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Concerning Nikodym-type sets in 3-dimensional curved spaces
Author(s):
Christopher
D.
Sogge
Journal:
J. Amer. Math. Soc.
12
(1999),
1-31.
MSC (1991):
Primary 42B25, 58J40
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Abstract:
We investigate maximal functions involving averages over geodesics 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
Email:
sogge@jhu.edu
DOI:
10.1090/S0894-0347-99-00289-1
PII:
S 0894-0347(99)00289-1
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
Copyright of article:
Copyright
1999,
American Mathematical Society
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