Characterize the endpoint criterion from vertical increments

Determine a necessary-and-sufficient endpoint regularity criterion expressed solely through vertical increments for bounded hyperbolic harmonic mappings, distinguishing the critical Besov classes B_{\infty,1}^{1} and B_{\infty,\infty}^{1} without assuming boundary regularity in advance.

Background

The paper proves subcritical transversal Hölder criteria for bounded hyperbolic harmonic mappings and shows that uniform vertical Lipschitz control alone does not imply global Lipschitz regularity at the endpoint. A lacunary Poisson extension provides a counterexample whose boundary trace lies in the Zygmund class B_{\infty,\infty}^{1} but not in the summable Besov class B_{\infty,1}^{1}.

The authors recover global Lipschitz regularity by imposing a Dini--Zygmund summability condition, which is sufficient but explicitly not claimed to be necessary for every Lipschitz trace. The unresolved problem is to identify a criterion based only on vertical increments that is both necessary and sufficient and correctly separates the two endpoint Besov behaviors, without presupposing boundary regularity.