Memoirs of the American Mathematical Society 2009; 123 pp; softcover Volume: 204 ISBN10: 0821849034 ISBN13: 9780821849033 List Price: US$69 Individual Members: US$41.40 Institutional Members: US$55.20 Order Code: MEMO/204/961
 In this work the authors deal with linear second order partial differential operators of the following type \[H=\partial_{t}L=\partial_{t}\sum_{i,j=1}^{q}a_{ij}(t,x) X_{i}X_{j}\sum_{k=1}^{q}a_{k}(t,x)X_{k}a_{0}(t,x)\] where \(X_{1},X_{2},\ldots,X_{q}\) is a system of real Hörmander's vector fields in some bounded domain \(\Omega\subseteq\mathbb{R}^{n}\), \(A=\left\{ a_{ij}\left( t,x\right) \right\} _{i,j=1}^{q}\) is a real symmetric uniformly positive definite matrix such that \[\lambda^{1}\vert\xi\vert^{2}\leq\sum_{i,j=1}^{q}a_{ij}(t,x) \xi_{i}\xi_{j}\leq\lambda\vert\xi\vert^{2}\forall\xi\in\mathbb{R}^{q}, x \in\Omega,t\in(T_{1},T_{2})\] for a suitable constant \(\lambda>0\) a for some real numbers \(T_{1} < T_{2}\). Table of Contents Part I: Operators with constant coefficients  Overview of Part I
 Global extension of Hörmander's vector fields and geometric properties of the CCdistance
 Global extension of the operator \(H_{A}\) and existence of a fundamental solution
 Uniform Gevray estimates and upper bounds of fundamental solutions for large \(d\left(x,y\right)\)
 Fractional integrals and uniform \(L^{2}\) bounds of fundamental solutions for large \(d\left(x,y\right)\)
 Uniform global upper bounds for fundamental solutions
 Uniform lower bounds for fundamental solutions
 Uniform upper bounds for the derivatives of the fundamental solutions
 Uniform upper bounds on the difference of the fundamental solutions of two operators
Part II: Fundamental solution for operators with Hölder continuous coefficients  Assumptions, main results and overview of Part II
 Fundamental solution for \(H\): the Levi method
 The Cauchy problem
 Lower bounds for fundamental solutions
 Regularity results
Part III: Harnack inequality for operators with Hölder continuous coefficients  Overview of Part III
 Green function for operators with smooth coefficients on regular domains
 Harnack inequality for operators with smooth coefficients
 Harnack inequality in the nonsmooth case
 Epilogue
 References
