Discrete Hardy Spaces and Heat Semigroup Associated with the Discrete Laplacian

Abstract

In this paper we study the behavior of some harmonic analysis operators associated with the discrete Laplacian ∆ d in discrete Hardy spaces H p (Z). We prove that the maximal operator and the Littlewood-Paley g function defined by the semigroup generated by ∆ d are bounded from H p (Z) into ℓ p (Z), 0 < p ≤ 1. Also, we establish that every ∆ d -spectral multiplier of Laplace transform type is a bounded operator from H p (Z) into itself, for every 0 < p ≤ 1.Chen and Fang ([5]) extended Eoff's result to higher dimensions when p = 1.The discrete Laplacian ∆ d on Z is defined by.(where f is a complex function defined on Z. For every 0 < p ≤ ∞, we denote by ℓ p (Z) the usual Lebesgue space on Z with respect to the counting measure µ. The operator ∆ d is bounded from ℓ p (Z) into itself, for every 0 < p ≤ ∞, and it is a nonnegative operator in ℓ 2 (Z). We define the function G(n, t) = e −2t I n (2t), n ∈ Z and t > 0.Here, for every n ∈ Z, I n represents the modified Bessel function of the first kind and order n. The main properties of I n can be encountered in [15, Chapter 5].For every t > 0 we consider the convolution operator W t defined byfor every f ∈ ℓ p (Z), 1 ≤ p ≤ ∞. In [6, Proposition 1] it was proved that the uniparametric family {W t } t>0 is a positive Markovian diffusion semigroup in the Stein's sense ([20]) in ℓ p (Z), 1 ≤ p ≤ ∞. Moreover, W t = e −t∆ d , t > 0, that is, −∆ d is the infinitesimal generator of {W t } t>0 .

Discrete diffusion semigroups associated with Jacobi-Dunkl and exceptional Jacobi polynomials

“…The degree of P [1] n is usually greater than n. Actually finite many ones are missing from the sequence of degrees, that is exceptional family of polynomials has finite codimension in the space of polynomials. Despite these facts, if the set of the gaps is admissible, {P [1] n } ∞ n=0 is a complete orthogonal system on I with respect to the weight…”

“…Besides the continuous operator semigroup, lately the study of discrete diffusion semigroups has come to the forefront of interest, see e.g. [9,1,2,4].…”

Discrete diffusion semigroups associated with Dunkl-Jacobi and exceptional Jacobi polynomials

“…The degree of P [1] n is usually greater than n. Actually finite many ones are missing from the sequence of degrees, that is exceptional family of polynomials has finite codimension in the space of polynomials. Despite these facts, if the set of the gaps is admissible,…”

“…where σ n := σ α,β n = P [1] n W,2 , and P [1] n = P α,β, [1] n = b(p α,β n ) ′ − bwp α,β n . Subsequently we assume that the admissibility condition mentioned above fulfils, that is the system is complete.…”

“…Besides the continuous operator semigroup, lately the study of discrete diffusion semigroups has come to the forefront of interest, see e.g. [7], [1], [2], [4].…”