Holomorphic L p -type for sub-Laplacians on connected Lie groups

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Holomorphic L ^p -type for sub-Laplacians on connected Lie groups

Jean Ludwig

, Detlef Müller

, Sofiane Souaifi

aUniversité de Metz, Mathématiques, Ile du Saulcy, 57045 Metz Cedex, France bMathematisches Seminar, C.A.-Universität Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany cIRMA, UFR de Mathématique et d’Informatique de Strasbourg 7, rue René Descartes, 67084 Strasbourg Cedex,

France

Received 2 April 2004; accepted 12 June 2008 Available online 26 July 2008 Communicated by P. Delorme Dedicated to the memory of Andrzej Hulanicki

Abstract

We study the problem of determining all connected Lie groupsGwhich have the following property (hlp): every sub-LaplacianLonGis of holomorphicL^p-type for 1p <∞,p=2. First we show that semi-simple non-compact Lie groups with finite center have this property, by using holomorphic families of representations in the class one principal series ofGand the Kunze–Stein phenomenon. We then apply an L^p-transference principle, essentially due to Anker, to show that every connected Lie groupGwhose semi- simple quotient by its radical is non-compact has property (hlp). For the convenience of the reader, we give a self-contained proof of this transference principle, which generalizes the well-known Coifman–Weiss principle. One is thus reduced to studying those groups for which the semi-simple quotient is compact, i.e. to compact extensions of solvable Lie groups. In this article, we consider semi-direct extensions of exponential solvable Lie groups by connected compact Lie groups. It had been proved in joint work by W. Hebisch and the two first named authors that every exponential solvable Lie groupS, which has a non-∗

regular co-adjoint orbit whose restriction to the nilradical is closed, has property (hlp), and we show here that (hlp) remains valid for compact extensions of these groups.

©2008 Elsevier Inc. All rights reserved.

✩ This work has been supported by the IHP network HARP “Harmonic Analysis and Related Problems” of the European Union.

* Corresponding author.

E-mail addresses:[email protected] (J. Ludwig), [email protected] (D. Müller), [email protected] (S. Souaifi).

0022-1236/$ – see front matter ©2008 Elsevier Inc. All rights reserved.

doi:10.1016/j.jfa.2008.06.014

Keywords:Connected Lie group; Sub-Laplacian; Functional calculus;L^p-spectral multiplier; Symmetry; Holomorphic L^p-type;L^p-transference by induced representations

Contents

1. Introduction . . . 1298

2. The semi-simple case . . . 1302

2.1. Preliminaries . . . 1302

2.2. A holomorphic family of representations ofGon mixedL^p-spaces . . . 1307

2.3. A holomorphic family of compact operators . . . 1311

2.4. Some consequences of the Kunze–Stein phenomenon . . . 1314

2.5. Proof of Theorem 1.1 . . . 1315

3. Transference forp-induced representations . . . 1318

3.1. p-Induced representations . . . 1318

3.2. A transference principle . . . 1322

4. The case of a non-compact semi-simple factor . . . 1327

5. Compact extensions of exponential solvable Lie groups . . . 1328

5.1. Compact operators arising in induced representations . . . 1328

5.2. Proof of Theorem 1.3 . . . 1333

Acknowledgment . . . 1336

Appendix A. On the spectra of sub-Laplacians of holomorphicL^p-type . . . 1336

References . . . 1337

1. Introduction

A comprehensive discussion of the problem studied in this article, background information and references to further literature can be found in [14]. We shall therefore content ourselves in this introduction by recalling some notation and results from [14].

Let (X, dμ) be a σ-finite measure space. If A is a self-adjoint linear operator on theL²- spaceL²(X, dμ), with spectral resolutionA=

Rλ dEλ, and ifF is a bounded Borel function onR, then we call F anL^p-multiplier for A(1p <∞), ifF (A):=

RF (λ) dEλ extends fromL^p∩L²(X, dμ)to a bounded linear operator onL^p(X, dμ). We shall denote byMp(A) the space of all L^p-multipliers forA, and by σ_p(A)the L^p-spectrum of A. We say thatAis of holomorphic L^p-type, if there exist some non-isolated pointλ₀ in the L²-spectrum σ₂(A) and an open complex neighborhoodUofλ₀inC, such that everyF ∈Mp(A)∩C_∞(R)extends holomorphically toU. Here,C_∞(R)denotes the space of all continuous functions onRvanishing at infinity.

