› › Interpolation Theorems And Applications To Singular Integrals

Interpolation Theorems and Applications to Singular Integrals

Author : Mervat Akram Madi

Publisher :

Page : 164 pages

File Size : 42,94 MB

Release : 2009

Category :

ISBN :

Get eBook

Book Description

A new area in mathematics has evolved out of interest in singular integrals. Att empts were made to bound singular integral operators with respect to certain Lp norms. Having various kinds of singular integrals that differ in the number of v ariables, the characteristics of the phase function, the values of the parameter s involved, etc bears witness for applying diverse methods as differentiation an d interpolation methods, and also affects the range of p's for which these opera tors are bounded. Meanwhile, the flexible properties of Lorentz norms allowed a great progress in real and complex interpolation methods which have always been a significant approach to the problem. Our plan is to show how both real and complex interpolation techniques can be ap plied to bound singular integral operators. After acquiring a sufficient idea ab out Lorentz spaces and their properties, we are going first to demonstrate a rea l interpolation method (Wolff interpolation theorem), and present Hardy's Lp ine quality as an application to it; and second, to prove a complex interpolation th eorem (Stein- Weiss complex interpolation theorem) and apply it to a more sophis ticated singular integral operator.



Extremal Problems in Interpolation Theory, Whitney-Besicovitch Coverings, and Singular Integrals

Author : Sergey Kislyakov

Publisher : Springer Science & Business Media

Page : 320 pages

File Size : 38,27 MB

Release : 2012-10-29

Category : Mathematics

ISBN : 3034804695

Get eBook

Book Description

In this book we suggest a unified method of constructing near-minimizers for certain important functionals arising in approximation, harmonic analysis and ill-posed problems and most widely used in interpolation theory. The constructions are based on far-reaching refinements of the classical Calderón–Zygmund decomposition. These new Calderón–Zygmund decompositions in turn are produced with the help of new covering theorems that combine many remarkable features of classical results established by Besicovitch, Whitney and Wiener. In many cases the minimizers constructed in the book are stable (i.e., remain near-minimizers) under the action of Calderón–Zygmund singular integral operators. The book is divided into two parts. While the new method is presented in great detail in the second part, the first is mainly devoted to the prerequisites needed for a self-contained presentation of the main topic. There we discuss the classical covering results mentioned above, various spectacular applications of the classical Calderón–Zygmund decompositions, and the relationship of all this to real interpolation. It also serves as a quick introduction to such important topics as spaces of smooth functions or singular integrals.



Singular Integrals and Related Topics

Author : Shanzhen Lu

Publisher : World Scientific

Page : 281 pages

File Size : 13,37 MB

Release : 2007

Category : Mathematics

ISBN : 9812706232

Get eBook

Book Description

This book introduces some important progress in the theory of Calderon-Zygmund singular integrals, oscillatory singular integrals, and Littlewood-Paley theory over the last decade. It includes some important research results by the authors and their cooperators, such as singular integrals with rough kernels on Block spaces and Hardy spaces, the criterion on boundedness of oscillatory singular integrals, and boundedness of the rough Marcinkiewicz integrals. These results have frequently been cited in many published papers.



Singular Integrals and Fourier Theory on Lipschitz Boundaries

Author : Tao Qian

Publisher : Springer

Page : 315 pages

File Size : 38,36 MB

Release : 2019-03-20

Category : Mathematics

ISBN : 9811365008

Get eBook

Book Description

The main purpose of this book is to provide a detailed and comprehensive survey of the theory of singular integrals and Fourier multipliers on Lipschitz curves and surfaces, an area that has been developed since the 1980s. The subject of singular integrals and the related Fourier multipliers on Lipschitz curves and surfaces has an extensive background in harmonic analysis and partial differential equations. The book elaborates on the basic framework, the Fourier methodology, and the main results in various contexts, especially addressing the following topics: singular integral operators with holomorphic kernels, fractional integral and differential operators with holomorphic kernels, holomorphic and monogenic Fourier multipliers, and Cauchy-Dunford functional calculi of the Dirac operators on Lipschitz curves and surfaces, and the high-dimensional Fueter mapping theorem with applications. The book offers a valuable resource for all graduate students and researchers interested in singular integrals and Fourier multipliers.





Interpolation of Operators

Author : Colin Bennett

Publisher : Academic Press

Page : 489 pages

File Size : 25,67 MB

Release : 1988-04-01

Category : Mathematics

ISBN : 0080874487

Get eBook

Book Description

This book presents interpolation theory from its classical roots beginning with Banach function spaces and equimeasurable rearrangements of functions, providing a thorough introduction to the theory of rearrangement-invariant Banach function spaces. At the same time, however, it clearly shows how the theory should be generalized in order to accommodate the more recent and powerful applications. Lebesgue, Lorentz, Zygmund, and Orlicz spaces receive detailed treatment, as do the classical interpolation theorems and their applications in harmonic analysis.The text includes a wide range of techniques and applications, and will serve as an amenable introduction and useful reference to the modern theory of interpolation of operators.



