The measures with $L^2$-bounded Riesz transform and the Painlevé problem for Lipschitz harmonic functions

In this talk I will explain a recent work, partially in collaboration with Damian Dabrowski, where we provide a geometric characterization of the measures $\mu$ in $\mathbb R^{n+1}$ with polynomial upper growth of degree $n$ such that the Riesz transform $R\mu (x) = \int \frac{x-y}{|x-y|^{n+1}}\,d\mu(y)$ belongs to $L^2(\mu)$. As a corollary, we obtain a characterization of the removable sets for Lipschitz harmonic functions in terms of a metric-geometric potential and we deduce that the class of removable sets for Lipschitz harmonic functions is invariant by bilipschitz mappings.