دانلود کتاب Linear and Quasilinear Parabolic Problems: Volume II: Function Spaces (Monographs in Mathematics)

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Preface Contents Notations and Conventions Chapter VI Auxiliary Material 1 Restriction-Extension Pairs 1.1 Smooth Functions on Corners Corners Restriction-Extension Operators Approximation by Test Functions 1.2 Tempered Distributions on Corners Duality Formulas The Main Theorem 1.3 Duality 1.4 Notes 2 Sequence Spaces 2.1 Duality of Sequence Spaces Definitions and Embeddings Duality Pairings 2.2 Weighted Sequence Spaces Image Spaces Embeddings and Duality 2.3 Interpolation Unweighted Spaces Weighted Spaces 2.4 Notes 3 Anisotropy 3.1 Anisotropic Dilations Weight Systems Dilations 3.2 Quasinorms 3.3 Parametric Augmentations Augmented Quasinorms Positive Homogeneity Differentiating Inverses Slowly Increasing Functions 3.4 Fourier Multipliers and Multiplier Spaces Elementary Fourier Multiplier Theorems Fourier Multiplier Spaces Resolvent Estimates 3.5 Multiplier Estimates Resolvent Estimates for Homogeneous Symbols Functions of Homogeneous Symbols Dunford Integral Representations Powers and Exponentials 3.6 Dyadic Partitions of Unity Preliminary Fourier Multiplier Theorems 3.7 Notes Chapter VII Function Spaces 1 Classical Spaces 1.1 Bounded Continuous Functions Banach Spaces of Bounded Continuous Functions Vector Measures 1.2 Sobolev Spaces Regular Distributions Basic Definitions 1.3 Restrictions and Extensions 1.4 Distributional Derivatives 1.5 Reflexivity 1.6 Embeddings 1.7 Notes 2 Besov Spaces 2.1 The Definition Preliminary Estimates A Retraction-Coretraction Pair The Final Definition 2.2 Embedding Theorems Little Besov Spaces Embeddings With Varying Target Spaces 2.3 Duality 2.4 Fourier Multiplier Theorems 2.5 Operators of Positive Type Resolvent Estimates A Representation Theorem Bounded Imaginary Powers Interpolation-Extrapolation Scales 2.6 Renorming by Derivatives Equivalent Norms Sandwich Theorems Sobolev Embeddings 2.7 Interpolation Real and Complex Interpolation Interpolation with Different Target Spaces Embeddings of Intersection Spaces Interpolation of Classical Spaces 2.8 Besov Spaces on Corners 2.9 Notes 3 Intrinsic Norms, Slobodeckii and Hölder Spaces 3.1 Commuting Semigroups 3.2 Semigroups and Interpolation Preliminary Estimates Renorming Intersections of Interpolation Spaces 3.3 Translation Semigroups 3.4 Renorming Besov Spaces 3.5 Intersection-Space Characterizations Intersection Space Representations Equivalent Norms NikoÍ skiǐ Spaces 3.6 Besov-Slobodeckii and Besov-Hölder Spaces Mixed Intersections Slobodeckii, Hölder, and Little Hölder Spaces 3.7 Little Hölder Spaces Very Little Hölder Spaces 3.8 Notes 4 Bessel Potential Spaces 4.1 Basic Facts, Embeddings, and Real Interpolation 4.2 A Marcinkiewicz Multiplier Theorem 4.3 Renorming by Derivatives 4.4 Duality 4.5 Complex Interpolation A Holomorphic Semigroup Interpolation with Different Target Spaces 4.6 Intersection-Space Characterizations 4.7 Notes 5 Triebel-Lizorkin Spaces 5.1 Maximal Inequalities Preliminary Estimates for Sequences Estimates for a Single Function 5.2 Definition and Basic Embeddings Equivalent Norms Embeddings Completeness 5.3 Fourier Multiplier Theorems 5.4 Interpolation 5.5 Renorming by Derivatives Sandwich Theorems 5.6 Sobolev Embeddings and Related Results Multiplicative Inequalities Optimal Sobolev-Type Embeddings Sharp Embeddings of Intersection Spaces 5.7 Gagliardo-Nirenberg Type Estimates Nonhomogeneous Inequalities Homogeneous Estimates Isotropic Multiplicative Inequalities Sobolev Inequality Parabolic Estimates 5.8 Notes 6 Point-Wise Multiplications 6.1 Preliminaries Continuity of Derivatives Point-Wise Products 6.2 Multiplications in Classical Spaces Spaces of Bounded Continuous Functions Sobolev Spaces Spaces of Negative Order 6.3 Multiplications in Besov Spaces of Positive Order 6.4 Multiplications in Besov Spaces of Negative Order The Reflexive Case The Non-Reflexive Case 6.5 Multiplications in