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Multidimensional van der Corput and sublevel set estimates

Author(s): Anthony Carbery; Michael Christ; James Wright
Journal: J. Amer. Math. Soc. 12 (1999), 981-1015.
MSC (1991): Primary 42B10; Secondary 26D10, 05D99
Posted: June 7, 1999
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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.

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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
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

DOI: 10.1090/S0894-0347-99-00309-4
PII: S 0894-0347(99)00309-4
Keywords: Oscillatory integrals, sublevel sets, van der Corput lemma, combinatorics
Received by editor(s): June 24, 1998
Posted: 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 of article: Copyright 1999, American Mathematical Society

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