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

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