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Concerning Nikodym-type sets in 3-dimensional curved spaces

Author: Christopher D. Sogge
Journal: J. Amer. Math. Soc. 12 (1999), 1-31
MSC (1991): Primary 42B25, 58J40
MathSciNet review: 1639543
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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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Additional Information

Christopher D. Sogge
Affiliation: Department of Mathematics, The Johns Hopkins University, Baltimore, Maryland 21218
MR Author ID: 164510

Keywords: Maximal functions, Riemannian manifolds, Nikodym sets
Received by editor(s): October 28, 1997
Additional Notes: The author was supported in part by the NSF
Article copyright: © Copyright 1999 American Mathematical Society