ON DIRICHLET TYPE SPACES AND α-BLOCH SPACES IN THE UNIT BALL OF Cn

2004,24B( 4):645-654

ON DIRICHLET TYPE SPACES AND a-BLOCH SPACES IN THE UNIT BALL OF c- 1 Li Eo ( -tvt)

Ouyang Caiheng ( iY:.fa tJtr

)

Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China

Abstract In this paper, the authors study the inclusion relations between Dirichlet type spaces Dr and a-Bloch spaces {3Q by means of higher radial derivative. The strictness and the best possibility of the inclusion relations are shown with constructive methods. FUrthermore, they sharpen one of the results when T = n, which proves that a conjecture in [7] is true. Key words

Dirichlet type spaces, a-Bloch spaces, inclusion relations

2000 MR Subject Classification

1

32A37, 32A18

Introduction

In 1966, G.D.Taylor defined Dirichlet type spaces Dr in the unit disk (see [14]), which take vector value in a given Hilbert space Hand T E R,

He studied the multipliers on Dr in that paper. As a special case, when H = C, Dirichlet type spaces attracted some interests of many researchers, and were studied in some different aspects, such as integral characterizations, multipliers and random power series (see [4]'[9]'[13]). In 1997, R.Aulaskari et al in [2] discussed the inclusion relations between Dirichlet type spaces and other well-known function spaces such as a-Bloch spaces. The main tool they used is the integral characterizations of Dr (d. [13]). In the case of several complex variables, Dirichlet type spaces have been studied recently. In [6], P.Y.Hu and J.H.Shi introduced the definition of Dirichlet type spaces with Taylor expansion coefficients for the unit ball of en. They generalized the results of [4] and [14] to higher dimensions (see [6] and [7]). In [5], S.J.Feng gave the integral characterizations of Dr with respect to invariant gradient, complex gradient and radial derivative. At the same time, he obtained some results for higher dimensions similar to [2], but there was a restriction to T < 2. We notice that the restriction for T can be weakened if we consider higher radial derivative. Therefore, the research on spaces Dr will be extended. Using it and combining the results of [10], in which we obtained the growth and the integral characterizations of a-Bloch functions 1 Received December 25, 2002; revised August 18, 2003. (10271117)

The research is supported by NNSF of China

646

ACTA MATHEMATICA SCIENTIA

Vol.24 Ser.B

in terms of their higher radial derivative, in this paper we focus on the study of the strictness and the best possibility of the inclusion relations between Dirichlet type spaces and a-Bloch spaces. The main results are as follows: (i) for 7 < 2 - 2a, 0 < a < 00, (30 eDT. (ii) for 7 < n + 2, D T C (31+ n;-, or alternatively for 0 < a < 00, D n + 2 - 20 C (30. Moreover, we prove the strictness and the best possibility of the above two inclusion relations by constructing some functions with Hadamard gaps. Combining (i) and (ii), we have D n +2 - 20 C (30 c D T « 2 - 20 ) . Thus, we give the exact location of Dirichlet type spaces in a-Bloch spaces similar to [16]. When 7 = n in (ii), we have D n C (31 =Bloch. Further, we sharpen this result to be D n CBMOA, which is still strict and best possible in the sense that the index n of D n cannot be smaller.

2

Preliminaries

Let measure measure denoted complex

B be the open unit ball of Cn(n > 1) with boundary 5. dv denotes the Lebesgue on B normalized so that v(B) = 1 and da the rotation-invariant positive normalized on 5, i.e. a(5) = 1. The class of all holomorphic functions with domain B will be by H(B). When n = 1, replace it by H(D), where D denotes the unit disk of the plane. For f E H(B), z E B, its complex gradient is \1 fez)

af ) af = ( -(z),···, -(z) aZ1 aZ n

and the radial derivative Rf(z) = (\1 fez), z) =

n

L::

j=1

*f(z)Zj. When mEN, Rm fez) = R(Rm-1 f) J

(z) denotes the higher radial derivative of fat z with order m. For f E H(B) with homogeneous 00 00 expansion fez) = L:: L:: aoz O , it is clear that Rmf(z) = L:: km L:: aoz o. k=O

We say that

f

E

lol=k

k=O

lol=k

H(B) is an a-Bloch function, denoted by f E (30 for 0 < a < Ilflll30 = sup l\1f(z)I(1-lzI 2)0 zEB

00,

if

< 00.

