Let
,
be Radon
measures on
, with
nonatomic and
doubling, and write
for the Lebesgue
decomposition of
relative to
.
For an interval
,
define
,
the Wasserstein distance of normalised blow-ups of
and
restricted
to
.
Let
be the square function
where
is the family of dyadic intervals of side-length at most 1. I prove that
is finite
almost everywhere
and infinite
almost
everywhere. I also prove a version of the result for a nondyadic variant of the square function
. The results answer
the simplest “”
case of a problem of J. Azzam, G. David and T. Toro.
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