The measures with $L^2$-bounded Riesz transform and the Painlev\'e problem for Lipschitz harmonic functions

arXiv2106.00680MaRDI QIDQ6369163

Xavier Tolsa

Publication date: 1 June 2021

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 $Rmu(x)=intfracx-y|x-y{|}^{n+1},dmu(y)$ belongs to $L^2(mu)$. More precisely, it is shown that ${\displaystyle }$|Rmu|_{L^2(mu)}^2 + |mu|approx int!!int_0^infty �eta_{2,mu}(x,r)^2,frac{mu(B(x,r))}{r^n},frac{dr}r,dmu(x) + |mu|,${\displaystyle }$ where $$ with the infimum taken over all affine $n$-planes $Lsubsetmathbb{R}^{n+1}$. 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.

Mathematics Subject Classification ID

Singular and oscillatory integrals (Calderón-Zygmund, etc.) (42B20) Geometric measure and integration theory, integral and normal currents in optimization (49Q15) Length, area, volume, other geometric measure theory (28A75)

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