The measures with $L^2$-bounded Riesz transform and the Painlevé problem

In this work we provide 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 $R\mu (x)=\int \frac{x-y}{\mathrm{\mid }x-y{\mathrm{\mid }}^{n+1}}d\mu (y)$ belongs to $L^2(\mu )$. More precisely, it is shown that $$\mathrm{\parallel }R\mu {\mathrm{\parallel }}_{L^2(\mu )}^2+\mathrm{\parallel }\mu \mathrm{\parallel }\approx \int \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{\int }_0^{\mathrm{\infty }}{\beta }_{2,\mu }(x,r{)}^2\frac{\mu (B(x,r))}{r^n}\frac{dr}{r}d\mu (x)+\mathrm{\parallel }\mu \mathrm{\parallel },$$ where ${\beta }_{\mu ,2}(x,r{)}^2={\inf }_L\frac{1}{r^n}{\int }_{B(x,r)}{\left(\frac{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(y,L)}{r}\right)}^{2}d\mu (y),$ with the infimum taken over all affine $n$-planes $L\subset {\mathbb{R}}^{n+1}$. 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.