TY - JOUR

T1 - L2-BOUNDEDNESS OF GRADIENTS OF SINGLE-LAYER POTENTIALS AND UNIFORM RECTIFIABILITY

AU - Prat, Laura

AU - Puliatti, Carmelo

AU - Tolsa, Xavier

N1 - Funding Information:
All the authors were partially supported by 2017-SGR-0395 (Catalonia) and MTM-2016-77635-P (MICINN, Spain). Tolsa was also partially supported by the ERC grant 320501 of the European Research Council (FP7/2007-2013). Puliatti is also partially supported by the grant MDM-2014-0445 (MINECO through the María de Maeztu Programme for Units of Excellence in R&D, Spain).
Publisher Copyright:
© 2021 Mathematical Sciences Publishers. All Rights Reserved.

PY - 2021

Y1 - 2021

N2 - Let A(·) be an (n+1)x(n+1) uniformly elliptic matrix with Holder continuous real coefficients and let εA(x, y) be the fundamental solution of the PDE div A(·)(Formula presented)u = 0 in ℝn+1. Let µ. be a compactly supported n-AD-regular measure in ℝn+1 and consider the associated operator (Formula presented) We show that if Tµ is bounded in L2 (µ), then p. 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 c ℝB+1 with finite Hausdorff measure Hn, if THn|E is bounded in L2(Hn |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.

AB - Let A(·) be an (n+1)x(n+1) uniformly elliptic matrix with Holder continuous real coefficients and let εA(x, y) be the fundamental solution of the PDE div A(·)(Formula presented)u = 0 in ℝn+1. Let µ. be a compactly supported n-AD-regular measure in ℝn+1 and consider the associated operator (Formula presented) We show that if Tµ is bounded in L2 (µ), then p. 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 c ℝB+1 with finite Hausdorff measure Hn, if THn|E is bounded in L2(Hn |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.

KW - David-Semmes problem

KW - elliptic measure

KW - gradient of the single-layer potential

KW - rectifiability

KW - uniform rectifiability

UR - http://www.scopus.com/inward/record.url?scp=85109956233&partnerID=8YFLogxK

U2 - 10.2140/apde.2021.14.717

DO - 10.2140/apde.2021.14.717

M3 - Article

AN - SCOPUS:85109956233

SN - 2157-5045

VL - 14

SP - 717

EP - 791

JO - Analysis and PDE

JF - Analysis and PDE

IS - 3

ER -