On smoothness of a fundamental solution to a second order hyperbolic equation

We consider the problem of constructing a fundamental solution to a second order hyperbolic linear equation with variable coefficients depending on the space variable $x\in\mathbb R^n$. Under the assumption of high but finite smoothness of the coefficients, we write out the structure of a fundamental solution, establish smoothness of the coefficients of the expansion of its singular part, and characterize smoothness of the regular part.