A fixed smooth-domain extension that preserves smooth parameters

Since $\bar{\mathrm{\Omega }}$ is bounded and closed [L2], it is compact by Heine-Borel [L2]. The family of all admissible smooth chart neighbourhoods of boundary points is an open cover of $\mathrm{\partial }\mathrm{\Omega }\subseteq \bar{\mathrm{\Omega }}$, and $\mathrm{\Omega }$ itself is open; hence the family formed by $\mathrm{\Omega }$ and all admissible smooth chart neighbourhoods covers the compact set $\bar{\mathrm{\Omega }}$, so it contains a finite subcover, say $W_1,\dots ,W_p$ together with $\mathrm{\Omega }$, where each $W_i$ carries chart data $(R_i,B_i,h_i,{\mathrm{\Phi }}_i)$ with $h_i\in C^{\mathrm{\infty }}$ and rigid motion $R_i$ [L1, L2]. Extracting the finitely many chart data is a finite choice, available in ZF; choose these chart neighbourhoods already inside $V$ and with flattened images equal to symmetric product boxes $B_i\times (-{\delta }_i,{\delta }_i)$: every boundary point has such a smaller neighbourhood, so use the cover by these smaller charts before taking the finite subcover. Fix these finitely many maps and data for the rest of the proof.

Put $b_k:=-2^k$ for $k\ge 0$, the integer powers of [L6]. For $N\ge k\ge 0$ define $a_k^{(N)}:={\prod }_{\begin{array}{c}0\le j\le N\\ j\mathrm{\ne }k\end{array}}\frac{1-b_j}{b_k-b_j}\mathrm{.}$ Here $1-b_j=1+2^j>0$; for $j<k$ one has $b_k-b_j=-(2^k-2^j)<0$, and for $j>k$ one has $b_k-b_j=2^j-2^k>0$. Hence $a_k^{(N)}\mathrm{\ne }0$ and its sign is $(-1{)}^k$ for every $N\ge k$.

For reals $x_0,\dots ,x_M\in [0,1)$ with ${\sum }_{j=0}^M{x}_j<1$ one has, by induction on $M$ using the field axioms, ${\prod }_{j=0}^M(1+x_j)\le (1-{\sum }_{j=0}^M{x}_j{)}^{-1}$ and ${\prod }_{j=0}^M(1-x_j)\ge 1-{\sum }_{j=0}^M{x}_j$. Applying this with $x_j=2^{-j}$ and the geometric sums ${\sum }_{j\ge 2}{2}^{-j}=\frac{1}{2}$ and ${\sum }_{j\ge 1}{2}^{-j}=1$ [L6] gives for every finite set $F\subseteq \mathbb{N}$ and every $M\ge 0$ the two bounds ${\prod }_{j\in F}(1+2^{-j})\le 6$ and ${\prod }_{l=1}^M(1-2^{-l})\ge \frac{1}{4}$: the first because the product over $\{0,1\}$ equals $2\cdot \frac{3}{2}=3$ and the product over $j\ge 2$ is at most $(1-\frac{1}{2}{)}^{-1}=2$, the second because the factor $l=1$ is $\frac{1}{2}$ and the remaining product is at least $1-{\sum }_{l\ge 2}{2}^{-l}=\frac{1}{2}$.

