Locally finite smooth partitions of unity on domains

If $\mathrm{\Omega }={\mathbb{C}}^n$, then $\delta \equiv +\mathrm{\infty }$ is constant. Otherwise ${\mathbb{C}}^n\setminus \mathrm{\Omega }\mathrm{\ne }\mathrm{\varnothing }$, and for $z,z^{\mathrm{\prime }}\in {\mathbb{C}}^n$ and every $w\in {\mathbb{C}}^n\setminus \mathrm{\Omega }$, $\mathrm{\mid }z-w\mathrm{\mid }\le \mathrm{\mid }z-z^{\mathrm{\prime }}\mathrm{\mid }+\mathrm{\mid }{z}^{\mathrm{\prime }}-w\mathrm{\mid }$, so taking infima and then interchanging $z,z^{\mathrm{\prime }}$ gives $\mathrm{\mid }\delta (z)-\delta (z^{\mathrm{\prime }})\mathrm{\mid }\le \mathrm{\mid }z-z^{\mathrm{\prime }}\mathrm{\mid }$. Thus $\delta$ is continuous in either case. For $j\ge 1$ put $K_j:=\{z\in \mathrm{\Omega }:\mathrm{\mid }z\mathrm{\mid }\le j\text{ and }\delta (z)\ge 1\mathrm{/}j\}$; then each $K_j$ is closed in ${\mathbb{C}}^n$ (intersection of the closed ball with the closed set $\{\delta \ge 1\mathrm{/}j\}$) and bounded, hence compact by [F3] read through [F4]; moreover $K_j\subseteq \mathrm{int}{K}_{j+1}$, because $\mathrm{\mid }z\mathrm{\mid }\le j<j+1$ and $\delta (z)\ge 1\mathrm{/}j>1\mathrm{/}(j+1)$ hold for $z\in K_j$ and persist on a small ball around $z$ by continuity of the modulus and of $\delta$; finally ${\bigcup }_j{K}_j=\mathrm{\Omega }$, since for $z\in \mathrm{\Omega }$ one has $\delta (z)>0$ (or $\delta (z)=+\mathrm{\infty }$) and $\mathrm{\mid }z\mathrm{\mid }<\mathrm{\infty }$, so some integer $j$ satisfies $j\ge \mathrm{\mid }z\mathrm{\mid }$ and $1\mathrm{/}j\le \delta (z)$.

Put $K_m:=\mathrm{\varnothing }$ for $m\le 0$ and, for $j\ge 1$, $S_j:=K_{j+1}\setminus \mathrm{int}{K}_{j-1}$ and $W_j:=\mathrm{int}{K}_{j+2}\setminus K_{j-2}$; then every $S_j$ is compact (a closed subset of the compact $K_{j+1}$), $S_j\subseteq W_j$ with $W_j$ open (because $K_{j+1}\subseteq \mathrm{int}{K}_{j+2}$ and $\mathrm{int}{K}_{j-1}\supseteq K_{j-2}$), and the $S_j$ cover $\mathrm{\Omega }$: for $z\in \mathrm{\Omega }$ let $m_0:=\min \{m:z\in K_m\}$, which exists by step 1.1, so $z\in K_{m_0}\subseteq K_{m_0+1}$ and $z\mathrm{\notin }{K}_{m_0-1}\supseteq \mathrm{int}{K}_{m_0-1}$, that is $z\in S_{m_0}$. The family $(W_j{)}_{j\ge 1}$ is locally finite: a neighbourhood of $z$ contained in $\mathrm{int}{K}_{m_0+1}$ misses every $W_j$ with $j\ge m_0+3$, while only finitely many smaller indices remain; thus the family is locally finite.

