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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Concerning Nikodym-type sets in 3-dimensional curved spaces
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by Christopher D. Sogge PDF
J. Amer. Math. Soc. 12 (1999), 1-31 Request permission

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
  • Email: sogge@jhu.edu
  • Received by editor(s): October 28, 1997
  • Additional Notes: The author was supported in part by the NSF
  • © Copyright 1999 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 12 (1999), 1-31
  • MSC (1991): Primary 42B25, 58J40
  • DOI: https://doi.org/10.1090/S0894-0347-99-00289-1
  • MathSciNet review: 1639543