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

Abstract

Let {X1,X2,…,Xm} be the basis of space of horizontal vector fields in a Carnot group G=(Rn;∘) (m<n). We prove high order Fefferman-Phong type inequalities in G. As applications, we derive a priori Lp(G) estimates for the nondivergence degenerate elliptic operators L=-∑i,j=1maij(x)XiXj+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.