  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
  $$
  \|D^{k-1}u\|_{\mathrm{L}^{\frac{n}{n-1}}(H^+)}\leq c\|\mathbb{A}
u\|_{\mathrm{L}^1(H^+)}\quad\text{for }u\in\mathrm{C}^\infty_c
(\mathbb{R}^{n},V)
  $$
  holds. This comes as a consequence of sharp trace theorems on $H=\partial
H^+$.

Analysis of PDEs;Functional Analysis