It is obvious that (30 is a normed linear space with II . 11130 (modulo constant functions), and

(30 1 C (30 2 for a1

< a2. In [6], fez) = L:: aoz E H(B) is said to be in Dirichlet type space O

o?:o

D T for 7 E R provided that

Ilfllb, = 2:)l al o?:o

Here[11] wo

+ nYwo la o l2 < 00.

r I 012 () = is ~ da~ =

(n - l)!a! (n+lal-1)!·

Specially, the space D) is called Dirichlet space. The spaces Do and D_ 1 are just Hardy space H 2(B) and Bergman space L~ (B), respectively. From the definition we know D T 1 C D T 2 when 71 > 72. D T is a Hilbert space with inner product (f, g) = L:: (Ial + nf aobowo where

fez) =

L::

o?:o

aoz o and g(z) =

L::

o?:o

bozO: both belong to D T •

o?:o

Li & Ouyang: DIRICHLET TYPE SPACES AND Q-BLOCH SPACES IN UNIT BALL

No.4

647

In [10], we characterized a-Bloch functions by higher radial derivative and obtained the following result: Lemma 2.1[10] For f(z) E H(B), 0 following conditions are equivalent (i) f(z) E (3U(B); (ii) sup IRm f(z)l(l _lzI 2)u+m-1 < 00; zEB

(iii) sup

aEB

IB IRmf(zW(1-lzI


00,

0


00,

2)p(u+m-1)(1-I
1


00,

mEN the

< 00,

where
k=l

> 0, f(z) E (3Q(D)

A holomorphic function f(z)

¢:::::}

00

lim sup laklnt-u

k-too

= I: Pnko(z) on B, k=l

>1

< 00.

Pnk being a homogeneous polynomial of

degree nk E N, is said to have Hadamard gaps if nk+t!nk ~ ,.\ > 1 for all k. In [16], a sufficient condition for a power series in B with Hadamard gaps belonging to a-Bloch spaces (3U(B) is given: 00 Lemma 2.3[16] Let f(z) = I: Pnk(z) be a power series on B with Hadamard gaps. k=l

Suppose that

= SUp{lPnk (01 : ~ E S}

IlPnk 1100 for all k ~ 1. Then

Lemma 2.4

f

E (3U(B) for 0

Let f(z)

~ n~-l

< a < 00. 00

= g(Zl) = I: akzf, i.e.

If f(z) E (jU(B), then g(Zl) E (3U(D).

k=O

the function depends only on one variable.

Proof It can be easily seen from the following relations:

sup l\7f(z)I(l-lzI 2)U = sup Ig'(Zl)I(1-lzI 2)U ~

zEB

zEB

sup

IZ 11< 1

Ig'(Zl)1(l-lzlI2)u.

Z2="'=Zn=O

In the folliowing, C denotes a positive constant which may be different from one occurrence to the next. The expression A ~ B means that there exists a positive constant C such that

C- 1A ~ B ~ CA. The unexplained notations can be found in [10].

3

Characterizations of Dirichlet Type Spaces Proposition 3.1

where

T

Let f(z)

= I: anz u E H(B). Then U::::o

< 2m and mEN.

