The measures with an associated square function operator bounded in $L^2$

In this paper we provide an extension of a theorem of David and Semmes (‘91) to general non-atomic measures. The result provides a geometric characterization of the non-atomic measures for which a certain class of square function operators, or singular integral operators, are bounded in $L^2(\mu )$. The description is given in terms of a modification of Jones’ $\beta$-coefficients.