Bochner-Riesz means on a conical singular manifold

We prove a sharp $L^p$-boundedness criterion for Bochner-Riesz multipliers on flat cones $X=(0,\mathrm{\infty })\times {\mathbb{S}}_{\sigma }^1$. The operator $S_{\lambda }^{\delta }({\mathrm{\Delta }}_X)$ is bounded on $L^p(X)$ for $1\le p\le \mathrm{\infty }$, $p\mathrm{\ne }2$, if and only if $\delta >{\delta }_c(p,2)=\max \{0,2\mid 1\mathrm{/}2-1\mathrm{/}p\mid -1\mathrm{/}2\}$. This result is also applicable to the infinite sector domain with Dirichlet or Neumann boundary, resolving the critical exponent problem in this wedge setting.