<span class="mathjax-formula">$L^p$</span>-Boundedness of Bergman projections in tube domains over homogeneous cones

L

^p

-Boundedness of Bergman projections in tube domains over homogeneous cones

CYRILLE NANA ANDBARTOSZTROJAN

Abstract. In this paper, we generalize to all tube domains over homogeneous conesL^p-continuity properties of the Bergman projection.

Mathematics Subject Classification (2010):32A20 (primary); 32A10, 32A25, 32A36 (secondary).

1.

Introduction

Let Dbe a domain in

C

^nanddvthe Lebesgue measure defined in

C

^{n.We denote} by P the Bergman projection i.e., the orthogonal projection of the Hilbert space L²(D,dv)onto its closed subspaceA²(D,dv)consisting of holomorphic functions on D. It is well-known that, under weak assumptions, P is an integral operator defined onL²(D,dv)by

P f(z)=

D

B(z, w)f(w)dv(w),

where B(·,·)is the Bergman kerneli.e., the reproducing kernel of A²(D,dv).In this work we consider the Bergman projection in tube domains over homogeneous cones and we are interested in the values ofp≥1 such that the Bergman projection

Pcan be extended to a bounded operator onL^p(D,dv).

The L^p-boundedness of Bergman projections on tube domains over cones has been studied by many authors. In [1], D. B´ekoll´e and A. Bonami considered the tube domain over the forward light cone; they obtained some sufficient conditions using Schur’s Lemma. They proved that this condition is necessary and sufficient for the positive Bergman operator, that is, the Bergman operator with kernel |B(·,·)|.

Jointly with M. M. Peloso and F. Ricci they improved this result in [5]. To take care of cancellations, they introduced the mixed-normed spaces L^p,q.The consid- eration of the case p = 2 of these spaces has been used in [2] by D. B´ekoll´e, C. Nana was partially supported by African Fellows Program of IH ´ES-Schlumberger Foundation.

Received October 19, 2009; accepted in revised form July 22, 2010.

A. Bonami and G. Garrig´os to generalize this improvement to the case of gen- eral symmetric cones via a Fourier transform in thex variables. Indeed, they have found values of pfor which the Bergman projection is bounded while the positive Bergman operator is unbounded. Together with the first author of this paper, they presented all these results with more details in the Lecture Notes [3] of the Work- shop “PDE, Classical Analysis and Applications” held in Yaound´e in December 2001. Moreover, D. Debertol in [10] obtained the generalization of the sufficient conditions above for general weighted measures. We follow the same direction in this paper.

On the other hand, D. B´ekoll´e and A. Temgoua in [7] generalized results in [1]

to the case of Siegel domains of type II, not necessarily symmetric; again they applied Schur’s Lemma to the positive Bergman operator.

As it is proved in [3] or [5], it is important to mention that all these sufficient conditions are far from being necessary when the rank of the cone is greater than 1. However, in [4], an improvement has been obtained in the case of the forward light cone. This is pursued in [14], where G. Garrig´os and A. Seeger improved previous work of T. Wolff on the cone multiplier.

The aim of our work is the generalization of all the theory to tube domains over convex homogeneous cones. More precisely, we shall consider general weighted measures, which coincide in the case of symmetric cones with those obtained by D.

Debertol [10]. A particular case of this work has been done in [6] by D. B´ekoll´e and the first author, who considered the tube domain over the Vinberg cone. This is the simplest example of a non self-dual cone. In this case and in the case of rank 2,the sufficient conditions obtained for the positive Bergman operator are also necessary. We do not know whether this is the case for any arbitrary open convex homogeneous cone.

In this paper, we prove all the results mentioned above in the case of tube domains over homogeneous cones. The main difficulty of this work is to develop for all homogeneous cones, the necessary tools that are well known in the case of symmetric cones and of the Vinberg cone. Once this is done for any arbitrary homogeneous cone, one can easily proceed as in the previous cases. We deeply rely on the Vinberg’s description of homogeneous cones presented in [20].

