41
IMHOTEP Vol.1, n°1 (1997)
THE DUALS OF BERGMAN SPACES IN SIEGEL DOMAINS OF TYPE II
Anatole TEMGOUA KAGOU
Abstract : In every homogeneous Siegel domain of type II , for some real number p0 > 2,
we characterize the dual of the weighted Bergman space A p ,r when 1  p < p0 . In the
symmetric case, we also characterize the dual of A
p,r
with p1 < p < 1 for some p1 (o,1),
and extend this to two homogeneous non symmetric Siegel domains of type II.
Résumé : Dans les domaines de Siegel homogènes de type II, nous caractérisons le dual de
l'espace de Bergman avec poids A p ,r , lorsque 1  p < p0 , où le nombre réel p0 dépend du
domaine. Dans le cas où le domaine est symétrique, nous caractérisons également le dual
de A p ,r lorsque p1 < p < 1 , avec p1 (o,1) , et nous généralisons ce résultat à deux
domaines de Siegel homogènes et non symétriques de type II.
I. INTRODUCTION.
Let D be a homogeneous Siegel domain of type II. Let d denote the Lebesgue
measure on D and let H(D) be the space of holomorphic functions in D endowed with the
topology of uniform convergence on compact subsets of D. The Bergman projection P of
D is the orthogonal projection of
2
2
L (D, d) onto its subspace A (D) consisting of
holomorphic functions. Moreover, P is the integral operator defined on L 2(D, d) by the
Bergman kernel B(, z) which, for D, was computed in [G].
_________
1991 AMS Subject Classification : 32A07, 32A25, 32H10, 32M15.
Keywords : Siegel domain of type II, weighted Bergman space, Bergman kernel,
Bergman projection, Bloch space.
The duals of Bergman spaces
Let r be a real number. Since D is homogeneous, the function   B( , ) does
not vanish on D, we can set:


Lp, r (D)  Lp D, Br (,  )d( ) , 0 < p <  .
Let p be a positive number. The weighted Bergman space A p ,r (D) is defined by
A p ,r (D) = Lp, r ( D)  H( D) .
If r = 0, then A p ,r (D) is simply denoted A p (D).
The weighted Bergman projection
P
is the orthogonal projection of
L2, ( D) onto A 2, ( D) . It is proved in [BT] that there exists a real number  D  0 such
that A 2, ( D) = {0} if    D and that for    D , P is the integral operator defined on
1 
(, z) . In all our work, we shall assume
L2, ( D) by the weighted Bergman kernel c B
that    D .
The "norm" .
p ,r
of A p ,r (D), with r   D , is defined by:
1/ p


p,r
p r


f

f (z) B (z, z)d(z)
, f A (D).
p, r  

D

Let  be a positive integer. S.G. Gindikin [G]
has defined a differential
polynomial   in D that satisfies the property:
   B(, z)  c B1   (, z)

  
( , z  D) .
A holomorphic function g in D is said to be a Bloch function in D if g satisfies the estimate:


g *  sup 
 (   ) z g ( z) B ( z, z)    .

zD


Let N = g  H(D) : g  0 . The Bloch space B  of D is defined by :
42
A. Temgoua Kagou
B  ={Bloch functions}/N.
p
For p  1, the space C ,r (D) is the quotient space by N of the space of holomorphic
functions g in D satisfying the estimate
1/ p
p





r


g p

B (z, z)g(z) B (z, z)d(z)
 .
C, r (D)  

D

p
p
For r = 0, the space C , r (D) is simply denoted C (D).
+
In the upper half-plane,  = {z C : Im z > 0}, R. Coifman and R. Rochberg
1 +
proved the following fact: the dual of the Bergman space A ( ) coincides with the Bloch
+
space of holomorphic functions in  , and can be realized as the Bergman projection of

+
L ( ). A few years later, D. Békollé in [B4] carried out the same study on symmetric
Siegel domains of type II. In fact, he proved that for homogeneous Siegel domains of type
1
II associated with self-dual cones, the dual space of A coincides with the Bloch space of
holomorphic functions, and for symmetric Siegel domains, this space can be realized as the
Bergman projection of L. On the other hand, for bounded symmetric domains, K. Zhu ([Z]
p
and [Z1]) studied the dual of the Bergman spaces A with small exponents (0 < p < 1) and
obtained that their dual is equal to the Bloch space.
Furthermore, in [B6] , D. Békollé proved that when p  (4/3 , 4) , the dual of
A p (R3 + i) , where  is the spherical cone in R3, is equal to A p' (R3 + i) ( p' is the
3
conjugate exponent of p ) and when p  (1 , 4/3] , the dual of A p (R + i) coincides with
p'
the space C  (R3 + i) with  = 1.
43
The duals of Bergman spaces
The purpose of this work is to extend these results to the weighted Bergman space
A p ,r (D) , 1  p < p0, in homogeneous Siegel domains of type II, where p 0 is a real
number greater than 2 and depends on the domain D ; when D is symmetric, we also show
that the dual of A p ,r (D) , p1 < p < 1 , where p1 depends on D , is equal the Bloch space.
This latter result is also true for two non symmetric domains.
The first aim of this work is to show that there exists 0 > 0 such that whenever
 > 0 ,
1° when D is homogeneous and 0 < p  1 , B  is isomorphic to a subspace of
the dual space ( A p ,r (D))* of A p ,r (D) and the two spaces are isomorphic when p = 1 ;
2°
the two spaces are equal when p1 < p < 1 when D is symmetric, and the
same is true for two particular non symmetric domains.
To show the first claim , let g be in B  and consider the linear functional n
A p ,r (D) defined by
1 r
1
p (z, z)d(z)


(f )  D (  )g (z)B (z, z)f (z)B
(f A p ,r (D)) .
1 1 r
p (z, z)d(z)   for all f  A p ,r (D), it follows that is bounded,
Since D f (z) B
hence belongs to ( A p ,r (D))* and is represented by g .
1,r
Conversely, assume p = 1 . To obtain B  = (A (D))*, let 

then there exists b  L (D) such that
(f )  D b(z)f (z)Br (z, z)d(z) , f A (D) .
1,r
Since f (z)  cr,  D B1  r   (z, )f ()Br   (, )d() , if we set
44
(A1,r(D))*;
A. Temgoua Kagou
g(z)  cr,  D B1  r   (z, )b()Br (, )d() ,
we easily get :
(f )  D g(z)f (z)Br   (z, z)d(z) .
In fact, since > , g is a holomorphic function on D satisfying the estimate


sup g( z) B ( z, z)  c b  .
z D
~
~
On the other hand , by a lemma of Trèves [Tr], there exists h  H( D) such that  h  g.
Let h be the equivalence class of all holomorphic solutions of this equation. Then h belongs
to B  , and the equality
(f )  D h(z)f (z)Br   (z, z)d(z)
1,r
yields the equality (A (D))*= B  .
We next assume that 0 < p < 1. Let us now define two homogeneous Siegel
domains D0 and D1. Set

 11 12



V0 =    12  22


0
 13



2
2
13 


12 13

0  : 11 

 0 ,  22  0 ,  33  0 .


