In the class of discontinuous unbounded initial functions with an integrable singularity, we consider the Cauchy problem for the pseudodifferential equation of local action of moving objects in the corresponding Riesz gravitational field. The fundamental solution to this problem is the Cauchy probability distribution of the force of local interaction between these objects. An explicit form of this solution is obtained, and the correct solvability of this problem is established. In this case, the form of the classical solution of the Cauchy problem is found, and the properties of its smoothness and behavior at infinity are studied