Friedrichs边值扩张引理及其在复分析中的应用
Friedrichs' extension lemma with boundary values and applications in complex analysis
摘要：
设$Q$是紧致光滑定向的具有光滑边界的黎曼流形上的一阶微分算子.然后，Friedrichs扩张引理指出$Q$在$L^p$ -空间（$1\leq p<\infty$）中的最小闭扩张$Q_{min}$（图的闭包）和最大闭扩张$Q_{max}$（在分布意义下）是一致的.在本文中，我们表明，同样是真实的关于$Q_{min}$和$Q_{max}$的边界值。这给出了弱边值的一个有用的特征，特别是对于$Q=d-bar$的Cauchy-Riemann算子。作为应用，我们导出了具有弱d-杆边值的$L^p$ -形式的Bochner-Martinelli-Koppelman公式.
英文摘要：
Let $Q$ be a first-order differential operator on a compact, smooth oriented Riemannian manifold with smooth boundary. Then, Friedrichs' extension lemma states that the minimal closed extension $Q_{min}$ (the closure of the graph) and the maximal closed extension $Q_{max}$ (in the sense of distributions) of $Q$ in $L^p$-spaces ($1\leq p<\infty$) coincide. In the present paper, we show that the same is true for boundary values with respect to $Q_{min}$ and $Q_{max}$. This gives a useful characterization of weak boundary values, particularly for $Q=d-bar$ the Cauchy-Riemann operator. As an application, we derive the Bochner-Martinelli-Koppelman formula for $L^p$-forms with weak d-bar-boundary values.
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