Fundamental Solutions and Local Solvability for Nonsmooth Hormander's Operators

ISBN-13: 9781470425593 / Angielski

Marco Bramanti;Luca Brandolini;Maria Manfredini

Opis

The authors consider operators of the form $L=sum_{i=1}^X_^+X_$ in a bounded domain of $mathbb^$ where $X_,X_,ldots,X_$ are nonsmooth Hormander's vector fields of step $r$ such that the highest order commutators are only Holder continuous. Applying Levi's parametrix method the authors construct a local fundamental solution $gamma$ for $L$ and provide growth estimates for $gamma$ and its first derivatives with respect to the vector fields. Requiring the existence of one more derivative of the coefficients the authors prove that $gamma$ also possesses second derivatives, and they deduce the local solvability of $L$, constructing, by means of $gamma$, a solution to $Lu=f$ with Holder continuous $f$. The authors also prove $C_{X,loc}^{2,alpha}$ estimates on this solution.

Spis treści

• Introduction
• Some known results about nonsmooth Hormander's vector fields
• Geometric estimates
• The parametrix method
• Further regularity of the fundamental solution and local solvability of $L$
• Appendix. Examples of nonsmooth Hormander's operators satisfying assumptions A or B
• Bibliography.