Riesz potential for $(k,1)$-generalized Fourier transform

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. We define the Riesz potential for the $(k,1)$-generalized Fourier transform and prove for it, a $(L^q,L^p)$-inequality with radial power weights, which is an analogue of the well-known Stein–Weiss inequality for the classical Riesz potential. For the Riesz potential we calculate the sharp value of the $L^p$-norm with radial power weights. The sharp value of the $L^p$-norm with radial power weights for the classical Riesz potential was obtained independently by W. Beckner and S. Samko.