such that

II f IIB:= sup{(1-l z I 2) 1!,(z)1 : zED} < +00 .

This is a semi-norm on B and it is Mobius invariant in the sense of lifo ep liB = II 1 liB for all fEB and ep E Aut(D), where Aut(D) is the Mobius group of bi-analytic mappings of D. The Bloch space B is a Banach space with the norm II 1 II = 1/(0)1 + II 1 liB. The little Bloch space of D, denoted Eo, is the closed subspace of B consisting of functions f with (1-lzI 2)j'(z) --40 (lzl --4 1-). The space of analytic functions on D of bounded mean oscillation, denoted by BMOA, is the set of functions 1 in ~ such that

II f IIBMoA := sup{1I 1 ⁰ epz - I(z) 112: zED} < +00 .

It is clear that 1/(0)1 ::;11 9 112 for every function 9 E H(D). Applying 9 = 1 ⁰ epz - I(z), it follows that (1- IzI 2 _)1!,(z)1 ::;11 10 epz - I(z) 112 for 1 ^E H(D) and zED. Thus it follows that BMOAcB.

Let a 2": 1. The a-Bloch space IP is defined to be the space offunctions f E H(D) such that

II 1 118<>:= sup{(1-lzI ² )0' I!'(z) I: ^{zED} < +00.}

And the little a-Bloch space .B(J is the closed subspace of IP consisting of functions 1 with (1-lzI ² )0'!,(z) --40 (Izl

1-). Note that B ¹ and Bd are the Bloch space and the little Bloch space, respectively.

Let w be analytic on {(; 11- (I < I}. Assume that IW(I-lzI2)1 --4 0 as zED and Izi --4 r. Then the weighted Bloch space B w is the space offunctions 1 ^E H(D) such that

II f IIB w := sup{lw(I-l z ^I 2

)11/'(z)1 : zED} < +00.

We define the following

Blog := U ^E H(D) : II 9 IIBlog:= :~b(I-/z/2) ^{(log 1_l} _lz12 ^{) jg'(z)/ < +oo},}

Blog,o:= U ^E H(D): lim_ (l_/z/2) (log - 1 11 1 2) Ig'(z)/ = OJ.

jzl-+l - Z

For a Banach space X, let S: X -+ X be a linear operator. Then the operator S is said to be compact if for every bounded sequence {xn} ⁱⁿ X, {S( xn)} has a convergent subsequence.

On the other hand, the operator S is said to be weakly compact if for every bounded sequence {Xn} in X, {S(x n)} has a weakly convergent subsequence. Then it is known that /)' is weakly compact if and only if S**(X**) c X, where S** is the second adjoint of S and X is identified with its image under the natural embedding into its second dual X**.

For 9 E H(D), the operator .1 ₉ is defined on the weighted Bloch space by the following:

.1 ₉ (f)(z):= faz f(()g'(C,)dC, (f E H(D)).

If g(z) = z, then.1 ₉ is the integration operator. If g(z) = log l~Z' then.1 ₉ is the Cesaro operator.

In [4], Ch. Pommerenke showed that .1

9 is a bounded operator on the Hardy space j{2 if and only if 9 belongs to BMOA, and this result was extended to the other Hardy spaces HP, 1 :s:; p < +00, in [11. In [2J, A. Aleman and A. G. Siskakis studied the operator .1 9 defined on the weighted (radial weight) Bergman space. Recently, in [5J, A. G. Siskakis and R. Zhao showed the following theorem:

Theorem A. The operatar .1 ₉ is boo1Uled on BMOA if and only if

(

(log 2 )2 )

sup II~ r ^{19'(z)1 2 (1-lzI 2 )dA(z) < +00,}

leaD JS(!)

and .1 ₉ is compact on BMOA if and only if

where S(I) = {z: I-III :s:; Izi ^< 1, fzT ^E I} far an arc I in aD.

In this paper, we shall study the boundedness and compactness of the operator .1 ₉ defined on the Bloch space, the a-Bloch space and the weighted Bloch space. Some of the techniques used to prove our theorems come from [2] and [5J.

Throughout this paper, positive constants C and K are not necessary the same as the one in at each occurrence.

§2. The boundedness and compactness of J g on the Bloch space

In this section, we study the boundedness and compactness of the operator .1 ₉ defined on the Bloch space.

