带两个Carleman位移的奇异积分方程的可解性问题

ON THE PROBLEM OF SOLVABILITY OF SINGULAR INTEGRAL EQUATIONS WITH TWO CARLEMAN'S SHIFTS

带Carleman位移的奇异积分方程理论近年来得到了很大发展。在[1],[2],[3]中完整地建立了这种奇异积分方程的Noether理论。所用的基本方法是建立所谓的对应方程组(是不带位移的奇异积分方程组,它的理论是已知的,参看[5],[6])。本文目的是利用类似方法解决带两个位移的奇异积分方程的可解性问题。

In this paper the problem of solvability of singular integral equations with two Carleman's shifts (eqution(1.1)) is considered using the method of corresponding system of equations. Suppose that Г is a closed simple Lyapunoff's curve and α(t), β(t), which both satisfy Carleman's canditions and α[β(t)]=β[α(t)], are two different homeomorphisms of Г onto itself. Coefficients a_k(t), b_k(t), k=0,1,2,3 and g(t) of equation (1.1) all belong to the space H_4μ(Г). The following main results are obtained. 1. Singular integral equation(1.1) is solvable, if the Noether's canditions def(P(t)±q(t))0 are satisfied. 2. The index of the singular integral equation(1.1) will be calculated by the formula Ind=1/8π{arg (det(p(t)-q(t)))/(det(p(t)+q(t)))}_Г where p(t) and q(t) are matrices of coefficents of the corresponding system of equations.