1 Smoothness and Function Spaces
1.1 Smooth Functions and Tempered Distributions
1.1.1 Space of Tempered Distributions Modulo Polynomials
1.1.2 Calder6n Reproducing Formula
Exercises
1.2 Laplacian, Riesz Potentials, and Bessel Potentials
1.2.1 Riesz Potentials
1.2.2 Bessel Potentials
Exercises
1.3 Sobolev Spaces
1.3.1 Definition and Basic Properties of General Sobolev
Spaces
1.3.2 Littlewood-Paley Characterization of Inhomogeneous
Sobolev Spaces
1.3.3 Littlewood-Paley Characterization of Homogeneous
Sobolev Spaces
Exercises
1.4 Lipschitz Spaces
1.4.1 Introduction to Lipschitz Spaces
1.4.2 Littlewood-Paley Characterization of Homogeneous
Lipschitz Spaces
1.4.3 Littlewood-Paley Characterization of Inhomogeneous
Lipschitz Spaces
Exercises
2 Hardy Spaces, Besov Spaces, and Triebel-Lizorkin Spaces
2.1 Hardy Spaces
2.1.1 Definition of Hardy Spaces
2.1.2 Quasi-norm Equivalence of Several Maximal Functions..
2.1.3 Consequences of the Characterizations of Hardy Spaces..
2.1.4 Vector-Valued Hp and Its Characterizations
2.1.5 Singular Integrals on vector-valued Hardy Spaces
Exercises
2.2 Function Spaces and the Square Function Characterization of Hardy
Spaces
2.2.1 Introduction to Function Spaces
2.2.2 Properties of Functions with Compactly Supported Fourier
Transforms
2.2.3 Equivalence of Function Space Norms
2.2.4 The Littlewood-Paley Characterization of Hardy Spaces ..
Exercises
2.3 Atomic Decomposition of Homogeneous Triebel-Lizorkin
Spaces
2.3.1 Embeddings and Completeness of Triebel-Lizorkin Spaces
2.3.2 The Space of Triebel-Lizorkin Sequences
2.3.3 The Smooth Atomic Decomposition of Homogeneous
Triebel-Lizorkin Spaces
2.3.4 The Nonsmooth Atomic Decomposition of Homogeneous
Triebel-Lizorkin Spaces
2.3.5 Atomic Decomposition of Hardy Spaces
Exercises