| 摘要: | 本篇文章主要證明奇異積分算子在加權哈弟空間上的有界性. 首先我們簡單介紹傳統哈弟空間的歷史及一些性質, 進而探討這些性質是否在加權哈弟空間上亦成立. 在此篇文章中加權函數皆為 A_p 權函數. 在第二章中即說明 A_p 權函數的起源,並給出 A_p 權的定義和一些基本性質. 值得一提的是, 這些性質在證明算子的加權有界性時給了我們莫大的幫助. 第三章定義了加權哈弟空間上的原子和分子, 並證明加權原子和加權分子保持某種平移不變性. 此外, 本章節亦給出加權哈弟空間上的原子分解和 分子刻劃, 並說明如何利用這兩個定理可以較容易地證明算子的加權有界性.第四, 五, 六和七章則利用此一特性分別針對特定的奇異積分算子來研究其加權有界性. 在第四章中, 假設捲積算子的核滿足較好的條件, 則我們得到捲積算子是H^p_w-H^p_w有界,0<p≦1. 因此, 我們證明了Hilbert 變換和 Riesz 變換為H^p_w-H^p_w 有界, 0<p≦1. 而此定理推廣[LL]中的結果.第五章所研究的算子為 Bochner-Riesz means. 我們證明了它是 H^p_w-H^p_w有界及它的極大算子是H^p_w-L^p_w有界. 而第六章的算子為齊次分數積分. 我們給出在某些 條件下, 它是H^p_{w^p}-L^q_{w^q}有界; 而在某些條件下, 它是H^p_{w^p}-H^q_{w^q}有界. 在第七章中,我們證明 Marcinkiewicz 積分算子是H^p_w-L^p_w 有界. In this thesis, we prove the boundedness of singular integral operator on weighted Hardy spaces. We first introduce the some properties of the classical hardy spaces and try to get the same properties on weighted Hardy spaces. In this article, the weight function is Ap weight. We introduce the definitions of Ap weight and give some properties of Ap in chapter 2. These properties play important roles to prove the boundedness of the operators on weighted Hardy spaces. In chapter 3, we define weighted atom and weighted molecule and prove some translation invariance. Moreover, we also give the atomic decomposition theory and the molecular characterization of weighted Hardy spaces. By the two theorems, the proof of H^p_w boundedness of the operator on weighted Hardy space becomes easier. In chapter 4, we show the convolution operator is bounded on weighted Hardy spaces. Moreover, we also show the Hilbert transform and the Riesz transforms are bounded on H^p_w, 0 < p≦1, provided w ε A1. In chapter 5, we show that the Bochner-Riesz means is bounded on H^p_w and its maximal operator is bounded from H^p_w to L^p_w. In chapter 6 we present the weighted (Hp,Lq) boundedness of homogeneous fractional integral operator and the weighted (Hp,Hq) boundedness of homogeneous fractional integral operator. In chapter 7, We show that the Marcinkiewicz integral operator is bounded from H^p_w to L^p_w. |
