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 $B\subset {\mathbb{R}}^{n+1}$: $${\int }_B{\int }_0^{\infty }{\left(\frac{\mu (B(x,r))}{r^{n-\frac{3}{8}}}\right)}^2\frac{dr}{r}d\mu (x)\le M(\frac{\mu (2B)}{r(B{)}^{n-\frac{3}{8}}}{)}^2\mu (2B).$$ More precisely, we show that $\mu$ satisfies the following estimate: $$\Vert \mathcal{R}\mu {\Vert }_{L^2(\mu )}^2+\Vert \mu \Vert \approx \int {\int }_0^{\infty }{\beta }_{\mu ,2}(x,r{)}^2\frac{\mu (B(x,r))}{r^n}\frac{dr}{r}d\mu (x)+\Vert \mu \Vert ,$$ where ${\beta }_{\mu ,2}(x,r{)}^2={\inf }_L\frac{1}{r^n}{\int }_{B(x,r)}{\left(\frac{\mathrm{dist}(y,L)}{r}\right)}^{2}d\mu (y),$ with the infimum taken over all affine $n$-planes $L\subset {\mathbb{R}}^{n+1}$. 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 Painlevé problem for Lipschitz harmonic functions.

111 pages

References in corpus (1)

• Geometric conditions for the $L^2$-boundedness of singular integral operators with odd kernels with respect to measures with polynomial growth in ${\mathbb{R}}^d$