22
33

 33 


5
Observe that V0 is a non self dual cone of rank 3. Define D 0 by D0 = R +i V0 . Then D0 is
a homogeneous,non symmetric, tubular domain. On the other hand, D 1 is the first example,
due to Piateckii- Chapiro , of a homogeneous non symmetric Siegel domain of type II, and



is defined as follows. Let V1 =   11 , 12 ,  22   R 3 :  22  0, 11 



45
2

12


 0 be the
 22



The duals of Bergman spaces
5
spherical cone in R , and consider the V1 - Hermitian form F1 in C defined by
F1: C  C  C 3


( u, v)  uv,0,0 .


zz


Then D1 = ( z, u)  C3  C :
 F( u, u)  V1 .
2
i




The restrictions on D, p and  are as follows : D is either a symmetric Siegel
domain of type II, or D = D0, or D = D1 . Then there exists p1  (0,1) such that for all p 
p, r
( p1 ,1) , we have (A
element of
(D))*
 B  . Here are the main steps of the proof. Let be an
(Ap, r(D))* and set p + r+1 , with large enough. Let f  Ap, r(D).
p
Then, in view of the molecular decomposition theorem [BT 1] , there exists {i } l such
that

1 r  
f (z)    i B p ( z, zi ) B p ( zi , zi ) ,
i
where {zi} is a lattice in D. Then we get


1
p (z, z)d(z).
 ( f )  D  ( B p (., z)f (z)B
46
A. Temgoua Kagou
 



Now , sin ce  B p (., z)  is holomorphic, then by the same lemma of Trèves, there exists g






 



p
 B such that   g (z)   B (., z)  by the choice of . Hence  is represented by a






Bloch function.
To go further, let h  L ( D) . Take 
large enough ; then the

function

z  G(z)  D B1 (z, )h()d() satisfies the estimate sup G ( z) B ( z, z)  c h 
zD
and G is holomorphic in D. Hence, by the same lemma of Trèves, there exists a function g
B  such that
(1)
(   )g  G .
Now, let P be the operator from L ( D) into B  which to each h  L ( D) assigns the

element g = Ph defined by (1). This operator P is called "Bergman projection" of L ( D)
into B  for the following reason : although P is not the integral operator P which is
associated with the Bergman kernel B( , z) , which has no meaning on L ( D) , it is easy
to show that for h  L2  L ( D) , the element Ph of B  can be represented by Ph .
The second aim of this paper is to show that the "Bergman projection" P is
bounded from L ( D) onto B  . In order to achieve this goal, we prove that for each real
number  sufficiently large and for each GH(D) such that
47
The duals of Bergman spaces


sup G ( z) B ( z, z)   ,
zD
one has the reproducing formula :
G()  c D B1 (, z)G( z) B ( z, z)dv( z)
(zD).
Hence, for each g  B  , if we set h( z)  (   )g( z) B ( z, z) , then h L ( D) and Ph
= g.
The third aim of this work is to determine on a particular Siegel domain D2 of type

II, a kernel K( , z) that determines P in the following way : for each hL (D2 ) , a
representative
of
the
element
Ph
of
B
is
given
by
the
function
  D K (, z)h (z)d(z) . D. Békollé has determined such a kernel K in three different
Siegel domains of type II, namely, the Cayley transform of the unit ball in C
n
[B1], the
tube over the spherical cone ([B2] and [B3] ), and finally, the tube over the cone of
symmetric positive-definite matrices [B5] . The domain we consider is


z  z*


D 2  (z, u )  M 2  M r,2 :
 u *u  V ,
2
i




where M2 (resp. Mr,2 is the set of complex matrices of order 2 (resp. with r lines and 2
rows) , and V the cone of Hermitian positive-definite matrices of order 2. We determine a
kernel B0 such that
(2)
(B  B0 )(( , v), (z, u ))  L1(D 2 , d(z, u ))
(3)
  B0 ((, v),(z, u))  0 .
Thus Ph can be represented by :
48
(( , v)  D2 );
A. Temgoua Kagou
P h(, v) = D (B- B0 )((, v),( z, u)) h( z, u)d( z, u)
2

(h L (D2 )).
Unfortunately, for Siegel domains of type II associated with cones of rank greater than 2,
the determination of a kernel B0 such that (2) and (3) simultaneously hold seems out of
reach.
The fourth aim of this work is to prove that there exists p 0  (2,) such that
whenever p'0 < p < p0 (p'0 is the conjugate exponent of p0 ) , the dual of A p ,r (D) is
p'
equal to A p ',r (D) , and when 1 < p < p0 , the dual of A p ,r (D) is equal to C,r (D) with
 > 0 .
The plan of this work is as follows. In section II, we recall some preliminary
results about affine-homogeneous Siegel domains of type II and we give precise statements
of our results. In section III, we prove that B  is contained in (A p , r (D))* when 0 < p
 1, and that B  = (A1, r ( D))* (Theorem II.7) ; under some additional assumptions on D,
p and , we prove that B  = (A p , r (D))* (Theorem II.8). In section IV, we show that
P L ( D) = B  (Theorem II.9). In section V, we determine a defining kernel B-B0 of a
representative of the Bergman projection of a bounded function in the particular domain
D0 (Theorem II.10). In section VI, we prove that (A p , r (D))* = Ap' , r (D) if p'0 < p < p0,
and (A p , r (D))* = C,r (D) if 1 < p < p0 (Theorem II.11).
p'
For p = 1 and r = 0, the above results were first presented in [T]. In the sequel, as
usual, the same letter c will denote constants that may be different from each other.
49
The duals of Bergman spaces
ACKNOWLEGMENT: We would like to thank D. Békollé and A. Bonami for
helpful conversations.
II. STATEMENTS OF RESULTS.
n
Let V  R , n  3, be an irreducible, open, convex homogeneous cone which
contains no straight line. We first recall the canonical decomposition of V as stated in [G].
NOTATIONS. (i) At the j th step, j =1,2,..., the real line will be denoted by Rjj ; at the k th
step, k = 1,2,..., Rk will stand for the nk- dimensional Euclidean space.
(ii) Let   R

be a convex homogeneous cone which contains no straight line


and let  be a homogeneous  - bilinear symmetric form defined on R  R . The real
homogeneous Siegel domain P = P (, ) is defined by
P = P(,) = {(y,t)  R  R : y - (t,t) }.
We shall denote by V(P) the homogeneous cone defined by:
V(P) = {(y,t,r)  R  R  R : r > 0 , (ry,t)  P }.
In order to describe the canonical decomposition of the cone V, we consider at the first
(1)
= (0,)  R11. At the second step, we associate with V
(1)
- bilinear symmetric form 
step, the cone V
homogeneous V
(2)
(1)
and with a
defined on R2 , the real Siegel domain
P(2) = P  V (1) ,  (2)   R11  R2 and then, the convex cone
V ( 2)  V( P (2) )  R11  R 2  R 22 .
At the k th step, we associate with the cone V
(k)
form 
(k-1)
and with a V
(k-1)
- bilinear symmetric
defined on Rk , a real Siegel domain P(k) =P  V ( k  1) ,  ( k )   R11 R2...Rk
50
A. Temgoua Kagou
and the cone
V ( k )  V( P ( k ) )  R11  R 2  R 22 ... R k  R kk .
It follows from the results of [G] that every homogeneous cone V which contains no
straight line can be decomposed in the form V
(l )
(up to a linear isomorphism). The
required number of steps to obtain V in this form is called the rank l of V, V= V
(l )
; this
n
yields the following decomposition of the space R that contains V :
(4)
l
n
R  R11  R 2  R 22 ... R l  R ll , n  l   ni .
i2
(k)
Furthermore, the projection  (iik ) of 
onto Rii (i < k) is a non-negative form. Thus
 (iik ) is positive definite on a subspace Rik of Rk with dim Rik = nik . We then have:
(5)
k 1
k 1
i 1
i 1
Rk =  R ik , n k   n ik .
We denote by G(V) the simply transitive group of linear transformations of V
described in [G]. With respect to its canonical decomposition, the cone V can be described
in the following quantitative manner : let x  V and let xj , j = 2,...,l (resp. xii , i = 1,...,l )
denote the projection of x onto Rj (resp. Rii ) ; then there exists a unique transformation h
 G(V) such that (h(x)) j = 0 , j = 1,...,l . We set ~
x  h(x) . The functions  j defined for j
= 1,...,l , by  j ( x)  ~
x jj , j = 1,...,l , define the cone V in the following sense : a point x of
n
R belongs to V if and only if  j (x) > 0, j = 1,...,l .
Since
the decomposition
(4)
of
R
n
n
decomposition of C :
Cn  C11  C2  C22 ...Cl  Cll ,
51
yields in a natural way the following
The duals of Bergman spaces
n
the functions  j , j = 1,...,l , can naturally be extended as rational functions on C .

