Variation for the Riesz transform and uniform rectifiability

Mas, Albert, and Tolsa, Xavier. "Variation for the Riesz transform and uniform rectifiability." Journal of the European Mathematical Society 016.11 (2014): 2267-2321. <http://eudml.org/doc/277336>.

@article{Mas2014,
abstract = {For $1 \le n < d$ integers and $\rho >2$, we prove that an $n$-dimensional Ahlfors-David regular measure $\mu $ in $\mathbb \{R\}^d$ is uniformly $n$-rectifiable if and only if the $\rho $-variation for the Riesz transform with respect to $\mu $ is a bounded operator in $L^2(\mu )$. 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 $L^2(\mu )$ boundedness of the Riesz transform to the uniform rectifiability of $\mu $.},
author = {Mas, Albert, Tolsa, Xavier},
journal = {Journal of the European Mathematical Society},
keywords = {$\rho $-variation and oscillation; Calderón-Zygmund singular integrals; Riesz transform; uniform rectifiability; Calderón-Zygmund singular integrals; Riesz transform; Radon measure; Ahlfors-David regularity; uniform rectifiability; -variation},
language = {eng},
number = {11},
pages = {2267-2321},
publisher = {European Mathematical Society Publishing House},
title = {Variation for the Riesz transform and uniform rectifiability},
url = {http://eudml.org/doc/277336},
volume = {016},
year = {2014},
}

TY - JOUR
AU - Mas, Albert
AU - Tolsa, Xavier
TI - Variation for the Riesz transform and uniform rectifiability
JO - Journal of the European Mathematical Society
PY - 2014
PB - European Mathematical Society Publishing House
VL - 016
IS - 11
SP - 2267
EP - 2321
AB - For $1 \le n < d$ integers and $\rho >2$, we prove that an $n$-dimensional Ahlfors-David regular measure $\mu $ in $\mathbb {R}^d$ is uniformly $n$-rectifiable if and only if the $\rho $-variation for the Riesz transform with respect to $\mu $ is a bounded operator in $L^2(\mu )$. 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 $L^2(\mu )$ boundedness of the Riesz transform to the uniform rectifiability of $\mu $.
LA - eng
KW - $\rho $-variation and oscillation; Calderón-Zygmund singular integrals; Riesz transform; uniform rectifiability; Calderón-Zygmund singular integrals; Riesz transform; Radon measure; Ahlfors-David regularity; uniform rectifiability; -variation
UR - http://eudml.org/doc/277336
ER -