A compactly supported common L²/H² extension from a bounded smooth domain

At each $x\in \mathrm{\partial }G$, choose a graph chart in the convention of F3, reversing its normal coordinate so that $G$ maps to $t>0$. Since its flattened image is open and contains the image of $x$, choose three nested doubled cylinders centered there, with successively larger positive tangential and normal radii, whose outer closure lies inside that image and whose inverse image lies compactly in $V$. For $n=1$ only the normal intervals occur. The inverse images of the innermost cylinders form a boundary cover; compactness of $\mathrm{\partial }G$ gives finitely many. The compact set left in $\bar{G}$ after removing these innermost patches misses $\mathrm{\partial }G$ and lies in $G$, so finitely many balls with closures in $G$ give a relatively compact interior open set covering it. Apply the partition supplier to the resulting finite cover of a neighborhood $N$ of $\bar{G}$. Choose a compact neighborhood of $\bar{G}$ in $N$; local finiteness leaves only finitely many partition pieces meeting it. Group those by cover member and extend their compactly supported sums by zero outside $N$. The resulting ambient smooth ${\rho }_0,\dots ,{\rho }_m$ sum to $1$ on $G$, with $\mathrm{supp}{\rho }_0⋐G$ and the other supports compactly in the innermost patches. The intervening and outer cylinders provide the positive margins required by the common reflection. Choose ${\psi }_i$ compactly supported in each outer chart and equal to $1$ near $\mathrm{supp}{\rho }_i$. All these choices precede $u$, and their finitely many supports lie compactly in $V$.

For $i>0$ flatten ${\rho }_{i}u$ to the positive half-cylinder and extend it by zero across only the lateral and top artificial edges, as a function $f_i$ on $H_{+}$. At order zero, change of variables bounds this map on $L^2$. At order two the weak chain rule has first derivatives bounded by a constant times the first input derivatives, and second derivatives bounded by a constant times the first and second input derivatives; these constants depend on the bounded first and second derivatives of the chart and its inverse on the larger closed patch. To handle the half-patch reaching $t=0$, apply the local chain formula on an exhaustion of matched interior subpatches. The same ambient bounds apply on every subpatch; all tests are covered and increasing the integrals gives the norm bound on the whole half-patch. The localized function vanishes near the artificial edges, so tests in $H_{+}$ prove that its zero extension across those edges has the corresponding weak derivatives extended by zero. No test in $H_{+}$ crosses the true boundary $t=0$.

Apply the common reflection to $f_i$, pull the result back to the larger doubled chart, multiply by ${\psi }_i$ and extend the product by zero outside the ambient chart. Call this $E_{i}u$. The support assertion of the flat lemma and the chosen chart margins keep the reflected support in the larger chart. Chart, reflection and cutoff bounds give $\mathrm{\parallel }{E}_{i}u{\mathrm{\parallel }}_2\le C_i\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(G)}$ and $\mathrm{\parallel }{E}_{i}u{\mathrm{\parallel }}_{W^{2,2}}\le C_i^{\mathrm{\prime }}\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{2,2}(G)}$. The same operations act on $L^2$ representatives and on their weakly differentiable subclass; all coordinate maps preserve null sets. On $G$, the reflected positive side is unchanged and ${\psi }_i{\rho }_i={\rho }_i$, so $r_G{E}_{i}u={\rho }_{i}u$.

Let $E_{0}u$ be the zero extension of ${\rho }_{0}u$ and define $E_{G}u=E_{0}u+{\sum }_{i=1}^m{E}_{i}u$. This is one linear formula on all of $L^2(G)$; the preceding estimates give both endpoint bounds for its restriction to $W^{2,2}(G)$. Restriction gives $r_G{E}_{G}u=u$. Its support is contained in the fixed union of the compact interior and ambient-chart cutoffs, so the same compact $K⋐V$ works for every input.

If $u=r_{G}F\in {\mathcal{R}}_2(G;\mathbb{K})$, F5 identifies $F$ with its weak $W^{2,2}$ class and F7 restricts that class and each of its derivatives to $G$, giving $\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{2,2}(G)}\le C\mathrm{\parallel }F{\mathrm{\parallel }}_{H^2}$. Conversely, if $u\in W^{2,2}(G;\mathbb{K})$, step 4.1 and F5 give the whole-space extension $E_{G}u\in H^2$ with $\mathrm{\parallel }{E}_{G}u{\mathrm{\parallel }}_{H^2}\le C^{\mathrm{\prime }}\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{2,2}(G)}$. In the real case the same construction preserves real values, so this extension also lies in the specified real subspace. Infimizing the first bound over exactly the extensions in the definition of ${\mathcal{R}}_2$ and using $E_{G}u$ for the second gives $C^{-1}\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{2,2}(G)}\le \mathrm{\parallel }u{\mathrm{\parallel }}_{{\mathcal{R}}_2}\le C^{\mathrm{\prime }}\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{2,2}(G)}$. This proves equality of the spaces and equivalence of norms; definiteness of the infimum follows also from $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(G)}\le \mathrm{\parallel }F{\mathrm{\parallel }}_{H^2}$, while homogeneity and the triangle inequality follow by scaling and adding extensions and then taking infima. At order zero F5 and F7 give $\mathrm{\parallel }u{\mathrm{\parallel }}_2\le \mathrm{\parallel }F{\mathrm{\parallel }}_{H^0}$ for every extension. Extension by zero of any $L^2(G)$ class has exactly the same $L^2=H^0$ norm, since its squared integral outside $G$ is zero. Thus every such class belongs to ${\mathcal{R}}_0$ and its infimum norm is exactly $\mathrm{\parallel }u{\mathrm{\parallel }}_2$.