Multidimensional van der Corput and sublevel set estimates |
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Authors: | Anthony Carbery Michael Christ James Wright |
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Affiliation: | Department of Mathematics & Statistics, University of Edinburgh, King's Buildings, Edinburgh EH9 3JZ, Scotland, United Kingdom ; Department of Mathematics, University of California, Berkeley, California 94720-3840 ; Department of Mathematics, University of New South Wales, 2052 Sydney, New South Wales, Australia |
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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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Keywords: | Oscillatory integrals sublevel sets van der Corput lemma combinatorics |
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