Admissibility and boundary values for Poletsky–Stessin Hardy spaces on general convex domains

Determine whether, for an arbitrary bounded convex domain Ω ⊂ ℂ^n and boundary Monge–Ampère measure μ_ρ defined by μ_ρ = lim_{r→0}(dd^c max{ρ,r})^n with ρ a convex exhaustion function of Ω, the space 𝒜(Ω) of holomorphic functions continuous up to the boundary is strongly admissible and whether functions in the associated Poletsky–Stessin Hardy space 𝔛(Ω, μ_ρ) admit boundary values in L^2(bΩ, μ_ρ).

Background

The paper discusses Hardy spaces on convex domains and their realization via integral transforms. For C^2-smooth convex domains with surface area measure, strong admissibility of 𝒜(Ω) and existence of boundary values are known, yielding a well-behaved reproducing kernel framework.

For general convex domains with boundary Monge–Ampère measures, the natural candidate for the Hardy space is the Poletsky–Stessin space. The authors note that it is unknown whether the strong admissibility of 𝒜(Ω) or boundary values for 𝔛(Ω, μ_ρ) hold in this general setting, which impacts the use of Leray and Laplace transforms in this broader context.