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

Nikodym sets and maximal functions associated with spheres