Non-Divergence Equations Structured on Hörmander Vector Fields : Heat Kernels and Harnack Inequalities.

Intro -- Contents -- Abstract -- Introduction -- Object and main results of the paper -- Previous results and bibliographic remarks -- Strategy and structure of the paper -- A motivation -- Part I: Operators with constant coefficients -- 1. Overview of Part I -- 2. Global extension of Hörmander's vector fields and geometric properties of the CC-distance -- 2.1. Some global geometric properties of CC-distances -- 2.2. Global extension of Hörmander's vector fields -- 3. Global extension of the operator HA and existence of a fundamental solution -- 4. Uniform Gevray estimates and upper bounds of fundamental solutions for large d( x,y) -- 5. Fractional integrals and uniform L2 bounds of fundamental solutions for large d( x,y) -- 6. Uniform global upper bounds for fundamental solutions -- 6.1. Preliminaries on homogeneous groups and the "lifting and approximation" technique -- Homogeneous groups -- Homogeneous Lie algebras -- Sublaplacians and homogeneous fundamental solutions -- General Hörmander's vector fields: "lifting and approximation" -- Parabolic context -- 6.2. Upper bounds on fundamental solutions -- 7. Uniform lower bounds for fundamental solutions -- 8. Uniform upper bounds for the derivatives of the fundamental solutions -- 9. Uniform upper bounds on the difference of the fundamental solutions of two operators -- Part II: Fundamental solution for operators with Hölder continuous coefficients -- 10. Assumptions, main results and overview of Part II -- Assumptions on the vector fields -- Function spaces -- Assumptions on the coefficients -- Main results -- Overview of Part II and relations with Part I -- Some auxiliary estimates -- Plan of Part II -- 11. Fundamental solution for H: the Levi method -- 12. The Cauchy problem -- 13. Lower bounds for fundamental solutions -- 14. Regularity results.

Part III: Harnack inequality for operators with Hölder continuous coefficients -- 15. Overview of Part III -- Plan of Part III -- 16. Green function for operators with smooth coefficients on regular domains -- 17. Harnack inequality for operators with smooth coefficients -- 18. Harnack inequality in the non-smooth case -- Epilogue -- 19. Applications to operators which are defined only locally -- 20. Further developments and open problems -- Fundamental solution for the stationary operator -- General Hörmander's vector fields defined on the whole Rn -- Time-dependent and rough vector fields -- References.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.