Riesz transform boundedness for a logarithmically corrected small-scale function

Determine whether the Riesz transform associated with a uniform local tree satisfying uniform volume growth, heat kernel estimates, and heat kernel gradient estimates is bounded on $L^p$ when the scale function satisfies $\Psi(r)\asymp r^2(\log(e/r))^{-1}$ near zero, a case not covered by the local Dini condition or the locally Gaussian condition.

Background

Theorem 1.1 establishes $L^p$ boundedness of the Riesz transform for all $p\geq 2$ under eventual Gaussianity together with either a local Dini condition on the scale function or local Gaussian behavior. The authors note that these two alternatives do not exhaust all possible small-scale behaviors.

As a borderline example, the scale function $\Psi(r)\asymp r^2(\log(e/r))^{-1}$ near zero satisfies neither of the sufficient hypotheses. The paper therefore leaves unresolved whether the corresponding Riesz transform is bounded in this case.