度量空间中多孔集的Packing维数与Ahlfors正则性
Packing dimension and Ahlfors regularity of porous sets in metric spaces
摘要：
设$X$是一个度量测度空间，它具有$s$ -正则测度$μ$ .我们证明了如果$A\subset X$是$\varrho$ -多孔的，则有$\dim_{\mathrm{p}}(A)\le s-c\varrho^s$，其中$\dim_{\mathrm{p}}$是堆积维数，$c$是一个正的常数，它依赖于$s$和$μ$的结构常数.这是一个模拟的一个众所周知的渐近尖锐的结果在欧几里德空间。通过一个例子说明，当$μ$是一个加倍测度时，相应的结果是不成立的。然而，在加倍的情况下，我们发现一个固定的$N\subset X$与$μ(N)=0$，使得$\dim_{\mathrm{p}}(A)\le\dim_{\mathrm{p}}(X)-c(\log\tfrac 1\varrho)^{-1}\varrho^t$对于所有的$\varrho$ -多孔集$A\subset X\setminus N$。这里，$c$和$t$是取决于$μ$的结构常数的常数。最后，我们用正则集刻画了完备$s$ -正则度量空间中的一致多孔集，证明了$A$是一致多孔的当且仅当存在$t<s$和$t$ -正则集$F$使得$A\subset F$ .
英文摘要：
Let $X$ be a metric measure space with an $s$-regular measure $μ$. We prove that if $A\subset X$ is $\varrho$-porous, then $\dim_{\mathrm{p}}(A)\le s-c\varrho^s$ where $\dim_{\mathrm{p}}$ is the packing dimension and $c$ is a positive constant which depends on $s$ and the structure constants of $μ$. This is an analogue of a well known asymptotically sharp result in Euclidean spaces. We illustrate by an example that the corresponding result is not valid if $μ$ is a doubling measure. However, in the doubling case we find a fixed $N\subset X$ with $μ(N)=0$ such that $\dim_{\mathrm{p}}(A)\le\dim_{\mathrm{p}}(X)-c(\log\tfrac 1\varrho)^{-1}\varrho^t$ for all $\varrho$-porous sets $A\subset X\setminus N$. Here $c$ and $t$ are constants which depend on the structure constant of $μ$. Finally, we characterize uniformly porous sets in complete $s$-regular metric spaces in terms of regular sets by verifying that $A$ is uniformly porous if and only if there is $t<s$ and a $t$-regular set $F$ such that $A\subset F$.
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