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Title of Journal: Math Z

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Abbravation: Mathematische Zeitschrift

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Springer-Verlag

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DOI

10.1016/j.apmr.2014.08.014

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1432-1823

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Packing dimension and Ahlfors regularity of porous

Authors: Esa Järvenpää Maarit Järvenpää Antti Käenmäki Tapio Rajala Sari Rogovin Ville Suomala
Publish Date: 2009/06/05
Volume: 266, Issue: 1, Pages: 83-105
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Abstract

Let X be a metric measure space with an sregular measure μ We prove that if Asubset X is varrho porous then rm dim pAle scvarrhos 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 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 Nsubset X with μN = 0 such that rm dim pAlerm dim pXclog tfrac1varrho1varrhot for all varrho porous sets A subset Xsetminus N Here c and t are constants which depend on the structure constant of μ Finally we characterize uniformly porous sets in complete sregular metric spaces in terms of regular sets by verifying that A is uniformly porous if and only if there is t  s and a tregular set F such that Asubset F

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