Muckenhoupt conditions for piecewise-power weights in Euclidean space with Dunkl measure

As a result of many years of research in the Fourier harmonic analysis, a class of linear integral Calderon–Sigmund operators was defined that are bounded in the spaces $L^p$ on $\mathbb{R}^d$ with the Lebesgue measure for $1<p<\infty$. B. Muckenhoupt found conditions on weight that are necessary and sufficient for the boundedness of the Calderon–Zygmund operators in $L^p$-spaces with one weight. They are now known as the Muckenhoupt $A_p$-conditions. G.H. Hardy and J.E. Littlewood $(d=1)$ and S.L. Sobolev $(d> 1)$ proved $(L^p,L^q)$-boundedness of the Riesz potential $I_{\ alpha}$ for $1 <p<q<\infty$, $\alpha=d\Bigl(\frac{1}{p}-\frac{1}{q}\Bigr)$. B. Muckenhoupt and R.L. Wheeden found $A_{p,q}$-weight condition for one weight $(L^p,L^q)$-boundedness of the Riesz potential. An important generalization of the Riesz potential has become the Dunkl–Riesz potential defined by S. Thangavelu and Yu. Xu in Euclidean space with the Dunkl measure. For the Dunkl–Riesz potential, we proved $(L^p,L^q)$-boundedness with two radial piecewise-power weights. In this paper, we define the Muckenhoupt $A_p$ and $A_{p,q}$-conditions for weights in Euclidean space with the Dunkl measure and find out when they are satisfied for piecewise-power weights. The obtained results show that the conditions of $(L^p,L^q)$-boundedness of the Dunkl–Riesz potential with one piecewise-power weight can be characterized using the $A_{p,q}$-condition. This suggests that the conditions of $(L^p,L^q)$-boundedness of the Dunkl–Riesz potential with one arbitrary weight can also be written using the $A_{p,q}$-condition.