The purpose of this paper is to investigate the harmonic analysis associated with Dunkl's operator on the real line to introduce a Besov type spaces generalizing the classical one. Different characterizations of these spaces are given using generalized translation operator and also by mean modulus of continuity.

For the Hardy space H ^p (ℝ₊), ((n/2)+(3/4))⁻¹<p≤1 and n an integer, we shall establish a Hardy's type inequality associated with the Hankel transform for over-critical exponent σ>2−p.

For the Hardy space H ^p (𝕂), 0<p≤1 and 𝕂 being the Laguerre hypergroup, we shall prove a Hardy's type inequality associated with the Laguerre Fourier transform.

In this paper, we introduce a new ring of ponderation functions which allowed us to describe and classify a large class of ring of Bessel integral operators. The main result is to give an explicit form of some ideals of such a ring of operators.