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2021 L2-boundedness of gradients of single-layer potentials and uniform rectifiability
Laura Prat, Carmelo Puliatti, Xavier Tolsa
Anal. PDE 14(3): 717-791 (2021). DOI: 10.2140/apde.2021.14.717

Abstract

Let A() be an (n+1)×(n+1) uniformly elliptic matrix with Hölder continuous real coefficients and let A(x,y) be the fundamental solution of the PDE divA()u=0 in n+1. Let μ be a compactly supported n-AD-regular measure in n+1 and consider the associated operator

Tμf(x)=xA(x,y)f(y)dμ(y).

We show that if Tμ is bounded in L2(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 En+1 with finite Hausdorff measure n, if Tn|E is bounded in L2(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.

Citation

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Laura Prat. Carmelo Puliatti. Xavier Tolsa. "L2-boundedness of gradients of single-layer potentials and uniform rectifiability." Anal. PDE 14 (3) 717 - 791, 2021. https://doi.org/10.2140/apde.2021.14.717

Information

Received: 26 November 2018; Revised: 5 September 2019; Accepted: 21 November 2019; Published: 2021
First available in Project Euclid: 25 June 2021

Digital Object Identifier: 10.2140/apde.2021.14.717

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

Rights: Copyright © 2021 Mathematical Sciences Publishers

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Vol.14 • No. 3 • 2021
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