-Version of Morgan’s Theorem for the n-Dimensional Euclidean Motion Group

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International Journal of Mathematics and Mathematical Sciences Volume 2007, Article ID 43834,9pages

doi:10.1155/2007/43834

Research Article

An L

^p

-L

^q

-Version of Morgan’s Theorem for the n-Dimensional Euclidean Motion Group

Sihem Ayadi and Kamel Mokni

Received 9 August 2006; Revised 11 January 2007; Accepted 15 January 2007 Recommended by Wolfgang zu Castell

We establish anL^p-L^q-version of Morgan’s theorem for the group Fourier transform on then-dimensional Euclidean motion groupM(n).

1. Introduction

An aspect of uncertainty principle in real classical analysis asserts that a function f and its Fourier transform fcannot decrease simultaneously very rapidly at infinity. As il- lustrations of this, one has Hardy’s theorem [1], Morgan’s theorem [2], and Beurling- H¨ormander’s theorem [3–5]. These theorems have been generalized to many other situ- ations; see, for example, [6–10].

In 1983, Cowling and Price [11] have proved anL^p-L^q-version of Hardy’s theorem. An L^p-L^q-version of Morgan’s theorem has been also proved by Ben Farah and Mokni [7].

To state theL^p-L^q-versions of Hardy’s and Morgan’s theorems more precisely, we propose the following.

Leta,b >0,p,q∈[1, +∞],α≥2, andβsuch that 1/α+ 1/β=1.

If we consider measurable functions f onRsuch that

e^a^|^x^|^αf ∈L^p(R), e^b^|^y^|^βf∈L^q(R), (1.1) we obtain the following.

(i) If (aα)^1/α(bβ)^1/β>(sin(π/2)(β−1))^1/β, then f =0 a.e.

(ii) If (aα)^1/α(bβ)^1/β≤(sin(π/2)(β−1))^1/β, then one has infinitely many suchf. The caseα=β=2,p=q=+∞corresponds to Hardy’s theorem.

The caseα=β=2, 1≤p,q <+∞corresponds to the Cowling-Price theorem.

The caseα >2,p=q=+∞corresponds to Morgan’s theorem.

The caseα >2, 1≤p,q <+∞corresponds to the Ben Farah-Mokni theorem.

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We remark that for each one of those cases there are further requirements for f if (aα)^1/α(bβ)^1/β=(sin(π/2)(β−1))^1/β.

In this paper, we give anL^p-L^q-version of Morgan’s theorem for then-dimensional Euclidean motion groupM(n), n≥2.

We can note that for the motion group, theorems of Beurling and Hardy have been studied by Sarkar and Thangavelu [12]. For example, the condition inTheorem 1.1below for f =0 a.e. for the caseα=2 follows from their work.

The motion groupM(n) is the semidirect product ofRⁿwithK=SO(n). As a set M(n)=Rⁿ×K, and the group law is given by

(x,k)(x,k)=(x+k·x,kk), (1.2) herek·xis the naturel action ofKonRⁿ. The Haar measure ofM(n) isdx dk, wheredx is the Lebesgue measure onRⁿanddkis the normalized Haar measure onK.

Denote byM(n) the unitary dual of the motion group. The abstract Plancherel theorem asserts that there is a unique measureμonM(n) such that for all f ∈L¹(M(n))∩ L²(M(n)),

M(n)

f(x,k)²dx dk=

M(n) trπ(f)π(f)^∗dμ(π), (1.3) whereπ(f)=

M(n)f(x,k)π(x,k)dx dkis the group Fourier transform of f atπ∈M(n).

It is well known that μis supported by the set of infinite-dimensional elements of M(n), which is parametrized by (r,λ) ∈]0,∞[×U, where U=SO(n−1) is the subgroup of SO(n) leaving fixed ε_n=(0,. . ., 0, 1) in Rⁿ. As such an element π_r,λ is realized in a Hilbert spaceHλ, we note that for f ∈L¹(M(n))∩L²(M(n)),πr,λ(f) is a Hilbert-Schmidt operator onHλ, moreover the restriction of the Plancherel measure on the part ]0,∞[×{λ} is given up to a constant depending only onn, byrⁿ⁻¹dr.

