The dichotomy of Nikodym sets and local smoothing estimates for wave equations

We show that Nikodym sets and local smoothing estimates for linear wave equations form a dichotomy: If Nikodym sets for a family of curves exist, then the related maximal operator is not bounded on $L^p({\mathbb{R}}^2)$ for any $p<\mathrm{\infty }$; if Nikodym sets do not exist, then local smoothing estimates hold, and the related maximal operator is bounded on $L^p({\mathbb{R}}^2)$ for some $p<\mathrm{\infty }$. Whenever the maximal operator is bounded on $L^p({\mathbb{R}}^2)$ for some $p<\mathrm{\infty }$, we also determine the sharp exponent for $L^p({\mathbb{R}}^2)$ bounds.