We prove that smooth $C^{\infty }$ functions are dense in weighted fractional Sobolev spaces on an arbitrary open set, under some mild conditions on the weight. We also obtain a similar result in non-weighted spaces defined by some kernel similar to $x\mapsto {\left|x\right|}^{-d-sp}$. One may consider the results to be a version of the Meyers–Serrin theorem.

中文翻译：

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加权分数次Sobolev空间中光滑函数的密度

我们证明那顺利 $C^{\infty }$在某些适度的权重条件下，函数在任意开放集上的加权分数Sobolev空间中都是密集的。我们在由一些类似于以下内容的内核定义的非加权空间中也获得了类似的结果$X\mapsto {\left|X\right|}^{-d-sp}$。可能会认为结果是迈耶斯-瑟林定理的一种形式。