Let $f$ be continuous on the strip $B\times (-\delta ,\delta )\times J$ (with $B\subseteq {\mathbb{R}}^2$ open, $\delta >0$) and suppose that every first partial derivative of $f$ exists and is continuous on each open half-strip $\{s>0\}$ and $\{s<0\}$ and extends continuously to the corresponding closed half-strip, the two extensions agreeing on $\{s=0\}$; let $E_i$ denote the resulting common continuous function on the whole strip, the index $i$ running over the four coordinates. Fix a point $p$ in the strip. If a segment $\beta \mapsto p+\beta e_i$, $\beta \in [0,h]$, stays inside one open half-strip, then $\beta \mapsto f(p+\beta e_i)$ is differentiable at every point of $[0,h]$ with derivative $E_i(p+\beta e_i)$ on $(0,h)$ and one-sided derivatives at the endpoints, and $E_i$ is continuous, hence integrable, on $[0,h]$ [L4, L5]; the second fundamental theorem gives $f(p+h{e}_i)-f(p)={\int }_0^h{E}_i(p+\beta e_i)d\beta$. If the segment meets $\{s=0\}$ at a parameter ${\beta }_0$, apply the identity on closed subintervals staying strictly in either open half-strip, then let their endpoints approach ${\beta }_0$. Continuity of $f$ gives convergence of the endpoint values, while the continuous extension of $E_i$ is bounded near ${\beta }_0$, so the omitted integrals tend to $0$. Additivity gives the same identity on the whole segment, including when ${\beta }_0$ is an endpoint; no derivative at the interface is assumed. For $h<0$, integrals are interpreted with reversed limits. If the segment lies entirely in the interface $s=0$ (a tangential or time direction), translate it by a positive normal distance $ϵ$ and use the identity just proved on the positive half-strip. Continuity of $f$ and of $E_i$ on the compact translated-segment strip gives uniform convergence as $ϵ\downarrow 0$, hence the same integral identity on $s=0$. Dividing by $h$ and letting $h\to 0$, continuity of $E_i$ at $p$ gives $(f(p+h{e}_i)-f(p))\mathrm{/}h=\frac{1}{h}{\int }_0^h{E}_i(p+\beta e_i)d\beta \to E_i(p)$; thus every first partial derivative of $f$ exists on the whole strip and equals the continuous function $E_i$. By [L4] $f$ is totally differentiable at every point with that Jacobian, and in particular all first partial derivatives are continuous, so $f$ is $C^1$ on the strip.

For $N\ge k$, rewriting each factor defined in step 1.2 as $1-b_j=2^j(1+2^{-j})$ and $b_k-b_j=\pm 2^{\max (j,k)}(1-2^{-\mathrm{\mid }j-k\mathrm{\mid }})$ gives $\mid a_k^{(N)}\mid =2^{-k(k+1)\mathrm{/}2}\frac{{\prod }_{j=0}^{k-1}(1+2^{-j}){\prod }_{j=k+1}^N(1+2^{-j})}{{\prod }_{l=1}^k(1-2^{-l}){\prod }_{m=1}^{N-k}(1-2^{-m})},$ where empty products are $1$; indeed the powers of $2$ contribute ${\sum }_{j<k}(j-k)=-k(k+1)\mathrm{/}2$, and the two denominator products collect the factors $1-2^{j-k}$ for $j<k$ and $1-2^{k-j}$ for $j>k$. By step 1.3 the numerator is at most $36$ and each denominator product is at least $\frac{1}{4}$, so $\mid a_k^{(N)}\mid \le 576\cdot 2^{-k(k+1)\mathrm{/}2}$ for all $N\ge k$. For fixed $k$ the ratio $\mid a_k^{(N+1)}\mid \mathrm{/}\mid a_k^{(N)}\mid =(1+2^{-(N+1)})\mathrm{/}(1-2^{-(N+1-k)})$ exceeds $1$ whenever $N+1>k$, because its numerator exceeds $1$ and its denominator lies in $(0,1)$; hence $N\mapsto \mid a_k^{(N)}\mid$ is nondecreasing and bounded above by $576\cdot 2^{-k(k+1)\mathrm{/}2}$, so the monotone convergence theorem [L6] gives a limit $a_k:={\lim }_{N\to \mathrm{\infty }}{a}_k^{(N)}$ with sign $(-1{)}^k$ and $\mid a_k\mid \le 576\cdot 2^{-k(k+1)\mathrm{/}2}$. Consequently, for every integer $r\ge 0$ and every $k\ge 2r+1$ one has $-\frac{k(k+1)}{2}+kr\le -k$, so $\mid a_k\mid \mid b_k{\mid }^r\le 576\cdot 2^{-k}$ for such $k$ and the series ${\sum }_{k=0}^{\mathrm{\infty }}\mid a_k\mid \mid b_k{\mid }^r$ converges, with tails bounded by ${\sum }_{k>M}\mid a_k\mid \mid b_k{\mid }^r\le 576\cdot 2^{-M}$ whenever $M\ge 2r+1$.

