Bochner-Riesz mean 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 $p\in [1,\mathrm{\infty }]\setminus \{2\}$, it has been conjectured that the Bochner-Riesz means $S_{\lambda }^{\delta }(\mathcal{L})f$ of order $\delta$ converges in $L^p$ for every $f\in L^p$ if and only if $\delta >\max (0,\mathrm{\mid }(p-2)\mathrm{/}p\mathrm{\mid }-1\mathrm{/}2)$. We prove the conjecture by obtaining uniform $L^p$ bounds on $S_{\lambda }^{\delta }(\mathcal{L})$ up to the sharp summability indices.