We prove an abstract form of Hardy's inequality for local and non-local regular Dirichlet forms on metric measure spaces, using the Green operator of the Dirichlet form in question. Under additional assumptions such as the volume doubling, the reverse volume doubling, and certain natural estimates of the Green function, we obtain the “classical” form of Hardy's inequality containing distance to a reference point or set.

中文翻译：

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公制度量空间上的Hardy不等式和Green函数

我们使用有问题的Dirichlet形式的Green运算符，证明了度量度量空间上局部和非局部常规Dirichlet形式的Hardy不等式的抽象形式。在其他假设（例如，体积加倍，反向体积加倍以及Green函数的某些自然估计）下，我们获得了Hardy不等式的“经典”形式，其中包含到参考点或集合的距离。