The Haseman Boundary Value Problem with Slowly Oscillating Coefficients and Shifts

The paper is devoted to studying the Haseman boundary value problem Φ+ ∘ α = GΦ− + g on a star-like Carleson curve Γ composed by logarithmic spirals in the setting of Lebesgue spaces, where Φ± are angular boundary values of an unknown analytic function Φ on Γ, G and g are given functions, and α is an orientation-preserving homeomorphism of Γ onto itself. This problem is reduced to the equivalent singular integral operator with a shift T = V _{α +} + GP₋ on a Lebesgue space Lp(Γ), where the operators P_± = 2−1(I±S_Γ) are related to the Cauchy singular integral operator S_Γ, and the shift operator V _α is given by V _α f = f ∘ α. Applying the theory of Mellin pseudodifferential operators with non-regular symbols of limited smoothness and essentially decreasing the smoothness of the shift α, we establish a Fredholm criterion and an index formula for the operator T provided that the shift derivative α’ and the coefficient G are slowly oscillating functions on Γ.