On the Solvability of Integral Operators with Bihomogeneous Kernels of the Compact Type and Variable Coefficients

In this paper, we consider the C ^? -algebra Ω_mult(? ⁿ ) of multiplicatively weakly oscillating functions and a new wide class of kernels of compact type, which includes the class of SO(n)-invariant kernels. For the Banach algebra generated by operators with kernels of this class and coefficients from Ω_mult(? ⁿ ), we construct a symbolic calculus, obtain necessary and sufficient conditions for the presence of the Fredholm property, and propose a method of calculating the index of families. Similar results are obtained for operators with bihomogeneous kernels of compact type and multiplicatively weakly oscillating coefficients, i.e., for operators from the tensor product \( {{\mathfrak{M}}_{p,n}}_{{_1}}\otimes {{\mathfrak{M}}_{p,n}}_{{_2}} \) .