Lebesgue boundedness of Riesz potential for $(k,1)$-generalized Fourier transform with radial piecewise power weights

In spaces with weight $|x|^{-1}v_k(x)$, where $v_k(x)$ is the Dunkl weight, there is the $(k,1)$-generalized Fourier transform. Harmonic analysis in these spaces is important, in particular, in problems of quantum mechanics. Recently, for the $(k,1)$-generalized Fourier transform, the Riesz potential was defined and the $(L^p,L^q)$-inequality with radial power weights was proved for it, which is an analogue of the well-known Stein–Weiss inequality for the classical Riesz potential and the Dunkl–Riesz potential. In the paper, this result is generalized to the case of radial piecewise power weights. Previously, a similar inequality was proved for the Dunkl–Riesz potential.