Characterization of bi-parametric potentials and rate of convergence of truncated hypersingular integrals in the Dunkl setting

In this work, we introduce the -semigroup for > 0, which unifies and extends the classical Poisson (for =1) and heat (for =2) semigroups within the Dunkl analysis framework. Leveraging this semigroup, we derive an explicit representation for the inverse of the Dunkl-Riesz potential and characterize the image of the function space Lₖᵖ (Rⁿ) for 1 p 0. To analyze the convergence of these approximations, we introduce the concept of -smoothness at a point x₀ in the Dunkl setting. We show that if a function f Lₖᵖ (Rⁿ) Lₖ² (Rⁿ), for 1 p, possesses -smoothness at x₀, then the truncated hypersingular approximations converge to f (x₀) as 0^+.