Some additions to the problem of multidimensional integral operators with oscillating coefficients

The paper presents multidimensional integral operators with homogeneous cores and radial oscillating coefficients of $|x|^{i\delta}$, acting in space $L_2(B_n)$. A symbol is defined for operators of this type in the terminology while necessary and sufficient conditions for noethericity and the computational formula for the index are obtained. A criterion for noethericity property and the index formula for a similar class of operators acting in $L_2(B_n)$ have been established. In this paper, we give some important additions and refinements to the main results published earlier. We have also considered a special case of an integral operator as a supplement.