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Book Cover
Author Abels, H. (Helmut)

Title Pseudodifferential and Singular Integral Operators : an Introduction with Applications
Published Berlin : De Gruyter, 2011


Description 1 online resource (232 pages)
Series De Gruyter graduate lectures
De Gruyter graduate.
Contents Preface; 1 Introduction; I Fourier Transformation and Pseudodifferential Operators; 2 Fourier Transformation and Tempered Distributions; 2.1 Definition and Basic Properties; 2.2 Rapidly Decreasing Functions -- P (Rn); 2.3 Inverse Fourier Transformation and Plancherel's Theorem; 2.4 Tempered Distributions and Fourier Transformation; 2.5 Fourier Transformation and Convolution of Tempered Distributions; 2.6 Convolution on on P'(Rn) and Fundamental Solutions; 2.7 Sobolev and Bessel Potential Spaces; 2.8 Vector-Valued Fourier-Transformation; 2.9 Final Remarks and Exercises; 2.9.1 Further Reading
2.9.2 Exercises3 Basic Calculus of Pseudodifferential Operators on Rn; 3.1 Symbol Classes and Basic Properties; 3.2 Composition of Pseudodifferential Operators: Motivation; 3.3 Oscillatory Integrals; 3.4 Double Symbols; 3.5 Composition of Pseudodifferential Operators; 3.6 Application: Elliptic Pseudodifferential Operators and Parametrices; 3.7 Boundedness on Cb8 (Rn) and Uniqueness of the Symbol; 3.8 Adjoints of Pseudodifferential Operators and Operators in (x, y)-Form; 3.9 Boundedness on L2(Rn) and L2-Bessel Potential Spaces; 3.10 Outlook: Coordinate Transformations and PsDOs on Manifolds
3.11 Final Remarks and Exercises3.11.1 Further Reading; 3.11.2 Exercises; II Singular Integral Operators; 4 Translation Invariant Singular Integral Operators; 4.1 Motivation; 4.2 Main Result in the Translation Invariant Case; 4.3 Calderón-Zygmund Decomposition and the Maximal Operator; 4.4 Proof of the Main Result in the Translation Invariant Case; 4.5 Examples of Singular Integral Operators; 4.6 Mikhlin Multiplier Theorem; 4.7 Outlook: Hardy spaces and BMO; 4.8 Final Remarks and Exercises; 4.8.1 Further Reading; 4.8.2 Exercises; 5 Non-Translation Invariant Singular Integral Operators
5.1 Motivation5.2 Extension to Non-Translation Invariant and Vector-Valued Singular Integral Operators; 5.3 Hilbert-Space-Valued Mikhlin Multiplier Theorem; 5.4 Kernel Representation of a Pseudodifferential Operator; 5.5 Consequences of the Kernel Representation; 5.6 Final Remarks and Exercises; 5.6.1 Further Reading; 5.6.2 Exercises; III Applications to Function Space and Differential Equations; 6 Introduction to Besov and Bessel Potential Spaces; 6.1 Motivation; 6.2 A Fourier-Analytic Characterization of Holder Continuity
6.3 Bessel Potential and Besov Spaces -- Definitions and Basic Properties6.4 Sobolev Embeddings; 6.5 Equivalent Norms; 6.6 Pseudodifferential Operators on Besov Spaces; 6.7 Final Remarks and Exercises; 6.7.1 Further Reading; 6.7.2 Exercises; 7 Applications to Elliptic and Parabolic Equations; 7.1 Applications of the Mikhlin Multiplier Theorem; 7.1.1 Resolvent of the Laplace Operator; 7.1.2 Spectrum of Multiplier Operators with Homogeneous Symbols; 7.1.3 Spectrum of a Constant Coefficient Differential Operator; 7.2 Applications of the Hilbert-Space-Valued Mikhlin Multiplier Theorem
Summary This book provides a self-contained and elementary introduction to the modern theory of pseudodifferential operators and their application to partial differential equations. It presents the necessary material on Fourier transformation and distribution theory, the basic calculus of pseudodifferential operators on the n-dimensional Euclidean space, an introduction to the theory of singular integral operators, the modern theory of Besov and Bessel potential spaces, and several applications to wellposedness and regularity question for elliptic and parabolic equations. The basic notation of functio
Notes 7.2.1 Maximal Regularity of Abstract ODEs in Hilbert Spaces
Bibliography Includes bibliographical references and index
Notes Print version record
Subject Pseudodifferential operators.
Integral operators.
MATHEMATICS -- Complex Analysis.
Integral operators
Pseudodifferential operators
Singulärer Integraloperator
Form Electronic book
ISBN 9783110250312