Construction of a cut-off function with special properties

Question

I would like to construct a $C^{\infty}$ cut-off function $\rho: \mathbb{R}^n \rightarrow [0,1]$ for every $r>0$ with the following properties:

1. $\rho (x) = 1$ for $|x| < r/2$
2. $\rho (x) = 0$ for $|x| > r$
3. $|D\rho(x)| < 4/r$ for $x\in \mathbb{R}^n$, where the norm should be supreme norm.

Context: I am reading a book ODEs with Applications by Chicone and this construction (Exercise 4.17) is used in proof of Hartman-Grobman theorem.

Thanks for any help!

Answer 1

Consider $f:\mathbb{R}\to [0,1]$ given by $$ f(t)= \begin{cases} 1, & 0\leq |t|\leq 5/8,\\ -4t+\frac{7}{2}, & 5/8<|t|\leq 7/8,\\ 0, & 7/8<|t|. \end{cases} $$ Setting $\rho_1(x):=f(|x|/r)$ gives all the properties you want except for the smoothness requirement. For this take $g\in C^\infty(\mathbb{R})$ supported on the unit ball, nonnegative, and $\int_{\mathbb{R}} g\ dx =1$. Define (this is just "smoothing" out $f$ at scales $\varepsilon$) $$ f_\varepsilon(t):= \int_{\mathbb{R}} f(t-s)g(s/\varepsilon)\varepsilon^{-1}\, ds. $$ For $\varepsilon<1/8$ we'll have $\rho(x):= f_\varepsilon(|x|/r)$ gives the desired properties.