Theorem 2.1. The operatar .1 ₉ is bOI11Uled on B if and only if 9 E Blog-

Proof. Suppose that 9 ^E Blog. Then /I 9 ^/llog= ~~E(l ^{-lzI 2 ) (log 1_I/} _z12 ^{) Ig'(z) I < +00.}

Let fEB. Then

(1-lz/ 2 )1 (Jgf)' (z)1 = (1-/zI 2 )lf(z)/lg'(z)1 = (1_/zI 2) (lOg 1 2) ^Ig'(z)1 If(z~1 ^.

l-lzl log l-lzl2 Since If(z)1 :::; e ^{/I f} liB _{log _ 1 _2} ^{(see [7,} Theorem 5.1.6 ]), we have

l-lzl

/I Jgf /lB :::; e~~E(I-lzI2) ^{(lOg 1_l} _lz12 ) Ig'(z)I/I f /lB

= ^{ell 9 IIBlogll} f liB .

To prove the converse, suppose that Jg is bounded on B. For a E D, put fa(z) = log l!az.

Then fa E B. For z E D(a,1'), we have log 1- ~a12 ^{:::; e !log I !} azl. Since the subhar- monicity of 19'(z)l, we see that (1 - laI 2 )2Ig'(a)12 :::; JD(a,r) 19'(z)12dA(z) (see [7, Proposition 4.3.8]) So by using the fact that there is a constant el > ⁰ (depending only on r) such that

f (1 ~ ^{1 2)2 dA (z) :::; el < 00, we have}

JD(a,r) - z

(1-la/ 2 l ^{(log 1 2)2} Ig'(a) 1 2 :::; (lOg 1 2)2 f 19'(z)/2dA(z)

l-Ial l-Ial JD(a,r)

:::; e f Ilog - 1 1_ 12Ig'(z)12dA(z) JD(a,r) - az

= e { ⁽¹ ~ ¹ 2)2 (1-lzI2)2Il0g - 1 1_ 12Ig'(z)12dA(z)

JD(a,r) - z - az

:::; e sup (1- l z I2)2!log - 1 1_ j2 1 g'(z)1 2

{ (1 ~ ^{1 2 )2 dA (z)}

zED(a,r) - az JD(a,r) - z

:::; eel su p (I- l z/ ² l/10g ^{- 1 1_} /2/g'(z)/2

zED - az

:::; eel sup /I Jgfa /I~

aED :::; eel II Jg 11 2

sup II fa II~ .

aED

Since II fa IIB= SUPZED(I-lzj2) 11!azllal :::; 2 < +00 for any a E D, we see sUPaED II fa /lB< ^00.

Hence we have SUPaED(I-laI 2

) (log 1_~aI2) ly(a)1 < 00. Thus 9 E Blog. 0

Lemma 2.2. ^Far f E lI(D) and 0 < r < 1, put fr(z) = f(rz), zED. Let f E Blog . Then _r->l- lim /I fr - f IIBlo _g = 0 if and only if f ^E B!og,o.

Proof. Suppose that f E Blog and lim II fr - f /lBlo = O. Then for any E > 0, there is

r->l- g

a 80 E (0,1) such that II fr - f IIBlog < ^E for 80 < r < 1. By using the fact la +bl 2 :::; 21al ² +21W

and the definition of II * IIBlog' we have

Since fr E Blog,a, we have (1-laI 2

)2 (lOg 1 2)2 Ifr(a)1 2 ---+ 0 ( lal---+ C). Hence we see 1- \a\

f ^E Blog,a.

To prove the converse, suppose I E Blog,G. Then for arbitrary enough small E > 0, there is a 8 E (0,1) such that (1-lz[2) log l-lzI21f'(z)/ < E for all 82 < fz/ < 1 and (1-lzI 2) log l-lzl2 is a decreasing function on 82 < Izi < 1. For 0 < r < 1, we have

II Ir - I IIBlog= sup{(1-lzI 2

) log 1 _11z121r 1'(rz) - j'(z)1 : zED}

:S sup{(1-lzI 2 ) log ~I 121r1'(rz) - 1'(z)1 : t5 < Izi < 1}

1- z

+sup{(1-lzI 2 )log 1112Ir 1'(rz)-J'(z)I:lz\:S8}.

1- z

Since r f'(rz) ---+ j'(z) uniformly for Izi :S 6, the second term in the above approaches to zero as r ---+ 1-. If 6 < r < 1 and 6 < Izl < 1, then we have 82 < rlzl < 1. Since (1- Iz12) log l-lzl 2 is a decreasing function on Izi E (6 2 ,1), for 62 < rlzl < Iz\ < 1,

(1- Iz1 2

) log 1 _11Z121r l' (rz)\ :S (1 - r21z12) log 1 _ :21z1 21f' (rz)1 < E.

