Introduction

1

3

  1.1. Symbols and oscillatory integrals

3

    1.1.1. Fourier transform and Sobolev spaces

3

    1.1.2. Symbol spaces

7

    1.1.3. Elements of the standard calculus

13

    1.1.4. Parameter-dependent operators

19

  1.2. Calculus with operator-valued symbols

22

    1.2.1. Pseudo-differential operators

22

    1.2.2. Examples

26

    1.2.3. Remarks on classical Sobolev spaces

29

2. Conormal asymptotics and meromorphic operator functions

35

  2.1. Operators of Fuchs type and the Mellin transform

35

    2.1.1. Mellin transform and associated Sobolev spaces

35

    2.1.2. Solutions with asymptotics

42

    2.1.3. A factorisation of meromorphic functions

49

    2.1.4. Quasi-homogeneity

53

  2.2. Pseudo-differential operator-valued meromorphic functions

61

   2.2.1. Kernel cut-off and Mellin quantisation

61

    2.2.2. Spaces of meromorphic operator functions

70

    2.2.3. Mellin pseudo-differential operators and Green operators with asymptotics

73

    2.2.4. The cone algebra

77

    2.2.5. Further remarks on Fuchs type operators

88

3. Edge pseudo-differential calculus

91

  3.1. Edge-degenerate operators

91

    3.1.1. Manifolds with edges

91

    3.1.2. The typical differential operators

92

    3.1.3. Weighted wedge Sobolev spaces

96

    3.1.4. Discrete edge asymptotics

101

  3.2. The edge algebra

103

    3.2.1. Edge-degenerate symbols

103

    3.2.2. Operator-valued edge symbols

105

    3.2.3. Edge pseudo-differential operators

115

Bibliography

121

List of Symbols

131

Index

133