Fundamental solutions and local solvability for nonsmooth Hörmander's operators
Author(s)
Bibliographic Information
Fundamental solutions and local solvability for nonsmooth Hörmander's operators
（Memoirs of the American Mathematical Society, no. 1182）
American Mathematical Society, 2017
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Note
"Volume 249, number 1182 (third of 8 numbers), September 2017"
Bibliography: p. 77-79
Description and Table of Contents
Description
The authors consider operators of the form $L=\sum_{i=1}^{n}X_{i}^{2}+X_{0}$ in a bounded domain of $\mathbb{R}^{p}$ where $X_{0},X_{1},\ldots,X_{n}$ 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.
Table of Contents
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.
by "Nielsen BookData"
