Smooth dyadic partition of a positive scale

Core idea

Choose a smooth nonincreasing $\rho:\mathbb R\to[0,1]$ with $\rho=1$ on $( -\infty,1]$ and $\rho=0$ on $[2,\infty)$ . Put

\chi(s)=\rho(s)-\rho(2s),\qquad Q_\ell=2^{-\ell},\qquad \chi_\ell(q)=\chi(q/Q_\ell).

For $q>0$ , the functions $\chi_\ell$ , $\ell\in\mathbb Z$ , form a smooth dyadic scale partition:

\chi_\ell\ge0,\qquad \sum_{\ell\in\mathbb Z}\chi_\ell(q)=1, \qquad \operatorname{supp}\chi_\ell\subset[Q_\ell/2,2Q_\ell].

The family is locally finite on $(0,\infty)$ ; at most three closed support intervals contain a given point.

Proof and derivative bounds

The finite sum from $\ell=-M$ to $N$ telescopes to $\rho(2^{-M}q)-\rho(2^{N+1}q)$ , which equals one for sufficiently large $M,N$ . Differentiating gives $|\partial_q^k\chi_\ell|\le C_k Q_\ell^{-k}$ ; on its support this is at most $C'_k q^{-k}$ . The constants are independent of $\ell$ . Hence every fixed power of $q\partial_q$ is uniformly bounded as well.