Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball

Adopt Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation and fix a dimension $q$. Smoothness of a function $\mathbb{R}^{q}\to\mathbb{R}$ is that of Smooth Functions on an Open Subset of ℝⁿ on $\mathbb{R}^{q}$, which is open by claim 1 of Euclidean Space Is Open in Itself; compact support is that of Compactly Supported Real-Valued Function for the topology of the open subsets of $\mathbb{R}^{q}$ (The Open Subsets of a Metric Space Form a Topology and Euclidean Openness Agrees with Metric Openness on $\mathbb{R}^n$); and $\Delta f=\sum_{i=1}^{q}\partial_{i}\partial_{i}f$ is the Laplacian of a function $f$ of class $C^{2}$ on $\mathbb{R}^{q}$. For $i,j\in[q]$, $\delta_{ij}$ denotes the entry in row $i$ and column $j$ of the identity matrix $I_{q}$, so $\delta_{ij}=1$ if $i=j$ and $\delta_{ij}=0$ otherwise; $\tfrac{t}{2}$ is as in Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (v); and $\iota(q)$, abbreviated $q$ where a real number is required, is the image of $q$ in $\mathbb{R}$ under the canonical map of The Canonical Map from the Natural Numbers to a Field, so that $qM=\iota(q)M$ for a real number $M$.

Let $\chi:\mathbb{R}^{q}\to\mathbb{R}$ be smooth with $0\le\chi(x)\le1$ for every $x$, $\chi(x)=1$ whenever $\lVert x\rVert\le1$ and $\chi(x)=0$ whenever $\lVert x\rVert\ge2$; such a function exists by Existence of a Smooth Plateau Function on Euclidean Space with $x_{0}=0_{\mathbb{R}^{q}}$, $r=1$ and $s=2$, since $\lVert x-0_{\mathbb{R}^{q}}\rVert=\lVert x\rVert$. For every positive real number $R$ define $\chi_{R},\psi_{R}:\mathbb{R}^{q}\to\mathbb{R}$ by

$$\chi_{R}(x)=\chi\bigl(R^{-1}x\bigr),\qquad \psi_{R}(x)=\tfrac{1}{2}\,\chi_{R}(x)\,\lVert x\rVert^{2}.$$

1. (Cutoffs)(i) For every positive $R$ the function $\chi_{R}$ is smooth and compactly supported, $0\le\chi_{R}(x)\le1$ for every $x$, $\chi_{R}(x)=1$ whenever $\lVert x\rVert\le R$, and $\chi_{R}(x)=0$ whenever $\lVert x\rVert\ge2R$. Moreover there are nonnegative real numbers $M_{1},M_{2}$, not depending on $R$, such that for every positive $R$, every $x\in\mathbb{R}^{q}$ and all $i,j\in[q]$,

$$|\partial_{i}\chi_{R}(x)|\le M_{1}R^{-1},\qquad |\partial_{j}\partial_{i}\chi_{R}(x)|\le M_{2}R^{-2}.$$

2. (Second-moment test functions)(ii) For every positive $R$ the function $\psi_{R}$ is smooth and compactly supported, and there is a nonnegative real number $M$, not depending on $R$, such that for every positive $R$, every $x\in\mathbb{R}^{q}$ and all $i,j\in[q]$,

$$|\partial_{i}\psi_{R}(x)|\le M\,\lVert x\rVert,\qquad |\partial_{j}\partial_{i}\psi_{R}(x)|\le M,\qquad |\Delta\psi_{R}(x)|\le qM .$$

3. (Agreement on the ball)(iii) For every positive $R$, every $x\in\mathbb{R}^{q}$ with $\lVert x\rVert<R$ and all $i,j\in[q]$,

$$\psi_{R}(x)=\tfrac{1}{2}\lVert x\rVert^{2},\qquad \partial_{i}\psi_{R}(x)=x_{i},\qquad \partial_{j}\partial_{i}\psi_{R}(x)=\delta_{ij},\qquad \Delta\psi_{R}(x)=q .$$