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

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.