n
Let   (1 ,...,  l )  C l ; we define the rational function (z) on C by :


l


n
(z) =   j ( z) j , z  C .
j1
n  mi
) , and let d denote the vector
For i = 1,...,l , set m j   n ji and d i  (1  i
2
i j
l
of R
whose components are di . In the sequel, e will denote the point of V whose
components are eii =1, ej = 0, i = 1,...,l , j = 2,...,l

Let us recall the definition of the conjugate cone V of V. Consider the inner
n
n
product < , > defined on R with respect to the canonical decomposition of R by :
( j)
<x,y> =  x ii y ii  2   ii ( x i , y j ) .
i j
i

Then V is defined by :



V = x  R n :  x, y   0, y  V  0 .

The adjoint group G (V) of G(V) with respect to < , > is the simply transitive group of


linear transformations of V . The cone V is an irreducible, convex, homogeneous cone
which contains no straight line, and it is also of rank l .

We shall denote by  j the defining functions of V . We have the following :
nij  n ij ( V )  n l  j1,l i 1
(1  i < j  l ).

For  C l , we define  by i  l  i  1 , i = 1,...,l , and we also define the function
52
A. Temgoua Kagou

 z 
n
on C by :
 z * 
j
l

   j ( z) .
j1
n
The Siegel domain of type II, associated with the homogeneous cone V of R and a
m
m
n
V-Hermitian, homogeneous form F : C  C  C , is defined by :
zz


 F( u, u)  V .
D=D(V,F) = ( z, u)  C n  C m :
2
i


The domain D is then an affine-homogeneous domain. Let Fii denote the projection of F
onto Cii , and C
(i)
the complex subpace of C
m
on which Fii is positive definite. Set
l
l (i )
m
. We shall denote by q the vector of
q i  dim C(i) ; then m   q i and C   C
C
i 1
i 1
l
N whose components are qi , i = 1,...,l .
We now recall the following two expressions of the Bergman kernel B((,v),(z,u))
of D=D(V,F) :
II.1 PROPOSITION [G] . The Bergman kernel B((,v),(z,u)) of D is given by
  z

B(, v), (z, u )   c
 F( v, u ) 
 2i

2d  q
z

  d  q
 c V exp    ,
 F( v, u )  ()
d.
2i


.
NOTATIONS. For   (1,...,  l )  R l , the notation 1+ stands for the vector
(1  1,...,1   l ) . Let  '  ( '1 ,..., ' l ) Rl ; we set  '  (1 '1 ,...,  l  ' l ) and by
53
The duals of Bergman spaces
n
 > ' , we mean that i > 'i for all i  {1,...,l}. For (,v) and (z,u) in D  C C
m
, we
let b denote the kernel
 z

b((,v),(z,u)) = 
 F( v, u)
 2i


Notice that B = cb. Moreover, b and b
2d  q
.
1+
 z


b ((,v),(z,u)) = 
 F( v, u)
 2i

l
,   R , will denote the expressions :
( 2d  q )
and
b
 z

 F( v, u)
((,v),(z,u)) = 
 2i

1+
2d  q  ( 2d  q ) 
.
Let r be a vector of R . For p  (0,) , we set Lp,r ( D)  Lp ( D, b  r ( z, z)dv( z)) and
l
p,r
p,r
p,r
define the weighted Bergman space A (D) by A (D) = L (D)H(D). We equip
p,r
p,r
A (D) with the L (D) - " norm" || || p,r. The weighted Bergman projection P r is the
2,r
2,r
2,r
orthogonal projection of L (D) onto A (D). Recall (cf. [BT] ) that A (D) = {0} when
ri 
ni  2
for some i {1,...,l } and otherwise, Pr is equal to the integral operator
2( 2d  q ) i
2,r
defined on L (D)) by the weighted Bergman kernel cr b
1+r
((,v),(z,u)) .
Let us now state some prerequisite results :
l
II.2 THEOREM [BT]. Let  and  be in R and (,v)D. Then we have :
1  ((, v),( z, u)) b   (( z, u),( z, u))d( z, u) <
D b
54
A. Temgoua Kagou
if and only if  i 
ni  2
ni
and  i   i 
, i = 1,...,l. In this case, the
2(2d  q ) i
2( 2d  q ) i
following equality holds :
1  ((, v),( z, u)) b   ( z, u),( z, u))d( z, u)  c
   ((, v),(, v)) .
D b
,  b
We shall need the following reproducing formulas which, indeed, improve those
obtained in [BT] (Theorem II.6.1, p.225), and whose proof, based on ideas of [BBR], will
be given in the appendix:
l
II.3 THEOREM. Let r be a vector of R such that ri 
ni  2
for all i = 1,...,l and p a
2(2d  q)i
 n  2(2d  q ) i (1  ri ) 
l
real number such that 1  p  min  i
 . Then for all   R such that
ni


i 
n i  2 p  1 ri

2( 2 d  q ) i p
p
(i = 1,...,l ), the reproducing formula Pf = f
holds for all f
p,r
A (D).
l
II.4 PROPOSITION [BT]. Let  R be such that i  0, i = 1,...,l . Then :
b ((, v),( z, u)  c b  ((, v),(, v))
and
b  (, v)  (', v'),( z, u)  ( z', v')  c b ((, v),(, v)) ,
for all (,v), (',v'), (z,u) and (z',u) in D.
p,r
II.5 LEMMA [R] . For all f  A (D) (p > 0) , we have the estimate :
55
The duals of Bergman spaces
f ( z, u)
p
p
 cb1 r (( z, u),( z, u)) f p,r .

DEFINITION : A vector  R is a V-integral vector if   is a polynomial in .
l
In the sequel,  will be a V-integral vector. We associate to  the differential
n
n
polynomial (   ) in C in the following way : for  C ,

(   )exp(<,>) =   exp(<,>)
n
( C ).
Let us now recall the following lemma due to F. Trèves [Tr] which is crucial in our
work.
II.6 LEMMA [Tr]. For each holomorphic function G in D, there exists a holomorphic
function g in D such that
(  ) g(,v)=G(,v) for all (,v)  D.
DEFINITION : A function g  H(D) is a Bloch function if :







2
d

q
g   sup  (   ) z g( z, u) b
(( z, u),( z, u))    .

