Symm's log kernel integral operators (Q1381909)

The author considers the integral equation with the logarithmic kernel \[ \pi^{-1} \int_{-1}^1 f(s) \ln | t-s| ds = g(t),\quad t \in (-1,1) \tag{1} \] which is called Symm's log integral equation in the paper. He describes the range of this operator in the space \(L_{p,\rho} (-1,1)\) with the power weight \(\rho =(1-x)^\alpha (1+x)^\beta\) where \(1<p<\infty\) and \(\alpha ,\beta \in (-1,1)\). The application to the solvability of the singular integral equation is also given. Rewiever's remark. It should be noted that the equation (1) is very well known in the theory of integral equations as the Carleman integral equation. \textit{T. Carleman} was the first who considered this equation [Zur Theorie der linearen Integralgleichungen, Math. Z. 9, 196-217 (1921; JFM 48.1249.01), pp. 196-197]. Equation (1) and more general (for example) \[ A(t)\int_a^t f(s) ds + B(t)\int_t^b f(s) ds + C(t) \int_a^b f(s) \ln | t-s| ds = g(t), \quad t \in (a,b) \] were widely investigated by \textit{S. G. Samko} [see: Metody Otobrazhenij, Groznyj 1976, 41-69 (1976; Zbl 0449.45002); Mathematical analysis and its applications, Work Collect., Rostov 1978, 103-121 (1978; Zbl 0449.45003)]. The first of the mentioned papers contains many references (more than 75) to previous investigations on the topic. Further investigation (including multidimensional case) may be found, for example, in the papers of \textit{A. A. Kilbas}, [Complex analysis and applications, Proc. Int. Conf., Varna/Bulg. 1981, 537-548 (1984; Zbl 0582.45010)] and \textit{S. I. Vasilets} [Dokl. Akad. Nauk BSSR 33, No. 9, 777-780 (1989; Zbl 0681.45002)]. See also \textit{A. A. Kilbas, M. Saigo} and \textit{S. I. Vasilets} [Solution of an integral equation of the first kind with a logarithmic kernel and internal coefficients, Fukuoka Univ. Sci. Reports, Vol. 25, No. 2, 69-87 (1995)] and references in this paper.