Non-divergence equations structured on Hörmander vector fields: heat kernels and Harnack inequalities

Abstract

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 Hormander'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 $\lambda^{-1}\vert\xi\vert^{2}\leq\sum_{i,j=1}^{q}a_{ij}(t,x) \xi_{i}\xi_{j}\leq\lambda\vert\xi\vert^{2}\text{}\forall\xi\in\mathbb{R}^{q}, x \in\Omega,t\in(T_{1},T_{2})$ for a suitable constant $\lambda>0$ a for some real numbers $T_{1} < T_{2}$. Table of Contents: Introduction. 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. (MEMO/204/961)