The duals of Bergman spaces in Siegel domains of type II

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 Ap r, when 1 £ p < p0 . In the

symmetric case, we also characterize the dual of

A

p r, with p1 < p < 1 for some p1Î (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 Ap 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 Ap r, lorsque p1 < p < 1 , avec p1Î (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 dn 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 L2(D, dn) onto its subspace A2(D) consisting of holomorphic functions. Moreover, P is the integral operator defined on L2(D, dn) by the Bergman kernel B(z, 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.

Let r be a real number. Since D is homogeneous, the function z B(z , z) does not vanish on D, we can set:

a

(

D,B ( , )d ( )

)

L ) D ( Lp,r = p -r z z n z , 0 < p < ¥ .

Let p be a positive number. The weighted Bergman space Ap r, (D) is defined by Ap r, (D) =Lp r, ( )D ÇH D( ).

If r = 0, then Ap r, (D) is simply denoted Ap(D).

The weighted Bergman projection is the orthogonal projection of onto A . It is proved in [BT] that there exists a real number such that = {0} if e £ e and that for ,

P

is the integral operator defined on by the weighted Bergman kernel . In all our work, we shall assume that .

P

_e eD e + L2,e( )D A2,e L2,e( )D e > D 2,e_{( )} ) eD< 0 D ( eD D e >

c B

_e 1 e

z

z

( , )

The "norm"

.

_{p r} of (D), with , is defined by:

, A p r, _r D > e , p / 1 D ) z ( d ) z , z ( r B p ) z ( f r , p f ÷ ÷ ø ö ç ç è æ ò - n = fÎ

A

p r, (D).

Let r be a positive integer. S.G. Gindikin [G] has defined a differential polynomial

L

_r in D that satisfies the property:

(z , z Î D) .

Lr z z r r z

æ

èç öø÷ B( , )z =c B1+ ( , )z

A holomorphic function g in D is said to be a Bloch function in D if g satisfies the estimate:

g

g z B

z

z D z *

=

sup (

ìí

_î

)

( )

( , )

üý

_þ

< ¥

Î

-L

_r r

z

.

B r={Bloch functions}/_N.

For p ³ 1, the space (D) is the quotient space by N of the space of holomorphic functions g in D satisfying the estimate

C

r,rp ¥ < ÷ ÷ ø ö ç ç è æ n -r L r -ò = r p / 1 ) z ( d ) z , z ( r B p ) z ( g ) z , z ( B D ) D ( p r , C g .

For r = 0, the space _Cr,rp (D) is simply denoted _{C (D). r}p

In the upper half-plane, p+ = {z ÎC : Im z > 0}, R. Coifman and R. Rochberg proved the following fact: the dual of the Bergman space A1(p+) coincides with the Bloch space of holomorphic functions in p+, and can be realized as the Bergman projection of L¥ (p+). 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 II associated with self-dual cones, the dual space of A1 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] and [Z₁]) studied the dual of the Bergman spaces Ap 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 Î (4/3 , 4) , the dual of (R

Ap 3+ iG) , where G is the spherical cone in R3

,

is equal to Ap'(R3 + iG) ( p' is the conjugate exponent of p ) and when p Î (1 , 4/3] , the dual of Ap(R3 + iG) coincides with the space _Crp' (R3+ iG) with r = 1.

The purpose of this work is to extend these results to the weighted Bergman space (D) , 1 £ p < p

Ap r, 0, in homogeneous Siegel domains of type II, where p0 is a real

number greater than 2 and depends on the domain D ; when D is symmetric, we also show that the dual of Ap r, (D) , p₁ < p < 1 , where p1depends 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 r₀> 0 such that whenever r > r0 ,

1° when D is homogeneous and 0 < p £ 1 , B r is isomorphic to a subspace of the dual space

(

Ap r, (D)

)*

of Ap 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 r and consider the linear functional j on} (D) defined by Ap r, ) z ( d ) z , z ( p r 1 1 B ) z ( f ) z , z ( B D( )g(z) ) f ( n + -r -ò L_r = j

(

fÎAp r, (D)

)

. Since n <¥ + -ò p (z,z)d (z) r 1 1 B Df(z) Ap r,

for all f Î (D), it follows that j is bounded, hence belongs to ( (D))*and is represented by g .

Ap r,

Conversely, assume p = 1 . To obtain _{B r =}

(

A1,r(D)

)

*

,

let j Î

(

A1,r(D)

)

*

;

then there exists b Î L¥(D) such that ) z ( d ) z , z ( r B ) z ( f Db(z) ) f ( =_ò - n j , fÎ A1,r(D) . Since f(z)=c_r,ròDB1+r+r(z,z)f(z)B-r-r(z,z)dn(z), if we set

, ) ( d ) , ( r B ) ( b ) , z ( DB1 r , r c ) z ( g = _rò + +r z z - zz nz we easily get : ) z ( d ) z , z ( r B ) z ( f Dg(z) ) f ( =_ò - -r n j .

In fact, since r > r0 , g is a holomorphic function on D satisfying the estimate

{

}

sup ( ) ( , ) . z D g z B z z c b Î - _£ ¥ r

On the other hand , by a lemma of Trèves [Tr], there exists . Let h be the equivalence class of all holomorphic solutions of this equation. Then h belongs to _B , and the equality

~ ( ) ~ hÎH D such that Lrh=g r ) z ( d ) z , z ( r B ) z ( f D h(z) ) f ( =_ò L_r - -r n j

yields the equality

(

A1,r(D)

)

*

=

B_r.

We next assume that 0 < p < 1. Let us now define two homogeneous Siegel domains D0 and D1. Set

V0

=

l l l l l l l l l l l l l l l æ è ç ç ç ö ø ÷ ÷ ÷ - - > > > ì ï î ï ï ü ý ïï þ ï ï 11 12 13 12 22 0 13 0 33 11 12 2 22 13 2 33 0 ₂₂ 0 ₃₃ 0 : , = í ï _,

_.

Observe that V0 is a non self dual cone of rank 3. Define D0 by D0 = R 5

+iV0 . Then D0 is

a homogeneous,non symmetric, tubular domain. On the other hand, D1 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 =

(

)

ï ï þ ïï ý ü ï ï î ïï í ì > l l -l > l Î l l l = l 0 22 2 12 11 , 0 22 : 3 R 22 , 12 , 11 be the

45

spherical cone in R5, and consider the V1 - Hermitian form F1 in C defined by

(

)

F C C C u v uv 1 3 0 0 : ( , ) , , . ´ ® a Then D1 = ( , )z u C C : z z ( , ) . i F u u V Î ´ - - Î ì í ï îï ü ý ï þï 3 2 1

The restrictions on D, p and r are as follows : D is either a symmetric Siegel domain of type II, or D = D₀

,

or D = D₁ . Then there exists p₁ Î (0,1) such that for all p Î ( p1 ,1) , we have

(

A

p, r

(D)

)*

Ì

B . Here are the main steps of the proof. Let j be an element of

(

A

r

p, r

(D)

)*

and set a = rp + r+1 , with r large enough. Let f Î Ap, r(D). Then, in view of the molecular decomposition theorem [BT1] , there exists {li }Îl

p such that f z B z z B z z i i p i r p i i ( )=å ( , ) ( , ) + -l a 1 a ,

where {zi} is a lattice in D. Then we get

j j a a n ( )f = ò_D (B p(., ) ( )z f z B p z z d z( , ) ( ) -1 .

. function Bloch a by d represente is Hence . of choice the by ) z (., p B ) z ( g that such g exists there , Trèves of lemma same the by then , c holomorphi is ) z (., p B ce sin , Now j a ÷ ÷ ÷ ÷ ø ö ç ç ç ç è æ a j = r L r Î ÷ ÷ ÷ ÷ ø ö ç ç ç ç è æ a j B

To go further, let . Take r large enough ; then the function satisfies the estimate

hÎL¥( )D ) ( d ) nz ( h ) , z ( B ) z ( G za =_òD 1+r z z

{

}

z D G z B z z c h Î -¥ £ sup ( ) r( , )

and G is holomorphic in D. Hence, by the same lemma of Trèves, there exists a function gÎ B r such that

(1) (L_r)g=G.

Now, let P be the operator from into B r which to each assigns the

element g = Ph defined by (1). This operator P is called "Bergman projection" of

into _B for the following reason : although P is not the integral operator _P which is associated with the Bergman kernel B(V , z) , which has no meaning on , it is easy to show that for , the element Ph of _{B r} can be represented by _Ph .

