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Author: David V. Cruz-Uribe
Publisher: Springer Science & Business Media
ISBN: 3034805489
Category : Mathematics
Languages : en
Pages : 316
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Book Description
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.
Author: David V. Cruz-Uribe
Publisher: Springer Science & Business Media
ISBN: 3034805489
Category : Mathematics
Languages : en
Pages : 316
Get Book Here
Book Description
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.
Author: Lars Diening
Publisher: Springer
ISBN: 3642183638
Category : Mathematics
Languages : en
Pages : 516
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Book Description
The field of variable exponent function spaces has witnessed an explosive growth in recent years. The standard reference article for basic properties is already 20 years old. Thus this self-contained monograph collecting all the basic properties of variable exponent Lebesgue and Sobolev spaces is timely and provides a much-needed accessible reference work utilizing consistent notation and terminology. Many results are also provided with new and improved proofs. The book also presents a number of applications to PDE and fluid dynamics.
Author: Rene Erlin Castillo
Publisher: Springer
ISBN: 3319300342
Category : Mathematics
Languages : en
Pages : 463
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Book Description
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.
Author: Lars Diening
Publisher: Springer Science & Business Media
ISBN: 364218362X
Category : Mathematics
Languages : en
Pages : 516
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Book Description
The field of variable exponent function spaces has witnessed an explosive growth in recent years. The standard reference article for basic properties is already 20 years old. Thus this self-contained monograph collecting all the basic properties of variable exponent Lebesgue and Sobolev spaces is timely and provides a much-needed accessible reference work utilizing consistent notation and terminology. Many results are also provided with new and improved proofs. The book also presents a number of applications to PDE and fluid dynamics.
Author: Springer
Publisher:
ISBN: 9783034805490
Category :
Languages : en
Pages : 324
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Book Description
Author: Vakhtang Kokilashvili
Publisher: Birkhäuser
ISBN: 3319210157
Category : Mathematics
Languages : en
Pages : 585
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Book Description
This book, the result of the authors' long and fruitful collaboration, focuses on integral operators in new, non-standard function spaces and presents a systematic study of the boundedness and compactness properties of basic, harmonic analysis integral operators in the following function spaces, among others: variable exponent Lebesgue and amalgam spaces, variable Hölder spaces, variable exponent Campanato, Morrey and Herz spaces, Iwaniec-Sbordone (grand Lebesgue) spaces, grand variable exponent Lebesgue spaces unifying the two spaces mentioned above, grand Morrey spaces, generalized grand Morrey spaces, and weighted analogues of some of them. The results obtained are widely applied to non-linear PDEs, singular integrals and PDO theory. One of the book's most distinctive features is that the majority of the statements proved here are in the form of criteria. The book is intended for a broad audience, ranging from researchers in the area to experts in applied mathematics and prospective students.
Author: Peter Quoc Hiep Nguyen
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
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.
Author: David Cruz-Uribe
Publisher: Springer
ISBN: 3034808402
Category : Mathematics
Languages : en
Pages : 173
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Book Description
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.
Author: Mauro Sanchiz Alonso
Publisher:
ISBN:
Category :
Languages : es
Pages : 0
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Book Description
Author: Matteo Pastorino
Publisher: John Wiley & Sons
ISBN: 0470602473
Category : Technology & Engineering
Languages : en
Pages : 367
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Book Description
An introduction to the most relevant theoretical and algorithmic aspects of modern microwave imaging approaches Microwave imaging—a technique used in sensing a given scene by means of interrogating microwaves—has recently proven its usefulness in providing excellent diagnostic capabilities in several areas, including civil and industrial engineering, nondestructive testing and evaluation, geophysical prospecting, and biomedical engineering. Microwave Imaging offers comprehensive descriptions of the most important techniques so far proposed for short-range microwave imaging—including reconstruction procedures and imaging systems and apparatus—enabling the reader to use microwaves for diagnostic purposes in a wide range of applications. This hands-on resource features: A review of the electromagnetic inverse scattering problem formulation, written from an engineering perspective and with notations The most effective reconstruction techniques based on diffracted waves, including time- and frequency-domain methods, as well as deterministic and stochastic space-domain procedures Currently proposed imaging apparatus, aimed at fast and accurate measurements of the scattered field data Insight on near field probes, microwave axial tomographs, and microwave cameras and scanners A discussion of practical applications with detailed descriptions and discussions of several specific examples (e.g., materials evaluation, crack detection, inspection of civil and industrial structures, subsurface detection, and medical applications) A look at emerging techniques and future trends Microwave Imaging is a practical resource for engineers, scientists, researchers, and professors in the fields of civil and industrial engineering, nondestructive testing and evaluation, geophysical prospecting, and biomedical engineering.