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

Suppose alpha is an orientation preserving diffeomorphism (shift) of R+ = (0, infinity) onto itself with the only fixed points 0 and a. We establish 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 0 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.