L¥( )D hÎL¥( )D

L¥( )D

L

¥

( )

D

r

hÎL2ÇL¥( )D

The second aim of this paper is to show that the "Bergman projection" P is bounded from onto B r . In order to achieve this goal, we prove that for each real number r sufficiently large and for each GÎH(D) such that

L¥( )D

{

}

z D G z B z z Î - _{< ¥} sup ( ) r( , ) , one has the reproducing formula :

(zÎD). G( )z =c_ròDB1+r( , ) ( )z z G z B-r( , )z z dv( )z

Hence, for each g ÎB , if we set , then hÎ and Ph = g.

r h z( )=(L_r) ( )g z B-r( , )z z L¥( )D

The third aim of this work is to determine on a particular Siegel domain D2of type

II, a kernel K(V , z) that determines P in the following way : for each hÎL¥(D2 ) , a

representative of the element Ph of B is given by the function . 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

r

ò z n

za _DK( ,z)h(z)d (z)

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

ïþ ï ý ü ïî ï í ì Î -Î = ´ u u V i 2 z z : M M ) u , z ( D * * 2 , r 2 2 ,

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)((z,v),(z,u))ÎL1(D₂,dn(z,u)) ((V , v) Î D2 );

(3)

( )

L_{r z}B₀(( , ), ( , ))zv z u º0 . Thus Ph can be represented by :

Ph( , v) = _D (B - B₀ (hÎ L

2

z ò )(( , ),( , )) ( , )zv z u h z u dn( , )z u ¥(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 p0 Î (2,¥) such that

whenever p'0 < p < p0 (p'0 is the conjugate exponent of p0 ) , the dual of (D) is

equal to (D) , and when 1 < p < p

Ap r,

r,p'r

Ap r', 0 , the dual of (D) is equal to (D) with r > r

Ap r, C

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 when 0 < p

£ 1, and that B = (Theorem II.7) ; under some additional assumptions on D,

p and r, we prove that _B = (Theorem II.8). In section IV, we show that P = _{B r} (Theorem II.9). In section V, we determine a defining kernel B-B

r ( (Ap r, ( ))D * ) D ( r , ' p A r )* D = (A1,r( ))D * r ( C_r,p'r , _{( ))}* Ap r D L¥( )D (Ap r, 0 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 Ap r, ( ))D *= if p'0 < p < p0,

and ( ) (D) if 1 < p < p0 (Theorem II.11).

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.

ACKNOWLEGMENT: We would like to thank D. Békollé and A. Bonami for helpful conversations.

II. STATEMENTS OF RESULTS.

Let V Ì Rn , n ³ 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,..., R_k will stand for the n_k- dimensional Euclidean space.

(ii) Let GÌ Rs be a convex homogeneous cone which contains no straight line and let j be a homogeneous G - bilinear symmetric form defined on Rt´ Rt . The real homogeneous Siegel domain P = P (G, j) is defined by

P = P(G,j) = {(y,t) Î Rs ´ Rt: y - j(t,t) ÎG}. We shall denote by V(P) the homogeneous cone defined by:

V(P) = {(y,t,r) Î Rs ´ Rt´ R : r > 0 , (ry,t) Î P }.

In order to describe the canonical decomposition of the cone V, we consider at the first step, the cone V(1) = (0,¥) Ì R11. At the second step, we associate with V(1) and with a

homogeneous V(1) - bilinear symmetric form j(2) defined on R2 , the real Siegel domain

P(2) = Pæ_èçV( )1 ,j( )2 ö_ø÷Ì R11 ´ R2 and then, the convex cone

V( )2 =V P( ( )2 )ÌR₁₁´R₂´R₂₂ .

At the k th step, we associate with the cone V(k-1) and with a V(k-1) - bilinear symmetric form j(k) defined on Rk , a real Siegel domain P(k) =Pæ_èçV(k-1),j( )k ö_ø÷Ì R11 ´R2´...´Rk

and the cone V( )k =V P( ( )k )ÌR11´R2´R22´ ´... Rk´Rkk l k jj .

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 yields the following decomposition of the space Rn that contains V :

(4)

R

n

=

R

₁₁

´

R

₂

´

R

₂₂

´ ´

...

R

_l

´

R

_l ,

n

n

_i . i

= + å

=

l

l 2

Furthermore, the projection j_ii( )k of j(k) onto Rii (i < k) is a non-negative form. Thus is positive definite on a subspace R

j_ii( )k

ik of Rk with dim Rik = nik. We then have:

(5) Rk = ik , . 1 k 1 i R P -= nk i ni k = = -å 1 1

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 Î 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

Î G(V) such that (h(x)) j = 0 , j = 1,...,l . We set . The functions j defined for j

= 1,...,l , by

~

_{( )}

x

=

h x

c

cj( )x =x~ , j = 1,...,l , define the cone V in the following sense : a point x of

Rn belongs to V if and only if c_j(x) > 0, j = 1,...,l .

Since the decomposition (4) of Rn yields in a natural way the following decomposition of Cn :

, Cn=C₁₁´C₂´C_{22 ...}´ ´C_l ´C_ll

the functions cj , j = 1,...,l , can naturally be extended as rational functions on C n

. Let r=(r₁,...,r_l)ÎCl ; we define the rational function (z)

i r on Cnby : (z)r = P

( )

, z Î C j j z j =1 l c ( )r n.

For i = 1,...,l , set m_j n_j and

i j = > å d_i= - +(1 ni+mi 2 ) )

, and let d denote the vector

of Rl 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 product < , > defined on Rn with respect to the canonical decomposition of Rn by :

<x,y> = åx y_ii x y . i ii ii j i j< i j å + 2 j( )( , Then V* is defined by : V* =

{

xÎRn: <x y, > > " Î -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 c*_j the defining functions of V* . We have the following : (1 £ i < j £l ).

n*_ij=n_ij(V*)=n_{l- + - +j 1,l i 1}

( )

z_*r* on Cn by : qi=di 1 .

( )

z j j z j * = Õ₌ æèç * öø÷ * r* _{c ( )} r 1 l

The Siegel domain of type II, associated with the homogeneous cone V of Rn and a V-Hermitian, homogeneous form F : Cm´ Cm® Cn , is defined by :

D=D(V,F) = ( , )z u C C : z z ( , ) i F u u V n m Î ´ - - Î ì î ü ý þ 2 í .

The domain D is then an affine-homogeneous domain. Let F_ii denote the projection of F onto C_ii , and C(i) the complex subpace of Cm on which F_ii is positive definite. Set

; then and . We shall denote by q the vector of

N C C i m ( )

m

_i i

= å

=1 l

q

C

m

C

i i

=

=1 l

P

( ) l

whose components are q_i , i = 1,...,l .

We now recall the following two expressions of the Bergman kernel B((z,v),(z,u)) of D=D(V,F) :

II.1 PROPOSITION [G] . The Bergman kernel B((z,v),(z,u)) of D is given by

(

)

. d V q d ) ( ) u , v ( F i 2 z , exp c q d 2 ) u , v ( F i 2 z c ) u , z ( ), v , ( B l ò * ÷l -_* *+ * ø ö ç è æ_-<l z- _{- >} = -÷ ø ö ç è æz- -= z .

NOTATIONS. For , the notation 1+a stands for the vector

. Let ÎR a=(a₁,...,a_l) Rl a'=( ' ,...,a₁ a_'l) Î ) a ( +a₁,...,1+ _l l ; we set aa'= 1 1(a a' ,...,a a_{l l}' ) and by

53

a > a' , we mean that ai > a'i for all i Î {1,...,l}. For (z,v) and (z,u) in D Ì C n

´C m , we let b denote the kernel

b((z,v),(z,u)) = z -æ -èç öø÷ -z i F v u d q 2 2 ( , ) .

Notice that B = cb. Moreover, ba and b1+a , a Î Rl , will denote the expressions :

ba((z,v),(z,u)) = z a - - ö ø÷ -z i F v u d q 2 2 ( , ) ( ) æ èç and b1+a((z,v),(z,u)) = z a -æ èç öø÷ - + -z i F v u d q d q 2 2 2 ( , ) ( ) .

Let r be a vector of Rl . For p Î (0,¥) , we set and define the weighted Bergman space A

Lp r, ( )D =Lp( ,D b-r( , )z z dv z( ))

p,r

(D) by Ap,r(D) = Lp,r(D)ÇH(D). We equip Ap,r(D) with the Lp,r(D) - " norm" || || _p,r. The weighted Bergman projection P_r is the orthogonal projection of L2,r(D) onto A2,r(D). Recall (cf. [BT] ) that A2,r(D) = {0} when

r n d q i i i £ + -2

2 2( ) for some i Î{1,...,l } and otherwise, Pr is equal to the integral operator defined on L2,r(D)) by the weighted Bergman kernel c_r b1+r((z,v),(z,u)) .

