Packing dimension and Ahlfors regularity of porous sets in metric spaces

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}$}$isthepackingdimensionand$c$ 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}$}(X)-c(\log\tfrac 1\varrho)^{-1}\varrho^t$forall$\varrho$-poroussets$A\subset X\setminus N$. 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 $A\subset F$.