奇異積分交換子的有界性

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邱子陽(Tzu-Yang Chiu) 查詢紙本館藏

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數學系

論文名稱

奇異積分交換子的有界性
(The boundedness of commutator for singular integrals)

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摘要(中)

本篇要討論的主題為與 Calderón-Zygmund 算子有關的交換子 L^p 有界的充分必要條件。令 K 為 Calderón-Zygmund 算子，若 b 為 BMO 函數，則交換子 [b,K] 的上界會被 b 的 BMO 範數所控制。相反的，若交換子內僅考慮 Riesz 變換 R_j, 則交換子 [b,R_j] 的下界也會被 b 的 BMO 範數所控制。本篇論文主要整理 Coifman, Rochberg 與 Weiss 發表在 Annals Math (Factorization theorems for Hardy spaces inseveral variables) 中的證明手法，並詳加描述使得讀者僅須具備實變知識即可讀懂。並在最後提供下界條件的另種證明方式。

摘要(英)

In this thesis, we study the necessary and sufficient conditions for the L^p-boundedness of the commutators related to Calderón-Zygmund operators. Let K be Calderón-Zygmund operators, if b is a BMO function, then the upper bound of commutators [b,K] is controlled by BMO norm of b. Conversely, if we only consider Riesz tranform R_j, then the lower bound of commutators [b,R_j] is also controlled by BMO norm of b. We mainly sort out the proof method published by Coifman, Rochbe and Weiss in Annals Math (Factorization theorems for Hardy spaces inseveral variables), and describes in detail so that readers can understand by the knowledge of real analysis. At the end, we provide another way to prove the condition of lower bound.

關鍵字(中)

★ 奇異積分交換子

關鍵字(英)

論文目次

摘要 i
Abstract ii
目錄 iii
1 引言與主要結果 1
2 預備知識 2
3 主要結果與證明 3
4 下界另一種證明方法 24

參考文獻

A. P. Calderón, Commutators of Singular Integral Operators, NAS(1965), 1092-1099.

R.R. Coifman and G. Weiss, Analysis Harmonique Non-Commutative sur Certains Espaces Homogènes, LNM(1971).

R. R. Coifman and Y. Meyer, On Commuators of Singulars and Bilinear Singular Integrals, AMS(1975), 315-311.

R. R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces inseveral variables, Annals of Mathematics(1976), 611-620.

C. Fefferman, Recent Progress in Classical Fourier Analysis, ICM(1974), 95-118.

S. Janson, Mean Oscillation and Commutators of Singular Integral Operators, Ark. Mat.(1978), 263-270.

F. John and L. Nirenberg, On Functions of Bounded Mean Oscillation, Commun. Pure Appl. Math(1961), 415-426.

C. J. Neugebauer, On The Hardy-Littlewood Maximal Function and Some Applications, AMS(1980), 99-105.

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, PMS(1970).

E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, PMS(1971).

A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Department of Mathematics, Indiana University Bloomington, Indiana(1986).

R. L. Wheeden and A. Zygmund, Measure and Integral, Marcel Dekker, Inc, New York-Basel(1977).

A. Uchiyama, On the compactness of operators of Hankel type, Tohoku Math. J.(1978), 163-171.

指導教授

李明憶(Ming-Yi Lee)

審核日期

2023-6-21

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