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

TL;DR

This paper characterizes conditions on differential operators ensuring a specific Sobolev inequality on halfspaces, deriving from sharp trace theorems, with implications for boundary ellipticity and $L^1$-estimates.

Contribution

It provides necessary and sufficient conditions on differential operators for boundary Sobolev inequalities on halfspaces, linking boundary ellipticity to limiting $L^1$-estimates.

Findings

Identifies conditions for Sobolev inequalities on halfspaces.

Establishes sharp trace theorems on boundary hyperplanes.

Connects boundary ellipticity with $L^1$-estimates.

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