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

摘要

Suppose α is an orientation-preserving diffeomorphism (shift) of ?_+ = (0, ∞) onto itself with the only fixed points 0 and ∞. 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_α)P_+ + (cI - dW_α)P_-acting on L~p (?_+ with 1 < p < ∞, where P_±/2,S is the Cauchy singular integral operator, and W_αf = f ? α is the shift operator, under the assumptions that the coefficients a, b, c, d and the derivative α′ of the shift are bounded and continuous on ?_+ and may admit discontinuities of slowly oscillating type at 0 and ∞. Now we prove that those conditions are also necessary.