Unboundedness of potential dependent Riesz transforms for totally irregular measures

We prove that, for totally irregular measures mu on R-d with d >= 3, the (d - 1)-dimensional Riesz transform T(A,mu)(V)f(x) = integral(Rd) del(1)epsilon(V)(A) (x,y) f(y) d mu(y) adapted to the Schrodinger operator L-A(V) = -divA del + V with fundamental solution epsilon(V)(A) 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 RHd. We achieve this by obtaining new exponential decay estimates for the kernel del(1)epsilon(V)(A) as well as Holder regularity estimates at local scales determined by the potential's critical radius function. (C) 2020 Elsevier Inc. All rights reserved.

Automated summary

This study demonstrates that the Riesz transform corresponding to a specific Schrodinger operator in spaces with totally irregular measures is not bounded in the L-2 space, even with the presence of potentials. New exponential decay estimates for the kernel of the Riesz transform were obtained, along with Holder regularity estimates at local scales determined by the critical radius function of the potential.