Unboundedness of potential dependent Riesz transforms for totally irregular measures

Abstract

We prove that, for totally irregular measures $\mu$ on ${\mathbb{R}}^d$ with $d\ge 3$, 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 Schrödinger 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 Hölder continuous coefficients $A$ since it allows for the presence of potentials $V$ in the reverse Hölder class $R{H}_d$. 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.