Proof By integration in polar coordinates, we have

648

ACTA MATHEMATICA SCIENTIA

= 2n

1 1 1

r 2n- 1(1 - r 2)2m-r- 1dr

o

00

= L k=O 00

= L k=O

00

r 2n- 1(1 - r 2)2m-r-1(L o k=O

lal 2m

laal 2Wa(n

L lal=k

lal 2m

1

laal 2Wan B(n

L

jal 2 m

L

L lal=k

aarkCI2d(T(~)

laal2r2kwa)dr

lal=k

r n+ k- 1(1 - r)2m-r- 1dr)

+ k, 2m - T)

lal=k

00

~ L lal 2m L k=O

1

m

k=O

S

1

= 2n

11f: lal

Vol.24 SeLS

laal 2wa(n

lal=k

+ kr-2m ~

00

L L'l aal 2Wa(n k=O lal=k

+ kr·

Here B(-,·) is the classical Beta function. In the above, the orthogonality of {~a} on S and the Stirling's formula are used, and the proof is completed by the definition of Dr. Next we give a necessary and sufficient condition for a function on B, with Hadamard gaps, to belong to Dirichlet type spaces Dr. We regard it as a proposition, which will be used later. 00 Proposition 3.2 Let !(z) = L: P nk (z) be a power series on B with Hadamard gaps. k=l

Then

00

!(Z) E Dr

where t; = {nj : 2 k ::; nj

< 2 k+l ,

{::::::::?

L 2 k=O

kT

nj E N}, IlPnjll~ =

Proof Applying Lemma 4.1 of [5] to !(z)

when nj E

L IlPnj II~ njEh 00

00

k=l

4

j=l

+ n)T

~ 2kT

00

L (nj k=O nj Eh

+ nrllPnj II~ ~

L 2 k=O

kT

L IlPnj II~· nj et,

Dirichlet Type Spaces and a-Bloch Spaces

Theorem 4.1 Let T < 2 - 2a, 0 < a and best possible. Proof V! E fla, by (ii) of Lemma 2.1

<

00.

Then fla C Dr' and the inclusion is strict

sup IRm!(z)I(1_lzI 2)a+m-l ::; C

zEB

then

and noting that (nj

00

11!llb = L(nj + nrllPnj II~ = L T

Is IPnjWI2d(T(~).

= L: Pnk(z)

h, we have

< 00

l.: = llR l

!(zW(l - IzI 2)2m-r- 1dv(t)

m !(zW(1_lzI 2)2(a+m-l)(1

::; C

< 00,

- IzI 2)-2a-1'+ldv(z)

(1 - IzI 2)-2a-r+ldv(z) ::; C· B(n, -2a -

T

+ 2).

649

Li &. Ouyang: DIRICHLET TYPE SPACES AND Q-BLOCH SPACES IN UNIT BALL

No.4

Here we use the fact that -20: -

T

+ 2 > a when

T

<2-

20: and Beta function is meaningful.

By Proposition 3.1, j(z) EDT, so

Next we construct some functions to prove the strictness and the best possibility of them. Let

h (z)

= 91 (zt} =

f

k=1

2- k; zt. Since 2 - 20: kr

k(

T

> 0,

)

2-2a-T

lim sup lakln1-a = lim sup2- T2 I-a = lim sup2-2-

k-+oo

k

k-+oo

k

k-+oo

=

00,

by Lemma 2.2 91(ZI) (j. (3a(D). Thus by Lemma 2.4 we know h(z) (j. (3a(B). On the other hand,

By Proposition 3.2,

h (z) EDT'

This means the strictness of (3a eDT'

About its best possibility, it is sufficient to prove (3a Let 12 (z)

00

=L

k=1

Pn k (z)

00

=L

k=1

rt D 2 - 2a

by the monotonicity of D T •

2k(a-1)W2 k (z), where {W2 k (z)} is a sequence of Ryll- Wojtaszczyk

polynomials with Hadamard gaps satisfying

IIW2kli00 = 1 and IIW2klip

;:::

C(n,p) (cf[12]). Since

then 12(z) E (3a(B) by Lemma 2.3. On the other hand,

00

L

2 k(2-2a) 11P2 k II~

=

k=1

00

L

2 k(2-2a)2 k(2a-2)

00

=

:L IIW

By Proposition 3.2, h(z) (j. D 2 -

II~

II~

;::: C(n, 2) :L 1 =

00.

k=1

2a ·

Let 0: = 1, we have (31 = Bloch eDT, T < O. Moreover Blochc

which can be compared with the well-known fact Blochc Remark 4.2

k

00

2k

k=1

Remark 4.1

IIW2

k=1

n

O
n Dr,

T
L~.

