# Calderón–Zygmund lemma

(Redirected from Calderón–Zygmund theory)

In mathematics, the Calderón–Zygmund lemma is a fundamental result in Fourier analysis, harmonic analysis, and singular integrals. It is named for the mathematicians Alberto Calderón and Antoni Zygmund.

Given an integrable function f  : RdC, where Rd denotes Euclidean space and C denotes the complex numbers, the lemma gives a precise way of partitioning Rd into two sets: one where f is essentially small; the other a countable collection of cubes where f is essentially large, but where some control of the function is retained.

This leads to the associated Calderón–Zygmund decomposition of f, wherein f is written as the sum of "good" and "bad" functions, using the above sets.

## Covering lemma

Let f  : RdC be integrable and α be a positive constant. Then there exists an open set Ω such that:

(1) Ω is a disjoint union of open cubes, Ω = ∪k Qk, such that for each Qk,
${\displaystyle \alpha \leq {\frac {1}{m(Q_{k})}}\int _{Q_{k}}|f(x)|\,dx\leq 2^{d}\alpha .}$
(2) | f (x)| ≤ α almost everywhere in the complement F of Ω.

## Calderón–Zygmund decomposition

Given f as above, we may write f as the sum of a "good" function g and a "bad" function b, f  = g + b. To do this, we define

${\displaystyle g(x)={\begin{cases}f(x),&x\in F,\\{\frac {1}{m(Q_{j})}}\int _{Q_{j}}f(t)\,dt,&x\in Q_{j},\end{cases}}}$

and let b =  f  − g. Consequently we have that

${\displaystyle b(x)=0,\ x\in F}$
${\displaystyle {\frac {1}{m(Q_{j})}}\int _{Q_{j}}b(x)\,dx=0}$
for each cube Qj.

The function b is thus supported on a collection of cubes where f is allowed to be "large", but has the beneficial property that its average value is zero on each of these cubes. Meanwhile, |g(x)| ≤ α for almost every x in F, and on each cube in Ω, g is equal to the average value of f over that cube, which by the covering chosen is not more than 2dα.