Journal of Lie Theory

Let $G$ be a homogeneous group and let $X_0$, $X_1$ , $X_2,\dots,X_{p_0}$ be left invariant real vector fields on $G$ satisfying H\"{o}rmander's rank condition. Assume that $X_1$, $X_2,\dots,X_{p_0}$ are homogeneous of degree one and $X_0$ is homogeneous of degree two. In this paper, we study the following equation with drift: $$ Lu\equiv\sum_{i,j=1}^{p_0}X_i(a_{ij}(x) X_ju) +a_0X_0u=\sum_{j=1}^{p_0}X_jF_j(x)\ , $$ 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 $a_0\in \mathbb{R}\backslash\{0\}$. Moreover, the coefficients $a_{ij}$ belong to the class $VMO$ (Vanishing Mean Oscillation) with respect to the subelliptic metric induced by the vector fields $X_0$, $X_1$, $X_2,\dots,X_{p_0}$. We derive local $L^p$ estimates for second order derivatives and H\"{o}lder estimates by establishing the representation formulas and higher order integrability of weak solutions to the above equation.