On the \(T(1)\)-theorem for the Cauchy integral
The author defines a weaker growth condition for the measure \(\mu\) than the standard doubling condition \(\mu (2D)\leq const\times\mu (D)\) for each disc \(D\), and gives an alternative proof, with respect to two recent proofs, of the \(L^2(\mu )\)-boundedness of the Cauchy integral operator, i.e., by the integral \(\int_{C}^{}| f| ^2d\mu\) (\(T(1)-\)theorem). Then he obtains the estimate from below of the analytic capacity of a compact subset contained in the contour \(C\) in terms of total Menger curvature. He shows that if the doubling condition fails then \(L^2\)-boundedness does not imply the \(L^\infty -\text{BMO}\) estimate, \(\text{BMO}(\mu )\) being the space of locally integrable functions with respect to \(\mu\).
scientific article; zbMATH DE number 2057365
scientific article