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Let $Q$ be a first-order differential operator on a compact, smooth orientedRiemannian manifold with smooth boundary. Then, Friedrichs' extension lemmastates 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 showthat 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, wederive the Bochner-Martinelli-Koppelman formula for $L^p$-forms with weakd-bar-boundary values.

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