Math. Nachr. zyxwvutsr 166 (1994) 113-133 Singular Integral Equations with zyxw PQC Coefficients and Freely Transformed Argument By A. BOTTCHER of Chemnitz, Yu. I. KARLOVICH’) of Odessa and B. SILBERMANN of Chemnitz (Received August 3, 1992) (Revised Version May 26, 1993) 1. Introduction This paper is concerned with equations of the form Here zyxwvut T is the complex unit circle and zyxwv cp E zyxwvut L2(T) is the unknown function. The given data are: the right-hand sidefE LZ(T), a compact operator K on L20, the coefficients zyx a,, ..., z a, and b,, ..., b,, and a finite collection g,, ..., g , of C1-diffeomorphisms (so-called shifts) of T onto itself. Under fairly weak assumptions on the nature of the shifts, the Fredholm theory of such equations has been elaborated by many people in case the coefficients are piecewise continuous or semi-almost periodic. We refer in this connection to the classical monograph [18] by LITVINCHUK and to the works by SEMENYUTA [22], KHEVELEV [17], MYASNIKOV and SAZONOV [19], [20], KARLOVICH and KRAVCHENKO [13], KARLOVICH [9], [lo], and SOLDATOV [24]. Surveys of this topic are [14], [15], [16]. For quasicontinuous (QC) or piecewise quasicontinuous (PQC) coefficients all we know is a Fredholm theory for the situation in which either no shifts occur (SARASON [21 I) or all the shifts belong to a finite cyclic group of self-maps of T (SILBERMANN [23], BOTTCHER, ROCH, SILBERMANN, SPITKOVSKY [2]); in the latter case the shifts are usually called Carleman shifts. This paper is a first contribution to the case of QC and PQC coefficients and non-Carleman shifts. It is well known that the theory is the more complicated the more massive are the sets of the fixed points of the shifts. Here we first consider equations with QC coefficients and topologically freely acting shifts, after which we study equations with PQC coefficients and shifts that act freely (i.e., have no fixed points at all). I) Partially supported by the German Research Foundation DFG within the Priority Research Programme “Boundary Element Methods”. 8 Math. Nachr., Bd. 166