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

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.