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

An aspect of uncertainty principle in real classical analysis asserts that a function f and its Fourier transform ̂ f cannot decrease simultaneously very rapidly at infinity. As illustrations of this, one has Hardy’s theorem [1], Morgan’s theorem [2], and BeurlingHörmander’s theorem [3–5]. These theorems have been generalized to many other situations; see, for example, [6–10]. In 1983, Cowling and Price [11] have proved an Lp-Lq-version of Hardy’s theorem. An Lp-Lq-version of Morgan’s theorem has been also proved by Ben Farah and Mokni [7]. To state the Lp-Lq-versions of Hardy’s and Morgan’s theorems more precisely, we propose the following. Let a,b > 0, p,q ∈ [1,+∞], α≥ 2, and β such that 1/α+ 1/β = 1. If we consider measurable functions f on R such that

In 1983, Cowling and Price [11] have proved an L 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 the L p -L q -versions of Hardy's and Morgan's theorems more precisely, we propose the following.
In this paper, we give an L p -L q -version of Morgan's theorem for the n-dimensional Euclidean motion group M(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 in Theorem 1.1 below for f = 0 a.e. for the case α = 2 follows from their work.
The motion group M(n) is the semidirect product of R n with K = SO(n). As a set M(n) = R n × K, and the group law is given by here k · x is the naturel action of K on R n . The Haar measure of M(n) is dx dk, where dx is the Lebesgue measure on R n and dk is the normalized Haar measure on K.
Denote by M(n) the unitary dual of the motion group. The abstract Plancherel theorem asserts that there is a unique measure μ on M(n) such that for all f ∈ L 1 (M(n)) ∩ L 2 (M(n)), where π( f ) = M(n) f (x,k)π(x,k)dx dk is the group Fourier transform of f at π ∈ M(n).
It is well known that μ is 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 R n . As such an element π r,λ is realized in a Hilbert space H λ , we note that for f ∈ L 1 (M(n)) ∩ L 2 (M(n)), π r,λ ( f ) is a Hilbert-Schmidt operator on H λ , moreover the restriction of the Plancherel measure on the part ]0, ∞[×{λ} is given up to a constant depending only on n, by r n−1 dr.

Description of the unitary dual of M(n)
We are going to describe the infinite-dimensional elements of M(n), which are sufficient for the Plancherel formula. We start by some notations.
For any integer m, let ·, · denote the Hermitian (resp., Euclidian) product on C m (resp., on R m ) and let · be the corresponding norm. For y = 0 in R n let U y be the stabilizer of y in K under its natural action on R n . U y is conjugate to the subgroup U = SO(n − 1) of SO(n) leaving fixed ε n = (0,...,0,1) in R n .
We remark that R n , the set of unitary characters of R n , is identified with R n . In fact any such character is of the form χ y , y ∈ R n , and is defined for all x ∈ R n by χ y (x) = e i x,y . The trivial character corresponds to y = 0.
To construct an infinite-dimensional irreducible unitary representation of the motion group M(n), we use the following steps.
Step 1. Take a nontrivial element χ y in R n . It is stabilized under the action of K by U y .
Step 2. Take λ ∈ U y and consider χ y ⊗ λ as a representation of the semidirect product of R n by U y denoted by R n U y .
Step 3. Induce χ y ⊗ λ from R n U y to M(n) to obtain a representation T y,λ of M(n).
We have then the following properties (see [13,14] for details).
(a) For y = 0 and any λ ∈ U y , the representation T y,λ is unitary and irreducible.
(b) Every infinite-dimensional irreducible unitary representation of M(n) is equivalent to T y,λ for some y and λ as above. (c) The representations T y1,λ1 and T y2,λ2 are equivalent if and only if y 1 = y 2 and λ 1 is equivalent to λ 2 under the obvious identification of U y1 with U y2 . In particular, when y = r > 0, T y,λ is equivalent to T rεn,λ , so the different classes of infinite-dimensional representations of M(n) can be parametrized by (r,λ) ∈]0, ∞[× U. We use the notation π r,λ for T rεn,λ and for its equivalence class in M(n). Let us make this representation explicit.
λ is an irreducible unitary representation of U = SO(n − 1), it is of finite dimension d λ and acts on C dλ . Let H λ be the vector space of all measurable function ψ : K → C dλ such that K ψ(k) 2 dk < ∞ and ψ(uk) = λ(u)(ψ(k)) for all u ∈ U, k ∈ K.H λ is a Hilbert space with respect to the inner product defined by for a ∈ R n , k,k 0 ∈ K. The Plancherel measure μ is then supported by the subset of M(n) given by {π r,λ : λ ∈ U,r ∈ R + }, and on each "piece" {π r,λ : r ∈ R + } with λ fixed in U, it is given by C n r n−1 dr, where C n is a constant depending only on n.

International Journal of Mathematics and Mathematical Sciences
The Fourier transform of a function f in L 1 (M(n)) is denoted as above by f . It is defined for (r,λ) ∈]0, ∞[× U by (the integral being interpreted suitably, see [15]). By the Plancherel theorem we know that for f ∈ L 1 (M(n)) ∩ L 2 (M(n)), f (r,λ) is a Hilbert-Schmidt operator. Let f (r,λ) HS be its Hilbert-Schmidt norm.

Morgan's theorem for the motion group
Before giving Morgan's theorem for the motion group M(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.
If g is an entire function on C satisfying the conditions g(x + iy) ≤ const e σ|y| ρ for any x, y ∈ R, e B|x| ρ g |R ∈ L q (R), We now give the L p -L q -version of Morgan's theorem.
Let λ ∈ U and let ϕ, ψ be K-finite vectors in H λ . We note that ϕ and ψ are continuous on K and thus bounded. On the other hand, for r ∈ R, (3.2) S. Ayadi and K. Mokni 5 Let Φ r (x,k) = (π r,λ (x,k)ϕ | ψ) for r ∈ R and (x,k) ∈ M(n). Then, by definition of π r,λ , we have Note that the integral on the right-hand side makes sense even if r ∈ C. Hence, with (x,k) fixed, the function Φ r (x,k) of the variable r extends to the whole complex plane. One can easily see that for fixed (x,k), z → Φ z (x,k) is an entire function on C. Moreover, for z ∈ C, where A is a constant depending only on λ, ϕ, and ψ. (Note that ϕ and ψ are continuous functions on K and hence are bounded.) Using the fact that dk 0 is a normalized measure on K, we obtain Since f satisfies hypothesis (i) of Theorem 3.2 and |(Φ z (x,k))| ≤ Ae |z|· x , we conclude that the function r → ( f (r,λ)ϕ | ψ) can be extended to the whole of C and indeed it can be proved that the function Further, from (3.6) and (3.7), we deduce that Let I =](bβ) −1/β (sin(π/2)(β − 1)) 1/β ,(aα) 1/α [, and C ∈ I. Applying the convex inequality |t y| ≤ (1/α)|t| α + (1/β)|y| β to the positive numbers C x and | Im z|/C, we obtain Using this inequality, hypothesis (i), the fact that dk is a normalized measure, and the inequality a > c α /α, we obtain as a function of z.
Since ϕ, ψ, and λ are arbitrary, then f (r,λ) ≡ 0 for all r ∈ R + and λ ∈ U. Hence, by the Plancherel formula, we get that f = 0 a.e. This completes the proof of the theorem.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall. This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos). We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/ according to the following timetable:

Manuscript Due
December 1, 2008 First Round of Reviews March 1, 2009