Cover
Series
Title page
Copyright page
Preface
Contents
I. FUNCTION SPACES AND ELLIPTIC DIFFERENTIAL EQUATIONS
1. Spaces of Entire Analytic Functions
1.1. Introduction
1.2. Preliminaries
1.2.1. Distributions
1.2.2. $L_p$-Spaces and Quasi-Banach Spaces
1.2.3. Maximal Inequalities
1.2.4. Admissible Borel Measures
1.3. Inequalities of Plancherel-Polya-Nikol'skij Type
1.3.1. A Maximal Inequality
1.3.2. Inequalities for the Lebesgue Measure
1.3.3. Inequalities for Atomic Measures
1.3.4. Inequalities for Admissible Borel Measures
1.3.5. A Representation Formula
1.4. $L_p$-Spaces of Analytic Functions
1.4.1. Definition and Main Inequalities
1.4.2. Basic Properties
1.4.3. Further Properties
1.5. Fourier Multipliers for $L_p$-Spaces
1.5.1. Definition and Criterion
1.5.2. A Multiplier Theorem
1.5.3. Convolution Algebras
1.5.4. Further Multiplier Assertions
1.6. $L_p(l_q)$-Spaees of Analytic Functions
1.6.1. Definition and Basic Properties
1.6.2. Maximal Inequalities
1.6.3. A Multiplier Theorem
1.6.4. Further Multiplier Assertions
2. Function Spaces on $R_n$
2.1. Introduction
2.2. The Historical Background, Motivations, and Principles
2.2.1. On the History of Function Spaces
2.2.2. The Constructive Spaces
2.2.3. The Criterion
2.2.4. Decomposition Method, the Principle
2.2.5. Approximation Procedures
2.3. Definition and Fundamental Properties
2.3.1. Definition
2.3.2. Equivalent Quasi-Norms and Elementary Embeddings
2.3.3. Basic Properties
2.3.4. The Spaces $F^s_{\infty,q}(R_n)$
2.3.5. An Orientation and Some Historical Remarks
2.3.6. Maximal Inequalities
2.3.7. A Fourier Multiplier Theorem
2.3.8. Lifting Property and Related Equivalent Quasi-Forms
2.3.9. Diversity of the Spaces $B^s_{p,q}(R_n)$ and $F^s_{p,q}(R_n)$
2.4. Interpolation
2.4.1. Preliminaries
2.4.2. Real Interpolation for the Spaces $B^s_{p,q}(R_n)$ and $B^s_{p,q}(R_n)$ with Fixed $p$
2.4.3. Real Interpolation for the Spaces $B^s_{p,p}(R_n)$
2.4.4. Complex Interpolation: Definitions
2.4.5. Complex Interpolation: Properties
2.4.6. Some Preparations
2.4.7. Complex Interpolation for the Spaces $B^s_{p,q}(R_n)$ and $F^s_{p,q}(R_n)$
2.4.8. Fourier Multipliers for the Spaces $F^s_{p,q}(R_n)$
2.4.9. The Spaces $L^\Omega_p(R_n,l_q)$: Complex Interpolation and Fourier Multipliers
2.5. Equivalent Quasi-Norms and Representations
2.5.1. An Orientation
2.5.2. Nikol'skij Representations
2.5.3. Characterizations by Approximation
2.5.4. Lizorkin Representations
2.5.5. Discrete Representations and Schauder Bases for $B^s_{p,q}(R_n)$
2.5.6. The Bessel-Potential Spaces $H^s_p(R_n)$ and the Soboiev Spaces $W^m_p(R_n)
2.5.7. The Besov Spaces $A^s_{p,q}(R_n)$ and the Zygmund Spaces $\mathcal{C}^s(R_n)$
2.5.8. The Local Hardy Spaces $h_p(R_n)$, the Space $bmo(R_n)$
2.5.9. Characterizations by Maximal Functions of Differences
2.5.10. Characterizations of the Spaces $F^s_{p,q}(R_n)$ by Differences
2.5.11. Characterizations of the Spaces $F^s_{p,q}(R_n)$ by Ball Means of Differences
2.5.12. Characterizations of the Spaces $B^s_{p,q}(R_n)$ by Differences; the Spaces $A^s_{p,q}(R_n)$ and $\mathcal{C}^s(R_n)$
2.5.13. Fubini Type Theorems
2.6. Fourier Multipliers
2.6.1. Definitions and Preliminaries
2.6.2. The Classes $\mathfrac{M}p$ and $\mathfrak{M}_p^H$
2.6.3. Properties of the Classes $\mathfrak{M}_p$
2.6.4. Properties of the Classes $\mathfrak{M}_p^H$
2.6.5. Convolution Algebras
2.6.6. The Classes $\mathfrak{M}{p,q}$
2.7. Embedding Theorems
2.7.1. Embedding Theorems for Different Metrics
2.7.2. Traces
2.8. Pointwise Multipliers
2.8.1. Definition and Preliminaries
2.8.2. General Multipliers
2.8.3. Multiplication Algebras
2.8.4. The Classes $P_{p,\alpha}(R_n)$
2.8.5. Special Multipliers for $B^s_{p,q}(R_n)$
2.8.6. Two Propositions
2.8.7. Characteristic Functions as Multipliers
2.8.8. Further Multipliers
2.9. Extensions
2.9.1. The Spaces $B^s_{p,q}(R_n^+)$ and $F^s_{p,q}(R_n^+)$
2.9.2. The Case $\min(p,q)>1$
2.9.3. The Case $0 < p \leq q < \infty$ and $n = 1$
2.9.4. The Extension Theorem
2.9.5. The Case $q<p$