This paper is divided into 8 sections. In Section 2, we give some geometric properties of homogeneous cones which are necessary to state our results. Section 3 is devoted to the statement of the results. In Section 4,we recall some useful results about homogeneous cones, such as the Whitney decomposition and the gamma function. Section 5 deals with Bergman spaces. In Sections 6 and 7,we give the proofs of results announced in the third section. The last section is devoted to some comments about necessary conditions.

ACKNOWLEDGEMENTS. We are grateful to A. Bonami for her critical observa- tions and fruitful discussions shared on this subject.

2.

Algebraic structure of homogeneous cones

LetV be an-dimensional real vector space andbe an open convex cone inV i.e., forx, y∈,andλ, µ >0,we haveλx∈andλx+µy ∈.We assume that does not contain straight lines and that it ishomogeneous, that is, the groupG() of all transformations of G L(V)which leave invariant acts transitively on . In [20], Vinberg described convex homogeneous cones as the cones of Hermitian positive matrices in aT-algebra. We recall the definition of aT-algebra.

Definition 2.1. A matrix algebra of rankr is a real algebra¹

U

bigraded by sub- spaces

U

i j, i,j =1, . . . ,r i.e.,

U

=

i,j

U

i j,such that

U

i j

U

j k ⊂

U

i k

and for j =l,

U

i j

U

lk =0.

As was recalled in [8], if we represent each a ∈

U

by the generalized matrix (a_{i j})^r_i,j=1,wherea_{i j} denotes the projection ofa onto

U

i j,then the representation ofabis given by the matrix product(a_{i j})(b_{i j}).

Definition 2.2. An involution of a matrix algebra

U

is a linear mapping:x →x of

U

onto itself that satisfies the following conditions:

(i) x=x; (ii) (x y)=yx;

(iii)

U

i j =

U

j i for allx, y∈

U

.

In its matrix representation, an involution corresponds to taking the transpose,i.e., (a)i j =a_{j i}.A consequence of the existence of an involution is that

n_{i j} =n_{j i}, (2.1)

where

n_{i j} =dim

U

i j.

Let

U

be an algebra with an involution.As in [19], we define the subspace of

“Hermitian matrices” in

U

^,

X

^{= {x ∈}

U

^{:x= x},} and

T

=

1≤i≤j≤r

U

i j,

the subalgebra of

U

consisting of upper triangular matrices.

1Associativity of the multiplication is not assumed.

We shall always assume that

U

ii =

R

^ciwherec2i =c_i.Letρdenote the unique isomorphism of

U

ii onto the algebra of real numbers

R

.For a matrixx ∈

U

,

x =^r

i=1

x_ii+

i=j

x_{i j}, we define its trace by

trx =^r

i=1

ρ(x_ii). (2.2)

Definition 2.3. A matrix algebra

U

with an involutionx →xis called aT-algebra if the following conditions are satisfied:

(i)

U

ii =

R

^{ci fori =1, . . . ,r;} (ii) forx_{i j} ∈

U

i j,c_ix_{i j} =x_{i j}c_j =x_{i j}; (iii) for allx,y∈

U

,tr(x y)=tr(yx);

(iv) for allx,y,z∈

U

,tr[x(yz)] =tr[(x y)z];

(v) ifx=0,then tr(x x) >0; (vi) for allt,u, v∈

T

,t(uv)=(tu)v;

(vii) for allt,u∈

T

^{,t(uu)=(tu)u.}

Remark 2.4. From (v) in the definition above the formula (x|y)=tr(x y)

defines a scalar product in

U

.Therefore a matrix algebra with an involution is Eu- clidean. Under this inner product,

U

^{i j} is orthogonal to

U

^{kl unless}(i,j)=(k,l).

We denote byethe unit element of the matrixT-algebra

U

,i.e., e=

r j=1

c_j. Let

H = {t ∈

T

:ρ(t_ii) >0, i =1, . . . ,r}

be the subgroup of upper triangular matrices whose diagonal elements are positive and let

(

X

)= {ss:s∈H} ⊂

X

.