For every $N\ge 0$ and every integer $r$ with $0\le r\le N$ one has ${\sum }_{k=0}^N{a}_k^{(N)}{b}_k^r=1.$ Indeed, expanding the $N$ linear factors of each ${\prod }_{j\mathrm{\ne }k}(z-b_j)$ by distributivity gives each function $z\mapsto {\prod }_{j\mathrm{\ne }k}(z-b_j)\mathrm{/}(b_k-b_j)$ the structure of a real polynomial of degree at most $N$ with explicit finite coefficient sums [L7], so $q(z):=z^r-{\sum }_{k=0}^N{b}_k^r{\prod }_{j\mathrm{\ne }k}(z-b_j)\mathrm{/}(b_k-b_j)$ is a real polynomial of degree at most $N$. At a node $z=b_i$ every product ${\prod }_{j\mathrm{\ne }k}(b_i-b_j)$ vanishes for $k\mathrm{\ne }i$ and equals ${\prod }_{j\mathrm{\ne }i}(b_i-b_j)$ for $k=i$, so $q(b_i)=b_i^r-b_i^r=0$ for all $i=0,\dots ,N$. Thus $q$ has $N+1$ distinct roots; were $q$ nonzero, its degree would be defined and at most $N$, contradicting the root bound [L7]. Hence $q$ is the zero polynomial, and evaluating at $z=1$, where ${\prod }_{j\mathrm{\ne }k}(1-b_j)\mathrm{/}(b_k-b_j)=a_k^{(N)}$, gives $0=q(1)=1-{\sum }_{k=0}^N{a}_k^{(N)}{b}_k^r$.

Let $f$ be continuous on the strip and suppose that for every word $(i_1,\dots ,i_m)$ of coordinate indices with $m\ge 1$ the iterated derivative ${\mathrm{\partial }}_{i_m}\cdots {\mathrm{\partial }}_{i_1}f$ exists and is continuous on each open half-strip and extends continuously to the closed half-strips, the two extensions agreeing on $\{s=0\}$; let $F_{i_1\dots i_m}$ be the common extension. Then $f$ is of class $C^{\mathrm{\infty }}$ on the strip, and every iterated derivative of $f$ on the strip equals the corresponding common extension. Indeed, induct on $m$: for $m=0$ this is the assumed continuity, and for $m\ge 1$ the iterated derivative $g:={\mathrm{\partial }}_{i_{m-1}}\cdots {\mathrm{\partial }}_{i_1}f$ has, by the induction hypothesis, the continuous extension $F_{i_1\dots i_{m-1}}$ to the strip, whose first partial derivatives on each open half-strip exist and are continuous and extend continuously to the closed half-strips with the extensions $F_{i_1\dots i_{m-1}i}$ agreeing on $\{s=0\}$; step 1.4 applied to $g$ shows that $g$ is $C^1$ on the strip with ${\mathrm{\partial }}_{i}g=F_{i_1\dots i_{m-1}i}$. Hence every iterated derivative of $f$ of every order exists and is continuous on the strip, that is, $f$ is $C^{\mathrm{\infty }}$ there, and each iterated derivative equals the corresponding common extension.

