Fundamental solutions and local solvability for nonsmooth H\"ormander's operators

TL;DR

This paper constructs local fundamental solutions for nonsmooth H"ormander's operators with H"older continuous coefficients, establishing local solvability and regularity results under minimal smoothness assumptions.

Contribution

It develops Levi's parametrix method for nonsmooth vector fields, proving existence of fundamental solutions and local solvability with regularity estimates.

Findings

Constructed local fundamental solutions for nonsmooth H"ormander operators.

Proved local solvability of the operator with H"older continuous right-hand side.

Established regularity estimates for solutions in H"older spaces.

Abstract

We consider operators of the form $L=\sum_{i=1}^{n}X_{i}^{2}+X_{0}$ in a bounded domain of R^p where X_0, X_1,...,X_n are nonsmooth H\"ormander's vector fields of step r such that the highest order commutators are only H\"older 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\"older continuous f. We also prove $C_{X,loc}^{2,\alpha}$ estimates on this solution.