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

Damian Dąbrowski, Xavier Tolsa

TL;DR

This work provides a precise geometric characterization of measures $\mu$ in $\mathbb{R}^{n+1}$ with polynomial growth for which the $n$-dimensional Riesz transform $R_\mu$ is bounded on $L^2(\mu)$. The authors establish a sharp energy identity linking $\|R_\mu\|_{L^2(\mu)}^2$ to a Wolff-type energy expressed through $β_{2,μ}$ and $θ_μ$, and deduce corollaries on removable sets for Lipschitz harmonic functions and bilipschitz invariance. The approach hinges on a refined David–Mattila lattice, a corona-type decomposition into ${\mathsf{DB}}(M)$-dominated regions, and a novel variational argument on carefully constructed approximating measures supported on tractable trees. By transferring lower bounds from these approximations to the original measure via Haar coefficients, the paper connects geometric density fluctuations to $L^2$-boundedness of singular integrals beyond AD-regular settings. These results advance the Painlevé-type program for Lipschitz harmonic capacities and illuminate the relationship between rectifiability-type geometric quantities and singular integral boundedness in higher codimension.

Abstract

In this work we provide a geometric characterization of the measures $μ$ in $\mathbb R^{n+1}$ with polynomial upper growth of degree $n$ such that the $n$-dimensional Riesz transform $Rμ(x) = \int \frac{x-y}{|x-y|^{n+1}}\,dμ(y)$ belongs to $L^2(μ)$. More precisely, it is shown that $$\|Rμ\|_{L^2(μ)}^2 + \|μ\|\approx \int\!\!\int_0^\infty β_{2,μ}(x,r)^2\,\frac{μ(B(x,r))}{r^n}\,\frac{dr}r\,dμ(x) + \|μ\|,$$ where $β_{μ,2}(x,r)^2 = \inf_L \frac1{r^n}\int_{B(x,r)} \left(\frac{\mathrm{dist}(y,L)}r\right)^2\,dμ(y),$ 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.

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

TL;DR

This work provides a precise geometric characterization of measures in with polynomial growth for which the -dimensional Riesz transform is bounded on . The authors establish a sharp energy identity linking to a Wolff-type energy expressed through and , and deduce corollaries on removable sets for Lipschitz harmonic functions and bilipschitz invariance. The approach hinges on a refined David–Mattila lattice, a corona-type decomposition into -dominated regions, and a novel variational argument on carefully constructed approximating measures supported on tractable trees. By transferring lower bounds from these approximations to the original measure via Haar coefficients, the paper connects geometric density fluctuations to -boundedness of singular integrals beyond AD-regular settings. These results advance the Painlevé-type program for Lipschitz harmonic capacities and illuminate the relationship between rectifiability-type geometric quantities and singular integral boundedness in higher codimension.

Abstract

In this work we provide a geometric characterization of the measures in with polynomial upper growth of degree such that the -dimensional Riesz transform belongs to . More precisely, it is shown that where with the infimum taken over all affine -planes . 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.
Paper Structure (37 sections, 79 theorems, 993 equations)

This paper contains 37 sections, 79 theorems, 993 equations.

Table of Contents

  1. Introduction
  2. Sketch of the proof
  3. The modified dyadic lattice of David and Mattila
  4. P-doubling cubes, cubes dominated from below, and the Main Propositions
  5. P-doubling cubes and the family hdk(Q)
  6. The energies ${\mathcal{E}}$, ${\mathcal{E}}^H$, ${\mathcal{E}}_\infty$, and the cubes dominated from below
  7. The Main Propositions and the proof of Theorem \ref{['teomain1']}
  8. The cubes with moderate decrement of Wolff energy and the associated tractable trees
  9. The family MDW and the enlarged cubes
  10. Generalized trees and negligible cubes
  11. Tractable trees
  12. The Riesz transform on the tractable trees: the approximating measures eta, nu, and the variational argument
  13. The suppressed Riesz transform and a Cotlar type inequality
  14. The family Reg(e'(R)) and the approximating measure eta
  15. The auxiliary family H
  16. ...and 22 more sections

Key Result

Theorem 1.1

Let $\mu$ be a Radon measure in ${\mathbb R}^{n+1}$ satisfying the polynomial growth condition and such that $\|{\mathcal{R}}_*\mu\|_{L^1(\mu)}<\infty$. Then where $C$ is an absolute constant.

Theorems & Definitions (151)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem : JNRT
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Lemma 2.1: David, Mattila
  • Lemma 2.3
  • Lemma 2.4
  • ...and 141 more