Fredholm property of composite two-dimensional integral operators with homogeneous singular-type kernels in L P space

The authors have previously studied two-dimensional Fredholm integral operators with homogeneous kernels of fiber-singular type. For this class of operators, the symbolic calculus is built using the theory of bilocal operators by V. Pilidi, and Fredholm criterion is formulated through the inversibility of two families: the family of one-dimensional convolution operators, and the family of one-dimensional singular integral operators with continuous coefficients. The aim of this work is to study composite two-dimensional integral operators with homogeneous kernels of fiber-singular type analogous to Simonenko’s continual convolution integral operators. This investigation is a part of a more general study of algebra of operators with homogeneous kernels which layers are singular operators with piecewise continuous coefficients. For the considered operators, the symbolic calculus and the necessary and sufficient Fredholm conditions are obtained.