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

Abstract

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

arXiv comment22 pages, 2 figures

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