Note that the product in His associative by (vi). The transformations

π(w):uu→(wu)(uw) (w,u∈H) (2.3) of(

X

⁾correspond to the left translations of (

X

⁾( [20, page 383]). Note that from properties (vi) and (vii) of Definition 2.3, forv, w∈H,

π(v)π(w)=π(vw). (2.4)

We have the following important result, due to Vinberg, which relates homogeneous cones toT-algebras.

Proposition 2.5 ([20, Proposition 1, page 384]). For every T -algebra

U

,the set (

X

)is a convex homogeneous cone in which, by(2.3), the group H acts linearly and transitively. Moreover, all convex homogeneous cones can be described as (

X

)for some T -algebra.

Therefore, in the sequel, we shall consider the open convex homogeneous cone defined by

= {ss:s ∈H}, where then-dimensional vector spaceV containingis

V = {x∈

U

:x=x}.

Since the mapping H s → ss ∈ is one-to-one, the group H acts simply transitively onby (2.3). Hence, by homogeneity, one can write=H·e,where we use the notation

π(t)e=t ·e

for allt ∈ H.As it is mentioned in [19], we have the factorization H = N A

where

N = {t ∈ H : ∀i, ρ(t_ii)=1}, A= {t ∈H : ∀i < j,t_{i j} =0}.

As in [13, page 14 and page 20], we introduce the following notation, related to the vector spaceV containing the homogeneous cone.We recall thatn_{i j} is the dimension of

U

i j;we define

n_i =

i−1

j=1

n_{j i} and

m_i = r j=i+1

n_{i j}; then

dimV =n=r + r

i=1

m_i =r+ r i=1

n_i. (2.5)

2.1. The equation of the cone

This is exactly what is done in [20, page 385]. In the T-algebra

U

^{of rankr},we consider the subspaces

U

^{k =}

1≤i,j≤k

U

^{i j,}

and with every elementx ∈V we associate a sequence of matrices tox^(k) ∈

U

^k,as follows

x^(r) = x x^(k−1) =

k−1

i,j=1

ρ(x_kk^(k))x_{i j}^(k)−x_{i k}^(k)x_{k j}^(k) ,

where we consider that the matrixx^(k−1)is formed from the second order “minors”

ofx^(k).We put

p_k(x)=ρ(x_kk^(k)), k =1, . . . ,r.

We notice that p_k(x) is a homogeneous polynomial of degree 2^r−k. In [15] the polynomials p_k are called thedeterminant-type polynomialsassociated to the cone andp₁is thecomposite determinant. Since the computation of the composite de- terminant is hard to carry, H. Ishi in [15, Proposition 1.4] gave recurrence relations between determinant-type polynomials p_k.Fork =1, . . . ,r andx∈,we put

Q_k(x)= _r p_k(x)

j=k+1p_j(x);

the functionsQ_k are homogeneous of degree 1.These functions are denoted byχk

in [13].

Lemma 2.6 ([20, Proposition 2, Chapter III]). The coneis determined by the inequalities

p_k(x) >0, k=1, . . . ,r. Also

= {x ∈V : Q_k(x) >0, k =1, . . . ,r}.

2.2. The adjoint cone

We consider the matrix algebra with involution

U

which differs from

U

only in its grading, and we put

U

i j =

U

r+1−i,r+1−j (i,j =1, . . . ,r).

It is proved in [20, Chapter 3, Section 6] that

U

^{is also aT}-algebra andV=V whereVis the subspace of

U

consisting of Hermitian matrices. The dual cone of the convex homogeneous coneis the set

^∗= {ξ∈V:(x|ξ) >0, ∀x∈\ {0}}.

The cone^∗is also convex and homogeneous and the group H= H= {t, t ∈H}

acts simply transitively in^∗.See [20, Chapter 1, Proposition 9]. Therefore, ^∗ = {tt,t ∈H}.

As previously, we write

(

U

^{)k =}

1≤i,j≤k

(

U

^{)i j}

and to every element ξ ∈ V we associate the determinant-type polynomials de- noted p^∗_k(ξ)of degree 2^r−kand the functions

Q^∗_k(ξ)= p^∗_k(ξ) _r

j=k+1p^∗_j(ξ). Thus

^∗= {ξ ∈V: Q^∗_k(ξ) >0, k=1, . . . ,r}.

