Packing dimension and Ahlfors regularity of porous sets in metric spaces

Abstract

Let $X$ be a metric measure space with an $s$-regular measure $\mu$. 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 $\mu$. 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 $\mu$ is a doubling measure. However, in the doubling case we find a fixed $N\subset X$ with $\mu (N)=0$ such that ${\dim }_{\mathrm{p}}(A)\le {\dim }_{\mathrm{p}}(X)-c(\log \frac{1}{\varrho }{)}^{-1}{\varrho }^t$ for all $\varrho$-porous sets $A\subset X\setminus N$. Here $c$ and $t$ are constants which depend on the structure constant of $\mu$. 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 $A\subset F$.

References in corpus (3)

• Ahlfors-David regular sets and bilipschitz maps
• Separation conditions on controlled Moran constructions
• Nonsymmetric conical upper density and $k$-porosity

Cited by in corpus (5)

• Nonsymmetric conical upper density and $k$-porosity
• On upper conical density results
• Compactness of $M$-uniform domains and optimal thermal insulation problems
• Mixed multifractal densities for quasi-Ahlfors vector-valued measures
• A mixed multifractal analysis for quasi Ahlfors vector-valued measures