On Variable Lebesgue Spaces
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Variable Lebesgue Spaces
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Author : David V. Cruz-Uribe
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-02-12
Variable Lebesgue Spaces written by David V. Cruz-Uribe and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-02-12 with Mathematics categories.
This book provides an accessible introduction to the theory of variable Lebesgue spaces. These spaces generalize the classical Lebesgue spaces by replacing the constant exponent p with a variable exponent p(x). They were introduced in the early 1930s but have become the focus of renewed interest since the early 1990s because of their connection with the calculus of variations and partial differential equations with nonstandard growth conditions, and for their applications to problems in physics and image processing. The book begins with the development of the basic function space properties. It avoids a more abstract, functional analysis approach, instead emphasizing an hands-on approach that makes clear the similarities and differences between the variable and classical Lebesgue spaces. The subsequent chapters are devoted to harmonic analysis on variable Lebesgue spaces. The theory of the Hardy-Littlewood maximal operator is completely developed, and the connections between variable Lebesgue spaces and the weighted norm inequalities are introduced. The other important operators in harmonic analysis - singular integrals, Riesz potentials, and approximate identities - are treated using a powerful generalization of the Rubio de Francia theory of extrapolation from the theory of weighted norm inequalities. The final chapter applies the results from previous chapters to prove basic results about variable Sobolev spaces.
An Introductory Course In Lebesgue Spaces
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Author : Rene Erlin Castillo
language : en
Publisher: Springer
Release Date : 2016-06-23
An Introductory Course In Lebesgue Spaces written by Rene Erlin Castillo and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-06-23 with Mathematics categories.
This book is devoted exclusively to Lebesgue spaces and their direct derived spaces. Unique in its sole dedication, this book explores Lebesgue spaces, distribution functions and nonincreasing rearrangement. Moreover, it also deals with weak, Lorentz and the more recent variable exponent and grand Lebesgue spaces with considerable detail to the proofs. The book also touches on basic harmonic analysis in the aforementioned spaces. An appendix is given at the end of the book giving it a self-contained character. This work is ideal for teachers, graduate students and researchers.
Variable Lebesgue Spaces
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Author : Springer
language : en
Publisher:
Release Date : 2013-02-12
Variable Lebesgue Spaces written by Springer and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-02-12 with categories.
On Variable Lebesgue Spaces
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Author : Peter Quoc Hiep Nguyen
language : en
Publisher:
Release Date : 2011
On Variable Lebesgue Spaces written by Peter Quoc Hiep Nguyen and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with categories.
The reader will recall that the classical $p$-Lebesgue spaces are those functions defined on a measure space $(X, \mu)$ whose modulus raised to the $p^{\rm th}$ power is integrable. This condition gives many quantitative measurements on the growth of the function, both locally and globally. Results and applications pertaining to such functions are ubiquitous. That said, the constancy of the exponent $p$ when computing $\int_X \abs{f}^p d\mu$ is limiting in the sense that it is intrinsically uniform in scope. Speaking loosely, there are instances in which one is concerned with the $p$ growth of a function in a region $A$ and its $q$ growth in another region $B$. As such, allowing the exponent to vary from region to region (or point to point) is a reasonable course of action. The task of developing such a theory was first taken up by Wladyslaw Orlicz in the 1930's. The theory he developed, of which variable Lebesgue spaces are a special case, was only intermittently studied and analyzed through the end of the century. However, at the turn of the millennium, several results and their applications sparked a focused and intense interest in variable $L^p$ spaces. It was found that with very few assumptions on the exponent function many of the classical structure and density theorems are valid in the variable-exponent case. Somewhat surprisingly, these results were largely proved using intuitive adaptations of well-established methods. In fact, this methodology set the tone for the first part of the decade, where a multitude of ``affirmative'' results emerged. While the successful adaptation of classical results persists to a large extent today, there are nontrivial situations in which one cannot hope to extend a result known for constant $L^p$. In this paper, we wish to explore both of the aforementioned directions of research. We will first establish the fundamentals for variable $L^p$. Afterwards, we will apply these fundamentals to some classical $L^p$ results that have been extended to the variable setting. We will conclude by shifting our attention to Littlewood-Paley theory, where we will furnish an example for which it is impossible to extend constant-exponent results to the variable case.
Structure And Operators On Variable Lebesgue Spaces
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Author : Mauro Sanchiz Alonso
language : es
Publisher:
Release Date : 2023
Structure And Operators On Variable Lebesgue Spaces written by Mauro Sanchiz Alonso and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023 with categories.
Orlicz Spaces And Generalized Orlicz Spaces
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Author : Petteri Harjulehto
language : en
Publisher: Springer
Release Date : 2019-05-07
Orlicz Spaces And Generalized Orlicz Spaces written by Petteri Harjulehto and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-05-07 with Mathematics categories.