Assume in addition that−Agenerates a symmetric diffusion semigroup (as in the case of the sub-Laplacians that we shall examine later). Then, ifAis of holomorphicL^p-type, one can show that

σ₂(A)=σ_p(A).

Moreover, if this semigroup is hypercontractive (as, for instance, in the case whereAis a sub- Laplacian on a unimodular Lie group), then the setUbelongs to theL^p-spectrum ofA, i.e.

U⊂σ_p(A), (1.1) and

σ2(A)σ_p(A) (1.2)

(see [8]).

Throughout this article,Gwill denote a connected Lie group.

Letdgbe aleft-invariantHaar measure onG. If(π,Hπ)is a unitary representation ofGon the Hilbert spaceH=Hπ, then we denote the integrated representation ofL¹(G)=L¹(G, dg) again byπ, i.e. π(f )ξ :=

Gf (g)π(g)ξ dg for every f ∈L¹(G),ξ ∈H. ForX∈g, we de- note by dπ(X) the infinitesimal generator of the one-parameter group of unitary operators t→π(expt X). Moreover, we shall often identifyXwith the correspondingright-invariantvec- tor fieldX^rf (g):=limt→01

t[f ((expt X)g)−f (g)]onGand writeX=X^r.

LetX1, . . . , Xkbe elements ofgwhich generategas a Lie algebra, which just means that the corresponding right-invariant vector fields satisfy Hörmander’s condition. The corresponding sub-LaplacianL:= −k

j=1X²_j is then essentially self-adjoint onD(G)⊂L²(G)and hypoel- liptic.

As has been pointed out to us by W. Hebisch, if such a right-invariant sub-LaplacianLis of holomorphicL^p-type, then properties (1.1) and (1.2) hold true forA:=L, even if the groupG is non-unimodular.

This is due to the existence of approximations to the identity by convolution from the right by smooth functions with compact support (cf. Lemma A.2 in Appendix A).

Denote by{e^{−t L}}t >0 the heat semigroup generated byL. Since Lis right G-invariant, for everyt >0,e^{−t L}admits a convolution kernelh_t such that

e^{−t L}f=ht∗f,

where∗denotes the usual convolution product inL¹(G). The function(t, g)→ht(g)is smooth on R>0×G, since the differential operator _∂t^∂ +L is hypoelliptic. Moreover, by [19, The- orems VIII.4.3 and V.4.2], the heat kernel ht as well as its right-invariant derivatives admit Gaussian type estimates in terms of the Carnot–Carathéodory distanceδassociated to the Hör- mander systemX1, . . . , Xk.

In particular, for every right-invariant differential operator D on G, there exist constants cD,t, CD,t>0, such that, for allg∈G,t >0,

Dh_t(g)C_D,te^{−cD,tδ(g,e)2}. (1.3) Let nowF₀∈Mp(L). By duality, we may assume that 1p2. WithF₀, also the function λ→F (λ):=e^−λF₀(λ)lies inMp(L), sinceF (L)=e^−LF₀(L), where the heat operatore^−Lis bounded on everyL^p(G) (1p <∞). Now by [14, Lemma 6.1], the operatorF₀(L)is bounded also on all the spacesL^q(G),pqp. Hence for every test functionf onG,

F (L)(f )=F₀(L)

e^−L(f )

=F₀(L)(h₁∗f )

=F₀(L)(h_1/2∗h_1/2∗f )=

F₀(L)h_1/2

∗h_1/2∗f,

by the right-invariance of the operator F₀(L). Since h_1/2 is contained in every L^q(G), 1 q∞, in particular inL¹(G), we see that the operatorF (L)acts by convolution from the left with the function(F₀(L)h_1/2)∗h_1/2which is contained in everyL^q(G),pqp, and so are all its derivatives from the right. We can thus identify the operatorF (L)with theC^∞-function F (L)δ:=(F₀(L)h_1/2)∗h_1/2, i.e.

F (L)(f )= F (L)δ

∗f, f ∈

pqp

L^q(G).

Recall that themodular functionΔ_GonGis defined by the equation

G

f (xg) dx=Δ_G(g)⁻¹

G

f (x) dx, g∈G.

Put, forg∈G:

f (g)ˇ :=f g⁻¹

, f^∗(g):=Δ⁻_G¹(g)f

g⁻¹ .

