On the Hausdorff-Young theorem

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On the Hausdorff-Young theorem

Wassim Nasserddine

To cite this version:

Wassim Nasserddine. On the Hausdorff-Young theorem. 2005. �hal-00129708�

Harmonic analysis/ Analyse harmonique

On the Hausdorff-Young theorem

Wassim Nasserddine¹

Abstract

Let Gmn =ax+b be the matricial group of a local field. The Hausdorff-Young theorem for G11was proved by Eymard-Terp [3] in 1978. We will establish here the Hausdorff-Young theorem forGnnfor alln∈N.

R´esum´e

SoitGmn=ax+bun groupe matriciel d’un corps local. En 1978 Eymard-Terp [3] ont prouv´e le th´eor`eme de Hausdorff-Young surG11. On ´etablit ici le th´eor`eme de Hausdorff-Young surGnn

pour toutn∈N.

Introduction

The Hausdorff-Young theorem was generalized by several authors as they are passing over from a locally compact Abelian group to a locally compact group which is not Abelian but which is unimodular. In this paper we will tackleGmn=ax+bthe matricial group of a local field, to be defined below, which is not unimodular and the theory will involve unbounded operators, as the case was before ifm=n= 1.

Let K be a local field, n, m ∈ N, m ≥ n ≥1. Let Mnm be the space of all n×m-matrices with elements from KandGLn be the multiplicative group of alln×n-inversible matrices with elements from K. Gnm denotes the group of pairs (b, a), where b ∈ Mnm and a ∈ GLn, with multiplication given by (b, a)(b⁰, a⁰) = (b+ab⁰, aa⁰). It is the semi-direct productMnmnGLn.

LetHbe the Hilbert spaceL²(GLn,_|det(u)|^du n). For allλinMmn, the formula [πλ(b, a)ξ](u) =τ(T r(bλu))ξ(ua),

(1)

defines a unitary representation of Gnm in H, where τ is a fixed additive unitary nontrivial character onK(see [4], p.224 and p.142). We defineLq(H), 1≤q <∞, to be the Banach space of bounded linear operators on HwithkAkq ={T r(|A|^q)}^1q <∞, where for a bounded operator A,|A|denotes the operator (A^∗A)¹² andkAk∞will denote the operator norm of A.δq denotes the unbounded operator inHdefined byδqξ(u) =|det(u)|^mqξ(u).

Let Gbess be the ”essential” dual of G= Gnm, 1 < p < 2, ¹_p +¹_q = 1. We will prove that if f ∈ D(G), the space of regular functions with compact support onG, then

Fp(f)∈L^q(Gbess,Lq(H)),

whereFp(f)λ:=Ff(λ)◦δ1/q:=πλ(f)◦δ1/q, and that ifn=m,f ∈L¹∩L^p(G), then the map f −→ Fp(f) extends uniquely to a linear map ofL^p(G) intoL^q(Gbess,Lq(H)) with norm≤1.

1Institut de Recherche Math´ematique Avanc´ee, Universit´e Louis Pasteur et CNRS, 7, Rue Ren´e Descartes, 67084 Strasbourg Cedex, France.

E-mail :nasserdd@math.u-strasbg.fr

1

1. PRELIMINARY

In 1912 Young proved the Hausdorff-Young theorem forG= the circle group whenp= 2k/(2k−

1), 2 ≤ k ∈ N. In 1923 Hausdorff generalized this result for all p, 1 < p < 2. In 1940 Weil established this theorem for any locally compact Abelian groups. For groups which are not Abelian but which are unimodular, Kunze [5] defined and proved the Hausdorff-Young theorem by using the theory of non-commutative integration. In 1974 Lipsman [6] gave a concrete realization if those unimodular groups are separable and of type I. In this last case, letGb be the set of unitary equivalence classes of continuous irreducible unitary representations of G and

Lp(G) =b {F is a measurable field of bounded operators onGb: kF(π)kp<∞, µalmost allπ∈G,b

