| Summary: | 碩士 === 國立中央大學 === 數學系 === 103 === 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.
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