Modern Fourier analysis

Preface
Acknowledgments
Contents
1 Smoothness and Function Spaces
	1.1 Smooth Functions and Tempered Distributions
		1.1.1 Space of Tempered Distributions Modulo Polynomials
		1.1.2 Calderón 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 SobolevSpaces
		1.3.2 Littlewood–Paley Characterization of InhomogeneousSobolev Spaces
		1.3.3 Littlewood–Paley Characterization of HomogeneousSobolev Spaces
	Exercises
	1.4 Lipschitz Spaces
		1.4.1 Introduction to Lipschitz Spaces
		1.4.2 Littlewood–Paley Characterization of HomogeneousLipschitz Spaces
		1.4.3 Littlewood–Paley Characterization of InhomogeneousLipschitz 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–LizorkinSpaces
		2.3.1 Embeddings and Completeness of Triebel–LizorkinSpaces
		2.3.2 The Space of Triebel–Lizorkin Sequences
		2.3.3 The Smooth Atomic Decomposition of HomogeneousTriebel–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 3.1.6
	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 MaximalFunction
		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 SharpFunction
	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 Calderón–Zygmund Operators Actingon Bounded Functions
	Exercises
	4.2 Consequences of L2 Boundedness
		4.2.1 Weak Type (1,1) and Lp Boundednessof Singular Integrals
		4.2.2 Boundedness of Maximal Singular Integrals
		4.2.3 H1→L1 and L∞→BMO Boundedness of SingularIntegrals
	Exercises
	4.3 The T(1) Theorem
		4.3.1 Preliminaries and Statement of the Theorem
		4.3.2 The Proof of Theorem 4.3.3
		4.3.3 An Application
	Exercises
	4.4 Paraproducts
		4.4.1 Introduction to Paraproducts
		4.4.2 L2 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 Calderón and the T(b) Theorem
		4.6.1 Introduction of the Cauchy Integral Operator alonga Lipschitz Curve
		4.6.2 Resolution of the Cauchy Integral and Reductionof Its L2 Boundedness 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 4.7.9
	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–Sjölin Theorem
		5.2.1 The Bochner–Riesz Kernel and Simple Estimates
		5.2.2 The Carleson–Sjölin Theorem
		5.2.3 The Kakeya Maximal Function
		5.2.4 Boundedness of a Square Function
		5.2.5 The Proof of Lemma 5.2.5
	Exercises
	5.3 Kakeya Maximal Operators
		5.3.1 Maximal Functions Associated with a Set of Directions
		5.3.2 The Boundedness of MΣN 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→q(Sn-1) to Hold
		5.4.2 A Restriction Theorem for the Fourier Transform
		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–RieszOperator
		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 LargeMass and Energy
		6.1.5 Proof of the Mass Lemma 6.1.8
		6.1.6 Proof of Energy Lemma 6.1.9
		6.1.7 Proof 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 6.2.2
	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 NonnegativeKernels
	Exercises
	7.2 Multilinear Interpolation
		7.2.1 Real Interpolation for Multilinear Operators
		7.2.2 Proof of Theorem 7.2.2
		7.2.3 Proofs of Lemmas 7.2.6 and 7.2.7
		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 Symbolsand Consequences
		7.3.4 Duality of Multilinear Multiplier Operators
	Exercises
	7.4 Calderón-Zygmund Operators of Several Functions
		7.4.1 Multilinear Calderón–Zygmund Theorem
		7.4.2 A Necessary and Sufficient Condition for the Boundednessof Multilinear Calderón–Zygmund Operators
	Exercises
	7.5 Multilinear Multiplier Theorems
		7.5.1 Some Preliminary Facts
		7.5.2 Coifman-Meyer Method
		7.5.3 Hörmander-Mihlin Multiplier Condition
		7.5.4 Proof of Main Result
	Exercises
	7.6 An Application Concerning the Leibniz Rule of FractionalDifferentiation
		7.6.1 Preliminary Lemma
		7.6.2 Proof of Theorem 7.6.1
	Exercises
Appendix 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
Appendix 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
Glossary
References
Index