Anisotropic estimates for sub-elliptic operators (Q943401)

In the 1970's Folland and Stein investigated a family os subelliptic scalar operators \(\mathcal{L}_\lambda\), arising naturally in the \(\overline{\partial}_b\)-complex. They introduced weighted Sobolev spaces as the natural spaces to deal with this complex, and then they obtained sharp estimates for \(\overline{\partial}_b\) in these spaces using integral kernels and approximate inverse. In the 1990's Rumin introduced a differential complex for compact contact manifolds showing that Folland-Stein operators play a central role in the analysis of the corresponding Laplacian and studied the estimates on the Laplacian using Folland-Stein analysis. In this paper the authors give a self-contained proof of sharp estimates in Folland-Stein spaces for the operators studied by Rumin; the essential tools in the proof are integration by parts and a modified bootstrapping method.