In the sequel, we will use the following notations: for allα = (α1, α2, . . . , αr) ∈

R

^{r,x ∈andξ∈∗,}

Q^α(x)= r j=1

Q^α_j^j(x) and (Q^∗)^α(ξ)= r j=1

(Q^∗_j)^αj(ξ).

We putτ =(τ1, τ2,· · ·, τr)∈

R

^{r with} τi =1+1

2(m_i +n_i).

Forx ∈,we havex =t ·ewheret ∈ H.Then from [20, Chapter 3, Section 3]

we have

Q_j(x)=t²_{j j} j =1, . . . ,r. (2.6) Lety∈.We have, for j =1, . . . ,r

Q_j(π(t)y)= Q_j(x)Q_j(y), (2.7)

since, for y=s·e,by (2.4) and (2.6) we can write

Q_j(π(t)y)= Q_j(π(ts)e)=t²_{j j}s²_{j j}= Q_j(x)Q_j(y).

Therefore, for anys∈H,

Q_j(π(s)x)= Q_j(s·e)Q_j(x) j =1, . . . ,r, (2.8) and

Q^τ(π(s)x)=detπ(s)Q^τ(x), (2.9) since

detπ(s)= Q^τ(s·e). (2.10) See [20, page 388].

The above properties are also valid if we replace Q_j by Q^∗_j andx ∈ by ξ ∈^∗.

Definition 2.7. LetC be an open cone. We say thatC isself-dualifC = C^∗.A homogeneous cone that is self-dual is said to be asymmetriccone.

In the following examples, that have been treated in [15], we compute the determinant-type polynomials.

Example 2.8. The cone of positive-definite symmetric matrices. This is a symmet- ric cone. We describe the above concepts for the cone=Sym₊(r,

R

^),contained in the vector spaceV =Sym(r,

R

^).The matrix algebra of rankr is the usual alge- bra

U

= V = V and the involution the transpose map V X →^tX. The unit element ofV is the usual identity matrixe= I andc_j = D_j are diagonal matrices whose entries are 0 except for the jth which is equal to 1.Obviously,n_{i j} = 1,for alli,j ∈ {1, . . . ,r}.

In this example, the group

T

consists in the upper triangular matrices in G L(r,

R

⁾ and the factorization y = t · I is precisely the Gauss decomposition of a positive symmetric matrix. See [11, Chapter VI, Section 3]. The subgroup N consists of all triangular matrices in G L(r,

R

⁾with 1s on the diagonal, while Ais given by the diagonal matrices P(a)=diag{a₁, . . . ,a_r}.

Finally, for each matrix X = (x_{i j})1≤i,j≤r ∈ V, we define the matrixξ = (x_{r+1−i,r+1−j})1≤i,j≤r ∈V.Then

Q_j(X)= r+1−j(ξ) r−j(ξ) and

Q^∗_j(X)= r+1−j(X) r−j(X)

where Q_j and Q^∗_j are the functions defined above andj are the usual principal minors from linear algebra, that is, the determinant of the j × j symmetric subma- trices obtained by restriction to the first jcoordinates. Observe that

detX = r j=1

Q_j(X)= r j=1

Q^∗_j(X).

Henceforth

Sym₊(r,

R

)= {X ∈Sym(r,

R

):j(X) >0, j =1, . . . ,r}.

Example 2.9. Vinberg’s Cone. This is the simplest example of a convex homoge- neous non selfadjoint cone of rank 3, given by Vinberg in [20, page 397]. Consider the Euclidean vector space

U

^{=V of 3×}3 matrices with real entries given by

x=



x₁₁ x₁₂ x₁₃ x₁₂ x₂₂ 0 x₁₃ 0 x₃₃



;

clearly,n₂₃ = 0, n₁₂ = n₁₃ = 1 so thatm₁ = 2,m₂ = m₃ = 0 andn₁ = 0,n₂=n₃ =1.As in the case of real symmetric matrices above, the unit element ofV is the usual identity matrixe= I andc_j = D_j are diagonal matrices whose entries are 0, except for the jth entry which is equal to 1.We have

