与$d$ -Hausdorff容度有关的集合的某些多孔型性质
Some porosity-type properties of sets related to the $d$-Hausdorff content
摘要：
设$S \subset \mathbb{R}^{n}$是非空集。给定$d \in [0,n)$和一个具有$l=l(\overline{Q}) \in (0,1]$的立方体$\overline{Q} \subset \mathbb{R}^{n}$，我们证明了如果某些$\overlineλ \in (0,1)$的$d$ -Hausdorff内容为$\mathcal{H}^{d}_{\infty}(\overline{Q} \cap S) < \overlineλl^{d}$，则集合$\overline{Q} \setminus S$包含一个特定的空腔.更确切地说，我们证明了存在一个伪度量$ρ=ρ_{S,d}$，使得对于每个足够小的$δ> 0$的$δ$ -邻域$U^ρ_{δl}(S)$的$S$在伪度量$ρ$不包含整个$\overline{Q}$。此外，我们建立了常数$\overlineδ=\overlineδ(n,d,\overlineλ)>0$和$\underlineγ=\underlineγ(n,d,\overlineλ)>0$的存在性，使得对所有的$δ\in (0,\overlineδ)$都有$\mathcal{L}^{n}(\overline{Q} \setminus U^ρ_{δl}(S)) \geq \underlineγ l^{n}$。如果，此外，集$S$是$d$ -低内容定期，我们证明存在一个常数$\underlineτ=\underlineτ(n,d,\overlineλ)>0$，使立方体$\overline{Q}$是$\underlineτ$ -多孔。几个例子说明了结果的清晰度。
英文摘要：
Let $S \subset \mathbb{R}^{n}$ be a nonempty set. Given $d \in [0,n)$ and a cube $\overline{Q} \subset \mathbb{R}^{n}$ with $l=l(\overline{Q}) \in (0,1]$, we show that if the $d$-Hausdorff content $\mathcal{H}^{d}_{\infty}(\overline{Q} \cap S) < \overlineλl^{d}$ for some $\overlineλ \in (0,1)$, then the set $\overline{Q} \setminus S$ contains a specific cavity. More precisely, we prove existence of a pseudometric $ρ=ρ_{S,d}$ such that for each sufficiently small $δ> 0$ the $δ$-neighborhood $U^ρ_{δl}(S)$ of $S$ in the pseudometric $ρ$ does not contain the whole $\overline{Q}$. Moreover, we establish the existence of constants $\overlineδ=\overlineδ(n,d,\overlineλ)>0$ and $\underlineγ=\underlineγ(n,d,\overlineλ)>0$ such that $\mathcal{L}^{n}(\overline{Q} \setminus U^ρ_{δl}(S)) \geq \underlineγ l^{n}$ for all $δ\in (0,\overlineδ)$. If, in addition, the set $S$ is $d$-lower content regular, we prove existence of a constant $\underlineτ=\underlineτ(n,d,\overlineλ)>0$ such that the cube $\overline{Q}$ is $\underlineτ$-porous. The sharpness of the results is illustrated by several examples.
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