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完整後設資料紀錄
DC 欄位 | 值 | 語言 |
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dc.contributor.advisor | 沈俊嚴(Chun-Yen Shen) | |
dc.contributor.author | Chih-Chieh Hung | en |
dc.contributor.author | 洪智捷 | zh_TW |
dc.date.accessioned | 2021-05-20T00:55:33Z | - |
dc.date.available | 2020-07-02 | |
dc.date.available | 2021-05-20T00:55:33Z | - |
dc.date.copyright | 2020-07-02 | |
dc.date.issued | 2020 | |
dc.date.submitted | 2020-07-01 | |
dc.identifier.citation | [1] Carlos Pe´rez, Sergei Treil, and Alexander Volberg. On A2 conjecture and corona decomposition of weights. Preprint, arXiv:1006.2630, 2010. [2] F. Nazarov, S. Treil, and A. Volberg. The T b-theorem on non-homogeneous spaces. Acta Math., 190(2):151–239, 2003. [3] T. Hytönen. The vector-valued nonhomogeneous Tb theorem. Int. Math. Res. Not. IMRN, (2):451–511, 2014. [4] F. Nazarov, S. Treil, and A. Volberg. Two weight inequalities for individual Haar multipliers and other well localized operators. Math. Res. Lett., 15(3): 583–597, 2008. [5] T. Hytönen, C. Pe´rez, S. Treil, and A. Volberg. Sharp weighted estimates for dyadic shifts and the A2 conjecture. J. Reine Angew. Math., 687:43– 86, 2014. [6] T. Hytönen. The sharp weighted bound for general Caldere´n-Zygmund operators. Ann. of Math. (2), 175(3):1473–1506, 2012. [7] T. Hytönen, M. T. Lacey, H. Martikainen, T. Orponen, M. C. Reguera, E. T. Sawyer, and I. Uriarte-Tuero. Weak and strong type estimates for maximal truncations of Caldere´nZygmund operators on Ap weighted spaces. J. Anal. Math., 118(1):177–220, 2012. 105 [8] B. Muckenhoupt and R. L. Wheeden. Some weighted weak-type inequali-ties for the HardyLittlewood maximal function and the Hilbert transform. Indiana Univ. Math. J., 26(5):801–816, 1977 [9] T. Hytönen and C. Pe´rez. Sharp weighted bounds involving A∞. Anal. PDE, 6(4):777–818, 2013. [10] M. T. Lacey, S. Petermichl, and M. C. Reguera. Sharp A2 inequality for Haar shift operators. Math. Ann., 348(1):127–141, 2010. [11] N. Fujii. Weighted bounded mean oscillation and singular integrals. Math. Japon., 22(5):529–534, 1977/78. [12] Stefanie Petermichl, Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol, Comptes Rendus de l’Académie des Sciences -Series I - Mathematics Volume 330, Issue 6, 15 March 2000, Pages 455-460 [13] Stefanie Petermichl, Sergei Treil, Alexander Volberg, Why the riesz transforms are averages of the dyadic shifts ?, Publicacions Matematiques, June 2002 [14] Jyh-Shing Roger Jang, Matlab 2015 ISBN 986347519X, 9789863475194 | |
dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/8480 | - |
dc.description.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)。 | zh_TW |
dc.description.abstract | 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. | en |
dc.description.provenance | Made available in DSpace on 2021-05-20T00:55:33Z (GMT). No. of bitstreams: 1 U0001-2906202001155900.pdf: 1517092 bytes, checksum: bef130cb0bd6ed27ba15714ac30b0557 (MD5) Previous issue date: 2020 | en |
dc.description.tableofcontents | Contents 1 Introduction 5 2 Hilbert transform 6 3 Riesz transform 10 4 An integral arising from dyadic average of Riesz transforms 26 5 General Calderon-Zygmund operators and sharp A2 bound 58 6 References 105 | |
dc.language.iso | en | |
dc.title | 奇異積分的二次平均以及最加加權上界 | zh_TW |
dc.title | Representing singular kernel as dyadic average and sharp A2 bound | en |
dc.type | Thesis | |
dc.date.schoolyear | 108-2 | |
dc.description.degree | 碩士 | |
dc.contributor.oralexamcommittee | 林欽誠(Chin-Cheng Lin),黃皓瑋(Hao-Wei Huang) | |
dc.subject.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, | zh_TW |
dc.subject.keyword | Calderon-Zygmund operator,Hilbert transform,Riesz transform,dyadic average,sharp A2 bound,Harr shift operator, | en |
dc.relation.page | 106 | |
dc.identifier.doi | 10.6342/NTU202001176 | |
dc.rights.note | 同意授權(全球公開) | |
dc.date.accepted | 2020-07-01 | |
dc.contributor.author-college | 理學院 | zh_TW |
dc.contributor.author-dept | 數學研究所 | zh_TW |
顯示於系所單位: | 數學系 |
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