An analogue of Beurling's theorem for the Laguerre hypergroup

In this paper, we prove an analogue of Beurling's theorem for the Laguerre hypergroup, then we derive some other versions of uncertainty principle.     

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 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.     

On the dual of a commutative signed hypergroup

Signed hypergroups are convolution structures similar to hypergroups, though being not necessarily positivity-preserving. We prove a generalized Plancherel theorem for positive definite measures on a commutative signed hypergroup, with an analogue of the classical Plancherel theorem as a special case. Moreover, signed hypergroups with subexponential growth are studied. As an application, the dual of the Laguerre convolution structure on ℝ₊ is determined.     

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.     

Pseudo-differential operators associated with Laguerre hypergroups

In this paper we consider the generalized shift operator generated from the Laguerre hypergroup; by means of this, pseudo-differential operators are investigated and Sobolev-boundedness results are obtained.     

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.     

Amenability notions of hypergroups and some applications to locally compact groups

Different notions of amenability on hypergroups and their relations are studied. Developing Leptin's theorem for discrete hypergroups, we characterize the existence of a bounded approximate identity for hypergroup Fourier algebras. We study the Leptin condition for discrete hypergroups derived from the representation theory of some classes of compact groups. Studying amenability of the hypergroup algebras for discrete commutative hypergroups, we obtain some results on amenability properties of some central Banach algebras on compact and discrete groups.     

Harmonic functions on hypergroups

We initiate a study of harmonic functions on hypergroups. In particular, we introduce the concept of a nilpotent hypergroup and show such hypergroup admits an invariant measure as well as a Liouville theorem for bounded harmonic functions. Further, positive harmonic functions on nilpotent hypergroups are shown to be integrals of exponential functions. For arbitrary hypergroups, we derive a Harnack inequality for positive harmonic functions and prove a Liouville theorem for compact hypergroups. We discuss an application to harmonic spherical functions.     

Isomorphism on Hypergroups

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Fuzzy幂群的基数定理

文（１）提出了幂群的概念，给出了幂群中各元素是等势的基数定理，文（２）提出了Ｆｕｚｚｙ幂群的概念，但没研究其中各元素的基数问题，本文深入研究这一问题，得到了由Ｄ．Ｄｕｂｏｉｓ等在文（３）中提出的和由李洪兴等在文（４）中提出的两种Ｆｕｚｚｙ集基数形式下的Ｆｕｚｚｙ幂群的基数定理，并给出了Ｆｕｚｚｙ幂群中与基数有关的若干结果。     

Uncertainty Inequalities on Laguerre Hypergroup

In this paper, we give analogues of local uncertainty inequality on ${\mathbb{R}^n}$ for stratified Laguerre hypergroup, 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.     

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.     

Higher order Riesz transforms,fractional derivatives,and Sobolev spaces for Laguerre expansions

We study different Sobolev spaces associated with multidimensional Laguerre expansions. To do this we establish an analogue of P.A. Meyer's multiplier theorem, prove some transference results between higher order Riesz–Hermite and Riesz–Laguerre transforms, and introduce fractional derivatives and integrals corresponding to the Laguerre setting. Hypercontractivity of the Laguerre semigroups and Calderón's reproduction formula are also discussed.     

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     

The octahedron theorem in Minkowski three-space a metric vector space proof of Miquel's theorem in the Laguerre plane

As is well known [1, p.480], the cycles and spears of the real Laguerre plane can be represented by the points and null planes of three-dimensional Minkowski space. Miquel's theorem in the Laguerre plane can then be expressed as an intrinsically interesting Minkowski space theorem the octahedron theorem. We outline the correspondence between the two theorems, and then give a metric vector space proof of the octahedron theorem, thereby providing an alternate proof of Miquel's theorem. We then discuss the generalization of both theorems to more general spaces.     

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.     

Uncertainty principles for the Cherednik transform

We shall investigate two uncertainty principles for the Cherednik transform on the Euclidean space $\mathfrak a$ ; Miyachi??s theorem and Beurling??s theorem. We give an analogue of Miyachi?? theorem for the Cherednik transform and under the assumption that $\mathfrak a$ has a hypergroup structure, an analogue of Beurling??s theorem for the Cherednik transform.     

A weighted transplantation theorem for Laguerre function expansions

We establish a generalized weighted transplantation theorem for Laguerre function expansions, which extends the corresponding result by G. Garrigós et al. “A sharp weighted transplantation theorem for Laguerre function expansions” (J. Funct. Anal. 244 (2007), pp. 247–276).     

More on Kleinewillinghöfer Types of Flat Laguerre Planes

Polster and Steinke [Result. Math., 46 (2004), 103–122] determined the possible Kleinewillingh?fer types of flat Laguerre planes. These types reflect transitivity properties of groups of certain central automorphisms. We exclude three more types from the list given there with respect to Laguerre homotheties. This yields a complete determination of all possible single types with respect to Laguerre homotheties that can occur in flat Laguerre planes. Building on results by M?urer and Hartmann to characterize ovoidal or miquelian Laguerre planes we further characterize certain flat Laguerre planes in terms of their Kleinewillingh?fer types. Received: January 16, 2007. Revised: July 26, 2007.