Friedrichs' extension lemma with boundary values and applications in complex analysis

Details Friedrichs' extension lemma with boundary values and applications in complex analysis Friedrichs' extension lemma with boundary values and applications in complex analysis

2008-03-03

arxiv.org/abs/0803.0092v2

Mathematics

Analysis of PDEs

20 pages

Scientific paper

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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