A method for excision of singularity for bisingular integral operators with continuous coefficients (Q1122101)

A complete bisingular integral operator A over Lyapunov contours \(\Gamma_ 1\), \(\Gamma_ 2\) in the complex plane is considered. Some conditions on the coefficients are assumed to justify the boundedness of A in \(L_ p(\Gamma_ 1\times \Gamma_ 2)\), \(1<p<\infty\). After excision of two singularities on these curves in the formula for A one obtains an integral operator \(A_{\alpha,\beta}\) with the bounded kernels. This family of operators \(A_{\alpha,\beta}\) tends to A in the strong operator topology as \(\alpha\) \(\to 0\), \(\beta\) \(\to 0.\) Assume that the considered operator A has an inverse. The author gives certain necessary and sufficient condition for the validity of the following assertion: There exist \(\alpha_ 0\), \(\beta_ 0>0\) such that the operators \(A_{\alpha,\beta}\) have inverse operators if \(0<\alpha <\alpha_ 0\), \(0<\beta <\beta_ 0\), and the family \(A^{- 1}_{\alpha,\beta}\) tends to \(A^{-1}\) in the strong operator topology as \(\alpha\) \(\to 0\), \(\beta\) \(\to 0\).