In the above we find that the higher radial derivative does not play the

role as expected and the result is the same as m = 1. But we know when T ;::: 2, \fa < 0: < 00, D T by the best possibility of the inclusion relation and the monotonicity of Dr, i.e., the inclusion relation in this drrection is no longer valid. So it is natural to consider the converse inclusion relation.

(3a

rt

T'heorern 4.2 possible.

Let

T

< n + 2. Then D C T

(31+ n;T , and the inclusion is strict and best

650

ACTA MATHEMATICA SCIENTIA

Proof Vj(z) E Dr and a E B, for 00

>

h

~

IB \R

=

2:

Vol.24 Ser.B

< 2m,m E N, by Proposition 3.1, we have

T

lR mj(z)1 2 (1 -lzI 2)2m-r- 1dv(z) m

l»: l»:

here we have used

f(z)\2(1- \z\2)2m-r-l(1_ \'Pa(zWt+1dv(z)

j(zW(1_lzI 2)2(a+m-l)(1 -lzI2)2+n-r-2a(1-I'Pa(zW)n+ld'\(z)

(4.1)

j(zW(l _lzI 2)2(a+m-l)(1 -1'Pa(zW)n+ld'\(z)

< n + 2 and

2: 1 +

2r.

Taking sup at the end of (4.1), this is the case aEB. in Lemma 2.1 (iii) for p = 2, q = ~. And by the monotonicity of (3a, we get T

0:

n

j(z) E (31+ n;~

.

Thus Next we prove its strictness. Let 00

00

h(z) = LPnk(z) = L2kn;~W2k(Z). k=l

Since IlFnkll oo By Lemma 2.3,

k=l

= SUp{lFnk(~)I: ~ E S} = 2k(1+n;~-1)IIW2kII00:S

13 (z)

2k(1+n;~-1).

E (31+ n;~ .

On the other hand, 00

00

L 2kr L k=l

njEh

IlFnj II~ = L 2kT2 k(n-r) IIW2k II~ k=l 00

00

= L2knllW2kll~ k=l

2: C(n,2) L2 kn = 00. k=l

By Proposition 3.2, h(z) ~ Dr. About the best possibility we only need prove

Dr

rt. (3a,

for

0:

n-T

< 1 + -2- and 00

14 (z) = 92 (zt} = L k=3 k > max] n2r ,O} when 0: < 1 + n;r. Since

by the monotonicity of (3a. Let suppose

0:

2N-l

'" k1-

k~

1 k1+z:..r. (in k)s

a .

k;

2N

=

>N -

-

1

1

T


1 +--r-(lnk)S

zt, s > ~. It is no matter to

1 ka+z:..r. (in k)s 1

> (2N)1+T- a

(2N)a+ ~;n (in 2N)s -

2(in 2N)s

Li & Ouyang: DIRICHLET TYPE SPACES AND Q-BLOCH SPACES IN UNIT BALL

No.4

651

When N -+ 00, the last term tends to infinity. By Remark 4 of Theorem 3.1 of [2], 92(Zl) (3°(D). Then by Lemma 2.4,

f/.

In the meantime,

Thus h(z) E D», Remark 4.3 The condition T < n + 2 is to make 1 + n 2r > 0 and so (30 meaningful. We can see that D n+2 - 20 C (30 is the alternative form of Dr C (31+ n2~. Combining Theorem 4.1 and Theorem 4.2, we get D n +2 - 20 C (30 C <2-20)

o.,

for 0

< a < 00, which is a conclusion similar to Theorem of [16].

