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.

中文翻译：

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一个Riesz引力场的柯西问题和局部涨落分布

在具有可积奇点的不连续无界初始函数类中，我们考虑了相应Riesz引力场中运动物体局部作用的伪微分方程的Cauchy问题。这个问题的根本解决方案是这些对象之间的局部相互作用力的柯西概率分布。得到了这个解的一个显式形式，并且建立了这个问题的正确可解性。在这种情况下，找到了柯西问题的经典解的形式，并研究了其在无穷远处的平滑性和行为。