Fundamental solutions and local solvability for nonsmooth Hormander's operators
Luca Brandolini, Marco Pedroni · IRIS UNIMORE (University of Modena and Reggio Emilia) · 2013
We 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 \textit{nonsmooth} H\"{o}rmander's vector fields of step $r$ such that the highest order commutators are only H\"{o}lder continuous. Applying Levi's parametrix method we 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 we prove that $\gamma$ also possesses second derivatives, and we deduce the local solvability of $L$, constructing, by means of $\gamma $, a solution to $Lu=f$ with H\"{o}lder continuous $f$. We also prove $C_{X,loc}^{2,\alpha}$ estimates on this solution.