Boundary ellipticity and limiting $L^1$-estimates on halfspaces

We identify necessary and sufficient conditions on $k$th order differential operators $\mathbb{A}$ in terms of a fixed halfspace $H^{+}\subset {\mathbb{R}}^n$ such that the Gagliardo–Nirenberg–Sobolev inequality $$\mathrm{\parallel }{D}^{k-1}u{\mathrm{\parallel }}_{{\mathrm{L}}^{\frac{n}{n-1}}(H^{+})}\le c\mathrm{\parallel }\mathbb{A}u{\mathrm{\parallel }}_{{\mathrm{L}}^1(H^{+})}\text{for }u\in {\mathrm{C}}_c^{\mathrm{\infty }}({\mathbb{R}}^n,V)$$ holds. This comes as a consequence of sharp trace theorems on $H=\mathrm{\partial }{H}^{+}$.