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Uncertainty Inequalities on Laguerre hypergroup
Rahmouni Atef
To cite this version:
Uncertainty Inequalities on Laguerre hypergroup
Rahmouni Atef
University of Carthage, Faculty of Sciences of Bizerte Department of Mathematics Bizerte 7021 Tunisia.
E-mail: Atef.Rahmouni@fsb.rnu.tn
Abstract
In this paper, we give analogues of local uncertainty inequality on Rnfor stratified Laguerre hyper-group, connected with the spectral analysis of a given homogeneous sublaplacian L, also indicate how local uncertainty inequalities imply global uncertainty inequalities. It would be interesting to note that we deduce the local uncertainty inequalities for the radial functions on the Heisenberg group. Finally, we extend Heisenberg-Pauli-Weyl uncertainty inequality by ultracontractive properties of the semigroups generated by the differential operator and on the estimate on the heat kernel.
2000 Mathematics Subject Classification: 42B10, 42B30, 33C45.
Key words and phrases: Uncertainty principle, Heisenberg-Pauli-Weyl Inequality, Laguerre hypergroup,
Laguerre Fourier transform.
1
Introduction
The serious question of certainty in science was high-lighted by Heisenberg, in 1927, via his uncertainty
principle (cf. [16]). He demonstrated, for instance, the impossibility of specifying simultaneously the
position and the speed (or the momentum) of an electron within an atom. In 1933, according to H. Hardy (cf. [14]).
A pair of transforms cannot both be very small. (1) This aspect of the uncertainty principle was already expounded by Norbert Wiener in a lecture in G¨ottingen in 1925. Unfortunately, no written record of this lecture seems to have survived, apart from the nontechnical account in Wiener’s autobiography (cf. [34], pp. 105-107), so one can only guess at what precise versions of (1) it might have contained. Whatever influence this lecture might have had on the physicists in the audience, however, the uncertainty principle did not really sink into the minds of signal analysts until Gabor’s fundamental work (cf. [11]) in 1946. Since then, it has become firmly embedded in the common culture.
On the mathematical side, there were sporadic developments relating to the uncertainty principle in the fifty years after the initial work in the 1920’s, followed by a steady stream of results in the last tow decades, we refer to the survey article (cf. [10]) for an overview of the history, and the book (cf. [15]) of Havin and J¨oricke for other forms of the uncertainty principle. These principles state that a function f and its Fourier transform bf cannot be simultaneously sharply localized. The most common quantitative formulation of the
uncertainty principle is the Heisenberg-Pauli-Weyl inequality. It says that, if f ∈ L2(Rn) Z Rn x2j| f (x)|2dxZ Rn ξ2j| bf (ξ )|2dξ ≥1 4 Z Rn | f (x)|2dx2, j ∈ {1, 2, ...n}. (2) Recently, many works have been consecrated to establish the Heisenberg-Pauli-Weyl inequality for various Fourier transforms. R¨osler ([26]) and Shimeno ([28]) have proved this inequality for the Dunkl transform, in ([27]) R¨osler and Voit have established the analogue of Heisenberg Pauli-Weyl-inequality for the generalized Hankel transform. Also, in ([3]) De Bruijn, using the Hermite polynomials gave a new proof of the Heisenberg-Pauli-Weyl inequality for the classical Fourier transform.
where∆denotes the Laplacian on Rn. This form of the Heisenberg-Pauli-Weyl inequality is better suited for extensions to other contexts, with the Laplacian replaced by a positive self-adjoint operator, and |x| by a distance function. The interpretation of uncertainty inequalities as spectral properties of differential operators is widely present in the literature (cf. [8], [9], [31]).
In this paper we are interested in the Laguerre hypergroup K = [0, +∞) × R which is the fundamental manifold of the radial function space for the Heisenberg group ([2], [18]). Let us recall that (K, ∗α) is
a commutative hypergroup (cf. [20], [32] pp. 243-263 ), on which the involution and the Haar measure are respectively given by the homeomorphism (x,t) → (x,t)−= (x, −t) and the Radon positive measure dmα(x,t) =
x2α+1
πΓ(α+1)dxdt. The unity element of (K, ∗α) is given by e = (0, 0), i.e. δ(x,t)∗αδ(0,0)= δ(0,0)∗α
δ(x,t)= δ(x,t)for all (x,t) ∈ K. The convolution product ∗α is defined for two bounded Radon measures µ
and ν on K as follows
hµ ∗αν , f i =
Z
K×K
T(x,t)(α)f (y, s)dµ(x,t)dν(y, s),
where α is a fixed nonnegative real number and {T(x,t)(α)}(x,t)∈Kare the translation operators on the Laguerre hypergroup (cf. [2], [20], [29], [32]), given by T(x,t)(α)f (y, s) = hδ(x,t)∗αδ(y,s), f i = α π R1 0 R2π 0 f ((ξ , η)r,θ)r(1 − r2)α −1dθ dr if α > 0, 1 2π R2π 0 f ((ξ , η)1,θ)dθ if α = 0, where (ξ , η)r,θ= ( p
x2+ y2+ 2xyr cos θ ,t + s + xyr sin θ ).
