Interior regularity for degenerate elliptic equations with drift on homogeneous groups (Q2856389)

Let \(G\) be a homogeneous group and let \(X_0, X_1, X_2, \dots, X_{p_0} (p_0<N)\) be left invariant real vector fields on \(G\) satisfying Hörmander's rank condition. Assume that \(X_0, X_1, X_2, \dots, X_{p_0}\) are homogeneous of degree one and \(X_0\) is homogeneous of degree two. The following equation with drift is studied: \(Lu\equiv \sum_{i,j=1}^{p_0} X_i(a_{ij}(x)X_ju)+ a_0X_0u=\sum_{j=1}^{p_0}X_jF_j(x)\) where \(a_{ij}(x)\) are real valued, bounded measurable functions defined in a domain \(\Omega\subset G, a_{ij}(x)=a_{ji}(x),\) satisfying the uniform ellipticity condition in \({\mathbb R}^{p_0}\) and \(a_0\in {\mathbb R}\setminus \{0\}.\) Moreover, the coefficients \(a_{ij}\) belong to the class VMO (vanishing mean oscillation) with respect to the subelliptic metric induced by the vector fields \(X_0, X_1, X_2, \dots, X_{p_0}\). Local \(L^p\) estimates are obtained for second order derivatives and Hölder estimates by establishing the representation formulas and higher order integrability of weak solutions to the above equation.