奇異積分的加權有界性

The weighted boundedness of singular integral operators

蔡孟哲(Meng-che Tsai)

指導教授 ： 林欽誠

國立中央大學/理學院/數學系/碩士(2008年)

摘要

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。

關鍵字

奇異積分 ； 有界性 ； 權

並列摘要

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 .

並列關鍵字

weight ； boundedness ； singular integral operators

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.

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,