Let D be a bounded domain in the complex plane with Lipschitz boundary. In the paper, we construct an integral solution operator T[f] for any <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2096_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\overline{\partial }$</EquationSource> <EquationSource Format="MATHML"><math> <mover> <mi>∂</mi> <mo>¯</mo> </mover> </math></EquationSource> </InlineEquation> closed (0, 1)-form <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2096_Article_IEq5.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="104" /> </InlineMediaObject> <EquationSource Format="TEX">$f\in L^p_{(0,1)}(D^n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <msubsup> <mi>L</mi> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mi>p</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>D</mi> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> solving the Cauchy-Riemain equation <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2096_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$\overline{\partial }u=f$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mi>∂</mi> <mo>¯</mo> </mover> <mi>u</mi> <mo>=</mo> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation> on the product domains <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2096_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$D^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>D</mi> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> and obtain the <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2096_Article_IEq8.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-estimates for all <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2096_Article_IEq9.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">$1<p\le \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo>≤</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>.

$\overline{\partial }$-Estimates on the Product of Bounded Lipschitz Domain

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发表日期：

2025年7月14日

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Song-Ying Li,
Sujuan Long,
Jie Luo

摘要

Let D be a bounded domain in the complex plane with Lipschitz boundary. In the paper, we construct an integral solution operator T[f] for any $\overline{\partial }$ $\bar{\partial}$ closed (0, 1)-form $f\in L^p_{(0,1)}(D^n)$ $f \in L_{(0, 1)}^{p} (D^{n})$ solving the Cauchy-Riemain equation $\overline{\partial }u=f$ $\bar{\partial} u = f$ on the product domains $D^n$ $D^{n}$ and obtain the $L^p$ $L^{p}$ -estimates for all $1<p\le \infty $ $1 < p \leq \infty$ .

\(\overline{\partial }\)-Estimates on the Product of Bounded Lipschitz Domain

摘要