For each $j\ge 1$ the set of finite lists (including the empty list when $S_j=\mathrm{\varnothing }$) $((z_1,r_1,i_1),\dots ,(z_N,r_N,i_N))$ with $z_k\in S_j$, $r_k>0$, $i_k\in I$, $\bar{B}(z_k,3r_k)\subseteq U_{i_k}\cap W_j$ and $S_j\subseteq {\bigcup }_{k}B(z_k,r_k)$ is nonempty: for every $z\in S_j\subseteq W_j$ the cover $\{U_i\}$ gives some $i(z)$ with $z\in U_{i(z)}$, the set $U_{i(z)}\cap W_j$ is open and contains $z$, so some radius $r(z)>0$ satisfies $\bar{B}(z,3r(z))\subseteq U_{i(z)}\cap W_j$ (choosing the pair $(i(z),r(z))$ by [F6]), and compactness of $S_j$ by step 2.1 lets the resulting open cover be reduced to a finite subcover; by [F5] select one such finite list for every $j\ge 1$ and enumerate the union of the selected lists as a sequence $(B_k,i_k,r_k{)}_{k\in \mathbb{N}}$ of balls $B_k=B(z_k,r_k)$. Then ${\bigcup }_k{B}_k\supseteq {\bigcup }_j{S}_j=\mathrm{\Omega }$ by step 2.1.

For each $k$, [F2] applied with the pair $0<r=r_k<R=3r_k\mathrm{/}2$ and the centre $z_k$ provides a smooth ${\chi }_k:{\mathbb{C}}^n\to [0,1]$ with ${\chi }_k=1$ on $\bar{B}(z_k,r_k)$ and $\mathrm{supp}{\chi }_k\subseteq B(z_k,3r_k\mathrm{/}2)$; the balls here are Euclidean balls of ${\mathbb{R}}^{2n}$ under the identification of [F4], so ${\chi }_k\in C^{\mathrm{\infty }}({\mathbb{C}}^n)$. By step 3.1, $\mathrm{supp}{\chi }_k\subset B(z_k,2r_k)\subseteq U_{i_k}\cap W_{j(k)}$; the countably many choices of the ${\chi }_k$ are read through [F5]. Since only finitely many selected balls occur for each $W_j$, each support lies in its assigned $W_j$, and the family $(W_j)$ is locally finite by step 2.1, every point of $\mathrm{\Omega }$ has a neighbourhood meeting only finitely many supports, so $\sigma :={\sum }_k{\chi }_k$ is a well-defined smooth function on $\mathrm{\Omega }$; finally $\sigma \ge 1$ at every point of $\mathrm{\Omega }$, because every point lies in some $S_j$ by step 2.1 and hence in some selected ball $B(z_k,r_k)$ on which ${\chi }_k=1$.

Define $V_k:=B(z_k,2r_k)\cap \mathrm{\Omega }$, so each $V_k$ is open with $V_k\subseteq U_{i(k)}$ and with $\mathrm{supp}{\chi }_k\subseteq V_k$ by step 4.1, and put ${\tilde{\chi }}_k:={\chi }_k\mathrm{/}\sigma$; then ${\tilde{\chi }}_k\in C^{\mathrm{\infty }}(\mathrm{\Omega })$, $0\le {\tilde{\chi }}_k\le 1$, $\mathrm{supp}{\tilde{\chi }}_k=\mathrm{supp}{\chi }_k\subseteq V_k$, and ${\sum }_k{\tilde{\chi }}_k=1$ because ${\sum }_k{\chi }_k=\sigma$. The family $(V_k)$ covers $\mathrm{\Omega }$, because ${\sum }_k{\tilde{\chi }}_k=1$ makes ${\tilde{\chi }}_k$ positive at every point for at least one $k$, and then that point lies in $V_k$; it refines $(U_i)$ by the map $k\mapsto i_k$, and is locally finite because $V_k\subseteq W_{j(k)}$ and $(W_j)$ is locally finite by step 2.1; the supports of the ${\tilde{\chi }}_k$ are locally finite for the same reason. Hence $({\tilde{\chi }}_k)$ is a smooth partition of unity subordinate to the locally finite refinement $(V_k)$ in the sense of [F1].