For the analogue of Morgan’s theorem onM(n) we propose the following version, where we use the notation f(r,λ)=π_r,λ(f).

Theorem 1.1. Letp,q∈[1, +∞],a,b∈]0, +∞[, andα,βpositive real numbers satisfying α >2 and 1/α+ 1/β=1.

Suppose thatf is inL²(M(n)) such that (i)e^a ^x ^αf(x,k)∈L^p(M(n)),

(ii)e^br^β f(r,λ) HS∈L^q(R⁺,rⁿ⁻¹dr) for all fixedλinU.

If (aα)^1/α(bβ)^1/β>(sin(π/2)(β−1))^1/β, thenf is null a.e.

If (aα)^1/α(bβ)^1/β≤(sin(π/2)(β−1))^1/β, then there are infinitely many suchf. This paper is organized as follows.

InSection 2, we give a description of the unitary dual of then-dimensional Euclidean motion groupM(n).Section 3is devoted to the above version of Morgan’s theorem for M(n).

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2. Description of the unitary dual ofM(n)

We are going to describe the infinite-dimensional elements ofM(n), which are suﬃcient for the Plancherel formula. We start by some notations.

For any integerm, let·,·denote the Hermitian (resp., Euclidian) product onC^m (resp., onR^m) and let · be the corresponding norm. Fory=0 inRⁿletUybe the stabilizer ofyinKunder its natural action onRⁿ.Uyis conjugate to the subgroupU= SO(n−1) of SO(n) leaving fixedε_n=(0,. . ., 0, 1) inRⁿ.

We remark thatRⁿ, the set of unitary characters ofRⁿ, is identified withRⁿ. In fact any such character is of the formχ_y,y∈Rⁿ, and is defined for allx∈Rⁿbyχ_y(x)=eⁱ^x,y. The trivial character corresponds toy=0.

To construct an infinite-dimensional irreducible unitary representation of the motion groupM(n), we use the following steps.

Step 1. Take a nontrivial elementχ_yinRⁿ. It is stabilized under the action ofKbyU_y. Step 2. Takeλ∈U_yand considerχ_y⊗λas a representation of the semidirect product of RⁿbyUydenoted byRⁿUy.

Step 3. Induceχy⊗λfromRⁿUytoM(n) to obtain a representationTy,λofM(n).

We have then the following properties (see [13,14] for details).

(a) Fory=0 and anyλ∈Uy, the representationTy,λis unitary and irreducible.

(b) Every infinite-dimensional irreducible unitary representation ofM(n) is equivalent toTy,λfor someyandλas above.

(c) The representations Ty1,λ1 andTy2,λ2 are equivalent if and only if y1 = y2

andλ1is equivalent toλ2under the obvious identification ofU_y1withU_y2. In particular, when y =r >0,Ty,λ is equivalent toTrεn,λ, so the diﬀerent classes of infinite-dimensional representations ofM(n) can be parametrized by (r,λ)∈]0,∞[×U.

We use the notationπ_r,λforT_rε_n_,λand for its equivalence class inM(n). Let us make this representation explicit.

λis an irreducible unitary representation ofU=SO(n−1), it is of finite dimensiondλ

and acts onC^d^λ. LetH_λbe the vector space of all measurable functionψ:K→C^d^λsuch that_K ψ(k) ²dk <∞andψ(uk)=λ(u)(ψ(k)) for allu∈U,k∈K.Hλis a Hilbert space with respect to the inner product defined by

ψ1|ψ2

=d_λ

K ψ1(k),ψ2(k)dk. (2.1)

πr,λacts onHλvia

π_r,λ(a,k)ψk0

=eⁱ^k⁻⁰¹^·^rεⁿ^,aψk0k, ψ∈H_λ, (2.2) fora∈Rⁿ,k,k0∈K.

