奇異積分的加權有界性

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蔡孟哲(Meng-che Tsai) 查詢紙本館藏

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論文名稱

奇異積分的加權有界性
(The weighted boundedness of singular integral operators)

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摘要(中)

在此篇文章中，我們給出一些方法去證明算子從到的有界性。當假設條件與Muckenhoupt權類有關時，我們可以了解到雙權模不等式的證明只依賴於單權模不等式。我們給出一些例子去說明如何證明它，那就是我們證明極大算子、奇異積分算子、極大奇異積分算子、Marcinkiewicz積分算子、Marcinkiewicz積分算子關於面積積分以及Marcinkiewicz積分算子關於Littlewood-Paley -函數都是從到有界。最後我們用另一個假設條件去證明Marcinkiewicz積分算子是從到有界。

摘要(英)

In this paper, we give some methods such that the operators are bounded from to .
Under the condition related to the Muckenhoupt weights class, we realize that the proof of two weighted norm inequality only depends on one-weighted norm inequality. We give some examples to describe how did we prove it; that is, we proved that the maximal operator , the singular integral operator , the maximal singular integral operator , the Marcinkiewicz integral operator ,the Marcinkiewicz integral operator related to the area integral , and the Marcinkiewicz integral operator related to the Littlewood-Paley -function operator are all bounded from to .
Finally, we prove that the Marcinkiewicz integral operator is bounded from to for another condition of .

關鍵字(中)

★ 奇異積分
★ 有界性
★ 權

關鍵字(英)

★ weight
★ boundedness
★ singular integral operators

論文目次

中文摘要...........................................i
英文摘要...........................................ii
Contents...........................................iii
Introduction.......................................p.2
Definitions and main results.......................p.3
Properties of weights..............................p.6
Proofs of Theorems.................................p.10
References.........................................p.22

參考文獻

1 E. Adams, On weighted norm inequalities for the Riesz transforms of functions with vanishing moments,
Studia Math. 78, (1984), 107-153.
2 J. Duoandikoetxea, Fourier Analysis, Grad. Stud. Math., vol. 29, Amer. Math. Soc., Providence, 2000.
3 J. Duoandikoetxea, Weighted norm inequalities for homogeneous singular integrals,
Trans. Amer. Math. Soc. 336, (1993), 869-880.
4 Y. Ding, D. Fan, and Y. Pan, Weighted boundedness for a class of rough Marcinkiewicz integral,
Indiana Univ. Math. J. 48, (1999), 1037-1055.
5 Y. Ding and C.-C. Lin, boundedness of some rough operators with different weights,
J. Math. Soc. Japan, 55, (2003), 209-230.
6 J. Garcia-Cuerva and J. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North Holland, 1985.
7 C. Neugebauer, Inserting -weights, Proc. Amer. Math. Soc. 87, (1983), 644-648.
8 Y. Rakotondratsimba, Two weight norm inequality for Calderon-Zygmund operators, Acta Math. Hungar. 80,
(1998), 39-54.
9 A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Academic Press, 1986.
10 A. Torchinsky and S. Wang, A note on the Marcinkiewicz integral, Coll. Math. 61-62, (1990), 235-243.
11 D. Watson, Weighted estimates for singular integrals via Fourier transform eatimates, Duke. Math. J. 60, (1990),
389-399.

指導教授

林欽誠(Chin-cheng Lin)

審核日期

2008-6-20

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