457
Internat. J. Math. & Math. Sci.
VOL. II NO. 3 (1988) 457-464
ON THE DUAL SPACE OF A WEIGHTED BERGMAN SPACE
n
ON THE UNIT BALL OF C
J.S. CHOA and H.O. KIM
Department of Applied Mathematics
KAIST
P.O. Box 150, Cheongryang
Seoul, KOREA
(Received January 30, 1987 and in revised form March 24, 1987)
ABSTRACT.
B
on the unit ball
C
of
n
A(Bn)(0
Berman space
The weighted
n
forms an F-space.
< p < I), of the holomorphic functions
We find the dual space of
A(Bn)
by determining its lckey topology.
KEY WORDS AND PHRASES.
Hardy space, Bergman space, Mackey topology.
30D55, 32A35.
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
INTRODUCTION.
n,
be the normalized Lebesgue measure and o
be the unit ball of C
Let B
n
with
be the rotation invariant positive Borel measure on S, the boundary of B
I.
o(S)
AP(Bn)(0a
The weighted Bergman space
B
tions holomorphic in
SUPOr<l MP(r;f)
p
If(r)IPdo().
HP(Bn)
MP(r;f)
where
p
in fact, the Hardy space
<
-I) consists of all func-
’
MP(r;f)(l-rd()2nr2n-ldr
P
0
P
< p <
for which
<
,
=,
-I,
-I,
Note that the weighted Bergman space
AP(B
a n)
is
-i (See [I]).
if
AP(Bn).
by determining the Mackey topology of
n
>
if
The purpose of this paper is to compute the dual space
case
if
(A(Bn))*
for
0 < p
The corresponding problems for the
are settled by Duren, Romberg and Shields [2], Shapiro [3] and Ahem [4].
Our computations are very similar to those of them.
Throughout this work,
a,
that the ratio
2.
Ca,g,.
denotes a positive constant depending only on
which may vary in the various places, and the notation
a(z)/b(z)
has a positive finite limit as
SOME PRELIMINARY RESULTS.
LEMMA 2.1.
If(z)
If
f
C
C
AP(Bn)(0a
n,p,
< p <
llfll p, (I
’
Izl)
-I), then
n+=+l
P
Izl
a(z)
I.
b(z)
means
J.S. CHOA and H.O. KIM
458
PROOF.
.
is proved in [I, Thm 7.2.5].
-I
The case
for
zl
-<_
For this range of
>
C
=>
C
C
C
since
is bounded
r)2nr2n-ldrd()
f fS If(r)IP(l r)ado()dr
MP(0;f)
p
n,p
f
MP(0;f)(I
p
n,p,a
a =-I
r)dr
(I
0)
a+l.
and the above result, we get
n
p
M (p;f)(ln,p p
Izl)
n
+i
C
If(z)
n,p p
By the result of the case
If(pz) =<
<
we have:
p
If(r) Ip(1
of fs
IifliP’P
Izl
--<
a > -I, it is enough to prove the result for
For the proof of the case
n,p,a
llfll p,
P
)
(i
p
Izl)
(I
Consequently we have
If(z)
If(drdr)l
r)
(z
a+l
C
(I
n,p,a llfll p,a
P
r)
n
(I
p
r)
n+a+l
-<COROLLARY 2.2.
C
llfll p,
n,p,
(a)
(I
P
r)
The convergence of
AP(B
n)a
P
llf gllp,
a
(0<p< I)
If
(P
with its invariant metric
d(f ,g)
gll p,a
>
I)
implies the uniform convergence on any compact subset of
(b)
AP(B
n)a
PROOF.
is an F-space if
0 < p <
B
n
and a Banach space if
(a) follows immediately from Lena 2.1.
p
I.
The proof of (b) is routine and
is omitted.
COROLLARY 2.3.
AP(B
n)a
C
A(B
n)D
if
0 < p < q
and
n+a+l
n+B+l
P
q
In particular,
APa (Bn) C Alo (Bn)
PROOF.
where
First, we prove the case
inequality of the following.
o
n+a+l
a > -I.
(n+ I).
We use Lemma 2.1 in the first
n
DUAL SPACE OF A WEIGHTED BRRGMAN SPACE ON I UNIT BALL OF C
f fS If(r)lq(1 r)B2nr2n-ldrdo()
q
0
f Sf if(r){Pcn,p,a,q
--<
0
(1
P
r)
I
q-p
r)f2nr2n-ldrdo(r.)
(1
n,p,a,q
C
I
-n+a+-----l
459
r)a2nr 2n-1 drdo()
If(r) IP(1
p,a
S
0
"[[f[[q
n,p,a,q
This completes the proof of the case
a > -I.
The remaining case is essentially a
result of Hardy and Littlewood, but we give a proof using Ahern’s technique in [5].
Let
f
C
HP(Bn).
