On a Weighted Singular Integral Operator with Shifts and Slowly Oscillating Data

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Let α,β be orientation-preserving diffeomorphism (shifts) of R+=(0,∞) onto itself with the only fixed points 0 and ∞ and Uα,Uβ be the isometric shift operators on Lᵖ(R+) given by Uαf=(α')¹/ᵖ(f∘α), Uβf=(β')¹/ᵖ(f∘β), and P₂±=(I± S₂)/2 where (S₂ f)(t):=1/πi∫₀∞ (tτ)¹/²⁻¹/ᵖf(τ)/τ-tdτ, t+, is the weighted Cauchy singular integral operator. We prove that if α',β' and c,d are continuous on R+ and slowly oscillating at 0 and ∞, and ₜ→ ₛ|c(t)|<1, ₜ→ ₛ|d(t)|<1, s∈{0,∞}, then the operator (I-cUα)P₂++(I-dUβ)P₂- is Fredholm on Lᵖ(R+) and its index is equal to zero. Moreover, its regularizers are described.