Uniform localized H2 algebra on a smooth exterior domain

Put $r=2\sqrt{n}$ and choose $R$ so large that $\mathrm{\Theta }\subset B_{R-4r}$. Set $G=\mathrm{\Omega }\cap B_R$ and $A=\{a\in {\mathbb{Z}}^n:\mathrm{\mid }a\mathrm{\mid }\ge R-3r\}$. Each $B(a,r)$ with $a\in A$ misses $\mathrm{\Theta }$. The sets $G$ and $B(a,r)$ cover $\mathrm{\Omega }$: outside $B_{R-2r}$ a nearest lattice point is at distance at most $r\mathrm{/}4$ and belongs to $A$, while points inside $B_R$ belong to $G$. Their overlap is bounded by a constant depending only on $n$, since the lattice points within distance $r$ of any point lie in a fixed-size cube. The boundary of $G$ consists of the separated smooth obstacle boundary and sphere. It is connected: an exterior path from any point of $G$ to a point outside $B_R$, stopped at its first sphere crossing, joins it within $G$ to the connected outer annulus; open connected Euclidean sets are path connected. Thus $G$ is a bounded smooth domain.

The cover also admits a smooth partition with uniformly bounded derivatives. Choose ${\chi }_0$ equal to one on $B_{R-2r}$ and supported in $B_{R-r}$. Take a nonnegative smooth bump $\rho$ supported in $B(0,r)$ and positive on $\overline{B(0,r\mathrm{/}2)}$, and set $S(x)={\sum }_{a\in A}\rho (x-a)$. The preceding nearest-lattice argument gives a positive uniform lower bound for $S$ where $1-{\chi }_0$ is nonzero. Define ${\chi }_a=(1-{\chi }_0)\rho (x-a)\mathrm{/}S(x)$ there and extend it by zero elsewhere. The locally finite sum is ${\chi }_0+{\sum }_a{\chi }_a=1$ on $\mathrm{\Omega }$; finite overlap, the lower bound for $S$, and the fixed bump give uniform bounds on derivatives through order two. Hence the associated localized $H^2$ norms are controlled by bounded overlap and the weak cutoff rule.

Restrict $u$ and $v$ to $G$ and to each exterior ball. By [F1], the local algebra and embedding constants are bounded by one fixed constant: use the constant for $G$ and the translation-invariant constant for $B(0,r)$. If $D_j$ denotes these domains and $a_j=\mathrm{\parallel }u{\mathrm{\parallel }}_{H^2(D_j)}$, $b_j=\mathrm{\parallel }v{\mathrm{\parallel }}_{H^2(D_j)}$, bounded overlap gives ${\sum }_j{a}_j^2\le N\mathrm{\parallel }u{\mathrm{\parallel }}_{H^2(\mathrm{\Omega })}^2$ and ${\sup }_j{b}_j\le \mathrm{\parallel }v{\mathrm{\parallel }}_{H^2(\mathrm{\Omega })}$. Local weak derivatives agree on overlaps, so the local products define the global weak derivatives. Consequently $\mathrm{\parallel }uv{\mathrm{\parallel }}_{H^2(\mathrm{\Omega })}^2\le C^2{\sum }_j{a}_j^2{b}_j^2\le C^{2}N\mathrm{\parallel }u{\mathrm{\parallel }}_{H^2(\mathrm{\Omega })}^2\mathrm{\parallel }v{\mathrm{\parallel }}_{H^2(\mathrm{\Omega })}^2\mathrm{.}$ The local supremum estimates and the cover likewise give $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathrm{\Omega })}\le C\mathrm{\parallel }u{\mathrm{\parallel }}_{H^2(\mathrm{\Omega })}$.

Complex conjugation commutes with real weak differentiation, as follows by conjugating the test-function identities, and preserves the $H^2$ norm. Since $m$ is odd, $g(u)=\lambda u^{(m+1)\mathrm{/}2}{\bar{u}}^{(m-1)\mathrm{/}2}$. Repeated multiplication gives the stated power bound. Replacing the factors one at a time in the difference of the two products gives a sum of $m$ products, each containing either $u-v$ or its conjugate and $m-1$ factors chosen from $u,v$ and their conjugates; the same algebra estimate proves the stated Lipschitz bound.

By [F1], $u{\mathrm{\mid }}_G$ has a continuous representative on $\bar{G}$; the same is true of $g(u){\mathrm{\mid }}_G$, whose representative is the polynomial applied to that of $u$. By [F3] and [F4], their obstacle traces are the corresponding boundary values as $L^2(\mathrm{\partial }\mathrm{\Theta },dS;\mathbb{C})$ classes. If the trace of $u$ vanishes $dS$-almost everywhere, then $g(0)=0$ shows that the trace of $g(u)$ does also. The chartwise trace is local, so enlarging $R$ does not change its restriction to $\mathrm{\partial }\mathrm{\Theta }$. This proves all assertions.