Thenf →f^∗is an isometric involution onL¹(G), and for any unitary representationπ ofG, we have

π(f )^∗=π(f^∗). (1.4)

The groupGis said to besymmetric, if the associated group algebraL¹(G)is symmetric, i.e. if every elementf ∈L¹(G)withf^∗=f has a real spectrum with respect to the involutive Banach algebraL¹(G).

In this paper we consider connected Lie groups for which every sub-Laplacian is of holo- morphic L^p-type. First, in Section 2, we consider connected semi-simple Lie groups G with finite center. We construct a holomorphic family of representations π_(z) of G on mixed L^p- spaces (see Section 2.2). Applying these representations toh₁, we obtain a holomorphic family of compact operators on these spaces (see Section 2.3). Using the Kunze–Stein phenomenon on semi-simple Lie groups (see Section 2.4), the eigenvectors of the operators π_(z)(h₁)allow us to construct a holomorphic family of L^p-functions on Gwhich are eigenvectors forF (L), if F ∈Mp(T )∩C_∞(R). From the corresponding holomorphic family of eigenvalues we can read off thatF admits a holomorphic extension in a neighborhood of some element in the spectrum ofL(see Section 2.5). This gives us:

Theorem 1.1.LetGbe a non-compact connected semi-simple Lie group with finite center. Then every sub-Laplacian onGis of holomorphicL^p-type, for1p <∞,p=2.

Remark 1.Even if at the end of the proof, we consider only ordinaryL^p-spaces, we need repre- sentations on mixedL^p-spaces. They are used to get some isometry property and then to apply the Kunze–Stein phenomenon.

In Sections 3.1 and 3.2, we discuss respectivelyp-induced representations and a generaliza- tion of the Coifman–Weiss transference principle [5]. We consider a separable locally compact groupG, and an isometric representationρof a closed subgroupSofGon spaces ofL^p-type, e.g.L^p-spacesL^p(Ω). Denote byπ_p:=ind^G_p,Sρthep-induced representation ofρ. We prove, among other results, that, for any functionf ∈L¹(G), the operator norm ofπ_p(f )is bounded by the norm of the convolution operatorλ_G(f )onL^p(G), provided the groupSis amenable. Here, λ_Gdenotes the left-regular representation. It should be noted that we do not require the groupG to be amenable. As an application we obtain theL^p-transference of a convolution operator onG to a convolution operator on the quotient groupG/S, in the case whereSis an amenable closed, normal subgroup.

When preparing this article, we were not aware of J.-Ph. Anker’s article [1] which, to a large extent, contains these transference results, and which we also recommend for further references to this topic. We are indebted to N. Lohoué for informing us on Anker’s work [1] as well as on the influence of C. Herz on the development of this field (compare [9]). For the convenience of the reader, we have nevertheless decided to include our approach to these transference results, since it differs from Anker’s by the use of a suitable cross section forG/S, which we feel makes the arguments a bit easier.

Applying this transference principle, we obtain the following generalization of Theorem 1.1 in Section 4.

Theorem 1.2.LetG=expgbe a connected Lie group, and denote byS=expsits radical. If G/S is not compact, then every sub-Laplacian onG is of holomorphicL^p-type, for any 1 p <∞,p=2.

It then suffices to study connected Lie groups for whichG/Sis compact. In Section 5, we shall consider groupsGwhich are the semi-direct product of a compact groupKwith a non-symmetric exponential solvable groupSfrom a certain class. The exponential solvable non-symmetric Lie groups have been completely classified by Poguntke [17] (with previous contributions by Lep- tin, Ludwig and Boidol) in terms of a purely Lie-algebraic condition (B). Let us describe this condition, which had been first introduced by Boidol in a different context [3].

Recall that the unitary dual ofS is in one to one correspondence with the space of coadjoint orbits in the dual spaces^∗ofsvia the Kirillov map, which associates with a given point∈s^∗ an irreducible unitary representationπ(see e.g. [8, Section 1]).

Ifis an element ofs^∗, denote by s():= X∈s|

[X, Y]

=0, for allY∈s

thestabilizer ofunder the coadjoint action ad^∗. Moreover, ifmis any Lie algebra, denote by m=m¹⊃m²⊃ · · ·

the descending central series ofm, i.e.m²= [m,m], andm^k+1= [m,m^k]. Put m^∞=

k

m^k.

Thenm^∞is the smallest idealkinmsuch thatm/kis nilpotent. Put m():=s()+ [s,s].