Hence

8up{(1-lzI 2 )log 1_1/zI2Ir1'(rz) - j'(z)l: 6 < Izi < 1}:S 2E

for all 6 < r < 1. So we see lim sup II fr - f IIBlog:S 2E. Thus we have lim_ II fr - I IIBlog = O. ⁰

T~l- r~l

Theorem 2.3. The operatCff J _g is compact on B if and cmly if lim (1_lzI2) (log ~I ¹ 2) Ig'(z)1 = O.

jzl----l- 1 - z

Proof. Since lJ(z) I :::; C ^1/ f liB log 1- Izl 1 ₂ for fEB, the unit ball of B is a normal family of analytic functions. By the normal family argument, J g is a compact operator on B if and only if every sequence {In} in B with II fn IIB:S 1 and fn ---+ 0 (n ---+ +00) uniformly on compact subsets of D has a subsequence {Ink} such that II Jgfnk IIB---+ 0 (n ---+ +00).

Suppose that lim (1 -lzI 2 ) log ~I 12Ig'(z)1 = ^O. By the proof of Theorem 2.1, we

!zl----l- 1- z

have II Jgf IIBS; C II 9 IIBlog ^{II f liB for fEB.} Then by Lemma 2.2, there exist polynomials P n such that II 9 - P n IIBIog ^--+ O. Since II (Jg - Jp

(f) IIBS; C II 9 - P n IIBIogli f liB, thus we have II Jg - Jp

lis; C II 9 - P n ^{IIBIog --+ 0} (n ^--+ 00). For any polynomials P, Jp is a compact operator on B (see [2, p.342]). Hence we see that J g is a compact operator on B.

To prove the converse, suppose that J _g is compact on B. Let an ^--+ a E aD and put fn(z) ^:= log 1 1 , fez) ^:= log - 1 1_ . Then fn(z) ^--+ fez) uniformly on compact subsets of

-anz - az

D. By the proof of Theorem 2.1 and the fact fa + W S; 21al2 + 2fW, we have (1- /anI ² )2 (lOg 1- ~an12 ^{) 2 Ig'(an W}

S; C sup (1- Iz12)2110g 1 1_ ¹ 2

\g'(z)1 2

zED(an,r) - anz

S; 2C sup IIog 1 1

- liz 12Ig'(Z)12(1_lzI2)2

zED(an,r) - anz

+2C sup /Iog_11_/2/g'(z)I2(1_lz/2?

zED(an,r) - az

S; 2C II Jg(fn - f) II~ ^+2C _zED(an,r) ^{sup [log - 1} _- ^1_ _az ^{/2 1g'(z)1 2(1 -lzI 2 )2}

=:Ml+M2.

By the compactness of Jg, ^{we have} Ml ^{--+ 0} (n ^--+ 00). Since Bo is a subspace of B and they share the same norm, the compactness of Jg on B implies its compactness on Bo (see [10, Lemma 8] or [5, Theorem 3.6]). Hence we see that Jg is weakly compact on Bo. Since (Bo)** = Band J;* = Jg, we have Jg(B) c Bo. Thus we have Jg(f) E Bo. Thus we have

M2 = _zED(an,r) sup /Iog ~/21g'(z)12(1_ _{- az} ^{Izl 2 )2 =} _zED(an,r) ^{sup (1_/zI 2 ) /(Jg(f»' (z)/r --+ 0 (n --+ 00).}

Hence we have lim (1-lzI2)Iog _ 1 _ ₂ Ig'(z) I = O. 0

. Izj->l- 1-lzl

§3. The boundedness and compactness of J g on the a-Bloch space

In this section, we study the boundedness and compactness of the operator J _g defined on the a-Bloch space for a> 1.

Theorem 3.1. Let a > 1. Then the operator J g is bounded an IJC' if and anly if sup(l- IzI ² )1g'(z)1 < +00,

zED

i.e 9 E B.

Proof. Suppose that sup(1-lzI ² )1g'(z)1 < +00. Let f E Ber.. Then we see

zED

Since If(z)1 ~ C II f liB" (1- JzI2)I-a (see [9, Proposition 7]), we have II Jgf IIB"~ ^{Csup(I-lzI 2} )lg' (z)[11 f liB" .

zED

To prove the converse, suppose that J g is a bounded operator on B ^a . For a E D, put fa(z) = (1 - az)l-a . Then it is clear that fa E Ea. By using the subharmonicity of Ig'(z)l, the fact that r ⁽¹ ~ ^12)2dA(z) ~ ^{C l < 00,} and the fact that 11- azl is comparable to

iD(a,,.) - z

(1_[z[2) on D(a, r), we have

(1_laj2)2Ig' (a)1 2 ~ r Ig'(z))2dA(z)

iD(a,,.)