Note that for the particular case µ = f mαand ν = gmα, f and g being two suitable functions on K, one
has µ ∗αν = ( f ∗αg)mα, where f ∗αg is the convolution product of f and g given by
f ∗αg(x,t) =
Z
K×K
T(−y,s)(α) f (x,t)g(y, s)dmα(x,t).
The dual (cf. p. 46, [2]) of Laguerre hypergroup, i.e. the space of all bounded continuous and multiplicative functions χ : K → C such that such that χ = χ , wheree χ (x, t) = χ (x, −t), (x, t) ∈ K, ise given ([21]) by bK = {ϕλ ,m; (λ , m) ∈ R∗× N} ∪ {ϕρ; ρ ≥ 0}, where ϕλ ,m(x,t) = e−iλtL
(α) m (|λ |x2) and ϕρ= jα(ρx);L (α) m (x) = e− x2 2 L(α)m (x)/L(α)m (0) and jα(x) = 2αΓ(α + 1)Jα(x)
xα , Jαbeing the Bessel function
of first kind and order α, and L(α)m being the Laguerre polynomial of degree m and order α (cf. [19], [30]). b
K which can be seen as a deformation of the hypergroup of radial functions on the Heisenberg group. The dual of the Laguerre hypergroup bK can be topologically identified with the so-called Heisenberg fan (cf. [6]), i.e., the subset embedded in R2given by
[
m∈N
{(λ , µ) ∈ R2: µ = |λ |(2m + α + 1), λ 6= 0}[{(0, µ) ∈ R2: µ ≥ 0}.
Moreover, the subset {(0, µ) ∈ R2; µ ≥ 0} has zero Plancherel measure; therefore it will usually be disregarded. Identifying bK and (R∗× N) ∪ [0, +∞[, the Fourier transform of a bounded Radon measure µ on the Laguerre hypergroup is then, by
F (µ)(λ,m) =Z K ϕ−λ ,m(x,t)dµ(x,t) and F (µ)(ρ) = Z K jα(ρ, x)dµ(x,t).
The Fourier Laguerre transform of a suitable function f : K → C is given by bf =F ( f dmα), so that
b f (λ , m) = Z K f (x,t)ϕ−λ ,m(x,t)dmα(x,t) and bf (ρ) = Z K f (x,t) jα(ρx)dmα(x,t).
isomorphism from the Schwartz space on K onto S( bK): the Schwartz space on bK. Its inverse is the operator f∨given by f∨(x,t) = Z R×N ϕλ ,m(x,t) f dγα,
where dγαis the Plancherel measure on bK given by dγα(λ , m) = Lαm(0)δm⊗ |λ |α +1dλ .
We introduce the following notations (cf. [20]) Lαp(K) (resp. L
p
α(R × N)) where 1 ≤ p ≤∞the p-th
Lebesgue space on K (resp. on R×N) formed by the measurable functions f : K → C (resp.Φ: R×N → C) such that k f kLp α(K)< +∞(resp. k f kL p α(R×N)< +∞) where k f kLp α(K)= (R K| f (x,t)| pdm α(x,t))1/p if p ∈ [1, +∞[, ess sup(x,t)∈K| f (x,t)| if p = +∞, and kΦkLp α(R×N)= (R R×N|Φ(λ , m)| pdγ α(λ , m))1/p if p ∈ [1, +∞[, ess sup(λ ,m)∈R×N|Φ(λ , m)| if p = +∞, we have the following Plancherel formula
k f kL2 α(K)= k bf kL2α( bK) , f ∈ L1α( bK) ∩ L2α( bK), and we have k bf kL∞α≤ k f kL1 α. (4)
This paper is organized as follows.
In section 2 we set some notations and collect some basic facts about the Laguerre hypergroup. In section 3 of central interest is the following slight sharpening of a local uncertainty inequality. Heisenberg’s inequality says that f is highly localized, then bf can not be concentrated near a single point, but it does not
preclude bf from being concentrated in a small neighborhood of two or more widely separated points. In
fact, the latter phenomenon cannot occur either, and it is the object of local uncertainty inequality to make this precise. The first such inequalities for the Fourier transform were obtained by Faris (cf. [7]), and they were subsequently sharpened and generalized by Price (cf. [22], [23]). Building on the ideas of Faris, Price, and Ricci (cf. [5]) we show the inequalities of the uncertainty on the local Laguerre hypergroup, it is the subject of the following results.
(a) If 0 < β <2α+42 , there is a constant K = K(α, β ) such that for every f ∈ L2
α(K), and every measurable
set E ⊂ bK; 0 < γα(E) <∞, Z b K | bf (λ , m)|2χEdγα(λ , m) ≤ Kγα(E) β 2α+4k |(x,t)|βf k2 2.
