Pseudodifferential and Singular Integral Operators: An Introduction with Applications

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. Highly motivated by problems arising in various applications Appropriate as a textbook for graduate courses or for independent study Several exercises and summaries of each chapter help beginners to understand the material 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 – ℘ (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 ℘ʹ(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 Exercises 3 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 Cb∞ (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 Exercises 3.11.1 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 Motivation 5.2 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 Properties 6.4 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 7.2.1 Maximal Regularity of Abstract ODEs in Hilbert Spaces 7.2.2 Hilbert-Space Valued Bessel Potential and Sobolev Spaces 7.3 Applications of Pseudodifferential Operators 7.3.1 Elliptic Regularity for Elliptic Pseudodifferential Operators 7.3.2 Resolvents of Parameter-Elliptic Differential Operators 7.3.3 Application of Resolvent Estimates to Parabolic Initial Value Problems 7.4 Final Remarks and Exercises 7.4.1 Further Reading 7.4.2 Exercises IV Appendix A Basic Results from Analysis A.1 Notation and Functions on ℝn A.2 Lebesgue Integral and Lp-Spaces A.3 Linear Operators and Dual Spaces A.4 Bochner Integral and Vector-Valued Lp-Spaces A.5 Fréchet Spaces A.6 Exercises Bibliography Index