H =



s =



s₁₁ s₁₂ s₁₃ 0 s₂₂ 0 0 0 s₃₃



: s_{j j} >0, j =1,2,3



. For everyx ∈V,

p₃(x)=x₃₃, p₂(x)=x₃₃x₂₂ and p₁(x)=x₃₃x₂₂(x₃₃x₁₁−x₁₃² )−x₃₃² x₁₂², so that

Q₃(x)=x₃₃, Q₂(x)=x₂₂and Q₁(x)=x₁₁− x₁₂² x₂₂ − x₁₃²

x₃₃; the convex homogeneous cone is then equal to the set

{x ∈V : Q_j(x) >0, j =1,2,3} = {x∈V : x is positive definite}.

Note that sincen₂₃=0,the spaceVis identified with the set

ξ =(ξ₍2), ξ₍3)):ξ₍k)=

ξ11 ξ1k

ξ1k ξkk

,k=2,3

,

and the cone^∗ is the set of elementsξ = (ξ₍2), ξ₍3))such that both components are positive definite; the group His the set

H=

t =(t₍₂₎,t₍₃₎):t_(k)=

t₁₁ 0 t_1k t_kk

; k =2,3; t_{j j} >0, j =1,2,3

,

the unit element is e = (e₍₁₎,e₍₂₎) with e₍₁₎ = e₍₂₎ = 1 0

0 1

. Therefore, for ξ ∈V,we have

p₃^∗(ξ)=ξ11, p^∗₂(ξ)=ξ11ξ22−ξ₁₂² and p^∗₁(ξ)=(ξ11ξ22−ξ₁₂²)(ξ11ξ33−ξ₁₂²) so that

Q^∗₃(ξ)=ξ11, Q^∗₂(ξ)=ξ22− ξ₁₂² ξ11

and Q^∗₁(ξ)=ξ33− ξ₁₃² ξ11; observe that detξ =₃

j=1Q^∗_j(ξ),henceξis not positive definite.

Remark 2.10. From [13, page 19], a necessary and sufficient condition for a cone to be self-conjugate or self-dual is that alln_{i j}are equal wheni = j.Letddenotes this common dimension for the spaces

U

i j.Then for every symmetric cone of rank r,we havem_i =(r−i)dandn_i =(i−1)dso that from (2.5) above, we obtain

(r−1)d 2 = n

r −1.

In particular,d=1 for the cone of positive-definite symmetric matrices.

3.

Statement of the results

LetT=V+ibe the tube domain over the open convex homogeneous cone. For eachw∈T,

Q^−2τ(mw)dv(w)

is the invariant measure with respect to the group of automorphisms of T. Let ν = (ν1, ν2,· · ·, νr) ∈

R

^r. We denote by L_ν^p(T), 1 ≤ p ≤ ∞,the Lebesgue spaceL^p(T,Q^ν−τ(mw)dv(w)).

Theweighted Bergman space A_ν^p(T)is the closed subspace ofL_ν^p(T)con- sisting of holomorphic functions. In order to have a non-trivial subspace, we take ν =(ν1, ν2,· · · , νr)∈

R

^{r such thatνi > mi+}2ⁿⁱ, i =1, . . . ,r.²

The orthogonal projection of the Hilbert space L²_ν(T)on its closed subspace A²_ν(T)is theweighted Bergman projection P_ν.We recall thatP_νis defined by the integral

P_νf(z)=

T

B_ν(z, w)f(w)Q^ν−τ(mw)dv(w),

2If there isk∈ {1, . . . ,r}such thatνk≤^m₂^k,thenA_ν^p(T)= {0}.

where

B_ν(z, w)=d_νQ^ν−τ

z− ¯w i

is theweighted Bergman kernel i.e., the reproducing kernel ofA²_ν(T).

In this paper, we discuss boundedness ofP_νonL_ν^p(T)for values of pdiffer- ent from 2.Let us consider the positive Bergman operator P_ν⁺ defined onL²_ν(T) by

P_ν⁺f(z)=

T

|B_ν(z, w)|f(w)Q^ν−τ(mw)dv(w).

