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

• Introduction

• Overview of Part I
• Global extension of Hörmander's vector fields and geometric properties of the CC-distance
• 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

• 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

• 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 non-smooth case
• Epilogue
• References