2.10. Diffeomorphic Maps
2.10.1. Preliminaries
2.10.2. The Main Theorem
2.11. Dual Spaces
2.11.1. Preliminaries
2.11.2. The Case $1 \leq p < \infty$
2.11.3. The Case $0 < p < 1$
2.12. Further Properties
2.12.1. Characterizations of $F^s_{p,q}(R_n)$ via Lusin and Littlewood-Paley Functions
2.12.2. Characterizations via Gauss-Weierstrass and Cauchy-Poisson Semi-Groups
2.12.3. Characterizations via Spline Functions
3. Function Spaces on Domains
3.1. Preliminaries, Motivations, and Methods
3.1.1. Motivations
3.1.2. The Problem of Inner Descriptions
3.1.3. The Localization Method
3.2. Definitions an«i Basic Properties
3.2.1. $C^\infty$-Domains
3.2.2. Definitions
3.2.3. Quasi-Banach Spaces
3.2.4. Elementary Embedding
3.3. Main Properties
3.3.1. Embedding
3.3.2. Pointwise Multipliers
3.3.3. Traces
3.3.4. Extensions
3.3.5. Equivalent Quasi-Norms
3.3.6. Interpolation
3.4. Further Properties
3.4.1. A Special Multiplication Property
3.4.2. Inner Descriptions, Equivalent Quasi-Norms
3.4.3. The Spaces $\r{B}^s_{p,q}(\Omega)$ and $\r{F}^s_{p,q}(\Omega)$
4. Regular Elliptic Differential Equations
4.1. Definitions aiA Preliminaries
4.1.1. Introduction
4.1.2. Definitions
4.1.3. Basic Properties of Elliptic Operators
4.1.4. Basic Properties of Regular Elliptic Systems
4.2. A Priori Estimates
4.2.1. Introduction and the Spaces $F^{s,r}{p,q}(R_n^+)$
4.2.2. A Priori Estimates, Part I: $R_n^+$, constant coefficients, Dirichlet problem
4.2.3. A Priori Estimates. Part II: $R_n^+$, constant coefficients, general boundary problem
4.2.4. A Priori Estimates. Part III: Bounded domain, variable coefficients, general boundary problem
4.3. Boundary Value Problems
4.3.1. Introduction and Hypothesis
4.3.2. The Basic Theorem
4.3.3. The Main Theorem
4.3.4. Boundary Value Problems in Zygmund Spaces
II. FURTHER TYPES OF FUNCTION SPACES
5. Homogeneous Function Spaces
5.1. Definitions and Basic Properties
5.1.1. Introduction (to Part II)
5.1.2. The Spaces $Z(R_n)$ and $Z'(R_n)$
5.1.3. Definitions
5.1.4. The Spaces $\r{F}^s{\infty,q}(R_n)$
5.1.5. Basic Properties
5.2. Further Properties
5.2.1. Maximal Inequalities
5.2.2. Fourier Multipliers
5.2.3. Equivalent Norms
5.2.4. The Hardy Spaces $H_p(R_n)$, the Spaces $BMO(R_n)$
5.2.5. Miscellaneous Properties
6. Ultra-Distributions and Weighted Spaces of Entire Analytic Functions
6.1. Ultra-Distributions
6.1.1. Introduction (to Chapters 6 and 7)
6.1.2. Definitions
6.1.3. Basic Properties
6.1.4. Paley-Wiener-Schwartz Theorems for Ultra-Distributions
6.2. Inequalities of Plancherel-PoIya-Nikol'skij Type
6.2.1. Admissible Weight Functions
6.2.2. Some Inequalities
6.2.3. The Basic Inequality
6.3. $L_p$-Spaces of Analytic Functions
6.3.1. Definition and Main Inequalities
6.3.2. Basic Properties
7. Weighted Function Spaces on $R_n$
7.1. Maximal Inequalities, Fourier Multipliers and Littlewood-Paley Theorems
7.1.1. The Spaces $L_p^\Omega(R_n, \rho(x), l_q)$
7.1.2. The Spaces $L_p(R_n, \rho(x), l_q)$
7.1.3. The Spaces $L_p(R_n, \rho(x))$
7.2. Weighted Spaces of Type $B^s_{p,q}$ and $F^s_{p,q}$
7.2.1. Definition
7.2.2. Basic Properties
7.2.3. Weighted Sobolev Spaces
7.2.4. Characterizations by Approximation
8. Weighted Function Spaces on Domains and Degenerate Elliptic Differential Equations
8.1. Weighted Function Spaces on Domains
8.1.1. Introduction and Definitions
8.1.2. Basic Properties
8.1.3. Inner Descriptions
8.2. Degenerate Elliptic Differential Equations
8.2.1. Definition and A Priori Estimate
8.2.2. A Mapping Property
9. Periodic Function Spaces
9.1. Introduction and Definitions
9.1.1. Introduction
9.1.2. Periodic Distributions on $R_n$
9.1.3. Function Spaces on $T_n$
9.1.4. Periodic Function Spaces on $R_n$
9.2. Properties
9.2.1. The Main Theorem
9.2.2. Fourier Multipliers and Maximal Inequalities
9.2.3. Periodic Sobolev Spaces
9.2.4. The Problem of Strong Summability
10. Further Types of Function Spaces
10.1. Anisotropic Function Spaces
10.2. Generalizations
10.3. Abstract Spaces and Spaces Related to Orthogonal Expansions
References
Index