Local Sobolev–Morrey estimates for nondivergence operators with drift on homogeneous groups

Abstract

Let $G$ be a homogeneous group, $X_0,X_1,X_2,\ldots ,X_{p_0}$ be left invariant real vector fields on $G$ and satisfy Hörmander’s rank condition. Assume that $X_1,X_2,\ldots ,X_{p_0}$ are homogeneous of degree one and $X_0$ is homogeneous of degree two. In this paper, we study the following operator with drift:

$$\begin{aligned} L=\sum _{i,j=1}^{p_0} a_{ij}(x)X_i X_j+a_0(x)X_0, \end{aligned}$$

where $a_{ij}(x)$ are real valued, bounded measurable functions defined in $G$, satisfying the uniform ellipticity condition and $a_0(x)$ is bounded away from zero in $G$. Moreover, we assume that the coefficients $a_{ij},\ a_0$ belong to the space ${ VMO}(G)$ (vanishing mean oscillation) with respect to the subelliptic metric induced by the vector fields $X_0,X_1,X_2\ldots ,X_{p_0}$. For this class of operators, we obtain the local Sobolev–Morrey estimates by establishing the boundedness on Morrey spaces for singular integrals under homogeneous type spaces and proving interpolation results on Sobolev–Morrey spaces.