Let us now state some prerequisite results :

II.2 THEOREM [BT]. Let a and e be in Rl and (z,v)ÎD. Then we have :

b v z u b z u z u d z u

D 1+

if and only if e_i i i n d q > + -2

2 2( ) and a e , i = 1,...,l. In this case, the following equality holds :

i i i i n d q - > -2 2( - ) b v z u b z u z u d z u D 1+ - c b v v ò a z(( , ),( , )) e( , ), ( , )) n( , )= _{a e,} a e z- (( , ), ( , )) . z 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:

II.3 THEOREM. Let r be a vector of Rl such that

i i i ) q d 2 ( 2 2 n r -+

> for all i = 1,...,l and p a

real number such that 1£ < ê -2 2 - 1+ ë ê ú û ú p n d q r n i i i

min ( ) ( i) . Then for all e Î Rl such that

ei i i i n d q p p r p > + - - + 2 2 2 1

( ) (i = 1,...,l ), the reproducing formula Pef = f holds for all f ÎAp,r(D).

II.4 PROPOSITION [BT]. Let aÎ Rl be such that a_i ³ 0, i = 1,...,l . Then : ba z(( , ), ( , )v z u £c b_a a z(( , ), ( , ))v z v

and

(

)

ba z( , )v +( ' , ' ), ( , )z v z u +( ' , ' )z v £c b_a a z(( , ), ( , ))v zv , for all (z,v), (z',v'), (z,u) and (z',u) in D.

II.5 LEMMA [R] . For all f Î Ap,r(D) (p > 0) , we have the estimate :

f z u p cb r z u z u f p r p ( , ) (( , ), ( , )) , £ 1+ _.

DEFINITION : A vector rÎ Rl is a V-integral vector if

( )

lr_** is a polynomial in l.

In the sequel, r will be a V-integral vector. We associate to r the differential polynomial (

L

_r)_z in Cn in the following way : for lÎ Cn,

(

L

_{r )z}exp(<l,z>) =

( )

lr_**exp(<l,z>) (zÎ Cn).

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 (L_r)_z g(z,v)=G(z,v) for all (z,v) Î D.

DEFINITION : A function g Î H(D) is a Bloch function if :

g z u D z g z u b d q z u z u * = Î -- + ì í ïï î ï ï ü ý ïï þ ï ï < ¥ sup ( , ) (Lr) ( , ) (( , ), ( , )) r 2 _.

Set N = {gÎH(D) : ( )g º 0 }. We define the Bloch space B r and the space (D) in the following manner, where we set

L_r

C

r,rp s= _{- +2d}r _q :

B r ={Bloch functions in D}

/

N ,

ï ï þ ïï ý ü ï ï î ïï í ì ¥ < ÷÷ø ö ççè æ _n ò Lr - s -= r Î p 1 ) u , z ( d )) u , z ( ), u , z (( D b p r p ) u , z ( f z ) ( p r f : ) D ( H f , C r

/

N. These two spaces have the following topological property:

.

_*

C

_r,rp _{. C ,rr}p ( )D r n d q i i i > + -2 2 2( ) , ,..., 1 i , 2 n_i i> = l r ( , )f g = ò_D(Lr)_{z g z u b}( , ) -s(( , ), ( , )) ( , )z u z u f z u b r z u- (( , ),( , ))z u dn( , )z u r = - +2d q.

LEMMA : (_{B , ) and (} (D) , ₎ are complex Banach spaces.

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 number and r a vector of Rl such that 0 < p £ 1 and , i = 1,...,l . Then the following assertions hold:

(i) B r is isomorphic to a subspace of (Ap,r(D))*,

(ii) _{B r is equal to A}1,r(D))* ) if r with equivalent norms.

The duality (A1,r(D)), _{B ) is given by}

, where s

II.8 THEOREM : Let D Ì CN be either a symmetric Siegel domain of type II, or D = D0, or D = D1. Let p1 =

2

2 1

N

N+ and let r be a vector of R such that r n d q i i i > + -2 2 2( ) , i = 1,...,l . Then for all p Î (p1 ,1) and all rÎ Rl such that

l ,..., 1 i , i ) q d 2 ( p i r 2 1 i ) q d 2 ( 2 2 i n i - + = + + + -+ + -> r

,

we have B r = (Ap,r(D))*, with equivalent norms. Moreover, the duality (Ap,r(D)), _{B r ) is given by}

) u , z ( d )) u , z ( ), u , z (( p r 1 1 b ) u , z ( f )) u , z ( ), u , z (( b ) u , z ( g z ) D ( ) g , f ( n + -s -ò L_r = , with . q d 2 + -r = s

REMARK: In the symmetric case, R.R. Coifman and R. Rochberg ([CR], p. 43-44) stated

that Ap,r(D)) and A r p 1 1, 1 ( ) - + +

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 Î L¥(D) and let r be a V-integral vector such that r_i > ni

2 (i = 1,...,l) . Then, in view of Theorem II.2, the following function G is holomorphic in D:

, G( , )zv =_òDb1+s(( , ),( , )) ( , )z v z u h z u dv z u( , )

where . q d 2 + -r = s ( , ) ( , ) g z u =G z u

By Lemma II.6, there exists such that

. Let g be the equivalence class of all holomorphic solutions of this equation. Then g Î B ; hence, we can define an operator P from L

~

_g

_Î

_{H D}

_{( )}

(L_{r z}) ~ r ¥(D) into B in the following way : r Ph = g.

P is called the "Bergman projection" of L¥(D) into _{B . Let us justify this name : the} Bergman projection P is usually the integral operator defined on L

r

2

(D) by : Ph(z ,v) =_òD B

(

(z,v),(z,u)

)

h(z,u)dn(z,u) (h Î L2(D)).

Now, let h Î _L2Ç _L¥(D) ; since (Lr z z) b(( , ),( , ))v z u =c b_r 1+s z(( , ),( , ))v z u (recall

s= r

-2d+q r

), one easily obtains that _{Ph is a representative of the element Ph = g of} B .

Our third result reads as follows:

II.9 THEOREM. Let D be a homogeneous Siegel domain of type II. Let r be a V-integral vector such that r_i > ni

2 (i=1,...,l). Then PL

¥_{(D) =}

B r and P has a bounded right inverse.

Theorem II.9 will be proved in section IV .

Our fourth result reads as follows :

II-10 THEOREM: Let D₂ z u M _{M r} z - z u u V 2i and = (5,5). =ì_íï Î ´ *- Î îï ü ý ï þï ( , ) _{2 2} * r Then

there exists a kernel b₀ in D₂ such that :

(i) with respect to (z , v) , b₀ ((z , v) , (z ,u)) is holomorphic in D₂ and (L_r)_z b0 ((z , v),(z,u)) º 0 ,

(ii) for all (z , v) ÎD₀ , (b- b₀ )((z , v) , (z ,u)) Î L1(D₂ , dn (z,u)) . Hence, for each h Î L¥(D2 ) , the function Ph defined on D2 by

(

)

(

)

Ph( ,v) = Dz ò ₂ b-b_{0 ( , ), ( , )}zv z u h z u d( , ) n( , )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 Rl such that (i=1,..., )

i q) 2(2d 2 1 n i r l -+ > and ïþ ï ý ü ïî ï í ì + - -i n i r i q) 2(2d 2 i 2n i min < p < 1 . Then we have the following assertions.

(i) If max i 2ni 2 2(2d q)i ri ni < p < mini 2ni 2 2(2d q)i ri ni + - -+ - -ì í ï îï ü ý ï þï + - -ì í ï îï ü ý ï þï 2 2 2( d _{q i ri}) , then (A p,r

(ii) (Ap,r (D))* = Cr,rp' ( )D whenever ri > n_i

2 , i = 1,...,l , with equivalent norms. The duality (Ap,r (D),C_r,rp' ( )D ) is given by

( , )f h _{D b}

(

( , ),( , )z u z u

)

( )

( , ) ( , )

(

( , ),( , )

)

( , ), zh z u f z u b r z u z u d z u

= ò -s Lr - n

where s = r -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

(i) Let g Î B_r and consider the linear functional j defined on Ap,r (D) by :

( )

(

)

(

)

r z z s z z z z z ( )f _D g( , )v b ( , ),( , ) ( , )v v f v b ( , ), ( , ) r p _{v v} = ò L - - + 1 1 j where s = r

-2d+q and 0 < p £ 1. By Lemma II.5, we have the estimate :

(

)

(

)

. r , p f c p 1 r , p f ) v , ( d ) v , ), v , ( r b p ) v , ( f D c ) v , ( d ) v , ( ), v , ( p r 1 1 b ) v , ( f D £ -z n ÷ø ö çè æ_{ò z} - _{z z} £ z n z z + -z ò

Hence j possesses the following property: | j(f)| £ c||g||_*

|

|f||_p,r ,

and thus ||j|| £ c||g||_* . Therefore j Î (A p,r

(D))

*

. This proves the inclusion of B_r into

(Ap,r (D))

*

.

