High Order Fefferman-Phong Type Inequalities in Carnot Groups and Regularity for Degenerate Elliptic Operators plus a Potential

Abstract

Let $\{X_1,X_2,\dots ,X_m\}$ be the basis of space of horizontal vector fields in a Carnot group $\mathbb{G}=({\mathbb{R}}^n;\circ )$ . We prove high order Fefferman-Phong type inequalities in $\mathbb{G}$. As applications, we derive a priori $L^p(\mathbb{G})$ estimates for the nondivergence degenerate elliptic operators $L=-{\sum }_{i,j=1}^m{a}_{ij}(x)X_i{X}_j+V(x)$ with $VMO$ coefficients and a potential $V$ belonging to an appropriate Stummel type class introduced in this paper. Some of our results are also new even for the usual Euclidean space.