  We study spherical analogues of Nikodym sets and related maximal functions.
In particular, we prove sharp $L^p$-estimates for Nikodym maximal functions
associated with spheres. As a corollary, any Nikodym set for spheres must have
full Hausdorff dimension. In addition, we consider a class of maximal functions
which contains the spherical maximal function as a special case. We show that
$L^p$-estimates for these maximal functions can be deduced from local smoothing
estimates for the wave equation relative to fractal measures.