By Lemma 2.1, we have
n
p
n,p(1 -[z[)
If(z)[
K
[[f[[p
Set
Mf()
[f(r)[
sup
0<r<l
Then we have
f f(r)[(I
r)
(- 1)n-12nr2n_ld r
(2.1)
0
[[fl[P 0f
If
Mf()
If
S f()
<
IIll p’
Kn Ilfll
p’
,p
K
n ,p
A)-n
(1-
n,p [[fl[ p
-<K
r)-n-ldr+Cn,pMf() Xf
(1
n
+C
n,p
.
by setting
Mf()
(1-
( )n(1
r)
dr
1_ 1)n
,)(
(2.2)
(- l)n
(2.2), (2.1) is dominated by
0
in
1-p
Mf(r.) p
C
n’p
by setting
in (2.2), (2.1) is dominated by
C
n,p
p
lifl[1-PMf(r.)
p
Hence, for any
(2.1)
C
S,
< C
n,p
Iil p
+ C n ,p
Integrating (2.3) with respect to
theorem [I, Thm. 5.6.5], we obtain
do(iO
over
S
(2.3)
and using the complex maximal
460
J.S. CHOA and H.O. KTM
2nr2n’Idrdo ()
1-p
+ Cn
[]ll p
c n,p lifll p
AP(Bn).
THE HACKEY TOPOLOGY OF
In this section, we will show that the Mackey topology of AP(B
n) is the
n_+l
restriction of the topology of
where o
(n+l).
p
First we give necessary definitions.
DEFINITION 3.1. The Hackey topology of a non-locally convex topological vector
space (X,x) is the unique locally convex topology m on X satisfying the follow3.
AI(Bn),
ing conditions:
(I)
(2)
is weaker than x,
m
the
x-closure of the absolutely convex hull of each x-neighborhood of the
origin contains an m-neighborhood of the origin (See [6, Thm I]).
DEFINITION 3.2. For B > -n and z, w C Bn, we define
1-<z, w>) B+n+
n
and
S,o(w)(,)
(
[wl2)’K(-,w).
The following proposition is useful in the sequel:
PROPOSITION 3.3. [I, p. 120] If B > -n, then K$(g,w) is a reproducing kernel
for the holomorphic functions in
In other words, if f is
holomorphic on Bn and integrable with respect to the measure (I
then
LI{(I-lwl2)dv(w)}.
lwl2)$dv(w),
LdA 3.4. [1, Prop. 1.4.10]
LEI4A 3.5. [7, Letma6]
For
I 0
gC B and c
n
and
r,0
f <x r)a(1 or)ldr s Ca,i(1
a
real, we define
i +
0, then
0)
0
for some positive constant
.
The next lemma is an easy application of the above two lemmas.
n_++1
LEMMA 3.6. Let 0 p
and fix
(n + I) m
Then
P
PROOF.
We only prove the case
-I.
Let w C Bn, and 0
r
1.
n
DUAL SPACE OF A WEIGHTED BERGMAN SPACE ON THE UNIT BALL OF C
461
Then we have, by Lemma 3.4 and 3.5,
p.
dr
C
n,p,o,B (I-
2 +n+l-(8+n+l)p
(8-)p (I-
[w[ 2
=< c n,p,o,8
Thus we have
sup{
P
il Js,o(w) lip,=
wCBn <.
The proofs of the following theorems are essentially the same as those of [3]
(Prop 4.4 and Prof. 45) and are omitted
n,+l
and 8 >
THEOREM 37 Let 0 < p <
P
C
n,p,,B
points in
<
f
such that for each
Bn
and a sequence
(lj)
AI(Bn
C
(n+l)
E
.
Then there exists
there exist a sequence
(wj)
of the
of the complex numbers such that
n,p,,8
and
[" jJ8 o(wj )’
f
(3.2)
j
where the last series converges in
THEOREM 3.8.
of
4.
-(Bn)
AloTHE DUAL
where
Al(Bn
)’o
The Mackey topology of
n++l
o
(n+l).
AP(B
n)_
is the restriction of the topology
P
AP(Bn).s
SPACE OF
Finally, we will find the dual space of
AP(Bn).
For the proof of this main
result, the following definition is needed:
(Radial fractional derivatives of holomorphic functions in
DEFINITION 4
Let
g(z)
k= 0 Gk(Z)
be the homogeneous expansion of
the radial fractional derivative of
Rqg(z)
k=0
(k+l)qGk(Z).
Let
f(z)
[ Fk(Z)
k=O
of order
g
.
k=0
[
lyl=k
c(y)z
Y,
q
g.
For any real number
is defined by
B
n
q,
J.S. CHOA and H.O. KIM
62
and
be the homogeneous expansions of
0
=<
0 <
f
We note that for
g, respectively
and
q > O,
I, we have
[
?
k--OIYl--k
2q
F (q)
(log)
0
2n(k+l)
(k+n) q
(n- l)!k!
