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 $f : R d \to C$ , where $R d$ denotes Euclidean space and $C$ denotes the complex numbers, the lemma gives a precise way of partitioning $R d$ 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 $f : R d \to C$ 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, $Ω = \cup k Q k$ , such that for each $Q k$ ,

$\alpha \leq {\frac {1}{m(Q_{k})}}\int _{{Q_{k}}}|f(x)|\,dx\leq 2^{d}\alpha .$

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

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

$b(x)=0,\ x\in F$

${\frac {1}{m(Q_{j})}}\int _{{Q_{j}}}b(x)\,dx=0$
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, $| g (x)| \leq α$ 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 $2 d α$ .

References

Calderon A. P., Zygmund, A. (1952), "On the existence of certain singular integrals", Acta Math, 88: 85–139
Hörmander, Lars (1990), The analysis of linear partial differential operators, I. Distribution theory and Fourier analysis (2nd ed.), Springer-Verlag, ISBN 3-540-52343-X
Stein, Elias (1970). "Chapters I–II". Singular Integrals and Differentiability Properties of Functions. Princeton University Press.

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Calderón–Zygmund lemma

Covering lemma

Calderón–Zygmund decomposition

See also

References