Distributional Boundary Values of Holomorphic Functions on Tubular Domains (Analysis in Theory and Applications, 2025)
The main purpose of this paper is to establish the distributional boundary values of functions in the weighted Hardy space, which improves the results ofCarmichael in [4] and [8], where the weight function is linear. As our main result, wewill prove that $f(z)$ in $H(ψ, Γ)$ has the $\mathcal{Z}'$ boundary value and can be expressed by theinverse Fourier transform of a distribution. Next, we will establish the $S'$ boundaryvalue under stronger assumptions and give more precise expression if $f(z)$ also converges to $U ∈ D'_{L^p}(\mathbb{R}^n),$ where $1 ≤ p ≤ 2.$ In addition, we will also study the inverseresult, in which we will prove that $f(z)$ is holomorphic on $T_Γ.$