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 → 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 → 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,

[math]\displaystyle{ \alpha\le \frac{1}{m(Q_k)} \int_{Q_k} |f(x)| \, dx \leq 2^d \alpha. }[/math]

(2) | f (x)| ≤ α almost everywhere in the complement F of Ω.

Here, [math]\displaystyle{ m(Q_k) }[/math] denotes the measure of the set [math]\displaystyle{ Q_k }[/math].

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

[math]\displaystyle{ 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} }[/math]

and let b =  f  − g. Consequently we have that

[math]\displaystyle{ b(x) = 0,\ x\in F }[/math]

[math]\displaystyle{ \frac{1}{m(Q_j)}\int_{Q_j} b(x)\, dx = 0 }[/math]

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)| ≤ 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^dα.

See also

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

References

• Calderon A. P., Zygmund, A. (1952), "On the existence of certain singular integrals", Acta Math 88: 85–139, doi:10.1007/BF02392130 
• 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. ISBN 9780691080796. https://archive.org/details/singularintegral0000stei.