Fredholmness and index of simplest singular integral operators with two slowly oscillating shifts

Let α and β be orientation-preserving diffeomorphisms (shifts) of R+ = (0,∞) onto itself with the only fixed points 0 and ∞, where the derivatives α and β may have discontinuities of slowly oscillating type at 0 and ∞. For p ∈ (1,∞) , we consider the weighted shift operators Uα and Uβ given on the Lebesgue space Lp(R+) by Uα f = (α)1/p( f ◦α) and Uβ f = (β )1/p( f ◦β ) . We apply the theory of Mellin pseudodifferential operators with symbols of limited smoothness to study the simplest singular integral operators with two shifts Ai j =UiαP++UjβP- on the space Lp(R+), where P± =(I±S)/2 are operators associated to the Cauchy singular integral operator S, and i, j ∈ Z. We prove that all Ai j are Fredholm operatorson Lp(R+) and have zero indices.