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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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Multidimensional van der Corput and sublevel set estimates
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by Anthony Carbery, Michael Christ and James Wright PDF
J. Amer. Math. Soc. 12 (1999), 981-1015 Request permission

Abstract:

Van der Corput’s lemma gives an upper bound for one-dimensional oscillatory integrals that depends only on a lower bound for some derivative of the phase, not on any upper bound of any sort. We establish generalizations to higher dimensions, in which the only hypothesis is that a partial derivative of the phase is assumed bounded below by a positive constant. Analogous upper bounds for measures of sublevel sets are also obtained. The analysis, particularly for the sublevel set estimates, has a more combinatorial flavour than in the one-dimensional case.
References
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Additional Information
  • Anthony Carbery
  • Affiliation: Department of Mathematics & Statistics, University of Edinburgh, King’s Buildings, Edinburgh EH9 3JZ, Scotland, United Kingdom
  • Email: carbery@maths.ed.ac.uk
  • Michael Christ
  • Affiliation: Department of Mathematics, University of California, Berkeley, California 94720-3840
  • MR Author ID: 48950
  • Email: mchrist@math.berkeley.edu
  • James Wright
  • Affiliation: Department of Mathematics, University of New South Wales, 2052 Sydney, New South Wales, Australia
  • Email: jimw@maths.unsw.edu.au
  • Received by editor(s): June 24, 1998
  • Published electronically: June 7, 1999
  • Additional Notes: This work was partially supported by EPSRC grants GR/L10024 and GR/L78574 (Carbery), NSF grant DMS 9623007 (Christ), ARC grants (Wright), and MSRI
  • © Copyright 1999 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 12 (1999), 981-1015
  • MSC (1991): Primary 42B10; Secondary 26D10, 05D99
  • DOI: https://doi.org/10.1090/S0894-0347-99-00309-4
  • MathSciNet review: 1683156