(ii) Let p = 1, and let j be in (A1,r (D))

*

. Then by the Hahn-Banach theorem, there exists a bounded function k in D such that

(

)

j( )f = ò_{D k z u f z u b}( , ) ( , ) -r ( , ), ( , )z u z u dn( , ).z u

On the other hand, by Theorem II.3, we have the following reproducing formula for every f Î A1,r (D) :

(

(z,u),( ,v)

)

f( ,v)b r

(

( ,v),( ,v)

)

d ( ,v). r 1 b D ) u , z ( f =_ò + +s z z - -s z z n z

Therefore by the Fubini theorem, we easily get

(

)

(

)

(

(z,u),(z,u)

)

d (z,u). r b ) u , z ( f ) v , ( d ) v , ( ), v , ( r b ) v , ( k ) u , z ( ), v , ( r 1 b D D ) f ( n s -- ÷ø ö çè æ_ò + +s _{z z} - _{z z nz} ò = j

Set g(z,u) = _{. Observe next that}

g Î H(D) and that by Theorem II.2, the following estimate holds :

(

)

(

)

D b r z u v k v b r v v d v ò 1+ +s ( , ), ( , )z ( , )z - ( , ),( , )z z n z( , )

(

)

g z u( , ) £C k_¥bs ( , ), ( , )z u z u

(

(z,u) Î D

),

since ,i 1,...,l. i ) q d 2 ( 2 i n i >_{- -} =

s Then in view of Lemma II.6, there exists h in _Br such

that L_rh = g. Therefore

( )

(

)

j( )f _{D r} ( , ) ( , ) s ( , ), ( , ) n( , ) zh z u f z u b r z u z u d z u

= ò L

-and thus, j is represented by the element g of B_r . This proves the reverse inclusion of (A1,r (D))* in _Br. ð

We shall now prove another reproducing formula whose proof relies on the dominated convergence theorem. In view of this formula, the Bergman weighted projection P_r also reproduces functions which satisfy certain uniform estimate.

III.2 PROPOSITION : Let r and e be two vectors of Rl such that

l.. ,..., 1 i , i i ) q d 2 ( 2 2 i n i r and i ) q d 2 ( 2 i n i> _{- -} > +_- +e =

e Let G be in H(D) such that

{

}

sup ( ) ( , ) . z D G z b z z Î -e _{< ¥ Then P} rG = G.

PROOF : Consider the sequence {G_n } defined by

G n z G z ie n b z n ie ( )= æ + , , èç öø÷ aæèç öø÷

where the positive exponent a is to be specified later. There exists c > 0 such that for every z in D and every positive integer n :

G z ie n cb z ie n z ie n cb ie n ie n + æ èç öø÷ £ æèç + + öø÷ £ æ_èç ö_ø÷ e e , , ,

where the latter inequality is yielded by Proposition II.4 since ei> 0 . Then , when we

keep n fixed, the function za G z ie n + æ

èç öø÷ is bounded and the following inequality holds :

D G n z b r z z d z Cn D b z n ie b r _{z z d z} ò ( )2 - ( , ) n( )£ ò 2a_èçæ , ö_ø÷ - ( , ) n( ).

63

We choose a so that 2 1

2 2

a_{i ri} ni

d _{q i}

- - >

- ( - ) for all i = 1,...,l. Hence by Theorem II.2, the latest integral converges and furthermore, PrGn = Gn for all n .

On the other hand, by Proposition II.4, since ai > 0 and ei > 0, we have :

. c ) z , z ( b n ie z ( G ) z , z ( b ) z ( n G -e £ + -e £

Also, by Theorem II.2, under our assumptions

i ) q d 2 ( 2 i n i> _- -e and l, ,..., 1 i , i i ) q d 2 ( 2 2 i n i r +e = -+ > we get : D b r z b r d ò 1+ ( , )z - +e z z n z( , ) ( )< ¥.

Henceforth, by the dominated convergence theorem, we easily obtain the equality PrG = G.

REMARK : This result extends the one obtained in [B2] 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 Î (0, 1] . Let a and rbe vectors of Rl such that

i r 1 i ) q d 2 ( 2 i n i> _{- -} + +

a and for all i 1,...,l,

) q d 2 ( 2 2 n r i i

i> +_- = Then for all

jÎ(Ap,r(D))

*

and for all z0 in D, we have

( )

( )

(

)

( )

D bp z z b p z b p z z d z b p z ò æ è ç ç çç ö ø ÷ ÷ ÷÷ -= æ è ç ç çç ö ø ÷ ÷ ÷÷ a j a a n j a 0 1 0 , ., , ) ( ) ., .

PROOF : By Proposition III.2, it is sufficient to show that (6) sup z DÎ

( )

( )

j a a b p z b r p _{z z} ., , . æ è ç ç çç ö ø ÷ ÷ ÷÷ + -< ¥ 1

But since jÎ (Ap,r(D))

*

, we obtain that

( )

( )

j a a a b p z c b p z p r c b r p _{z z} ., (., ) , ' , æ è ç ç çç ö ø ÷ ÷ ÷÷ £ £ - -1 ,

where the latest estimate follows from Theorem II.2. This proves Corollary III.3 .

ÿ

III.4 PROOF OF THEOREM II.8

Let D Ì CN be a symmetric Siegel domain of type II, or let D be equal to D₀or

D1 . Remark that for p Î 2

2 11 2 2 2 N N and r R satisfying ri ni d _{q i} + æ èç öø÷ Î > + -, ( ) , l the domain

D satisfies the hypotheses of the molecular decomposition theorem [BT1] for functions in

Bergman spaces Ap,r (D) ; more precisely, for such values of p and r, there exist constants c = c(p,r) and C = C(p,r) such that for every f Î Ap,r(D), there exists an lp -sequence {li}

such that

(

)

f z _ibp z z_{i b} r p i zi zi ( )= ( , ) , + -= ¥ å l a 1 a 0 (zÎ D) ,

where {z_i} is a lattice in D and the following estimate holds :

c f _{p r}p C f

ip _{p r}p

, £ål < , .

Moreover, the vector a of Rl is defined as follows. Let r denote a V-integral vector in Rl such that when we sets= r

- +2d q, the following condition is satisfied:

l. ,..., 1 i p i r 2 1 1 i ) q d 2 ( 2 2 i n i = + + + -+ > s

The vector a is given by a = sp+1+r.

Let j Î(Ap,r (D))

*

. For f Î Ap, r (D), define the sequence {f_N} of functions in D by (7) f z _ibp z z_{i b}

(

)

r p i N zi zi N( )= ( , ) , + -= å l a 1 a 0 (zÎ D).

Then {f_N} converges to f in Ap, r (D). Hence j(f_N) goes to j(f) as N tends to infinity. Now, in view of Corollary III.3, we get :

( )

( )

( )

(

)

j a a j a a n b p .,_{zi D b}p z_{i z}, b p .,z b p z z d z, ) ( ) æ è ç ç çç ö ø ÷ ÷ ÷÷= ò æ è ç ç çç ö ø ÷ ÷ ÷÷ -1 (i Î Z₊),

and combining this with (7) yields for every positive integer N :

( )

( )

p(z,z)d (z). 1 b z ., p b ) z ( N f D N f n a -÷ ÷ ÷ ÷ ø ö ç ç ç ç è æ a j ò = j .

