Embedding theorems into Lipschitz and BMO spaces and applications to quasilinear subelliptic differential equations

Abstract

This paper proves Harnack's inequality for solutions to a class of quasilinear subelliptic differential equations.The proof relies on various embedding theorems into nonisotropic Lipschitz and BMO spaces associated with the vector fields X 1 , . . ., Xm satisfying Hrmander's condition.The nonlinear subelliptic equations under study include the important p-sub-Laplacian equation, e.g.,and A is a constant; B, C and D can be in appropriate function spaces.We note that A can be nonzero.Recently, Franchi, Wheeden and the author showed in [FLW] that a Poincar inequality holds when p = 1 and q = Q Q-1 (when p = q = 1, the result was due to Jerison [Jer]).We mention that this endpoint result for p = 1 contains certain important geometric information.Indeed, applying this Poincar inequality, we also derived a relative isoperimetric