The weighted Muckenhoupt conjecture for the Carleson–Dunkl operator

Let $\alpha\geq -1/2$, $p\in(1,\infty)$, and let $w\colon\mathbb{R}\to[0,\infty)$ satisfy

$$\sup_{B}\left(\frac{1}{|B|}\int_B w(x)|x|^{2\alpha+1}\,\mathrm{d}x\right)\left(\frac{1}{|B|}\int_B w(x)^{-p'/p}|x|^{2\alpha+1}\,\mathrm{d}x\right)^{p/p'}<\infty,$$

where the supremum is over all intervals $B\subset\mathbb{R}$ and $1/p+1/p'=1$. The weighted Carleson–Dunkl conjecture. Under this condition, there is a constant $C_{p,\alpha,w}$ such that

$$\big\|\mathcal{C}_{\ast}^{\alpha}f\big\|_{L^p(\mathbb{R},w|x|^{2\alpha+1}\,\mathrm{d}x)}\leq C_{p,\alpha,w}\|f\|_{L^p(\mathbb{R},w|x|^{2\alpha+1}\,\mathrm{d}x)}$$

for every $f\in L^p(\mathbb{R},w|x|^{2\alpha+1}\,\mathrm{d}x)$. The preceding discussion shows that the known condition $w\in A_p^\alpha$ is not optimal even when $p=2$, motivating this Muckenhoupt-type condition with the measure $|x|^{2\alpha+1}\,\mathrm{d}x$; whether it implies the weighted inequality remains open.