Then we say that , respectively the associated coadjoint orbit Ω():=Ad^∗(G), satisfies Boidol’s condition(B), if

|m()^∞=0. (B)

According to [17], the group S is non-symmetric if and only if there exists a coadjoint orbit satisfying Boidol’s condition.

IfΩ is a coadjoint orbit, and ifnis the nilradical ofs, then Ω|n:= {|n|∈Ω} ⊂n^∗ will denote the restriction ofΩton.

We show that the methods developed in [8] can also be applied to the case of a compact extension of an exponential solvable group and thus obtain:

Theorem 1.3. Let G=KS be a semi-direct product of a compact Lie group K with an exponential solvable Lie group S, and assume that there exists a coadjoint orbit Ω()⊂s^∗ satisfying Boidol’s condition, whose restriction to the nilradical nis closed inn^∗. Then every sub-Laplacian onGis of holomorphicL^p-type, for1p <∞, p=2.

Remarks.(a) A sub-LaplacianLonGis of holomorphicL^p-type if and only if every continuous bounded multiplier F ∈Mp(L)extends holomorphically to an open neighborhood of a non- isolated point inσ2(L).

(b) If the restriction of a coadjoint orbit to the nilradical is closed, then the orbit itself is closed (see [8, Theorem 2.2]).

(c) What we really use in the proof is the following property of the orbitΩ:

Ωis closed, and for every real characterνofswhich does not vanish ons(), there exists a sequence{τ_n}nof real numbers such thatlimn→∞(Ω+τ_nν)= ∞in the orbit space.

This property is a consequence of the closedness ofΩ|n. There are, however, many examples where the condition above is satisfied, so that the conclusion of the theorem still holds, even though the restriction of Ω to the nilradical is not closed (see e.g. [8, Section 7]). We do not know whether the condition above automatically holds whenever the orbitΩis closed.

Observe that, contrary to the semisimple case, we need to consider representations on mixed L^p-spaces till the end of the proof.

2. The semi-simple case

2.1. Preliminaries

In the following, ifMis a topological space,C₀(M)will mean the space of compactly sup- ported continuous functions onM.

As usual, ifSis a Lie group,swill denote its Lie algebra.

IfEis a vector space, denote byE^∗its algebraic dual. If it is real,E_Cdenotes its complexi- fication. LetF be a vector subspace ofE. We identify the restrictionλ_|F ofλ∈E^∗orE_C^∗ with an element of respectivelyF^∗orF_C^∗.

LetGbe a connected semisimple real Lie group with finite center andgits Lie algebra. Fix a Cartan involutionθofGand denote byKthe fixed point group forθ. The Cartan decomposition of the Lie algebragofGwith respect toθis given by

g=k⊕p,

wherekis the Lie algebra ofKandpthe−1-eigenspace ingfor the differential ofθ, denoted again byθ. We fix a subspaceaofpwhich is maximal with respect to the condition that it is an abelian subalgebra ofg. It is endowed with the scalar product(·,·)given by the Killing formB, which is positive definite onp. By duality, we endowa^∗with the corresponding, induced scalar product, which we also denote by(·,·). Let| · |be the associated norm onpanda^∗.

For any root α∈a^∗, we denote by g_α the corresponding root space, i.e. g_α := {X ∈g| [H, X] =α(X), H∈a}. We fix a setR⁺of positive roots of aing. LetP denote the corre- sponding minimal parabolic subgroup ofG, containingA:=expa, andP =MANits Langlands decomposition.

Denote byρthe linear form onagiven by ρ(X):=1

2tr(adX_|n), X∈a, wherenis the Lie algebra ofN.

Let · denote the “norm” onGdefined in [2, §2]. Recall that, forg∈G,gis the operator norm of Adgconsidered as an operator ong, endowed with the real Hilbert structure,(X, Y )→

−B(X, θ Y )as scalar product. This norm isK-biinvariant and, according to [2, Lemma 2.1], and satisfies the following properties:

· is a continuous and proper function onG, g =θ (g)=g⁻¹1;

xyxy;

there existsc₁, c₂>0 such that,

forY ∈p, thene^c1|Y|expYe^c2|Y|;

for alla∈A, n∈N, aan. (2.1)

We choose a basis fora^∗, following, for example, [6, p. 220].