( z, u )  D 


Set N = {gH(D) : (  )g  0 }. We define the Bloch space B  and the space
p
C ,r (D) in the following manner, where we set  
/
B  ={Bloch functions in D} N ,
while
p
C ,r (D) is the quotient space:
56

:
2d  q
A. Temgoua Kagou
1




p  p  r

p
(( z, u ), (z, u )) d(z, u )    N.
f  H(D) : f C p   D ( ) z f (z, u ) b
, r 





/
These two spaces have the following topological property:
p
LEMMA : (B  , . ) and ( C ,r (D) , . p
are complex Banach spaces.
C , r ( D) )

Our first two results read as follows:
II.7 THEOREM. Let D be a homogeneous Siegel domain of type II. Let p be a real
l
number and r a vector of R such that 0 < p  1 and ri 
ni  2
, i = 1,...,l . Then the
2( 2d  q ) i
following assertions hold:
p,r

(i) B  is isomorphic to a subspace of (A (D)) ,
n
1,r

(ii) B  is equal to A (D)) ) if i  i , i  1,..., l , with equivalent norms.
2
1,r
The duality (A (D)), B  ) is given by
( f , g)  D (   ) z g( z, u) b   (( z, u),( z, u)) f ( z, u) b  r (( z, u),( z, u))d( z, u) ,
where  

.
2d  q
57
The duals of Bergman spaces
N
II.8 THEOREM : Let D  C
or D = D1. Let p1 =
be either a symmetric Siegel domain of type II, or D = D0,
ni  2
2N
and let r be a vector of R such that ri 
, i = 1,...,l .
2( 2d  q ) i
2N  1
Then for all p  (p1 ,1) and all  Rl such that
n 2
1  2ri
i   i
 (2d  q)i 
(2d  q)i , i  1,..., l ,
2
p

p,r
we have B  = (A (D)) , with equivalent norms.
p,r
Moreover, the duality (A (D)), B  ) is given by
1 r
1
p (( z, u ), (z, u )) d(z, u ) ,
(f , g)  D (  ) z g(z, u )b   (( z, u ), (z, u )) f (z, u )b
with  

.
 2d  q
REMARK: In the symmetric case, R.R. Coifman and R. Rochberg ([CR], p. 43-44) stated
1,1
p,r
that A (D)) and A
1 r
p ( D) have the same dual. The proof of Theorem II.8 relies on
their atomic decomposition theorem . For a proof of the atomic decomposition theorem, cf.
[BT1].
Theorems II.7 and II.8 will be proved in section III.

Let h  L (D) and let  be a V-integral vector such that i >
ni
(i = 1,...,l) . Then,
2
in view of Theorem II.2, the following function G is holomorphic in D:
G(, v)  D b1 ((, v),( z, u)) h( z, u)dv( z, u) ,
58
A. Temgoua Kagou
where


.
 2d  q
By
Lemma
II.6,
there
exists
~
g  H (D)
such
that
(  ) z ~
g( z, u)  G( z, u) . Let g be the equivalence class of all holomorphic solutions of this

equation. Then g  B  ; hence, we can define an operator P from L (D) into B  in the
following way :
Ph = g.

P is called the "Bergman projection" of L (D) into B  . Let us justify this name : the
2
Bergman projection P is usually the integral operator defined on L (D) by :
Ph( ,v) = D B(, v), (z, u ) h (z, u )d(z, u )
2
(h  L (D)).
2

Now, let h  L  L (D) ; since (   ) b((, v),( z, u))  c b1   ((, v),( z, u)) (recall


), one easily obtains that Ph is a representative of the element Ph = g of
2d  q
B .
Our third result reads as follows:
II.9 THEOREM. Let D be a homogeneous Siegel domain of type II. Let  be a V-integral
vector such that i >
ni

(i=1,...,l). Then PL (D) = B  and P has a bounded right
2
inverse.
Theorem II.9 will be proved in section IV .
Our fourth result reads as follows :
59
The duals of Bergman spaces


II-10 THEOREM: Let D2  ( z, u)  M 2  M r 2



z - z

 u * u  V and  = (5,5). Then
2i


there exists a kernel b0 in D2 such that :
(i) with respect to ( , v) , b0 (( , v) , (z ,u)) is holomorphic in D 2 and
() b0 (( , v),(z,u))  0 ,
1
(ii) for all ( , v) D0 , (b- b0 ) (( , v) , (z ,u))  L (D2 , d (z,u)) .

Hence, for each h  L (D2 ) , the function Ph defined on D2 by
Ph(, v) = D
2
 b  b0  (, v),( z, u) h( z, u)d( z, u)
is a representative of Ph.
This result will be proved in section V.
Our fifth result focuses on the case p > 1 and reads as follows:
II-11 THEOREM : Let D be a homogeneous Siegel domain of type II. Let p be a real
number
and
r
a
vector
of
R
l
such
that
n1  2
ri 
(i = 1,..., l ) and
2(2d  q)i

 2n  2  2(2d  q)i ri 

1 < p < min  i
.
n


i 
i

Then we have the following assertions.


 2n  2  2(2d  q) i ri 

 2n i  2  2(2d  q) i ri 

p,r
(i) If max i
 < p < min
, then (A
n

2

2
(
2
d

q
)
r
n
 i



i 
i 
i i 
i

p',r
(D))* = A
(D) (p' is the conjugate exponent of p).
60
A. Temgoua Kagou
p,r
(ii) (A
p,r
The duality (A
p' ( D) whenever  > n i , i = 1,...,l , with equivalent norms.
(D))* = C ,
i
r
2
p' ( D) ) is given by
(D), C ,
r
 
( f , h)  D b   ( z, u),( z, u)   h( z, u)f ( z, u) b  r  ( z, u),( z, u) d( z, u),
z
where  =

.
2d  q
This theorem will be proved in section VI.
III. PROOFS OF THEOREMS II.7 AND II.8.
III.1 PROOF OF THEOREM II.7
p,r
(i) Let g  B and consider the linear functional  defined on A
 
( f )   D  

where  =
(D) by :
1 r
1
p (, v),(, v)
g(, v)b   (, v),(, v) f (, v) b



and 0 < p  1. By Lemma II.5, we have the estimate :
2d  q
1 r
1
p (, v), (, v)d(, v) 
D f (, v) b
p
1 p
 c D f (, v) b  r (, v), , v) d(, v) f
p, r


cf
.
p, r
Hence
 possesses the following property:
| (f)|  c||g||||f||p,r ,
p,r
and thus ||||  c||g|| . Therefore   (A
(D))*. This proves the inclusion of B into
61
The duals of Bergman spaces
p,r
(A
(D))*.
1,r
(ii) Let p = 1, and let  be in (A
(D))*. Then by the Hahn-Banach theorem,
there exists a bounded function k in D such that
( f )  D k( z, u)f ( z, u) b  r  ( z, u),( z, u) d( z, u).
On the other hand, by Theorem II.3, we have the following reproducing formula for every f
1,r
A
(D) :
f (z, u )  D b1  r   (z, u ), (, v)f (, v)b r   (, v), (, v) d(, v).
Therefore by the Fubini theorem, we easily get
(f )  D  D b1  r   (, v), (z, u ) k (, v)b  r (, v), (, v) d(, v) 


f (z, u )b  r   (z, u ), (z, u ) d(z, u ).
Set g(z,u) = D b1  r    ( z, u),(, v) k(, v) b  r  (, v),(, v) d(, v) . Observe next that
g  H(D) and that by Theorem II.2, the following estimate holds :
g( z, u)  C k  b   ( z, u),( z, u)
((z,u)  D),
ni
, i  1,..., l. Then in view of Lemma II.6, there exists h in B such
since i 
 2(2d  q)i
that h = g. Therefore
 
( f )  D   h( z, u)f ( z, u) b  r    ( z, u),( z, u) d( z, u)
z
and thus,  is represented by the element g of B . This proves the reverse inclusion of
(A1,r (D))* in B.