Pseudodifferential and Singular Integral Operators

Author : Helmut Abels

Publisher : Walter de Gruyter

Page : 233 pages

File Size : 28,15 MB

Release : 2011-12-23

Category : Mathematics

ISBN : 3110250314

Get eBook

Book Description

This textbook provides a self-contained and elementary introduction to the modern theory of pseudodifferential operators and their applications to partial differential equations. In the first chapters, the necessary material on Fourier transformation and distribution theory is presented. Subsequently the basic calculus of pseudodifferential operators on the n-dimensional Euclidean space is developed. In order to present the deep results on regularity questions for partial differential equations, an introduction to the theory of singular integral operators is given - which is of interest for its own. Moreover, to get a wide range of applications, one chapter is devoted to the modern theory of Besov and Bessel potential spaces. In order to demonstrate some fundamental approaches and the power of the theory, several applications to wellposedness and regularity question for elliptic and parabolic equations are presented throughout the book. The basic notation of functional analysis needed in the book is introduced and summarized in the appendix. The text is comprehensible for students of mathematics and physics with a basic education in analysis.



Singular Integrals and Function Spaces

Author : The Anh Bui

Publisher :

Page : 140 pages

File Size : 43,1 MB

Release : 2012

Category : Hardy spaces

ISBN :

Get eBook

Book Description

The main aim of this thesis is to study the boundedness of some singular integrals on various function spaces. The main results of this thesis are presented in three parts. -- In the first part, two criteria on the Lp-weighted norm inequalities of singular integral operators with non-smooth kernels and the endpoint estimates of the commutators of these operators with BMO functions are obtained. As applications, we first studied the weighted norm inequalities of Riesz transforms associated to Schrödinger operators, Green functions and spectral multipliers and then endpoint estimates of commutators of these singular integrals with BMO functions such as the Riesz transforms, the square functions and the spectral multipliers. -- The second part is dedicated to study the Hardy spaces associated to the discrete Laplacians on graphs and applications. Some characterizations of Hardy spaces associated to operators such as the atomic characterization and the square function characterization are obtained. Then we consider the boundedness of singular integrals on these Hardy spaces. -- In the third part, we develop the theory of Hardy spaces, RBMO spaces and Calder on-Zygmund operators in the setting of nonhomogeneous spaces. Some important results are addressed in this part such as the Interpolation Theorem between Hardy spaces and RBMO spaces, the boundedness of Calder on-Zygmund operators on Hardy spaces and RBMO spaces and the Calderón-Zygmund decomposition.



Singular Integrals and Differentiability Properties of Functions (PMS-30), Volume 30

Author : Elias M. Stein

Publisher : Princeton University Press

Page : 306 pages

File Size : 13,92 MB

Release : 2016-06-02

Category : Mathematics

ISBN : 1400883881

Get eBook

Book Description

Singular integrals are among the most interesting and important objects of study in analysis, one of the three main branches of mathematics. They deal with real and complex numbers and their functions. In this book, Princeton professor Elias Stein, a leading mathematical innovator as well as a gifted expositor, produced what has been called the most influential mathematics text in the last thirty-five years. One reason for its success as a text is its almost legendary presentation: Stein takes arcane material, previously understood only by specialists, and makes it accessible even to beginning graduate students. Readers have reflected that when you read this book, not only do you see that the greats of the past have done exciting work, but you also feel inspired that you can master the subject and contribute to it yourself. Singular integrals were known to only a few specialists when Stein's book was first published. Over time, however, the book has inspired a whole generation of researchers to apply its methods to a broad range of problems in many disciplines, including engineering, biology, and finance. Stein has received numerous awards for his research, including the Wolf Prize of Israel, the Steele Prize, and the National Medal of Science. He has published eight books with Princeton, including Real Analysis in 2005.



Singular Integral Equations

Author : E.G. Ladopoulos

Publisher : Springer Science & Business Media

Page : 569 pages

File Size : 13,90 MB

Release : 2013-03-09

Category : Technology & Engineering

ISBN : 3662042916

Get eBook

Book Description

The present book deals with the finite-part singular integral equations, the multidimensional singular integral equations and the non-linear singular integral equations, which are currently used in many fields of engineering mechanics with applied character, like elasticity, plasticity, thermoelastoplasticity, viscoelasticity, viscoplasticity, fracture mechanics, structural analysis, fluid mechanics, aerodynamics and elastodynamics. These types of singular integral equations form the latest high technology on the solution of very important problems of solid and fluid mechanics and therefore special attention should be given by the reader of the present book, who is interested for the new technology of the twentieth-one century. Chapter 1 is devoted with a historical report and an extended outline of References, for the finite-part singular integral equations, the multidimensional singular integral equations and the non-linear singular integral equations. Chapter 2 provides a finite-part singular integral representation analysis in Lp spaces and in general Hilbert spaces. In the same Chapter are investigated all possible approximation methods for the numerical evaluation of the finite-part singular integral equations, as closed form solutions for the above type of integral equations are available only in simple cases. Also, Chapter 2 provides further a generalization of the well known Sokhotski-Plemelj formulae and the Nother theorems, for the case of a finite-part singular integral equation.