This easily implies that for every positive integer N :

(8)

( )

p(z,z)d (z). 1 b z ., p b ) z ( N f D ) N f ( n a -÷ ÷ ÷ ÷ ø ö ç ç ç ç è æ a j ò = j

other hand, it follows from (6) that

(

)

( )

(

)

(

f_N f

)

(z)pb r(z,z)d (z) D ' C ) z ( d ) z , z ( p r 1 1 b ) z ( f N f D C ) z ( d ) z , z ( p 1 b z ., p b ) z ( f N f D n -ò £ n + -ò £ n a -÷ ÷ ÷ ÷ ø ö ç ç ç ç è æ a j -ò

where the latest inequality follows from Lemma II.5. Henceforth, since {fN} converges to f

in Ap,r(D), we conclude that the right hand side of (8) tends to

( )

(

D f z b p z b p z z d z ò æ è ç ç çç ö ø ÷ ÷ ÷÷ -( )j ., , ) ( ) a a n 1

)

. We have then proved that for every f in Ap, r (D) :

( )

p(z,z)d (z). 1 b z ., p b ) z ( f D ) f ( n a -÷ ÷ ÷ ÷ ø ö ç ç ç ç è æ a j ò = j

Now,by Lemma II.6, since the function za j b p z

a (., ) æ è ç ç çç ö ø ÷ ÷

÷÷is holomorphic in D, there exists

g in H(D) such that L_rg(z) = bp(.,z) . Furthermore, with the choice of a and with

÷ ÷ ÷ ÷ ø ö ç ç ç ç è æ a j s=a- -1 r

p , we deduce from (6) that for every zÎ D, the following estimate holds :

sup z DÎ

( )

( )

j a s b p .,z b z z, . æ è ç ç çç ö ø ÷ ÷ ÷÷ - _{< ¥}

Hence g is a Bloch function in D, and g clearly represents the bounded linear functional

jÎ(Ap,r (D))

*

in the following manner :

j( )f _{D r}g z b( ) s(( ), ( ) ( )z z f z b ( , ) n( ). r

p _{z z d z}

= ò L - - +

1 1

Moreover, g_{* £ j . This proves the inclusion of (A}c p,r (D))

*

in B_r. Theorem II.8 is entirely proved. 

IV. PROOF OF THEOREM II.9.

Since PL¥(D) Ì B_r , let us prove that B_r Ì PL¥(D). Let g Î B_r and set

( )

(

)

h v g v b v v with d q ( , )z ( , ) ( , ),( , ) . r z z s z z s r = - = - + L 2 Then h Î L ¥_{(D) . We are}

going to show that Ph = g. It suffices to prove the equality :

( )

L

(

)

( )

L

(

)

r z zg( , )v = ò_{D b}1+s z( , ),( , )v z u r zg z u b( , ) -s ( , ), ( , )z u z u dn( , ).z u Setting e = s , this equality follows from Proposition III.2. To complete the proof, let us determine a right inverse of P. Define R as follows:

Rg( , )zv =

( )

L_{r z z}g( , )v b-s z

(

( , ), ( , )v zv

)

(

gÎB_r

)

.

Then PRg = g and Rg Î L¥(D). Furthemore Rg _L¥_{( )D} £ g_* and thus R £1 This . completes the proof of Theorem II.9. 

V. PROOF OF THEOREM II.10. V.1 Preliminaries on D₂.

The cone V =

{

z Î M₂, z Hermitian , z > 0

}

is of rank 2 and linearly equivalent to the spherical cone in R4 ; moreover, it is self-conjugate with respect to the inner product

< z ,z > = z₁₁z₁₁+ z₁₂z₁₂+ z₂₁z₂₁ + z₂₂ z_{22 ,}

where Set We also

have n . 2 M to belong 22 z 21 z 12 z 11 z z and 22 21 12 11 ÷÷ø ö ççè æ = ÷÷ø ö ççè æ z z z z = z e=æ è ç ö ø ÷ 1 0 0 1 .

12 = 2 , n1 = 0 , n2= 2 , d = (-2,-2) . The functions that define the cone V are

defined by , ₂( ) ₂₂

(

₂

)

22 21 12 11 ) ( 1 z =z -z_zz c z =z zÎM . c

Let F: M_r,2 ´ M_r,2 ® M ₂ be defined by F(v,u) = u

*

v where u = (u₁, u₂) and v = (v₁, v₂) are in Mr,2 with v₁ ,v2 in Cr (i = 1,2) . Hence F11 (v,u) = u1*v₁, F₂₂ (v,u) = u2* v2 and it is clear that F_iiis concentrated on

Cr ´ C r and is positive definite on Cr

(i = 1,2). Therefore q = (r,r).

Let r = (5,5) be a V-integral vector. It is straightforward that

( )

_{Lr z} ¶ ¶z ¶z ¶ ¶z ¶z =æ -è ç ç ö ø ÷ ÷ 2 11 22 2 12 21 5 and

( )

_{Lr z z}

(

)

r

(

)

r z B( , ), ( , )v z u =c B d q ( , ),( , )v z u + - + 1 2 _,

69

where B is the Bergman kernel of D₂. It then follows from Theorem II.2 that is integrable with respect to (z,u) in D

( )

_{Lr z z}B

(

( , ),( , )v z u

)

₂ since r = (r₁, r₂), r₁> 0, r2> 1. We are now ready to state the following lemma whose proof is just computational.

V.2 LEMMA : Let a1 and a₂be two integers such that a₁Î

{

-4,-3,-2,-1,0

}

or a₂Î

{

-3, -2,-1,0,1

}

. Then

( )

Lr zc₂a c a z2 ₁ 1 2 0 - - æ - -èç öø÷º z i u v *

* , where (z,u) and (z,v) belong to

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) u r r ie z i 2 4 1 5 2 8 2 c- + c- + æ -èç öø÷ ( ) ( ) * ; (b) u r r ie z i 25 1 5 2 9 2 c- +( )c- +( )æ_èç - *ö_ø÷ ; (c) u z r r ie z i 24 12 1 5 2 9 2 c- +( )c- +( )æ_èç - *ö_ø÷ ; (d) u z r r ie z i 2 4 21 1 5 2 9 2 c- +( )c- +( )æ_èç - *ö_ø÷ ; (e) u z r r ie z i 2 5 21 1 5 2 9 2 c- +( )c- +( )æ_èç - *ö_ø÷ ;

(f) u u r r ie z i 1 2 4 1 5 2 9 2 c- +( )c- +( )æ_èç - *ö_ø÷ ; (g) u u r r ie z i 1 25 1 5 2 9 2 c- +( )c- +( )æ_èç - *ö_ø÷ ; (h) u u r r ie z i 1 2 4 1 5 2 9 2 c- +( )c- +( )æ_èç - *ö_ø÷ .

We shall also use the following lemma.

V.4. LEMMA : Let (z, v) be in D₀. Then there exists two positive constants c₁and c2 such that for all (z,u) Î D₂, the following estimates hold :

a) c ie z i z i u v c ie z i 1 2c _èçæ -₂ ö_ø÷ £ c z2_èçæ -₂ - ö_ø÷ £ 2c2æ_èç -₂ ö_ø÷ * * * * ; b) c ie z i z i u v c ie z i 1 1c _èçæ -₂ ö_ø÷ £ c z1æ_èç -₂ - _ø÷ö £ 2 1c æ_èç -₂ ö_ø÷ * * * * .

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 C_D such that for all z , z ,

z' in D , B z B z CDd z z ( , ) ( , ' ) ( , ' ) z z - £1 whenever d(z,z') £ 10.

We are now ready to handle the proof of Theorem II.10 :

PROOF OF THEOREM II.10. We can define b0 in the following manner :

(

)

(

)

, 3 v * u i 2 * z 2 i 2 * z ie 2 i 2 * z ie ) r 7 ( 2 2 v * u i 2 * z 2 i 2 * z ie 2 i 2 * z ie ) r 6 ( 1 i 2 * z ie ) r 4 ( 2 v * u i 2 * z ) r 4 ( 2 i 2 * z ie ) r 5 ( 2 ) r 4 ( i 2 * z ie ) r 4 ( 2 v * u i 2 * z ) r 4 ( 2 ) 1 ( v * u i 2 * z ) 1 ( ) r 4 ( 1 v * u i 2 * z ) r 4 ( 1 ) u , z ( ), v , ( 0 b b ú ú û ù ÷÷ø ö ççè æ ÷ ø ö ç è æz- -c -÷ ø ö ç è æ -c ÷ ø ö ç è æ -+ -bc -÷÷ø ö ççè æ ÷ ø ö ç è æz- -c -÷ ø ö ç è æ -c ÷ ø ö ç è æ -+ -ac -÷÷ø ö ççè æ ÷ ø ö ç è æ -+ -c -÷ ø ö ç è æz- -+ -c ÷ ø ö ç è æ -+ -c + -÷ ø ö ç è æ -+ -c -ê ë é ÷ ø ö ç è æz- -+ -c ÷÷ ÷ ø ö çç ç è æ ÷ ÷ ø ö ç ç è æ -z + -c -÷ ø ö ç è æz- -+ -c = z -where

(

0,v₂

)

. 22 0 0 i ) 1 ( v , ) 1 ( and 2 ) r 6 )( r 5 )( r 4 ( , 2 ) r 5 )( r 4 ( ÷ ÷ ø ö ç ç è æ ÷÷ø ö ççè æ z = ÷ø ö çè æz + + + = b + + = a

Then b₀ ((z,v),(z,u)) is a holomorphic function of (z,v) in D₂. Furthermore, in view of

Lemma V.2 and the fact that c z₁ 1 2 1 ( ) _* * ( ) - -æ è ç ç ö ø ÷ ÷ z

i u v does not depend on z11 , z21 and z12 , we have

( )

Lr zb0

(

( , ),( , )z v z u

)

º This proves the first assertion. 0.