Z

Gb

kF(π)k^p_pdµ(π)<∞,1≤p <∞}

If we writeF1∼F2 whenF1(π) =F2(π) forµ-almost allπ, thenLp(G) =b Lp(G)/b ∼is a Banach space under the norm

kFkp= Z

Gb

kF(π)k^p_pdµ(π)

!¹_p

, 1≤p <∞, and the Hausdorff-Young theorem ([6], th.2.2, p.214) asserts that if

1< p <2, 1 p+1

q = 1, f ∈L1∩Lp(G), thenF(f) := ˆf ∈Lq(G) andb f −→fˆ, where ˆf(π) :=R

Gf(g)π(g)dg, extends uniquely to a map ofLp(G) into Lq(G) with normb ≤1.

If the group G is not unimodular, new problemes appear to thoroughly deal with the Fourier transformation inL^p(G). In our case, that isG=Gnm, like Eymard-Terp’s case [3] which helped us as a model, we are led to temper the Fourier transformation of L¹(G) by the unbounded operatorsδ¹

q.

Gnmis a locally compact group on which we fix the left Haar measure to bed(b, a) = _|det(a)|^dbdan+m, where db (resp. da) is the canonical measure on Mnm 'K^nm (resp. onGLn ⊂Mnn) deduced from the Haar measure onK, and|.|is the module inK(fora∈K,|a|is defined byd(ax) =|a|dx, wheredx is a Haar measure on (K,+) (see [8], p.4) ). Gnm is not unimodular (_|det(a)|^dbdan is a right Haar measure) and its modular function is ∆nm(b, a) =_|det(a)|¹ m.

Letting Grass(m, k) (0 ≤ k ≤ min(n, m)) denote the set of k-finite dimensional vector sub- spaces inK^m,Mmk(k) denote the set of allm×k-matrices with rank k andSmk([1], p.63) denote the canonical realization ofGrass(m, k) inMmk(k). Smk is a compact subspace ofMmk(k).

There exists ([1], prop.3.1.2, p.85) only one measure sonSmk (m, k≥1,k ≤m) such that, for allf ∈L¹(Mmk, dλ), we have:

Z

Mmk

f(λ)dλ= Z

Smk

Z

GLk

f(µa)|deta|^m da

|det(a)|^k

! ds(µ) (2)

s is called the canonical measure of Smk. From now on, G and S denote respectively Gnm and Smn. The representation πλ defined by (1) is only irreducible if rang(λ) = n, and the role of essential dual of G is played by S via the connectionS 3λ−→πλ. Let us recall two important results (which are in [1], p.96, p.118) for the groupsG=Gnm, m≥n:

Theorem 1 ( Plancherel Theorem). Letf ∈ D(G), then F(f)◦δ1/2 ∈L²(S,L2(H))and f −→

F(f)◦δ1/2 extends uniquely to an isometric, isomorphism ofL² ontoL²(S,L2(H)).

Theorem 2 ( Fourier Inversion Theorem). Let A(G) be the Fourier algebra of G andf ∈L¹∩ A(G), then F(f)◦δ1∈L¹(S,L1(H))and , for allx∈G, we have:

f(x) = Z

S

T r(πλ(x⁻¹)F(f)◦δ1)ds(λ).

2. The Hausdorff-Young theorem for the matricial groupsGnn=ax+b Lemma 3([1], p.94). LetKbe a disconnected local field. Iff ∈ D(G), then the function(λ, a)−→

fˆ(., a)λvanishes off a compact subset ofMmn×GLn, wherefˆ(., a)(λ) =R

Mnmf(b, a)τ(T r(bλ))db.

Definition. We call the Fourier transform off ∈L¹(G) and we denote byFf, the function λ−→ Ff(λ) =πλ(f)

defined in S with values inL∞(H) the space of bounded operators onH.

