Fredholmness and index of simplest weighted 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 Uαf = (α')1/p(f o α) and Uβf = (β')1/p(f o β). For i, j ∈ Z we study the simplest weighted singular integral operators with two shifts Aij = Uα iP+ γ + Uβ jP- γ on Lp(R+), where P± γ = (I ± Sγ)/2 are operators associated to theweighted Cauchy singular integral operator with γ ∈ C satisfying 0 < 1/p + Rγ < 1. We prove that the operator Aij is a Fredholm operator on Lp(R+) and has zero index if where wij (t) = log[αi (β-j (t))/t] and αi, β-j are iterations of α, β. This statement extends an earlier result obtained by the author, Yuri Karlovich, and Amarino Lebre for γ = 0.