Unboundedness of potential dependent Riesz transforms for totally irregular measures

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

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