SOME CHARACTERIZATIONS OF α-BLOCH SPACES ON THE UNIT BALL OF Cn

1997,17(4):471-477

SOME CHARACTERIZATIONS OF a-BLOCH SPACES ON THE UNIT BALL OF en 1 Yang Weisheng (

~_!t

)

Young Scientists Lab. of Mathematical Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, P.O.Box 71010, Wuhan 430071, China

Abstract In this paper,we get the invariant gradient characterizations and ,B-Carleson measure characterization of a-Bloch functions.Two examples are also given to show that the condition about a is the best. Key words invariant gradient, a-Bloch, ,B-carleson measure

1 Introduction Let H(B) denote the class of all holomorphic functions in the unit ball B. We say that

f

E BO ,if

sup IVf(z)/(1 -

zEB

IzI 2 )O < 00,

0
00.

When n=l,replace them by H(D) and BO(D). Hardy and Littlewood proved that([3,2]): BO(D) = Lip(1 - a). We know that Lip,B can be used t? describe the dual space of Hardy space HP(D) for 0 < p < 1 ([2]). So BO are important in the theory of Hardy spaces. Several authors have been working in this field(see [4,10,12,13]). In [11], we have proved that in the unit ball of (1) For f E H(B) and 0 < a < 00, .f E BO ¢:> sUPzEB l'Rf(z)I(1 - IzI 2 )O < 00; as a corallary, BO = A(B)nLip(1 - a) for

en,

o<

a < 1. (2) BO l+n., Bg< <1+.1 C ~ c BO >1 .!l±.!.' for n > 1 and _ pOp 0_ + p o < p < oo,where HP, ~ denote the Hardy spaces and Bergman spaces, respectively. (3) 8° C VMOA for 0 < a < 1 and BMOA C BO for 1 :::; a < 00. In this paper, we give some characterizations of the a-Bloch spaces(see the Theorem and the Corallary in Section 3). In Section 2, we give some necessary notations and basic results about them.

!

In Section 4,two examples are given to show that the condition a > in the Theorem can not be improved. Usually, it is difficult to give a good example in several complex variables. A remark(Remark 3) is also given to explain the difference of the restriction about a between in one complex variable and in several complex variables. 1 Received

Nov.l,1996; revised Jan.15,1997.

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ACTA MATHEMATICA SCIENTIA

Vol.17

2 Notations Let B denote the unit ball in

en, and for a E B, cpa (z) is the Mobius transformation ° = v:'. CPa E Aut(B), Aut(B) is the group of

of B satisfying CPa(O) = a, CPa (a) = and CPa biholomorphic automorphisms of B, cf.[7].

=

Let V/(z) "\l(f 0 CPz)(O) denote the invariant gradient of f (see [5]), and 6.f(z) = 6.(1 0 cpz )(0) denote the invariant Laplacian of 1 (see [7]). By a direct computation, we get that for

1 E H(B), IVf(zW

For

f

= (1-lzI 2)(IVf(zW -1'R.f(zW). = ~z;.lfI2(z). 4

E H(B), define the complex normal gradient of

"\lNf=

{

f as the following:

< "\l f, z/Izl > z/lzl, z:j:. 0, z = 0,

0,

and the complex tangential gradient of

f as the following:

Let JRCPa denote the real Jacobian of CPa. For (3 E (-00,00), let dVf3(z) = K(z, z)l- f3dv(z), where K(z, w) (1- < z, w > )-n-l is the Berman kernel and v is the normalized Lebesgue

=

measure on B(see [7]). For a E B, r E (0,1], let r B = {z E B : Izi < r}, E{a, r) Lemma 1 When a E B, r E (0,1], f E L 1(dvf3(B)), (3 E (-00,00),

1 rB

When r

f(z)dv13(z) = (

JE(a,r)

= CPa.(r B).

f(lfJa (w)) (JRlfJa (w))13dv13 (w).

= 1,{3 = 1, it is the usual integral transformation(see [7]).

When r = 1,{3 = 0,

it is the M-invariant integral transformation(see [5]). Proof Let z = CPa(w), by 2.2.6(6) of [7] we get

1 rB

f(z)dvf3(z)

1 =1 =

rB

l(z)K(z,z)l-

f(z)(l

rB

f3dv(z)

-l zI 2 )(f3 - 1)(n+l )dv(z)

= (

f(lfJa(w))(l - IlfJa(wW)(13-1)(n+l)JRlfJa(w)dv(w)

=f

f(lfJa(W))(1-llfJa(~)12)(13-1)(n+l\1_lwI2)(13-1)(n+1)

JE(a,r)

1-

JE(a,r)

Iwl

JRCPa(w)dv(w) = (

f(lfJa(w))(JRlfJa(W))13 K(W,W)l- 13dv(w)

= (

f(lfJa (w))(JRlfJa (w)) 13 dv13 (w).

