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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.

Brownian motion processes. --- Hardy spaces. --- Bounded mean oscillation. --- BMO (Mathematics) --- Function spaces --- Spaces, Hardy --- Functional analysis --- Functions of complex variables --- Wiener processes --- Brownian movements --- Fluctuations (Physics) --- Markov processes

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Functional analysis --- 517.986.6 --- Hardy spaces --- Spaces, Hardy --- Functions of complex variables --- Harmonic analysis of functions of groups and homogeneous spaces --- Hardy spaces. --- 517.986.6 Harmonic analysis of functions of groups and homogeneous spaces

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If H is a Hilbert space and T : H ? H is a continuous linear operator, a natural question to ask is: What are the closed subspaces M of H for which T M ? M? Of course the famous invariant subspace problem asks whether or not T has any non-trivial invariant subspaces. This monograph is part of a long line of study of the invariant subspaces of the operator T = M (multiplication by the independent variable z, i. e. , M f = zf )on a z z Hilbert space of analytic functions on a bounded domain G in C. The characterization of these M -invariant subspaces is particularly interesting since it entails both the properties z of the functions inside the domain G, their zero sets for example, as well as the behavior of the functions near the boundary of G. The operator M is not only interesting in its z own right but often serves as a model operator for certain classes of linear operators. By this we mean that given an operator T on H with certain properties (certain subnormal operators or two-isometric operators with the right spectral properties, etc. ), there is a Hilbert space of analytic functions on a domain G for which T is unitarity equivalent to M .

Hardy spaces. --- Hardy spaces --- Mathematics --- Calculus --- Physical Sciences & Mathematics --- Mathematics. --- Math --- Spaces, Hardy --- Functions of complex variables. --- Functions of a Complex Variable. --- Science --- Complex variables --- Elliptic functions --- Functions of real variables --- Functional analysis --- Functions of complex variables

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The subject of this book is operator theory on the Hardy space H2, also called the Hardy-Hilbert space. This is a popular area, partially because the Hardy-Hilbert space is the most natural setting for operator theory. A reader who masters the material covered in this book will have acquired a firm foundation for the study of all spaces of analytic functions and of operators on them. The goal is to provide an elementary and engaging introduction to this subject that will be readable by everyone who has understood introductory courses in complex analysis and in functional analysis. The exposition, blending techniques from "soft" and "hard" analysis, is intended to be as clear and instructive as possible. Many of the proofs are very elegant. This book evolved from a graduate course that was taught at the University of Toronto. It should prove suitable as a textbook for beginning graduate students, or even for well-prepared advanced undergraduates, as well as for independent study. There are numerous exercises at the end of each chapter, along with a brief guide for further study which includes references to applications to topics in engineering.

Operator theory. --- Hardy spaces. --- Hilbert space. --- Functional analysis --- Banach spaces --- Hyperspace --- Inner product spaces --- Spaces, Hardy --- Functions of complex variables --- Functional analysis. --- Functions of complex variables. --- Operator Theory. --- Functional Analysis. --- Functions of a Complex Variable. --- Complex variables --- Elliptic functions --- Functions of real variables --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations

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Analytical spaces --- Harmonic analysis. Fourier analysis --- 517.518 --- Metric theory of functions --- 517.518 Metric theory of functions --- Hardy spaces. --- Wavelets (Mathematics) --- Hardy, Espaces de --- Ondelettes --- Hardy spaces --- Wavelet analysis --- Harmonic analysis --- Spaces, Hardy --- Functional analysis --- Functions of complex variables --- Hardy, Espaces de. --- Ondelettes.