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

Abstract

In this work we obtain 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 $\mathcal{R}\mu (x)=\int \frac{x-y}{|x-y{|}^{n+1}}d\mu (y)$ belongs to $L^2(\mu )$. More precisely, we show that $$|\mathcal{R}\mu|{L^2(\mu)}^2 + |\mu|\approx \int\int_0^\infty \beta{\mu,2}(x,r)^2\frac{\mu(B(x,r))}{r^n}\frac{dr}r d\mu(x) + |\mu|,$$ where ${\beta }_{\mu ,2}(x,r{)}^2=\underset{L}{inf}\frac{1}{r^n}{\int }_{B(x,r)}{\left(\frac{\mathrm{dist}(y,L)}{r}\right)}^{2}d\mu (y),$ 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.