Multilinear operators in harmonic analysis: Methods and applications
Dong, Dong
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https://hdl.handle.net/2142/101503
Description
Title
Multilinear operators in harmonic analysis: Methods and applications
Author(s)
Dong, Dong
Issue Date
2018-06-22
Director of Research (if dissertation) or Advisor (if thesis)
Li, Xiaochun
Doctoral Committee Chair(s)
Erdogan, M. Burak
Committee Member(s)
Ford, Kevin
Boca, Florin
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
multilinear Hilbert transforms
discrete operators
Roth theorem
Abstract
We first give a survey on multilinear Hilbert transforms. Then we study several variants of bilinear Hilbert transform such as bilinear Hilbert transform along two polynomials, discrete (integer) bilinear Hilbert transform along polynomials, finite field version of bilinear Radon transform, a hybrid of bilinear Hilbert transform and the paraproduct, etc. Our aim is to find operator norms of these operators: showing that they are finite or have certain decay. Applications in Roth type theorems will also be given.
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