Letα₁, . . . , α_r denote the simple roots inR⁺. By the Gram–Schmidt process, one constructs from the basis{α₁, . . . , α_r}of a^∗ an orthonormal basis{β₁, . . . , β_r}ofa^∗ in a such way that, for everyj =1, . . . , r, the vector space Vect{β₁, . . . , β_j}spanned by{β₁, . . . , β_j}agrees with Vect{α₁, . . . , α_j}, and, for every 1k < j r,(β_j, α_k)=0. DefineH_j (j=1, . . . , r) as the element ofagiven byβ_k(H_j)=δ_{j k}(k=1, . . . , r), and put:

a_j:=RH_j, a^j:=

j

k=1

a_k;

Rj:=R^j\R^j−1, withR⁰:= ∅, R^j:=R⁺∩Vect{α1, . . . , αj};

n^j:=

α∈R^j

gα, nj:=

α∈Rj

gα.

We define, forj=1, . . . , r, the reductive Lie subalgebram^j ofgby setting:

m^j:=θ n^j

+m+a^j+n^j.

In this way, we obtain a finite sequence of reductive Lie subalgebras ofg, m=:m⁰⊂m¹⊂ · · · ⊂m^r=g,

such that

m^j=θ (n_j)+m^j−1+a_j+n_j (j=1, . . . , r).

Thenm^j−1⊕aj⊕njis a parabolic subalgebra ofm^j.

Observe thatGis a real reductive Lie group in the Harish-Chandra class (see e.g. [7, p. 58], for the definition of this class of reductive Lie groups). We can then inductively define a decreasing sequence of reductive real Lie groupsM^j in the Harish-Chandra class, starting fromM^r =G, in the following way.

Let Pj denote the parabolic subgroup of M^j corresponding to the parabolic subalgebra m^j−1⊕aj⊕nj, andP_j=M^j−1A_jN_jits Langlands decomposition. HereA_j (respectivelyN_j) is the analytic subgroup ofM^j with Lie algebraa_j(respectivelyn_j),M^j−1A_jis the centralizer inM^jofA_j, and

M^j−1:=

χ∈Hom(M^j−1A_j,R^×₊)

Kerχ

(see e.g. [7, Theorem 2.3.1]).

Moreover,M^j−1AjnormalizesNjandθ (Nj), andM^j−1is a reductive Lie subgroup ofM^j, in the Harish-Chandra class, with Lie algebram^j−1(see [7, Proposition 2.1.5]).

PutK^j:=M^j∩K(j=1, . . . , r). ThenK^j is the maximal compact subgroup ofM^j related to the Cartan involutionθ_|Mj ofM^j(see e.g. [7, Theorem 2.3.2, p. 68]). Hence,M^jis the product

K^jP_j=K^jM^j−1A_jN_j.

Fix invariant measuresdk,dm,da,dn,dm_j,dk_j,da_j,dn_j for respectivelyK,M,A,N,M^j, K^j,A_j,N_j.

Choose an invariant measuredxonGsuch that

G

ϕ(x) dx=

K×A×N

a^2ρϕ(kan) dk da dn, for allϕ∈C0(G) (2.2)

(see e.g. [7, Proposition 2.4.2]).

We shall next recall an integral formula. LetSbe a reductive Lie group in the Harish-Chandra class, and letS=Kexppbe its Cartan decomposition, whereKis a maximal compact subgroup of S. LetQbe a parabolic subgroup ofS related to the above Cartan decomposition, and let Q=M_QA_QN_Qbe its Langlands decomposition.

LetK_Q:=K∩Q=K∩M_Q, and put, for k∈K,[k] :=kK_Q inK/K_Q. We extend this notation toSby putting, fors=kman,(k, m, a, n)∈K×MQ×AQ×NQ,[s] := [k]. This is still well defined even though the representation ofsinKMQAQNQis not unique. In fact,

s=kman=kman

if and only ifa=a,n=n, andk=kk_Q,m=k_Q⁻¹mfor somek_Q∈K_Q(see e.g. [7, Theo- rem 2.3.3]). From this we see that the decomposition above becomes unique, if we requiremto be in exp(mQ∩p).

Everys∈Sthus admits a unique decompositions=kman, with(k, m, a, n)∈K×exp(mQ∩ p)×A_Q×N_Q. We then writek_Q(s):=k,m_Q(s):=m,a_Q(s):=aandn_Q(s):=n, i.e.

s=kQ(s)mQ(s)aQ(s)nQ(s).

In particular,[s] =k_Q(s)K_Q.

