We study the density of functions which are holomorphic in a neighbourhood of the closure <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3105_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\overline{\Omega }$</EquationSource> <EquationSource Format="MATHML"><math> <mover> <mi mathvariant="normal">Ω</mi> <mo>¯</mo> </mover> </math></EquationSource> </InlineEquation> of a bounded non-smooth pseudoconvex domain <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3105_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>, in the Bergman space <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3105_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$ H^2(\Omega ,\varphi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with a plurisubharmonic weight function <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3105_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation>. As an application, we show that the Hartogs domain <Equation ID="Equ26"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3105_Article_Equ26.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="341" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \Omega _\alpha : = \{(z,w) \in D\times \mathbb {C}: |w|< \delta ^\alpha _D(z) \}, \ \ \ \alpha >0, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi mathvariant="normal">Ω</mi> <mi>α</mi> </msub> <mo>:</mo> <mo>=</mo> <mrow> <mo stretchy="false">{</mo> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <mi>D</mi> <mo>×</mo> <mi mathvariant="double-struck">C</mi> <mo>:</mo> <mo stretchy="false">|</mo> <mi>w</mi> <mo stretchy="false">|</mo> <mo><</mo> </mrow> <msubsup> <mi>δ</mi> <mi>D</mi> <mi>α</mi> </msubsup> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">}</mo> <mo>,</mo> <mspace width="4pt" /> <mspace width="4pt" /> <mspace width="4pt" /> <mi>α</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3105_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="63" /> </InlineMediaObject> <EquationSource Format="TEX">$D\subset \subset \mathbb {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mo>⊂</mo> <mo>⊂</mo> <mi mathvariant="double-struck">C</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3105_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta _D$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>δ</mi> <mi>D</mi> </msub> </math></EquationSource> </InlineEquation> denotes the boundary distance to D, is Bergman complete if and only if every boundary point of D is non-isolated.

Density in weighted Bergman spaces and Bergman completeness of Hartogs domains

期刊论文
发表日期：

2025年2月7日

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Bo-Yong Chen,
John Erik Fornæss,
Jujie Wu

摘要

We study the density of functions which are holomorphic in a neighbourhood of the closure $\overline{\Omega }$ $\bar{Ω}$ of a bounded non-smooth pseudoconvex domain $\Omega $ $Ω$ , in the Bergman space $ H^2(\Omega ,\varphi )$ $H^{2} (Ω, φ)$ with a plurisubharmonic weight function $\varphi $ $φ$ . As an application, we show that the Hartogs domain $\begin{aligned} \Omega _\alpha : = \{(z,w) \in D\times \mathbb {C}: |w|< \delta ^\alpha _D(z) \}, \ \ \ \alpha >0, \end{aligned}$ $\begin{matrix} Ω_{α} : = {(z, w) \in D \times C : | w | < δ_{D}^{α} (z)}, α > 0, \end{matrix}$ where $D\subset \subset \mathbb {C}$ $D \subset \subset C$ and $\delta _D$ $δ_{D}$ denotes the boundary distance to D, is Bergman complete if and only if every boundary point of D is non-isolated.