Vol. 14, No. 3, 2021

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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
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 E n+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.

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