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SHARP BOUNDS FOR MULTILINEAR CURVED KAKEYA, RESTRICTION AND OSCILLATORY INTEGRAL ESTIMATES AWAY FROM THE ENDPOINT
Author(s) -
Tao Terence
Publication year - 2020
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/mtk.12029
Subject(s) - multilinear map , mathematics , cartesian product , curvature , monotonic function , flow (mathematics) , product (mathematics) , cartesian coordinate system , scale (ratio) , mathematical analysis , pure mathematics , combinatorics , geometry , physics , quantum mechanics
We revisit the multilinear Kakeya, curved Kakeya, restriction and oscillatory integral estimates that were obtained in a paper of Bennett, Carbery, and the author using a heat flow monotonicity method applied to a fractional Cartesian product, together with induction on scales arguments. Many of these estimates contained losses of the form R ε (orlog O ( 1 ) R ) for some scale factor R . By further developing the heat flow method, and applying it directly for the first time to the multilinear curved Kakeya and restriction settings, we are able to eliminate these losses, as long as the exponent p stays away from the endpoint. In particular, we establish global multilinear restriction estimates away from the endpoint, without any curvature hypotheses on the hypersurfaces.