SOLVABILITY OF A CLASS OF CONVOLUTION INTEGRAL EQUATIONS WITH SINGULAR INTEGRAL–DIFFERENTIAL OPERATORS

In this paper, we consider a class of convolution integral equations with singular integral-differential operators. First, we establish the relation between Fourier analysis theory and Riemann boundary value problems, and investigate the theory of Noether solvability and some properties of Cauchy integral operators. Via using Fourier transform, we convert such equations into complex boundary value problems. By means of the regularity theory of the classical Riemann-Hilbert problems and of the theory of complex analysis, we obtain the conditions of Noether solvability and analytical solutions. In addition, we also study the analytical property of solution near nodes. Thus, this article is significant for the study of developing complex analysis, functional analysis, integral equations and complex boundary value problems, and it also provides theoretical support to quantum field theory and Ising model.