The maximal Bochner–Riesz conjecture on Métivier groups

Let $G$ be a Métivier group with sub-Laplacian $\mathcal{L}$, and let $S_*^{\alpha}(\mathcal{L})$ denote the maximal Bochner–Riesz operator associated with $\mathcal{L}$. Write $\alpha_d(p)=\max\{d(\tfrac12-\tfrac1p)-\tfrac12,0\}$. Maximal Bochner–Riesz conjecture on Métivier groups. For $2\leq p\leq\infty$ and $\alpha>\alpha_d(p)$,

$$\|S_*^{\alpha}(\mathcal{L})f\|_{L^p(G)}\leq C\|f\|_{L^p(G)}.$$

The preceding corollary establishes related bounds in part of this range and sharpness in the range covered there; the conjecture asks for the stated full range on Métivier groups.