Let be an uniformly elliptic matrix with Hölder continuous real coefficients and let be the fundamental solution of the PDE in . Let be a compactly supported -AD-regular measure in and consider the associated operator
We show that if is bounded in , then is uniformly -rectifiable. This extends the solution of the codimension- 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 with finite Hausdorff measure , if is bounded in , then is -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.
"-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