Remark. Iff ∈L¹(G), then [Ff(λ)]ξ(u) =

Z

Mnm

Z

GLn

τ(T r(bλu))ξ(ua)f(b, a) dbda

|det(a)|^n+m, ξ∈ H, (3)

in fact, letξ andη inH. Then

|<Ff(λ)ξ, η >|=|< πλ(f)ξ, η >|

=| Z

G

< πλ(g)ξ, η > f(g)dg|

≤ Z

G

Z

GLn

|τ(T r(bλu))ξ(ua)η(u)| du

|det(u)|ⁿ|f(b, a)|d(b, a)

≤ Z

G

|f(b, a)|d(b, a) Z

GLn

|ξ(ua)||η(u)| du

|det(u)|ⁿ

<∞, by Fubini’s theorem, the integral R

Gτ(T r(bλu))ξ(ua)η(u)f(b, a)d(b, a) exists, for almost all u∈ GLn, and we have

Z

GLn

Z

G

τ(T r(bλu))ξ(ua)f(b, a)d(b, a)

!

η(u) du

|det(u)|ⁿ

=<Ff(λ)ξ, η >

= Z

GLn

Ff(λ)ξ(u)η(u) du

|det(u)|ⁿ,

sinceηis arbitrary andd(b, a) =_|det(a)|^dbdan+m, we see that (3) is valid.

Lemma 4. Let K be a local field, and let 1 ≤p≤2, 1/p+ 1/q = 1, f ∈ D(G). Then, for all λ∈ Mmn(n), the operator Fp(f)λ:= Ff(λ)◦δ1/q := πλ(f)◦δ1/q extends to a Hilbert-Schmidt operator on H that we denote againFf(λ)◦δ1/q.

Proof. Letξ∈dom(δ1/q), then [Fp(f)λ]ξ(u) = [πλ(f)◦δ_1/q]ξ(u)

= Z

Mnm

Z

GLn

πλ(b, a)(δ1/qξ)(u)f(b, a) dbda

|det(a)|^n+m

= Z

M_nm

Z

GL_n

τ(T r(bλu))δ1/qξ(ua)f(b, a) dbda

|det(a)|^n+m

= Z Z

τ(T r(bλu))|det(a)|^mqξ(a)f(b, u⁻¹a) dbd(u⁻¹a)

|det(au⁻¹)|^n+m (a→u⁻¹a)

= Z

GL_n

fˆ(., u⁻¹a)(λu)|det(a)|^mp−p1

|det(au⁻¹)|^mξ(a) da

|det(a)|ⁿ

= Z

GLn

K_f,p^λ (u, a)ξ(a) da

|det(a)|ⁿ HenceFp(f)λacts inHwith the kernel

K_f,p^λ (u, a) = ˆf(., u⁻¹a)(λu)|det(a)|^mp−1p

|det(au⁻¹)|^m This kernel is Hilbert-Schmidt, in fact

Z

GLn

Z

GLn

|K_f,p^λ (u, a)|² duda

|det(u)|ⁿ|det(a)|ⁿ

= Z

Mnn

Z

GLn

|fˆ(., u⁻¹a)(λu)|²|det(a)|^2mp−1p

|det(au⁻¹)|^2m

duda

|det(u)|ⁿ|det(a)|ⁿ

= Z Z

|fˆ(., a)(λu)|²|det(ua)|^2mp−1p

|det(a)|^2m

dud(au)

|det(u)|ⁿ|det(au)|ⁿ (a→ua)

= Z Z

|fˆ(., a)(λu)|²|det(a)|^{2mp−p1−n−2m}|det(u)|^2mp−p1−nduda

IfKis disconnected, this last integral is finite since by lemma 3 (µ, a)−→fˆ(., a)µvanishes off a compact subset ofMmn×GLn. IfK=RorC, letKn be the projection ofsuppf onGLn. For allu, ifa /∈Kn, we have f(b, a) = 0, ∀b, from which ˆf(., a)(λu) =R

