Abstract
Let the operator N be defined by\(N_\phi (x): = \frac{1}{{i\pi }}\mathop \smallint \limits_ \circ ^\infty (x + y)^{ - 1} \). It is shown that in the spaces LP(Rü;h) (h(x) = xβo|x+i|β∞; -1<βo<p-1;-1<βo+β∞<p-1; 1<p<∞), N is contained in the closed algebra generated by the one-sided Hilbert transformation. This result is used to simplify the construction of a symbol for singular integral operators with orientation reversing Carleman shift and piecewise continuous coefficients given by Gohberg and Krupnik in [8].
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Costabel, M. A contribution to the theory of singular integral equations with carleman shift. Integr equ oper theory 2, 11–24 (1979). https://doi.org/10.1007/BF01729358
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DOI: https://doi.org/10.1007/BF01729358