We prove a Leibniz rule for \({\mathrm {BV}}\) functions in a complete metric space that is equipped with a doubling measure and supports a Poincaré inequality. Unlike in previous versions of the rule, we do not assume the functions to be locally essentially bounded and the end result does not involve a constant \(C\ge 1\), and so our result seems to be essentially the best possible. In order to obtain the rule in such generality, we first study the weak* convergence of the variation measure of \({\mathrm {BV}}\) functions, with quasi semicontinuous test functions.

中文翻译：

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度量空间中$$ {\ mathrm {BV}} $$ BV函数的清晰Leibniz规则

我们证明了在完整度量空间中用于\（{\ mathrm {BV}} \）函数的Leibniz规则，该规则空间具有加倍的度量并支持Poincaré不等式。与以前的规则版本不同，我们不假定函数在局部上是有界的，并且最终结果不包含常数\（C \ ge 1 \），因此我们的结果似乎基本上是最好的。为了获得这种通用性的规则，我们首先使用准半连续检验函数研究\（{\ mathrm {BV}} \）函数的变化量度的弱*收敛性。