多項式調變不變奇異積分算子

Polynomial Modulation Invariant Singular Integral Operator

徐永昌(Yung-Chang Hsu)

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/NTU202100160

摘要

本文針對多項式卡爾松算子高維推廣在勒貝格空間下的有界性作深入探討。相比於Victor Lie與Pavel Zorin-Kranich之前的工作，該文章的主要貢獻包含：以具體的構造法來確認細節論證、用稀疏算子的語言來重新詮釋部分證明、及提供一個具教學啟發性的完整說明。

關鍵字

時頻分析 ； 多重解析度分析 ； CZ算子 ； 稀疏壓制 ； TT*-T*T方法

並列摘要

We deeply study the Lp boundedness of the generalization of Polynomial Carleson Operator. Our main contributions, comparing to previous works done by Victor Lie and by Pavel Zorin-Kranich, are to verify de- tails with explicit constructions, modify some part with language of Sparse Dominance, and provide a heuristic interpretation about the whole treatment in general.

並列關鍵字

Time-Frequency Analysis ； Multi-Resolution Analysis ； CZO ； Sparse Dominance ； TT* method

參考文獻

Lennart Carleson. “On convergence and growth of partial sums of Fourier series”. In: Acta Math. 116 (1966), pp. 135–157. doi: 10. 1007/BF02392815. url: https://doi.org/10.1007/BF02392815.

Charles Fefferman. “Pointwise Convergence of Fourier Series”. In: Annals of Mathematics 98.3 (1973), pp. 551–571. issn: 0003486X. url: http://www.jstor.org/stable/1970917.

Michael Lacey and Christoph Thiele. “Lp Estimates on the Bilinear Hilbert Transform for 2 < p < ∞”. In: Annals of Mathematics 146.3 (1997), pp. 693–724. issn: 0003486X. url: http://www.jstor.org/ stable/2952458.

Michael Lacey and Christoph Thiele. “A proof of boundedness of the Carleson operator”. In: Mathematical Research Letters 7 (July 2000), pp. 361–370. doi: 10.4310/MRL.2000.v7.n4.a1.

J. Duoandikoetxea et al. Fourier Analysis. American Mathematical Society, 2001. isbn: 9780821821725. url: https://books.google. com.tw/books?id=6fgRCgAAQBAJ.