Calderón-Zygmund lemma

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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.

1 Calderón-Zygmund lemma
- 1.1 Covering lemma
- 1.2 Calderón-Zygmund decomposition
2 References

[edit] Calderón-Zygmund lemma

[edit] Covering lemma

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

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

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

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

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

[edit] 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

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

$\int_{Q_j} b(x)\, dx = 0$ 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 $Ω$ , g is equal to the average value of f over that cube, which by the covering chosen is not more than $2 d α$ .