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

Abstract

In this work, we introduce the $\beta$-semigroup for $\beta >0$, which unifies and extends the classical Poisson (for $\beta =1$) and heat (for $\beta =2$) 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 $L_k^p({\mathbb{R}}^n)$ for $1\le p<\frac{n+2\gamma }{\alpha }$. We further define the bi-parametric potential of order $\alpha$ by $${\mathfrak{S}}_k^{(\alpha ,\beta )}={\left(I+(-{\Delta }_k{)}^{\beta /2}\right)}^{-\alpha /\beta }$$ 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 $\epsilon >0$. To analyze the convergence of these approximations, we introduce the concept of $\eta$-smoothness at a point $x_0$ in the Dunkl setting. We show that if a function $f\in L_k^p({\mathbb{R}}^n)\cap L_k^2({\mathbb{R}}^n)$, for $1\le p\le \infty$, possesses $\eta$-smoothness at $x_0$, then the truncated hypersingular approximations converge to $f(x_0)$ as $\epsilon \to 0^{+}$.

26 pages