Existence of C^2-smooth domains in ℂ^2 where the Laplace transform range differs from A^2(ℂ^2, ωΩ)

Determine whether there exist bounded C^2-smooth domains Ω ⊂ ℂ^2 for which the range of the Laplace transform on the Hardy space 𝓗^2(Ω, μ_Ω), with μ_Ω the boundary Monge–Ampère measure associated to the Minkowski functional of Ω, is not equal to the weighted Bergman space A^2(ℂ^2, ω_Ω), where ω_Ω(z) = ∥e^{⟨z,·⟩}∥^{-2}_{L^2(bΩ, μ_Ω)} (dd^c H_Ω)^2(z).

Background

The paper provides a counterexample for the ℓ^1 ball in ℂ^2 showing strict inclusions among spaces defined via different measures, highlighting that the Lindholm-type characterization does not extend universally in higher dimensions.

Despite this, the authors state they do not know of any C^2-smooth domains in ℂ^2 where the image of the Laplace transform on 𝓗^2(Ω, μΩ) fails to coincide with A^2(ℂ^2, ωΩ), leaving open whether such domains exist or whether equality might hold universally within this regularity class.