The Plancherel measureμis then supported by the subset ofM(n) given by {πr,λ:λ∈ U,r ∈R⁺}, and on each “piece”{πr,λ:r∈R⁺}withλfixed inU, it is given by Cnrⁿ⁻¹dr, whereCnis a constant depending only onn.

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The Fourier transform of a function f in L¹(M(n)) is denoted as above by f. It is defined for (r,λ)∈]0,∞[×U by

f(r,λ)=πr,λ(f)=

Rⁿ

K f(a,k)πr,λ(a,k)dk da (2.3) (the integral being interpreted suitably, see [15]).

By the Plancherel theorem we know that for f ∈L¹(M(n))∩L²(M(n)), f(r,λ) is a Hilbert-Schmidt operator. Let f(r,λ) _HSbe its Hilbert-Schmidt norm.

3. Morgan’s theorem for the motion group

Before giving Morgan’s theorem for the motion groupM(n), we state the following complex analysis lemma proved by Ben Farah and Mokni [7]. This lemma plays a crucial role in the proof of our main theorem.

Lemma 3.1. Supposeρ∈]1, 2[,q∈[1, +∞],σ >0, andB > σsin(π/2)(ρ−1).

Ifgis an entire function onCsatisfying the conditions

g(x+iy)≤conste^σ^|^y^|^ρ for anyx,y∈R, e^B^|^x^|^ρg_|R∈L^q(R),

(3.1)

theng=0.

We now give theL^p-L^q-version of Morgan’s theorem.

Theorem 3.2. Let p,q∈[1, +∞],a,b∈]0, +∞[, andα,βpositive real numbers satisfying α >2 and 1/α+ 1/β=1.

Suppose thatf is a measurable function onM(n) such that (i)e^a ^x ^αf(x,k)∈L^p(M(n)),

(ii)e^br^β f(r,λ) _HS∈L^q(R⁺,rⁿ⁻¹dr) for all fixedλinU.

If (aα)^1/α(bβ)^1/β>(sin(π/2)(β−1))^1/β, thenf is null a.e.

Proof. To prove that f =0, we are going to prove that f(r,λ)=0. For this, it suﬃces to show that for fixedλ∈Uand for any fixedK-finite vectorsϕandψinHλ, the condition (aα)^1/α(bβ)^1/β>(sin(π/2)(β−1))^1/βimplies that (f(r,λ)ϕ|ψ)≡0 as a function ofrand λ.

Letλ∈Uand letϕ,ψbeK-finite vectors inHλ. We note thatϕandψare continuous onKand thus bounded. On the other hand, forr∈R,

f(r,λ)ϕ|ψ=

K

Rⁿ f(x,k)π_r,λ(x,k)ϕ|ψdx dk. (3.2)

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LetΦr(x,k)=(πr,λ(x,k)ϕ|ψ) forr∈Rand (x,k)∈M(n). Then, by definition ofπr,λ, we have

Φr(x,k)=dλ

K πr,λ(x,k)ϕk0 ,ψk0

dk0

=dλ

Keⁱ^k⁰⁻¹^·^rεⁿ^,x ϕk0k,ψk0

dk0

=dλ

Keⁱ^rεⁿ^,k⁰^x ϕk0k,ψk0

dk0.

(3.3)

Note that the integral on the right-hand side makes sense even ifr∈C. Hence, with (x,k) fixed, the functionΦr(x,k) of the variablerextends to the whole complex plane. One can easily see that for fixed (x,k),z→Φz(x,k) is an entire function onC. Moreover, forz∈C,

Φz(x,k)≤dλ

K

eⁱ^zεⁿ^,k⁰^a·ϕk0k·ψk0dk0. (3.4) Then

Φz(x,k)≤A

Ke⁻ ^(Imz)εⁿ^,k⁰^x

dk0, (3.5)

whereAis a constant depending only onλ,ϕ, andψ. (Note thatϕandψare continuous functions onKand hence are bounded.)

