Non-divergence equations structured on Hörmander vector fields : heat kernels and Harnack inequalities
著者
書誌事項
Non-divergence equations structured on Hörmander vector fields : heat kernels and Harnack inequalities
（Memoirs of the American Mathematical Society, no. 961）
American Mathematical Society, c2009
大学図書館所蔵 全11件
注記
"Volume 204, number 961 (end of volume)."
Includes bibliographical references (p. 121-123)
内容説明・目次
内容説明
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)
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