# 臺灣博碩士論文加值系統

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 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。
 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.
 X iAbstract ii iiiContents v1 Four Definitions of the Real Hardy Space H1(Rn) 11.1 Real Hardy Spaces with Their Different Forms . . . . . . . . . . . . . . . . . . . . 11.2 kfkH1at kfkH1at1;2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 kfkH1M kfkH1at , Littlewood-Paley Theorem . . . . . . . . . . . . . . . . . . . . . 81.4 kfkH1M kfkH1R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 Dual Space of H1, BMO 632.1 Bounded Mean Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632.2 Grafakos’s Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642.3 Stein-Fefferman’s Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703 The John-Nirenberg Theorem 793.1 The John-Nirenberg Theorem, Exponential Integrability . . . . . . . . . . . . . . . 793.2 The pth Order Version of BMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854 Carleson Measures 874.1 The Whitney Decomposition of Open Sets in Rn . . . . . . . . . . . . . . . . . . . 874.2 Definition of Carleson Measures and Some of Its Related Estimates . . . . . . . . . 894.3 BMO Functions and Carleson Measures . . . . . . . . . . . . . . . . . . . . . . . . 955 Singular Integrals of Nonconvolution Type 1005.1 Singular Integrals of Nonconvolution Type . . . . . . . . . . . . . . . . . . . . . . . 1005.2 Standard Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.3 Singular Integral Operators Associated with Standard Kernels, L2 Boundedness . . 1115.4 Consequences of L2 Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256 The T(1) Theorem 1417 A Continuous Version of Littlewood-Paley Theorem 2727.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2727.2 Proof of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421v8 Applications of the Continuous Version of Littlewood-Paley Theorem 4228.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4228.2 Proof of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5328.3 Littlewood-Paley Theorem II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5348.4 A Short Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5469 Some Constructions of Certain Smooth Functions 54910 Hormander’s Schwartz Kernel Theorem 55611 Three Interpolation Theorems 56611.1 A Basic Interpolation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56611.2 The Marcinkiewicz Interpolation Theorem . . . . . . . . . . . . . . . . . . . . . . . 57011.3 The Riesz-Thorin Interpolation Theorem . . . . . . . . . . . . . . . . . . . . . . . . 578References 586
 [1] Uchiyama, Hardy Spaces on the Euclidean Space, Springer-Verlag, New York, 2001.[2] Grafakos, Classical Fourier Analysis, Springer-Verlag, New York, 2008.[3] Grafakos, Modern Fourier Analysis, Springer-Verlag, New York, 2008.[4] Grafakos, Classical Fourier Analysis, Springer-Verlag, New York, 2014.[5] Grafakos, Modern Fourier Analysis, Springer-Verlag, New York, 2014.[6] Francia, Weighted Norm Inequalities and Related Topics, North Holland, 1985.[7] Meyer, Calderon-Zygmund and Multilinear Operators, Cambridge University Press,United Kingdom, 1997.[8] Triebel, The Structure of Functions, Birkhauser Verlag, Berlin, 2000.[9] Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals,Princeton University Press, New Jersey, 1993.[10] Stein, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press,New Jersey, 1993.[11] Stein, Singular Integrals and Dierentiability Properties of Functions, Princeton UniversityPress, New Jersey, 1993.[12] Duistermaat, Distributions, Birkhauser, Berlin, 2010.[13] Treves, Topological Vector Spaces, Distributions, and Kernels, Academic Press, SanDiego, 1967.[14] Narici, Topological Vector Spaces, CRC Press, London, 2011.[15] Weidmann, Linear Operators in Hilbert Spaces, Springer-Verlag, New York, 1980.[16] Megginson, An Introduction to Banach Space Theory, Springer-Verlag, New York, 1998.[17] Blackandar, Operator Algebras, Springer-Verlag, New York, 2006.[18] Takesaki, Theory of Operator Algebra I, Springer-Verlag, New York, 1979.[19] Yeh, Theory of Measure and Integration, World Scientic, Singapore, 2006.[20] Wilansky, Topology for Analysis, Dover, United States, 1970.[21] Ciesielski, Set Theory for the Working Mathematician, Cambridge University Press,United Kingdom, 1997.[22] Roman, Advanced Linear Algebra, Springer-Verlag, New York, 2007.
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