L2-boundedness of gradients of single-layer potentials and uniform rectifiability

Prat, Laura; Puliatti, Carmelo; Tolsa, Xavier

doi:10.2140/apde.2021.14.717

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$L^2$-boundedness of gradients of single-layer potentials and uniform rectifiability

Laura Prat, Carmelo Puliatti and Xavier Tolsa

Vol. 14 (2021), No. 3, 717–791

DOI: 10.2140/apde.2021.14.717

Abstract

Let $A (\cdot)$ be an $(n + 1) \times (n + 1)$ uniformly elliptic matrix with Hölder continuous real coefficients and let $ℰ_{A} (x, y)$ be the fundamental solution of the PDE $div A (\cdot) \nabla u = 0$ in $ℝ^{n + 1}$ . Let $μ$ be a compactly supported $n$ -AD-regular measure in $ℝ^{n + 1}$ and consider the associated operator

T_{μ} f (x) = \int \nabla_{x} ℰ_{A} (x, y) f (y) d μ (y) .

We show that if $T_{μ}$ is bounded in $L^{2} (μ)$ , then $μ$ is uniformly $n$ -rectifiable. This extends the solution of the codimension- $1$ David–Semmes problem for the Riesz transform to the gradient of the single-layer potential. Together with a previous result of Conde-Alonso, Mourgoglou and Tolsa, this shows that, given $E \subset ℝ^{n + 1}$ with finite Hausdorff measure $ℋ^{n}$ , if $T_{ℋ^{n} |_{E}}$ is bounded in $L^{2} (ℋ^{n} |_{E})$ , then $E$ is $n$ -rectifiable. Further, as an application we show that if the elliptic measure associated to the above PDE is absolutely continuous with respect to surface measure, then it must be rectifiable, analogously to what happens with harmonic measure.

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Keywords

David–Semmes problem, uniform rectifiability, gradient of the single-layer potential, elliptic measure, rectifiability

Mathematical Subject Classification 2010

Primary: 28A75, 31B15, 35J15, 42B37

Milestones

Received: 26 November 2018

Revised: 5 September 2019

Accepted: 21 November 2019

Published: 18 May 2021

Authors

Laura Prat
Departament de Matemàtiques Universitat Autònoma de Barcelona and Centre de Recerca Matemàtica Bellaterra (Barcelona) Catalonia

Carmelo Puliatti
Departament de Matemàtiques and BGSMath Universitat Autònoma de Barcelona Bellaterra (Barcelona) Catalonia

Xavier Tolsa
ICREA Bellaterra (Barcelona) Catalonia	Departament de Matemàtiques and BGSMath Universitat Autònoma de Barcelona Bellaterra (Barcelona) Catalonia

Vol. 14, No. 3, 2021