Interior a priori estimates for supersolutions of fully nonlinear subelliptic equations under geometric conditions

Abstract

In this note, we prove interior a priori first- and second-order estimates for solutions of fully nonlinear degenerate elliptic inequalities structured over the vector fields of Carnot groups, under the main assumption that $u$ is semiconvex along the fields. These estimates for supersolutions are new even for linear subelliptic inequalities in nondivergence form, whereas in the nonlinear setting they do not require neither convexity nor concavity on the second derivatives. We complement the analysis exhibiting an explicit example showing that horizontal $W^{2,q}$ regularity of Calderón-Zygmund type for fully nonlinear subelliptic equations posed on the Heisenberg group cannot be in general expected in the range $q<Q$, $Q$ being the homogeneous dimension of the group.

arXiv commentThis is the accepted version of the following article: Bull. London Math. Soc. 2024 56; 1385-1398, which has been published in final form at https://doi.org/10.1112/blms.13001