We prove that, for totally irregular measures $\mu$ on $\mathbb{R}^{d}$ with $d\geq3$, the $(d-1)$-dimensional Riesz transform $$ T_{A,\mu}^{V}f(x) = \int_{\mathbb{R}^d} \nabla_{1}\mathcal{E}_{A}^{V}(x,y) f(y) \, d \mu(y) $$ adapted to the Schrodinger operator $L_{A}^{V} = -\mathrm{div} A \nabla + V$ with fundamental solution $\mathcal{E}_{A}^{V}$ is not bounded on $L^{2}(\mu)$. This generalises recent results obtained by Conde-Alonso, Mourgoglou and Tolsa for free-space elliptic operators with Holder continuous coefficients $A$ since it allows for the presence of potentials $V$ in the reverse Holder class $RH_{d}$. We achieve this by obtaining new exponential decay estimates for the kernel $\nabla_{1} \mathcal{E}_{A}^{V}$ as well as Holder regularity estimates at local scales determined by the potential's critical radius function.

中文翻译：

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完全不规则测度的势能相关 Riesz 变换的无界性

我们证明，对于 $\mathbb{R}^{d}$ 和 $d\geq3$ 上的完全不规则测度 $\mu$，$(d-1)$ 维 Riesz 变换 $$ T_{A,\ mu}^{V}f(x) = \int_{\mathbb{R}^d} \nabla_{1}\mathcal{E}_{A}^{V}(x,y) f(y) \ , d \mu(y) $$ 适应薛定谔算子 $L_{A}^{V} = -\mathrm{div} A \nabla + V$ 基本解 $\mathcal{E}_{A}^ {V}$ 不受 $L^{2}(\mu)$ 的限制。这概括了 Conde-Alonso、Mourgoglou 和 Tolsa 最近获得的具有 Holder 连续系数 $A$ 的自由空间椭圆算子的结果，因为它允许在反向 Holder 类 $RH_{d}$ 中存在势 $V$。我们通过获得内核 $\nabla_{1} \mathcal{E}_{A}^{V}$ 的新指数衰减估计以及由潜在临界半径函数确定的局部尺度的 Holder 正则性估计来实现这一点。