Elliptic operators on refined Sobolev scales on vector bundles

Abstract

We introduce a refined Sobolev scale on a vector bundle over a closed infinitely smooth manifold. This scale consists of inner product Hörmander spaces parametrized with a real number and a function varying slowly at infinity in the sense of Karamata. We prove that these spaces are obtained by the interpolation with a function parameter between inner product Sobolev spaces. An arbitrary classical elliptic pseudodifferential operator acting between vector bundles of the same rank is investigated on this scale. We prove that this operator is bounded and Fredholm on pairs of appropriate Hörmander spaces. We also prove that the solutions to the corresponding elliptic equation satisfy a certain a priori estimate on these spaces. The local regularity of these solutions is investigated on the refined Sobolev scale. We find new sufficient conditions for the solutions to have continuous derivatives of a given order.

arXiv comment22 pages

References in corpus (8)

• Interpolation with a function parameter and refined scale of spaces
• Interpolation Hilbert spaces between Sobolev spaces
• Parabolic problems and interpolation with a function parameter
• Spectral gaps of the one-dimensional Schrödinger operators with singular periodic potentials
• Parameter-elliptic problems and interpolation with a function parameter
• Parameter-elliptic operators on the extended Sobolev scale
• Elliptic boundary-value problems in Hörmander spaces
• Hill's potentials in Hörmander spaces and their spectral gaps