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The main goal of this paper is to present an alternative, real variable proof of the$T$(1)-theorem for the Cauchy integral. We then prove that the estimate from below of analytic capacity in terms of total Menger curvature is a direct consequence of the$T$(1)-theorem. An example shows that the$L$^{∞}-BMO estimate for the Cauchy integral does not follow from$L$^{2}boundedness when the underlying measure is not doubling.

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We show that the Luzin area integral or the square function on the unit ball of ℂ^{$n$}, regarded as an operator in the weighted space$L$^{2}($w$) has a linear bound in terms of the invariant$A$_{2}characteristic of the weight. We show a dimension-free estimate for the “area-integral” associated with the weighted$L$^{2}($w$) norm of the square function. We prove the equivalence of the classical and the invariant$A$_{2}classes.