Marco Bramanti 
Fundamental Solutions and Local Solvability for Nonsmooth Hoermander’s Operators [PDF ebook] 

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 Hoermander’s vector fields of step $r$ such that the highest order commutators are only Hoelder 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 Hoelder continuous $f$. The authors also prove $C_{X, loc}^{2, /alpha}$ estimates on this solution.