厄米克利福德分析中的Szeg $\ddot{o}$投影和矩阵希尔伯特变换
Szeg$\ddot{o}$ projection and matrix Hilbert transform in Hermitean Clifford analysis
摘要：
研究了Hermite-Clifford分析中两个复Hermite-Dirac算子的同时零解问题，提出了矩阵希尔伯特变换，并证明了它满足经典分析和正交Clifford分析中的希尔伯特变换的类似性质.在此背景下，我们将为定义在偶数维欧几里得空间有界子域上的厄米单演函数的Hardy空间引入Szeg $\ddot{o}$投影算子，建立将Szeg $\ddot{o}$投影算子与定义在偶数维欧几里得空间有界子域上的厄米单演函数的Hardy空间上的Hardy投影算子紧密联系起来的Kerzman-Stein公式，利用Hardy投影算子及其伴随算子得到Szeg $\ddot{o}$投影算子。进一步给出了Hermite-Clifford分析中矩阵希尔伯特变换酉性的代数和几何刻画。
英文摘要：
The simultaneous null solutions of the two complex Hermitean Dirac operators are focused on in Hermitean Clifford analysis, where the matrix Hilbert transform was presented and proved to satisfy the analogous properties of the Hilbert transform in classical analysis and in orthogonal Clifford analysis. Under this setting we will introduce the Szeg$\ddot{o}$ projection operator for the Hardy space of Hermitean monogenic functions defined on a bounded subdomain of even dimensional Euclidean space, establish the Kerzman-Stein formula which closely connects the Szeg$\ddot{o}$ projection operator with the Hardy projection operator onto the Hardy space of Hermitean monogenic functions defined on a bounded subdomain of even dimensional Euclidean space, and get the Szeg$\ddot{o}$ projection operator in terms of the Hardy projection operator and its adjoint. Further we will give the algebraic and geometric characterizations for the matrix Hilbert transform to be unitary in Hermitean Clifford analysis.
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