For every <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_150_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">$0<r<\frac{1}{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>r</mi> <mo><</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, we will construct a flat Kähler manifold M and a relatively compact domain with smooth boundary <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_150_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega \subset M$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <mi>M</mi> </mrow> </math></EquationSource> </InlineEquation> that is Stein but not hyperconvex such that the Bergman projection P on <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_150_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is regular in the <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_150_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> Sobolev space <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_150_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$W^s(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>W</mi> <mi>s</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_150_Article_IEq6.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">$0\le s<r$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>s</mi> <mo><</mo> <mi>r</mi> </mrow> </math></EquationSource> </InlineEquation> but irregular in <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_150_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$W^r(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>W</mi> <mi>r</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. On these domains, we will also construct <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_150_Article_IEq8.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">$f\in C^\infty ({\overline{\Omega }})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <msup> <mi>C</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mover> <mi mathvariant="normal">Ω</mi> <mo>¯</mo> </mover> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_150_Article_IEq9.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="96" /> </InlineMediaObject> <EquationSource Format="TEX">$Pf\notin C^\infty ({\overline{\Omega }})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mi>f</mi> <mo>∉</mo> <msup> <mi>C</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mover> <mi mathvariant="normal">Ω</mi> <mo>¯</mo> </mover> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We will prove the same result for the invariant Bergman projection on (2, 0)-forms. These domains are modeled on a construction of Diederich and Ohsawa.

Sobolev regularity of the Bergman projection on a smoothly bounded Stein domain that is not hyperconvex

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2025年3月11日

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Phillip S. Harrington

摘要

For every $0<r<\frac{1}{2}$ $0 < r < \frac{1}{2}$ , we will construct a flat Kähler manifold M and a relatively compact domain with smooth boundary $\Omega \subset M$ $Ω \subset M$ that is Stein but not hyperconvex such that the Bergman projection P on $\Omega $ $Ω$ is regular in the $L^2$ $L^{2}$ Sobolev space $W^s(\Omega )$ $W^{s} (Ω)$ for all $0\le s<r$ $0 \leq s < r$ but irregular in $W^r(\Omega )$ $W^{r} (Ω)$ . On these domains, we will also construct $f\in C^\infty ({\overline{\Omega }})$ $f \in C^{\infty} (\bar{Ω})$ such that $Pf\notin C^\infty ({\overline{\Omega }})$ $P f \notin C^{\infty} (\bar{Ω})$ . We will prove the same result for the invariant Bergman projection on (2, 0)-forms. These domains are modeled on a construction of Diederich and Ohsawa.