Anisotropic estimates for sub-elliptic operators

In the 1970’s,Folland and Stein studied a family of subelliptic scalar operators Lwhich arise naturally in the（?）-complex.They introduced weighted Sobolev spaces as the natural spaces for this complex,and then obtained sharp estimates for（?）b in these spaces using integral kernels and approximate inverses.In the 1990’s,Rumin introduced a differential complex for compact contact manifolds,showed that the Folland-Stein operators are central to the analysis for the corresponding Laplace operator,and derived the necessary estimates for the Laplacian from the Folland Stein analysis. In this paper,we give a self-contained derivation of sharp estimates in the anisotropic Folland-Stein spaces for the operators studied by Rumin using integration by parts and a modified approach to bootstrapping.