<p>We construct a global homotopy formula for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2122_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$a_q$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>a</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation> domains in a complex manifold. The homotopy operators in the formula will gain 1/2 derivative in Hölder-Zygmund spaces <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2122_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$\Lambda ^{r}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="normal">Λ</mi> <mi>r</mi> </msup> </math></EquationSource> </InlineEquation> when the boundaries of the domains are in <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2122_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">$\Lambda ^{r+3}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="normal">Λ</mi> <mrow> <mi>r</mi> <mo>+</mo> <mn>3</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2122_Article_IEq7.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$r>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

A Homotopy Formula for $a_q$ Domains in Complex Manifolds

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2025年8月7日

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Xianghong Gong,
Ziming Shi

摘要

We construct a global homotopy formula for $a_q$ $a_{q}$ domains in a complex manifold. The homotopy operators in the formula will gain 1/2 derivative in Hölder-Zygmund spaces $\Lambda ^{r}$ $Λ^{r}$ when the boundaries of the domains are in $\Lambda ^{r+3}$ $Λ^{r + 3}$ with $r>0$ $r > 0$ .

A Homotopy Formula for \(a_q\) Domains in Complex Manifolds

摘要