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 nat...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.展开更多
基金This work was supported by NSERC(Grant No.RGPIN/9319-2005)
文摘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.
