New criteria for the rectifiability of Radon measures in terms of Riesz transforms

Paper

Abstract

In this paper we explore the connection between quantitative rectifiability of measures and the

boundedness of the codimension one Riesz transform. Among other things, we prove the following. Let

be a Radon measure in

with growth of degree

such that the

-dimensional Riesz transform

is bounded in

, and let

be a suitably doubling ball such that: (i) There exists some (small) ball

centered in

with

such that, for some constant

(ii) For some

is small enough, depending on

and

, and

is small enough, then there exists a uniformly

-rectifiable set

and some

such that