Optimality of the quantitative Domar bounds

Ascertain whether the quantitative bounds on M(x) obtained in Theorem \ref{thm-Domar-2-quant}, namely M(x) ≤ a^{\varphi^{-}(D^{-1} dist(x, ∂Ω))} with \varphi(t) = \delta(t)\psi(t) defined from the distribution function of F, and in Theorem \ref{thm-Domar-3-quant}, namely M(x) ≤ a^{\rho^{-}(D^{-1} dist(x, ∂Ω))} with \rho(t) = \int_0^{\mu_{q_*}(t−\lambda)} (\mu_{q_*}^{−}(s) − t + 1 + \lambda) ds^{1/q_*}, are optimal in their dependence on dist(x, ∂Ω) and the majorant F under the stated integrability assumptions, or whether they can be uniformly improved.

Background

Domar’s classical theorems provide conditions ensuring boundedness of the extremal subharmonic envelope M(x) defined by a majorant F. The authors derive quantitative versions (Theorems \ref{thm-Domar-2-quant} and \ref{thm-Domar-3-quant}) that give explicit control of M(x) in terms of distance to the boundary and distributional data of F.

The open question seeks to determine whether these explicit rates are sharp or admit improvements while retaining the same assumptions.