Generalized Sobolev–Morrey estimates for hypoelliptic operators on homogeneous groups

Let G= (RN, ∘ , δλ) be a homogeneous group, Q is the homogeneous dimension of G, X, X1, … , Xm be left invariant real vector fields on G and satisfy Hörmander’s rank condition on RN. Assume that X1, … , Xm(m≤ N- 1) are homogeneous of degree one and X is homogeneous of degree two with respect to the family of dilations (δλ) λ>. Consider the following hypoelliptic operator with drift on GL=∑i,j=1maijXiXj+a0X0,where (aij) is a m× m constant matrix satisfying the elliptic condition in Rm and a≠ 0. In this paper, for this class of operators, we obtain the generalized Sobolev–Morrey estimates by establishing boundedness of a large class of sublinear operators Tα, α∈ [0 , Q) generated by Calderón–Zygmund operators (α= 0) and generated by fractional integral operator (α> 0) on generalized Morrey spaces and proving interpolation results on generalized Sobolev–Morrey spaces on G. The sublinear operators under consideration contain integral operators of harmonic analysis such as Hardy–Littlewood and fractional maximal operators, Calderón–Zygmund operators, fractional integral operators on homogeneous groups, etc. © 2021, The Royal Academy of Sciences, Madrid.