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 $&thinsp;f&thinsp; : R^{d} → C$, 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 $&thinsp;f&thinsp;$ is essentially small; the other a countable collection of cubes where $&thinsp;f&thinsp;$ is essentially large, but where some control of the function is retained.

This leads to the associated Calderón–Zygmund decomposition of $&thinsp;f&thinsp;$, wherein $&thinsp;f&thinsp;$ is written as the sum of "good" and "bad" functions, using the above sets.

Covering lemma
Let $&thinsp;f&thinsp; : R^{d} → C$ 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, $Ω = ∪_{k} Q_{k}$, such that for each $Q_{k}$,


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


 * (2) $|&thinsp;f&thinsp;(x)| ≤ α$ almost everywhere in the complement $F$ of $Ω$.

Here, $$m(Q_k)$$ denotes the measure of the set $$Q_k$$.

Calderón–Zygmund decomposition
Given $&thinsp;f&thinsp;$ as above, we may write $&thinsp;f&thinsp;$ as the sum of a "good" function $g$ and a "bad" function $b$, $&thinsp;f&thinsp; = g + b$. 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 $b = &thinsp;f&thinsp; − g$. 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 $&thinsp;f&thinsp;$ is allowed to be "large", but has the beneficial property that its average value is zero on each of these cubes. Meanwhile, $|g(x)| ≤ α$ for almost every $x$ in $F$, and on each cube in $Ω$, $g$ is equal to the average value of $&thinsp;f&thinsp;$ over that cube, which by the covering chosen is not more than $2^{d}α$.