Dyadic annulus - Knowlpedia

Definition

For an integer $k$ , a dyadic annulus in $\mathbb R^d$ is a set of the form

A_k=\{x\in\mathbb R^d:2^k\le |x|<2^{k+1}\}.

Changing open or closed endpoints, or multiplying both radii by a fixed constant, gives an equivalent convention for most estimates.

Scale decomposition

The annuli are pairwise disjoint and cover $\mathbb R^d\setminus\{0\}$ . A sum or integral over large radii can therefore be estimated by summing its contributions over $k$ .

Smooth localization

Scaled bump functions produce a smooth dyadic partition in which each point belongs to only boundedly many enlarged annuli. Derivatives of a cutoff at radius $2^k$ gain factors of $2^{-k}$ .

References

Loukas Grafakos, Classical Fourier Analysis, 3rd ed., Springer, 2014. DOI record.