Mellin pseudodifferential operators with slowly varying symbols and singular integrals on Carleson curves with Muckenhoupt weights

We show that by using Mellin pseudodifferential operators whose double symbol depends analytically on the co-variable we can rather quickly arrive at descriptions of the local spectra of the Cauchy singular integral operator over a large class of Carleson curves with Muckenhoupt weights. The approach of this paper extends some recent results of the spectral theory of singular integral operators to the case of piecewise slowly varying coefficients and also yields new and surprising interpretations of these results.