Variation for the Riesz transform and uniform rectifiability

Details Variation for the Riesz transform and uniform rectifiability Variation for the Riesz transform and uniform rectifiability

2011-09-02

arxiv.org/abs/1109.0466v1

Mathematics

Classical Analysis and ODEs

46 pages

Scientific paper

For 02, we prove that an n-dimensional Ahlfors-David regular measure M in R^d is uniformly n-rectifiable if and only if the r-variation for the Riesz transform with respect to M is a bounded operator in L^2(M). This result can be considered as a partial solution to a well known open problem posed by G. David and S. Semmes which relates the L^2(M) boundedness of the Riesz transform to the uniform rectifiability of M.

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