Curvatures of the Melnikov type, Hausdorff dimension, rectifiability, and singular integrals on \({\mathbb{R}}^n\) (Q5929043)

An active area of research in harmonic and complex analysis is the question of which subsets of \(C\) (or \(R^n\)) have zero analytic capacity. This is closely related to whether the associated Cauchy integral operator is bounded on \(L^2(\mu)\) for a suitable measure \(\mu\) on the set. In the case of the Cauchy integral on subsets of \(C\) there is a remarkable identity of Melnikov which shows that the \(L^2(\mu)\) norm of the Cauchy integral of \(d\mu\) is the integral of a \textit{non-negative} geometric quantity, namely the square of the Menger curvature of triples of points on the set. This is crucial in much of the recent work on the problem. NEWLINENEWLINENEWLINEIn this paper the author shows how such an identity in fact holds for a more general class of singular integrals on one-dimensional subsets of the plane, for instance when the Euclidean metric is replaced by an \(l^p\) metric. The resulting curvature is greater than or comparable to the Menger curvature, and the author can thus generalize various results in this area, for instance giving a \(T(1)\)-type characterization of the sets on which the Cauchy integral is bounded in \(L^2\). NEWLINENEWLINENEWLINEIn higher dimensions the author has negative results which show rather conclusively that no such formula can be expected.