Usual Square Function on “weakly flat” one-dimensional sets

In 1993 David and Semmes proved that an Ahlfors-David regular d-dimensional set in \mathbb{R}^d+1 is uniformly rectifiable if and only if its associated usual square function is bounded on L². We make a first step towards an analogous statement without the lower bound of the AD-regularity and show that a one-dimensional set in \mathbb{R}² satisfying a “weak flattening” condition is rectifiable if its usual square function is bounded on L².
No background in geometric measure theory will be assumed; an introduction to the area will be provided.