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

Abstract EN

Let { X 1 , X 2 , … , X m } be the basis of space of horizontal vector fields in a Carnot group G = ( R n ; ∘ ) ( m < n ) .

We prove high order Fefferman-Phong type inequalities in G .

As applications, we derive a priori L p ( G ) estimates for the nondivergence degenerate elliptic operators L = - ∑ i , j = 1 m a i j ( x ) X i X j + V ( x ) with V M O 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.