Mnmf(b, a)τ(T r(bλu))db= 0 follows. On the other hand, the functions µ−→fˆ(., a)µ are, uniformly on a, rapidly decreasing onMmn, hence

|det(µ^∗µ)|^{m(p−1)p −n2}|fˆ(., a)(µ)|²≤ C0

1 +kµk⁴ⁿ,

for allµ∈Mmn, where the constantC0 is independent ofa. Settingµ=λu, then

|det(u)|²

!^m(p−1)_p −ⁿ₂

=|det(µ^∗µ)

det(λ^∗λ)|^{m(p−1)p −n2} It follows that

Z

GLn

Z

GLn

|K_f,p^λ (u, a)|² duda

|det(u)|ⁿ|det(a)|ⁿ

≤ C0

det(λ^∗λ)^{m(p−1)p −n2} Z

Kn

|det(a)|^{2m(p−1)p −n−2m}da Z

Mnn

du 1 +kλuk⁴ⁿ

<∞

Lemma 5. Let us define f^∗p byf^∗p(g) =|det(a)|^m−n+mpf¯(g⁻¹), where g= (b, a). Then, for all f inD(G), we have:

Fp(f^∗p) = [Fp(f)]^∗

Proof. One only has to show that [Fp(f)λ]^∗ acts in H = L²(GLn,_|det(u)|^du n) with the kernel K_f^λ∗p,p(u, a) =K_f,p^λ (a, u) which is Hilbert-Schmidt by the previous lemma.

Lettingg= (b, a), theng⁻¹= (−a⁻¹b, a⁻¹), so that [f^∗p]^∧(., au⁻¹)(λu) =

Z

Mnm

f^∗p(b, u⁻¹a)τ(T r(bλu))db

= Z

|det(au⁻¹)|^m−n+mpf¯(− b u⁻¹a, 1

u⁻¹a)τ(T r(bλu))db

=|det(au⁻¹)|^m−n+mp|det(u⁻¹a)|ⁿ

Z f¯(−b, a⁻¹u)τ(T r(bλa))db (b→u⁻¹ab)

=|det(au⁻¹)|^m+mp

Z f¯(−b, a⁻¹u)τ(T r(bλa))db

Thus

K_f^λ∗p_,p(u, a) = [f^∗p]^∧(., u⁻¹a)(λu)|det(a)|^mp−1p

|det(au⁻¹)|^m

=|det(au⁻¹)|^m+mpf¯ˆ(., a⁻¹u)(λa)|det(a)|^mp−1p

|det(au⁻¹)|^m

=|det(u)|^m−mp−m|det(a)|^{m+mp+m(p−1)p −m}f(., a¯ˆ ⁻¹u)(λa)

=f(., a¯ˆ ⁻¹u)(λa)|det(u)|^mp−1p

|det(ua⁻¹)|^m

=K_f,p^λ (a, u)

Theorem 6 ([7], p.177). Let1< p < 2, 1/p+ 1/q = 1, and let k ∈L²(X×X), where X is a σ-finite measure space. If K is the integral operator with kernel k, then

kKkq ≤(kkkp,qkk^∗kp,q)¹², wherekkkp,q= (R

(R

|k(x, y)|^pdx)^qpdy)^1q etk^∗(x, y) =k(y, x).

The main result of this paper is theorem 7 which follows. Ifm =n in its last statement, the annouced result is obtained. Let us give some simple formulae to use afterwards.

Letλ∈Mmn andC(λ) :=inf{kλxk, x∈Mnn,kxk= 1}, then ([1] lemme3.1.1, p.84) we have

|C(λ)−C(µ)| ≤ kλ−µk, (4)

for allλet µinMmn. In fact, for allx∈Mnn withkxk= 1, we have

C(λ)≤ kλxk ≤ kλx−µxk+kµxk=⇒C(λ)≤ kλ−µk+C(µ), similarly,C(µ)≤ kλ−µk+C(λ), from which (4) follows.

Letλ0be fixed inMmn, by the definition ofC(λ), we have, for all 06=u∈Mnn,C(λ)≤ kλ_kuk^u k and

kλuk ≥C(λ)kuk

If λis very close toλ0, then kλ−λ0k ≤ ^C(λ₂⁰⁾, it follows from (4) that |C(λ)−C(λ0)| ≤ ^C(λ₂⁰⁾ andC(λ)≥ ^C(λ₂⁰⁾. Therefore

kλuk ≥ C(λ0) 2 kuk (5)

kλ0uk ≥ C(λ0) 2 kuk (6)

Theorem 7. Letf ∈ D(G), and letKn be the projection of suppf onGLn, then (i) For allλ∈Mmn(n), the operator Fp(f)λis inLq(H).

