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This third volume of the monograph examines potential theory. The first chapter develops potential theory with respect to a single kernel (or discrete time semigroup). All the essential ideas of the theory are presented: excessive functions, reductions, sweeping, maximum principle. The second chapter begins with a study of the notion of reduction in the most general situation possible - the ``gambling house'' of Dubins and Savage. The beautiful results presented have never been made accessible to a wide public. These are then connected with the theory of sweeping with respect to a cone of cont

Probabilities. --- Potential theory (Mathematics) --- Semigroups. --- Group theory --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk

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In this volume two topics are discussed: the construction of Feller and Lp-sub-Markovian semigroups by starting with a pseudo-differential operator, and the potential theory of these semigroups and their generators. The first part of the text essentially discusses the analysis of pseudo-differential operators with negative definite symbols and develops a symbolic calculus; in addition, it deals with special approaches, such as subordination in the sense of Bochner. The second part handles capacities, function spaces associated with continuous negative definite functions, Lp -sub-Markovian sem

Semigroups of operators. --- Pseudodifferential operators. --- Potential theory (Mathematics) --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Operators, Pseudodifferential --- Pseudo-differential operators --- Operator theory --- Operators, Semigroups of

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Potential theory (Mathematics) --- Functional analysis --- Probabilities --- Geometry --- Potentiel, Théorie du --- Analyse fonctionnelle --- Probabilités --- Géométrie --- Functional calculus --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Calculus of variations --- Functional equations --- Integral equations --- Mathematical analysis --- Mechanics --- Mathematical Sciences --- Applied Mathematics --- General and Others

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The book The E. M. Stein Lectures on Hardy Spaces is based on a graduate course on real variable Hardy spaces which was given by E.M. Stein at Princeton University in the academic year 1973-1974. Stein, along with C. Fefferman and G. Weiss, pioneered this subject area, removing the theory of Hardy spaces from its traditional dependence on complex variables, and to reveal its real-variable underpinnings. This book is based on Steven G. Krantz’s notes from the course given by Stein. The text builds on Fefferman's theorem that BMO is the dual of the Hardy space. Using maximal functions, singular integrals, and related ideas, Stein offers many new characterizations of the Hardy spaces. The result is a rich tapestry of ideas that develops the theory of singular integrals to a new level. The final chapter describes the major developments since 1974. This monograph is of broad interest to graduate students and researchers in mathematical analysis. Prerequisites for the book include a solid understanding of real variable theory and complex variable theory. A basic knowledge of functional analysis would also be useful.

Fourier analysis. --- Potential theory (Mathematics). --- Fourier Analysis. --- Potential Theory. --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Analysis, Fourier --- Bounded mean oscillation. --- Hardy spaces. --- Stein, Elias M., --- Spaces, Hardy --- Functional analysis --- Functions of complex variables --- BMO (Mathematics) --- Function spaces --- Stein, E. M. --- Steĭn, Ilaĭes M., --- Стейн, Илайес М., --- Espais de Hardy --- Espais funcionals

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This monograph presents the state of the art of convexity, with an emphasis to integral representation. The exposition is focused on Choquet's theory of function spaces with a link to compact convex sets. An important feature of the book is an interplay between various mathematical subjects, such as functional analysis, measure theory, descriptive set theory, Banach spaces theory and potential theory. A substantial part of the material is of fairly recent origin and many results appear in the book form for the first time. The text is self-contained and covers a wide range of applications. From the contents: Geometry of convex sets Choquet theory of function spaces Affine functions on compact convex sets Perfect classes of functions and representation of affine functions Simplicial function spaces Choquet's theory of function cones Topologies on boundaries Several results on function spaces and compact convex sets Continuous and measurable selectors Construction of function spaces Function spaces in potential theory and Dirichlet problem Applications

Functional analysis. --- Convex domains. --- Banach spaces. --- Potential theory (Mathematics) --- Integral representations. --- Representations, Integral --- Algebraic number theory --- Crystallography, Mathematical --- Representations of groups --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Functions of complex variables --- Generalized spaces --- Topology --- Convex regions --- Convexity --- Calculus of variations --- Convex geometry --- Point set theory --- Functional calculus --- Functional equations --- Integral equations --- Convex Analysis. --- Dirichlet Problem. --- Functional Analysis. --- Partial Differential Equation. --- Potential Theory. --- Functional analysis --- Convex domains --- Banach spaces --- Integral representations