Abstract

One of the most fundamental steps leading to the solution of the analytic capacity problem (for 1-sets) was the discovery by Melnikov of an identity relating the sum of permutations of products of the Cauchy kernel to the three-point Menger curvature. We here undertake the study of analogues of this so-called Menger-Melnikov curvature, as a nonnegative function defined on certain copies of Rⁿ, in relation to some natural singular integral operators on subsets of Rⁿ of various Hausdorff dimensions. In recent work we proved that the Riesz kernels x^−m−1 (m in N∖) do not admit identities like that of Melnikov in any L^k norm ( k in N). In this paper we extend these investigations in various ways. Mainly, we replace the Euclidean norm by equivalent metrics δ(•,•) and we consider all possible k,m,n,δ(•,•). We do this in hopes of finding better algebraic properties which may allow extending the ideas to higher dimensional sets. On the one hand, we show that for m > 1 no such identities are admissible at least when δ is a norm that is invariant under reflections and permutations of the coordinates. On the other hand, for m = 1, we show that for each choice of metric, one gets an identity and a curvature like those of Melnikov. This allows us to generalize those parts of the recent singular integral and rectifiability theories for the Cauchy kernel that depend on curvature to these much more general kernels, and provides a more general framework for the curvature approach.

Authors