= r (1_/zI2)2a(1_/z/2)2(I-a)/g'(z)/2 dA(z)

iD(a,,.) (1-lzI 2)2

~ ^C r ^{(1-lzI 2 )2a 11-} azI 2(1-a) 19'(z)12 dA(z~

i ^{D(a,,.) (1 - Izi} )2

1 ^{dA(z) ( 2 l_a'I)2}

~ C (1 _)

2)2 SUp (1 - Izi )a 11 - azl Ig (z)1 D(a,,.) Z zED(a,,.)

~ ^{CCI (SU P(I- l} z/ 2)a /1- az/I-a /g'(Z)/)

2

zED

~ ^{CCI SUp} II Jgfa 1/1"

aED

~ ^{CCI II J g 11 2 SUp} II fa 111" .

aED

Since II fa IIB"= sUPZED(1 -lzj2)a \~=~:g[ ~ ^{(0' - 1)2 a < +00 for any a ED,} we see sUPaED II fa IIB"< +00. Hence we have sup(1-laI2)jg'(a)\ < +00. 0

aED

Theorem 3.2. Let 0' > 1. Then the operator J g is compact on B ^a if and only if lim (1-lzI 2 )[g'(z)1 = 0, i.e 9 E Bo.

!zl... l -

Proof. Since [f(z)1 ~ C II f liB" (1- IzI 2)I-a for f ^E B

the unit ball of B a is a normal family of analytic functions. By the normal family argument, J _g is a compact operator on B ^a if and only if every sequence {fn} in BD. with II fn IIB"~ 1 and In -+ 0 (n -+ +(0) uniformly on compact subsets of D has a subsequence {ink} such that II Jgfnk IIB"-+ 0 (n -+ +(0).

Suppose that lim (1 -lzI 2)lg' (z)1 = 0. By the proof of Theorem 3.1, we have

!zl... l -

II Jgf _IIB"~ C II 9 IIBII f liB"

for f E BD.. Then by [7, Theorem 5.2.2.]' there exist polynomials Pn such that II g- Pn ^{IIB-+ O.}

Since II (J g - Jp

(f) _IIB"~ C II 9 - P n IIBII f liB'" we have

II Jg - Jp

II~ C II 9 - P n IIB-+ 0 (n -+ (0).

For any polynomials P, Jp is a compact operator on B ^a (see [2 , p.342]). Hence we see that Jg is a compact operator on EO.

To prove the converse, suppose that J g is compact on EO. Let an

a E () D and put fn(z) := (l-anz)l-a , fez) := (1- az)l-a. Then fn(z)

fez) uniformly on compact subsets of D. By the proof of Theorem 3.1 and the fact la + W s 21al2 + 21W, we have

(1-lanI2)2Ig'(an)12 s ^{C ( 11- a} n zI2 (1-a) Ig'(z)12(1-lzI2 )2a dA(z~

J D(an,r) (1- Izi )2

s ^2C r ^{1(1- a} nz)(1-a) - (1- az)(1-a)j2Ig'(z)12(1_lzI2)2a dA(z~

J D(an,r) (1- Izi )2

+2C r ^11- azI 2(1-a) 19'(z)12(1_lzI2)2a dA(z~

JD(an,r) (1-lzl )2

S K II Jg(Jn - f) lI~a + K sup 11- azI 2(1-a) Ig'(z) /2(1 - /z/2)2a

zED(an,r)

=: It +h

By the compactness of J _{g ,} we have It ^{-40 (n -4} 00). Since B8 is a subspace of ~ and they share the same norm, the compactness of J _g on EO implies its compactness on BO. Hence we see that J g is weakly compact on BO. Since (BO)** = ~ (see [9]) and J;* = Jg, we have Jg(EO) C BO. Thus we have Jg(J) E B8. Thus we have

h = sup 11- azI2(1-a) Ig'(z»)2(1-lzI2)2a = sup ((1_lzj2)a j(Jg(J»' (z)l)2

zED(an,r) zED(an,r)

= ;~b ^{(XD(an,r) (z)(1 -lzI 2} )a I(Jg(J»/ (z)l/ -40 (n -4 00).

