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 Lᵖ(R²) for any p<∞; if Nikodym sets do not exist, then local smoothing estimates hold, and the related maximal operator is bounded on Lᵖ(R²) for some p<∞. Whenever the maximal operator is bounded on Lᵖ(R²) for some p<∞, we also determine the sharp exponent for Lᵖ(R²) bounds.