Interior estimates for divergence degenerate elliptic systems in Carnot groups

Let X 1 , … , X q be the basis of the space of horizontal vector fields on a homogeneous Carnot group G = ( R n , ∘ ) ( q < n ) . We consider the following divergence degenerate elliptic system ∑ β = 1 N ∑ i , j = 1 q X i ( a α β i j ( x ) X j u β ) = ∑ i = 1 q X i f α i , α = 1 , 2 , … , N where the coefficients a α β i j are real valued bounded measurable functions defined in Ω ⊂ G , satisfying the strong Legendre condition and belonging to the space V M O l o c ( Ω ) (defined by the Carnot–Carathéodory distance induced by the X i ’s). We prove interior H W 1 , p estimates ( 2 ≤ p < ∞ ) for weak solutions to the system.