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In harmonic analysis, a branch of mathematics, the space of functions of bounded mean oscillation (BMO), introduced by John & Nirenberg (1961), plays the same role in the theory of Hardy spaces that the space L of bounded functions plays in the theory of Lp-spaces. A general reference for functions of bounded mean oscillation is (Stein 1993, chapter IV).The space VMO of functions of vanishing mean oscillation is the closure in BMO of the continuous functions that vanish at infinity. It can also be defined as the space of functions whose "mean oscillations" on balls Q are not only bounded, but also tend to zero uniformly as the radius of the ball Q tends to 0 or infinity. The space VMO is a sort of Hardy space analogue of the space of continuous functions vanishing at infinity, and in particular the Hardy space H1 is the dual of VMO. (Stein 1993, p. 180)

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'Still waters run deep.' This proverb expresses exactly how a mathematician Akihito Uchiyama and his works were. He was not celebrated except in the field of harmonic analysis, and indeed he never wanted that. He suddenly passed away in summer of 1997 at the age of 48. However, nowadays his contributions to the fields of harmonic analysis and real analysis are permeating through various fields of analysis deep and wide. One could write several papers explaining his contributions and how they have been absorbed into these fields, developed, and used in further breakthroughs. Peter W. Jones (Professor of Yale University) says in his special contribution to this book that Uchiyama's decomposition of BMO functions is considered to be the Mount Everest of Hardy space theory. This book is based on the draft, which the author Akihito Uchiyama had completed by 1990. It deals with the theory of real Hardy spaces on the n-dimensional Euclidean space. Here the author explains scrupulously some of important results on Hardy spaces by real-variable methods, in particular, the atomic decomposition of elements in Hardy spaces and his constructive proof of the Fefferman-Stein decomposition of BMO functions into the sum of a bounded?function and Riesz transforms of bounded functions.

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This book consists of four chapters. In first chapter there is a short review of theory of Backward Stochastic Differential Equations (BSDEs) and Bounded Mean Oscillation (BMO) martingales. In second chapter an interesting connections between theory of BSDEs and BMO martingales is studied. Using the BSDE tool a new proofs of some classical results on BMO martingales are provided. In Third chapter we have studied Backward Stochastic Differential Equations with a convex generator of quadratic growth. Existence and uniqueness of a solution is proved for such equations driven by continuous martingale with unbounded characteristic. Results on the existence and uniqueness for BSDEs with quadratic growth we have used in fourth chapter, to solve the linear-quadratic regulator (LQR) problem in general martingale setting. We derived the corresponding BSDE for LQR problem and expressed the optimal strategy of LQR problem in terms of the unique solution of corresponding BSDE.

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This exposition of research on the martingale and analytic inequalities associated with Hardy spaces and functions of bounded mean oscillation (BMO) introduces the subject by concentrating on the connection between the probabilistic and analytic approaches. Short surveys of classical results on the maximal, square and Littlewood-Paley functions and the theory of Brownian motion introduce a detailed discussion of the Burkholder-Gundy-Silverstein characterization of HP in terms of maximal functions. The book examines the basis of the abstract martingale definitions of HP and BMO, makes generally available for the first time work of Gundy et al. on characterizations of BMO, and includes a probabilistic proof of the Fefferman-Stein Theorem on the duality of H11 and BMO.

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Various applications of equimeasurable function rearrangements to the 'best constant'-type problems are considered in this volume. Several classical theorems are presented along with some very recent results. In particular, the text includes a product-space extension of the Rising Sun lemma, a product-space version of the John-Nirenberg inequality for bounded mean oscillation (BMO) functions with sharp exponent, a refinement of the Gurov-Reshetnyak lemma, sharp embedding theorems for Muckenhoupt, Gurov-Reshetnyak, reverse Hölder, and Gehring classes, etc. This volume is interesting for graduate students and mathematicians involved with these topics. TOC:Preface.- 1.Preliminaries and auxilliary results.- 2. Properties of oscillations and the definition of the BMO-class.- 3.Estimates of rearrangements and the John-Nirenberg theorem.- 4.The BMO-estimates of the Hardy-type transforms.- 5.The Gurov-Reshetnyak class of functions.- Appendix: A.The boundedness of the Hardy-Littlewood maximal operator from BMO into BLO.- B.The weighted analogs of the Riesz lemma and the Gurov-Reshetnyak theorem.- C.Classes of functions satisfying the reverse Hölder inequality.- References.- Index.

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Various applications of equimeasurable function rearrangements to the ''best constant'-type problems are considered in this volume. Several classical theorems are presented along with some very recent results. In particular, the text includes a product-space extension of the Rising Sun lemma, a product-space version of the John-Nirenberg inequality for bounded mean oscillation (BMO) functions with sharp exponent, a refinement of the Gurov-Reshetnyak lemma, sharp embedding theorems for Muckenhoupt, Gurov-Reshetnyak, reverse Hölder, and Gehring classes, etc. This volume is interesting for graduate students and mathematicians involved with these topics.

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