Abstract. In this paper we introduce Hardy-Lorentz spaces with variable exponents associated to dilations in R n . We establish maximal characterizations and atomic decompositions for our variable exponent anisotropic Hardy-Lorentz spaces.

Abstract. In this paper we define variable exponent Sobolev spaces associated with Jacobi expansions. We prove that our generalized Sobolev spaces can be characterized as variable exponent potential spaces and as variable exponent Triebel-Lizorkin type spaces.Mathematics subject classification (2010): Primary 42C10, Secondary 42C05, 42C20.

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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 .

In this paper we introduce variable exponent local Hardy spaces h p(·) L (R n ) associated with a non-negative self-adjoint operator L. We assume that, for every t > 0, the operator e −tL has an integral representation whose kernel satisfies a Gaussian upper bound. We define h p(·) L (R n ) by using an area square integral involving the semigroup {e −tL } t>0 . A molecular characterization of h p(·) L (R n ) is established. As an application of the molecular characterization we prove that h p(·) L (R n ) coincides with the (global) Hardy space H p(·) L (R n ) provided that 0 does not belong to the spectrum of L. Also, we show that h p(·) L (R n ) = H p(·) L+I (R n ).