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Two problems on doubling measures. (English) Zbl 0862.28005

A measure \(\mu\) on a metric space \(X\) is doubling if \(\mu (B(x,2\rho)) \leq C \mu (B(x,\rho))\) for all open balls \(B(x,2\rho)\), \(B(x,\rho)\) and some \(C>0\).
Doubling measures \(\neq 0\) exist on any compact metric space whose metric satisfies a condition of uniform branching, according to a theorem of Vol’berg and Konyagin. If a doubling measure on a compact metric space is continuous, there is a second one, singular with the first (a problem in the work cited). There are compact subsets of \(R^1\) carrying a purely atomic doubling measure and also one which is partly continuous.
Besides doubling measures on \(R^1\) there are measures which are “doubling” for adjacent dyadic intervals whose union is also dyadic. There is a closed set \(S\) in \([0,1]\) which isn’t a null set for dyadic doubling measures; but for almost all real \(t\), \(t+S\) is a null set for that class of measures.

MSC:

28A75 Length, area, volume, other geometric measure theory
54E45 Compact (locally compact) metric spaces
28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
30C62 Quasiconformal mappings in the complex plane
42B25 Maximal functions, Littlewood-Paley theory
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