线丛Shilov--Poisson变换在有界对称域上的边界刚性
Boundary Rigidity of Line-Bundle Shilov--Poisson Transforms on Bounded Symmetric Domains
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中文总结 AI 辅助
研究有界对称域上线丛Shilov--Poisson变换的边界刚性，确定有限正则性在内部强制为零或满足协变微分方程，并给出Graham型定理。
中文摘要 AI 辅助
设$\Omega=G/K$为除单位圆盘外的不可约有界对称域，$S$为其Shilov边界。我们研究$S$与$\Omega$上齐次线丛之间的双参数Shilov--Poisson变换族。对于$L^2$边界数据的变换$F$，我们确定何种有限欧几里得正则性直至全边界$\partial\Omega$会在内部强制成立。在五个显式例外参数族之外，足够高的有限正则性迫使$F=0$。在每个例外族中，结论是内部协变微分方程。在标量Poisson--Szegő情形下，它产生有限阶Graham型定理：所述边界正则性迫使Bergman调和函数为多重调和函数。
英文摘要
Let $Ω=G/K$ be an irreducible bounded symmetric domain other than the unit disk, and let $S$ be its Shilov boundary. We study the two-parameter family of Shilov--Poisson transforms between homogeneous line bundles over $S$ and $Ω$. For a transform $F$ of $L^2$ boundary data, we determine what finite Euclidean regularity up to the full boundary $\partialΩ$ forces in the interior. Outside five explicit exceptional parameter families, sufficiently high finite regularity forces $F=0$. In each exceptional family, the conclusion is an interior covariant differential equation. In the scalar Poisson--Szegő case, it yields a finite-order Graham type theorem: the stated boundary regularity forces a Bergman-harmonic function to be pluriharmonic.
发表机构
- School of Mathematics, Tianjin University（天津大学数学学院）
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