Remark 4.4 From Theorem 4.1 and Theorem 4.2, we can conclude that for -00 < T < 2, when 1 - ~ ::; a < 1 + n 2r, (30 and Dr are not included in each other. This is similar to (ii) of Theorem 3.2 of [2].

5

Dirichlet Type Spaces and BMOA

In Dirichlet type spaces of one variable, Dirichlet space D 1 = AD is a very small function space since we know AD C Qp,o from Theorem 1 of [1]. In the case of several variables,

n

O
.

the corresponding Dirichlet space D 1 ItBloch from Theorem 4.2, which shows that D 1 is a much larger space. Let T = n in Theorem 4.2, we get D n cBloch. Further, we can prove

o; CBMOA. Lemma

5.1[6]

for any T E R. (ii) When

(i) For fixed z E B, the series J(z) =

L: (Ial +n)-r w;;1lzol2 is convergent a~O

T

> n, J(z) = 0(1) and T

< n,

T=n.

Lemma

5.2[6]

Let k, j be nonnegative integers, a multi-index and [o] = k, then

L

i3+z.'=a lill=j

Wa w/3w v

~

(j

+ n)n-l (k - j + n)n-l (k + n)n-l

652

ACTA MATHEMATICA SCIENTIA

Vol.24 Ser.B

Here j3 and v are multi-indexes. Theorem 5.1 D n CBMOA, and the inclusion is strict and best possible with respect to Proof Let f(z)

=L

"'2:0

a",z'" E D n , then from the definition of D n we have

IIfl11

n

= ~)Iod "'2:0

+ n)n w",la",1 2 < 00.

In order to prove the theorem, from [3] it is sufficient to prove sup

aEB

2

;; B

IRf(z)1 (1 -Iz

12 )

2 (1 - lal )n 2 11 _ (z,a )1 2 n dv(z) < CllfllD

In fact,

By applying Schwarz's inequality, the last term of the above

n '

No.4

653

Li &. Ouyang: DIRICHLET TYPE SPACES AND a-BLOCH SPACES IN UNIT BALL

(k-j)! . v!

~

2(n+k-j) v2 " pr } la 1 ] LJ LJ v!(n + j)nW(3(n + k _ j)n-l(k _ j)! . j=O

(5.3)

1131=,

f3+v=o:

When 10:1 = k, we have k

(n

+ k)2 f;:o I~j v!(n + j)n w(3(n + k - j)n-l(k - j)!

Wet

.8+v=o

k

=L L j=O

""

1131=j .8+..,=0:

pr(n + k - j)(n - 1)! W(3W v (n + k)2(n + j)n(n + k - j)n-l(k - j)! Wet

~ pf(n + k - j) " ~ LJ (n + j)n(n + k - j)n-l(k - j)!(n + k)2 LJ

1131=,

j=O

Wet

wr.>w v ·

.B+v=o:

I-'

By Lemma 5.2, the above becomes

pr(n + k _ j) (j + n)n-l(k _ j + n)n-l j)n-l(k - j)!(n + k)2 (k + n)n-l

k

~ ~ (n + j)n(n + k -

k j2(n + k _ j)n-l ~ "LJ (n + k)2(n + j)n(n + k j=O

-

k

L

j=O

sv::

j2(n + k < (n+k)n+l(n+j) -

i: j=O

ir:'

(j + n)n-l(n + k _ j)n-l

(k + 1)k (n+k)2 - (n+k)2 j

(5.4)

(k + n)n-l

<

<

1

.

Combining (5.3) and (5.4), we get

r

2 2 (1 - lal 2)n (1-lzl )11_(z,a)1 2 n dv(z)

JB1R!(z)1

s; e(l - lal')" 2

:S C(l -laI )n

t, I~k (t "~o

(f L

(n + j)"W alaal'( n +k - j)"-1 (k

(ri + k)nWetlaetI2) (f(n

k=O letl=k

k=O

+ k)n-l

~!

j)l la"I')

L =: laet l2)

letl=k

By Lemma 5.1 and the definition of D n , the above

Taking sup, this means !(z) EBMOA. Thus aEB

D«

C BMOA.

