论文标题
当地链条规则的故障分数变化
Failure of the local chain rule for the fractional variation
论文作者
论文摘要
我们证明，链条规则的本地版本无法满足Arxiv中定义的分数变化：1809.08575。在$ n = 1 $的情况下，我们证明了更强的结果，在bv^α（\ mathbb {r}）中表现出功能$ f \，这样$ | f | \ notin bv^α（\ mathbb {r}）$。局部链条规则的失败是由于分数变化的非负功能的某些令人惊讶的刚度特性的结果，而分数变化又是从本地定位到半个空间的分数硬质不平等得出的。我们的方法利用了Arxiv的结果：2111.13942和先前论文的分布方法Arxiv：1809.08575，Arxiv：1910.13419，Arxiv：2011.03928，Arxiv，Arxiv：2109.15263。作为副产品，我们完善了在Arxiv中获得的分数硬度不平等：1611.07204，Arxiv：1806.07588，我们证明了密切相关的Meyers-Ziemer痕迹的分数版本。
We prove that the local version of the chain rule cannot hold for the fractional variation defined in arXiv:1809.08575. In the case $n = 1$, we prove a stronger result, exhibiting a function $f \in BV^α(\mathbb{R})$ such that $|f| \notin BV^α(\mathbb{R})$. The failure of the local chain rule is a consequence of some surprising rigidity properties for non-negative functions with bounded fractional variation which, in turn, are derived from a fractional Hardy inequality localized to half-spaces. Our approach exploits the results of arXiv:2111.13942 and the distributional approach of the previous papers arXiv:1809.08575, arXiv:1910.13419, arXiv:2011.03928, arXiv:2109.15263. As a byproduct, we refine the fractional Hardy inequality obtained in arXiv:1611.07204, arXiv:1806.07588 and we prove a fractional version of the closely related Meyers-Ziemer trace inequality.
