Concerning Nikodym-type sets in 3-dimensional curved spaces

Sogge, Christopher

doi:10.1090/S0894-0347-99-00289-1

Concerning Nikodym-type sets in 3-dimensional curved spaces
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by Christopher D. Sogge

J. Amer. Math. Soc. 12 (1999), 1-31

DOI: https://doi.org/10.1090/S0894-0347-99-00289-1

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Abstract:

We investigate maximal functions involving averages over geo- desics in three-dimensional Riemannian manifolds. We first show that one can easily extend the Euclidean results of Bourgain and Wolff if one assumes constant curvature. These results need not hold if this assumption is dropped. Nonetheless, we formulate a generic geometric condition which allows favorable estimates. Curiously, this condition ensures that one is in some sense as far as possible from the constant curvature case. Assuming this, one can prove dimensional estimates for Nikodym-type sets which are essentially optimal. Optimal estimates for the related maximal functions are still open though.

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