Bessel Potential Spaces 6.6 Space-Dependent Bilinear Maps 6.7 Notes 7 Compactness 7.1 Equicontinuity Compact Sets in BUC Compact Sets in Lq 7.2 Compact Embeddings Compact Embeddings of Besov Spaces Compact Embeddings of Hölder, Sobolev-Slobodeckii, and Bessel Potential Spaces 7.3 Function Spaces on Intervals Classical Spaces Besov Spaces A Retraction-Coretraction Theorem Interpolations and Embeddings The Rellich-Kondrachov Theorem 7.4 Aubin-Lions Type Theorems The General Result Limit Cases Applications 7.5 Notes 8 Parameter-Dependent Spaces 8.1 Sobolev Spaces and Bounded Continuous Functions 8.2 Besov and Bessel Potential Spaces 8.3 Intersection-Space Characterizations 8.4 Fourier Multipliers 8.5 Notes Chapter VIII Traces and Boundary Operators 1 Traces 1.1 Trace Operators 1.2 The Retraction Theorem 1.3 Traces on Half-Spaces Parameter-Dependence General Besov spaces 1.4 Spaces of Vanishing Traces Sobolev-Slobodeckii Spaces 1.5 Weighted Spaces Weighted Lebesgue Spaces Hardy Inequalities Weighted Space Characterizations of Sobolev-Slobodeckii Spaces 1.6 Further Characterizations of Spaces with Vanishing Traces General Besov Spaces Bessel Potential Spaces Hölder Spaces 1.7 Representation Theorems for Spaces of Negative Order Spaces of Mildly Negative Order Spaces of Strongly Negative Order Duality of Sums and Intersections Weighted Space Representations 1.8 Traces for Corners Traces on a Single Face Vanishing Traces on Corners Faces of Higher Codimensions Compatibility Conditions The Retraction Theorem for Corners 1.9 Notes 2 Boundary Operators 2.1 Boundary Operators on Half-Spaces Normal Boundary Operators 2.2 Systems of Boundary Operators The Boundary Operator Retraction Theorem Embeddings with Boundary Conditions 2.3 Transmission Operators Patching Together Half-Spaces 2.4 Interpolation With Boundary Conditions Preliminaries The Main Theorem Generalizations Complex Interpolation of Bessel Potential Spaces 2.5 Notes Appendix Vector-Valued Distributions 1 Tensor Products and Convolutions 1.1 Locally Convex Topologies The Uniform Boundedness Principle Hypocontinuity Montel Spaces Strict Inductive Limits Smooth Functions Test Functions Rapidly Decreasing Smooth Functions Slowly Increasing Smooth Functions Spaces of Vector-Valued Distributions 1.2 Convolutions Convolutions of Distributions and Test Functions Translation-Invariant Operators Convolutions of Two Distributions Elementary Properties of Convolutions Convolutions of Temperate Distributions 1.3 Approximations Multiplications Leibniz' Rule Approximation by Test Functions Density by Iteration Approximation by Tensor Products Approximation by Polynomials Separability 1.4 Topological Tensor Products and the Kernel Theorem Algebraic Tensor Products Basic Examples Projective Tensor Products Nuclear Maps and Spaces Projective Tensor Products and Maps of Finite Rank Approximation by Maps of Finite Rank Completeness of Spaces of Linear Operators The Abstract Kernel Theorem Tensor Product Characterizations of Some Distribution Spaces 1.5 Extending Bilinear Maps General Hypothesis 1.6 Point-Wise Multiplication A Characterization of OM The General Theorem Basic Properties of Multiplications 1.7 Scalar Products and Duality Pairings Parseval's Formula Duality Pairings 1.8 Tensor Products of Distributions and Kernel Theorems Approximation by Tensor Products Tensor Products of Distributions Basic Properties Examples Topological Tensor Products of Distributions Kernel Theorems 1.9 Convolutions of Vector-Valued Distributions The Basic Theorem Lp Functions with Compact Supports Convolutions of Regular Distributions Tensor Products and Convolutions Basic Properties Convolution Algebras Convolutions of Bounded and Integrable Functions The Convolution Theorem 2 Vector Measures and the Riesz Representation Theorem Measures of Bounded Variation Integrals with Respect to Vector Measures Vector Measures as Distributions Convolutions Involving Vector Measures The Riesz Representation Theorem Bibliography List of Symbols Index