(b) If β > 2α+42 , there is a constant eK = eK(α, β ) such that for every f ∈ L2α(K), and every measurable
set E ⊂ bK; 0 < γα(E) <∞, Z b K | bf (λ , m)|2χEdγα≤Kγe α(E)k f k 2−2α+4 β 2 k |(x,t)| βf k 2α+4 β 2 .
(c) Cases (a) and (b) fail for β =2α+42 .
Finally (Section 4), building on the ideas of (cf. [4]) to establish an inequality analogous to (3), and variants of it, for the Laguerre hypergroup. Our purpose is to prove a general form of the Heisenberg-Pauli-Weyl inequality in Laguerre hypergroup. More precisely, using the spectral theory, ultracontractive properties of the semigroups generated by partial differential operator and the estimate on the heat kernel on Laguerre hypergroup, that is for all f ∈ L2α(K), a, b ∈ R; a, b ≥ 1 and η ∈ R such that ηa = (1 − η)b, we have Z K |(x,t)|2a| f (x,t)|2dm α(x,t) !η2 Z R×N |(λ , m)|b/2| bf (λ , m)|2dγ α(λ , m) !1−η2 ≥ Ck f k2. Throughout this paper, C will always represent a positive constant, not necessarily the same in each occur-rence.
2
Preliminaries
To describe the harmonic analysis in our setting we begin with introducing the operator
L = −∂ 2 ∂ x2+ 2α + 1 x ∂ ∂ x+ x 2∂2 ∂ t2 ,
which is positive, symmetric in L2α(K), we endow the space K with homogeneous of degree one norm (with respect to the family dilations (δρ)ρ >0),
N(x,t) = |(x,t)| = (x4+ 4t2)1/4, (x,t) ∈ K.
For α = n − 1, n being a positive integer, the operator is the radial part of the sub-Laplacian on the Heisen-berg group Hn(cf. [29]).
Also, we introduce the operatorΛ=Λ21− (2Λ2+ 2∂ λ∂ )2
defined on bK, whereΛ=|λ |1(m∆+∆−+
(α + 1)∆+) andΛ=2|λ |−1((m + α + 1)∆++ m∆−).
∆± are given for a suitable functionΦby:∆+Φ(λ , m) =Φ(λ , m+1)−Φ(λ , m),∆−Φ(λ , m) =Φ(λ , m)−
Φ(λ , m − 1), if m ≥ 1 and∆−Φ(λ , 0) =Φ(λ , 0), and the quasinorm
N (λ,m) = |(λ,m)| = 4|λ|(m +α + 1
2 ), (λ , m) ∈ bK.
These operators satisfy some basic properties which can be found in (cf. [1], [20], [21]) namely one has
Lϕλ ,m= −|(λ , m)|ϕλ ,mandΛϕλ ,m= |(x,t)|4ϕλ ,m, by the properties of L and the symmetry one can observe
that cL f (λ , m) = 4|λ |(m +α +1
2 ) bf (λ , m) (cf. [29]), and we define the following operator [
Lb/2f (λ , m) = |(λ , m)|b/2 b
f (λ , m), (λ , m) ∈ bK.
We will denote by δρ(x,t) = (ρx, ρ2t), the dilated of (x,t) ∈ K and by Br(x,t) the ball centered at (x,t)
of radius r, i.e., the set Br(x,t) = {(y, s) ∈ K : |(x − y,t − s)|K< r}, and ωα=
Γ(α +1 2 ) 2√πΓ(α 2+1)Γ(α+1) its surface area of Br= Br(0, 0) (cf. [12]). Denote by fρ(x,t) = ρ(2α+4)f (δ1 ρ (x,t))
the dilated of the function f defined on K preserving the mean of f with respect to the measure dmα, in the
sense that
k fρkL1
α(K)= k f kL1α(K), ∀ f ∈ L
1
α(K), ρ > 0.
Throughout this paper, let us fix the notation of the norm by k − kLp
3
The Local Uncertainty Principle
The purpose of this section we develop a family of inequalities in their sharpest forms, which constitute the principle of local uncertainty. It is the subject of the following theorem.
Theorem 1.
(a) If 0 < β <2α+42 , there is a constant K = K(α, β ) such that for every f ∈ L2
α(K), and every
measur-able set E ⊂ bK; 0 < γα(E) <∞,
Z b K | bf (λ , m)|2χEdγα(λ , m) ≤ Kγα(E) β 2α+4k |(x,t)|βf k2 2, (5) where K(α, β ) = α + 2 α + 2 − β ω α β2 2α+4β . (6)
(b) If β >2α+42 , there is a constant eK = eK(α, β ) such that for every f ∈ L2α(K), and every measurable
set E ⊂ bK; 0 < γα(E) <∞, Z b K | bf (λ , m)|2χEdγα≤Kγe α(E)k f k 2−2α+4 β 2 k |(x,t)|βf k 2α+4 β 2 , (7) where e K(α, β ) = π ωα (β − (α + 2)) sin((α+2)πβ ) α + 2 β − (α + 2) − α +2 β −(α +2) .