(ii) The functionλ−→ Fp(f)λis continuous of Mmn(n)into the Banach space Lq(H).

In particular thatFp is a linear map of D(G) intoL^q(S,Lq(H)).

(iii)kFp(f)kq ≤C

1 2

fkfkp, whereCf = supa∈Kn|det(a⁻¹)|^(m−n)p

!¹_p .

Proof. Let us show that Fp(f)λ is inLq(H), in fact, since by lemma 4Fp(f)λacts in Hwith the kernel of Hilbert-Schmidt

K_f,p^λ (u, a) = ˆf(., u⁻¹a)(λu)|det(a)|^mp−1p

|det(au⁻¹)|^m, it follows from theorem 6 that

kFp(f)λkq ≤ kK_f,p^λ kp,qk(K_f,p^λ )^∗kp,q

!¹₂ ,

wherekK_f,p^λ kp,q= R

GLn

R

GLn|K_f,p^λ (u, a)|^p_|det(u)|^du n

!^q_p

da

|det(a)|ⁿ

!¹_q

and (K_f,p^λ )^∗(u, a) =K_f,p^λ (a, u) Now

k(K_f,p^λ )^∗k^q_p,q= Z

GLn

Z

GLn

|(K_f,p^λ )^∗(a, u)|^p da

|det(a)|ⁿ

!^q_p du

|det(u)|ⁿ

= Z

GLn

Z

GLn

|K_f,p^λ (u, a)|^p da

|det(a)|ⁿ

!^q_p du

|det(u)|ⁿ

= Z Z

|fˆ(., u⁻¹a)(λu)|^p|det(a)|^m(p−1)

|det(au⁻¹)|^mp da

|det(a)|ⁿ

!^q_p du

|det(u)|ⁿ

= Z Z

|fˆ(., a)(λu)|^p|det(au)|^m(p−1)d(ua)

|det(a)|^mp|det(ua)|ⁿ

!^q_p du

|det(u)|ⁿ (a→ua)

= Z

Mnn

Z

GLn

|fˆ(., a)(λu)|^p|det(a)|^−m−n|det(u)|^{m(p−1)−npq} da

!_p^q du (7)

IfKis disconnected, then this last integral is finite by lemma 3 since f ∈ D(G) andλis fixed.

IfK=Ror C, letKn be the projection ofsuppf onGLn andµ=λu. Then

|det(µ^∗µ)|^{m(p−1)2 −np2q}|fˆ(., a)(µ)| ≤ C0

1 +kµk⁴ⁿ, and consequently

k(K_f,p^λ )^∗k^q_p,q≤ C0

|det(λ^∗λ)|^{m(p−1)2 −np2q}

!^q_pZ Z

|det(a)|^−m−nda 1 +kλuk⁴ⁿ

!^q_p du

≤ C0

|det(λ^∗λ)|^{m(p−1)2 −np2q}

!^q_p Z

Kn

|det(a)|^−m−nda

!^q_pZ

Mnn

du (1 +kλuk⁴ⁿ)^qp

<∞ because q p >1)

By the proof of lemma 5, we have

K_f^λ∗p,p(u, a) =K_f,p^λ (a, u),

hence

kK_f,p^λ k^q_p,q= Z

GLn

Z

GLn

|K_f,p^λ (a, u)|^p da

|det(a)|ⁿ

!^q_p du

|det(u)|ⁿ

= Z

GLn

Z

GLn

|K_f^λ∗p_,p(u, a)|^p da

|det(a)|ⁿ

!_p^q du

|det(u)|ⁿ

On the other hand, the function (b, a)−→ |det(a)|^m−n+mp is, infinitely differentiable ifK=R orC, locally constant whenKis disconnected since the module|.|inKis locally constant. There- foref^∗p∈ D(G) and the previous reasoning implies thatkK_f,p^λ k^q_p,q<∞, and thusFp(f)λ∈ Lq(H).

