奇異積分算子的加權模不等式

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李明憶(Ming-Yi Lee) 查詢紙本館藏

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數學系

論文名稱

奇異積分算子的加權模不等式
(Weighted Norm Inequalities ofSingular Integral Operators)

相關論文

★ Marcinkiewicz積分交換子的有界性	★ 帶變量核之奇異積分算子
★ 加權赫茲形式哈弟空間上的郝曼德乘算子	★ 奇異積分的加權有界性
★ 乘積空間上離散型Littlewood-Paley理論	★ Hardy-Hilbert型式的不等式和Cauchy加法映射的穩定性
★ 關於section的哈代空間上的分子刻畫	★ Hardy spaces associated to para-accrective functions
★ 哈地空間在開集合上的極大函數刻畫

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摘要(中)

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 ="" 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.

摘要(英)

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.

關鍵字(中)

★ 加權哈弟空間
★ 分子刻劃

關鍵字(英)

★ molecular characterization
★ weighted Hardy spaces

論文目次

Chapter 1.
Introduction . . . . . . . . . .. . . . . . . . . . . . . . 1
Chapter 2.
Ap Weight . . . . . . . . . . . . . . . . . . . . . . .. . . 5
Chapter 3.
Weighted Hardy spaces. . . . . . . . . . . . .. . . . . . . 10
Chapter 4.
Hilbert transform and Riesztransform . . . . . . . . . . . . 16
Chapter 5.
Bochner-Riesz means . . . . . . . . . . . . . .. . . . . . . 22
Chapter 6.
Riesz potential operators . . . . . . . . . . .. . . . . . . 30
Chapter 7.
Marcinkiewicz integral . . . . . . . . . . . . . . . . . . . 41
References . . . . . . . . . . . . . . . . . . . . . . . . . 49

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指導教授

林欽誠(chin-cheng Lin)

審核日期

2002-6-5

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