# H\"ormander spaces on manifolds, and their application to elliptic boundary-value problems

## TL;DR

This paper develops an extended Sobolev scale of H"ormander spaces on manifolds with boundary, which are used to analyze elliptic boundary-value problems and establish their Fredholm properties and regularity of solutions.

## Contribution

It introduces a new scale of H"ormander spaces on manifolds that are interpolation spaces independent of local charts, and applies this to elliptic boundary-value problems.

## Key findings

- Established Fredholm property for elliptic boundary-value problems in the new scale.
- Derived conditions for solutions to belong to spaces of continuously differentiable functions.
- Extended the functional framework for elliptic problems on manifolds.

## Abstract

We introduce an extended Sobolev scale on a smooth compact manifold with boundary. The scale is formed by inner-product H\"ormander spaces for which an RO-varying radial function serves as a regularity index. These spaces do not depend on a choice of local charts on the manifold. The scale consists of all Hilbert spaces that are interpolation ones for pairs of inner-product Sobolev spaces, is obtained by the interpolation with a function parameter of these pairs, and is closed with respect to this interpolation. As an application of the scale introduced, we give a theorem on the Fredholm property of a general elliptic boundary-value problem on appropriate H\"ormander spaces and find sufficient conditions under which its generalized solutions belong to the space of $p\geq0$ times continuously differential functions.