Anisotropic estimates for sub-elliptic operators

Abstract

In the 1970’s, Folland and Stein studied a family of subelliptic scalar operators $\mathcal{L}_\lambda$ which arise naturally in the $\bar \partial _b$-complex. They introduced weighted Sobolev spaces as the natural spaces for this complex, and then obtained sharp estimates for $\bar \partial _b$ 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.