This book presents a systematic treatment of generalized Orlicz spaces (also known as Musielak–Orlicz spaces) with minimal assumptions on the generating Φ-function. It introduces and develops a technique centered on the use of equivalent Φ-functions. Results from classical functional analysis are presented in detail and new material is included on harmonic analysis. Extrapolation is used to prove, for example, the boundedness of Calderón–Zygmund operators. Finally, central results are provided for Sobolev spaces, including Poincaré and Sobolev–Poincaré inequalities in norm and modular forms. Primarily aimed at researchers and PhD students interested in Orlicz spaces or generalized Orlicz spaces, this book can be used as a basis for advanced graduate courses in analysis.
Lebesgue Integration On Euclidean Space
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Author : Frank Jones
language : en
Publisher: Jones & Bartlett Learning
Release Date : 2001
Lebesgue Integration On Euclidean Space written by Frank Jones and has been published by Jones & Bartlett Learning this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001 with Computers categories.
"'Lebesgue Integration on Euclidean Space' contains a concrete, intuitive, and patient derivation of Lebesgue measure and integration on Rn. It contains many exercises that are incorporated throughout the text, enabling the reader to apply immediately the new ideas that have been presented" --
Variable Lebesgue Spaces And Hyperbolic Systems
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Author : David Cruz-Uribe
language : en
Publisher: Springer
Release Date : 2014-07-22
Variable Lebesgue Spaces And Hyperbolic Systems written by David Cruz-Uribe and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-07-22 with Mathematics categories.
This book targets graduate students and researchers who want to learn about Lebesgue spaces and solutions to hyperbolic equations. It is divided into two parts. Part 1 provides an introduction to the theory of variable Lebesgue spaces: Banach function spaces like the classical Lebesgue spaces but with the constant exponent replaced by an exponent function. These spaces arise naturally from the study of partial differential equations and variational integrals with non-standard growth conditions. They have applications to electrorheological fluids in physics and to image reconstruction. After an introduction that sketches history and motivation, the authors develop the function space properties of variable Lebesgue spaces; proofs are modeled on the classical theory. Subsequently, the Hardy-Littlewood maximal operator is discussed. In the last chapter, other operators from harmonic analysis are considered, such as convolution operators and singular integrals. The text is mostly self-contained, with only some more technical proofs and background material omitted. Part 2 gives an overview of the asymptotic properties of solutions to hyperbolic equations and systems with time-dependent coefficients. First, an overview of known results is given for general scalar hyperbolic equations of higher order with constant coefficients. Then strongly hyperbolic systems with time-dependent coefficients are considered. A feature of the described approach is that oscillations in coefficients are allowed. Propagators for the Cauchy problems are constructed as oscillatory integrals by working in appropriate time-frequency symbol classes. A number of examples is considered and the sharpness of results is discussed. An exemplary treatment of dissipative terms shows how effective lower order terms can change asymptotic properties and thus complements the exposition.
Functional Analysis Sobolev Spaces And Partial Differential Equations
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Author : Haim Brezis
language : en
Publisher: Springer Science & Business Media
Release Date : 2010-11-10
Functional Analysis Sobolev Spaces And Partial Differential Equations written by Haim Brezis and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-11-10 with Mathematics categories.
This textbook is a completely revised, updated, and expanded English edition of the important Analyse fonctionnelle (1983). In addition, it contains a wealth of problems and exercises (with solutions) to guide the reader. Uniquely, this book presents in a coherent, concise and unified way the main results from functional analysis together with the main results from the theory of partial differential equations (PDEs). Although there are many books on functional analysis and many on PDEs, this is the first to cover both of these closely connected topics. Since the French book was first published, it has been translated into Spanish, Italian, Japanese, Korean, Romanian, Greek and Chinese. The English edition makes a welcome addition to this list.
Lectures On Analysis On Metric Spaces
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Author : Juha Heinonen
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Lectures On Analysis On Metric Spaces written by Juha Heinonen and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
Analysis in spaces with no a priori smooth structure has progressed to include concepts from the first order calculus. In particular, there have been important advances in understanding the infinitesimal versus global behavior of Lipschitz functions and quasiconformal mappings in rather general settings; abstract Sobolev space theories have been instrumental in this development. The purpose of this book is to communicate some of the recent work in the area while preparing the reader to study more substantial, related articles. The material can be roughly divided into three different types: classical, standard but sometimes with a new twist, and recent. The author first studies basic covering theorems and their applications to analysis in metric measure spaces. This is followed by a discussion on Sobolev spaces emphasizing principles that are valid in larger contexts. The last few sections of the book present a basic theory of quasisymmetric maps between metric spaces. Much of the material is relatively recent and appears for the first time in book format. There are plenty of exercises. The book is well suited for self-study, or as a text in a graduate course or seminar. The material is relevant to anyone who is interested in analysis and geometry in nonsmooth settings.