Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

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
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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 42B10, 26D10, 05D99

Retrieve articles in all journals with MSC (1991): 42B10, 26D10, 05D99


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: http://dx.doi.org/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
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