Small cap decouplings
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GAFA Geometric And Functional Analysis
SMALL CAP DECOUPLINGS Ciprian Demeter, Larry Guth and Hong Wang To the memory of Jean Bourgain
Abstract. We develop a toolbox for proving decouplings into boxes with diameter smaller than the canonical scale. As an application of this new technique, we solve three problems for which earlier methods have failed. We start by verifying the small cap decoupling for the parabola. Then we find sharp estimates for exponential sums with small frequency separation on the moment curve in R3 . This part of the work relies on recent improved Kakeya-type estimates for planar tubes, as well as on new multilinear incidence bounds for plates and planks. We also combine our method with the recent advance on the reverse square function estimate, in order to prove small cap decoupling into square-like caps for the two dimensional cone. The Appendix by Roger Heath-Brown contains an application of the new exponential sum estimates for the moment curve, to the Riemann zeta-function.
Contents 1 2 3 4 5
A Brief Overview of “Old” and “New” Decouplings . . . . . . . . A Few Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . The New Results and the Methods of Proof . . . . . . . . . . . . A Refined Flat Decoupling . . . . . . . . . . . . . . . . . . . . . The Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . 5.1 An initial bilinear reduction for Theorem 3.1. . . . . . . . . 5.2 A refined planar Kakeya inequality. . . . . . . . . . . . . . . 5.3 A refined lp decoupling for boxes of canonical scale. . . . . . 5.4 Proof of Theorem 5.1. . . . . . . . . . . . . . . . . . . . . . 6 Improved Incidences for Vinogradov Plates . . . . . . . . . . . . 6.1 The case 13 < α ≤ 12 . . . . . . . . . . . . . . . . . . . . . . . 6.2 The case 12 < α < 23 . . . . . . . . . . . . . . . . . . . . . . . 7 Refined lp Decoupling at Canonical Scale for the Moment Curve 8 Proof of Theorem 3.3 in the Range 0 < β ≤ 1 . . . . . . . . . . . 8.1 Proof of Proposition 8.5. . . . . . . . . . . . . . . . . . . . . 8.2 Proof of Proposition 8.6. . . . . . . . . . . . . . . . . . . . .
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The first author is partially supported by the NSF Grant DMS-1800305. The second author is partially supported by a Simons Investigator Award. The third author was partially supported by the Simons Foundation grant of David Jerison while she was at MIT, and supported by the S.S. Chern Foundation for Mathematics Research Fund and by the NSF while at IAS
C. DEMETER ET AL.
GAFA
8.3 Plank incidences. . . . . . . . . . . . . . . . . . Proof of Theorem 3.3 in the Range 1 < β ≤ 32 . . . . 9.1 Proof of Proposition 9.3. . . . . . . . . . . . . . 9.2 Proof of Theorem 9.4. . . . . . . . . . . . . . . 9.3 Plank incidences. . . . . . . . . . . . . . .
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