Revistes Catalanes amb Accés Obert (RACO)

The Riesz kernels do not give rise to higher dimensional analogues of the Menger-Melnikov curvature

H. M. Farag

Resum


Ever since the discovery of the connection between the Menger-Melnikov curvature and the Cauchy kernel in the $L^{2}$ 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 $\bold{R}^{n}$, even in any $L^{k}$ norm $(k\in\bold{N})$, would probably carry nontrivial information on whether the boundedness of these kernels in the appropriate norm implies rectifiability properties of the underlying sets or measures. Answering such questions would also have an impact on another important problem, namely whether totally unrectifiable $m$-sets are removable for Lipschitz harmonic functions in $\bold{R}^{m+1}.$ It has generally been believed that some such expressions should exist at least for some choices of $m$, $k$, or $n$, but the apparent complexity involved made the search rather difficult, even with the aid of computers. However, our rather surprising result is that, in fact, not a single higher dimensional analogue of this useful curvature can be derived from the Riesz kernels in the same fashion, and that, even for 1-sets, the Menger-Melnikov curvature is unique in a certain sense.

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