The main purpose of this paper is to prove Hörmander’s $L^p$–$L^q$ boundedness of Fourier multipliers on commutative hypergroups. We carry out this objective by establishing the Paley inequality and Hausdorff–Young–Paley inequality for commutative hypergroups. We show the $L^p$–$L^q$ boundedness of the spectral multipliers for the generalised radial Laplacian by examining our results on Chébli–Trimèche hypergroups. As a consequence, we obtain embedding theorems and time asymptotics for the $L^p$–$L^q$ norms of the heat kernel for generalised radial Laplacian.

Let H be an ultraspherical hypergroup and let $A(H)$ be the Fourier algebra associated with $H.$ In this paper, we study the dual and the double dual of $A(H).$ We prove among other things that the subspace of all uniformly continuous functionals on $A(H)$ forms a $C^*$-algebra. We also prove that the double dual $A(H)^{\ast \ast }$ is neither commutative nor semisimple with respect to the Arens product, unless the underlying hypergroup H is finite. Finally, we study the unit elements of $A(H)^{\ast \ast }.$

Let $K$ be an ultraspherical hypergroup associated with a locally compact group $G$ and a spherical projector $\pi$ and let $\text{VN}(K)$ denote the dual of the Fourier algebra $A(K)$ corresponding to $K$. In this note, we show that the set of invariant means on $\text{VN}(K)$ is singleton if and only if $K$ is discrete. Here $K$ need not be second countable. We also study invariant means on the dual of the Fourier algebra ${{A}_{0}}(K)$, the closure of $A(K)$ in the cb-multiplier norm. Finally, we consider generalized translations and generalized invariant means.

We characterize dual spaces and compute hyperdimensions of irreducible representations for two classes of compact hypergroups namely conjugacy classes of compact groups and compact hypergroups constructed by joining compact and finite hypergroups. Also, studying the representation theory of finite hypergroups, we highlight some interesting differences and similarities between the representation theories of finite hypergroups and finite groups. Finally, we compute the Heisenberg inequality for compact hypergroups.

In this paperwe present a fixed point property for amenable hypergroups that is analogous to Rickert’s fixed point theorem for semigroups. It equates the existence of a left invariant mean on the space of weakly right uniformly continuous functions to the existence of a fixed point for any action of the hypergroup. Using this fixed point property, certain hypergroups are shown to have a left Haar measure.

In this paper we first show that for a locally compact amenable group $G$, every proper abstract Segal algebra of the Fourier algebra on $G$ is not approximately amenable; consequently, every proper Segal algebra on a locally compact abelian group is not approximately amenable. Then using the hypergroup generated by the dual of a compact group, it is shown that all proper Segal algebras of a class of compact groups including the $2\times 2$ special unitary group, $\mathrm{SU} (2)$, are not approximately amenable.

In this paper, we discuss various maximal functions on the Laguerre hypergroup $\mathbf{K}$ including the heat maximal function, the Poisson maximal function, and the Hardy–Littlewood maximal function which is consistent with the structure of hypergroup of $\mathbf{K}$. We shall establish the weak type (1, 1) estimates for these maximal functions. The ${{L}^{p}}$ estimates for $p\,>\,1$ follow fromthe interpolation. Some applications are included.

In this paper we introduce probability-preserving convolution algebras on cones of positive semidefinite matrices over one of the division algebras ${\mathbb F} = {\mathbb R}, {\mathbb C}$ or ${\mathbb H}$ which interpolate the convolution algebras of radial bounded Borel measures on a matrix space $M_{p,q}({\mathbb F})$ with $p\geq q$. Radiality in this context means invariance under the action of the unitary group $U_p({\mathbb F})$ from the left. We obtain a continuous series of commutative hypergroups whose characters are given by Bessel functions of matrix argument. Our results generalize well-known structures in the rank-one case, namely the Bessel–Kingman hypergroups on the positive real line, to a higher rank setting. In a second part of the paper we study structures depending only on the matrix spectra. Under the mapping $r\mapsto \text{spec}(r)$, the convolutions on the underlying matrix cone induce a continuous series of hypergroup convolutions on a Weyl chamber of type $B_q$. The characters are now Dunkl-type Bessel functions. These convolution algebras on the Weyl chamber naturally extend the harmonic analysis for Cartan motion groups associated with the Grassmann manifolds $U(p,q)/(U_p\times U_q)$ over ${\mathbb F}$.

If $G$ is a closed subgroup of a commutative hypergroup $K$, then the coset space $K/G$ carries a quotient hypergroup structure. In this paper, we study related convolution structures on $K/G$ coming fromdeformations of the quotient hypergroup structure by certain functions on $K$ which we call partial characters with respect to $G$. They are usually not probability-preserving, but lead to so-called signed hypergroups on $K/G$. A first example is provided by the Laguerre convolution on $[0,\infty [$, which is interpreted as a signed quotient hypergroup convolution derived from the Heisenberg group. Moreover, signed hypergroups associated with the Gelfand pair $\left( U\left( n,1 \right),\,U\left( n \right) \right)$ are discussed.

In this paper we consider Fourier multipliers on local Hardy spaces ${{\mathbf{h}}^{\mathbf{p}}}(0<p\le 1)$ for Chébli-Trimèche hypergroups. The molecular characterization is investigated which allows us to prove a version of Hörmander’s multiplier theorem.

The distributions of nearest neighbour random walks on hypercubes in continuous time t 0 can be expressed in terms of binomial distributions; their limit behaviour for t, N → ∞ is well-known. We study here these random walks in discrete time and derive explicit bounds for the deviation of their distribution from their counterparts in continuous time with respect to the total variation norm. Our results lead to a recent asymptotic result of Diaconis, Graham and Morrison for the deviation from uniformity for N →∞. Our proofs use Krawtchouk polynomials and a version of the Diaconis–Shahshahani upper bound lemma. We also apply our methods to certain birth-and-death random walks associated with Krawtchouk polynomials.

A compact hypergroup is called almost discrete if it is homeomorphic to the one-point-compactification of a countably infinite discrete set. If the group Up of all p-adic units acts multiplicatively on the p-adic integers, then the associated compact orbit hypergroup has this property. In this paper we start with an exact projective sequence of finite hypergroups and use successive substitution to construct a new surjective projective system of finite hypergroups whose limit is almost discrete. We prove that all compact almost discrete hypergroups appear in this way—up to isomorphism and up to a technical restriction. We also determine the duals of these hypergroups, and we present some examples coming from partitions of compact totally disconnected groups.