We generalize some characterizations of uniformly rectifiable (UR) sets to sets whose
Hausdorff content is lower regular (and, in particular, is not necessarily Ahlfors
regular). For example, David and Semmes showed that, given an Ahlfors
-regular set
, if we consider
the set
of surface cubes (in the sense of Christ and David) near which
does not look approximately like a union of planes, then
is UR if
and only if
satisfies a Carleson packing condition, that is, for any surface cube
,
We show that, for lower content regular sets that aren’t necessarily Ahlfors regular, if
denotes the square
sum of
-numbers
over subcubes of
as in the traveling salesman theorem for higher-dimensional sets, presented by Azzam
and Schul, then
We prove similar results for other uniform rectifiability criteria, such as the
local symmetry, local convexity, and generalized weak exterior convexity
conditions.
En route, we show how to construct a corona decomposition of any lower
content regular set by Ahlfors regular sets, similar to the classical corona
decomposition of UR sets by Lipschitz graphs developed by David and Semmes.
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