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Project Details
Funding Scheme : General Research Fund
Project Number : 14303817
Project Title(English) : Problems in harmonic analysis related to Carelson's operator 
Project Title(Chinese) : 關於 Carleson 算子的調和分析 
Principal Investigator(English) : Prof Feng, De-Jun 
Principal Investigator(Chinese) :  
Department : Dept of Mathematics
Institution : The Chinese University of Hong Kong
E-mail Address : djfeng@math.cuhk.edu.hk 
Tel : 26097965 
Co - Investigator(s) :
Prof Yung, Po-lam
Panel : Physical Sciences
Subject Area : Mathematics
Exercise Year : 2017 / 18
Fund Approved : 472,351
Project Status : Completed
Completion Date : 31-8-2021
Project Objectives :
To further develop higher order wave packet analysis in dimension 2.
To extend Lie's result about the quadratic Carleson operator to dimension 2.
To better understand polynomial Carleson operators in dimension 2 when we blend it with a Radon transform over the parabola.
To obtain a variation norm estimate of the oscillatory integrals considered by Stein and Wainger.
Abstract as per original application
(English/Chinese):
我們研究 Carleson 算子的一些推廣, 用的主要是振蕩積分和時空/頻率的同步分解.
Realisation of objectives: The central object of investigation in this proposal is Carleson's operator. Its importance lies in the fact that it is one of the simplest modulation invariant operators in harmonic analysis, and the powerful tools developed to tackle this modulation invariance has since flourished into a sub-field called time-frequency (or wave packet) analysis. In our investigation, we are especially interested in the polynomial Carleson operator, the Radon-type Carleson operator, and the Stein-Wainger operator. We manage to prove sharp 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 made use of various local smoothing and square function estimates for the solution of the Schrodinger equation. The paper above also found applications in discrete harmonic analysis, such as an alternative proof of an earlier result of Krause and Lacey. We also investigate the maximal operators associated with families of Hilbert transforms along non-flat homogeneous curves. We provide an optimal result for the L^p operator norms of these maximal operators for p>2 and also for 1
Summary of objectives addressed:
Objectives Addressed Percentage achieved
1.To further develop higher order wave packet analysis in dimension 2.Yes100%
2.To extend Lie's result about the quadratic Carleson operator to dimension 2.Yes100%
3.To better understand polynomial Carleson operators in dimension 2 when we blend it with a Radon transform over the parabola. Yes100%
4.To obtain a variation norm estimate of the oscillatory integrals considered by Stein and Wainger.Yes100%
N.A.
Research Outcome
Major findings and research outcome: In [1] 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 [5] we investigate the maximal operators associated with families of Hilbert transforms along non-flat homogeneous curves. We provide an optimal result for the L^p operator norms of these maximal operators for p>2. In particular we show that the operator norm for the maximal operator depends on the minimal number of intervals of the form (R,2R) that is needed to cover the parametrised space. In [2] we complete this result by estimating the L^p norm for 1
Potential for further development of the research
and the proposed course of action:		 To further understand the polynomial Carleson operators in higher dimensions.
Layman's Summary of
Completion Report:		 In 1966, L. Carleson proved a celebrated theorem about pointwise convergence of Fourier series. This result can be reformulated in terms of the boundedness of the so-called Carleson's operator. In our project, we provide some sharp estimations (including the variational norms and Lp norms) on several variants of Carleson's operator (in particular those that involve features of Radon transforms, and certain polynomial phases), by developing techniques in oscillatory integrals and time-frequency analysis. We also provide a short and elementary proof of the l^2 decoupling inequality for some curves. The techniques we developed in this project would pave a way towards a better understanding of wave packet analysis in higher dimensions, and be applicable to other related problems.
Research Output
Peer-reviewed journal publication(s)
arising directly from this research project :
(* denotes the corresponding author)
Year of
Publication		 Author(s) Title and Journal/Book Accessible from Institution Repository
2020 Shaoming Guo*, Joris Roos, Po-Lam Yung  Sharp variation-norm estimates for oscillatory integrals related to Carleson's theorem  Yes 
2020 SHAOMING GUO, JORIS ROOS, ANDREAS SEEGER AND PO-LAM YUNG  Maximal functions associated with families of homogeneous curves: L^p bounds for p ≤ 2  Yes 
2021 Shaoming Guo*, Zane Kun Li, Po-Lam Yung, Pavel Zorin-Kranich  A short proof of l^2 decoupling for the moment curve  Yes 
2021 SHAOMING GUO, ZANE KUN LI, AND PO-LAM YUNG  A bilinear proof of decoupling for the cubic moment curve  Yes 
2020 Shaoming Guo*, Joris Roos, Andreas Seeger and Po-Lam Yung  A maximal function for families of Hilbert transforms along homogeneous curves  Yes 
2021 Philip T. Gressman*, Shaoming Guo, Lillian B. Pierce, Joris Roos, Po-Lam Yung  Reversing a Philosophy: From Counting to Square Functions and Decoupling  Yes 
Recognized international conference(s)
in which paper(s) related to this research
project was/were delivered :
Month/Year/City Title Conference Name
Porto Alegre Variational norm estimates for some oscillatory integrals related to Carleson's operator  International Congress of Mathematicians (ICM) Satellite Conference in Harmonic Analysis 
Stockholm Dimension estimates for C^1 iterated function systems and repellers  (Online) New frontiers in dimension theory of dynamical systems - Applications in metric number theory, 
Tianjing Estimates of the dimension of self-similar measures with overlaps  International Conference on Analysis and PDEs on Manifolds and Fractals 
Shanghai Dimension of attractors of nonlinear iterated function systems  Topological and probabilistic methods in low-dimensional dynamics 
Other impact
(e.g. award of patents or prizes,
collaboration with other research institutions,
technology transfer, etc.):
  SCREEN ID: SCRRM00542