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

TL;DR

This paper extends a theorem to characterize non-atomic measures where certain square function operators are bounded in L^2, using a modified geometric measure condition based on Jones' beta-coefficients.

Contribution

It provides a new geometric characterization of measures for which square function operators are bounded in L^2, generalizing previous results to broader measures.

Findings

Characterization of non-atomic measures with bounded square function operators

Extension of David and Semmes' theorem to general measures

Use of modified Jones' beta-coefficients for measure analysis

Abstract

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.