Unboundedness of potential dependent Riesz transforms for totally irregular measures

Abstract

We prove that, for totally irregular measures μ on Rd with d≥3, the (d−1)-dimensional Riesz transformTA,μVf(x)=∫Rd∇1EAV(x,y)f(y)dμ(y) adapted to the Schrödinger operator LAV=−divA∇+V with fundamental solution EAV is not bounded on L2(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 RHd. We achieve this by obtaining new exponential decay estimates for the kernel ∇1EAV as well as Hölder regularity estimates at local scales determined by the potential's critical radius function.