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

Abstract

In this work we provide a geometric characterization of the measures $\hat{I}¼$ in ${\mathbb{R}}^{n+1}$ with polynomial upper growth of degree $n$ such that the $n$-dimensional Riesz transform $R\hat{I}¼(x)=\int \frac{x-y}{|x-y{|}^{n+1}}d\hat{I}¼(y)$ belongs to $L^2(\hat{I}¼)$. More precisely, it is shown that $$\Vert R\hat{I}¼{\Vert }_{L^2(\hat{I}¼)}^2+\Vert \hat{I}¼\Vert \approx \int {\int }_0^{\infty }{\hat{I}}_{2,\hat{I}¼}^2(x,r{)}^2\frac{\hat{I}¼(B(x,r))}{r^n}\frac{dr}{r}d\hat{I}¼(x)+\Vert \hat{I}¼\Vert ,$$ where ${\hat{I}}_{\hat{I}¼,2}^2(x,r{)}^2={\inf }_L\frac{1}{r^n}{\int }_{B(x,r)}{\left(\frac{\mathrm{dist}(y,L)}{r}\right)}^{2}d\hat{I}¼(y),$ 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.

Minor typos corrected. Added a final appendix summarizing notation