دانلود کتاب Pseudodifferential and Singular Integral Operators: An Introduction with Applications

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Preface\n1 Introduction\nI Fourier Transformation and Pseudodifferential Operators\n 2 Fourier Transformation and Tempered Distributions\n 2.1 Definition and Basic Properties\n 2.2 Rapidly Decreasing Functions – ℘ (ℝn)\n 2.3 Inverse Fourier Transformation and Plancherel’s Theorem\n 2.4 Tempered Distributions and Fourier Transformation\n 2.5 Fourier Transformation and Convolution of Tempered Distributions\n 2.6 Convolution on on ℘ʹ(ℝn) and Fundamental Solutions\n 2.7 Sobolev and Bessel Potential Spaces\n 2.8 Vector-Valued Fourier-Transformation\n 2.9 Final Remarks and Exercises\n 2.9.1 Further Reading\n 2.9.2 Exercises\n 3 Basic Calculus of Pseudodifferential Operators on ℝn\n 3.1 Symbol Classes and Basic Properties\n 3.2 Composition of Pseudodifferential Operators: Motivation\n 3.3 Oscillatory Integrals\n 3.4 Double Symbols\n 3.5 Composition of Pseudodifferential Operators\n 3.6 Application: Elliptic Pseudodifferential Operators and Parametrices\n 3.7 Boundedness on Cb∞ (ℝn) and Uniqueness of the Symbol\n 3.8 Adjoints of Pseudodifferential Operators and Operators in (x, y )-Form\n 3.9 Boundedness on L2(ℝn) and L2-Bessel Potential Spaces\n 3.10 Outlook: Coordinate Transformations and PsDOs on Manifolds\n 3.11 Final Remarks and Exercises\n 3.11.1 Further Reading\n 3.11.2 Exercises\nII Singular Integral Operators\n 4 Translation Invariant Singular Integral Operators\n 4.1 Motivation\n 4.2 Main Result in the Translation Invariant Case\n 4.3 Calderón-Zygmund Decomposition and the Maximal Operator\n 4.4 Proof of the Main Result in the Translation Invariant Case\n 4.5 Examples of Singular Integral Operators\n 4.6 Mikhlin Multiplier Theorem\n 4.7 Outlook: Hardy spaces and BMO\n 4.8 Final Remarks and Exercises\n 4.8.1 Further Reading\n 4.8.2 Exercises\n 5 Non-Translation Invariant Singular Integral Operators\n 5.1 Motivation\n 5.2 Extension to Non-Translation Invariant and Vector-Valued Singular Integral Operators\n 5.3 Hilbert-Space-Valued Mikhlin Multiplier Theorem\n 5.4 Kernel Representation of a Pseudodifferential Operator\n 5.5 Consequences of the Kernel Representation\n 5.6 Final Remarks and Exercises\n 5.6.1 Further Reading\n 5.6.2 Exercises\nIII Applications to Function Space and Differential Equations\n 6 Introduction to Besov and Bessel Potential Spaces\n 6.1 Motivation\n 6.2 A Fourier-Analytic Characterization of Holder Continuity\n 6.3 Bessel Potential and Besov Spaces – Definitions and Basic Properties\n 6.4 Sobolev Embeddings\n 6.5 Equivalent Norms\n 6.6 Pseudodifferential Operators on Besov Spaces\n 6.7 Final Remarks and Exercises\n 6.7.1 Further Reading\n 6.7.2 Exercises\n 7 Applications to Elliptic and Parabolic Equations\n 7.1 Applications of the Mikhlin Multiplier Theorem\n 7.1.1 Resolvent of the Laplace Operator\n 7.1.2 Spectrum of Multiplier Operators with Homogeneous Symbols\n 7.1.3 Spectrum of a Constant Coefficient Differential Operator\n 7.2 Applications of the Hilbert-Space-Valued Mikhlin Multiplier Theorem\n 7.2.1 Maximal Regularity of Abstract ODEs in Hilbert Spaces\n 7.2.2 Hilbert-Space Valued Bessel Potential and Sobolev Spaces\n 7.3 Applications of Pseudodifferential Operators\n 7.3.1 Elliptic Regularity for Elliptic Pseudodifferential Operators\n 7.3.2 Resolvents of Parameter-Elliptic Differential Operators\n 7.3.3 Application of Resolvent Estimates to Parabolic Initial Value Problems\n 7.4 Final Remarks and Exercises\n 7.4.1 Further Reading\n 7.4.2 Exercises\nIV Appendix\n A Basic Results from Analysis\n A.1 Notation and Functions on ℝn\n A.2 Lebesgue Integral and Lp-Spaces\n A.3 Linear Operators and Dual Spaces\n A.4 Bochner Integral and Vector-Valued Lp-Spaces\n A.5 Fréchet Spaces\n A.6 Exercises\nBibliography\nIndex