Multidimensional van der Corput and sublevel set estimates
Anthony CarberyDepartment of Mathematics & Statistics, University of Edinburgh, Kingâs Buildings, Edinburgh EH9 3JZ, Scotland, United KingdomMichael ChristDepartment of Mathematics, University of California, Berkeley, California 94720-3840James WrightDepartment of Mathematics, University of New South Wales, 2052 Sydney, New South Wales, Australia
1999de
ABI
Annotatsiya
Van der Corputâs 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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