Singular Integrals and Geometric Measure Theory: towards a solution of David--Semmes problem.
Seminar Room 2, Newton Institute Gatehouse
AbstractThe boundedness of the Riesz operator (whose kernel is the gradient of the fundamental solution for Laplacian in R^d) in L^2 with respect to d-1 dimensional Hausdorff measure must imply the rectifiability of this measure. This statement became known as David--Semmes problem. Guy David and Steven Semmes devoted two books to it. But it has been proved only for d=2, first by Mattila--Melnikov--Verdera for the case of homogeneous set, and later by Tolsa in a non-homogeneous situation. The non-homogeneous situation for d=2 also involves relations between beta numbers of Peter Jones and Menger's curvature. However, Menger's curvature is ``cruelly missing" (by the expression of Guy David) in dimensions d>2. In a recent work of Nazarov--Tolsa--Volberg the conjecture of David and Semmes has been validated. The proof (which does not involve Menger's curvature) gives a new and completely different proofs of the abovementioned results also in the case d=2. It is a long and not-so-easy paper. The result can be cast in the language of the existence of bounded harmonic vector fields in certain (infinitely connected) domains. In fact, our result is a certain co-dimension 1 claim. In higher co-dimensions the problem (which we will explain) rests open.
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