This paper is aimed to establish Hardy and Cowling-Price type theorems for the Fourier transform tied to a generalized Cherednik operator on the real line.

Let {Tt}t>0 be the semigroup of linear operators generated by a Schrödinger operator -A = Δ - V, where V is a nonnegative potential that belongs to a certain reverse Hölder class. We define a Hardy space HA1 by means of a maximal function associated with the semigroup {Tt}t>0. Atomic and Riesz transforms characterizations of HA1 are shown.

For a Schrödinger operator A = -Δ + V, where V is a nonnegative polynomial, we define a Hardy $H_A^1$ space associated with A. An atomic characterization of $H_A^1$ is shown.

Our main result is a Hardy type inequality with respect to the two-parameter Vilenkin system (*) $({\sum }_{k=1}^{\infty }{\sum }_{j=1}^{\infty }{\left|f̂(k,j)\right|}^p{\left(kj\right)}^{p-2}{)}^{1/p}\le C_p\parallel f{\parallel }_{H_{**}^p}$ (1/2 < p≤2) where f belongs to the Hardy space $H_{**}^p\left(G_m\times G_s\right)$ defined by means of a maximal function. This inequality is extended to p > 2 if the Vilenkin-Fourier coefficients of f form a monotone sequence. We show that the converse of (*) also holds for all p > 0 under the monotonicity assumption.

Let G be a semisimple Lie group with Iwasawa decomposition G = KAN. Let X = G/K be the associated symmetric space and assume that X is of rank one. Let M be the centraliser of A in K and consider an orthonormal basis $Y_{\delta ,j}:\delta \in K̂₀,1\le j\le d_{\delta }$ of L²(K/M) consisting of K-finite functions of type δ on K/M. For a function f on X let f̃(λ,b), λ ∈ ℂ, be the Helgason Fourier transform. Let $h_t$ be the heat kernel associated to the Laplace-Beltrami operator and let $Q_{\delta }(i\lambda +\varrho )$ be the Kostant polynomials. We establish the following version...

We develop the L² harmonic analysis for (Dirac) spinors on the real hyperbolic space Hⁿ(ℝ) and give the analogue of the classical notions and results known for functions and differential forms: we investigate the Poisson transform, spherical function theory, spherical Fourier transform and Fourier transform. Very explicit expressions and statements are obtained by reduction to Jacobi analysis on L²(ℝ). As applications, we describe the exact spectrum of the Dirac operator, study the Abel transform...

Denote by $L^1(K\setminus G/K)$ the algebra of spherical integrable functions on $SU(1,1)$, with convolution as multiplication. This is a commutative semi-simple algebra, and we use its Gelfand transform to study the ideals in $L^1(K\setminus G/K)$. In particular, we are interested in conditions on an ideal that ensure that it is all of $L^1(K\setminus G/K)$, or that it is $L_0^1(K\setminus G/K)$. Spherical functions on $SU(1,1)$ are naturally represented as radial functions on the unit disk $D$ in the complex plane. Using this representation, these results are applied to characterize harmonic...

This note attempts to understand graph limits as defined by Lovasz and Szegedy in terms of harmonic analysis on semigroups. This is done by representing probability distributions of random exchangeable graphs as mixtures of characters on the semigroup of unlabeled graphs with node-disjoint union, thereby providing an alternative derivation of de Finetti's theorem for random exchangeable graphs.