Using the identity a n b n b a

k n a k b n + + = -= å - + + -( 1) ( 1) _{( )} ( ) ( 0 1 _{1 k) and}

in view of Lemma V.4 , we have the estimate :

(

)

(

)

(

_{12 21 12} z_{21 21}z₁₂ 2 ₁₂ v₂ 2 ₁₂ u₂ v₁ 2v₁z₂₁ u₂ 4u₂ u₁v₁ v₂

)

. i 2 * z ie 2 4 1 1 v 1 u i 2 i 11 2 2 v 4 2 u i 2 i 22 i 2 * z ie ) r 8 ( 2 ) r 5 ( 1 c ) u , z ( ), v , ( 0 b b + + z + z + z + z + z z ÷ ø ö ç è æ -c + + -z ÷ ÷ ø ö ç ç è æ + -z ÷ ø ö ç è æ -+ -c + -c £ z

-Hence in view of Lemma V.3 , (b-b₀ ) ((z,v) ,(z,u)) is in L1(D₂, dn(z,u)). This completes the proof of Theorem II.10. ð

VI. PROOF OF THEOREM II.11.

VI.I. THEOREM [BT] . Let e and r be in Rl such that

i ) q d 2 ( 1 i> -e and , i ) q d 2 ( 2 2 i n i r -+

> i=1,…,l. Then P_e is bounded from Lp,r (D) into Ap,r (D) if

max ,..., , ( ) ( ) min_,..., ( ) . i ni d _{q i ri} ni d _{q i i} p _i ni d _{q i ri} ni = + - -+ - -ì í ï îï ü ý ï þï< < = + - -1 12 2 2 2 2 2 2 ₁ 2 2 2 2 l e l

I.2 PROOF OF THEOREM II.11. (i) In view of Theorems II.3 and VI.1, we have Pr f =

f for every f Î Ap,r (D), and P g_r c g

p r, £ p r, p r, . One easily obtains the announced

results from the Hahn-Banach and Riesz representation theorems.

(ii) Let jÎ (Ap,r (D))

*

. By the Hahn- Banach theorem , there exists g Î Lp',r (D) such that for all f Î Ap,r(D) , we have

(

)

j( )f = ò_{D g z u f z u b}( , ) ( , ) -r ( , ), ( , )z u z u dn( , ).z u Set s= r - +2d q. By Theorem II.3, Ps+rf = f , f Î A p,r (D) , then

(

)

(

(z,u),(z,u)

)

d (z,u) r b ) u , z ( f ) u , z ( g r , r T D ) u , z ( d ) u , z ( ), u , z ( r b ) u , z ( f r P ) u , z ( g D ) f ( n -+ s ò = n -+ s ò = j

where T_{s+r ,r} is the adjoint operator of P_s+rwith respect to the inner product < f , g >2, r =ò_{D f z u g z u b}( , ) ( , ) -r

(

( , ),( , )z u z u d

)

n( , ),z u

and it is expressed in the following way :

(

)

). D w ( ) u , z ( d ) u , z ( ), u , z ( r b ) u , z ( f )) u , z ( , w ( r 1 b D ) w , w ( b ) w ( f r , r T Î n -+ s + ò s -= + s Since r_i ni and p n d q i ri ni d _{q i ri} > > + - -+ - -æ è çç ö_ø÷÷ 2 2 2 2 2 2 2 2 ' max ( )

( ) , it follows from Theorem VI.1 that T

r r g p r cp r gp r

s + , _{' ,} £ , _{' ,} .

Clearly, the function za bs(z ,z ) T_s+r,rg(z) belongs to H(D) , then by Lemma II.6 , there exists h ÎH(D) such that L_rh(z)= bs(z,z) T_s+r,rg(z) for every z in D. Then h

ÎC_rp'_{, r}( )D and

Crp', r( )D

h £ jc . Thus, finally, h represents j in the following way :

(

)

( )

(

)

j( )f = ò_{D b}-s ( , ), ( , )z u z u L_r h z u f z u b r z u( , ) ( , ) - ( , ), ( , )z u dn( , ).z u

Conversely, let h Î C_rp'_{, r}( )D ; thenthe function za b- s(z ,z ) L_rh (z) belongs to

Lp',r (D) and the linear functional jdefined by

(

)

( )

(

)

j( )f = ò_{D b}-s ( , ),( , )z u z u L_r h z u f z u b r z u( , ) ( , ) - ( , ),( , )z u dn( , )z u (fÎAp,r(D)) is such that h p' _(D). r , c r £

j _C Hence jÎ (Ap,r (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

r be a vector of Rl such that ri >ni

2 for all i = 1,...,l. Then Cr,r is equivalent to p

Ap,r(D) in the following manner : every equivalence class in _C contains one and only one representative f belonging to A

r,rp ( )D

p,r

(D) , and conversely, every fÎAp,r(D) is a representative of an equivalence class in C . In fact, first let g Î . Set h(z)=b

r,rp ( )D

)

Cr,rp ( )D

s

(z,z)L_r g(z) and f = Pr

h ;

then by Theorem VI.I

,

f belongs to A p,r

(D) under our assumptions on p, since h belongs to Lp,r(D). Therefore L_rf = Pr+s

(

Lrg

). On the other

hand,

L_r gÎ Ap, sp+r(D) ; then Pr+s

(

Lrg

) =

Lr g . Hence Lrg = Lrf and then, f is a

representative of the class g in _Cr,rp (D .

), ( r 1 b ) , z ( h + h d ) , ( r r 1 b n h z + s + + _{z) dn(h)} ÷ø ö d ) z , z ( n s -rp, r(D)

Next, let f ÎAp,r(D) be such that L_rf º 0. Then f = Prf , and this implies that

Tr+s,rf(z) = b --s

(

z,,z) L_rf(z) º 0. By the Fubini theorem, one easily obtains that

(

)

1 b ) , ( r b ) z ( f D c ( r b ) z , ( D ) , ( r b ) ( f D c ) ( f r , r T r P h h -ò = çè æ_{ò z} -h h -h ò = z s +

where the latest equality follows from the reproducing formula P_r+s(b1+r(. , z) ) (h)= b1+r(h , z) for all h and z in D, since the function b1+r(. , z) belongs to Ap,r(D) and

ri > ni

2 for all i = 1,...,l . Finally, we conclude that Pr(Tr+s,rf) =Pr (f) = f, and since T_r+s,r f º 0, this implies that f º 0. Hence each class in C has only one representative in Ap,r(D) .

Conversely , let f Î Ap,r(D) ; then P_rf = f and for all z in D. Hence, by Theorem VI.1, T

Lr z s z zf( )b- ( , )=_Tr+s_{,r f}( )z , r+s,r f belongs to L p,r (D) since p d _{q i ri} and i ni > + - -+ - -ì í ï îï ü ý ï þï > max i 2ni 2 2(2d q)i ri ni 2 2 2( ) r 2 rp, r( )D

for all i = 1,...,l. Therefore f is a

representative of a class in C .

APPENDIX

PROOF OF LEMMA V. 3 .

We first remark that for all (z, u) Î D₂ :

(*) u ie z i 2 2 2 ₂ < c ( - *) .

By this remark (*), we have the estimate :

u r r ie z i r r ie z i 24 1 5 2 8 2 1 5 2 6 2 c- + c- + æ - c c èç öø÷ £ - + - + -æ èç öø÷ ( ) ( ) * ( ) ( ) * _.

