Emergence, persistence, and disappearance of logarithmic spirals in the spectra of singular integral operators

This paper is devoted to the surprising metamorphosis of the spectrum of the Cauchy singular integral over composed Carleson curves in L(p) spaces with Muckenhoupt weights. In the absence of a weight, the spectrum may contain certain circular arcs if the curve is sufficiently nice, these circular arcs metamorphose into logarithmic double spirals for more complicated curves, and in the case of a general composed Carleson curve these double spirals may blow up to heavy sets whose boundaries are nevertheless comprised of pieces of logarithmic spirals. Muckenhoupt weights may further thicken the spectrum: until some point the weights are unable to destroy the circular arcs and logarithmic spirals in the spectrum, but beyond this point some kind of interference between the curve and the weight results in a complete disappearance of spirality ... Here we present two approaches to the problem of understanding the emergence, persistence, and disappearance of logarithmic spirals in this context: Mellin convolution techniques on the one hand and Wiener-Hopf factorization methods on the other. Our focus is on pointing out the critical stages in the spectral metamorphosis for the Cauchy singular integral, on comparing the two approaches mentioned, and on their implications for the symbol calculus of the closed algebra of singular integral operators with piecewise continuous coefficients.