Existence of a regular g with stronger-than-power singularity yielding a direct distance-to-B bound

Construct a regular function g: (0, ∞) → (0, ∞) whose singularity at 0^+ dominates every power t^{−b} (for all b > 0), and identify a nontrivial class of sets A ⊂ Ω for which the growth control u(x) ≤ g(dist(x, A ∪ B)) on Ω(A ∪ B) implies the direct bound u(x) ≤ g(v dist(x, B)) on Ω \ B for some constant v ∈ (0,1).

Background

Example \ref{ex:fast-growth-g} and Remark \ref{rem:worse-s-b} indicate that when g grows faster than any power near 0+, quantitative Domar methods typically produce bounds that are much worse than the desired form g(v dist(x,B)), and may fail to yield such a direct bound.

The authors ask whether there nevertheless exists some regular g with stronger-than-power singularity for which the desired direct distance-to-B estimate holds for a meaningful class of sets A.

References

We end the article with the following open questions. In particular, does there exist a regular function g(t), whose singularity at 0+ is stronger than any power t{-b}, such that, for a class of sets A, eq:estim-u1 implies an estimate u(x)\le g(v dist(x,B)), where v\in (0,1) is a constant?

eq:estim-u1:

u(x)≤g(dist(x,A∪B)),x∈Ω(A∪B),u(x)\le g(dist(x,A\cup B)), \quad x\in\Omega(A\cup B),

— Self-improving estimates of growth of subharmonic and analytic functions (2508.04496 - Bello et al., 6 Aug 2025) in Question 4, end of Section 5 (Application of quantitative Domar's results to our problem)