Abstract
In [8] Naor and Tao extended to the metric setting the O(dlogd) bounds given by Stein and Strömberg for Lebesgue measure in Rd, deriving these bounds first from a localization result, and second, from a random Vitali lemma. Here we show that the Stein–Strömberg original argument can also be adapted to the metric setting, giving a third proof. We also weaken the hypotheses, and additionally, we sharpen the estimates for Lebesgue measure.
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