Abstract
We apply the Bennett–Carbery–Tao multilinear restriction estimate in order to bound restriction operators and more general oscillatory integral operators. We get improved L p estimates in the Stein restriction problem for dimension at least 5 and a small improvement in dimension 3. We prove similar estimates for Hörmander-type oscillatory integral operators when the quadratic term in the phase function is positive definite, getting improvements in dimension at least 5. We also prove estimates for Hörmander-type oscillatory integral operators in even dimensions. These last oscillatory estimates are related to improved bounds on the dimensions of curved Kakeya sets in even dimensions.
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Bourgain, J., Guth, L. Bounds on Oscillatory Integral Operators Based on Multilinear Estimates. Geom. Funct. Anal. 21, 1239–1295 (2011). https://doi.org/10.1007/s00039-011-0140-9
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DOI: https://doi.org/10.1007/s00039-011-0140-9