PACKING DIMENSION AND AHLFORS REGULARITY OF POROUS SETS IN METRIC SPACES

Abstract. Let X be a metric measure space with an s-regular measure µ. We prove that if A ⊂ X is ̺-porous, then dimp(A) ≤ s − c̺ s where dimp 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 asymp-totically 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 ⊂ X with µ(N) = 0 such that dimp(A) ≤ dimp(X) − c(log 1 ̺)−1̺t for all ̺-porous sets A ⊂ X \ 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 ⊂ F. 1