Let $\mathcal{U}$ be the finite family consisting of the members $W_1,\dots ,W_p$ of step 1.1 and the member $\mathrm{\Omega }$. For each $x\in \bar{\mathrm{\Omega }}$ choose a member of $\mathcal{U}$ containing $x$ and then an open Euclidean ball with rational centre and rational radius containing $x$ and whose closure is contained in that member; the countable, explicitly defined family $\mathcal{B}$ of all open balls with rational centre and rational radius whose closure is contained in at least one member of $\mathcal{U}$ covers the compact set $\bar{\mathrm{\Omega }}$, so compactness [L2] yields finitely many balls $B_1,\dots ,B_q\in \mathcal{B}$ covering $\bar{\mathrm{\Omega }}$. For each $l\le q$ choose a member $U_{\sigma (l)}$ of $\mathcal{U}$ containing $\overline{B_l}$; this is a finite choice in ZF. By [L3] applied to the compact sets $\overline{B_l}\subseteq U_{\sigma (l)}$ there are ${\psi }_l\in C_c^{\mathrm{\infty }}(U_{\sigma (l)})$ with $0\le {\psi }_l\le 1$ and ${\psi }_l=1$ on a neighbourhood of $\overline{B_l}$. Then $S:={\sum }_{l=1}^q{\psi }_l$ is smooth and positive on a neighbourhood of $\bar{\mathrm{\Omega }}$ because each $\overline{B_l}$ carries such a neighbourhood; apply [L3] again, with compact $K=\bar{\mathrm{\Omega }}$ and open $O=\{S>0\}$, to obtain $\chi \in C_c^{\mathrm{\infty }}(\{S>0\})$ with $0\le \chi \le 1$ and $\chi =1$ on a neighbourhood of $\bar{\mathrm{\Omega }}$. Since the support of $\chi$ is compact and contained in $\{S>0\}$, the function $\chi \mathrm{/}S$, defined to be $\chi \cdot (1\mathrm{/}S)$ on $\{S>0\}$ and $0$ elsewhere, is smooth with compact support in $\{S>0\}$. For $j=1,\dots ,p$ put ${\chi }_j:=\frac{\chi }{S}{\sum }_{\begin{array}{c}1\le l\le q\\ \sigma (l)=j\end{array}}{\psi }_l$ and put ${\chi }_0:=\frac{\chi }{S}{\sum }_{\begin{array}{c}1\le l\le q\\ \sigma (l)=0\end{array}}{\psi }_l$, assigning index $0$ to the member $\mathrm{\Omega }$. Each ${\chi }_j$ is smooth, compactly supported and nonnegative, with $\mathrm{supp}{\chi }_j\subseteq W_j$ for $j\ge 1$ and $\mathrm{supp}{\chi }_0\subseteq \mathrm{\Omega }$, and ${\sum }_{j=0}^p{\chi }_j=\chi =1$ on a neighbourhood of $\bar{\mathrm{\Omega }}$, in particular on $\mathrm{\Omega }$.

For every integer $r\ge 0$ the series ${\sum }_{k=0}^{\mathrm{\infty }}{a}_k{b}_k^r$ converges absolutely with sum $1$. Indeed, fix $M\ge 2r+1$ and $N\ge \max (r,M)$; by step 2.2, $1={\sum }_{k=0}^N{a}_k^{(N)}{b}_k^r$, so $1-{\sum }_{k=0}^M{a}_k{b}_k^r={\sum }_{k=0}^M(a_k^{(N)}-a_k)b_k^r+{\sum }_{k=M+1}^N{a}_k^{(N)}{b}_k^r$; the first sum tends to $0$ as $N\to \mathrm{\infty }$ because there are finitely many coefficients and each converges to $a_k$ by step 2.1, and the second sum is bounded in absolute value by $576\cdot 2^{-M}$ uniformly in $N$ by step 2.1. Letting first $N\to \mathrm{\infty }$ and then $M\to \mathrm{\infty }$ gives ${\sum }_{k=0}^{\mathrm{\infty }}{a}_k{b}_k^r=1$, the absolute convergence being part of step 2.1.

