A HEAT KERNEL VERSION OF HARDY’S THEOREM FOR THE LAGUERRE HYPERGROUP

In this article, we prove a heat kernel version of Hardy’s theorem for the Laguerre hypergroup.     

A local central limit theorem on the Laguerre hypergroup

We consider here the Laguerre hypergroup (K,α_*), where K=[0,+∞[×R and α_* a convolution product on K coming from the product formula satisfied by the Laguerre functions (m∈N, α?0). We set on this hypergroup a local central limit theorem which consists to give a weakly estimate of the asymptotic behavior of the convolution powers μα_*k=μα_*?α_*μ (k times), μ being a given probability measure satisfying some regularity conditions on this hypergroup. It is also given a central local limit theorem for some particular radial probability measures on the (2n+1)-dimensional Heisenberg group Hn.     

Beurling's theorem for Riemannian symmetric spaces II

We prove two versions of Beurling's theorem for Riemannian symmetric spaces of arbitrary rank. One of them uses the group Fourier transform and the other uses the Helgason Fourier transform. This is the master theorem in the quantitative uncertainty principle.

     

A characterization of inverse Radon transform on the Laguerre hypergroup

Let K=[0,∞)×R be the Laguerre hypergroup which is the fundamental manifold of the radial function space for the Heisenberg group. In this note we give another characterization for a subspace of S(K) (Schwartz space) such that the Radon transform Rα on K is a bijection. We show that this characterization is equivalent to that in [M.M. Nessibi, K. Trimèche, Inversion of the Radon transform on the Laguerre hypergroup by using generalized wavelets, J. Math. Anal. Appl. 208 (1997) 337-363]. In addition, we establish an inversion formula of the Radon transform Rα in the weak sense.     

A Wiener-Tauberian and a Pompeiu type theorems on the Laguerre hypergroup

A Wiener-Tauberian theorem is proven on the Laguerre hypergroup [M.M. Nessibi, K. Trimèche, Inversion of the Radon transform on the Laguerre hypergroup by using generalized wavelets, J. Math. Anal. Appl. 208 (1997) 337-363]. As consequence of this theorem we establish a Pompeiu type-theorem and we study some of its applications.     

Generalized Besov type spaces on the dual of the Laguerre hypergroup

In this paper we introduce the homogeneous Besov type spaces ∧p,q^γ（K） on the dual of Laguerre hypergroup K and we establish some new harmonic analysis results. We give some character- izations of these spaces using equivalent seminorms. Also we study the non-homogeneous Besov type spaces ∧p,q^γ（K）. We give some properties of these spaces and embeddings results with respect to their parameters p, q and γ.     

On fractional maximal function and fractional integral on the Laguerre hypergroup

Let be the Laguerre hypergroup which is the fundamental manifold of the radial function space for the Heisenberg group. In this paper we obtain necessary and sufficient conditions on the parameters for the boundedness of the fractional maximal operator and the fractional integral operator on the Laguerre hypergroup from the spaces to the spaces and from the spaces to the weak spaces .     

A localization theorem for Laguerre expansions

Regularity properties of Laguerre means are studied in terms of certain Sobolev spaces defined using Laguerre functions. As an application we prove a localization theorem for Laguerre expansions.     

A local two radii theorem on the Chébli-Trimèche hypergroup

Let 0<r₁<r₂<r₁+r₂<R. It is given a necessary and sufficient condition so that the null function is the unique solution f∈C^∞(]−R,+R[) of the system

(1)     

A heat kernel version of Cowling-Price theorem for the Laguerre hypergroup

In this paper, we prove a heat kernel version of Cowling-Price theorem for the Laguerre hypergroup.     

A sharp weighted transplantation theorem for Laguerre function expansions

We find the sharp range of boundedness for transplantation operators associated with Laguerre function expansions in Lp spaces with power weights. Namely, the operators interchanging and are bounded in Lp(yδp) if and only if , where ρ=min{α,β}. This improves a previous partial result by Stempak and Trebels, which was only sharp for ρ?0. Our approach is based on new multiplier estimates for Hermite expansions, weighted inequalities for local singular integrals and a careful analysis of Kanjin's original proof of the unweighted case. As a consequence we obtain new results on multipliers, Riesz transforms and g-functions for Laguerre expansions in Lp(yδp).     

An infinite color analogue of Rado's theorem

Let R be a subring of the complex numbers and a be a cardinal. A system L of linear homogeneous equations with coefficients in R is called a-regular over R if, for every a-coloring of the nonzero elements of R, there is a monochromatic solution to L in distinct variables. In 1943, Rado classified those finite systems of linear homogeneous equations that are a-regular over R for all positive integers a. For every infinite cardinal a, we classify those finite systems of linear homogeneous equations that are a-regular over R. As a corollary, for every positive integer s, we have ℵ₀²>ℵs if and only if the equation x₀+sx₁=x₂+?+xs+2 is ℵ₀-regular over R. This generalizes the case s=1 due to Erd?s.     

A non-miquelian Laguerre plane satisfying a theorem of miquelian type

This note shows that a theorem of miquelian type known as (M2) holds in a certain non miquelian Laguerre plane of shear type as defined by Löwen and Pfüller[1].Dedicated to Professor H. Karzel on the occasion of his 70th birthday     

Amenability of compact hypergroup algebras

We show that for a compact hypergroup K, the hypergroup algebra is amenable as a Banach algebra if the set of hyperdimensions of irreducible representations of K is bounded above. Conversely if is amenable, the set of ratios of the hyperdimension to the dimension of irreducible representations of K is bounded above. These are equivalent for compact commutative hypergroups.     

Laguerre Geometry of Surfaces in R^3

Let f ： M → R3 be an oriented surface with non-degenerate second fundamental form. We denote by H and K its mean curvature and Gauss curvature. Then the Laguerre volume of f, defined by L（f） = f（H2 - K）/KdM, is an invariant under the Laguerre transformations. The critical surfaces of the functional L are called Laguerre minimal surfaces. In this paper we study the Laguerre minimal surfaces in R^3 by using the Laguerre Gauss map. It is known that a generic Laguerre minimal surface has a dual Laguerre minimal surface with the same Gauss map. In this paper we show that any surface which is not Laguerre minimal is uniquely determined by its Laguerre Gauss map. We show also that round spheres are the only compact Laguerre minimal surfaces in R^3. And we give a classification theorem of surfaces in R^3 with vanishing Laguerre form.     

R~6中的Laguerre等参超曲面的分类

若超曲面的Laguerre形式为零且Laguerre第二基本形式的特征值(称为Laguerre主曲率)为常数,则称超曲面为Laguerre等参超曲面.对R~6中的Laguerre等参超曲面进行了研究,得到了分类定理.     

Invariance principles for random walks on hypergroups on ℝ2 and ℕ

Let (X _n:n) be i.i.d. with finite variance and values in a hypergroupK:=₊ or and _j=1 ⁿ X _j be the randomized sum of these random variables. It is shown that the processes converge in distribution to a Gaussian process in the caseK=₊, that the processes converge towards a Bessel process on ₊ in the case of polynomial growth of the hypergroupK=₊ or , and that in the case of exponential growth converges towards a Brownian motion asn.