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