Theorem 3.1. The operator P_ν⁺is bounded on L_ν^p(T)when 1+ max

1≤i≤r ni

2

νi −^m₂ⁱ < p<1+ min

1≤i≤r

νi −^m₂ⁱ

ni 2

. Hence P_νis bounded for this range of p.

Recall that this theorem has been proved by D. B´ekoll´e and A. Temgoua in [7]. We give a new proof of this theorem within the framework ofT-algebra construction of convex homogeneous cones. This sufficient condition is also necessary for some open convex homogeneous cones, for example when the rank is 2 and for the case of Vinberg cone and its dual. (See [6, Theorem 1.1].) Moreover, for general symmetric cones, if we assume thatν=(ν, . . . , ν)∈

R

^r,then this sufficient condition is also necessary. (See [3, Theorem 4.10].)

Moreover, for the case of tube domains over symmetric cones and the tube domain over the Vinberg cone, the authors of [3] and [6] respectively established that there are values ofpfor whichP_νis bounded, butP_ν⁺is unbounded. We extend this result to the tube domain over open convex homogeneous cones. We have the following theorem, which is the main result of this paper.

Theorem 3.2.

i) When the Bergman projector is bounded from L_ν^p(T)to A_ν^p(T), we have 1+ max

1≤i≤r

n_i 2

νi+1+^m₂ⁱ +ⁿ₂ⁱ < p<1+ min

1≤i≤r

νi +1+ ^m₂ⁱ +ⁿ₂ⁱ

n_i 2

.

ii) The Bergman projector P_νextends to a bounded operator on L_ν^p(T)for 1+ max

1≤i≤r

ni 2

νi −^m₂ⁱ +ⁿ₂ⁱ < p<1+ min

1≤i≤r

νi − ^m₂ⁱ + ⁿ₂ⁱ

n_i 2

.

The necessary condition is not hard to prove. We describe the main ideas in the proof of the sufficient condition. As in [3] and [6], we must take advantage of the oscillations of the Bergman kernel. Hence, we are induced to use the Fourier

transform in thex variables and consequently to focus onL²-norms in these vari- ables. For this reason, we introduce mixed norms spaces. For 1 ≤ p,q ≤ ∞, letL_ν^p,q(T)=L^q(,Q^ν−τ(y)d y;L^p(V,d x))be the space of functions f onT such that

f_Lν^p,q_(T) :=

V

|f(x+i y)|^pd x ^q

p

Q^ν−τ(y)d y ¹_q

is finite (with obvious modification if p = ∞.) As before, we call A_ν^p,q(T)the closed subspace ofL_ν^p,q(T)consisting of holomorphic functions.

For p=2,we prove thatP_νis bounded onL²_ν^,q(T)when 2

1+ max

1≤i≤r n_i

2

νi− ^m₂ⁱ

<q <2

1+ min

1≤i≤r

νi −^m₂ⁱ

n_i 2

.

Then Theorem 3.2 follows by interpolation with Theorem 3.1. Note that in the case of symmetric cones, which Debertol considered, the necessary condition of the L²_ν^,q(T)-boundedness of the weighted Bergman projector P_ν has been left open.

We still have the same difficulty here. Nevertheless, we observe that, for the case of rank 2 and the Vinberg cone [6], the sufficient condition above is also necessary.

4.

Some useful results in a convex homogeneous cone

In this section, we recall some important facts about homogeneous cones such as the Riemannian structure that yields an isometry between the cone and its dual and the Whitney decomposition of the cone. Most of these results have been established in [3] and [6].

4.1. The Riemannian structureand its dual

We denote byϕ(respectivelyϕ_∗) thecharacteristic functionof the cone(respec- tively^∗); then forx∈andξ ∈^∗,

ϕ(x)=

^∗e^−(x|ξ)dξ and ϕ_∗(ξ)=

e^−(ξ|x)d x.

Recall that the gradient of a differentiable function f at the pointx ∈

R

^{nis defined} by

(∇f(x)|u)= D_uf(x)= d

dt f(x+tu) t=0

for allu∈

R

^n.

Forx ∈we definex∈^∗by

x = −∇logϕ(x).

