Contents
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 \I/#DOs 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
\IHDOS
65
4.1. Almost homogeneous approach to the Heisenberg calculus 65
4.2. Holomorphic families of \I//DOs 67
4.3. Composition of holomorphic families of \I/#DOs 69
4.4. Kernel characterization of holomorphic families of \IDOs 73
4.5. Holomorphic families of \IDOs on a general Heisenberg manifold 76
4.6. Transposes and adjoints of holomorphic families of
SS?HDOS
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
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