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2020 Sharp variation-norm estimates for oscillatory integrals related to Carleson's theorem
Shaoming Guo, Joris Roos, Po-Lam Yung
Anal. PDE 13(5): 1457-1500 (2020). DOI: 10.2140/apde.2020.13.1457

Abstract

We prove variation-norm estimates for certain oscillatory integrals related to Carleson’s theorem. Bounds for the corresponding maximal operators were first proven by Stein and Wainger. Our estimates are sharp in the range of exponents, up to endpoints. Such variation-norm estimates have applications to discrete analogues and ergodic theory. The proof relies on square function estimates for Schrödinger-like equations due to Lee, Rogers and Seeger. In dimension 1, our proof additionally relies on a local smoothing estimate. Though the known endpoint local smoothing estimate by Rogers and Seeger is more than sufficient for our purpose, we also give a proof of certain local smoothing estimates using Bourgain–Guth iteration and the Bourgain–Demeter 2 decoupling theorem. This may be of independent interest, because it improves the previously known range of exponents for spatial dimensions n4.

Citation

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Shaoming Guo. Joris Roos. Po-Lam Yung. "Sharp variation-norm estimates for oscillatory integrals related to Carleson's theorem." Anal. PDE 13 (5) 1457 - 1500, 2020. https://doi.org/10.2140/apde.2020.13.1457

Information

Received: 24 April 2018; Revised: 20 April 2019; Accepted: 31 May 2019; Published: 2020
First available in Project Euclid: 17 September 2020

zbMATH: 07271835
MathSciNet: MR4149067
Digital Object Identifier: 10.2140/apde.2020.13.1457

Subjects:
Primary: 42B20
Secondary: 42B25

Keywords: polynomial Carleson , variation-norm

Rights: Copyright © 2020 Mathematical Sciences Publishers

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Vol.13 • No. 5 • 2020
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