JE(a,r) JE(a,r)

Yang: SOME CHARACTERIZATIONS OF a-BLOCH SPACES

No.4

473

A finite positive Borel measure J-L on B is called a ,8-Carleson measure if for any fixed r

> 0 and 0 < ,8 < 00, J-L(E(a, r)) sup, f3 < aEB v(E(a, r))

see [6]. Let

n == B

00,

in Theorem A of [6], we get

Lemma 2 Let It be a finite positive Borel measure on B, 1 - n~l < ,8 < 00, then Jl is a ,B-Carleson measure if and only if

Throughout this paper C denote a finite constant, it is not necessarily same at its each appearance.

3 Characterizations of a-Bloch Functions Theorem equivalent:

When

i

<

a

< 00,0 <

p

< 00,1 - n~l < ,8 < 00, the following are

(i)f E B":

(ii)(l - IzI2)(a-l) IV f(z) I is bounded in B;

L

(1 - IzI 2)p(O-1)IVf(z)IP(JR(ii): By the definition, IVfl 2 == IVNfl 2 + IV

(iii)suPaEB

Proof IzI 2IV Nf (z) 12, we have

Tfl

2

and l'R,f(z)1 2

IVf(z)1 2 == (1-lzI2)(IVf(z)12 -1'R,f(z)12)

+ IVTf(z)1 2 -lzI 2IVNf(z)1 2 ) == (1- rz12)21VN f(z)1 2 + (1-lzI 2)IVTf(z)12 ::; (1 -1zI2)2IVf(z)12 + (1-lzI 2)IVTf(z)12

== (1-lzI2)(IVNf(z)12

(1)

so

By Lemma 2.2 of [1], when

!'

f E Ba,a >

then

(1 -lzI 2) 2a By (2), sup(1-lzI2)(a-l)IVf(z)1 < zEB

00.

1IVTf(z)1 2

::;

C < 00.

(i)=>(ii) is proved.

(ii)==:>(iii):Suppose sup{IVf(z)I(1-lzI 2)
< 00.

By,8 > 1- n~l' we know

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ACTA MATHEMATIC A SCIENTIA

((3 - l)(n + 1) > -1, so let r

1

Vol.17

= 1 in Lemma 1, we get

(1- IzI 2)p(a-l)I Vf(z)I P(JR
~

sup{ IV f(z )IP(l - IzI 2)p(a-l)}

zEB

=MP =MP <

1

I

r(J R
JB

dv,B(w)

(1 -lzI 2)(,B-l)(n+l)dv(z)

00.

It shows that (ii)==:>(iii).

1 < l U(
(1 - \zI2)p(a-l)IV f( z )\P(JR
(iii):=::}(i): Suppose SUPaEB

00.

Let u be M-subharmonic, by (2.4) of [8], for each a E B,

u(a)

o < t < 1.

Therefore for a fixed r E (0,1)

2n

S 2n

r

t 2n- 1(1 - t 2)(,B-l)(n+l)dt · u(a)

1 JO r

t 2n- 1(1_ t 2)(,B-l)(n+l)dt

By Lemma 1

u(a)

f

E

~

V{3

/) rB

r

JE(a,r)

l

U(
u(Z)(JR
H(B) and epa E Aut(B) imply lV'flP is M-subharmonic , thus IVf(a)IP (1 -l aI 2)paIV f(a)IP

~

/ B)

vf3 r

r

~ (1- lal 2)pa

IVf(z)IP(JR
t

JE(a,r)

vf3 r

B)

r

JE(a,r)

IVf(z)IP(JR
For z E E(a, r), we have CPa(z) E r B; therefore, 1_

2

r <

1_

()12 _ (1 - Iz I2 )(1 - lal 2 ) (1 - IzI2 )(1 - lal 2 ) 4(1 - Iz1 2 ) 12 < (1 - lal)2 ~ 1 - lal 2 '

1

1-

Because IVf(z)1

2

lal 2 :::; - -4(21 - Izl'-). ?

1- r

.

= (1-lzI2)(IV'j(z)12 -IRj(z)1 ) 2: (1-l zI2 )21V'j (z)12, we have

(1-laI 2)paIVf(z)IP

2

~ (1 ~r2ya(1-lzI2)paIVf(z)IP ~ (1 ~ r2ya(1-lzI2)p(a-l)IVf(z)IP.

(3)

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Yang: SOME CHARACTERIZATIONS OF a-BLOCH SPACES

No.4

By (3) and the hypothesis, we get

(1- laI 2)paIVf(a)IP

<

r

(1 B)(1 4 2)P'"

vf3 r

- r

JE(a,r)

IV!(z)IP(1-lzI 2)p(a-l)(JRlt'a(Z)),8dv,8(z) <

00.

So

sup(1-laI2)a/Vf(a)/ < 00.

aEB

(iii)=>(i) is proved.