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 d

(

d = 0 for (c) and (d) and d = 0.5 for (e)

),

the following function is integrable in D₂ :

g z u z r r ie z i d( , )= c- +( )c- + +( d)æ *- . èç öø÷ 12 1 5 2 7 2

To this aim, we shall use computational techniques due to Békollé in [B₂] . Let a1 and a2 be two real numbers such that 0 < a1 < 1 and 1 < a₂ < 3 + 2d, and consider the sets D₃ and D₄ defined by : D z u D z u r r ie z i D D D 3 2 1 4 ₁ 2 4 ₂ 2 4 2 3 = Î £ - + + - + + æ -èç öø÷ ì í î ü ý þ = ( , ) ( , ) ( ) ( ) * \ . : gd c a c a

In D₁ , |g_d (z, u)| is dominated by an integrable function in D₂by our assumption on a₁and a₂ and Theorem II. 2. In D₄ , it is obvious that

, ) u , z ( h c ) u , z ( g_d £ _d 2 ) u , z ( h where _d is equal to : h z u z r r ie z i d( , )= c( + -a ) / c-( -a + + d) / æ - * èç öø÷ 12 1 6 ₁ 2 2 10 ₂ 2 2 2 .

From the very definition of c₁and c₂ , we have :

h z u c z r r ie z i d( , )= _¶¶ c- -( a + ) / c- -( a + + d) / æ - * èç öø÷ ì í î ü ý þ 1 1 4 2 2 8 ₂ 2 2 2 .

Therefore one easily gets :

V. = V* of functions defining two are 11 ) ( * 2 and 11 21 12 22 ) ( * 1 where , d ) ( V 2 / ) 1 r ( * 2 ) ( 2 / ) 2 r 2 4 ( * 1 12 i 2 * z ie , exp c ) u , z ( h l = l c l l l -l = l c l l ò ÷l c -a + + d l c -a ø ö ç è æ_-<l - _> = d

Hence, by the Plancherel-Gindikin formula [BT] with Cm=M_r2, we get :

(

)

(

)( )

(

)

( )

. r ) ( 2 ) ( 1 c * q c ) u ( d u * u , 2 exp m C ce sin , d 2 12 2 2 2 2 12 22 11 4 2 1 2 11 Vexp 11 22 ' c V 1 r 2( )exp( 2 ,u*u d r * 2 2 r 2 2 * 1 2 12 e , exp m C d (u) c ) u , z ( d 2 ) u , z ( h 2 D -÷ø ö çè æ_c*_{l c}* _l = -* l = n > l < -ò l l d + a -÷ ø ö ç è æ_{l l - l} -d -a -a l ò l +l = ÷ ø ö ç è æ ò <l > l c -a + + dc -a + - l - <l > l ò n = n d ò

By a polar coordinate change of variables, the integration with respect to l12

converges if a2< 3 + 2d, and we have :

h z u d z u d d D d( , ) n( , ) exp( (l l ))l a l a d l l . 2 11 22 111 22 4 ₂ 2 11 22 0 0 2 = - + - - + ¥ ò ¥ ò ò

Henceforth, this integral converges by our assumption on a1 and a2 . This proves

assertions (c), (d) and (e) .

Following the same pattern, it is sufficient to prove that the function f_d(z, u) defined by : ÷ ø ö ç è æ -c c = - + - + +d d(z,u) u ie_2iz* f ) r 6 ( 2 ) r 5 ( 1 1

is integrable in D₂

(

for (f), take d = 1; for (g), take d = 0. 5 , and for (h), take d = 0

)

. The techniques used before imply that it is sufficient to prove that if 0 < a1 < 1 and 1 < a2 < 2 + 2d , we have : |f_d (z,u)| £ |K_d(z,u) |2 where K z u c u r r ie z i d c a c a d ( , )= - * - + _- - + + æ è ç ç ç ö ø ÷ ÷ ÷ -æ èç öø÷ 1 1 6 ₁ 2 2 8 ₂ 2 2 2 is square integrable in D2. It is clear that :

2 ( )d ; r 1 2 2 2 2 r 2 4 1 i 2 * z ie exp V 1 u c ) u , z ( K l l ÷ ÷ ÷ ÷ ø ö ç ç ç ç è æ -a + * c d + + a -* c ÷ ø ö ç è æ_-<l - _> ò = d then : ) u ( d d ) ( r 1 2 ) ( 2 r 2 2 1 2 1 u e , exp( c ) u , z ( d 2 ) u , z ( K 2 D n l l + a -* c l d + + a -òò -<l > c* = n d ò

(

)

(

)

(

( _{11 22})

)

₁₁ 1 ₂₂3 2 2d ₁₁d ₂₂. exp 0 0 c d 2 1 1 2 12 . 22 11 1 2 2 2 11 22 Vexp 11 22 c l l a -d + l a -l l + l -ò¥ -ò¥ = l d + a -÷ ø ö ç è æ _l -l l a -d -a + -l l ò - l +l =

This converges by our assumption on a1 and a₂ . 

PROOF OF THEOREM II.3.

We first prove the following lemma:

A LEMMA. Let pÎ[ 1, ¥ ) and let r be a vector of Rl such that r_i ³ -1 (i = 1,...,l). Let f Î Ap,r(D). Then for every (y,u) Î V ´ Cm such that y - F(u,u) Î V, the holomorphic function f_{y, u} defined on Rn +iV by

f_{y, u} (z) = f (z+iy,u)

belongs to the Hardy space Hp (Rn+iV). Moreover, there is a constant C = C(p,r) such that for all f Î Ap,r(D) and (y,u) Î V ´ Cm satisfying y - F(u,u) Î V, then

. f )) u , iy ( ), u , iy (( r 1 Cb p ) iV n R ( p H u , y f p r , p + £ +

Proof of Lemma A. First take (y,u) = (e,0). Let P be a polydisc centered at (ie,0) with

closure contained in D. Let (R1 ,...,R_n+m) be the multiradius of P. Then : (1) f x( +ie, )0 p£_{C P xò +( , )0} f(x h+i , )v pd d dx h n ) . (v

If P’ is the projection of P on iV ´ Cm , we obtain :

(2) _P' f( i ,v)pd d (v) d . 1 R x ) v ( d d d p ) v , i ( f ) 0 , x ( P x+ h x h n £òx- < çèæò x+ h h n ö÷ø x ò +

Combining (1) and (2) yields

( )

(

)

Rn f x ie pdx C R n Rn P f i v pd d v d

ò ( + , )0 £ ₁ ò ò _' (x h+ , ) h n( ) x.

Since P' is a compact set contained in

{

(ih,v)ÎiV´Cm:h-F(u,u)ÎV

}

(

c'_r£b-r (z h+i , ),(v z h+i , )v £c_r

, there are two positive constants c_r and c'_r such that

)

for all (ih,v) Î

P’ and xÎ Rn. Hence, for every f Î H(D) , the following holds :

(

(

)

)

. p r , p f r C p ) v , i ( f ' P n R r C dx p ) 0 , ie x ( f n R £ h + x ò ò £ + ò d ) v ( d d ) v , i ( ), v , i ( r _{x+ h x+ h h n x} -b

Next assume that (iy,u) is an arbitrary point of D. Since D is affine-homogeneous, there is g

Î G(D) such that g-1(x+iy,u) = (x+ie,0) for every x in Rn . Set h = fog. Then

(

( i ,v),( i ,v)

)

d d d (v). b ) v , i ( h C dx ) ie x ( h dx ) u , iy x ( f r p D r p R p Rn n n h x h + x h + x h + x ò £ + ò = + ò

-Make the change of variables x = x‘ , (ih,v) = g-1(ih‘ ,v’). Then the equality

(

(x iy,u),(x iy,u)

)

cb g det -2= + + yields that

(

) (

)

(

)

(

)

b i v i v b i v i v g cb i v i v b x iy u x iy u ( , ), ( , ) ( ' ' , ' ),( ' ' , ' ) det ( ' ' , ' ),( ' ' , ' ) ( , ), ( , ) . x h x h x h x h x h x h + + = + + = + + - + + 2 1 Furthermore, for r Î Rl, one can prove that

b-r

(

(x h+i , ),(v x h+i , )v

)

=_{cr b}-r

(

( 'x h+i ' , ' ), ( 'v x h+i ' , ' )v

)

br

(

(x+iy u, ), (x+iy u, )

)

. Hence:

(

)

R n f x iy u pdx c r b r x iy u x iy u f p r p ò ( + , ) £ ' + ( + , ), ( + , ) , 1 _. So, dx p ) u ), y ( i x ( f n R V sup p ) iV n R ( p H u , y f _ò + +h Î h = +

(

)

(

(x iy,u),(x iy,u)

)

f p_p,r. r 1 b r ' c p r , p f ) u ), y ( i x ( ), u ), y ( i x ( r 1 b n R V sup r ' c + + + = h + + h + + + ò Î h £

The last equality follows from Proposition II.4 since ri ³ -1 (i = 1,...,l ). 

The following corollary can be deduced from results in [SW] :

B COROLLARY. Let pÎ[ 1, ¥ ) and let r be a vector of Rl such that ri³ -1 (i = 1,...,l ).

