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# Calderón-Zygmund lemma

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 $f: mathbf\left\{R\right\}^\left\{d\right\} to mathbf\left\{C\right\}$, where $mathbf\left\{R\right\}^d$ denotes Euclidean space and $mathbf\left\{C\right\}$ denotes the complex numbers, the lemma gives a precise way of partitioning $mathbf\left\{R\right\}^d$ 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.

## Calderón-Zygmund lemma

### Covering lemma

Let $f: mathbf\left\{R\right\}^\left\{d\right\} to mathbf\left\{C\right\}$ be integrable and α be a positive constant. Then there exist sets F and $Omega$ such that:

1) $mathbf\left\{R\right\}^d = F cup Omega$ with $Fcap Omega = varnothing;$

2) $|f\left(x\right)| leq alpha$ almost everywhere in F;

3) $Omega$ is a union of cubes, $Omega = cup_k Q_k$, whose interiors are mutually disjoint, and so that for each $Q_k,$

$alpha < frac\left\{1\right\}\left\{m\left(Q_k\right)\right\} int_\left\{Q_k\right\} f\left(x\right), dx leq 2^d alpha.$

### 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

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

where $Q_j^o$ denotes the interior of $Q_j$, and let $b = f - g$. Consequently we have that

$b\left(x\right) = 0, xin F$

$int_\left\{Q_j\right\} b\left(x\right), dx = 0$ 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 $|g\left(x\right)| leq alpha$ 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 $2^d alpha$.

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