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.

Calderón-Zygmund lemma

Covering lemma

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

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

2) $|f(x)|\; leq\; alpha$ 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 $Q\_k,$

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(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

$b(x)\; =\; 0,\; xin\; F$

$int\_\{Q\_j\}\; b(x),\; dx\; =\; 0$ 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 $2^d\; alpha$.

References

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