(c) Cases (a) and (b) fail for β =2α+42 .
Proof. Let Br denote the closed unit ball in K and Bcr its complement. Denote by χBr and χBcr the
charac-teristic functions.
Part(a), let f ∈ L2α(K). By Minkowski’s inequality and (4), for all r > 0, we have k bf χEk2 ≤ k( f χBrb)χEk2+ k( f χBcrb)χEk2
≤ (γα(E))
1/2k( f χ
Brb)k∞+ k( f χBcrb)χEk2
≤ (γα(E))1/2k f χBrk1+ k( f χBcrb)k2. (8)
On the other hand, by H¨older’s inequality and hypothesis β < 2α+42 , we have k f χBrk1 ≤ k |(x,t)| −β χBrk2 k |(x,t)| βf k 2 ≤ Γ(α +1 2 ) 2√πΓ(α 2+1)Γ(α+1) α + 2 − β !1/2 rα +2−βk |(x,t)|βf k 2, ≤ ωα α + 2 − β 1/2 rα +2−βk |(x,t)|βf k 2. (9)
Combining the relations (8), (9) and (10), by choosing r0> 0 to satisfy r02α+4= β 2 γα(E)ωα , we minimize the quantity on the right side to obtain
Z b K | bf (λ , m)|2χEdγα 1/2 ≤ K(α, β )γα(E) 2α+4β k |(x,t)|βf k 2, where K(α, β ) = α + 2 α + 2 − β ω α β2 2α+4β . Part (b), from the hypothesis β >2α+42 , we deduce that the function
(x,t) 7→ (1 + |(x,t)|2β)−1 belongs to L1
α(K) ∩ L
2
α(K) and by H¨older’s inequality, we have
k f k2 1 ≤ (1 + |(x,t)| 2β)1/2f 2 2 (1 + |(x,t)| 2β)−1/2 2 2 ≤ k f k22+ k |(x,t)|2βf k22 (1 + |(x,t)| 2β)−1/2 2 2. (11)
However, by standard calculus and (cf. [13], page 322), we have (1 + |(x,t)| 2β)−1/2 2 2= π ωα β sin((α+2)π β ) . (12)
For ρ > 0, we put as above fρ(x,t) = ρ−(2α+4)f (ρx,ρt2), then we have k fρk22= ρ−(2α+4)k f k22 and k |(x,t)|βf ρk 2 2= ρ2β −(2α+4)k |(x,t)|βf k22. Replacing f by fρ in the relation (11), and by equation (12) we deduce
k f k2 1≤ π ωα β sin((α+2)πβ ) ρ−2(α+2)k f k22+ ρ2(β −(α+2))k |(x,t)|βf k22 . In particular for ρ02β= α + 2 β − (α + 2) k f k2 2 k |(x,t)|βf k2 2 , we get k f k2 1≤ π ωα (β − (α + 2))β sin((α+2)πβ ) α + 2 β − (α + 2) − α +2 β −(α +2) k f k2− 2α+4 β 2 k |(x,t)| βf k 2α+4 β 2 . (13)
Now, according the relations (13), the function f ∈ L1α(K) we have
where e K(α, β ) = π ωα (β − (α + 2)) sin((α+2)πβ ) α + 2 β − (α + 2) −β −(α +2)α +2 .
Part (c), we collect together the counterexamples necessary to establish Theorem 1.
Counterexample1: Necessary to establish Theorem 1(a). We consider the function f1(x,t) = h1(t)g1(x) with h1(t) = |t|−1/4 if |t| ≤ 1, 0 elsewhere and g1(x) = |x|2(32β −32−α) if |x| ≤ 1, 0 elsewhere, where2α+42 ≥ β >2α+4 6 .
The function f1∈ L2α(K), we shall prove that if β =
2α+4 2 then k bf1(λ , m)χEk22= +∞. Let Cm=πΓ(α+1)L1 α m(0). Then we have b f1(λ , m) = Cm |λ |12(3β +12) Z |λ | 0 u−1/4cos(u)du ! Z |λ | 0 e−u/2Lα m(u)u 3 2(β −1)du ! .
