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

TL;DR

This paper extends Friedrichs' extension lemma to boundary values for first-order differential operators on manifolds, providing new characterizations of weak boundary values and applications to complex analysis, including a formula for $L^p$-forms.

Contribution

It proves that the minimal and maximal extensions coincide for boundary values of first-order operators, especially for the d-bar operator, enhancing understanding of weak boundary values.

Findings

Minimal and maximal extensions coincide for boundary values.

Characterization of weak boundary values for the d-bar operator.

Derivation of Bochner-Martinelli-Koppelman formula for $L^p$-forms.

Abstract

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.