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Sharp variation-norm
estimates for oscillatory integrals related to Carleson's
theorem
Shaoming Guo, Joris Roos and Po-Lam Yung |
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Vol. 13 (2020), No. 5, 1457–1500
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Abstract |
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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 decoupling theorem. This may be of independent interest, because it improves the previously known range of exponents for spatial dimensions . |
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Keywords
variation-norm, polynomial Carleson
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Mathematical Subject Classification 2010
Primary: 42B20
Secondary: 42B25
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Milestones
Received: 24 April 2018
Revised: 20 April 2019
Accepted: 31 May 2019
Published: 27 July 2020
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Authors |
| Shaoming Guo | |
| Department of Mathematics University of Wisconsin Madison, WI United States |
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| Joris Roos | |
| Department of Mathematics University of Wisconsin Madison, WI United States |
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| Po-Lam Yung | |
| Department of Mathematics The Chinese University of Hong Kong Shatin Hong Kong |
Mathematical Sciences
Institute Australian National University Canberra Australia |
