Optimality of the estimate in Theorem \ref{thm-Domar-A-B}

Determine whether, under the assumptions of Theorem \ref{thm-Domar-A-B} (bounded open Ω ⊂ R^k; closed B ⊂ R^k; compact p_*‑admissible A ⊂ Ω with m(A)=0; continuously differentiable g; and the constructed functions \mu_{ad}, \mu_{ad}^{−}, and \rho_{ad}), the bound u(x) ≤ a^{\rho_{ad}^{−}(dist(x, B)/(3D))} for x ∈ Ω \ B is optimal, or whether it can be improved under the same hypotheses.

Background

Theorem \ref{thm-Domar-A-B} applies quantitative Domar methods to the paper’s main growth-transfer problem, producing an explicit bound for subharmonic u in terms of distance to B with a function \rho_{ad} derived from the admissibility of A and the growth of g.

The authors question whether this derived bound represents a sharp estimate or if stronger bounds (e.g., closer to g(C dist(x,B))) are attainable without strengthening assumptions.