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0．Background 1

0.1．Fourier Transform 1

0.2．Basic Real Variable Theory 9

0.3．Fractional Integration and Sobolev Embedding Theorems 22

0.4．Wave Front Sets and the Cotangent Bundle 28

0.5．Oscillatory Integrals 36

Notes 39

1．Stationary Phase 40

1.1．Stationary Phase Estimates 40

1.2．Fourier Transform of Surface-carried Measures 49

Notes 54

2．Non-homogeneous Oscillatory Integral Operators 55

2.1．Non-degenerate Oscillatory Integral Operators 56

2.2．Oscillatory Integral Operators Related to the Restriction Theorem 58

2.3．Riesz Means in Rn 65

2.4．Kakeya Maximal Functions and Maximal Riesz Means in R2 71

Notes 92

3．Pseudo-differential Operators 93

3.1．Some Basics 93

3.2．Equivalence of Phase Functions 100

3.3．Self-adjoint Elliptic Pseudo-differential Operators on Compact Manifolds 106

Notes 112

4．The Half-wave Operator and Functions of Pseudo-differential Operators 113

4.1．The Half-wave Operator 114

4.2．The Sharp Weyl Formula 124

4.3．Smooth Functions of Pseudo-differential Operators 131

Notes 133

5．Lp Estimates of Eigenfunctions 135

5.1．The Discrete L2 Restriction Theorem 136

5.2．Estimates for Riesz Means 149

5.3．More General Multiplier Theorems 153

Notes 158

6．Fourier Integral Operators 160

6.1．Lagrangian Distributions 161

6.2．Regularity Properties 168

6.3．Spherical Maximal Theorems:Take 1 186

Notes 193

7．Local Smoothing of Fourier Integral Operators 194

7.1．Local Smoothing in Two Dimensions and Variable Coefficient Kakeya Maximal Theorems 195

7.2．Local Smoothing in Higher Dimensions 214

7.3．Spherical Maximal Theorems Revisited 224

Notes 227

Appendix:Lagrangian Subspaces of T*IRn 228

Bibliography 230

Index 237

Index of Notation 238