A Primer on BMO

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姓名

王權豪(Ooi Keng Hao) 查詢紙本館藏

畢業系所

數學系

論文名稱

(A Primer on BMO)

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摘要(中)

我們將會探究BMO函數的性質以及實Hardy空間H^1與它的聯係。所需定義與定理會給予充分的介紹與解釋以及證明。然而我們的目的是證明H^1的對偶可以相對應到BMO函數。我們也證明了稍微不一樣的Littlewood-Paley定理。由於我們證明的版本有著濃厚的連續味道，我們將之取名為Continuous Version of Littlewood-Paley定理。證明這定理我們用了向量值函數的積分理論，爲了使得我們的討論更加嚴謹，向量值函數相應有關的可測性討論也被包括其中。然而，上述兩大定理得證後，我們將之應用於證明調和分析界最有影響力的T(1)定理。讀者輕易發現，於證明T(1)定理中，文獻中所介紹的引理及定理與結果皆派上用場。除此，爲了方便讀者，我們將幾乎所有有關調和分析基礎的定義與定理都給説明以及證明一遍。無論如何，因爲頁數的考量，我們排除了更基本的數學知識，也就是實變。讀者在這裡被假設為有一定程度的實變技術，最至少，研究所一年級的訓練與要求。

摘要(英)

We are going to study the behaviour of functions of bounded mean oscillation and the properties of real Hardy space $H^{1}$. The definitions and proofs of theorems would be given in full details, where the goal is to prove that the dual space of $H^{1}$ could be corresponding to functions of bounded mean oscillation. We also proved a slightly different version of Littlewood-Paley theorem that one could easily find in traditional text of harmonic analysis. Since such a version has a heavy continuous sense comparing to the old one, we named it as Continuous Version of Littlewood-Paley Theorem. We believe that none of any harmonic analyst give rise to such a name, the reader is suggested not to stick heavily to the taste of the term that we have created, of course, any suggestion of better name would be appreciated though. For the technique in dealing the continuous version of Littlewood-Paley theorem, we have tacitly used the idea of vector integrations that such approach is also suggested by Rubio de Francia and Loukas Grafakos. Nevertheless, we have conjoined the measurability aspect in vector integrations in order to make our discussions more accurately and precisely. A huge application of the continuous version of Littlewood-Paley theorem is by no doubt to prove the most significant result in harmonic analysis that called $T(1)$ theorem. It is the proof of $T(1)$ theorem which utilize almost all the theorems and results in the following discourse. We note that the idea of the proof of $T(1)$ indicated in the discourse is due to Meyer and Coifman. Apart from these main aims, the fundamental knowledge of harmonic analysis is needed for both readers and author. One may consult the related classical texts of harmonic analysis in order to understand better the corresponding notions or any notations. However, for the sake of simplicity in reading the discourse, we have included almost all the fundamental notations and theorems in harmonic analysis. Due to the massive volume that we have written, we do not provide the fundamental knowledge about measure theory, or equivalently, real analysis in the following discourse. The readers are assumed to have a sufficient knowledge about measure theory for at least first year training in graduate course.

關鍵字(中)

★ 奇異積分

關鍵字(英)

★ BMO
★ Hardy Space
★ Littlewood
★ Paley
★ T1
★ Singular Integral
★ Schwartz
★ Distribution
★ Fourier Multiplier

論文目次

X i
Abstract ii
iii
Contents v
1 Four Definitions of the Real Hardy Space H1(Rn) 1
1.1 Real Hardy Spaces with Their Different Forms . . . . . . . . . . . . . . . . . . . . 1
1.2 kfkH1
at
kfkH1
at1;2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 kfkH1
M
kfkH1
at , Littlewood-Paley Theorem . . . . . . . . . . . . . . . . . . . . . 8
1.4 kfkH1
M
kfkH1
R
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2 Dual Space of H1, BMO 63
2.1 Bounded Mean Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.2 Grafakos’s Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.3 Stein-Fefferman’s Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3 The John-Nirenberg Theorem 79
3.1 The John-Nirenberg Theorem, Exponential Integrability . . . . . . . . . . . . . . . 79
3.2 The pth Order Version of BMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4 Carleson Measures 87
4.1 The Whitney Decomposition of Open Sets in Rn . . . . . . . . . . . . . . . . . . . 87
4.2 Definition of Carleson Measures and Some of Its Related Estimates . . . . . . . . . 89
4.3 BMO Functions and Carleson Measures . . . . . . . . . . . . . . . . . . . . . . . . 95
5 Singular Integrals of Nonconvolution Type 100
5.1 Singular Integrals of Nonconvolution Type . . . . . . . . . . . . . . . . . . . . . . . 100
5.2 Standard Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.3 Singular Integral Operators Associated with Standard Kernels, L2 Boundedness . . 111
5.4 Consequences of L2 Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6 The T(1) Theorem 141
7 A Continuous Version of Littlewood-Paley Theorem 272
7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
7.2 Proof of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
v
8 Applications of the Continuous Version of Littlewood-Paley Theorem 422
8.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
8.2 Proof of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
8.3 Littlewood-Paley Theorem II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534
8.4 A Short Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
9 Some Constructions of Certain Smooth Functions 549
10 Hormander’s Schwartz Kernel Theorem 556
11 Three Interpolation Theorems 566
11.1 A Basic Interpolation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566
11.2 The Marcinkiewicz Interpolation Theorem . . . . . . . . . . . . . . . . . . . . . . . 570
11.3 The Riesz-Thorin Interpolation Theorem . . . . . . . . . . . . . . . . . . . . . . . . 578
References 586

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指導教授

方向(Fang Xiang)

審核日期

2015-8-24

推文