Bochner–Riesz means for the twisted Laplacian in $\mathbb{R}^{2}$

We study the Bochner–Riesz problem for the twisted Laplacian \mathcal{L} on \mathbb{R}^{2} . For p\in [1, \infty]\setminus\{2\} , it has been conjectured that the Bochner–Riesz means S_{\lambda}^{\delta}(\mathcal{L}) f of order \delta converge in L^{p} for every f\in L^{p} if and only if \delta> \max(0, |(p-2)/p|-1/2) . We prove the conjecture by obtaining uniform L^{p} bounds on S_{\lambda}^{\delta}(\mathcal{L}) up to the sharp summability indices.