Similarly, forξ ∈^∗we define

ξ = −∇logϕ_∗(ξ).

Note that for each x ∈ andξ ∈ ^∗, we havex = x andξ = ξ.(See [20, Chapter 1, Section 4].)

Forx =t ·e,

(x|π(t)y)=(−∇logϕ(x)|π(t)y)= − d

du logϕ(x+uπ(t)y) u=0

= − d

du logϕ(π(t)(e+uy)) u=0

= − d

du log(ϕ◦π(t))(e+uy) u=0

= − d

du logϕ(e+uy) u=0

= (e|y) so that

x =t⁻¹·e. Moreover, for allt ∈H,we have

Q_j(t·e)Q^∗_j(t⁻¹·e)=1,

where j = 1, . . . ,r. Let e₀ be the unique fixed point of the map σ : x → x, (cf.[11, Proposition I.3.5]). Sinceis a homogeneous cone, everyx ∈can be written asx =π(t)e₀;therefore,x=π(t⁻¹)e₀and by (2.8) we have

Q_j(x)Q^∗_j(x)=Q_j(e₀)Q^∗_j(e₀)=constant (4.1) for j =1, . . . ,r.

Since the function logϕis strictly convex (cf.[11, Proposition I.3.3]), the sym- metric bilinear form on

R

ⁿ

G_x(u, v)= D_uD_vlogϕ(x) (respectively G_ξ(u, v)= D_uD_vlogϕ_∗(ξ)) whereu, v ∈

R

^{n defines on}(respectively^∗) a structure of Riemannian mani- fold. The corresponding Riemannian distances are given by

d(x,y)=inf

γ

1 0

G_{γ (t)}(γ (˙ t),γ (˙ t))dt

and

d_∗(ξ, η)=inf

γ^∗

₁

0

G^∗_γ∗(t)(γ˙^∗(t),γ˙^∗(t))dt

,

where the infimum is taken on the smooth pathsγ : [0,1] →(respectivelyγ^∗ : [0,1] →^∗) such thatγ (0)=x, γ (1)=y(respectivelyγ^∗(0)=ξ, γ^∗(1)=η).

The Riemannian distancesdandd_∗are invariant under the action ofG()and G(^∗)respectively,i.e.,

∀x, y∈, ∀g∈G(), d(gx,gy)=d(x,y) and

∀ξ, η∈^∗,∀g∈G(), d_∗(g^∗ξ,g^∗η)=d_∗(ξ, η).

(See [11, pages 15-16].) We have the following:

Theorem 4.1. The mapσ : x →xbetween the Riemannian manifoldsand^∗ is an isometry; that is

d_∗(x, y)=d(x,y).

4.2. The invariant measure on

Since we also have the identification^∗ ≡ H·e,we deduce from (2.9) that the measure

dm(x)= Q^−τ(x)d x (respectivelydm_∗(ξ)=(Q^∗)^−τ(ξ)dξ) is H-invariant on(respectivelyH-invariant on^∗).

Lemma 4.2. Givenλ >0,there is a constant C =C(λ) >0such that:

i) if d(y,t)≤λthen_C¹ ≤ ^Q_Q^j_j⁽₍^y_t)⁾ ≤C for all j =1, . . . ,r and x, y ∈; ii) if d_∗(ξ, η)≤λthen _C¹ ≤ ^Q_Q^∗∗^j(ξ)

j(η) ≤C for all j =1, . . . ,r andξ, η∈^∗. Letλ >0, y∈(respectivelyξ∈^∗) andd(respectivelyd_∗) theG()-invariant (respectivelyG(^∗)-invariant) distance defined in(respectively^∗). We denote by

B_λ(y)= {x ∈:d(y, x) < λ}

and

B_λ^∗(ξ)= {η∈^∗ :d_∗(η, ξ) < λ}

thed-ball (respectivelyd_∗-ball) centered at the pointy(respectivelyξ) with radius λ.

Lemma 4.3. Let0< λ <1.Then

m(B_λ(y))∼λⁿ and m_∗(B_λ^∗(ξ))∼λⁿ.

