Elliptic boundary problems and the Boutet de Monvel calculus in Besov and Triebel--Lizorkin spaces

TL;DR

This paper extends the Boutet de Monvel calculus to Besov and Triebel--Lizorkin spaces, establishing continuity and Fredholm properties for boundary operators with improved results over previous work.

Contribution

It generalizes the calculus to broader function spaces and refines the understanding of elliptic boundary operators' properties in these contexts.

Findings

Extended the calculus to Besov and Triebel--Lizorkin spaces.

Proved continuity and Fredholm properties for boundary operators.

Improved results on the range complements of elliptic Green operators.

Abstract

The Boutet de Monvel calculus of pseudo-differential boundary operators is generalised to the full scales of Besov and Triebel--Lizorkin spaces (though with finite integral exponents for the latter). The continuity and Fredholm properties proved here extend those previously obtained by Franke and Grubb, and the results on range complements of surjectively elliptic Green operators improve the earlier known, even for the classical spaces with $1<p<\infty$. The symbol classes treated are the uniformly estimated ones. Some precisions are given for the general definitions of trace and singular Green operators of class 0.