Studies In Potential Theory And Function Spaces
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Studies in Potential Theory and Function Spaces
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Author | : Torbjörn Kolsrud |
Publsiher | : Unknown |
Total Pages | : 6 |
Release | : 1983 |
Genre | : Electronic Book |
ISBN | : 9171462473 |
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Function Spaces and Potential Theory
Author | : David R. Adams,Lars I. Hedberg |
Publsiher | : Springer Science & Business Media |
Total Pages | : 372 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 9783662032824 |
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"..carefully and thoughtfully written and prepared with, in my opinion, just the right amount of detail included...will certainly be a primary source that I shall turn to." Proceedings of the Edinburgh Mathematical Society
Nonlinear Potential Theory on Metric Spaces
Author | : Anders Björn,Jana Björn |
Publsiher | : European Mathematical Society |
Total Pages | : 422 |
Release | : 2011 |
Genre | : Harmonic functions |
ISBN | : 303719099X |
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The $p$-Laplace equation is the main prototype for nonlinear elliptic problems and forms a basis for various applications, such as injection moulding of plastics, nonlinear elasticity theory, and image processing. Its solutions, called p-harmonic functions, have been studied in various contexts since the 1960s, first on Euclidean spaces and later on Riemannian manifolds, graphs, and Heisenberg groups. Nonlinear potential theory of p-harmonic functions on metric spaces has been developing since the 1990s and generalizes and unites these earlier theories. This monograph gives a unified treatment of the subject and covers most of the available results in the field, so far scattered over a large number of research papers. The aim is to serve both as an introduction to the area for interested readers and as a reference text for active researchers. The presentation is rather self contained, but it is assumed that readers know measure theory and functional analysis. The first half of the book deals with Sobolev type spaces, so-called Newtonian spaces, based on upper gradients on general metric spaces. In the second half, these spaces are used to study p-harmonic functions on metric spaces, and a nonlinear potential theory is developed under some additional, but natural, assumptions on the underlying metric space. Each chapter contains historical notes with relevant references, and an extensive index is provided at the end of the book.
Integral Representation Theory
Author | : Jaroslav Lukeš |
Publsiher | : Walter de Gruyter |
Total Pages | : 732 |
Release | : 2010 |
Genre | : Mathematics |
ISBN | : 9783110203202 |
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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
Nonlinear Potential Theory and Weighted Sobolev Spaces
Author | : Bengt O. Turesson |
Publsiher | : Springer |
Total Pages | : 188 |
Release | : 2007-05-06 |
Genre | : Mathematics |
ISBN | : 9783540451686 |
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The book systematically develops the nonlinear potential theory connected with the weighted Sobolev spaces, where the weight usually belongs to Muckenhoupt's class of Ap weights. These spaces occur as solutions spaces for degenerate elliptic partial differential equations. The Sobolev space theory covers results concerning approximation, extension, and interpolation, Sobolev and Poincaré inequalities, Maz'ya type embedding theorems, and isoperimetric inequalities. In the chapter devoted to potential theory, several weighted capacities are investigated. Moreover, "Kellogg lemmas" are established for various concepts of thinness. Applications of potential theory to weighted Sobolev spaces include quasi continuity of Sobolev functions, Poincaré inequalities, and spectral synthesis theorems.
Theory of Function Spaces
Author | : Hans Triebel |
Publsiher | : Birkhäuser |
Total Pages | : 292 |
Release | : 1983-01-01 |
Genre | : Science |
ISBN | : UCAL:B5008768 |
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The book deals with the two scales Bsp,q and Fsp,q of spaces of distributions, where ‐∞s∞ and 0p,q≤∞, which include many classical and modern spaces, such as Hölder spaces, Zygmund classes, Sobolev spaces, Besov spaces, Bessel-potential spaces, Hardy spaces and spaces of BMO-type. It is the main aim of this book to give a unified treatment of the corresponding spaces on the Euclidean n-space Rsubn
Potential Theory on Locally Compact Abelian Groups
Author | : C. van den Berg,G. Forst |
Publsiher | : Springer Science & Business Media |
Total Pages | : 205 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 9783642661280 |
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Classical potential theory can be roughly characterized as the study of Newtonian potentials and the Laplace operator on the Euclidean space JR3. It was discovered around 1930 that there is a profound connection between classical potential 3 theory and the theory of Brownian motion in JR . The Brownian motion is determined by its semigroup of transition probabilities, the Brownian semigroup, and the connection between classical potential theory and the theory of Brownian motion can be described analytically in the following way: The Laplace operator is the infinitesimal generator for the Brownian semigroup and the Newtonian potential kernel is the" integral" of the Brownian semigroup with respect to time. This connection between classical potential theory and the theory of Brownian motion led Hunt (cf. Hunt [2]) to consider general "potential theories" defined in terms of certain stochastic processes or equivalently in terms of certain semi groups of operators on spaces of functions. The purpose of the present exposition is to study such general potential theories where the following aspects of classical potential theory are preserved: (i) The theory is defined on a locally compact abelian group. (ii) The theory is translation invariant in the sense that any translate of a potential or a harmonic function is again a potential, respectively a harmonic function; this property of classical potential theory can also be expressed by saying that the Laplace operator is a differential operator with constant co efficients.
Littlewood Paley Theory and the Study of Function Spaces
Author | : Michael Frazier,Björn Jawerth,Guido Weiss |
Publsiher | : American Mathematical Soc. |
Total Pages | : 142 |
Release | : 1991 |
Genre | : Mathematics |
ISBN | : 9780821807316 |
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Littlewood-Paley theory was developed to study function spaces in harmonic analysis and partial differential equations. Recently, it has contributed to the development of the *q-transform and wavelet decompositions. Based on lectures presented at the NSF-CBMS Regional Research Conference on Harmonic Analysis and Function Spaces, held at Auburn University in July 1989, this book is aimed at mathematicians, as well as mathematically literate scientists and engineers interested in harmonic analysis or wavelets. The authors provide not only a general understanding of the area of harmonic analysis relating to Littlewood-Paley theory and atomic and wavelet decompositions, but also some motivation and background helpful in understanding the recent theory of wavelets. The book begins with some simple examples which provide an overview of the classical Littlewood-Paley theory. The *q-transform, wavelet, and smooth atomic expansions are presented as natural extensions of the classical theory. Finally, applications to harmonic analysis (Calderon-Zygmund operators), signal processing (compression), and mathematical physics (potential theory) are discussed.