(ii) Letλ0 be fixed inMmn(n), and let λbe very close to λ0 inMmn(n). We define K_f,p^λ,λ0 to be the kernel of the operatorFp(f)λ− Fp(f)λ0. Then (repeat the same computation which gave (7))

k(K_f,p^λ,λ0)^∗k^q_p,q= Z

Mnn

Z

GLn

|fˆ(., a)(λu)−fˆ(., a)(λ0u)|^p|det(a)|^−m−n|det(u)|^{m(p−1)−npq} da

!^q_p du

If K is disconnected, then, by lemma 3, and by (5),(6), we can integrate, in a neighbor- hood of λ0, on a compact K1×K2 of Mn×GLn which is independent of λ, so |fˆ(., a)(λu)| ≤ 1K1×K2(u, a)2|fˆ(., a)(λ0u) for all λvery close to λ0, and by Lebesgue’s dominated convergence theorem, we have

k(K_f,p^λ,λ0)^∗k^q_p,q−→0 whenλ−→λ0.

IfK=RorC, there exists an independent constantC1 ofasuch that, for allµandν inMmn, we have

|fˆ(., a)(µ)−fˆ(., a)(ν)| ≤C1kµ−νk,

since the functionsb−→f(b, a) vanish off a compact part ofMnmwhich is independent ofa.

LetKn be the projection of suppf onGLn. For any >0, we choose a compact subset K of Mnnsuch that

Z

Mnn\K

2

1 + (^C(λ₂⁰⁾)⁴ⁿkuk⁴ⁿ

!_p^q du≤

Then

k(K_f,p^λ,λ0)^∗k^q_p,q= Z

K

Z

Kn

|fˆ(., a)(λu)−fˆ(., a)(λ0u)|^p|det(a)|^−m−n|det(u)|^{m(p−1)−npq} da

!_p^q du

+ Z

Mnn\K

Z

Kn

|fˆ(., a)(λu)−fˆ(., a)(λ0u)|^p|det(a)|^−m−n|det(u)|^{m(p−1)−npq} da

!^q_p du

≤[(C1kλ−λ0k)^p]^qp Z

K

Z

Kn

kuk^p|det(a)|^−m−n|det(u)|^{m(p−1)−npq} da

!^q_p du

+C

q p

0Cλ0

Z

Mnn\K

Z

Kn

(|det(a)|^−m−n

1 +kλuk⁴ⁿ +|det(a)|^−m−n 1 +kλ0uk⁴ⁿ )da

!^q_p du whereCλ0 is a constant depending onλ0

≤C

q p

1kλ−λ0k^qCK ( whereCK is a constant depending on K ) +C

q p

0Cλ0

Z

Kn

|det(a)|^−m−nda

!^q_pZ

Mnn\K

1

1 +kλuk⁴ⁿ + 1 1 +kλ0uk⁴ⁿ

!^q_p du Since, by (5) and (6), we have

Z

Mnn\K

1

1 +kλuk⁴ⁿ + 1 1 +kλ0uk⁴ⁿ

!^q_p du

≤ Z

Mnn\K

2

1 + (^C(λ₂⁰⁾)⁴ⁿkuk⁴ⁿ

!^q_p du

≤,

it follows thatk(K_f,p^λ,λ0)^∗kp,q−→0 whenλ−→λ0. Similarly, we prove that

kK_f,p^λ,λ0kp,q −→0 whenλ−→λ0, and the inequality of theorem 6 completes the proof of (ii).

