Non-Divergence Equations Structured on Hormander Vector Fields

by Bramanti, Marco; Brandolini, Luca; Lanconelli, Ermanno; Uguzzoni, Francesco

• ISBN13: 9780821849033
• ISBN10: 0821849034
• Format: Paperback
• Copyright: 2010-03-30
• Publisher: Amer Mathematical Society

Summary

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

Table of Contents

Introduction

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

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

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

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