Extensions of divergence-free fields in $\mathrm{L}^{1}$-based function spaces

Abstract

We establish the first extension results for divergence-free (or solenoidal) elements of $\mathrm{L}^{1}$-based function spaces. Here, the key point is to preserve the solenoidality constraint while simultaneously keeping the underlying $\mathrm{L}^{1}$-boundedness. While previous results as in Kato et al. [Extension and representation of divergence-free vector fields on bounded domains, Math. Res. Lett., 2000] for $\mathrm{L}^{p}$-based function spaces, $1<p<\infty$, rely on PDE approaches, basic principles from harmonic analysis rule out such strategies in the $\mathrm{L}^{1}$-context. By means of a novel method adapted to the divergence-free constraint via differential forms, we establish the existence of such extension operators in the $\mathrm{L}^{1}$-based situation. This applies both to the case of convex domains, where a global extensions can be achieved, as well as to the Lipschitz case, where a local extension can be achieved. Being applicable to $1<p<\infty$ too, our method provides a unifying approach to the cases $p\in\{1,\infty\}$ and $1<p<\infty$. Specifically, covering the exponents $p\in\{1,\infty\}$, this answers a borderline case left open by Kato et al. [Extension and representation of divergence-free vector fields on bounded domains, Math. Res. Lett., 2000] in the affirmative. By use of explicit examples, the assumptions on the underlying domains are shown to be almost optimal.