Index formulae for pseudodifferential operators with discontinuous symbols (Q1357878)

The authors discuss Fredholm property and index formula for pseudodifferential operators with discontinuous symbol on a compact \(n\)-dimensional manifold \(X\). Two cases are considered. First, results are given for symbols with isolated singularities, recollecting and generalizing previous results of \textit{G. V. Rozenblyum} and \textit{B. A. Plamenevskiĭ} [Funct. Anal. Appl. 26, No. 4, 266--275 (1992); translation from Funkts. Anal. Prilozh. 26, No. 4, 45--56 (1992; Zbl 0879.58073)]. Attention is then put on operators with symbols discontinuous along an \(n-1\)-dimensional submanifold \(Y\) of \(X\). In local coordinates \(x=(y,t)\), \(y\in Y\), \(t\in (-1,1)\) and dual variables \(\xi= (\eta, \tau)\); symbols \(a(x,\xi)\) are considered, zero order homogeneous with respect to \(\xi\), such that there exist smooth limits \[ A_\pm (y, \eta, \tau)= \lim_{t\to 0} a(y,\pm t,\eta,\tau). \] Fredholm property and index formula are expressed in terms of the symbol \(a(x,\xi)\), considered in \(X/Y\), and suitable operator-valued pseudodifferential operators on \(Y\), defined in terms of \(A_\pm\).