Regularity theory of Fourier integral operators with complex phases and singularities of affine fibrations

Title Page......Page 3
Copyright Page......Page 4
Contents......Page 5
Introduction......Page 7
1 Fourier integral operators......Page 11
1.1 Fourier integral operators with real phases......Page 12
1.2 Fourier integral operators with complex phases......Page 15
1.3 Spaces of functions......Page 18
1.4 Overview of the regularity theory for the real phase......Page 21
1.5.1 Smooth factorization condition......Page 23
1.5.2 Parametric fibrations......Page 27
1.5.3 Operators, commuting with translations......Page 30
1.6 Fourier integral operators with complex phase functions......Page 31
1.7 Estimates for operators with complex phases......Page 33
1.8 A relation between real and complex phases......Page 36
1.9 Proofs......Page 37
1.10 Estimates in other spaces......Page 45
1.11.1 Essentially homogeneous case......Page 47
1.11.2 A representation formula for continuous operators of small negative orders......Page 49
1.12 Smooth factorization condition for complex phases......Page 50
2.1 Introduction......Page 59
2.2.1 Affine fibrations setting......Page 60
2.2.2 Jacobian affine fibrations setting......Page 62
2.3.2 Jacobian affine fibration setting......Page 64
2.4 Main results......Page 65
2.5 Methods of complex analytic geometry......Page 66
2.6 Some properties of affine parametric fibrations......Page 70
2.7 General properties of affine fibrations......Page 71
2.8 Affine fibrations of Jacobian type......Page 80
2.9 Limiting properties of the fibers......Page 88
3.1 Fibrations in local coordinates......Page 91
3.2 Fibrations of gradient type......Page 95
3.3 Reconstruction of the phase function......Page 98
3.4 Existence of singular fibrations of gradient type......Page 104
4.1 Complexification......Page 107
4.2 Estimates for analytic Fourier integral operators......Page 109
5.1 Introduction......Page 113
5.2 Cauchy problem for a class of strictly hyperbolic equations......Page 116
5.3 Monge-Ampere equation......Page 120
5.4 Cauchy problems with complex characteristic roots......Page 121
5.5 A parametrix for a first order complex partial differential operator......Page 126
Bibliography......Page 129
Summary......Page 135