Fix $B\subseteq {\mathbb{R}}^2$ open, $\delta >0$ and an interval $J$, and let $w:(B\times (-\delta ,0])\times J\to \mathbb{R}$ be continuous, of class $C^{\mathrm{\infty }}$ on $B\times (-\delta ,0)\times J$, with every mixed derivative $D_y^{\alpha }{\mathrm{\partial }}_s^r{\mathrm{\partial }}_{\tau }^{j}w$ extending continuously to $B\times (-\delta ,0]\times J$, and suppose $w$ vanishes outside a compact subset of $B\times (-\delta ,0]$. Choosing a cutoff $\rho \in C_c^{\mathrm{\infty }}((-\delta \mathrm{/}2,\delta \mathrm{/}2))$ with $0\le \rho \le 1$ and $\rho =1$ near $0$ [L3], define $(\mathcal{E}w)(y,s,\tau ):=w(y,s,\tau )$ for $s\le 0$ and $(\mathcal{E}w)(y,s,\tau ):={\sum }_{k=0}^{\mathrm{\infty }}{a}_k\rho (b_{k}s)w(y,b_{k}s,\tau )$ for $s>0$, with the limiting coefficients $a_k$ of step 2.1. In this formula $\rho (z)w(y,z,\tau )$ and each of its derivatives are extended by zero for $z\le -\delta$; this extension is smooth there because $\rho$ vanishes on a neighbourhood of $-\delta$. For fixed $s>0$ the series is finite: $\mathrm{\mid }{b}_{k}s\mathrm{\mid }=2^{k}s\ge \delta \mathrm{/}2$ for all large $k$, where $\rho$ vanishes, so only finitely many terms are nonzero; hence $\mathcal{E}w$ is well defined on $B\times (-\delta ,\delta )\times J$. As $s\downarrow 0$ each term tends to $a_{k}w(y,0,\tau )$ by continuity of $w$ and $\rho (0)=1$, and the tail is bounded by $(\sup \mathrm{\mid }w\mathrm{\mid }){\sum }_{k>M}\mathrm{\mid }{a}_k\mathrm{\mid }$, which tends to $0$ uniformly on compact sets by step 2.1; by step 3.1 with $r=0$ the lower and upper limits both equal $w(y,0,\tau )$, so $\mathcal{E}w$ is continuous on the strip. Moreover $\mathcal{E}w$ vanishes on the complement of the fixed compact spatial support of $w$ on $s\le 0$ together with the projection of that support in the $y$-variable times $[0,\delta \mathrm{/}2]$, with arbitrary $\tau \in J$, so it is spatially supported in a fixed compact subset of $B\times (-\delta ,\delta )$.

$\mathcal{E}w$ is of class $C^{\mathrm{\infty }}$ on each of the two open half-strips, with the following derivative formula on $s>0$, valid for every multi-index $\alpha$ and all $r,j\ge 0$: $D_y^{\alpha }{\mathrm{\partial }}_s^r{\mathrm{\partial }}_{\tau }^j(\mathcal{E}w)(y,s,\tau )={\sum }_{k=0}^{\mathrm{\infty }}{a}_k{b}_k^r\left[D_z^r{D}_y^{\alpha }{\mathrm{\partial }}_{\tau }^j\right(\rho (z)w(y,z,\tau )){]}_{z=b_{k}s},$ a finite sum for each fixed $(y,s,\tau )$ because only the finitely many $k$ with $\mathrm{\mid }{b}_{k}s\mathrm{\mid }<\delta \mathrm{/}2$ contribute. On $s<0$ the function $\mathcal{E}w$ equals $w$, which is smooth there. Both assertions follow by applying the product and chain rules [L4] termwise to a finite sum: on each set $\{s>\epsilon \}$ with $\epsilon >0$ the constraint $2^{k}s<\delta \mathrm{/}2$ admits only finitely many $k$, so $\mathcal{E}w$ is a finite sum of smooth functions there, and every point of $\{s>0\}$ has a neighbourhood with such an $\epsilon >0$. Moreover, for every compact $Y^{\mathrm{\prime }}\times K\subseteq B\times J$ and all $\alpha ,r,j$ there is a finite constant $C_{\alpha ,r,j}$, independent of $w$, with $\sup \{\mathrm{\mid }{D}_y^{\alpha }{\mathrm{\partial }}_s^r{\mathrm{\partial }}_{\tau }^j(\mathcal{E}w)\mathrm{\mid }:y\in Y^{\mathrm{\prime }},\mathrm{\mid }s\mathrm{\mid }<\delta \mathrm{/}2,\tau \in K\}\le C_{\alpha ,r,j}{\sum }_{\mathrm{\mid }\beta \mathrm{\mid }\le \mathrm{\mid }\alpha \mathrm{\mid }}{\sum }_{q\le r}\sup \{\mathrm{\mid }{D}_y^{\beta }{\mathrm{\partial }}_z^q{\mathrm{\partial }}_{\tau }^{j}w(y,z,\tau )\mathrm{\mid }:y\in Y^{\mathrm{\prime }\mathrm{\prime }},-\delta \mathrm{/}2\le z\le 0,\tau \in K\}$ for some open $Y^{\mathrm{\prime }\mathrm{\prime }}$ with $Y^{\mathrm{\prime }}\subseteq \overline{Y^{\mathrm{\prime }\mathrm{\prime }}}\subseteq B$: each term contributes a factor $\mathrm{\mid }{a}_k\mathrm{\mid }\mathrm{\mid }{b}_k{\mathrm{\mid }}^r$ times a fixed bound for the derivatives of $\rho$ of order at most $r$, and ${\sum }_k\mathrm{\mid }{a}_k\mathrm{\mid }\mathrm{\mid }{b}_k{\mathrm{\mid }}^r<\mathrm{\infty }$ by step 2.1.

