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Title: | 奇異積分的二次平均以及最加加權上界 Representing singular kernel as dyadic average and sharp A2 bound |
Authors: | Chih-Chieh Hung 洪智捷 |
Advisor: | 沈俊嚴(Chun-Yen Shen) |
Keyword: | 卡德隆-吉格曼算子,希爾伯特轉換,里斯轉換,二次平均,最佳加權上界,哈爾偏移算子, Calderon-Zygmund operator,Hilbert transform,Riesz transform,dyadic average,sharp A2 bound,Harr shift operator, |
Publication Year : | 2020 |
Degree: | 碩士 |
Abstract: | 在調和分析中一個重要的核心問題是研究奇異積分算子的最佳加權上界問題,而此問題相當於研究奇異積分算子在L2加權的有界性。在2000年,S. Petermichl使用哈爾小波平均來表示希爾伯特轉換的核,此方法後來被發現是研究此問題的重大突破,爾後里斯轉換(Riesz transform)的核,甚至一般奇異積分算子的核也被找出類似的表示方法。在此基礎之上,S.Petermichl於2007解決希爾伯特轉換的最佳加權上界問題,T. Hytonen則於2012解決一般奇異積分最佳加權上界問題。本篇論文會先介紹如何使用哈爾小波平均來表示希爾伯特轉換的核(2000, S.Petermichl),此方法雖然簡單卻隱含對希爾伯特轉換深刻的觀察。接著我們會介紹如何使用哈爾小波平均來表示里斯轉換的核(2002, S. Petermichl, S. Treil and A.Volberg),這不單單只是推廣希爾伯特轉換的結果到高維度,而是將前方法作一個統整與重新表示,找出一個推廣到高維度的方式,而這證明過程中,出現一個特殊積分不等於零的假設,雖然最後作者提出另一條路徑解決,但原本特殊積分不等於0的問題在維度大於2還是未知的,本篇論文中我們解決積分非零的問題在維度等於3的時候。最後我們介紹如何表示一般的奇異積分算子,並解決最佳加權上界的問題(2012, T. Hytonen)。 A central research problem in the area of Harmonic analysis isto prove the sharp weighted bound for singular integrals. In 2000 S.Petermichlused dyadic averages of Haar shifts to represent the kernel of Hilbert transformwhich in turn enabled her to obtain the sharp A2 bound for Hilbert transform.Shortly after, the kernels of Riesz transforms were also obtained via averages ofHaar shifts and finally the full generality was made by T. Hytonen who solvedthe longstanding A2 conjecture for singular integrals. In this dissertation, wefirst introduce how to use the averages of Haar shifts to represent the kernelof Hilbert transform (2000, S.Petermichl). Second, we will introduce how torepresent the kernels of Riesz transforms via dyadic averages of Haar shifts(2002,S. Petermichl, S. Treil and A. Volberg). This result not only extendsPetermichl’s ideas to higher dimensions, but also explicitly constructs the Haarshifts for Riesz transforms. However in order to make the result nondegeneratean integral that arises in the process of averaging Haar shifts must be nonzero.S. Petermichl, S. Treil and A. Volberg provided a proof to show the integralis nonzero in dimension two but for other dimensions the problem remainsunknown. A new part of this dissertation is to prove the integral is nonzero indimension three. Finally we also discuss the breakthrough work of T. Hytonenin 2012 that solves the A2 conjecture for singular integrals. |
URI: | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/8480 |
DOI: | 10.6342/NTU202001176 |
Fulltext Rights: | 同意授權(全球公開) |
Appears in Collections: | 數學系 |
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U0001-2906202001155900.pdf | 1.48 MB | Adobe PDF | View/Open |
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