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.
|Number of pages||75|
|Journal||Analysis and PDE|
|Publication status||Published - 2021|
- David-Semmes problem
- elliptic measure
- gradient of the single-layer potential
- uniform rectifiability