GENERALIZED WEIGHTED SOBOLEV–MORREY ESTIMATES FOR HYPOELLIPTIC OPERATORS WITH DRIFT ON HOMOGENEOUS GROUPS

Let G = ( RN,◦,δλ ) be a homogeneous group, Q be the homogeneous dimension of G, X0,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 X0 is homogeneous of degree two with respect to the family of dilations ( δλ )λ>0. Consider the following hypoelliptic operator with drift on G (Formula Presented), where (aij) is a constant matrix satisfying the elliptic condition in Rm and a0 ≠ 0. In this paper, for this class of operators we obtain generalized weighted 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 weighted Morrey spaces and proving interpolation results in generalized weighted Sobolev-Morrey spaces on G © 2022. Journal of Mathematical Inequalities.All Rights Reserved.