Fredholm and regularity theory of Douglis–Nirenberg elliptic systems on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^{N}}$$\end{document}

We give a fairly complete exposition of the Fredholm properties of the Douglis–Nirenberg elliptic systems on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^{N}}$$\end{document} in the classical (unweighted) Lp Sobolev spaces and under “minimal” assumptions about the coefficients. These assumptions rule out the use of classical pseudodifferential operator theory, although it is indirectly of assistance in places. After generalizing a necessary and sufficient condition for Fredholmness, already known in special cases, various invariance properties are established (index, null space, etc.), with respect to p and the Douglis–Nirenberg numbers. Among other things, this requires getting around the problem that the Lp spaces are not ordered by inclusion. In turn, with some work, invariance leads to a regularity theory more general than what can be obtained by the method of differential quotients.