For every multi-index $\alpha$, all $r,j\ge 0$ and every compact $Y^{\mathrm{\prime }}\times K\subseteq B\times J$, the derivatives in step 5.1 converge uniformly as $s\downarrow 0$ to $D_y^{\alpha }{\mathrm{\partial }}_s^r{\mathrm{\partial }}_{\tau }^{j}w(y,0,\tau )$, and from the other side $D_y^{\alpha }{\mathrm{\partial }}_s^r{\mathrm{\partial }}_{\tau }^j(\mathcal{E}w)(y,s,\tau )=D_y^{\alpha }{\mathrm{\partial }}_s^r{\mathrm{\partial }}_{\tau }^{j}w(y,s,\tau )\to D_y^{\alpha }{\mathrm{\partial }}_s^r{\mathrm{\partial }}_{\tau }^{j}w(y,0,\tau )$. Indeed, choose an open $Y^{\mathrm{\prime }\mathrm{\prime }}$ with $Y^{\mathrm{\prime }}\subseteq \overline{Y^{\mathrm{\prime }\mathrm{\prime }}}\subseteq B$; all derivatives of $\rho w$ through order $r+\mathrm{\mid }\alpha \mathrm{\mid }+j$ appearing in step 5.1 are continuous on the compact set $\overline{Y^{\mathrm{\prime }\mathrm{\prime }}}\times [-\delta \mathrm{/}2,0]\times K$, hence bounded by a finite constant $C$; for $M\ge 2r+1$ the tail is bounded by $C{\sum }_{k>M}\mathrm{\mid }{a}_k\mathrm{\mid }\mathrm{\mid }{b}_k{\mathrm{\mid }}^r\le 576C\cdot 2^{-M}$ by step 2.1, uniformly in $(y,s,\tau )$, while the finitely many head terms converge as $s\downarrow 0$ because $\rho =1$ near $0$ and all positive-order derivatives of $\rho$ vanish at $0$, so each head term tends to $a_k{b}_k^r{D}_y^{\alpha }{\mathrm{\partial }}_s^r{\mathrm{\partial }}_{\tau }^{j}w(y,0,\tau )$; the moment identity of step 3.1 makes the total limit equal to $D_y^{\alpha }{\mathrm{\partial }}_s^r{\mathrm{\partial }}_{\tau }^{j}w(y,0,\tau )$, uniformly on $Y^{\mathrm{\prime }}\times K$. Consequently every mixed derivative of $\mathcal{E}w$ on either side extends continuously to $s=0$, and the two one-sided continuous extensions agree there.

Applying step 2.3 to the function $\mathcal{E}w$ on the strip $B\times (-\delta ,\delta )\times J$: it is continuous by step 4.1, and by step 6.1 every word of iterated derivatives of $\mathcal{E}w$ has matching continuous one-sided extensions; hence $\mathcal{E}w$ is of class $C^{\mathrm{\infty }}$ on the strip, agrees with $w$ on $\{s\le 0\}$, and every mixed derivative of $\mathcal{E}w$ at a point with $s=0$ equals the corresponding boundary value of the mixed derivative of $w$.