62
A. Temgoua Kagou
We shall now prove another reproducing formula whose proof relies on the
dominated convergence theorem. In view
of this formula, the Bergman weighted
projection Pr also reproduces functions which satisfy certain uniform estimate.
III.2
i 
PROPOSITION
:
Let
r
be
and
ni 2
and ri 
 i , i  1,..., l..
 2(2d  q)i
2(2d  q)i
sup
z D
ni
two
vectors
of
Rl
such
that
Let G be in H(D) such that
 G(z) b (z, z)  . Then P G = G.
r
PROOF : Consider the sequence {Gn } defined by
ie   z 

G n ( z)  G z   b  , ie ,

n  n 
where the positive exponent is to be specified later. There exists c > 0 such that for every
z in D and every positive integer n :
ie 
ie
ie 


G  z    cb   z  , z  


n
n
n
 ie ie 
 cb   ,  ,
 n n
where the latter inequality is yielded by Proposition II.4 since i> 0 . Then , when we
ie 

keep n fixed, the function z  G z   is bounded and the following inequality holds :

n
2 r
2  z   r
D G n ( z) b ( z, z)d( z)  C n D b  , ie b ( z, z)d( z).
n 
63
The duals of Bergman spaces
ni
We choose so that 2 i  1  ri 
for all i = 1,...,l. Hence by Theorem II.2,
2(2d  q ) i
the latest integral converges and furthermore, P r Gn = Gn for all n .
On the other hand, by Proposition II.4, since i andiwe have :
Also,
G n ( z ) b   ( z, z )  G ( z 
ie  
b (z, z)  c.
n
by
under
Theorem
II.2,
our
assumptions
i 
ni
 2( 2d  q ) i
and
ni 2
ri 
  i , i  1,..., l, we get :
2( 2d  q ) i
1  r ( z, ) b  r   (, )d()  .
D b
Henceforth, by the dominated convergence theorem, we easily obtain the equality P rG = G.
REMARK : This result extends the one obtained in [B 2] for symmetric Siegel domains of
type II, via the bounded circular realization of the domain. Our proof is straight and
includes all homogeneous Siegel domains of type II.
III.3 COROLLARY : Let p  (0, 1] . Let and rbe vectors of Rl such that
i 
ni
 2(2d  q)i
 1  ri
and
ri 
ni  2
for all i  1,..., l,
2(2d  q)i
p,r
(A (D))* and for all z0 in D, we have


 

 


 1


p
p
p
p
 z, z)d( z)   b ., z0  .
D b  z 0 , z b  ., z b








64
Then
for
all
A. Temgoua Kagou
PROOF : By Proposition III.2, it is sufficient to show that
(6)
 
 1 r  


p
sup  b p  ., z b
 z, z  .


zD 



p,r
But since   (A (D))*, we obtain that

 1 r
 



p
 b p  ., z  c b p (., z)
 c' b
 z, z  ,




p, r
where the latest estimate follows from Theorem II.2. This proves Corollary III.3 .

III.4 PROOF OF THEOREM II.8
N
Let D  C be a symmetric Siegel domain of type II, or let D be equal to D 0 or
l
ni  2
 2N 
,1 and r  R satisfying ri 
, the domain
D1 . Remark that for p  
 2N  1 
2( 2d  q ) i
D satisfies the hypotheses of the molecular decomposition theorem [BT 1] for functions in
p,r
Bergman spaces A
(D) ; more precisely, for such values of p and r, there exist constants
p,r
c = c(p,r) and C = C(p,r) such that for every f  A (D), there exists an l p -sequence {i}
such that

1 r  

p
f ( z)    i b p ( z, z i ) b
 zi , zi 
i0
where {zi} is a lattice in D and the following estimate holds :
c f pp,r    i
p
p
 C f p,r
.
65
(z D) ,
The duals of Bergman spaces
l
Moreover, the vector of R is defined as follows. Let denote a V-integral vector in R
such that when we set  
l

, the following condition is satisfied:
2d  q
ni  2
1  2ri
i 
1
2(2d  q)i
p
i  1,..., l.
The vector is given byp+1+r.
p,r
Let  (A
p, r
(D))*. For f  A
(D), define the sequence {fN} of functions in
D by
(7)
Then

1 r  
N
p
f N ( z)    i b p ( z, z i ) b
 zi , zi 
i0
{fN} converges to f in A
p, r
(z D).
(D). Hence (fN) goes to (f) as N tends to infinity.
Now, in view of Corollary III.3, we get :


 

 




 1
p
p
p
 b ., z i   D b  z i , z b  ., z b p  z, z)d( z)








(i  Z+),
and combining this with (7) yields for every positive integer N :

 


 1
p (z, z)d(z). .
f N   D f N (z) b p ., z b






This easily implies that for every positive integer N :
(8)

 


 1
p
(f N )  D f N (z) b ., z b p (z, z)d(z).






Now, let N tend to infinity in identity (8). On the one hand, ( fN) goes to (f). On the
66
A. Temgoua Kagou
other hand, it follows from (6) that

 


 1
p
p (z, z)d(z)  C
D f N  f (z) b ., z b
D






 C' D
1 r
1
p (z, z)d(z)
f N f (z) b
f N  f (z) p b  r (z, z)d(z)
where the latest inequality follows from Lemma II.5. Henceforth, since {fN} converges to f
in
p,r
A (D),
we
conclude
that
the
right
hand
side
of
(8)
tends
to

 


 1
p
p z, z)d( z) . We have then proved that for every f in Ap, r (D) :


D f ( z) b  ., z b





 


 1
p
p (z, z)d(z).
(f )  D f (z) b ., z b






 



Now,by Lemma II.6, since the function z   b p (., z) is holomorphic in D, there exists




 



g in H(D) such that g(z) =  b p (., z ) . Furthermore, with the choice of  and with







 1 r
, we deduce from (6) that for every z D, the following estimate holds :
p
67
The duals of Bergman spaces
 



sup  b p  ., z b    z, z  .

zD 


Hence g is a Bloch function in D, and g clearly represents the bounded linear functional
p,r
(A
(D))* in the following manner :
1 r
1


p ( z, z)d( z).
( f )  D   g( z) b (( z),( z) f ( z) b
Moreover, g   c  . This proves the inclusion of (A
entirely proved.
p,r
(D))* in B . Theorem II.8 is


IV. PROOF OF THEOREM II.9.


Since PL (D)  B , let us prove that B  PL (D). Let g  B and set
  g(, v)b (, v),(, v)
h(, v)  
with  


. Then h  L (D) . We are
2d  q
going to show that Ph = g. It suffices to prove the equality :
  g(, v)  D b1    (, v),(z, u)  g(z, u)b  (z, u),(z, u)d(z, u).
Setting  =  , this equality follows from Proposition III.2. To complete the proof, let us
determine a right inverse of P. Define R as follows:
 
Rg(, v)   g(, v) b   (, v),(, v)


g B.
Then PRg = g and Rg  L (D). Furthemore Rg L ( D)  g  and thus R  1. This
completes the proof of Theorem II.9.

