<p>For plurisubharmonic <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_166_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> in a bounded pseudoconvex <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_166_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_166_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb C^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="double-struck">C</mi> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>, we analyze the lower bound for the Bergman kernel (on the diagonal, with weight <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_166_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">$e^{-\varphi }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>φ</mi> </mrow> </msup> </math></EquationSource> </InlineEquation>) of the form <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_166_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="98" /> </InlineMediaObject> <EquationSource Format="TEX">$K_{\Omega ,\varphi }\ge e^{\varphi }/C$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>K</mi> <mrow> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi>φ</mi> </mrow> </msub> <mo>≥</mo> <msup> <mi>e</mi> <mi>φ</mi> </msup> <mo stretchy="false">/</mo> <mi>C</mi> </mrow> </math></EquationSource> </InlineEquation>. By the Ohsawa–Takegoshi extension theorem, there exists such a constant depending on <i>n</i> and the diameter of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_166_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>. We give a proof of this using Hörmander’s <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_166_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> estimate for <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_166_Article_IEq8.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\bar{\partial }$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mrow> <mi>∂</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> </math></EquationSource> </InlineEquation> directly. We also show that one can improve the constant to <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_166_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">$C=|\Omega |$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mo>=</mo> <mo stretchy="false">|</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation> if <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_166_Article_IEq10.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$n=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and to <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_166_Article_IEq11.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">$C=2\,|\Omega |$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mo>=</mo> <mn>2</mn> <mspace width="0.166667em" /> <mo stretchy="false">|</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation> if <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_166_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is convex, where <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_166_Article_IEq13.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">$|\Omega |$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation> denotes the volume (or area) of <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_166_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>. We conjecture that the former holds in arbitrary dimension (it trivially does in the unweighted case).</p>

On the pointwise Ohsawa–Takegoshi theorem

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2025年9月24日

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Zbigniew Błocki

摘要

For plurisubharmonic $\varphi $ $φ$ in a bounded pseudoconvex $\Omega $ $Ω$ in $\mathbb C^n$ $C^{n}$ , we analyze the lower bound for the Bergman kernel (on the diagonal, with weight $e^{-\varphi }$ $e^{- φ}$ ) of the form $K_{\Omega ,\varphi }\ge e^{\varphi }/C$ $K_{Ω, φ} \geq e^{φ} / C$ . By the Ohsawa–Takegoshi extension theorem, there exists such a constant depending on n and the diameter of $\Omega $ $Ω$ . We give a proof of this using Hörmander’s $L^2$ $L^{2}$ estimate for $\bar{\partial }$ $\bar{\partial}$ directly. We also show that one can improve the constant to $C=|\Omega |$ $C = | Ω |$ if $n=1$ $n = 1$ and to $C=2\,|\Omega |$ $C = 2 | Ω |$ if $\Omega $ $Ω$ is convex, where $|\Omega |$ $| Ω |$ denotes the volume (or area) of $\Omega $ $Ω$ . We conjecture that the former holds in arbitrary dimension (it trivially does in the unweighted case).