In these notes we survey some new results concerning the $\rho $-variation for singular integral operators defined on Lipschitz graphs. Moreover, we investigate the relationship between variational inequalities for singular integrals on AD regular measures and geometric properties of these measures. An overview of the main results and applications, as well as some ideas of the proofs, are given.

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 ${L}^{2}\left(\mu \right)$. 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 ${L}^{2}\left(\mu \right)$ boundedness of the Riesz transform to the uniform rectifiability of $\mu $.

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