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

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 ℝ^B+1 with finite Hausdorff measure Hⁿ, if T_Hn|_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.