Elliptic problems and H\"ormander spaces

TL;DR

This paper surveys modern results on elliptic problems within the refined framework of H"ormander function spaces, highlighting their properties, local smoothness, and generalizations of classical theorems.

Contribution

It introduces a refined Hilbert scale of H"ormander spaces that extends Sobolev spaces, preserving Fredholm properties and enabling new generalizations of elliptic theory.

Findings

Fredholm property holds in H"ormander spaces

Refined scale is closed under interpolation

Generalizations of Lions-Magenes theorems provided

Abstract

The paper gives a survey of the modern results on elliptic problems on the H\"ormander function spaces. More precisely, elliptic problems are studied on a Hilbert scale of the isotropic H\"ormander spaces parametrized by a real number and a function slowly varying at $+\infty$ in the Karamata sense. This refined scale is finer than the Sobolev scale and is closed with respect to the interpolation with a function parameter. The Fredholm property of elliptic operators and elliptic boundary-value problems is preserved for this scale. A local refined smoothness of the elliptic problem solution is studied. An abstract construction of classes of function spaces in which the elliptic problem is a Fredholm one is found. In particular, some generalizations of the Lions-Magenes theorems are given.