The measures with $L^2$-bounded Riesz transform satisfying a subcritical Wolff-type energy condition

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 $L^2(\mu)$, under the assumption that $\mu$ satisfies the following Wolff energy estimate, for any ball $B\subset\mathbb{R}^{n+1}$: $$\int_B \int_0^\infty \left(\frac{\mu(B(x,r))}{r^{n-\frac38}}\right)^2\,\frac{dr}r\,d\mu(x)\leq M\,\bigg(\frac{\mu(2B)}{r(B)^{n-\frac38}}\bigg)^2\,\mu(2B).$$ More precisely, we show that $\mu$ satisfies the following estimate: $$\|\mathcal{R}\mu\|_{L^2(\mu)}^2 + \|\mu\|\approx \int\!\!\int_0^\infty \beta_{\mu,2}(x,r)^2\,\frac{\mu(B(x,r))}{r^n}\,\frac{dr}r\,d\mu(x) + \|\mu\|,$$ where $\beta_{\mu,2}(x,r)^2 = \inf_L \frac1{r^n}\int_{B(x,r)} \left(\frac{\mathrm{dist}(y,L)}r\right)^2\,d\mu(y),$ with the infimum taken over all affine $n$-planes $L\subset\mathbb{R}^{n+1}$. In a companion paper which relies on the results obtained in this work it is shown that the same result holds without the above assumption regarding the Wolff energy of $\mu$. This result has important consequences for the Painlev\'e problem for Lipschitz harmonic functions.