A Fefferman-Phong Type Inequality and Applications to Quasilinear Subelliptic Equations

We establish a nonlocal generalization of a well-known inequality by C. Fefferman and D. H. Phong (equation presented) for u ∈ C^∞₀(ℝⁿ) and V belonging to the Morrey space M^s,2s_n with 1 < s ≤ n/2, when the gradient in the right-hand side is replaced by the energy associated to an arbitrary system of Lipschitz continuous vector fields. Accordingly, the multiplier V is taken in an appropriate Morrey space defined using the Carnot-Carathéodory metric generated by the vector fields. As an application, we prove the Harnack inequality and the Hölder continuity of solutions for a wide class of second order quasilinear subelliptic equations.