Multidimensional van der Corput

and sublevel set estimates

Authors:
Anthony Carbery, Michael Christ and James Wright

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

Published electronically:
June 7, 1999

MathSciNet review:
1683156

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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:
https://doi.org/10.1090/S0894-0347-99-00309-4

Keywords:
Oscillatory integrals,
sublevel sets,
van der Corput lemma,
combinatorics

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

Article copyright:
© Copyright 1999
American Mathematical Society