On the real line we consider singular integral operators with a linear Carleman shift and complex conjugation, acting in
, the real space of all Lebesgue measurable complex value functions on ℝ with
= 2 power. We show that the original singular integral operator with shift and conjugation is, after extension, equivalent to a singular integral operator without shift and with a 4 × 4 matrix coefficients. By exploiting the properties of the factorization of the symbol of this last operator, it is possible to describe the solution of a generalized Riemann boundary value problem with a Carleman shift.
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