Propagation estimates for perturbed fractional operators below the current threshold

Establish weighted propagation estimates for $H_\kappa=L_\kappa-V$ when $0<\alpha<1$ and $V$ is a decaying external potential, in particular estimates that would support the nonexistence analysis for long-range perturbations without relying on the restriction $\alpha\geq 1/2$.

Background

For α<1\alpha<1, the paper proves weighted propagation bounds only for the unperturbed Hardy operator LκL_\kappa. The proof for an external potential would require a suitable analogue of the resolvent estimates used for α≥1\alpha\geq1.

The missing estimates are relevant to extending the relative-wave-operator nonexistence theorem to smaller values of α\alpha. The authors also indicate that the threshold α>1/2\alpha>1/2, with the endpoint α=1/2\alpha=1/2 available in dimensions d≥2d\geq2, is likely technical rather than intrinsic.

References

We note that a slightly longer argument yields analogous estimates for $\alpha\in(0,1)$ and $\mu\in(0,\alpha)$. However, we were unable to use it to prove corresponding propagation bounds for $\alpha<1$; in fact, in Lemma~\ref{lem9-small-alpha}, we will prove weighted propagation bounds for $L_\kappa$ whose proof proceeds rather differently.

— On wave operators for Hardy operators with and without perturbations  (2610.08594 - Ishida et al., 6 Oct 2026) in Section 1, subsection “Scattering theory for more general fractional operators and Hardy potentials” and Section 1, subsection “Weighted propagation estimates”