Let $A\left(\cdot \right)$ be an $\left(n+1\right)\times \left(n+1\right)$ uniformly elliptic matrix with Hölder continuous real coefficients and let ${\mathcal{ℰ}}_A\left(x,y\right)$ be the fundamental solution of the PDE $divA\left(\cdot \right)\nabla u=0$ in ${ℝ}^{n+1}$. Let $\mu$ be a compactly supported $n$-AD-regular measure in ${ℝ}^{n+1}$ and consider the associated operator

We show that if $T_{\mu }$ is bounded in $L^2\left(\mu \right)$, then $\mu$ 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 $E\subset {ℝ}^{n+1}$ with finite Hausdorff measure ${\mathcal{\mathscr{H}}}^n$, if $T_{{\mathcal{\mathscr{H}}}^n{|}_E}$ is bounded in $L^2\left({\mathcal{\mathscr{H}}}^n{|}_E\right)$, 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.

让 $\mathrm{一种}（\cdot ）$ 豆角，扁豆 $（ñ+\mathrm{1个}）\times （ñ+\mathrm{1个}）$ 具有Hölder连续实系数的均匀椭圆矩阵，并令 ${\mathcal{ℰ}}_{\mathrm{一种}}（X，ÿ）$ 是PDE的基本解决方案 $股利\mathrm{一种}（\cdot ）\nabla ü=0$ 在 ${ℝ}^{ñ+\mathrm{1个}}$。让$\mu$ 得到紧凑的支持 $ñ$-AD-常规测量 ${ℝ}^{ñ+\mathrm{1个}}$ 并考虑关联的运算符

我们证明，如果 ${Ť}_{\mu }$ 受到限制 ${\mathrm{大号}}^{\mathrm{2个}}（\mu ）$， 然后 $\mu$ 均匀地 $ñ$可纠正的。这扩展了余维的解决方案-$\mathrm{1个}$Riesz变换的David–Semmes问题转化为单层电势的梯度。加上Conde-Alonso，Mourgoglou和Tolsa的先前结果，这表明$E\subset {ℝ}^{ñ+\mathrm{1个}}$ 用有限的Hausdorff测度 ${\mathcal{\mathscr{H}}}^{ñ}$， 如果 ${Ť}_{{\mathcal{\mathscr{H}}}^{ñ}{|}_E}$ 受到限制 ${\mathrm{大号}}^{\mathrm{2个}}（{\mathcal{\mathscr{H}}}^{ñ}{|}_E）$， 然后 $E$ 是 $ñ$可纠正的。此外，作为一项应用，我们表明，如果与上述PDE关联的椭圆度量相对于表面度量绝对是连续的，则类似于谐波度量，它必须是可纠正的。