For $j\ge 1$ put $w_j:={\chi }_{j}u$ restricted to $\mathrm{\Omega }\times J$; its support is compactly contained in $W_j\cap \bar{\mathrm{\Omega }}$. Pulling back by the smooth diffeomorphism ${\mathrm{\Phi }}_j$ [L1] gives a function, again written $w_j$, on the open half-strip ${\mathrm{\Phi }}_j(W_j)\cap \{t<0\}$ times $J$ that is smooth up to $t=0$ with all mixed derivatives extending continuously, and whose support is compact in ${\mathrm{\Phi }}_j(W_j)\cap \{t\le 0\}$. Choose open boxes $B_j\times (-{\delta }_j,{\delta }_j)\subseteq {\mathrm{\Phi }}_j(W_j)$ containing that compact support and extend $w_j$ by zero where needed on the half-strip $B_j\times (-{\delta }_j,0]\times J$, and let ${\mathcal{E}}_j$ be the construction of steps 4.1, 5.1 and 7.1 for these data. Then ${\mathcal{E}}_j{w}_j$ is $C^{\mathrm{\infty }}$ on $B_j\times (-{\delta }_j,{\delta }_j)\times J$ by step 7.1, is supported in a compact subset of that box, and agrees with $w_j$ wherever $t\le 0$. Pulling back by ${\mathrm{\Phi }}_j^{-1}$ and extending by zero outside the box gives a function $g_j\in C^{\mathrm{\infty }}({\mathbb{R}}^3\times J)$ with spatial support in a fixed compact subset of $W_j\subseteq V$. For $j=0$ extend ${\chi }_{0}u$ by zero on $({\mathbb{R}}^3\setminus \mathrm{\Omega })\times J$; since $\mathrm{supp}{\chi }_0$ is compact and contained in $\mathrm{\Omega }$, the result $g_0$ is $C^{\mathrm{\infty }}$ on ${\mathbb{R}}^3\times J$ with compact spatial support in $\mathrm{\Omega }$. Define $E_{\mathrm{\Omega }}u:={\sum }_{j=0}^p{g}_j\mathrm{.}$ Each step is linear in $u$ (pullback, multiplication by the fixed cutoffs ${\chi }_j$ and the fixed coefficient series are linear), and the data ${\mathrm{\Phi }}_j,{\chi }_j,{\rho }_j,B_j,{\delta }_j,a_k$ depend only on $\mathrm{\Omega }$ and its fixed atlas and cutoffs, not on $J$, on $u$, or on $\tau$; hence $E_{\mathrm{\Omega }}$ is a real-linear operator determined only by these fixed data, and it commutes with ${\mathrm{\partial }}_{\tau }$ because no piece depends on $\tau$. Its values are spatially supported in the fixed compact spatial set supplied by the finitely many chart boxes and cutoff supports, contained in $V$.

By step 7.1 each $g_j$ with $j\ge 1$ is $C^{\mathrm{\infty }}$ on ${\mathbb{R}}^3\times J$, and $g_0$ is $C^{\mathrm{\infty }}$ by construction, so the finite sum $E_{\mathrm{\Omega }}u$ is $C^{\mathrm{\infty }}$ on ${\mathbb{R}}^3\times J$ [L4]. Its derivatives on ${\mathbb{R}}^3\times J$ are the corresponding sums of the derivatives of the $g_j$, and for every multi-index $\alpha$ and $j\ge 0$ the boundary values on $\mathrm{\partial }\mathrm{\Omega }$ from the exterior side of the piece $g_i$ equal the corresponding boundary values of ${\chi }_{i}u$ by steps 6.1 and 7.1.

