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<p>The <i>k</i>-Cauchy–Fueter complex is the quaternionic counterpart of the Cauchy–Riemann complex in several complex variables, which plays a fundamental role in quaternionic analysis. In this work, we investigate the regularity of solutions to the non-homogeneous <i>k</i>-Cauchy–Fueter equation. Based on the Bourgain–Brezis inequalities, we extend the Limiting Sobolev inequalities to <i>k</i>-Cauchy–Fueter complex. In particular, we get the Gagliardo–Nirenberg inequality for the <i>k</i>-CF operator. In certain cases, the Hardy space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\mathcal {H}}^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">H</mi> </mrow> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> is used in place of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^1.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>1</mn> </msup> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>

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Limiting Sobolev Inequalities for k-Cauchy–Fueter Complex

  • 期刊论文

  • 发表日期：

    2026年4月27日

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  • Xin Luo,
  • Sihan Ning

摘要

The k-Cauchy–Fueter complex is the quaternionic counterpart of the Cauchy–Riemann complex in several complex variables, which plays a fundamental role in quaternionic analysis. In this work, we investigate the regularity of solutions to the non-homogeneous k-Cauchy–Fueter equation. Based on the Bourgain–Brezis inequalities, we extend the Limiting Sobolev inequalities to k-Cauchy–Fueter complex. In particular, we get the Gagliardo–Nirenberg inequality for the k-CF operator. In certain cases, the Hardy space \({\mathcal {H}}^1\) H 1 is used in place of \(L^1.\) L 1 .