Radon measures and Lipschitz graphs

Abstract

For all 1⩽m⩽n−1, we investigate the interaction of locally finite measures in Rn 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 normalΓ1,normalΓ2,⋯ such that μ(double-struckRn∖⋃1∞normalΓi)=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 Hm to E⊆double-struckRn with 0<scriptHmfalse(Efalse)<∞. 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.

Cones, rectifiability, and singular integral operators

“…finiteness of their conical Dini function implies that is n-rectifiable. We would like to stress, however, that neither Theorem 1.4 implies the results from [7], nor the other way around.…”

“…In Theorem D of [45], Naples showed that a modified version of (1.2) can be used to characterize pointwise doubling measures carried by Lipschitz graphs, that is, measures vanishing outside of a countable union of n-dimensional Lipschitz graphs. In an even more recent paper [7], Badger and Naples completely describe measures carried by n-dimensional Lipschitz graphs on R d . They use a Dini condition imposed on the socalled conical defect, and their condition is closely related to (1.4).…”

“…; x/ < 1 for -a.e. x 2 R d , then it actually follows from [7] that -a.e. finiteness of their conical Dini function implies that is n-rectifiable.…”

Identifying 1-rectifiable measures in Carnot groups

“…It turns out that detecting m-dimensional rectifiability is more subtle for measures of Hausdorff dimension less than m than it is for measures of Hausdorff dimension m. Pointwise characterizations of locally finite measures on X that are carried by a family A (without restriction on dimension, doubling properties, or null sets of µ!) are presently available in two situations: (i) for measures on R n carried by rectifiable curves [15], and (ii) for measures on R n carried by m-dimensional Lipschitz graphs [11]. These results are made possible by a thorough understanding of subsets of rectifiable curves or Lipschitz graphs in R n and the incorporation of ideas from harmonic analysis.…”