Derivatives of a cutoff composed with a controlled scale

Statement

Let $q:\Omega\to(0,1]$ be smooth and suppose for a fixed $c>0$ that

|\partial^\alpha q|\le C_\alpha q^{1-c|\alpha|}\qquad(|\alpha|\ge1).

If $\chi$ is smooth, equals one on $[0,1/2]$ , and vanishes on $[1,\infty)$ , then for $a>0$ ,

|\partial^\alpha[\chi(aq)]|\le C_{\alpha,\chi}q^{-c|\alpha|},

with constants independent of $a$ . This is the chain rule on the cutoff transition.

Cancellation of the cutoff parameter

For positive derivative order, every term has the form $a^k\chi^{(k)}(aq)\prod_{r=1}^k\partial^{\beta_r}q$ , with $\sum_r|\beta_r|=|\alpha|$ . Its support has $1/2\le aq\le1$ . The derivatives of $q$ contribute $q^{k-c|\alpha|}$ , and $a^kq^k$ is bounded there. Order zero uses only boundedness of $\chi$ . An anisotropic assumption $|\partial^\alpha q|\le C_\alpha q^{1-\sum_i c_i\alpha_i}$ gives the corresponding anisotropic loss by the same proof.