Integr. Equ. Oper. Theory 70 (2011), 63–99 DOI 10.1007/s00020-011-1871-6 Published online March 15, 2011 c  Springer Basel AG 2011 Integral Equations and Operator Theory On the Solvability of Singular Integral Equations with Reflection on the Unit Circle L. P. Castro and E. M. Rojas Dedicated to the memory of Professor Di´ omedes B´ arcenas Abstract. The solvability of a class of singular integral equations with reflection in weighted Lebesgue spaces is analyzed, and the correspond- ing solutions are obtained. The main techniques are based on the con- sideration of certain complementary projections and operator identities. Therefore, the equations under study are associated with systems of pure singular integral equations. These systems will be then analyzed by means of a corresponding Riemann boundary value problem. As a consequence of such a procedure, the solutions of the initial equations are constructed from the solutions of Riemann boundary value problems. In the final part of the paper, the method is also applied to singular inte- gral equations with the so-called commutative and anti-commutative weighted Carleman shifts. Mathematics Subject Classification (2010). Primary 45E05; Secondary 30E20, 30E25, 45E10, 47A68, 47G10. Keywords. Singular integral equation, reflection, shift, solvability, Riemann boundary value problem. 1. Introduction The formulation of linear boundary values problems (BVP’s) for analytic functions has a very long history which is usually considered to start with B. Riemann’s work [26, 27]. It is also clear that D. Hilbert [14], C. Haseman (1907), T. Carleman [4], N.I. Muskhelishvili, F.D. Gakhov, I.N. Vekua (and their students) carried out significant research on problems of this type. His- torically, the paper by Haseman [13] was the first work in which the boundary This work was supported in part by Center for R&D in Mathematics and Applications, University of Aveiro, Portugal, through FCT—Portuguese Foundation for Science and Technology. .