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→L1and 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 L' Boundedness of Paraproducts 4.4.3 Fundamental Properties of Paraproducts Exercises 4.5 An Almost Orthogonality Lemma and Applications 4.5.1 The Cotlar-Knapp-Stein Almost Orthogonality Lemma 4.5.2 An Application 4.5.3 Almost Orthogonality and the T (1) Theorem 4.5.4 Pseudodifferential Operators Exercises 4.6 The Cauchy Integral of Calderon and the T (b) Theorem 4.6.1 Introduction of the Cauchy Integral Operator along a Lipschitz Curve 4.6.2 Resolution of the Cauchy Integral and Reduction of Its L2Boundedness to a Quadratic Estimate 4.6.3 A Quadratic T (1) Type Theorem
4.6.4 A T (b) Theorem and the L2 Boundedness of the Cauchy Integral Exercises 4.7 Square Roots of Elliptic Operators 4.7.1 Preliminaries and Statement of the Main Result 4.7.2 Estimates for Elliptic Operators on Rn 4.7.3 Reduction to a Quadratic Estimate 4.7.4 Reduction to a Carleson Measure Estimate 4.7.5 The T (b) Argument 4.7.6 Proof of Lemma 4 Exercises 5 Boundedness and Convergence of Fourier Integrals 5.1 The Multiplier Problem for the Ball 5.1.1 Sprouting of Triangles 5.1.2 The counterexample Exercises 5.2 Bochner-Riesz Means and the Carleson-Sjlin Theorem 5.2.1 The Bochner-Riesz Kernel and Simple Estimates 5.2.2 The Carleson-Sjolin Theorem 5.2.3 The Kakeya Maximal Function 5.2.4 Boundedness of a Square Function 5.2.5 The Proof of Lemma 5 Exercises 5.3 Kakeya Maximal Operators 5.3.1 Maximal Functions Associated with a Set of Directions 5.3.2 The Boundedness of tzy on LP(R2) 5.3.3 The Higher-Dimensional Kakeya Maximal Operator Exercises 5.4 Fourier Transform Restriction and Bochner-Riesz Means 5.4.1 Necessary Conditions for Rp→g(Sn-1) to Hold 5.4.2 A Restriction Theorem for the Fourier Transfornm 5.4.3 Applications to Bochner-Riesz Multipliers 5.4.4 The Full Restriction Theorem on R2 Exercises 5.5 Almost Everywhere Convergence of Bochner-Riesz Means 5.5.1 A Counterexample for the Maximal Bochner-Riesz Operator 5.5.2 Almost Everywhere Summability of the Bochner-Riesz Means 5.5.3 Estimates for Radial Multipliers Exercises 6 Time-Frequency Analysis and the Carleson-Hunt Theorem 6.1 Almost Everywhere Convergence of Fourier Integrals 6.1.1 Preliminaries 6.1.2 Discretization of the Carleson Operator 6.1.3 Linearization of a Maximal Dyadic Sum 6.1.4 Iterative Selection of Sets of Tiles with Large Mass and Energy 6.1.5 Proof of the Mass Lemma 6 6.1.6 Proof of Energy Lemma 6 6.1.7Proof of the Basic Estimate Lemma 6.1.10 Exercises 6.2 Distributional Estimates for the Carleson Operator 6.2.1 The Main Theorem and Preliminary Reductions
6.2.2 The Proof of Estimate(6.2.18) 6.2.3 The Proof of Estimate(6.2.19) 6.2.4 The Proof of Lemma 6 Exercises 6.3 The Maximal Carleson Operator and Weighted Estimates Exercises 7 Multilinear Harmonic Analysis 7.1 Multilinear Operators 7.1.1 Examples and initial results 7.1.2 Kernels and Duality of m-linear Operators 7.1.3 Multilinear Convolution Operators with Nonnegative Kernels Exercises 7.2 Multilinear Interpolation 7.2.1 Real Interpolation for Multilinear Operators 7.2.2 Proof of Theorem 7 7.2.3 Proofs of Lemmas 7.2.6 and 7 7.2.4 Multilinear Complex Interpolation 7.2.5 Multilinear Interpolation between Adjoint Operators Exercises 7.3 Vector-valued Estimates and Multilinear Convolution Operators 7.3.1 Multilinear Vector-valued Inequalities 7.3.2 Multilinear Convolution and Multiplier Operators 7.3.3 Regularizations of Multilinear Symbols and Consequences 7.3.4 Duality of Multilinear Multiplier Operators Exercises 7.4 Calderon-Zygmund Operators of Several Functions 7.4.1 Multilinear Calderon-Zygmund Theorem 7.4.2 A Necessary and Sufficient Condition for the Boundedness of Multilinear Calder6n-Zygmund Operators Exercises 7.5 Multilinear Multiplier Theorems 7.5.1 Some Preliminary Facts 7.5.2 Coifman-Meyer Method 7.5.3 Hormander-Mihlin Multiplier Condition 7.5.4 Proof of Main Result Exercises 7.6 An Application Concerning the Leibniz Rule of Fractional Differentiation 7.6.1 Preliminary Lemma 7.6.2 Proof of Theorem 7 Exercises A The Schur Lemma A.1 The Classical Schur Lemma A.2 Schur's Lemma for Positive Operators A.3 An Example A.4 Historical Remarks B Smoothness and Vanishing Moments B.1 The Case of No Cancellation B.2 One Function has Cancellation B.3 One Function has Cancellation:An Example B.4 Both Functions have Cancellation: An Example B.5 The Case of Three Factors with No Cancellation
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