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arXiv 2109.07553math.FA
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Sobolev $W_{p}^{1}(\mathbb{R}^{n})$ -空间到$\mathbb{R}^{n}$的任意紧子集的迹的几乎精确刻画.案例$p \in (1,n]$

Almost sharp descriptions of traces of Sobolev $W_{p}^{1}(\mathbb{R}^{n})$-spaces to arbitrary compact subsets of $\mathbb{R}^{n}$. The case $p \in (1,n]$

Alexander Tyulenev

备注：
The final version accepted in Ann.~Sc.~Norm.~Super.~Pisa Cl.~Sci.~(5) 26, issue 1 (2025)
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摘要：

设$S \subset \mathbb{R}^{n}$是一个任意的非空紧集，使得对于某个$d \in (0,n]$，$d$ -Hausdorff容度为$\mathcal{H}^{d}_{\infty}(S) > 0$.对于每个$p \in (\max\{1,n-d\},n]$，得到了Sobolev空间$W_{p}^{1}(\mathbb{R}^{n})$的迹空间$W_{p}^{1}(\mathbb{R}^{n})|_{S}$对集合$S$的几乎尖锐的本征描述.进一步，对于每个$p \in (\max\{1,n-d\},n]$和$\varepsilon \in (0, \min\{p-(n-d),p-1\})$，构造了从迹空间$W_{p}^{1}(\mathbb{R}^{n})|_{S}$到空间$W_{p-\varepsilon}^{1}(\mathbb{R}^{n})$的新的有界线性扩张算子.

英文摘要：

Let $S \subset \mathbb{R}^{n}$ be an arbitrary nonempty compact set such that the $d$-Hausdorff content $\mathcal{H}^{d}_{\infty}(S) > 0$ for some $d \in (0,n]$. For each $p \in (\max\{1,n-d\},n]$, an almost sharp intrinsic description of the trace space $W_{p}^{1}(\mathbb{R}^{n})|_{S}$ of the Sobolev space $W_{p}^{1}(\mathbb{R}^{n})$ to the set $S$ is obtained. Furthermore, for each $p \in (\max\{1,n-d\},n]$ and $\varepsilon \in (0, \min\{p-(n-d),p-1\})$, new bounded linear extension operators from the trace space $W_{p}^{1}(\mathbb{R}^{n})|_{S}$ into the space $W_{p-\varepsilon}^{1}(\mathbb{R}^{n})$ are constructed.

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