Using the fact thatdk0is a normalized measure onK, we obtain

Φz(x,k)≤Ae^|^Imz^|·^x . (3.6)

By definition ofΦz(x,k), we have f(z,λ)ϕ|ψ=

K

z−→ f(z,λ)ϕ|ψ is an entire function. (3.8) Further, from (3.6) and (3.7), we deduce that

f(z,λ)ϕ|ψ≤A

K

Rⁿ

f(x,k)e^|^Imz^|·||^x^||dx dk. (3.9) LetI=](bβ)⁻^1/β(sin(π/2)(β−1))^1/β, (aα)^1/α[, andC∈I. Applying the convex inequality

|t y| ≤(1/α)|t|^α+ (1/β)|y|^βto the positive numbersC x and|Imz|/C, we obtain

|Imz| ·x≤C^α

α x^α+ 1

βC^β^|Imz|^β, (3.10)

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thus

f(z,λ)ϕ|ψ≤Ae^(1/βC^β⁾^|^Imz^|^β

K

Rⁿ

f(x,k)e^(C^α^/α) ^x ^αdx dk. (3.11)

Then

f(z,λ)ϕ|ψ≤Ae^(1/βC^β⁾^|^Imz^|^β

K

Rⁿe^a ^x ^αf(x,k)e^(C^α^/α⁻^a)^x ^αdx dk. (3.12) Using this inequality, hypothesis (i), the fact thatdk is a normalized measure, and the inequalitya > c^α/α, we obtain

f(z,λ)ϕ|ψ≤conste^(1/βC^β⁾^|^Imz^|^β. (3.13) On the other hand, sinceπ₋_r,λandπ_r,λare equivalent as representations ofM(n),

f(−r,λ)_HS= f(r,λ)_HS. (3.14) Hypothesis (ii) ofTheorem 3.2and the inequality (3.14) imply that the function

r−→e^br^β f(r,λ)_HS belongs toL^q(R), (3.15) thus

r−→e^br^β f(r,λ)ϕ|ψ)_L²_(H_λ) belongs toL^q(R). (3.16) It is clear from (3.8), (3.13), (3.16) that the function z→(f(z,λ)ϕ,ψ) satisfies the hypothesis ofLemma 3.1, and so

f(z,λ)ϕ|ψ≡0 (3.17)

as a function of z.

Sinceϕ,ψ, andλare arbitrary, thenf(r,λ)≡0 for allr∈R+andλ∈U. Hence, by the Plancherel formula, we get that f =0 a.e. This completes the proof of the theorem.

In order to prove that our version respects the analogy with Morgan’s theorem, let us now establish the sharpness of the condition

(aα)^1/α(bβ)^1/β>sin(π/2)(β₋1)^1/β (3.18) inTheorem 3.2.

Proposition 3.3. Letp,q∈[1, +∞],a,b∈]0, +∞[, andα,βpositive real numbers satisfy- ingα >2 and 1/α+ 1/β=1.

If (aα)^1/α(bβ)^1/β≤(sin(π/2)(β−1))^1/β, then there are infinitely many measurable func- tions onM(n) satisfying

(i)e^a ^x ^αf(x,k)∈L^p(M(n)),

(ii)e^br^β f(r,λ) HS∈L^q(R⁺,rⁿ⁻¹dr) for anyλfixed inU.

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To prove this proposition, we use the following lemma fora,b,α,βas above.

Lemma 3.4. If (aα)^1/α(bβ)^1/β=(sin(π/2)(β−1))^1/β, then for allm∈Randm=(2m+ d(2−α))/(2α−2), there exists a nonzero measurable function onM(n) satisfying

(i) (1 + x )⁻^me^a ^x ^αf ∈L^∞(M(n)),

(ii) (1 +r)⁻^me^br^β f(r,λ) HS∈L^∞(R⁺,rⁿ⁻¹dr) for any fixedλinU.

