調和分析中的扁平與非扁平算子:時頻分析與震盪積分

Flat and NonFlat Operators in Harmonic Analysis: Time-Frequency Analysis and Oscillatory Integral

林友翔(Yu-Hsiang Lin)

指導教授 ： 沈俊嚴

國立臺灣大學/理學院/數學研究所/碩士(2021年)

https://doi.org/10.6342/NTU202100959

摘要

本文討論調和分析中用來估計扁平與非扁平算子勒貝格範數有界性的兩種工具，時頻分析與震盪積分。在本文中，我們以自洽的方式探討五個與此領域相關的工作。

關鍵字

時頻分析 ； 卡爾松算子 ； 雙線性希爾伯特變換 ； 勒貝格外測度理論 ； 震盪積分

並列摘要

We study two kinds of tools, time-frequency analysis and oscillatory integral, dealing with the Lp boundedness of some prototype of flat and nonflat operaotors in Harmonic analysis, respectively. We verified five important works in detail and present them in a self-contained way.

並列關鍵字

Time-Frequency Analysis ； Carleson Operator ； Bilinear Hilbert Transform ； Lebesgue Outer Measure Theory ； Oscillatory Integral

參考文獻

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[Hun68]Richard A Hunt.On the convergence of Fourier series, Orthogo-nal Expansions and their Continuous Analogues (Proc. Conf., Ed-wardsville, Ill., 1967). 1968.

[Sj ̈o71]Per Sj ̈olin. “Convergence almost everywhere of certain singular in-tegrals and multiple Fourier series”. In:Arkiv f ̈or Matematik9.1(1971), pp. 65–90.

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