On almost everywhere convergence of planar Bochner-Riesz means

We demonstrate the almost everywhere convergence of the planar Bochner-Riesz means for $L^p$ functions in the optimal range when $5\mathrm{/}3\le p\le 2$. This is achieved by establishing a sharp $L^{5\mathrm{/}3}$ estimate for a maximal operator closely associated with the Bochner-Riesz multiplier operator. The estimate depends on a new refined $L^2$ estimate, which may be of independent interest.