The measures with $L^2$-bounded Riesz transform and the Painlevé problem for Lipschitz harmonic functions

Abstract

This work provides 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 $R\mu (x)=\int \frac{x-y}{|x-y{|}^{n+1}}d\mu (y)$ belongs to $L^2(\mu )$. More precisely, it is shown that $$\Vert R\mu {\Vert }_{L^2(\mu )}^2+\Vert \mu \Vert \approx \int {\int }_0^{\infty }{\beta }_{2,\mu }(x,r{)}^2\frac{\mu (B(x,r))}{r^n}\frac{dr}{r}d\mu (x)+\Vert \mu \Vert ,$$ where ${\beta }_{\mu ,2}(x,r{)}^2={\inf }_L\frac{1}{r^n}{\int }_{B(x,r)}{\left(\frac{dist(y,L)}{r}\right)}^{2}d\mu (y),$ with the infimum taken over all affine $n$-planes $L\subset {\mathbb{R}}^{n+1}$. As a corollary, one obtains a characterization of the removable sets for Lipschitz harmonic functions in terms of a metric-geometric potential and one deduces that the class of removable sets for Lipschitz harmonic functions is invariant by bilipschitz mappings.

An additional corollary is written in the Introduction. arXiv admin note: text overlap with arXiv:2106.00303

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 ${\mathbb{R}}^d$