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For integers and , we prove that an -dimensional Ahlfors-David regular measure in is uniformly -rectifiable if and only if the -variation for the Riesz transform with respect to is a bounded operator in . 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 boundedness of the Riesz transform to the uniform rectifiability of .
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 -
