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

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