SOBOLEV SPACES ON LIE MANIFOLDS AND POLYHEDRAL DOMAINS

BERND AMMANN ALEXANDRU IONESCU AND VICTOR NISTOR

Abstract. We study Sobolev spaces on Lie manifolds, which we define as a class of manifolds described by vector fields. The class of Lie manifolds includes the Euclidean spaces R^{n}, asymptotically flat manifolds, conformally compact manifolds, and manifolds with cylindrical ends. As in the classical case of R^{n}, we define Sobolev spaces using derivatives, powers of the Laplacian,or a suitable class of partitions of unity, and discuss their properties on Lie manifolds. The results include the definition of the trace map, a characterization of its range, the extension theorem, the density of smooth functions, and interpolation properties.