Vol. 13, No. 5, 2020

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Sharp variation-norm estimates for oscillatory integrals related to Carleson's theorem

Shaoming Guo, Joris Roos and Po-Lam Yung

Vol. 13 (2020), No. 5, 1457–1500
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 n 4.

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Keywords
variation-norm, polynomial Carleson
Mathematical Subject Classification 2010
Primary: 42B20
Secondary: 42B25
Milestones
Received: 24 April 2018
Revised: 20 April 2019
Accepted: 31 May 2019
Published: 27 July 2020
Authors
Shaoming Guo
Department of Mathematics
University of Wisconsin
Madison, WI
United States
Joris Roos
Department of Mathematics
University of Wisconsin
Madison, WI
United States
Po-Lam Yung
Department of Mathematics
The Chinese University of Hong Kong
Shatin
Hong Kong
Mathematical Sciences Institute
Australian National University
Canberra
Australia