Optimality of the reflection-loss factor

Determine whether the reflection-loss factor M_k in the general-data weak-type (1,1) estimate for the vector Dunkl–Riesz transform is optimal.

Background

For the vector Dunkl–Riesz transform $\mathcal R_k$, the paper proves the general weak-type estimate $$\|\mathcal R_k f\|_{L^{1,\infty}(\nu_k;\ell^2)}\le (M_k+2)\|f\|_{L^1(\nu_k)}$$, where $M_k$ is the number of positive roots with strictly positive multiplicity. The factor $M_k$ arises from enlarging the positivity set by the reflected sets required by the differential–difference structure of the Dunkl operators.

The author explicitly declines to assert that this reflection loss is optimal. Thus, it remains unresolved whether the factor $M_k$ is genuinely necessary for arbitrary, non-reflection-invariant data, or whether a sharper cancellation or localization argument can improve the bound. The paper notes that removing the reflected enlargement would require an additional principle not provided by the direct variational method.