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

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, $1<p<\mathrm{\infty }$, 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 $1<p<\mathrm{\infty }$ too, our method provides a unifying approach to the cases $p\in \{1,\mathrm{\infty }\}$ and $1<p<\mathrm{\infty }$. Specifically, covering the exponents $p\in \{1,\mathrm{\infty }\}$, 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.