Derivative estimates for scaled cutoffs

Statement

Let $\chi\in C_c^\infty(\mathbb R^n)$ , $a\in\mathbb R^n$ , and $\delta>0$ . Set $\chi_{a,\delta}(x)=\chi((x-a)/\delta)$ . Then

\partial^\alpha\chi_{a,\delta}(x) =\delta^{-|\alpha|}(\partial^\alpha\chi)((x-a)/\delta), \qquad \|\partial^\alpha\chi_{a,\delta}\|_\infty =\delta^{-|\alpha|}\|\partial^\alpha\chi\|_\infty.

Thus the support shrinks with $\delta$ , while derivatives grow with inverse powers of $\delta$ . The identity follows by repeated use of the chain rule.

Anisotropic scaling

For positive widths $\delta_1,\ldots,\delta_n$ , replacing the argument by $((x_i-a_i)/\delta_i)_i$ gives the factor $\prod_i\delta_i^{-\alpha_i}$ . Different directions can therefore incur different derivative costs.

Product errors

The Leibniz formula records exactly which derivatives strike the cutoff when differentiating $\chi_{a,\delta}f$ . These terms are supported in its transition region when at least one derivative strikes a cutoff that is constant on its plateau and outside its support.