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Volume 41, Issue 1
Distributional Boundary Values of Holomorphic Functions on Tubular Domains

Guantie Deng & Weiwei Wang

Anal. Theory Appl., 41 (2025), pp. 35-51.

Published online: 2025-04

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  • Abstract

The main purpose of this paper is to establish the distributional boundary values of functions in the weighted Hardy space, which improves the results of Carmichael in [4] and [8], where the weight function is linear. As our main result, we will prove that $f(z)$ in $H(ψ, Γ)$ has the $\mathcal{Z}'$ boundary value and can be expressed by the inverse Fourier transform of a distribution. Next, we will establish the $S'$ boundary value 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 inverse result, in which we will prove that $f(z)$ is holomorphic on $T_Γ.$

  • AMS Subject Headings

32A07, 32A40, 42B25, 42B30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{ATA-41-35, author = {Deng , Guantie and Wang , Weiwei}, title = {Distributional Boundary Values of Holomorphic Functions on Tubular Domains}, journal = {Analysis in Theory and Applications}, year = {2025}, volume = {41}, number = {1}, pages = {35--51}, abstract = {

The main purpose of this paper is to establish the distributional boundary values of functions in the weighted Hardy space, which improves the results of Carmichael in [4] and [8], where the weight function is linear. As our main result, we will prove that $f(z)$ in $H(ψ, Γ)$ has the $\mathcal{Z}'$ boundary value and can be expressed by the inverse Fourier transform of a distribution. Next, we will establish the $S'$ boundary value 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 inverse result, in which we will prove that $f(z)$ is holomorphic on $T_Γ.$

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-2022-0017}, url = {http://global-sci.org/intro/article_detail/ata/23957.html} }
TY - JOUR T1 - Distributional Boundary Values of Holomorphic Functions on Tubular Domains AU - Deng , Guantie AU - Wang , Weiwei JO - Analysis in Theory and Applications VL - 1 SP - 35 EP - 51 PY - 2025 DA - 2025/04 SN - 41 DO - http://doi.org/10.4208/ata.OA-2022-0017 UR - https://global-sci.org/intro/article_detail/ata/23957.html KW - The weighted Hardy space, distributional boundary values, tubular domains. AB -

The main purpose of this paper is to establish the distributional boundary values of functions in the weighted Hardy space, which improves the results of Carmichael in [4] and [8], where the weight function is linear. As our main result, we will prove that $f(z)$ in $H(ψ, Γ)$ has the $\mathcal{Z}'$ boundary value and can be expressed by the inverse Fourier transform of a distribution. Next, we will establish the $S'$ boundary value 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 inverse result, in which we will prove that $f(z)$ is holomorphic on $T_Γ.$

Deng , Guantie and Wang , Weiwei. (2025). Distributional Boundary Values of Holomorphic Functions on Tubular Domains. Analysis in Theory and Applications. 41 (1). 35-51. doi:10.4208/ata.OA-2022-0017
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