For possibly discontinuous functions including, for instance, Sobolev functions, we present new Blaschke-Privaloff-type criteria for superharmonicity and harmonicity. This opens the way for introduction of a substantial generalization of the Laplace operator. These potential-theoretic considerations lead to a new kind of non-absolutely convergent integral where the integrand may be a highly oscillating pointwise function or even a distribution-valued function. In turn, this integral gives a precise meaning to some generalized Poisson equations with a wild right hand side.

中文翻译：

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非绝对收敛的广义拉普拉斯算子

对于可能不连续的函数，例如Sobolev函数，我们提出了新的Blaschke-Privaloff型超谐波和谐波标准。这为引入大量Laplace运算符开辟了道路。这些潜在的理论考虑导致了一种新型的非绝对收敛积分，其中被积物可能是一个高度振荡的逐点函数，甚至是一个分布值函数。反过来，该积分为带有广义右手边的一些广义泊松方程提供了精确的含义。