68
A. Temgoua Kagou
V. PROOF OF THEOREM II.10.
V.1 Preliminaries on D2.
The cone V = {z  M2 , z Hermitian , z > 0} is of rank 2 and linearly equivalent
4
to the spherical cone in R ; moreover, it is self-conjugate with respect to the inner
product
<  ,z > = 11 z11 + 12 z12 + 21 z21 + 22 z22 ,
12 

z
 and z   11
where    11

z


22 
 21
 21
have
z12 
 1 0
 belong to M 2 . Set e  
 . We also
z 22 
 0 1
n12 = 2 , n1 = 0 , n2 = 2 , d = (-2,-2) . The functions that define the cone V are
defined by
 
1( )  11  12 21 ,  2 ( )   22
 22
Let F: Mr, 2
 Mr,2
and v = (v1 , v2) are in Mr,2
   2 .
 M 2 be defined by F(v,u) = u* v where u = (u1, u2)
r
with v1 ,v2 in C (i = 1,2) . Hence F11 (v,u) = u1*v1,
r
F22 (v,u) = u2* v2 and it is clear that Fii is concentrated on C  C
definite on C
r
(i = 1,2). Therefore q = (r,r).
Let  = (5,5) be a V-integral vector. It is straightforward that
 



2
2



  11 22 12  21 
5
and

1
  B (, v),( z, u)  c B 2d  q  (, v),( z, u) ,

 
69
r
and is positive
The duals of Bergman spaces
where B is the Bergman kernel of D2. It then follows from Theorem II.2 that
  B (, v),(z, u) is integrable with respect to (z,u) in D2
since  = (1, 2), 1 > 0,
2 > 1. We are now ready to state the following lemma whose proof is just computational.
V.2 LEMMA : Let 1 and 2 be two integers such that 1  {-4,-3,-2,-1,0} or 2{-3,
  22 11   2iz *  u * v  0 , where (z,u) and (,v) belong to
-2,-1,0,1}. Then  
D2 .
The proof of the next lemma is somewhat lengthy and will be given in the
appendix.
V.3. LEMMA : The following functions are integrable in D2 :
(a)
4  (5  r )  (8  r )  ie  z * 
u 2 1
2

 ;
 2i 
(b)
5
 ie  z * 
u 2 1 (5  r )  2 (9  r ) 
 ;
 2i 
(c)
4
 (5  r )  (9  r )  ie  z *
u 2 z12 1
2

 ;
 2i 
(d)
4
 (5  r )  (9  r )  ie  z *
u 2 z 211
2

 ;
 2i 
(e)
5
 (5  r )  (9  r )  ie  z * 
u 2 z 211
2

 ;
 2i 
70
A. Temgoua Kagou
(f)
4  (5  r )  (9  r )  ie  z * 
u1 u 2 1
2

 ;
 2i 
(g)
5  (5  r )  (9  r )  ie  z * 
u1 u 2 1
2

 ;
 2i 
(h)
4  (5  r )  (9  r )  ie  z * 
u1 u 2 1
2

 .
 2i 
We shall also use the following lemma.
V.4. LEMMA : Let (, v) be in D0. Then there exists two positive constants c1 and c2
such that for all (z,u)  D2 , the following estimates hold :
a)
   z*

 ie  z *
 ie  z * 
c1  2 
 u * v  c 2  2 
  2 
;
 2i 
 2i 
 2i

b)
   z*

 ie  z *
 ie  z *
c1 1
 u * v  c 2 1
  1
.
 2i 
 2i 
 2i

PROOF : This is a straightforward consequence of the following lemma due to A.
Koranyi:
LEMMA ([CR] , cf. also [BT1 ] ) Let D be a symmetric Siegel domain of type II and let d
denote the Bergman distance on D2. Then there exists a constant CD such that for all  , z ,
z' in D ,
B(, z)
 1  C D d ( z, z') whenever d(z,z')  10.
B(, z')
We are now ready to handle the proof of Theorem II.10 :
71
The duals of Bergman spaces
PROOF OF THEOREM II.10. We can define b0 in the following manner :

 (1)  z *


 u * v (1)  

2i


b  b 0 (, v), (z, u )   1 (4  r)    z *  u * v   1 (4  r)  



2i

  (4  r )    z *
  (4  r )  ie  z * 
 u * v   2



 2
2
i


 2i 

  (4  r )  ie  z *  
 (5  r )  ie  z *   (4  r )    z *
 ( 4  r ) 2
 u * v   2

  2


 
 2i 
 2i

 2i  
2
   z*

 (6  r )  ie  z *   ie  z * 
 1
 u * v  

  2 
  2 
 2i   2i 
 2i

3
   z*

 (7  r )  ie  z *   ie  z * 
  2
 u * v   ,

  2 
  2 
 2i   2i 
 2i
 

where

 i

0 
(4  r )(5  r )
(4  r )(5  r )( 6  r )
0, v 2 .
, 
and   (1) , v(1)    




   0  22 
2
2

Then b0 ((,v),(z,u)) is a holomorphic function of (,v) in D2. Furthermore, in view of
  (1)  z *

Lemma V.2 and the fact that 1
 u * v (1)  does not depend on 11 , 21 and


2i


 
12 , we have   b 0  (, v),( z, u)  0. This proves the first assertion.

n
Using the identity a ( n  1)  b ( n  1)  ( b  a )  a ( k  1) b ( n  1  k ) and
k0
in view of Lemma V.4 , we have the estimate :
b  b0 (, v), (z, u)
1
 ie  z * 
4
2   i
 (5  r )  (8  r )  ie  z *    22  i
 c 1
2
 u 2 v 2  11
 u1 v1   2 



2
i
2
i
4
 2i  
 2i 

 12  21  12

z 21   21z12  2 12 v 2  2 12 u 2 v1  2 v1 z 21 u 2  4 u 2 u1 v1 v 2 .
72
A. Temgoua Kagou
1
Hence in view of Lemma V.3 , (b-b0 ) ((,v) ,(z,u)) is in L (D2, d(z,u)). This completes
the proof of Theorem II.10.

VI. PROOF OF THEOREM II.11.
1
l
VI.I. THEOREM [BT] . Let  and r be in R such that i 
and
(2d  q )i
ni  2
ri 
, i=1,…,l. Then P is bounded from Lp,r (D) into Ap,r (D) if
2 ( 2d  q ) i

2ni  2  2(2d  q) i ri
 2n  2  2(2d  q) i ri 

max 1, i
.
  p  min
ni
i  1,..., l 
i  1,..., l
 ni  2  2(2d  q) i  i 

I.2 PROOF OF THEOREM II.11. (i) In view of Theorems II.3 and VI.1, we have P r f =
p,r
f for every f  A
(D), and Pr g
 cp, r g p, r . One easily obtains the announced
p, r
results from the Hahn-Banach and Riesz representation theorems.
p,r
(ii) Let  (A
p',r
(D))* . By the Hahn- Banach theorem , there exists g  L
p,r
such that for all f  A (D) , we have
( f )  D g( z, u)f ( z, u) b  r  ( z, u),( z, u) d( z, u).
Set  

. By Theorem II.3, P+r f = f , f  Ap,r (D) , then
2d  q
(f )  D g(z, u )P  r f (z, u )b r (z, u ), (z, u ) d(z, u )
 D T  r, r g(z, u )f (z, u )b  r (z, u ), (z, u ) d(z, u )
where T+r ,r is the adjoint operator of P+r with respect to the inner product
< f , g >2, r = D f ( z, u) g( z, u)b  r  ( z, u),( z, u) d( z, u),
73
(D)
The duals of Bergman spaces
and it is expressed in the following way :
T  r, r f (w )  b   (w, w )D b1    r (w, (z, u ))f (z, u )b  r (z, u ), (z, u )d(z, u )
(w  D).
 2 n  2  2(2d  q ) i ri 
n
 , it follows from Theorem VI.1 that
Since i  i and p '  max

2
 n i  2  2(2d  q ) i ri 
T  r , r g
 c p, r g p ' , r .
p' , r

Clearly, the function z  b (z ,z ) T+r,rg(z) belongs to H(D) , then by Lemma

II.6 , there exists h H(D) such that h(z)= b (z,z) T+r,rg(z) for every z in D. Then h
p' ( D) and h p'
 C ,
r
C , r ( D)  c  . Thus, finally, h represents  in the following way :
 
( f )  D b   ( z, u),( z, u)   h( z, u)f ( z, u) b  r  ( z, u),( z, u) d( z, u).
-
p' ( D) ; then the function z
Conversely, let h  C ,
 b (z ,z ) h (z) belongs to
r
p',r
L
(D) and the linear functional  defined by
 
