Global Sobolev-Morrey estimates for hypoelliptic operators with drift on homogeneous groups

Let G be a homogeneous group and X0,X1,X2,...,Xp0 be left invariant real vector fields on G satisfying the H¨ormander’s rank condition. Assume that X1,X2,...,Xp0 are homogeneous of degree one and X0 is homogeneous of degree two. In this paper, we study the following hypoelliptic operator with drift: L =∑ p0 i,j=1 aijXiXj + a0X0, where (aij) is a constant matric satisfying the uniform ellipticity condition and a0 is a constant away from zero, and obtain the global Sobolev-Morrey estimates on G by establishing the Morrey boundedness of the singular integrals on homogeneous spaces and interpolation inequalities depending on vector fields.