Proof. By theG()-invariance of the distance, we have, for allt ∈ H, B_λ(e) = t ·B_λ(e),so thatm(B_λ(y))=m(B_λ(e))for ally∈.We have

m(B_λ(e))=

B_λ(e)dm(y)=

B_λ(e)Q(y)^−τd y∼

B_λ(e)d y.

It is well known that the distancedis equivalent to the Euclidean distance on com- pact subsets ofV (cf.[17]); hence there are two positive constantsc₁andc₂ such that

{y∈V : |y−e| ≤c₁λ} ⊂ B_λ(e)⊂ {y ∈V : |y−e| ≤c₂λ}

and the result follows.

4.3. The Whitney decomposition of the cone

We give now the Whitney decomposition of the cone , which is obtained, for instance, as in Lemma 3.5 of [6].

Lemma 4.4. Given0 < λ <1,there exists a sequence{y_j}j of points ofsuch that the following three properties hold:

i) the balls Bλ

2(y_j)are pairwise disjoint;

ii) the balls B_λ(y_j)form a covering of;

iii) there is an integer N = N()such that every y ∈ belongs to at most N balls B_λ(y_j).

Remark 4.5. This lemma is also true for the dual cone^∗.

Definition 4.6. Sequences {y_j}j (respectively {ξj}j) of points of (respectively ^∗) that satisfy properties of Lemma 4.4 are calledλ-lattices of (respectively ^∗.).

The family{B_λ(y_j)}j (respectively{B_λ^∗(ξj)}j) is called theWhitney decompo- sitionof the cone(respectively^∗).

Proposition 4.7. The sequence{y_j}j is a λ-lattice of if and only if {y_j}j is a λ-lattice in^∗.The sequence{y_j}j is called the dual lattice of theλ-lattice{y_j}j. Lemma 4.8. Let(y₀, ξ0)∈×^∗;then

|B_λ(y₀)| =C_λQ^τ(y₀) and |B_λ^∗(ξ0)| =C_λ(Q^∗)^τ(ξ0). (4.2) Proof. We know thaty₀ =t·ewitht ∈ H;if we use the change of variablesy= π(t)x, d y = Q^τ(y₀)d x and since the distanced is G()-invariant, d(y, y₀) = d(π(t)x, π(t)e)=d(x,e).Hence,

|B_λ(y₀)| = Q^τ(y₀)

B_λ(e)d x =C_λQ^τ(y₀).

The same argument holds forB_λ^∗(ξ0).

Proposition 4.9. Let y ∈ (respectively ξ ∈ ^∗). There is a constant γ = γ (, ^∗)≥1such that

1

γ < (y|ξ) (y|ξ0) < γ

respectively 1

γ < (y|ξ) (y₀|ξ) < γ

wheneverξ ∈B_λ^∗(ξ0)(respectively y ∈B_λ(y₀)).

Corollary 4.10. Let(y₀, ξ0)∈×^∗.There is a constantγ >0such that n

γ ≤(y|ξ)≤nγ for all(y, ξ)∈B_λ(y0)×B_λ^∗(ξ0).

Lemma 4.11. There is a constant c>0such that for all t ∈H, π(t)x ≤c(t·e|e)x,

where x ∈.

Proof. Let us first remark that, since the function(x,y) ∈

U

×

U

→ x y is continuous, there isC >0 such that, for allx,y∈

U

,

x y ≤Cxy. (4.3)

Letx∈.Thenx =s·ewiths∈ H.Then, by (2.4) and (4.3),

π(t)x = π(ts)e = (ts)(ts) ≤C³t²s²=C³(t·e|e)(s·e|e).

Applying the Cauchy-Schwarz inequality, we obtain (s·e|e)=(x|e)≤√

rx; hence

π(t)x ≤C³√

r(t ·e|e)x. 4.4. The gamma function of a homogeneous cone

The following lemmas are given in order to define the holomorphic determination of the logarithm of Q^α_j^j and hence define the gamma function of and^∗. We consider once more theT-subalgebras

U

^k =

1≤i,j≤k

U

i j, with the units

e_k =c₁+ · · · +c_k,

and we denote by^(k)the associated open convex homogeneous cone.