Fory∈Sandk∈K, we definey[k] ∈K/K_Qas follows:

y[k] := [yk].

Moreover, for anyγ∈a^∗_CandY∈a, we put(expY )^γ :=e^{γ (Y )}.

One can deduce, for example from [20, Lemma 2.4.1], the following lemma.

Lemma 2.1. Fix an invariant measure dk on K and let d[k] denote the corresponding left- invariant measure onK/K_Q. For anyy∈S, we then have

d y[k]

=a_Q(yk)^−2ρQd[k],

whereρ_Q∈a^∗_Qis given byρ_Q(X)=¹₂tr(adX_|n_Q)(X∈a_Q);that is, for anyf∈C(K/K_Q),

K/K_Q

f [k]

d[k] =

K/K_Q

aQ(yk)^−2ρQf y[k]

d[k].

We return now to our semisimple Lie groupG. In the following, we shall use another basis ofa^∗, given as follows. Forj=1, . . . , r, letρ_j denote the element ofa^∗_jdefined by

ρj(X):=1

2tr(adX_|n_j) for allX∈aj. Notice that we can identifyρ_jwith the restrictionρ_|aj ofρtoa_j.

By [6, Lemma 4.1],ρ_j andβ_j are scalar multiples of each other. In particular, the family {ρ_j}is an orthogonal basis ofa^∗, and therefore ofa^∗_C. For everyν∈a^∗(respectivelyν∈a^∗_C) and j =1, . . . , r, we defineν_j∈R(respectivelyC) by the following:

ν= r

j=1

ν_jρ_j.

Recall that, for j =1, . . . , r,P_j is a parabolic subgroup of the real reductive Lie group in the Harish-Chandra class, Mj. Therefore, by taking(S, K, Q):=(M^j, K^j, Pj)in the discussion above, putkj :=kP_j,mj−1:=mP_j,aj:=aP_j andnj:=nP_j. Thus, anyg∈M^j has a unique decomposition g =k_j(g)m_j−1(g)a_j(g)n_j(g), with k_j(g)∈K^j, m_j−1(g)∈exp(m^j−1∩p), a_j(g)∈A_j andn_j(g)∈N_j. Notice thatm₀∩p= {0}, i.e.m₀(g)=e.

Lemma 2.2.Denote byr_y the right multiplication withy∈G. Letj ∈ {1, . . . , r},g∈M^j and k_l∈K^l (l=1, . . . , j ).

We define recursively the elementg_lofM^l,l=1, . . . , j, starting froml=j, by puttingg_j:=

gandgl−1:=ml−1(glkl), i.e.

g_l=m_l◦(r_kl+1◦m_l+1)◦ · · · ◦(r_kj◦m_j)(g), 1lj−1.

Then, the following estimate holds:

1 l=j

al

glk^l g.

Proof. We first show that, for 1pj, g =

p

l=j

al(glkl)·mp−1(gpkp)· j

l=p

nl(glkl)k_l⁻¹

. (2.3)

(Here the products are non-commutative products, in which the order of the factors is indicated by the order of indices.) We use an induction, starting fromp=j. Ifp=j andg∈M^j, then

g =gkjk_j⁻¹=kj(gkj)mj−1(gkj)aj(gkj)nj(gkj)k⁻_j¹.

Using the leftK-invariance of the norm and the fact thataj(gkj)∈Aj andmj−1(gkj)∈M^j−1 commute, we find that

g =a_j(gk_j)m_j−1(gk_j)n_j(gk_j)k_j⁻¹,

so that (2.3) holds forp=j. Assume now, by induction, that (2.3) is true forp+1 in place ofp, i.e.

g =

p+1 l=j

al(glkl)·mp(gp+1kp+1)· j

l=p+1

nl(glkl)k_l⁻¹ .

We then decompose:

mp(gp+1kp+1)kp=gpkp=kp(gpkp)mp−1(gpkp)ap(gpkp)np(gpkp).

Sincek_p(g_pk_p)∈K^p⊂M^l, forplj, it commutes witha_l(g_lk_l),forl=p+1, . . . , j, and therefore, because of theK-invariance of · , we have:

g =

p+1 l=j

a_l(g_lk_l)·m_p−1(g_pk_p)a_p(g_pk_p)n_p(g_pk_p)k⁻_p¹· j

l=p+1

n_l(g_lk_l)k⁻_l ¹ .