Next we construct some functions to prove the strictness. Let !5(Z) From Proposition 4 of [3] we know

!5(Z) E BMOA(B) {::::::} 93(Zt} E Bloch(D). Since

k

= 93(Zl) = L: Z[ 00

k=O

.

654

ACTA MATHEMATICA SCI ENTIA

then by Proposition 3.2, f5(Z) f/. D n . In the meantime, lim sup lakl = 1 < k-+oo

00,

Vol.24 Ser.B

by Lemma 2.2, 93(zd E Bloch(D). Thus

f5(Z) E BMOA(B). By the best possibility of D n CBloch and the monotonicity of Dr' we know Dr ~BMOA for T < n, which means the above inclusion relation is best possible with respect to Dr. 2n Remark 5.1 In Lemma 2.3 of [7], Dr C H~(B) for 0 < T < n. The authors especially pointed out that D n ~ Hoo(B), i.e., the inclusion relation is invalid for T = n. So they conjecture D n CBMOA. Here we give a proof of the conjecture. Remark 5.2 It is well known that BMOAc H 2 = Do. From the above theorem it is natural to ask whether BMOA is included in Dr for 0 < T < n. By Theorem 2.3 of [5] we know Qp C D n ( l - p) for n~l < p ::; 1 and this inclusion is strict and best possible. So it means that BMOA= Q1 ~ Dr(O < T). Summarizing all we discussed, we know that there is no inclusion relation between BMOA and Dr for 0 < T < n. Acknowledgement The authors would like to express their thanks to the referee for suggesting a brief proof of Proposition 3.2. References 1 Aulaskari R, Csordas G. Besov spaces and the Qq.o classes. Acta Sci Math, 1995, 60: 31-48 2 Aulaskari R, Lappan P, Xiao J, Zhao R H. On a-Bloch spaces and multipliers of Dirichlet spaces. J Math Anal Appl, 1997, 209: 103-121 3 Choa J S. Some properties of analytic functions on the unit ball with Hadamard gaps. Complex Variables, 1996, 20: 277-285 4 Cochran W G. Shapiro J H, Ullrich D C. Random Dirichlet functions: multipliers and smoothness. Canad J Math, 1993, 45(2): 255-268 Analysis, 2001, 5 Feng S J. On Dirichlet type spaces, a-Bloch spaces and Qp spaces on the unit ball of 21: 41-52 6 Hu P Y, Shi J H. Multipliers of Dirichlet type spaces. Acta Math Sinica (English Series), 2001,17: 263-272 Chin Ann of Math, 1999, 7 Hu P Y, Shi J H. Random Dirichlet type functions on the unit ball of 20B(3): 369-376 8 Jevtic M, Pavlovic M. LP-behavior of power series with positive coefficients and Hardy spaces. Proc Arner Math Soc, 1983, 87: 309-316 9 Kerman R, Sawyer E. Carleson measures and multipliers of Dirichlet-type spaces. Trans Amer Math Soc, 1988,309(1): 87-98 10 Li B, Ouyang C H. Higher radial derivative of Bloch type functions. Acta Math Sci, 2002, 22B(3): 433-445 11 Rudin W. Function theory in the unit ball of New York: Springer-Verlag, 1980 12 Ryll J, Wojtaszczyk P. On homogeneous polynomials on a complex ball. Trans Arner Math Soc, 1983, 276: 107-116 13 Stegenga D A. Multipliers of the Dirichlet spaces. Illinois J Math, 1980, 24(1): 113-139 14 Taylor G D. Multipliers on Do. Trans Arner Math Soc, 1966, 123: 229-240 15 Yamashita S. Gap series and a-Bloch functions. Yokohama Math J, 1980, 28: 31-36 16 Yang W S, Ouyang C H. Exact location of a-Bloch spaces in L~ and HP of a complex unit ball. Rocky Mountain J Math, 2000, 30: 1151-1169

en.

en.

en.