(iii) In fact, we have Z

S

kFp(f)λk^q_qds(λ)≤ Z

S

kK_f,p^λ kp,q^q2 k(K_f,p^λ )^∗kp,q^q2 ds(λ) by theorem 6

≤ Z

S

kK_f,p^λ k^qds(λ)

!¹₂ Z

S

k(K_f,p^λ )^∗k^qds(λ)

!¹₂

by H¨older inequality

On the other hand, from the calculation which gave (7), it follows that Z

S

k(K_f,p^λ )^∗k^qds(λ)

= Z

S

Z

GLn

Z

GLn

|fˆ(., a)(λu)|^p|det(a)|^−m−n|det(u)|^{m(p−1)−npq} da

!^q_p

duds(λ)

≤ Z

GLn

Z

S

Z

GLn

|fˆ(., a)(λu)|^qdet(u)|^m−nduds(λ)

!^p_q da

|det(a)|^m+n

!^q_p

We used the Minkowski’s generalized inequality (α≥1):

Z Z (

Z

|φ(a, λ, u)|dµ(a))^αdν(u)ds(λ)

!_α¹

≤

Z Z Z

|φ(a, λ, u)|^αdν(u)ds(λ)

!_α¹ dµ(a), with

φ(a, λ, u) =|fˆ(., a)(λu)|^p|det(u)|^{m(p−1)−npq} , and

α= q

p, dµ(a) = da

|det(a)|^m+n, dν(u) =du.

From (2), fork=n, we have Z

GLn

Z

S

Z

GLn

|fˆ(., a)(λu)|^qdet(u)|^m−nduds(λ)

!^p_q da

|det(a)|^m+n

!_p^q

= Z

GLn

Z

Mmn

|f(., a)(θ)|ˆ ^qdθ

!^p_q da

|det(a)|^m+n

!^q_p (8)

≤ Z

GL_n

Z

M_nm

|f(b, a)|^p dbda

|det(a)|^m+n

!^q_p

=kfk^q_p,

by applying to (8) the Hausdorff-Young theorem for the additive group Mnm since the formula τλ(b) = τ(T r(bλ)) defines a unitary character of the additive group Mnm and λ −→ τλ is an isomorphism of the additive groupMmn onto the dual of (Mnm,+).

Similarly, sinceK_f^λ∗p,p(u, a) =K_f,p^λ (a, u) by the proof of lemma 5, andf^∗p∈ D, we show that Z

S

kK_f,p^λ k^qds(λ)≤ kf^∗pk^q_p

Now Z

G

|f^∗p|^pdg= Z

G

|det(a)|^(m−n)p+m|f¯(g⁻¹)|^pdg

= Z

G

|det(a⁻¹)|^(m−n)p+m|f¯(g)|^p∆(g⁻¹)dg (g→g⁻¹)

= Z

G

|det(a⁻¹)|^(m−n)p|f¯(g)|^pdg

≤(Cf)^pkfk^p_p

Hence Z

S

kK_f,p^λ k^qds(λ)≤ kf^∗pk^q_p≤(Cfkfkp)^q, and therefore

kFp(f)kq = Z

S

kFp(f)λk^q_qds(λ)

!¹_q

≤(Cf)¹²kfkp

Remarks. If m < n, the theorems 1, 2 are already true, but in this case you have to alter the definitions ofS,H, πλ,δ1/q (see [1]); and it may not be difficult to prove an analogy of theorem 7 with another Cf in (iii). We tackled the case wherem≥n to be able to reach the valuem=n and obtain the Hausdorff-Young theorem.

Theorem 8 ( Hausdorff-Young Theorem). If n=m, 1< p <2, 1

p+1

q = 1, f ∈L¹∩L^p(G),

then Fp(f)∈L^q(S,Lq(H)) andf −→ Fp(f) extends uniquely to a, norm-decreasing, linear map of L^p(G)intoL^q(S,Lq(H))with norm ≤1.

Proof. In fact, by theorem 7 (even if m≥n) we have Fp(f)∈L^q(S,Lq(H)) for anyf ∈ D(G).

Ifm =n, we get (Cf)¹² = 1, hencef −→ Fp(f) is continuous of (D(G),k.kp) intoL^q(S,Lq(H)) with norm≤1, it extends uniquely toL^p(G), in particular, iff ∈L¹∩L^p(G), we take a sequence (fk)∈ D(G) which tends to f in norm ofL¹(G) and in norm of L^p(G).