Take E = BR(0, 0) (cf. Lemma 2) and as Lα0(0) = 1. We have k bf1(λ , m)χEk22 = ∞
∑
m=0 Lα m(0) Z R | bf1(λ , m)χE|2|λ |α +1dλ . = ∞∑
m=0 Lα m(0) Z R m+ α+12 −R m+ α+12 | bf1(λ , m)|2|λ |α +1 dλ . ≥ 2 [πΓ(α + 1)]2 Z R α +1 2 0 |ge1(λ )|2|λ |α −3β +12 dλ , whereeg1(λ ) = (R|λ | 0 u−1/4cos(u)du)( R|λ | 0 e−u/2u 3 2(β −1)du). But we haveR∞ 0 u−1/4cos(u)du =Γ(34) cos( 3π 8) and R∞ 0 e−u/2u 3 2(β −1)du = 232β −12Γ(3β 2 − 1 2). So, there exist C > 0, R > 0, and r > 0, such that for R ≥ |λ | ≥ r we have |ge1(λ )| ≥ C and k bf1(λ , m)χEk22≥ CR R α +1 2 r |λ |α −3β + 1 2dλ → +∞as r → 0+, if α 3+ 1 2< β . In particular for β = 2α+42 , this provides a contra-diction since the left side of (5) is bounded. Indeed, by (8) we have,
k bf1χEk2≤ (γα(E)) 1/2k f 1χBrk1+ k( f1χBcrb)k2. So, k( f1χBc rb)k2≤ r3β +1/2 4β +1/2B( 3 8, β +2α −1 4 ) and k f1χBrk1= r3β +1/2 3β +1/2B( 3 8, 3β −1 4 ), whereas k bf1χEk2≤ B(3 8, β +2α −1 4 ) 4β +1/2 + B(38,3β −14 ) 3β +1/2 r3β +1/2−→ 0 as r −→ 0+.
The function f2∈ L2α(K), we shall prove that if β = 2α+4 2 then k bf2(λ , m)χEk 2 2= +∞. Let Cm=πΓ(α+1)L1 α m(0). Then we have b f2(λ , m) = Cm |λ |β −12 Z |λ | 0 u−1/4cos(u)du ! Z |λ | 0 e−u/2Lα m(u)u β 2− 5 2du ! .
Take E = BR(0, 0) (cf. Lemma 2) and as Lα0(0) = 1. We have k bf2(λ , m)χEk22 = ∞
∑
m=0 Lα m(0) Z R | bf2(λ , m)χE|2|λ |α +1dλ . = ∞∑
m=0 Lα m(0) Z R m+ α+12 −R m+ α+12 | bf2(λ , m)|2|λ |α +1 dλ . ≥ 2 [πΓ(α + 1)]2 Z R α +1 2 1 |ge2(λ )|2|λ |α −β +2 dλ , whereeg2(λ ) = (R|λ | 1 u−1/4cos(u)du)( R|λ | 1 e−u/2u β 2−52du). But we haveR∞ 0 u−1/4cos(u)du =Γ(34) cos( 3π 8) and R∞ 0 e−u/2u β 2−52du = 2β2−32Γ(β 2− 32). So, there exist C > 0, R > 1, such that for R ≥ |λ | ≥ 1 we have |ge2(λ )| ≥ C and k bf2(λ , m)χEk
2 2≥ C R R α +1 2 1 |λ |α −β +2dλ → +∞as R →∞, if 2α+62 > β . In particular for β =2α+42 , this provides a contradiction since the left side of (7) is bounded. Indeed, by (14) we have,
k bf1χEk2≤ γα(E)k f1k1≤ 2C 2β + 3B( 3 8, β 4− 1 8). Remarks 1.
1. The relations among the exponents in these inequalities are forced by homogeneity considerations
Q = 2α + 4 of Laguerre hypergroup K.
2. Price and Sitaram ([24], Remark 5.2) proved the following local uncertainty inequality on the Heisen-berg group Hn: Given 0 < θ < 12, for each f ∈ L1(Hn) ∩ L2(Hn) and E with Lebesgue measure m(E) <∞, Z E Tr(πλ( f ) ∗ πλ( f ))dµ(λ ) !1/2 ≤ Kθm(E)θk | . |θf k2, where Kθ= (2θ2)−θ(1 − 2θ )−1(1 − 2θ )θ.
Using Theorem 1a), let θ =2α+4β which satisfies 0 < θ <12. For each f ∈ Lrad1 (Hn) ∩ L2rad(Hn), we can replace their Kθby the sharper one
Kθ ,rad(n − 1, β ) = (2θ2)−θ(1 − 2θ )−1 ωn−1 2(n + 1)2 θ , and ωn−1= Γ( n 2) 2√πΓ(n+12 )Γ(n).
We shall use the local uncertainty inequality (5) to prove an analogue of Heisenberg-Pauli-Weyl uncer-tainty inequality, therefore we are so led to impose the most important lemma that we need for the sequel.
For this purpose we define the ball in bK with center (λ , m) and radius r > 0 ( for shortness Br) to be the
set
Br= {(µ, n) ∈ bK, N (λ − µ , max(n − m, 0) +α + 1 2 ) < r}. Examples: Figs of two balls with center (0, 0) and (2, 6) respectively and radius r = 1.
Fig 1: Ball B1(0, 0) Fig 2: Ball B1(2, 6)
λ = 0 λ = 2 m = 6 N R R N 6 6 -
-Lemma 1. The measure of Brwith respect to the Plancherel measure dγαis finite, in the sense that for all
r > 0 and (λ , m) ∈ bK
γα(Br) <∞. Proof. For fixed (λ , m) ∈ bK and r > 0 one has
Since Lα n(0) ∼ n α Γ(α+1), it followsL α n(0) α +2[( r n+α +1 2 + λ )α +2− ( r n+α +1 2 − λ )α +2] ∼Cr n2, and that dγα(Br) <∞,
where we have computed the result above for the case N (λ,m) < r for all m ≥ 0. An analogous result follows for the complement case.
