平面上Sobolev函数乘积集的可移除性

IF 0.8 4区数学 Q2 MATHEMATICS

Arkiv for Matematik Pub Date : 2021-11-05 DOI:10.4310/arkiv.2023.v61.n1.a4

T. Rajala, Ugo Bindini

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引用次数: 0

摘要

我们研究闭集$C，F\subet\mathbb｛R｝$使乘积$C\times F$对于$W^｛1，p｝$可移动或不可移动的条件。主要结果表明，小维分量$C$的Hausdorff维数确定了一个临界指数，在该指数之上，乘积对于某些正测度集$F$是可移除的，而在该指数之下，乘积对于另一个正测度完全不连通集$F$$是不可移除的。此外，如果集合$C$是Ahlfors正则的，则上述可移除性适用于任何完全不连通的$F$。

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Removability of product sets for Sobolev functions in the plane

We study conditions on closed sets $C,F \subset \mathbb{R}$ making the product $C \times F$ removable or non-removable for $W^{1,p}$. The main results show that the Hausdorff-dimension of the smaller dimensional component $C$ determines a critical exponent above which the product is removable for some positive measure sets $F$, but below which the product is not removable for another collection of positive measure totally disconnected sets $F$. Moreover, if the set $C$ is Ahlfors-regular, the above removability holds for any totally disconnected $F$.

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