Hence we have lim (1-lzI ² )1g'(z)1 = o. ⁰

Izl--+l-

§4. The boundedness and compactness of J _g on the weighted Bloch space B _w

In this section, we study the boundedness and compactness of J g on the weighted Bloch space B _{w •}

Theorem 4.1. ^{Let 0} < r < +00. Let w be analytic, non-vanishing on {( : 11-(1 < I}, and Iw(I-lzI ² )1-4 0 as Izl-4 1-. Suppose that

(i) sUPZED Iw(I-lzI ² )1 JJzl IW(1~s2)1 ^{< +00,}

( ..) Iw(1-lzI ² )1 n sUPz,aED Iw(l-az)! < +00,

( "" ") nz JO r 1zl IW(1-s2)\ _ds < + 00 f or any z E D ,

(iv) .k1zllw(1~s2)1

(Iz1-4 1-),

(v) for any a E D there is a constant C> 0 (independent on a) such that /:g ^{-=-I~~j) Is C}

for all z E D(a, r),

(vi) there is a constant K > O(independent on z) such that io r ^lzl I ( w 1 - ds 2) , ::; s K I io r w ⁽¹ 1 - ZT/ ^{) d} 1J ¹ far all zED.

Then the opel'atar Jg is bounded on Bw if and ^only if

l ^{lz, ds}

II ^g IIw:= sup Iw(I-lzI ² )/ I ( 2)11g'(z)/ < +00.

zED 0 w 1- s

Proof. Suppose that II g Ilw< ^{00. Let} f E B w . Then we see If(a) - f(O)1 = la 1 ^{1 j'(at)dtl}

::; /a/1

1

/w(1-/at/ 2

)/IJ'(at)/IW(I_I,atI2 )1 dt

r ^{1 jaj rial 1}

::;11 f IIB

io Iw(1_\at\2)!dt =11 f IIB

io ^Iw(l- s2)l ds . Thus If(a)1 ::; C II f IIB

fd ^al IW(1~s2)lds. ^Then

Iw(1-lzI 2 )11 (Jgf)' (z)1 = IW(1-lzI 2)llf(z)IIg'(z)1

2 r ^1zl ds ,

::; Clw(I-lzl )1 i _o jw(l- s2)llg (z)111 f IIB

::; C II g IIwll f liB", . Hence we have II Jgf liB",::; ^C II f IIB

To prove the converse, suppose that Jg is bounded on B w. ^Put ha(z) := f~ w(l~a'l})dT/' By the assumptions that there is a constant C > 0 (independent on a) such that l:g-=-l~iJ) I::; ^{C for}

all z E D(a, r), and that there is a constant K > 0 (independent on z) such that fci ^zl IW(1~s211 ^::;

K /fo ^z w(1~Z1)) ^{d17/ for all zED, by using} the subharmonicity of (lw(I-az)llf~w(1~aT/)d17IIg'(z)1)2,

we have

(

2 rial ds ' ) ²

Iw(I-lal )1 i _o Iw(l- s2)llg (a)1

::; (K

^w(1-laI2 )11J: ^w(l _~ ^{aT/) d1J !lg'(a)l) 2}

::; K 2 (1_~~12)2 ^{k(a,r) (\W(l- az)\} liZ ^w(1 ~ ^{aT/) d 1J}

^{\g'(z)l) 2 dA(z)}

::; K ² C ² ( ~112)2 r (lw(1-lzI2)llha(z)llg'(z)I)2 dA(z)

1- a iD(a,r)

::; K ² C ² CIC2 sup Iw(1-lzI2)12Iha(z)j2Ig'(z)12

zED(a,r)

::; K 2 _C 2 _C

1 ^{C2 II} Jgha 111

Iw(1 - I z1 ² )1

Since sup I ( )1 < +00, we see h a E B w . By the boundedness of J g on B w , we have

z,aED W 1- az

l ^{lz, ds ,}

sup Iw(I - Iz/ ^{2 )/} I ^{(1 2)llg} (z)/ < +00. 0

zED 0 W - S

For example, w«) := (and w«) := (0 and w«() := (log( satisfy the conditions (i)

(vi) of Theorem 4.1.

We define the following

l ^{lz, ds}

Bw ^:= {f E H(D) : II 9 IIBw:= sup Iw(I-lzI2)\ I ( 2),lg'(z)\ < +oo},

zED 0 w 1- s

• 2 1!Z! ^{ds ,}

Bw,o := {f E H(D): lIm Iw(I - Izl )/ I ( 2)llu (z)/ = OJ.