Remark 1. The volume of the ball Brdepends not only on its radius r, but also it is largely close to its center (λ , m) which means that the Plancherel measure dγαis not invariant under the standard translation over R2.
Throughout subsequently, we take the measurable set in bK by Er= {(λ , m) ∈ bK, |(λ , m)| = 4|λ |(m +α + 1
2 ) < r 2},
that is the ball in bK with center (0, 0) and radius r2.
Lemma 2. The measure of Erwith respect to the Plancherel measure dγαis finite, in the sense that for all
r > 0 γα(Er) <∞. More precisely, γα(Er) = 2 r2(α+2) α + 2 m≥0
∑
Lα m(0) (m +α +1 2 )α +2 . (16)Proof. For fixed r > 0 one has
γα(Er) = Z Er dγα(λ , m) =
∑
m≥0 Lα m(0) Z r2 m+ α+12 −r2 m+ α+12 |λ |α +1dλ = 2r 2(α+2) α + 2 m≥0∑
Lα m(0) (m +α +1 2 )α +2 .One can remark easily that the volume of the set Eris finite since Lαm(0) ∼ m α Γ(α+1), so L α m(0) (m+α +1 2 )α +2 ∼ C m2 and hence γα(Er) <∞.
Corollary 1. For all f ∈ L2α(K) and β > 0 one has
Z K |(x,t)|2β| f (x,t)|2dmα(x,t) ! Z b K |(λ , m)|β /2| b f (λ , m)|2dγα(λ , m) ! ≥ Ck f k42, where C is a constant.
Proof. First, suppose that 0 < β <2α+42 .
Consider Z Ec r | bf (λ , m)|2dγα = Z Ec r | bf (λ , m)|2|(λ , m)|β|(λ , m)|−βdγ α ≤ r−2β Z Ec r |(λ , m)| β /2| bf (λ , m)| 2 dγα ≤ Cr−2β Z b K |Lβ /2\f (λ , m)|2dγ α. (18)
By the local uncertainty inequality (a), with 0 < 2α+4β <12, we get
Z Er | bf (λ , m)|2dγα≤ Cr 2βZ K |(x,t)|2β| f (x,t)|2dmα. (19)
Combining the relations (17), (18) and (19), we obtained
k f k22≤ C r2β Z K |(x,t)|2β| f (x,t)|2dmα+ r−2β Z b K |Lβ /2\f (λ , m)|2dγ α ! .
However, let g be the function defined on ]0, +∞[ by g(r) = r2βk |(x,t)|βf k2
2+ r−2βk |(λ , m)|β /2bf k22, then, the minimum of the function g is attained of the point
r0= k |(λ , m)|β /2 b f k2 2 k |(x,t)|βf k2 2 !4β1 , and g(r0) = 2k |(x,t)|βf k 2k |(λ , m)|β /2bf k2. Then we have the desired inequality
k f k42≤ C Z K |(x,t)|2β| f (x,t)|2dmα(x,t) ! Z b K |Lβ /2\f (λ , m)|2dγ α(λ , m) ! . (20)
Second, for β ≥2α+42 . By H¨older’s inequality, we have
Z K | f (x,t)|2|(x,t)|2βdmα(x,t) ≤ k f k2 Z K | f (x,t)|2|(x,t)|4βdmα(x,t) !1/2 , and Z b K |Lβ /2f (λ , m)|2dγ α ≤ Z b K |(λ , m)|β | bf (λ , m)| | bf (λ , m)|dγ α ≤ Z b K | bf (λ , m)|2dγα !1/2 Z b K |Lβf (λ , m)|2dγ α !1/2 ≤ k f k2 Z b K |Lβ f (λ , m)|2dγα !1/2
4
Heisenberg-Pauli-Weyl inequality
In this section, we extend the inequality of the uncertainty of Heisenberg-Pauli-Weyl (Corollary 1) to the more general case. We need to use another method based on ultracontractive semigroups generated by the differential operator L and the estimation of the heat kernel, the result gives by
Theorem 2. Let a, b ≥ 1 and η ∈ R such that ηa = (1 − η)b, then for all f ∈ L2α(K), we have
Z K |(x,t)|2a| f (x,t)|2dmα(x,t) !η2 Z b K |(λ , m)|b/2f (λ , m)|b 2dγα(λ , m) !1−η2 ≥ Cαk f k2 (21) where Cαis a constant.
Remark 2. In the particular case when a = b = β and η = 12, the previous result gives us the Heisenberg-Pauli-Weyl inequality ( cf. Corollary 1)
k |(x,t)|2βf k2
2 k |(λ , m)|β /2bf k22≥ Ck f k42. In our proof of Theorem 2, the heat kernel hsplays an important role.
4.1
Characterization of the heat kernel.
The heat kernel on K is an analogue of the Gauss kernel pson Rn, there is associated to the operator L.
