Extended Sobolev scale for vector bundles, and its applications

Abstract: We study an extended Sobolev scale for smooth vector bundles over a smooth closed manifold. This scale is built on the base of inner product distribution spaces of generalized smoothness given by an arbitrary positive function OR-varying at infinity. We show that this scale is obtained by the quadratic interpolation (with a function parameter) between inner product Sobolev spaces, is closed with respect to the quadratic interpolation, and consists of all Hilbert spaces that are interpolation spaces between inner product Sobolev spaces. Embedding theorems and a duality theorem are proved for this scale. We give applications of the extended Sobolev scale to mixed-order (Douglis--Nirenberg) elliptic pseudodifferential operators acting between vector bundles of the same rank. We prove their Fredholm property on appropriate pairs of spaces on the scale, give a sufficient and necessary condition for the local generalized smoothness of solutions to a mixed-order elliptic system and provide a corresponding a priori estimate of the solutions. We also give a sufficient condition for a chosen component of the solution to be $q$ times continuously differentiable on a subset of the manifold.