Wavelet-based inversion and analysis of Flett, Riesz and bi-parametric potentials in $(k,1)$ generalized Fourier framework

TL;DR

This paper develops wavelet-based inversion formulas and analyzes Bessel, Flett, Riesz, and bi-parametric potentials within the $(k,1)$-generalized Fourier framework, introducing new potentials, inversion methods, and function spaces.

Contribution

It introduces a unified approach to potentials in the $(k,1)$-generalized Fourier setting, including explicit inversion formulas and bi-parametric generalizations.

Findings

Derived explicit inversion formulas for Flett and Riesz potentials.

Introduced bi-parametric potentials generalizing Bessel and Flett potentials.

Defined new function spaces associated with these potentials.

Abstract

In this paper, we construct and analyze Bessel and Flett potentials associated with the heat and Poisson semigroups in the framework of the $(k,1)$-generalized Fourier transform. We establish fundamental properties of these potentials and derive an explicit inversion formula for the Flett potential using a wavelet-like transform. Furthermore, we introduce a $\beta$-semigroup $\mathcal{B}_k^{(\beta,t)}$, defined via $W_k^{(\beta, t)}$, which enables the formulation of an inversion formula for the Riesz potential. As a unifying extension, we define and investigate bi-parametric potentials $\mathfrak{J}_k^{(\alpha,\beta)}$, which generalize both the Bessel potential and the Flett potential. In addition, we define the associated function spaces.