Let {Hs: s > 0} = {e−sL: s > 0} defines a semigroup (heat-diffusion semigroup) of operators such that for
any φ ∈ C0∞(K), Hsφ is a solution of
L(x,t)+
∂ ∂ s = 0
and Hsφ −→ φ a.e. as s −→ 0. For every s > 0, Hs is an integral operator with kernel hs, i.e. for any φ ∈ C0∞(K),
Hsφ = φ ∗ hs.
Then hs, s > 0 are bi-invariant functions and h as a function of the variables s ∈ R+satisfying the properties
(cf. [29]).
Proposition 1. The heat kernel hssatisfies that
(1) {hs: s > 0} form a semigroup under convolution ∗. That is, hs∗ ht= hs+tfor s,t > 0, (2) hs(x, −t) = hs(x,t), Z K hs(x,t)dmα(x,t) = 1. (3) hsis a fundamental solution of (L +∂ s∂)hs= 0, on K × (0, +∞). (4) hρ2s(ρs, ρ2t) = ρ−(2α+4)hs(x,t). Furthermore, bhs(λ , m) = e−4|λ |(m+ α +1 2 )s. The pointwise estimate of the heat kernel hs(x,t) is given by (cf. [17]).
Lemma 3. There are positive constants C and A such that
Lemma 4. Let a > 0. Then for all f ∈ Lα2(K), we have
ke−sLf k2≤ Cs−a/2k |(x,t)|af k2 (24)
where C is a constant.
Proof. For r > 0, let denote the characteristic function of {(x,t) : |(x,t)| < r}.
We set fr = f χBr, and f
r= f − f
r. Then, since | fr(x,t)| ≤ r−a|(x,t)|af (x,t)| and e−sLis a semigroup of
contractions,
ke−sLfrk2≤ k frk2≤ r−ak |(x,t)|af k2. On the other hand, we have
ke−sLfrk2= k fr∗ hsk2 ≤ k frk1khsk2 ≤ khsk2 Z K |(x,t)|2a| f r(x,t)|2dmα(x,t) !1/2 Z K |(x,t)|−2aχBrdmα(x,t) !1/2 since, Z K |(x,t)|−2aχBrdmα(x,t) = ωαr (2α+4)−2a, which gives, ke−sLfrk2≤ Cr−a+(α+2)k |(x,t)|af k2khsk2. By (23), we have khsk2≤ Cs− (α+1) 2 . Hence, ke−sLf k2 ≤ ke−sLfrk2+ ke−sLfrk2 ≤ Cr−a1 + r(α+2)s−(α+1)2 k |(x,t)|af k2. Choosing r = s1/2, we obtain (24).
4.2
Proof of Theorem 2.
Let f ∈ L2α(K) satisfy the hypothesis. For all s > 0, by Lemma 4 and spectral Theorem we have k f k2 ≤ ke−sLf k2+ k(1 − e−sL) f k2
≤ Cs−a/2k |(x,t)|af k2+ k(1 − e−sL)(sL)−b/2(sL)b/2f k2
.
The last term is controlled, using spectral Theorem (Theorem VIII.5 p.262, [25]) by sb/2kLb/2f k 2since (1 − e−s)s−b/2is bounded for s ≥ 0 if b ≤ 2. Hence, we obtain
However, let g be the function defined on ]0, +∞[ by
g(s) = s−a/2k |(x,t)|af k2+ sb/2k |(λ , m)|b/2bf k2, then, the minimum of the function g is attained of the point
s0= a b k |(x,t)|af k 2 k |(λ , m)|b/2f kb 2 a+b2 and g(s0) = a b 2η−1 k |(x,t)|af kη2 k |(λ , m)|b/2bf k 1−η 2 . From which optimizing in s, we obtain the result
k f k2≤ C k |(x,t)|af kη 2 k |(λ , m)|b/2bf k 1−η 2 , for b ≤ 2.
If b > 2, let b0≤ 2. For u ≥ 0 and b0< b, ub0 ≤ 1 + ubwhich for u =|(λ ,m)| ε
1/2
gives the inequality |(λ ,m)| ε b0/2 ≤ 1 +|(λ ,m)| ε b/2
, for all ε > 0. It follows that k |(λ , m)|b0/2 b f k2≤ εb 0 /2k f k 2+ ε b0−b 2 k |(λ , m)|b/2bf k2. Let g(ε) = εb0/2k f k2+ ε b0−b 2 k |(λ , m)|b/2bf k2 then, the minimum of the function g is attained of the point
ε0= b − b0 b0 k |(λ , m)|b/2 b f k2 k f k2 2/b . Optimizing in ε0, we get k |(λ , m)|b0/2 b f k2≤ Ck f k1−b 0/b 2 k |(λ , m)|b/2bf k b0/b
2 . Plugging this into (21) with is
replaced by b, we get the result for b > 2.
