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. 
Benjamin Jaye, Fedor Nazarov and Xavier Tolsa
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