Izl--+l- 0 w 1- s

Lemma 4.2. Letw be as in Thearem4.1. Mareauer suppose that Iw(I-lz12 )1 Id ^zl \W(ldss2j \1

o ^{(lzl -1-). Then far f E Bw, lim} II fr - f IIw= 0 if and only if f E Bw,o.

r--+l-

Proof. Suppose that IE Bw and lim II Ir - I ^{IIw= O.} Then for any 1O > 0, there is a

r--+l-

l50 E (0,1) such that II Ir - I Ilw< 1O for l50 < r < 1. By using the fact la + bl ^{2 S;} 21 al ² + 21bf

and the definition of II * ^1/ w, we have

(

2 r ^{lzl ds ) 2}

^{2 ( 2} r ^{lzl ds ) 2}

²

Iw(I -lz[ )1 10 ^{Iw(l- s2)1 II (z)[ s; 21O + 2 [w(I - Izl >[ io Iw(I- s2)\ [lr(z)l·}

Since Ir ^{E Bw;o, we have} I ^{E Bw,o.}

To prove the converse, suppose I E Bw;o. Then for arbitrary small 1O > 0, there is a 0 E (0, 1) such that Iw(I-lzI2 )1 Id ^zl lW(1~82j! ^11'(z)1 < 1O for all 0 ² < Izi < 1 and Iw(I-lzI 2

)1 Id ^{zl \w(lds82)!}

is a decreasing function on 0 ² < Iz) < 1. We have

2 r ^{lzl ds , ,}

II fr - f IIwS; sup{ Iw(I-lzl )1 io Iw(I- s2)llrI (rz) - I (z)/ : 8 < Izl < 1 }

r ^{lzl ds ,}

+ sup{ Iw(I-lzI 2 )1 i

o Iw(I- s2)llr !,(rz) - I (z)1 : Izi S; 0 }.

Since r I'(rz) - I'(z) uniformly for Izi ^{S; 0,} the second term in the above approaches to zero as

r -1-. H 0 < r < 1 and 0 < Izl < 1, then we have 0 ² < rlzl < 1. For 0 ² < rlzl < Izi < 1,

2 r ^{lzl d s , 2} r ^{lrzl ds}

Iw(1 -Izl )\ 10 lweI _ s2)llr I (rz)1 S; Iw(l- Irzl )1 i _o Iwel- s2)111' (rz) \ < 1O.

Hence

2 r ^{lzl ds , ,}

sup{ Iw(I-lzl )110 Iw(l- s2)llr I (rz) - I (z)1 : 0 < Izi < 1 } S; 21O

for all 0 < r < 1. So we see lim sup II Ir - I ^{IIwS; 21O. Thus we have lim II} Ir - I ^{Ilw= O. 0}

r--+l- r--+l-

We also see that examples w«() := ( and w«() := (0 and w«() = (log( satisfy the condition of Lemma 4.2.

Proposition 4.3. Let w be as in Thearem 4.1. Suppose that lar any a E D,

Iw(1 - lal2)1 is comparable to Iw(1 - Iz12)1 on D(a, r), and that Iw(I-lzI2 )1 Id ^z1 IW(1~82)1 ¹

o (jzj --+ 1-). If lim/z/--->1-jw(l-jzI2)j Jdzllu;(l~52)11J/(Z)1 = 0, then the operatar J _g is compact on Bu;.

Proof. By the proof of Theorem 4.1, we have II Jgf IIB

C II g Ilwll f IIB

Suppose

. 2 llZ' ^ds

that lim Iw(1 -izi )\ ) (1 2)) Ig ^(z)\ = O. Then by Lemma 4.2, there eXIst polynollll-

)zj-->l- 0 w - s

als P n such that II ^{g - P} n IIw--+ O. Since II (J g - Jpn) (J) IIB

C II ^{g - P} n IIwll f IIB

we have

II J g - Jp

1/:::; C II g - P _n IIw--+ 0 (n --+ CXJ). For any polynomials P, Jp is a compact operator on B w (see [2, p.342]). Hence we see that J g is a compact operator on B _{w .} 0

Acknowledgment. The author wishes to express his sincere gratitude to Professor Takahiko Nakazi for his many helpful suggestions and advices.

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Tokyo Metropolitan College of Technology

Tokyo 140-0011 , Japan

[email protected]

Received Ivlarch 7, 2001 Revised July 13, 2001