Non-Divergence Equations Structured on Hormander Vector Fields

Introduction Part I: Operators with constant coefficients: Overview of Part I Global extension of Hormander'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 Part II: Fundamental solution for operators with Holder 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 Holder 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 non-smooth case Epilogue References