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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$69
Individual Members: US$41.40
Institutional Members: US$55.20
Order Code: MEMO/204/961
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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
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
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