Proof. We put for (x,k) inM(n)

f(x,k)= −i

Cz^νe^z^q⁻^qA^x ²^zdz, (3.19) whereq=α/(α−2),A^α=(1/4)((α−2)a)²,ν=(2m+ 4−α)/2(α−2), andCis the path which lies in the half-plane Rez >0, and goes to infinity, in the directions argz= ±θ0, π/2q < θ0< π/q.

According to Morgan (see [2, page 190]), for x → ∞, we have f(x,k)∼(α−2)

(α−2)a 2

m/α π α

x ^me⁻^a ^x ^α. (3.20) On the other hand, forλfixed inU, ( f(r,λ)ϕ|ψ) is equal to

−id_λ

K

Rⁿ

C

Kz^νe^z^q⁻^qA ^a²^zeⁱ^rεⁿ^,k⁰^a ϕk0k,ψk0

dk0dz da dk, (3.21)

which by a change of variablesx=k⁻0¹ais equal to

−id_λ

K

Rⁿ

C

Kz^νe^z^q⁻^qA^x ²^zeⁱ^rεⁿ^,x ϕk0k,ψk0

dk0dz dx dk. (3.22)

Using this equality and Fubini’s theorem, we obtain the following expression for (f(r, λ)ϕ|ψ):

−id_λ

K ϕk0k,ψk0

dk0dk

C

Rⁿz^νe^z^q⁻^qA^x ²^zeⁱ^rεⁿ^,xdx dz. (3.23) Since

Rⁿe⁻^qA^x ²^zeⁱ^k⁰⁻¹^rεⁿ^,xdx= π

qAz n/2

e⁻^r²^/4qaz, (3.24)

we deduce that

f(r,λ)ϕ|ψ= −idλ

π qA

n/2

K ϕk0k,ψk0 dk0dk

Cz^ν⁻^n/2e^z^q⁻^r²^/4aqzdz.

(3.25) Now, we fix an orthonormal basis{ej;j∈N}ofHλ. Taking into account that f(r,λ) is a Hilbert-Schmidt operator, we then replaceϕbyei,ψbyejand take the sum oni,j∈Nto

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obtain

f(r,λ)_HS=const.

Cz^ν⁻^n/2e^z^q⁻^r²^/4aqzdz a.e. (3.26) Adapting the method of Morgan (see [2, page 191]), we obtain

f(r,λ)_HS=O(r^me⁻^br^β) (3.27) withm=(2m+n(2−α))/(2α−2). We conclude by using the estimations (3.20) and

(3.27).

Proof ofProposition 3.3. It suﬃces to prove the proposition for (aα)^1/α(bβ)^1/β=

sinπ

2(β−1) 1/β

, (3.28)

and the rest is a deduction. Letmbe a real number verifying m <min

−n

p,n(1−α)

q +n(α−2) 2

(3.29) with the convention 1/r=0 whenr= ∞. If m=(2m+n(2−α))/(2α−2), thenm<

−n/q.

For fixedλinU, Lemma 3.4gives a nonzero measurable function f onM(n) satisfying the inequalities

e^a ^x^αf(x,k)≤const.1 + x ^m, e^br^β f(r,λ)_HS≤const.(1 +r)^m.

(3.30) The conditionsm <−n/ pandm<−n/qand the fact that dk is a normalized measure imply thate^a^||^x^||^αf belongs toL^p(M(n)) ande^br^β||f(r,λ)||HSbelongs toL^q(R⁺,Cnrⁿ⁻¹dr)

for fixedλinU.

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Sihem Ayadi: Département de Mathématiques, Faculté des Sciences de Monastir, Université de Monastir, Monastir 5019, Tunisia

Email address:sihem [email protected]

Kamel Mokni: Département de Mathématiques, Faculté des Sciences de Monastir, Université de Monastir, Monastir 5019, Tunisia

Email address:[email protected]