The restriction to the boundary of holomorphic functions on a domain are CR functions, which are annihilated by tangential Cauchy-Riemann operator and useful for studying holomorphic functions. The restriction to the boundary <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\partial \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation> of k-regular functions on a domain <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> are called k-CF functions, which are annihilated by tangential k-Cauchy-Fueter operators. The Szegö projection is the orthogonal projection from the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L^2(\partial \Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>-space to the space of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> k-CF functions on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\partial \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation>, whose kernel is called Szegö kernel. We find the explicit forms of tangential k-Cauchy-Fueter operators and the associated tangential Laplacians <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Box _b$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mo>□</mo> <mi>b</mi> </msub> </math></EquationSource> </InlineEquation> on nondegenerate quadratic rigid hypersurfaces in <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathbb {H}^{n+1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">H</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>, which have the structure of nilpotent Lie groups of step two. By using the Laguerre calculus on the associated nilpotent Lie groups of step two, we analyze the kernel of the associated tangential Laplacians <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Box _b$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mo>□</mo> <mi>b</mi> </msub> </math></EquationSource> </InlineEquation> and construct Szegö kernels on the nondegenerate quadratic rigid hypersurfaces.

The Szegö kernel for k-CF functions on nondegenerate quadratic rigid hypersurfaces in the quaternionic space

期刊论文
发表日期：

2026年2月28日

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Qianqian Kang,
Der-Chen Chang,
Wei Wang

摘要

The restriction to the boundary of holomorphic functions on a domain are CR functions, which are annihilated by tangential Cauchy-Riemann operator and useful for studying holomorphic functions. The restriction to the boundary $\partial \Omega $ $\partial Ω$ of k-regular functions on a domain $\Omega $ $Ω$ are called k-CF functions, which are annihilated by tangential k-Cauchy-Fueter operators. The Szegö projection is the orthogonal projection from the $L^2(\partial \Omega )$ $L^{2} (\partial Ω)$ -space to the space of $L^2$ $L^{2}$ k-CF functions on $\partial \Omega $ $\partial Ω$ , whose kernel is called Szegö kernel. We find the explicit forms of tangential k-Cauchy-Fueter operators and the associated tangential Laplacians $\Box _b$ $□_{b}$ on nondegenerate quadratic rigid hypersurfaces in $\mathbb {H}^{n+1}$ $H^{n + 1}$ , which have the structure of nilpotent Lie groups of step two. By using the Laguerre calculus on the associated nilpotent Lie groups of step two, we analyze the kernel of the associated tangential Laplacians $\Box _b$ $□_{b}$ and construct Szegö kernels on the nondegenerate quadratic rigid hypersurfaces.