Acknowledgements. I want to express my sincere gratitude to Professor Gerald B. Folland for useful
discussions which have influenced some parts of the text for detecting and removing some misprints. Also, I would like to thank Professor Burglind Juhl-J¨oricke for his advice, and his general remarks.
References
[1] M. Assal, Pseudo-differential operators associated with Laguerre hypergroups. J. Comp. Appl. Math., 233 (2009), 617–620.
[2] W. Bloom, H. Heyer, Harmonic Analysis of Probability Measures on Hypergroups. De Gruyter Stud. Math., vol. 20, Berlin/New York, 1994.
[3] N. G. De Bruijn, Uncertainty principles in Fourier analysis. Inequalities (Proc. Sympos. Wright-Patterson Air Force Base, Ohio, 1965), Academic Press, New York, (1967), 57–71.
[4] P. Ciatti, F. Ricci, M. Sundari, Heisenberg-Pauli-Weyl uncertainty inequalities and polynomial volume growth. Adv. Math., 215 (2007), 616–625.
[5] P. Ciatti, F. Ricci, M. Sundari, A local uncertainty inequality for step-two nilpotent Lie groups. In: Proc. Internat. Conf. in Harmonic Analysis and Quantum Groups, Bull. Kerala Math. Ass. 3 (2006), 53–72.
[7] W. G. Faris, Inequalities and uncertainty inequalities, J. Math. Phys., 19 (1978), 461–466.
[8] C. Fefferman, D. H. Phong, The uncertainty principle and sharp G˚arding inequality. Comm. Pure Appl. Math., 34 (1981), 285–331.
[9] C. Fefferman, The uncertainty principle. Bull. Amer. Math. Soc., 9 (1983), 129–206.
[10] G. Folland, A. Sitaram, The uncertainty principle: A mathematical survey. J. Fourier An. Appl., 3 (1997), 207– 238.
[11] D. Gabor, Theory of communication. J. Inst. Elec. Engr., 93 (1946), 429–457.
[12] S. Guliyev, Polar coordinates in Laguerre hypergroup. Khazar J. Math., 2 (2006), 11–20.
[13] I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals. Series, and Products Seventh Edition, Elsevier Inc, 2007. [14] G. H. Hardy, A theorem concerning Fourier transforms. J. London Math. Soc 8 (1933), 227–231
[15] V. Havin, B. J¨oricke, The Uncertainty Principle in Harmonic Analysis. Springer, Berlin 1994.
[16] W. Heisenberg, ¨Uber den anschaulichen Inhalt der quantentheoretischen Kinematic und Mechanik. Zeit. Physik,
43, 172 (1927); The Physical Principles of the Quantum Theory (Dover, New York, 1949; The Univ. Chicago Press, 1930).
[17] J. Huang, H. Liu, A heat Kernel version of Hardy’s theorem for the Laguerre Hypergroup. Acta. Math. Scientia. 30 B(2) 2010.
[18] R. I. Jewett, Spaces with an abstract convolution of measures. Adv. Math., 18 (1973), 1–101. [19] N. N. Lebedev, Special Functions and Thier Applications. Dover Publications, New York, 1972.
[20] M. M. Nessibi, K. Trim`eche, Inversion of the Radon Transform on the Laguerre Hypergroup by using Generalized
Wavelets. J. Math. Anal. Appl., 208 (1997), 337–363.
[21] M. M. Nessibi, B. Selmi, A Wiener-Tauberian and a Pompeiu type theorems on the Laguerre hypergroup. J. Math. Anal. Appl., 351 (2009), 232–243.
[22] J. F. Price, Inequalities and local uncertainty principles. J. Math. Phys., 24 (1983), 1711–1714. [23] J. F. Price, Sharp local uncertainty principles. Studia Math., 85 (1987), 37–45.
[24] J. F. Price, A. Sitaram, Local uncertainty principles for locally compact groups. Trans. Amer. Math. Soc., 308 (1988), 105–114.
[25] M. Reed, B. Simon, Methods of Modern Mathematical Physics: Functional Analysis. Vol.1, Academic Press, 1980.
[26] M. R¨osler, An uncertainty principle for the Dunkl transform. Bull. Austral Math. Soc. 59 (1999), 353–360. [27] M. R¨osler and M. Voit, An uncertainty principle for Hankel Transforms. Amer. Math. Society, 127 (1999), 183–
194.
[28] N. Shimeno, A note on the uncertainty principle for the Dunkl transform. J. Math. Sci. Univ. Tokyo, 8 (2001), 33–42.
[29] K. Stempak, An algebra associated with the generalized sublaplacian. Studia. Math., 88 (1988), 245–256. [30] G. Szego, Orthogonal Ploynomials. Amer. Math. Society, Providence, RI, 1939.
[31] S. Thangavelu, Some uncertainty inequalities. Proc. Indian Acad. Sci. (Math. Sci.), 100 (1990), 137–145. [32] K. Trim`eche, Generalized wavelets and hypergroups. Gordon And Breach Science Publishers, 1997. [33] N. Wiener, The Fourier Integral and Certain of its Applications. Cambridge, 1933.
