# Variation for the Riesz transform and uniform rectifiability

Journal of the European Mathematical Society (2014)

- Volume: 016, Issue: 11, page 2267-2321
- ISSN: 1435-9855

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topMas, 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 -

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