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

In this work, we introduce the β-semigroup for β > 0, which unifies and extends the classical Poisson (for β=1) and heat (for β=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 < n + 2γ/α. We further define the bi-parametric potential of order α by Sₖ(α,β) = (I + (-Δₖ)β/2)-α/β and establish its inverse along with a detailed description of the associated range space. Our approach employs a wavelet-based method that represents the inverse as the limit of truncated hypersingular integrals parameterized by ε > 0. To analyze the convergence of these approximations, we introduce the concept of η-smoothness at a point x₀ in the Dunkl setting. We show that if a function f ∈ Lₖᵖ(Rⁿ) ∩ Lₖ²(Rⁿ), for 1 ≤ p ≤ ∞, possesses η-smoothness at x₀, then the truncated hypersingular approximations converge to f(x₀) as ε → 0⁺.