Corollary 9. Setting1< p <2, ¹_p+¹_q = 1, x∈R. Then (i) The domain of δxFp(f)is dense in Hfor anyf ∈ D(G).

(ii) Ifn=m, f ∈L^p(G)and∆^xf ∈L^p(G), thenDom(δxFp(f)is also dense inH.

Proof. (i) Let f ∈ D(G), since ∆^xf is also ∈ D(G), from the proof of lemma 4 it follows that Fp(∆^xf)λacts inHwith the kernel of Hilbert-Schmidt

K_∆^λxf,p(u, a) = [∆^xf]^∧(., u⁻¹a)(λu)|det(a)|^mp−p1

|det(au⁻¹)|^m

= Z

Mnm

∆^xf(b, u⁻¹a)τ(T r(bλu))db

=|det(u⁻¹a)|^−mxK_f,p^λ (u, a) Letξ∈Dom(δx), then

[Fp(∆^xf)λ]δxξ(u) = Z

GL_n

K_∆^λxf,p(u, a)δxξ(a) da

|det(a)|ⁿ

= Z

GLn

|det(u⁻¹a)|^−mxK_f,p^λ (u, a)|det(a)|^mxξ(a) da

|det(a)|ⁿ

=|det(u)|^mx Z

GL_n

K_f,p^λ (u, a)ξ(a) da

|det(a)|ⁿ

=δxFp(f)λξ(u).

Hence

Fp(∆^xf)δx⊆δxFp(f),

and consequently the domain ofδxis stable by Fp(f) and thus, sinceDom(δx) is dense in H, (i) follows.

(ii) Let us take a sequence (fk)∈ D(G) such thatfk −→f and ∆^xfk−→∆^xf inL^p(G). Then, for anyξ∈Dom(δx), and for almost allu, we have

Fp(∆^xf)δxξ(u) = lim

k Fp(∆^xfk)δxξ(u) by theorem 8

= lim

k |det(u)|^mxFp(fk)ξ(u)

=|det(u)|^mxFp(f)ξ(u) by theorem 8

=δxFp(f)ξ(u).

ThusFp(∆^xf)δx⊆δxFp(f), which completes the proof of corollary 9.

If n divides m, the essential dual of G amounts to a single point by ([1], p.80) which always bears the Plancherel measure and the Fourier inverse theorem (see th.1 and th.2). Whether the essential dual is very large or a single point, the proof of theorem 7 asserts that the constantCf

only depends on f, m and n. So let us raise the following problem

Problem 1. What happens to the last two statements ifm > n?

Remarks. In Eymard-Terp’s article ([3], p.227), ifm=n= 1, the following result appears kFp(f)kq ≤Apkfkp, Ap =p^2p1 q^−12q, f ∈L¹∩L^p(G).

Their proof relies on Babenko-Beckner’s inequality ([2], th.1, p.162) which is valid in the case where K =R (or K= Rⁿ, Ap → Aⁿ_p). If K is disconnected, this inequality fails by ([7], th.B, p.180).

If Kis connected, in theorem 8, the norm of Fp is strictly <1, kFpk ≤A^nm_p if K=R, and kFpk ≤A^2nm_p whenK=C, in fact, you just have to apply ([2], th.1, p.162) to (8).

References

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[2] BECKNER, W. Inequalities in Fourier analysis, Ann. of Math., 102 (1975), 159-182.

[3] EYMARD, P.et TERP, M. La transformation de Fourier et son inverse sur le groupe desax+bd’un corps local. (French) Analyse harmonique sur les groupes de Lie (Sm., Nancy-Strasbourg 1976–1978), II, pp. 207–248, Lecture Notes in Math.n◦739, Springer, Berlin, 1979.

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Saunders Company, 1969.

[5] KUNZE, R. A.L

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[8] WEIL, A. Basic number theory. Die Grundlehren der mathematischen Wissenschaften, Band 144 Springer- Verlag New York, Inc., New York 1967