Douglis--Nirenberg elliptic systems in H\"ormander spaces

TL;DR

This paper studies Douglis--Nirenberg elliptic systems within H"ormander spaces, establishing a priori estimates, interior regularity, and conditions for the Fredholm property, advancing understanding of these systems in advanced functional frameworks.

Contribution

It introduces analysis of elliptic systems in H"ormander spaces with a radial parameter, providing new a priori estimates and criteria for Fredholm properties.

Findings

Established a priori estimates for solutions.

Analyzed interior regularity of solutions.

Provided a sufficient condition for the Fredholm property.

Abstract

We investigate Douglis--Nirenberg uniformly elliptic systems in $\mathbb{R}^{n}$ on a class of H\"ormander inner product spaces. They are parametrized with a radial function parameter which is RO-varying at $+\infty$, considered as a function of $(1+|\xi|^{2})^{1/2}$ with $\xi\in\mathbb{R}^{n}$. An a'priori estimate for solutions is proved, and their interior regularity is studied. A sufficient condition for the systems to have the Fredholm property is given.