Weak Carleson Conditions in Uniformly Rectifiable Metric Spaces: The WCD and Alpha Numbers
Year: 2025
Type: article
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 that 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.
DOI: 10.1093/imrn/rnaf352
Author: Jared Krandel
Institution: University of California, Davis
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $4,534
OpenAlex ID: W4417088317