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

Figures (2)

• Figure 1: The geometric argument underlying \ref{['eq:maingeometric']} in the proof of Theorem \ref{['thm:main']}. When $|y'|>c_{2}$, so $(y',1)\in\mathcal{M}:=\{(x',1)\colon\;|x'|>c_{2}\}$, then $\pi(y',1)$, the projection of $(y',1)$ onto $\mathbb{S}^{n-1}$, belongs to $\mathcal{N}$.
• Figure 2: Shifting holomorphic maps along $\mathbb R^{n-2}$. To construct domains for which there is no trace operator $\operatorname{BV}^{\mathbb{A}}(\Omega)\to\mathrm{T}_{k}(\partial\Omega,V)$ in absence of $\mathbb{C}$-ellipticity, one picks $\xi\in\mathbb{C}^{n}\setminus\{0\}$ and $v\in (V+\mathrm{i}V)\setminus\{0\}$ such that $\mathbb{A}(\xi)v=0$. For suitable holomorphic functions $f\colon \mathbb{C}\supset\mathbb{D}\to\mathbb{C}$ (e.g. with $f^{(k-1)}(z)=\frac{1}{z-1}$ and the complex disk $\mathbb{D}$), either the real or the imaginary part of $u(x):=f(x\cdot\mathrm{Re}(\xi)+\operatorname{i} x\cdot\mathrm{Im}(\xi))v$ violate the trace estimate over a set $\Omega$ that up to a rotation coincides with $\{t_{1}\mathrm{Re}(\xi)+t_{2}\mathrm{Im}(\xi)+(0,z")\colon\;t_{1}^{2}+t_{2}^{2}<1,\;z"\in\mathbb R^{n-2}\}$ (figure to the left); see BDG, GRVS. In the same way, one can come up with domains that violate the the trace estimate for non-boundary-elliptic operators by use of Lemma \ref{['lem:nec_bdry_ell']} (figure to the right). Including a straight piece orthogonal to $\mathrm{Im}(\xi)$, one sees the necessity of the boundary ellipticity even more directly.

Theorems & Definitions (36)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1
• Theorem 2.2
• Proposition 2.3
• proof
• Proposition 2.4: PW1
• proof
• proof : Proof of Proposition \ref{['thm:extension_many_derivatives']}