Connections between Hankel transforms of different order for $L^p$-functions are examined. Well known are the results of Guy [Guy] and Schindler [Sch]. Further relations result from projection formulae for Bessel functions of different order. Consequences for Hankel multipliers are exhibited and implications for radial Fourier multipliers on Euclidean spaces of different dimensions indicated.

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.

We investigate heat-diffusion and Poisson integrals associated with Laguerre and special Hermite expansions on weighted $L^p$ spaces with $A_p$ weights.

The aim of this paper is to extend the study of Riesz transforms associated to Dunkl Ornstein-Uhlenbeck operator considered by A. Nowak, L. Roncal and K. Stempak to higher order.