Moreover,ap(gpkp)commutes withmp−1(gpkp), and so (2.3) follows.

Applying now (2.3) forp=1, we obtain

1 l=j

a_l(g_lk_l) j

l=1

n_l(g_lk_l)k⁻_l ¹

= g. (2.4)

By rightK-invariance of the norm, the left-hand side of this equation is equal to

1 l=j

al(glkl) j

l=1

nl(glkl)k_l⁻¹ 1 l=j

k_l

.

Notice that we can writej

l=1n_l(g_lk_l)k_l⁻¹1

l=jk_las follows:

n₁(g₁k₁)

k⁻₁¹n₂(g₂k₂)k₁

(k₂k₁)⁻¹n₃(g₃k₃)k₂k₁

· · · ₁

l=j−1

k_{l −}1

n_j(g_jk_j) 1 l=j−1

k_l

.

For 2pj,(1

l=p−1k_l)⁻¹lies inK^p−1⊂M^p−1and thus normalizesN_p. Hence, we get that

j

l=1

n_l(g_lk_l)k_l⁻¹ 1 l=j

k_l∈N.

Using the last property of the norm given in (2.1), the left-hand side of (2.4) is then greater or equal to₁

l=ja_l(g_lk_l), which proves the lemma. 2

2.2. A holomorphic family of representations ofGon mixedL^p-spaces

Forν∈a^∗_C, letM(G, P , ν)denote the space of complex-valued measurable functionsf on Gsatisfying the following covariance property:

f (gman)=a^−(ν+ρ)f (g) for allg∈G, m∈M, a∈A, n∈N.

The space M(G, P , ν) is endowed with the left regular action of G, denoted by π˜_ν, i.e.

[ ˜π_ν(g)f](g)=f (g⁻¹g). The representationsπ˜_ν form the class-one principal series.

LetM(K/M)be the space of rightM-invariant measurable functions onK.

The restriction toKof functions onGgives us a linear isomorphism fromM(G, P , ν)onto M(K/M). Denote byI_ν:f→f_ν the inverse mapping. Thenf_ν∈M(G, P , ν)is given by

f_ν(kan):=a^−(ν+ρ)f (k) for allk∈K, a∈A, n∈N, ifG=KAN denotes the Iwasawa decomposition ofG.

If we intertwine the representation π˜_ν with I_ν, we obtain a representation π_ν of G on M(K/M), given by

π_ν(g)f

ν= ˜π_ν(g)f_ν, iff ∈M(K/M), g∈G.

For j =1, . . . , r, denote by dk˙_j the quotient measure on K^j/K^j−1 coming from dk_j. It is invariant by left translations. Notice thatK^j−1=K^j∩M^j−1.

We choose a left-invariant measuredk˙onK/Msuch that, for anyf ∈C(K/M),

K/M

f (k) dk˙=

K^r/K^r−1

. . .

K¹/M

f (k_r· · ·k₁) dk˙₁. . . dk˙_r. (2.5)

Letp=(p₁, . . . , p_r)∈ [1,+∞[^r. One can easily see that, for everyf∈M(K/M),k∈K, the function onK^j given by

k→

K^j−1/K^j−2

. . .

K¹/M

f (kkkj−1. . . k1)^p1dk˙1

p2/p1

. . . dk˙j−1

1/pj−1

,

is rightK^j−1-invariant.

We can thus define the mixedL^p-space,L^p(K/M), as the space of all (equivalent classes of) functionsf inM(K/M)whose mixedL^p-norm:

fp:=

K^r/K^r−1

. . .

K¹/M

f (k_r. . . k₁)^p1dk˙₁ p₂/p₁

. . . dk˙_r 1/pr

is finite, endowed with this norm. This definition extends to the case where some of thep_j are infinite, by the usual modifications.

Letd denote the rightG- and leftK-invariant metric onG, given by d(g, g):= 1

c₁loggg⁻¹ (g, g∈G),

wherec₁is the positive constant appearing in (2.1). Notice thatd(g, e)=0 if and only ifglies in the center ofG. In particular,dis not separating.

Then, fora=expY, withY ∈a⊂p, andγ∈a^∗, we have, in view of the fourth property of · in (2.1), that

a^γ=e^{γ (Y )}e^|γ||Y|=

e^c1|Y|_|γ|/c1a^|γ|c1 =e^|γ|d(a,e). (2.6) Proposition 2.1.For everyf ∈L^p(K/M)andg∈G, we have:

π_ν(g)f

pe^|

j(_pj² −Reνj−1)ρj|d(g,e)

fp.

