On combinations of the circle shifts and some one-dimensional integral operator

The diffeomorphism ζ=ζ(eis) of the unit circle and the operator Ψφ(t)=1/πi∫Γ[ζ′(τ)\ζ(τ)-ζ(t)-1\τ-t]φ(τ)dτ are under consideration. The main results can be stated as follows: If ζ(t)∈C1,α(Γ), 00,β(Γ), 0μ(Γ) for μμ(Γ)≤const∥φ(t)∥C0,β(Γ), where the constant depends on ∥ζ∥C1,α(Γ) only. If μ=1, then Ψφ(t)∈Cμ-ε(Γ) for all 01, then Ψφ(t)∈C1,μ-1(Γ), and ∥Ψφ(t)∥C1,μ-1(Γ)≤const∥φ(t)∥C0,β(Γ), where the constant depends on ∥ζ∥C1,α(Γ) only. If ζ(t)∈C1,α(Γ), 01,β(Γ), 01,α(Γ), and ∥Ψφ(t)∥C1,α(Γ)≤const∥φ(t)∥C0,1(Γ)≤const∥φ(t)∥C1,β(Γ), where the constant depends on ∥ζ∥C1,α(Γ) only. The index α in the left-hand side of the last inequality can not be improved. The appropriate example is given.