(n- I+
c(y)d(y)
q-I
q
O
k
f R-a f (r) Rq+ag (r0 ) 2nr2n-ldrdo ().
(4.1)
S
We can now prove the duality relation.
We use the idea of Ahern [4] in the proof
of the following.
0 < p <
Let
THEOREM 4.2.
(A(Bn))*
f
"
H(Bn)
By Theorem 3 8
0.
simplicity we assume o
and let
f
Take
P
Then
Izl) IR+2f(z)
sup(l
g
(n + I).
llfll^
It suffices to compute
<
(R)}
(A) *
For
such that
n
Then by (4.1)
be a polynomial.
I
k=O]y!=k
=2
n4x+l
o
(A)* (A)*
PROOF.
zB
and
V
c(y)d(Y)
(n- I) !k!
(n- I+
f Sf
-1
R f (r
R2g
rp g
2n(k + I) q
(k + n) q
)2nr 2n-
k
drdo
0
(log
7
f(r)R2g(rp)2nr2n-ldrd(e)"
Hence
lim
p/l
y
[
k=OIy
(n 1) !k!
c()d(’)’) (n+k)
T =k
log
f(r)l
r
Since
log
l-r
c k,= II f II
A II g II
lim
0 "+1
.
Ia2z(==)l 2nr2n-ldrdo()
zBn
^
Since polynomials are dense in
,(f)
(l-r)
sup
ok)
I, (4.2) is dominated by
r
as
2n(k + 1) q
(k+n) q
k=OIyl=k
AO, the mapping
c(Y)d(y)
(n- 1)!k! 2n(k+l) q
(n- + k)
(k+n)q
O
k
(4.2)
DUAL SPACE OF A WEIGHTED BERGt SPACE ON THE UNIT BALL OF Cn
AO.
extends to be a bounded linear functional on
01(2
A
Ll(2nr2n-ldrdo()),
G
L(R)(2nr2n-ldrdo())
in
f
(f)
for each
Ll(2nr2n-ldrdo())
on the space
functional
f
A
in
But
Since
extends to be a bounded linear
since
(LI) *
L
there exists
such that
f(r)G(r)2nr2n-ldrdo()
0
S
O.
Let
0
S
G(w)
H(z)
(l-<z w
be the holomorphic projection of
(f)
C (A)
Conversely, let
by the Hahn-Banach theorem
463
>)n+l
G.
2no
If
2n-
Idodo(n)
(w
On)
is a holomorphic polynomial, then
f
f Sf
f(r)G(r5)2nr 2n-I drdo()
f Sf
f (r) H
0
(rK)2nr2n-ldrdo (K)
0
f sfr
f (r) RH(r)
2nr2n-ldrdo ()
0
f f
0 S
where
g
R
-1
R-IH.
is defined to be
sup
f(r)R2g(r)2nr2n-ldrdo(),
Izl)lR1H(z)l
(I-
The proof will be complete if we can show that
.
<
n
Since
DH(r)
.[ .[
Dr
0
S
(n+l)
r
< r, 0n >G(on)
(I- <r, On >) n+2
2nD2n-I dDdo(),
(z
r)
we have
do(n)
r,o>l n+2
IIGII
=< c
RIH(r)
sup
zgB
n
(I
DH(r)
--r.
do
(1 -or)
(4.3)
l-r"
n
By (4.3) and
f
0
Dr
I-I)IRIH(z)I
+H(r)
<
do
we have
J.S. CHOA and H.O. KIM
464
REFERENCES
I.
2.
n
Sprlnger-Verlag, New York, 1980
Theory in the Unit Ball of C
p
DUREN, P.L. ROMBERG, W. and SHIELDS, A.L. Linear Functlonals on H space with
0 < p < I, J. Reine Angew. Math. 38 (1969), 32-60.
RUDIN, W.
Function
Some F-spaces of Harmonic Functions for which Orllcz-Pettls Theorem
Fails, Proc. London Math. Soc. 50 (1985), 299-313.
3.
SHAPIRO, J.H.
4.
AHERN, P.R. and
5.
KIM, H.O. Derivatives of Blaschke Products, Pacific J. of Math. 114 (1984),
175-190.
6.
SHAPIRO, J.H. Mackey Topologies, reproducing Kernels and Diagonal Maps on the
Hardy and Bergman Spaces, Duke Math. J. 43 (1976), 187-202.
7.
SHIELDS, A.L. and WILLIAMS, D.L. Bounded Projections, Duality, and Multipliers
in Space of Analytic Functions, Trans. Amer. Math. Soc. 162 (1971),
287-302.
JEVTI,
M. Duality and Multipliers for Mixed Normed Spaces,
Michigan Math. J. 30 (1983), 53-64.