Abstract
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 L p(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.
Mathematics Subject Classification (2010). Primary 45E05, 47A53; Secondary 30E25, 47G10, 47G30.
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Karlovich, Y.I. (2017). The Haseman Boundary Value Problem with Slowly Oscillating Coefficients and Shifts. In: Bini, D., Ehrhardt, T., Karlovich, A., Spitkovsky, I. (eds) Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics. Operator Theory: Advances and Applications, vol 259. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-49182-0_19
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DOI: https://doi.org/10.1007/978-3-319-49182-0_19
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Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-49180-6
Online ISBN: 978-3-319-49182-0
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