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,

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

(2) | f (x)| ≤ 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

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

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

${\displaystyle b(x)=0,\ x\in F}$

${\displaystyle {\frac {1}{m(Q_{j})}}\int _{Q_{j}}b(x)\,dx=0}$

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)| ≤ 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.

References

• Calderon A. P., Zygmund, A. (1952), "On the existence of certain singular integrals", Acta Math, 88: 85–139CS1 maint: multiple names: authors list (link)
• 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.