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This work provides a geometric characterization of the measures $\mu$ in$\mathbb R^{n+1}$ with polynomial upper growth of degree $n$ such that the$n$-dimensional Riesz transform $R\mu (x) = \int\frac{x-y}{|x-y|^{n+1}}\,d\mu(y)$ belongs to $L^2(\mu)$. More precisely, it isshown that $$\|R\mu\|_{L^2(\mu)}^2 + \|\mu\|\approx \int\!\!\int_0^\infty\beta_{2,\mu}(x,r)^2\,\frac{\mu(B(x,r))}{r^n}\,\frac{dr}r\,d\mu(x) + \|\mu\|,$$where $\beta_{\mu,2}(x,r)^2 = \inf_L \frac1{r^n}\int_{B(x,r)}\left(\frac{dist(y,L)}r\right)^2\,d\mu(y),$ with the infimum taken over allaffine $n$-planes $L\subset \mathbb R^{n+1}$. As a corollary, one obtains acharacterization of the removable sets for Lipschitz harmonic functions interms of a metric-geometric potential and one deduces that the class ofremovable sets for Lipschitz harmonic functions is invariant by bilipschitzmappings.

The measures with $L^2$-bounded Riesz transform and the Painlev\'e problem for Lipschitz harmonic functions