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 Template:Math, where Template:Math denotes Euclidean space and Template:Math denotes the complex numbers, the lemma gives a precise way of partitioning Template:Math into two sets: one where Template:Math is essentially small; the other a countable collection of cubes where Template:Math is essentially large, but where some control of the function is retained.

This leads to the associated Calderón–Zygmund decomposition of Template:Math, wherein Template:Math is written as the sum of "good" and "bad" functions, using the above sets.

Covering lemma

Let Template:Math be integrable and Template:Mvar be a positive constant. Then there exists an open set Template:Math such that:

(1) Template:Math is a disjoint union of open cubes, Template:Math, such that for each Template:Math,

$\alpha \le \frac{1}{m(Q_k)}\underset{Q_k}{\int }|f(x)|dx\le 2^d\alpha .$

(2) Template:Math almost everywhere in the complement Template:Mvar of Template:Math.

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

Calderón–Zygmund decomposition

Given Template:Math as above, we may write Template:Math as the sum of a "good" function Template:Mvar and a "bad" function Template:Mvar, Template:Math. To do this, we define

$g(x)=\{\begin{array}{cc}f(x),& x\in F,\\ \frac{1}{m(Q_j)}\underset{Q_j}{\int }f(t)dt,& x\in Q_j,\end{array}$

and let Template:Math. Consequently we have that

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

$\frac{1}{m(Q_j)}\underset{Q_j}{\int }b(x)dx=0$

for each cube Template:Math.

The function Template:Mvar is thus supported on a collection of cubes where Template:Math is allowed to be "large", but has the beneficial property that its average value is zero on each of these cubes. Meanwhile, Template:Math for almost every Template:Mvar in Template:Mvar, and on each cube in Template:Math, Template:Mvar is equal to the average value of Template:Math over that cube, which by the covering chosen is not more than Template:Math.

See also

• Singular integral operators of convolution type, for a proof and application of the lemma in one dimension.
• Rising sun lemma

References

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