( f )  D b    ( z, u),( z, u)   h( z, u)f ( z, u) b  r  ( z, u),( z, u) d( z, u)
(fAp,r(D))
is such that   c h
Cp', r (D)
p,r
. Hence  (A
(D))* . Therefore this proves assertion (ii)
and Theorem II.11 is entirely proved.
REMARK : Let r and p satisfy the hypotheses of assertion (i) of Theorem II.11, and let
n
p
l
be a vector of R such that i  i for all i = 1,...,l. Then C, r ( D) is equivalent to
2
74
A. Temgoua Kagou
p
p,r
A (D) in the following manner : every equivalence class in C, r ( D) contains one and
p,r
p,r
only one representative f belonging to A (D) , and conversely, every fA (D) is a
p
p
representative of an equivalence class in C, r ( D) . In fact, first let g  C, r ( D) . Set

p,r
h(z)=b (z,z) g(z) and f = Pr h ; then by Theorem VI.I, f belongs to A (D) under our
assumptions on p, since h belongs to L
p,r
(D). Therefore f = Pr+( g ). On the other
hand,  g Ap, p+r(D) ; then Pr+( g ) = g . Hence  g =  f and then, f is a
p
representative of the class g in C, r ( D) .
p,r
Next, let f A (D) be such that  f  0. Then f
Tr+,rf(z) = b-


= Pr f , and this implies that
-
(z,,z) f(z)  0. By the Fubini theorem, one easily obtains that
Pr Tr  , r f ( )
 c D f ()b  r (, ) D b1  r   (z, )b1  r (, z)b  r   (z, z)d(z) d()



r
1

r
 c D f (z)b (, )b
(, )d(),
where the latest equality follows from the reproducing formula Pr+ (b
b
1+r
( , ) for all  andin D, since the function b
1+r
1+r
(. , ) ) ( =
(. , ) belongs to A
p,r
(D) and
n
i  i for all i = 1,...,l .Finally, we conclude that Pr(Tr+,r f) = Pr (f) = f, and since
2
p
Tr+,r f  0, this implies that f  0. Hence each class in C, r ( D) has only one
representative in A
p,r
(D) .
75
The duals of Bergman spaces
Conversely , let f  A
p,r
(D) ; then Pr f = f and   f()b - (, )  Tr  , r f () ,
p,r
for all  in D. Hence, by Theorem VI.1, T r+,r f belongs to L (D) since
 2n  2  2(2d  q) i ri 


p  max i

i 
 n i  2  2(2d  q) i ri 

and
n
i  i
2
for all i = 1,...,l. Therefore f is a
p
representative of a class in C, r ( D) .
APPENDIX
PROOF OF LEMMA V. 3 .
We first remark that for all (z, u)  D2 :
(*)
u2
2
 2 (
ie  z *
).
2i
By this remark (*), we have the estimate :
u2
4 (5  r ) (8  r )  ie  z *
1
2


 2i 
(5  r ) (6  r )  ie  z *
 1
2

.
 2i 
This latest function is integrable by Theorem II.2. Hence (a) is proved . The same estimate
permits to conclude that (b) is true . For the next three integrals, by the previous remark (*),
it is sufficient to prove that for a suitable choice of  ( = 0 for (c) and (d) and  = 0.5 for
(e) ), the following function is integrable in D2 :
g ( z, u)  z12 
(5  r ) (7  r  )  ie *  z 
2

.
1
 2i 
76
A. Temgoua Kagou
To this aim, we shall use computational techniques due to Békollé in [B 2] . Let 1
and 2 be two real numbers such that 0 < 1 < 1 and 1 < 2 < 3 + 2, and consider the sets
D3 and D4 defined by :

 (4  r  1 )  (4  r   2 )  ie  z *  
D 3  ( z, u)  D 2 : g  ( z, u)  1
2


 2i  

D 4  D 2 \ D 3.
In D1 , g (z, u) is dominated by an integrable function in D2 by our assumption
on 1 and 2 and Theorem II. 2. In D4 , it is obvious that
2
g  (z, u )  c h  (z, u ) ,
where h  (z, u) is equal to :
(6  r  1) / 2 (10   2  r  2) / 2  ie  z *
h ( z, u)  z12 1
2

.
 2i 
From the very definition of 1 and 2 , we have :
h  ( z, u )  c
  ( r  1  4) / 2 (8   2  r  2) / 2  ie  z * 
2

.

 2i  
z  1
Therefore one easily gets :
ie  z * 
* (4   2  r  2) / 2
* (r  1) / 2

h  (z, u )  c V exp    ,
 12 1
( ) 2
()d,
2i


 
where 1* ()   22  12 21 and *2 ()  11 are two defining functions of V* = V.
11
Hence, by the Plancherel-Gindikin formula [BT] with Cm  M r 2 , we get :
77
The duals of Bergman spaces
2
D h  (z, u ) d(z, u )  c C m d(u )
2
2 * 2   2  r  2 * r  1  r  2


2
() exp( 2  , u * u  d 
 V exp  , e   12 1


2   2  2
2
2

 c' V exp 11   22 11  2  1  2  4  11 22  12 
12 d,


r
q*
sin ce C m exp  2  , u * u  d(u )  c 
 c 1 ()2 ()  .


By a polar coordinate change of variables, the integration with respect to 12
converges if 2 < 3 + 2, and we have :
 4    2
2
 
d11d 22 .
D h ( z, u) d( z, u)  0 0 exp( (11   22 ))11 1  22 2
2
Henceforth, this integral converges by our assumption on 1 and 2 . This proves
assertions (c), (d) and (e) .
Following the same pattern, it is sufficient to prove that the function f(z, u)
defined by :
f  (z, u )  u1 1 (5 r )  2
is integrable in D2
 ( 6  r  )
( for (f), take  = 1;
 ie  z * 


 2i 
for (g), take  = 0. 5 , and for (h), take  = 0 ).
The techniques used before imply that it is sufficient to prove that if 0 < 1 < 1 and 1 < 2
< 2 + 2 , we have :
|f(z,uK(z,u

8   2  r  2 

6  1  r

 
  ie  z *
2
2

where K  ( z, u)  c u1  1
2
 is square integrable in

  2i 


D2. It is clear that :
78
A. Temgoua Kagou
 4   2  r  2 2    r
1
ie  z *  

2
2
K  (z, u )  c u1 V exp    
  1
2
2i






()d ;



then :
2
D K  (z, u ) d(z, u )
2
 1  r
2 2   2  r  2
 c  exp(   , e  u1 1
()2
(  ) d d ( u )
1  1  2
2

 c V exp  11   22  22 11 2   2  2  1  11 22 .  12 
d


 c 0 0 exp  (11   22 ) 11 1  22 3  2   2 d11d 22 .
This converges by our assumption on 1 and 2 .

PROOF OF THEOREM II.3.
We first prove the following lemma:
l
A LEMMA. Let p[ 1,  ) and let r be a vector of R such that ri  -1 (i = 1,...,l). Let f 
p,r
m
A (D). Then for every (y,u)  V  C
such that y - F(u,u)  V, the holomorphic
n
function fy, u defined on R +iV by
fy, u (z) = f (z+iy,u)
p
belongs to the Hardy space H (Rn+iV). Moreover, there is a constant C = C(p,r) such that
p,r
m
for all f  A (D) and (y,u)  V  C
f y, u
satisfying y - F(u,u)  V, then
p
 Cb1  r ((iy , u ), (iy , u )) f p, r .
p
n
H (R  iV )
p
Proof of Lemma A. First take (y,u) = (e,0). Let P be a polydisc centered at (ie,0) with
79
The duals of Bergman spaces
closure contained in D. Let (R1 ,...,Rn+m) be the multiradius of P. Then :
(1)
p
f ( x  ie,0) p  CP  ( x,0) f (  i, v) ddd( v) .
m
If P’ is the projection of P on iV  C , we obtain :
(2)
p
p


P  (x,0) f (  i, v) ddd(v)     x  R  P' f (  i, v) dd(v) d.
1 

Combining (1) and (2) yields
n
p
R n f ( x  ie,0) dx  C R1 R n


p
P' f (  i, v) dd ( v) d.

