Introduction .................................................... 1
Part I: Operators with constant coefficients .................. 7
1 Overview of Part I ........................................... 7
2 Global extension of Hörmander's vector fields and geometric
properties of the CC-distance ................................ 9
2.1 Some global geometric properties of CC-distances ....... 10
2.2 Global extension of Hörmander's vector fields .......... 13
3 Global extension of the operator HA and existence of
a fundamental solution ...................................... 15
4 Uniform Gevray estimates and upper bounds of fundamental
solutions for large d(x,y) .................................. 18
5 Fractional integrals and uniform L2 bounds of fundamental
solutions for large d(x,y) .................................. 25
6 Uniform global upper bounds for fundamental solutions ....... 30
6.1 Homogeneous groups ..................................... 31
6.2. Upper bounds on fundamental solutions .................. 37
7 Uniform lower bounds for fundamental solutions .............. 54
8 Uniform upper bounds for the derivatives of the
fundamental solutions ....................................... 57
9 Uniform upper bounds on the difference of the fundamental
solutions of two operators .................................. 60
Part II: Fundamental solution for operators with Hölder
continuous coefficients .............................. 67
10 Assumptions, main results and overview of Part II ........... 67
11 Fundamental solution for H: the Levi method ................. 74
12 The Cauchy problem .......................................... 86
13 Lower bounds for fundamental solutions ...................... 89
14 Regularity results .......................................... 93
Part III: Harnack inequality for operators with Hölder
continuous coefficients .............................. 99
15 Overview of Part III ........................................ 99
16 Green function for operators with smooth coefficients on
regular domains ............................................ 101
17 Harnack inequality for operators with smooth
coefficients ............................................... 108
18 Harnack inequality in the non-smooth case .................. 111
Epilogue ..................................................... 115
19 Applications to operators which are defined only locally ... 115
20 Further developments and open problems ..................... 117
References .................................................... 121
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