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.

Elliptic problems with nonclassical coundary conditions in an extended Sobolev scale

“…In the case where ϕ varies regularly at infinity [32,8], Theorems 7.1-7.5 are proved in [37]. If the vector bundles are trivial, Theorems 7.1, 7.2, 7.5 and 7.6 are covered by [36] (see also [17] and [20,Section 2.4.3], where Theorem 7.1 is established in the p = 1 case).…”

“…We apply these results to the study of elliptic pseudodifferential operators on this scale. Note that the refined Sobolev scale for vector bundles was introduced and investigated by Zinchenko [37].…”

Elliptic Problems with Boundary Conditions of Higher Orders in Hörmander Spaces

“…Using this interpolation method, Mikhailets and Murach extended all basic theorems on properties of regular elliptic problems from the Sobolev spaces over the indicated Hörmander spaces. This theory also contains theorems on solvability of elliptic systems on manifolds in Hörmander spaces [48,76]. It is set forth in monograph [46] and surveys [43,44].…”