# 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 $f: \mathbf{R}^{d} \to \mathbf{C}$, where $\mathbf{R}^d$ denotes Euclidean space and $\mathbf{C}$ denotes the complex numbers, the lemma gives a precise way of partitioning $\mathbf{R}^d$ 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 $f: \mathbf{R}^{d} \to \mathbf{C}$ be integrable and α be a positive constant. Then there exist an open set Ω such that:

1) $\Omega$ is a disjoint union of open cubes, $\Omega = \cup_k Q_k$, such that for each $Q_k,$
$\alpha\le \frac{1}{m(Q_k)} \int_{Q_k} |f(x)|\, dx \leq 2^d \alpha.$
2) $|f(x)| \leq \alpha$ 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

$g(x) = \left\{\begin{array}{cc}f(x), & x \in F, \\ \frac{1}{m(Q_j)}\int_{Q_j}f(t)\,dt, & x \in Q_j,\end{array}\right.$

and let $b = f - g$. Consequently we have that

$b(x) = 0,\ x\in F$
$\int_{Q_j} b(x)\, dx = 0$ for each cube $Q_j.$

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 $|g(x)| \leq \alpha$ for almost every x in F, and on each cube in $\Omega$, g is equal to the average value of f over that cube, which by the covering chosen is not more than $2^d \alpha$.