Weighted inequalities for Dunkl–Riesz potential

For the classical Riesz potential or the fractional integral $I_{\alpha}$, the Hardy–Littlewood– Sobolev–Stein–Weiss $(L^p, L^q)$-boundedness conditions with power weights are well known. Using the Fourier transform $\mathcal{F}$, the Riesz potential is determined by the equality $\mathcal{F}(I_{\alpha}f)(y)=$ $=|y|^{-\alpha}\mathcal{F}(f)(y)$. An important generalization of the Fourier transform became the Dunkl transform $\mathcal{F}_k(f)$, acting in Lebesgue spaces with Dunkl's weight, defined by the root system $R\subset \mathbb{R}^d$, its reflection group $G$ and a non-negative multiplicity function $k$ on $R$, invariant with respect to $G$. S. Thangavelu and Yu. Xu using the equality $\mathcal{F}_k (I_{\alpha}^kf)(y)=|y|^{-\alpha}\mathcal{F}_k(f)(y)$ determined the $D$-Riesz potential $I_{\alpha}^k$. For the $D$-Riesz potential, the boundedness conditions in Lebesgue spaces with Dunkl weight and power weights, similar to the conditions for the Riesz potential, were also proved. At the conference "Follow-up Approximation Theory and Function Spaces"   in the Centre de Recerca Matemàtica (CRM, Barcelona, 2017) M. L. Goldman raised the question about $(L_p,L_q) $-boundedness conditions of the D-Riesz potential with piecewise-power weights. Consideration of piecewise-power weights makes it possible to reveal the influence of the behavior of weights at zero and infinity on the boundedness of the $D$-Riesz potential. This paper provides a complete answer to this question. In particular, in the case of the Riesz potential, necessary and sufficient conditions are obtained. As auxiliary results, necessary and sufficient conditions for the boundedness of the Hardy and Bellman operators are proved in Lebesgue spaces with Dunkl weight and piecewise-power weights.