Since P' is a compact set contained in (i, v)  iV  C m :   F(u, u )  V , there are two
positive constants cr and c'r such that c' r  b  r  (  i, v),(  i, v)  c r for all (i,v) 
n
P’ and   R . Hence, for every f  H(D) , the following holds :
p
R n f ( x  ie ,0) dx

 C r R n
 Cr f
p r
P' f (  i, v) b (  i, v), (  i, v) dd ( v) d
p
.
p, r
Next assume that (iy,u) is an arbitrary point of D. Since D is affine-homogeneous, there is g
-1
n
 G(D) such that g (x+iy,u) = (x+ie,0) for every x in R . Set h = fog. Then
p
p
R n f ( x  iy , u ) dx  R n h ( x  ie ) dx
 C r D h (  i, v) b  r (  i, v), (  i, v) ddd( v).
p
Make the change of variables  = ‘ , (i,v) = g-1(i‘ ,v’). Then the equality
det g
2
 cb( x  iy , u ), ( x  iy , u ) 
yields that
80
A. Temgoua Kagou
b (  i, v),(  i, v)  b (' i', v'),(' i', v') det g
2
 cb (' i', v'),(' i', v') b 1 ( x  iy, u),( x  iy, u) .
l
Furthermore, for r  R , one can prove that
b r  (  i, v),(  i, v)  c r b r  (' i', v'),(' i', v') b r  ( x  iy, u),( x  iy, u).
Hence:
p
1  r ( x  iy, u),( x  iy, u) f p .

 p, r
R n f ( x  iy, u) dx  c' r b
So,
f y, u
p
H p (R n  iV )
p
 sup R n f ( x  i( y  ), u) dx
 V
p
 c'r sup R n b1  r ( x  i( y  ), u ), ( x  i( y  ), u )  f
p, r
 V
p
 c'r b1  r ( x  iy , u ), ( x  iy , u )  f
.
p, r
The last equality follows from Proposition II.4 since r i  -1 (i = 1,...,l ).

The following corollary can be deduced from results in [SW] :
l
B COROLLARY. Let p[ 1,  ) and let r be a vector of R such that ri  -1 (i = 1,...,l ).
p,r
Let (iy,u)  D and y’ V. Then for all f  A
(D) ,
p
p
R n f ( x  i( y  y' ), u) dx  R n f ( x  iy) dx
and
81
The duals of Bergman spaces
lim  n f ( x  i( y  y'), u)  f ( x  iy, u) p dx  0.
y'0 R
y'
l
PROOF OF THEOREM II.3 : Let  be a vector of R such that i >
2, 
Recall that for every f  A
ni  2
(i = 1,...,l ).
2( 2d  q ) i
(D) , P f is given by the following formula:
P f ( z, u)  c  D b1    ( z, u),(, v)) f (, v) b   (, v), , v) d(, v)
((z,u)  D).
Hence P extends as an operator on Lp,r (D) if

  
p
'
1   ( z, u),(, v) b 


D b
r
 p'
p
 (, v), , v)d(, v)  
when p > 1 (resp. if

   r
sup  b1    ( z, u),(, v) b 
 (, v), , v)   


(,v)D
when p = 1) for all (z,u)  D. By Theorem II.2
(resp. by Proposition
II.4) , this is the
case if and only if
i 
 n  2(2d  q ) i (1  ri ) 
n i  2 p  1 ri

(i = 1,...,l ) and p  min  i
 when p > 1
2( 2d  q ) i p
p
ni
i 1,...,l 

[resp. if 1+i  i-ri  0 (i = 1,...,l ) when p = 1 ]. For p = 1, this reduces to the two
conditions ri  -1 and i  ri (i = 1,...,l ).
It suffices to show that for every ( ,v)  D, under our assumptions on , r and p ,
p,r
the bounded linear functional (,v) defined on A
(D) by
 (, v) ( f )  f (, v)  c  D b1    (, v),( z, u) f ( z, u) b    ( z, u),( z, u) d( z, u)
which is identically 0 on A
p, r
2, 
(D)  A
(D) , vanishes identically on A
82
p, r
(D). The
A. Temgoua Kagou
desired conclusion follows from the following lemma :
l
C LEMMA : Let  and r be two vectors of R such that i >
p, r
(i = 1,...,l ). Then for every p  [1,), the subspace A
ni  2
ni  2
and ri >
2( 2d  q ) i
2( 2 d  q ) i
2, 
(D)A
p, r
(D) is dense in A
(D).
Proof of Lemma C. We use the following notations : for z = (x+iy)  D , we write
 x  iy u 
z
p, r
for 
,
(D). Let  be a positive
 , and we write ie for (ie,0). Let f  A
n
 n
n
number to be chosen later. Consider the sequence {fq} defined by

ie   z 
fq ( z)  c f  z   b  , ie
q q 

where c = b
-
(z D ) ,
p, r
(0, ie). We will show that for  large , fq  A
positive integer q, and that
2, 
(D)  A
(D) for every
lim f  fq
 0.
p, r
q

ie 
The function z  f  z   is bounded by Lemma II.5 and Proposition II.4. Take
q

z 
p, r
 big enough so that the function z  b   , ie  c b  ( z, qie) belongs to A
(D) 
q 
2, 
A
(D) for every q N. Next, by the Minkowski inequality, we have the estimate :
83
The duals of Bergman spaces
1
p

p
z 
p
f  fq
  D f (z) c  b   , ie   1 d(z)


p, r
q 


1
p
p

p
z  
ie 
  D c b   , ie  f  z    f (z) b  r (z, z)d(z) .


q
q  


By Lemma II.4, there is a positive constant A such that for all z  D and q  N, we have
z 
c b  , ie  A  . Hence, by the dominated convergence theorem, the first integral on
q 
the right side of the estimate goes to 0 as q goes to infinity. Now, to study the second
integral, it suffices to prove that
p

ie 
I q  D f  z    f ( z) b  r ( z, z)d( z)
q

goes to 0 when q goes to infinity. Set z = (x+iy, u)  D and obtain that Iq is equal to


p 




 e 



(
2
d

q
)
r
dy d(u ).
R n V  F(u, u )  R n f  x  iy   , u   f ( x  iy , u ) dx y  F(u, u ) 
q

 







Observe that Corollary B yields the following two facts :
p


e  
p
p

R n f  x  i y  , u   f ( x  iy , u ) dx  2 R n f ( x  iy ) dx
q 


and
p


e 
lim R n f  x  i y   , u  f ( x  iy, u) dx  0.
q 


q
p, r
On the other hand, since f  A
p
(D), the function ( y, u)  R n f ( x  iy, u dx
84
A. Temgoua Kagou
m
is integrable on {(y,u) : y  V+F(u,u) , u  C }
with
respect
to
the
measure
(y-F(u,u))-(2d-q)r dyd(u) . Hence, by the dominated convergence theorem, it follows that
Iq goes to 0 as q tends to infinity. Lemma C is proved.

Thus Theorem II.3 is entirely proved.
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1
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1
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
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85
The duals of Bergman spaces
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Department of Mathematics
Faculty of Science P.O. Box 812
University of Yaounde I
Yaounde (Cameroon)
86
A. Temgoua Kagou
87