●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
2.4 Singular Integrals on Function Spaces
2.4.1 Singular Integrals on the Hardy Space H1
2.4.2 Singular Integrals on Besov-Lipschitz Spaces
2.4.3 Singular Integrals on HP（Rn)
2.4.4 A Singular Integral Characterization of H1(Rn)
Exercises
3 BMO and Carleson Measures
3.1 Functions of Bounded Mean Oscillation
3.1.1 Definition and Basic Properties of BMO
3.1.2 The John-Nirenberg Theorem
3.1.3 Consequences of Theorem 3
Exercises
3.2 Duality between H1 and BMO
Exercises
3.3 Nontangential Maximal Functions and Carleson Measures
3.3.1 Definition and Basic Properties of Carleson Measures
3.3.2 BMO Functions and Carleson Measures
Exercises
3.4 The Sharp Maximal Function
3.4.1 Definition and Basic Properties of the Sharp Maximal Function
3.4.2 A Good Lambda Estimate for the Sharp Function
3.4.3 Interpolation Using BMO
3.4.4 Estimates for Singular Integrals Involving the Sharp Function
Exercises
3.5 Commutators of Singular Integrals with BMO Functions
3.5.1 An Orlicz-Type Maximal Function
3.5.2 A Pointwise Estimate for the Commutator
3.5.3 LP Boundedness of the Commutator
Exercises
4 Singular Integrals of Nonconvolution Type
4.1 General Background and the Role of BMO
4.1.1 Standard Kernels
4.1.2 Operators Associated with Standard Kernels
4.1.3 Calderon-Zygmund Operators Acting on Bounded Functions
Exercises
4.2 Consequences of L2Boundedness
4.2.1 mWeak Type(1,1) and LP Boundedness of Singular Integrals、
4.2.2 Boundedness of Maximal Singular Integrals
4.2.3 H1→L1 and L →BMO Boundedness of Singular Integrals
Exercises
4.3 The T(1) Theorem
4.3.1 Preliminaries and Statement of the Theorem
4.3.2 The Proof of Theorem 4
4.3.3 An Application
Exercises
4.4 Paraproducts
4.4.1 Introduction to Paraproducts
4.4.2 L2 Boundedness of Paraproducts
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