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BL
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| Record Number
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860637
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| Main Entry
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Urbina-Romero, Wilfredo
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| Title & Author
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Gaussian harmonic analysis /\ Wilfredo Urbina-Romero.
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| Publication Statement
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Cham, Switzerland :: Springer,, [2019]
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, ©2019
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| Series Statement
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Springer monographs in mathematics
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| Page. NO
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1 online resource
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| ISBN
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3030055973
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: 9783030055974
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3030055965
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9783030055967
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| Bibliographies/Indexes
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Includes bibliographical references and index.
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| Contents
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Intro; Foreword; Preface; Contents; 1 Preliminary Results: The Gaussian Measure and HermitePolynomials; 1.1 The Gaussian Measure; 1.2 Estimates for the Gaussian Measure of Balls in Rd and the Doubling Condition; 1.3 Hermite Polynomials; Hermite Polynomials in One Variable; Hermite Polynomials in d Variables; 1.4 Notes and Further Results; 2 The Ornstein-Uhlenbeck Operator and the Ornstein-Uhlenbeck Semigroup; 2.1 The Ornstein-Uhlenbeck Operator; 2.2 Definition and Basic Properties of the Ornstein-Uhlenbeck Semigroup
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2.3 The Hypercontractivity Property for the Ornstein-Uhlenbeck Semigroup and the Logarithmic Sobolev Inequality2.4 Applications of the Hypercontractivity Property; 2.5 Notes and Further Results; 3 The Poisson-Hermite Semigroup; 3.1 Definition and Basic Properties; 3.2 Characterization of ∂2∂t2 + L-Harmonic Functions; 3.3 Generalized Poisson-Hermite Semigroups; 3.4 Conjugate Poisson-Hermite Semigroup; 3.5 Notes and Further Results; 4 Covering Lemmas, Gaussian Maximal Functions, and Calderón-Zygmund Operators; 4.1 Covering Lemmas with Respect to the Gaussian Measure
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4.2 Hardy-Littlewood Maximal Function with Respect to the Gaussian Measure and Its Variants4.3 The Maximal Functions of the Ornstein-Uhlenbeck and Poisson-Hermite Semigroups; The Continuity Properties of the Ornstein-Uhlenbeck Maximal Function; The Continuity Properties of the Poisson-Hermite Maximal Function; 4.4 The Local and Global Regions; 4.5 Calderón-Zygmund Operators and the Gaussian Measure; 4.6 The Non-tangential Maximal Functions for the Ornstein-Uhlenbeck and Poisson-Hermite Semigroups; The Non-tangential Ornstein-Uhlenbeck Maximal Function
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The Non-tangential Poisson-Hermite Maximal Function4.7 Radial and Non-tangential Convergence of the Ornstein-Uhlenbeck and Poisson-Hermite Semigroups; 4.8 Notes and Further Results; 5 Littlewood-Paley-Stein Theory with Respect to theGaussian Measure; 5.1 The Gaussian Littlewood-Paley g Function and Its Variants; 5.2 The Higher Order Gaussian Littlewood-Paley g Functions; 5.3 The Gaussian Lusin Area Function; 5.4 Notes and Further Results; 6 Spectral Multiplier Operators with Respect to theGaussian Measure; 6.1 Gaussian Spectral Multiplier Operators; 6.2 Meyer's Multipliers
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| Abstract
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Authored by a ranking authority in Gaussian harmonic analysis, this book embodies a state-of-the-art entrée at the intersection of two important fields of research: harmonic analysis and probability. The book is intended for a very diverse audience, from graduate students all the way to researchers working in a broad spectrum of areas in analysis. Written with the graduate student in mind, it is assumed that the reader has familiarity with the basics of real analysis as well as with classical harmonic analysis, including Calderón-Zygmund theory; also some knowledge of basic orthogonal polynomials theory would be convenient. The monograph develops the main topics of classical harmonic analysis (semigroups, covering lemmas, maximal functions, Littlewood-Paley functions, spectral multipliers, fractional integrals and fractional derivatives, singular integrals) with respect to the Gaussian measure. The text provide an updated exposition, as self-contained as possible, of all the topics in Gaussian harmonic analysis that up to now are mostly scattered in research papers and sections of books; also an exhaustive bibliography for further reading. Each chapter ends with a section of notes and further results where connections between Gaussian harmonic analysis and other connected fields, points of view and alternative techniques are given. Mathematicians and researchers in several areas will find the breadth and depth of the treatment of the subject highly useful.
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| Subject
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Harmonic analysis.
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Probabilities.
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| Subject
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Harmonic analysis.
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| Subject
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MATHEMATICS-- Calculus.
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| Subject
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MATHEMATICS-- Mathematical Analysis.
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| Subject
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Probabilities.
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| Dewey Classification
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515.2433
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| LC Classification
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QA403
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