From the introduction: Let be vector fields in satisfying Hörmander’s condition for hypoellipticity: rank Lie at every point . Denote by the formal adjoint of . The linear operator is the subelliptic Laplacian associated to the vector fields .
Given an open set , and a function , denote by the subelliptic gradient of . For we consider the functional
and define to be the completion of in the norm generated by . The Euler equation of is
We call the operator in (1) the subelliptic -Laplacian. Critical points of are (weak) solutions of (1), and vice-versa.
In this paper, we propose to study a general class of nonlinear subelliptic equations, whose prototype is constituted by (1) above. Our objectives are: a) To establish an optimal embedding result of Sobolev type for the subelliptic spaces ; b) To prove a Harnack type inequality for nonnegative solutions. From b) the Hölder continuity of solutions with respect to the -control distance will follow.