Let (iy,u) Î D and y’ ÎV. Then for all f Î Ap,r(D) ,

R n f x i y y u pdx R n f x iy pdx

ò ( + ( + ' ), ) _{£ ò} ( + )

and

lim ' ' ( ( ' ), ) ( , ) y y Rn f x i y y u f x iy u pdx ® Î ò + + - + = 0 0 G .

PROOF OF THEOREMII.3 : Let e be a vector of Rl such that ei > n

d q i i + -2 2 2( ) (i = 1,...,l ). Recall that for every f Î A2, e(D) , P_ef is given by the following formula:

P f z u_e ( , )=c_{eòD b}1+e

(

( , ), ( , )) ( , )z u zv

)

f zv b-e z z

(

( , ), , )v v d

)

n z( , )v

(

(z,u) Î D

).

Hence

P_e extends as an operator on Lp,r(D) if

(

)

(

)

D b z u v p b r p p v v d v ò + - -æ è ç ö ø ÷ < ¥ 1 e _{( , ), ( , )z} ' e ' _{( , ), , )z z n z( , )} when p > 1 (resp. if sup

(

) (

)(

)

( , ) ( , ), ( , ) ( , ), , ) z e _z e _{z z} v D b z u v b r v v Î + -ì í î ü ý þ< ¥ 1

when p = 1) for all (z,u) Î D. By Theorem II.2

(

resp. by Proposition II.4

)

, this is the case if and only if

ei i i i n d q p p r p > + - - + 2 2 2 1 ( ) (i = 1,...,l ) and p when p > 1

[

resp. if 1+e i n d q r n i i i < = - - + ì í î ü ý þ min ,..., ( ) ( 1 2 2 1 l i)

i³ ei-ri ³ 0 (i = 1,...,l ) when p = 1

].

For p = 1, this reduces to the two

conditions r_i³ -1 and ei ³ r_i (i = 1,...,l ).

It suffices to show that for every (z ,v) Î D, under our assumptions on e, r and p , the bounded linear functional j_(z,v)defined on Ap,r(D) by

(

)

(

)

j z_{( , ) ( )v f} =f( , )zv - òc_{e D b}1+e z( , ), ( , ) ( , )v z u f z u b-e ( , ),( , )z u z u dn( , )z u which is identically 0 on Ap, r(D) Ç A2, e(D) , vanishes identically on Ap, r(D). The

desired conclusion followsfrom the following lemma :

C LEMMA : Let e and r be two vectors of Rl such that e_i> n

d q i i + -2 2 2( ) and ri > n d q i i + -2 2 2( ) (i = 1,...,l ). Then for every p Î [1,¥), the subspace Ap, r(D)ÇA2, e(D) is dense in Ap, r(D).

Proof of Lemma C. We use the following notations : for z = (x+iy) Î D , we write z n for x iy n u n + æ è ç ö ø ÷

, , and we write ie for (ie,0). Let f Î Ap, r(D). Let a be a positive

number to be chosen later. Consider the sequence {f_q} defined by f z c f z ie q b z q ie q( )= æ + , è ç ö ø ÷ æ è ç ö ø ÷ a a (z ÎD ) ,

where c_a = b- a (0, ie). We will show that for a large , f_q Î Ap, r(D) Ç A2, e(D) for every positive integer q, and that lim

, . q f _fq p r ® ¥ - = 0 The function z f z ie q a æ + è ç ö ø

÷ is bounded by Lemma II.5 and Proposition II.4. Take

a big enough so that the function z b z

q ie c b z qie a aæ , _a a( è ç ö ø ÷ = , ) belongs to Ap, r(D) Ç

A2, e(D) for every qÎ N. Next, by the Minkowski inequality, we have the estimate :

. p 1 ) z ( d ) z , z ( r b p ) z ( f q ie z f p ie , q z b c D p 1 ) z ( d p 1 ie , q z b c p ) z ( f D r , p q f f ú ú ú û ù ê ê ê ë é n -÷÷ø ö ççè æ ₊ ÷÷ø ö ççè æ a a ò + ú ú ú û ù ê ê ê ë é n -÷÷ø ö ççè æ a a ò £

-By Lemma II.4, there is a positive constant A_a such that for all z Î D and q Î N, we have

c b z q ie A a aæ , a è ç ö ø

÷ £ . Hence, by the dominated convergence theorem, the first integral on

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

I _{D f z} ie q f z p b r z z d z q= ò æ + è ç ö_ø÷ - ( ) - ( , ) n( )

goes to 0 when q goes to infinity. Set z = (x+iy, u) Î D and obtain that Iq is equal to

(

)

. n R V F(u,u) Rn dx y F(u,u) (2d q)rdy d (u) p ) u , iy x ( f u , q e iy x f ò _{ò +} ò + + - + - - - n ú ú ú û ù ê ê ê ë é ÷ ÷ ÷ ø ö ç ç ç è æ ÷÷ø ö ççè æ ÷÷ø ö ççè æ

Observe that Corollary B yields the following two facts :

dx p ) iy x ( f n R p 2 dx p ) u , iy x ( f u , q e y i x f n R ÷÷ø- + £ ò + ö ççè æ ÷÷ø ö ççè æ ₊ + ò and lim , ( , ) . q R n f x i y e q u f x iy u p dx ® ¥ò + + æ è ç ö ø ÷ æ è ç ö ø ÷ - + = 0

is integrable on {(y,u) : y Î V+F(u,u) , u Î Cm} with respect to the measure

(

y-F(u,u)

)

-(2d-q)r dydn(u) . Hence, by the dominated convergence theorem, it follows that I_q goes to 0 as q tends to infinity. Lemma C is proved. •

Thus Theorem II.3 is entirely proved.

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[B₁] D. Békollé : Le dual de la classe de Bergman A1 dans le transformé de Cayley de la boule unité de Cn . C. R. Acad. Sci. Paris 296 (1983), pp. 377-380 .

[B₂] D. Békollé : Le dual de la classe de Bergman dans le complexifié du cône sphérique, C. R. Acad. Sci. Paris 296 (1983), pp. 581-583 .

[B₃] D. Békollé : Le dual de l'espace des fonctions holomorphes intégrables dans des domaines de Siegel. Ann. Inst. Fourier (Grenoble) 34 (1984), pp. 125-154 .

[B₄] D. Békollé : The dual of the Bergman space A1 in symmetric Siegel domains of type II. Trans. Amer. Soc. 296 (1986), pp. 607-619 .

[B₅] D. Békollé : The Bergman projection of L¥in tubes over cones of real, symmetric, positive-definite matrices. Trans. Amer. Soc. 296 (1986), pp. 621-639.

[B₆] D. Békollé : Solutions avec estimations de l'équation des ondes . Séminaire d'Analyse Harmonique, 1983-1984, Publi. Math. Orsay (1985) pp.113-125.

[BBR] D.Békollé, A.Bonami & F. Ricci : Work in progress.

[BT] D. Békollé & A. Temgoua Kagou : Reproducing properties and Lp- estimates for Bergman projections in Siegel domains of type II. Studia Math. 115 (3) (1995), pp. 219-239.

[BT₁] D. Békollé & A. Temgoua Kagou : Molecular decompositions and interpolation. Integral Equations and Operator Theory (to appear)

[CR] R. R Coifman & R. Rochberg : Representation theorems for holomorphic and harmonic fonctions in Lp. Astérisque 77 (1980), pp. 11-66, Soc. Math. France .

[G] S. G. Gindikin : Analysis in homogeneous domains. Russian Math. Surveys 19 (4) (1964), pp. 1-89 .

[R] R. Rochberg : Interpolation in Bergman spaces. Mich. Math. J. 29 (1982), pp. 229-236 . [SW] E. M. Stein & G. Weiss : Introduction to Fourier Analysis on Euclidean spaces.Princeton Univ. Press, Princeton, New Jersey (1971) .

[T] A. Temgoua Kagou : Domaine de Siegel de type II : noyau de Bergman. Thèse de Doctorat de 3ème Cycle, Université de Yaoundé I (1993) .

[Tr] F.Trèves : Linear partial differential equations with constant coefficients. Mathematics and its applications, Vol. 6, Gordon Breach, New York, 1966 .

[Z] K. Zhu : Bergman and Hardy spaces with small exponents. Pacific J. Math. Vol. 162, No1 (1994), pp 189-199.

[Z1] K.Zhu : Holomorphic Besov spaces on bounded symmetric domains II. Indiana Univ. Math. J. 44 (4) (1995), pp 1017-1031.

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