Seminorm bound. Let $m\ge 0$, $j\ge 0$ and compact $K\subseteq J$ be given. If $K=\mathrm{\varnothing }$, both time seminorms are $0$ by the stated convention, so the estimate is $0\le C_{m}0$. Suppose now that $K\mathrm{\ne }\mathrm{\varnothing }$. For $\mathrm{\mid }\alpha \mathrm{\mid }\le m$ and $\tau \in K$, step 5.1 bounds each $\mathrm{\mid }{D}^{\alpha }{\mathrm{\partial }}_{\tau }^j{g}_i(x,\tau )\mathrm{\mid }$ by a finite constant times ${\sum }_{\mathrm{\mid }\beta \mathrm{\mid }\le m}{\sup }_{\bar{\mathrm{\Omega }}}\mathrm{\mid }{D}^{\beta }{\mathrm{\partial }}_{\tau }^{j}u(\cdot ,\tau )\mathrm{\mid }$: for $i=0$ this is immediate from the Leibniz rule and compact support, and for $i\ge 1$ the pullback introduces the bounded derivatives of ${\mathrm{\Phi }}_i^{-1}$ on the compact piece, the local cutoff contributes finitely many bounded terms, and step 5.1 supplies the coefficient sum ${\sum }_k\mathrm{\mid }{a}_k\mathrm{\mid }\mathrm{\mid }{b}_k{\mathrm{\mid }}^m<\mathrm{\infty }$; the spatial derivatives of order at most $m$ and the fixed time order $j$ enter only through $D^{\beta }{\mathrm{\partial }}_{\tau }^{j}u$ with $\mathrm{\mid }\beta \mathrm{\mid }\le m$, because the normal argument is sampled at $b_{k}t$ and differentiated at most $r\le m$ times. Summing over the finitely many pieces and over $\mathrm{\mid }\alpha \mathrm{\mid }\le m$ gives a constant $C_m$, independent of $u$, $j$, $K$ and $\tau$, with $\mathrm{\parallel }{\mathrm{\partial }}_{\tau }^j(E_{\mathrm{\Omega }}u)(\cdot ,\tau ){\mathrm{\parallel }}_{C^m({\mathbb{R}}^3)}\le C_m\mathrm{\parallel }{\mathrm{\partial }}_{\tau }^{j}u(\cdot ,\tau ){\mathrm{\parallel }}_{C^m(\bar{\mathrm{\Omega }})}$; the right-hand side is bounded on $K$ because the finitely many mixed derivatives are continuous on the compact set $\bar{\mathrm{\Omega }}\times K$, and taking the supremum over $\tau \in K$ gives the displayed estimate. The commutation identity below holds for every $\tau \in J$, independently of $K$. Since each piece commutes with ${\mathrm{\partial }}_{\tau }$, ${\mathrm{\partial }}_{\tau }^j(E_{\mathrm{\Omega }}u)=E_{\mathrm{\Omega }}({\mathrm{\partial }}_{\tau }^{j}u)$.

On $\mathrm{\Omega }\times J$ one has $E_{\mathrm{\Omega }}u={\sum }_{j=0}^p{\chi }_{j}u=u$ because ${\sum }_j{\chi }_j=1$ there and each piece agrees with ${\chi }_{j}u$ on the physical side; by continuity this identity extends to $\bar{\mathrm{\Omega }}\times J$. More generally, for every multi-index $\alpha$ and every $j\ge 0$, the continuous extension of $D^{\alpha }{\mathrm{\partial }}_{\tau }^{j}u$ to $\bar{\mathrm{\Omega }}\times J$ equals $D^{\alpha }{\mathrm{\partial }}_{\tau }^j(E_{\mathrm{\Omega }}u)$ on $\bar{\mathrm{\Omega }}\times J$: on the open set $\mathrm{\Omega }\times J$ this is the Leibniz rule for the finite sum ${\sum }_i{\chi }_{i}u$ [L4], and on $\mathrm{\partial }\mathrm{\Omega }$ it follows from step 9.1 because the exterior boundary values of the pieces add up to the boundary values of $u$, the partition identity ${\sum }_i{\chi }_i=1$ holding on a neighbourhood of $\bar{\mathrm{\Omega }}$.

For a finite-dimensional vector-valued input $u=(u_1,\dots ,u_N)$ the construction applied componentwise produces $E_{\mathrm{\Omega }}u:=(E_{\mathrm{\Omega }}{u}_1,\dots ,E_{\mathrm{\Omega }}{u}_N)$, and all bounds and agreement statements hold in each component. As to choice: the finite atlas is extracted by compactness from the definable family of all admissible smooth chart patches, the finitely many cutoffs and indices are finite choices in ZF, and the coefficients $a_k$ are fixed by the explicit formula of step 1.2 and the limits of step 2.1; no infinite selection is made, so the construction adds no choice principle beyond those carried by the cited suppliers. This completes the construction and proves every assertion of the statement.