Thusπ_ν defines a representationπ_ν,pofGonL^p(K/M). Furthermore, this gives an analytic family(πν,p)_ν∈a∗

C of representations ofGonL^p(K/M).

Before giving the proof, we show the following statement. We keep the same notations as in Lemma 2.2.

Lemma 2.3.Letg∈M^j, k∈Kandf_ν∈M(G, P , ν). Then:

K^j/K^j−1

. . . K¹/M

f_ν(kgk_j. . . k₁)^p1dk˙₁ p₂/p₁

. . . dk˙_j 1/p_j

=

K^j/K^j−1

. . . K¹/M

1 l=j

a_l(g_lk_l)^{−(Reνl+1)ρl}f_ν

k 1 l=j

k_l(g_lk_l)

p₁ dk˙₁

p₂/p₁ . . . dk˙_j

1/p_j .

Proof. We use induction onj. Forj =0, one has, by right M-invariance off and sinceg∈ M⁰=M, that

fν(kg)=fν(k).

Assume that the statement is true forj−1. Observe thata_j(gk_j)commutes withk_j−1. . . k₁∈ M^j−1, and that(kj−1. . . k1)⁻¹nj(gkj)kj−1. . . k1∈N. Therefore, the covariance property offν

applied to the integration overK^j/K^j−1, implies:

K^j/K^j−1

. . . K¹/M

f_ν(kgk_j. . . k₁)^p1dk˙₁ p₂/p₁

. . . dk˙_j 1/p_j

=

K^j/K^j−1 . . .

K¹/M

a_j(gkj)^{−(Reνj+1)ρj}fν

kkj(gkj)m_j−1(gkj)k_j−1. . . k1^p1dk˙1 p₂/p₁

. . . dk˙j 1/p_j

.

Sinceg=g_j andm_j−1(gk_j)=g_j−1∈M^j−1, the statement holds, using the induction hypothe- sis. 2

Proof of Proposition 2.1. If we apply (2.6) toγ :=_r

j=1(_p²

j −Reν_j−1)ρj and notice that thea_l’s are pairwise orthogonal with respect to(·,·), we get, in view of Lemma 2.2:

sup

kj∈K^j, j=1,...,r

r

j=1

a_j(g_jk_j)⁽

2

pj−Reν_j−1)ρ_j

g^|cγ1|=e^|γ|d(g,e).

On the other hand, according to the above lemma and Lemma 2.1, applied successively to the integrations overK^j/K^j−1,j=1, . . . , r, we have:

π_ν g⁻¹

f

p sup

kj∈K^j, j=1,...,r

₁

j=r

a_j(g_jk_j)⁽

2

pj−Reν_j−1)ρ_j

fp.

Sinced(g⁻¹, e)=d(g, e), the first assertion of the proposition follows.

In order to prove the analyticity of the family of representations π_ν,p, choose p = (p1, . . . , pr)∈ [1,∞[^r and denote by p=(p₁, . . . , p_r)∈ ]1,∞]^r the tuple of conjugate ex- ponents, i.e., 1/pj+1/p_j=1. Then, forf ∈L^p(K/M),u∈L^p(K/M)=(L^p(K/M)) and g∈G, we have:

πν,p(g)f, u

=

K/M

πν,p(g)f

(k)u(k) dk˙

=

K/M

a

g⁻¹k₋(ν+ρ)

f κ

g⁻¹k u(k) dk.˙

Here, the functionsκ(·), a(·), n(·)onGare given by the unique factorizationg=κ(g)a(g)n(g) ofg, according to the Iwasawa decompositionG=KAN.

Obviously, the expression above is analytic inν∈a^∗_C, which finishes the proof. 2 Fort=(t₁, . . . , t_r)∈ ]0,+∞[^r, let

Ω_t:= ν∈a^∗_C|Reν_j|< t_j, j=1, . . . , r . Moreover, forp0, let

p:=(p, . . . , p)∈R^r.

According to our choice of measure onK/M(cf. (2.5)), notice that L^p(K/M)=L^p(K/M), · p= · L^p(K/M). Proposition 2.2.

(i) For allp∈ [1,+∞[^r,f ∈L^p(K/M),ν∈Ωt,g∈G, we have π_ν,p(g)f

pe^{j(tj+1)|ρj|d(g,e)}fp.