$C^*$-Algebra of Integral Operators with Homogeneous Kernels and Oscillating Coefficients

We consider the $C^*$-algebra generated by multidimensional integral operators with $(-n)$th-order homogeneous kernels and by the operators of multiplication by oscillating coefficients of the form $|x|^{i\alpha}$. For this algebra, we construct an operator symbolic calculus and obtain necessary and sufficient conditions for the Fredholm property of an operator in terms of this calculus.