Interior HW^{1,p} estimates for divergence degenerate elliptic systems in Carnot groups

TL;DR

This paper establishes interior Hardy–W^{1,p} estimates for solutions to divergence-form degenerate elliptic systems structured on horizontal vector fields in Carnot groups, advancing regularity theory in sub-Riemannian geometry.

Contribution

It proves the first interior Hardy–W^{1,p} estimates for degenerate elliptic systems on Carnot groups with VMO coefficients, extending regularity results to sub-Riemannian settings.

Findings

Proved interior HW^{1,p} estimates for 2<p<ty.

Extended regularity theory to degenerate elliptic systems in Carnot groups.

Handled coefficients in VMO_{loc} with strong Legendre condition.

Abstract

Let X_1,...,X_q be the basis of the space of horizontal vector fields on a homogeneous Carnot group in R^n (q<n). We consider a degenerate elliptic system of N equations, in divergence form, structured on these vector fields, where the coefficients a_{ab}^{ij} (i,j=1,2,...,q, a,b=1,2,...,N) are real valued bounded measurable functions defined in a bounded domain A of R^n, satisfying the strong Legendre condition and belonging to the space VMO_{loc}(A) (defined by the Carnot-Caratheodory distance induced by the X_i's). We prove interior HW^{1,p} estimates (2<p<\infty) for weak solutions to the system.