Besicovitch's circle pairs, analytic capacity, and Cauchy integrals for a class of unrectifiable 1-sets.
On the 1/2 Problem of Besicovitch: quasi-arcs do not contain sharp saw-teeth.
In this paper we give an alternative proof of our recent result that totally unrectifiable 1-sets which satisfy a measure-theoretic flatness condition at almost every point and sufficiently small scales, satisfy Besicovitch's 1/2-Conjecture which states that the lower spherical density for totally unrectifiable 1-sets should be bounded above by 1/2 at almost every point. This is in contrast to rectifiable 1-sets which actually possess a density equal to unity at almost every point. Our present method...
The Riesz kernels do not give rise to higher dimensional analogues of the Menger-Melnikov curvature.
Ever since the discovery of the connection between the Menger-Melnikov curvature and the Cauchy kernel in the L norm, and its impressive utility in the analytic capacity problem, higher dimensional analogues have been coveted. The lesson from 1-sets was that any such (nontrivial, nonnegative) expression, using the Riesz kernels for m-sets in R, even in any L norm (k ∈ N), would probably carry nontrivial information on whether the boundedness of these kernels in the appropriate norm implies rectifiability...
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