The index of weighted singular integral operators with shifts and slowly oscillating data

Let α and β be orientation-preserving diffeomorphism (shifts) of R+=(0,∞) onto itself with the only fixed points 0 and ∞. We establish a Fredholm criterion and calculate the index of the weighted singular integral operator with shifts(aI−bUα)Pγ++(cI−dUβ)Pγ−, acting on the space Lp(R+), where Pγ±=(I±Sγ)/2 are the operators associated to the weighted Cauchy singular integral operator Sγ given by(Sγf)(t)=1πi∫R+(tτ)γf(τ)τ−tdτ with γ∈C satisfying 0<1/p+ℜγ<1, and Uα,Uβ are the isometric shift operators given byUαf=(α′)1/p(f∘α),Uβf=(β′)1/p(f∘β), under the assumptions that the coefficients a,b,c,d and the derivatives α′,β′ of the shifts are bounded and continuous on R+ and admit discontinuities of slowly oscillating type at 0 and ∞.