Estimates for the Maximal Cauchy Integral on Chord-arc Curves

We study the chord-arc Jordan curves that satisfy the Cotlar-type inequality $T_{\ast }(f)\lesssim M^2(Tf),$ where $T$ is the Cauchy transform, $T_{\ast }$ is the maximal Cauchy transform and $M$ is the Hardy-Littlewood maximal function. Under the background assumption of asymptotic quasi-conformality we find a characterization of such curves in terms of the smoothness of a parametrization of the curve.