Geodesic
Variation for the Riesz transform and uniform rectifiability
Journal of the European Mathematical Society, Tome 16 (2014) no. 11, pp. 2267-2321.

Voir la notice de l'article provenant de la source EMS Press

For 1≤nand ρ>2, we prove that an n-dimensional Ahlfors–David regular measure μ in Rd is uniformly n-rectifiable if and only if the ρ-variation for the Riesz transform with respect to μ is a bounded operator in L2(μ). 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 L2(μ) boundedness of the Riesz transform to the uniform rectifiability of μ.
DOI : 10.4171/jems/487
Classification : 42-XX, 00-XX
Keywords: ρ-variation and oscillation, Calderón-Zygmund singular integrals, Riesz transform, uniform rectifiability 
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     title = {Variation for the {Riesz} transform and uniform rectifiability},
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Albert Mas; Xavier Tolsa. Variation for the Riesz transform and uniform rectifiability. Journal of the European Mathematical Society, Tome 16 (2014) no. 11, pp. 2267-2321. doi : 10.4171/jems/487. http://geodesic.mathdoc.fr/articles/10.4171/jems/487/

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