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 ∈ N∖) do not
admit identities like that of Melnikov in any Lk norm ( k ∈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.
by equivalent metrics 