Necessity of the Carleson condition for Hölder density transformations

Determine whether the Carleson condition for the transformed square function \(\Delta^F_\mu\) can fail for a uniformly \(n\)-rectifiable measure \(\mu\) and a Hölder-continuous function \(F\) that is not Lipschitz on \(J_0=[c_0^{-1},c_0]\).

Background

For an $n$-AD-regular measure with infinite total mass, the paper proves that a uniformly $n$-rectifiable measure satisfies an $L^{2/\alpha}$ estimate for $\Delta^F_\mu$ when $F$ is strictly increasing and $\alpha$-Hölder on the density interval $J_0$. This does not yield the desired exponent-2 Carleson estimate when $\alpha<1$, because the known Carleson estimate for $\Delta_\mu$ does not control the corresponding lower power of $|\Delta_\mu|$. The authors explicitly leave unresolved whether the exponent-2 condition can fail in this setting.