TY - JOUR
T1 - Small cap decouplings
AU - Demeter, Ciprian
AU - Guth, Larry
AU - Wang, Hong
N1 - Funding Information:
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.
Publisher Copyright:
© 2020, Springer Nature Switzerland AG.
PY - 2020/8/1
Y1 - 2020/8/1
N2 - 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.
AB - 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.
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U2 - 10.1007/s00039-020-00541-5
DO - 10.1007/s00039-020-00541-5
M3 - Article
AN - SCOPUS:85089063955
SN - 1016-443X
VL - 30
SP - 989
EP - 1062
JO - Geometric and Functional Analysis
JF - Geometric and Functional Analysis
IS - 4
ER -