Necessary Conditions for Fredholmness of Singular Integral Operators with Shifts and Slowly Oscillating Data

Suppose alpha is an orientation-preserving diffeomorphism (shift) of R+ = (0, alpha) onto itself with the only fixed points 0 and alpha. In Karlovich et al. (Integr Equ Oper Theory 2011, doi:10.1007/s00020-010-1861-0) we found sufficient conditions for the Fredholmness of the singular integral operator with shift (aI - bW(alpha))P++(cI - dW(alpha))P- acting on L-p (R+) with 1 < p < infinity, where P-+/- = (I +/- S)/2, S is the Cauchy singular integral operator, and W(alpha)f = f.alpha is the shift operator, under the assumptions that the coefficients a, b, c, d and the derivative alpha' of the shift are bounded and continuous on R+ and may admit discontinuities of slowly oscillating type at 0 and infinity. Now we prove that those conditions are also necessary.