Corollary For ~ < a < 00,0 < p < 00,1- n~l < {3 < 00, f E B" ¢=::> IVf(z)IP(1-lzI2)p(a-l)dvf3(z) is f3-Carleson measure. Proof Using Lemma 2, and (i){::}(iii) of the Theorem,we can get the result easily.

=

=

=

Remark 1 When a 1,{3 1 and B" B, the usual Bloch spaces(see [9]), from the results of the Theorem and the Corallary,we can easily get the main results of [1] with noticing /Vf(z)/2 = ~LilfI2(z) and Remark 3.2 of [6].

4 Examples Example 1 Let f(Zl,Z2) = (1- Zl)l-~, then IVf(z)1 = 11- a111- Zl/-a, therefore

f

E

e: for °< a

< 00. IV f(z)/2

= (1 -lz/ 2)(IVf(z)12 -1R,f(z)1 2) = (1 - IzI2)(1- IZlI 2)11- zll- 2al1 - al 2

( 1 -lzI2)2(a-l)IVf(z)12 But when a

<

!' let

Z

< C (1_lzI

-

2

) 2a - l

(1 - IZlI2)2a-l

< 00'

'

= (0, Y),.O < Y -+ 1, then

at this time, the Theorem is not true.

Example 2 Let f(Zl' Z2) = (1 - Zl)~

of

+ z2Iog(1- zl),then,

1 _1. = --(1 - Zl) 2 OZl 2

-

~!

UZ2 (1-lzI 2) t

-

Z2

--,

1 - Zl

= log(1 - zt},

Z I_Of I < (1-lzI2)t(~11- zfr t + _1_ 2_ 1 ) OZl 2 /l-Zl/

1

IZ21(1 -lzI 2)t 11 - zll (1 -lzlI2)~(1- IzI2)~

<-+----- 2

1

~ 2" +

11 - zll

< 3,

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ACTA MATHEMATIC A SCIENTIA

af I = (1-lzl~)211og(1 ') 1 (1 - Izi 2)21 IOZ2 - zl)1 so

f

E

Vol.l7

< 00,

Bt.

But we can prove that (1-lzI2)2(t-1)IVf(z)12 is unbounded on B.

Let z == (y, 0),0 < y < 1, then

(1 - IzI 2 ) 2( t- 1) IV f(z) 12

==

(~(1

- y)-l + 10g2(1 _ y) _ ~y2(1 _ y)-l)

== 10g2(1 - y) + ~(1 + y), when y

-+

1,

!.

It means that the results of the Theorem are not true when a == Remark 2 From the above examples, we know the results of the Theorem are not

!.

!

true for a ~ So the condition a > in the Theorem can not be improved. Actually, Example 2 is the best one in the following sense:

(4)

!

Let a == in (2) of this paper, and use (6) of [1], we can get (4). Remark 3 In (A)¢:>(D) of Theorem 1 in [13]:

II f II~~¢} sup f

aEDJD

let p == 2p', q == 2, then

f E Ba

¢} sUPaED =

If'(z)IP(1 - IzI2)pa-2(1 - ISOa(zW)Qdo-(z),

L L L

sUPaED

= sUPaED =

sUPaED

<

00.

(1f'(zW)P' (1_lzI2)2p'a(1

~ ~1;~W)2dO-(z)

[(If'(zW)(1- IzI2 ) 2JP' (1 -lzn 2p'a-2p'JRSOa(z)do-(z) IVf(z)1 2P' (1-l z I 2)2 p'(a-l)1RSOa(z)do-(z)

L

IVf(z)IP(1-l z 2)p(a-1)JRSOa(z)do-(z) I

en

It is coincide with our result in when f3 == 1. But in Theorem 1 of [13], the result is true for 0 < a < restriction a > What is the reason?

!.

00,

and there is no the

No.4

477

Yang: SOME CHARACTERIZATIONS OF a-BLOCH SPACES

In fact, in D, when z #- 0, \IT/(Z) == 0, so the term (1-lzI 2)1\l'T/(z)1 2 in (1) vanishes; when z == 0, \l/(z) == VT/(z), therefore in D,

In view of the above, we can give the following

Open Question. In the unit ball of Cn(n > 1), about sa for 0 < a :S expression of integral characterization associated with the invariant gradient

t, what is the V/(z)?

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London Math.

Soc., 1980,12:

241-267 Yamashita S. Gap series and a-Bloch functions. Yokohama Math. J.,1980,28:31-36 Yang W S, Ouyang C H. Relationship between a-Bloch spaces and other classes of holomorphic functions(to appear) Yu Y D. a-Bloch functions and Carleson measure. J. Beijing Inst. Tech.,1992,12:11-19 Zhao R H. a-Bloch functions and VMOA. Acta Mathematica Scientia,1996,16(3): 349-360