Radon measures and Lipschitz graphs. Issue 3 (23rd February 2021)

• Record Type: Journal Article
• Authors: Badger, Matthew; Naples, Lisa
• Is Part Of: Bulletin of the London Mathematical Society. Volume 53:Issue 3(2021)
• Page Start: 921
• Page End: 936
• Publication Date: 2021-02-23
• Subjects: 28A75 (primary); Mathematics -- Periodicals; 510
• DOI: 10.1112/blms.12473
• Languages: English
• ISSNs: 0024-6093

Abstract

Abstract: For all 1 ⩽ m ⩽ n − 1, we investigate the interaction of locally finite measures in R n with the family of m ‐dimensional Lipschitz graphs. For instance, we characterize Radon measures μ, which are carried by Lipschitz graphs in the sense that there exist graphs Γ 1, Γ 2, ⋯ such that μ ( R n ∖ ⋃ 1 ∞ Γ i ) = 0, using only countably many evaluations of the measure. This problem in geometric measure theory was classically studied within smaller classes of measures, for example, for the restrictions of m ‐dimensional Hausdorff measure H m to E ⊆ R n with 0 < H m ( E ) < ∞ . However, an example of Csörnyei, Käenmäki, Rajala, and Suomala shows that classical methods are insufficient to detect when a general measure charges a Lipschitz graph. To develop a characterization of Lipschitz graph rectifiability for arbitrary Radon measures, we look at the behavior of coarse doubling ratios of the measure on dyadic cubes that intersect conical annuli. This extends a characterization of graph rectifiability for pointwise doubling measures by Naples by mimicking the approach used in the characterization of Radon measures carried by rectifiable curves by Badger and Schul.