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Non-Divergence Equations Structured on Hörmander Vector Fields: Heat Kernels and Harnack Inequalities
Marco Bramanti, Politecnico di Milano, Italy, Luca Brandolini, Università di Bergamo, Bologna, Italy, and Ermanno Lanconelli and Francesco Uguzzoni, Università di Bologna, Italy
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Memoirs of the American Mathematical Society
2009; 123 pp; softcover
Volume: 204
ISBN-10: 0-8218-4903-4
ISBN-13: 978-0-8218-4903-3
List Price: US$73 Individual Members: US$43.80
Institutional Members: US\$58.40
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 $$T_{1} < T_{2}$$.

• Introduction
Part I: Operators with constant coefficients
• 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
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 non-smooth case
• Epilogue
• References