Multidimensional van der Corput and sublevel set estimates

Carbery, Anthony; Christ, Michael; Wright, James

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

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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
Posted: June 7, 1999
MathSciNet review: 1683156
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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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