Weak Carleson conditions in uniformly rectifiable metric spaces: the WCD and alpha numbers

Abstract

We investigate characterizations of uniformly rectifiable (UR) metric spaces by so-called weak Carleson conditions for flatness coefficients which measure the extent to which Hausdorff measure on the metric space differs from Hausdorff measure on a normed space. First, we show that UR metric spaces satisfy David and Semmes's weak constant density condition, a quantitative regularity property which implies most balls in the space support a measure with nearly constant density in a neighborhood of scales and locations. Second, we introduce a metric space variant of Tolsa's alpha numbers that measure a local normalized $L_1$ mass transport cost between the space's Hausdorff measure and Hausdorff measure on a normed space. We show that a weak Carleson condition for these alpha numbers gives a characterization of metric uniform rectifiability. We derive both results as corollaries of a more general abstract result which gives a tool for transferring weak Carleson conditions to spaces with very big pieces of spaces with a given weak Carleson condition.