Multidimensional van der Corput and sublevel set estimates

Carbery, Anthony; Christ, Michael; Wright, James

doi:10.1090/S0894-0347-99-00309-4

Multidimensional van der Corput and sublevel set estimates
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by Anthony Carbery, Michael Christ and James Wright

J. Amer. Math. Soc. 12 (1999), 981-1015

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

Published electronically: 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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