Small cap decouplings

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 estimate...

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Bibliographic Details
Published inGeometric and functional analysis Vol. 30; no. 4; pp. 989 - 1062
Main Authors Demeter, Ciprian, Guth, Larry, Wang, Hong
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.08.2020
Springer Nature B.V
Subjects
Analysis
Decoupling
Diameters
Estimates
Mathematics
Mathematics and Statistics
Tubes
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Summary: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 R 3 . 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.
ISSN:1016-443X
1420-8970
DOI:10.1007/s00039-020-00541-5