Weighted Sobolev–Morrey estimates for hypoelliptic operators with drift on homogeneous groups

Abstract

Let X0,X1,…,Xq be left invariant real vector fields on the homogeneous group G and assume that X1,…,Xq are homogeneous of degree one and X0 is homogeneous of degree two. In this paper we consider the following hypoelliptic operator with driftL=∑i,j=1qaijXiXj+a0X0 where (aij) is a constant matrix satisfying the elliptic condition in Rq and a0≠0, and obtain weighted Sobolev–Morrey estimates and Morrey estimates with two weights of L by establishing boundedness of several singular integrals and interpolation lemmas on G.