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<p>We obtain uniform estimates for the canonical solution to <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1953_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">\(\bar{\partial }u=f\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mrow> <mi>∂</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mi>u</mi> <mo>=</mo> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation> on the Cartesian product of bounded planar domains with <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1953_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(C^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> boundaries, when <i>f</i> is continuous up to the boundary. This generalizes Landucci’s result for the bidisc toward higher dimensional product domains. In particular, it answers an open question of Kerzman for continuous datum.</p>

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Uniform Estimates for the Canonical Solution to the \(\bar{\partial }\)-equation on Product Domains

  • 期刊论文

  • 发表日期：

    2025年3月18日

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  • Robert Xin Dong,
  • Yifei Pan,
  • Yuan Zhang

摘要

We obtain uniform estimates for the canonical solution to \(\bar{\partial }u=f\) ∂ ¯ u = f on the Cartesian product of bounded planar domains with \(C^2\) C 2 boundaries, when f is continuous up to the boundary. This generalizes Landucci’s result for the bidisc toward higher dimensional product domains. In particular, it answers an open question of Kerzman for continuous datum.