Chapter 1. Introduction ......................................... 1
1.1. Heisenberg manifolds and their main differential
operators .................................................. 2
1.2. Intrinsic approach to the Heisenberg calculus .............. 3
1.3. Holomorphic families of ΨHDOs .............................. 8
1.4. Heat equation and complex powers of hypoelliptic
operators .................................................. 9
1.5. Spectral asymptotics for hypoelliptic operators ........... 13
1.6. Weyl asymptotics and CR geometry .......................... 14
1.7. Weyl asymptotics and contact geometry ..................... 15
1.8. Organization of the memoir ................................ 15
Chapter 2. Heisenberg manifolds and their main differential
operators ................................................. 17
2.1. Heisenberg manifolds ...................................... 17
2.2. Main differential operators on Heisenberg manifolds ....... 22
Chapter 3. Intrinsic Approach to the Heisenberg Calculus ....... 29
3.1. Heisenberg calculus ....................................... 29
3.2. Principal symbol and model operators ...................... 37
3.3. Hypoellipticity and Rockland condition .................... 42
3.4. Invertibility criteria for sublaplacians .................. 52
3.5. Invertibility criteria for the main differential
operators ................................................. 56
Chapter 4. Holomorphic families of ΨHDOs ....................... 65
4.1. Almost homogeneous approach to the Heisenberg calculus .... 65
4.2. Holomorphic families of ΨHDOs ............................. 67
4.3. Composition of holomorphic families of ΨHDOs .............. 69
4.4. Kernel characterization of holomorphic families of ΨDOs ... 73
4.5. Holomorphic families of ΨDOs on a general Heisenberg
manifold .................................................. 76
4.6. Transposes and adjoints of holomorphic families of
ΨHDOs ..................................................... 78
Chapter 5. Heat Equation and Complex Powers of Hypoelliptic
Operators ................................................. 81
5.1. Pseudodifferential representation of the heat kernel ...... 81
5.2. Heat equation and sublaplacians ........................... 87
5.3. Complex powers of hypoelliptic differential operators ..... 93
5.4. Rockland condition and the heat equation .................. 97
5.5. Weighted Sobolev Spaces .................................. 102
Chapter 6. Spectral Asymptotics for Hypoelliptic Operators .... 107
6.1. Spectral asymptotics for hypoelliptic operators .......... 107
6.2. Weyl asymptotics and CR geometry ......................... 111
6.3. Weyl asymptotics and contact geometry .................... 119
Appendix A. Proof of Proposition 3.1.18 ....................... 123
Appendix B. Proof of Proposition 3.1.21 ....................... 127
Appendix. Bibliography ........................................ 131
References .................................................. 131
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